Georges Pagis
email: georges.pagis@ifma.fr

Nicolas Bouton
email: nicolas.bouton@ifma.fr

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

Philippe Martinet
email: philippe.martinet@irccyn.ec-nantes.fr

Design of a Controller for Enlarging Parallel Robots Workspace through Type 2 Singularity Crossing

In order to increase the workspace size of parallel robots (largely reduced by the presence of singularities) several solutions have been proposed. One promising solution consists in the definition of optimal trajectories that ensure the non degeneracy of the dynamic model in the singularity and therefore are able to cross the Type 2 singularities. Those works are based on the computation of the optimal trajectories and assume that the robot can perfectly track the desired trajectory. Nevertheless, this assumption cannot be verified in reality due to modelling errors which largely impact the control law used to follow the desired trajectory. Therefore, if the optimal trajectory is not perfectly tracked, the dynamic model can degenerate near the Type 2 singularities and the robot might stay blocked.

In order to solve that problem, this paper proposes a multimodel approach that allows parallel robots to cross the Type 2 singularities without any torque discontinuity. The main idea is to shift near singularities from the full robot dynamic model to another simplified one that can never degenerate. The proposed control law is then coupled with an optimal trajectory planning methodology that makes the singularity crossing more robust to modelling errors. The proposed approach is validated experimentally on a prototype of Five-bar planar parallel mechanism.

I. INTRODUCTION

Parallel manipulators have many advantages in terms of acceleration capacities and payload-to-weight ratio, but one of their main drawbacks concerns the presence of singularities [START_REF] Arakelian | Increase of singularityfree zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF], [START_REF] Conconi | A new assessment of singularities of parallel kinematic chains[END_REF], [START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF] which divide the workspace into different aspects, each aspect corresponding to one (or more) assembly mode [START_REF] Merlet | Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data. and Automation[END_REF]. For a global overview of the singularity problem, the reader is referred to [START_REF] Conconi | A new assessment of singularities of parallel kinematic chains[END_REF].

Type 2 (or parallel) singularities [START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF] are probably the most constraining singularities of parallel manipulators. In those singularities, one (or more) manipulator's degree of freedom becomes uncontrollable. Moreover, Type 2 singularities divide the manipulator's workspace in different aspects, resulting in the decrease of the reachable workspace size. Therefore, several approaches have been envisaged in the literature in order to increase the workspace size such as:

• The design of parallel robots without singularities [START_REF] Briot | Optimal technologyoriented design of parallel robots for high-speed machining applications[END_REF], [START_REF] Gogu | Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations[END_REF]. • The use of redundancy [START_REF] Nahon | Force optimization in redundantly actuated closed kinematic chains[END_REF], [START_REF] Kurtz | Multiple-goal kinematic optimization of a parallel spherical mechanism with actuator redundancy[END_REF] or of mechanisms with variable actuation modes [START_REF] Arakelian | Increase of singularityfree zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF], [START_REF] Rakotomanga | Kinetostatic performance of a planar parallel mechanism with variable actuation[END_REF]. • Planning assembly mode changing trajectories by either by-passing a cusp point [START_REF] Zein | Non singular assemblymode changing motions for 3-RPR parallel manipulators[END_REF] or directly going through a Type 2 singularity [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF], [START_REF] Ider | Inverse dynamics of parallel manipulators in the presence of drive singularities[END_REF]. In [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF], a physical criterion, obtained through the analysis of the dynamic model, is presented. Based on the mentioned criterion, it is possible to plane a singularity crossing trajectory which (i) avoids the degeneracy of the dynamic model (i.e. obtaining infinite joint reactions during the motion) and (ii) allows the robot to cross the singularity.

This last solution is promising since it can considerably increase the workspace size of any parallel mechanism. However, in previous works, it has been considered that the controller allowed the mechanism to perfectly track the desired trajectory. This is obviously impossible due to modelling uncertainties. In order to fill this gap, the aim of this paper is to propose an advanced control law dedicated to the Type 2 singularity crossing.

To correctly track trajectories, several control approaches can be used [START_REF] Khalil | Modeling, identification & control of robots[END_REF], but the most effective is probably the Computed Torque Control (CTC) law [START_REF] Khalil | Modeling, identification & control of robots[END_REF], [START_REF] Craig | Introduction to Robotics[END_REF], [START_REF] Spong | Robot dynamics and control[END_REF]. More precisely, this control law permits to compute the actuators torque of the robot based on its dynamic model and the desired trajectory to follow. Obviously, CTC is sensitive to the robot dynamic model that (i) must be well identified [START_REF] Gautier | Dynamic identification of robots with power model[END_REF], [START_REF] Briot | Global Identification of Drive Gains and Dynamic Parameters of Parallel Robots[END_REF] and (ii) must not degenerate near singularities, even if the trajectory does not perfectly respect the physical criterion mentioned above.

Therefore, in order to avoid the dynamic model degeneracy near the singularity, a multi-model CTC (e.g. see [START_REF] Craig | Introduction to Robotics[END_REF], [START_REF] Spong | Robot dynamics and control[END_REF]) is proposed in the present paper. This controller is combined with the definition of new singularity-crossing criteria developed for increasing the robustness of the control law along the desired trajectory.

II. TRAJECTORY GENERATION FOR CROSSING A TYPE 2 SINGULARITY

A. Dynamic modelling of parallel mechanisms

This section will briefly recall the dynamic equations of a parallel manipulator composed of m links, n degrees of freedom (dof ) and driven by n actuators. The position and the speed of the manipulator can be fully described using:

• q = [q 1 , q 2 , ..., q n ] T and q = [ q1 , q2 , ..., qn ] T that represent respectively the vector of active joints variables and active joints velocities, • x = [x, y, z, φ , ψ, θ ] T and v = ẋ, ẏ, ż, φ , ψ, θ T that are the end-effector pose parameters and their derivatives with respect to time, respectively. x, y and z represent the position of the platform controlled point and φ , ψ and θ the orientation parameters of the platform about three axes a φ , a ψ and a θ .

Using the Lagrangian formalism, the mechanism's dynamic model can be written as:

τ = w b + B T λ , (1)
w p = A T λ (2)
where • τ is the vector of the input efforts, • λ is the vector of the Lagrange multipliers,

• A and B are two matrices deduced from the mechanism loop-closure equations, such that Av = B q [4],

• w b and w p are related to the Lagrangian L of the system by:

w b = d dt ∂ L ∂ q - ∂ L ∂ q , w p = d dt ∂ L ∂ v - ∂ L ∂ x (3)
In this expression, w p is the wrench applied to the platform by the legs and external forces [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF].

Then, assuming that the matrix A can be inverted, by substituting (2) into (1) the general dynamic model of parallel manipulators is obtained [START_REF] Khalil | Modeling, identification & control of robots[END_REF]:

τ = w b + J T 0 w p , (4)
where • 0 w p is the expression of the wrench w p in the base frame, i.e. 0 w p = Dw p with D the matrix relating the platform twist t (expressed in the base frame) to the vector v by t = Dv [START_REF] Merlet | Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data. and Automation[END_REF],

• J = 0 A -1 B is the Jacobian matrix between the platform twist t and q, with 0 A is the expression of the matrix A in the base frame, i.e. 0 A = AD -1 .

B. Type 2 singularity crossing

Based on the analysis of the kinematic model, the authors of [START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF] proposed a classification of the singularities in three different types: Type 1 or serial singularities occur when the mechanism is in a position such as the kinematic matrix B becomes rank deficient. In such configurations, the mechanism loses the ability to move in one given direction. Type 2 singularities or parallel singularities occur when the kinematic matrix 0 A becomes rank deficient. In Type 2 singularities, one (or more) robot degree of freedom becomes uncontrollable. Such singularities divide the workspace in different aspects, resulting in a reduction of the manipulator's workspace. Moreover, in the presence of such singularities, the robot may also not be able to resist to an external wrench applied on the platform and the reactions in joints grow to infinity. Type 3 singularities are configurations where both Type 1 and Type 2 singular configurations appear at the same time. They are discarded in the following of the paper as they appear if both Types 1 and 2 singularity exist.

Finally parallel mechanism with less than 6 dof can have another type of singularity such as the constraint singularities [START_REF] Zlatanov | Constraint singularities of parallel mechanisms[END_REF], [START_REF] Conconi | A new assessment of singularities of parallel kinematic chains[END_REF].

If a parallel mechanism is in a singular position of Type 2, the matrix A T cannot be inverted in Equation ( 2). The dynamic model degenerates and therefore cannot be solved. However, as explained in the introduction, it has been proven in [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF] that a mechanism can cross a Type 2 singularity without torque discontinuity. Indeed, on a Type 2 singularity, the columns of 0 A are linearly dependent, i.e. there exist a vector t s such that:

0 At s = 0 ⇔ t s T 0 A T = 0 (5)
The vector t s represents the twist of the uncontrollable motion of the platform in the singularity locus [START_REF] Merlet | Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data. and Automation[END_REF]. Thus, multiplying (2) by t T s leads to:

t T s 0 A T λ = 0 (6)
In that case, the following condition must also be satisfied:

t T s 0 w p = 0 (7)
which is the condition for the non-degeneracy of the dynamic model [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF]. As a result, if the desired manipulator motion doesn't guarantee the achievement of a wrench w p that respects the condition ( 7), the dynamic model is degenerated and the manipulator desired input efforts must grow to infinity to produce the desired platform motion. Physically, this condition means that the parallel manipulator can cross the Type 2 singularity if and only if the wrench 0 w p exerted by the legs and external efforts on the platform is reciprocal to the twist t s of the uncontrollable motion in the Type 2 singularity.

C. Generation of a robust trajectory for crossing Type 2 singularity

In order to cross a singularity without torque discontinuity, the mechanism has to follow a trajectory that respects the criterion [START_REF] Figielski | Towards Development of a 2-DOF Planar Parallel Robot with Optimal Workspace Use[END_REF] on the singularity locus. Theoretically, the dynamic model degenerates only on the singularity locus, however numerically the matrix A is singular on a space around the singularity locus. Therefore the criterion [START_REF] Figielski | Towards Development of a 2-DOF Planar Parallel Robot with Optimal Workspace Use[END_REF] has to be respected in this space around the singularity locus in order to prevent the dynamic model from degenerating.

The trajectory generation is achieved using polynomials, which degree can vary. Indeed, in order to guarantee that the criterion ( 7) is respected around the singularity locus, it is proposed in this work to vanish the criterion [START_REF] Figielski | Towards Development of a 2-DOF Planar Parallel Robot with Optimal Workspace Use[END_REF] and n of its derivatives:

t T s d i w p dt i = 0 i = 1, ..., n, (8)
To the best of our knowledge, this is the first time that such criteria are proposed. Generating a trajectory based on these criteria [START_REF] Kurtz | Multiple-goal kinematic optimization of a parallel spherical mechanism with actuator redundancy[END_REF] allows to increase the robustness to model uncertainties and control error around the singularity. Experimental results shows that nullifying the 2 first derivatives of the criterion might be enough.

Next section will present the control law used to allow the singularity crossing. The Computed Torque Control (CTC) [START_REF] Spong | Robot dynamics and control[END_REF] is a control law that computes the input torques that the actuators must apply on the mechanism in order to follow a given trajectory. It is based on the dynamic model presented in Part II. As for any type of control law in robotics, the aim of the controller CTC is to minimize the error in either joint or task space. Since at proximity of a Type 2 singularity the kinematic matrix A is singular, it is not possible to compute the Cartesian velocities from the joint velocities using the DKM (Direct kinematic Model). Sensors being usually measuring joint space values, only the joint space control law should be used to cross Type 2 singularities,.

The control error e to minimize can be expressed as:

e = q d -q ⇒ ė = qd -q ⇒ ë = qd -q (9)
where • q d (respectively qd and qd ) is the desired joint position (respectively velocity and acceleration), • q (respectively q and q) is the current measured joint position (respectively velocity and acceleration). In order to force the error to tend to zero, a second order differential equation on e is imposed by:

ë + K d ė + K p e = 0 (10)
By substituting Eq. ( 9) into Eq. ( 10) one can obtain:

q = qd + K d ė + K p e (11)
Let us rewrite the dynamic model of the mechanism as [START_REF] Spong | Robot dynamics and control[END_REF], [START_REF] Ghorbel | A reduced model for constrained rigid bodies with application to parallel robots[END_REF]:

τ = w b + J T w p = M q + H(q, q) (12)
Consequently, CTC (Fig. ( 1)) computes the input torques by substituting Eq. ( 11) into the dynamic model presented in Eq. ( 12):

τ = M ( qd + K d ė + K p e) + H(q, q) (13)
It should be noted that the vector of positions in the task space x is necessary to compute matrices M and H. However, most mechanisms have sensors measuring the vector of positions in the joint space q. Therefore the DKM is needed.

Unfortunately, when planning a Type 2 singularity crossing trajectory, the mechanism changes its assembly mode, and so the solution of the DKM has to change. To do so, the controller need the information that the mechanism changed its assembly mode which cannot be deduced from the joint positions. Experimentally, the most reliable solution is to choose the DKM solution based on the desired trajectory.

It should be mentioned that, when using the proposed CTC for crossing Type 2 singularities, the trajectory planned for respecting the crossing criteria (cf. Part II) will be different from the real one due to errors in the dynamic model. As a result, speaking numerically, the crossing criteria (7) will never be respected and the computed torque control could send infinite torques to the robot that will prevent the singularity crossing due to the inversion of the matrix A.

Next section presents a multi-model control law allowing the controller to avoid this issue.

B. Multi-model control law

To avoid these numeric issues, the proposed solution is to plan a trajectory respecting around the singularity locus the criterion:

w p = 0 ( 14
)
This new criterion still guarantees that the dynamic criterion t s T w p = 0 is respected. Moreover, the first n derivatives of this new criterion are also nullified. The direction t s being time independent, this guarantees that the first n derivatives of the dynamic criterion are also null as presented in Section II-C. Of course during the real robot displacement, numerically w p will not be null, but such a new criterion allows the implementation of a multi-model control law. The multi-model CTC law presented in this paper consists in using two models :

• Model 1 -The complete dynamic model as long as A is invertible:

τ = w b + J T w p ( 15
)
• Model 2 -A reduced dynamic model that cannot degenerate when the mechanism is close to a singular position:

τ = w b (16)
The second dynamic model is used to compute input torques only when the trajectory has been planned in order to have w p = 0. Considering that the control law is correctly adjusted, the effective trajectory is close enough to the desired one and therefore the hypothesis w p = 0 is acceptable.

Once the mechanism is far enough from the singularity locus, i.e. the matrix A is numerically invertible, the control switches back to the complete dynamic model and the mechanism can finish its trajectory.

Moreover, in order to cross the singularity locus, the controller need a metric that defines the moment when the Model 2 has to be used. The discussion about the best indicator of singularity proximity is a well known problematic [START_REF] Merlet | Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data. and Automation[END_REF], [START_REF] Glazunov | Speed and Force Criteria for the Proximity to Singularities of Parallel Structure Manipulators[END_REF], [START_REF] Voglewede | Measuring closeness to singularities for parallel manipulators[END_REF]. Here, as the controller may become unstable due to numerical problems linked to the inversion of the matrix A, it has been decided to use the numerical condition number of that matrix to define the space where the dynamic model might degenerates and therefore where the control law has to switch between the two models.

Next sections present experimental results of Type 2 singularity crossing using this multi-model control law on a planar Five-bar mechanism.

IV. CASE STUDY

A. Presentation of the Five-bar mechanism

A Five-bar mechanism is a planar parallel mechanism composed of two actuators located at the revolute joints positioned in points A and E and 3 passive revolute joints in points B, C and D (Fig. 2). The mechanism used in this work has been designed so that it can reach all the workspace positions without collision between the proximal and the distal legs.

The mechanism and its parameters are presented in Fig. 2. The links dimensions have been calibrated using a Laser Tracker (Table I).

B. Gain tunning

The proportional and derivative gains have been tuned based on the mechanism's natural frequency [START_REF] Khalil | Modeling, identification & control of robots[END_REF]. This frequency was retrieved using a ring-out procedure. The mechanism was excited using an impedance hammer, and its response was recorded using 5 accelerometers. The first natural frequency of the Five-bar mechanism in its isotropic configuration (when links BC and CD are perpendicular) is at 34.2 Hz.

For a given control bandwidth fixed by a frequency ω both gains are adjusted as :

K p = ω 2 , K d = 2ξ ω ( 17
)
where ξ is a a damping coefficient usually fixed to 1 to have a critically damped system. To guarantee that the gains do not bring the system in the neighbourhood of the instability domain, the chosen frequency must be smaller than the natural resonance frequency [START_REF] Khalil | Modeling, identification & control of robots[END_REF]. Therefore a frequency of ω = ω r /3 has been chosen, resulting in gains values :

K p = 1150, K d = 70 (18)

C. Dynamic model

A full dynamic model of the robot has been computed using the methodology presented in [START_REF] Briot | Global Identification of Drive Gains and Dynamic Parameters of Parallel Robots[END_REF] and its identification was made using a weight least square method based on the use of exciting trajectories followed with a classic geometrical control law [START_REF] Gautier | Dynamic identification of robots with power model[END_REF]. The identification results in the following model that fully describes the robot dynamics of the studied mechanism:

τ = m 3 J T ẍ ÿ + zz 1 q1 zz 2 q2 + f v1 q1 f v2 q2 + f s1 sign( q1 ) f s2 sign( q2 ) (19)
where :

• m 3 is a mass equivalent located on the end effector; m 3 = 0.40 ± 0.02kg • zz 1 and zz 2 are rotational equivalent inertial terms respectively on the first and second actuator; zz 1 = 1.83 • 10 -2 ± 6.97 • 10 -4 kg.m 2 ; zz 2 = 1.96 • 10 -2 ± 6.60 • 10 -4 kg.m 2 ; • f s1 is a Coulomb friction term on the first actuator (respectively f s2 on the second actuator); f s1 = 2.94 ± 0.10N.m; f s2 = 2.95 ± 0.09N.m; • f v1 is a viscous friction term on the first actuator (respectively f v2 on the second actuator); f v1 = 6.76 ± 0.018N.m.s f v2 = 6.75 ± 0.17N.m.s. This identified dynamic model is related to the Eq. ( 2) by:

w p = m 3 ẍ ÿ , w b = zz 1 q1 zz 2 q2 + f v1 q1 f v2 q2 + f s1 sign( q1 ) f s2 sign( q2 ) (20)
It should be noted that the friction terms in both passive joints are insignificant and therefore the identification routine returned null values.

D. Control law implementation

The Five-bar mechanism is controlled by an industrial control architecture developed by ADEPT with an open architecture. This control architecture allows the user to control the mechanism either in position, speed or torques, using a C/C++ software developed by ADEPT France: CIDE. This software was designed mostly for position control, therefore every securities preventing mostly physical damage had to be developed for the computed torque control law.

The dynamic model identified contain both accelerations in the joint space and in the task space. Therefore, in order to express the dynamic model as in Eq. ( 12), the task space acceleration has to be expressed as a function of the joint space acceleration. This can be done by differentiating the kinematic model:

v = J q ⇒ v = J q + J q (21)
By substituting [START_REF] Gautier | Dynamic identification of robots with power model[END_REF] into [START_REF] Glazunov | Speed and Force Criteria for the Proximity to Singularities of Parallel Structure Manipulators[END_REF] one can obtain the dynamic model used for the computed torque control law as presented in Equation ( 12):

τ = M q1 q2 + H, where (22)
M = m 3 J T J + zz 1 0 0 zz 4 (23)
H = m 3 J T J q1 q2 + f v1 q1 + f s1 sign( q1 ) f v2 q2 + f s2 sign( q2 ) (24)
Finally, in order to compute on-line the dynamic model of the robot and the CTC control law, the actuator's positions and speeds have been filtered by using the oversampling method at 1Khz (the control law turning at 250 Hz).

V. EXPERIMENTAL RESULTS

A. Generation of a crossing trajectory

The crossing trajectories are generated using two polynomials P x and P y such as :

x = P x (x f -x 0 ) + x 0 , y = P y (y f -y 0 ) + y 0 ( 25
)
where

• P x (t 0 ) = P y (t 0 ) = 0, • P x (t f ) = P y (t f ) = 1,
Each of them are 8 th order polynomial, corresponding to 8 conditions on each axes: two conditions for the initial position and speed, two for the final position and speed, one for the singular position and three to guarantee that the singularity crossing criteria ( 14) is respected around the singularity locus [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF].

Figure 3 represents a crossing trajectory in the task space as well as the evolution of the task space coordinates along this trajectory and the evolution of the dynamic criteria [START_REF] Khalil | Modeling, identification & control of robots[END_REF] for:

x 0 = 0.1, y 0 = 0.34, x s = 0.05475, y s = 0.2,

x f = 0, y f = 0.1, (26)

B. Type 2 singularity crossing results and process repeatability

This section presents the results of Type 2 singularity crossing for different trajectories computed according to the method presented in Section II-C and in the previous paragraph. Figure 4 represents the input torques computed by the computed torque control law along different crossing trajectories from one assembly mode to another, the desired trajectory and the control error. For each trajectory, the mechanism crosses the singularity without torque discontinuity, with a conditioning number limit set to 30 for switching between the two models.

The multi-model control law engenders an increase of the error around the singularity locus. Therefore, when the control law switches back to the complete dynamic model, the input torques can significantly increase in order to nullify this error. This can be seen on the first trajectory at 0.7s.

All three trajectories represented in Figure 4 were planned to cross the singularity at 0.5s and end at 1s. For each trajectory, the first figure represents the desired trajectory in the task space and the Type 2 singularity of the mechanism.

For testing the robustness of the proposed controller, each trajectory is run five times. Moreover, the starting point and the ending point were chosen randomly and neither those points nor the crossing direction have any effect on the singularity crossing. During all our experiments using this controller, the robot has always successfully crossed the singularity loci without any difficulty. Thus, our controller is totally robust to the desired trajectory.

VI. CONCLUSION

One of the most important drawbacks of parallel robots is the small size of their workspace, which is moreover reduced by the presence of singularities. In order to increase the workspace size, several solutions have been proposed. One promising approach is the definition of optimal trajectories able to cross the Type 2 singularities. However, this solution assumes that the controller is not subject to modelling errors and that the robot is able to perfectly track Different crossing trajectories and corresponding input torque applied the desired trajectory. Nevertheless, due to modelling errors, this assumption cannot be verified. As a consequence, if the optimal trajectory is not perfectly tracked, the dynamic model can degenerate near the Type 2 singularities, this can cause the computation of infinite torques and the robot might stay blocked into the singularity.

In order to avoid this problem, this paper has proposed a controller dedicated to the Type 2 singularity crossing. This controller was based on a multi-model approach that allowed the parallel robots to cross the Type 2 singularities without any torque discontinuity. The main idea was to shift near singularities from the full robot dynamic model to another simplified one that can never degenerate. This controller was coupled with an optimal trajectory planning methodology that makes the singularity crossing more robust to modelling errors.

Such a controller has been validated experimentally on a prototype of a Five-bar planar parallel mechanism. The results have shown the robustness and the relevancy of the controller dedicated to parallel robot Type 2 singularity crossing.

Fig. 1 .

 1 Fig. 1. Computed torque control law

2 Fig. 2 .FIVE

 22 Fig. 2. Five-bar mechanism designed and manufactured at IFMATABLE I FIVE-BAR MECHANISM'S GEOMETRIC PARAMETERS Parameter a L 1 L 2 L 3 L 4 Value (m) 0.2822 0.2130 0.1888 0.1878 0.2130 Precision (m) 1.10 -5 1.10 -5 1.10 -5 1.10 -5 1.10 -5

Fig. 3 .

 3 Fig. 3. Example of 8 th order polynomial trajectory crossing the Type 2 singularity locus

  Fig. 4.Different crossing trajectories and corresponding input torque applied


S Briot

I A Bonev

C M Gosselin
email: gosselin@gmc.ulaval.ca

V Arakelian
email: vigen.arakelyan@insa-rennes.fr

Complete shaking force and shaking moment balancing of planar parallel manipulators with prismatic pairs

Keywords: Shaking force, shaking moment, balancing, planar parallel robots with prismatic pairs

This paper deals with the complete shaking force and shaking moment balancing of planar parallel manipulators with prismatic pairs. The cancellation of the dynamic loads transmitted to the ground is a challenge for these types of manipulators.

2 It is obvious that the classical methods based on the optimal redistribution of movable masses and additional counter-rotations can be used to cancel shaking force and shaking moment.

However, such a balancing of parallel manipulators with prismatic pairs is only attained via a considerably complicated design. This paper shows that it is possible to balance planar parallel mechanisms using Scott-Russell mechanisms. Such an approach enables a division of the number of counter-rotations by two. Numerical simulations carried out using ADAMS software validate the obtained results and illustrate that the suggested balancing enables the creation of a parallel manipulator transmitting no inertia load to its base.

INTRODUCTION

Shaking force balancing is mostly obtained via an optimal redistribution of movable masses [1-10] or adjustment of kinematic parameters [START_REF] Ouyang | Force Balancing of Robotic Mechanisms based on Adjustment of Kinematic Parameters[END_REF]. The cancellation of the shaking moment is a more complicated task and can be obtained using three main different methods: (i) shaking moment balancing using counter-rotations [12-17] (Fig. 1a) (ii) shaking moment balancing by adding four bar linkages [18-22] (Fig. 1b) and (iii) shaking moment balancing by optimal trajectory planning [17, 23, 24].

Previous works have been devoted to the study of parallel manipulators with revolute joints and until now, to our knowledge, no study has been carried out on complete shaking force and shaking moment balancing of parallel manipulators with prismatic pairs. In this paper, for the first time, we propose solutions for complete shaking force and shaking moment balancing of planar parallel manipulators with prismatic pairs. We illustrate these solutions via the 3-RPR parallel manipulator. All obtained results are validated using ADAMS software simulations.

COMPLETE SHAKING MOMENT AND SHAKING FORCE BALANCING BY ADDING AN IDLER LOOP BETWEEN THE BASE AND THE PLATFORM

Inertia force balancing by adding an idler loop is known to be used for 1-degree-of-freedom (DOF) mechanisms [START_REF] Bagci | Complete shaking force and shaking moment balancing of link mechanisms using balancing idler loops[END_REF][START_REF] Frolov | Theory of mechanisms and machines[END_REF][START_REF] Doronin | Balanced slider-crank mechanism[END_REF][START_REF] Hilpert | Weight balancing of precision mechanical instruments[END_REF][START_REF] Arakelian | Equilibrage dynamique complet des mécanismes[END_REF]. With regard to planar manipulators, such an approach has only been used in the balancing of gravitational and inertia forces [9, 10, 30, 31].

In this section, the complete shaking force and shaking moment balancing of planar manipulators by adding an idler loop is discussed. The added balancing loop is mounted between the base and the platform of the mechanism. We illustrate the suggested balancing technique on a 3-RPR mechanism (Fig. 2). Please note that we do not mention the type of actuation of the mechanism as it has no influence on the balancing.

Firstly, let us analyze the cancellation of the dynamic reactions of the 3-RPR planar parallel mechanism (Fig. 2a). Such a mechanism has 3 DOF (two translations in the Oxy plane and one rotation of the moving platform around an axis perpendicular to Oxy) and is composed of three identical legs, each being composed of a revolute joint attached to the base at point A i (in the remainder of this report, i = 1, 2, 3), one moving prismatic guide, located at point B i , and another revolute joint attached to the platform at C i . The base and platform triangles, denoted A 1 A 2 A 3 and Considering that the x axis is directed along the line A 1 A 2 , the y axis being perpendicular to the x axis and the origin of the base frame located at point O, the centre of the circumcircle of triangle A 1 A 2 A 3 , one can define the coordinate x, y and  of the platform, as being respectively the coordinates of point P along the x and y axes and the angle between the line C 1 C 2 and A 1 A 2 .

C 1 C 2 C 3 ,
The length B i C i is denoted L 1 . Let us also denote as S ji the centre of mass of link ij (j = 1, 2), which has a mass m j and an axial moment of inertia I j . The centre of the mass of the platform is located at point P. The mass of the platform is m p and its axial moment of inertia I p .

In order to cancel the shaking forces and shaking moment of the manipulator, an idler loop is added between the base and the platform (Fig. 2b). The lengths EF and FP are denoted L 5 and L 6 respectively. The centre of mass of elements 5 and 6 of the idler loop are denoted S 3 and S 4 with masses m 5 and m 6 and axial moments of inertia I 5 and I 6 , respectively. The positions of the centre of masses are Thus, considering the shaking force F of leg 1, its expression is:

d AiS1i = r 1 L 1 u i , d CiS2i = (r 2 -1) L 1 u i , d ES5 =
6 6 5 5 3 1 2 1 S S P p i j Sji j m m m m d d d d F                (1)
where respectively.

After a simple development of expression (1), it can be seen that the shaking force F can be expressed as:

        F p p i i m r m m m r m m m r m r m d a a F   6 5 5 2 4 6 6 2 3 1 2 2 1 1 3 3 1             (2) with                       i i i i i i i L       sin cos cos sin 2 1    a , (3a)
                      6 6 2 6 6 6 6 6 4 sin cos cos sin          L a , (3b)
and

F d   is the acceleration of point F.
At this step, only five counterweights are needed in the cancellation of the shaking force, but it could be seen after more derivations that three others are necessary for the cancellation of the shaking moment. Therefore, we propose directly adding three supplementary counterweights (Fig. 3). The positions of the eight masses are

d AiMcp1i = r cp1 L 1 u i , d BiMcp2i = r cp2 L 1 u i , d EMcp5 = r cp5 d EF ,
      F cp cp cp cp cp i i cp cp cp cp bal r m m r m r m r m d a a F F   5 5 6 4 6 6 3 1 2 2 1 1 1          (4)
Thus, the shaking force is cancelled if:

1 1 1 1 cp cp r r m m   , ( 5a
)   2 2 2 2 1 1 cp cp r r m m     , ( 5b
)   6 6 6 2 2 6 3 cp p cp cp r m r m m m m      , (5c) and 
 5 5 5 6 6 2 2 5 3 cp cp p cp cp r m r m m m m m m        . ( 5d
)
The expression of the shaking moment M O of the modified structure (expressed at point O) can be written as:

dt dH M O O  , (6)
where H O is the angular momentum of the leg (expressed at point O). Thus, in order to cancel the shaking moment, the angular momentum is held constant over time.

The expression of the angular momentum H O is:

                                 6 5 3 1 2 1 3 1 2 1 j j j OMj OMj OMj OMj cpj OSj OSj OSj OSj j p i j OMji OMji OMji OMji cpj i j i j OSji OSji OSji OSji j O I x y y x m x y y x m I x y y x m I x y y x m H               (7)
where x OQ , y OQ , OQ

x  and OQ y  are the position and velocities of any point Q along x and y axes, respectively, Q being either point S ji , M ji, (j = 1, 2), S j or M j (j = 5, 6).

Developing and introducing ( 5) into (7) yields

                          5 2 5 2 2 5 2 5 6 6 2 5 5 2 5 5 5 6 2 6 2 2 2 6 6 2 6 6 6 2 2 2 3 1 2 1 2 2 2 3 1 2 1 2 2 2 2 1 1 2 1 1 2 1 3 3 3 1 1             L m m L m m m r m r m I L m m m r m r m I R m m I L r m L r m r m r m I I H cp p cp cp cp cp p cp cp p cp p i i cp cp i i cp cp O                              (8)
After such modifications of the RRR chain, the angular momentum of the legs of the mechanism and of the RRR chain can be balanced using six counter-rotations (Fig. 3), which have an axial moment of inertia equal to:

      2 1 2 2 2 2 2 2 2 1 1 2 1 1 2 1 1 1 1 L r m r m r m r m I I I cp cp cp cp cr         (9a)   2 2 2 2 3 p cp p cr R m m I I    (9b)     6 2

Numerical application

Let us illustrate the suggested balancing approach using numerical simulations carried out with ADAMS software. For this purpose, non balanced and balanced 3-RPR parallel manipulators will be compared.

The chosen trajectory for simulations is a straight line of the controlled point of the platform, achieved in t f = 0.25 s, between P 0 = (x 0 , y 0 ) = (-0.05 m, 0) and P f = (x f , y f ) = (-0.2 m, 0) with a rotation of the platform from  0 = 0° to  f = 30°. For the displacement of the mechanism, fifth order polynomial laws are used and therefore the trajectory is defined by the following expressions:

            5 4 3 0 0 / 6 / 15 / 10 f f f f t t t t t t x x x t x      , ( 10a
)   0  t y , ( 10b
)             5 4 3 0 0 / 6 / 15 / 10 f f f f t t t t t t t          . ( 10c
)
The parameters used for the simulations are the following:

-radii of the circumcircles of the base triangle A 1 A 2 A 3 and the platform triangle

C 1 C 2 C 3 ,R b = 0.35 m, R p = 0.1 m; -L 1 = 0.

COMPLETE SHAKING FORCE AND SHAKING MOMENT BALANCING VIA

SCOTT-RUSSELL MECHANISM

In this section another approach for complete shaking force and shaking moment balancing is developed, which consists of adding Scott-Russell mechanisms to the initial architecture of a manipulator. This approach enables a reduction in the number of counter-rotations.

Properties of the Scott-Russell mechanism.

Firstly, let us observe a simple slider-crank mechanism (Fig. 5). Let us denote the lengths AB Let us consider the balancing of this mechanism. The expression of the shaking force F of a slider-crank mechanism can be written as:

   3 1 i Si i m d F   (11)
where Si d   is the acceleration of the centre of mass S i .

Developing [START_REF] Ouyang | Force Balancing of Robotic Mechanisms based on Adjustment of Kinematic Parameters[END_REF], the expression of F becomes:

   a d F 3 2 2 2 1 1 m r m m r m B       (12)
with

                      2 2 2 2 2 2 2 2 sin cos cos sin          L a . ( 13
) B d   is the acceleration of point B.
The constant terms of eq. ( 12) can be cancelled by the addition of two counterweights M j , (j = 1, 2) (Fig. 5), of which the masses are m cpj . Their positions are:

d AMcp1 = r cp1 d AB , d BMcp2 = r cp2 d BC ,
r cp1 and r cp2 being dimensionless coefficients. With the addition of the counterweights, the shaking force becomes:

  a d F F 2 2 2 1 1 cp cp B cp cp cp bal r m m r m       (14)
Thus, the shaking force vanishes if:

2 3 2 2 2 cp cp r m r m m    and 1 3 2 2 1 1 1 cp cp cp r m m m r m m      . ( 15
)
The expression of the angular momentum H A (expressed at point A) is:

                 2 1 3 1 j MSj MSj MSj MSj cpj j j j ASj ASj ASj ASj j A y x y x m I y x y x m H       (16)
where x AQ , y AQ , AQ

x  and AQ y  are the position and velocities of any point Q along x and y axes, respectively, Q being either point S j or M j (j = 1, 2, 3).

Developing and introducing ( 15) into ( 16),

        2 2 2 3 2 2 2 2 2 2 2 1 2 1 3 2 2 2 1 1 2 1 1 1     L m r m r m I L m m m r m r m I H cp cp cp cp cp A           (17)
with

     2 2 2 L x x y a x x y AB AC AB AB AC AB            . ( 18
)
where x AC , x AB , y AB are the coordinates of points C and B, respectively, and AC x  , AB x  , AB y  their velocities.

In order to cancel the shaking moment M A , the angular momentum has to be constant or null.

Developing [START_REF] Gosselin | On the development of reactionless parallel manipulators[END_REF], one notices that this can be obtained if:

a = 0 and L 1 = L 2 . ( 19
)
In such a case,

2 1       . Therefore, the shaking moment is cancelled if:     0 2 1 2 2 2 2 2 2 2 2 1 2 2 2 1 1 2 1 1 1         L r m r m I L m m r m r m I cp cp cp cp cp (20)

Balancing of a manipulator's leg using a Scott-Russell mechanism.

Now let us consider a manipulator's leg with an added Scott-Russell mechanism (Fig. 6

      2 3 2 2 2 2 1 4 4 3 3 3 2 2 1 1 1 1 a a d F m r m r m r m r m m m m r m r m cp cp B cp cp cp             (21)
with [START_REF] Wu | Design of reactionless 3-DOF and 6-DOF parallel manipulators using parallelepiped mechanisms[END_REF] and

                      03 03 2 03 03 03 03 3 1 sin cos cos sin          L a
                                        
                  L a . ( 23
)
At this step, only one counterweight is needed for the cancellation of the shaking force, but it could be seen after more derivations that another is necessary for the cancellation of the shaking moment. Therefore, we propose adding this supplementary counterweight directly. The positions of the two masses are: d AMcp3 = r cp3 L 3 u, d CMcp4 = r cp4 L 3 u, r cp3 and r cp4 being dimensionless coefficients. Their masses are respectively denoted m cp3 and m cp4 . With the addition of the counterweights, the shaking force becomes:

  2 4 4 3 3 1 3 3 a a d F F cp cp cp cp cp B cp bal r m r m m m        . (24)
Thus, the shaking force is cancelled if:

4 4 4 4 cp cp r r m m   , ( 25a
) 3 3 3 3 cp cp r r m m   , ( 25b
) 2 3 3 2 2 2 cp cp cp r m m r m m     (25c)
and

1 3 3 2 2 1 1 1 cp cp cp cp r m m m m r m m       . ( 25d
)
Developing and simplifying, the expression of the angular momentum is:

41 2 03 1     eq eq A I I H   (26)
with

        2 3 2 4 4 2 4 4 2 3 3 2 3 3 4 3 2 1 2 1 2 2 2 2 2 2 2 1 1 2 1 1 1 1 1 L r m r m r m r m I I I I L r m r m r m r m I cp cp cp cp cp cp cp cp eq               (27)
    2 1 2 2 2 2 2 2 2 2 1 2 2 2 1 2 1 1 1 2 L r m r m I L m m r m r m I I cp cp cp cp cp eq         . (28)
From [START_REF] Gosselin | Synthesis and design of reactionless three-degree-offreedom parallel mechanisms[END_REF], I eq2 = 0. Therefore, the shaking moment of the slider-crank can be cancelled using a simple counter-rotation I cr with an axial moment of inertia equal to I eq1 .

Shaking moment and shaking force balancing of the 3-RPR manipulator

Now, let us apply such an approach to the 3-RPR mechanism. First of all, let us substitute the platform mass by three points masses located at C 1 , C 2 and C 3 , with the values of mass equal to m p1 , m p2 and m p3 respectively [13, 31, 32]. Such a condition can be obtained if:

3 / p pi m m  and 2 3 p pi p R m I  . ( 29
)
where R p is the radius of the circumcircle of C 1 C 2 C 3 . Such a decomposition of the platform enables us to consider the shaking force and shaking moment balancing of each leg of the mechanism. Then, modifying each leg in order to obtain a mechanism similar to a slider-crank linkage, (i.e., by adding an idler loop to each leg), the shaking force and shaking moment are cancelled if:

4 4 4 4 cp cp r r m m   , ( 30a
) 3 3 3 3 cp pi cp r m r m m    , ( 30b
) 2 3 3 2 2 2 cp pi cp cp r m m m r m m      , ( 30c
) 1 3 3 2 2 1 1 1 cp pi cp cp cp r m m m m m r m m        , (30d)
    2 1 2 2 2 2 2 2 2 2 1 2 2 2 1 1 2 1 1 1 0 L r m r m I L m m r m r m I cp cp cp cp cp         , (30e)
               (30f)
taking into account that I cr is the axial moment of inertia of the counter-rotations (Fig. 7). Thus, with this approach it is possible to create a fully-balanced shaking force and shaking moment 3-RPR mechanism with only three counter-rotations (Fig. 7), i.e., this method enables a reduction in the number of counter-rotations by a factor of two.

Numerical application

The parameters used for the simulations are the followings:

-radii of the circumcircles of the base triangle A 1 A 2 A 3 and the platform triangle For these new parameters and for the trajectory used previously, taking into account that the position coefficients of the counterweights are equal to r cpj = -0.5 (j = 1, 3, 4), r cp2 = -1, the new values of the counterweights are: m cp1 = 3.17 kg, m cp2 = 11.71 kg, m cp3 = 0.33 kg, m cp4 = 0.75 kg.

The shaking force and shaking moment are then computed (dashed line in Fig. 8). It is possible to see that, with the counterweights, the shaking efforts are cancelled, while the maximal value of the shaking moment is increased by a factor 28. Finally, we add the counter-rotations. Their values are equal to I cr = 1.56907 kg.m². With such counter-rotations, the shaking moment is balanced (gray line in Fig. 8c).

CONCLUSIONS

This paper presents the complete shaking force and shaking moment balancing of planar parallel manipulators with prismatic pairs. Two approaches are discussed: balancing via adding an idler loop mounted between the platform and the base of the manipulator and balancing via the Scott-Russell mechanism, which enables a reduction in the number of counter-rotations by a factor of two. All studied balancing techniques are validated by simulations carried out using ADAMS software. The obtained results show that parallel manipulators balanced using the suggested methods transmit no inertia loads to their bases, i.e., the sum of all ground forces and their moments are zero.

Finally, we would like to mention that using Scott-Russel mechanisms remains using some modified 3-RRR manipulator. Thus, if we do not consider the type of actuation, the prismatic guides could be suppressed and our work could be of no interest. But the goal of our study is to propose the complete shaking force and shaking moment balancing of manipulators for applications where an actuation via a prismatic motor is needed, such as in high load carrying (using hydraulic devices). Therefore, we think the proposed solutions are of great interest to the scientific community.

List of figures

Figure 1 .

 1 Figure 1. Complete shaking force and shaking moment balanced 3-RRR planar parallel

  are equilateral. On this manipulator, typically, the actuated joints are the first revolute joint at A i or the linear guide at B i . (a) Schematics of the 3-RPR mechanism. (b) Schematics of the added idler loop (RRR chain).

Figure 2 .

 2 Figure 2. Schematic of the 3-RPR robot under study.

r 5 d

 5 EF , and d FS6 = r 6 d FP , r 1 , r 2 , r 5 , and r 6 being dimensionless coefficients, and u i a unit vector directed along B i C i .

Figure 3 .

 3 Figure 3. Schematics of the 3-RPR mechanism with the added RRR chain used for the

Figure 4 .

 4 Figure 4. Shaking force and shaking moment before (solid line) and after (dashed line) the

and

  BC as respectively L 1 and L 2 , and the centre of masses of link i (i = 1, 2, 3) as S i , which has a mass m i and an axial moment of inertia I i . The positions of the centres of mass are d AS1 = r 1 d AB , d AS2 = d AB + r 2 d BC , d AS3 = d AC + d CS3 , r 1 and r 2 being dimensionless coefficients, d CS3 = L 3 r 3 x, (L 3 is a constant).

Figure 5 .

 5 Figure 5. A general slider-crank mechanism.

  ). Let us denote as S 4 the centre of mass of link 4, which has a mass m 4 and an axial moment of inertia I 4 . The position of S 4 is: d CS4 = L 3 r 4 u, r 4 being a dimensionless coefficient and u a unit vector along d CS3 .

Figure 6 .

 6 Figure 6. A manipulator leg with added Scott-Russell mechanism.

Figure 7 .

 7 Figure 7. Schematics of a shaking force and shaking moment balanced 3-RPR mechanism.

  C 1 C 2 C 3 R b = 0.35 m, R p = 0.1 m; -L 1 = L 2 = 0.25 m; L 3 = 0.025 m -r 1 = r 2 = 0.5; r 3 = 0; r 4 = 4; -m 1 = 1.09 kg; m 2 = 1.1 kg; m 3 = 0.37 kg; m 4 = 0.75 kg; m p = 1 kg; -I 1 = 0.00738 kg.m²; I 2 = 0.58389 kg.m²; I 3 = 0.00344 kg.m²; I 6 = 0.00025 kg.m²; I p = 0.01 kg.m².

Finally, it shouldFigure 8 .Figure 9 .Figure 10 .

 8910 Figure 8. Shaking force and shaking moment, before (solid line) and after (dashed line) the

Figure 1 .

 1 Figure 1. Complete shaking force and shaking moment balanced 3-RRR planar parallel

Figure 2 .

 2 Figure 2. Schematic of the 3-RPR robot under study.

Figure 3 .

 3 Figure 3. Schematics of the 3-RPR mechanism with the added RRR chain used for the

Figure 4 .

 4 Figure 4. Shaking force and shaking moment before (solid line) and after (dashed line) the

Figure 5 .

 5 Figure 5. A general slider-crank mechanism.

Figure 6 .

 6 Figure 6. A manipulator leg with added Scott-Russell mechanism.

Figure 7 .

 7 Figure 7. Schematics of a shaking force and shaking moment balanced 3-RPR mechanism.

Figure 8 .

 8 Figure 8. Shaking force and shaking moment, before (solid line) and after (dashed line) the

Figure 9 .

 9 Figure 9. Complete shaking force and shaking moment balancing of planar manipulators with

Figure 10 .

 10 Figure 10. Complete shaking force and shaking moment balancing of planar manipulators with


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

Ilian A Bonev
email: ilian.bonev@etsmtl.ca

Pantopteron: a New Fully-Decoupled 3-DOF Translational Parallel Robot for Pick-and-Place Applications

In this paper, a novel 3-DOF fully decoupled translational parallel robot, called the Pantopteron, is presented. This manipulator is similar to the Tripteron Cartesian parallel manipulator, but due to the use of three pantograph linkages, an amplification of the actuators displacements is achieved. Therefore, equipped with the same actuators, the mobile platform of the Pantopteron moves many-times faster than that of the Tripteron. This amplification is defined by the magnification factor of the pantograph linkages. The kinematics, workspace and constraint singularities of the proposed parallel robot are studied in detail. Design considerations are also discussed and a possible prototype is illustrated. .

Introduction

Less than a decade ago, any known parallel robot with three or more degrees of freedom (DOF) was inevitably associated with nonlinear highly-coupled kinematics, singularities, and a complex-shaped workspace. In May 2001, a revolutionary simple 3-DOF translational parallel robot, with fully-decoupled inputoutput equations, was disclosed by Gosselin and Kong in a Canadian provisional patent application. Its simplest design, illustrated in Fig. 1(a), is basically a Cartesian robot and is therefore isotropic (its Jacobian matrix is diagonal and constant). After careful investigation, Kong and Gosselin, helped by the second author of this paper, drafted a patent application that covers all possible design variations. Each of these variations, named Tripteron, is now patent protected in many countries, including USA [START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF].

Later in 2002, Carricato and Parenti-Castelli [START_REF] Carricato | Singularity-Free Fully-Isotropic Translational Parallel Manipulators[END_REF] and Kong and Gosselin [START_REF] Kong | Type Synthesis of Linear Translational Parallel Manipulators[END_REF] [START_REF] Kong | A Class of 3-DOF Translational Parallel Manipulators with Linear Input-Output Equations[END_REF] proposed separately a large family of decoupled 3-DOF translational parallel mechanisms, all covered by the above-mentioned patent. At the same time, Kim and Tsai [START_REF] Kim | Evaluation of a Cartesian Parallel Manipulator[END_REF] presented independently the simplest member of this family (the one shown in Fig. 1(a)).

Who among these three groups of researchers was the first to invent the 3-DOF isotropic parallel robot is not the subject of this paper. In fact, a synthesis of possible legs for the mechanism in Fig. 1(a) was presented in [START_REF] Hervé | Structural Synthesis of Parallel Robots Generating Spatial Translation[END_REF] as early as in 1991. What is obvious is that the year 2002 marked the beginning of worldwide research activities on isotropic 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. Among these papers, reference [START_REF] Gogu | Structural Synthesis of Fully-Isotropic Translational Parallel Robots via Theory of Linear Transformations[END_REF] proposes other variations of the Tripteron family. Specific members from the Tripteron family were studied in detail in many works (e.g, [START_REF] Carricato | A Novel Fully Decoupled 2-DOF Parallel Wrist[END_REF][10] [START_REF] Ruggiu | Kinematic Analysis of the CUR Translational Manipulator[END_REF][12] [START_REF] Li | P-CUBE, A Decoupled Parallel Robot Only with Prismatic Pairs[END_REF]).

The basic Tripteron parallel robot consists of three identical legs. Each leg has a base-mounted actuator, allowing translation along a fixed direction, and a planar chain. To make the mechanism isotropic, the planes of motion of the three planar chains should be orthogonal. Some of the joints of the planar chains may actually be replaced with higher-degree pairs, such as a universal joint, but this has no effect on the kinematics of the mechanism.

In the basic Tripteron, the displacement of the mobile platform along a given Cartesian axis is directly proportional to the displacement of each linear actuator. When the plane of the planar chain in a leg is normal to the direction of the corresponding linear actuator (as in the mechanism of Fig. 1(a)), there is a one-toone relationship, and the Jacobian matrix is the identity matrix. Otherwise, larger displacements are needed from the actuators to produce smaller motions at the mobile platform (as in the mechanism of Fig. 1b).

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 Robot -the Story of Success[END_REF] is to offer an even faster design. Hence, it would have been great if we could build a Tripteron 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 simplest Tripteron of Fig. 1(a), but this seems to be a reasonable price to pay. Indeed, the proposed design is the result of a large study on the synthesis of parallel manipulator using pantographs [START_REF] Briot | PAMINSA: a New Family of Decoupled Parallel Manipulators[END_REF]. One such manipulator was already successfully built and proves 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, is presented. The structure is described and its movements are analyzed, as well as their singularities. Then, its workspace is studied and various design considerations are given. Finally, conclusions are drawn.

Kinematic Analysis

Description of the architecture

The architecture of the Pantopteron is schematized in Fig. 2. It is composed of three identical 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 another 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.

For the Pantopteron, the actuators are located at the linear pairs (1i) (Fig. 3). These three pairs are connected to the base so that their axes 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 each joint (12i) being orthogonal. They also are connected to actuator (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 . Such an architecture allows three fully-decoupled translational DOFs. This will be now proved.

Mobility analysis

Let x, y, z be the axes of the base frame (Fig. 2). Without loss of generality, let us consider the displacements of the platform when legs 2 and 3 are disconnected, as well as the actuator M 1 located at pair [START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF][START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF]. A simple analysis could show that the platform has five passive DOFs, three translations and two rotations (one about the axis of pair [START_REF] Ruggiu | Kinematic Analysis of the CUR Translational Manipulator[END_REF][START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF] and another about the axis of pair (12,1)). Therefore, leg 1 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 e j (j = 1 to 5) the unit screw corresponding to the passive displacement of the platform. The expression of these screws, expressed in the base frame at point C 1 , can be written as: , where θ 1 is the angle between a 11 and y axes, and γ 1 represents the rotation between vector a 31 and the axis of pair [START_REF] Lee | Cartesian Parallel Manipulators with Pseudoplanar Limbs[END_REF][START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF]. The Plücker coordinates of the unit screws can be described in matrix E 1 as

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 0 cos
The wrench r 11 , transmitted to the platform by the leg, is orthogonal to the twists composing the lines of matrix (2) with 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 0 0 0 0 0 0 13 13 13
The twists defining the passive displacements of the platform are orthogonal to this matrix of rank equal to 3. In the general case, there are three independent passive displacements, which are the three translations about x, y and z axes. Thus, the platform is constrained by the legs to have only translational displacements.

Note that, in some cases where the three wrenches are linearly dependant, the platform gains one additional DOF. This case will be treated later, in the section about singularities.

Let us now consider that the actuator M 1 located at pair (1,1) is fixed. So, due to the decoupling properties of the pantograph linkages, the position about x axis of point C 1 is fixed. Thus, the platform has now two passive translational DOFs, which are orthogonal to the x axis. 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 i are fixed, six wrenches (r 11 , r 21 , r 12 , r 22 , r 13 , r 23 ) are applied on it. Let us denote by R the matrix composed of these wrenches applied on the platform by the legs. The expression of R in the base frame, and expressed at point O, is:

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - - = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 0 0 0 0 0 0 0 sin
In this expression, θ 1 , θ 2 and θ 3 are the angles between vectors a 11 , a 12 and a 13 and y, z and x axes, respectively. Note that, analyzing the condition of orthogonality on the axes of pairs (12i), it could be proven that angles γ i are constrained to be equal to 0.

Therefore, these terms disappear from Eq. ( 5). 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 five wrenches applied on the platform,

[ ] T z y x z y x v v v ω ω ω = 1 t (6)
where ω x , ω y , and ω 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. = T [START_REF] Gogu | Structural Synthesis of Parallel Robots, Part 1 -Methodology[END_REF] As, in Eq. ( 7), there are five equations for six unknowns (the components of t 1 ), the system admits an infinity of solutions.

If
Thus, without loss of generality, let us fix the value of v z to 1. Rearranging Eq. ( 7) yields: [START_REF] Gogu | Structural Synthesis of Fully-Isotropic Translational Parallel Robots via Theory of Linear Transformations[END_REF] from which, if the matrix is non-singular, t 1 can be found as

0 R' = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
[ ] T 1 0 0 0 0 0 = 1 t (9)
We would like to mention that the case where the matrix R' is singular will be studied later, in the section about singularities. 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 becomes obvious 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 proved 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 on a displacement of the platform along the same direction, but amplified by the pantograph linkage.

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

Geometric and kinematic models, and singularity analysis

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, the following trivial system of decoupled linear equations governs the movement of the Pantopteron: (

)

1 1 1 a x k x G - - = ρ (10) ( ) 2 2 2 b y k y G - - = ρ (11) ( ) 3 3 3 c z k z G - - = ρ (12)
where k is the magnification factor of the pantograph linkages, x, y and z are the coordinates of point P of the platform along x, y and z axes respectively and ρ 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 (see Eq. ( 22)).

Since k ≠ 0, the above system of independent equations can be inverted to give the trivial solution to the inverse kinematics of the Pantopteron.

Though simple as it is, the system of independent equations (10-12) can be rewritten in matrix form as c q J x + = (13) where x = [x, y, z] T is the vector of output Cartesian coordinates and q = [ρ 1 , ρ 2 , ρ 3 ] T is the vector of input coordinates, and:

J (14a) [ ] T G G G c z k b y k a x k 3 3 2 2 1 1 + + + - = c
(14b) Differentiating Eq. ( 13) leads to

q J x & & =
(15) Hence, J is the Jacobian matrix of the Pantopteron. Recall that the terms of the Jacobian matrix of the fastest Tripteron are equal to 1. Therefore, the Pantopteron displaces k times faster than the Tripteron (where k is obviously greater than 1). 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.

One type of singularities occurring in the proposed mechanism is 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 are aligned along the same axis (Fig. 4); in such a case, given one position of the platform, there are infinity of orientations for the pantograph linkage. Moreover, if during a displacement of the mechanism, a leg comes close from 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 pair (9i). These two kinds of singularity define the boundaries of the workspace. They are similar to the singular configurations that we can meet in the Tripteron.

A second case of singularities appears if the system of wrenches applied on the platform degenerates, i.e., if the matrix R of Eq. ( 5) becomes singular. Such a singularity is called a constraint singularity [START_REF] Zlatanov | Constraint Singularities of Parallel Mechanisms[END_REF]. This happens if: 0 sin sin sin cos cos cos ) det( R [START_REF] Briot | PAMINSA: a New Family of Decoupled Parallel Manipulators[END_REF] In such a case, the three moments applied on 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 observe the example presented on Fig. 5. The axis a 11 is parallel to the y axis and axes a 12 and a 13 are parallel to the x axis. Thus, the gained DOF of the platform is a rotation about an axis parallel to the z axis.

Expressing equation ( 16) in the Cartesian space, it becomes:

0 ) ( ) ( ) ( ) ( ) ( ) ( ) det( 3 2 1 3 3 2 2 1 1 3 2 1 3 3 + Γ Γ Γ - - - = G C G C G C G C G C G C y y x x z z x x z z y y R (17) where 2 1 1 2 1 1 1 ) ( ) ( G C G C z z y y - + - = Γ (18) 2 2 2 2 2 2 2 ) ( ) ( G C G C x x z z - + - = Γ ( 19
) 2 3 3 2 3 3 3 ) ( ) ( G C G C y y x x - + - = Γ (20)
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 x, y and z axes, respectively. Disregarding the case where Γ 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 ( 21
)
Taking into account that the terms x Gi , y Gi , z Gi appearing in ( 21) are constant and that where coefficient p i are constant terms depending on the position of points G i and of the shape of the platform. Fixing the altitude z of the platform, Eq. ( 23) is the expression of a hyperbola, whose coefficients depend on the altitude of the platform and on the geometric parameters of the mechanism. -

T i i i T i c b a z y x ] , , [ ] , , [ + = OC (
y G1 = z G1 = x G2 = z G2 = x G3 = y G3 = 1 m; -
the parameters of the pantograph linkages are irrelevant, but allow a workspace comprised in the interval x, y ∈ [0 m, 2 m]. We would like to mention that our mechanism, contrary to the Tripteron, has constraints singularities. This is due to the fact that each leg of the Tripteron is 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 has singularities, they can easily be removed from its workspace.

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 about some other possible architectures based on this mechanism.

Geometric workspace analysis

Many parameters influence the size of the workspace of the Pantopteron. As 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; -the singular configurations. Using the geometric approach, we will compute the workspace of the Pantopteron. As the Pantopteron is a translational parallel mechanism, its workspace 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 α 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 (α i ) max and (α 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 pair (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 (ρ 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.

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

i EiCi AiEi AiEi EiCi AiCi l l l l l α cos 2 2 2 - + = (24)
k l l AiCi AiBi / =
(25) 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 α i , the displacement locus of point C i is an ellipse E (Fig. 7).

Thus, considering the extremes (α i ) max and (α i ) min of angle α i , the boundaries of the workspace are given by the ellipses E min and E max (Fig. 8(a)).

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. 8(a)). The displacement of point C i when the actuator M i is at its maximal stroke (ρ 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 guide (9i) (Fig. 8(a)).

The entire vertex space is represented at Fig. 8(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 out of our workspace by a proper design of the stroke of the linear guides.

As researchers, the first thing on our mind was to implement in Matlab our geometric method in order to be able to optimize the workspace of the Pantopteron 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]. Figure 9(a) shows an example of the workspace of a Pantopteron 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.

Obviously, it would be a mistake to design a Cartesian parallel mechanism with 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 with sufficiently long legs has to become a box whose sides are of length k ∆ρ i (∆ρ i being the stroke of actuator M i ), as shown in Fig. 9(b).

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 has not to be so complicated, and can be reduced to a hollow cylinder (Fig. 8(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. 8(a)).

The size of the workspace of the Pantopteron is the other main advantage of the proposed robot. Indeed, the maximal volume of the workspace of the Tripteron is V = ∆ρ 1 ∆ρ 2 ∆ρ 3 while that of the Pantopteron is V = k 3 ∆ρ 1 ∆ρ 2 ∆ρ 3 , i.e., for the same set of given actuators, the workspace of the Pantopteron is k 3 times bigger than that of the Tripteron.

Moreover, it is well known that the actuators represent at least 80% of the global cost of for a robot. For creating a fast mechanism with actuated prismatic pairs, it is now preferable to use electric linear actuators that reach higher velocities. However, the main drawback of such actuators is their price, which is proportional to the length of their stroke. For a given maximal workspace, the stroke of the actuators of the Tripteron is k times greater than that of the motors of the Pantopteron. Therefore, even if the Pantopteron is more complicated to design than a Tripteron, its manufacturing cost would likely be lower.

Singularity-free workspace

It is impossible to speak about the workspace of a parallel mechanism without dealing with singularities. As seen previously from Eq. ( 23), the singular configurations depend on the position of the mobile platform, on the locations of the axes of the basemounted revolute joints, and on the shape of the platform. Thus, analyzing Eq. ( 23), there are twelve design parameters which are y G1 , z G1 , x G2 , z G2 , x G3 , y G3 , b 1 , c 1 , a 2 , c 2 , a 3 and b 3 (we do not consider the lengths of the links of the pantograph linkages as they do not influence these singular configurations). So, there is too much parameter in order to perform a complete analysis of the singular configurations. However, it is possible to restrict our analysis to some particular designs, which will decrease the number of parameters.

Thus, we will consider in this part a mechanism which will have a platform with two axes concurrent (for example for pairs (12,1) and (12,3)), and a base of which two axes of actuators are also concurrent (for example those of actuators M 1 and M 2 ). Therefore, considering that the intersection point of the axes of actuators M 1 and M 2 is the origin of the base frame, and that point P is at the intersection of the two axes of the platform, only x G3 , y G3 , a 3 and b 3 stay variable, the other ten parameters being equal to zero.

In such a case, equation (23) becomes:

( ) ( ) ( ) 0 2 3 3 3 3 = - + - + G G y b x x a y y x z
(26) 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: which, in 3D, represent two planes P 2 and P 3 . Thus, the Cartesian space may be separated into eight regions (Fig. 10). In regions I, III, V and VII, there are no singularities and it will be quite easy todevelop a manipulator of which the workspace is located in these regions, taking into account the previous geometric approach. In regions II, IV, VI and VIII, even if there are singular configurations, they are quite close from the planes P 2 and P 3 , and thus, it is also quite easy to inscribe a cube representing the workspace of the mechanism in these regions. Please note that in the special cases where all parameters are equal to zero, or a

Other possible architectures

Finally, we would like to mention that the design of the Pantopteron presented here is not the only solution for creating such a mechanism. First, as the leg is made up of a pantograph linkage, several design are possible, which are presented in [START_REF] Lu | Synthesis of Planar Five-Bar Pantograph Configurations by a Geometric Method[END_REF]. However, we think 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) could be removed and replaced by any kinematic chain able to perform a planar displacement, as planar RRR, RPR, PPR or PRR chains. Moreover, using such chains, points H i and G i need not be aligned.

Note also that other architectures with various DOFs are possible by modifying our Pantopteron, such as the mechanism with 4 decoupled DOFs represented it Fig. 12.

Conclusions

In this paper, a novel 3-DOF fully-decoupled isotropic translational parallel mechanism, named the Pantopteron, was presented. This mechanism is very similar to the Tripteron Cartesian parallel mechanism, 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, the Pantopteron displaces k times faster than the Tripteron (k being the magnification factor of the pantograph linkages). Moreover, for a given set of actuators, its workspace is k 3 times bigger than that of the Tripteron. Due to this property, if the size of the workspace is given, the stroke of the actuators of the Pantopteron is k times smaller than that of the Tripteron, which allows reducing the manufacturing cost. 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.

  (a) the directions of the actuators are orthogonal. (b) the directions of the actuators are parallel.

Figure 1 .

 1 Figure 1. Two versions of the Tripteron.

Figure 2 .

 2 Figure 2. Schematics of the Pantopteron.

Figure 3 .

 3 Figure 3. Schematics of a leg of the Pantopteron.

  rotations about the axes of pairs (12,1) and (11,1),

  Thus, r 11 is a wrench of zero pitch (a pure moment).

Figure 4 .

 4 Figure 4. Example of leg singularities.

Fig. 6 Figure 6 .

 66 Figure 6. Example of constraint singularity loci.

Figure 5 .

 5 Figure 5. Example of a constraint singularity.

Figure 7 .

 7 Figure 7. Displacement of C i when α i is fixed.

Figure 8 .

 8 Figure 8. Schematics of the vertex space of a leg from the Pantopteron.

Figure 9 .

 9 Figure 9. Workspace of the Pantopteron.

  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,

  3 = x G3 and b 3 = y G3 , Eq. (26hyperbolas degenerates into two straight lines of equations x = 0 and y = 0. Therefore, the eight regions delimited by planes P i have no singularities. A possible version of a prototype of a Pantopteron is represented at Fig. 11. Its geometric parameters are: -l AiEi = 0.2 m, l EiCi = 0.3 m, k = 3; -y G1 = z G1 = x G2 = z G2 = x G3 = y G3 = 0 m, b 1 = c 1 = a 2 = c 2 = a 3 = b 3 = 0 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) ; -(α i ) min = 25°, (α i ) max = 155° ; Its design is achieved such as its workspace is a cube whose side is equal to 0.18 m.

Figure 10 .

 10 Figure 10. Singularity free workspaces.

Figure 11 .

 11 Figure 11. CAD view of the prototype of Pantopteron.

Figure 12 .

 12 Figure 12. Pantopteron with 4 DOF.

  [START_REF] Lu | Synthesis of Planar Five-Bar Pantograph Configurations by a Geometric Method[END_REF] where a i , b i , c i are constant terms, Eq. (21) can be rewritten under the form:

	1 p	x	y	z	+	p	2	x	y	+	p	3	x	z	+	p	4	y	z	+	p	5	x	+	p	6	y	+	p	7	z	+	p	8	=	0	(23)

2 1 1 = Γ Γ Γ ---