L Kaci

C Boudaud

S Briot

P Martinet

Elastostatic Modelling of a Wooden Parallel Robot

Keywords: Eco-design, parallel robot, elastostatic model, wood

Eco-design of industrial robots is a field of research which has been rarely explored in the past. In order to considerably decrease the environmental impact of robot during the design phase, metal or carbon composite parts can be replaced by bio-sourced materials, such as wood. Indeed, wood has interesting mechanical properties, but its performance / dimensions will vary with the atmospheric conditions / external solicitations and with the conditions in which trees have grown. In order to be able to design a stiff industrial robot, robust design approaches must be used. These approaches must be fed with elastostatic models that are able to predict the variability in the robot deformations due to the variability of the wood mechanical properties. In this paper, we develop an elastostatic model for a wooden parallel robot which is able to cope with the variations of the wood mechanical properties. The prediction of this model in terms of deformations are compared with experimental measurements made on a wooden parallel robot mockup. Results show that there is a good correlation between the measurement displacements and the computed ones.

Introduction

The Climate Change Mitigation (CCM) has become a priority in the world, as shown during the COP21. It becomes urgent to decrease the Environmental Impact (EI) of the human activities. Robotics, which takes an increasing place in our everyday life, must also be part of this effort.

A recent study [START_REF]Fizians report[END_REF] shows that, contrary to what could be imagined, a large percentage (around 50 %) of the EI during the entire robot life is due to the use of metallic materials (steel or aluminum, even if they are recycled) for the design of their links. The percentage is even worth if carbon composites are used. The rest of the EI is globally due to the robot energy consumption during use.

A method to reduce EI of robot design, that is rarely used, is to replace the polluting metals and carbon composites with bio-sourced materials (BSM) that have a little (even no) EI [START_REF] Thakur | Green Composites from Natural Resources[END_REF][START_REF] Laurent | Vers un robot en bois: Première partie[END_REF]. Moreover, BSM have good stiffness performance, and their quantity is only limited by their growing time [START_REF] Kretschmann | Ch. 5: Mechanical properties of Wood[END_REF]. In this work, developped in the scope of the French project RobEcolo [START_REF]Project RobEcolo[END_REF], we are interested in using wood instead of metals to design the robot links.

Using wood in machines is not a new idea. For instance, it is used in chassis of cars [1]. It found also applications in the design of buildings due to its stiffness and low cost [START_REF] Kretschmann | Ch. 5: Mechanical properties of Wood[END_REF]. It is also used in Robotics for the design of mock-ups and proof-of-concept prototypes [START_REF] Roennau | Design and kinematics of a biologically-inspired leg for a six-legged walking machine[END_REF][START_REF] Quigley | A low-cost compliant 7-DOF robotic manipulator[END_REF]. However, a detailed study of these works shows that wood is never used in critical parts ensuring accuracy. Indeed, the wood performance / dimensions will vary with the atmospheric conditions / external solicitations and with the conditions they have grown [START_REF] Kretschmann | Ch. 5: Mechanical properties of Wood[END_REF][START_REF] Wang | From hydrophilicity to hydrophobicity: a critical review[END_REF]; thus, new robot design issues appear: How to be sure that an industrial robot made with wood can be accurate and stiff even if wood properties vary?

A first attempt to introduce wood in industrial robot design was presented in [START_REF] Laurent | Vers un robot en bois: Première partie[END_REF]. The results showed that the approach was valid enough to compete with usual materials. However, this study did not deal with the aforementioned issues.

Indeed, the accuracy issue can be treated through proper control approaches: external sensors combined with proper controllers can be used to accurately control the platform pose [START_REF] Chaumette | Handbook of robotics, chap. 24: Visual Servoing and Visual Tracking[END_REF]. Regarding the robot stiffness, we believe that this issue can be handled through robust design approaches [START_REF] Wang | Multi-objective robust optimization using a postoptimality sensitivity analysis technique: Application to a wind turbine design[END_REF]. However, these design approaches must be fed with stiffness models which are able to predict the variability of the robot deformations due to the variability in the material properties.

In this vein, the present paper aims at proposing an elastostic model of a wooden parallel robot being able to predict the deformations and their variability.

Mechanical Properties of Wood

Wood is an organic product structure of infinite variation of detail and design. It is on this account that no two woods are alike: in reality no two specimens from the same log are identical [START_REF] Kretschmann | Ch. 7: Stress Grades and Design properties for Lumber, Round Timber, and Ties[END_REF]. Several studies giving the mechanical properties (e.g. Young's modulus, Poisson's ratios) for different pieces of wood can be found in the literature [START_REF] Kollmann | Ch. 7: Mechanics and rheology of wood[END_REF][START_REF] Kretschmann | Ch. 5: Mechanical properties of Wood[END_REF]. The properties are defined by repeated tests, that provide mean values and rarely the standard deviation (it is known that the coefficient of variation can reach up to 30 % for some characteristics [START_REF] Kretschmann | Ch. 5: Mechanical properties of Wood[END_REF]).

We are investigating the wood in the design of industrial parallel robots, hence it is necessary to know the mechanical properties of the wood to define the elastic deformation of the robot due to the loads applied on it. However wood is subject to a scale effect: impact of singularities (e.g. knots, direction of grain) are linked to the size of the pieces of wood. Therefore new tests have to be performed to define the mechanical properties corresponding to the dimension of the robot links. Indeed, values given in [START_REF] Kollmann | Ch. 7: Mechanics and rheology of wood[END_REF][START_REF] Kretschmann | Ch. 5: Mechanical properties of Wood[END_REF] are usually obtained for beams of large dimensions unlike the links of our robot which will be much smaller. Moreover, unlike usual material used in robot design, wood is an orthotropic material. The three different directions are longitudinal L, radial R, and tangential T . According to the theory of elasticity, the wood is parameterizes by three Young's moduli

(E L , E R and E T ), six Poisson's ratios (ν LR , ν LT , ν RT , ν RL , ν T L , ν T R ) and three moduli of rigidity (G LR , G LT , G RT ).
The links of parallel robots have cross-sections with smaller dimensions than the link lengths [START_REF] Merlet | Parallel Robots[END_REF], and can be modeled according to the beam theory (Euler-Bernoulli model). For this reason, in this work only the longitudinal Young's modulus E L and Poisson's ratio ν = (ν LR +ν LL ) 2 are necessary. In order to decrease the dimensional variability of wood due to humidity, we decided to design our links with a novel type of acetylated wood [START_REF] Wang | From hydrophilicity to hydrophobicity: a critical review[END_REF][START_REF] Hill | Wood Modification: Chemical, Thermal and Other Processes[END_REF] Accoya Pine. This wood is known to have many advantages over raw wood species in terms of durability and dimensional stability. This choice implied that we had to made mechanical tests because no data are provided in literature for this new type of wood. Hence, to define the Young's modulus E L of Accoya Pine, bending tests were made (testing for the Poisson's ratio is not useful, as it has a relatively small impact on the bending strain [START_REF] Kollmann | Ch. 7: Mechanics and rheology of wood[END_REF]). Thirty specimens are used, the dimensions of the specimens are the same as the dimensions of a standard link of the robot we plan to build (length of 30 cm and square section of 3 cm × 3 cm). Cantilever tests were performed using loads of 1 kg, 2 kg and 5 kg (Fig. 1(a)) and the measure of the deflection is used to compute E L (the computation was inspired by the usual EN 408 standard). It is very important to mention that the preconditioning of the specimens was made at a temperature of 20 Celsius deg. and at a relative humidity of 65 %,which for this specific acetylated wood implies a moisture content of 3.1 %.

As shown on Fig. 1(b), we obtain a variable Young's modulus which follows a normal distribution with mean value equal to 9732 MPa, standard deviation of 2462 MPa. This data will be used as input of the model presented in the next section.

3 Estimation of the robot elastic displacement

Stiffness modeling of a five-bar mechanism

The planar five-bar mechanism (Fig. 2(a)) is a two degrees-of-freedom (dof ) parallel robot able to achieve two translations in the plane (x 0 Oy 0 ). The end-effector is located at point C.

The analytical stiffness model presented in this section is based on matrix structural analysis method. This model was used in [START_REF] Deblaise | A systematic analytical method for pkm stiffness matrix calculation[END_REF] in order to calculate the displacement of the end-effector of a parallel robot, under some external loadings.

The five-bar mechanism under study is modelled with four 3D flexible beams, each beam having two nodes at their extremity corresponding of the characteristic Fig. 2 The five-bar mechanism and its stiffness model points (Fig. 2(b)). Nodes 1 and 8 are clampled onto the groud while some kinematic constraint relations will be imposed to the other nodes in order to parameterize the free rotations of the passive revolute joints.

The stiffness matrix i K i associated with the beam i (i = 1 to 4 (Fig. 2(b))) and expressed in its local frame R i is a 12-dimensional constant square symmetric matrix and depends on the geometrical and mechanical parameters [START_REF] Imbert | Analyse des structures par éléments finis[END_REF].

i K i can be expressed in the base frame of the mechanism R 0 by using the following relation:

0 K i = T -1 i K i T with: T =     R 0 3 0 3 0 3 0 3 R 0 3 0 3 0 3 0 3 R 0 3 0 3 0 3 0 3 R     (1)
where R is the rotation matrix from the local frame R i to the base frame R 0 , and 0 3 is a (3 × 3) zero matrix. 0 K i can be decomposed into the four following (6 × 6) sub-matrices:

0 K i = 0 K i 11 0 K i 12 0 K i 21 0 K i 22 (2)
Stiffness assembly [START_REF] Imbert | Analyse des structures par éléments finis[END_REF] is a technique used to define the stiffness matrix of the a whole structure. For the five-bar mechanism, all the beam stiffness matrices are assembled in the base frame according to the nodes as follows:

0 K t =         0 K 1 22 0 6 0 6 0 6 0 6 0 6 0 6 0 K 2 11 0 K 2 12 0 6 0 6 0 6 0 6 0 K 2 21 0 K 2 22 0 6 0 6 0 6 0 6 0 6 0 0 6 K 3 11 0 K 3 12 0 6 0 6 0 6 0 6 0 K 3 21 0 K 3 22 0 6 0 6 0 6 0 6 0 6 0 6 0 K 4 11         (36,36) (3)
The structure of the mechanism is composed by three passive revolute joints, the characteristics of a revolute joint is that all movements of two adjacent nodes are the same except the rotation around joint axis. Those joint properties can be described by using the kinematic relation between two nodal displacements [START_REF] Deblaise | A systematic analytical method for pkm stiffness matrix calculation[END_REF]. Assembling all these kinematic relations into a unique expression gives:

0 A t ∆ x t = 0 (36,1) (4)
where, in our problem, A t is of dimension (15×36) and

∆ x t = [∆ x T 1 , ∆ x T 2 , ... , ∆ x T 7 ]
T is the vector gathering all the nodal displacement ∆ x i with i = 1,2,... ,7.

In the same way we defined ∆ x t , the corresponding vector that gathers all the nodal wrenches is given by

f t = [f T 1 , f T 2 , ... , f T 7 ] T
, where f j is the wrench applied at node j. Taking into account the constraint expressed in equations ( 4), the displacement of the structure can be obtained using the stiffness relation (relation wrench/displacement) [START_REF] Imbert | Analyse des structures par éléments finis[END_REF]:

K G ∆ x G = f G (5)
where:

K G = 0 K t A T t A t 0 15 (51,51) , ∆ x G = ∆ x t λ λ λ t (51,1)
, and

f G = f t 0 15 (51,1) (6)
λ λ λ t is the Lagrange multipliers vector necessary to take into account the kinematic constraints. The displacement is calculated as follows:

∆ x G = K -1 G f G (7)
from which the nodal displacement ∆ x t can be extracted.

Predictive model for the wooden robot deformation

At this step, we have introduced the stiffness model of the five-bar mechanism taking into account that the Young's modulus and Poisson's ratio are known. However the experimental results described in section 2 showed how important is the variability of the mechanical properties of the wood. Monte Carlo method [START_REF] Cools | Monte Carlo and quasi-Monte Carlo methods[END_REF] can be used to solve any problem having random variables as inputs. The main purpose of this model is to study the influence of the variation of the mechanical properties of the wooden links onto the robot deformation. As robot stiffness depends on its configuration, the study of the displacement in different configurations of the robot is necessary. Hence, we decided to test the robot properties in the workspace which is discretized by the grid shown in Fig. 2(a). For each point of the grid, the method is performed as follows:

1. For each link of the robot, a value of Young's modulus is randomly generated based on the normal distribution identified in section 2. 2. External wrenches are applied on the end-effector. Then the end-effector displacement is obtained through relation (5). 3. The previous two steps are repeated n = 40000 times which ensure the stability of the results (identical mean value at 0.1 µm). This method is applied on a case study in the next section. A MATLAB code was written to compute the robot displacement using the modeling procedure presented in section 3.1. The computation of all displacements took around 35 sec. per robot configuration (for a Pentium 4 2.70 GHz, 16 GB of RAM).

4 Case study

Description of wooden five-bar mechanism mockup

To validate the model proposed, a wooden five-bar mechanism mockup was developed (Fig. 3). The mockup is composed of four identical links (length of 28 cm and square cross section of 3 cm × 3 cm) made of Accoya Pine wood connected with three passive revolute joints. The distance between the fixed axes on the base is equal to 27 cm.

Experiments

The experimental measuring setup is made of two micrometers of 0.01 mm resolution as presented in Fig. 3. A force of 10 N acting along x and y axis respectively is applied on the end-effector, and the resulting displacements at the end-effector along x, y are measured. Tests were carried out for the nine configurations of the robot shown in Fig. 2(a). For a given configuration the experiment is repeated 10 times. Then all the experimental results are compared to the results obtained with a numerical analysis.

The results are summarized in Fig. 4. Concerning the displacements along x-axis and y-axis, the graphs show that the maximum and minimum experimental displacements are in the range of deformations predicted by the model. Results show that the deflections predicted by the model are in a much wider range than the experimental measurements. This was expected because numerically, 40000 values of rigidity were simulated while, experimentally, only one robot could be produced. Experimental results are globally in between the numerical limits. Nevertheless, • The passive joint stiffness was not considered • Viscoelasticity characteristics of the wood was not taken into account.

However, from those experiments, we can claim that the theoretical model is satisfactory and the proposed modeling procedure is efficient for predicting a realistic behavior of the wooden robot.

Conclusion

Robot eco-design is a field of research in Robotics that should be deeper investigated: this could contribute to the Climate Change Mitigation.

In order to considerably decrease the environmental impact of robot during the design phase, we proposed to replace metal or carbon composite parts by biosourced materials, such as wood. Wood has interesting mechanical properties but also drawbacks: its mechanical performance encounters a large variability. In order to be able to design a stiff industrial robot, robust design approaches must be used. These approaches must be fed with elastostatic models that are able to predict the variability in the robot deformations due to the variability of the wood mechanical properties.

In this paper, we developped an elastostatic model for a wooden parallel robot which was able to cope with the variations of the wood mechanical properties. The prediction of this model in terms of deformations were compared with experimental measurements made on a wooden parallel robot mockup. Results have shown that there is a good correlation between the measurement displacements and the computed ones. However, differences occured between the theoretical and experimental results that may differ because of unmodelled phenomena.

Our future works will concern the refining of the proposed elastic model, the computation of the natural frequencies and the definition of a robust design methodology for wooden industrial robots.

  named

Fig. 1

 1 Fig. 1 Experimental setup to define the mechanical properties of Accoya Pine

  (a) Schematic of the five-bar mechanism (actuators in gray) (b) Beam representation

Fig. 3

 3 Fig. 3 Experimental setup

Fig. 4

 4 Fig. 4 Measured and simulated displacements

Acknowledgements This work is supported by the French Région Pays de la Loire in the frame of the project RobEcolo (Convention No. 2015-10773).


Damien Six
email: damien.six@ls2n.fr

S Ébastien Briot
email: sebastien.briot@ls2n.fr

Abdelhamid Chriette
email: abdelhamid.chriette@ls2n.fr

Philippe Martinet
email: philippe.martinet@ls2n.fr

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

Parallel robots present singular configurations that divide the operational workspace into several aspects. It was proven that Type 2 and Leg Passive Joint Twist System singularities can be crossed with a trajectory respecting a given dynamic criterion. However, the practical implementation of a controller able to track such trajectories is up to now limited to restrictive cases of Type 2 singularities crossing. Analyzing the structure of the inverse dynamic model, this paper proposes a global solution allowing the tracking of trajectories respecting the general criterion for any singularity that leads to potential issues of dynamic model degeneracy. The tracking is operated in the robot joint space. Experimental results on a five-bar mechanism showed the controller ability to successfully cross Type 2 singularities.

Introduction

Parallel robots present many advantages over serial robots as higher acceleration capacities, better stiffness and payload-to-weight ratio. However, their use in an industrial context remains limited due, in part, to the division of their workspace in several aspects by singularities [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 | Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics[END_REF][START_REF] Zlatanov | Constraint singularities of parallel mechanisms[END_REF][START_REF] Conconi | A New Assessment of Singularities of Parallel Kinematic Chains[END_REF]. In some singularities, the inverse dynamic model of the mechanism does not admit a finite solution without the respect of a dynamic criterion [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF]. Those singularities will be referred as Type 2 [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] and Leg Passive Joint Twist System (LPJTS) singularities, which are a kind of leg singularities [START_REF] Conconi | A New Assessment of Singularities of Parallel Kinematic Chains[END_REF]. The scope of this paper is restricted to the above mentioned singularities. The other singularities such as constraint singularities are not covered.

Several approaches have been explored in the literature to increase the size of the operational workspace of parallel robot, which is usually limited by the presence of singularities. The main approaches are:

• The design of parallel robots without singularities [START_REF] Liu | Kinematics, singularity and workspace of planar 5R symmetrical parallel mechanisms[END_REF][START_REF] Gogu | Structural synthesis of fully-isotropic parallel robots via theory of linear transformations and evolutionary morphology[END_REF][START_REF] Briot | On the Dynamic Properties of Rigid-Link Flexible-Joint Parallel Manipulators in the Presence of Type 2 Singularities[END_REF]. This solution is the most usual one, but it usually leads to the design of robots with a small workspace size or robot architectures with very low practicability. • 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][START_REF] Rakotomanga | Kinetostatic Performance of a Planar Parallel Mechanism with Variable Actuation[END_REF], which increases the actuation cost. • Mechanisms with variable actuation modes [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF]. This solution is generally carried out when the mechanism is stopped. • The modification of the Type 2 singularities locus by changing the robot working mode. The main way to proceed is to cross a Type 1 singularity by reaching the workspace boundary and changing the leg configuration [START_REF] Campos | Development of a Five-Bar Parallel Robot Paper JMR-17-1046 -Corresponding Author: SIX 10 With Large Workspace[END_REF]. This method however may not grant an access to the complete workspace for all robot architecture. • The planning of assembly mode changing trajectories encircling cusp points (for Type 2 singularities) [START_REF] Zein | Non-Singular Assembly-mode Changing Motions for 3-RPR Parallel Manipulators[END_REF]. This solution requires the design of complex trajectories, which will not be the most efficient for fast operations, and only few mechanism have cusp points. • The planning of assembly mode changing trajectories directly through the singularity [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF][START_REF] Ider | Inverse dynamics of parallel manipulators in the presence of drive singularities[END_REF] with respect to a dynamic criterion when crossing the singularity.

Crossing the singularity seems a promising solution because a direct path between two points can be generated on any robot structure as long as this path respects the dynamic criterion for crossing. In practical implementation, the precision required on the trajectory tracking is not obtained with a simple PID controller and the effective crossing of the sin-Paper JMR-17-1046 -Corresponding Author: SIX 1 gularity is not guaranteed. Then, the use of a dynamic controller as computed-torque control have been explored. Unfortunately, the computation of the inverse dynamic model in such controller require the inversion of matrices that become singular in Type 2 or LPJTS singularities. This issue was partially solved in [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] and [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF]. However, both solutions are limited to Type 2 singularities. Also the solution in [START_REF] Pagis | Enlarging parallel robot workspace through Type-2 singularity crossing[END_REF] requires a restrictive trajectories and the solution in [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF] is limited to a control of the Cartesian coordinates of the robot tool, which are rarely available on industrial robots. This paper proposes a global solution allowing the crossing of Type 2 and LPJTS singularities with a computedtorque control, valid for any trajectory that crosses the singularity, as long as the dynamic criterion established in [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF][START_REF] Ider | Inverse dynamics of parallel manipulators in the presence of drive singularities[END_REF] is respected. To obtain this solution, the following improvements are performed from existing solutions:

1. The proof that the dynamic criterion expressed in the previous studies is not only a necessary condition to cross the singularity but guarantee a solution to the inverse dynamic model, which is expressed for Type 2 and LPJTS singularities. 2. A projection of the desired wrench when solving the inverse dynamic model in the control law is developed to avoid unbounded input torques from numerical uncertainties and tracking errors. 3. The extension of the second order kinematic equations to singular configurations is obtained to allow a feedback control law in joint space.

The present paper is structured as follows. Section 2 discusses the expression of the inverse dynamic model that remains valid in singularity. Section 3 presents a control law designed to track a trajectory crossing a singularity. The validation of the designed controller is discussed through experimentation in section 4. Conclusions are given in section 5.

Dynamic model at singularity locus 2.1 Dynamic modeling of parallel mechanisms

In this section, the dynamic equations of a parallel manipulator are briefly recalled. For a more detailed analysis, the reader is referred to [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF][START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF]. 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. n d passive joints are presents in the manipulator legs. 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. • q d and qd two n d -dimensional vectors of passive joint variables and passive joint velocities respectively. • x and ẋ two n do f -dimensional vectors of the independent platform coordinates and their time derivative respectively.

The relations between those coordinates are found by writing the closed-loop equations h(x, q a ) = 0 and h a (x, q a , q d ) = 0.

The time derivative of those closed-loop equations gives the kinematic constraints equations

Aẋ + B qa = 0 (1) J x ẋ -J a qa -J d qd = 0 (2)
with A, B, J a , J x and J d of respective size

(n do f × n do f ), (n do f × n do f ), (n d × n do f ), (n d × n do f ) and (n d × n d ).
Let us also consider the Lagrangian L of the system and τ a , τ d and w x respectively n do f , n d and n do f -dimensional vectors related to the Lagrangian by

τ a = d dt ∂L ∂ qa T - ∂L ∂q a T (3)
τ d = d dt ∂L ∂ qd T - ∂L ∂q d T (4)
w x = d dt ∂L ∂ẋ T - ∂L ∂x T (5)
Using method of Lagrange multipliers, the inverse dynamic model of the mechanism can be written as [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF]

τ = τ a -B T λ 1 -J T a λ 2 (6)
A T λ 1 -J T x λ 2 = w x (7)
J T d λ 2 = τ d (8)
where

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

• λ 1 and λ 2 are a n do f -dimensional and n d -dimensional vectors of Lagrange multipliers respectively, From the expression of the Lagrangian of the mechanical system, equations (3)-( 5) can be obtained under the form [START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF] τ a = M a qa qd + c a (9)

τ d = M d qa qd + c d (10)
w x = M x ẍ + c x (11)
where

• M a is a (n do f × (n do f + n p )
) matrix and M d a (n p × (n do f + n p )) matrix depending on the joint coordinates q a and q d , • c a is a n do f -dimensional vector and c d a n d -dimensional vector depending on the joint coordinates q a and q d and velocities qa and qd , • M x is a (n do f × n do f ) definite positive matrix depending on the platform coordinates x, • c r a n do f -dimensional vector depending on the platform coordinates x and their time derivative ẋ.

P(x,y)

O x 0 y 0 f 2 f 1 x s A 11 A 21 A 12 A 22
Figure 1: Example of a five-bar mechanism in a Type 2 singularity configuration. The vector ẋs represents the direction of the uncontrolled platform motion. f 1 and f 2 are the forces that can be generated through actuation on the platform. In this configuration, the resultant force that can be applied by the system on the platform is collinear to f 1 and f 2 . Two actuators are used to generate an effort along one direction, an overconstraint is consequently generated in the mechanism.

Solution to the inverse dynamic model and degeneracy conditions

In this section, the main results concerning the general solution to the inverse dynamic model equations ( 6)-( 8) and the degeneracy conditions of this model are recalled. For a detailed study of the dynamic model and its degeneracy conditions, the reader is referred to [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF]. The general solution to the inverse dynamic model of a parallel robot can be computed by expressing λ 2 from (8) and λ 1 from [START_REF] Liu | Kinematics, singularity and workspace of planar 5R symmetrical parallel mechanisms[END_REF] and by introducing them into (6)

τ = τ a -J T a J -T d τ d -B T A -T w d (12)
with w d = w x + J T x J -T d τ d . The expression of the dynamic model under this form requires a full rank of the matrices A and J d [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF]. However, this requirement is not always met. The rank degeneracy of those matrices is associated to kinematic singularities of parallel mechanisms identified in the literature:

• The degeneracy of the matrix A corresponds to a kinematic singularity named Type 2 singularity [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. In such singularity, one (or more) of the degrees of freedom of the platform becomes uncontrollable (see Fig. 1). • The degeneracy of the matrix J d corresponds to a kinematic singularity named LPJTS singularity [START_REF] Conconi | A New Assessment of Singularities of Parallel Kinematic Chains[END_REF][START_REF] Briot | Dynamics of Parallel Robots: From Rigid Bodies to Flexible Elements[END_REF]. In such singularity, one (or more) of the leg gains an uncontrollable motion on a passive joint, even if the platform is fully controlled.

For clarity purposes, it is assumed that only one direction is uncontrollable at singularity locus, corresponding to rank degeneracy of the matrix A or J d of order one1 . It was previously proved that the inverse dynamic model remains consistent under the respect of a dynamic criterion when cross- 1. Proofs and considerations on those criteria can be found in [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF] and extended to LPJTS singularities in [START_REF] Briot | Degeneracy conditions of the dynamic model of parallel robots[END_REF].

Solution to the inverse dynamic model at singularity

locus Equation ( 12) gives a solution to the dynamic model away from singularities. In this section, a solution to the inverse dynamic model in a singularity is given under the respect of the criteria described in Table 1.

Type 2 singularity

Let us assume a parallel robot on a Type 2 singularity with no other singularity on the same locus2 . Then, the inverse dynamic equations ( 6)-( 8) can be written under the form

τ = τ a -B T λ 1 -J T a J -T d τ d (13)
A T λ 1 = w d (14)
with

w d = w x + J T x J -T d τ d .
The rank degeneracy of the matrix A implies that the equation ( 14) will give at least one exact solution λ 1 if and only if w d is included in the range of the matrix A T . The range of A T represents all the total wrenches w d that can be applied by the legs through actuation and external forces on the platform. From linear algebra, it is known that the range of the matrix A T is the orthogonal complement of the kernel of the matrix A

ker(A) ⊥ = Im(A T ) (15)
If the dynamic criterion expressed in table 1 is respected, then w d is in the range of the matrix A and the inverse dynamic equations admits at least an exact solution. In order to find it, the inverse dynamic model is projected on the platform efforts by pre-multiplying equation ( 13) by

A T B -T A T B -T τ = A T B -T τ a -A T λ 1 -A T B -T J T a J -T d τ d (16)
Then, A T λ 1 can be replaced in ( 16) from ( 14), giving

J T T 2 τ = w T 2 ( 17
)
with

J T 2 = B -1 A and w T 2 = J T T 2 τ a -w x -(J T x + J T T 2 J T a )J -T d τ d .
Knowing that a solution to the inverse dynamic model ( 13)-( 14) is garanteed if the dynamic criterion is respected, one solution can be expressed using the Moore-Penrose pseudoinverse

τ = J T T 2 + w T 2 (18)
Since the matrix J T T 2 is not full rank, its kernel is not empty, and so there is an infinite number of solutions to the inverse dynamic model including [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF]. An illustration is given Fig. 1, on a five-bar mechanism. The wrenches generated by the system on the platform at singularity are two collinear forces. Their opposite action generates an overconstraint in the system. Several combination of those forces (and so several input torques) can generate a desired acceleration of the platform. Equation [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF] gives the solution to the inverse dynamic model minimizing the input torques, which is usually the desired solution to limit the power consumption of the mechanical system. However, for specific applications, it is possible to express a general solution to the inverse dynamic model under the form

τ = J T T 2 + w T 2 + (I n do f -J T T 2 + J T T 2 )ν (19)
with ν a n do f -dimensional arbitrary vector called the overconstraint. Among the possible solutions to the inverse dynamic model, the solution minimizing the input torques, given by equation [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF], will be retained. Note also that, out of a singularity, the following equality is obtained J T T 2

+ = J -T T 2 .
Then, the solution to the inverse dynamic model under the form ( 18) is also valid out of the singularity (equivalent to equation ( 12)).

LPJTS singularity

An equivalent inverse dynamic model can be expressed for a LPJTS singularity. In this section, a solution to the inverse dynamic model on a LPJTS singularity is expressed under the respect of the criterion given in Table 1. The inverse dynamic equations ( 6)-( 8) can be written under the form

τ = τ a -B T A -T w x -J T da λ 2 ( 20
)
J T d λ 2 = τ d (21)
with J da = J x A -1 B + J a . Note that J da is a Jacobian matrix linking the velocity of the active joints to the velocity of the passive joints

J da qa + J d qd = 0 (22)
The degeneracy of the matrix J da corresponds to a rare singularity where the mechanism can generate a non-null active joint velocity qa with no motion of the passive joints. Let us consider that the LPJTS singularity does not coincide with such singularity or a Type 2 singularity 3 . The rank degeneracy of the matrix J T d implies that the equation ( 21) admits at least one exact solution λ 2 if and only if τ d is included in the range of J T d which is an equivalent condition to the criterion previously defined to cross a LPJTS singularity in Table 1. Projecting the inverse dynamic model ( 20)-( 21) on the efforts exerted on the passive joints gives

J T p τ = w p (23)
with

J p = J -1 da J d and w p = -τ d + J T d J -T da (τ a -B T A -T w x )
With the respect of the criterion [START_REF] Briot | Optimal Force Generation in Parallel Manipulators for Passing through the Singular Positions[END_REF], the exact solution to the inverse dynamic model minimizing the input torques is

τ = J T p + w p (24)
As in the case of a Type 2 singularity, the general solution to the inverse dynamic model ( 20)-( 21) can be expressed under the form

τ = J T p + w p + (I n do f -J T p + J T p )ν (25)
with ν an arbitrary vector of overconstraint.

Control law

In this section, the design of a computed torque control law in order to track trajectories crossing a singularity is discussed. As industrial robot sensors usually measure the active joint coordinates, the control designed is based on the joint coordinates of the mechanical system.

Extended second-order loop-closure equations in the neighborhood of a singularity

The inverse dynamic model of a parallel mechanism contains terms function of the platform, passive and active joints accelerations. To design a computed torque control in joint space, it is necessary to express those accelerations as function of the controlled coordinates acceleration, i.e. the active joint accelerations. The time derivative of loopclosure equations ( 1) and [START_REF] Zlatanov | Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics[END_REF] gives

Aẍ + B qa + b = 0 (26) J x ẍ -J a qa -J d qd + d = 0 ( 27
)
with b = Ȧt r + Ḃ qa and d = Jx ẋ -Jta qa -Jtd qd . Out of a Type 2 singularity, those two equations can be combined giving

J da qa + J d qd + n = 0 (28) with J da = J x A -1 B + J a and n = J x A -1 b -d.
Then, out of singularities, the passive joints acceleration and platform acceleration can be expressed as function of the active joint acceleration with

ẍ = -A -1 B qa -A -1 b (29) qd = -J -1 d J da qa -J -1 d n (30)
However, those expressions are not valid in singularities, as the matrices A and J d are singular. To compute those expressions in singularity, it is necessary to extend the second-order loop-closure equations.

Type 2 singularity

When crossing a Type 2 singularity the inversion of the matrix A required to solve equation ( 26) is not possible anymore. In a Type 2 singularity the appearance of one uncontrolled degree of freedom on the platform is related to a kinematic constraint on the joint coordinates. This constraint is respected if and only if -B qa -b is included in the range of A. Let us consider this constraint respected, then equation [START_REF] Gautier | Identification of the dynamic parameters of a closed loop robot[END_REF] will give an infinite number of solutions

ẍ = -A + (B qa + b) + (I n a -A + A)ν (31)
with ν a n a -dimensional arbitrary vector. This is due to the fact that the platform coordinates acceleration vector ẍ do not only depends on joint accelerations but also on free dynamics associated to the uncontrollable platform motion. The dynamic criterion expressed in Table 1 ẋT

s w d = 0 (32)
is not only a necessary criterion to be respected by a trajectory that crosses the singularity but also an expression of the free dynamics of the platform. This free dynamics can be developed as function of ẍ. Let us consider the Lagrangian of the robot as function of the active joint and platform coordinates and velocities L ax = f ( qa , q a , ẋ, x). Then w d is given by

w d = d dt ∂L ax ∂ẋ T - ∂L ax ∂x T = M ax ẍ + c ax (33)
with M ax a definite positive matrix (n a × n a ). Equation (33) introduced in equation (32) gives

ẋT s M ax ẍ + ẋT s c ax = 0 ( 34
)
The second-order loop-closure equation ( 26) can be completed with equation (34) giving

A e ẍ = -B e qa -b e (35)
with

A e = A ẋT s M ax B e = B 0 1×n do f b e = b ẋT s c ax
A e is a (n do f + 1 × n do f ) matrix. Even at singularity, matrix A e rank remains equal to n do f (see proof below). If a solution to this system exists, it is unique and given by the expression

ẍ = -A + e (B e qa + b e ) (36)
The existence of the solution can be guaranteed for a trajectory designed in Cartesian space and then mapped to joint space. However, when using this expression in the controller design, the existence of an exact solution is not guaranteed. The computation of the solution through the Moore-Penrose pseudo-inverse will ensure the computation of a solution minimizing the quadratic error in ẍ, thus leading to a good estimation of the inverse dynamic model.

Proof.

The solutions to equation ( 26) expressed in equation ( 31) span an affine subspace S of the platform accelerations of direction ker A (considered dimension 1). We will prove that the affine subspace spanned by the solutions in ẍ to the dynamic criterion equation (34) intersect in one unique point with S. The solutions to equation ( 34) is an affine subspace of dimension n do f -1. The direction of this affine subspace is given by

ẋT s M ax ẍ = 0 (37)
As M ax is a positive definite matrices, ẋT s M ax ẋs > 0. Thus, any ẋs in ker A is not solution to the equation (37), meaning that ker A is not included in the direction given by (37). Then, the affine subspaces spanned by the solutions in ẋ to the equations ( 26) and (34) are two supplementary subspaces. Their intersection is a point in the platform accelerations space. This point is the unique possible platform acceleration that can be performed by the robot in the singularity, completing the proof.

The solution to the extended loop-closure equations (35) gives an exact computation of the platform acceleration if and only if the dynamic criterion is respected by the platform acceleration vector. However, in the design of our trajectories, the criterion is only respected in the neighborhood of a singularity. Away from it, it will be necessary to use the classical loop-closure equations [START_REF] Gautier | Identification of the dynamic parameters of a closed loop robot[END_REF]. For the purpose of designing a controller in experimental conditions, the transition between those two equations have to be continuous in the neighborhood of the singularity. A Type 2 singularity is characterized by the degeneracy of the matrix A. The matrix condition number κ(A) = e max (A) e min (A) is chosen as an indicator of the proximity to the singularity, with e max () and e min () being respectively the maximal and minimal singular values of a matrix, leading to

ẍ = -(σA -1 B + (1 -σ)A + e B e ) qa -σA -1 b -(1 -σ)A + e b e (38)
with σ a continuous transition function equal to 0 in the neighborhood of a singularity and to 1 away from it.

σ =                  1 if 1 κ(A) > 2ε, 1 -κ(A)ε κ(A)ε if ε < 1 κ(A) < 2ε, 0 if 1 κ(A) < ε. (39)
with ε a threshold characterizing the proximity of the singularity determined experimentally.

LPJTS singularity

A similar result can be obtained for a LPJTS singularity. The equation [START_REF] Renaud | Kinematic and dynamic identification of parallel mechanisms[END_REF] cannot be solved at singularity locus. In this case, the extended second-order loop-closure equation will take the form J d,e qd = -J da,e qa -n e (40)

with

J d,e = J d qT ds M de J da,e = J da 0 1×n do f n e = n qT ds c de M de = M d -J -1 da J d 1 n p ×1 c de = c d + M d -J -1 da n 0 n p ×1
With a reasoning similar to the one applied for Type 2 singularities (section (3.1.1)), it can be proved that the system (40) will give a unique solution in LPJTS singularity. The computation of the passive joint acceleration is then a combination of the classical loop-closure equations (30) and the extended loop-closure equation ( 40

), qd = -(σ p J -1 d J da -(1 -σ p )J + d,e J da,e ) qa -σ p J -1 d n -(1 -σ p )J + td,e n e (41)
where

σ p =                  1 if 1 κ(J d ) > 2ε p , 1 -κ(J d )ε p κ(J d )ε p if ε < 1 κ(J d ) < 2ε p , 0 if 1 κ(J d ) < ε p . (42)
with ε p a threshold characterizing the proximity of the singularity.

Expression of the inverse dynamic model for a control in joint space

In section 2.3.1, general solutions to the inverse dynamic model that are valid in and out of a singularity are given by equation ( 18) for Type 2 singularities and ( 24) for LPJTS singularities. Those models will be used for the design of a the computed torque control law to cross the singularity. The inverse dynamic model is expressed as function of the vectors τ a , τ d and w x issued from the Lagrangian of the system by equations ( 3)-( 5). For encoder-based computed torque control purpose, this inverse dynamic model must be expressed as a function of the acceleration of the measured coordinates qa . This expression will be computed using equations ( 9)-( 11) and the extended second-order loop-closure equations expressed in section 3.1.

Type 2 singularity

Combining the second order kinematic equations ( 27) and (38) with the equations issued from the Lagrangian ( 11)- [START_REF] Nahon | Force optimization in redundantly-actuated closed kinematic chains[END_REF] and the inverse dynamic model [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF] leads to the expression of the model as function of the joint acceleration qa . The model can be expressed under the form

τ = M T 2 qa + h T 2 (43)
where

• M T 2 is the (n do f × n do f
) inertia matrix of the system. M T 2 depends on the joints coordinates q and platform coordinates x. Skipping all mathematical derivations, it comes

M T 2 = J T T 2 + J T T 2 M a J σ + M x (σA -1 B + (1 -σ)A + e B e ) -(J T x + J T T 2 J T a )J -T d J σ (44) with J σ = I n do f -J -1 d (J x (σA -1 B + (1 -σ)A + e B e ) + J a )
.

• h T 2 is a n do f -dimensional vector regrouping the gravitational, centrifugal and Coriolis terms. h T 2 depends on the joints coordinates q and velocities q and platform Paper JMR-17-1046 -Corresponding Author: SIX coordinates x and their time derivative ẋ. Skipping all mathematical derivations, one can obtain

h T 2 = J T T 2 + J T T 2 (M a n σ + c a ) + M x (σA -1 b + (1 -σ)A + e b e ) -c x -(J T x + J T T 2 J T a )J -T d (M d n σ -c d ) (45)
with

n σ = 0 n do f ×1 -J -1 d J x (σA -1 b + (1 -σ)A + e b e ) + d .

LPJTS singularity

An expression similar to equation ( 43) can be derived from the inverse dynamic model obtained for a LPJTS singularity. Combining the second order kinematic equations ( 29) and ( 41) with the equations issued from the Lagrangian ( 11)- [START_REF] Nahon | Force optimization in redundantly-actuated closed kinematic chains[END_REF] and the inverse dynamic model equation ( 24) leads to the expression of the model as function of the joint acceleration qa .

τ = M p qa + h p (46)
Skipping the mathematical derivations, M p is given by

M p = J T p + -M d J σ p +J T p M a J σ p + B T A -T M x A -1 B (47)
with

J σp = I n do f -σ p J -1 d J da -(1 -σ p )J +
td,e J e and h p is given by

h p = J T p + -M d n σ p + c d + J T p M a n σ p + c a +B T A -T (M x A -1 b -c r ) (48)
with

n σp = 0 n do f ×1 -σ p J -1 d n -(1 -σ p )J +
td,e n e

Wrench projection in the neighborhood of a singularity

Around the singularity locus, the dynamic criterion established (Table 1) will not be exactly respected by a controller in real conditions. Unfortunately, only the strict respect of this criterion guarantee an exact solution to the inverse dynamic model. The existence of numerical and tracking errors may lead to the computation of huge torques in the neighborhood of a singularity. Error amplification around the singularity is due to the presence of a singular value near zero in the inverted matrix. Based on the assumption that the criterion is respected in the neighborhood of the singularity, a projection of the desired wrench in the dynamic model is performed. The design of the error inhibitor is given in this section for a Type 2 singularity from equation [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF]. The design of a similar inhibitor for a LPJTS singularity can be obtained from equation [START_REF] Özgür | A vision-based generic dynamic model of PKMs and its experimental validation on the Quattro parallel robot[END_REF].

Let us consider the inverse dynamic model equation to solve under the form (18)

A T B -T τ = w T 2 (49)
The matrix A degenerates in a Type 2 singularity. Let consider its singular value decomposition

A = UΣV T ( 50
)
where Σ is the diagonal matrix of ordered positive singular values (e 1 , . . . , e n ) and U, V two unitary matrices. At singularity locus, the last singular value is null e n = 0. Multiplying equation ( 49) by the matrix V T gives

ΣU T B -T τ = V T w T 2 (51)
In order for the dynamic model to give an exact solution, the last component of the vector w s = V T w T 2 must be equal to zero. This is an other formulation of the dynamic criterion necessary to cross the singularity (note that the last line of the matrix V T is included in ker A). As the dynamic criterion is not exactly respected, the last component of the vector w s has to be driven toward zero in the neighborhood of the singularity to avoid the numerical degeneracy of the inverse dynamic model. As in section 3.1.1, the proximity to the singularity is characterized by the condition number using the norm-2 of the matrix A. A function σ s is defined by

σ s =                  1 if 1 κ(A) > 2ε s , 1 -κ(A)ε s κ(A)ε s if ε s < 1 κ(A) < 2ε s , 0 if 1 κ(A) < ε s . (52)
with ε s threshold determined experimentally. σ s is a function equals to 1 and continuously driven towards 0 as the system get closer to the singularity. Multiplying the last component of the vector w s by σ s ensures that the desired wrench in equation (49) respects the dynamic criterion. This will avoid exponential increase of the torques consequent to minimal errors on the trajectory tracking.

Computed torque control law

The computed torque control (CTC) is an advanced control law for non-linear systems where the inverse dynamic model is used to realize an exact linearization of the system. Computed torque control allows the design of considerably more precise, energy-efficient and compliant controls for robots. The inclusion of the mechanism dynamic model into the controller is desirable to enforce a dynamical criterion such as the criterion required for singularity crossing (Table 1). The convergence and the stability of such controllers is well known [START_REF] Samson | Robust control of a class of nonlinear systems and applications to robotics[END_REF]. For all aforementioned reasons, this control law is an excellent candidate to track trajectories crossing the singularity. The following developments are suited both for Type 2 and LPJTS singularities.

From the previous section results, the inverse dynamic model of the robot can be expressed as a function of the active joint accelerations even at singular configuration qa :

τ = M qa + h (53)
The computed torque control law is obtained with the design of an auxiliary control signal u corresponding to the joint acceleration qa from which the input torques are computed

τ = Mu + h (54)
The following PD control law is applied

u = qt + K d ( qt -qa ) + K p (q t -q a ) (55)
where • q t , qt and qt are respectively the desired active joints position, velocity and acceleration along the tracked trajectory, • K d and K p are positive definite gain matrices. This control law ensures a second-order convergence of the joint position error e = q t -q a . Fig. 2 represents the classical CTC controller that can be applied to a parallel mechanism in order to track a trajectory. The computation of the matrices M and h requires an estimation of the platform and passive joints position and velocity. Several techniques may be used to get this estimation such as a computation through the geometric model, interval analysis method [START_REF] Merlet | Solving the Forward Kinematics of a Gough-Type Parallel Manipulator with Interval Analysis[END_REF] or exteroceptive measures [START_REF] Özgür | A vision-based generic dynamic model of PKMs and its experimental validation on the Quattro parallel robot[END_REF]. The estimation may also be computed from the desired trajectory, as long as the tracking is reliable [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. A computed torque control in Cartesian coordinates may also be designed to limit the issues with solutions to the second-order kinematic equations for Type 2 singularities (see section 3.1.1). However, to be efficient, such controller requires a precise external measure of the end-effector position, which is currently rarely available on robots. For a review of dynamic control of parallel robots, the reader is referred to [START_REF] Paccot | A Review on the Dynamic Control of Parallel Kinematic Machines: Theory and Experiments[END_REF].

Case study

The proposed approach was validated on an experimental platform. As the models and controllers are similar for Type 2 and LPJTS singularities, the experimental validation is only presented for the crossing of Type 2 singularities with a planar five-bar mechanism.

A five-bar mechanism is a planar parallel mechanism composed of two actuated joints located points A 11 and A 21 and three passive joints located at points A 12 , A 22 and A 13 (see Fig. 3). This mechanism is able to generate a motion of the end-effector situated in A 13 (x = (x, y) T ) through the actuation of the joints q 11 and q 21 . The passive joints coordinates are respectively noted q 12 on the first arm, q 22 on the second arm and q 13 for the joint linking those two arms.

Identified dynamic model

A full dynamic model identification was computed using a weighted least square method applied on exciting trajectories described in [START_REF] Gautier | Identification of the dynamic parameters of a closed loop robot[END_REF][START_REF] Renaud | Kinematic and dynamic identification of parallel mechanisms[END_REF]. The base parameters identified for the five-bar mechanism were The base parameters lead to the computation of the dynamic model ( 6)-( 8) under the form • Z a positive diagonal matrix

τ = Z qa + f -B T λ 1 A T λ 1 = m 3R ẍ (
Z = zz 11R 0 0 zz 21R (57)
• f the active joint friction terms

f s11 sign( q11 ) f s21 sign( q21 ) + f o f f 11 f o f f 21 (58)
with sign a mathematical function extracting the sign of a real number.

The five-bar mechanism presents Type 2 singularities corresponding to the degeneracy of the matrix A. Type 2 singularity configurations happen when the two distal links of the five-bar mechanism are aligned. In this case, the uncontrolled motion is orthogonal to this alignment (see Fig. 1).

The dynamic criterion to be applied when crossing a Type 2 singularity (Table 1), related to the dynamic model (56), is then given by the equation ÿ = ẍ tan(q i1,s + q i2,s )

with q i1,s and q i2,s joint i1 and i2 positions of arm i in singular configuration. The expression of the inverse dynamic model that does not degenerate at singularity locus [START_REF] Six | A Controller for Avoiding Dynamic Model Degeneracy of Parallel Robots During Type 2 Singularity Crossing[END_REF], as long as the criterion (59) is respected, is given by

τ = (J T T 2 ) + (J T T 2 Z qa -m 3R ẍ) (60)

Trajectory generation

To test the controller a trajectory crossing twice a Type 2 singularity has been generated with the five-bar mechanism. This trajectory (shown in Fig. 4) is a return trip from a point A = [0; 0.15] m to a point B = [0; 0.02] m. The trajectory crosses twice the singularity point S = [-0.048; 0.067] m. A point to point trajectory is computed in Cartesian coordinates and composed of

Experimental results

This section presents the experimental results of the trajectory defined in section 4.2. Figure 5 shows the torque computed by the controller and the joint tracking error during the trajectory. The thresholds ε and ε s are both set to 0.1 for this experiment. With the dynamic model computed, there is no discontinuity in torque computation when crossing the singularity. The tracking error may locally increase just after the singularity crossing. Several factors generate this effect. First, the dynamic model around the singularity is an approximation which considers that the criterion established (59) is strictly respected. In real conditions, a strict respect of the criterion cannot be ensured, leading to a less efficient estimation of the dynamic model around the singularity and a deterioration of the tracking quality. It must also be considered that locally, a Type 2 singularity constraints the gap between the two proximal legs of the system, if the calibration of the proximal joint positions is not of perfect accuracy, an error will be generated by this constraint when crossing. And finally, it must be considered that in the neighborhood of a singularity, the system is locally underactuated. This aspect makes a part of the tracking errors locally impossible to be corrected by the controller. The appropriate precautions taken make that the torque input remain bounded at singularity, at the cost of a local angular tracking error. Nevertheless, a trajectory have been generated with several singularity points and accelerations. The trajectory have been tested at least 10 times with success in crossing the singularities each time. Experiment videos are available at https://www.youtube.com/watch?v=xVAaF4ycfho

Conclusion

The presence of singularities in the workspace of parallel robot reduces the effective workspace. To increase this workspace, a promising solution that consists in designing trajectories able to cross singularities has been developed in a previous study. However, the controllers developed to track those trajectories were limited to restrictive cases of Type 2 singularities crossing. This paper proposes new expressions of the inverse dynamic model, adapted for computation even in the singularity. The inverse dynamic model developed also handle the local overconstraint that appears in the singularity. Combined with the extension of the second order loopclosure equations in the singularity, it allows the design of a computed torque control able to track the desired trajectories. The theoretical contributions of this paper also extended the results to the crossing of LPJTS singularities. The controller was validated experimentally on a five-bar mechanism.

Future work on this subject includes the development of techniques to validate the operating mode change when crossing a singularity without the use of exteroceptive sensors. Also, one limitation of the controller is its dependency to a threshold determined experimentally. An improvement of the approach would be to develop a formal method to compute an appropriate value. Work related to a better consider-ation of the underactuation into the controller when crossing the singularity is also in progress.

Figure 2 :

 2 Figure 2: Computed Torque controller in joint space

Figure 3 :

 3 Figure 3: Five-bar mechanism and parametrization scheme

•

  grouped inertia for the two proximal links zz 11R = 0.0133 kg/m 2 and zz 21R = 0.0142 kg/m 2 , • grouped point mass at the end-effector m 3R = 0.537 kg, • frictions (coulomb model) static f o f f 11 = 0.0855 Nm, f o f f 21 = 0 Nm and kinetic f s11 = 0.345 Nm/s, f s21 = 0.430 Nm/s parameters for the active joints.

Figure 4 :

 4 Figure 4: Tracked trajectory in Cartesian space (to scale)

Figure 5 :

 5 Figure 5: Input torque applied, σ and σ s values and tracking error on a singularity crossing trajectory. σ = σ s = 0 means that the robot is close to a singularity and the wrench projection in the controller is active.

Table 1 :

 1 Dynamic criterion to avoid degeneracy of the inverse dynamic model in a singularity = 0 ẋs ∈ ker(A) qds ∈ ker(J d ) ing a singularity. The criteria established are summarized in Table

	Singularity Type 2 [21]	LPJTS [5]
	Criterion	ẋT s w d = 0	qT ds τ d

Table 2 :

 2 • A trajectory during a time interval [t si ;t so ] around the singularity with constant acceleration ( ẍs , ÿs ) that respects the criterion (59), given by (with t the time variable) , α y , β x and β y are coefficients computed such that the trajectory respect a desired position and velocity at singularity point. • Two five degree polynomials ensuring a continuous transition form the starting point to the singularity trajectory and from the singularity trajectory to the end point. Those polynomials are defined by initial and final position, a null velocity and null acceleration of the endeffector at those points and by the the trajectory duration t f .Table2gives the parameters used to define the trajectory from point A to point B and from point B to point A. The trajectory in joint space is computed through the inverse geometric model of the robot. Trajectory parameters

	x = y =	ẍs 2 2 ÿs	t 2 + α x t + β x t 2 + α y t + β y	(61)
	α x

Paper JMR-17-1046 -Corresponding Author: SIX

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

This hypothesis is taken given that the case of the coincidence of two singularities is extremely rare and generally avoided in the design of a parallel robot.Paper JMR-17-1046 -Corresponding Author: SIX

This hypothesis is taken given that the case of the coincidence of two singularities is extremely rare and generally avoided in the design of a parallel robot.Paper JMR-17-1046 -Corresponding Author: SIX


Sébastien Briot
email: sebastien.briot@ens.insa-rennes.fr

G C G M - Ea3913

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

Sylvain Guégan
email: sylvain.guegan@insa-rennes.fr

-Load Carrying Capacity

Design and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipulator with High

Introduction

The robots with varied architectures generate Schönflies motions (alternatively called SCARA motions or 3T1R motions), in which the end-effector carries out three independent translations and rotates around a fixed axis. The first robot of this type was developed at Japon's Yamanashi University [START_REF] Makino | SCARA robot and its family[END_REF]. This robot, named SCARA (Selective Compliance Assembly Robot Arm) is a high-performance system and nowadays, there are about 240 types of SCARA robots developed by 20 Companies. With the emergence of parallel manipulators, new structures have been proposed for carrying out Schönflies motions. One of the earliest parallel manipulators generating these motions was the Delta robot [START_REF] Clavel | DELTA, a fast robot with parallel geometry[END_REF][START_REF] Clavel | Device for the movement and positioning of an element in space[END_REF]. In this parallel architecture with only three-translation degrees of freedom, the rotation of the gripper is achieved by an additional actuator mounted on the frame. Later, the Kanuk and the Manta architectures were developed [START_REF] Rolland | The Manta and the Kanuk: Novel 4-DOF Parallel Manipulators for Industrial Handling[END_REF]. A contribution to the further development of the Delta robot was the manipulator H4 [START_REF] Pierrot | Four-degree-of-freedom parallel robot[END_REF][START_REF] Pierrot | H4 Parallel Robot: Modeling, Design and Preliminary Experiments[END_REF], in which four legs are mounted on the frame making it possible to carry out 3T1R motions with 4 coupled actuators. At McGill University, several architectures based on parallelogram mechanisms were developed [START_REF] Angeles | A Novel Manipulator Architecture for the Production of SCARA motions[END_REF][START_REF] Angeles | Four-degree-offreedom parallel manipulator for producing schonflies motions[END_REF][START_REF] Angeles | The Morphology Design for a Parallel Schönflies-Motion Generator[END_REF]. New architectures were proposed on the basis of systematic synthesis procedures [START_REF] Yang | Structure Synthesis of 4-DOF (3-Translation and 1-Rotation) Parallel Robot Mechanisms Based on the Units of Single-Opened-Chain[END_REF][START_REF] Angeles | The Qualitative Synthesis of Parallel Manipulators[END_REF][START_REF] Zhao | Geometric Synthesis of Spatial Parallel Manipulators With Fewer Than Six Degrees of Freedom[END_REF][START_REF] Li | Type Synthesis of 4-DOF Parallel Manipulators[END_REF]. However, both for the other parallel manipulators and for these architectures, a drawback is the high coupling of kinematics and dynamics. This non-linearity of the kinematic and dynamic models of parallel manipulators is not attractive for industrial applications. In order to solve this problem new structures have been developed over the last few years [START_REF] Carricato | On the topological and geometrical synthesis and classification of translational parallel mechanisms[END_REF][START_REF] Carricato | Fully Isotropic Four-Degrees-of-Freedom Parallel Mechanisms for Schoenflies Motion[END_REF][START_REF] Gosselin | A Fully Decoupled 3-DOF Translational Parallel Mechanism[END_REF][START_REF] Kong | Type Synthesis of Input-Output Decoupled Parallel Manipulators[END_REF][START_REF] Gogu | Fully-Isotropic T3R1-Type Parallel Manipulators[END_REF][START_REF] Gogu | Singularity-Free Fully-Isotropic Parallel Manipulators with Schönflies Motions[END_REF] in which the decoupling of the displacements in regard to all degrees of freedom of the platform is achieved.

Our observations have shown that despite rather-encouraging results, it is not easy to develop a parallel architecture with fullydecoupled motions and to conserve its principal advantages: the higher stiffness of the structure with light links. In the case of the fully decoupled parallel manipulators, the platform is connected to the base by several legs but each of them operates only in one direction, i.e. one leg carries the whole load. In other words, the parallel manipulator becomes a structure in which the payload is connected to several chains operating separately. There were also efforts to create partially decoupled parallel manipulators [START_REF] Richard | Kinematic Analysis and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipualtor[END_REF] but the noted drawbacks are the same.

Which is why we have tried to find a compromise between the decoupling of the movements and the architectural particularity of the parallel structures, i.e. the higher stiffness of the structure with light links. In other words, we have changed the statement of the problem: it is not essential that a parallel architecture will be fully-decoupled, it may also be partially decoupled but it is important to obtain a mechanical architecture with a good payload-to-weight ratio.

New concept of the partially decoupled parallel manipulator

An energetic analysis shows that the gravity work of a body moving in the horizontal plane is equal to zero (the gravitational forces are always perpendicular to the displacements). But the work of the same force moving along the vertical axis is other than zero (the gravitational forces are parallel to the displacements). This phenomenon is used in the design of the hand operated manipulators [START_REF] Arakelian | Equilibrage des manipulateurs manuels[END_REF][START_REF] Arakelian | The history of the creation and development of hand-operated balanced manipulators[END_REF], in which the horizontal displacements of the payload are carried out manually and the vertical displacements are actuated. This principle is applied in the design of the new parallel manipulators called PAMINSA 1 .

Let us consider the mechanical architecture of the suggested manipulators. The aim of this innovation was to develop a parallel architecture in which displacements of the platform in the horizontal plane are independent of its vertical displacements. For this purpose, the pantograph linkage is used as a leg. The pantograph is a mechanical system with two input points A i and B i and one output point C i (Fig. 1a). These input points linearly control the displacement of the output point C i . Thus, one linear actuator connected with input point B i can control the vertical displacement of the output point C i and one other linear actuator with horizontal axis is able to control its horizontal displacements. Please note that these motions are completely decoupled, i.e. they can be carried out independently. Now let us connect three Scheiner pantograph linkages with the base and the platform as shown in Fig. 1b. In the obtained structure, one vertical actuator M v controls the vertical displacement of point B i of the pantograph linkages, resulting in the vertical displacement of joints C i of the moving platform. The generation of motion in the horizontal plane is achieved by the actuators M 1 , M 2 and M 3 moving the input joints A i . Fig. 2 shows the first hand-operated model of PAMINSA. It is easy to see that, in the suggested architecture, the vertical translation of the platform along z axis is decoupled from its displacements in the horizontal plane (translations along x and y axes and rotation φ around z axis).

Among the obvious advantages of the suggested manipulator architecture, we would like to note the following: (i) the decoupling of the control powers into two parts, making it possible to raise an important payload to a fixed altitude with powerful actuators and then to displace it on the horizontal plane with less powerful actuators; (ii) a great accuracy in the horizontal positioning because the payload can be locked in the horizontal plane using mechanical architecture of the manipulator (in other words, if the position of the vertical actuator is fixed, the altitude of the platform cannot change); (iii) the cancellation of the loads of gravity on the rotating actuators which displace the platform in the horizontal plane; (iv) the simplification of the vertical control which is based on linear input/output relationships. The next section will present the kinematics and statics of the suggested manipulator.

Kinematic, singularity and static analysis

Kinematic and singularity analysis

The kinematics of this manipulator is simple because its analysis can be decomposed into two parts. The displacement along the vertical axis is described by the well known copying properties of the pantograph:

Z k z =
, where k is the magnification factor of the pantograph, Z is the displacement of the input point B i and z is the displacement of the output point C i of the pantograph (Fig. 1a). With regard to the displacements in the horizontal plane, the projection of the manipulator in the horizontal plane is similar to a 3-RPR parallel manipulator (Fig. 3). In this case, the horizontal motions of the platform are produced by three actuators M 1 , M 2 and M 3 . The position and orientation of the mobile coordinate system of the platform in the fixed coordinate system xOy can be represented by the vector The relations between the input

( ) T , , 3 2 1 θ θ θ = q
and output ( ) T

x,y,φ = x parameters can be found from the closure loop equations:

0 2 / 3 ) 3 2 sin( sin 2 / ) 3 2 cos( cos 1 1 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - + - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - + - x R R y R R b pl b pl π φ θ π φ θ (1) 0 2 / 3 ) 3 2 sin( sin 2 / ) 3 2 cos( cos 2 2 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - - - x R R y R R b pl b pl π φ θ π φ θ (2) ( ) ( ) 0 sin sin cos cos 3 3 = - - - + - x R y R R pl b pl φ θ φ θ (3) where 3 2 1 OO OO OO b l l l R = = = and 3 2 1 SC SC SC pl l l l R = = = .
The Jacobian matrix J, expressing the relation between the vector of the time derivatives of the input joint coordinates and the times derivatives of the position of the platform, can be derived from the expressions (1-3) [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Zlatanov | Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics[END_REF][START_REF] Bonev | Singularity analysis of 3-DOF planar parallel mechanisms via Screw Theory[END_REF][START_REF] Briot | Self Motions of General 3-RPR Planar Parallel Robots[END_REF]:

B A J 1 - = (4) with ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = k 0 0 0 0 0 0 0 0 0 0 0 0 3 2 1 ρ ρ ρ A , i i i C O = ρ (5) ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - - - - - + - = 1 0 0 0 0 ) sin( cos sin 0 ) 3 2 sin( cos sin 0 ) 3 2 sin( cos sin 3 3 3 2 2 2 1 1 1 θ φ θ θ θ π φ θ θ θ π φ θ θ pl pl pl R R R B (6)
Thus, the singularity of parallel manipulators appears when these matrices are rank-deficient, i.e. when 0 ) det(

3 2 1 = = ρ ρ ρ k A (7) ( ) ( ) ( ) ( ) 0 cos 2 cos ) det( 3 2 1 2 2 2 2 = - + - + - - = ρ ρ ρ φ φ b pl b pl b pl R R R R y x R R B (8)
= - + - + φ n b n b R R R R y x
. The manipulator configuration corresponding to this singularity is shown in Fig. 5. In this configuration, the center S of the platform is on a circle of radius

φ cos 2 2 2 n b n b R R R R - +
and for the fixed active-joint variables, the platform may generate infinitesimal rotation around center O * . Other types of singularities, which are not described in the previous approach, are due to the degeneracy of the pantograph linkages. They can be found by analyzing the Jacobian matrix J (LEG)i , defining the relations between the time derivatives of the coordinates of points C i (expressed in the plane of the pantograph) and the time derivatives of the angles i 1 ϕ and i 2 ϕ , where, i 1 ϕ is the angle between the directions of the prismatic pair H i and segment B i D i and i 2 ϕ is the angle between directions of segments B i D i and D i C i (Fig. 1a):

( ) ( ) ( ) ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + + + - + - - = i i i C i D i i i C i D i i D i B i i i C i D i i i C i D i i D i B i l l l l l l 2 1 2 1 1 2 1 2 1 1 ) ( cos cos cos sin sin sin ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ LEG J , (i = 1, 2, 3) (9)
Thus, the singularity appears when this matrix is rankdeficient, i.e. when 0 sin ) det(

2 (LEG) = = i i C i D i D i B i l l ϕ J ( 10
)
These conditions will be reached when

0 sin 2 = i i C i D i D i B l l ϕ , i.e.

when link B i D i is either fully stretched out or folded back on link D i C i .

It should be noted that there are some particular cases of these singular configurations when finite translations or rotations can be produced (self-motions) [START_REF] Briot | Self Motions of General 3-RPR Planar Parallel Robots[END_REF].

Static analysis

With regard to the static analysis of the manipulator, the input torques/force st j Q applied to the actuators M j (j = 1, 2, 3, 4) due to the force of gravity of links, joints and platform of the studied manipulator can be expressed as:

) 4 , 3 , 2 , 1 ( , 3 1 1 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ∑ ∑ = = j Q Q Q p n i st ipj st plj st j , ( 11
)
where st ipj Q is the load applied to the actuator j due to the gravity of the i-th link or bearings of the p-th leg (p = 1, 2, 3),

st plj
Q is the load applied to the actuator j due to the gravity of the platform (see detailed expression in [START_REF] Arakelian | Static and Dynamic Analysis of the PAMINSA[END_REF]). These loads can be represented in the form:

j p ip T ip st ipj z y x z y x Q ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ = 3 1 ) , , , ( ) , , , ( G J φ φ (12)
( ) j pl T plj Q G J = (13)
where J ip is the Jacobian matrix between the point P ip and the actuated variables [θ 1 , θ 2 , θ 3 , Z] (i.e. it relates the velocity of point P ip to the articular joints velocities), P ip is the center of masses of the i-th link, J is the general Jacobian matrix of the robot between the point P and the actuated variables, P is the center of masses of the platform, G and G ip are respectively the forces of gravity of the platform and the links (or bearings).

It is easy to see that:

( ) j pl T plj Q G J = =0, for j = 1, 2, 3. (14)
i.e. the input torques of the rotating actuators due to the gravity of the platform are cancelled because the gravitational forces are always perpendicular to the displacements (the platform carry out the displacements in the horizontal plane).

It should be noted that as the studied architecture of the manipulator is symmetrical, the values of the input torques for the actuators are also symmetrical but they are situated in different zones (rotations of ±120°).

Thus, the input torques of the rotating actuators taking into account the mass distribution of each leg can be written as following:

),

= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ ∑ = = j z y x Q Q p n i st ipj st j φ ( 3 , 2 , 1 ( , ) , , , ( 3 1 1
) 15
where

j p ip T ik st ipj z y x z y x Q ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ = 3 1 ) , , , ( ) , , , ( G J φ φ (16)
It was also shown in [START_REF] Arakelian | Equilibrage des manipulateurs manuels[END_REF] that the input torques due to the movable masses of the pantograph linkage may be cancelled by its optimal redistribution. Thus, by means of complete static balancing of legs, it is possible to cancel the loads due to the movable masses of the legs on the rotating actuators. Fig. 6 shows the variations of the actuator torques before and after mass balancing. After complete static balancing the potential energy of the manipulator is constant for any configuration and zero actuator torques are required. For example, the added masses are located at axes F i (Fig. 1a) and their values are 2.8 kg.

Prototype and experimental validation of the suggested design approach

In order to validate the properties of a new parallel manipulator, we have developed a prototype (Fig. 7) at the I.N.S.A. in Rennes. The displacements on the horizontal plane of the developed prototype are obtained by Harmonic Drive motors connected with the legs by means of toothed-belt transmission.

As an actuator, which controls the linear displacement of the vertical axes of the manipulator's legs, a PARVEX motor system was chosen. The pantograph linkage has been carried out with double rods in order to increase the stiffness of the mechanical system. The pantograph links of the built prototype are hollow tubes with a thickness of 1.5 mm. It is obvious that, for an industrial application, creating a pantograph with single and more rigid links is more attractive.

In order to validate the numerical simulations with experimental tests, we have measured the input torques/efforts of the actuators with and without a payload of 200 N (Fig. 7a,b) for the trajectory given in figure 8. The static balancing of the manipulator is experimentally accomplished by adding counterweights of 2.8 kg at the axes F i to the pantograph linkages (Fig. 9).

The experimental bench could be seen on figure 10. It is composed of the PAMINSA robot with its control system, a computer to interact with the user and a dSPACE 1103 card. To control the system, the softwares Matlab/Simulink/RTI/ ControlDesk have been used. The sampling period was 1 ms. For the motions in the horizontal plane, the PAMINSA is composed of three identical Harmonic Drive DC motors (3557012CR) in combination with Fulhaber incremental encoders (HEDS5500) and Fulhaber motion controllers (MCDC2805). A Parvex DC motor, (RX320DR1200) including an encoder and a brake system is used for the translation along the vertical axis, in combination

N

with an e-motion servo-amplifier (SCA-SS-70-10). The dSPACE 1103 carries out the interface between the PC and the servosystems. This card allows the control of the 4 motorized axes and has specific entries to which we have directly connected the 4 incremental encoders. We have also connected three LEM (HX10NP) current sensors to observe the torque of motor 1, 2 and 3. The robot is controlled by the classical PI law directly implemented under Simulink/ControlDesk. In order to make it, each motion controller has been configured to be only used as power amplifiers. Thus, the motor voltage is proportional to the voltage at the analog input of each motion controller. Optics Sensors have been added for the initialization of the robot and for the setting up safety measures.

The obtained results are presented in figure 11. The numerical simulations were validated via these experimental tests. The curves with and without payload for the 3 rotating actuators (Fig. 11a,b,c) are superposed. We can see that they are similar, i.e. the loads on these actuators are cancelled. The small differences might result from friction in the joints, manufacturing errors, elasticity of the links or tracking errors.

Regarding the vertical actuator (Fig. 11d), it supports the payload and the increase of the input force is significant.

Thus, it can be noted that the obtained measures validate the proposed design approach.

Conclusion

In this paper, we have presented a novel motion decoupling technique for parallel mechanisms and its application to the design of a new partially decoupled parallel manipulator called PAMINSA. The architecture of this parallel manipulator is composed of three legs, which are pantograph linkages.

Figure 1 .

 1 Figure 1. PAMINSA: a) kinematic chain of each leg; b) 3D view.

Figure 2 .

 2 Figure 2. The first model of PAMINSA.

  ( ) T x,y,φ = x , where x, y are the coordinates of the center of the isosceles triangle C 1 C 2 C 3 , φ is the angle of rotation around this center. The input parameters are expressed by the vector

Figure 3 .

 3 Figure 3. Projection of the manipulator in the horizontal plane.

..

  Figure 4 . Two singular positions corresponding to ( ) b pl R R 1 2 , 1 cos - ± = φ

Figure 5 .

 5 Figure 5. Singular configuration of the manipulator when the center S of the platform is on a circle of radius φ cos 2 2 2 n b n b R R R R -+

Fig. 6 .

 6 Fig. 6. Variations of the actuator torques for z = -0.6 m and φ  = 0° before (dark grey) and after (bright grey) static balancing of legs.

Figure 7 .

 7 Figure 7. The prototype of PAMINSA developed in the I.N.S.A. of Rennes.

Figure 8 .

 8 Figure 8. Position of the platform for z = -0.6 m and φ = 0 °.

Figure 9 .

 9 Figure 9. Counterweights mounted into the pantograph linkages.

Fig. 10 .

 10 Fig. 10. Experimental bench.

PAMINSA: Parallel Manipulator of the I.N.S.A.

Acknowledgments

The authors would like to acknowledge the financial support of Bretagne Valorisation, as well as Professor Jean Le Flecher for the advice during the development of the prototype.

Such a design allows the cancellation of the loads of gravity on the actuators, displacing the platform with the payload in the horizontal plane. This property has been validated by experimental tests carried out on the built prototype. It was shown that the input torques of the actuators displacing the platform in the horizontal plane with and without a payload of 20 kg are similar.

The authors believe that the proposed manipulators may be used in industrial applications for the manipulation of heavy equipment with little power consumption. Use in various fields are possible, depending on the type of industrial application.

Finally, we would like to note that the proposed manipulators have been patented [START_REF] Arakelian | Parallel robot comprising means for setting in motion a mobile element split in two separate subassemblies[END_REF] and additional information is available upon request. It should be also noted that on the basis of this design approach similar parallel manipulators with 3 to 6 degrees of freedom and having the same properties may be developed [START_REF] Arakelian | Design and Prototyping of New 4, 5 and 6 Degrees of Freedom Parallel Manipulators Based on the Copying Properties of the Pantograph Linkage[END_REF][START_REF] Briot | Analyze and Optimization of a New Class of Parallel Manipulators with decoupled motions[END_REF].