D Dumont

F Lebon
email: lebon@lmgc.univ-montp2.fr

A Ould Khaoua

A Numerical Tool for Periodic Heterogeneous Media: Application to Interface in AlÕSiC Composites

A wavelet-Galerkin method for periodic heterogeneous media is presented. The advantages are to remove the mesh and to make adaptivity easier. Numerical results are presented. A specific study of interfaces in a Al-SiC composite is given.

Introduction

A great number of recent papers are concerned by the solution of partial differential equations by wavelet bases ͓͑1,2͔͒. Mainly, these works deal with one-dimensional or scalar two-dimensional problems. The solution of the elastostatics system by this kind of method is not usual ͓͑3,4͔͒. Boundary problems on open bounded sets are very difficult to treat ͓͑5͔͒. Nevertheless, periodic conditions on elementary bounded sets are natural for the use of wavelet transform. In this paper, we show how to use such a technique and we give applications to interfaces in Al-SiC composite. In the first section we give the notations and the necessary mathematical background. In the second section we present the mechanical problem: the homogenization of periodic heterogeneous media. The third section is concerned with the algorithm: a wavelet-Galerkin method using Daubechies wavelets ͓͑6͔͒. The determination of the macroscopic coefficients is treated in the fourth section. Applications and numerical results are described in the fifth section. Concluding remarks are given.

Notations and Mathematical Background

In this section, we present the notations used in the following of the paper. The sets of kinematically and statically admissible fields are denoted H, L, and S. Let

Lϭ͑L 2 ͑ Y ͒͒ 3 and Hϭ͑H p 1 ͑ Y ͒͒ 2 Sϭ͕vL loc 2 ͑ Y ͒,v͑x 1 ϩk 1 ,x 2 ϩk 2 ͒ϭv͑x 1 ,x 2 ͒a.e.,k 1 ,k 2 K͖ (1) H p 1 ͑ Y ͒ϭ͕vS, v ,i L 2 ͑ Y ͒, iϭ1,2͖.
u ,i denotes the ith partial derivative of the function u. We denote C the fourth-order elasticity tensor, e the strain tensor, and the stress tensor. In the following Y ϭ]0,1͓ 2 . To construct a wavelet basis of H, we use the compactly supported wavelets introduced by I. Daubechies ͓6͔ which is a basis of L 2 (R). These wavelets are periodized in order to obtain bases of S. By tensorial and cartesian products wavelet bases of H are obtained ͓͑7͔͒. We denote ⌿ l , lϭ1, 2, 3, and ⌿ 0 the wavelets and the scale functions ͑six degrees-of-freedom for each point͒, N j ϭ2 j Ϫ1 and ⌳ j ϭ͓0,N j ͔ 2 . Let V j be the subspace of dimension 2 2 jϩ1 of H generated by this wavelet at approximation level j. An element of V j max is thus written as

u͑x 1 ,x 2 ͒ϭ͑ u 1 ͑ x 1 ,x 2 ͒,u 2 ͑ x 1 ,x 2 ͒͒ u d ϭ ͚ ⌳ j 0 u j 0 d0 ⌿ j 0 0 ϩ ͚ lϭ1 lϭ3 ͚ jϭ j 0 jϭ j max ͚ ⌳ j u j dl ⌿ j l .
(2) j 0 is a given integer, dϭ1 or 2 and ϭ(k 1 ,k 2 ).

The Mechanical Problem

We consider a multiphase isotropic elastic composite ͑Fig. 1͒ and we intend to study the behavior of this heterogeneous media. We introduce the notion of equivalent material, i.e., we mean that under the same loadings, this equivalent material has globally the same response. In former papers ͓͑8,9͔͒ bounds for the bulk and shear moduli of a two-phase composite have been given. Without going into further detail, these bounds depend on the shear and bulk moduli of the two phases and on the volumic fraction of the two phases in the composite. In the same way, the theory of periodic homogenization ͓͑10͔͒ focuses on an idealized composite

Remarks.

͑i͒ The problem ͑P͒ is solved classically by a finite element method or by fast Fourier transform ͓͑11͔͒. We have chosen to introduce wavelet methods in order to eliminate the notion of mesh and to eliminate Gibbs phenomena.

͑ii͒ If the discretization of problem ͑P͒ in a orthonormal wave- let basis leads to the system KUϭB, then the discretization of problem ( P ) leads to (KϩId)UϭB, where Id is the identity matrix.

͑iii͒ The tensor C could be given by the image ͑pixels͒ of the microstructure.

Wavelet-Galerkin Method

The variational problem ( P ) is discretized by a Galerkin method. We have introduced a wavelet basis because of their localization and adaptivity properties. The projection of the plane elasticity operator into the wavelet basis ͓͑7͔͒ is given by a stiffness matrix K where the ''elementary matrix'' of order 2 is K j j Ј Ј llЈ ϭ ͫ F 1111 ϩF 1321 ϩF 3112 ϩF 3322 F 1221 ϩF 1311 ϩF 3222 ϩF 3312 F 1212 ϩF 2322 ϩF 3111 ϩF 3321 F 2222 ϩF 2312 ϩF 3221 ϩF 3311 ͬ

(3)

where

F pq␣␤ ϭ ͵ Y C pq ⌿ j,␣ l ⌿ j Ј Ј ,␤ lЈ dx 1 dx 2 .
We have chosen to decompose the tensor C on a wavelet basis at level J noted . In the numerical applications, Haar wavelet is used with its compact support equal to the square ͓l 1 /2 j ,(l 1 ϩ1)/2 j ͔ϫ͓l 2 /2 j ,(l 2 ϩ1)/2 j ͔. This wavelet is constant on this support which is a pixel of the image representation. Thus, the wavelet coefficient jl is equal to the value of the tensor C on this pixel. Due to the form of the wavelets ͑Cartesian and tensorial products of one dimensional wavelets͒ the computation of the coefficients of the matrix K leads to the determination of elementary terms which are integral of products of three one-dimensional wavelets and their derivatives:

͵ 0 1 Jr d m ⌿ js dx m d n ⌿ jt dx n , m,nϭ0,1. ( 4
)
These terms are obtained by the determination of eigenvectors of a low-order matrix ͓͑7,12͔͒. The right-hand side of the problem corresponding to the term l(v) in problem ( P ) is computed by a similar technique ͓͑4,5͔͒. Classically, the matrix K is a sparse matrix ͑Fig. 2͒. Because of the form of the wavelets bases, it seems natural to solve the linear system which is a discretized version of problem ( P ) by multigrid techniques ͓͑13,14͔͒. Nevertheless, we have chosen to use a conjugate gradient method.

Determination of the Macroscopic Coefficients

The determination of the elastic macroscopic coefficients corresponds to the computation of the macroscopic stress tensor ⌺:

⌺ϭ ͵ Y ͑ CEϩCe͑u ͒͒dx 1 dx 2 .
(5)

The computation of these terms is in the same way as the matrix and the right-hand side ͓4͔.

Numerical Results

We present the example of a three-phase fiber-matrix composite ͑Fig. 3͒: SiC for the fiber, Al for the matrix and an interface. The Lame ´coefficients associated to the interface are ␣ and ␤ where ␥ is the thickness of the interface. ␣ and ␤ are real positive parameters and ␥ is a given function with a sufficient regularity. We have shown in former papers ͓͑15-17͔͒ that when tends to zero, i.e., the thickness and the rigidity parameters tend to zero, we obtain an elastostatic limit problem with an interface law. This interface law keeps in memory the mechanical and geometrical properties of the layer. The interface law is given in Table 1 with respect to the value of the parameters ␣ and ␤. ␣ and ␤ determine how the thickness and the rigidity tend to zero. It is necessary to quantify the limit, in other words we seek an interval in which the initial problem could be approximated by the limit problem for which the solution is more easy to obtain. On the other hand, it is very important to quantify the influence

/⑀→0 u N ϭ0 T ϭ0 N ϭ ␥ u N T ϭ0 n ϭ0 /⑀→ ¯uN ϭ0 T ϭ ␥ u T N ϭͩ ␥ ϩ2 ␥ͪu N N ϭ2 ␥ u N T ϭ ␥ u T T ϭ ␥ u T /⑀→ϱ uϭ0 uϭ0 uϭ0 /⑀→ϱ /⑀→ ¯/⑀→0
of the interface on the macroscopic coefficients, i.e., on the elastic behavior of a structure. Due to the shape and the thickness of the interface this problem is very difficult to treat by classical techniques.

We present, in Fig. 4, a study of the convergence of the jump of displacement in the interface for the case ␥ϭ1, for two values of ␣ and ␤. and are chosen as Aluminum coefficients. In this case the jump is equal to zero in the interface law. We have found that for values of smaller than 0.4 p.c. of the structure the interface law could be considered as valid. Note that the displacement in the interface has the form ͓͑16͔͒ u(r,)Ӎru()ϩu 0 . Figure 5 shows the influence of the thickness parameter on the first component of the homogenized elasticity tensor for different values of ␣ and ␤. For small values of the thickness ͑ smaller than 0.2 p.c. of the structure͒ it is convenient to neglect the interface. Note that for values of the thickness larger than 0.02, the coefficient depends linearly on the thickness.

Concluding Remarks

In this paper, we have shown a robust tool to compute the overall response of a composite. In particular, our method is able to compute the influence of an interface even at a very small level. In the future, we want to investigate more complex materials such random materials ͓͑18͔͒ or other kind of interfaces ͓͑19,20͔͒.

  consisting of the juxtaposition of identical heterogeneities and classically, we need to solve an elastostatics problem on a representative volume Y ͑Problem P͒: Problem. P EL be given, find uH such that a(u,v)ϭl(v) ᭙vH with a͑u,v ͒ϭ ͵ Y ͑u ͒:e͑v ͒dyϭ ͵ Y C͑ y ͒e͑ u ͒:e͑ v ͒dy and l͑v ͒ϭϪ ͵ Y CE:D͑v ͒dy Because of the nonuniqueness of the solution of problem ͑P͒ ͑defined within a translation͒, problem ͑P͒ is replaced by problem ( P ) ͑''viscous'' problem͒: Problem. P EL be given, find uH such that a (u,v)ϭl(v) ᭙vH with a ͑ u,v ͒ϭ ͵ Y ͑u ͒:e͑ v ͒dyϩ ͵ Y uvdy It can be shown that the solution of this problem converges toward the solution of problem ͑P͒ with average equal to zero ͓͑4͔͒.

Fig. 1

 1 Fig. 1 An example of a composite and its representative volume

Fig. 3

 3 Fig. 3 Al-SiC composite with an interfacial zone "thickness …

Fig. 5

 5 Fig. 5 Homogenized coefficient for different values of ␣ and ␤ "MÄ3…

Table 1 Interface laws

 1


Damien Chablat
email: damien.chablat@cnrs.fr

Luc Rolland
email: luc.rolland@uws.ac.uk

NAVARO II, A NEW PARALLEL ROBOT WITH EIGHT ACTUATION MODES

This article presents a new variable actuation mechanism based on the 3-RPR parallel robot. This mechanism is an evolution of the NaVARo robot, a 3-RRR parallel robot, for which the second revolute joint of the three legs is replaced by a scissor to obtain a larger working space and avoid the use of parallelograms to operate the second revolute joint. To obtain a better spatial rigidity, the leg mechanism is constructed by placing the scissors in an orthogonal plane to the displacement. Unlike the first NaVARO robot, the kinematic model is simpler because there is only one solution to the inverse kinematic model. Surfaces of singularity can be calculated and presented in a compact form. The singularity equations are presented for a robot with a similar base and mobile platform.

INTRODUCTION

A major drawback of serial and parallel mechanisms is the inhomogeneity of kinetostatic performance in their workspace. For example, dexterity, accuracy and stiffness are generally poor in the vicinity of the singularities that may appear in the working space of these mechanisms. For parallel robots, their inverse kinematics problem often has several solutions, which can be called "working mode" [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF]. However, it is difficult to achieve a large workspace without singularity for a given working mode. Therefore, it is necessary to plan a change path of the working mode to avoid parallel singularities [START_REF] Chablat | Moveability and Collision Analysis for Fully-Parallel Manipulators[END_REF][START_REF] Wenger | Kinematic Analysis of a New Parallel Machine Tool: the Orthoglide[END_REF]. In such a case, the initial trajectory would not be followed.

One solution to this problem is to introduce activation re-dundancy, which involves force control algorithms [START_REF] Alba-Gomez | Consistent Kinetostatic Indices for Planar 3-DOF Parallel Manipulators, Application to the Optimal Kinematic Inversion[END_REF]. Another approach is to use the concept of joint coupling as proposed by [START_REF] Theingin | Management of parallel-manipulator singularities using joint-coupling[END_REF] or to select the articulation actuated in each leg in relation to the placement of the end-effector, [START_REF] Arakelian | Increase of Singularity-Free Zones in the Workspace of Parallel Manipulators Using Mechanisms of Variable Structure[END_REF], as emphasized in this article.

To solve this problem, a first variable actuation mechanism (VAM) was introduced in 2008 [START_REF] Rakotomanga | Kinetostatic performance of a planar parallel mechanism with variable actuation[END_REF], called NaVARo for Nantes Variable Actuation Robot. This mechanism has eight actuation modes and is based on a 3-RRR parallel robot with either the first or second revolute joint actuated. As this mechanism has eight solutions to the inverse kinematic model, the determination of singularities and separation according to the current working mode is very difficult algebraically [START_REF] Bonev | Singularity Loci of Planar Parallel Manipulators with Revolute Joints[END_REF]. In addition, the volume swept by the robot's legs is large and parallelograms reduce the workspace. A framework has been developed to pilot the prototype of this robot. The main problem comes from the position of the position sensor on the motor, which is not always connected to the robot base. Additional sensors may separate assembly modes, but a slight slip in the couplings may disturb the location of the mobile platform [START_REF] Chablat | A framework for the control of a parallel manipulator with several actuation modes[END_REF].

The aim of the article is to propose a new mechanism, based on the 3-RPR parallel robot for which singularities are easier to calculate for all actuation modes. The outline of this article is as follows. The first section presents the architecture of the NaVARo II robot with its eight actuation modes. The second section presents the study of kinematics and presents the algebraic equations of parallel singularities as well as the limits of the workspace. Depending on these limits, the singular surfaces are reduced to present only the singularities in the workspace.

F p ε FIGURE 2. 3-RPR with variable actuation

Mechanism architecture of the NaVARo II

The VAM concept was examined in [START_REF] Theingin | Management of parallel-manipulator singularities using joint-coupling[END_REF][START_REF] Arakelian | Increase of Singularity-Free Zones in the Workspace of Parallel Manipulators Using Mechanisms of Variable Structure[END_REF]. They derived a VAM from the architecture of the 3-RPR planar parallel manipulator by actuating either the first revolute joint or the prismatic joint of its legs. The same concept was introduced in [START_REF] Rakotomanga | Kinetostatic performance of a planar parallel mechanism with variable actuation[END_REF] based on the 3-RRR and a first prototype was created in [START_REF] Caro | Kinematic and dynamic modeling of a parallel manipulator with eight actuation modes[END_REF].

The new 3-RPR robot concept with variable actuation is shown in Fig. 1. The use of scissors makes it possible to limit the space requirement during movements in contrast to previous designs and improves rigidity. The number of scissors can be optimized according to the possible height of the mobile platform, the desired stiffness or the desired maximum length of the equivalent prismatic joint [START_REF] Takesue | Scissor lift with real-time self-adjustment ability based on variable gravity compensation mechanism[END_REF][START_REF] Islam | Dynamic analysis of Scissor Lift mechanism through bond graph modeling[END_REF][START_REF] Rolland | Kinematics Synthesis of a New Generation of Rapid Linear Actuators for High Velocity Robotics[END_REF].

This mechanism can be represented on a projection like all 3-RPR mechanisms. The pose of the mobile platform is determined by the Cartesian coordinates (x, y) of the operating point P expressed in the basic frame F b and the angle α, i.e. the angle between the reference frames F b and F p (Figure 2). To illustrate this article, the following dimensions have been fixed,

A 1 A 2 = A 1 A 3 = A 2 A 3 = 90, B 1 B 2 = B 1 B 3 = B 2 B 3 = 30, ε = π/3 and 8 ≤ ρ i ≤ 59 for i = 1, 2, 3.
A new transmission system has been developed and installed in each branch of the NaVARo II so that the manipulator can easily switch from one actuation mode to another. As for NaVARo I, it consists of one motor with a shaft connected to two clutches to make the pivot connection between the base and the leg or to The NAVARO II transmission system make a prismatic joint to operate the scissor. Figure 3 shows a actuation diagram of the NaVARo II. This system can be considered as a double clutch and contains: (i) an electric motor (pink), (ii) a main shaft (pink), (iii) a base (blue), (iv) the first axis of the leg (green), and (v) two electromechanical clutches (red) which connects to the shaft of the first revolute joint (orange) or to the prismatic link shaft (yellow). Two position sensors give the angular position of the leg relative to the base (sensor 1) and the length of the prismatic joint via the ball screw position (sensor 2). The values of the two sensors, combined with the joint limits, allow us to know the current assembly mode of the robot.

The actuation modes are slightly different from the NaVARo I. Each transmission system has four actuation schemes, that are defined thereafter:

1. None of clutches 1 and 2 are active. The main shaft can move freely in relation to the base. In this case, neither the pivot joint nor the prismatic joint is actuated. The leg can move freely, i. e. θ i or ρ i are passive, i = 1, 2, 3. 2. Clutch 1 is active while clutch 2 is not. The first leg axis (green) is driven by the rotation of the motor shaft. In this case, the angle θ i is active while ρ i is passive, i = 1,2,3. 3. Clutch 2 is active while clutch 1 is not. The first leg joint is free but the rotation of the motor shaft leads to a displacement of the slider, which activates the scissor. In this case, the θ i is passive and ρ i is active, i = 1, 2, 3. 4. Both clutches 1 and 2 are active. Both joints cause a synchronized rotation and translation motion. The end of the leg will make a spiral motion.

The latter actuation mode differs from the NaVARo I. Only the second and third actuation modes are used in our study. Thus, NaVARo II has eight actuation modes, as shown in Table 1. For example, the first actuation mode corresponds to the 3-RPR mechanism, also referred to as the RPR 1 -RPR 2 -RPR 3 mechanism, since the first revolute joint (located at point A i ) of its leg are actuated. Similarly, the eighth actuation mode corresponds to the 3-RPR manipulator, also known as the RPR 1 -RPR 2 -RPR 3 mechanism, since the prismatic joints of its leg are actuated. Actuating mode number active joints

1 RPR 1 -RPR 2 -RPR 3 θ 1 , θ 2 , θ 3 2 RPR 1 -RPR 2 -RPR 3 θ 1 , θ 2 , ρ 3 3 RPR 1 -RPR 2 -RPR 3 θ 1 , ρ 2 , θ 3 4 RRP 1 -RPR 2 -RPR 3 ρ 1 , θ 2 , θ 3 5 RPR 1 -RPR 2 -RPR 3 θ 1 , ρ 2 , ρ 3 6 RRP 1 -RPR 2 -RPR 3 ρ 1 , ρ 2 , θ 3 7 RPR 1 -RPR 2 -RPR 3 ρ 1 , θ 2 , ρ 3 8 RPR 1 -RPR 2 -RPR 3 ρ 1 , ρ 2 , ρ 3

Kinematic modeling of the NaVARo II

In this section, we present the kinematic model that is commonly used to define geometrically singular configurations, then the constraint equations, the workspace boundaries and surfaces that define the singularity loci.

Kinematic modeling

The velocity ṗ of point P can be obtained in three different forms, depending on which leg is traversed, namely,

ṗ = θi E(b i -a i ) + ρi b i -a i ||b i -a i || + αE(p -b i ) (1)
with matrix E defined as

E = 0 -1 1 0 (2)
Thus, p, b i , a i are the position vectors of points P, A i and B i , respectively, and α is the rate of angle α.

To compute the kinematic model, we have to eliminate the idle joints θ i or ρ i as a function of the actuation mode. For θi , we have to dot-multiply by

h i = (b i -a i ) T (3)
and for h i = ρi by

E b i -a i ||b i -a i || . ( 4
) A 1 B 1 B 2 B 3 A 2 A 3 L 2 L 3 L 1 FIGURE 4.
Example of singular configuration for the first actuation mode when the lines L 1 , L 2 and L 3 intersect at one point

The kinematic model of the VAM can now be cast in vector form, namely,

At = B q with t = [ ṗ α] T and q = [ q1 q2 q3 ] T (5)
with qi thus being the vector of actuated joint rates where qi = θi when the first revolute joint is driven and qi = ρi when the prismatic joint is driven, for i = 1, 2, 3. A and B are respectively, the direct and the inverse Jacobian matrices of the mechanism, defined as

A =   h 1 h 1 E(p -b 1 ) h 2 h 2 E(p -b 2 ) h 1 h 3 E(p -b 3 )   (6)
B = diag[ρ 1 ρ 2 ρ 3 ] (7)
The geometric conditions for parallel singularities are well known in the literature for the first and eighth actuation modes.

For the first actuation mode, it is when the lines L i , normal to the axis (A i B i ) are intersecting at one point, see Fig. 4. For the eighth actuation mode, it is when the lines M i , passing through the axis (A i B i ) are intersecting at one point, see Fig. 5. For the other modes, it is just necessary to consider either the L i or M i lines according to the actuated joints, i.e. L i when the i th revolute joint is actuated and M i when the i th prismatic joint is actuated.

Constraint equations

To maintain the symmetry of the robot, the position of the end-effector is in the center of the mobile platform. The constraint equations for all actuation modes can be written by traversing the closed loops of the mechanism. Equations 8-11

A 1 B 1 B 2 B 3 A 2 A 3 M 2 M 3 M 1 FIGURE 5.
Example of singular configuration for the eighth actuation mode when the lines L 1 , L 2 and L 3 intersect at one point describe the two closed loops and equations 12-13 define the position and orientation of the mobile platform.

ρ 1 C θ 1 + 30C α -ρ 2 C θ 2 -90 = 0 (8) ρ 1 S θ 1 + 30S α -ρ 2 S θ 2 = 0 (9) ρ 1 C θ 1 + 15(C α - √ 3S α -3) -ρ 3 C θ 3 = 0 ( 10
)
ρ 1 S θ 1 + 15(S α + √ 3C α -3 √ 3) -ρ 3 S θ 3 = 0 (11) x -ρ 1 C θ 1 -15C α + 5 √ 3S α = 0 (12) y -ρ 1 S θ 1 -15S α -5 √ 3C α = 0 ( 13
)
with

C α = cos(α), S α = sin(α), C θ i = cos(θ i ), S θ i = sin(θ i ) for i = 1, 2, 3.
To make these equations algebraic, we use a substitution of all trigonometric functions as well as the square root function with √ 3 = S3 and S3 2 = 3 cos(β ) = C β and sin(β ) = S β for any angles β . We obtain a system with 11 equations, four for loop closures, two for the position and orientation of the mobile platform, four for trigonometric functions and one for the square root function with 14 unknowns. In this case, the manipulation of equations is not trivial and powerful algebraic tools must be used like the Siropa library implemented in Maple [START_REF] Siropa | Algebraic and robotic functions[END_REF][START_REF] Jha | Joint space and Singularities of a family of Delta-Like Robot Mechanism and Machine Theory[END_REF].

Workspace boundaries

If the revolute joints have no limits, the boundary of the workspace is given by the minimum and maximum extension of the scissors as shown in Fig. 6. The minimum value of ρ i permits to avoid serial singular configuration where ρ i = 0. Us- FIGURE 6. Minimum and maximum lengths of the scissors ing the constraint equations and ranges limits of prismatic joints (8 ≤ ρ i ≤ 59), we find six surface equations that describe the boundary of the working space. These limits mean that there is no collision between the legs and the mobile platform.

10 √ 3(xS α -yC α ) + x 2 -30xC α + y 2 -30yS α -3181 = 0 (14) (-10C α y + 10S α (x -90)) √ 3 + (30x -2700)C α + x 2 + y 2 + 30yS α -180x + 4919 = 0 (15) (20C α y + (-20x + 900)S α -90y) √ 3 + x 2 + y 2 -90x -2700C α + 4919 = 0 (16) 10 √ 3(xS α -yC α ) + x 2 -30xC α + y 2 -30yS α + 236 = 0 (17) (-10C α y + 10S α (x -90)) √ 3 + (30x -2700)C α + x 2 + y 2 + 30yS α -180x + 8336 = 0 (18) (20C α y + (-20x + 900)S α -90y) √ 3 + x 2 + y 2 -90x -2700C α + 8336 = 0 (19)
One of the functions of the Siropa library is to display surfaces that can be limited by inequality equations. The surfaces in blue, red, green represent the minimum and maximum limits of leg one, two and three, respectively in Fig. 7. The projections onto the planes (xy), (xα) and (yα) are used to estimate the main dimensions of the workspace. A cylindrical algebraic decomposition (CAD) can also be performed to have a partition of the workspace for each actuation mode [START_REF] Collins | Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition[END_REF].

Singular configurations

From the constraint equations, it is possible to write the determinant of matrix A. These determinants depend on the positions of the mobile platform and the positions of the passive and active joints. Only an elimination method like Groebner's basis can successfully obtain the representation of singularities in the Cartesian workspace. Note that for only the first and eighth actuation modes, the determinant of A is factorized to form two parallel planes (for the eighth actuation mode, an unrepresented singularity exists for α = π). The equations of the singularities for the eight actuation modes are given in the Appendix. As there is only one working mode, the equations of these surfaces are simpler than for the NAVARO I robot for which it is not possible to simply describe these equations.

Figures 8 and9 show all singularities for the eight actuation modes without and with the joint limit conditions. As none of them are superposed, it is possible to completely move through the workspace by choosing a non singular actuation mode for any position of the mobile platform. The same motion planning algorithm introduced for NaVARo I can be used to select the actuation mode able to avoid singular configurations [START_REF] Caro | Algorithm for the Actuation Mode Selection of the Parallel Manipulator NAVARO[END_REF].

Figures 10-17 represent the singularities of each actuation mode with on the left the singularities without joint limits and on the right the one included in the workspace. The three actuation modes where a single prismatic joint is actuated are similar by a rotation of 120 degrees (Figs. 111213). Similarly, the actuation modes where only one revolute joint actuated are also similar (Figs. 141516).

Conclusions

In this article, a new version of the NaVARo robot was introduced. Thanks to the change of actuation mode, the entire Cartesian workspace can be used. By eliminating the parallelograms that allowed the first NaVARO robot to have an actuation on the second pivot joint, the Cartesian workspace is larger. In addition, it is possible to place sensors on both actuated joints of each leg to locate the mobile platform when we solve the direct kinematic problem. The use of scissors makes it possible to have a greater rigidity in the transverse direction of the robot movement as well as a variation of displacement which can be increased according to the number of bars. Unlike the NaVARo I, which is based on a 3-RRR robot, the NaVARo II is based on the architecture of the 3-RPR, which has only one solution with the inverse kinematic model for any actuation mode. This property allows a complete writing of singularity equations whereas for the robot 3-RRR, in the literature, these equations can be written only for a given orientation of the mobile platform. Future works will be carried out to evaluate the stiffness of the robot based on the size and the number of scissors and the number of solutions to the direct kinematic model to determine if there are actuation modes for which the robot is cuspidal.

FIGURE 1 .

 1 FIGURE 1. Isometric view of NaVARo II

  FIGURE 3. The NAVARO II transmission system

FIGURE 7 .

 7 FIGURE 7. Workspace of the NaVARo II in isometric view and three projections onto the planes (xy), (xα) and (yα)

FIGURE 8 .

 8 FIGURE 8. The singularity surfaces for the eight actuation modes

FIGURE 9 .

 9 FIGURE 9. The singularity surfaces for the eight actuation modes in the robot workspace

FIGURE 10 . 1 FIGURE 11 .

 10111 FIGURE 10. Singularity surfaces for actuation modes 1

FIGURE 12 .FIGURE 13 .

 1213 FIGURE 12. Singularity surfaces for actuation modes 3

FIGURE 14 . 5 FIGURE 15 .

 14515 FIGURE 14. Singularity surfaces for actuation modes 5

FIGURE 16 .FIGURE 17 .

 1617 FIGURE 16. Singularity surfaces for actuation modes 7

TABLE 1 .

 1 The eight actuating modes of the 3-RRR VAM

Actuation mode 1


D Chablat
email: damien.chablat@cnrs.fr

E Ottaviano
email: ottaviano@unicas.it

S Venkateswaran
email: swaminath.venkateswaran@ls2n.fr

Self-Motion of the 3-PPPS Parallel Robot with Delta-Shaped Base

Keywords: Parallel robots, 3-PPPS, singularity analysis, kinematics, self-motion

This paper presents the kinematic analysis of the 3-PPPS parallel robot with an equilateral mobile platform and an equilateral-shaped base. Like the other 3-PPPS robots studied in the literature, it is proved that the parallel singularities depend only on the orientation of the endeffector. The quaternion parameters are used to represent the singularity surfaces. The study of the direct kinematic model shows that this robot admits a self-motion of the Cardanic type. This explains why the direct kinematic model admits an infinite number of solutions in the center of the workspace at the "home" position but has never been studied until now.

Introduction

It has been shown that by applying simplifications in parallel robot design parameters, self-motions may appear [START_REF] Husty | A special type of singular Stewart-Gough platform[END_REF][START_REF] Karger | Singularities and self motions of a special type of platforms[END_REF][START_REF] Wohlhart | Synthesis of architecturally mobile double-planar platforms[END_REF]. For example, Bonev et al. demonstrated that all singular orientations of the popular 3-RRR spherical parallel robot design (known as the Agile Eye) correspond to self motions [START_REF] Bonev | Working and assembly modes of the Agile Eye[END_REF], but arguably this design has the "best" spherical wrist. The Cardanic motion can be found as self-motion for two robots in literature, the 3-RPR parallel robot [START_REF] Chablat | Self Motions of a Special 3-RPR Planar Parallel Robot[END_REF] and the PamInsa robot [START_REF] Briot | Self-Motions of General 3-RPR Planar Parallel Robots[END_REF].

Most of the examples for a fully parallel 6-DOF manipulator can be categorized by the type of their six identical serial chains namely UPS [START_REF] Merlet | Parallel robots[END_REF][START_REF] Pierrot | From hexa to hexam[END_REF][START_REF] Corbel | Optimal Design of a 6-dof Parallel Measurement Mechanism Integrated in a 3-dof Parallel Machine-Tool[END_REF][START_REF] Stoughton | Modified Stewart platform manipulator with improved dexterity[END_REF][START_REF] Ji | Identification of placement parameters for modular platform manipulators[END_REF], RUS [START_REF] Honegger | Adaptive control of the Hexaglide, a 6 dof parallel manipulator[END_REF] and PUS [START_REF] Hunt | Structural kinematics of in parallel actuated robot arms[END_REF]. However for all these robots, the orientation of the workspace is rather limited due to the interferences between the legs.

To solve this problem, new parallel robot designs with six degrees of freedom appeared recently having only three legs with two actuators per leg. The Monash Epicyclic-Parallel Manipulator (MEPaM), called 3-PPPS is a six DOF parallel manipulator with all actuators mounted on the base [START_REF] Chen | A Six-Dof Epicyclic-Parallel Manipulator[END_REF]. Several variants of this robot have been studied were the three legs are made with three orthogonal prismatic joints and one spherical joint in series. The first two prismatic joints of each legs are actuated. In the first design, the three legs are orthogonal [START_REF] Caro | Non-Singular Assembly Mode Changing Trajectories of a 6-DOF Parallel Robot[END_REF]. For this design, 1 the robot can have up to six solutions to the Direct Kinematic Problem (DKP) and is capable of making non-singular assembly mode change trajectories [START_REF] Caro | Non-Singular Assembly Mode Changing Trajectories of a 6-DOF Parallel Robot[END_REF]. In [START_REF] Chen | A Six-Dof Epicyclic-Parallel Manipulator[END_REF], the robot was made with an equilateral mobile platform and an equilateral-shape base. In [START_REF] Chablat | Kinematics and workspace analysis of a 3-PPPS parallel robot with U-shaped base[END_REF][START_REF] Chablat | The 3-PPPS parallel robot with U-shape Base, a 6-DOF parallel robot with simple kinematics[END_REF], the robot was designed with an equilateral mobile platform and a U-shaped base. For this design, the Direct Kinematic Model (DKM) is simple and can be solved with only quadratic equations.

The common point of these three variants is that the parallel singularity is independent of the position of the end-effector. This paper presents the self-motion of the 3-PPPS parallel robot derived from [START_REF] Chen | A Six-Dof Epicyclic-Parallel Manipulator[END_REF] with an equilateral mobile platform and an equilateral-shaped base.

The outline of this article is as follows. In the next section, the manipulator architecture as well as its associated constraint equations are explained. Followed by that, calculations for parallel singularities as well as self-motions are presented. The article then ends with conclusions.

Mechanism Architecture

The robot under study is a simplified version of the MEPaM that has been developed at the Monash University [START_REF] Chen | A Six-Dof Epicyclic-Parallel Manipulator[END_REF][START_REF] Caro | Singularity Analysis of a Sixdof Parallel Manipulator using Grassmann-Cayley Algebra and Gröbner Bases[END_REF]. This architecture is derived from the 3-PPSP that was introduced earlier [START_REF] Byun | Analysis of a novel 6-dof,3-PPSP parallel manipulator[END_REF].

Geometric Parameters

For this parallel robot, the three legs are identical and it consists of two actuated prismatic joints, a passive prismatic joint and a spherical joint (Figure 1.1). The axes of first three joints of each leg form an orthogonal reference frame.

The coordinates of the point C 1 are ρ 1x , ρ 1y and ρ 1z , wherein the last two are actuated. We assume an origin A i for each legs and the equations are given by:

A 1 = [2, 0, 0] T (1.1) A 2 = [-1, √ 3, 0] T (1.2) A 3 = [-1, - √ 3, 0] T (1.3)
The coordinates of C 2 and C 3 are obtained by a rotation around the z axis by 2π/3 and -2π/3 respectively and the equations are as follows:

C 1 = [ρ 1x , ρ 1y , ρ 1z ] T (1.4) C 2 = [-ρ 2x /2 - √ 3ρ 2y /2, √ 3ρ 2x /2 -ρ 2y /2, ρ 2z ] T (1.5) C 3 = [-ρ 3x /2 + √ 3ρ 3y /2, - √ 3ρ 3x /2 -ρ 3y /2, ρ 3z ] T (1.6) A 1 P B 1 C 1 A 3 B 3 C 3 A 2 B 2 C 2 r 1x r 1z r 1y r 2x r 3x r 2z r 3z r 2y r 3y
x y z Fig. 1.1 A scheme for the 3-PPPS parallel robot and its parameters in its "home" position with the actuated prismatic joints in blue, the passive joints in white and the mobile platform drawn in green with x = 1/ √ 3, y = 0, z = 0, q 1 = 1, q 2 = 0, q 3 = 0, q 4 = 0 Three locations are now written to describe the mobile platform in the moving frame for an equilateral triangle whose edge lengths are set to one.

V 1 = [0, 0, 0] T (1.7) V 2 = [- √ 3/2, 1/2, 0] T (1.8) V 3 = [- √ 3/2, -1/2, 0] T (1.9)
Generally in the robotics community, Euler or Tilt-and-Torsion angles are used to represent the orientation of the mobile platform [START_REF] Caro | Non-Singular Assembly Mode Changing Trajectories of a 6-DOF Parallel Robot[END_REF]. These methods have a physical meaning, but there are singularities to represent certain orientations. The unit quaternions give a redundant representation to define the orientation but at the same time it gives a unique definition for all orientations. The rotation matrix R is described by:

R =   2q 2 1 + 2q 2 2 -1 -2q 1 q 4 + 2q 2 q 3 2q 1 q 3 + 2q 2 q 4 2q 1 q 4 + 2q 2 q 3 2q 2 1 + 2q 2 3 -1 -2q 1 q 2 + 2q 3 q 4 -2q 1 q 3 + 2q 2 q 4 2q 1 q 2 + 2q 3 q 4 2q 2 1 + 2q 2 4 -1   (1.10)
Here q 1 ≥ 0 and q 2 1 + q 2 2 + q 2 3 + q 2 4 = 1. We can write the coordinates of the mobile platform using the previous rotation matrix as:

C i = RV i + P where P = [x, y, z] T (1.11)
Thus, we can write the set of constraint equations with the position of C i in the both reference frames by the relations:

ρ 1y = y (1.12) ρ 1z = z (1.13) (2q 1 q 4 -x) √ 3 + 2q 1 2 + 3q 2 2 -q 3 2 -y -2ρ 2y = 1 (1.14) - √ 3q 1 q 3 + √ 3q 2 q 4 -q 1 q 2 -q 3 q 4 + ρ 2z = z (1.15) (2q 1 q 4 + x) √ 3 -2q 1 2 -3q 2 2 + q 3 2 -y -2ρ 3y = -1 (1.16)
-√ 3q 1 q 3 + √ 3q 2 q 4 + q 1 q 2 + q 3 q 4 + ρ 3z = z (1.17)

Constraint equations

The main problem is to find the location of the mobile platform by looking for the values of the passive prismatic joints [ρ 1x , ρ 1y , ρ 1z ] as proposed in [START_REF] Parenti-Castelli | Direct displacement analysis for some classes of spatial parallel mechanisms[END_REF]. The distances between any couple of points C i are given by:

||C 1 -C 2 || = ||C 1 -C 3 || = ||C 2 -C 3 || = 1 (1.18)
This method is also used by [START_REF] Chen | A Six-Dof Epicyclic-Parallel Manipulator[END_REF] for the 3-PPPS by using the dialytic elimination [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms[END_REF].

A fourth degree polynomial equation with complicate coefficients is then obtained. However, no self-motion was detected.

Unfortunately, when we want to solve the DKM for the "home" position, i.e. ρ iy = 0 and ρ iz = 0 and i = 1, 2, 3, we have an infinite number of solutions, which correspond to the self-motion. This result remains the same if we set ρ 1z = ρ 2z = ρ 3z . Assuming that this motion is in a plane parallel to the plane (0xy), we write the system coefficients for ρ iz = 0 with i = 1, 2, 3. The fourth degree polynomial then degenerates and a quadratic polynomial equation is obtained which is of the form:

-3) = 0
Here all the terms of the equation cancel out each other when (ρ 1y + ρ 2y + ρ 3y = 0). With these conditions, the robot becomes similar to the 3-RPR parallel robot for which its self-motion produces a Cardanic motion. Since the self-motion corresponds to singular configurations, the objective now will be to locate these singularities with respect to other singularities. Geometrically, the self-motion has been described when the motion axes of passive joints intersect at a single point and an angle of 2π/3 is formed between each axis. An example is shown in Fig. 1.2 where the green mobile platform is one assembly mode obtained in the "home" position. This phenomenon has already been characterized for the 3-RPR parallel robot [START_REF] Chablat | Self Motions of a Special 3-RPR Planar Parallel Robot[END_REF] and the PamInsa robot in [START_REF] Briot | Self-Motions of General 3-RPR Planar Parallel Robots[END_REF]. To validate this result, we use the Siropa library programmed under Maple and its "InfiniteEquations" function [START_REF] Jha | Joint space and Singularities of a family of Delta-Like Robot Mechanism and Machine Theory[END_REF]. Three condition are obtained which are (ρ 1y + ρ 2y + ρ 3y = 0), ρ 1z = ρ 2z and ρ 2z = ρ 3z .

x y z x y z Fig. 1.2 Set of postures to describe the Cardanic motion starting from the "home" position

Singularity Analysis

The singular configurations of the 3-PPPS robot have been studied in several articles with either a parametrization of orientations using Euler angles or Quaternions [START_REF] Chablat | Kinematics and workspace analysis of a 3-PPPS parallel robot with U-shaped base[END_REF]. Serial and parallel Jacobian matrices can be computed by differentiating the constraint equations with respect to time [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Sefrioui | Singularity analysis and representation of planar parallel manipulators[END_REF][START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF]. These Jacobian serial and parallel matrices must satisfy the following relationship

At + B ρ = 0 (1. 20
)
where t is the twist of the moving platform and ρ is the vector of the active joint velocities.

According to the leg topology of the 3-PPPS robot, there is no serial singularity because the determinant of the B matrix does not vanish. Using the same approach as in [START_REF] Caro | Singularity Analysis of a Sixdof Parallel Manipulator using Grassmann-Cayley Algebra and Gröbner Bases[END_REF], we can determine the matrix A and its determinant can be factorized as follows:

q 1 2q 2 2q 3 2 + q 4 2 (q 1q 4 ) (q 1 + q 4 ) = 0 (1.21)

To calculate this result, we use the "ParallelSingularties" function from the Siropa library [START_REF] Jha | Joint space and Singularities of a family of Delta-Like Robot Mechanism and Machine Theory[END_REF]. To represent this surface, we eliminate q 1 thanks to the relation on the quaternions

q 2 2 + q 3 2 + 2q 4 2 -1 2q 2 2 + 2q 3 2 -1 = 0 (1.22)
One of the surface represents a cylinder and the other an ellipsoid. Figure 1.3 depicts these surfaces bounded by the unit sphere. By writing the conditions (ρ 1y + ρ 2y + ρ 3y = 0) and ρ 1z = ρ 2z = ρ 3z with the constraint equations, the Groebner basis elimination method makes it possible to obtain a set of equations that depends on q 2 , q 3 and q 4 . q 4 q 3 q 2 Fig. 1.3 Parallel singularity of the 3-PPPS robot with quaternion representation q 2 q 4 = 0

q 3 q 4 = 0 q 2 q 2 2 + q 3 2 -1 = 0 q 3 q 2 2 + q 3 2 -1 = 0 q 4 3 -q 4 = 0 q 2 4 + q 3 2 + q 4 2 -1 q 2 2 + q 3 2 q 4 2 = 0 (1.23)
The intersection of these six equations and the equation 1.22 is a unit radius circle centered at the origin of the plane (0q 2 q 3 ) as shown in Fig. 1.4. This circle is tangential to the parallel singularity shown in Fig. 1.3. As the singularity does not depend on the position, the self-motion exists for an infinite number of positions of the mobile platform. This phenomenon may not be found if numerical methods are used to solve the DKM because the relations (ρ 1y + ρ 2y + ρ 3y = 0) and ρ 1z = ρ 2z = ρ 3z are never completely satisfied.

q 4 q 3 q 2

Conclusions and Perspectives

In this article, we studied the parallel robot 3PPPS to explain the presence of a selfmotion at the "home" position. This movement is a Cardanic movement that has already been studied on the 3-RPR parallel robot or the PamInsa robot. Self-motions can often explain the problems of solving the DKM when using algebraic methods. The calculation of a Groebner base makes it possible to detect this problem but the characterization of this movement for robots with six degrees of freedom is difficult because of the size of the equations. Other robots, such as the CaPaMan at the University of Cassino have the same type of singularity despite the phenomenon never being studied. The objective of future work will be to identify architectures with passive prismatic articulations connected to the mobile platform which is capable of having the same singularity conditions.

Fig. 1 . 4

 14 Fig. 1.4 Location of self-motions in the workspace.

Self-Motion of the 3-PPPS Parallel Robot with Delta-Shaped Base


Shanti Ulfsbjorninn

Mohamed Lahrouchi

The typology of the distribution of Edge: the propensity for bipositionality

We discuss the grammatical conditions that can be imposed between segmental content (features) and syllable structure (positions) and how a representational preference can influence diachronic development. The discussion centers on the co--distribution of two properties: occlusivity and bipositionality. The first is the phonological feature that induces occlusivity and reduces amplitude (that is: |ʔ|, which we will refer to as Edge(*)), the second is the autosegmental structural property of belonging to multiple positions (which we refer to as 'C.C'). Edge(*) and bipositionality have a universal affinity but they are not reducible to each other. Instead, the inherent diachronic tendency to preserve Edge(*) in bipositional structures can become grammaticalised through licensing conditions that dictate the alignment of the two properties. This can be expressed bidirectionally, forming two major language types. Type A has the condition stated from the featural perspective (Edge(*) must be found in C.C). While, Type B comes from the other direction (C.C must contain Edge(*)). Crucially, the same structure is diachronically stable: (Edge(*)--C.C). What varies is the distribution of those properties elsewhere (given the direction of licensing condition). Type A excludes Edge(*) from {#__,V_V}, while Type B excludes C.Cs without Edge(*). Although there is variation on this point, there is a UG component, because there are no anti--Type A/B languages where Edge(*) repels bipositionality.

Introduction

This paper is broadly about the grammatical relationship that can be imposed between segmental content (features) and syllable structure (positions). In its broadest terms, we present a representational model where well--formedness is defined in terms of licensing. The discussion centers on the co--distribution of two properties: occlusivity and bipositionality. By occlusivity we mean a phonological feature that induces occlusivity and reduces amplitude. In Element Theory (Backley 2011) the occlusivity feature is: |ʔ| and it may be headed or headless. For expository purposes we will be referring to this as Edge(*) -in section (3) we will introduce the feature in a little more detail. By 'bipositionality' (C.C) we mean the autosegmental property of belonging to multiple positions in syllable structure (as in geminates, heterosyllabic consonant clusters etc). We will show that there is a universal affinity between Edge(*) and bipositionality, but that neither property can be reduced to the other. The distribution of Edge(*) shows in a particularly clear way the Honeyboneian principle of 'sharing makes us stronger'.

The structure of the paper is as follows. We begin the discussion with a description of Ontena Gadsup, the only known language to violate a universal against underlying oral stops. This fact is a product of an analysis and it is itself reducible to another highly unusual state of affairs: the unified lenition environment of {#__, V__V}. We will show that comparison of a closely related dialect suggests that diachronically there was no such unified environment and that initial weakening was a secondary and separate process from intervocalic spirantisation. The actual distribution of stopness in Ontena Gadsup is shown to be regulated by the licensing of Edge* in relation to bipositionality. We show that it forms a linguistic type (Type A) that also finds expression in the spirantising Berber languages. The analysis simultaneously predicts another major linguistic type (Type B), one which is attested in the so called 'Prince languages' such as the Kingi dialect of Soninké. In Type A languages Edge* is licensed by being contained in a bipositional syllable structure, while in Type B languages it is the bipositional structure that is licensed by containing Edge. Although there is variation on this point, there is a UG component because there are no anti--Type A/B languages where Edge(*) repels bipositionality. Such different language types must be able to arise diachronically, and we consider this below, suggesting that with a diachronic reanalysis one could make a Type B into a Type A language.

Ontena Gadsup lenition and violation of universals

Gadsup refers to a cluster of Trans--New Guinea languages spoken in the Eastern Highlands Province of Papua New Guinea. Of these, the Ontena variety of Gadsup is a 'rarissima' language. It appears unique in defying the absolute phonological universal that the inventories of all languages must contain oral stop consonants. This statement is, however, a matter of analysis. In Ontena Gadsup stops have an extremely limited distribution. They are not found word-initially or intervocalically. They occur only after a homorganic nasal or a glottal stop: {[N._], [ʔ._]}.

Due to the extremely limited distribution of oral stop consonants, formal economy suggests that they should be derived from underlying fricatives by a fortition rule.

(1) [-son , +cont ]

→ [-cont ] / [-cont] __
According to this analysis, the underlying set of consonants in Ontena Gadsup reduces to the inventory shown in (2).

(2) Ontena Gadsup inventory [START_REF] Frantz | Sketch Phonology of Ontena Village[END_REF] Group 1 ɸ β s ɾ x Group 2 m n j (ʔ) From the application of the rule in (1) to the inventory of consonants in (2), the consonants of Group 1 are obliged to surface as stops: [p, b, t, d, k]

Problem

The part of this analysis that is clear is the reason why [-cont] consonants retain their non--continuancy when following non-continuant consonants. The problem with this analysis is that it suggests a story where the word--initial and intervocalic environments were unified as one weak environment. The implied weakening hypothesis is shown in (4).

(4) Lenition environment: Weak {#__ , V__V} Strong {N.__, C.__} In fact, because there are no reported alternations in the language, another diachronic pathway suggests itself, one which does not take {#__ , V__V} as a unified weak environment. The origin of the Gadsup pattern could be explained as a two--step process where the first step is post--vocal spirantisation which is then followed by initial weakening.

(5) Two step weakening hypothesis (Post)--Intervocalic spirantisation followed by Initial spirantisation As shown in (5), a Stage I language (with stops in all positions) could introduce a phonological process of intervocalic spirantisation. 1 This would eliminate stops from all positions, except word--initially and post--consonantally (Stage II). Then there would be a second process of initial spirantisation that would remove the stops from initial position also, leaving them exclusively in post--consonantal environment, this pattern (Stage III) is what is attested in Ontena Gadsup.

If this were the proper diachronic account for the synchronic distribution of stops in Gadsup it would not require unification of initial and intervocalic environments into one weak environment. Therefore, fricatives would share these two environments but initial weakening and the intervocalic weakening would not have the same cause. The intervocalic pattern would be true lenition, while the initial weakening would be caused by another factor, perhaps not even technically lenition.

Beyond wanting to know 'what actually happened' there is a theoretically significant reason for wanting to distinguish between these two hypotheses. The normal expectation is that the initial 1 We do not discuss stops in 'coda' position because they are irrelevant for the discussion of Ontena.

1.1'Problem

The) part) of) this) analysis) that) is) clear) is) the) reason) why) [Bcont]) consonants) retain) their) nonB continuancy) when) following) nonBcontinuant) consonants.) The) problem) with) this) analysis) is that)it)suggests)a)story)where)the)wordBinitial)and)intervocalic)environments)were)uniUied)as one)weak)environment.)The)implied)weakening)hypothesis)is)shown)in)(4).

(

))Lenition)environment: Weak) {#__),)V__V} Strong) {N.__,)C.__} 4
In) fact,) because) there) are) no) reported) alternations) in) the) language,) another) diachronic pathway)suggest)itself,)one)which)does)not)take){#__),)V__V})as)a)uniUied)weak)environment. The)origin)of)the)Gadsup)pattern)could)be)explained)as)a)twoBstep)process)where)the)Uirst)step is)post%vocal*spirantisation)which)is)then)followed)by)initial*weakening.

(5))Two)step)weakening)hypothesis (Post)BIntervocalic)spirantisation)followed)by)Initial)spirantisation

Stage'I ' Stage'II '''''''''''''Stage'III kiki))) kixi) xixi kinki kinki xinki kiɁki kiɁki xiɁki )))))))(Post)BIntervocalic ))))Initial ))))))))))))spirantisation ))))))))))spirantisation
As) shown) in) (5),) a) Stage) I) language) (with) stops) in) all) positions)) could) regularize) a phonological) process) of) intervocalic) spirantisation.2 ) This) would) eliminate) stops) from) all positions,) except) wordBinitially) and) postBconsonantally) (Stage) II).) Then) there) would) be) a second)process)of)initial)spirantisation)that)would)remove)the)stops)from)initial)position)also,

5

The typology of the distribution of Edge position is phonologically strong -there is evidence for this from experimental work, diachronic change and synchronic alternations.

Experimental work suggests the universality of initial strength [START_REF] Becker | Asymmetries in generalizing alternations to and from initial syllables[END_REF]Becker et al. in Luckily there is an argument in favour of the non--lenition, two step analysis of initial weakening.

Proof of the two step process

The confirmation of the two--step diachronic analysis, sketched in (5) in the previous section, comes from a related dialect. In a charming analysis--driven punchline, while Ontena Gadsup has no underlying stops, Akuna Gadsup has plenty of stops but no underlying fricatives. 2The fricatives of Akuna are allophonic and are generated by a rule of intervocalic spirantisation. (6) Akuna Gadsup inventory (Frantz and Frantz 1966) Group 1 p t d k Group 2 β m n j (ʔ) In fact, the Akuna dialect appears to have what is presumably also the Proto--Ontena Gadsup phonological system. In Akuna, intervocalic stops have all lenited to fricatives, but they have been preserved post-consonantally, that is in precisely the context where they are still found in Ontena Gadsup: {[N._], [ʔ._]}. Crucially, unlike modern Ontena Gadsup, the stops have also (categorically) remained in initial position. We show the difference between these two varieties in (7) beneath.

(7) Distribution of Stops Initial Intervocalic Post--consonantal Akuna yes no yes Ontena no no yes Interestingly, Akuna is reported to have the beginnings of 'initial weakening'. At the time that it was described, it had already reduced all instances of word--initial historic *b to [β], but it had also begun an optional and variable process of word--initial weakening: the oral stop consonants /p, t, d, k/ are also beginning to surface as: [ɸ, s, ɾ, x]. Crucially, the intervocalic spirantisation was fully completed in the Akuna variety before the initial weakening began toward the path of becoming fixed. Based on this, we are comfortable in claiming that the diachronic path behind the loss of stops in Gadsup was not achieved through a phonological unification of initial and intervocalic environments as one weak environment. The synchronic pattern is a product of a two--step diachronic model (shown in (5) and repeated in (8) below), where spirantisation in intervocalic and initial environments do not share one cause. We label Stage III as Ontena Gadsup and place Akuna above the arrow because it is clearly categorically in Stage II, but apparently moving toward Stage III. (8) Weakening in Gadsup (Post)--Intervocalic spirantisation followed by initial spirantisation

Distribution of stopness in Gadsup

The structural description of Ontena Gadsup's unusual distribution of stopness leads to an interesting discussion with regard to the grammar of segment and syllable structure interaction. Before we describe the pattern in phonological terms, we will present a lightning introduction to Element Theory (ET), the framework the analysis is set in (the analysis could be translated into other featural systems).

toward' the' path' of ' becoming' 2ixed.' Based' on' this,' we' are' comfortable' in' claiming' that' the diachronic'path'behind'the'loss'of'stops'in'Gadsup'was not'achieved'through'a'phonological uni2ication'of'initial'and'intervocalic'environments'as'one'weak'environment.'The'synchronic pattern' is' a' product' of' a' two?step' diachronic' model' (shown' in' (5)' and' repeated' in' (8) beneath),' where' spirantisation' in' intervocalic' and' initial' environments' do' not' share' one cause.' We' label' Stage' III' as' Ontena' Gadsup' and' place' Akuna' above' the' arrow' because' it' is clearly'categorically'in'Stage'II,'but'apparently'moving'toward'Stage'III.

(8)'Weakening'in'Gadsup (Post)?Intervocalic'spirantisation'followed'by'initial'spirantisation

Stage&I & Stage&II &&&&Akuna Ontena&Gadsup kiki''' kixi' xixi kinki kinki xinki kiɁki kiɁki xiɁki '''''''(Post)?Intervocalic '''''Initial ''''''''''''spirantisation ''''''''''''spirantisation 2&Distribution&of&stopness&in&Gadsup
The' structural' description' of' Ontena' Gadsup's' unusual' distribution' of' stopness' leads' to' an interesting'discussion'with'regard'the'grammar'of'segment'and'syllable'structure'interaction. Before'we'describe'the'pattern'in'phonological'terms,'we'will'present'a'lightning'introduction to'Element'Theory'(ET),'the'framework'the'analysis'is'set'in'(the'analysis'could'be'translated into'other'featural'systems).

ET ' is' a' system' of' representation' based' on' equipollent,' non?articulatory' features (neutral'between'speaker'and'hearer)' (Harris'and'Lindsey'1995).'For'a'recent'comprehensive introduction'see 'Backley'(2011).'The'elements'broadly'split'into'a''place''vs.''manner/voicing' split.'|A|,'|I|,'|U|'are'the''place',''colours''or'resonance'elements,'while'|H|'stands'for'aperiodic noise,'voicelessness'and'high'tone,'|L|'for'murmur,'voicing'and'low'tone,'and'|ʔ|'for'creakiness and' a' sustained' drop' in' overall' amplitude.' These' last' three' elements' can' be' referred' to' by name: Noise, Murmur'and Edge'respectively.'Each'element'can'be'headed'or'unheaded,'this'is marked' by' underlyning' the' element' in' its' brackets' |H|' vs.' |H|,' or' when' named' by' adding' a star: Noise' vs. Noise*.' We' will' assume' that' the' elemental' make' up' of' sounds' if' roughly' as shown'in'(9). 4 (9)'Element'make'up'by'class'of'sound

Edge &&&&&| Noise &&&&&&&&| Murmur&&&&&|& Place

The typology of the distribution of Edge ET is a system of representation based on equipollent, non-articulatory features (neutral between speaker and hearer) [START_REF] Harris | The Elements of Phonological Representation[END_REF]. For a recent comprehensive introduction see Backley (2011). The elements broadly split into a 'place' vs. 'manner/voicing' split. |A|, |I|, |U| are the 'place', 'colours' or resonance elements, while |H| stands for aperiodic noise, voicelessness and high tone, |L| for murmur, voicing and low tone, and |ʔ| for creakiness and a sustained drop in overall amplitude. These last three elements can be referred to by name: Noise, Murmur and Edge respectively. Each element can be headed or unheaded -this is marked by underlining the element in its brackets (|H| vs. |H|), or when referred to by 'name' the headed variant of the feature is shown with a superscript asterisk (Noise vs. Noise*). We will assume that the elemental make up of sounds is roughly as shown in (9).3 (9) Element composition by class of sound Headed elements are shown with an asterisk, and in curly brackets in the example column (as underlining is not clearly seen in tables). Also, for simplicity, features for voicing contrasts have not been shown. Because Gadsup requires a discussion of the distribution of stopness, it is the element Edge(*) that needs explaining. With respect to Edge(*), the sound classes line up as follows: stops are distinguished by this abrupt and sustained drop in amplitude, though they also contain some noise in their burst and some spectral modulation corresponding to place of articulation. Affricates are similar but they also emphasise their Noise component. In both Stops and Affricates Edge(*) is headed (Edge*) (9a). Oral fricatives on the other hand do not contain Edge(*) at all -they are made up of Noise to correspond to 3 In fact, all varieties of GP/Strict CV assume that phonological specification of features is broadly language specific. In some languages stops may or may not contain laryngeal features. Laterals may or may not contain Edge. Etc…

7"

The$typology$of$the$distribution$of$Edge$ ET" is" a" system" of" representation" based" on" equipollent," non7 articulatory"features"(neutral"between"speaker"and"hearer)" [START_REF] Harris | The Elements of Phonological Representation[END_REF]." For" a" recent" comprehensive" introduction" see " Backley" (2011)." The" elements" broadly" split" into" a" 'place'" vs." 'manner/voicing'" split." |A|," |I|," |U|" are" the" 'place'," 'colours'" or" resonance" elements," while" |H|" stands" for" aperiodic" noise," voicelessness" and" high" tone," |L|" for" murmur," voicing" and" low" tone," and" |ʔ|" for" creakiness" and" a" sustained" drop"in"overall"amplitude."These"last"three"elements"can"be"referred"to" by" name:" Noise," Murmur" and" Edge" respectively." Each" element" can" be" headed"or"unheaded"-"this"is"marked"by"underlining"the"element"in"its" brackets"(|H|"vs."|H|),"or"when"referred"to"by"'name'"the"headed"variant" of" the" feature" is" shown" with" a" superscript" asterisk" (Noise" vs." Noise*)." We" will" assume" that" the" elemental" make" up" of" sounds" is" roughly" as" shown"in"(9). 3 "" " " (9)" Element"composition"by"class"of"sound" Headed$ elements$ are$ shown$ with$ asterisk,$ and$ in$ curly$ brackets$ in$ the$ example$ column$ (as$ underining$ is$ not$ clearly$ seen$ in$ tables).$ Also,$ for$ simplicity,$features$for$voicing$contrasts$have$not$been$shown.$ $ $ " " Edge" Noise" Murmur" Place" Example" Oral"stop"

""|ʔ|*" |H|" " A,"I,"U" /q/" ""({ʔ},"H,"U,"A)" Oral"Afficate" ""|ʔ|*" ""|H|*" " A,"I,"U" /q͡ χ/"""({ʔ,"H},"U,"A)" Oral"Fricative" " ""|H|*" " A,"I,"U" /f/"" ""({H},"U)" Nasal"

|ʔ|" " |L|" A,"I,"U" /m/" ""(ʔ,"L,"U)" Lateral"

|ʔ|" " " A,"I,"U" /ɫ/"" ""(ʔ,"A,"U)" Rhotic" " " " A,"I,"U" /ɹ w /" ""(A,"U)" Glide" " " " A,"I,"U" /w/""""(U)" Glottal"fricative" " |H|" " " /h/"" ""(H)" Glottal"stop"

|ʔ|" " " " /"ʔ/"" ""(ʔ)" " " Because"Gadsup"requires"a"discussion"of"the"distribution"of"stopness,"it" is" the" element" Edge(*)" that" needs" explaining." With" respect" to" Edge(*)," the" sound" classes" line" up" as" follows:" stops" are" distinguished" by" this" abrupt"and"sustained"drop"in"amplitude,"though"they"also"contain"some" noise" in" their" burst" and" some" spectral" modulation" corresponding" to" """""""""""""""""""""""""""""""""""""""""""""""""""""""" their aperiodic energy in addition to some spectral modulation (9b). Nasals, on the other hand (9c), do contain Edge reflecting their sustained drop in amplitude, however, this is not their headed property. Laterals (9d) also may or may not have an Edge element. Rhotics, (9e) have never been shown to have one, and that's also the case for the rest of the semi--consonantal noises (glides etc) (9f).

The peculiarity of Gadsup Ontena revolves around the distribution of |ʔ|, that is, Edge*. Recall that in Ontena Gadsup, stops cannot be found word--initially or intervocalically, instead they must be preceded by either a nasal or a glottal stop. This is equivalent to saying that Edge* is only licensed in structures where it branches across two positions: once as a head (on the right). The headedness of the branching edge is expressed on the headedness of the heterosyllabic cluster, it is on the right (cf. [START_REF] Kaye | Constituent structure and government in phonology[END_REF][START_REF] Charette | License to Govern[END_REF]). ( 10) Ontena Gadsup 4 To describe the well--formedness of words in Ontena Gadsup one could write a condition on licensing such as is expressed in (11). 4 While Edge* must branch to a dependent position, the headed part of Edge* is manifested only on the headed part of the C.C cluster, the rightmost part. We show the following with the hypothetical minimal pair: 'umanti' and 'umacti'.

distinguished by this abrupt and sustained drop in amplitude, though they also contain some noise i their burst and some spectral modulation corresponding to place of articulation. Affricates are simila but they also emphasise their Noise component. In both Stops and Affricates Edge(*) is heade (Edge*) (9a). Oral fricatives on the other hand do not contain Edge(*) at all, they are made up o Noise to correspond to their aperiodic energy in addition to some spectral modulation (9b). Nasal on the other hand (9c), do contain Edge reflecting their sustained drop in amplitude, however, this not their headed property. Laterals (9d) also may or may not have an Edge element. Rhotics, (9e have never been shown to have one, and that's the case for the rest of the semi-consonantal noise (glides etc…) (9f).

The peculiarity of Gadsup Ontena revolves around the distribution of |ʔ|, Edge*. Recall tha in Ontena Gadsup, stops cannot be found word-initially or intervocalically, instead they must b preceded by either a nasal of a glottal stop. This is equivalent to saying that Edge* is only licensed i structures where it branches across two positions: once as a head (on the right). The headedness o the branching edge is expressed on the headedness of the heterosyllabic cluster, it is on the right (c [START_REF] Kaye | Constituent structure and government in phonology[END_REF][START_REF] Charette | License to Govern[END_REF]).

(10) Ontena Gadsup5 (a) Headed edge found after homorganic nasal [umanti] 'example'

Dep. Head | | C V C V C v C V | | | | | | u m a L ʔ H Place i (b) Headed edge found after glottal stop [umaʔki] Dep. Head | | C V C V C v C V | | | | | u m a ʔ H Place i
To describe the well-formedness of words in Ontena Gadsup one could write a condition on licensin such as is expressed in (11).

The typology of the distribution of Edge

(11) Licensing condition on Edge* in Ontena Gadsup

• Edge* (|ʔ|) must branch to a dependent position.

• A feature Edge* is licensed iff it is contained by two C positions (Or: Edge* must branch) The licensing condition in ( 11) is reminiscient of [START_REF] Charette | Licensing constraints and vowel harmony in Turkic languages[END_REF] licensing constraints except that it checks well-formedness across two tiers of representation, the syllable structure tier and the melodic tier (features).

The licensing condition in ( 11) is stated as a restriction on melody. Therefore, as it is relevant to Ontena Gadsup, Edge* is only found in structures where it can also branch to a dependent position. As a consequence of this condition, stops are restricted to post--consonantal positions, the relationship between the two properties Edge* and bipositionality are presented in the table in (12). ( 12) Edge* distribution and bipositionality in Ontena

The connection between the properties of Edge and bipositionality already has an expression in the Government Phonology literature. [START_REF] Jensen | Is ? and element ? Towards a non--segmental phonology[END_REF], [START_REF] Pöchtrager | The Structure of Length[END_REF], [START_REF] Pöchtrager | GP 2.0[END_REF] The way that the argument is written works in the following way that we have listed in beneath. The/ way/ that/ the/ argument/ is/ written/ works/ in/ the/ following/ way/ that/ we/ have/ listed/ in (13)/beneath.

Initial

(13)/Logical/structure/of/argument ! In/a/given/language/there/will/be/certain/syllable/structures,/x,/y,/z…. ! In/this/language/there/are/also/certain/features:/α,/β,/γ… ! The/grammar/includes/a/set/of/licensing/conditions:/1,/2,/3… ! 1/says/that/feature/α/must/be/found/in/x ! Therefore/feature/α/cannot/be/found/in/y/(because/y/is/not/x). ! However,/anything/else/may/be/contained/by/y/(as/far/as/1/is/concerned)./// The/way/this/worked/for/Ontena/Gadsup/is/schematized/in/( 14). This argument relies on a licensing condition that regulates the co-occurrence restrictions of features and syllable structures. The way it works is laid out in (13) beneath. ( 13) Logical structure of licensing condition

• In a given language there will be certain syllable structures, x, y, z….

• In this language there are also certain features: α, β, γ… • The grammar includes a set of licensing conditions: 1, 2, 3…

• 1 says that feature α must be found in x

• Therefore feature α cannot be found in y (because y is not x).

• However, anything else may be contained by y (as far as 1 is concerned). The way this works for Ontena Gadsup is schematized in ( 14). ( 14 [START_REF] Jensen | Is ? and element ? Towards a non--segmental phonology[END_REF] and [START_REF] Pöchtrager | The Structure of Length[END_REF] are not correct that Edge* is reducible to a consonantal structure with multiple positions, there is nonetheless a positive (and we will see, universal) relationship between these two properties, one of which is a feature and the other a syllable structure, and we were able to capture this relationship through a licensing condition.

The typology of the distribution of Edge

In fact, the licensing condition stated in (11) -applied to either headed or headless edge (Edge* or Edge) -and the logic laid out in (13) unravel into a typology.

The typology of the distribution of Edge

When it comes to the distribution of Edge(*) there are in fact many types of languages. Some have no restriction, or few if any restrictions on Edge(*); some have very specific conditions, where the distribution of Edge(*) is tied to other linguistic objects/licensing forces (such as Edge* and Government in Tuscan Italian (cf. [START_REF] Bafile | La spirantizzazione toscana nell'ambito della teoria degli elementi[END_REF][START_REF] Marotta | Lenition in Tuscan Italian (Gorgia Toscana)[END_REF]). However, the hypotheses proposed to account for Ontena Gadsup in the previous section extend into a very broad typology of two grand classes. These two major types of Edge(*)--distribution are interesting when put next to each other, as they shed light on the nature of phonology, and may even be diachronically connected, as we show in the final section, so it is these two grand classes that will be discussed in this paper. They are shown in ( 15). ( 15

Type A

Type A is the kind of language that includes Ontena Gadsup and where we kicked off the discussion. Ontena Gadsup is indeed very rare in its patterning but it is not, typologically speaking, an isolate. Some Berber languages manifest the same distribution of stopness and can easily be analysed as further instances of Type A. Type A is the kind of language you get when the distribution of Edge* is positively connected with bipositionality and, crucially, the grammar is focused on the distribution of the feature, almost as if it were tracking the co--indexation of these two properties on these two different tiers from the perspective of the feature. It is a feature--centric expression of the licensing condition that marries Edge* and bipositionality. ( 16

Type A

Type A is the kind of language that contains Ontena Gadsup and where we kicked off the discussion. Ontena Gadsup is indeed very rare but it is not, typologically speaking, an isolate. Some Berber languages manifest the same distribution of stopness and can easily be analysed as further instances of Type A.

Type A is the kind of language you get when the property of Edge* is positively connected with bipositionality and, crucially, the grammar is focused on the distribution of the feature. Almost as if it were tracking the co-indexation of these two properties on these two different tiers from the perspective of the feature. It is a feature-centric expression of the licensing condition that marries Edge* and bipositionality.

C … C *# C | | Melody ʔ ʔ
In this language type, there are no Edge*-based restrictions on the consonant that populate bipositional structures. Rime-onset sequences and geminates can (in principle, and all things being equal) contain any consonantal types, but singleton onsets cannot contain Edge*.

In certain Berber varieties, oral stops spirantize in many contexts except when they are geminated (17a-b). Word-initial stops also spirantize in these varieties. Elsewhere, 't, d' resist

Type A

Type A is the kind of language that contains Ontena Gadsup and where we kicked off the discussion. Ontena Gadsup is indeed very rare but it is not, typologically speaking, an isolate. Some Berber languages manifest the same distribution of stopness and can easily be analysed as further instances of Type A.

Type A is the kind of language you get when the property of Edge* is positively connected with bipositionality and, crucially, the grammar is focused on the distribution of the feature. Almost as if it were tracking the co-indexation of these two properties on these two different tiers from the perspective of the feature. It is a feature-centric expression of the licensing condition that marries Edge* and bipositionality.

C … C *# C | | Melody ʔ ʔ
In this language type, there are no Edge*-based restrictions on the consonant that populate bipositional structures. Rime-onset sequences and geminates can (in principle, and all things being equal) contain any consonantal types, but singleton onsets cannot contain Edge*.

In certain Berber varieties, oral stops spirantize in many contexts except when they are geminated (17a-b). Word-initial stops also spirantize in these varieties. Elsewhere, 't, d' resist

In this language type, there are no Edge*--based restrictions on the consonants that populate bipositional structures. Rime--onset sequences and geminates can (in principle, and all things being equal) contain any consonantal types, but singleton onsets cannot contain Edge*.

In certain varieties of Berber like Tamazight, Tarifit and Kabyle, oral stops spirantise in many contexts except when they are geminated (17a--b). Word--initial stops also spirantise in these varieties. Elsewhere, 't, d' resist spirantisation but only when they are preceded by 'm, n, l' (18) -see El [START_REF] Kirat | Spirantization in the Beni Iznassen dialect, diachrony and synchrony[END_REF], and [START_REF] Maarten | Berber Phonology[END_REF] In (17a), we see Tamazight Berber spirantizing singleton stops in various positions, including intervocalic and word--initial. These can be contrasted with their cognates in Tashlhiyt where these stops remain unchanged. (17b) contrasts singleton stops that have become spirantized in the aorist and their geminated counterparts in the imperfective which remain non--spirantised. Data in (18) further show cases where 't, d' resist spirantization when preceded by 'm, n, l'.

In the spirantising Berber languages, whatever other conditions hold, Edge* as a feature (stopness) is only found in bipositional structures, either as a geminate or as part of N.C and L.C sequences:

[tirgin] > [θirʒin] 'embers' vs. [fəttəl] 'roll', [θaħnint] 'tender woman', [ultma] 'my sister'. The fact that Edge* is shared across two positions is crucial. Merely being post--consonantal is not sufficient to block spirantisation: [agurdi] vs. [açurði] 'kind of bug'. In our model, this is an expected outcome because, as we show in (9), rhotics do not contain occlusivity. Therefore, in R.C (and unlike N.C and L.C) sequences Edge* cannot branch across the two positions. Consequently, R.C structures do not meet the licensing condition discussed in this section, and the stops in these configurations must spirantise.

Type B

Meanwhile, Type B languages share the positive relationship between Edge and bipositionality, but they express the condition from the other direction, from the syllable structure. Indeed, the licensing condition in Type B languages is phrased as in (19) As we see in ( 19), the condition forces bipositional syllable structure to contain a certain feature. Notice here that the condition is a little less strict as it allows Edge in both its incarnations (headed and headless). 6Many of what are sometimes called the 'Prince' languages would be of this type [START_REF] Prince | Phonology with tiers[END_REF][START_REF] Piggott | The phonotactics of a "Prince" language: a case study[END_REF]. These languages have consonant clusters but they are restricted to two--member, heterosyllabic (rime--onset) clusters which contain either geminates or N.C clusters. 7 A good example of a Type B language is the Kingi variety of Soninké [START_REF] Creissels | Le rôle du ton dans la flexion du Soninké[END_REF]. The basic segmental inventory of this language is given in ( 20). ( 20 8 Voiced stops are very marginally attested in geminates, b, d, j. Because, they are in fact possible, albeit low frequency, we take them to be categorically permitted and their rarity is due to substantive factors. g is not attested, we take this to be an accidental gap. 9 The h was diachronically f (Creissels p.c.).

5 Diachronic and theoretical consequences

Diachronic consequences

We have presented Ontena Gadsup as a Type A language and Kingi Soninké as a Type B language with regard Edge(*) and C.C. In what way can these two systems interact in diachrony? A language like Kingi Soninké forces the bipositional structures to contain Edge. Some of this condition is mimicked by Akuna Gadsup where (at the very least) intervocalic spirantisation creates paradigmatic syllable structure contrasts where singletons never have Edge, while bipositional structures do. Consider stage II in ( 27). ( 27) Diachrony in Edge and C.C licensing (repeated from ( 8))

(Post)--Intervocalic spirantisation followed by initial spirantisation At Stage I there is nothing remarkable because in the middle of the word the child language learner sees stops everywhere. But if the language begins to be spirantise stops, so that it moves towards Stage II, the child learner will begin to receive (admittedly very partial) paradigmatic contrasts between fricatives and stops.

#Gadsup#system.#It#is#therefore#conceivably#possible to#go#from#a#system like#Kingi#Soninké#to#something like#Ontena#Gadsup,#the#Hirst#ingredient#is
At Stage I there is nothing remarkable because in the middle of the word the child language learner sees stops everywhere. But if the language begins to be spirantise stops, so that it moves towards Stage II, the child learner will begin to receive (admittedly very partial) paradigmatic contrasts between fricatives and stops.

(28) Towards Stage II Weak | No Edge* | (a) k i x i (b) k i n k i | Edge* | Strong
Specifically, the child sees a very partial pattern where stopness cannot be found in intervocalic context except where it is hosted by a bipositional structure.

In keeping with Honeybone's (2005) 'sharing makes us stronger', if Edge* is never removed when it is shared across two positions, then the child could posit the licensing condition that stopness is positively related to bipositionality.

Specifically, the child sees a very partial pattern where stopness cannot be found in intervocalic context except where it is hosted by a bipositional structure.

In keeping with [START_REF] Honeybone | Headhood, Elements, Specification and Contrastivity: Phonological papers in honour of John Anderson[END_REF] 'sharing makes us stronger', if Edge* is never removed when it is shared across two positions, then the child could posit the licensing condition that stopness is positively related to bipositionality, such as what is formulated in (11).

This positive correlation of Edge* and bipositionality can become progressively stronger (as more and more stops are lenited). In so doing, there will be more and more examples of this contrast. Somewhere (presumably on an s--curve of change) the child will set a categorical licensing constraint banning Edge* from singleton consonants. If that is phonologised as a licensing condition like: Edge* can only be found in C.C, then the change will be categorical and one will obtain the Ontena Gadsup system. It is therefore conceivably possible to go from a system like Kingi Soninké to something like Ontena Gadsup, the first ingredient is intervocalic spirantisation.

In fact, many paths of change are probably possible -and that is only considering the phonological properties of Edge(*) and bipositionality. The precise diachrony, showing how this relates to the life cycle of phonological processes [START_REF] Bermudez | Amphichronic explanation and the life cycle of phonological processes[END_REF], would need to be carefully worked out. But it seems from our licensing conditions that such segment / syllable structure mappings would be a fruitful area for future study.

Theoretical consequences

We have seen a positive relationship between Edge(*) and bipositionality. These two properties can be found without each other, meaning that their distribution is not universally and bidirectionally co-extensive. They are in a very important sense independent (they are on different tiers and cannot be reduced to each other). There is variation therefore in how these two properties are related to each other in grammars. But crucially, we argue that there is a UG component to the distribution of Edge in relation to bipositionality. The structure shown in (26) which is created bottom up in Type A languages and top down in Type B languages reflects a deep affinity between these two separate properties. We state this observation in (29).10 (29) Edge(*) and bipositionality go together

The typology of the distribution of Edge Indeed, in no language with both monopositional and bipositional structures could one restrict the distribution of Edge* to singleton stops, while simultaneously banning its presence in its bipositional structures. In no language does spirantisation target only geminates and not singletons. Or perhaps better (less likely to be reduced to substantive phonetic factors), there will not be languages with systematic gaps in rime--onset sequences (where Edge in N.C sequences and geminates are systematically banned). Discovering what underlies the affinity behind the observation in (29) would itself make a valuable research project.

Conclusion

This paper has considered the distribution of two linguistic properties: Edge(*) and bipositionality (C.C). The former is a featural property, the latter is a syllable--structure property. In this paper, we have demonstrated that there is a positive relationship between them. In extreme cases this creates the phonological patterns of Kingi Soninké and Ontena Gadsup, where both languages have phonologised licensing conditions which juxtapose Edge(*) and bipositionality. We showed that this was achieved in two basic ways which formed two basic classes of languages: Type A, which restricts the presence of the feature Edge* to the bipositional syllable structures, and Type B, which obliges a bipositional syllable structure to contain Edge across its two positions. The effects of these licensing conditions on the Edge(*) feature and on bipositionality when they are not coextensive produces cross--linguistic variation. Type A languages end up banning stops from singleton onset position, both initially and intervocalically. While Type B languages end up banning anything but stops, nasals and laterals from consonant clusters and geminates. While it is true that Edge(*) and bi--positionality are not universally coextensive (the two properties cannot be reduced to each other), the variation in Edge--licensing is neither random nor logically exhaustive, therefore it does have value in terms of phonological universal grammar. Indeed, in no language is Edge(*) explicitly restricted to mono--positional structures. The inherent strength of initial positions is also preserved (even in a language like Ontena Gadsup where stops are banned word--initially) because the apparent initial weakening can be reanalysed as a condition on bipositionality for a certain element -it is not the product of a generalised weak environment that unifies the word--initial and intervocalic environments. We have speculated on the origin of this

  in post--consonantal contexts. This post--consonantal hardening is shown in (3

  and in line with the facts, bipositional structures are free to host any other consonantal sequences: e.g. [onsena].5

  ) Type A Edge* is licensed by being bi--positional (C.C) (Ontena Gadsup, Berber) Type B C.C is licensed by containing Edge ('Prince languages', Soninké)

  ) Type A (a) Licensing status: Edge* is licensed by being bi--positional (C.C) (b) Example: Ontena Gadsup, Berber. (c) Syllable structure (d) Consequences: Melody

  status: Edge* is licensed by being bi-positional (C

  status: Edge* is licensed by being bi-positional (C

  #I#there#is#nothing#remarkable#because#in#the#middle#of#the#word#the#child#language learner# sees# stops# everywhere.# But# if# the# language# begins# to# be# spirantise# stops,# so# that# it moves# towards# Stage# II,# the# child# learner# will# begin# to# receive# (admittedly# very# partial) paradigmatic#contrasts#between#fricatives#and#stops. the# child# sees# a# very# partial# pattern# where# stopness# cannot# be# found# in intervocalic#context#except*where*it*is*hosted*by*a*bipositional*structure.In# keeping# with# Honeybone's# (2005)# 'sharing# makes# us# stronger',# if Edge*# is# never removed# when# it# is# shared# across# two# positions,# then# the# child# could# posit# the# licensing condition#that#stopness#is#positively#related#to#bipositionality. This# positive# correlation# of Edge*# and# bipositionality# can# become# progressively stronger# (as# more# and# more# stops# are# lenited).# In# so# doing,# there# will# be# more# and# more examples#of#this#contrast.#Somewhere#(presumably#on#an#sNcurve#of#change)#the#child#will#set a# categorical# licensing# constraint# banning Edge*# from# singleton# consonants.# If# that# is phonologised#as#a#licensing#condition#like: Edge* must*be*found*in*C.C,#then#the#change#will#be categorical#and#one#will#obtain#the#Ontena

  and other work in what is called 'GP 2.0' actually reduce these two properties as if they were expressions of each other: Bipositionality = Edge. In this framework, the feature Edge is expressed solely as a structural configuration of essentially two positions.It is formally important to note that in our work the two properties are not being reduced to each other. In fact, our observation that in Ontena Gadsup Edge* and Bipositionality are related through licensing goes only one way. It is an expression limiting the distribution of the feature Edge*. According to the formulation of the licensing condition in (11) Licensing condition on Edge*: Edge* (|ʔ|) must branch to a dependent position.

	 A feature Edge* is licensed iff it is contained by two C positions (Or: Edge* must branc
	Syllable tier
	C	…	C
			|
	Melody tier		|ʔ|
	The licensing condition in (11) is reminiscient of Charette and Göksel's (1998) licensing const
	except that it checks well-formedness across two tiers of representation, the syllable structure tie
	the melodic tier (features).
	Initial		Intervocalic	Post-consonantal
	no		no	yes
	Monopositional		Bipositional
	The connection between the properties of Edge and bipositionality already has an expression
	GP literature. Jensen (1994), Pöchtrager (2006), Pöchtrager and Kaye (2013) and other work in
	is called 'GP 2.0' actually reduce these two properties as if they were expressions of each o
	Bipositionality = Edge. In this framework, the feature Edge is expressed solely as a stru
	configuration of essentially two positions.
	It is formally important to note that in our work the two properties are not being reduc
	each other. In fact, our observation that in Ontena Gadsup Edge* and Bipositionality are re

The licensing condition in (

11

) is stated such as it is a restriction on melody. Therefore is relevant to Ontena Gadsup, Edge* is only found in structures where it can also branch dependent position. As a consequence of this condition, stops are restricted to post-conson positions, the relationship between the two properties Edge* and bipositionality are presented table in (12). (12) Edge* distribution and bipositionality in Ontena through licensing goes only one way. It is an expression limiting the distribution of the feature E According to the formulation of the licenning condition in (11), and in line with the facts, biposi structures are free to host any other consonantal sequences: e.g. [onsena]. 6

  being reduced/ to/ each/ other./ In/ fact,/ our/ observation/ that/ in/ Ontena/ Gadsup Edge*/ and Bipositionality/are/related/through/licensing/goes/only/one/way./It/is/an/expression/limiting/the distribution/ of/ the/ feature Edge*./ According/ to/ the/ formulation/ of/ the/ licenning/ condition/ in (11),/and/in/line/with/the/facts,/bipositional/structures/are/free/to/host/any/other/consonantal sequences:/e.g./[onsena]. 6

		Intervocalic	Post.consonantal
	no	no	yes
	Monopositional		Bipositional

The/connection/between/the/properties/of Edge/and/bipositionality/already/has/an/expression in/the/GP/literature.

[START_REF] Jensen | Is ? and element ? Towards a non--segmental phonology[END_REF]

,/Pöchtrager/(2006),/Pöchtrager/and/Kaye/(2013)/and/other work/ in/ what/ is/ called/ 'GP/ 2.0'/ actually/ reduce/ these/ two/ properties/ as/ if/ they/ were expressions/ of/ each/ other:/ Bipositionality/ = Edge./ In/ this/ framework,/ the/ feature Edge/ is expressed/solely/as/a/structural/conPiguration/of/essentially/two/positions./ It/ is/ formally/ important/ to/ note/ that/ in/ our/ work/ the/ two/ properties/ are not/

  . It is checking the positive relationship of bipositional syllable structure and the feature Edge in terms of what the syllable structure contains. (19) Type B (cf. 15): C.C is licensed by containing Edge in both positions.

We do not discuss stops in 'coda' position because they are irrelevant for the discussion of Ontena.

Apart from [β] which is significant as we will show soon.

"In"fact,"all"varieties"of"GP/Strict"CV"assume"that"phonological"specification"of"features" is" broadly" language" specific." In" some" languages" stops" may" or" may" not" contain" laryngeal"features."Laterals"may"or"may"not"contain"Edge."Etc…"

While Edge* must branch to a dependent position the headed part of Edge* is manifested only on the headed part of the C.C cluster, the rightmost part. We show the following with the hypothetical minimal pair: 'umanti' and 'umacti'.

Unless any other conditions are stated.

Whatever applies to Edge applies to Edge*, but not everything that applies to Edge* applies also to Edge.

Not prenasalised consonants, but actual N+C sequences.

It's not a principle, definition or condition. It's merely an observation.

Acknowledgements

Thanks to Larry Hyman, Eva Zimmerman, Jochen Trommer, Paul Smolensky, Tobias Scheer, Patrick Honeybone, Jean Lowenstamm and Sam Hellmuth, and to audiences at LAGB 2016 and RFP Nice for their questions and comments; needless to say, the people mentioned here do not necessarily agree with all or any part of what is expressed in this paper.

The typology of the distribution of Edge A preceding nasal consonant triggers these hardening alternations. Their structural description in Element Theory is one where singleton onsets, which can contain any consonant (with or without Edge), are 'suddenly' forced into the head of a bipositional structure. The head of the bipositional structure in Kingi must always contain Edge, and so the consonants harden to meet this licensing condition. Edge becomes part of their description. We express this in (23), as a licensing condition, just as we did for Ontena Gadsup.

Bipositionality and Edge

As we have seen a few times now. Both Type A and Type B languages there is pressure from the grammar to positively combine the feature Edge(*) (either in only its headed form or in its headed and unheaded form) with a bipositional syllable structure (C.C). The description that both these grammar types are aiming to produce is identical, repeated in (25) beneath.

(25) Alignment of C.C and Edge(*)

Shanti Ulfsbjorninn & Mohamed Lahrouchi 16

Bipositionality and Edge

As we have seen a few times now, there is pressure from the grammar in both Type A and Type B languages to positively combine the feature Edge(*) (either in only its headed form or in its headed and unheaded form) with a bipositional syllable structure (C.C). The description that both these types of grammar are aiming to produce is identical, and is repeated in ( 25). ( 25) Alignment of C.C and Edge(*)

The languages diverge, however, in terms of the direction from which the licensing condition is stated. In Type A languages the condition is on the feature, while in Type B languages the condition is on the syllable structure. From a UG perspective, licensing conditions across tiers are bidirectional, as shown in ( 26). ( 26) Condition for Type A and Type B Type A 'is my feature in the right place' Type B 'does the structure contain the right feature' The way that this creates variation, hence typology, is from what conditions are generated elsewhere. Type A languages enforce a ban on Edge* occurring in singleton stops (initially and intervocalically). Type B languages allow no set of features to be present in C.C unless they include Edge.

In all instances the structure that is being positively associated through licensing is one where material is shared. In the next section we will show some potential diachronic interactions of C.C and Edge(*) and how they relate to [START_REF] Honeybone | Headhood, Elements, Specification and Contrastivity: Phonological papers in honour of John Anderson[END_REF] notion that 'sharing makes us stronger'.

Bipositionality and Edge

As we have seen a few times now. Both Type A and Type B languages there is pressure from th grammar to positively combine the feature Edge(*) (either in only its headed form or in its heade and unheaded form) with a bipositional syllable structure (C.C). The description that both the grammar types are aiming to produce is identical, repeated in (25) beneath. The languages diverge, however, in terms of the direction from which the licensing condition is stated.

In Type A languages the condition is on the feature, while in Type B languages the condition is on the syllable structure. From a UG perspective, licensing conditions across tiers are bidirectional as shown in ( 26). The way that this creates variation, hence typology, is from what conditions are generated elsewhere.

Type A enforces a ban on Edge* occurring in singleton stops (initially and intervocalically). While, Type B languages force no features to be present in C.C lest they also contain Edge.

In all instances the structure that is being positively associated through licensing is one where material is shared. In the next section we will show some potential diachronic interactions of C.C and Edge(*) and how they relate to [START_REF] Honeybone | Headhood, Elements, Specification and Contrastivity: Phonological papers in honour of John Anderson[END_REF] s.ulfsbjorninn@ucl.ac.uk, mohamed.lahrouchi@univ--paris8.fr