Mazen Zein

Philippe Wenger

Damien Chablat Irccyn

Singularity Surfaces and Maximal Singularity-Free Boxes in the Joint Space of Planar 3-RPR Parallel Manipulators

Keywords: 3-RPR parallel manipulators, singularity, singularity-free zones, joint space, joint limits. I

In this paper, a method to compute joint space singularity surfaces of 3-RPR planar parallel manipulators is first presented. Then, a procedure to determine maximal joint space singularity-free boxes is introduced. Numerical examples are given in order to illustrate graphically the results. This study is of high interest for planning trajectories in the joint space of 3-RPR parallel manipulators and for manipulators design as it may constitute a tool for choosing appropriate joint limits and thus for sizing the link lengths of the manipulator.

Introduction

Most parallel manipulators have singularities that limit the motion of the moving platform. The most dangerous ones are the singularities associated with the direct kinematics, where two assembly-modes coalesce. Indeed, approaching such a singularity results in large actuator torques or forces, and in a loss of stiffness. Hence, these singularities are undesirable. There exists three main ways of coping with singularities, which have their own merits. A first approach consists in eliminating the singularities at the design stage by properly determining the kinematic architecture, the geometric parameters and the joint limits [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF][START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF][START_REF] Gallant | The Synthesis of Planar Parallel Manipulators with Prismatic Joints for an Optimal, Singularity-Free Workspace[END_REF]. This approach is safe but difficult to apply in general and restricts the design possibilities. A second approach is the determination of the singularity-free regions in the workspace [START_REF] Constantinescu | Détermination de régions exemptes de singularités pour des mécanismes parallèles plans et sphériques à 3 degrés de liberté[END_REF][START_REF] Kong | Determination of the uniqueness domains of 3-RPR planar parallel manipulators with similar platforms[END_REF][START_REF] Li | Determination of maximal singularity-free zones in the workspace of planar three-degreeof-freedom parallel mechanisms[END_REF][START_REF] Li | Determination of maximal singularity-free zones in the six-dimensional workspace of the general Gough-Stewart platform[END_REF][START_REF] Merlet | Determination of the presence of singularities in 6D workspace of a Gough parallel manipulator[END_REF][START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF]. This solution does not involve a priori design restrictions but, because of the complexity of the singularity surfaces, it may be difficult to determine definitely safe regions. Finally, a third way consists in planning singularity-free trajectories in the manipulator workspace [START_REF] Bhattacharya | Comparison of an exact and an approximate method of singularity avoidance in platform type parallel manipulators[END_REF][START_REF] Dash | Singularity-free path planning of parallel manipulators using clustering algorithm and line geometry[END_REF][START_REF] Merlet | Trajectory verification in the workspace for parallel manipulators[END_REF]. With this solution one is also faced with the complexity of the singularity equations but larger zones of the workspace may be exploited. In this paper, we choose to use the second approach by determining maximal joint space singularity-free boxes. This approach will help us determine appropriate joint limits and link dimensions. Planar parallel manipulators and particularly manipulators with three extensible leg rods, referred to as 3-RPR, have received a lot of attention because they have interesting potential applications in planar motion systems [START_REF] Gosselin | Solution polynomiale au problème de la cinématique directe des manipulateurs parallèles plans à 3 degrés de liberté[END_REF][START_REF] Merlet | Parallel Robots[END_REF]. As shown in [START_REF] Mcaree | An explanation of neverspecial assembly changing motions for 3-3 parallel manipulators[END_REF], moreover, the study of the 3-RPR planar manipulator may help better understand the kinematic behavior of its more complex spatial counterpart, the 6dof octahedral manipulator, which has also triangular base and platforms. The singularities of these manipulators have been most often represented in their workspace [START_REF] Husty | The General Singularity Surface of Planar Three-Legged Platforms[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF][START_REF] Mcaree | An explanation of neverspecial assembly changing motions for 3-3 parallel manipulators[END_REF] but more rarely in their joint space [START_REF] Mcaree | An explanation of neverspecial assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF][START_REF] Zein | Singular Curves and Cusp Points in The Joint Space of 3-RPR Parallel Manipulators[END_REF]. Hunt and Primrose showed that 3-RPR planar manipulator could have up to 6 assembly-modes [START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF]. Mcaree and Daniel analyzed the joint space singularities through slices to explain non-singular changing trajectories [188], and Zein et al analyzed the topology of these slices in [START_REF] Zein | Singular Curves and Cusp Points in The Joint Space of 3-RPR Parallel Manipulators[END_REF]. It was shown in [START_REF] Mcaree | An explanation of neverspecial assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF][START_REF] Wenger | Workspace and assembly modes in fully parallel manipulators: A descriptive study[END_REF] that, to change its assembly mode without meeting a singularity, a 3-RPR manipulator should encircle a cusp point in its joint space. In this paper, a method to compute and to represent joint space singularities of 3-RPR planar parallel manipulators is first proposed. A procedure is then provided to determine maximal joint space singularity-free boxes. This work is of a high interest for the determination of appropriate joint limits and for planning trajectories in the joint space.

II. Manipulators under study

The manipulators under study are 3-DOF planar parallel manipulators with three extensible leg rods (Fig. 1). These manipulators have been frequently studied and have interesting potential applications in planar motion systems. The geometric parameters are the three sides of the moving platform d 1 , d 2 , d 3 and the position of the base revolute joint centers defined by A 1 , A 2 and A 3 . The reference frame is centered at A 1 and the x-axis passes through A 2 . Thus, A 1 = (0, 0), A 2 = (A 2x , 0) and

A 3 = (A 3x , A 3y ). The parameter  is function of d 1 , d 2 and d 3 .
The joint space Q is defined by the vectors of the lengths of the three actuated extensible links

  T 1 2 3     q     .
12th IFToMM World Congress, Besançon (France), June18-21, 2007

x A 1 A 2 B 2 B 3 B 1 q q 2 d 1 d 3 d 2 q 1 y A 3 q 3 a   2  3  1
Fig. 1. A 3-RPR parallel manipulator

III. Kinematics of 3-RPR parallel manipulators

The relation between the joint space Q and the workspace W can be expressed as a system of non-linear algebraic equations, which can be written as:

( ) 0 F  x , q (1)
where x and q are respectively the vectors of the workspace and joint space variables. Differentiating equation ( 1) with respect to time leads to the velocity model:

 At + Bq 0  (2)
where

  T w  t
c  (w is the scalar angular velocity and c  is the two-dimensional velocity vector of the operational point B 1 of the platform if we used the first workspace parameters), A and B are 33 Jacobian matrices which are configuration dependent, and

  T 1 2 3     q    
is the joint velocity vector.

IV. Joint space singularities of 3-RPR parallel manipulators

The singularities of 3-DOF planar parallel manipulators have been extensively studied (see for example [START_REF] Bonev | Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory[END_REF][START_REF] Gosselin | Singularity analysis of closed loop kinematic chains[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF][START_REF] Mcaree | An explanation of neverspecial assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Sefrioui | On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators[END_REF]). They were defined in the workspace (x, y, a and to the author"s knowledge, there exist a small number of works dealing with the singular configurations in the manipulators joint space ( 1 ,  2 ,  3 ).

In a parallel singularity, matrix A is singular. To derive the singularity equations, it is usual to expand the determinant of A. We use rather a geometric approach that does not involve complicated algebraic calculus. The 3-RPR parallel manipulator is in a singular configuration whenever the axes of its three legs are concurrent or parallel [START_REF] Hunt | Geometry of Mechanisms[END_REF] (Fig. 2).

x

A 1 A 2 B 1 q 2 q 1 y A 3 q 3 B 2 B 3 d 1 d 3 d 2 a  Axis 1 Axis 3 Axis 2  2  1  3
Fig. 2. A 3-RPR parallel manipulator on a singular configuration.

In order to derive this geometric condition, we derive the condition for the three leg axes to intersect at a common point (possibly at infinity). We first write the equations of the three leg axes:

x xy yx y x A y x A A qq qq q q q           (3)
Eliminating x and y yields the following singularity equation in the task parameters (q 1 , q 2 , q 3 ):

  2 2 31 3 3 3 3 12 0 x x y A s s A s A c s    (4)
where

  sin ii s q  ,   cos ii c q  and   sin ij i j s qq  .
It is possible to express Eq. ( 4) as a function of the joint space parameters  1 ,  2 and  3 by using the constraint equations of the 3-RPR manipulator. However, the resulting equation would be too complicated to yield real insights, and difficult to handle. Our approach to compute the singular configurations in the joint space consists in reducing the dimension of the problem by first considering two-dimensional slices of the configuration space by fixing the first leg rod length  1 .

The singular surfaces in the full joint space are then calculated by "stacking" the slices.

Step 1: We rewrite Eq. ( 4) as a function of  1 , a and q 1 using the constraint equations of the manipulator.

        2 2 2 1 1 1 2 2 1 1 1 3 3 3 1 1 3 3 3 3 1 1 3 c c cos 0 s s sin 0 c c cos 0 s s sin 0 x x y Ad d Ad Ad   a   a   a    a                          (5)
The first (respectively last) two equations make it possible to express  2 (respectively  3 ) as function of  1 , a and q 1 . Then, c 2 and s 2 (respectively c 3 and s 3 ) are calculated as function of  1 , a and q 1 from the first (respectively last) 12th IFToMM World Congress, Besançon (France), June18-21, 2007 two equations of [START_REF] Constantinescu | Détermination de régions exemptes de singularités pour des mécanismes parallèles plans et sphériques à 3 degrés de liberté[END_REF] and their expressions are input in Eq. ( 4), which, now, depend only on L 1 , a and q 1 .

Step 2: We fix a value for  1 , so Eq. ( 4) depends now only on a and q 1 . By varying a or q 1 , we compute the roots of the equation, to obtain the singular configurations (a s , q 1s ) for a fixed  1s .

Step 3: For every singular configuration computed in the space (a, q 1 ) in the second step of the approach, we calculate the corresponding values  2s and  3s using the equation system [START_REF] Dash | Singularity-free path planning of parallel manipulators using clustering algorithm and line geometry[END_REF]. We have thus the singular configurations curves in a slice of the joint space ( 2 ,  3 ) with  1 fixed.

        2 2 1 1 1 2 2 1 1 1 2 3 1 1 3 3 2 3 1 1 3 cos( ) cos( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) x x y Ad d Ad Ad  q a   q a  q a    q a                         (6)
Figure 3 shows a slice of the joint space singular configurations for  1 =17 obtained for the same 3-RPR manipulator used in [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fullyparallel manipulators[END_REF][START_REF] Mcaree | An explanation of neverspecial assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Merlet | Parallel Robots[END_REF]. We refer only to this manipulator in this paper in order to illustrate our work. The geometric parameters of this manipulator are recalled below in an arbitrary length unit: Step 4: We compute the joint space singularity slices for a number of  1 values, to do this we have to repeat steps 2 and 3 while varying  1 .

A 1 =(0, 0) d 1 =17.04 A 2 =(
Finally, we collect all the computed slices in one file to obtain the singularities in the joint space ( 1 , 2 ,  3 ).

Figure 4 represents the singularities in the joint space of the manipulator studied when  1 varies from 0 to 50. To obtain this surface, we have imported the solutions obtained in step 4 into a CAD software, and we have meshed them together. Obviously, there is continuity between the singularities slices, one can claim, without any doubt, that there is singularity between the different slices. The surface depicted in Fig. 4 is of interest: i. for planning trajectories in the joint space because it shows clearly the joint space regions that are free of singularities.

ii. for manipulator design, because it constitutes a tool for defining the values of the joint limits such that the joint space is a singularity-free box.

V. Maximal joint space singularity-free boxes

In a context of design and/or trajectory planning, an important problem is to find singularity-free zones in the joint space.

In this section, we introduce a new procedure to determine maximal singularity-free boxes in the joint space of 3-RPR manipulators. These singularity-free boxes will help us fix the manipulator joints limits.

Two numerical examples are provided to illustrate the effectiveness of the procedure.

A. Procedure

Step 1: We choose an initial joint space configuration Q 0 ( 10 , 20 , 30 ). This configuration can be chosen according to several considerations, for example choosing Q 0 as the image through the inverse kinematics of a prescribed workspace center, or choosing it directly in the joint space as the center of a large singularity-free zone.

Step 2: We calculate the largest singularity-free cube centered at Q 0 ( 10 , 20 , 30 ).

12th IFToMM World Congress, Besançon (France), June18-21, 2007 To do this, we calculate the infinity norm distance d, also known as Chebyshev distance, between the center point Q 0 ( 10 , 20 , 30 ) and each of the joint space singular points Q s ( 1s , 2s , 3s ) computed in Section IV:

  10 1 20 2 30 3 max , , s s s d           (7)
and we keep the minimal distance d min found over all, because we are searching for the distance between the closest joint space singularity configuration Q s from the center point Q 0 .

The length of the singularity-free cube edge a will be:

min 2 ad  (8)
Step 3: The choice of the initial center point Q 0 ( 10 , 20 , 30 ) does not lead to an optimized solution, in other words varying lightly the center point position may lead to a largest singularity-free cube. Thus, the position of the initial point must be optimized, which we have done using a Hooke and Jeeves optimization scheme [START_REF] Hooke | Direct search solution of numerical and statistical problems[END_REF]. Note that the solution found is a local optimum.

Step 4: The cube found in step 3 touches the closest singular configuration to the center point. In order that the cube does not touch the singularities surface and for more security we subtract a small security value s from the distance d min . Such a value can be related to laws of command to stop the motion when the joint velocity is maximum.

The manipulator joint limits corresponding to the cube found can be easily computed as follows:

    min 0 min max 0 min ii ii ds ds              with i=1,2,3 (9)

B. Application of the procedure

In this section, two examples are provided in order illustrate the application of the procedure.

Example 1:

For the same manipulator studied, we consider the center point Q 0 (35,25,45) in the joint space. This point was chosen in the center of a large singularity-free zone in the joint space. By computing the Chebyshev distances between each joint space singular point Q s computed in section IV, and Q 0 , the minimal distance obtained is d min =5.3, so the edge length of the singularity-free cube is a = 10.6. By running the optimization algorithm, we find a maximal value d min =7.175 for a center point Q(41.625, 24.875, 44.125).

We subtract a security value of 0.1 from d min which becomes d min =7.075.

Figure 5 shows the joint space singularity surface and the maximal joint space singularity-free cube centered at Q(41.625, 24.875, 44.125) for the manipulator studied. Figures 6 shows the images through the direct kinematics of the maximal joint space singularity-free cube, which are two separate singularity-free components, each of them being located in an aspect of the workspace [START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF]. The projections of these two components onto the (x,y) plane are plotted in gray in Figure 6.

Example 2:

For the same manipulator, we consider in this example the center point Q 0 (30,50,35) in the joint space. By computing the Chebyshev distances between each joint space singular point Q s computed in Section IV and Q 0 , the minimal distance obtained is d min =4, so the edge length of the singularity-free cube is a = 8. By running the optimization algorithm, we find a maximal value d min =5.794 for a center point Q(38.125, 50, 33). We subtract a security value of 0.1 from d min , which becomes d min =5.694. Figure 8 shows the joint space singularity surface and the maximal joint space singularity-free cube centered at Q(38.125, 50, 33) for the manipulator studied. Figures 9 displays the images by the direct kinematics of the maximal joint space singularity-free cube, which are two separate singularity-free components, one in each aspect of the workspace. The projections of these two components onto the (x,y) plane are plotted in gray in Figure 9. This figure is displayed with the same viewing angle as in Figure 6 to show the difference between the two examples.

x y a Fig. 9. Images by direct kinematics of the joint space singularity-free cube (in black), and their projection on the (x,y) plane (in gray).

C. Future works

We can see in figures 6 and 9 that the components in the workspace do not have regular forms. Because this study is carried out in the joint space only, it cannot take into account any of the properties, in the workspace, of the image by direct kinematics of the singularity-free cube found. This work will be extended by taking into account the largest regular volume (cube, cylinder…) inside the workspace components images of the singularity-free cube. The idea will then be to optimize the location of the initial point Q 0 ( 10 , 20 , 30 ) such that the image of the maximal singularity-free cube in the workspace generates a regular volume of maximal size.

VI. Conclusion

A procedure for computing joint space singularities of 3-RPR parallel manipulators has been presented firstly in this paper. Secondly, a procedure for the determination of maximal joint space singularity-free boxes has been provided. These two procedures are of interest for planning trajectories in the joint space, and for manipulators design because they provide a tool for choosing the values of the joint limits. Future work will optimize the choice of the cube center point Q 0 in the joint space in order to maximize the volumes of the workspace components images of the singularity-free cube.

2 Fig. 3 .

 23 Fig. 3. Singular configurations in (2, 3) for 1=17.

Fig. 4 .

 4 Fig. 4. Joint space singularity surfaces of the 3-RPR manipulator studied when 1 varies from 0 to 50.

Fig. 5 .

 5 Fig. 5. Joint space singularity surfaces and maximal joint space singularity-free cube centered at Q(41.625, 24.875, 44.125).

Fig. 6 .

 6 Fig. 6. Images by direct kinematics of the joint space singularity-free cube (in black), and their projection on the (x,y) plane (in gray).

Figures 7 aFig. 7 .

 77 Figures 7 shows the two workspace components and the workspace singularities of the 3-RPR manipulator studied, the singularities are plotted in color.

Fig. 8 .

 8 Fig. 8. Joint space singularity curves and maximal joint space singularity-free cube centered at Q(38.125, 50, 33).


Myriam Polette

Christine Gilles

Béatrice Nawrocki-Raby

Jouko Lohi

Walter Hunziker

Jean-Michel Foidart

Philippe Birembaut

Membrane-Type 1 Matrix Metalloproteinase Expression Is Regulated by Zonula Occludens-1 in Human Breast Cancer Cells

The acquisition of a migratory/invasive phenotype by tumor cells is characterized by the loss of cell-cell adhesion contacts and the expression of degradative properties. In this study, we examined the effect of the disorganization of occludin/zonula occludens (ZO)-1 tight junction (TJ) complexes on the expression of membrane-type 1 matrix metalloproteinase (MT1-MMP). We first compared the expression of MT1-MMP and the localization of occludin/ZO-1 complexes in breast tumor cell lines displaying various degrees of invasiveness. We showed that the expression of MT1-MMP in invasive breast tumor cell lines correlates with the absence of occludin and with a cytoplasmic localization of ZO-1. In contrast, noninvasive cell lines displayed a membrane staining for both ZO-1 and occludin and did not express MT1-MMP. In vivo, cytoplasmic ZO-1 and MT1-MMP could be detected in invasive tumor clusters of human breast carcinomas. We then used RNA interference strategy to inhibit ZO-1 expression in invasive BT549 cells and to evaluate the effect of ZO-1 downregulation on MT1-MMP expression. We observed that ZO-1 small interfering RNA transfection down-regulates MT1-MMP mRNAs and proteins and subsequently decreases the ability of tumor cells to invade a reconstituted basement membrane in a Boyden chamber assay. Inversely, transfection of expression vectors encoding wildtype ZO-1 or the NH 2 -terminal fragment of ZO-1 comprising the PSD95/DLG/ZO-1 domains in BT549 activated a human MT1-MMP promoter luciferase reporter construct and increased cell invasiveness. Such transfections concomitantly activated the β-catenin/TCF/LEF pathway. Our results therefore show that ZO-1, besides its structural role in TJ assembly, can intervene in signaling events promoting tumor cell invasion.

INTRODUCTION

The acquisition of an invasive phenotype by epithelial tumor cells requires the disruption of intercellular adhesion. Although the reorganization of E-cadherin/β-catenin complexes of adherens junctions during tumor cell migration and invasion has been largely documented, little is known about the implication of tight junction (TJ) molecules in tumor progression [START_REF] Conacci-Sorrell | The cadherin-catenin adhesion system in signaling and cancer[END_REF]. In normal epithelial cells, TJs are composed of transmembrane molecules (occludin, claudin, and junctional adhesion molecules) linked to the actin cytoskeleton through cytoplasmic submembranous components, including zonula occludens (ZO)-l, ZO-2, and ZO-3 [START_REF] Gonzalez-Mariscal | Tight junction proteins[END_REF][START_REF] Itoh | The 220-kD protein colocalizing with cadherins in nonepithelial cells is identical to ZO-1, a tight junction-associated protein in epithelial cells: cDNA cloning and immunoelectron microscopy[END_REF]. ZO-1 (220 kDa) interacts directly with the others TJ proteins through its different structural conserved domains, three PSD95/ DLG/ZO-1 (PDZ) domains, one Src homology 3 domain, and one guanylate kinase homologous (GK) domain. ZO-1 is thus able to bind occludin via the GK domain, some claudins via the first PDZ domain, ZO-2 or ZO-3 via the second PDZ domain, and actin via the COOH-terminal end [START_REF] Gonzalez-Mariscal | Tight junction proteins[END_REF]. ZO-1 also binds other TJ proteins, such as AF-6 and cingulin [START_REF] Gonzalez-Mariscal | Tight junction proteins[END_REF]. Because of its interactions with many TJ components, ZO-1 acts as a crucial central regulator of the structural organization of the TJs at the plasma membrane. In carcinomas, ZO-1 is generally considered a tumor suppressor [START_REF] Tsukita | Submembranous junctional plaque proteins include potential tumor suppressor molecules[END_REF]. Indeed, several studies have reported that a reduction of ZO-1 expression correlates with tumor dedifferentiation and progression in breast and colorectal carcinomas [START_REF] Hoover | Loss of the tight junction MAGUK ZO-1 in breast cancer: relationship to glandular differentiation and loss of heterozygosity[END_REF][START_REF] Kaihara | Dedifferent-iation and decreased expression of adhesion molecules, E-cadherin and ZO-1, in colorectal cancer are closely related to liver metastasis[END_REF][START_REF] Martin | Loss of tight junction plaque molecules in breast cancer tissues is associated with a poor prognosis in patients with breast cancer[END_REF]. It has also been shown that ZO-1 displayed a cytoplasmic distribution in breast carcinomas [START_REF] Martin | Loss of tight junction plaque molecules in breast cancer tissues is associated with a poor prognosis in patients with breast cancer[END_REF]. In addition, ZO-1 has been found overexpressed in primary and metastatic pancreatic cancers [START_REF] Kleeff | Altered expression and localization of the tight junction protein ZO-1 in primary and metastatic pancreatic cancer[END_REF]. Generating different ZO-1 deletion constructs, it has been shown that a form of ZO-1, which lacks the GK domain and thus does not bind occludin and fails to be recruited to the plasma membrane, is able to confer tumorigenicity to transfected Madin-Darby canine kidney I (MDCK I) cells injected into nude mice [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF]. In addition, migrating epithelial MDCK I cells have been reported to display a ZO-1 nuclear localization [START_REF] Gottardi | The junction-associated protein, zonula occludens-1, localizes to the nucleus before the maturation and during the remodeling of cell-cell contacts[END_REF]. It has also been reported previously that ZO-1 contains several nuclear sorting signals, suggesting that ZO-1 shuttles between the plasma membrane and the nucleus [START_REF] Gonzalez-Mariscal | Molecular characterization of the tight junction protein ZO-1 in MDCK cells[END_REF]. Based on these results, the concept has emerged that, in addition to its structural role in TJ organization, ZO-1, once delocalized from TJs, may be involved in signaling pathways favoring epithelial cell migration and invasion.

Besides the loss of cell-cell contacts, tumor invasion is also characterized by the expression of degradative properties by tumor cells. Among the proteases implicated in this process, matrix metalloproteinases (MMP) occupy a central role. MMPs are a family of zinc endopeptidases comprising 25 members that degrade many extracellular matrix (ECM) components. Although the majority of MMPs are secreted as soluble enzymes, six membrane-bound MMPs have also been identified [START_REF] Polette | Tumour invasion and matrix metalloproteinases[END_REF]. Among the six membrane-type MMPs, membrane-type 1 MMP (MT1-MMP), which possesses a transmembrane domain, has been particularly implicated in pericellular proteolysis associated with cell migration [START_REF] Seiki | Roles of pericellular proteolysis by membrane type-1 matrix metalloproteinase in cancer invasion and angiogenesis[END_REF]. In addition to the degradation of structural ECM components, such as fibronectin, vitronectin, laminin-1, laminin-5, fibrin, and collagen type I, II, and III, MT1-MMP can also degrade cell adhesion molecules, such as CD44 and α v , integrin, as well as some cytokines, including MCP-3, stromal cell-derived factor-1α, and pro-tumor necrosis factor-α [START_REF] Seiki | Membrane-type 1 matrix metalloproteinase: a key enzyme for tumor invasion[END_REF]. MT1-MMP is also known to be an activator of other MMPs, particularly of MMP-2. Owning these properties, MT1-MMP regulates various cell functions that contribute to enhanced metastatic abilities. MT1-MMP has indeed been shown to be associated with tumor progression, particularly in cell invasion, angiogenesis, and metastasis. Accordingly, several studies have described an over-expression of MT1-MMP in many types of human carcinomas [START_REF] Seiki | Membrane-type 1 matrix metalloproteinase: a key enzyme for tumor invasion[END_REF]. Moreover, numerous reports have indicated that MT1-MMP is produced by invasive tumor cells in vitro as well as in vivo [START_REF] Gilles | Matrix metallop roteinases and epithelia-to-mesenchymal transition: implications for carcinoma metastasis[END_REF]. Interestingly, MT1-MMP expression in epithelial tumor cells has been shown to correlate with and be regulated by a reorganization of E-cadherin/β-catenin complexes [START_REF] Kqubek | β-Catenin activates a coordinated expression of the proinvasive factors laminin-5 72 chain and MTl-MMP in colorectal carcinomas[END_REF][START_REF] Takahashi | Identification of membrane-type matrix metalloproteinase-1 as a target of the β-catenin/Tcf4 complex in human colorectal cancers[END_REF]. These data emphasize a role of cell-cell contact molecules in the regulation of tumorpromoting genes, such as MMP.

In the present study, we investigated the potential role of ZO-1 in the regulation of MT1-MMP in invasive tumor cells. We document an association between the reorganization of occludin/ZO-1 complexes and the expression of MT1-MMP in invasive breast tumor cell lines. We also report that ZO-1 small interfering RNA (siRNA) transfection in invasive cells decreases MT1-MMP expression and invasive properties. In addition, transfection of ZO-1 cDNA was found to concomitantly activate the MT1-MMP promoter and the β-catenin/TCF/LEF pathway.

MATERIALS AND METHODS

Cell culture

All human mammary epithelial cells were obtained from the American Type Culture Collection (Rockville, MD). MCF-7, BT20, MDA-435, BT549, and Hs578T cells were cultured in DMEM containing 10% FCS.

Transfection of small interfering RNA

Three 19-nucleotide specific sequences were selected in the coding sequence of ZO-1 (Genbank accession no. L14837) to generate 21-nucleotide sense and 21-nucleotide antisense strands of the type (19N) TT (N, any nucleotide). The sense and antisense strands were then annealed to obtain duplexes with identical 3' overhangs. The sequences were submitted to a BLAST search against the human genome to ensure the specificity of the siRNA to the targeted sequence. Three corresponding scramble duplexes, which do not recognize any sequence in the human genome, were used as controls. The 19-nucleotide specific sequences for the three ZO-1 siRNAs are as follows: ZO-1 Sil, 5'-GUUAUACGAGCGAUCUCAU-3'; ZO-1 Si2, 5'-GGAGGAAACAG-CUAUAUGG-3'; and ZO-1 Si3, 5'-GACGAGAUAAUCCUCAUUU-3'. For transfection of the siRNA duplexes, 75,000 cells were plated in six-well plates in 2 mL/well of culture medium. Twenty-four hours after plating, the cells were transfected by phosphate calcium precipitation by adding in each well 200 µL of a mixture containing the siRNA duplexes (50, 20, or 2 nmol/L), 140 mmol/L NaCl, 0.75 mmol/L Na 2 HPO 4 , 6 mmol/L glucose, 5 mmol/L KC1, 25 mmol/L HEPES, and 125 mmol/L CaCl 2 . Twenty-four hours after transfection, the cells were extensively washed with PBS and incubated for 48 hours in culture medium before they were harvested for reverse transcription-PCR (RT-PCR) analyses, Western blotting analyses, or Boyden chamber assays.

Transient transfections of zonula occludens-1 expression vectors and reporter assay

The expression vectors encoding wild-type ZO-1 (ZO-1-WT) or the NH 2 -terminal fragment of ZO-1 comprising the PDZ domains, ZO-1-PDZ (1-3), have been characterized previously [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF]. The MT1-MMP promoter luciferase reporter vector was constructed by subcloning a 1.2-kb fragment of the 5' promoter region of MT1-MMP [START_REF] Lohi | Structural analysis and promoter characterization of the human membrane-type matrix metalloproteinase-1 {MTl-MMP) gene[END_REF] into the firefly luciferase reporter plasmid pGL-3 (Promega, Madison, WI). The TOP-FLASH and FOP-FLASH plasmids containing three wild-type (TOP-FLASH) or mutated (FOP-FLASH) copies of the β-catenin/TCF binding sites upstream of a minimal herpesvirus thymidine kinase promoter driving the firefly luciferase expression were a kind gift from H.C. Clevers (University Hospital, Utrecht, The Netherlands; ref. [START_REF] Korinek | Constitutive transcriptional activation by a β-catenin-Tcf complex in APC-/-colon carcinoma[END_REF].

Transient transfections were done with Fugene transfection reagent (Roche, Indianapolis, IN). For immunofluorescence analyses, Boyden chamber assays, or preparation of cytosolic versus nuclear extracts, cells were plated in six-well plates or 10 cm Petri dishes 24 hours before the addition of the DNA/Fugene mixture. The cells were transfected according to the manufacturer's instructions with a mixture (100 µL for six-well plates and 400 µL for 10 cm Petri dish) containing 30 µL Fugene per 1 mL serum-free DMEM and 20 µg ZO-l-WT or ZO-l-PDZ expression vector (or the corresponding pLNCX empty vector). The cells were collected 24 or 48 hours after transfection. For promoter luciferase assays, 50,000 cells were plated in 24-well plates 30 minutes before the addition of the DNA/Fugene mixture. Each well was incubated with a mixture containing 20 µL serum-free DMEM, 0.6 µL Fugene, 0.2 µg firefly promoter luciferase reporter plasmid (either the MT1-MMP promoter, the TOP-FLASH or the FOP-FLASH reporter construct), 0.2 µg ZO-l-WT or ZO-1-PDZ expression vector (or the corresponding pLNCX empty vector), and 0.8 ng Renilla luciferase expression vector phRG-TK (Promega). Twenty-four hours after transfection, the cells were lysed in 50 µL passive lysis buffer and the luciferase activity was determined with a luminometer using the Dual Luciferase Assay System (Promega) on 20 µL lysate. For each experiment, the firefly luciferase activity was normalized to the activity of the Renilla luciferase used as internal control. The results were expressed as fold induction determined by normalizing each firefly luciferase value to the Renilla luciferase internal control value and by dividing these normalized values with the mean normalized value of the corresponding reporter construct transfected with the empty expression vectors. To assess the β-catenin/ TCF/LEF activities, the normalized values obtained with the FOP-FLASH reporter plasmid were subtracted from the normalized values obtained with the TOP-FLASH reporter plasmid. The resulting values were then divided with the mean normalized value of the condition transfected with the empty expression vector. Each experiment was done at least thrice in triplicate. Data are expressed as means ± SD. A nonparametric test (Mann-Whitney U test) was done and P < 0.05 was considered significant.

Immunofluorescence

Confluent monolayers of cells cultured on glass coverslips were fixed with methanol for 10 minutes at -20°C. For confocal microscopy analyses of ZO-1, cells were fixed in 2.5% paraformaldehyde in PBS for 30 minutes and then permeabilized with 0.2% Triton X-100 for 15 minutes at room temperature. The coverslips were then saturated for 10 minutes with 3% bovine serum albumin (BSA) in PBS. Monolayers were next (after intermediate washes in PBS) incubated for 1 hour with a monoclonal antibody to ZO-1 (Z01-1A12, Zymed Laboratories, San Francisco, CA) or occludin (OC-3F10, Zymed Laboratories) and then with a biotinylated antimouse IgG (Amersham, Arlington Heights, IL) for 1 hour. Coverslips were exposed to Alexa Fluor 492-coupled streptavidin for 30 minutes (Molecular Probes, Eugene, OR). The coverslips were then mounted with Aquapolymount antifading solution (Agar, Essex, United Kingdom) onto glass slides and observed under a Zeiss fluorescence microscope or with a MRC-600 confocal laser scanning microscope (Bio-Rad, Richmond, CA).

Western blotting analyses

For ZO-1, occludin, and MT1-MMP detection, cells were rinsed twice in PBS and extracted in lysis buffer [50 mmol/L Tris (pH 7.4), 150 mmol/L NaCl, 1% (v/v) Igepal, 1% (w/v) sodium deoxycholate, 5 mmol/L iodoacetamide, 0.1% (w/v) SDS] containing Complete protease inhibitor cocktail (Roche). To examine the subcellular distribution of β-catenin, cytosolic and nuclear extracts of BT549 cells transfected with the ZO-1-WT or ZO-l-PDZ expression vectors (or the corresponding pLNCX empty vector) were done using the ProteoExtract Subcellular Proteome Extraction kit (Calbiochem, La Jolla, CA). Protein concentration was determined with the detergent-compatible protein assay (Bio-Rad). Total protein (10 µg) was separated on 7.5% and 12% SDS-PAGE gels for ZO-1 and for occludin and MT1-MMP analyses, respectively. Cytosolic extracts (4 µg) or nuclear extracts (2 µg) were separated on 10% SDS-PAGE to analyze the expression of β-catenin. The proteins were transferred to polyvinylidene difluoride membranes (NEN, Boston, MA), which were then blocked with 5% milk (w/v), 0.1% Tween 20 (w/v) in PBS for 2 hours before exposure to the primary antibody overnight at 4°C: a monoclonal antibody to ZO-1 (ZO1-1A12), a monoclonal antibody to occludin (OC-3F10), a monoclonal antibody directed against the hemopexin-like domain of MT1-MMP (clone 2D7, kindly provided by Dr. Rio, Institut de Génétique et de Biologie Moléculaire et Cellulaire, Illkirch, France), and a monoclonal antibody directed against the unphosphorylated form of β-catenin (clone 8E4, A.G. Scientific, San Diego, CA). The filters were then incubated with either a horseradish peroxidase-conjugated goat anti-rabbit or swine anti-mouse antibody (DAKO, Glostrup, Denmark). Signals were detected with an Enhanced Chemiluminescence Plus kit (Amersham Pharmacia Biotech, Buckinghamshire, United Kingdom). Subsequent detection of actin (using a rabbit antibody to actin, clone A2066, Sigma-Aldrich, St. Louis, MO) or c-myc (using a monoclonal antibody to c-myc, clone 9E10, Sigma-Aldrich) was done on the same filters as a control. The values obtained for MT1-MMP were normalized to values obtained for actin in three independent transfection experiments. Results were expressed as fold induction determined by dividing the normalized value with the normalized value of the corresponding scrambled siRNA control. Data are expressed as means ± SD. A nonparametric test (Mann-Whitney U test) was done and P < 0.05 was considered significant.

Reverse transcription-PCR analyses

Total RNA extraction was done with the RNA isolation kit (Roche). RT-PCR was done using 10 ng total RNA the GeneAmp Thermostable RNA PCR kit (Perkin-Elmer, Foster City, CA), and pairs of primers for human MT1-MMP and for 28S rRNA (Eurogentec, Seraing, Belgium). Forward and reverse primers for human MT1-MMP and 28S were designed as follows: MT1-MMP primers (forward 5'-GGATACCCAATGCCCATTGGCCA-3' and reverse 5'-CCATTGGGCATCCAGAAGAGAGC-3') and 28S primers (forward 5'-GTTCACCCACTAATAGGGAACGTGA-3' and reverse 5'-GGATTCTGACTTAGAGGCGTTCAGT-3'). Reverse transcription was done at 70°C for 15 minutes. Amplification cycles were as follows: 15 seconds at 94°C, 20 seconds at 68°C, and 10 seconds at 72°C. Twenty cycles were allowed for MT1-MMP amplification and 15 cycles for 28S amplification. Products were separated on acrylamide gels, stained with SYBR Gold (Molecular Probes), and quantified by fluorimetric scanning (LAS-1000, Fuji, Stamford, CT). The values obtained for MT1-MMP amplification were normalized to the values obtained for 28S amplification in three independent transfection experiments. Results were expressed as fold induction determined by dividing the normalized value of a given siRNA with the normalized value of the corresponding scrambled siRNA control. Data are expressed as means ± SD. A nonparametric test (Mann-Whitney U test) was done and P < 0.05 was considered significant.

Modified Boyden chamber invasion assay

The in vitro invasive properties of transfected cells were assessed using a modified Boyden chamber assay. Cells (10 4 ) were suspended in 30 µL serum-free medium supplemented with 0.1% BSA and placed in the upper compartment of a 48-well invasion chamber (Neuroprobe, Pleasanton, CA). The lower compartment of the chamber was filled with 30 µL medium containing 10% FCS and 1% BSA The two compartments were separated by a porous filter (8 µm pore, Nucleopore, Pleasanton, CA) coated with Matrigel (1 mg/10 mL used to coat the filter). After 24 hours of incubation at 37°C, the filters were fixed in methanol for 10 minutes and stained with hematoxylin for 5 minutes. The cells at the upper surface of the filters were wiped away with a cotton swab. Quantification of the invasion assay was done by counting the number of cells at the lower surface of the filters (10 fields at 400-fold magnification). Experiments were done at least thrice in triplicate. Data are expressed as means ± SD. A nonparametric test (Mann-Whitney U test) was done and P < 0.05 was considered significant.

Immunohistochemistry on breast tissues

Human tissues were obtained from biopsies of 15 invasive breast ductal carcinomas. The samples were fixed in formalin and embedded in paraffin. Immunohistochemistry for ZO-1 and MT1-MMP was done on 5 µm serial sections. The sections were deparaffinized, rehydrated, and treated with 0.3% H 2 O 2 for 5 minutes to quench endogenous peroxidase activity. Slides were then incubated for 30 minutes with a monoclonal antibody to either ZO-1 (ZO1-1A12) or MT1-MMP (clone 113-B7, Oncogene Science, Cambridge, MA). After several washes, the slides were successively incubated with biotin-coupled anti-mouse antibody (DAKO) and then with a horseradish peroxidase-conjugated streptavidin (DAKO). The peroxidase activity was revealed using the LSAB kit (DAKO). The slides were briefly counterstained with Mayer's hematoxylin, mounted, and observed under a Zeiss Axiophot microscope.

RESULTS

Membrane-type 1 matrix metalloproteinase is expressed by invasive, occludin-negative breast tumor cell lines

To examine the relationship between the presence or localization of occludin/ ZO-1 complexes and MT1-MMP expression, we first examined the expression of these three molecules in two noninvasive (MCF-7 and BT20) and three invasive (MDA-435, BT549, and Hs578T) epithelial breast cancer cell lines. The invasive phenotype of these breast cancer cells, assessed by a modified Boyden chamber assay, has been characterized in previous studies and was confirmed in our present study (data not shown; refs. [START_REF] Sommers | Differentiation state and invasiveness of human breast cancer cell lines[END_REF][START_REF] Thompson | Association of increased basement membrane invasiveness with absence of estrogen receptor and expression of vimentin in human breast cancer cell lines[END_REF][START_REF] Nawrocki | Quantitative cell dispersion analysis: new test to measure tumor cell aggressiveness[END_REF]. By Western blotting, we observed an inverse correlation between occludin and MT1-MMP expression. Occludin was only detected in noninvasive cell lines, whereas MT1-MMP was only found in invasive cell lines. In contrast, ZO-1 was expressed in all the cell lines analyzed regardless of their invasive properties (Fig. 1).

The cytoplasmic localization of zonula occludens-1 correlates with invasiveness of tumor cells

We then examined more closely the subcellular localization of occludin and ZO-1 in the different tumor cell lines. By immunofluorescence microscopy, we found that occludin was present at the cell membrane of noninvasive tumor cells (as illustrated in MCF-7 in Fig. 2A) and was absent in invasive tumor cells (as illustrated in BT549 in Fig. 2A ). Regarding ZO-1, we observed that ZO-1 was mainly localized at the membrane of noninvasive cells, whereas it showed a predominantly diffuse cytoplasmic staining in invasive tumor cells (Fig. 2A andB).

ZO-1 and MT1-MMP expression was also observed in several cases of human ductal breast carcinomas. Staining of serial sections of breast carcinomas with ZO-1 and MT1-MMP antibodies revealed the concomitant expression of MT1-MMP and ZO-1 in invasive tumor clusters, where ZO-1 showed a heterogenous cytoplasmic localization (Fig. 3). Although in all cases ZO-1 was expressed in tumor cells staining for MT1-MMP, not all MT1-MMP-positive cells showed ZO-1 labeling.

Figure 1. Expression of occludin, ZO-1, and MT1-MMP in human breast cancer cell lines. Western blot analyses of ZO-1, occludin, and MT1-MMP were done on five human tumor cell lines (MCF-7, BT20, MDA-435, BT549, and Hs578T

). Actin was used as a loading control.

Figure 2. Localization of occludin/ZO-1 in human tumor cell lines. A, ZO-1 and occludin immunolabeling in MCF-7 and BT549 monolayers. Bar, 50 µm. B, confocal microscopy analyses of ZO-1 localization in MCF-7 and BT549 cell monolayers. Optical sections were taken in the basal or the intermediate region of the cells. Bar, 16 µm.

Regulation of membrane-type 1 matrix metalloproteinase expression by zonula occludens-1

To evaluate the potential implication of ZO-1 in the regulation of MT1-MMP expression, we used a RNA silencing strategy to down-regulate the endogenous ZO-1 levels in three invasive tumor cell lines. Transfection of a siRNA against ZO-1 down-regulated ZO-1 expression in the three cell lines at the protein level (Fig. 4). We concomitantly observed a clear diminution of MT1-MMP protein in tumor cells treated with the ZO-1 siRNA compared with cells treated with scrambled control siRNA (Fig. 4). Because the siRNA-mediated downregulation of ZO-1 worked best in BT549 cells, we used this cell line to further investigate the regulation of MT1-MMP expression by ZO-1. To exclude a potential nonspecific effect of the ZO-1 siRNA used above, we first confirmed our results using two different siRNAs against ZO-1. As revealed by RT-PCR analyses, all three ZO-1 siRNAs significantly reduced MT1-MMP mRNA expression in BT549 tumor cells compared with scrambled control siRNAs (0.46 ± 0.06, 0.52 ± 0.05, and 0.71 ± 0.04 for Si-1, Si-2, and Si-3, respectively; P < 0.05; Fig. 5B andC). Accordingly, Western blotting also revealed a significant decrease of the 60-kDa active form of MT1-MMP (0.5 ± 0.05, 0.28 ± 0.05, and 0.22 ± 0.05 for Sil, Si-2, and Si-3, respectively; P < 0.001; Fig. 5D andE). Concomitantly, we also observed a down-regulation of the 45-kDa form of MT1-MMP (Fig. 5E). As shown in Fig. 6, the inhibiting effect of ZO-1 Si3 siRNA on MT1-MMP expression is observed at various concentrations of siRNA. We also tested the effect of ZO-1 silencing on cell invasion in the Boyden chamber assay and observed that ZO-1 Si3 siRNA at 2, 20, and 50 nmol/L significantly reduced the invasive properties of the BT549 cells (435 ± 40 versus 150 ± 15, 468 ± 69 versus 70 ± 9, and 453 ± 43 versus 55 ± 2.9 cells per 10 fields for 2, 20, and 50 nmol/L concentrations of scrambled Si3 siRNA versus ZO-1 Si3 siRNA, respectively; P < 0.001; Fig. 6B).

Figure 3. Cytoplasmic ZO-1 and MT1-MMP expression in ductal breast carcinomas. ZO-1 (A) and MT1-MMP (B) immunolabelings are detected on invasive tumor cells (T) in ductal breast carcinomas.

Zonula occludens-1 regulates the membrane-type 1 matrix metalloproteinase promoter and activates the β-catenin/TCF/ LEF transcriptional activity

It has been shown previously that the NH 2 -terminal part of ZO-1 encompassing the three PDZ domains is able to activate the β-catenin/TCF pathway in MDCK I cells [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF]. β-catenin, once delocalized from the adherens junction, can indeed accumulate in the cytoplasm, translocate to the nucleus, and, through binding to TCF/LEF transcription factors, directly regulate the transcription of genes implicated in tumor invasion [START_REF] Conacci-Sorrell | The cadherin-catenin adhesion system in signaling and cancer[END_REF]. We thus examined the possibility that the regulation of MT1-MMP by ZO-1 could be associated with the activation of this pathway. For this purpose, BT549 tumor cells were transiently cotransfected with expression vectors encoding the NH 2 -terminal portion of ZO-1, including the three PDZ domains (ZO-1-PDZ) or ZO-l-WT and a MT1-MMP promoter reporter plasmid or the TOP-FLASH/FOP-FLASH reporters. First, we observed that transient transfection of ZO-1-WT and ZO-1-PDZ expression vectors increased the cytoplasmic labeling of ZO-1 in BT549 cells and their invasiveness in Boyden chamber assays (1.97 + 0.25-and 1.89 + 0.16-fold induction for ZO-l-WT and ZO-1-PDZ transfection, respectively; Fig. 1A andB). We also found that both ZO-1-WT and ZO-1-PDZ expression vectors significantly increased the activity of the MT1-MMP promoter in BT549 tumor cells (3.11 + 0.3-and 2.48 + 0.3-fold induction for ZO-l-WT and ZO-l-PDZ transfection, respectively; P < 0.001; Fig. 7C). This was accompanied by an increased activity of the TOP-FLASH reporter system (9.12 + 3.6-and 5.85 + 1.96-fold induction for ZO-l-WT and ZO-l-PDZ transfection, respectively), attesting of the activation of the βcatenin/TCF/LEF pathway following ZO-l-WT and ZO-l-PDZ transfection (P < 0.001; Fig. 7D). It has been shown that the NH 2 -terminal unphosphorylated form of β-catenin is involved in the transcriptional activity of βcatenin [START_REF] Staal | Wnt signals are transmitted through N-terminally dephos-phorylated β-catenin[END_REF]. Using an antibody specific for the unphosphorylated form of β-catenin, we therefore examined by Western blotting the presence of this particular form of β-catenin in cytosolic or nuclear extracts of cells transfected with ZO-1-WT and ZO-1-PDZ expression vectors. Consistent with the results obtained with the TOP-FLASH reporter assays, we found an accumulation of the unphosphorylated form of β-catenin in the nucleus of ZO-1-WT or ZO-1-PDZ-transfected cells compared with control vector transfected cells (Fig. 7E).

Figure 6. Effect of ZO-1 siRNAs on in vitro cell invasion. BT549 tumor cells were transfected with either ZO-1

Si3 siRNA or the scrambled Si3 control siRNA at 2, 20, or 50 nmol/L. A, Western blot analyses of MT1-MMP expression in the transfected cells. Actin was used as a loading control. B, analysis of the invasive capacity of ZO-1 siRNA-transfected BT549 cells compared with scrambled control siRNA transfectants in a modified Boyden chamber assay (P < 0.001).

DISCUSSION

In the present study, we have shown that (a) the lack of occludin expression and the cytoplasmic localization of ZO-1 correlates with the invasive phenotype of tumor cells, (b) MT1-MMP is expressed only in the occludinnegative invasive tumor cell lines, (c) ZO-1 and MT1-MMP can be detected in invasive tumor clusters in breast carcinomas, (d) the reduction of ZO-1 in invasive tumor cells decreases the expression of MT1-MMP and their invasive properties, and (e) the activation of the MT1-MMP promoter by ZO-1 is associated with the activation of the β-catenin/TCF/LEF signaling pathway.

We first showed that occludin was detected only in noninvasive tumor cells as a membrane staining. This agrees with several reports showing that the loss of occludin expression correlates with tumor progression in human endometrial and prostate carcinomas [START_REF] Busch | Down-regulation of CEACAMl in human prostate cancer: correlation with loss of cell polarity, increased proliferation rate, and Gleason grade 3 to 4 transition[END_REF][START_REF] Tobioka | Occludin expression decreases with the progression of human endometrial carcinoma[END_REF]. Similarly, in vitro, occludin was also observed in welldifferentiated epithelial cell lines but not detected in invasive tumor cells and in dedifferentiated epithelial cell lines [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF][START_REF] Hoevel | Expression and targeting of the tight junction protein CLDN1 in CLDN1negative human breast tumor cells[END_REF][START_REF] Wan | Tight junction properties of the immortalized human bronchial epithelial cell lines Calu-3 and 16HBE14o[END_REF]. These findings strengthened the hypothesis that occludin expression could be a hallmark of a noninvasive status. Interestingly, we also found that, although ZO-1 was present in all cell lines, it was present in the cytoplasm of invasive tumor cells, whereas in noninvasive cells it localized to the plasma membrane. Accordingly, Gottardi et al. have described that ZO-1 delocalizes from the plasma membrane in migrating epithelial cells at the edge of wounded monolayers, correlating with dedifferentiation of these cells and suggesting that ZO-1 may have a role in signaling events [START_REF] Gottardi | The junction-associated protein, zonula occludens-1, localizes to the nucleus before the maturation and during the remodeling of cell-cell contacts[END_REF]. In vivo, we found a diffuse cytoplasmic distribution of ZO-1 in breast tumors as also reported by Martin et al. [START_REF] Martin | Loss of tight junction plaque molecules in breast cancer tissues is associated with a poor prognosis in patients with breast cancer[END_REF]. Taken together, these data show that a reorganization of membrane-associated occludin/ZO-1 complexes is associated with enhanced migratory and invasive properties of epithelial tumor cells and agree with the general concept describing that the acquisition of an effective invasive phenotype is related to a dedifferentiation of the epithelial tumor cells.

The redistribution of ZO-1 in invasive tumor cells suggests that, besides its role in cell-cell adhesion, ZO-1 is implicated in the regulation of invasive genes. Accordingly, we show that the accumulation of ZO-1 in the cytoplasm is associated with the expression of MT1-MMP in invasive tumor cell lines, a correlation that can also be observed in invasive tumor clusters of breast carcinomas. In the literature, it has been repeatedly reported that MT1-MMP is associated with the migratory and invasive phenotype of tumor cells; in particular, MT1-MMP has been directly implicated in the migration/invasion of several dedifferentiated cell lines [START_REF] Gilles | Matrix metallop roteinases and epithelia-to-mesenchymal transition: implications for carcinoma metastasis[END_REF][START_REF] Gilles | Contribution of MT1-MMP and of human laminin-5 γ2 chain degradation to mammary epithelial cell migration[END_REF][START_REF] Ueda | Sequence-specific silencing of MTl-MMP expression suppresses tumor cell migration and invasion: importance of MTl-MMP as a therapeutic target for invasive tumors[END_REF][START_REF] Sounni | MTl-MMP expression promotes tumor growth and angiogenesis through an up-regulation of vascular endothelial growth factor expression[END_REF][START_REF] Sato | A matrix metalloproteinase expressed on the surface of invasive tumour cells [see comments[END_REF]. More than a correlation, we now show that ZO-1 can modulate MT1-MMP levels. Indeed, the down-regulation of ZO-1 levels by siRNAs clearly decreased MT1-MMP expression in invasive tumor cells as well as their in vitro invasive capacity. Inversely, the transfection of ZO-1 into BT549 cells increased the MT1-MMP promoter activity and their invasive properties. Only a few studies have implicated ZO-1 in the regulation of gene expression. The Drosophilia orthologue of ZO-1, tamou, has been linked to the regulation of the extramacrochaetae gene that participates in neural development [START_REF] Takahisa | The Drosophila tamou gene, a component of the activating pathway of extramacrochaetae expression, encodes a protein homologous to mammalian cell-cell junction-associated protein ZO-1[END_REF]. Balda and Matter have shown that ZO-1 via its interaction with a Y-box transcription factor, ZO-1-associated nucleic acid-binding protein, regulates ErbB-2 expression, thereby modulating epithelial cell differentiation and morphogenesis [START_REF] Balda | The tight junction protein ZO-1 and an interacting transcription factor regulate ErbB-2 expression[END_REF]. In addition, the NH 2 -terminal three PDZ domains of ZO-1, which by themselves are unable to localize to the plasma membrane and accumulate in the cytoplasm, promote a phenotypic conversion characterized by the repression of epithelial markers and the induction of mesenchymal features in transfected MDCK I and corneal epithelial cells [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF][START_REF] Ryeom | Truncation mutants of the tight junction protein ZO-1 disrupt corneal epithelial cell morphology[END_REF]. In these studies, transfection of full-length ZO-1 led to its recruitment to the plasma membrane in MDCK I cells and did not induce any phenotypic conversion. This emphasized that ZO-1, once delocalized from the cell membrane, could play a role in the regulation of genes controlling cell differentiation [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF][START_REF] Ryeom | Truncation mutants of the tight junction protein ZO-1 disrupt corneal epithelial cell morphology[END_REF]. Interestingly, in our experiments, transient transfection of either the PDZ domains or the full-length ZO-1 construct activated the human MT1-MMP promoter in invasive BT549 tumor cells. Our cells do not express occludin, which is required to maintain ZO-1 to the TJ plasma membrane, likely explaining why, in contrast to MDCK I cells, full-length ZO-1 also mediates an effect. Taken together, our results provide evidence that ZO-1 can regulate MT1-MMP expression and could thereby modulate the invasive properties of epithelial tumor cells.

The mechanisms by which ZO-1 affects the expression of tumor-promoting genes, such as MT1-MMP, remain unclear. Because of its delocalization in dedifferentiated epithelial cells, it would be pertinent to believe that ZO-1, once delocalized from the cell membrane, could modulate specific transcriptional pathways that control the expression of invasion-associated genes.

An interaction between ZO-1 and the β-catenin/TCF/LEF pathway has accordingly been documented. Reichert et al. have indeed reported the activation of the β-catenin/TCF/LEF transcription pathway following the transfection of MDCK I cells with the ZO-1-PDZ construct [START_REF] Reichert | The PDZ domains of zonula occludens-1 induce an epithelial to mesenchymal transition of Madin-Darby canine kidney I cells. Evidence for a role of β-catenin/Tcf/Lef signaling[END_REF]. In agreement with the results in MDCK I cells, transfection of BT549 cells with the Z0-1-PDZ and the Z0-1-WT construct activated the β-catenin/TCF/LEF pathway concomitantly with the MT1-MMP promoter. Consistent with the activation of β-catenin/TCF/LEF transcriptional activity, the unphosphorylated form of β-catenin accumulated in the nucleus of BT549 cells transfected with Z0-1-WT and ZO-l-PDZ compared with vector transfected cells. This agrees with the literature data showing that the unphosphorylated form of β-catenin is implicated in the transcriptional activity of β-catenin in the nucleus [START_REF] Staal | Wnt signals are transmitted through N-terminally dephos-phorylated β-catenin[END_REF], although the selective binding of β-catenin to E-cadherin or TCF/LEF factors does only rely on the NH 2 -terminal unphosphorylated status of β-catenin [START_REF] Gottardi | Distinct molecular forms of β-catenin are targeted to adhesive or transcriptional complexes[END_REF]. Thus, although the implication of β-catenin-independent regulatory mechanisms cannot be excluded, our data suggest that the delocalization of ZO-1 could activate the β-catenin/TCF/LEF pathway, which could in turn induce MT1-MMP expression. Indeed, MT1-MMP has recently been reported to be a target gene of the β-catenin/TCF/ LEF pathway and β-catenin was shown to activate MT1-MMP transcription via its interaction with a TCF-binding element present in the MT1-MMP promoter [START_REF] Kqubek | β-Catenin activates a coordinated expression of the proinvasive factors laminin-5 72 chain and MTl-MMP in colorectal carcinomas[END_REF][START_REF] Takahashi | Identification of membrane-type matrix metalloproteinase-1 as a target of the β-catenin/Tcf4 complex in human colorectal cancers[END_REF]. The mechanism by which ZO-1 affects the β-catenin pathway is still unknown and a direct link between ZO-1 and β-catenin has not been established. Nevertheless, ZO-1 could interact with other molecules of the Wnt signaling pathway or of adherens junctions, thereby modulating the βcatenin signaling functions. For instance, in cells lacking TJs or in early stages of cell-cell adhesion of epithelial cells, ZO-1 has been reported to directly bind to α-catenin, a molecule that links the actin cytoskeleton to the E-cadherin/β-catenin complex, suggesting, that under certain conditions, α-catenin acts as a linker between ZO-1 and β-catenin [START_REF] Imamura | Functional domains of α-catenin required for the strong state of cadherin-based cell adhesion[END_REF][START_REF] Itoh | Involvement of ZO-1 in cadherin-based cell adhesion through its direct binding to α catenin and actin filaments[END_REF]. More recently, ZO-1 and ZO-2 have been implicated in localizing ARVCF, which, like βcatenin, is an armadillo repeat protein and also binds to E-cadherin, to the plasma membrane or the nucleus [START_REF] Kausalya | Association of ARVCF with ZO-1 and ZO-2: binding to PDZ-domain proteins and cell-cell adhesion regulate plasma membrane and nuclear localization of ARVCF[END_REF].

Irrespective of the precise mechanism involved, our data are consistent with a scenario where ZO-1, if delocalized from TJ, can activate the β-catenin/TCF/LEF pathway, which in turn could regulate the expression of MT1-MMP.

In conclusion, our data establish a correlation between MT1-MMP expression and a cytoplasmic deloalization of ZO-1 in invasive occludin-negative tumor cells. Furthermore, we show that ZO-1 contributes to tumor cell invasion by modulating MT1-MMP expression likely via activation of β-catenin/TCF/LEF transcriptional activity. These findings emphasize the dual role of adhesion molecules, such as ZO-1, as structural components of cell-cell junctions in noninvasive epithelial cells and as contributing factors to tumor progression through their ability to induce the expression of tumor-promoting genes in invasive cells. analyses of the localization of ZO-1 in BT549 tumor cells transfected with the empty expression vector (pLNCX), the ZO-1-WT expression vector (ZO-1), or the NH 2 -terminal ZO-1-PDZ expression vector (PDZ). B, Boyden chamber analyses of the invasive capacity of BT549 cells transfected with ZO-1 -WT (ZO-1 ) or ZO-1 -PDZ (PDZ) or the control vector (pLNCX). Data are fold induction relative to the transfection of the empty expression vector (P < 0.001). C, MT1-MMP promoter reporter assay. BT549 tumor cells were cotransfected with a human MT1-MMP promoter luciferase reporter plasmid and either the empty expression vector (pLNCX), the ZO-1-WT expression vector (ZO-1), or the NH 2 -terminal ZO-1-PDZ expression vector (PDZ). Data are fold induction relative to the cotransfection of the MT1-MMP reporter plasmid with the empty expression vector (P < 0.001). D, TOP-FLASH/FOP-FLASH reporter assay. The TOP-FLASH or FOP-FLASH reporter plasmids were cotransfected with the empty expression vector (pLNCX), the ZO-1 -WT expression vector (ZO-1 ), or the ZO-1 -PDZ expression vector (PDZ) in BT549 tumor cells. The normalized FOP values were subtracted from the normalized TOP values. Data are fold induction relative to the values obtained in the controls transfected with the empty expression vector (P < 0.001). E, Western blot analyses of the unphosphorylated form of β-catenin in cytosolic and nuclear fractions of BT549 cells transfected with either the empty expression vector (pLNCX), the ZO-1-WT expression vector (ZO-1), or the NH 2 -terminal ZO-1-PDZ expression vector (PDZ). Actin and cmyc expression were analyzed as controls.

Belgium), the Fédération Belge Contre le Cancer, the Fonds spéciaux de la Recherche (University of Liège), the Centre Anticancéreux près l'Université de Liège, the Fortis Banque Assurances, the Fondation Léon Frédéricq (University of Liège), the DGTRE from the Région Wallonne, the Fonds d'Investissements de la Recherche Scientifique (CHU, Liège, Belgium), the Interuniversity Attraction Poles Programme-Belgian Science Policy (Brussels, Belgium), and the Agency for Science, Technology and Research, Singapore.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

We thank Dr. H.C. Clevers for the TOP-FLASH, FOP-FLASH, and TCF-4 expression vector.

Figure 4 .

 4 Figure 4. Down-regulation of MT1-MMP expression in invasive tumor cell lines by ZO-1 siRNA. Western blot analyses of ZO-1, MT1-MMP, and actin expression were done on total extracts of MDA-435, BT549, and Hs578T transfected with the ZO-1 Si3 siRNA or its corresponding scrambled control siRNA at a 50 nmol/L concentration. HT1080 cells treated with concavalin A (HT1080+) were used as a control to reveal the two forms of MT1-MMP at 60 and 45 kDa.

Figure 5 .

 5 Figure 5. Down-regulation of MT1-MMP expression by ZO-1 siRNAs. BT549 tumor cells were transfected with three different ZO-1 siRNAs (Si1-Si3) or corresponding scrambled control siRNAs (ScrSi1-ScrSi3) at 50 nmol/L. A, Western blot analyses of ZO-1 expression in BT549 cells transfected with the siRNAs. B, RT-PCR analyses of MT1-MMP and 28S expression in BT549 cells transfected with the siRNAs. C, quantification of RT-PCR analyses of MT1-MMP mRNA normalized with 28S values. Data are fold induction relative to the scrambled siRNA controls (P < 0.05). D, Western blot analyses of MT1-MMP and actin expression in BT549 cells transfected with the ZO-1 siRNAs. HT1080 cells treated with concavalin A (HT1080+) were used as a control to reveal the two forms of MT1-MMP at 60 and 45 kDa. E, quantification of the level of the 60 kDa activated MT1-MMP protein normalized to actin levels. Data are fold induction relative to the scrambled siRNA controls (P <0.001).

Figure 7 .

 7 Figure 7. Activation of β-catenin/TCF/LEF pathway and MT1-MMP promoter by ZO-1. A, immunofluorescence

Published in: Cancer Research (2005), vol. 65, iss. 17, pp. 7691-7698 Status: Postprint (Author's version)

Acknowledgments

Grant support: CGRI-FNRS-INSERM coopération, the Lions Club of Soissons, the Communauté française de Belgique (Actions de Recherches Concertées), the Commission of European Communities (FP6 Brecosm), the Fonds de la Recherche Scientifique Médicale, the Fonds National de la Recherche Scientifique (FNRS,


Mazen Zein
email: mazen.zein@irccyn.ec-nantes.fr

Philippe Wenger

Damien Chablat

An Algorithm for Computing Cusp Points in the Joint Space of 3-RPR Parallel Manipulators

Keywords: Cusp points, 3-RPR parallel manipulator, singularity, joint space, assembly mode

This paper presents an algorithm for detecting and computing the cusp points in the joint space of 3-RPR planar parallel manipulators. In manipulator kinematics, cusp points are special points, which appear on the singular curves of the manipulators. The nonsingular change of assembly mode of 3-RPR parallel manipulators was shown to be associated with the existence of cusp points. At each of these points, three direct kinematic solutions coincide. In the literature, a condition for the existence of three coincident direct kinematic solutions was established, but has never been exploited, because the algebra involved was too complicated to be solved. The algorithm presented in this paper solves this equation and detects all the cusp points in the joint space of these manipulators.

I. Introduction

Critical points such as bifurcation points, turning points and cusp points can be considered of particular interest in applied sciences. The application of cusp points on manipulators kinematics is what we will be considering here. A serial (resp. parallel) manipulator with cusp points in its workspace (resp. in its joint space) can change posture (resp. assembly mode) without crossing a singularity. When no cusp points exist, such a singular-free motion is not possible. Regarding serial manipulators, it had been widely believed that they all should meet a singularity when changing posture. " Parenti-Castelli and Innocenti (1988)" were the first to prove the existence of non-singular posture changing motions on in 6-DOF serial manipulators. Similar results were found for 3R serial manipulators in " [START_REF] Burdick | A Classification of 3R regional manipulator Singularities and Geometries[END_REF][START_REF] Ph | A New General Formalism for the Kinematic Analysis for all Non-Redundant Manipulators[END_REF]". " Elomri and Wenger (1995)" 2 have shown that serial manipulators having the ability to change posture without meeting a singularity are characterised by the existence of cusp points in their workspace, where three inverse kinematic solutions coincide. Cusp points in serial manipulators can be determined by looking for the triple roots of the inverse kinematics polynomial "Elomri and [START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF]" or from the equation of the workspace boundary " [START_REF] Ottaviano | A cartesian representation for the boundary workspace of 3R manipulators[END_REF]".

Most fully parallel manipulators have multiple direct kinematic solutions, which are associated with the different assembly modes. " [START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF]" first showed that to move from one assembly mode to another, a fully parallel manipulator had to cross a singularity. But, "Innocenti and Parenti-Castelli (1998)" found a 3-RPR parallel manipulator able to change its assembly mode without crossing a singularity. One year later, " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" pointed out that 3-RPR and octahedral manipulators can undertake non-singular assembly changing motions, if a point with triple direct kinematic solutions exists in their joint space, this point is "a cusp point" in a section of the joint space. He established a condition for the existence of cusp points. But this condition has never been exploited, because the algebra involved in this condition was found to be too complicated. [START_REF] Wenger | Workspace and Assembly modes in Fully-Parallel Manipulators : A Descriptive Study[END_REF]" showed that to accomplish a non-singular assembly-mode changing motion, a 3-RPR manipulator platform should encircle a cusp point in its joint space. Thus, the determination of the cusp points is of interest for planning trajectories. In this paper, an algorithm for detecting all cusp points and computing their coordinates in the joint space of 3-RPR parallel manipulators is established; it is based on the abovementioned condition. This work finds application in both design and trajectory planning. In the following sections, we present the 3-RPR parallel manipulators studied and their constraint equations, we explain briefly the cusp points existence condition established by " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]", then the algorithm is described and run on two different 3-RPR manipulators.

II. Preliminaries

II.1 Manipulators studied

The manipulators under study are 3-DOF planar parallel manipulators with three extensible leg rods (Fig. 1). These manipulators have been frequently studied, for example by " [START_REF] Sefrioui | On the quadratic nature of the singularity curves of planar three-degree-offreedom parallel manipulators[END_REF][START_REF] Sefrioui | On the quadratic nature of the singularity curves of planar three-degree-offreedom parallel manipulators[END_REF]"Merlet (2000)". Each of the three extensible leg rods is actuated with a prismatic joint. The geometric parameters of the manipulators are the three sides of the moving platform d 1 , d 2 , d 3 and the position of the base revolute joint centres defined by A 1 , A 2 and A 3 . The reference frame is centred at A 1 and the x-axis passes through A 2 . Thus, A 1 = (0, 0), A 2 = ( A 2x , 0) and

A 3 = ( A 3x , A 3y ). x A 1 A 2 B 2 B 3 B 1  L 1 L 2   d 1 d 3 d 2   y A 3 L 3     h Fig 1.
The 3-RPR parallel manipulator under study.

II.2 Constraint equations

Let L  (L 1 , L 2 , L 3 ) define the lengths of the three leg rods and let   ( 1 ,  2 ,  3 ) define the three angles between the leg rods and the x-axis. The six parameters (L, ) can be regarded as a configuration of the manipulator but only three of them are independent, so that the configuration space is a 3-dimensional manifold embedded in a 6-dimensional space. The dependency between (L, ) can be identified by writing the fixed distances between the three vertices of the mobile platform B 1 , B 2 , B 3 , which yield the following constraint equations

            2 1 2 1 2 1 1 2 2 3 2 3 2 2 2 3 1 3 1 3 3 ( , ) ( , ) ( , ) ( , ) ( , ) 0 (
, ) ( , ) ( , ) ( , ) ( , ) 0 ( , ) ( , ) ( , ) ( , ) 0 ( , ) T T T d d d                          L θ b L θ b L θ b L θ b L θ L θ b L θ b L θ b L θ b L θ L θ b L θ b L θ b L θ b L θ (1)
where b i is the vector defining the coordinates of B i in the reference frame as function of L and . For more simplicity, (L, ) will be omitted in the following equations.

Expanding each  i as a series about the configuration (L, ) yields

2 3 3 3 3 1 1 1 1 33 11 2 .. 0 1 ... ! 1 ! i i i i i i i j j j j j j j j n i i i jj jj LL LL L L n                                                               (2)
If one keeps only the first-order and second-order terms, Eq. ( 2) can be written in matrix form as follows 3) can be used to describe an arbitrary local motion at a given configuration of the manipulator " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]". When first order terms of Eq. ( 3) are sufficient to describe the motion, the manipulator is in a regular configuration and the following equation can be used instead of Eq. ( 3)

T TT T T T TT T                                                                                                    L θ θ L L θL θL θL (3) Equation (
    ,, 0       Γ L θ Γ L θ θL θL (4)
Otherwise the configuration (L, ) is special and the manipulator meets a singularity. This happens when the constraint Jacobian  Γ/ θ drops rank so that the second order terms of the equation ( 3) are needed to describe the constraints. The three vertices of the moving platform have the following coordinates in the fixed reference frame

            1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 cos sin ; cos sin ; cos sin . TT x T xy L L A L L A L A L                      bb b
Thus, the constraint Jacobian can be put in the following form

        1 2 1 2 12 2 1 21 2 2 2 2 3 2 3 2 3 3 3 23 3 2 2 23 3 3 1 3 1 3 31 3 3 3 1 31 3 1 3 3 0 ( ) ( ) 20 (( 0 )) xx x x x x yy xx yy L A s L s L L s A s L A A s L A A s L s A c L s A c L A s L s L A s L s A c A c                            Γ θ (5)
where

  sin ii s   ,   cos ii c   and   sin ij i j s   .

III. Existence condition of cusp points

To determine the cusp points, we need first to characterise the singular configurations. " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" determined the singularities of the 3-RPR manipulator by looking for the configurations where det( Γ/ θ) vanishes. In our work, we have used a geometric approach that is much more direct than the calculation of the determinant, which does not simplify easily. It is well known that a 3-RPR manipulator is in a singular configuration whenever the axes of its three leg rods intersect (possibly at infinity). The derivation of this geometric condition is straightforward and yields the following equation

  2 2 31 3 3 3 3 12 0 x x y A s s A s A c s    (6)
For serial 3-DOF manipulators, the cusp points can be determined by deriving the condition under which the inverse kinematics polynomial admits three identical roots (Elomri 1995). However this approach is much more complicated when applied to the direct kinematics polynomial of 3-RPR manipulators because this polynomial is of degree 6. An interesting alternative approach was proposed in " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" by writing the condition under which the manipulator loses first and second order constraints. The resulting condition for triple coalescence of assembly modes was shown to take the following form

2 22 3 12 1 2 3 2 2 2 0 T u u u               vv θ θ θ ( 7
)
where v is a unit vector in the right kernel of matrix  Γ/ θ , and u 1 , u 2 , u 3 are the three components of the unit vector u that spans the left kernel. Vectors u and v can be chosen in the set of nonzero rows and columns of the adjoint of matrix  Γ/ θ (i.e. the matrix of cofactors of the transpose of  Γ/ θ ), respectively.

Calculating the adjoint of  Γ/ θ from Eq. ( 5) yields adj 1 2 2 5 3 5 3 4 2 6 3 6

1 4 4 5 1 6 k k k k k k k k k k k k k k k k k k              Γ θ (8)
where

          1 2 3 2 2 3 23 3 2 2 3 1 13 3 3 3 3 3 3 3 2 3 2 23 3 3 2 2 2 x x y xy x x y k L A A s L s A c k L L s A s A c k L A A s L s A c                    4 1 3 13 3 1 3 1 5 2 1 12 2 2 6 1 2 12 2 1 2 2 2 xy x x k L L s A s A c k L L s A s k L L s A s        (9)
Taking u (resp. v) as the first row (resp. column) of ( 9), the equation ( 7) can be written as

  12 2 2 2 3 1 2 1 2 3 4 1 4 1 2 2 5 3 5 3 4 2 2 2 14 0 kk k k k k k k k k k k k k k k kk                         θ θ θ (10)
where

          2 1 2 1 2 21 1 2 21 1 1 2 21 2 2 2 1 21 2 2 2 2 3 2 2 2 3 23 2 3 23 3 2 3 2 3 3 2 3 23 2 23 3 3 2 1 3 1 3 31 3 1 1 3 3 2 0 2 0 0 0 0 0 0 0 () 20
() 0 0 2 x x xx y xx y x y L A c L c L L c L L c L A c L c L A A c L L c L c A s L A A c L L c L c A s L A c L c A s L L                                                   θ θ θ   31 1 3 31 3 1 31 3 3 3 3 0 0 0 0 xy c L L c L L c A c A s              (11)
" [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" left equation ( 10) as such and no information was provided on how to use it in a computer program. He noted that the expansion of this equation was too complicated to yield any real insight.

We have developed an algorithm to solve this equation for any 3-RPR manipulator and we have implemented it in Maple. This algorithm detects all the cusp points inside the joint space of any 3-RPR manipulators and computes their coordinates. We present it in the next section, and we run it on two different 3-RPR manipulators.

IV. Algorithm for calculating cusp points

The existence of cusp points allows the 3-RPR manipulator to undertake non-singular assembly mode changing trajectories, these special trajectories can be executed by encircling a cusp point. " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" stated that cusp points are pernicious and should be avoided or designed out by judicious dimensioning.

The configuration of the 3-RPR manipulator is given by six parameters: the three rod lengths (L 1 , L 2 ,

L 3 ), and the platform position variables ( 1 ,  2 ,  3 ). Only three of these parameters are independent. In order to reduce the dimension of our problem, " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF])" shows that it is possible to consider two-dimensional slices of the configuration space by fixing one of the leg rod lengths. By doing so, the manipulator configuration can be fully defined by only two parameters. For example, for a fixed value of L 1 , a configuration may be fully defined by either (, 1 ) or (L 2 , L 3 ). Note that in the first case, the configuration is defined in the output space by the position and the orientation of the moving platform (L 1 and  1 define the position of B 1 in the plane and  defines the orientation of the moving platform in the plane). In the second case, the configuration is defined in the joint space by the three leg rod lengths.

In our work, we have always taken L 1 as the fixed parameter. After fixing the value of L 1 , we first calculate the singularity curves in (L 2 , L 3 ), and then we compute all the cusp points of this twodimensional slice.

IV.1 Algorithm

If we consider equation ( 6), we notice that it is a function of ( 1 ,  2 ,  3 ). The existence condition of cusp points ( 10) is a function of (L 1 , L 2 , L 3 ) and ( 1 ,  2 ,  3 ). Our first goal is to establish an equation, which is a function of (L 1 , ,  1 ), and then to solve it to obtain the cusp points coordinates. Thus, we first consider the following set of equations computed from the geometry of the manipulator.

                        2 1 1 1 2 2 3 1 1 3 3 3 1 1 1 2 2 3 1 1 3 3 3 cos cos cos cos cos s sin sin sin sin sin sin x x y A L d L A L d co L Ld L A L d L                                            (12)
The algorithm for detecting cusp points is implemented in MAPLE; its steps are presented below:

1. First, the expression of cos( 2 ), cos( 3 ), sin( 2 ) and sin( 3 ) in ( 12) are substituted into the singularities equations ( 6). Then, sin(), cos(), cos( 1 ) and sin( 1 ) are replaced by the tangents of their half angles tan(/2) and tan( 1 /2), as a consequence we obtain an equation of the form:

  1 1 1 , , 0 F L t t  (13) where   tan t   and   11 tan t   . 2.
Then, the expression of cos( 2 ), cos( 3 ), sin( 2 ) and sin( 3 ) in ( 12) are substituted into equations ( 9) and ( 11) and sin(), cos(), cos( 1 ) and sin( 1 ) are replaced by the tangents of their half angles tan(/2) and tan( 1 /2). We get an equation of the form:

  1 1 1 , , 0 E L t t  (14)
So, we notice that the two equations ( 6) and ( 10) are written now as function of three parameters only.

3. We fix now L 1 , and we input the manipulator parameters d 1 , d 2 , d 3 , A 2x , A 3x and A 3y . We have noticed that the direct substitution of the real values of

 

sin  and   cos  into the equations ( 13) and ( 14) make the equations resolution very complicated in the following steps. Thus, we write   sin  and   cos  as a function of an intermediate parameter h, which is the altitude of the moving triangle. 4. The Maple resultant function is used to eliminate

  tan t  
from the two equations ( 13) and ( 14). The resulting equation is a polynomial of degree 96 in

  ( 15
)
where Q is a 24 th -order univariate polynomial in t 1 and P 1 , P 2 ,…,P n are quadratic and quartic polynomials in t 1 . Note that the factor form cannot be obtained without the intermediate parameter h. 5. We input the parameter h value. We solve equation ( 15). Each real root t 1i is back-substituted into (13), which is then solved for t. For every t 1i , we obtain different t ij . Finally, we get a number of solution couples (t ij ,t 1i ). 6. We substitute the values of each solution couple (t ij ,t 1i ) into ( 14), and we keep only those that satisfy this equation. 7. The solutions (t ij ,t 1i ) kept in the last step should give the coordinates of the cusp points. To verify this, we calculate the direct kinematic solutions for each solution (t ij ,t 1i ). In many instances, we have found that some solutions do not yield three coincident solutions, which means that they are not associated with cusp points. So we reject them and we keep only those solutions that give three coincident direct kinematic solutions. These couples are the coordinates of the cusp points, we call them ( ij , 1i ) cusp . After executing our algorithm hundreds of times, we have noticed that in each case all cusp points were determined by the 24 th -order polynomial Q of equation ( 15), that is, all remaining factors provided spurious solutions. Thus, we may conjecture that the cusp points are determined by Q, although we have no mathematical proof for this fact. All the real roots of Q are the cusp points. With this conjecture, our algorithm simplifies significantly because instead of solving (16) (a 96 th order polynomial) we just have to solve polynomial Q (a 24 th order polynomial). To implement this result in the algorithm, we must change steps 5 and 6 into the following steps 5' and 6', and eliminate step 7: 5'. We input the parameter h real value. We solve the polynomial Q. We substitute every real root 13), and we solve it for tan(/2). For every  1i , we obtain different values  ij . Finally, we get a number of couples ( ij , 1i ).

 1i of Q into equation (
6'. We substitute the values of each couple ( ij , 1i ) into equation ( 14). The couples that satisfy this equation are the cusp points coordinates. We call them ( ij , 1i ) cusp . Finally, to obtain the coordinates of the cusp points in the joint space (L 1 , L 2 , L 3 ), we use the following equations computed from the geometry of the manipulator:

            2 2 2 2 1 1 1 1 1 1 cos cos sin sin x L A L b L b           (17)                             2 3 1 1 3 3 2 3 1 1 3 cos cos cos sin sin sin sin cos cos sin x y A L b L A L b                   (18)
and we obtain the cusp points in a slice of the joint space for a fixed value of L 1 .

IV.2 Important conclusion

In step 7 of the algorithm, we have noticed that the cusp existence condition generates solutions that do not provide triple direct kinematic solutions. This means that the cusp existence condition established by " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" is not a necessary and sufficient condition but only a sufficient condition.

IV.3 Algorithm execution

In this paragraph, we present the results of some executions of the algorithm for two different 3-RPR parallel manipulators. Only some slices of the joint space are presented. However, for both manipulators, we have run the algorithm for L 1 varying from 0 to 50 with a scanning step of 0.1. We have noticed that the number of cusp points varies from one slice to another. Note that in serial manipulators, the number of cusp points does not depend on the workspace section (if we consider just the sections which passes through the axis of the first revolute joint).

On the other hand, the number of cusp points stabilizes for sufficiently large values of L 1 . For example, there are always four cusp points for the first manipulators as soon as L 1 >31. Finally, the maximal number of cusp points depends on the geometry of the manipulator. For example, the second manipulator may have at most 6 cusp points whereas the first one may has 8 cusp points.

We have also found manipulators with only 0, 2 or 4 cusp points. A symmetric manipulator with two similar platform has 0 cusp points as this manipulator is non-cuspidal "Mcaree (1999)". We have not been able to find manipulators with more than 8 cusp points.

The algorithm execution time slightly depends on the value of L 1 and of the 3-RPR manipulator parameters. It highly depends on the number of digits required for the calculation. For 90 digits (which is necessary to guaranty a good accuracy), it is about two minutes on a computer equipped with a 3GHz-Pentium 4 with 512 Mo of Ram.

IV.3.1 3-RPR parallel manipulator used in "Mcaree (1999)"

First, we begin with the 3-RPR parallel manipulator used in " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF]". The geometric parameters of this manipulator are recalled below in an arbitrary length unit:

A 1 =(0, 0) d 1 =17.04 A 2 =(15.91, 0) d 2 =16.54 A 3 =(0, 10) d 3 =20.84
 A slice for L 1 =14.98 For the same fixed value L 1 =14.98, as in " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" and "Innocenti (1998)", the algorithm detects six cusp points instead of five identified in " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]". Figure 2 shows the singular curves of the 3-RPR manipulator for L 1 =14.98, and the six cusp points pinpointed with circles. The sixth point missed by " [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]" is the point A, it is circled with bold lines and in-boxed in a separate view. The zoomed view shows that it is really a cusp point.  A slice for L 1 =34 For the same manipulator with L 1 =34, four cusp points are found. Figure 3 shows the singularity curves and the four cusps in the slice of the joint space for L 1 =34.  A slice for L 1 =27 For the same manipulator with L 1 =27, eight cusp points are found. Figure 4 shows the singularity curves and the eight cusps in the slice of the joint space for L 1 =27.

L 2 L 3 A B C D E F Fig 2.
L 2 L 3 A B C D E F G H

IV.3.2 Another 3-RPR parallel manipulator

The geometric parameters of the second manipulator are given below in an arbitrary length unit:

A 1 =(0, 0) (11,27) d 3 =4

d 1 =13 A 2 =(30, 0) d 2 =9 A 3 =
 For L 1 =3: In this manipulator joint space section, the algorithm detects four cusp points. Figure 5 shows the singularity curves and the four cusps in the slice of the joint space for L 1 =3.

V. Conclusions

In this paper, we have reviewed and exploited the cusp points existence condition defined by "Mcaree (1999)", we have found that it is only a sufficient condition and not a necessary one. An algorithm, able to detect and to compute cusp points inside any section of the joint space of any 3-RPR parallel manipulator, has been established. To the best of the authors' knowledge, such an algorithm had never been proposed before. The algorithm results in a 96 th degree univariate polynomial that can be put in a factored form. We have showed with intensive numerical experiments that the cusp points coordinates are the real roots of a 24 th degree univariate polynomial, which is one of the factors of the 96 th polynomial. Finally, the results of four numerical executions of the algorithm on two different manipulators has been exposed. Determination of the cusp points is an important issue for planning non-singular assembly mode changing trajectories in parallel manipulators. Contrary to serial manipulators, the number of cusp points is not the same in all sections. Future work will investigate the transition slices of the joint space where the number of cusp points changes, and its physical meaning for the behaviour of the manipulator.

Fig 3 .

 3 Fig 3. Singular curves and cusp points in a slice of the 3-RPR manipulator joint space (L 2 ,L 3 ) for L 1 =34. The coordinates of the four cusp points are given in the table 2 below

Fig 4 .

 4 Fig 4. Singular curves and cusp points in a slice of the 3-RPR manipulator joint space (L 2 ,L 3 ) for L 1 =27.

Fig 5 .

 5 Fig 5. Singular curves and cusp points in a slice of the joint space (L 2 ,L 3 ) of the second 3-RPR manipulator studied for L 1 =3.

  The coordinates of the six cusp points are given in the table1 below

					2 ,L 3 ) for
			L 1 =14.98.
			 1	L 2	L 3
	Cusp A	50.67 deg	-69.12 deg	0.84	3.77
	Cusp B	-2.59 deg	177.32 deg	13.85	6.26
	Cusp C	-122.89 deg	114.05 deg	31.27	16.17
	Cusp D	57.48 deg	133.77 deg	30.44	26.61
	Cusp E	-0.59 deg	15.46 deg	16.02	29.56
	Cusp F	170.37 deg	-10.65 deg	17.98	26.44

Singular curves and cusp points in a slice of the 3-RPR manipulator joint space (L Tab 1. Coordinates of the six cusp points for L 1 =14.98.

  The coordinates of the four cusp points are given in the table2 below

			 1	L 2	L 3
	Cusp A	-3.84 deg	-167.01 deg	33.22	19.00
	Cusp B	52.71 deg	-61.76 deg	19.46	22.68
	Cusp C	-1.07 deg	15.43 deg	35.00	48.64
	Cusp D	55.85 deg	128.19 deg	49.14	45.52
		Tab 2. Coordinates of the four cusp points for L 1 =34.

  Coordinates of the eight cusp points for L 1 =27.

			 1	L 2	L 3
	Cusp A	-0.95 deg	15.47 deg	28.01	41.63
	Cusp B	56.20 deg	129.36 deg	42.21	38.54
	Cusp C	-5.11 deg	-168.45 deg	26.31	11.84
	Cusp D	52.23 deg	-63.22 deg	12.56	15.71
	Cusp E	-168.17 deg	8.70 deg	5.92	29.74
	Cusp F	-125.54 deg	43.86 deg	7.98	27.36
	Cusp G	-113.95 deg	63.88 deg	13.96	21.66
	Cusp H	-129.36 deg	103.65 deg	35.57	4.80
		Tab 3.

The coordinates of the eight cusp points are given in the table

3 below

  The coordinates of the four cusp points are given in the table 4 below Tab 4. Coordinates of the four cusp points of the second manipulator studied for L 1 =3.

			 1	L 2	L 3
	Cusp A	12.52 deg	-145.11 deg	19.80	29.43
	Cusp B	156.76 deg	-63.48 deg	40.68	31.11
	Cusp C	-168.74 deg	26.38 deg	40.08	29.14
	Cusp D	-20.32 deg	114.01 deg	19.11	27.01