Sébastien Briot
email: sebastien.briot@irccyn.ec-nantes.fr

Nicolas Bouton
email: nicolas.bouton@sigma-clermont.fr

Pascal Bigras
email: pascal.bigras@etsmtl.ca

Controlling Parallel Robots during Singular Assembly Mode Changing

In order to increase the reachable workspace of parallel robots, a promising solution consists of the definition of optimal trajectories that ensure the non-degeneracy of the dynamic model in the Type 2 (or parallel) singularity and allow to cross it. However, this assumes that the control law can perfectly track the desired trajectory, which is impossible due to modeling errors. This paper improves the results obtained previously by two of the authors of the present work that defined a robust multi-model approach allowing parallel robots to cross Type 2 singularities. The main idea was to shift near singularities the general dynamic model to a simplified one that can never degenerate. However, because in Type 2 singularity, the robot is locally overconstrained, we will show that the simplified dynamic model admits infinity of exact solutions. We propose to use the solution leading to a null overconstraint in order to avoid undesirable effects near singularities, such as buckling or unstability. The mentioned multi-model control approach is modified accordingly and it is then applied on a prototype of five-bar mechanism to show the effects of correctly and not-correctly managed overconstraint.

Introduction

Probably, the most important drawback of parallel robots is the presence of Type 2 (or parallel) singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] in their workspace which divide it into different aspects (each aspect corresponding to one or more assembly modes [START_REF] Merlet | Parallel Robots[END_REF]) and near which their performance is drastically reduced. Many strategies for enlarging the size of the operational workspace of parallel robots have been defined (e.g. optimal design [START_REF] Kong | A class of 3-dof translational parallel manipulators with linear input-output equations[END_REF], redundancy [START_REF] Muller | Internal preload control of redundantly actuated parallel manipulators -its application to backlash avoiding control[END_REF][START_REF] Kotlarski | Optimization strategies for additional actuators of kinematically redundant parallel kinematic machines[END_REF], working mode changing [START_REF] Bourbonais | Minimum-time trajectory planning and control of a reconfigurable pick-and-place parallel robot[END_REF]). However, all these methods have some limitations: they require the architectural modifications or they cannot necessary lead to the access of the total workspace.

In order to increase the size of the reachable workspace, methods for working mode or assembly mode changing have been more recently proposed [START_REF] Bourbonais | Minimum-time trajectory planning and control of a reconfigurable pick-and-place parallel robot[END_REF][START_REF] Zein | Non-singular assembly-mode changing motions for 3-RPR parallel manipulators[END_REF][START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF]. One of them consists in crossing the Type 2 singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] by defining optimal trajectories respecting a criterion based on the analysis of the degeneracy conditions of the dynamic model [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF]. This last approach is based on the following considerations.

As shown in [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF], the inverse dynamic model, which allows the computation of the motor input efforts, depends on the Jacobian matrix. In Type 2 singularity, the determinant of the inverse Jacobian matrix tends to zero [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] (and thus, the Jacobian matrix cannot be computed). As a result, along a trajectory passing through a Type 2 singularity, the desired motor input efforts may tend to infinity near the singularity, leading to motion infeasibility. However, the authors of [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF] were the first to provide a physical criterion for changing assembly modes by passing through the Type 2 singularities, which, mathematically speaking, can be summarized as follows: in order to avoid the dynamic model degeneracy, the trajectory must be planned so that the efforts applied by the legs, the external system and the internal dynamics to the platform are reciprocal to the null space of the matrix inverse Jacobian matrix.

Thanks to the proposed approach, it was possible to plan optimal desired trajectories for crossing the singularities. However, due to modeling errors, a robot will never be able to perfectly follow the optimal trajectory whatever x 0 y 0 O Uncontrollable motion along this direction

τ 1 τ 2 f 1 f 2
Fig. 1: A five-bar mechanism in a Type 2 singularity: in absence of any other effects, the torque τ i of the motor i leads to the application of a force f i on the end-effector. In singularity, f 1 is collinear to f 2 . As a result, a motion is gained in the direction reciprocal to f i (i.e. the end-effector is underconstrained along the direction of this uncontrollable motion) and, to produce a given force f along the axis x 0 , there is an infinity of solutions for the forces f i , i.e. the mechanism is overconstrained along x 0 .

the controller efficiency, thus potentially leading to failure of the Type 2 singularity crossing approach. In order to solve this issue, an advanced control law has been defined in [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] to finally ensure the change of assembly mode. This controller is based on the definition of a multi-model computed torque control law (e.g. see [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]) including two types of dynamic models:

1. a general dynamic model which is used for ensuring the robot trajectory tracking far from singularity, 2. a second simplified dynamic model which can never degenerate (due to the explicit deletion of the terms that may degenerate in singularity), and which is only used when the robot is close from a Type 2 singularity. This second dynamic model remain a good representation of the robot dynamics based on the fact that deleted terms stay close to zero near the singularity by planning an optimal trajectory. The switch between the two models in the controller was made near singularity by using a sigmoid function [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF].

All these results, applied either on planar [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF] or spatial robots [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF], showed that it was possible to successfully cross the Type 2 singularities and thus to enlarge the robot operational workspace.

Despite these rather encouraging results, some questions are still (partially or in totality) unanswered. One of them concerns the management of the local overconstraints in singularity. Indeed, in Type 2 singularities, the robot gains instantaneously an uncontrollable motion. This is due to the fact that the system of actuation wrenches on the end-effector is degenerated. As a result, the end-effector is underconstrained along the direction of the uncontrollable motion, but overconstrained along the other directions (Fig. 1). This overconstraint along the controllable directions of the motion may lead to some undesirable effects on the robot, such as buckling or instability of the end-effector [START_REF] Pashkevich | Enhanced stiffness modeling of manipulators with passive joints[END_REF].

To avoid the problems due to the local overconstraint, we propose in this work to modify the multi-model computed torque control law [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] so that, near singularity, the part of the control input signal leading to mechanism overconstraint is canceled.

The present paper is decomposed as follows. The next section makes necessary recalls on the parallel robot dynamic model degeneracy conditions and gives all the possible solution of the inverse dynamic model in Type 2 singularities. Then, Section 3 presents a modification of the control law [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] in order to cancel the overconstraint when crossing the singularity. In Section 4, we provide experimental results on a five-bar mechanism (DexTAR robot from Mecademic) in order to show the relevance and the robustness of the proposed advanced control law and the effect of an overconstraint when crossing the singularity. Finally, in Section 5, conclusions are drawn.

2 Recalls and solutions of the dynamic model in Type 2 singularities

Recalls on the dynamic model degeneracy conditions in Type 2 singularities

As proven in [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF], the inverse dynamic model of a general parallel robots takes the following form if there is no (serial) leg singularities (out of the scope of the present analysis):

τ = w b -B T λ (1)
A T λ = w p (2)
where

• τ are the input torques / forces,

• λ is a vector of Lagrange multipliers,

• matrices A and B are defined though the first-order input-output kinematic constraint relation given by

Aẋ + B q = 0 (3)
in which

-A = ∂ h ∂ x , B = ∂ h ∂ q
where h is the vector of the loop-closure equations [START_REF] Merlet | Parallel Robots[END_REF]; matrices A and B are square, -q is the vector of the active joint coordinates ( q being its derivative with respect to time), -x are the platform coordinates (ẋ being the derivative of the components of x with respect to time),

• w b and w p can be defined such that:

w b = d dt ∂ L ∂ q - ∂ L ∂ q (4)
w p = d dt ∂ L ∂ ẋ - ∂ L ∂ x (5)
in which L is the Lagrangian of the system which depends explicitly on q, x, q and ẋ (only).

It is known from [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] that Type 2 singularities appear when the matrix A defined in (3) is rank-deficient. Outside from Type 2 singularities, this matrix is invertible and the inverse dynamic model is given by

τ = w b + J -T inv w p (6)
where J inv = -B -1 A is the inverse Jacobian matrix. Expression (6) can be obtained by finding the value of λ = A -T w p thanks to (2) and then substituting it in [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. In Type 2 singularities, the matrix A, and thus the inverse Jacobian matrix J inv , are rank deficient. As a result, the model ( 6) is no more valid. In order to avoid the dynamic model degeneracy, the following facts must be considered. First, as the matrix A is rank deficient 1 , it exists a non-zero vector t s belonging to the null-space of A such that:

At s = 0 ⇔ t T s A T = 0 (7)
t s represents the direction of the uncontrollable motion inside the singularity [START_REF] Merlet | Parallel Robots[END_REF]. Thus, multiplying the left-hand side of (2) by t T s , we must obtain

t T s A T λ = 0 ( 8
)
As a result, in order to have consistency in the dynamic equations, by also multiplying the right-hand side of (2) by t T s , the robot motion must ensure that, in the singularity, we have

t T s w p = 0 (9)
This criterion was already provided in [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF] and, physically speaking, it can be summarized as follows: in order to avoid the dynamic model degeneracy, the trajectory must be planned so that the efforts applied by the legs, the external system and the internal dynamics to the platform (represented by the vector w p ) are reciprocal to the null space of the inverse Jacobian matrix (represented by the vector t s ). By planning a trajectory so that ( 9) is respected, the robot can cross a Type 2 singularity, as shown in several examples validated through experimentations [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF].

Solutions of the inverse dynamic model in Type 2 singularities

A question arises: even if the criterion ( 9) is respected, what could be the solution(s) to the inverse dynamic model given in ( 1) and ( 2)?

Mathematically speaking, if ( 9) is respected, it exists (at least) an exact and finite solution λ = λ * to the equation ( 2) (see Appendix A). On the contrary, if the criterion ( 9) is not respected, there is no solution to this equation. All the possible solutions λ * are given by:

λ * = A T + w p + I n -A T + A T ξ (10)
where A T + is the Moore-Penrose pseudo-inverse of A T , I n is the (n × n) identity matrix and ξ is a vector of dimension n taking any values. This results in an infinity of possible values for λ * , and thus from (1), for the input efforts τ = w b -B T λ * . Physically speaking, this means that, in Type 2 singularity, the system is overconstrained.

From this last expression of τ , it is quite difficult to estimate the value of the overconstraint. This is why it is preferable to rewrite the dynamic model equations ( 6) as follows:

J T inv τ = J T inv w b + w p (11)
Expression [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF] can be obtained by finding the value of λ = -B -T (τ -w b ) thanks to (1) and then substituting it in [START_REF] Merlet | Parallel Robots[END_REF]. Matrix B is of full rank in Type 2 singularities, while matrix J T inv is not. If the criterion ( 9) is respected in Type 2 singularities, then all possible exact solutions τ * for the value of the motor input efforts are given by:

τ * = J T + inv J T inv w b + w p + I n -J T + inv J T inv η (12)
where J T + inv is the Moore-Penrose pseudo-inverse of J T inv and η is a vector of dimension n taking any values. η is called the overconstraint [START_REF] Muller | Internal prestress control of redundantly actuated parallel manipulators its application to backlash avoiding control[END_REF] and τ * is of minimal norm when η = 0, i.e. when

τ * = Gw b + J T + inv w p (13)
with G = J T + inv J T inv . This solution is appealing because a too high overconstraint along the controllable directions of the motion may lead to some undesirable effects on the robot, such as buckling or instability of the end-effector [START_REF] Pashkevich | Enhanced stiffness modeling of manipulators with passive joints[END_REF]. Therefore this solution should be considered when developing a controller for crossing Type 2 singularities. This is what we did in the next Section.

3 Design of a controller for crossing Type 2 singularities

Recalls on the previous works

As previously mentioned, since the robot will never be able to perfectly follow the optimal trajectory that should be planned in order to respect the criterion (9), an advanced control law was defined in [START_REF] Pagis | Design of a controller for enlarging parallel robots workspace through Type 2 singularity crossing[END_REF] to finally ensure the change of assembly mode. This controller was based on the definition of a multi-model computed torque control law (e.g. see [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]) including two types of dynamic models:

1. a general dynamic model [START_REF] Bourbonais | Minimum-time trajectory planning and control of a reconfigurable pick-and-place parallel robot[END_REF] which is used for ensuring the robot trajectory tracking far from singularity, 2. a second simplified dynamic model equal to τ = w b which can never degenerate (due to the explicit deletion of the terms J -T inv w p from ( 6)), and which is only used when the robot is close from a Type 2 singularity.

This second dynamic model remain a good representation of the robot dynamics based on the fact that the desired trajectory was planned so that the term w p and its m derivatives with respect to time are zero in singularity, thus resulting in the fact that:

• w p = 0 respects the criterion (9),

• w p and its m derivatives with respect to time are zero allows w p to be zero not only in the singularity but also around it.

Computed torque control law outside of Type 2 singularities

Starting from the definitions (4) and ( 5) and the fact that the Lagragian depends explicitly on q, x, q and ẋ only, we know from [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF] that the terms w b and w p can be written under the following form:

w b = d dt ∂ L ∂ q - ∂ L ∂ q = M q (q, x) q + c q ( q, q, ẋ, x) (14)
w p = d dt ∂ L ∂ ẋ - ∂ L ∂ x = M x (q, x)ẍ + c x ( q, q, ẋ, x) (15)
It should be mentioned that the expressions of matrices M q and M x and vectors c q and c x do not degenerate in Type 2 singularities even if the criterion ( 9) is not respected [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF]. By using the second-order kinematic constraint relations, obtained by differentiating (3) with respect to time and given by Aẍ

+ B q = b ⇒ ẍ = -A -1 B q + b = J -1 inv q + b ( 16
)
where b = -Ȧẋ -Ḃ q and b = A -1 b, and then introducing them into (6), we get a general dynamic model of the form:

τ = M q + c (17)
where

• M = M q + J -T inv M x J -1
inv is the robot matrix of inertia

• c = c q + J -T inv c x + J -T inv M x b
is the vector of Coriolis, centrifugal effects, gravity and friction efforts.

Even though this model is not linear regarding the position and the velocities of the mechanism, it is linear regarding its acceleration. Therefore, by replacing the angular acceleration q in (17) by an adapted control signal u, the dynamics of the system is linear with respect to the control variable:

τ = Mu + c (18)
A double integrator between the control signal and the joint variables appears and thus, only a PD control law is used to impose the control signal (it can be a PID if static friction is high) [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]:

u = qd + K v ( q -qd ) + K p (q -q d ) ⇒ ë + K v ė + K p e = 0 (19)
where

• q d , qd and qd are the desired joint position, velocity and acceleration, respectively,

• q, q and q are the current and measured joint position, velocity and acceleration, respectively,

• e = qq d is the current tracking error in the joint space,

• K v and K p are two constant gain matrices.

This control signal is a classic second-order control law. The control input converts a complicated nonlinear controller design problem into a linear system consisting of n decoupled subsystems. Of course, this controller is based on the dynamic model of the mechanism. If this model is not accurate, the tracking error can therefore be important, but the control signal still guarantees that this tracking error respects Eq. ( 19) and therefore tends to zero with the desired second order dynamics. Consequently, the computed torque controller (CTC) computes the input efforts that ensure the second order dynamics on the tracking error:

τ = M( qd + K v ( q -qd ) + K p (q -q d )) + c (20)

Computed torque control law around Type 2 singularities

As previously mentioned, thanks to an optimal trajectory planning around singularity, we impose the fact that w p = 0 (and more specifically, that M x ẍ = 0 and c x = 0). As a result, the dynamic model is simplified into the new model τ = w b which can not degenerate in Type 2 singularities even if the criterion ( 9) is not respected [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF] and which takes the form, from ( 14), τ = w b = M q q + c q (21) Therefore, by replacing the angular acceleration q in (21) by an adapted control signal u, the dynamics of the system is linear with respect to the control variable:

τ = M q u + c q (22)
Then, as previously, imposing a PD control law (19) to the control signal u leads to τ = M q ( qd + K v ( qqd ) + K p (q -q d )) + c q (23)

Multi-model control law

Finally, the multi-model control law defined in Fig. 2 is used. In this scheme, we switch from the general model τ = M q + c (used far from singularities) to the simplified model τ = M q q + c q (used in the neighborhood of singularities). The switch between the two models in the controller is made near singularity by using a sigmoid function σ which depends on a criterion δ characterizing the proximity of singularity, as shown in [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF][START_REF] Pagis | Design of a controller for enlarging parallel robots workspace through Type 2 singularity crossing[END_REF]. In what follows, it is decided that, if ε is a threshold (positive) for the metrics δ characterizing the proximity to singularity, and that |δ | < ε characterizes the presence of singularity. The control input sent to the robot is finally:

τ f = (1 -σ )τ + σ τ ( 24
)
where σ is a piecewise continuous function defined such that:

• σ = 0 "far" from singularity (|δ | ≥ αε, where α is a positive number chosen by the user)

• σ = 1 "near" singularity (|δ | < ε) • σ ∈ [0 1] in between (ε ≤ |δ | < αε)
, where σ is a smooth monotonic increasing function.

+ - + + + + + + + + + x d IGM q d d/dt d/dt d/dt d/dt K p K v M c ROBOT q u 1-σ σ M q c q e q .
τ ' τ f τ f =(1-σ) τ+σ τ ' σ=1 near sing., σ=0 if not τ Fig. 2: The multi-model control scheme able to allow the singularity crossing. Far from singularity, σ = 0, while if the robot is close to a singularity, σ = 1. The change of the values in σ is not discontinuous and there is a transition phase during which 0 < σ < 1.

+ -

+ + + + + + + + + x d IGM q d d/dt d/dt d/dt d/dt K p K v M c ROBOT q u 1-σ σ M q c q e q . G τ ' τ * τ f τ f =(1-σ) τ+σ τ * σ=1 near sing., σ=0 if not τ Fig. 3:
The modified multi-model control scheme able to suppress the overconstraint during the singularity crossing.

Modification of the controller for canceling the overconstraint

The solution of the dynamic model that was chosen in the previous work was τ = w b (for w p = 0) which is not necessarily the solution leading to zero overconstraint in the system. From [START_REF] Muller | Internal prestress control of redundantly actuated parallel manipulators its application to backlash avoiding control[END_REF], this solution should be

τ * = Gw b = G(M q q + c q ) = Gτ (25)
w p being considered null. In order to impose this solution in the controller, we multiply the signal τ by the matrix G (Fig. 3), so that the control signal when we are close from singularity becomes:

τ * = G(M q u + c q ) = G(M q ( qd + K v ( q -qd ) + K p (q -q d )) + c q ) ( 26
)
The control input sent to the robot is finally:

τ f = (1 -σ )τ + σ τ * (27)
where the function σ was defined in the previous section.

Note on the computation of the term G near singularities

We recall from ( 13) that G = J T + inv J T inv . J T + inv is the Moore-Penrose pseudo-inverse of J T inv and it can be computed through SVD by the suppression or modification of the smallest singular value of J T inv . In order to avoid computational issues due to imperfect trajectory tracking, it is preferable to ensure the cancellation of the smallest singular value of J T inv when σ = 1, while it is not necessary to suppress it when σ = 0. During the transition phase, in order to avoid discontinuities, it is better to use a transition function so that, if s is the modified value of the smallest singular value of J T inv , and s 0 is the value when |δ | = αε, we have

• s = s 0 "far" from singularity (|δ | ≥ αε,
where α is a positive number chosen by the user)

• s = 0 "near" singularity (|δ | < ε)

• s ∈ [0 s 0 ] in between (ε ≤ |δ | < αε)
, where s is a smooth monotonic increasing function.

Case study

In this section, we provide experimental results on a five-bar mechanism (DexTAR robot from Mecademic) in order to show the relevance and the robustness of the proposed advanced control law and the effect of an overconstraint when crossing the singularity. First, we presents the benchmark, and then show the experimental results.

Benchmark

The DexTAR robot (Fig. 4(a)) is a five-bar mechanism (Fig. 4(b)), i.e. a planar parallel robot composed of two actuators located at the revolute joints positioned at points A and E and 3 passive revolute joints at points B, C and D. The prototype of the five-bar was designed such that i = 90 mm (i = 1, ..., 4) and a = 118 mm. To control the robot, the motors amplifiers receive a command proportional to the torques to deliver by the actuator. To send this control command, a dSPACE 1104 control board was used. The controller sampling period is 1 ms. The multi-model computed torque controller presented in the present paper was implemented on the robot. To design the robot controller, the software Matlab/Simulink combined with the software dSPACE ControlDesk has been used.

For implementing the multi-model computed torque controller, the dynamic model of the robot was necessary. It was identified by using the procedure shown in [START_REF] Briot | Global identification of joint drive gains and dynamic parameters of parallel robots[END_REF] and it takes the following form:

τ = zz 1R 0 0 zz 2R q1 q2 + f s 1 sign( q1 ) f s 2 sign( q2 ) + m R J -T inv ẍ ÿ (28)
where

• zz 1R = 0.0137 kgm 2 ± 0.51 % and zz 2R = 0.0145 kgm 2 ± 0.54 % are grouped moments of inertia around z,

• f s 1 = 0.30 Nm ±1.43 % and f s 2 = 0.37 Nm ±1.34 % are Coulomb friction terms in the actuated joints,

• m R = 0.51 kg ±1.26 % is a grouped mass on the end-effector,

• J inv is the inverse Jacobian matrix,

• sign(u) is the sign function (sign(u) = +1 if u > 0, sign(u) = -1 if u < 0, and sign(u) = 0 if u = 0),

• x and y are the Cartesian coordinates of the end-effector along the axes x and y, respectively,

• q 1 is the joint angle of motor located at point A while q 2 is the joint angle of motor located at point E.

The terms zz iR , f s i and m R are the essential parameters [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF], i.e. the parameters that are well-identified because they represent the large majority of the robot dynamics effects. As a result, the model ( 28) is able to predict the input torques of the DexTAR with an error lower than 10 %.

It should be mentioned that the computation of the platform accelerations and robot Jacobian matrix are detailed in [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF].

From [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF], the model (28) can be written under the following matrix form:

τ = M q1 q2 + c (29)
where

• M = M q + J -T inv M x J -1 inv , with M q = zz 1R 0 0 zz 2R and M x = m R I 2 • c = c q + J -T inv c x + J -T inv M x b
, where c q = f s 1 sign( q1 ) f s 2 sign( q2 ) , c x = 0, and b is a bi-dimensional vector that can be obtained from the second-order kinematics [START_REF] Briot | Global identification of joint drive gains and dynamic parameters of parallel robots[END_REF].

By identification with the simplified dynamic model given in (22), the simplified dynamics of the five-bar that will be used in the controller near singularities is given by:

τ * = G(M q q + c q ) = G zz 1R 0 0 zz 2R q1 q2 + f s 1 sign( q1 ) f s 2 sign( q2 ) (30)
Let us recall that G = J T + inv J T inv , in which J T inv is rank deficient near singularity (see the section 3.3 for the computation of J T + inv ).

Experimental results

Based on the criterion w p = 0 ⇒ ẍ = ÿ = 0 (which respects the criterion (9) and allows to use the control approach developed in Section 3) and the cancellation of its first and second derivatives with respect to time, it was possible to generate desired trajectory which avoids the degeneracy of the dynamic model in and around Type 2 singularities. We planned a trajectory through singularity (Fig. 5) where the initial point is P 0 = [0 0.15 m] T , the final point is P f = [0 0.02 m] T and the point at which we decide to cross the singularity is P s = [0 0.084 m] T . Once point P f is attained, the robot do the reverse path. The duration for the full trajectory is of 0.8 sec while the robot must cross the singularity at t = 0.3 sec and t = 0.5 sec. Results along this trajectory, when the controller shown in Fig. 3 is implemented, are presented in Fig. 6. They showed that the robots was able to cross the singularity without any problem. 2Then, we decided to analyze the effect of a too high unmanaged overconstraint when crossing a singularity. For that, instead of using the solution ( 13) of the dynamic model inside the singularity, we chose a more general solution [START_REF] Pashkevich | Enhanced stiffness modeling of manipulators with passive joints[END_REF] where the overconstraint term η is different from zero and is equal η = [-2 2] T . Results along the same trajectory are presented in Fig. 7. Around t = 0.3 sec, the input efforts are saturated (saturation fixed at ±8 Nm) and the robot is no more able to follow the trajectory. This results in growing tracking errors (up to 15 deg) as the robots was indeed blocked into the Type 2 singularity.

Conclusion

Increasing the size of the operational workspace of parallel robots is a keypoint in order to favor their industrial development. A promising solution consists in crossing Type 2 singularities in order to access other workspace aspects. This can be done by the definition of optimal trajectories that ensure the non-degeneracy of the dynamic model in the Type 2 (or parallel) singularity. However, this assumes that the control law can perfectly track the desired trajectory, which is impossible due to modeling errors.

In their previous works, two of the authors of the present paper defined a robust multi-model approach allowing parallel robots to cross Type 2 singularities. The main idea was to shift near singularities the general dynamic model to a simplified dynamic model that can never degenerate. However, because in Type 2 singularity, the robot is locally overconstrained, we showed in the present paper that the simplified dynamic model admits infinity of exact solutions, among which it is necessary to select one. We proposed to use the solution leading to a null overconstraint in order to avoid undesirable effects near singularities, such as buckling or unstability.

Accordingly, the mentioned multi-model control approach was modified and it was then applied on a prototype of five-bar mechanism. We made two types of experiments: (i) tracking a trajectory by using the new controller where the overconstraint is cancelled, and (ii) tracking a trajectory when the overconstraint is too high. In the first case, the robot was able to cross the singularity without any problem while in the second case, the robot stay blocked in it, which showed the efficiency of the proposed approach. So, let us project this equation in another basis B which contains as a principal vector the unit vector t s . As A is of dimension (n × n), it is possible to define n -1 vectors t ⊥ j s such that

• t ⊥ j s = 1 • t T s t ⊥ j s = 0
• (t ⊥k s ) T t ⊥ j s = 0 for any k = j ( j, k = 1, ..., n -1).

Let us group all vector t ⊥ j s in a matrix T ⊥ s such that:

T ⊥ s = t ⊥1 s . . . t ⊥n-1 s (32)
Passing from the Cartesian frame to the basis B can be done by using the transformation R B defined such that

R B = T ⊥ s t s (33)
R B is an orthogonal matrix: as a result R -1 B = R T B and det(R B ) = ±1. Thus, if we project the equation (31) in B, we have

R T B A T λ = R T B w p ⇒ (T ⊥ s ) T A T t T s A T λ = (T ⊥ s ) T w p t T s w p (34)
If the criterion ( 9) is respected, and as t s is in the null-space of A, the equation (34) can be rewritten as

(T ⊥ s ) T A T 0 1×n λ = (T ⊥ s ) T w p 0 ⇔ (T ⊥ s ) T A T λ = (T ⊥ s ) T w p (35)
λ being a vector with n independent variables, and (35) containing n -1 independent equations, there is an infinity of exact solutions to the equation (35), and as a result, to the equation (31).

If the criterion (9) is not respected, i.e. t T s w p = 0, the equation (34) can be rewritten as

(T ⊥ s ) T A T 0 1×n λ = (T ⊥ s ) T w p t T s w p ( = 0) (36)
Here, the dynamic equations are no more consistent, and there is no exact solutions to the equation (36). However, we can find an approximate solution λ a which minimizes the norm of the error e = A T λw p .

Fig. 4 :

 4 Fig. 4: The five-bar mechanism under study.

Fig. 5 :

 5 Fig. 5: Starting point P 0 and ending point P f of the Type 2 singularity crossing trajectory

Fig. 6 :

 6 Fig.6: Results in terms of input torques and tracking errors when using the controller of Fig.3.

Fig. 7 :

 7 Fig.7: Results in terms of input torques and tracking errors when the overconstraint is too high.

We consider here a loss of rank equal to 1, but a loss of rank of higher order could be considered.

The video of the DexTAR moving along this trajectory can be downloaded in: https://youtu.be/UR9b5ouFy0o.

We consider once again a loss of rank equal to 1, but a loss of rank of higher order could be considered.

Acknowledgements

This work was supported by the French ANR project ARROW (ANR-2011BS3-006-01).

A Appendix

Let us consider that the robot is in Type 2 singularity. As a result, the matrix A of (3) is rank deficient and we can define non-zero unit vector t s belonging to the null-space of A 3 .

The dynamics equation (2), which is recalled here for reasons of clarity,

are expressed in the Cartesian frame (because the term w p is the derivative of the Lagrangian with respect to x and ẋ) but they can also be projected in another basis B.


D Six

S Briot

A Chriette

P Martinet

A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots during Type 2 Singularity Crossing

Keywords:

Parallel robots presents singular configurations that divide the operational workspace into several aspects. It was proven that such singularities can be crossed under the constraint of a dynamic criterion. However, the development of a controller able to track such trajectories is up to now limited by a restrictive criterion, which decreases the total number of possible trajectories for crossing a singularity. In this paper, by finding a solution to the inverse dynamic model at the singularity locus, we were able to implement a controller in Cartesian space and track a trajectory that crosses a Type 2 singularity.

Introduction

Parallel robots present many advantages over serial robots, such as higher acceleration capacities, better stiffness and higher payload-to-weight ratios. However, their use in an industrial context remains limited partly due to the division of their workspace by singularities into several aspects [START_REF] Merlet | Parallel robots[END_REF]. Numerous singularities are prone to appear on a parallel mechanism. For a global overview the reader is referred to [START_REF] Zlatanov | Constraint singularities of parallel mechanisms[END_REF][START_REF] Conconi | A New Assessment of Singularities of Parallel Kinematic Chains[END_REF]. This paper will focus on a specific singularity, called Type 2 singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF], where the inverse dynamic model (IDM) of the mechanism does not admit a finite solution without respecting a dynamic criterion [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF].

Previous studies [START_REF] Ider | Inverse dynamics of parallel manipulators in the presence of drive singularities[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF] proposed a solution to increase the workspace of parallel robots by planning assembly mode changing trajectories directly through the singularity. This solution seems promising because a direct path between two points crossing a Type 2 singularity can be generated on any robot structure as long as the dynamic criterion is respected by the trajectory. However, the tracking of a trajectory, and so respecting the dynamic criterion is not always perfectly ensured in reality. Then, the design of a stable controller to cross the singularity becomes problematic. This issue was partially solved in [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] by the design of a multi-model computed torque control. However, the solution proposed requires a more restrictive criterion on the designed trajectory than the dynamic criterion computed in previous studies [START_REF] Ider | Inverse dynamics of parallel manipulators in the presence of drive singularities[END_REF][START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF], decreasing the total number of possible trajectories for crossing the singularity. This paper aims to extend the solution proposed in [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] by the design of a generic controller in Cartesian space able to carry out the crossing of a Type 2 singularity as long as the trajectory design respects the crossing dynamic criterion. The validation of the designed controller is discussed through experiments in Section 4.

Dynamic model at singularity locus

In this section, we will briefly recall the dynamic equations of a parallel manipulator and discuss its degeneracy on a Type 2 singularity. For a more detailed analysis of the dynamic modeling of a parallel manipulator, the reader is referred to [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF][START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. We will then propose a general solution to the dynamic model that does not degenerate at singularity locus.

Dynamic modelling of parallel mechanisms, Type 2 singularity degeneracy condition and criteria for singularity crossing

The studied manipulator is composed of a fixed base, linked by several kinematic chains (the legs) to a mobile platform actuated along n do f independent coordinates. Actuation is provided by n do f active joints. The configuration and velocity of the manipulator can be described using:

• q a and qa two n do f -dimensional vectors of active joint variables and active joint velocities, respectively. • x and ẋ two n do f -dimensional vectors of the independent platform coordinates and their time derivatives, respectively.

Relations between these coordinates are found by writing the closed-loop equations. Using Lagrangian formalism, the IDM of the mechanism can be written as [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF] τ = w b -B T λ and

w d = A T r λ (1)
Where,

• τ is the n do f -dimensional vector of the input efforts,

• λ is a n do f -dimensional vector of Lagrange multipliers,

• A r and B are two (n do f ×n do f ) matrices characterizing the first-order input/output kinematic constraints

A r ẋ + B qa = 0 (2)
• w b and w d are respectively n do f -dimensional vectors related to the Lagrangian L of the system (which can be explicitly expressed as a function of q a , x, qa , and ẋ) by [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF]

w b = d dt ∂ L ∂ qa T - ∂ L ∂ q a T and w d = d dt ∂ L ∂ ẋ T - ∂ L ∂ x T (3)
Fig. 1: Example of a five-bar mechanism in a Type 2 singularity configuration. The vector t s represents the uncontrolled motion.

From equations (1), considering the matrix A r to be full rank, the dynamic model of a parallel manipulator is obtained

τ = w b -B T A -T r w d (4)
This model remains valid as long as the matrix A r is invertible. The degeneracy of the matrix A r corresponds to a kinematic singularity, named Type 2 singularity [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF].

In such a singularity, one (or more) of the degrees of freedom of the platform becomes uncontrollable (see Fig. 1). For clarity on this article, we will limit the study to one uncontrollable degree of freedom 1 . Approaching a singularity, the determinant of the matrix A T r tends toward zero. The computed input efforts (Eq. ( 4)) are related to the inverse of this determinant and may tend towards infinite values in the neighborhood of a singularity. A previous study [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF], showed this issue can be avoided by respecting a dynamical criterion at the singularity locus. As the matrix A r is degenerated, a non-null vector t s exists in its kernel. Multiplying the equation ( 1) by t T s gives us

t T s A T r λ = 0 ⇒ t T s w d = 0 (5)
This equation is a necessary condition for the inverse dynamic equations (1) to remain consistent in a Type 2 singularity. w d represents the sum of the wrenches applied on the platform by the legs, inertia/gravitational effects, and the external environment and t s represents the direction of the uncontrollable motion of the platform inside the singularity in Cartesian space (see Fig. 1). Equation [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] implies that these two vectors must be reciprocal when crossing a Type 2 singularity so that the input torque remains finite.

Solution to the inverse dynamic model at singularity locus

Equation (4) gives a solution to the dynamic model away from singularities. We show below that it is possible to find solutions to the IDM at a singularity if and only if the criterion (5) is respected. From basic knowledge in linear algebra, the rank degeneracy of the matrix A r implies that the equation ( 1) admits at least one exact solution λ if and only if w d is included in the image of the matrix A T r . The image of A T r is spanned by all the total wrenches w d that can be applied by the legs through actuation and external forces on the platform. From linear algebra, we know also that the image of the matrix A T r is the orthogonal complement of the kernel of the matrix A r

ker(A r ) ⊥ = Im(A T r ) (6)
This equation can be interpreted as "Any total wrench that can be generated on the platform by the legs and the external forces is reciprocal to the uncontrolled motion". It is equivalent to the criterion already expressed to avoid the degeneracy of the IDM on a Type 2 singularity [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. If the criterion is respected the inverse dynamic equations admits at least an exact solution. In order to find it, let us consider the dynamic equations under an other form. Multiplying (1) by A T r B -T gives2

A T r B -T τ = A T r B -T w b -A T r λ (7)
Then, A T r λ can be replaced in equation ( 7) from equation (1).

J T inv τ = J T inv w b + w d (8)
with J inv = -B -1 A r . Knowing that the IDM (1) admits at least an exact solution, if the criterion ( 5) is respected, the solution minimizing the input torques can be expressed using the Moore-Penrose pseudo-inverse of J T inv :

τ = J T inv + (J T inv w b + w d ) (9)
3 Design of a computed torque control law in Cartesian space

The computed torque control (CTC) [START_REF] Paccot | A Review on the Dynamic Control of Parallel Kinematic Machines: Theory and Experiments[END_REF] is a natural controller to enforce a dynamical criterion such as the criterion computed for singularity crossing [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. To implement it, the IDM (9) of the robot must be expressed as a function of the acceleration of the controlled coordinates. From [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF], w b and w d can be expressed as function of the robot active joint and platform accelerations

w b = M a qa + c a and w d = M x ẍ + c x (10)
where

• M a and M x are (n do f × n do f ) matrices depending on the robot configuration coordinates q a and x. • c a and c x are n do f -dimensional vectors depending on the robot configuration coordinates q a , x and their time derivative qa , ẋ.

Moreover, the time derivative of the loop-closure equations (2) gives

A r ẍ + B qa + b = 0 (11)
with b = Ȧr ẋ + Ḃ qa . Eq. ( 11) links the acceleration of the active joints to the platform acceleration.

Robot controllers are usually established in joint space. However, in a Type 2 singularity, the matrix A r is not invertible anymore, and then w d cannot be expressed as a function of the joint accelerations qa . In order to avoid this issue, a previous study [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] chose to impose a criterion w d = 0 to cross the singularity , allowing the computation of the IDM as function of the joints accelerations only. However, this criterion is more restrictive than t T s w d = 0 (5), thus limiting the number of achievable trajectories. In order allow all trajectories respecting the criterion t T s w d = 0, in this paper, we will express the IDM as function of the platform acceleration and, consequently, design a CTC controller in Cartesian space. Even at singularity locus, from [START_REF] Samson | Robust control of a class of non-linear systems and applications to robotics[END_REF] the joint accelerations can be expressed as a function of the platform acceleration qa

= -B -1 A r ẍ -B -1 b (12)
Introducing ( 10) into the IDM [START_REF] Paccot | A Review on the Dynamic Control of Parallel Kinematic Machines: Theory and Experiments[END_REF] gives

τ = J T inv + J T inv M a qa + J T inv + M x ẍ + J T inv + (J T inv c a + c x ) (13)
Then combining ( 13) and ( 12) gives the expression of the IDM as a function of the platform acceleration

τ = Mẍ + h ( 14
)
where

M = J T inv + (-J T inv M a B -1 A r +M x ) and h = J T inv + (J T inv (c a -B -1 b)+c x ). Away from a singularity J T inv is full rank, so J T inv + = J -T inv .
In the close neighborhood of a singularity locus, the computation of the torques from the IDM given in ( 14) remains bounded only if the dynamic criterion established ( 5) is exactly respected. Unfortunately, tracking errors make the exact respect of the criterion impossible in a real experiment. In our CTC application, the IDM is computed from an auxiliary control signal u corresponding to a desired platform acceleration. The enforcement of the criterion is carried out by projecting w u = M x u + c x in the image of the matrix A T r . This projection ensures that the IDM admits a unique solution and that the desired input efforts remain bounded.

From the expression (14) of the dynamic model, a classical CTC controller in Cartesian space, as described in Fig. 2, is established for the tracking of a trajectory with a parallel mechanism. The proof of convergence and stability of such controllers in Cartesian space is discussed in [START_REF] Samson | Robust control of a class of non-linear systems and applications to robotics[END_REF]. An exact linearization is performed by the inner loop of the controller and a double integrator is obtained between the auxiliary control signal u and the joints coordinates qa . Then, a PD-control loop is sufficient to ensure the convergence. An integral term may be added to compensate residual modeling errors. It should be noted that the controlled system output is the platform coordinates vector. This output may be directly measured with an external sensor or estimated via computation, this point is discussed in section 4.2.

x t ẍt - + K p + + + d dt K d IDM Mu + h + enforced criterion u ROBOT τ d dt x PD Controller
Linearized system Fig. 2: Computed Torque controller in Cartesian space.

5bar_kin-eps-converted-to.pdf 4 Case study

Experiment

The proposed approach was validated on the crossing of Type 2 singularities with a planar five-bar mechanism designed by Mecademics (see Fig. 3). This mechanism is able to generate a motion of the end-effector located at A 13 (x = (x, y) T ) through the actuation of the joints q 11 and q 21 . To test the controller a trajectory that cross a Type 2 singularity twice has been generated for the five-bar mechanism. This trajectory is a return trip between point A and point B (see Fig. 4), optimized in order to respect the dynamic criterion to be enforced when crossing a Type 2 singularity (5) for this mechanism. A controller was implemented as defined in section 3. However, only active joint coordinates were available on our experimental platform. Hence, an estimation of the platform coordinates, using the computation of the direct geometric model (DGM) of the robot, has been introduced in the feedback loop [START_REF] Tachi | Dynamic Control of a Manipulator with Passive Joints in Operational Space[END_REF]. The controller implemented enabled the crossing of a Type 2 singularity in experimental condition. The input torque and tracking error on the end-effector pose estimation are given in Fig. 4.

Discussion about the practical estimation of the platform coordinates for the controller

For our implementation, we used the DGM as an estimator of the platform coordinates in the implementation of our controller. This implementation generates several issues:

• The estimator uncertainties, issued from calibration errors and DGM modeling errors, are not corrected by the controller, leading to a steady state error of the end-effector position in Cartesian space. • Around a Type 2 singularity, any error in joint position is amplified in Cartesian space along the direction of the uncontrollable motion. Specific precautions have been taken in the estimation of the end-effector pose around the singularity at the cost of estimation uncertainty around the singularity locus. Those uncertainties in the estimator causes an increase of the tracking error just after the singularity is crossed by the robot (see Fig. 4).

A better implementation of this controller should be based on an external measure of the end-effector position. However, the ability of our controller to cross the singularity even with those uncertainties demonstrated the robustness of our implementation. The repeatability of the process was tested on a trajectory crossing the Type 2 singularities several times on a five-bar mechanism at different points with a successful crossing at every point. A video of the DexTAR robot following this trajectory is available online at http://www.irccyn.ec-nantes.fr/∼six/videos.

Conclusions

In a Type 2 singular configuration, the IDM of a parallel robot cannot be computed in its general form due to the degeneracy in its equation system. In this paper we showed that, under the constraint of a dynamic criterion, the IDM equations can be solved even at singular configurations. This expression allowed us to design a controller able to track a trajectory respecting the dynamic criterion defined while crossing a Type 2 singularity. The theoretical results were confirmed through experimentation. The ability to cross Type 2 singularities is promising to increase the reachable workspace of any parallel robot that has such singularities in its workspace. However, the designed controller can only be implemented in Cartesian space. The implementation of a joint space CTC controller is currently under investigation.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Prototype of the five-bar mechanism and parametrization scheme

Even if the case rarely appears on existing parallel robots, the results can be extended to a higher order degeneracy.

We consider that B is full rank. This hypothesis is taken as the case of the coincidence of two singularities is extremely rare and generally avoided in the design of a parallel robot. Note that the computation of B -T is not necessary in the inverse dynamic model away from a type 2 singularity.


Clément Boudaud

Sébastien Briot

Lila Kaci

Philippe Martinet

Philippe Martinet Robecolo

RobEcolo : Conception de Robots industriels à base de matériaux à faible impact Écologique

RobEcolo

Objectifs

L'industrie européenne est engagée dans un processus de robotisation afin de rester compétitive. Cependant, il est méconnu que l'utilisation des robots (sans même considérer le procédé industriel pour lequel ils seront utilisés) impacte considérablement l'environnement (Fig. 1). On estime l'impact de la fabrication (seulement) du stock mondial de robots industriels, à 11 millions de tonnes en équivalent CO 2 rejetées dans l'atmosphère (équivalent à 36 millions de camions parcourant la distance Nantes-Paris). Ceci est grandement dû aux matériaux métalliques, voire composites de carbone, dont la manufacture est extrêmement polluante (même s'ils sont recyclés).

Montrer qu'il est possible de réduire considérablement l'impact environnemental des robots industriels (2 ddl, Fig. 2) en remplaçant la grande majorité des matériaux métalliques/composites de carbone par des matériaux bio-sourcés à faible impact écologique, tout en garantissant la fiabilité du robot en termes de durée de vie et de performances mécaniques (critères : répétabilité < 0,5 mm et déformation sous charge de 1 kg inférieure à 0,5 mm) même si les propriétés / dimensions du bois varient.

Premiers résultats

Méthode

Seule la partie matériau est abordée ici (une partie du projet concerne des aspects spécifiques à la robotique (capteurs extéroceptifs, algorithme de conception robuste, commande). Ici la méthode consiste à :

• Développer un modèle mécanique numérique du robot, fiable, précis et rapide, qui sera utilisé par les algorithmes d'optimisation de la conception du robot.

• Caractériser précisément les caractéristiques mécaniques du bois retenu (on commence par étudier le bois acétylé, pour sa stabilité dimensionnelle).

• Réaliser un prototype pour valider le modèle numérique.

Rigidité

Deux méthodes d'essais, l'une destructive (Fig. 3), l'autre non (Fig. 4). Les dimensions des éprouvettes sont de l'ordre de grandeur de celles des bras du robot : 30 x 30 x 350 mm.

Essais préliminaires sur

Stabilité dimensionnelle

Nécessité de connaître avec précision le coefficient de dilatation hydrique longitudinal (données non existante), de plus l'acétylation :

• réduit beaucoup les dilatations hydriques • n'a à priori pas d'impact sur les dilatations thermiques Essais en cours (Fig. 7

  Fig. 1. Impact environnemental d'un robot Kuka KR 270 sur toute sa durée de vie (fabrication + 12 ans de service) / Consommation énergétique durant une phase d'utilisation de 12 ans.

Fig. 2 .

 2 Fig. 2. Robot plan à 2 ddl qui sert de base à l'étude

Fig. 3 .

 3 Fig. 3. Flexion 4 points (EN 408) Fig. 4. Flexion porte-à-faux

Fig. 5 .

 5 Fig. 5. Modules de Young en flexion Fig. 6. Modules de Young en fonction de la densité

) 1 .Fig. 8 .

 18 Fig. 7. Conditions des essais en cours


Sébastien Briot
email: sebastien.briot.1@ens.etsmtl.cailian.bonev@etsmtl.ca

Ilian A Bonev

Pantopteron-4: a New 3T1R Decoupled Parallel Manipulator for Pick-and-Place Applications

Keywords: parallel manipulator, decoupling, kinematics, Schoenflies motions, singularity, workspace, design

In this paper, a novel 4-DOF decoupled parallel manipulator with Schoenflies motions, called the Pantopteron-4, is presented. This manipulator is able to perform the same movements as the Isoglide4 or the Quadrupteron, but, due to its architecture which is made of three pantograph linkages, an amplification of the movements between the actuators and the platform displacements is achieved.

Therefore, having the same actuators for both robots, the Pantopteron-4 displaces (theoretically) manytimes faster than the Isoglide4 or the Quadrupteron, depending on the magnification factor of the pantograph linkages. Thus, this mechanism is foreseen to be used in applications where the velocities and accelerations have to be high, as in pick-and-place. First, the kinematics of the Pantopteron-4 is presented. Then, its workspace is analyzed. Finally, a prototype of the mechanism is shown and conclusions are given.

I. INTRODUCTION

Less than a decade ago, any known parallel robot, whatever its number of degrees of freedom (DOF), was inevitably associated with nonlinear highly-coupled kinematics, singularities, and a complex-shaped workspace. However, in May 2001, this fact was refuted by the discovery of a revolutionary simple 3-DOF translational parallel robot, with fully-decoupled input-output equations, disclosed by Gosselin and Kong in a Canadian provisional patent application [START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF]. Its simplest design is basically a Cartesian robot and is therefore isotropic (its Jacobian matrix is diagonal and constant). Later in 2002, many researchers proposed separately a large family of decoupled 3-DOF translational parallel mechanisms, all covered by the above-mentioned patent [START_REF] Carricato | Singularity-Free Fully-Isotropic Translational Parallel Manipulators[END_REF]- [START_REF] Kim | Evaluation of a Cartesian Parallel Manipulator[END_REF]. These works cleared the way for the creation of various decoupled parallel mechanisms.

The most prolific author on this subject, Gogu, wrote dozens of papers and even a 700-page manuscript [START_REF] Gogu | Structural Synthesis of Parallel Robots, Part 1 -Methodology[END_REF] proposing isotropic architectures for nearly all combinations of translational and rotational degrees of freedom. Specifically, many efforts have been done in creating decoupled robots with Schoenflies motions [START_REF] Hervé | The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design[END_REF]- [START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot with uncoupled kinematics[END_REF] for pick-and-place applications, driven by the commercial success of the Delta [START_REF] Bonev | Delta Robotthe Story of Success[END_REF] and Quattro [START_REF] Pierrot | Optimal Design of a 4-dof Parallel Manipulator: From Academia to Industry[END_REF] robots. Examples of decoupled 3T1R (three translational DOFs and one rotational DOF) structures are the Quadrupteron [START_REF] Gogu | Structural Synthesis of Fully-Isotropic Parallel Robots with Schoenflies Motions via Theory of Linear Transformations and Evolutionary Morphology[END_REF] and the Isoglide4 [START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot with uncoupled kinematics[END_REF], shown in Fig. 1.

The basic Quadrupteron or Isoglide4, which are very similar, consists of four identical legs. Each leg has a base-mounted actuator, allowing translation along a fixed direction, and a planar chain. In these basic robots, linear actuators are employed and the displacements of three of them are directly proportional to the translational displacements of the mobile platform along a given Cartesian axis. The orientation of the end-effector is obtained by a scissors-like motion of the actuators. However, as we recently witnessed with the commercialization of the Quattro robot by Adept Technology [START_REF] Pierrot | Optimal Design of a 4-dof Parallel Manipulator: From Academia to Industry[END_REF], the only way to compete the hugely successful Delta pick-and-place robot [START_REF] Bonev | Delta Robotthe Story of Success[END_REF] is to offer an even faster design. Hence, it would have been great if we could build a Quadrupteron or an Isoglide4 with an amplification factor. Not only would this robot be isotropic, but it may move several times faster than its linear actuators. This paper is the first to provide such a solution through the use of pantographs. Of course, the proposed design is more complicated than the simple Quadrupteron or Isoglide4 of Fig. 1, but this seems to be a reasonable price to pay. Moreover, the new robot is only made of three identical legs, in contrast to other 3T1R decoupled parallel robots, which is a great advantage in terms of workspace volume and acceleration capacities. Indeed, the proposed design is the result of a large study on the synthesis of parallel manipulators using pantographs [START_REF] Briot | PAMINSA: a New Family of Decoupled Parallel Manipulators[END_REF], [START_REF] Briot | Pantopteron: a New Fully-Decoupled 3-DOF Translational Parallel Robot for Pick-and-Place Applications[END_REF]. One such manipulator was already successfully built and proved the viability of using pantographs [START_REF] Briot | Design and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipulator with High-Load Carrying Capacity[END_REF].

The paper is organized as follows. Next, the kinematics of the proposed design, named the Pantopteron-4, is presented. The structure is described, its mobility analyzed, and its singularities described. Then, its workspace is studied and various design considerations are given. Finally, conclusions are drawn.

II. KINEMATIC ANALYSIS

A. Description of the architecture

The architecture of the manipulator is illustrated in Fig. 2. It is composed of three legs which correspond to pantograph linkages (Fig. 3).

The pantograph is a mechanical system with two input points A i and B i and one output point C i (in the remainder of this paper, i = 1, 2, 3). These input points linearly control the displacement of the output point C i . A kinematic analysis shows that a linear actuator connected with input point B i controls the vertical displacement of the output point C i and one other linear actuator with an axis parallel to a 1i controls the displacements along the same axis. Note that these motions are completely decoupled, i.e., they can be carried out independently. The input/output relationships for displacements are linear and are determined by the magnification factor k of the pantograph (k = A i C i /A i B i ). These properties of the pantograph mechanism are used in the Pantopteron-4 manipulator.

For the Pantopteron-4, the actuators which allow the translational displacements are located at the prismatic joints 1i (Fig. 3), and the actuator that controls the orientation of the platform is located at the revolute joint 10,3. The directions of the prismatic joints 1i are orthogonal. All other joints are passive.

Each pantograph linkage is attached to the platform at point C i via a Cardan joint, the axes of each joint 12i being orthogonal. They are also connected to actuators 1i via a revolute joint, which allows the leg to have five DOFs: three translations and two rotations about the axes of the Cardan joint located at C i . The platform of the mechanism is not rigid, but made of two elements connected via a revolute joint. Such an architecture allows the manipulator to have four decoupled DOF. This will be now proved.

B. Mobility analysis

Let x, y, z be the axes of the base frame (Fig. 2) and a 1i , a 2i and a 3i the local frame attached to leg i (Fig. 3). As mentioned in the previous paragraph, when all actuators are disconnected, each leg of the mechanism has five passive DOFs, three translations and two rotations (one about the axis of joint 11,1 and another about the axis of joint 12,1). Therefore, each leg applies one wrench on the platform that constrains its displacements. This wrench is the reciprocal screw to the twists of each passive displacement of the platform. We denote as ) (i j e (j = 1 to 5) the unit screw corresponding to one of the passive displacements of the leg j. For leg 1, these screws, expressed in the base frame at point C 1 , can be written as [START_REF] Lee | Cartesian Parallel Manipulators with Pseudoplanar Limbs[END_REF]:

-for the translations along x, y and z,

  T 0 0 1 0 0 0 ) 1 ( 1  e ,   T 0 1 0 0 0 0 ) 1 ( 2  e and   T 1 0 0 0 0 0 ) 1 (
  T 0 0 0 sin sin sin cos cos 1 1 1 1 1 ) 1 ( 4       e and   T 0 0 0 cos sin 0 1 1 ) 1 ( 5     e ,
where  1 is the angle between the a 11 axis and the y axis, and  1 represents the angle between vector a 31 and the axis of joint 12,1.

The Plücker coordinates of the unit screws can be described in matrix E 1 as Similarly, it is possible to find that the wrenches r 1i transmitted to the platform by the legs when all actuators are disconnected are all pure moments. Let Q be the matrix composed of these wrenches applied on the platform by the legs. The expression of Q in the base frame, and expressed at point O, is:

                                   0 0 0 cos sin 0 0 0 0 sin sin sin os cos 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 ) 1 ( 5 ) 1 ( 4 ) 1 ( 3 ) 1 ( 2 ) 1 ( 1        c
The expressions of wrenches r 12 and r 13 can be obtained using approaches similar to the previous one.

Analyzing the condition of orthogonality on the axes of joints 12i, it could be proven that angles  i are constrained to be equal to 0. Therefore, these terms disappear from Eq. ( 4), which becomes

           0 0 0 0 sin cos 0 0 0 cos 0 sin 0 0 0 sin cos 0 3 3 2 2 1 1       Q (5)
Because the platform is not rigid (it is composed of two elements linked by a passive revolute joint whose axis is vertical), matrix R, which is composed of the wrenches transmitted through the platform to the element pl 2 (Fig. 2) can be written under the form:

           0 0 0 0 sin cos 0 0 0 0 0 sin 0 0 0 0 cos 0 3 3 2 1     R (6)
The twists defining the passive displacements of the platform are orthogonal to this matrix of rank equal to 2. In the general case, there are four independent passive displacements, which are the three translations about the x, y and z axes and one rotation about the z axis. Thus, the platform is constrained by the legs to have only Schoenflies motions.

Let us now consider that the actuator M 1 located at joint 1,1 is fixed. Due to the decoupling properties of the pantograph linkages, the position of point C 1 along the x axis is fixed. Thus, the platform has now two passive translational DOFs, which are orthogonal to the x axis, and still one rotational DOF.

Therefore, a supplementary constraint is applied on the platform, which restrains its displacement.

Using an approach similar to the previous one, the second wrench applied by the leg on the platform,

expressed at point C 1 , is   T 0 0 1 0 0 0  21 r
.

By a similar analysis, is can be seen that, when the three legs are connected to the platform and the actuators M 1 , M 2 , M 3 are fixed, six wrenches (r 11 , r 21 , r 12 , r 22 , r 13 , r 23 ) are applied on the platform. Finally, let us now consider that actuator M 4 located at joint 10,3 is fixed. Due to the decoupling properties of the pantograph linkages, the position of point C 3 along the a 23 axis is fixed. Therefore, a supplementary constraint is applied on the platform, which restrains its displacement. This supplementary wrench applied by leg 3 on the platform, expressed at point

C 3 , is   T 0 cos sin 0 0 0 3 3     33 r .
Let us denote by S the matrix composed of seven wrenches applied on the platform by the legs. The expression of S in the base frame, and expressed at point P, is:

                                                  0 cos sin 0 0 1 0 0 0 0 0 0 0 sin cos 0 1 0 0 0 0 0 cos 0 sin 0 0 1 0 0 0 0 sin cos 0
with

Ci PCi x x x   , Ci PCi y y y   , Ci PCi z z z  
, (x, y and z are the coordinates of point P of the platform along x, y and z axes respectively and x Ci , y Ci and z Ci are the coordinates of point C i of the platform along x, y and z axes) and

3 3 3 3 63 sin cos   PC PC y x s   .
Because the platform is not rigid, matrix T, which is composed of the wrenches transmitted through the platform to the element pl 2 can be written under the form:

                          0 cos sin 0 0 1 0 0 0 0 0 0 0 sin cos 0 1 0 0 0 0 0 0 0 0 sin 0 0 1 0 0 0 0 0 0 cos 0 3 3 63 3 3 3 3 2 2 1 1       s y x z z PC PC PC PC T (8)
Now, let us analyze the passive displacement of the platform when actuators M 1 , M 2 or M 3 are disconnected. Without loss of generality, let us consider that actuator M 3 is disconnected. Thus, the manipulator gains one passive DOF. The twist corresponding to this passive DOF is the screw t 1 which is orthogonal to the six wrenches applied on the element pl 2 ,

  T z y x z y x v v v     1 t (9)
where  x ,  y , and  z correspond to the rotational velocities of the platform about x, y and z axes, and v x , v y and v z to its translational velocities along x, y and z axes. If t 1 is a passive motion, the following relation must hold: [START_REF] Gogu | Fully-Isotropic Parallel Manipulators with Schoenflies Motions and Complex Legs with Rhombus Loops[END_REF] From Eq. ( 10), it is quite trivial to find the expression of t 1 :

0 t 1                        0 cos sin 0 0 0 0 0 0 sin cos 0 1 0 0 0 0 0 0 0 0 sin 0 0 1 0 0 0 0 0 0 cos 0 3 3 63 3 3 2 2 1 1       s z z PC PC
  T 1 0 0 0 0 0  1 t (11)
Thus, throughout the workspace of the mechanism, the permitted passive motion of the platform when actuator M 3 is disconnected is a free translation along the z axis. Thus, actuator M 3 controls the translation of the platform along the z axis. Moreover, as the axis of actuator M 3 is also directed along the z axis, it comes that, due to the copying properties of the pantograph linkage, a displacement of actuator M 3 is transformed on a displacement of the platform along the same direction, but amplified by the pantograph linkage.

By similar analyses, it could be proven that actuator M 1 (resp. M 2 ) controls the translation of the platform along the x axis (resp. the y axis). Moreover, a displacement of actuator M 1 (resp. M 2 ) is transformed into a displacement of the platform along the same direction, but amplified by the pantograph linkage.

Thus, the input-output relations for the translational displacements of this manipulator are linear. Let us now analyze the permitted displacement when actuator M 4 is disconnected. In such a case, the passive twist t 2 of the platform pl 2 can be found via the equation:

0 t 2                        1 0 0 0 0 0 0 0 sin cos 0 1 0 0 0 0 0 0 0 0 sin 0 0 1 0 0 0 0 0 0 cos 0 3 3 3 3 2 2 1 1 PC PC PC PC y x z z     (12)
From Eq. ( 12), it is quite trivial to find the expression of t 2 :

  T 0 0 0 1 0 0  2 t ( 13
)
Throughout the workspace of the mechanism, the permitted passive motion of the platform when actuator M 4 is disconnected is a free rotation around the z axis. Thus, actuator M 4 controls the rotation of the platform along the z axis.

Thus, the input-output relations for this manipulator are decoupled, and it belongs to the family of the decoupled 3T1R parallel mechanisms.

C. Geometric and kinematic models

The origin O of the base frame is fixed such that it coincides with point P of the platform when all linear actuators have zero length. It is also considered that an increasing actuator's length displaces the platform along the positive part of the corresponding base frame axis. Therefore, taking into account that

T i i i T i c b a z y x ] , , [ ] , , [   OC
, the following trivial system of decoupled linear equations governs the translational movements of the Pantopteron-4:

  1 1 1 a x k x G     (14)   2 2 2 b y k y G     (15)
  3 3 3 c z k z G     ( 16
)
where k is the magnification factor of the pantograph linkages,  i is the length of actuator i, x G1 , y G2 and z G3 are coordinates of points G i of the platform along x, y and z axes respectively and a 1 , b 2 and c 3 are constant terms defining the shape of the platform.

One additional relationship can be derived to define the orientation of the platform, which can be found from the following loop-closure equation:

3 3 3 3 C G OG OC   . (17)
Developing and simplifying, one can find:

    3 3 3 3 sin cos cos sin 0     G G x r x y r y       ( 18
)
where r is the length of element pl 2 . Equation ( 18) leads to:

              3 3 1 3 cos sin tan G G x r x y r y    (19)
Since k ≠ 0, the above system of independent equations can be easily inverted to give the two solutions to the inverse kinematics of the Pantopteron-4.

Differentiating Eqs. ( 14), ( 15), ( 16) and ( 18) leads to:

0 q B v A    , (20)
where

  T z y x      , , ,  v ,   T 3 3 2 1 , , ,           q and                  3 3 3 cos 0 cos sin 0 1 0 0 0 0 1 0 0 0 0 1     r A (21)                  3 0 0 0 0 0 0 0 0 0 0 0 0 l k k k B ( 22
)
with

    2 3 2 3 3 sin cos G G y r y x r x l         .
Thus, one can define the Jacobian matrix J of the Pantopteron-4 by:

                            3 3 3 3 3 3 1 cos 0 cos cos cos sin 0 0 0 0 0 0 0 0 0         r l r k r k k k k B A J (23)
Recall that the three first diagonal terms of the Jacobian matrix of the Isoglide4 or the Quadrupteron are equal to 1. Therefore, the Pantopteron-4 displaces k times faster than the Isoglide4 or the Quadrupteron (where k is obviously greater than 1). Moreover, the use of three legs in the Pantopteron-4 instead of four in the other robots allows enlarging the workspace of the mechanism and improves its acceleration capacities. It is also clear that due to this property, and to the greater number of joints in comparison with the Tripteron, the accuracy of the proposed robot will be lower. However, the purpose of this robot is not to be more accurate, but to be much faster.

D. Singularity analysis

In this section, we analyze the singularities of the Pantopteron-4. It will be shown that the robot may have Type 1 and 2 singular configurations, as well as constraint singularities. However, as it will be presented later, the manipulator may be designed in such a way that its workspace does not contain any singularities.

Type 1 singularities.

Analyzing matrix B, it can be found that Type 1 singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] appear when l 3 = 0, which implies that points A 3 , B 3 and C 3 are aligned along the same axis (Fig. 4). In such a case, given one position of the platform, there are infinitely many orientations for the pantograph linkage. Other kinds of Type 1 singularities occurring in the mechanism are due to the degeneracy of the kinematics of the pantograph legs. Such singularities appear when:

-the parallelograms B i D i E i F i degenerates into a line; near such case of singularity, the efforts in the revolute joints located at E i , F i , D i and B i grow considerably, so it has to be avoided by limiting the angle between links (A i E i ) and (E i C i );

-points A i , B i and C i of any leg are collinear (Fig. 4); in such a case, given one position of the platform, there are infinitely many orientations for the pantograph linkage. Moreover, if during a displacement of the mechanism, a leg comes close to this singularity, the angular velocity of the pantograph linkage around the axis defined by segment (G i B i ) becomes very high. Therefore, the neighbourhood of such configurations should be avoided by limiting the displacement of joint 9i.

These two kinds of singularity define the boundaries of the workspace. They are similar to the singular configurations present in the Quadrupteron of Isoglide4.

Type 2 singularities.

Type 2 singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] of the mechanism are also quite simple to analyze. They appear when  =  3 + /2. In such a case, the wrench r 33 is directed along the direction of the platform pl 2 (Fig. 5). This is the reason why, when moving actuator M 4 , rotations of the platform around the vertical axis are impossible. On the other hand, fixing the position of actuator M 4 , the platform can encounter a small rotation around point P.

Introducing  =  3 + /2 into Eq. ( 18) leads to:

  r x y x y G G          cos sin cos sin 0 3 3 . ( 24
)
Thus, fixing the orientation  of the platform, for any altitude z, the singularity loci are defined by a straight line in the horizontal plane Oxy (or a vertical plane in 3D).

Constraint singularities.

Other cases of singularities appear if the system of wrenches applied on the platform degenerates. The degeneracy of the system of wrenches can be analyzed using the Newton-Euler theorem.

f f f f . ( 25b
)
where f ij (i = 1, 2, 3, j = 1, 2) are the norms of vector f ij (Fig. 6).

Fig. 6. Constraints applied to the platform of the Pantopteron-4.

Rewriting this system of equations into matrix form yields:

    T T T f f f f f f f p p p p p 6 1 0 f M   
where

                      
            r M , (28)
For  ≠  3 + /2, the mechanism is in a constraint singularity if and only if:

0 sin sin sin cos cos cos 3 2 1 3 2 1          h (29)
In such a case, the three moments r 1i applied to the platform are linearly dependant, i.e., their axes are parallel or coplanar. Thus, the platform becomes unconstrained and it gains one supplementary DOF. Let us study the example presented on Fig. 7. Axis a 11 is parallel to the y axis and axes a 12 and a 13 are parallel to the x axis. Thus, the DOF gained by the platform is a rotation about an axis parallel to the z axis. Expressing Eq. ( 29) in the Cartesian space yields:

0 ) ( ) ( ) ( ) ( ) ( ) ( 3 2 1 3 3 2 2 1 1 3 2 1 3 3 2 2 1 1                G C G C G C G C G C G C y y x x z z x x z z y y h ( 30
)
where

2 1 1 2 1 1 1 ) ( ) ( G C G C z z y y      , ( 31a
) 2 2 2 2 2 2 2 ) ( ) ( G C G C x x z z      , ( 31b
) 2 3 3 2 3 3 3 ) ( ) ( G C G C y y x x      . ( 31c
)
In these expressions, x Ci , y Ci , z Ci , x Gi , y Gi , z Gi correspond to the coordinates of points C i and G i about the

x, y and z axes, respectively. Disregarding the case where  i tends to infinity, singularities appear when:

0 ) ( ) ( ) ( ) ( ) ( ) ( 3 3 2 2 1 1 3 3 2 2 1 1         G C G C G C G C G C G C y y x x z z x x z z y y ( 32
)
Taking into account that the terms x Gi , y Gi , z Gi appearing in (32) are constant and that

T i i i T i c b a z y x ] , , [ ] , , [   OC ( 33
)
where a i , b i , c i are either constants (for i = 1, 2) or variables depending on angle , for i = 3.

Equation (32) can be rewritten under the form:

0 8 7 6 5 4 3 2 1         p z p y p x p z y p z x p y x p z y x p ( 34
)
where coefficients p i are terms depending on the angle , the position of points G i and of the shape of the platform. Fixing the altitude z of the platform, Eq. ( 34) is the expression of a hyperbola, of which the coefficients depend on the altitude of the platform, on its orientation and on the geometric parameters of the mechanism.

Thus, contrary to the Isoglide4 or the Quadrupteron, our mechanism has constraints singularities. This is due to the fact that some legs of the Isoglide4 or of the Quadrupteron are attached to the platform by a revolute joint, instead of a Cardan joint, which overconstrains the displacement of the platform and allows avoiding such singular configurations. However, it will be shown in the following section that, even if the Pantopteron-4 has singularities, they can be easily removed from its workspace.

III. DESIGN CONSIDERATIONS

In this part, we will perform the analysis of the workspace of the mechanism, taking into account the geometric limitations and singular configurations, and discuss some other possible architectures based on this mechanism.

A. Geometric workspace analysis

Many parameters influence the size of the workspace of the Pantopteron-4. Among the main parameters, we can mention:

-the lengths of the links of the pantograph;

-on the locations of the axes of the base-mounted revolute joints;

-the shape of the platform;

-the maximal stroke of the actuators and of the passive linear guide;

-the interference between the links.

Using a geometrical approach, we will compute the workspace of the Pantopteron-4. As the Pantopteron-4 is a 3T1R parallel mechanism, its workspace for a given orientation of the platform can be found as the intersection of three so-called vertex spaces.

Analyzing the vertex space of the leg i, it only depends on:

-the lengths of the links of the pantograph;

-the maximal and minimal strokes of the actuators and of the passive linear guide;

-the interferences between the links;

-the singular configurations.

In a first step, let us concentrate on the boundaries of the workspace due to the interference of the links and of the singular configurations. As mentioned previously, for a leg, there are two types of singularities:

a. when the parallelogram B i D i E i F i degenerates into a line; such a singularity can be avoided by limiting the angle  i between the links (A i E i ) and (E i C i ) of the parallelogram, which, in the same time, allows limiting some inferences between the links. The maximal and minimal angles will be denoted ( i ) max and ( i ) min , respectively. b. when points A i , B i and C i are aligned along the same axis; such a case can easily be avoided by limiting the stroke of the passive prismatic joint 9i. This minimal stroke will be denoted (s i ) min .

To avoid interference between the links and the base, a maximal stroke of the actuator has to be fixed

at ( i ) max .
Each leg is mounted in rotation around one axis parallel to a 3i . Thus, the problem of finding the vertex space can be limited to a planar analysis of the minimal and maximal displacements of point C i , the entire vertex space being found by symmetry of revolution of these displacements. Cases (b) and (c) are much simpler to analyze. The displacement of point C i when the passive guide (9i) is at its minimal stroke (s i ) min is a vertical line L 1 located at (k-1) times the distance (s i ) min from the vertical axis (G i B i ) (Fig. 9(a)). The displacement of point C i when the actuator M i is at its maximal stroke ( i ) max is a horizontal line L 2 located at k times the distance between the maximal position of point B i and the position of point A i along the axis a 3i , from the axis of the horizontal passive joint 9i (Fig. 9(a)).

The entire vertex space is represented at Fig. 9(b). On all of these figures, two boundaries due to two constraints, which are the maximal strokes of the actuated and passive linear joints, are not represented.

These boundaries are vertical and horizontal straight lines. However, in a first step, it is preferable to have the largest vertex space for the legs and, thus, to remove these two boundaries from our workspace by a proper selection of the stroke of the linear guides.

We can implement in Matlab our geometric method in order to be able to optimize the workspace of the Pantopteron-4 by minimizing the lengths of the pantograph's links in each leg. This could be done more promptly in a commercial CAD system, such as CATIA [START_REF] Bonev | A Geometrical Method for Computing the Constant-Orientation Workspace of 6-PRRS Parallel Manipulators[END_REF]. However, it would obviously be a mistake to design a 3T1R parallel mechanism with such a complex workspace. Thus, our decision is to keep the links as long as it takes, so that the workspace of the mechanism becomes a simple geometric form, namely a rectangular parallelepiped. In other words, the workspace of a Pantopteron-4 with sufficiently long legs has to become a box whose sides are of length k  i ( i being the stroke of actuator M i , i = 1, 2, 3), as shown in Fig. 10(b) (see the example in the next section).

In order to obtain such a simple volume, when the three vertex spaces are intersected, it is the planar caps that limit the workspace and not the other surfaces. Of course, we still try to minimize the length of the links, by carefully locating the prismatic actuators on the base and properly choosing the dimensions of the mobile platform and of the stroke of the actuators. Furthermore, if the workspace of the mechanism has to be a parallelepiped, the shape of the vertex space should not be too complicated, and can be reduced to a hollow cylinder (Fig. 9(c)). This can be accomplished by properly constraining the maximal stroke of the active and passive linear guides in order to obtain, in the planar projection of the workspace, a rectangle denoted as the useful vertex space (two possible examples of the useful vertex space are presented in Fig. 9(a)).

The workspace volume of the Pantopteron-4 is the other main advantage of the proposed robot.

Indeed, the maximal volume of the workspace of the Quadrupteron or Isoglide4 is V =  1  2  3 while that of the Pantopteron-4 is V = k 3  1  2  3 , i.e., for the same set of given actuators, the workspace of the Pantopteron-4 is k 3 times bigger than that of the other robots.

Moreover, it is well known that the actuators represent a major portion of the cost of a robot. For creating a fast mechanism with actuated prismatic joint, it is preferable to use linear motors that reach higher velocities. However, the main drawback of such actuators is their price, which is directly proportional to the length of their stroke. For a given maximal workspace, the stroke of the actuators of the Quadrupteron or Isoglide4 is k times greater than that of the motors of the Pantopteron-4. Therefore, even if the Pantopteron-4 is more complicated to design than a Quadrupteron or an Isoglide4, its manufacturing cost would likely be lower.

B. Singularity-free workspace

It is impossible to speak about the workspace of a parallel mechanism without dealing with singularities. As observed from Eq. (34), the constraint singularities depend on the position of the mobile platform, on the locations of the axes of the base-mounted revolute joints, and on the shape of the platform. Thus, analyzing Eq. ( 34), there are nine design parameters which are y G1 , z G1 , x G2 , z G2 , x G3 , y G3 , a 1 , b 2 , and c 3 (we do not consider the lengths of the links of the pantograph linkages as they do not influence these singular configurations). So, there are too many parameters for a complete analysis of the singular configurations. Therefore, we will restrict our analysis to some particular designs.

We will consider in this part a mechanism which has a platform with two concurrent axes (for example the ones of joints 12,1 and 12,2), and a base whose three pantograph axes of rotations are also concurrent. Therefore, considering that the intersection point of the pantograph axes is the origin of the base frame, and that point P is at the intersection of the two axes of the platform, only c 3 (= r) stay variable, the other ten parameters being equal to zero.

In such a case, Eq. (34) becomes:

  0 sin cos 2      r x r y y x z (37)
Thus, singular configurations will appear if the platform of the mechanism is located in the plane P 1 (z = 0), or if it is located on a hyperbola H whose expression is:

0 sin cos 2      r x r y y x ( 37
)
Please note that this expression does not depend on the altitude z of the platform. It is well known that such a hyperbola has two asymptotes,

/ ) cos (  r x   (39) 2 / ) sin (  r y   2
which, in 3D, represent two planes which we will denote by P 2 () and P 3 (). These planes, projected in the horizontal plane P 1 , are represented in Fig. 11a for several values of angle .

Let us now take into account the Type 2 singular configurations. These singularities are described by Eq. ( 24) and are represented on Fig. 11a Thus, for any angle , the singularity-free workspace does not exist. This is due to the fact that, if we represent the all planes P 2 () and P 3 () and all lines L() for any angle  between 0° and 360° on the same figure, the entire workspace will be filled with singularities. However, this major drawback can be easily suppressed by limiting the possible rotation of the platform in an interval of 90°. The workspace with the boundaries of the singularity loci for   [0, 90°] is represented on Fig. 11b. It is clear on this picture that the regions IV and VIII are completely free of singular configurations. Obviously, several applications need a rotation superior to 90°. However, such a drawback can be compensated by using an amplification device, such as the one presented on Fig. 12, in order to transform the limited rotation of the platform into large rotations of the end-effector. This system is composed of two gears, one fixed on the rotating link pl 2 of the platform, the other on the orientation-fixed part pl 1 . The manipulated device will be located on the smallest gear that will permit the amplification of the rotation of the link pl 2 . A pulley belt mechanism may also be used instead of gears.

A possible version of a prototype of a Pantopteron-4 is represented at Fig. 13. Its geometric parameters are:

-l AiEi = 0.2 m, l EiCi = 0.

C. Other possible architectures

Finally, we would like to mention that the design of the Pantopteron-4 presented here is not the only solution for creating such a mechanism. First, as the leg is made up of a pantograph linkage, several designs are possible, which are presented in [START_REF] Lu | Synthesis of Planar Five-Bar Pantograph Configurations by a Geometric Method[END_REF]. However, we believe that the architecture we proposed is the most practical one. Moreover, note that the planar RP chain composed of the revolute joint 10i and the prismatic joint 9i may be removed and replaced by any kinematic chain able to perform a planar displacement, such as planar RRR, RPR, PPR or PRR chains (Fig. 14). Using such chains, points H i and G i need not be aligned. However, such changes in the design will lead to different singular configurations.

Fig. 1 .

 1 Fig. 1. Some partially decoupled 3T1R parallel manipulators.

Fig. 2 .

 2 Fig. 2. Schematics of the Pantopteron-4 manipulator.

Fig. 3 .

 3 Fig. 3. Schematics of one leg of the Pantopteron-4 (i = 1, 2, 3).

  the platform by the leg 1 is orthogonal to the twists composing the lines of matrix is a wrench of zero pitch (a pure moment).

Fig. 4 .

 4 Fig. 4. Example of Type 1 singularities.

Fig. 5 .

 5 Fig. 5. Example of Type 2 singularity.

Figure 6

 6 Figure 6 represents the forces applied to the platform by the legs. Let us suppose that a wrench f is applied on the platform pl 2 at point P. Let us also denote by p = [p 1 , p 2 , p 3 , p 4 , p 5 , 0] T the reaction

  are constraint singularities if matrix M degenerates, i.e., if:

Fig. 7 .

 7 Fig. 7. Example of a constraint singularity.

Fig. 9 .

 9 Fig. 9. Schematics of the vertex space of a leg from the Pantopteron-4.

Figure 10 (

 10 a) shows an example of the workspace of a Pantopteron-4 with relatively short legs. We can obtain the best ratio between the lengths of the links and the volume of the workspace. A relatively large increase of the link lengths will result in only a negligible gain in the workspace volume. (a) with relatively short legs. (b) with relatively long legs.

Fig. 10 .

 10 Fig. 10. Orientation workspace of the Pantopteron-4.

  by the lines denoted L(). It is also possible to represent the curve tangent to all these lines, which represents the workspace without Type 2 singular configurations: it is the circle C of radius r, centred in O. (a) Singularities in the workspace of the Pantopteron-4 for any angle  (planar projection) (b) Singularities in the workspace of the Pantopteron-4 for   [0, 90°] (3D).

Fig. 11 .

 11 Fig. 11. Singularity free workspaces.

Fig. 12 .

 12 Fig. 12. Amplification device for the rotation of the platform.

  3 m, k = 3; -y G1 = z G1 = x G2 = z G2 = x G3 = y G3 = 0 m, a 1 = b 2 = 0 m, r = 0.05 m; -actuator strokes = 0.06 m ((z i ) min = -0.22 m, (z i ) max = -0.16 m) -passive linear guide strokes = 0.14 m ((s i ) min = 0.01 m, (s i ) max = 0.15 m) ;-( i ) min = 25°, ( i ) max = 155°.Its design is achieved such that its workspace is a cube whose side is equal to 0.18 m for any value of angle   [0, 90°].

Fig. 13 .

 13 Fig. 13. CAD model of a possible prototype of Pantopteron-4.

  (a) the RP chain is replaced by a RPR chain. (b) the RP chain is replaced by a RRR chain. (c) the RP chain is replaced by a PRR chain. (d) the RP chain is replaced by a PPR chain.

Fig. 14 .

 14 Fig. 14. Other possible legs for the Pantopteron-4.

Considering case (a), we have to find the boundaries of the leg when angle  i is fixed. Fixing angle  i is equivalent to fixing the lengths of segments (A i B i ) and (A i C i ). These lengths are equal to:

(36) Fig. 8. Displacement of C i when  i is fixed.

Displacing the prismatic guides, segments (A i B i ) and (B i C i ) describe Cardanic motions [START_REF] Sekulie | Method of Synthesis of Cardanic Motion[END_REF]- [START_REF] Tischler | A Spatial Extension of Cardanic Movement: Its Geometry and Some Derived Mechanisms[END_REF]. As a result, for a given angle  i , the displacement locus of point C i is an ellipse E (Fig. 8). Thus, considering the extremes ( i ) max and ( i ) min of angle  i , the boundaries of the workspace are given by the ellipses E min and E max (Fig. 9(a)).

IV. CONCLUSIONS

In this paper, a novel 4-DOF decoupled 3T1R parallel mechanism, named the Pantopteron-4, was presented. The Pantopteron-4 end-effector displaces k times faster than its linear motors (k being the magnification factor of the pantograph linkages). Moreover, for a given set of actuators, its workspace is k 3 times bigger than the stroke of its actuators. Though the mechanism proposed has several singular configurations, it is easy to choose proper design parameters that lead to a large singularity-free workspace. This novel mechanism is foreseen to be used in applications where the velocities and accelerations have to be high, such as in pick-and-place.