Raza Ur-Rehman
email: ur-rehman@irccyn.ec-nantes.fr

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

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

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

Path Placement Optimization of Manipulators Based on Energy Consumption: Application to the Orthoglide 3-axis

Keywords: Path placement, Energy consumption, Optimization, Parallel manipulators Placement de trajectoires, Consommation énergétique, Optimisation, Manipulateur parallèles

This paper deals with the optimal path placement for a manipulator based on energy consumption. It proposes a methodology to determine the optimal location of a given test path within the workspace of a manipulator with minimal electric energy used by the actuators while taking into account the geometric, kinematic and dynamic constraints. The proposed methodology is applied to the Orthoglide 3-axis, a three-degree-of-freedom translational parallel kinematic machine (PKM), as an illustrative example.

INTRODUCTION

Optimal trajectory planning of manipulators has been a relevant area for roboticists for many years. Indeed, several authors have worked on trajectory planning based on different optimization objectives. A review of trajectory planning techniques is given in [START_REF] Ata | Optimal trajectory planning of manipulators: A review[END_REF]. Trajectory planning deals with the determination of the path and velocity/acceleration profiles (or the time history of the robot's joints), the start and end points of the trajectory being predefined and fixed in the workspace. As a matter of fact, this approach is suitable for most of robotic applications. A path is a continuous curve in the configuration space connecting the initial configuration of the manipulator to its final configuration [START_REF] Rajan | Minimum Time Trajectory Planning[END_REF]. Trajectory planning usually aims at minimizing the travel distance [START_REF] Pires | An evolutionary approach to robot structure and trajectory optimization[END_REF][START_REF] Tian | Motion planning for redundant manipulators using a floating point genetic algorithm[END_REF], travel time [START_REF] Chan | Genetic-based minimum time trajectory planning of articulated manipulators with torque constraints[END_REF][START_REF] Cao | Time optimal and smooth constrained path planning for robot manipulators[END_REF][START_REF] Pledel | Actuator constraints in point to point motion planning of manipulators[END_REF] and/or the energy consumed [START_REF] Shugen | Real-time algorithm for quasi-minimum energy control of robotic manipulators[END_REF][START_REF] Field | Iterative dynamic programming, an approach to minimum energy trajectory planning for robotic manipulators[END_REF][START_REF] Hirakawa | Trajectory planning of redundant manipulators for minimum energy consumption without matrix inversion[END_REF][START_REF] Khoukhi | A projected gradient augmented Lagrangian approach to multi-objective trajectory planning of redundant robots[END_REF], while satisfying several geometric, kinematic and dynamic constraints.

Another less explored aspect of trajectory planning is the placement of a given path within the workspace. It aims at determining the optimum location of a predefined path to be followed by the end-effector of the manipulator within its workspace with respect to one or many given objective(s) and constraint(s). This path can be the shape of a component to be machined, a welded profile or an artistic/decorative profile etc. In such situations, the trajectory planner cannot alter the shape of the path but he/she can only play with the location of that path within the workspace of the manipulator in order to optimize one or several criterion(a). Such an approach can be very interesting in many robotic applications. For example, in machining, a workpiece can be better located within the workspace of the robot to perform a given operation more efficiently with respect to the energy consumed.

The path placement problem has not been extensively studied in the past. Nevertheless, some researchers proposed to solve it with respect to various optimization objectives. Several performance criteria for path location problems can be considered simultaneously (multiobjective) or individually, such as travel time, different kinetostatic performance indices (manipulability or conditioning number), kinematic performance (velocity, acceleration), avoiding obstacles, reduced wear or vibration, energy consumption etc. In the following paragraphs, a brief survey of the work of different researchers to solve the problem of path placement optimization for various applications is presented.

Nelson and Donath [START_REF] Nelson | Optimization of the location of assembly tasks in a manipulator workspace[END_REF] proposed an algorithm for the optimum location of an assembly task in the manipulator workspace while taking the manipulability measure as the optimization criterion. They considered that the location of the assembly task within the workspace that results in the highest manipulability is a locally optimal position for performing the assembly. However, Aspragathos [START_REF] Aspragathos | Optimal location of path following tasks in the workspace of a manipulator using genetic algorithm[END_REF][START_REF] Aspragathos | Optimal location of a robot path when considering velocity performance[END_REF] considered that the manipulability index and the dexterity, usually quantified by the condition number of the Jacobian matrix of the manipulator, can characterize the motion ability of the manipulator, but these criteria cannot depict the ability of a manipulator to move in a given direction. Hence they introduced a criterion to characterize the best velocity performance of the robot end-effector with the path location. They used the concept of the orientation of the manipulability ellipsoid relative to the desired path and used genetic algorithm to come up with an optimal solution.

Fardanesh et al [START_REF] Fardanesh | Minimum cycle time location of a task in the workspace of a robot arm[END_REF] proposed an approach for optimal positioning of a prescribed task in the workspace of a 2R-manipulator. Optimal location of the task is considered to be the location that yields the minimum cycle time for the task. In another study, Feddema [START_REF] Feddema | Kinematically optimal robot placement for minimum time coordinated motion[END_REF] formulated and solved a problem of robot base placement for minimum time joint coordinated motion within a work cell. The proposed algorithm considers only the kinematics and the maximum acceleration of each joint in order to obtain a 25% cycle time improvement for a typical example.

Hemmerle [START_REF] Hemmerle | Optimal path placement for kinematically redundant manipulators[END_REF] presented an algorithm for optimum path placement of a redundant manipulator by defining a cost function related to robot joints motion and limits. The proposed approach did not consider the path as a whole but points along the entire path, hence cost function considers the performance only at the node points and not the path in-between the nodes.

Chou and Sadler [START_REF] Chou | Optimal location of robot trajectories for minimization of actuator torque[END_REF] developed an optimization technique for the optimum placement of a robotic manipulator based on the actuators torque requirements. Pamanes and Zeghloul [START_REF] Pamanes | A criterion for optimal placement of robotic manipulators[END_REF][START_REF] Pamanes | Optimal placement of robotic manipulators using multiple kinematic criteria[END_REF] considered multiple kinematic indices to find the optimal placement of a manipulator by specifying the path with a number of points and then assigning an optimization criterion to each point. The objective was to find the path location in order to have optimal values of all the criteria assigned to the path's points. In [START_REF] Pamanes | On the optimal placement and task compatibility of manipulators[END_REF], the problem of optimal placement with joint-limits and obstacle avoidance is addressed. Lately, a general formulation was presented to determine the optimal location of a path for a redundant manipulator while dealing with mono-and multi-objective problems [START_REF] Pamanes | Single and multiobjective optimization of path placement for redundant robotic manipulators[END_REF]. The goal of this research work was to keep the joint variables within their limits and to minimize the magnitude of their displacements [START_REF] Pamanes | Multi-Criteria Optimal Placement of Manipulators in Cluttered Environments[END_REF][START_REF] Pamanes | On The Task Feasibility of non Redundant Manipulators[END_REF][START_REF] Zeghloul | Reachable workspace in the presence of obstacles. Application to robot path planning in cartesian space[END_REF][START_REF] Zeghloul | Optimal Placement of Manipulators in Presence of Obstacles[END_REF][START_REF] Zeghloul | Multi-criteria optimal placement of robots in constrained environements[END_REF][START_REF] Zeghloul | Multi-Criteria optimal placement of robot manipulators[END_REF][START_REF] Zeghloul | Optimization of kinematics performances of manipulators under specified task conditions[END_REF].

With a general literature survey, it comes out that although several performance indices are introduced or considered, there is very little emphasis on the dynamic aspects reflecting the energy consumption by the robot actuators. The increasing number of the robotic applications emphasize the importance of the energy saving not only to enhance efficiency but also to face the world energy problems. Therefore, it is pertinent to locate well the path to be followed by the end-effector of a robot within its workspace in order to minimize the energy consumed by its actuators. Accordingly, the major contribution of this paper are i ) the minimization of the energy requirements by optimum path placement and ii ) use of electric energy consumed by the actuators instead of treating mechanical energy relations. Hence, we propose an approach to optimize the location of a given path within the workspace of a manipulator in order to minimize the electric energy consumed by its actuators. It should be noted that energy optimal poses can affect various performance indices such as manipulability, dexterity, stiffness etc. In the scope of this paper, we have only considered the electric energy consumption as an optimization criterion (objective function) in order to highlight its influence on motion planning. However, others optimization criteria , such as the manipulability, dexterity, stiffness, the motors torque required, will be taken into account in future work. Accordingly, we will come up with multiobjective path placement optimization problem that may not be convex. For that matter, we will use other optimization tools such as Genetic Algorithms to solve it.

The paper is organized as follows. In section II, we propose a minimum energy path placement optimization problem that includes basic problem formulation, electric energy calculations and the algorithm proposed to solve the problem. In section III, the Orthoglide 3axis: a three-degree-of-freedom translational Parallel Kinematics Machine (PKM), is used as a case study to implement the proposed optimization methodology and results are presented for rectangular test paths.

The problem aims at determining the optimal location of a predefined path in order to minimize the electric energy used by the actuators. The optimization problem is subject to geometric, kinematics and dynamics constraints. Geometric constraints include joint limits and the boundaries of the workspace. Kinematic constraints deal with the maximum actuator velocities whereas dynamic constraints are related to actuator wrenches. Contrary to the trajectories usually defined with start and end point configurations, the entire path is supposed to be known within the framework of this research work. The path location can be defined in a similar way as to define the location of a workpiece with respect to a manipulator reference point.

Path Localization

In order to formulate and describe the problem, two reference frames are defined: i) the path frame F p and ii) the base frame F b , as shown in Fig. 1. The path frame F p , is attached to the given/required path at a suitable point such as geometric center of the path. As F p is attached to the path, the end-effector trajectory parameters remains constant in this reference frame, no matter where it is located. In other words, the path is fully defined and constant in F p . It can also be named workpiece frame as it characterizes the position and the orientation of the workpiece within the manipulator workspace. The base frame F b can also be called global or manipulator frame. It is attached to the manipulator base and is used to locate a workpiece (or F p ) with respect to the manipulator coordinate system. The location and orientation of F p with respect to F b can be defined in such a way that the whole path lies within the workspace. The position of F p with respect to F b is defined with As a matter of fact, any trajectory defined in F p can be transformed in the base frame F b by a transformation matrix. For instance, point P , of Cartesian coordinates x Pp , y Pp , z Pp in F p can be expressed in F b as follows:

x X b Y b Z b O b F b F b X p Y p Z p O p F p P p Fp P X b Y b Z b O X p Y p Z p u v w φ φ ψ ψ θ θ φ ψ θ (b)
p F b = b T p p Fp (1)
namely, (3) The path placement is specified with b T p . Let x = [x Op y Op z Op φ θ ψ] T define the path placement within the workspace of the manipulator, in the reference frame F b . Consequently, the components of x are the decision variables of the optimization problem at hand. In the context of a general machining process like milling operation, the feature to be machined on the workpiece is defined with respect to the frame attached to the workpiece, namely F p . Likewise, the machining operation conditions such as machining velocity and acceleration are fully defined in F p . Finally, the part to be machined is defined by the designer and located in F b whereas the machining operation conditions and robot trajectory planning are defined by the production engineer with respect to the the corresponding part, namely F p . Here, we introduce a methodology to help the production engineer well locate the workpiece, namely F p , within the robot base frame F b in order to minimize the actuators electric energy consumption.

x P b y P b z P b 1 T F b = b T p x Pp y Pp z Pp 1 T Fp (2

Path Placement Optimization Problem Formulation

The goal of this research work is to help the path planner find the best location of the path to be followed by the robot in order to minimize the energy used by its actuators. It can be formulated as an optimization problem, namely, "For a predefined path in F p , find the optimum location and orientation of F p with respect to F b , defined by the decision variables x, in order to minimize the electric energy used by the manipulator actuators to generate that path, while respecting the geometric, kinematic and dynamic constraints of the manipulator." It can also be formulated mathematically as follows:

min x E t = n i=1 E i (x)
subject to:

   q il ≤ q i ≤ q iu | qi | ≤ qiu (i = 1 • • • n) |τ i | ≤ τ iu (4)
x is the path placement vector corresponding to the transformation matrix b T p . E t is the total electric energy required by the n actuators whereas E i is the total electric energy required by the i th actuator to follow the path. q i , qi , τ i are respectively the i th actuator displacement, rate and torque. q il is the lower bound and q iu (resp. qiu and τ iu ) is the upper bound of i th actuator displacement (resp. rate and torque). For a given path placement vector x, these constraints can be evaluated by means of the manipulator kinematic, velocity and dynamic models. It is noteworthy that the manipulator geometric constraints guarantee that the whole path lies inside the prescribed workspace. Similarly, the bounds on actuator rates ( qiu ) and torques (τ iu ) ensure that the manipulator will not go through any singular configuration while following the path. The electric energy used by the actuators is formulated in the next section.

Objective Function: Electric Energy

The energy used by the motors depends on their corresponding velocities and torques. As a matter of fact, the electric current in the motors varies with motor velocities and torques. Accordingly, the motor's self-inductance phenomenon appears. The current I drawn by the motors and the motor electromotive potential V e can be calculated as a function of the required torque τ and the angular velocity ω of the actuators, namely,

I = τ K t (5)
V e = K e ω (6)
K t being the torque sensitivity factor or motor constant expressed in [Nm/A] and K e the back electromotive force constant expressed in [V.(rad/sec) -1 ].

The total electric power P T is composed of [START_REF] Lacroux | Les actionneurs électriques pour la robotique et les asservissements[END_REF]:

• The resistive power loss (Joule effect):

P J = RI 2 (7)
• the inductive power loss:

P L = LI dI dt (8)
• the power used to produce the electromotive force:

P EM = V e I (9)
Accordingly, the total electric power P T can be expressed as follows:

P T = P J + P L + P EM (10)
R being the motor winding resistance expressed in Ohm [Ω] and L the motor inductance coefficient expressed in Henry [H].

Finally, the energy E consumed by a motor can be evaluated by integrating P T over the total trajectory time T, namely,

E = T 0 P T dt (11)
P T being the instantaneous electric power at time t, defined in Eq. [START_REF] Hirakawa | Trajectory planning of redundant manipulators for minimum energy consumption without matrix inversion[END_REF].

It should be noted that Eq. ( 5) allows us to consider the energy used by the actuators when they do not move but still produce a torque to keep the manipulator at a certain stationary configuration (with respect to that particular direction or actuator), like resisting the gravity.

It is noteworthy that energy calculation model presented in this section is suitable for the brushless motors, which are generally used for PKMs. However, depending on the motors/derives in application, energy calculation model can be developed accordingly.

Resolution of the Path Placement Optimization Problem

To solve the problem, a general optimization approach is proposed as illustrated in Fig. 2. The approach can be summarized in three constituent elements or phases:

1. Preparation Phase: Manipulator geometric, dynamic and electric parameters along with the definition of the required path are used as the known input data of the optimization problem. The base frame F b is defined. The path to be followed by the end-effector of the robot is defined in the path frame F p . The terms of the transformation matrix between F p and F b are the decision variables of the optimization problem.

2. Evaluation Phase: At this stage, the inverse kinematic model (IKM), the inverse velocity model (IVM) and the inverse dynamic model (IDM) of the manipulator are determined for each set of design parameters obtained from the optimization routine. Accordingly, the objective function and the constraints of the optimization problem are evaluated.

3. Optimization Phase: The objective function and constraints evaluated at the previous step are analyzed by means of the optimization algorithm. Once the constraints are satisfied, the objective function is tested for its optimum value. The convergence criteria are checked. If latter are respected, the optimization algorithm stops. Otherwise, other iterations are run as long as the convergence criteria are not satisfied.

Finally, the optimum path placement is obtained by means of the position and orientation of the origin of F p with respect to F b defined by the terms of the transformation matrix between F p and F b , namely, the location vector

x * =[x * Op y * Op z * Op φ * θ * ψ * ] T .

CASE STUDY: APPLICATION TO THE ORHTOGLIDE

Description of the Orthoglide

The Orthoglide is a Delta-type PKM [START_REF] Clavel | Delta, a fast robot with parallel geometry[END_REF] dedicated to 3-axis rapid machining applications developed in IRCCyN [START_REF] Wenger | Kinematic analysis of a new parallel machine tool: The Orthoglide[END_REF]. It gathers the advantages of both serial and parallel kinematic

Objective and Constraints Evaluation

Optimization Phase Table 1: Orthoglide workspace parameters Cartesian workspace, defined by the length of the cube sides, namely, L workspace [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]. The base frame F b is defined with the unit vector e i in the direction of the i th prismatic joint, namely, X b , Y b and Z b , the origin O b of F b being the intersecting point of e i . Two points Q + and Q -are defined in such a way that the velocity transmission factor is 1/2 and 2 at these two points [START_REF] Chablat | Architecture Optimization, of a 3-DOF Parallel Mechanism for Machining Applications, The Orthoglide[END_REF]. A cube is then constructed with Q + Q -as its diagonal. It should be noted that the cubic workspace center, i.e., point C, and the origin O b of the reference frame F b do not coincide, as shown in Fig. 3(b). In the scope of this study, L workspace is equal to 0.200m. Accordingly, the coordinates of points Q + , Q -and C for the said workspace are given in Table 1. Similarly, the prismatic actuator bounds, ρ min and ρ max , can be calculated [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]. Table 2 shows the lower and upper bounds of the prismatic joints displacements and their maximum allowable velocity and torque for the Orthoglide. The geometric, kinematic and dynamic parameters of the Orthoglide are defined in [START_REF] Wenger | Kinematic analysis of a new parallel machine tool: The Orthoglide[END_REF][START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF][START_REF] Chablat | Architecture Optimization, of a 3-DOF Parallel Mechanism for Machining Applications, The Orthoglide[END_REF][START_REF] Guégan | Contribution la modélisation et l'identification dynamique des robots parallèles[END_REF].

q il ≤ q i ≤ q iu | qi | ≤ qiu |τ i | ≤ τ iu E t = E min x j = x j+1
Electric energy E i used by each actuator is calculated by means of Eqs. ( 5) to [START_REF] Khoukhi | A projected gradient augmented Lagrangian approach to multi-objective trajectory planning of redundant robots[END_REF]. As the Orthoglide 3-axis has three 3-phase Sanyo Denki synchronous servo motors (ref erence : P 30B0604D), Eq. ( 10) is multiplied by 3 to cater for the power consumed by the each phase of the motor in order to calculate the electric power P T i used by each actuator, i.e.,

P T i = 3(RI 2 + LI dI dt + V e I) (12)

Trajectory Planning and External Forces

In order to apply the methodology proposed for path placement optimization, a rectangular test path is considered. The test path is defined by the length L and the width W of the rectangle, as shown in Fig. 4(a). Path reference frame F p is located at the geometric center of the rectangle. This type of path can be the example of the generation of a rectangular pocket like that of Fig. 4(b). The position of F p in the base frame F b is defined with the Cartesian coordinates of the origin of F p , O p (x Op , y Op , z Op ) and the orientation of F p with respect to F b is given by Euler's angles, as depicted in Fig. 1(b). For the sake of simplicity, only one of the three rotation angles is considered i.e, rotation about Z b -axis while X b Y b and X p Y p planes are considered to be always parallel. Accordingly, there are four path placement variables, i.e., x Op , y Op , z Op and φ, as illustrated in Fig. 4. The magnitude of the end-effector velocity is supposed to be constant along the path. Hence, for given path dimensions, position vector p Fp = [x P p y P p z P p ] T and velocity vector v Fp = [ ẋP p ẏP p żP p ] T in the path frame can be evaluated as a function of time. Figure 5 shows the position and velocity profiles in F p for a 0.05 × 0.10 m rectangular path and for a constant end-effector velocity of 1.0 m.s -1 . Position and velocity vectors defined in F p can be expressed in F b by means of the transformation matrix defined in Eq. ( 3), namely, T being the decision variables vector of the optimization problem. For a matter of simplicity and not to deal with tangent and curvature discontinuities, we consider that the path is composed of four independent line segments. Therefore, we do not pay attention to the discontinuities between the segments. In order to analyze the effect of external cutting/machining forces in the generation of a given path, a groove milling operation is considered as shown in Fig. 6, [START_REF] Majou | Parametric stiffness analysis of the Orthoglide[END_REF]. With constant feed rate or end-effector velocity v p of magnitude 0.66 m.s -1 , i.e, 40 m.min -1 , the following components of cutting forces are considered: F f =component in the feed direction = 10 N F a =component along the axis of cutting tool = 25 N F r =component perpendicular to F f and F a = 215 N

    x P b y P b z P b 1     =     cos φ -sin φ 0 x Op sin φ cos φ 0 y Op 0 0 1 z Op 0 0 0 1         x Pp y Pp z Pp 1    

Path Placement Optimization Problem

The path placement optimization problem for the Orthoglide 3-axis is formulated in order to minimize the total electric energy used by the three prismatic actuators. The kinematic, velocity and dynamic models of the manipulator are used to evaluate the required actuator displacements, velocities and torques. The constraints of the optimization problem are the geometric, kinematic and dynamic ones. It should be noted that within the prescribed workspace, the Orthoglide 3-axis is free of internal collisions and that there is not any limit on the passive joints. Therefore, the geometric constraints for the path placement problem are the upper and lower limits of the prismatic joints variables. The kinematic constraints are the maximum velocities that the prismatic actuators can produce whereas the dynamic constraints are the limits of the torque/force that the actuators can produce. The kinematic and dynamic constraints are obtained from the catalogue, as shown in Table 2. As already mentioned, the decision variables are the Cartesian coordinates of the origin of F p and the orientation angle of F p with respect to F b .

The optimization problem can be formulated as follows,

min x E t = n i=1 E i (x) subject to:    ρ min ≤ ρ x,y,z ≤ ρ max |v x,y,z | ≤ v max |τ x,y,z | ≤ τ max (13)
where x = [x Op , y Op , z Op , φ] T . The subscripts x, y and z are used for three prismatic actuators or three Cartesian directions. ρ min and ρ max are respectively the minimum and maximum displacements of the prismatic joints as presented in Table 1 The optimization problem was solved by using the MATLAB fmincon function, which is a general constrained optimization solver using the derivative-based search algorithms. The optimization process was performed with different starting points and it turned out that MATLAB fmincon function always converges to the same solution no matter the starting point. Furthermore, to study the variation pattern of energy requirements at different points within the workspace, a workspace discretisation is carried out with respect to the path placement variables and the energy is calculated for each of the discrete point for a given path while verifying the constraints.

X b Y b Z b F a F r F f V p

Results and Discussions of the Path Placement Optimization Problem

The path placement process introduced in this paper is highlighted by means of the rectangular path shown in Fig. 4. As a matter of fact, the path placement process is performed for different rectangles with constant aspect ratio of 2, i.e., L/W = 2. With the help of the optimization algorithm, the location of the path corresponding to minimum and maximum energy consumption is obtained, i.e., the best and the worst path locations with respect to the electric energy consumed. Figures 7 and8 show the location of different rectangular paths with the minimum and maximum energy consumption in the Orthoglide 3-axis cubic workspace. The magnitude of the energy used for both best and worst cases and the corresponding gain of energy is given in Table 3 and is illustrated in Fig. 9. In Fig. 9, % saving is the percent energy saving between the best(minimum) and the worst(maximum) energy consumption. • and is a maximum when the path is located in the vicinity of point Q -with φ = 45 • . From Fig. 9, it can be noticed that the smaller the path, the higher the energy saving. This higher gain for the smaller path is due to the higher range of displacement of the path within the manipulator workspace.

X b [mm] Y b [mm] Z b [mm] C O b Q + Q - Y b Z b X b 20 × 40
X b [mm] Y b [mm] Z b [mm] C O b Q + Q - X b Y b Z b 80
In order to view the energy variation trends in the workspace, a test path of size 30mm × 60mm is taken and the energy required for the generation of this test path is evaluated for several positions and orientations. Figure 10 shows the variations in the energy used with respect to x Op and z Op for a constant orientation φ of 0 • and for three different values of y Op . Figure 11 illustrates the isocontours of the energy required by the motors with respect to x Op and y Op for given values of z Op and φ, namely, z Op = 0 and φ = 0 • . From Fig. 10, it is apparent that the energy required is not sensitive to variations in z Op . The reason being that the path lies in the X b Y b -plane and the actuator displacements along Z b -axis is not significant. Figures 10 and 11 also show that the energy required by the motors is a minimum when the path is located in the neighbourhood of the isotropic configuration and is a maximum when the latter is located in the neighbourhood of singularities.

Figures 12 and13 show the variations in the energy used with the path orientation in different areas of the cubic workspace. It can be seen that the energy used is usually a maximum when φ = 45 • . However, the energy consumption is a maximum for a path orientation different than 45 • , for some path locations. For example, the energy required at the upper right corner of the workspace is higher for φ = 30 • than φ = 45 • . Figure 14 shows the comparison of trajectory parameters of 30mm × 60mm test path for minimum and maximum energy locations. It makes sense that when E = E max the range of actuator displacements is larger than when E = E min . Similarly actuators experience higher values of maximum velocities and torques when E = E max as shown in Fig. 14(b-c), which results in higher energy consumption for each actuator, as shown in Fig. 14(d). These results mean that the actuators may reach their performance limits due to an inappropriate location of the path in the workspace.

E min E max E x E y E z τ x τ y τ z V x V y V z X Y Z 0

Acknowledgment

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

CONCLUSIONS

A methodology for path placement optimization was proposed in this paper. The electric energy required by the actuators to follow a predefined path is considered as the optimization criterion. Electric energy requirement is calculated with the help of the required actuator torques and velocities along with motors electric parameters. To verify the feasibility of the solutions, actuators performance limits such as their joint limits, maximum velocities and torque were used as the constraints of the optimization problem. The kinematic, velocity and dynamic models were used to come up with the objective function and constraints.

The proposed methodology was applied to the Orthoglide 3-axis, a three-degree-of-freedom translational parallel manipulator with a quasi-cubic workspace. Rectangular test paths of different sizes were considered as illustrative examples. These paths are similar to those used to realize pocketing operations.

The use of the electric energy instead of mechanical energy as an optimization criterion is pertinent. Although actuator electric energy consumption depends on the mechanical energy requirements, the electric energy evaluation is more comprehensive than its mechanical counterpart. General approach to calculate the mechanical energy with the help of manipulator velocity and dynamic models, i.e., by using actuator torques and velocities, may lead to an under estimation of the energy requirements in the case where actuators are experiencing torques with zero velocities. Besides, usual mechanical energy calculations do not consider the resistive energy loss in the motor windings as well as the energy loss due to the variations in the actuator velocities. Those variations affect the current requirements and hence induce electromotive forces in the actuators. Accordingly, the electric energy formulation takes into account all these energy losses.

The energy required to perform a given task depends on the position and the orientation of the task within the workspace of the manipulator. Accordingly, some electric energy can be saved by properly selecting the position and the orientation of the task. Indeed, a misplaced task can cause excessive energy consumption and can force the actuators to go over their performance limits.

For the Orthoglide 3-axis, the optimum path location is found to be in the neighbourhood of the isotropic configuration but there is no general rule to predict the exact optimal position and orientation of a task particularly for a complicated three dimensional task or for an irregular workspace. However, a detailed analysis of the energy variation within the workspace for a given task can lead to the optimal position/orientation of that particular task. Numerical optimization algorithms are useful for such a comprehensive analysis in which all the problem constraints and performance measures can be considered simultaneously.

In the future work, the path placement optimization problem will be dealt as a multiobjective one. For example, along with energy requirements, the manipulator dexterity and stiffness can be considered as optimization objectives.

Figure 1 :

 1 Figure 1: (a). Path P to be followed by the end-effector P of a manipulator, F b and F p being the base and path frames, (b). Euler angles the Cartesian coordinates of the origin of F p and the relative orientation of the two frames is characterized by means of Euler angles. However, keeping in view the constraints of the manipulator wrist, Euler angles are uniquely defined in the context of milling operation with the parameterization given in Fig. 1(b). It allows to avoid the singularity of Euler parameters.

  ) b T p being the transformation matrix from F p to F b . Let O p (x Op , y Op , z Op ) be the origin of the path frame expressed in frame F b and (φ, θ, ψ) the Euler angles characterizing the orientation of frame F p with respect to frame F b . Accordingly, b T p =     cos φ cos θ cos φ sin θ sin ψsin φ cos ψ cos φ sin θ cos ψ + sin φ sin ψ x Op sin φ cos θ sin φ sin θ sin ψ + cos φ cos ψ sin φ sin θ cos ψcos φ sin ψ y Op sin θ cos θ sin ψ cos θ cos ψ z Op

  Parameters Definition of base frame F b Electric motors parameters Path definition Definition of Path frame F p Trajectory Definition in F p Initial Guess x 0 x 0 = [x Op 0 y Op 0 z Op 0 φ 0 θ 0 ψ 0 ] T Evaluation Phase Transformation Matrix b T p Trajectory Definition in F b

Figure 2 :Figure 3 :

 23 Figure 2: Flowchart of the path placement optimization process

  Figure 4: Rectangular test path

Figure 5 :

 5 Figure 5: Test trajectory for a rectangular path of size 0.05m × 0.10m

Figure 6 :

 6 Figure 6: Cutting forces

Figure 7 :

 7 Figure 7: Locations of rectangular path of different sizes (W mm × Lmm) that yield a minimum energy consumption

Figure 8 :

 8 Figure 8: Locations of rectangular path of different sizes (W mm × Lmm) that yield a maximum energy consumption

Figure 9 :Figure 10 :

 910 Figure 9: E min and E max and percentage saving as a function of the rectangular path width (L = 2W )

Figure 11 :Figure 12 :

 1112 Figure 11: Energy vs x Op and y Op for 30mm × 60mm rectangular path

xFigure 13 :

 13 Figure 13: Energy as a function of x Op and φ for y Op = z Op = 0

Figure 14 :

 14 Figure 14: Comparison of 30mm × 60mm trajectory parameters for E min and E max locations

Table 2 :

 2

	ρ i min	0.126 m
	ρ imax	0.383 m
	v imax	1.00 m.s -1
	τ imax	1.274 Nm

Orthoglide actuators parameters (i = x, y, z)

Table 3 :

 3 It can be seen from Figs. 7 and 8 that the energy consumption is a Minimum and maximum energy used for a given rectangular path minimum when the path is located in the vicinity of the isotropic configuration with velocity

	Rectangular path dimensions [mm] E min [J] E max [J] % gain Width (W ) Length (L)
	20	40	15.26	44.46	65.68
	30	60	22.88	61.35	62.71
	40	80	30.41	76.31	60.15
	50	100	38.55	89.80	57.07
	60	120	46.83	102.11	54.13
	70	140	56.82	113.46	49.92
	80	160	65.94	121.17	46.89


FROM BINARY FEATURES TO ELEMENTS: IMPLICATIONS FOR MARKEDNESS THEORY AND PHONOLOGICAL ACQUISITION 1

JOAQUIM BRANDÃO DE CARVALHO (UNIVERSITY OF PARIS 8,CNRS UMR 7023) This article aims to assess the two competing models of features that have dominated the history of phonological theory for almost one century: the first, which originates from Trubetzkoy, allows both "privative" and "equipollent" contrasts; the second comes from Jakobson's strict binarism, and has been continued in generative phonology up to the present day. Firstly, it will be argued that the latter model suffers from at least three problems that follow from overgeneration, and make it impossible to provide a natural and unified account of systemic and processual markedness. Secondly, it will be shown how these problems can be given a straightforward solution through an approach based on privative elements, in which markedness becomes the result of operations acting on primary segments. Interestingly, such a view turns out to be remarkably compatible with recent findings in L1 acquisition, namely Kuhl's Native Language Magnet Theory.

INTRODUCTION

hat is universal and what is particular about phonological representations? The answer provided by structuralism, which prevailed in the language sciences for much of the 20th century, emphasised relativism. Partly derived from the Saussurean concept of "value", itself based on the form/substance dichotomy, and then reinforced by Prague phonology, the long-dominant views favoured a definition of linguistic objects that was both arbitrary and differential: any element of a given system is what the other elements of the system are not. Hence, even though two elements of different systems are substantially identical, they do not necessarily constitute the same object if these systems differ. The advent of generative linguistics did not fundamentally change this state of affairs: while generative phonology did seek to identify general principles and constraints in support of the hypothesis of a Universal Grammar, of which the inventory of features was one of the attributes, the epistemological break with the various structuralisms was not so clear-cut with regard to phonemic representations. W I will try to show here ( §1) that this relativism is largely based on a strictly binarist conception of phonological distinctions dating back to Jakobson, which has largely replaced, in scientific practice, the much richer typology of oppositions due to Trubetzkoy. It will then be seen ( §2) that binarist models suffer from several problems, which support their replacement with the so-called "unarist" models, where the old Trubetzkoyan idea of the distinction between "privative" and "equipollent" oppositions is revived. The consequence ( §3) is a radical change of perspective concerning both the nature of markedness and the structure of phonemic systems. It is remarkable -and this will be stressed in the conclusion -that this evolution of theoretical phonology, defended by certain schools of thought for some forty years, is in line with results

PAGE 347

RADICAL: A JOURNAL OF PHONOLOGY,3 obtained quite independently, during the same period, by research on the categories acquired by children learning their "mother tongue", both in the domain of production and in the recently renewed field of perception.

TRUBETZKOY VERSUS JAKOBSON

PRIVATIVE OPPOSITIONS

Phonology was born in Europe from the work of Trubetzkoy, whose "logical classification of distinctive oppositions" forms the core of his Principles of Phonology (1939Phonology ( [1976: 68-93]: 68-93]). In this "pluralist" perspective, in view of the subsequent theoretical drift, any opposition can be classified, from a strictly systemic point of view, according to three criteria:

(1) a. Its relation to the whole system of oppositions in the language:

bilateral or multilateral, proportional or isolated oppositions.

b. The relationship between its terms:

privative, equipollent or gradual oppositions.

c. The extent of its distinctive power:

constant or neutralizable oppositions.

Of these three criteria, (1b) is the more important for our purposes. An opposition between two phonemes is said to be privative if the distinction is based on the presence in one of them of a quality -a feature -which is absent in the other. The one with this feature is said to be marked, this distinctive feature being called a mark; the one without it is unmarked. An oral consonant or vowel such as /b/ or /o/, without the nasal mark, is unmarked compared to its marked nasal counterpart /m/ or /õ/; the opposition PAGE 348

RADICAL: A JOURNAL OF PHONOLOGY,3 of nasality is therefore privative. The same is true of voiceless consonants such as /p/, /t/ or /k/ vis-à-vis voiced consonants such as /b/, /d/ or /g/: the voicing mark is absent in the former but present in the latter. A privative opposition thus involves a relationship between something (the mark) and its absence, i.e. zero. The relationship between /b/ and /m/ could be expressed algebraically as follows:

( Quite different is the case of the so-called equipollent opposition, in which two phonemes contrast through two different features. If we take the example of the contrast between the plosives /p/ and /t/, it would be wrong to say that /p/ differs from /t/ in that it lacks the latter's "coronal" mark, or that /t/ is a "non-labial": what, then, would the "dorsal /k/" be? We should then say that the phonemes /p/, /t/ and /k/ are distinguished from each other in that their place-of-articulation features are in an equipollent relation: "labial" ~ "coronal" ~ "dorsal". To take a semantic analogy, if the opposition between "tree" and "apple tree" is privative (the former being unmarked), the opposition between "apple tree" and "plum tree" will be equipollent, since it involves two equally present features: "tree bearing apples" ~ "tree bearing plums".

Finally, a gradual or scalar opposition involves at least three phonemes and is based on the variable weight of a given feature. If they really exist, these oppositions seem to play a role in phonology limited to vowel height contrasts.

PAGE 349

BINARY OPPOSITIONS

It was against this typology that the binarist thesis developed after the war. Conceived by Jakobson as early as the 1930s, it was to flourish in the United States in [START_REF] Jakobson | Preliminaries to speech analysis[END_REF], and was to be pursued by the generative phonology of [START_REF] Chomsky | The sound pattern of English[END_REF]. This "totalitarian" conception reduces any phonological opposition to a single type based on an antithetical, antagonistic relationship between its two terms. Thus, the equipollent opposition structuring, in Trubetzkoy, the triplet /p t k/ is broken down, according to Jakobson, into two binary oppositions: (a) /k/ "compact" is opposed to /p/ and /t/ "diffuse"; (b) /p/ "grave" is opposed to /t/ "acute". These oppositions are formally and conceptually indistinguishable from the opposition between /m/ and /õ/, which are "nasal", and /b/ and /o/, which are "non-nasal". In all cases, the two terms of the antinomy are present and thus have an equivalent status: any opposition is of the type "F" ~ "¬F" (F = feature); privativity disappears with equipollence. 2 Corollary: the difference between two phonemes of a given system is now based on the number of features that distinguish them from each other: thus the oppositions /p/ ~ /b/ and /t/ ~ /d/ are based on 1 feature, whereas /p/ ~ /d/ is based on 2 features (Martinet 1964(Martinet [1974: 86]: 86]). 3 The extralinguistic motivations for Jakobsonian binarism combined quick allusions to Wallon (1945) and his theses on "the initial stages of thought and speech" in children, with later studies on speech perception and communication theory (cf. Jakobson 1973: 153-155, 172). But whatever its justifications or sources of inspira-2 Contrary to a frequent usage in the literature written in English (e.g., [START_REF] Kim | Phonological features: privative or equipollent? AB thesis[END_REF], I will stick here to the strict meaning of the term "equipollent" as found in Trubetzkoy, which does not entail any antagonistic relationship between the terms of the opposition, unlike the contrasts that are called "binary" here.

3

Note that, even if they reject the principle of binarism (e.g., Martinet 1964Martinet [1974: 83-89]: 83-89]), later structuralist trends often admit its practice: cf. the oppositions "oral" ~ "nasal", "anterior" ~ "posterior", and, w i t h in the F r e n c h functionalist school, "sourd" ~ "sonore", "arrondi" ~ "étiré".

PAGE 350

RADICAL: A JOURNAL OF PHONOLOGY, 3 tion, binarism owes much of its survival to its adoption by generative phonology in the 1960s. The latter will return to articulatory features in which consonantal and vowel qualities are once again strongly differentiated. It is well known that, apart from its strictly binary aspect, the two other original features of the feature system proposed by [START_REF] Jakobson | Preliminaries to speech analysis[END_REF]

THREE PROBLEMS WITH BINARISM

CONTRASTIVE SPECIFICATION

A classic justification for binary feature systems is provided by the theory of "contrastive specification", which is supposed to be based on both typological and ontogenetic facts. Let us compare the value of the three closed vowels of French with the three closed vowels of Korean. The distinctive features in each system are shown in (3,4); the [+high] feature has been omitted, as it is not relevant for our purposes. and lost by the aphasic after /y/ and / / ɯ . 4 Note that the same classic feature calculation very often demonstrates the remarkable character of /a/. As it is the only low vowel in a majority of systems, the feature [+low] is sufficient to distinguish it from all other vowels; it has therefore only this distinctive feature. Interestingly, /a/ is the only universal vowel. 5 It is also said to be the first vowel to stabilise in children's production, and the last to disappear in aphasics. 6 The structure of systems would thus reflect the typological and ontogenetic complexity of their elements. Binarist models thus seem to account for systemic markedness: a phoneme X is more marked than Y if, and only if, X contains more features than Y. In short, "who can do more can do less". However, let us note in passing that the application of the principle of contrastive specification to binary features in (3, 4) renders the distinctive features relative to the system following the Saussurean viewpoint: in the same way as /y/ and / /, ɯ /i/ and /u/ differ in French and Korean even though there would be, in this case, substantial identity between the two languages.

4

It seems that there is no need to reconsider what Jakobson (1941Jakobson ( [1969: 61]: 61]) said about the acquisition of /y/: "The rounded palatal vowels, which Rousselot very appropriately called "secondary", appear in child language only after the "primary" vowels, i.e. after the rounded velar vowel and after the unrounded palatal vowel with the same degree of aperture." The implicational relationships brought into play by the loss of distinctions being the reciprocal of those governing their acquisition (ibid.: 64), we deduce that the disappearance of /i/ and /u/ in aphasics presupposes that of /y/. The studies quoted by Jakobson for French (ibidem: 66, n. 80) are confirmed today by Levelt (1994: §6, esp. p. 133) and [START_REF] Beers | The phonology of normally developing and language-impaired children[END_REF] for Dutch, by [START_REF] Elsen | Erstspracherwerb: Der Erwerb des deutschen Lautsystems[END_REF] and [START_REF] Geilmann | Markedness and underspecification: the case of German schwa in L1 acquisition[END_REF] for German, by [START_REF] Fee | Underspecification, parameters, and the acquisition of vowels[END_REF] and Zajdó (2002) for Hungarian. Concerning the acquisition of /ɯ/ in Korean, cf. [START_REF] Jun | Phonological Development of Korean: A Case Study[END_REF].

5

Ubykh and Kabardian, which, with only two phonemes, have the minimal vowel system among the world's languages, show /a/, which has a "closed" (featureless) counterpart / /: cf., for Ka ɨ bardian, [START_REF] Kuipers | Phoneme and Morpheme in Kabardian[END_REF] reformulated by [START_REF] Anderson | Kabardian disemvowelled, again[END_REF].

6

On the acquisitional priority of height distinctions by the child, Jakobson's classic theses (1941 [1969: 51 ff.]) are confirmed today by various studies, a summary of which can be found in Bernhardt & Stemberger (1998: 343-353, 364-366).

PAGE 353

RADICAL: A JOURNAL OF PHONOLOGY, 3

SYSTEMIC AND PROCESSUAL MARKEDNESS

Despite lengthy discussions on the problems posed by feature (under)specification, which are inherent to the adoption of a binarist a priori (Carvalho 1997: 14-17), this line of reasoning has persisted: cf. e.g. [START_REF] Steriade | Redundant values[END_REF]Steriade ( , 1995)), [START_REF] Clements | Toward a substantive theory of feature specification[END_REF]), Mester & Itô (1989). Apart from the unarist theses discussed below, the only developments challenging the general model illustrated above -the Radical Underspecification Theory [START_REF] Archangeli | Underspecification in Yawelmani phonology and morphology[END_REF][START_REF] Archangeli | Aspects of underspecification theory[END_REF] and the Contrastive Hierarchy Theory [START_REF] Dresher | The contrastive hierarchy in phonology[END_REF])are based on the consideration of processual markedness. For example, from the fact that, while being a lexical vowel like any other, / / ɯ is also, in Korean, the "neutral" vowel by its propensity to syncope and/or epenthesis in particular, the proponents of Radical Underspecification Theory deduce that the features of / / ɯ are underspecified in the whole Korean vowel system, hence the subsystem in ( 5), where only one value per feature ever appears.

( /i/ /y/ /u/ Both theories are a timely reminder of the importance of the behaviour of segments, i.e. the processual aspect, in addition to their place in the system, the systemic aspect; processual evidence is indeed far much richer than Trubetzkoy's neutralization.

However, I would like to mention one important point: the reformulations in (5, 6b)

eliminate the formal basis of systemic markedness provided by ( 3) and ( 4): / / ɯ has, in (5), fewer features than /i/ and /u/ whereas it is an incomparably rarer vowel than the latter; /i/ has, in (6b), as many features as /y/ whereas the latter is also infinitely less frequent than /i/. Thereby, Radical Underspecification Theory and Contrastive Hierarchy Theory erase the most interesting aspect of the classical binarist approach. My thesis is that it is possible and desirable to account for both systemic and processual markedness, while getting rid of the problems associated with binary features. The representations in (3, 4) pose, indeed, three problems; one would like to believe they could be overcome in isolation, but their accumulation makes it more difficult.

NATURALNESS

A first problem with binarist models of phonological features is that they lead to PAGE 355

RADICAL: A JOURNAL OF PHONOLOGY, 3 b. /i/ /y/ /u/ -BACK -BACK +BACK -ROUNDED +ROUNDED
the postulation of objects whose existence seems counterintuitive. It is therefore the naturalness of certain features that is at issue here. For example, what is [-rounded] or, to use French terminology, "étiré"? What is [-voiced] (or "sourd")? It can be rightly objected that these terms equate default postures with articulatory gestures. Can it seriously be argued that one is purposely not rounding one's lips to make /i/ or /e/?

Are we really making an articulatory "gesture" by adopting a configuration of the larynx that has many points in common with the normal state of the glottis outside phonation? Structural linguistics has certainly taught us that, in a system of values, absence is also a value, but, precisely, the absence of a given feature F is much more naturally formalised by privative oppositions, where we have F ~ Ø, than by binary contrasts.

As was recalled in §2.2, only specified features are supposed to be active, by being, for example, the vector of harmony or assimilation phenomena (Martinet 1974: 112-113;[START_REF] Steriade | Redundant values[END_REF]Steriade , 1995;;[START_REF] Dresher | The contrastive hierarchy in phonology[END_REF]). Thus, French il se bat, il se gare are realized

[izba] and [iz a ], ɡ ʁ where the contrastive voicing of /b/ and /ɡ/ spreads onto the preceding /s/, but il se noie, il se lave, il se rase are pronounced, without assimilation, [isnwa],

[islav], [is az], since ʁ the voicing of the sonorants is not distinctive: there are no voiceless /n/, /l/ and /r/ phonemes in French. Now, as noted by Van der Hulst (2016: 96), it is impossible to prove both the existence and the non-existence of phenomena like unrounding harmony, devoicing or orality assimilation. Also, whereas palatalizations, involving [-back], abound in the world's languages, nowhere is there a proven case of velarization, i.e. propagation of [+back]; more precisely, no language makes use of the contrast between velarization and pharyngealization (Ladefoged 1971: 63 ff). Interestingly, to confine ourselves to purely phonetic studies, it follows from [START_REF] Esling | There Are No Back Vowels: The Laryngeal Articulator Model[END_REF] Laryngeal Articulator Model that "there are no back vowels".

PAGE 356

RADICAL: A JOURNAL OF PHONOLOGY, 3 I am aware that there is apparent evidence for binary features, though. [START_REF] Kim | Phonological features: privative or equipollent? AB thesis[END_REF] review of this article) mentions several cases where [-voice], [+back] and [-nasal] seem to play a role in phonological processes. However, I do not think that the reported facts bring decisive evidence for the existence of such features, for five reasons. Two of them, that are closely linked together, pertain to the theoretical background of my proposal, and apply to all the mentioned cases. Firstly, contrary to what is assumed by [START_REF] Kim | Phonological features: privative or equipollent? AB thesis[END_REF] 27-33), I disagree that there could be "phonetic evidence" for binary (or monovalent)

features. Secondly, why can there be "three-way contrasts" (involving

[+T] ~ Ø ~ [-T]
and thus binary features) if they are not phonemic, as Kim herself acknowledges (p.

33)? (See also Raimy (2021), although his primes are assumed to be privative.) Without this condition, [+T], Ø and [-T] cannot be "categorially distinct" (p. 32). The reason for these two remarks is the same: features are inherently phonological objects, and phonology is primarily based on contrast, even though phonological theory should explain and unify both systemic ( §2.1) and processual ( §2.2) aspects, as will be seen in §3.

My three other arguments are more specific, depending on the property or the example at stake. can also disassociate from the onset, hence the so-called "devoicing assimilations" (as in French [me s ] d̥ ɛ médecin 'doctor'), which do not require propagation of any feature. My last arguments refer to the nature of the processes involved, more precisely to whether we deal with lexicalized generalizations (much like the well-known "velar softening") or with the opposite case of low-level phenomena. As regards [back], I contend that Turkish vowel harmony can be usefully analysed in terms of propagation of a "front" feature, since it has the effect of palatalizing certain consonants [START_REF] Clements | Vowel and Consonant Disharmony in Turkish[END_REF]. Admittedly, as pointed out by Kim, some suffixes containing "back" vowels resist front harmony (Van der Hulst & Van de Weijer 1991). However, vowel harmony is probably the legacy of an earlier stage of Turkish phonology. Possibly like final devoicing (cf.

[etyd] etüd 'study'), it is no longer phonologically driven in modern Turkish, as is also shown by the existence of many non-harmonic stems, and its purely morphological conditioning [START_REF] Bacanlı | Vowel harmony and other morphological processes in Turkish[END_REF].

On dence over [±rounded]. However, it is questionable why a presumed non-distinctive feature such as [±rounded] participates in several phonological processes (cf. §2.3):

Yawelmani, for example, has a rounding harmony while lacking rounded front vowels.

I could play the devil's advocate by claiming, à la [START_REF] Chao | The non-uniqueness of phonemic solutions of phonetic systems[END_REF], that it is precisely in this uncertainty that the unmarkedness of such systems lies: Spanish is not faced with a choice between (8a) and (8b). If we want at all costs an invariant that distinguishes /i/ and /u/, this should be the phonetic content common to [-rounded] and

[-back], on the one hand, and to [+rounded] and [+back], on the other hand: let us say, to use Jakobson's terminology, the "sharpened" value of /i/ versus the "flattened" value of /u/. Thus, the undecidability in (8a,b) would be due to the inventory of features adopted, not to their binary character, which would remain in the contrast "sharpened"

~ "flattened". Why not? Let us simply point out that, as will be seen below, the problem posed by the non-uniqueness of the representations of /i/ ~ /u/ in Spanish can be solved with the same feature system used for French and Korean but within a non-binarist framework. Thus, the undecidability in (8a,b) adds up to the unnaturalness of [-rounded] and the like ( §2.3) in weakening the scope of the binarist representations seen in (3, 4).

CONTRASTIVE SPECIFICATION AND MARKEDNESS

The third problem with binarist models is that they lead to an empirically false prediction. Let us take seriously the assumption underlying (3, 4) that the structure of phonemic systems reflects the typological and ontogenetic complexity of their elements, with the higher number of distinctive features in /y/ and / /, ɯ i.e. the markedness of these vowels, expressing their rarity and the universally late nature of their acquisition ( §2.1). If this hypothesis were true, then a system with the four closed vowels /i/ and PAGE 360

RADICAL: A JOURNAL OF PHONOLOGY, 3

/u/ plus /y/ and / / ɯ should not exist, since it would not entail any definite order in the acquisition of these vowels by the child (or in their loss by the aphasic), all the features being necessarily distinctive as in (9). However, such systems do exist, notably in almost all Turkish languages. And, although there are, to my knowledge, no detailed studies on the acquisition of Turkish vowels, /y/ and / / ɯ are expected to be acquired by the Turkish-speaking child after /i/ and /u/.

(

RETURN TO TRUBETZKOY

BINARISM AND OVERGENERATION

I admit there seems to be a residue of binarism in two cases: vowel height and tone,

where both H and L are necessary so far, especially in 3-tone systems for which no single privative tone [START_REF] Hyman | Crosslinguistic studies of tonal phenomena: Tonogenesis, Japanese accentology, and other topics[END_REF]) can be invoked. However, I remain hopeful that these issues can be given a representational (1963) and Stanley (1967) pointed out very early on, the speciously trivalent character of a binary system admitting underpecification is responsible for the overgeneration of segmental representations. While these problems did give rise to objections in the 1980s and 1990s [START_REF] Goldsmith | Vowel harmony in Khalkha Mongolian, Yaka, Finnish and Hungarian[END_REF][START_REF] Steriade | Redundant values[END_REF]Steriade , 1995;;Mester & Itô 1989;[START_REF] Lombardi | Laryngeal features and laryngeal neutralization[END_REF][START_REF] Lombardi | Postlexical rules and the status of privative features[END_REF], they did not lead to a systematic critique of binarism, leaving room either for the postulation of ad hoc "marking conventions" or for a preference for fully specified feature matrices that preclude any possibility of formal expression of markedness (cf. Carvalho 1997). 7

Let us therefore turn back to Trubetzkoy's typology of oppositions where any quality is opposed either to its absence (privative opposition) or to another (non-antinomic) quality (equipollent opposition). This is the basis of the so-called unarist or monovalent hypothesis which, in the form of various theories and models -Particle Phonology [START_REF] Schane | The fundamentals of particle phonology[END_REF][START_REF] Carvalho | What are vowels made of? The "no-rule" approach and particle phonology[END_REF], Element and Government Theory [START_REF] Kaye | The internal structure of phonological elements: a theory of charm and government[END_REF][START_REF] Gibbon | Lingual activity in two speech-disordered children's attempts to produce velar and alveolar stop consonants: evidence from electropalatographic (EPG) data[END_REF], Dependency Phonology (Anderson & Ewen 1987) -has developed over the last forty years alongside the neo-Jakobsonian binarism of classical 7

Incidentally, [START_REF] Halle | The sound pattern of Russian[END_REF] famous "argument against the phoneme" only holds in a binarist framework, not in a privative framework that excludes 3-way contrasts (cf. Anderson 2021: §5.5). One of the reasons for the attachment of generative phonology to binary features may have been its desire to deny the relevance of the classical phonemic level.

MONOVALENCY AND MARKEDNESS

We know that the existence of /y/ in the languages of the world presupposes the existence of both /i/ and /u/. Let us therefore accept the following equation:

(10) /y/ = /i/  /u/ It is also known that:

(11) /y/ = {front, rounded}

We can therefore deduce:

(12) /i/ = {front} and /u/ = {rounded} where [back] disappears.

What, then, is / /? ɯ We know that this vowel is the chromatic complement of /y/.

So, if /y/ = /i/  /u/ = {front, rounded}, then:

(13) / / ɯ = /i/  /u/ = Ø
The so-called feature [back] is therefore nothing else than the default realisation of 8 I am leaving aside here Van der Hulst's (2020) Radical CV Theory, whose primitives, albeit privative, are organized in antagonistic pairs, like binary features. This "hybrid" model would deserve a specific review, in particular with regard to markedness-related aspects.

PAGE 363

RADICAL: A JOURNAL OF PHONOLOGY, 3

the feature [high], i.e. when it is not associated with other features, as in /i, u, y/. 9

The vowel subsystems examined so far have thus the following representations:

( There are no less than four points to note which show the interest of (14-17) compared to the binarist representations seen in §2.

Firstly -and this is the major contribution of this work to the theory of phonological primitives -, systemic markedness is not an arithmetic variable based on the number of features (cf. §2.1). "Unmarked", or primary, phonemes are those that show "stand-alone pronounceability". "That is, each element on its own characterizes a possible phonological segment" (Van der Hulst 2016: 88), like |I|, |U| and |A| in /i/, /u/ and /a/. "Marked" phonemes are those that either lack elements or imply an operation on primary phonemes, like /y/ and / / ɯ as defined in ( 10) and ( 13). It follows that the existence and acquisition of /y/ presupposes those of /i/ and /u/; there is no need to resort to arbitrary markedness conventions like those in Chomsky & Halle (1968: §9) to explain this.

Secondly, there is a privative opposition between /y/ and /i u/, and between /i u/ and / /, while ɯ /i/, [front], and /u/, [rounded], are in an equipollent relation. 11 The "unrounded" and "back" features therefore have no phonological existence, as can be proven by reductio ad absurdum: if "front" were the unmarked term, then the primary vowels would be /y/ = {rounded} and / / ɯ = {back}, and the derived vowels, /u/ = /y/  / / = {rounded, back} and /i/ = /y/ ɯ  / / = Ø, which is empirically false. ɯ

Thirdly, /i/ and /u/ have consistently the same representation in (14-17), whereas this was only the case for /y/ and / / ɯ in the binarist model; thus, two unmarked phonemes from two different systems should always have the same representation, the 10 Given element combination, further gradual oppositions emerge from headedness (cf. /i/ ~ /e/ ~ / / ɛ ~ /a/), and can be generalized beyond vowel height (cf. Swedish /i/ ~ /y/ ~ / / ~ /u/). ʉ 11 More specifically, |I| and |U| are in an equipollent opposition in the (non-antagonistic) sense that this word has been defined in fn. 2. It follows that these elements do not necessarily share the same tier, which precisely allows for the existence of front rounded vowels (cf. [START_REF] Kaye | The internal structure of phonological elements: a theory of charm and government[END_REF].

PAGE 365

RADICAL: A JOURNAL OF PHONOLOGY,3 differences between languages consisting in the combination of features, i.e., as we have seen for /y/ and / /, ɯ in the existence or non-existence of operations on primary phonemes.

Fourthly, the representation of Korean vowels in ( 16) turns out to be identical to that given in ( 5), obtained on the basis of their behaviour. The emptiness of / / ɯ is thus both a reflection of its systemically marked character by virtue of ( 13) and of its processual unmarkedness, whereby, for the same reasons as many schwas, it is an underspecified vowel. 12 The systemic and processual facets of markedness are thus unified with the same representations.

FINAL CONSIDERATIONS

I would like to draw three conclusions from the comparison between binarism and unarism in phonology. Firstly, on the epistemological level, the problem of knowing whether features are binary or unary is not a purely scholastic matter of notation, despite certain appearances, since the scope of a theory depends closely on the "alphabet", the primitives, that it uses. As we have just seen, the abandonment of a system of binary features in favour of a unary system allows us to move from relativistic representations, where, in the Saussurean tradition, the value of an element is dictated by its place in a given system, to universal representations; universalism follows from unarism as relativism follows from binarism.

Secondly, as is often the case in the history of science, two observations must 12 Not all default epenthetic vowels are underspecified as in Korean. It is well-known that many of them are "coloured": cf. Brazilian Portuguese and Finnish [i], Spanish or Polish [e], and even the " e muet" of modern standard French, which has the same melodic content as the lexical vowel /ø/ [START_REF] Carvalho | Entre phonologie et lexique : l'alternance schwa/è en français[END_REF]. Leaving aside possible cases of lexicalization, I think there should still be a formal basis for identifying such vowels. The easiest way resorts to their autosegmental representation. Especially if they alternate with zero, either they have floating material, or they receive their neighbour's melody; in both cases, they can thereby be argued to be underlyingly empty. Otherwise, the very concept of autosegmental association should be revised accordingly (see [START_REF] Cavirani | Modelling phonologization. Vowel reduction and epenthesis in Lunigiana dialects[END_REF].

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RADICAL: A JOURNAL OF PHONOLOGY, 3 be made: on the one hand, an erroneous theory is not necessarily useless; on the other hand, we often end up making something new out of something old. Indeed, if the criticism of the predominant binarist views will have provoked a return to Trubetzkoy, it will also have made it possible to arrive, on purely linguistic grounds, at conclusions that Trubetzkoy would perhaps not have accepted. For example, as a system based on privative oppositions rules out three-way contrasts, there is nothing to distinguish the unmarked phoneme from what he called an archiphoneme.

Finally, and above all, through the universality of the representations and the reformulation of markedness that it entails, this evolution of phonological theory over the last decades is in line with results obtained quite independently, during the same period, by research on the acquisition of the L1. This domain has been marked, if not by the discovery, at least by the theoretical recognition of a fact that was absent from the work, essentially based on speech production, to which Jakobson ([1941] 1969) had access and which supported the binarist theory of segmental markedness (cf. §2.1): to take up the subtitle of Kornfeld & Goehl's article (1974), "kids know more than they say". In other words, there is a fundamental asymmetry between production and perception in the process of L1 acquisition (cf., for example, [START_REF] Kornfeld | A new twist to an old observation: kids know more than they say[END_REF][START_REF] Pinker | Language learnability and language development[END_REF][START_REF] Gibbon | Lingual activity in two speech-disordered children's attempts to produce velar and alveolar stop consonants: evidence from electropalatographic (EPG) data[END_REF][START_REF] Faber | The perceptual infrastructure of early phonological development[END_REF][START_REF] Hale | Formal and empirical arguments concerning phonological acquisition[END_REF]). On the one hand, classical implicational views, where /e/ implies /i/ ~ /a/ and /y/ implies /i/ ~ /u/, etc., still seem to be supported by research on speech production. The development of speech in children would therefore involve markedness as defined above, i.e. the ability to perform operations on primitive unary objects. By this means, the ability to produce both the underspecified vowel / / and the overs ɯ pecified vowel /y/ implies the existence of the unary categories I and U, and thus the ability to produce the primary vowels of which I and U are the constituent categories: one does not acquire zero ex nihilo, since there are no "negative evidences" [START_REF] Hale | Phonology as cognition[END_REF]; so it PAGE 367

RADICAL: A JOURNAL OF PHONOLOGY, 3

is not surprising that / / -like ɯ /ə/, /ʔ/ and /h/, other underspecified segments resulting from intersections -is a late-acquired and marked phoneme, and thus not universal.

On the other hand, however, several studies have revealed surprising facts about the perception of the very young learner. According to some, children under 3 months of age show an extremely early discrimination ability and perceive phonic differences that are absent in their linguistic environment [START_REF] Trehub | The discrimination of foreign speech contrasts by infants and adults[END_REF][START_REF] Tees | Perceptual flexibility: Maintenance or recovery of the ability to discriminate non-native speech sounds[END_REF][START_REF] Goodman | The development of speech perception[END_REF]. One could certainly object that these are not "phonemes" at this stage; this early perception would not be linguistic, and the notion of markedness would remain alien to it, like, I would suggest, clicks, which are both highly marked consonants in a phonological system and possibly universal as sounds with paralinguistic value. However, these results concerning the universality of early perception are completed by further research on the cognitive foundations of perception, which may support the universal character of phonemic representations based on unary primitives.

From Kuhl's theory of "magnet effects", it emerges that the perceptual space progressively contracts around a certain number of phonetic "prototypes", a phenomenon which would exert a detectable influence on the perception of speech from the age of 6 months. 13 Interestingly, this seems to be closely associated with the child's "specialisation" in the categories of his native language, which can be assumed to reflect progressive assignment of adult-like phonological representations.

The place of these prototypes in the acoustic space is a function of the environment and the child's experience; they therefore differ somewhat from language to language -an /i/, an /u/ or an /a/ are not realised in the same way everywhere -and even more clearly according to the distribution -and hence the value, distinctive or not -of each sound in the different languages. However, the "magnetic effect" is universal and 13

For the original version of the theory, see [START_REF] Kuhl | Human adults and human infants show a "perceptual magnet effect" for the prototypes of speech categories, monkeys do not[END_REF][START_REF] Kuhl | Psychoacoustics and speech perception: internal standards, perceptual anchors, and prototypes[END_REF]Kuhl ( , 2000b)), [START_REF] Kuhl | Linguistic experience alters phonetic perception in infants by 6 months of age[END_REF], [START_REF] Iverson | Perceptual magnet and phoneme boundary effects in speech perception: Do they arise from a common mechanism[END_REF] or the summaries in Kuhl (2000a[START_REF] Kuhl | Early language acquisition: cracking the speech code[END_REF]. A later, extended version is proposed in [START_REF] Kuhl | Phonetic learning as a pathway to language: new data and native language PAGE[END_REF].

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RADICAL: A JOURNAL OF PHONOLOGY,3 the emergence of prototypes underlying categorial perception is remarkably consistent with the idea of unarist approaches, according to which, contrary to classical relativistic views of contrastive specification, the representation of a given segment is based on a small number of universal elements. The first prototypes to emerge would then be the primary segments, those that contain only one of the elements such as |I|, |A|, |U|, these thus playing the role of "attractors" in the perceptual space of vowels (other elements concern consonants). The subsequent emergence of other prototypes would only involve interactions between pre-existing objects, through operations on primary prototypes. Thus, after long and separate journeys, we might now witness, if not a general convergence between phonology and acquisition theories -some usagebased and probabilistic trends leave little room for the very idea of representation -, at least a punctual and remarkable concordance on the issue of phonemic categories.

COMMENTS

Carvalho's article is a welcome exploration of whether the primitives of phonological representation are privative or binary, a topic that has received undeservedly little attention in the recent literature (with some notable exceptions such as van der Hulst 2016;

Purnell, Raimy & Salmons 2019;and Raimy 2021). Following Carvalho (and contra, e.g., Kim 2002), I will reserve the term 'equipollent' for its original Trubetzkoyan meaning of contrast along a dimension consisting of distinct properties (like labial, coronal, and dorsal), and use 'binary' to refer to the idea that a single property, such as nasality, has antagonistic (plus and minus) values of equal formal status.

The debate between privative and binary primitives gets slippery because it is multidimensional: privative features are a different type of object from privative elements, although both have differences, some the same and some different, when compared to binary features; and since 'binary elements' is an oxymoron, the systemic comparison is asymmetrical. And this is to say nothing of differences in computational systems, or in broader representational geometries, that can generate varying predictions from the same representational primitives. Still, I very much agree with Carvalho's assessment that the issue of privativity versus binarity is a substantive matter that goes beyond mere notation: namely, it is a channel for the formal encoding of answers to fundamental empirical and typological questions, and so throughout the discussion we must stick close- The main argument for privativity has usually been the overgeneration one covered by Carvalho, namely that certain feature values like [-nasal] and [-round] seem to be phonologically inactive, squaring nicely with the fact that they can be seen as not involving any active articulatory gesture or deviation from the resting state. As noted by van der Hulst (2016: 96), it is impossible to prove the non-existence of phenomena like unrounding harmony, devoicing assimilation, or assimilation of orality. Even in apparent cases, it is possible that an analysis can be found that does not actually involve propagation of a minus value, as Carvalho proposes for the devoicing assimilations discussed by [START_REF] Kim | Phonological features: privative or equipollent? AB thesis[END_REF]. He also suggests that Turkish, where it is has been argued that both values of [±back] are needed (see [START_REF] Kim | Phonological features: privative or equipollent? AB thesis[END_REF], based on Clements & Sezer 1982), can be dealt PAGE 378

RADICAL: A JOURNAL OF PHONOLOGY, 3 with using privative frontness as attempted by van der Hulst & van de Weijer (1991), but with some other way of dealing with invariant back vowels that resist harmony (which have been the main obstacle to this type of analysis).

I am not clear on what this some other way would be, but it is obviously a matter for future research. In general, many previous arguments for binarity rest on the need for underlying prespecification of [+F], [-F], and Ø in order to deal with three-way contrasts in phonological behavior (Stanley 1967). Carvalho rejects this line of analysis as conflating the processual with the phonemic/systemic. I think this is a very interesting argument, but cannot immediately think of another way to encode these kinds of fundamentally lexical differences in underlying representations, without falling afoul of the same criticism.

Carvalho and I disagree over whether it is valid to argue for binary features on the basis of surface underspecification, as diagnosed through phonetic evidence (Keating 1988, Cohn 1990, Myers 1998). It is true that not everyone agrees on where phonological representations end and 'The Phonetics' beings, so precise phonetic realizations may not always be relevant to establishing phonological representations; but in cases where surface distinctions between minus and zero values exist within a single language, the phonology which yields that result deserves careful scrutiny. I suspect that further pursuit of this point would lead back to the rich and complex topic of processual versus phonemic/systemic aspects of phonology.

Something that I found quite stimulating was the idea that Front and Round can be thought of as being in an equipollent opposition. The feature-theoretic concept of autosegmental tiers requires fixed ideas about what constitutes a single 'dimension' for purposes of defining an opposition. But how do we define similarity versus difference in a property, when two ends of a putative continuum will always be different in some ways? It is also abundantly clear from parasitic harmony, consonant harmony, and other PAGE 379

RADICAL: A JOURNAL OF PHONOLOGY,3 phenomena that some featural interactions are dependent on certain values being present on other tiers; the autonomy of autosegmental tiers is, in reality, limited. Binary feature theory minutiously parcels out phonological properties into separate tiers -below the Element Theory standard of stand-alone pronounceability -but then has virtually no theory of how the tiers relate to each other and interact; in my opinion, even Feature Geometry had rather little to say about this.

Acquisition is yet another dimension of what is classically seen as 'markedness'. I had lots of questions about this aspect of the paper, as I endeavored to disentangle the different levels and senses of acquisition: as Carvalho notes, perception and production proceed on different timelines, and presumably, so too does the process of mapping an acoustic category to a phonological representation. For example, I did not completely understand the argument that acquisition of the Turkish vowel system should be impossible if the vowels consist of binary features. It is true that there is no obvious hierarchy of simplicity in the representations, but are there not other factors that influence the path of acquisition? And, is it really impossible to acquire more than one segment at the same time, such that the absence of a clear linear sequence causes acquisition to fail?

Carvalho cites Kuhl's work on perceptual magnets in acquisition as a potential source of evidence for elements. As categories are acquired, infants 'warp' the perceptual space to minimize within-category differences, so there is proportionally much higher sensitivity to differences of comparable acoustic magnitude when they straddle (2) The classical types of phonological opposition as found in Trubetzkoy's Principles of Phonology, (3) The different assumptions of the pair "marked/unmarked" throughout the 20 th century, (4) The postulate of phonological primitives in different phonological schools of thought, and (5) The parametric nature of phonemes as confirmed by recent experimental and data-driven approaches. Departing from the interrelation of questions such as these, the author provides us with solid arguments in favour of a view of phonology as the module that regulates subsegmental and segmental units as their crucial components.

Together with some previous work by the author ("De quoi sont faites les voyelles?" (1993), [START_REF] Dresher | The contrastive hierarchy in phonology[END_REF] work on phonemes and their relation to the Contrastive Hierarchy, and Durand's (2005) chapter on phonological primitives, this paper will remain as a fundamental reference for our understanding of the essential nature of segments and their constituents either as ontological objects and as epistemological constructions. A better understanding of the pertinence of keeping some traditional distinctions -such as the distinction between vowels and consonants, for instance -will also be attained thanks to this work.

  " ~ Ø or "voice" ~ Ø amounts to denying any phonological existence to [-nasal] or[-voiced]. Therefore, as we shall see, such features cannot be actors in any phonological process.

  kba/ → [kba] or [gba] b. /gpa/ → [gpa] or [kpa] As a result, all cases considered by Wetzels & Mascaró (2001) and Kim (2002: 43) are possible, depending on the "governing strength" of N 2 (which needs further research).

  JOURNAL OFPHONOLOGY, 3 generative phonology. 8 This step backwards is all the more interesting as the question of whether phonological primitives are privative or binary has received little attention in the recent literature (some exceptions being[START_REF] Kim | Phonological features: privative or equipollent? AB thesis[END_REF], Van der Hulst 2016, Purnell, Raimy & Salmons 2019, and Raimy 2021).

  JOURNAL OF PHONOLOGY, 3 ly to these. What structural types of oppositions are possible along a single phonological dimension, as evidenced by contrast, natural-class behavior, and other phonological activity? To what extent are the behaviors predicted by different types of oppositions attested in the world's languages? What is the relationship between various types of complexity -acquisitional, articulatory -and segments' patterning within phonological systems?Carvalho observes that the axiomatically privative approaches to phonological representations include Particle Phonology, Element Theory, Government Phonology, and Dependency Phonology. These are contrasted with the strict binarism of generative phonology starting in the 1960s. Within the neo-Jakobsonian tradition, however, rumblings of discontent with binary features appeared in the 1980s[START_REF] Goldsmith | Vowel harmony in Khalkha Mongolian, Yaka, Finnish and Hungarian[END_REF][START_REF] Steriade | Redundant values[END_REF], Mester & Itô 1989) and gathered steam throughout the 1990s[START_REF] Lombardi | Laryngeal features and laryngeal neutralization[END_REF][START_REF] Lombardi | Postlexical rules and the status of privative features[END_REF] Steriade 1995; Ewen & van der Hulst 2001: §2.1.1) before fizzing out with the ascent of Optimality Theory. Even Clements & Hume (1995), whose feature system was arguably conservative in allowing both plus and minus values of features such as [nasal] and [round], replaced binary [±back] with the articulator features [coronal] and [dorsal].

  category boundaries. Again, I wonder at what stage representations emerge relative to the formation of perceptual categories, and I think there could be a lot of complexity in trying to disentangle the predictions of elements versus features (privative or binary) here. I am thinking, for example, of work by Paul Boersma and colleagues (e.g., Boersma, Escudero & Hayes 2003; Boersma, Chládková & Benders 2021). If acquisition of a perceptual category precedes (or is in back-and-forth dialogue with) the assignment PAGE 380 RADICAL: A JOURNAL OF PHONOLOGY, 3of a phonological representation, then there may be ample room for varying interpretations of acquisitional studies vis-a-vis phonological theory. However, if acquisition of a perceptual category -as defined by the presence of warping effects -is assumed to entail assignment of an adult-like phonological representation, then I see how magnet effects around prototypes of [i], [a], and [u] might be taken as evidence for elements. Possibly, one type of counterevidence, in favor of features, would be if the emergence of warping effects around those prototypes was preceded by other kinds of sensitivity to the endpoints of acoustic continua corresponding to peripheral points in the vowel space, which in many languages would also be [i], [a], and [u] -making the possibilities tricky (but not impossible) to distinguish. In other words, are the acquisitional and phonological facts simply independent consequences of [i], [a] and [u] lying at the peripheries of the acoustic/articulatory space, or is there really something else going on? Boersma, Paul, Kateřina Chládková & Titia Benders. 2021. Phonological features emerge substance-freely from the phonetics and the morphology. Manuscript available online at https://www.fon.hum.uva.nl/paul/papers/BoeChlaBen23public.pdf. Boersma, Paul, Paola Escudero & Rachel Hayes. 2003. Learning abstract phonological from auditory phonetic categories: an integrated model for the acquisition of language-specific sound categories. Proceedings of ICPhS 15, 1013-1016. Clements, G. N. & Elizabeth Hume. 1995. The internal organization of speech sounds. In J. Goldsmith (ed.), The Handbook of Phonological

  consisted, on the one hand, in their acoustic-perceptual nature, and, on the other hand, in the assumption of qualities common to consonants

	and vowels: thus /p/, /t/ and /k/ were supposed to be distinguished by means of the
	same features as /u/, /i/ and /a/ respectively. Although the abandonment of these two
	aspects by generative phonology led it to reintroduce equipollent consonantal place
	features, Jakobson's antinomies were retained. Notationally, they have simply been
	replaced with [±F], an expression that I will adopt from now on although it has the dis-
	advantage of suggesting a privative type of opposition; in reality, as we shall see, [-F]
	does not in any way mean "absence of F", since [+F] and [-F] can behave in the
	same way, by propagating for example.

  the other hand, as regards nasality, no [-nasal] feature is required. The 'shielding' observed between an oral vowel and the preceding nasal onset, whereby /na/ → [n d a] vs. /nã/ → [nã] (cf., e.g., Stanton 2017), appears to me simply for what it is according to most interpretations of the phenomenon: a low-level strategy of contrast enhancement; nasality does not generally spread from onsets to nuclei. A coplanar model -as the one in (7) -may provide a reason for such enhancement by allowing two types of nasality in the same language: one on the C-plane, as in most languages, the other on the V-plane, as in the Min dialects of Taiwan (cf.Chang 1985: 19), which spreads onto the onset.

	(8) a. Spanish I	/i/	/u/
		-BACK	+BACK
	b. Spanish II	/i/	/u/
		-ROUNDED +ROUNDED
	Generative phonology got rid of this problem by proclaiming the redundant status
	of [±rounded] in any system without rounded front vowels. The basis for what
	could be called a theoretical sleight of hand was the observation that in m o s t sys-
	tems distinguishing two low vowels, it is an unrounded back / / or a central /a/ th ɑ	at
	contrasts with /a/ or a front /ae/ respectively; hence, the [±back] feature takes prece-

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2.4 UNDECIDABILITY

The second problem with binarist models -with the exception of Radical Underspecification and Contrastive Hierarchy theories (cf. §2.2) -is the representational undecidability of high vowels in most of the world's languages, which lack phonemes similar to /y/ in French or / / ɯ in Korean. Take one of the countless languages that show only /i/ and /u/ as high vowels: do we have (8a) or (8b) in Spanish for example?

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  Leaving aside these two points for further research, I contend that multiplying ¬T features leads to an overgeneration of phonological processes, most of which lack solid empirical evidence.Let us consider the set of high vowels examined so far. The two binary features used to differentiate them, plus the zero corresponding to the non-distinctive value, imply 9 combinations, i.e. 9 possible matrices for only 4 phonemes. Hence such unnatural

	specifications as [-rounded], [-nasal] or [-voiced] (cf. §2.3), hence the undecidabil-
	ity noted in (8) (cf. §2.4), and the disappearance of the markedness hierarchy in (9),
	where /i/ and /u/ can have as many distinctive features as /y/ and / / ɯ (cf. §2.5). It is the
	binarist bias that, through the overgeneration of features and therefore of representa-
	tions it entails, is directly at the source of the problems mentioned so far.
	The issues associated with this overgeneration had been glimpsed by
	Chomsky & Halle (1968: §9): for example, why are /y a / ɯ or /ae / forbi ɨ ɒ	dden vowel
	patterns despite being as simple as /i a u/ in terms of number of features? As Lightner
	PAGE 361
	RADICAL: A JOURNAL OF PHONOLOGY, 3

solution, for example by assuming that vowel height and ATR/RTRness -like voice (cf. §2.3) -are not based on features but on structure (cf. Pöchtrager 2006; Carvalho & Faust 2017; Cavirani & van Oostendorp 2020), probably in association with headedness (cf. Ulfsbjorninn 2021).

  Theory, 245-306. Oxford: Blackwell. Clements, G. N. & Engin Sezer. 1982. Vowel and consonant disharmony in Turkish. In H. van der Hulst & N. Smith (eds.), The Structure of Phonological Representations, Part II, 213-255. Cohn, Abigail. 1990. Phonetics and Phonological Rules of Nasalization. UCLA Working Papers in Phonetics 76. Ewen, Colin & Harry van der Hulst. 2001. The Phonological Structure of Words. Cambridge University Press. Goldsmith, John. 1985. Vowel harmony in Khalkha Mongolian, Yaka, Finnish and Hungarian. Phonology 2: 253-275. Inkelas, Sharon, C. Orhan Orgun & Cheryl Zoll. 1997. The implications of lexical exceptions for the nature of grammar. In I. Roca (ed.), Derivations and Constraints in Phonology, 393-418. Oxford: Clarendon Press. Keating, Patricia. 1988. Underspecification in phonetics. Phonology 5: 275-292. Kim, Yuni. 2002. Phonological features: privative or equipollent? AB thesis, Harvard University. Lombardi, Linda. 1991. Laryngeal Features and Laryngeal Neutralization. Amherst, MA: GLSA Publications. Lombardi, Linda. 1996. Postlexical rules and the status of privative features. Phonology 13: 1-38. Mester, Armin & Junko Itô. 1989. Feature predictability and underspecification: palatal prosody in Japanese mimetics. Language 65(2): 258-293. Myers, Scott.

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PHONOLOGY, 3

  Carvalho, J. B. (1993). De quoi sont faites les voyelles? Phonologie tridimensionnelle des particules et harmonie vocalique, in Laks, B., Plénat, M. (eds), De natura sonorum: Essais de phonologie. Saint Denis: Presses Universitaires de Vincennes, pp. 65-100. Dresher, B. E. (2009) The contrastive hierarchy in phonology. Cambridge: Cambridge University Press. Durand, J. (2005). Les primitives phonologiques: des traits distinctifs aux éléments, in Nguyen, N. et al. (eds). Phonologie et phonétique: Forme et substance. Paris: Hermès, pp. 63-89.

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DISCUSSION WITH JOÃO VELOSO (UNIVERSITY OF PORTO)

Veloso, João. 2023. discussion in: Carvalho, Joaquim Brandão de (auth.) "From binary features to elements: Implications for markedness theory and phonological acquisition". Radical: A Journal of Phonology,3,[383][384].

COMMENTS

Current phonological research often obliterates some "less young" issues and topics that have been abandoned my most recent perspectives. The price of this is costly: to mention but a few of the main limitations of this kind of hasty approach, one could refer to (i) working on hypotheses that previous, ignored research has already deemed inappropriate, or (ii) insisting redundantly on results that previous, ignored research has already proven worthy and robust. Ignoring the study of the history of the discipline -I mean, the main results of prominent phonologist who are not read anymore -shows a lack of linguistic culture and is a risky way to waste time and energy. This is not the case, obviously, with this paper. With a bullet-proof amount of knowledge of the most important scientific results of different phonological schools throughout the history of the discipline, the author develops an insightful review of those questions that still are at the very core of phonological theory: How can we conceive of the "phoneme" and "contrast" one century after Trubetzkoy? What is the real feedstock of phonological units? How tenable is the notion of markedness still nowadays?

In this paper, the author enters the very core of these fundamental questions and allows us to think of the phonological component of the grammar as an ever-growing jigsaw. The main achievement of this paper is the ability to interrelate aspects like: (1) The "oppositional nature" of linguistic units as we have been taught since Saussure's Cours, PAGE 383 RADICAL: A JOURNAL OF PHONOLOGY,3