Serge Dumont
email: serge.dumont@u-picardie.fr

On enhanced descend algorithms for solving frictional multi-contact problems : applications to the Discrete Element Method

Keywords: Granular materials, Contact mechanics, Newton algorithms, Bi-potential, Augmented Lagrangian

. Following the ideas of Feng et al. [3], a new Newton method is developed to improve these classical algorithms and numerical experiments are presented to show that these methods are faster than the previous ones and provides results with a better quality.

Introduction

This is a first draft of a paper that will be submitted in a near future.

The paper is organized as follow: in the next part, we present the equations to be solved for the Discrete Element Method, and the frictional contact law considered. In the third part, we first present two classical methods to numerically solve the full problem, the first one based on the bi-potential theory, and the second one on the Augmented Lagrangian theory. Then, we show how these methods can be enhanced using an appropriate Newton method. The last part on this article is devoted to the numerical experiments in order to show the main properties of these algorithms.

2 Problem Setting 2.1 The equations of motion of a multi-contact system Classically (see for example [START_REF] Jean | The non smooth contact dynamics method[END_REF][START_REF] Jean | Unilaterality and dry friction in the dynamics of rigid bodies collection[END_REF][START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF]), the motion of a multi-contact system is described using a global generalized coordinate q (for N p particles, q ∈ R d×Np , where d = 6 for a 3D problem and d = 3 for a 2D problem). Due to the possible shocks between particles, the equations of motion has to be formulated in term of differential measure equation:

Md q + F int (t, q, q)dt = F ext (t, q, q)dt + dR [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution method[END_REF] where

• M represents the generalized mass matrix;

• F int and F ext represent the internal and external forces respectively;

• dR is a non-negative real measure, representing the reaction forces and impulses between particles in contact.

For the sake of simplicity and without lost of generality, only the external forces are considered in the following. The internal forces are neglected because the general case can be easily derived through a linearizing procedure.

Then, for the numerics, the equation ( 1) is integrated on each time interval [t k , t k+1 ], and approximated using a θ-method with θ ∈] 1 2 , 1] for stability reason (see [START_REF] Renouf | Conjugate gradient type algorithms for frictional multicontact problems: applications to granular materials[END_REF]). Therefore, the classical approximation of equation (1) yields M( qk+1 -qn ) = ∆t(θF k+1 + (1 -θ)F k ) + R k+1 q k+1 = q k + ∆tθ qk+1 + ∆t(1 -θ) qk

(2)

We will denote qfree k = qk + M -1 ∆t(θF k+1 + (1 -θ)F k ) the free velocity (velocity when the contact forces vanish). Then, the first equation in [START_REF] Saxcé | New inequality and functional for contact with friction: the implicit standard material approach[END_REF] becomes

qk+1 = qfree k + M -1 R k+1 . (3)
In order to write the contact law, for a contact c between two particles (1 ≤ c ≤ N c , where N c is is the total number of contact), we define the local-global mapping

u c = P * (q, c) q R = P (q, c)r c (4)
where u c is the local relative velocity between the two bodies in contact and r c is the local contact forces (u c , r c ∈ R d where d is the dimension of the problem, and P * is the transpose of matrix P ). We also denote P(q) the total-global mapping, for u and r in R d×Nc (vectors composed of all relative velocity and contact forces respectively):

u = P * (q) q R = P(q)r (5)
In the discretization, a prediction of q is computed to estimate the mapping P(q) (see equations [START_REF] Renouf | Numerical simulation of twodimensional steady granular flows in rotating drum: On surface flow rheology[END_REF] and [START_REF] Rockafellar | Convex Analysis[END_REF] in the following).

Using the equations ( 2) and ( 5), the discretization of the motion of a multi-contact system, with frictional contact between particles can be written:

ũk+1 = ũfree k + Wr k+1 law c (ũ c k+1 , r c k+1 ) = .true. ∀c ∈ {1, 2, ..., N c } (6)
where W = P * M -1 P is the Delassus operator, and ũfree k = P * qfree k is the relative free velocity. Notice that a Newton impact law is also considered (see [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF] and equation [START_REF] Ramaniraka | Thermomémcanique des contacts entre deux solides déformables[END_REF] in the following), that modify u k and u f ree k by ũk and ũfree k respectively.

The second equation in [START_REF] Fortin | Numerical Simulation of Granular Materials by an improved Discrete Element Method[END_REF] is the implicit frictional contact law that is in our case the classical Signorini condition and Coulomb's friction law.

The frictional contact law

In the local coordinates system defined by the local normal vector n and the tangential vector t ⊥ n, any element u and r can be uniquely decomposed as u = u n n + u t and r = r n n + r t respectively. In these coordinates, the unilateral contact law can be stated using the Signorini's conditions (see figure 1 for a graphical representation):

u n ≥ 0, r n ≥ 0, u n r n = 0. (7)
u n

Contact

No contact r n

Figure 1: The Signorini conditions

On the other hand, the Coulomb's law of friction can be stated using the algorithmic form (see figure 2 for a graphical representation):

      If r n = 0 then u n ≥ 0 ! No contact
Else if r n > 0 and r t < µr n then u = 0 ! Sticking Else r n > 0 and r t = µr n then ∃λ ≥ 0 such that u t = λ rt rt ! Sliding For a given friction coefficient µ, let K µ be the isotropic Coulomb's cone, which defines the set of admissible forces (see figure 3):

K µ = {r = r n n + r t : r t -µr n ≤ 0} (9) R R n t K µ K µ * Figure 3: The Coulomb's cone
The previous law can be also written:

      If r n = 0 then u n ≥ 0 ! No contact Else if r ∈ I(K µ ) then u = 0 ! Sticking Else r n > 0 and r ∈ B(K µ ) then ∃λ ≥ 0 such that u t = λ rt rt ! Sliding ( 10
)
where I(K µ ) and B(K µ ) are respectively the interior and the boundary of the cone K µ .

Numerical Resolution of the contact/friction problems

We will describe in this section the numerical algorithms that will be considered in the following. Generally, to solve the problem (6), the numerical algorithms considered are based on two levels: the global level where the equations of motion are solved, and the local level devoted to the resolution of the contact law.

Resolution of the global problem : the Non Linear Gauss Seidel Method (NLGS)

In this paragraph, we describe the algorithm used at the global level to solve the problem [START_REF] Fortin | Numerical Simulation of Granular Materials by an improved Discrete Element Method[END_REF]. Following the ideas of Jean and Moreau [START_REF] Jean | The non smooth contact dynamics method[END_REF][START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF], we use the non-linear Gauss-Seidel algorithm which is the most commonly used. It consists in considering successively each contact until the convergence. The numerical criterion used to state the convergence will be studied latter in the paper. This method is intrinsically sequential but it is possible to used a simple multi-threading technique which consists in splitting the contact loop into several threads. This method has been studied in [START_REF] Renouf | A parallel version of the Non Smooth Contact Dynamics algorithm applied to the simulation of granular media[END_REF] in the case where the local algorithm is based on the Augmented Lagrangian method.

Notice that it is also possible to consider at this stage more sophisticated method such as a conjugate gradient type method (see for example [START_REF] Renouf | Conjugate gradient type algorithms for frictional multicontact problems: applications to granular materials[END_REF]).

The standard bi-potential based method (SBP)

In this paragraph, we provide a first method to solve the contact problem, at the local level (contact point between two particles). The method is based on the notion of bi-potential, introduced by de Saxcé et al. [START_REF] Saxcé | New inequality and functional for contact with friction: the implicit standard material approach[END_REF].

Using the bi-potential framework, it can be shown (see for example [START_REF] Saxcé | New inequality and functional for contact with friction: the implicit standard material approach[END_REF][START_REF] Fortin | Simulation numérique de la dynamique des systèmes multicorps appliquée aux milieux granulaires[END_REF][START_REF] Fortin | Numerical Simulation of Granular Materials by an improved Discrete Element Method[END_REF][START_REF] Sanni | Modélisation et simulation bi et tri-dimensionnelles de la dynamique unilatérale des systèmes multi-corps de grandes tailles: application aux milieux granulaires[END_REF]) that a couple (u, r) verifies the Signorini-Coulomb contact rules if

b c (v, s) + v • s ≥ b c (u, r) + u • r = 0 ∀v, s (11)
where b c is the bi-potential

b c (-u, r) = Ψ R + (u n ) + Ψ Kµ (r) + µr n u t ( 12
)
and Ψ C stands for the indicatrix function of the set C:

Ψ C (x) = 0 if x ∈ C, Ψ C (x) = +∞ if x / ∈ C.
Consequently, the contact law can be written in a compact form of an implicit subnormality rule (or a differential inclusion rule):

-u ∈ ∂ r b c (-u, r). (13)
Then, for a contact c, at a NLGS iteration i, knowing the relative velocity ũc,i , the algorithm to compute r c,i+1 from r c,i is based on the minimization of the bi-potential (see for exemple [START_REF] Fortin | Simulation numérique de la dynamique des systèmes multicorps appliquée aux milieux granulaires[END_REF], page 51), using the inequality:

b c (-ũ c,i , r) + ũc,i • r ≥ b c (-ũ c,i , r c,i+1 ) + ũk,i • r c,i+1
∀r ∈ K µ [START_REF] Renouf | Conjugate gradient type algorithms for frictional multicontact problems: applications to granular materials[END_REF] or g(r) ≥ g(r c,i+1 ), ∀r ∈ K µ , if we denote

g(r) = Ψ R + (ũ c,i n ) + Ψ Kµ (r) + µr n ũc,i t + ũc,i • r. ( 15
)
The minimization of ( 14) is classically realized using a projected gradient projection (Uzawa method) without considering the singular term Ψ R + (ũ c,i n ). This minimization can also be viewed as the proximal point of the augmented force r -ρũ, with respect to the function r → ρb c (-ũ, r) (see for example [START_REF] Saxcé | New inequality and functional for contact with friction: the implicit standard material approach[END_REF][START_REF] Fortin | Simulation numérique de la dynamique des systèmes multicorps appliquée aux milieux granulaires[END_REF][START_REF] Fortin | Numerical Simulation of Granular Materials by an improved Discrete Element Method[END_REF]):

r = prox(r -ρũ, ρb c (-ũ, r)).
More precisely, the Uzawa method leads to compute the augmented force τ c,i+1 = r c,i -ρ∇g(r c,i ), where g is the differential part of g:

∇g(r c,i ) = ∇ r (µr n ũc,i t + ũc,i • r) = µ ũc,i t n + ũc,i ,
and to consider the force at next step as a projection of the augmented force onto the set of admissible force r c,i+1 = proj(τ c,i+1 , K µ ), that provides equations ( 21) and (22) in the resolution algorithm of the global problem. The proj(τ c,i+1 , K µ ) stands for the orthogonal projection over the convex K µ , that can be computed exactly (see [START_REF] Fortin | Simulation numérique de la dynamique des systèmes multicorps appliquée aux milieux granulaires[END_REF]).

This algorithm will be referred as the SBP (Standard Bi-Potential) method above and throughout.

For a sake of simplicity, we denote hereafter the descent direction

D c,i = µ ũc,i t n + ũc,i .
Remark 1 A first improvement of this method could be to compute the optimal step ρ c,i . To do so, we have to minimize

ρ → g(r c,i -ρD c,i ), (16)
or, more precisely,

ρ → Ψ R + (ũ c,i n ) + Ψ Kµ (r c,i -ρD c,i ) + µ(r c,i n -ρD c,i • n) ũc,i t + ũc,i • (r c,i -ρD c,i ) = Ψ R + (ũ c,i n ) + Ψ Kµ (r c,i -ρD c,i ) -ρD c,i • (µ ũc,i t n + ũc,i ) + Cte = Ψ R + (ũ c,i n ) + Ψ Kµ (r c,i -ρD c,i ) -ρ D c,i 2 + Cte. ( 17
)
We can observe that this method do not permit to choose an optimal parameter ρ since g, as a function of ρ, is linear, excepted in the case where D c,i / ∈ K µ . A solution could be to modify the function g, for example by replacing ũc,i by a prediction of ũc,i+1 using the equations of the dynamics. Unfortunately, this method do not provides good numerical results.

Then, the standard bi-potential based algorithm (SBP) can be written (see [START_REF] Sanni | Modélisation et simulation bi et tri-dimensionnelles de la dynamique unilatérale des systèmes multi-corps de grandes tailles: application aux milieux granulaires[END_REF] for example):

• Loop on the step time k -Prediction of a position (for the computation of the local-global mapping):

q k+ 1 2 = q k + ∆t 2 qk ; (18)
-Initialization of the motion: q0 k+1 = qfree k (initialization of the contact forces with R = 0).

-Loop on i ≥ 0 (NLGS), until convergence * Loop on the contacts c:

• Computation of the local-global mapping u-= P * (q k+ 1 2 , c) qk ; uc,+i = P t (q k+ 1 2 , c) qi k+1 (19)
• Newton shock law

ũc,i n = u c,+i n + e n u - n 1 + e n ; ũc,i t = u c,+i t + e n u - t 1 + e t (20)
• Prediction of the reaction:

τ c,i+1 = r c,i -ρ ũc,i t + (ũ c,i n + µ ũc,i t )n (21)
• Correction of the reaction:

r c,i+1 = proj(τ c,i+1 , K µ ) (22)
• Actualization of the generalized displacement:

qi+1 k+1 = qfree k + M -1 ( α≤c P (q k+ 1 2 , α)r α,i+1 + α>c P (q k+ 1 2 , α)r α,i ) (23) * End of the loop on contacts c.
-End of the loop on i of NLGS when the convergence is reached: qk+1 = qi+1 k+1 -Actualization of the generalized displacements:

q k+1 = q k+ 1 2 + ∆t 2 qk+1
• End of the loop on the step time k.

Remark 2 Notice that only one iteration of the Uzawa algorithm at the local level is considered. Various previous studies (see for example [START_REF] Joli | Uzawa and Newton algorithms to solve frictional contact problems within the bi-potential framework[END_REF]) show that there is no significant improvement of the method if several iterations of the Uzawa algorithm are considered at this stage.

Newton method and enhanced bi-potential method (EBP)

We introduce in this section a Newton method in order to speed up the convergence of the computation of the solution. This method has been already used, especially in the case of the augmented lagrangian method developed by Alart et al. [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution method[END_REF], and the ideas presented in this article follows those of Feng et al. [START_REF] Feng | The bi-potential method applied to the modeling of dynamics problems with friction[END_REF] and have been adapted to the problem of the discrete element method. The main idea of this technique is to find the solution of the optimization problem, not as a minimum of a functional, but rather as a zero of a function, using the Euler equation of the problem. Then a standard Newton method can be developed to solve this Euler equation. The technique is first described in the case of the bi-potential framework, and will adapted to the augmented lagrangian method farther.

We recall that the local problem that has to be solved, for each contact c can be written

     ũc k+1 = ũc,free k + Nc α=1 W cα r α r c = proj(τ c , K µ ) ∀c = 1, ..., N c (24)
where τ c = r c -ρ(µ ũc t n + ũ) is the augmented reaction (see 21), and W cα = P * (q k+ 1 2 ,c )M -1 P (q k+ 1 2 ,α ) is the local Delassus operator. This problem can be written equivalently

         ũc k+1 -ũc,free k - Nc α=1 W cα r α = 0 r c -proj(τ c , K µ ) = 0 ∀c = 1, ..., N c (25)
Reminding now that we want to use a Newton algorithm to solve theses equations inside the Non Linear Gauss Seidel loop on the variable i, we define now, for each contact c = 1, ..., N c , the function

f i c (χ) =      ũc,i -ũc,free k - Nc α=1 W cα r α,i Z c,i     
where :

• the vector Z c is the error on the prediction of the reaction

Z c,i (r c,i , ũc,i ) = r c,i -proj(τ c,i , K µ ), (26)
• χ c = (r c,i , ũc,i ) t , • χ = (χ 1 , χ 2 , ..., χ Nc ) t
Remark 3 The first equality in the relation f (χ) = 0 is the equation of motion for the bodies in contact, and the second relation is the frictional Coulomb law between the bodies in contact, written within the bipotential framework.

Then we have to write a Newton algorithm to solve the problem f (χ) = 0. This algorithm can be written, for a contact c, by substituting equations ( 21) and ( 22) in algorithm (SBP) by the followings:

• Initialization: χ 0 c = r 0 = r c,i , v 0 = ũc,i t , = 0
• Loop on , until convergence:

-τ c = r -ρ(µ v t n + v ) -Resolution: ∂f c ∂χ c (χ ) ∆χ c = -f c (χ ) (27)
-Actualization:

χ +1 c = χ c + ∆χ c
• End of the loop on until convergence, ũc,i+1 = v and r c,i+1 = r .

Remark 4 This algorithm needs more than one iteration at each Non Linear Gauss Seidel iteration to be efficient. As a consequence and compared to the Uzawa algorithm, the solution in the Newton algorithm is controlled by both the local (iteration ) and global convergence criteria (iteration i, see [START_REF] Feng | The bi-potential method applied to the modeling of dynamics problems with friction[END_REF][START_REF] Joli | Uzawa and Newton algorithms to solve frictional contact problems within the bi-potential framework[END_REF]).

The local convergence criterion for the Newton algorithm is defined by:

ε c N ewt (χ ) = v -u c,f ree k -W r + r -proj(r , K µ ) (28)
This criterion measure f c (χ ) that has to be sufficiently small.

The matrix ∂fc ∂χc (χ) represents the tangential matrix of the local equations for the contact c. This matrix is of dimension 6 × 6 for a 3 dimensional problem, and 3 × 3 for a 2 dimensional problem. For a 3 dimensional problem, the general form of this matrix is the following:

∂f c ∂χ c (χ) = -W Id 3×3 A c B c (29)
where

A c = ∂Z c ∂r n ∂Z c ∂r t 1 ∂Z c ∂r t 2 B c = ∂Z c ∂v n ∂Z c ∂v t 1 ∂Z c ∂v t 2 (30)
The matrices A c and B c takes different forms according to the contact status:

• First case: sliding contact.

In that case, we have

µ τ t ≥ -τ n τ t ≥ µτ n then P roj(τ, K µ ) = τ - τ t -µτ n 1 + µ 2 τ t τ t -µn and
Z c = ρ(µ v k t n + v k ) + τ t -µτ n 1 + µ 2 τ t τ t -µn
The computation of the derivatives of Z c provides the matrices A c and B c :

- ∂Z c ∂r n = - µ 1 + µ 2 τ t τ t -µn - ∂Z c ∂r t 1 = τ t 1 (1 + µ 2 ) τ t τ t τ t -µn + τ t -µτ n 1 + µ 2 t 1 τ t - τ t 1 τ t 3 τ t - ∂Z c ∂r t 2 = τ t 2 (1 + µ 2 ) τ t τ t τ t -µn + τ t -µτ n 1 + µ 2 t 2 τ t - τ t 2 τ t 3 τ t - ∂Z c ∂v n = ρn + ρµ 1 + µ 2 τ t τ t -µn - ∂Z c ∂v t 1 = ρ t 1 + µ v t 1 v t n - ρ 1 + µ 2   τt 1 τt - µ 2 vt 1 vt τt τt -µn + ( τ t -µτ n ) t 1 τt - τt 1 τt 3 τ t   - ∂Z c ∂v t 2 = ρ t 2 + µ v t 2 v t n - ρ 1 + µ 2   τt 2 τt - µ 2 vt 2 vt τt τt -µn + ( τ t -µτ n ) t 2 τt - τ t 2 τt 3 τ t  
For a 2D problem, these computations yields:

-

∂Z c ∂r n = µ 1 + µ 2 (µn -θ r t) - ∂Z c ∂r t = 1 (1 + µ 2 ) (-µθ r n + t) - ∂Z c ∂v n = ρ 1 + µ 2 (n + µθ r t) - ∂Z c ∂v t = ρµ 1 + µ 2 (θ v + θ r )n + µ(1 -θ r θ v )t
where θ v = sign(v t ) and θ r = sign(τ t ).

• Second case: sticking contact.

In that case, we have

µ τ t ≥ -τ n τ t < µτ n then Z c = ρ(µ v k t n + v k
) and the computation of the derivatives of Z c reads:

-A c = 0 3×3 - ∂Z c ∂v n = ρn - ∂Z c ∂v t 1 = ρµ v t 1 v t n + ρt 1 - ∂Z c ∂v t 2 = ρµ v t 2 v t n + ρt 2
For a 2D problem, these computations leads to:

-A c = 0 2×2 - ∂Z c ∂v n = ρn - ∂Z c ∂v t = ρµθ v n + ρt
• Third case: no contact.

In that case, the matrices A c = Id 3×3 and B c vanishes, and

χ +1 c = 0 v k

Resolution of the linear system

Generally, the drawback of a Newton is the computational cost of the linear system to be solved at each iteration. Here, the particular form of the tangent matrix allows the use of a condensation technique. More precisely, the linear system to be solved can be written:

-W Id 3×3 A c B c δr δv = -f -g . ( 31
)
The first equation yields δv = -f + W δr, and introducing this equality is the second equation leads to solve the linear system

(A c + B c W )δr = -g + B c f . ( 32
)
This properties halves the size of the linear system to be solved.

Remark 5 A drawback of the bi-potential framework is that, due to is specificity, it is rather difficult to consider fully coupled problems, where the contact law and another phenomena, such as electricity or thermic effects are strongly coupled. The other method presented in this paper has a better property from this point of view because it is based on a more standard mathematical background in the theory of optimization.

Newton method and enhanced augmented lagrangian method, (SAL) and (EAL)

In [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution method[END_REF], Alart et al. propose another method to solve the frictional contact problem. This method has been also used with various improvement (parallelization, conjugate gradient method for example) to solve multi-contact problems [START_REF] Renouf | Conjugate gradient type algorithms for frictional multicontact problems: applications to granular materials[END_REF][START_REF] Renouf | Gradient type algorithms for 2d/3d frictionless/frictional multicontact problems[END_REF][START_REF] Renouf | A parallel version of the Non Smooth Contact Dynamics algorithm applied to the simulation of granular media[END_REF][START_REF] Renouf | Comparison of algorithms for collisions, contact and friction in view of real-time applications[END_REF][START_REF] Renouf | Numerical simulation of twodimensional steady granular flows in rotating drum: On surface flow rheology[END_REF]. Even if the coupled frictional contact problem is not an optimization problem anymore, it is always possible to formally formulate a "quasi"-optimization problem, for which the constraint set depends on the normal components of the solution as a parameter. The solution is then searched as a saddle point of a "quasi" augmented Lagrangian of the problem. More precisely, the global problem on all unknowns that has to be solved at each time step (in place of equation ( 24)) has the following form:

   u = u f ree + Wr r ≥ 0, u ≥ 0, r • u = 0. (33)
In order to solve this problem, for a given r ∈ R 3×Nc , one can define the cartesian product of infinite half cylinder with section equal to the ball B(0, µr c ) of radius µr c by:

C(µr) = Nc c=1 R + × B(0, µr c )
and then, the granular type frictional contact problem is given by r ∈ argmin r∈C(µr)

1 2 r • Wr + u f ree • r = argmin r∈C(µr) J(r), (34)
and the projected gradient method the minimize this problem reads (for each iteration i of the NLGS algorithm):

r i+1 = proj(r i -ρ(u f ree + Wr i ), C(µr i+1 )), (35)
or r i+1 = proj(τ i+1 , C(µr i+1 )), with τ i+1 = r i -ρu i , u i = u f ree + Wr i . This algorithm will be referred to hereinafter as the SAL (Simple Augmented Lagrangian) method.

Notice that this method is very closed to the SBP method. More precisely, for a contact c, only the descent direction ũc,i + µ ũc,i t n in ( 21) is replaced by ũc,i and the projection r c,i+1 = proj(τ c,i+1 , K µ ) in ( 23) is replaced by

     r c,i+1 n = max(0, τ c,i+1 n ) r c,i+1 t = τ c,i+1 t τ c,i+1 t µr c,i+1 n .
Remark 6 On the contrary, it is possible to see the algorithm developed from the bipotential formalism as a slight modification of the algorithm above. Indeed, it is only necessary to change the set C(r) by K = Nc c=1 K µ , and to change the descent direction ũc,i by ũc,i + µ ũc,i t n which remains a descent direction for the SAL method, since

∇J(r c,i+1 ) • D c,i = -ũc,i 2 -ũc,i • (µ ũc,i t )n = -ũc,i 2 -µu c,i n ũc,i t which is negative since µ ∈ [0, 1].
Then, acting by analogy, we can develop a Newton method to find the minimum of J by seeking the solution as a zero of the function f (χ) where, for a contact c

fc (χ) =      ũc k+1 -ũc,free k - Nc α=1 W cα r α Zc     
, the vector Zc is the error on the prediction of the reaction

Zc (r c , ũc k+1 ) = r c -proj(τ c k+1 , C c (µτ c k+1 )), (36)
and the set C c (µr c ) is the set of admissible forces C c (µr c ) = R + × B(0, r c ). This method will be refered as the EAL (Enhanced Augmented Lagrangian) method hereafter.

Then, as bellow, we have three cases in the computation of the tangent matrix ∂ f ∂χ c (χ ) :

• First case: sliding contact (τ n > 0, τ t ≥ µτ n )

We have: proj(τ c , C c (µτ c )) = τ n n + τt τt µτ n t and Zc = ρv n nτt τt µτ n + r t . The computation of the derivatives of Zc provides the matrices A c and B c :

- ∂ Zc ∂r n = -µ τ t τ t - ∂ Zc ∂r t 1 = t 1 -µτ n t 1 τ t - τ t 1 τ t 3 τ t - ∂ Zc ∂r t 2 = t 2 -µτ n t 2 τ t - τ t 2 τ t 3 τ t - ∂ Zc ∂v n = ρ n + µ τ t τ t - ∂ Zc ∂v t 1 = -ρµτ n t 1 τ t - τ t 1 τ t 3 τ t - ∂ Zc ∂v t 2 = -ρµτ n t 2 τ t - τ t 2 τ t 3 τ t
For a two dimensional problem, these computations yields

A c = 0 0 -µθ r 1 B c = 1 0 µθ r 0 .
• Second case: sticking contact (τ n > 0, τ t < µτ n ) proj(τ c , C c (µτ c )) = τ c and the computation of the derivatives of Z c reads

-A c = 0 3×3 -B c = ρId 3×3 • Third case: no contact (τ n ≤ 0) proj(τ c , C c (µτ c )) = 0,

The global stopping (convergence) criterion

We present in this paragraph the convergence criterion on the global non linear Gauss-Seidel iterations. This criterion, developed from that proposed in [START_REF] Fortin | An improved discrete element method based on a variational formulation of the contact law[END_REF] has been extended in the case of the Newton and bi-potential (EBP) method, where some term are naturally vanishing in the original Uzawa and bi-potential (SBP) method. This criterion ε glob has been written in such a way that if the solution verify that ε glob is sufficiently small, then this solution has good properties on the equation of motion and Signorini Coulomb contact law. Consequently, this criterion stays valid for the methods developed with the augmented lagrangian (SAL and EAL methods). This criterion can be stated:

ε glob = 1 N c Nc c=1 ε c motion + ε c proj + ε bc + ε c pen ( 37
)
where:

• ε c motion = ũcũc m where ũc m = ũc,i + Nc α=1 W cα r α , so ε motion measures the error on the equation of motion (see equation ( 24), this term vanishes for the SBP and SAL method);

• ε c proj = r c -proj(r c , K µ ) 2
is the error for the projection on the Coulomb cone (vanishing for the SBP method);

• ε bc = ũc • r c + µr c n ũc t
is the absolute value of the bi-potential that has to vanish if and only if the couple (ũ c , r c ) verifies the Signorini Coulomb contact law (see formula 11);

• ε c pen = -min(0, ũc n ) is the value of the penetration.

Remark 7

One can notice that is absolutely necessary to verify in the criterion that there is no penetration, because nothing in the presented algorithm ensures that is condition is verify at the end of the loop. Moreover, if this condition is not satisfied, the rest of bipotential can be negative or equal to zero, even if the couple (ũ, r) is not a solution.

Numerical results

We present in the section three numerical examples with an increasing complexity. In these computations, the descent parameter ρ is taken in such a way that the result is optimal, in terms of time computing. Denoting ρ = m i m j m i +m j 1 ∆t , for the SBP and the SAL methods, we have chosen ρ = 0.6ρ, whereas for the EBP and the EAL methods, it is better the take ρ = ρ. We recall that it has been show that, for the the bi-potential method (see for example [START_REF] Feng | The bi-potential method applied to the modeling of dynamics problems with friction[END_REF]) and the augmented lagrangian method (see for example [START_REF] Renouf | Conjugate gradient type algorithms for frictional multicontact problems: applications to granular materials[END_REF]), the parameter ρ has to verify ρ < 2ρ in order to ensure the convergence. Generally, for these two methods, the convergence is very sensitive on this parameter. We will show in the last paragraph of this study that for the EBP method, the parameter ρ can be taken in a large range around the value ρ without changing dramatically the convergence of the method.

At each iteration of the NLGS algorithm, the Newton algorithm is stopped either if the convergence is obtained (ε c N ewt ≤ 10 -5 ), or if the number of iteration of the Newton algorithm reached 100 when there is no convergence.

Ball sliding on a plane

In this first example, we consider a ball placed on a table with an initial horizontal velocity equal to 1.5 m• s -1 . The ray of the ball is equal to 5 • 10 -3 m, and the friction coefficient between wall and ball is equal to µ = 0.7. The time step of discretization is equal to 10 -4 s. In this experiment, the ball first slides on the table, and then the ball rolls without sliding. The global stoping criterion is equal to ε glob = 10 -10 . Figure 4: Example 1 -A ball is launched with an initial horizontal velocity (left). First, the ball slides. Then, the ball rolls without slipping (right).

We can observe from these numerical results that the error coming from the projection is very small for the four methods. The Standard Bi-Potential (SBP) method and the Standard Augmented Lagrangian method (SAL) give very closed results, both in term of quality (see figures 5 and 6) and in term of time computing (see table 1). Nevertheless, we can notice that the time computing is smaller with the SAL method, because there is less computations at each iteration (no term such as ũt and projection easier to compute for example). The Enhanced Bi-Potential method provides better results, both in term of quality (see figure 7) and in term of time computing (6.5% better). The Enhanced Augmented Lagrangian method converges after the first Non Linear Gauss Seidel iteration for every time steps, and consequently, this is the faster method on this example (7% faster than the SBP method).

Method

Sedimentation of 4 balls in a box

In this second experiment, we consider the sedimentation of 4 balls of radius ranging from 4 • 10 -4 m to 5 • 10 -4 m. For the computations, the time step of discretization is equal to ∆t = 10 -4 s., and the Non linear Gauss-Seidel loop is stopped either if the the global stopping criterion on the NLGS method is equal to ε glob = 10 -10 , or after 5000 iterations if there is no convergence (this case never occurs in this experiment). The friction coefficient between the balls and between the balls and the walls is equal to µ = 0.3. Like in the previous simulation" the SBP and the SAL method methods provide very similar results (see figures 9 and 10). For these two methods, we can notice that here the global error is essentially due to the penetrations. The SAL method is 2% faster than the SBP method (table 2).

Results obtained by the EBP method are better (figure 11), and here the overall error is governed by the error on the equations of the motion. The EBP method is 11% faster than the SBP method, and the penetration is 5 times smaller. In this example the EAL is the faster method (16,8% faster than the SBP method), and the penetration is very small (see figure 12).

Sedimentation of 500 balls

In this example, we consider the sedimentation of 500 balls (see figure 13) of radii ranging from 2.5•10 -4 m to 5•10 -4 m, the time step of discretization is equal to ∆t = 5•10 -5 s, and the Non linear Gauss-Seidel loop is stopped if the global estimator (37) verifies ε glob ≤ 10 -12 or after after 5000 iterations if there is no convergence. The friction coefficient between the balls and between the balls and the walls is equal to µ = 0.3. The results in table 3 are obtained after 1000 time steps.

In this example, the difference between methods SBP and SAL on the one hand, and the method EBP and EAL on the other hand is larger (see table 3). We can notice that the SAL method is 10.95% faster than the SBP method, and the EBP is 21.83% faster than the SBP method. Here, the EAL method is no longer the faster one, but the penetration is very small. Again, for the two first methods the global error is essentially due to the penetrations whereas the two last methods, the error is essentially due to the failure to follow precisely the equations of motion.

Discussion on the descent parameter ρ

We consider again the third example solved by the Newton and bi-potential method (ε tot = 10 -8 , maximal number of iterations of Newton method equal to 100, ε N ewt = 10 These results show one of the main advantage of the EBP method. Indeed, one can notice that in table 4, the CPU time and the quality of the solution are very similar if α is equal to 1 or 2. Even if α is equal to five, the convergence is not to damaged. In that case, one remain the the SBP and the SAL are no longer convergent. If the parameter α is small, the method converges but the convergence rate is very small. One can notice that the EAL method is much more sensitive about the parameter α, essentially in the convergence of the Newton method.

Conclusion

The results presented show that, using an appropriate Newton method, it is possible to improve the computational time over that 20% compared to the standard methods. Moreover, one principal drawback of that type of methods, that is the dependance of the results on the parameter ρ does not exist anymore.

In the future, this method will be extended to the case of a contact law with adhesion. This improvement will be realized in a near future.

Figure 2 :

 2 Figure 2: The Coulomb conditions

  then the matrices A c = Id 3×3 and B c vanishes, and χ +1 c

Figure 5 :Figure 6 :

 56 Figure 5: Example 1 -Convergence for the standard bi-potential based method, 5 th iteration

Figure 7 :

 7 Figure 7: Example 1 -Convergence for the Newton and bi-potential method, 5 th iteration

Figure 8 :

 8 Figure 8: Example 2 -Sedimentation of four balls under the gravity effect.

Figure 9 :

 9 Figure 9: Example 2 -Convergence of the non-linear Gauss-Seidel iterations for the standard bi-potential based method (1000th time step) The two last curves overlaps, showing that the global error is governed by the error of penetration.

Figure 10 :Figure 11 :

 1011 Figure 10: Example 2 -Convergence of the non-linear Gauss-Seidel iterations for the standard augmented lagrangian method (1000th time step). The two last curves overlaps, showing that the global error is governed by the error of penetration.

Figure 12 :

 12 Figure 12: Example 2 -Convergence of the non-linear Gauss-Seidel iterations for the Newton and Augmented Lagrangian method (1000th time step). The two last curves overlaps, and the other ones does not appear on the figure because the corresponding errors are lower than 10 -16 .

Figure 13 :

 13 Figure 13: Example 3 -Zoom on balls falling under the gravity effect. Initial configuration on the left, final configuration on the right.

Figure 14 :

 14 Figure 14: Example 3 -Convergence of the non-linear Gauss-Seidel iterations for the standard bi-potential based method (1000th time step). The two last curves collapse.

Figure 15 :

 15 Figure 15: Example 3 -Convergence of the non-linear Gauss-Seidel iterations for the standard augmented lagrangian method (1000th time step). The two last curves collapse.

Figure 16 :

 16 Figure 16: Example 3 -Convergence of the non-linear Gauss-Seidel iterations for the Newton and bi-potential method (1000th time step). The two last curves collapse.

Figure 17 :m i m j m i +m j 1 ∆t

 171 Figure 17: Example 3 -Convergence of the non-linear Gauss-Seidel iterations for the Newton and Augmented Lagragian method (1000th time step). The two last curves collapse.

Table 1 :

 1 Comparison of the results obtained by the four methods on the first example (after the 2000th time step).

		Number of	Error	Total
		NLGS iterations	ε glob	CPU time (s)
		(last time step) (last time step)
	SBP	18	0.384 • 10 -10	9.44
	SAL	18	0.384 • 10 -10	9.28
	EBP	1	0	8.83
	EAL	1	0.175 • 10 -13	8.78

Table 2 :

 2 Comparison of the results obtained by the four methods on the second example (after the 1000th time step)

	Method	Number of	Error	Maximal	Total
		NLGS iterations	ε glob	penetration	CPU time (s)
		(last time step) (last time step) (last time step)
	SBP	305	0.949 • 10 -12	0.310 • 10 -11	2.92
	SAL	301	0.980 • 10 -12	0.340 • 10 -11	2.87
	EBP	161	0.635 • 10 -12	0.641 • 10 -12	2.59
	EAL	158	0.973 • 10 -12	0.208 • 10 -19	2.43

Table 3 :

 3 -5 , 500 th Comparison of the results obtained by the four methods on the third example (after the 1000th iteration, N max = 5000 iterations)

	Method	Number of	Error	Maximal	Total
		NLGS iterations	ε glob	penetration	CPU time (s)
		(last time step) (last time step) (last time step)
	SBP	5000	0.119 • 10 -6	0.213 • 10 -5	1092.95
	SAL	5000	0.135 • 10 -6	0.533 • 10 -5	973.31
	EBP	5000	0.156 • 10 -6	0.286 • 10 -6	854.31
	EAL	5000	0.101 • 10 -6	0.390 • 10 -17	916.65

Table 4 :

 4 Comparison of the results obtained for various values of ρ = αρ on the third example (after the 500th iteration, N max = 5000 iterations)

	α Number of NLGS	Maximal	Total CPU
		iterations	penetration	time (s)
		(last time step)	(last time step)
	5	652	0.110 • 10 -6	65.08
	2	414	0.177 • 10 -6	55.86
	1	750	0.149 • 10 -6	53.95
	1 2 1 5	812 667	0.634 • 10 -6 0.219 • 10 -5	78.43 176.14

Acknowledgment: This work was supported by the french CNRS grant "PEPS Opticontact", and by the project "TRIBAL" supported by the Picardy region and the European funds FEDER.


PhD Pascal-Yan Sayegh
email: yansayegh@ymail.com

The Rule of the Sacred: Religion, Nationalism and Secularity

Introduction

The title of the panel in which this paper is being presented is "Nationalism and its Others", and in this context religion appears a priori as one such "Other". In this paper, I would like to take the other side of the same coin -which is the question of the relation between nationalism and its others -and begin by considering what religion and nationalism have in common. In a recent article Rogers Brubaker discusses different approaches of the study of the relation between religion and nationalism. The first of these four approaches is precisely "the strategy of treating religion and nationalism as analogous phenomena" (Brubaker, 2012:3). How should we consider an analogy between two complex things which we already formally, terminologically, contextually, have predicated as different? One could say it is precisely because they are presented to us as different that we can consider the relation between them. What Brubaker suggests is connecting "both phenomenon to more general social structures and processes" and considering "religion and nationalism […] under more encompassing conceptual rubrics: as a mode of identification, a mode of social organization, and a way of framing political claims" (2012:4). With this in mind, I suggest defining religion and nationalism as sociohistorical imaginaries, and consider this basis as their formal relation.

A sociohistorical imaginary is what institutes a given society and simultaneously provides the tools to interpret its reality [START_REF] Castoriadis | The Imaginary Institution of Society[END_REF].

In this paper, I first discuss aspects of the linguistic forms which is the prism from which I approach the relation between religion and nationalism. By considering the terms themselves, we can interpret elements of the inherited code, which is the English language. Substituting signification to the Aristotelian category of substance, I make use of the associated categories of form and matter to elaborate on the relation between religion and nationalism. Considered as necessarily correlated, these categories articulate a dynamics between language and significations, which appears as the common modus operandi of religion and nationalism. I further discuss how their significations are negotiated sociohistorically, instituting specific times and spaces, and corresponding temporal and spiritual institutions. The paper concludes on the historical role of secularity in the imaginary distinction between religion and nationalism.

Elementary forms

Brubakers conceptual rubrics relate to the way Cornelius Castoriadis characterises the social imaginary, for "the imaginary of the society or of the period considered" is: "This element -which gives a specific orientation to every institutional system, which overdetermines the choice and the connections of symbolic networks, which is the creation of each historical period, its singular manner of living, of seeing and of conducting its own existence, its world, and its relations with this world, this originary structuring component, this central signifying signified, the source of that which presents itself in every instance as an indisputable and undisputed meaning, the basis for articulating what does matter and what does not, the origin of the surplus of being of the objects of practical, affective and intellectual investment, whether individual or collective […]." (Castoriadis, 1997:243) The social imaginary is a dynamics of grids of significations, hierarchies in the etymological sense, ordering and selecting what is pertinent (Castoriadis, 1986:465); what Brubaker terms "ontologies and structures of justification " (2012:17). Such significations can be seen as universal in the sense that we, as complex cultural animals, necessarily produce and reproduce such significations to make sense of and in reality. In De Anima, Aristotle examines the nature of the soul and elaborates his ontology of substance.1 Substance and soul can be equated as they both represent the cause and origin of existence (De Anima II.4, Hicks 1907:49), the reality of reality so to say. By analogy, paradigmatic imaginary significations can be considered the substance or the soul of a given culture.

Aristotle presents three senses of substance: form, matter and the form-matter compound (De Anima II.1, Hicks 1907:65). The aim of the present explanation of these categories is to provide working definitions before elaborating on the relation between these categories.

Extracted from the complexity of Aristotle's ontology, they appear here as necessarily simplified. The distinction form-matter is one of the main categorical distinctions established by Aristotle (Bernardt, 1979:109). The term "form" translates two different terms: morphè and eidos. The former designates the aspect of things, their superficial forms, whereas the latter, which is the one referred to in the form-matter distinction, refers to formal reality, understood as the rational structure of reality (Pellegrin 2009:48). The term "matter" translates hylè ("wood"), but as a concept, it is yet not reducible to the physical, material aspect of things. More generally, matter designates what is subjected to change, what can potentially take form and lose its form (Cohen, 2009b). The complex relation Aristotle elaborates between these categories is best summed up by the following exegesis:

"Form may be accidental to the matter that it informs, but it is essential to the compound substance (i.e., the compound of matter and form) that it is the form of. Form is what makes the individual plants and animals what they are. Therefore, it is the substance of those individuals." [START_REF] Cohen | Aristotle on Substance, Matter, and Form[END_REF] What this suggests is that the relation between substance (signification), form and matter operates as a kind of cycle. The logical nexus this relation creates shows us that the form of a substance is necessarily correlated to the latter, while matter may first appear of a secondary importance. Nevertheless, as social imaginaries can be primarily defined as instituting a (cultural, symbolical) time and a space, the framework here derives fundamentally from Aristotle's perspective whereby substance is postulated as a universal. By instituting a spacetime, the sociohistorical imaginary institutes form and matter that is relative to its frame.

While a signification may well be conceived as universal, form and therefore matter are predicated from a given perspective.2 However complex, they are limited and limiting. As a relative process, formulation necessarily mirrors materialisation, or there would be no realisation. And while form determines or predicates matter, it is the conjunction/disjunction of the two which is significant.

In the imaginary space-time, codetermination appears as defining the relation between form and matter appears codetermination of form and matter. The reason why Aristotle states form as the cause, the origin of the cycle, is because the formal dimension is the first conscious, rational, logical dimension of the social imaginary we apprehend. As form determines matter, matter is the second dimension, the dimension where experience is formed. And it is only through this dynamic codetermination, a speech-act so to say, that a sociohistorical sense can emerge and significations be instituted. The two different perspectives are nevertheless articulated together. The manner in which this is operated is through the formation of a code, a language, conceived in time as the repetition of the formulation of experience and the signification of forms.3 These different codes are our only means of communication within a cultural or imaginary space-time, as there is no social world outside language (in reference to Paul Ricoeur, Langdridge, 2006:646). In short, we reach an understanding of the reality of this process by imagining a chronology, a hierarchy of forms. And again, looking from the perspective of the signification, rather than the form which institutes a cultural origin, Castoriadis defines the concept of signification as "an indefinite skein of interminable referrals to something other than (than what would appear to be stated directly) " (1997:253).

These referrals are the forms used in the different languages (linguistic, but also artistic, physical, sensual, etc.) and the experiences tied to them. Castoriadis then points to the central role of language: "These other things can be both significations and non-significations -that to which significations relate or refer. The lexicon of the significations of a language does not revolve around itself, is not closed in upon itself, as has flatly been stated. What is closed in upon itself, fictively, is the code, the lexicon of ensemblist-identitary signifieds, each of which can take on one or more sufficient definitions. But the lexicon of significations is always open; for the full signification of a word is everything that can be socially stated, thought, represented or done on the basis of this word. " (1997:253) The phrase "the full signification of a word" reduces language down to linguistic language.

The reason Castoriadis exemplifies the relation between signification and language in this way is because linguistic language is considered language par excellence. On the other hand this reduction opens up a complexity, which already appears formally in the definition that follows the reductive statement: "everything that can be socially stated, thought, represented or done." The situation may still appear as paradoxical: in general, words (or signs) allow us to give form, to formulate, to predicate our reality; and in particular, they inform, encrypt, make this reality more complex. This disjunction is only resolved on a higher level, through a process also involving the dimension of conjunction.

One might wonder here how a categorisation Aristotle applies to plants or animals, also applies to the much more abstract significations we are concerned with, those of religion and nationalism. But even if a form is considered necessary in its relation to a signification, the actual form of a signification is accidental: on the higher level we mention above, matter is form and vice versa. The interrogation is therefore epistemological and should thus be rephrased as follows: how does the process of signification which applies to forms which matter in the natural world, applies to those which matter in the cultural world?

Forms of matter

By considering religion and nationalism as sociohistorical imaginaries, religion and nationalism are conceived as forms relative to the paradigmatic significations that they are the forms of. This infers in return that the signification to which these forms are relative to are restricted by the sociohistorical matter framed in the imaginary space-time continuum. So the first reason this categorisation applies is because religion and nationalism are tied to social experiences, or rather they are social experiences. I have started by defining religion and nationalism as social imaginaries suggesting the potential analogy between the two concepts.

If we look up their respective definitions in the Oxford Dictionary, the definitions we find are fairly abstract. The first entries of the definitions of religion and nationalism read, respectively: "the belief in and worship of a superhuman controlling power, especially a personal God or gods" and "patriotic feeling, principles, or efforts" (Oxford Dictionaries Online, ODO, 2010). The primary sense in which they are abstract is because neither definitions provide a (particular) when nor where. At first sight, they are each conceived as equal to themselves, as identical, in time and space. 4 And yet, both words are themselves, as linguistic formations, set in a given frame. In the case of religion, for instance, the last entry on the "origin" of the word in the Oxford Dictionary reads: "Middle English (originally in the sense 'life under monastic vows'): from Old French, or from Latin religio(n-) 'obligation, bond, reverence', perhaps based on Latin religare 'to bind'". The first information places the term in its original time frame of Middle English: a period between the 11 th and 15 th centuries (which corresponds to the High and Late Middle Ages), in a corner of the world loosely referred to as England. Although in brackets, the second information is perhaps just as much important. The original meaning, "life under monastic vows", greatly restricts the signification of the term religion compared to the contemporary sense provided by the first entry of the definition. The evolution of the same form, meaning a precise hierarchical and normative relation with the institution of the Church, to the contemporary meaning encompassing all cultural relations to a God figure comes under the poetic or metaphoric aspect of language [START_REF] Ricoeur | The Rule of Metaphor: The Creation of Meaning in Language[END_REF]. In this particular case, this translation can be described as a transhistorical synecdoche.

The image of the initial signification of religion refers to the regular clergy, those members of the Church of Rome who follow, literally, the rules of a religious order. Traditionally, the regular clergy is contrasted with the secular clergy, which means those members of the Church who live "in the world" (ODO, 2010). In the Middle Ages, the world of the secular clergy of the Church of Rome is medieval Europe. Described as feudal,5 the societies of medieval Europe themselves are described as organised according to a set of cultural rules, and are therefore societies of order. 6 In addition, the contemporary sense of the term secular, from which the concept of secularity referred to in this paper derives, appears as also resulting from a transhistorical synecdoche of a similar nature as the term religion, where a part is used to describe the whole. 7A synecdoche is a common figure of speech, a particular type of metaphor, were a part is substituted for the whole or the whole for a part. Contrary to other common figures of speech, such as comparisons, metaphors do not present a semiotic sign which signifies the process of analogy or substitution.8 But most importantly, although certain metaphors have become worn out, in the The Rule of Metaphor, Paul Ricoeur shows how meaning is produced and reproduced across levels and categories of discourse through the living power of the metaphor as "the conjunction of fiction and redescription" (Ricoeur, 2003:291). The process of the "metaphoricity of metaphors" thus corresponds to the creation of social significations. The fictional aspect of metaphors relates to the way human beings experience reality: as this reality is elusive, fictional elements are necessary to organise it in the form of narratives.

Redescription refines the notion of identity: a metaphor is not simply the copy of what it refers to; it is a transfer of the same to the other (in time, in form, etc.).

History is always taken in medias res, which means that in matter, continuity and discontinuity are always interrelated. Once sociohistorical matter is predicated by a form (through redefinition, reformulation, translation, etc.) it has the potential to become real, to be imagined as real, and concurrently, because it becomes transmittable, it becomes social. The languages we use have been transformed semantically, grammatically and graphically.

Writing itself fixates forms for posterity; it is therefore not fortuitous that the invention of the printing machine also participated in the homogenisation of languages and their formation into national languages [START_REF] Anderson | Imagined Communities: Reflections on the Origin and Spread of Nationalism[END_REF]. Ricoeur argues that language is in fact part of the process of experiencing human reality because of what he terms the expressibility of experience: "To bring [experience] into language is not to change it into something else, but, in articulating and developing it, to make it become itself" (1981:115). On a phenomenological level, matter also determines form: its indeterminacy renders its reformulation necessary for its human, social reality. In time and space, this means that formulations do not necessarily correspond to the socio-historical matter they describe. Indeed, our experience of things always appears to us as a tension between similarity and difference. Our modern perception of experience realigns and fixates form and matter in time; perhaps for reasons of clarity, but also partly because we have cultural tendency to perceive things in a linear fashion, and because it essentially is an ideological perspective on things. 9 In this particular instance it is a perspective on history, a view of history, which processes historical material into linear, positivist narratives. And such linearity is one of the characteristics of national historiography, thought and developed from the perspective of positivism (Geary, 2002). But nationalist historiography would not come into being before the nineteenth century, in the secular universities of Europe (and by extension, in the rest of the world), and genealogical linearity is not exclusive to nationalist narratives. Of course, most narratives are indeed linear, the traditional form is chronological as the various narratives found in the Bible illustrate. Both cases, of national historiography and in the biblical narratives, are representative of significations of authority, providing myths that found the spaces and the times to be reproduced in their sociohistorical realization. 10The two imaginaries of religion and nationalism are thus analogous as far as they determine matter in a linear way. And yet, they appear as different on many other levels. Formally of course, but more fundamentally on the level of the actual spaces and times were and when they operate these determinations. In European history, the frames they institute are different: most national histories set the origins of their nation in the early medieval ages -the dimension of the time of nations is a much shorter time span that is instituted by the book of Genesis; furthermore, the space of nations is a usually clearly defined and fixed territory, the homeland. A priori, this space-time does not challenge the space-time of biblical creation. In other words, the incipit of nationalist significations (the origins) is not as paradigmatic as that of religious significations (the creation). An illustration on how this is significant is that there are no national ideologies which promote a different calendar for instance, a different organization of the week in seven days, etc. Indeed, after being reproduced by national cultures, this has now logically become part of a contemporary global culture. Challenging our calendar may seem absurd, partly because it is reproduced across the globe as the international calendar, and more significantly because we believe that its basic form corresponds to the most natural. And yet, even within Christianity, the topic has been much debated, and different branches still do not have identical calendars. And a calendar, is a form which predicates natural cycles, it is not a natural signification. More precisely, it is a cultural paradigmatic signification, of the measuring and remembering of human time, that has been negotiated throughout history and across cultural spaces.11

Sociohistorical significations

The contemporary secularized form of the Gregorian calendar predicates our global contemporary time scale, but it also signifies the transformation, or translation of its historical forms. The different forms are negotiated against other forms, which are sometimes forgotten and generally considered obsolete, or primitive. Nevertheless they are part of what their relations, their relative negotiations result in: a form of forms, a transhistorical signification.

Nationalism can be considered to be secular in the sense that it does not challenge a certain number of "regular" paradigmatic significations of religion, as illustrated in the present case with Christianity. Nationalism negotiates relative spaces between such significations to exert its own cultural hegemony. Moreover, as suggested in the above illustration, cultural forms correspond to sociohistorical experiences. The illustration also presents how forms are signified through other forms in relative dimensions. Geopolitical institutions are such discursive forms. Indeed, they signify a certain spatial imaginary. For example, national institutions and national territories (and histories and languages) are ideally tied together.

Similarly, religious institutions are also ideally tied to the elements of the world they While negotiations surrounding our calendar time scale appear to be relatively stale lately, negotiations surrounding our political institutions within this time scale occupy a generally more turbulent space. While nationalism writes linear and coherent national stories, the story of nationalism follows a formally more chaotic history. The period of transition between medieval institutions and modern institutions has been determining for the formation of nationalism (Brubakers, 2012:6). In other words, different stages in the negotiations for cultural power between state institutions and the institutions of the Church of Rome have in time and in space made nationalism, the ideal of nationalism, possible. It is not fortuitous that this transition relates to linguistic transitions as well. Middle English is an expression that does not capture the depth of the its history. From a subaltern group of languages and dialects, one particular version of Middle English was renegotiated as a political and legal (albeit chiefly oral) language in the transition between the High and Late Middle Ages [START_REF] Crystal | The Stories of English[END_REF]. The high point of this transition on the part of political institution is the Pleading in English Act (or Statute of Pleading) of 1362 under the reign of Edward III. It is certainly significant that this occurred during the reign of Edward III who was the king who initiated the Hundred Years War for the crown of the kingdom of France. The decline of Anglo-Norman, or Norman French, is thus concurrent to the political recognition of an English language during a major political and military struggle between two feudal states.12 And yet, this foundational stage in the becoming of the English language into a national language is not the result of a nationalist ideology: as the original text of the Act (written in the Legal French of the period) reads, it is rather in terms of usage, or customs, that the act is justified. 13 At the same time, we have to recognise how such developments have participated in informing (although perhaps a priori in an arbitrary way) modern significations of people, of popular sovereignty, and of course of standardised languages, all participating at their level to the gestation of national imaginaries.

At a later stage, another high point in the "magmatic" evolution of the English language has certainly been the Reformation which presents us with more substantial elements of the Church and state, but also with elements of more fundamental "cosmological", changes (Szporluk 1988:86): it presents us with the conjunction of challenges against the doctrines but also against the hierarchies of power associated with it. 14 The reformation in the Kingdom of England was a process which originated under the rule of king Henry VIII, who was instituted supreme head of the newly formed Church of England, effectively -but not entirely dogmatically -separating it from the Holy Roman Catholic Church by a series of Parliamentary acts in the 1530s (Pendrill, 2000:88-94; see also [START_REF] Jones | The English Reformation: Religious and Cultural Adaptation[END_REF]. The Protestant reformation which was gaining momentum in mainland Europe exerted great influence on the process of the English reformation, but it was not until Elizabeth I's accession to the throne that protestant dogmas became dominant.

The theological principle of the sola scriptura ("by scripture alone") was the ultimate principle in protestant theology vindicated in the Reformation. It is a formal principle which establishes the Bible as the only source of legitimate authority (McGarth, 2007:59). 15 One of the main consequences is the depreciation of the other sources of power: the clergy and tradition. Their successful discursive deprecation has resulted, in time, in the relativisation of their signification. Another consequence is the promotion of a less restricted access to the Bible, which entailed the development of popular, vernacular versions of the Bible. 16 The cosmos which was promoted through the Reformation was radically different from the then traditional order which established the Holy See as a central authority in medieval Europe. 17 In consequence, papal power and its signification(s) diminished in pace with the spread of the Reformation across Europe. The new order was finally settled in 1648 with the signing of the Peace of Westphalia treaties, which put an end to the religious violence and wars which had torn Europe during the many preceding decades. The most relevant element in regards to the Peace of Westphalia is that it established the recognition of state sovereignty, signifying a new form of sovereignty. The recognition of the absolute sovereignty of states, de facto and de jure, effectively disintegrated the significance of the papal power of recognition of kingship in favour of power of the kings and princes of Europe [START_REF] Straumann | The Peace of Westphalia (1648) as a Secular Constitution[END_REF][START_REF] Suter | Globalization and the New World Order[END_REF].

14 There were other radical innuendos of imaginary change in the catholic space, notably in the cosmopolitanism and humanism of theologians and scholars such as Michel de Montaigne, Desiderius Erasmus Roterodamus and Thomas More, which were the first "beams" of the Enlightenment period and constitute the beginning of what is called the "Republic of Letters". 15 The second general category in Christian theology being tradition, or "material principles". 16 At first, Henry VIII was opposed to the introduction of an English translation of the Bible. Eventually, the Tyndale Bible translation, even if incomplete, would be the first to benefit from printing and will be considered 17 In continuity of the already authoritative symbol of the city of Rome. The conversion of Frankish king Clovis in the fifth century A.D. effectively established the medieval power association between Rome and European states carried out through the clergy which answered both to their respective monarchs and to the Pope. This association was further expressed in the papal recognitions of kingship which can be assimilated to the symbolic act of incarnation, establishing the "divine right" of the kings. The first such explicit recognition took place on behalf of Frankish king Pippin in 757. At the same time, the king also decreed the first legal settlement of the question of the temporal power of the Papacy (see Saebø, 2000).

What these readjustments and mutations signify is the modification of the social order: the secular clergy was loosing its prime position. Without being entirely relativised, the Early Modern period in Europe is marked by the mutlidimensional negotiations which the demoting of the political cum cultural power of the Church of Rome represents. This was of course no sudden change. This power had already been severely hampered in the century which separated the early stages of the Reformation and the Peace of Westphalia. The well established divine right of kings was one of the elements which would justify the instituting of King Henry VIII as supreme head of the Church of England. This signified that the king was recognised as being invested with kingly power directly by the form which represented the ultimate and primary signification: the figure of God. 18 By the time the Peace of Westphalia was being negotiated, this signification had informed all the European heads of state (Blin, 2006:56).19

Conclusion

I have briefly examined fragments of the political and ideological contestation of the medieval order, primarily the relation between the Church of Rome and the Kingdom of England to illustrate the multidimensional relations between religion, nationalism and by extension secularity. It is in the transitions between significations that spaces of negotiations are formulated. Although nationalism, as a doctrine, will not be formulated before late modernity, it is during this transition from feudal to modern society that many of the significations which would recompose into nationalism have been instituted. This brief examination has been set in a framework based on categories of form and matter. A form is an authority when it establishes a hierarchy of the forms that inform it. The way it establishes such a hierarchy is determined by sociohistorical matter, that is, in time and space.

An authoritative form is thus a performative cultural discourse, in a similar way certain utterances are speech-acts [START_REF] Austin | How to do Things with Words: The William James Lectures delivered at Harvard University in[END_REF]. But this is not inherent to the form, if there is no sociohistorical matter from which we can abstract reproducing forms; if there is no experience, there is no performance. A form can be described as authoritative only a posteriori, that is after the phenomenological experience which will justify its position, regardless whether the form was already formulated or inherited (reproduction and distortion of an instituted signification) or whether it is a new form. We cannot access the experience of others, and reality in general, without languages, linguistic or otherwise. Furthermore, all languages are elaborated on the basis of imaginary forms through an elaborate and repetitive process of symbolling (the act of adding meaning). In order to create such an access, formulation becomes a necessity; in order to maintain this access (and to refrain from opening other accesses), reformulation becomes a necessity: as is suggested in the ODO definition, the word religion was redefined through reformulation, that is, with other forms used to add and eventually change the meaning of the term, refining the form and its meaning into the fixed and abstract signification we use today.

This theoretical elaboration thus appears to be a sort of détournement (Debord, 1970: §204-209), a hijacking of Aristotelean ontological categories for the purposes of an epistemological reflection for the study of the relation between religion and nationalism as social imaginaries.

The aim was less a demonstration and rather the formulation of epistemological interrogations for the study of religion and nationalism, and by extension secularity. Indeed, this reflection leads to many interrogations for further inquiry into the signification of nationalism, which does not appear as fundamental as religion and its instituting of a world, but rather as the endeavour, the justification for modern institutions and social groups to define a certain place within this world. The dimensions signified in religious and national narratives and histories, albeit performed from different space-time perspectives, raises the question of the extent to which nationalist histories, at least in the European context, is a reformulation of the historiographical narratives of the Old Testament and, on a mythical level, represent versions or adaptations of how other peoples than the initial chosen people can themselves have a shared sense of uniqueness. 20 What of the concept and signification of secularity? As it derives from religious discourse, the signification of the term appears as more ambivalent as simply that which is not religious. In consequence, and bearing in mind the implicit idea that religion and nationalism follow an analoguous modus operandi, the interrogation of their relation is indeed the interrogation of the signification of secularity. The question could be formulated as follows: do secular rules follow a substantially different process of signification than "regular" rules? This further leads 20 There are numerous elements across different nationalisms which illustrate this: the expression "manifest destiny" in American nationalism is perhaps one of the most obvious illustration. On another note, the age-old and widespread antisemistism also suggests how a cultural, cosmological complex of inferiority has been central in European cultures (although not exclusively), leading in the age of nationalism to the most extreme form of exclusion. The Nazi regime is the obvious high point of this evolution, but it would be simplistic to ascribe such motives to the national-socialism and other emergences of fascism alone.

to questioning how radical and revolutionary modernity actually is if we take into account the long historical moment within which modernity, with nationalism, occurs. The extent to which nationalism profoundly opposes traditional religious frameworks is debatable, contrary for instance to elements of the development of the scientific method. To conclude on a contemporary note, what is the signification in contemporary political discourse, where "revolutions" -a scientific term -are deprecated in favour of more consensual "reforms" -a religious term?

  formulate. The ideal being what the institutions tend to, what they promote, what their positions are, and not simply the representation of an a-temporal universal. The ideal thus defines the interests of the corresponding institution and justifies the experience.

The category of substance has many dimensions for Aristotle and is central to his ontology. The simplest working definition is to define substance as reality which could translate of the initial Greek term ousia(Cohen,

2009:37).

The question of perspective is fundamental in Albert Einstein's theory of relativity which considers precisely space-time as a continuum.

The repetition of these processes is accompanied by other processes: the memorisation and transmission of the forms.

We could elaborate further on the similarities between the two definitions. E.g. each of them describes experiences which in relation to a particular external figure: God or patrie (literally, fatherland), etc.

For specialists, Feudalism is a debated concept. While indeed the Middle Ages present us with feudal societies, feudalism, as a social organising principle, is not a particularity of the Middle Ages.

I am using this expression in a literal sense, and not in reference to the debate between society of order and society of class. A society of class is in fact, in our context, a society of order. The continuities as well as the discontinuities between pre-industrial societies described as societies of order and industrial societies of class oversimplify the complexities and the qualities of social significations in the transition between these two models.

A determining period for the transformation of terms was certainly, as Eric Hobsbawm points out for the terms used to describe society, the "age of the dual revolution"[START_REF] Hobsbawm | The Age of Revolutions[END_REF]. Such is the case for order and class mentioned in the previous note.

The synecdoche is related to another type of metaphor, the metonymy which operates a transfer either from a whole entity to one of its attributes (as in metonymy; 'the blue' standing for 'the sky') or vice versa. Synecdoches are frequent in political and media discourses: names of states are regularly substituted for the names of governments, political representatives or sports teams. Conversely, the names of capital cities often replace the names of government representatives, or, if this particular illustration is considered a metonymy, it transfers the meaning of a political or economic institution situated in the city to the name of the city itself.

It could be one of the imaginary responses to the chaos of every-day life, where experience takes place without our possibility to mediate it through a formulation. Immediate experience is the experience of chaos. It could be interesting to discuss this in the light of the acceleration of time.

The now global seven days week further illustrates how such frames can be extracted from their original justification.

In relation to the acceleration of time, we could add that the rationalisation of time, in hours, seconds and more, has reduced the sequences at which we experience time. This could partly explain how the acceleration of time is being experienced.

The Hundred Years War also corresponds to the actual end of the crusades which had signified the consolidation of the Christian commonwealth.

A reproduction of the statute alongside a modern English translation can be accessed on the following webpage: http://www.languageandlaw.org/texts/stats/pleading.htm.

This was also supported by the notions of kinship and blood which had been of particular significance to the nobles of Europe since long before.

Looking Western Schism and its origins, we get a deeper and longer sense of the negotiations of political or institutional power between Kings and Popes, and indeed between Popes as well. In the historical novel by Maurice Druon, The Accursed Kings, the thought of a cardinal expresses the double face (temporal and spiritual) position of the head of the Church of Rome: "Being a priest is not enough to become Pope; one also has to know how to be a prince." (personal translation,Druon, 1977:19).


Tej Dallej
email: tej.dallej@lasmea.univ-bpclermont.fr

Marc Gouttefarde

Nicolas Andreff

Micaël Michelin

Philippe Martinet

Towards vision-based control of cable-driven parallel robots

This paper deals with the vision-based control of cable-driven parallel robots. First, a 3D pose visual servoing is proposed, where the end-effector pose is indirectly measured and used for regulation. This method is illustrated and validated on a cable-driven parallel robot prototype. Second, to take into account the dynamics of the platform and using a Cartesian pose and velocity estimator, a vision-based computed torque control is developed and validated in simulation.

I. INTRODUCTION

Parallel mechanisms have a closed-loop structures, with several kinematic chains connecting the base and the mobile platform [START_REF] Merlet | Parallel robots[END_REF], [START_REF] Bonev | The true Origins of Parallel Robots[END_REF]. Most existing parallel robots are designed with rigid legs which cannot exceed a certain length. A major drawback of such designs is their limited workspace.

Parallel cable-driven robots are a particular type of parallel robots which consist essentially of a moving platform connected in parallel to a base by cables [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF]. The length of the cables is adjusted to control the position and orientation of the platform. The cable tensions are related to forces and torques at the moving platform. Parallel cable-driven robots have several advantages over more conventional rigid-link robots. The main one is maybe scalability: Cables with small to very large lengths are easily stored on spools or drums allowing to build parallel cable robots with a workspace of global dimension ranging from a few centimeters to tens of meters or more [START_REF] Riehl | On the determination of cable characteristics for large dimension cable-driven parallel mechanisms[END_REF]. Depending on their size and practical realization, they may also be less expensive, easier to build, reconfigure and, when thin cables or wires are used, safer and non intrusive. These useful properties make cable-driven parallel robots good candidates for several applications, e.g., robotic cranes [START_REF] Albus | The nist robocrane[END_REF] or automated construction systems [START_REF] Williams | Contour-crafting-cartesian cable robot system concepts: Workspace and stiffness comparisons[END_REF] and human-scale force-feedback haptic systems [START_REF] Ishii | A 3d spatial interface device using tensed strings[END_REF]. Parallel cable-driven robots also have drawbacks such as the unilateral character of actuation (the cables can only pull) which, for some applications, make actuation redundancy compulsory [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF].

The control and the modeling of cable-driven parallel robots are complex due to the flexible nature of the cables. In [START_REF] Alp | Cable suspended robots: design, planning and control[END_REF], the authors proposed controllers for the cable suspended robot using Lyapunov theory and feedback linearization. In this work, there is no measure of Cartesian position. The pose (position and orientation) of the mobile platform can be estimated through the forward kinematics but the latter is generally difficult to solve and the measures of cable lengths by means of active joint variables may be inaccurate due notably to cable elongations. Another solution is to use ad-Fig. 1. The LIRMM/Tecnalia ReelAx8 robot ditional passive cables setting up an independent metrology [START_REF] Williams | Seven-dof cable-suspended robot with independent metrology[END_REF] and whose layout is such that the forward kinematics has closed-loop solutions. This solution increases however the probability of cable collisions and adds constraints to the mechanical design.

Another control strategy was presented in [START_REF] Fang | Motion control of a tendon-based parallel manipulator using optimal tension distribution[END_REF]. The proposed method uses a nonlinear feed-forward control laws in cable length coordinates and compute an optimal tension distribution.

In [START_REF] Oh | Generation of feasible set points and control of a cable robot[END_REF], the authors proposed to control a 6-DOF cable robot by means of recursive algorithm to generate reachable domains for motion of the robot end-effector taking into account input constraints. However, continuous tendon forces are not guaranteed. Moreover, due to the complexity of the dynamics of the robot general motion, rotations are not considered in this work. Ref. [START_REF] Mikelsons | A realtime capable force calculation algorithm for redundant tendon-based parallel manipulators[END_REF] proposes a real-time algorithm for tendon force calculation, without iterative steps. This approach guarantees the continuity of force distribution.

Several control laws are proposed in [START_REF] Dallej | 3d pose visual servoing relieves parallel robot control from joint sensing[END_REF], [START_REF] Andreff | Image-based visual servoing of Gough-Stewart parallel manipulators using legs observation[END_REF], [START_REF] Paccot | A review on dynamic control of parallel kinematic machine: theory and experiments[END_REF] using visual servoing techniques [START_REF] Weiss | Dynamic sensor-based control of robots with visual feedback[END_REF], [START_REF] Espiau | A new approach to visual servoing in robotics[END_REF], [START_REF] Hutchinson | A tutorial on visual servo control[END_REF]. They prove that the use of computer vision in the feedback control loop simplifies the kinematic models by introducing additional sensors into the parallel robots and thus yields easier control.

For the control of cable-driven parallel robots, visual servoing methods can also be a good alternative, since it closes the control loop over the vision sensor. External sensing of the end-effector pose in the feedback signal replaces advantageously the forward kinematic model. The perception models are simpler than the kinematic models and contain less unmodelled physical phenomena.This is especially useful in cable-driven robots because the flexibility of the cable.

Therefore, the contribution of this paper is to present vision-based control of a cable-driven parallel robot. Section II presents the kinematic modeling of the generic configuration of cable-driven mechanisms. This method is applied to the ReelAx8 parallel robot (Fig. 1). Section III-A presents a first method for the vision-based control of cable-driven parallel robots which is 3D pose visual servoing. Then, the vision-based computed torque control approach is presented in Section III-B. Section IV presents an experimental validation of the 3D pose visual servoing as applied to the ReelAx8 robot. Section V shows simulation results of the vision-based computed torque control. Conclusions are finally given in Section VI.

II. VISION-BASED KINEMATICS

In the kinematic modeling the notations given in Table I are used. • i v is vector v expressed in Fi.

• qi defines the motorized joint angle i.

• li defines the length of cable i.

• ϑi is the tension applied by cable i.

• i Tj = i Rj i tj 0 1
is the homogeneous matrix associated to the rigid transformation from Fi to Fj.

• i τj is the cartesian velocity of the origin of Fj expressed in Fi. • X, Ẋ and Ẍ are a representation of the end-effector pose, velocity and acceleration, respectively. • Γ is the vector of motor torques.

• M + is the pseudo-inverse of M.

• M is the estimation of M.

• [a]× is the cross-product matrix of vector a.

TABLE I NOTATIONS USED THROUGHOUT THE PAPER.

ReelAx8 is a 6-DOF cable-driven parallel robot (Fig. 1). It has a moving platform (end-effector) connected to a fixed base by 8 driving cables of varying length l i and cable tension ϑ i , i ∈ 1..8 (k = 8). Each cable (Fig. 2) is attached to the moving platform at point B i and extends from the base at point A i .

We can imagine that ReelAx8 robot as a Gough-Stewart platform [START_REF] Gough | Universal tyre test machine[END_REF], [START_REF] Stewart | A platform with six degrees of freedom[END_REF] which is turned upside down. When the cables are tensed and considered massless and inextensible, the kinematic modeling of Gough-Stewart platforms and of parallel cable-driven robot are very similar.

Fig. 2 shows a generic configuration of cable i. The cabledriven parallel robot is equipped with a camera defined by its reference frame R c and a pattern defined by frame R p .

We consider R f a fixed reference frame attached to the base and R m a mobile reference frame attached to the endeffector. There are two visual servoing configurations. In the first, (eye-in-hand), the camera attached to the end-effector (R c = R m ) observes a pattern fixed with respect to the base frame (R p = R f ). In the second one (eye-to-hand), R p is a mobile reference frame (R p = R m ) and R c is the fixed reference frame (R c = R f ). Introducing u i the unit vector pointing from A i to B i (Fig. 2), one can write:

Mobile platform

m ---→ A i B i = l i m u i (1)
This equation can be expressed in any Euclidean reference frame. Hence, it can be expressed in the mobile frame R m , which simplifies the adequation between the instantaneous inverse kinematic model and the regulated error in the visual servoing task. In the sequel, we consider the ReelAx8 robot equipped with a camera and a pattern in eye-to-hand configuration (Fig. 3).

Time differentiating (1) and using m u T i m ui = 0 yield:

li = d( m ---→ A i B i T ) dt m u i = d( m B i -m A i ) T dt m u i (2)
where B i is time invariant in the mobile frame R m , (2) yields:

li = - d( m A i ) T dt m u i = - d( m R f f A i + m t f ) T dt m u i (3)
Thus, according e.g. to [START_REF] Dallej | 3d pose visual servoing relieves parallel robot control from joint sensing[END_REF], [START_REF] Andreff | Kinematic modelling of some parallel manipulators for control purposes[END_REF], the instantaneous inverse kinematic model of a parallel cable-driven robot, relating the Cartesian velocity m τ m = m τ m/m * = ( m V m m Ω m ) T of the mobile frame to the time derivative of cable length vector ˙l = ( l 1 .. lk ) T is obtained:

˙l = m D inv m m τ m (4)
with

m D inv m =    m u T 1 ( m B 1 × m u 1 ) T . . . . . . m u T k ( m B k × m u k ) T    (5)
where the m B i is a constant (calibration) parameter. The motorized joint angle vector q of the robot are linearly related to the time derivatives of cable lengths and can be computed as:

q = 1 r c ˙l ( 6
)
where r c is the radius of the drums collecting the cables.

Note that the direction of each cable can be written as

m u i = m ---→ A i B i m ---→ A i B i = m B i -m A i m ---→ A i B i (7)
Being given that m A i = ( m R f f A i + m t f ), the instantaneous inverse kinematic model ( 4) depends on the pose m T f , the rigid transformation between camera and pattern, and on the constant (calibration) parameters f A i and m B i .

III. VISION-BASED CONTROL

A. 3D pose visual servoing 1) Interaction matrix associated to the pose: Consider R m and R m * the desired and the current mobile frame positions, respectively. For the control of the robot end-effector pose, in [START_REF] Wilson | Relative endeffector control using cartesian position-based visual servoing[END_REF], [START_REF] Martinet | Robot control using 3d monocular pose estimation[END_REF], [START_REF] Daucher | Speed command of a robotic system by monocular pose estimate[END_REF], [START_REF] Thuilot | Position based visual servoing: keeping the object in the field of vision[END_REF] the authors present the following visual primitive s:

s = ( s t s w ) (8)
In our case, s t = m t m * is the translation error between R m and R m * and s w = uθ, where u is the axis and θ is the angle of the rotation matrix m R m * . The visual primitive s is thus 0 when the current frame reaches the desired one (s * = 0).

As shown in [START_REF] Thuilot | Position based visual servoing: keeping the object in the field of vision[END_REF], [START_REF] Malis | 2 1/2 d visual servoing[END_REF], we can write:

   ṡt = -m V m/m * + [ m t m * ] × m Ω m/m * d(uθ) dt = -L w m Ω m/m * (9)
where

• L w = I 3 -θ 2 [u] × + 1 -sinc(θ) sinc 2 ( θ 2 ) [u] 2 × • sinc(θ) = sin(θ) θ
The time derivative of s is:

ṡ = L s m τ m ( 10
)
where m τ m is the end-effector Cartesian velocity and L s is the interaction matrix expressed as:

L s = -I 3 [ m t m * ] × 0 3 -L w (11)
Notice that in the case of s = 0, the interaction matrix simplifies into L 0 = -I 6 .

2) Control law:

To regulate the error between the current primitive vector s and the desired one s * = 0, one can consider the exponential decay ṡ = -λs. The vision-based control (Fig. 4) can then be expressed as:

m τ m = -λ L s + s (12)
Let us note that only information from the vision sensor are used to define the relation between s and the end-effector Cartesian velocity m τ m [START_REF] Malis | 2 1/2 d visual servoing[END_REF].

Note also that the instantaneous inverse kinematic model can be written as:

˙l = m D inv

B. Vision-based computed torque control

In section III-A, we presented a vision-based kinematic control, where the end-effector pose is measured using a camera and used for regulation. This method simplifies the control and replaces joint sensors. However, it does not take into account the dynamics of the robot.

The contribution of this section is to present a vision-based computed torque control for ReelAx8 cable-driven parallel robot (Fig. 5). In this section, we consider that the center of mass corresponds to the end-effector origin E.

1) Simplified dynamics: In this section, we take into account only the dynamic of the moving platform. The actual ReelAx8 prototype is not a large dimension mechanism. So, the mass of each cable can be neglected to have a simplified dynamics.

The gravitational force applied on the end-effector at the point E is m p g. m p and g are respectively the mass and the gravity acceleration vector.

Let us denote f I e the moment of inertia of the mobile platform about its center of mass expressed in the fixed frame R f .

Using the Newton-Euler's laws, one can write the applied and inertia force wrench exerted at point E and expressed in the fixed frame R f as:

F e = Fp ηp R f = mp(g-at) -f Ie f Ωe- f Ωe×( f Ie f Ωe) (14)
where F p is the sum of the applied and inertia forces on the end-effector, η p is the inertia moment, a t is the linear acceleration and f Ω e is the angular velocity of the endeffector expressed in the fixed frame R f .

2) Equation of motion: Noting that a cable can only pull the end-effector (Fig. 2), one can write the cable tension at point B i :

ϑ i = -ϑ i f u i (15)
where ϑ i = ϑ i is tension of cable i which is to be nonnegative.

The wrench applied by cable i at the reference point E is:

i = ϑi η Bi + f --→ EBi×ϑi R f (16)
Assuming that the moment of ϑ i at point B i is η Bi = 0, we obtain using ( 15) an ( 16):

i = ϑ i W i (17)
where

W i = -f u i -f --→ EBi× f u i = -f u i -f Re e Bi× f u i (18)
To keep platform in equilibrium, one can write the relation between the applied and inertia forces wrench F e (from ( 14)) and cable tensions wrenches as:

F e + k i=1 ( i ) = 0 (19)
It can be also written as:

F e + Wϑ = 0 (20)
where W = (W 1 , ..., W k ), ϑ = (ϑ 1 , ..., ϑ k ) T and k is the number of cables. Finally, one can compute the vector of cable tensions as:

ϑ = -W + F e (21)
Then, neglecting the inertia of the drums, the motor torques can be computed using:

Γ = r c ϑ (22)
where r c is the reel motor radius of the drums. As shown in Fig. 5, a vector of forces f null lying in the nullspace of matrix W can be added to the tension vector computed in [START_REF] Andreff | Kinematic modelling of some parallel manipulators for control purposes[END_REF] in order to try to keep all tensions positive [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF] during motion but without changing this latter. The computation of f null may be an issue which is beyond the scope of this paper. In the simulation of Section V, f null is taken equal to the zero vector.

3) Combining measure and computed torque control: Using a fast exteroceptive measure [START_REF] Paccot | A review on dynamic control of parallel kinematic machine: theory and experiments[END_REF], [START_REF] Dahmouche | 3d pose and velocity visual tracking based on sequential region of interest acquisition[END_REF], we can reduce the complexity of control schemes. The proposed computed torque control in Fig. 5 needs the estimation of cartesian pose and velocity of the moving platform. They can be measured by vision, which is used directly in the control loop to compensate dynamics in real time.

The controller is a standard computed torque controller in the Cartesian space, namely a PD and a feedforward controller (see Fig. 5). The output of which u c is homogeneous to an acceleration. The control output can thus be fed into [START_REF] Andreff | Image-based visual servoing of Gough-Stewart parallel manipulators using legs observation[END_REF] to compute the desired wrench.

The cable tensions are related to the moving platform dynamics by W, which can be estimated only by using information from vision (the direction f u i and f R e ) and constant calibration parameter e B i .

IV. EXPERIMENTAL VALIDATION

The 3D pose kinematic visual servoing approach of section III-A has been implemented and applied to ReelAx8 cable-driven parallel robot. An off-the-shelf P hotonf ocusM V -D1024 -T rackCam camera is used to estimate the pose [START_REF] Dahmouche | 3d pose and velocity visual tracking based on sequential region of interest acquisition[END_REF] of the mobile platform.

The camera is attached to the base frame and a pattern is fixed to the end-effector (eye-to-hand configuration). The camera is placed in front of the robot so that it can cover a large amount of the workspace (Fig. 3). In an experiment, the ReelAx8 robot is asked to reach the desired position from the initial configuration (Fig. 3) c t m = (-0.0181, -0.0744, 1.4248) T (m) as the pattern position and θ = 0 as the rotation angle. The desired position is obtained from the reference position by a translation of -20 mm along the y c axis of the camera.

Fig. 6 and Fig. 7 show that the errors converge to 0 as expected, from an initial cartesian error to a steady state one ( 1.5 mm).

Notice that the error curves are exponentials and that the 3D trajectory (Fig. 8) of the pattern is linear in camera frame, which confirms the properties of the 3D pose visual servoing.

V. SIMULATION RESULTS

The vision-based computed torque control is validated by means of a ReelAx8 cable-driven parallel robot simulation. All kinematic parameters are expressed in the fixed frame R f . In the simulations presented below, we choose the cartesian initial position b t e = (0, 0, 0) T and orientations α = β = γ = 0.

Fig. 9 and Fig. 10 show the reference rectangular motion. It is composed of translations along the three axis x b , y b and z b . The last part of the reference trajectory (Fig. 10) is composed of rotations around the three axis x b , y b and z b .

To show the robustness of the approach, we added noise to the estimation of the cartesian pose b T e and velocity. Consequently, we define at each time a sample random noise for translation and operational velocity, with maximal value of 0.2 mm and 2 mm/s, respectively. We also added noise to the rotation angle and angular velocity, with maximal amplitude of 0.01 • and of 0.1 • /s, respectively. Fig. 11, Fig. 12 and Fig. 13 show a potentially good robustness.

VI. CONCLUSION

This paper proposed methods to control a cable-driven parallel robot using a visual target and a calibrated camera. The use of computer vision in the feedback simplifies the kinematic modeling by introducing additional sensors into the robot and thus yields easier control.

Using a 3D pose kinematic visual servoing method, the end-effector pose is used for regulation. This method validated on a cable robot prototype, ReelAx8, robot differs from standard control in the joint space by the fact that the forward kinematic model is never used in the control.

It has to be noticed that the pose is estimated in the task space which is the Cartesian space of the robot. Consequently, we wanted to fuse a 3D pose and velocity visual tracking [START_REF] Dahmouche | 3d pose and velocity visual tracking based on sequential region of interest acquisition[END_REF] and a dynamic control of cable-driven parallel robots.

Doing so, we have introduced a second approach: a visionbased computed torque control. The presented method was validated on a ReelAx8 robot simulator. It is performed with noise added to show the validity of the approach.

The next step in this work is to implement the visionbased computed torque control on the ReelAx8 robot and to extend the method to large dimension cable-driven parallel robots.

•

  i = 1..k denotes the driving cables, where k is the number of cables. • Boldface characters denote vectors. Unit vectors are underlined. • F b = (O, x b , y b , z b ), Fe = (E, x e , y e , z e ) and Fc = (Oc, x c , y c , z c ) denote the base, end-effector and camera reference frames, respectively.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Generic configuration of cable i

Fig. 4 .

 4 Fig. 4. 3D pose visual servoing of a cable-driven parallel robot This method allows the execution of a classic and simple robotic task in the Cartesian space.

Fig. 5 .

 5 Fig. 5. Vision-based computed torque control of a cable-driven parallel robot

Fig. 6 .

 6 Fig. 6. Evolution of the cartesian error.

Fig. 7 .

 7 Fig. 7. Evolution of the cartesian velocities.

Fig. 8 .

 8 Fig. 8. Evolution of the pattern trajectory.

Fig. 9 .

 9 Fig. 9. Evolution of the end-effector trajectory.

Fig. 10 .

 10 Fig. 10. Desired position and orientation of the end-effector.

Fig. 11 .

 11 Fig. 11. Evolution of the translation errors.

Fig. 12 .

 12 Fig. 12. Evolution of the orientation errors.

Fig. 13 .

 13 Fig. 13. Evolution of motor torques.

  Table II presents mean values and standard variances (deviations) of the error vector norm.

		Mean value	Standard variance
	Translation error (mm)	0.4	0.1
	Orientation error (°)	0.0225	0.0038
	Operational velocity error (mm/s)	1.1	3.6
	Orientation velocity error (°/s)	0.0616	0.2069

TABLE II MEAN

 II VALUES AND STANDARD VARIANCES OF CARTESIAN ERRORS

VII. ACKNOWLEDGMENT

The financial support of the ANR (grant 2009 SEGI 018, CoGiRo project) is greatly acknowledged.