Maher Baili
email: maher.baili@irccyn.ec-nantes.fr

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

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

Classification of one family of 3R positioning Manipulators

Keywords: Cuspidal Manipulator, Joint Space, Workspace, Singularity, Aspect, Homotopy class, Genericity

The aim of this paper is to classify one family of 3R serial positioning manipulators. This categorization is based on the number of cusp points of quaternary, binary, generic and non-generic manipulators. It was found three subsets of manipulators with 0, 2 or 4 cusp points and one homotopy class for generic quaternary manipulators. This classification allows us to define the design parameters for which the manipulator is cuspidal or not, i.e., for which the manipulator can or cannot change posture without meeting a singularity, respectively.

INTRODUCTION

There is a strong relationship between global kinematic properties and the topology of singularities. Few authors are interested in global kinematic analysis of manipulators. Most of them work on control and trajectory planning or on manipulator workspace analysis. However, the geometry and topology of the singularities are a very important way for the analysis and classification of the kinematic properties of manipulators. In [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF], we have a detailed analysis of 3R manipulator singularities. In [START_REF] Pai | Generic Singularities of Robot Manipulator[END_REF], the notion of genericity was introduced. A manipulator is generic if its singularities are generic (they do not intersect in the joint space). Non-generic manipulators form hypersurfaces dividing the space of manipulators into different sets of generic ones. Consequently, most manipulators are generic. In [START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF], a categorization of all quaternary generic 3R positioning manipulators was done using homotopy classes. It was found exactly eight classes of homotopic quaternary generic 3R manipulators. The goal of this paper is to classify one family of positioning serial 3R manipulator. Moreover, an exhaustive classification of non-generic and binary manipulators is given. This study serves as an efficient tool for the categorization of cuspidal and non-cuspidal manipulators. This paper is organized as follow. Section 2 recalls some notions like singularities, cuspidality, genericity and homotopy class. Section 3 describes the family of 3R serial manipulator, which will be used all along this paper to make the categorization. In section 4, we analyse the results found. Some examples are given in section 5. Finally, last section concludes this paper.

PRELIMINARIES

SINGULARITIES

This paper deals with serial 3R positioning manipulators and only positioning singularities (referred to as "singularity" in the rest of the paper) are considered. A singularity can be characterized by a set of joint configurations that nullifies the determinant of the Jacobian matrix. They divide the joint space into at least two domains called aspects [START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF]. The aspects are the maximal free-singularity domains in the joint space.

[1] defines the critical point surfaces as the connected and continuous subset of singularities. Their corresponding images in the workspace are defined as critical value surfaces. The critical value surfaces divide the workspace into different regions with different number of inverse kinematic solutions or postures [START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF].

For a 3R manipulator, the joint space has the structure of a 3-dimentionnal torus, which can be reduced to a 2dimentionnal-torus (θ 2 ,θ 3 ) because the manipulators do not depend on θ 1 . In order to clarify Figure 2, the torus is cut along its generators, so the singularities are plotted in a 2π dimensional space. We must identify the opposite side of the square to keep the topology of the torus.

CUSPIDAL MANIPULATORS

A cuspidal manipulator is one that can change posture without meeting a singularity. The existence of such manipulators was discovered simultaneously by [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF] and [START_REF] Parenti | Spatial Open Kinematic Chains: Singularities, Regions and Subregions[END_REF]. In [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of all Non-Redundant Manipulators[END_REF], a theory and methodology were introduced to characterize new uniqueness domains in the joint space of cuspidal manipulators. Some examples of these manipulators are studied and analysed in [START_REF] Wenger | Changing Posture For Cuspidal Robot Manipulators[END_REF]. The only possible region of the workspace where a cuspidal manipulator can change posture without meeting singularity, is a region with four inverse kinematic solutions. Characterization of cuspidal manipulators has been a serious difficulty. Obviously, observation of several examples of manipulators gave rise to some conjectures by authors. In fact, some of them think that manipulators with simplifying geometric conditions like intersecting, orthogonal or parallel joint axes cannot avoid singularities when changing posture [START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF][START_REF] Parenti | Spatial Open Kinematic Chains: Singularities, Regions and Subregions[END_REF][START_REF] Wenger | A New General Formalism for the Kinematic Analysis of all Non-Redundant Manipulators[END_REF][START_REF] Wenger | Changing Posture For Cuspidal Robot Manipulators[END_REF][START_REF] Wenger | How to Recognize Simply a Non-Singular Posture Changing 3-DOF Manipulator[END_REF][START_REF] Khalil | A New Geometric Notation for Open and Closed Loop Robots[END_REF][START_REF] El Omri | Kinematic Analysis of Robot Manipulator[END_REF][START_REF] Smith | Design of Solvable 6 R manipulators[END_REF]. Others think that manipulators with arbitrary kinematic parameters are cuspidal [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Pai | Generic Singularities of Robot Manipulator[END_REF][START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF][START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF][START_REF] Parenti | Spatial Open Kinematic Chains: Singularities, Regions and Subregions[END_REF][START_REF] Wenger | A New General Formalism for the Kinematic Analysis of all Non-Redundant Manipulators[END_REF][START_REF] Wenger | Changing Posture For Cuspidal Robot Manipulators[END_REF][START_REF] Wenger | How to Recognize Simply a Non-Singular Posture Changing 3-DOF Manipulator[END_REF][START_REF] Khalil | A New Geometric Notation for Open and Closed Loop Robots[END_REF][START_REF] El Omri | Kinematic Analysis of Robot Manipulator[END_REF][START_REF] Smith | Design of Solvable 6 R manipulators[END_REF][START_REF] Burdick | A Classification of 3R Positioning Manipulators Singularities and Geometries[END_REF]. Neither the first nor the second idea can be stated in a general way. In [START_REF] Wenger | How to Recognize Simply a Non-Singular Posture Changing 3-DOF Manipulator[END_REF], a new characterization of cuspidal manipulators was done: a 3-DOF positioning manipulator can change posture without meeting a singularity if and only if there exists at least one point in its workspace with exactly three coincident inverse kinematic solutions and such a point is called a cusp point. Figure 1 shows in a cross-section of the workspace, the critical value surfaces for one cuspidal manipulator (D-H parameters modified [START_REF] Khalil | A New Geometric Notation for Open and Closed Loop Robots[END_REF]:

d 2 = 1, d 3 = 2, d 4 = 1.5, r 2 = 1, r 3 = 0, α 2 =
-90deg and α 3 = 90deg). There are four cusp points and two regions with four and two possible postures, respectively. Numeric and graphic methods are used to check the existence conditions of a cusp point. Consequently, it provides a useful tool for the purpose of manipulator design. In general cases, it is not possible to write the existence conditions of cusp points in an explicit, amenable expression of the DH-parameters [START_REF] El Omri | Kinematic Analysis of Robot Manipulator[END_REF].

GENERIC MANIPULATOR

In [START_REF] Pai | Generic Singularities of Robot Manipulator[END_REF], a generic manipulator is defined as one having no intersection of its smooth singularity surfaces in the joint space. Furthermore, a generic 3R manipulator must satisfy the following two conditions:

(1) The jacobian matrix has rank 2 at all critical points.

(2) All singular points, s θ , must satisfy the following condition:

[ det ( ( ))] 0 for at least one, =1, 2 θ

s i i θ ∂ ≠ ∂ J (1)
Simplifications in manipulator geometry (like intersecting or parallel joint axis) often lead to non-genericity and most of industrial manipulators are non-generic. However, many non-generic manipulators have complicated DHparameters [START_REF] Smith | Design of Solvable 6 R manipulators[END_REF][START_REF] Burdick | A Classification of 3R Positioning Manipulators Singularities and Geometries[END_REF]. Generic manipulators have stable global kinematic properties under small changes in their design parameters.

HOMOTOPY CLASSES

Homotopy classes were defined in [START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF] only for generic quaternary manipulators. A quaternary manipulator is defined as one having 4 inverse kinematic solutions. A binary manipulator has only 2 solutions. Two quaternary generic manipulators are homotopic if the singularity surfaces of one manipulator can be smoothly deformed to the singularity surfaces of the other. [START_REF] Burdick | A Classification of 3R Positioning Manipulators Singularities and Geometries[END_REF] shows that two homotopic manipulators have the same multiplicity of their kinematic maps. This result can be used to say that homotopic manipulators have the same maximum number of inverse kinematic solution by aspect. Thus, if one manipulator is cuspidal (resp. non-cuspidal), all manipulators homotopic to it are cuspidal (resp. noncuspidal).

One singularity surface forms a loop on the surface of the torus (θ 2 ,θ 3 ). Consequently, there are as many homotopy classes as ways of encircling the two generators of the torus. Figure 2 shows three different homotopy classes. Lines L1 and L2 represented in the square (

2 π θ π -≤ ≤ , 3 π θ π -≤ ≤ ) correspond to circles along the θ 2 -generator
and the θ 3 -generator of the torus respectively. However, L3 does not encircle any of the two generators, so it is homotopic to one point.

The homotopy class of a manipulator is denoted n (n 2 , n 3 ), where:

n: number of singularity surfaces in joint space.

n 2 : number of time the singularity surface encircles the θ 2 -generator.

n 3 : number of time the singularity surface encircles the θ 3 -generator.

To identify the homotopy class of one manipulator, the idea is to take one singularity surface, to count le number of "jumps" between two opposite sides of the square representation. At each jump, n 2 and n 3 are increased or decreased according to whether a jump occurs from to or from to

π π π π - -. θ 2 θ 3 (θ 2 , θ 3 )-torus L3 L2 L1 L1 θ 2 θ 3 L2 θ 3 θ 2 L3 θ 3 θ 2

Figure 2: Some loops of homotopy classes on the torus

For example, the homotopy class of L1 (resp. L2, L3) is (1,0), (resp. (0,1), (0,0)).

Submitted to ICAR 2003

3/6

The number and the homotopy class of the singularity surfaces define a set of homotopic generic manipulators.

The set of all 3R positioning manipulators is divided into subsets of homotopic generic manipulators separated by subsets of non-generic manipulators [START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF].

PROBLEM FORMULATION

The kinematic structures of most industrial manipulators are frequently decoupled into positioning and orientation devices. Their positioning structures are generally such that α 2 = ± 90 deg, α 3 = 0 deg, r 2 =0 or d 2 =0 and r 3 =0 (like the "PUMA" manipulator, all the manipulators such that α 3 = 0 are non-generic, quaternary and non-cuspidal [START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF][START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF][START_REF] Parenti | Spatial Open Kinematic Chains: Singularities, Regions and Subregions[END_REF][START_REF] Wenger | A New General Formalism for the Kinematic Analysis of all Non-Redundant Manipulators[END_REF][START_REF] Wenger | Changing Posture For Cuspidal Robot Manipulators[END_REF][START_REF] Wenger | How to Recognize Simply a Non-Singular Posture Changing 3-DOF Manipulator[END_REF][START_REF] Khalil | A New Geometric Notation for Open and Closed Loop Robots[END_REF][START_REF] El Omri | Kinematic Analysis of Robot Manipulator[END_REF][START_REF] Smith | Design of Solvable 6 R manipulators[END_REF]). In this paper, however, we study a family of positioning manipulators such that α 2 = ± 90deg, α 3 = ± 90 deg and r 3 =0 (Figure 3). This is a more interesting family because such manipulators can be binary or quaternary, generic or non-generic, cuspidal or non-cuspidal. We normalise d 3 , d 4 and r 2 by d 2 . So, we have only 3 parameters to consider:

= + ⎧ ⎪ -+ + ⎪ ⎪ = + ⎨ ⎪ + + + ⎪ ⎪ = -+ ⎩ (2)
Our goal is to express a condition to know if the manipulator is cuspidal or not. Equations ( 2) can be written in θ 3 only [START_REF] Pieper | The Kinematics of Manipulators under Computer Control[END_REF]. We obtain the equation below: ( 1 )

2 4 2 ( 1) and ( 1)
( 1 )

R L m x y r m rd d m rd L R d r m d r m L R d d m d d ⎧ + - = --+ + ⎧ ⎪ = ⎪ ⎪ ⎪ ⎪ = + -- = + ⎨ ⎨ ⎪ ⎪ = -- = ⎪ ⎪ ⎩ ⎪ ⎩ With:
= -+ ⎧ = + ⎧ ⎪ = - + ⎨ ⎨ = + + ⎩ ⎪ = - + + ⎩
To say if the manipulator is cuspidal or not, the polynomial (4) (of degree 4 in t and with coefficients function of x, y, z, d 3 , d 4 , r 2 ) must admit real triple roots. This is equivalent to solve the system:

, , , , , ) 0

Where: , , arethevariables.

, , are the parameters.

P t d d r R z P t d d r R z t P t d d r R z t P t d d r R z t t R z d d r = ⎧ ⎪ ∂ ⎪ = ⎪ ∂ ⎪ ⎨∂ = ⎪ ∂ ⎪ ∂ ⎪ ≠ ⎪ ∂ ⎩ - -
We note that one solution in t lifts unique 3-uplet (θ 1 ,θ 2 ,θ 3 ) except when z = 0, that can be treated like a particular case.

[15] computed a partition of the space of parameters, in which the number of real solutions is constant in each cell. This permits us to select one manipulator in each cell that can be considered like a representative manipulator of this cell. Under the hypotheses 2 0 and 0

Z z R Z = > -> ,
elimination of the variables t, Z, R after many operations and using Groebner Basis allows us to find one triangular system. Its regular roots represent solutions of the problem. Analysis of this system leads to only two equation surfaces dividing the parameter space into different zones [START_REF] Corvez | Using computer algebra tools to classify serial manipulators[END_REF].

These surfaces equations are defined by: Here, we can see 5 zones in plane (d 3 , r 2 ) and over everyone (in direction of d 4 ), we have 7 cells. For example, we have 35 cells in zone1 and 28 in zone2.

ANALYSIS WITH NUMBER OF CUSP POINTS

As it was said in the previous paragraph, the idea is to take one test point in each cell and to draw its workspace in order to know whether it is cuspidal or not. If the corresponding manipulator is cuspidal, it has the same number of cusp points and kinematic properties as all other manipulators in the cell.

We distinct two kinds of manipulators:

-Manipulator having four solutions for the inverse kinematic problem (quaternary manipulator).

-Manipulator having two solutions for the inverse kinematic problem (binary manipulator).

Classification into homotopy classes, correspond only for quaternary manipulators.

There exist three subsets of manipulators with 0, 2 and 4 cusp points, respectively. Table 1 shows that all noncuspidal manipulators (0 cusp point) are binary and generic. In addition, these manipulators may have 2 or 4 aspects. Table 2 shows that all manipulators having two cusp points are quaternary, non-generic and have 4 aspects. On the other hand, Table 3 shows that manipulators with four cusp points are all quaternary and either generic with homotopy class 2(1, 0) and with 2 aspects, or quaternary non-generic with 4 aspects.

Only one homotopy class 2(1, 0) is found in this work among the eight possible homotopy classes found in [START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF] which are: {(1(0,0), 2(0,0), 1(0,0)+2(1,0), 2(1,0), 4(1,0), 2(0,1), 2(1,1), 2(2,1)}. The two cells [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF] and (4,1,4) and all cells of Table 2 contain non-generic manipulators.

Binary generic manipulators with 2 aspects Binary generic manipulators with 4 aspects (1,1,5); [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Pai | Generic Singularities of Robot Manipulator[END_REF][START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF]; [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Wenger | Classification of 3R Positioning Manipulators[END_REF][START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF]; [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF]; [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF][START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF]; [START_REF] Burdick | Kinematic Analysis and Design of Redundant Manipulators[END_REF][START_REF] Tsai | Admissible Motion in Manipulator's Workspace[END_REF][START_REF] Borrel | A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination[END_REF].

( ). The number of inverse kinematic solutions in each region of the workspace is given. Figure 5 shows all possible shapes of joint spaces and workspaces met in this classification. Table 4 illustrates some information about proprieties of manipulators like their homotopy class (only for generic quaternary ones) and their cell belonging.

AKNOWLEDGMENTS

Figure 1 :

 1 Figure 1: Cusp points in the workspace section of a cuspidal manipulator.

Figure 3 :

 3 Figure 3: 3R manipulators studied. The direct kinematic equations are defined by: 3 4 3 1 2 2 4 3 1 1 3 4 3 1 2 2 4 3 1 1 3 4 3 2

  Algebraic Decomposition) adapted to such polynomial makes representation of possible cells. Our work is limited only to rational test points because we are interested in cells of maximal dimension (no computation with real algebraic numbers).Finally, we obtain one test point by cell, with positive rational coordinates and a set of hypersurfaces containing the other cells of the CAD. In practice, 105 maximal dimension cells are provided. Figure4depicts a CAD projection in the plane (d 3 , r 2 ) of the partition of parameter space (d 3 , r 2 , d 4 ).

Figure 4 :

 4 Figure 4: projection of the partition of parameters' space.

( 1 ,

 1 1,1); (1,1,2); (1,2,1); (1,2,7); (1,3,1); (1,4,1); (1,5,1). (1,1,6);(1,1,7); (1,2,6); (1,3,6); (1,3,7); (1,4,6); (1,4,7); (1,5,5); (1,5,6); (1,5,7). (2,1,1); (2,1,2); (2,2,1); (2,3,1); (2,4,1).(2,1,7); (2,2,7); (2,3,7); (2,4,7).

  Fig. 5

Figure 5 :

 5 Figure 5: Some examples of manipulators6. CONCLUSIONSIn this paper, we have provided a classification of one family of positioning manipulators such that α 2 = ± 90 deg, α 3 = ± 90 deg and r 3 =0. This classification was based on the number of cusp points. Parameter space was divided into 105 cells. In each cell, all manipulators have either 0, 2 or 4 cusp points. In addition to that, a complete classification of all manipulators of the family studied was provided using different criteria such as number of aspects, generic/non-generic and binary/quaternary manipulators.First, all binary manipulators are generic and noncuspidal and can have 2 or 4 aspects. Second, all quaternary non-generic manipulators are cuspidal, have 4 aspects and can have 2 or 4 cusp points. Finally, quaternary generic manipulators have 2 aspects and belong to the homotopy class 2(1, 0): the only homotopy class founded among the eight possible.

Future

  research work is to include the complete classification of a larger family of manipulators by relaxing r 3 (r 3 ≠ 0).

Table 1 :

 1 Cells with 0 cusp point

	Quaternary non-generic manipulators with 4 aspects

Table 2 :

 2 Cells with 2 cusp points

	Submitted to ICAR 2003

Table 3 :

 3 Cells with 4 cusp points

	5. EXAMPLES
	In this section, some examples of manipulators are
	provided. Figure 5 depicts the singularity surfaces
	(resp. critical value surface in a cross section of workspace) in 2 3 , θ θ (resp. workspace section defined
	by plane	ρ =	2 x y z 2 , +

Table 4 :

 4 Some examples of manipulators

This research was partially supported by CNRS (Project MATHSTIC "Cuspidal Robots and Triple Roots).


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

Philippe Wenger

A New Concept of Modular Parallel Mechanism for Machining Applications

Keywords: Parallel Machine Tool, Isotropic Design, and Singularity

The subject of this paper is the design of a new concept of modular parallel mechanisms for three, four or five-axis machining applications. Most parallel mechanisms are designed for three-or six-axis machining applications. In the last case, the position and the orientation of the tool are coupled and the shape of the workspace is complex. The aim of this paper is to use a simple parallel mechanism with two-degree-of-freedom (dof) for translation motions and to add one or two legs to add one or two-dofs for rotation motions. The kinematics and singular configurations are studied for each mechanism.

Introduction

Parallel kinematic machines (PKM) are commonly claimed to offer several advantages over their serial counterparts, like high structural rigidity, high dynamic capacities and high accuracy [START_REF] Treib | Similarity laws of serial and parallel manipulators for machine tools[END_REF]. Thus, PKM are interesting alternative designs for high-speed machining applications. The first industrial application of PKMs was the Gough platform, designed in 1957 to test tyres [START_REF] Gough | Contribution to discussion of papers on research in automobile stability, control an tyre performance[END_REF]. PKMs have then been used for many years in flight simulators and robotic applications [START_REF] Merlet | The parallel robot[END_REF] because of their low moving mass and high dynamic performances [START_REF] Treib | Similarity laws of serial and parallel manipulators for machine tools[END_REF]. This is why parallel kinematic machine tools attract the interest of most researchers and companies. Since the first prototype presented in 1994 during the IMTS in Chicago by Gidding&Lewis (the VARIAX), many other prototypes have appeared.

To design a parallel mechanism, two important problems must be solved. The first one is the singular configurations, which can be located inside the workspace. For a six-dof parallel mechanism, like the Gough-Stewart platform, the location of the singular configurations is very difficult to characterize and can change as a function of small variations of the design parameters [START_REF] Merlet | The parallel robot[END_REF]. The second problem is the nonhomogeneity of its performance indices (condition number, stiffness...) throughout the workspace [START_REF] Treib | Similarity laws of serial and parallel manipulators for machine tools[END_REF]. To the authors' knowledge, only one parallel mechanism is isotropic throughout the workspace [START_REF] Kong | Generation of parallel manipulators with three translational degrees of freedom based on screw theory[END_REF] but its stiffness is insufficient to be used in machining applications because its legs are subject to bending. Unfortunately, this concept is limited to three-dof mechanisms and cannot be extended to four or five-dof parallel mechanisms. Numerous papers deal with the design of parallel mechanisms [START_REF] Kong | Generation of parallel manipulators with three translational degrees of freedom based on screw theory[END_REF][START_REF] Hervé | Structural Synthesis of Parallel Robots Generating Spatial Translation[END_REF]. However, there is a lack of four-or five-dof parallel mechanisms, which are especially needed for machining applications [START_REF] Gao | New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs[END_REF]. To decrease the cost of industrialization of new PKM and to reduce the problems of design, a modular strategy can be applied. The translation and rotation motions can be divided into two separated parts to produce a mechanism where the direct kinematic problem is decoupled. This simplification yields also some simplifications in the definition of the singular configurations. The organization of this paper is as follows. Next section presents design problems of parallel mechanisms. The kinematic description and singularity analysis of the parallel mechanism used, are reported in section 3. 2 About parallel kinematic machines

General remarks

In a PKM, the tool is connected to the base through several kinematic chains or legs that are mounted in parallel. The legs are generally made of telescopic struts with fixed foot points (Figure 1a), or fixed length struts with moveable foot points (Figure 1b).

P P

Figure 1a: A bipod PKM Figure 1b: A biglide PKM For machining applications, the second architecture is more appropriate because the masses in motion are lower.

The linear joints can be actuated by means of linear motors or by conventional rotary motors with ball screws. A classification of the legs suitable to produce motions for parallel kinematic machines is given by [START_REF] Gao | New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs[END_REF] with their degrees of freedom and constraints. The connection of identical or different kinematic legs permits the authors to define two-, three-, four-and five-dof parallel mechanisms. However, it is not possible to remove one leg from a four-dof to produce a three-dof mechanism because no modular approach is used.

Singularities

The singular configurations (also called singularities) of a PKM may appear inside the workspace or at its boundaries. There are two 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. These configurations are located at the boundary of the workspace. A configuration where the tool cannot resist any effort and in turn, becomes uncontrollable is called a parallel singularity. Parallel singularities are particularly undesirable because they induce the following problems (i) a high increase in forces in joints and links, that may damage the structure, and (ii) a decrease of the mechanism stiffness that can lead to uncontrolled motions of the tool though actuated joints are locked. Figures 2a and2b show the singularities for the biglide mechanism of Fig. 1b. In Fig. 2a, we have a serial singularity. The velocity amplification factor along the vertical direction is null and the force amplification factor is infinite.

Figure 2b 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 determination of the singular configurations for twodof mechanisms is very simple; conversely, for a six-dof mechanism like Gough-Stewart platform, a mechanism with six-dof, the problem is very difficult [START_REF] Merlet | The parallel robot[END_REF]. With a modular architecture, when position and orientation of the mobile platform are decoupled, the determination of singularities is easier.

P P

Kinetostatic performance of parallel mechanism

Various performance indices have been devised to assess the kinetostatic performances of serial and parallel mechanisms. The literature on performance indices is extremely rich to fit in the limits of this paper [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF] (service angle, dexterous workspace and manipulability…). The main problem of these performance indices is that they do not take into account the location of the tool frame. However, the Jacobian determinant depends on this location [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF] and this location depends on the tool used.

Another problem is that to the authors' knowledge there is no parallel mechanism, suitable for machining, for which the kinetostatic performance indices are constant throughout the workspace (like the condition number or the stiffness…). For a serial three-axis machine tool, a motion of an actuated joint yields the same motion of the tool (the transmission factors are equal to one). For a parallel machine, these motions are generally not equivalent. When the mechanism is close to a parallel singularity, a small joint rate can generate a large velocity of the tool. This means that the positioning accuracy of the tool is lower in some directions for some configurations close to parallel singularities because the 3/6 encoder resolution is amplified. In addition, a high velocity amplification factor in one direction is equivalent to a loss of stiffness in this direction. The manipulability ellipsoid of the Jacobian matrix of robotic manipulators was defined two decades ago [START_REF] Salisbury | Articulated Hands: Force Control and Kinematic Issues[END_REF]. Unfortunately, this concept is quite difficult to apply when the tool frame can produce rotation and translation motions. Indeed, in this case, the Jacobian matrix is not homogeneous [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF]. The first way to solve this problem is its normalization by computing its characteristic length [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF]. The second one is to change the form of the Jacobian matrix. The first part of the mechanism for translational motion is optimized using homogeneous matrix. Then, the part dedicated to rotation motion can be optimized using the method introduced by [START_REF] Chablat | On the Kinetostatic Optimization of Revolute-Coupled Planar Manipulators[END_REF].

3 Kinematics of mechanisms for translation motions. This mechanism can be optimized to have a workspace whose shape is close to a square workspace and the velocity amplification factors are bounded [START_REF] Chablat | Conception Isotropique d'une morphologie parallèle: Application à l'usinage[END_REF]. The joint variables 1  and 2  are associated with the two prismatic joints. The output variables are the Cartesian coordinates of the tool center point

T y x P ] [  .
To control the orientation of the reference frame attached to P, two parallelograms can be used, which also increase the rigidity of the structure, Figure 3. To produce the third translational motion, it is possible to place orthogonally a third prismatic joint. This one can be located as in the case of Figure 4. Another solution is to use the Orthoglide mechanism, an isotropic three-dof mechanism [START_REF] Chablat | Design of a Three-Axis Isotropic Parallel Manipulator for Machining Applications: The Orthoglide[END_REF]. The choice between these two solutions depends on the main application of the milling machine. For example, for aeronautical pieces, the solution of fig. 4 is more appropriate because long and fine pieces are built. Conversely, for rapid prototyping of compact parts, we can choose the Orthoglide.

a b i a p         (1a) ) ( 2 2 2 2 a b i a p         (1b)
where

    e a
, respectively. For an isotropic configuration to exist where the velocity amplification factors are equal to one, we must have 0 . 2 1  e e [START_REF] Chablat | Conception Isotropique d'une morphologie parallèle: Application à l'usinage[END_REF] (Figure 5). Two square useful workspaces can be used. The first one has horizontal and vertical sides. The second one has oblique sides but its size is higher. , thus obtaining

1 1 1 1 1 1 ) ( ) (       e a b p a b T T (2a) 2 2 2 2 2 2 ) ( ) (       e a b p a b T T (2b)
Equations (2a-b) can be cast in vector form, namely    B p A 

, with A and B denoted, respectively, as the parallel and serial Jacobian matrices, where   is defined as the vector of actuated joint rates and p  is the velocity of point P, i.e.,  e b a . In [12], the range limits are defined to avoid these two singularities in using suitable bounds on the velocity factor amplification.

         T T ) ( ) ( 2 2 1 1 a b a b A          2 2 2
         2 1     and        y x    p When A

Kinematics of mechanisms for translation and rotation motions

The aim of this section is to define the kinematics of two mechanisms with one and two dofs of rotation, respectively. To be modular, the direct kinematic problem must be decoupled between position and orientation equations. A decoupled version of the Gough-Stewart Platform exists but it is very difficult to build because three spherical joints must coincide [START_REF] Khalil | Kinematic Analysis and Singular configurations of a class of parallel robots[END_REF]. Thus, it cannot be used to perform milling applications. The main idea of the proposed architecture is to attach a new body with the tool frame to the mobile platform of the two dofs defined in the previous section. The new joint admits one or two-dof according to the prescribed tasks.

Kinematics of a spatial parallel mechanism with one-dof of rotation

To add one-dof on the mechanism defined in section 3, we introduce one revolute joint between the previous mobile platform and the tool frame. Only one leg is necessary to hold the tool frame in position. Figure 6 shows the mechanism obtained with two translational dofs and one rotational dof. The architecture of the leg added is PUU where P and U stand for Prismatic and Universal joints, respectively [START_REF] Gao | New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs[END_REF].

The new prismatic joint is located orthogonaly to the first two prismatic joints. This location can be easily justified because on this configuration, i.e. when 1 1

2 2    b a b a and 3 3 3    b a
b p , the third leg is far away from serial and parallel singularities. Let   be referred to as the vector of actuated joint rates and p  as the velocity vector of point P,

1 2 2 [] T          and [] T xy  p  
Due to the architecture of the two-dof mechanism and the location of P, its velocity on the z-axis is equal to zero. p  can be written in three different ways by traversing the three chains

P B A i i , 1 1 1 1 ()
     p a i b a   (3a) 2 2 2 2 ()
     p a i b a   (3b) 3 3 3 3 3 ( ) ( ) ( )
           p a j k b a j p b    (3c)
where , respectively. Moreover, the velocities 1 a  , 2 a  and 3 a  of 1 A , 2 A and 3 A are given by

1 1 1     e a , 2 2 2     e a and 3 3 3
  ae  , respectively. We want to eliminate the passive joint rates i   and  from Eqs. (3a-c), which we do upon dot-multiplying Eqs. (3a-c) by Equations (4a-c) can be cast in vector form, namely,

i i a b  , 1 1 1 1 1 1 ( ) ( ) TT     b a p b a e  (4a) 2 2 2 2 2 2 ( ) ( ) TT     b a p b a e  (4b) 3 3 3 3 3 3 3 3 3 ( ) ( ) ( ) ( )
 tJ   with B A J 1   and T y x ] [      t
where A and B are the parallel and serial Jacobian matrices, respectively, e . However, these singular configurations are simple and can be avoided by proper limits on the actuated joints.

Kinematics of a spatial mechanism with two-dof of rotation

To add two dofs on the mechanism defined in section 3, we introduce a universal joint, between the previous mobile platform and the tool frame. Two legs are necessary to hold the tool frame in position and we use the same archtitecture than for the previous mechanism, i.e. the PUU mechanism. Figure 7 depicts a parallel mechanism with two translational dofs and two rotational dofs. Let   be referred to as the vector of actuated joint rates and p  as the velocity vector of point P,

  T 4 3 2 1            and [] T xy  p  
p  can be written in four different ways by traversing the three chains

P B A i i , 1 1 1 1 ()      p a i b a   (5a) 2 2 2 2 ()      p a i b a   (5b) 3 3 3 3 3 3 3 ( ) ( ) ( ) ( )              p a i k b a j i p b      (5c) 4 4 4 4 4 4 4 ( ) ( ) ( ) ( )              p a i k b a j i p b    (
( ) 0 0 ( ) 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T T T T T T T T                             4

Discussion

The problem of the optimal design of the two mechanisms defined in the previous sections is not addressed in this paper. In future works, for both mechanisms, it is possible to define a configuration for which the Jacobian matrix is isotropic. This result permits us to define the location of the prismatic joints and to define the condition length to normalize the Jacobian matrix. A method to define the range limits is explained in [START_REF] Chablat | On the Kinetostatic Optimization of Revolute-Coupled Planar Manipulators[END_REF], via a distance to the isotropic configuration.

Conclusions

In this paper, a new class of modular mechanisms is introduced with two, three, four and five dofs. All the actuated joints are prismatic joints, which can be actuated by means of linear motors or by conventional rotary motors with ball screws. The topology of the legs used to add one or two dof is the same. Only three types of joints are used, i.e., prismatic, revolute and universal joints. All the singularities are characterized easily because position and orientation are decoupled for the direct kinematic problem and can be avoid by proper design. In the future, for the modular architecture, the lengths of the legs as well as their positions will be optimized, to take into account the velocity amplification factors.

  Section 4 is devoted to design of two new architectures of parallel mechanisms with one or two dofs of rotation. D. Chablat and Ph. Wenger 2/6

Figure

  Figure 2a: A serial singularity

Figure 3

 3 Figure3shows a PKM with two dofs. The output body is connected to the linear joints through a set of two parallelograms of equal lengths ii L A B  , so that it can move only in translation.

Figure 3 :

 3 Figure 3: Parallel mechanism with two-dof The two legs are PPa identical chains, where P and Pa stand for Prismatic and Parallelogram joints, respectively.This mechanism can be optimized to have a workspace whose shape is close to a square workspace and the velocity amplification factors are bounded[START_REF] Chablat | Conception Isotropique d'une morphologie parallèle: Application à l'usinage[END_REF]. The joint variables 1  and 2  are associated with the two prismatic joints. The output variables are the Cartesian coordinates of the tool center point

Figure 4 :

 4 Figure 4: Hybrid mechanism with three-dof The velocity p  of point P can be expressed in two different ways. By traversing the closed loop ) ( 2 2 1 1 P B A P B A  in two possible directions, we obtain ) ( 1 1 1 1

Figure 5 :

 5 Figure 5: Cartesian workspace and isotropic configuration

Figure 6 :

 6 Figure 6: Parallel mechanism with two-dof of translation and one-dof of rotation

T

  There are two new singularities when one leg is added. The first one is a parallel singularity when

Figure 7 :

 7 Figure 7: Parallel mechanism with two-dof of translation and two-dof of rotationWe want to eliminate the passive joint rates

  When two legs are added, the same singular configurations as the previous mechanism occur, a parallel singularity occurs when the lines ) (

  and B are not singular, we obtain the relations,

	 p 	  J	with	J		A	B 1 
	Parallel singularities occur whenever the lines ( 11 AB ) and
	( 22 AB ) are colinear, i.e. when 12   	k 	,	for k = 1,2,....
	Serial singularities occur whenever	1 e b a or 1 1 
	2	2	2

Acknowledgments

This research was partially supported by the CNRS (Project ROBEA "Machine à Architecture compleXe"). The authors would like to thank Mr. Caro for his valuable remarks on this paper.


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

Bennis Fouad
email: fouad.bennis@irccyn.ec-nantes.fr

Realistic Rendering of Kinetostatic Indices of Mechanisms

Keywords: Kinetostatic properties, Mechanisms, Virtual reality devices, Simulation, Virtual manufacturing

The work presented in this paper is related to the use of a haptic device in an environment of robotic simulation. Such device introduces a new approach to feel and to understand the boundaries of the workspace of mechanisms as well as its kinetostatic properties. Indeed, these concepts are abstract and thus often difficult to understand for the end-users. To catch his attention, we propose to amplify the problems of the mechanisms in order to help him to take the good decisions.

1-Introduction

This article presents a new approach for the interpretation and the analysis of boundaries of the workspace and the kinematics properties of mechanisms.

The simulation tools provide powerful solutions for planning and designing of complex assembly facilities, lines and workplaces. In the digital mockup context, the haptic interfaces can be used to reduce the development cycle of products.

The simulation tools have taken, for a few years, a great place in the design process of robotized environments. In order to define a manipulator task, in the CAO-Robotics applications, we have tools to place the end-effector of the robot, according to its joint co-ordinates (direct geometric model) or to know the manipulator postures, according to the localization of its end-effector (inverse geometric model).

These two tools are used [START_REF] Chablat | Haptic device and object/robot/mannequin simulation in a CAD/CAM software: eM-Virtual desktop[END_REF] to make trajectory planning, [START_REF]GHOST SDK Programmer's Guide[END_REF] to make optimal placement of mechanisms while being coupled with collision detection tools and (3) to know the boundaries of the mechanism workspace. For this last case, it is not possible to represent them when the manipulator has more than 3 degrees of freedom or only in a degraded way by representing the dexterous workspace. Other tools must be used in order to understand the behavior real of the manipulator. The detection of the singular configurations is an important problem because in this case, exist some directions, for which the robot cannot move or can become difficult to control (vibrations).

The problem of these tools (when it exists) is the representation and the interpretation of the results by the engineer. The important for the engineer is not only the value on the point but its variation and the information about the most favorable directions. The exploitation of the results is not obvious and the link with the performance value of the real phenomenon is not trivial.

When one analyzes the communication between the operator and the computer, he can perceive that the operator immersion in the digital model is very weak. In the majority of the cases, the operator only has a keyboard, a classic mouse or a 3D mouse to manage the displacements of the objects. For the visual side, the use of stereoscopic glasses adds the necessary three-dimensional vision to the apprehension of the distances (and therefore of proximity). The displacement of objects or manipulators in the space is a difficult problem when rotations and translations are coupled. Only the use of haptic device as a Phantom desktop allows a realistic displacement of these objects. These displacements can be limited by the collisions between the manipulator and its environment or between the segments of the manipulator. Several works have already been achieved in this sense, but they are limited to the reproduction of merely material phenomena [START_REF] Chablat | Haptic device and object/robot/mannequin simulation in a CAD/CAM software: eM-Virtual desktop[END_REF].

In this paper, we show a paradigm how a haptic device with 3 or 6 degrees of freedom can allow the operator to interpret the limit of the workspace of the manipulators (including position and orientation). Then, we will add the interpretation of the manipulator mechanical conditioning while introducing some stimuli (viscosity, vibration…) in its displacements. When the conditioning is high or that one approach of the joint limits the operator will be able to feel, while displacing the manipulator's effector, the proximity of a singular configuration. Thus, we will be able to materialize virtual phenomena (theoretical performances) while using a peripheral of virtual reality and we enrich the digital mockup by additional semantics. Indeed, the indications of mechanical performances are often difficult to represent using traditional means. Thanks to the integration of the haptic feedback in the environment of simulation, the user will have an additional tool for the materialization of these properties. The integration of the peripheral haptic feedback in the environment of simulation eM-Workplace was the subject of a specific development for the Tecnomatix [START_REF] Chablat | Haptic device and object/robot/mannequin simulation in a CAD/CAM software: eM-Virtual desktop[END_REF]. The work presented in [START_REF] Chablat | Haptic device and object/robot/mannequin simulation in a CAD/CAM software: eM-Virtual desktop[END_REF] is related to the manipulation of objects, robots and models in a cluttered workspace. The work that we present in this paper is an extension of the results gotten.

2-Collision detection and haptic force-feedback generation

2.1-Collision detection

Desktop virtual prototyping can reduce development costs and cycle times, increase design flexibility and facilitate a more interactive, concurrent engineering process. However, trying to manipulate parts in these scenarios by visual feedback alone is generally very difficult. Providing the operator with "sense of touch" feedback going well beyond mere collision detection. Ideally, this sense of touch should include force and torque feedback in the same six degrees of freedom that the object would have in free space-x, y, z, roll, pitch and yaw.

In order to perform the operation of "collision detection", several tools can be find. For example, the Voxel PointShell (VPS) software is used by Sensable in GHOST software developer's kit [START_REF]GHOST SDK Programmer's Guide[END_REF][START_REF] Sensable | [END_REF][START_REF]Virtual Technologies[END_REF]. There are several semi-automatic or automatics methods, for the detection and the avoidance of collision in generation of trajectory. These methods are not yet sufficiently mature for a complete integration and a respect of the real time constraint [START_REF] Chedmail | Object manipulation and mannequin driving based on multi-agent architecture[END_REF][START_REF] Chedmail | A distributed Approach for Access and Visbility Task with a Manikin and a Robot in a Virtual Reality Environment[END_REF][START_REF] Cohen | Six Degree of Freedom Hapatic System for Desktop Virtual Prototyping Applications[END_REF].

In our application, we use the collision detection integrated in eM-Workspace to avoid the implementation of new algorithm. This function gives us several data. The characteristic of our approach is strongly related to the direct use of the models available in marketed software. Virtual a reality application should not require a new programming of all the models already offered in this software. Several aspects can be exploited from these models, such that:

 CAD models (geometrical aspect): These models are generally relatively heavy and can indeed decrease the real times performances in a VR environment. On the other hand, the transfer of these models in software not dedicated leads to a lost of data and are very times consuming.

 Non-geometrical data relating to the kinematics and dynamics behavior of the objects, robots and mechanisms. The CAO-Robotics environments integrate and manage these data complementarily with the geometrical data.

 The behavior models are also a specificity of CAO-Robotics software (kinematic and dynamic models, workspace, collision detection...). The fact that this software can manage the majority of the industrial robots is important and provides a considerable advantage to integrate new VR functions inside.

Conversely, each model introduces additional constraints into the VR environment by increasing the times response of the user interface. Our choice is a necessary compromise, which preserves the entirety and the exactitude of the models and which exploits simplified or approximate functions (example collision detection).

When the user activates force feedback, he must select an environment of collisions. This popup appears when "Env." button has been clicked. It enables to choose the pair, which could be in collision with selected object. Here, the idea is that two objects are needed to have a collision. Then with selected object and environment, the object could be in collision with environment. The Toolbox/collisions command can check for collisions and near misses either between all of the components in the workcell, or between the items in the lists prepared by these commands. Checking collisions and near misses among prepared lists of components is more efficient, if the lists are small, than checking among every component in the workcell, since it reduces the computational overhead that would otherwise be required to check for collisions and near misses between irrelevant and uninteresting components.

2.2-Force-feedback

The user interface of virtual reality software can be divided into two main classes (i) the input and (ii) the output. In the context of CAD/CAM software, the number of these devices is limited. In the context of virtual reality applications, the haptic sense is a powerful tool to believe that something is "real". The visual display is the main device used in CAD/CAM software. However, for a virtual reality software, there exist five categories of visual display, (i) monitor-based VR, (ii) projector VR (stationary displays), (iii) occlusive HMDs, (iv) nonoclusive HMDs (head-based displays), and (v) palm VR (hand based displays).

There are three types of haptic displays, (i) tactile devices, (ii) end-effector displays, and (iii) robotically operated shape displays. The first one provides information to the user in response to touching, grasping, feeling surfaces, or sensing the temperature of an object. The second one provides resistance and pressure to achieve these effects. The last one uses robots to present physical object to the user fingertips and provides information about shape, texture, and location to the user. The haptic device used provides 6 degree of freedom positional sensing but only tree degree force feedback and a small workspace. To avoid these problems, an intuitive strategy is used to lock the rotation of the stylus when an object is in the vicinity of environment and to manage the stylus/object attachment. The Phantom desktop workspace is (16 cm x 13 cm x 13 cm). The relative precision of position is 0.02 mm and the force-feedback is 6.4 N maxi and 1.4 N in continuous.

3-eM-Virtual Desktop application

The eM-Worplace provides powerful solutions for planning and designing of complex assembly facilities, lines and workplaces. In the digital mockup context, the haptic interfaces can be used to reduce the development cycle of products. In the literature, we can find several prototypes of software that integrate the haptic devices but, currently, not any software exist in the context of CAD/CAM software. Thus, to reduce the development cycle of products is necessary to limit the number of software to avoid the limitations of data exchanges.

Two haptic devices are implemented in eM-Virtual Desktop. The first one is the Phantom desktop device (Figure 1), a forcefeedback haptic device of SensAble Technologies [START_REF] Sensable | [END_REF]. The second is the CyberGlove device (Figure 2) [START_REF]Virtual Technologies[END_REF]. In this article, only the implementation of the first device is illustrated.

All the Phantom products, from the Sensable society [START_REF] Sensable | [END_REF] have the same architecture. The Phantom is grasped via a stylus on which extremity a switch is located. The device used in our application is the Phantom desktop. It is a six-dof input device since both the position and the orientation of the user's hand are sensed, but is a tree-dof force-feedback device, since only the forces along the x-, y-and z-axes are sent back to the user. Thus, it cannot mechanically return any torque information to the user. For the exploration of application areas requiring force feedback in six degrees of freedom (6DOF), there is phantom 1.5/6DOF but is not yet implemented in our software [START_REF] Chedmail | A distributed Approach for Access and Visbility Task with a Manikin and a Robot in a Virtual Reality Environment[END_REF]. The software eM-Virtual desktop in completely integrated in eM-Workplace. It is an addition application that can be addressed on the list of available applications. Figure 3 depicts de communication protocol used for this application (see figure 3 and [START_REF] Chablat | Haptic device and object/robot/mannequin simulation in a CAD/CAM software: eM-Virtual desktop[END_REF]).

The General Haptic Open Software Toolkit developed by SenAble (GHOST) includes several tools to make the collision detection in a simple environment. The implemented objects are simple shape like cylinder, sphere, or vrml geometry objects. In an industrial environment, such tool cannot be used due to the complexity of digital mockup and its object motions on the cell. As a matter of fact, several processes can carry out in the same time to manage an assembly cell for example. To solve this problem, we will use the collision algorithm detection of eM-Workplace. In fact, not any geometric entities are replicated and we work on a uniqueness database. The main problem of this architecture is the implementation of robust and fast interface between the GHOST and eM-Workplace, which is presented in the article. Three different loops are used to manage the graphics; the collision detection and the haptic feedback according to theirs own frequencies.

Fig.3: Communication protocol

There are three levels of sensibility (Rough, medium and fine) on the translation movement. It enables to choose the sensibility of the relation between the screen and the movement of Phantom desktop. The option "screen" is a special sensibility: Following the dimension of the view (zoom-/zoom+), the sensibility is automatically calculated proportionally with the zoom of the view.

So the user can define coefficients for all items (Rough, Medium, Fine).

3.1-Managing objects, robots and mannequin

Three types of object can be managed by the Phantom desktop via the motion of a stylus in a cell. For each type, a mechanical comportment is associated. The simplest is an object, which can move with respect to its geometric center or its self-origin or a frame defined by the operator (fig. 4).

The second one is a robot, which can be moved via its endeffector (TCPF) in using the inverse kinematic model or its base when the robot is compared with a rigid body (fig. 5). The third one is a mannequin, which can manage either its left hand or right hand. The inverse kinematic model of the mannequin can managed its 56 degree of freedom with respect to ergonomic constraints and lock either one hand or the knees or the torso (Fig. 6).

The application eM-Human provides a virtual environment for interactive design and optimization of manual tasks. In a 3D model of the actual manufacturing environment, you can define the work sequence using a virtual human model. Comprehensive functions allow for accurate analysis of the workplace in regards to execution time and ergonomics of the human task. The impact of modifications can be checked instantly, thus enabling the planner to optimize the work system prior to implementation.

eM-Human provides a 3D virtual environment wherein you can design and optimize manual operations. A library of human models of different gender and sizes, based on international standards, ensures that the workplace design is suitable for a broad range of workers. The human models provide inverse kinematics and posture calculations for the complete body, enabling detailed, accurate and efficient design of human tasks. Different grasping and walking macros allow fast and simple definition of human motions.

eM-Human provides capabilities to detect collisions between the human and the environment, and to analyze reachability, ensuring the feasibility of human tasks. A separate window on the screen, showing the worker´s field-of-vision, allows close examination of tasks from the worker´s point of view.

With the application eM-Virtual Desktop, all the displacement of the manikin are available via the stylus of the Phantom desktop (Fig. 6).

Fig. 6: A mannequin in the eM-Virtual Desktop and its 56 degrees of freedom

4-Mechanism properties

Haptic device in the environment of robotic simulation introduces a new approach to feel and to understand the boundaries of the workspace of mechanisms as well as its kinetostatic properties. In the following, the application of the haptic force-feedback device corresponding to these properties is presented.

4.1-Mechanism definitions

The mechanisms (or robots) are widely used in various industrial applications. Since last decade, other areas of application have emerged: medical, service, transport, underwater, entertainment… To make path planning, two methods are possible, on line or off line. The first stops the mechanism during its programming whereas the second brings more flexibility because it can be to realize on another site without stopping the production.

A mechanism consists of two distinct subsystems, one (or more) end-effector and an articulated mechanical structure.

There exist various types of mechanism (serial, tree structured and closed chains), but a great part of its properties are common.

4.2-Direct and inverse geometric model and workspace

The control of a mechanism requires the computation of two mathematical models. The transformation models between the joint space and the task space. Indeed, these transformation models are very important since the mechanism is controlled in the joint space whereas tasks are defined in the task space.

Two classes of models are studied, (i) direct and inverse geometric model, and (ii) direct and inverse kinematic models. In the context of the simulation, it is often the first one, which is used.

To select or to program a mechanism, the workspace and its singularity branches must be studied.

Let

 

1 n qq  q  be an element of the joint space and let  

1 m xx  X 
is the corresponding element in the task space, with

  f  Xq .
The joint domain Q is define as the set of all reachable configuration taking into account the joints limits:

  min max | , 1,..., i i i q q q i n      Qq
The image of Q by the direct geometric model defines the workspace W of the robot, with

  f  WQ .
The workspace W is the set of the position and the orientation reachable by the end-effector. Its shape depends on the architecture and its boundary is defined by singularities and the joints limits. The singular branches are defined by calculated the determinant of the Jacobian matrix J, where i e are associated with each revolute of prismatic joints and i r is defined as that joining i O with P, directed from the former to the latter (Fig. 7).

   n-1 e    e  e n  
 n e n-1

 n-1 P Fig. 7: A general n-revolute spherical writs However, when there is an obstacle in its workspace, additional boundaries limiting the reachable zones appear [START_REF] Vinogradov | Details of Kinematics of Manipulators with the Method of Volumes[END_REF][START_REF] Kumar | The Workspace of a Mechanical Manipulator[END_REF]. In general case, the workspace of a robot is a 6dimmensionnal space (position and orientation of the endeffector), which is difficult to handle. So usually, we must study its projection in the 3-dimmensional-position space (fig. 8). In our first study only the last type of boundary are introduced in the simulation software. However, to be complete, we need to introduce all the boundary types to feel all the properties of the mechanism.

4.3-Kinetostatic indices

Various performance indices have been devised to assess the kinetostatic performance of serial manipulators. Among these, the concepts of service angle [START_REF] Vinogradov | Details of Kinematics of Manipulators with the Method of Volumes[END_REF], dexterous workspace [START_REF] Kumar | The Workspace of a Mechanical Manipulator[END_REF] and manipulability [START_REF] Yoshikawa | Manipulability of Robotic Mechanisms[END_REF] are worth mentioning. All these different concepts allow the definition of the kinetostatic performance of a manipulator from correspondingly different viewpoints. However, with the exception of Yoshikawa's manipulability index [START_REF] Yoshikawa | Manipulability of Robotic Mechanisms[END_REF], none of these considers the invertibility of the Jacobian matrix. A dimensionless quality index was recently introduced by Lee [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 in the end-effector, for the Jacobian determinant is independent of this location. The proof of the foregoing fact 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 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    J of the Jacobian matrix was proposed by Salisbury [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,

  l s     J ,
where l  is the largest singular value of the Jacobian matrix J and s  the smallest one.

In fact, the condition number gives, for a square matrix, a measure of the relative round off error amplification of the computed results [START_REF] Golub | Matrix Computations[END_REF] with respect to the data round off 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, noted L, was introduced in [START_REF] Angeles | Kinematic Isotropy and the Conditioning Index of Serial Manipulators[END_REF] to cope with the abovementioned non-homogeneity. Apparently, nevertheless, this concept has found strong opposition within some circles, mainly because of the lack of a direct geometric interpretation of the concept. However, recent papers are introduced for serial and parallel mechanism, a novel performance index that lends itself to a straightforward manipulation and leads to sound geometric relations. Briefly stated, the performance index proposed here is based on the concept of distance in the space of m  n matrices, which is based, in turn, on the concept of inner product of this space. The performance index underlying this paper thus measures the distance of a given Jacobian matrix from an isotropic matrix of the same gestalt. With the purpose of rendering the Jacobian matrix dimensionally homogeneous, we resort to the concept of posture-dependent conditioning length. Thus, given an arbitrary serial manipulator in an arbitrary posture, it is possible to define a unique length that renders this matrix dimensionally homogeneous and of minimum distance to isotropy. The characteristic length of the manipulator is then defined as the conditioning length corresponding to the posture that renders the above-mentioned distance a minimum over all possible manipulator postures.

5-Introduction of the realistic rendering concept

The CAD-CAM software allows the simulation of mechanism through their direct and inverse geometric models. In the first case, it can exist of the dialog boxes with circular or linear sliders in which a change of color appears when one approaches these limits. In the second case, i.e. the use of the inverse geometrical model, we can represent only boundary spaces of dimensions 3.

5.1-Force-feedback corresponding to joint limits

When the mechanism reaches its joint limits or the workspace limits, two types of information can be given to the user: visual or resonant. However, no information allows him to find a solution to continue his displacement [START_REF] Kumar | The Workspace of a Mechanical Manipulator[END_REF]. In the working space, it is difficult to feel the proximity of the boundaries limiting the reachable zones.

-1 8 0 + 1 8 0 This problem of joint limits of the manipulator is resolved in our application by making direct analogy with a physical collision. So when one of the links reaches its joint limit, the user must feel a force-feedback indicating him a collision. That, by sensing the force-feedback the user is not obliged to supervise all visual information on the screen. He immediately feels the joint limits.

J o i n t 1 ( a ) -1 8 0 + 1 8 0 J o i n t 1 ( b )
In order to inform the user before arriving to the joint limit, we introduced a threshold security value close to the joint limit (Fig. 10). Let us define: q i_max and q i_min two extreme joint limits corresponding to the joint q i and  qi the threshold security value of q i . The force-feedback felt by the user is null in the interval [(q imin +  qi ) (q imax - qi )]. This force increases linearly until a maximal value. Thus the user can adapt his strategy in order to generate the trajectory. maximum Force feedback q min q max Force feedback Fig. 10: Smooth Force feedback for the joint limits

5.2-Boundaries of the workspace and singular configurations

For uncoupled architectures or for architectures with three dof, one can visualize the limit of the prismatic workspace (for example the workspace of the robot Puma and the Orthoglide). In the case of uncoupled architecture (Diestro and platform of flight (fig. 11 and12), one can only materialize their workspace corresponding to a given orientation. From Figure 8 one can see that the boundaries of the workspace is not easy to analyse and to takes into account.

One can notice that the complexity increase for robots with internal boundaries of the workspace and for Gough-Stewart. This workspace is also function of the orientation.

In our application, we consider this workspace as a physical obstacle and we provide the same tools, presented in section 2, in order to avoid these limits. The conditioning number is used to take into account sensibility analysis. To prevent the critical situations with "bad" condition number, a threshold security value of the conditioning number is introduced. The force-feedback value is deduced from the condition number. Its value is maximal when the condition number is greater than the threshold security value (fig. 13). Since:

  l s     J ,
where l  is the largest singular value of the Jacobian matrix J and s  the smallest one then:

0 ≤ 1/κ (J) ≤ 1
The limits of the workspace are obtained when 1/κ (J) is close to zero. The best performances are in the region where 1/κ is close to 1. One can also notice that if 1/κ (J) decrease very quickly, the robustness of the performances of the configuration in the corresponding situation is very sensible to the local displacements.

6-Application to a 2-dof parallel robot

Parallel kinematic machines (PKM) are commonly claimed to offer several advantages over their serial counterpart, like high structural rigidity, high dynamic capacities and high accuracy. On the other hand, they generally suffer from a reduced operational workspace due to the presence of internal singularities or self-collisions. The mechanism under study is a simple 2-dof parallel mechanism for which not any kinetostatic performance index is constant throughout its workspace. In next section, the kinematic equations are presented and the value of the kinetostatic performance index used, i.e. the condition number of two matrices.

6.1-Kinematic Relations

The manipulator under study is a five-bar, revolute (R)coupled linkage, as displayed in Fig. 14. The actuated joint variables are 1  and 2  , while the Cartesian variables are the (x, y) coordinates of the revolute center P.

Fig. 14: A two-dof closed-chain manipulator

Lengths 0 L , 1 L , 2 L , 3
L , and 4 L define the geometry of this manipulator entirely. However, in this paper we focus on a symmetric manipulators, with [START_REF] Golub | Matrix Computations[END_REF] LL  and 24 LL  . The symmetric architecture of the manipulator at hand is justified for general tasks. In manipulator design, then, one is interested in obtaining values of 0 L , 1 L , and 2 L that optimize a given objective function under some prescribed constraints.

The velocity p  of point P, of position vector p, can be obtained in two different forms, depending on the direction in which the loop is traversed, namely, inverse-kinematic singularities). The direct-kinematic singularities are located inside the Cartesian workspace and the inverse-kinematic singularities are often located on its boundaries. In a direct-kinematic singularity, the mobile platform wins one or several degrees of freedom and thus become incontrollable. On such configurations, the forces inside the mechanism tend to infinity. However, this kind of behavior is not trivial to detect without a VR interface. In the next sections, the condition number of both matrices is derived to provide us specific indices to render these boundaries. These indices are used to generate the value of force feedback to define a user friendly interface.

6.2-Direct-Kinematics Matrix

To calculate the condition number of matrix A, we need the product     holds are thus the most accurate for purposes of the direct kinematics of the manipulator. Correspondingly, the locus of points whereby matrix A is isotropic is called the isotropy locus in the Cartesian workspace. On the other hand, manipulator postures whereby 34 k     denote a singular matrix A. Such singularities occur at the boundary of the Joint space of the manipulator, and hence, the locus of P whereby these singularities occur, namely, the singularity locus in the Joint space, defines this boundary. Interestingly, isotropy can be obtained regardless of the dimensions of the manipulator, as long as (i) it is symmetric and (ii) 2 0 L  .

6.3-Inverse-Kinematics Matrix

By virtue of the diagonal form of matrix B, its singular values, 1  and 2  , are simply the absolute values of its diagonal entries, namely, . Interestingly, B can be rendered isotropic regardless of the dimensions of the manipulator, as long as (i) it is symmetric and (ii) 1 0 L  and 2 0 L  .

6.4-Example with the direct kinematic matrix

We assume here the dimensions 0 6 L  , 1 8 L  , and 2 5 L  , in certain units of length that we need not specify. The isoconditioning curves for the direct-kinematic matrix both in the Cartesian and in the joint spaces are displayed in Figs. [START_REF] Angeles | Kinematic Isotropy and the Conditioning Index of Serial Manipulators[END_REF]. A better representation of iso-conditioning curves can be obtained in the Cartesian space by displaying these curves for every working mode [START_REF] Chablat | The Isoconditioning Loci of A Class of Closed-Chain Manipulators[END_REF], which we do in Fig. 16.

Fig. 16: The four working modes and their iso-conditioning curves in the Cartesian space

According to the working mode chosen, the behavior of the mechanism studied change.

7-Conclusions

We have shown in this study that some complex concept can be simplified in order to be popularized. Furthermore, in a general case, the visualization of complex concept is not possible. Thus, the utilization of user interface seems to be the most appropriated solution for this problem.

We have introduced in this paper, new tool for the engineer to fell and to understand the properties of mechanisms. Such tolls can be used in the industry as well as in the engineering school to illustrate some theoretical notions like singular configuration or condition number.

Fig. 1 :

 1 Fig.1: Phantom desktop Fig. 2: Cyberglove

Fig. 4 :Fig. 5 :

 45 Fig. 4: Reference frames associated to the solid

Fig. 8 :

 8 Fig. 8: Workspace of a 6-dof mechanism

Fig. 9 :

 9 Fig. 9: Joint limits representation (a) with classic limits (b) with smooth limits

Fig. 11 :

 11 Fig. 11: The Diestro robot Fig. 12: The Gough-Stewart platform

Fig. 13 :

 13 Fig. 13: Force feedback corresponding to the conditioning number performance

E 1 B

 1 and c and d denoting the position vectors, in the frame indicated in Fig.14, of points C and D, respectively. After simplification, we write the kinematic equations in vector form, such that,  Ap Bθ  with θ  defined as the vector of actuated joint rates, of components  and 2   . Moreover A and B are, respectively, the direct-kinematics and the inverse-kinematics matrices of the manipulator, defined as,The singular configurations associated to A (resp. B) are called direct-kinematic singularities (resp.

  above expression for the condition number of the Jacobian matrix B, it is apparent that  . The locus of points where   1  B, and hence, where B is isotropic, is called the isotropy locus of the manipulator in the joint space. At the other end of the spectrum,    B tends to of B. These singularities are associated with the inverse kinematics of the manipulator, and hence, lie within its Cartesian workspace, not at the boundary of this one. The singularity locus of B thus defines the Cartesian workspace of the manipulator. Therefore, the Cartesian workspace of the manipulator is bounded by the singularity locus of B, i.e., the locus of points where

Fig. 15 :

 15 Fig. 15: The iso-conditioning curves in the Cartesian and joint space