Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations

Ngoc-Tu Trinh, Vincent Andrieu, and Cheng-Zhong Xu

Abstract-The paper deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi-linear hyperbolic partial differential equations. Both the control input and the measured output are located on the boundary. The closed-loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the Lyapunov direct method. Based on the elaborated Lyapunov function we prove local exponential stability of the nonlinear closed-loop system with the same controller. The output regulation to the set-point with zero static error by the integral controller is shown upon the nonlinear system. Numerical simulations by the Preissmann scheme are carried out to validatethe robustness performance of the designed controller to face tounknown constant disturbances.

Index Terms-Boundary control, PI controller, hyperbolic system, Lyapunov function, partial differential equations, exponential stability, numerical simulation.

I. INTRODUCTION

The paper is concerned with the control of nonlinear systems governed by scalar quasi-linear hyperbolic partial differential equations. This type of systems appear in many industrial applications and in study of traffic flow. For instance, quasi-linear hyperbolic equations include Burgers equations [START_REF] Burgers | A mathematical model illustrating the theory of turbulence[END_REF] which are employed in modeling turbulent fluid motion. Another example is given by the equation employed by Lighthill-Whitham in [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF] to describe traffic flow on long crowded roads. Finally scalar conservation laws can also be regarded as a particular simpler case of quasi-linear hyperbolic systems under some regularity assumptions (see for instance [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF], [START_REF] Bressan | Hyperbolic systems of conservation laws: the onedimensional Cauchy problem[END_REF]).

The problem of controlling systems governed by hyperbolic partial differential equations (PDE) with both inputs and outputs on the boundary has attracted a considerable amount of studies [START_REF] Georges | Infinite-dimensional nonlinear predictive controller design for open channel hydraulic systems[END_REF], [START_REF] Bounit | Regulation of an irrigation channel through the semigroup approach[END_REF], [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF]. Interested readers can find a nice literature review on the fields in the section 2 of [START_REF] Coron | Dissipative boundary conditions for onedimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF] and in [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF]. Many available results have established appropriate boundary conditions to ensure asymptotic or exponential stability of the equilibrium state as in [START_REF] Li | Boundary value problems for quasi linear hyperbolic equations[END_REF], [START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equation[END_REF], [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF], [START_REF] Coron | Dissipative boundary conditions for onedimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF], [START_REF] Krstic | Backstepping boundary control for firstorder hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF] and [START_REF] Xu | Symmetric hyperbolic systems and applications to exponential stability of heat exchanges and irrigation canals[END_REF] or [START_REF] Xu | Linearization method to stability analysis for nonlinear hyperbolic systems[END_REF] (most of them in C 1 topology but also in the H 2 topology, see [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]).

In these papers the boundary conditions are given as a function of the output. In other words, it is the static control law that has been investigated. As it is well-known, one of the drawbacks of this type of controllers is the lack of regulation effect in the presence of constant perturbations. This motivates the introduction of integral actions in the control law by using a dynamic output feedback.

In this paper, our objective is to design an integral controller to guarantee asymptotic stability of the equilibrium of the nonlinear closed-loop system and the output regulation to a given set-point. The idea of using dynamic output feedback control for infinite dimensional systems is inspired by the works of [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF], [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF] and [START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF], with implementation of integral action for infinite-dimensional linear systems. These results have been further developed in recent publications for linearized hyperbolic systems, by using Lyapunov techniques in [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] and [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments[END_REF], by using Laplace transformation in frequency domain in [START_REF] Bastin | Stability of linear density-flow hyperbolic systems under PI boundary control[END_REF] and by using a semi-group approach in [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] and [START_REF] Trinh | PI regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF]. The backstepping method has been exploited in [START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking[END_REF] to elaborate PI (proportional and integral) controllers in the same context. For nonlinear systems, the work of Tamasoiu [START_REF] Tamasoiu | Stabilization and boundary control for balance and conservation laws[END_REF] considers a PI controller with damping for a scalar hyperbolic system. However it loses the regulation effect because of the damping required in the PI controller. It is also interesting to note that asymptotic stabilization of entropy solutions to scalar conservation laws has been recently studied by Perrollaz in [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF] and by Balandin et al. in [START_REF] Blandin | Regularity and Lyapunov stabilization of weak solutions to scalar conservation laws[END_REF]. In particular they have considered the stabilization problem of weak entropy solutions by boundary control and internal control around a constant equilibrium state for a scalar 1D conservation law with strictly convex flux. In [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF] an internal state feedback control law has been designed to asymptotically stabilize the entropy solutions around a constant equilibrium in the topologies L 1 and L ∞ . A stabilizing nonlocal boundary control law (depending on time and the whole initial data) has been proposed in [START_REF] Blandin | Regularity and Lyapunov stabilization of weak solutions to scalar conservation laws[END_REF] to get asymptotic stability of the constant equilibrium in the L 2 topology.

In our work, we consider a 1D scalar conservation law with strictly increasing flux. We are interested in the classical solutions of the system around an equilibrium state. We propose a boundary integral output feedback controller to asymptotically stabilize the system. The local stability and the regulation effect for the nonlinear closed-loop system are proven in the H 2 topology by using Lyapunov techniques. The contribution of our paper is threefold: (i) inspired from [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] we construct a new Lyapunov functional to prove exponential stabilization of the closed-loop system with the designed integral controller; (ii) we provide a mathematical proof of the regulation effect with zero static error offered by the integral action of the dynamical control law; (iii) the Preissmann scheme is implemented to realize numerical simulations for the nonlinear closed-loop PDE system.

The paper is organized as follows. Section II is devoted to the statement of the regulation problem and the announcement of the main result. Section III presents the proof of the main result. More precisely, in Section III-A asymptotic stabilization of the linearized system is considered by proposing a Lyapunov functional; In Section III-B the stabilization problem is solved for the nonlinear system by extending the proposed Lyapunov functional. Numerical simulations to validating the theoretical results are implemented in Section IV by applying the Preissmann numerical scheme. Finally Section V is devoted to our conclusions.

II. STATEMENT OF THE PROBLEM AND MAIN RESULT

In this paper, we consider a 1D quasi-linear hyperbolic system of the form :

∂ψ ∂t (x, t) + F (ψ(x, t)) ∂ψ ∂x (x, t) = 0, x ∈ (0, L), t ∈ R + , (1) where
L is a positive constant, ψ : (0, L) × R + → R is the state in C([0, ∞), H 2 (0, L)), and F : R → R is a C 2 function such that F (σ) > 0 ∀ σ ∈ R.
The initial condition is given by ψ(•, 0) = ψ 0 ∈ H 2 (0, L). Notice that H 2 (0, L) is the usual Sobolev space defined by

H 2 (0, L) = {f ∈ L 2 (0, L) | f , f ∈ L 2 (0, L)}
where L 2 (0, L) denotes the usual Hilbert space of square summable functions on the open set (0, L). The Sobolev space H 2 (0, L) is normed by

f 2 H 2 = L 0 (|f (x)| 2 + |f (x)| 2 + |f (x)| 2 )dx ∀ f ∈ H 2 (0, L).
We consider the control u on the boundary x = 0, i.e.,

ψ(0, t) = u(t) , t ∈ R + .
The output we wish to regulate is also located on the boundary and eventually corrupted by an additive unknown disturbance, i.e.

y(t) = ψ(L, t) + w o , t ∈ R + ,
where w o ∈ R is an unknown constant. Our control objective is to design a dynamic output feedback control law in order to achieve asymptotic stabilization of the closed loop system and to ensure that the output y(t) converges to a desired set-point y r ∈ R, as t → ∞.

In our study, the control action u(t) has the structure of an integral controller. We assume that an unknown constant disturbance may corrupt the control. Hence we write the control law as follows

u(t) = -k I ζ(t) + w c , ζ(t) = y(t) -y r
where w c ∈ R is an unknown constant and k I is a positive constant called tuning parameter.

To summarize, the closed-loop system with disturbances is governed by the following PDE :

               ∂ψ ∂t (x, t) = -F (ψ(x, t)) ∂ψ ∂x (x, t) ζ(t) = ψ(L, t) -y r + w o ψ(0, t) = -k I ζ(t) + w c ψ(x, 0) = ψ 0 (x), ζ(0) = ζ 0 . (2)
We are studying a nonlinear infinite-dimensional system controlled by an integral controller faced with unknown constant disturbances on the control and the output. The purpose of the paper is to find sufficient conditions on the control parameter k I > 0 such that the three objectives are realized : (a) the closed-loop system (2) is well posed; (b) asymptotic stability of the closed-loop system is guaranteed; and (c) the regulation property holds lim

t→∞ |y(t) -y r | = 0. (3)
As only the classical solutions are considered, in the following we restrict ourselves to study the solutions from the initial data (ψ 0 , ζ 0 ) in H 2 (0, L) × R which satisfy the C 0 and C 1 compatibility conditions :

ψ 0 (0) = -k I ζ 0 + w c F (ψ 0 (0))ψ 0 (0) = k I (ψ 0 (L) -y r + w o ). (4)
To be simple the initial data with the compatibility condition satisfied up to the required order are called compatible initial data throughout the paper. From now on, the state space X for (2) is the Hilbert space

X = H 2 (0, L) × R equipped with the norm (f, z) 2 X = f 2 H 2 + z 2 .
Note that due to the constants w 0 , w c and y r , (ψ, ζ) = (0, 0) is not a steady state of the closed-loop system. In fact the equilibrium denoted

(ψ ∞ , ζ ∞ ) is defined as follows ψ ∞ = y r -w o , ζ ∞ = k -1 I (w o + w c -y r ). ( 5
) Let B X ((f, z), δ) denote the open ball in X centered at (f, z) with radius δ > 0, i.e., B X ((f, z), δ) = {(ψ, ζ) ∈ X | (ψ, ζ) -(f, z) X < δ} .
Then the main result of the paper is stated as follows.

Theorem 1: There exist positive real constants k * I and δ such that, for each k I ∈ (0, k * I ), and for every (y r , w o , w c ) ∈ R 3 and every compatible (ψ 0 , ζ 0 ) ∈ B X ((ψ ∞ , ζ ∞ ), δ), the following assertions hold true :

1) The closed-loop system (2) has a unique solution

(ψ, ζ) ∈ C([0, ∞), X); 2)
The solution of the closed-loop system (2) converges exponentially to the equilibrium state (ψ ∞ , ζ ∞ ) in the state space X as t → ∞, and the disturbed output is regulated to the desired set-point y r , i.e.,

lim t→∞ |y(t) -y r | = 0 .
3) There exist real constants M > 0 and ω > 0 such that

(ψ(•, t) -ψ ∞ , ζ(t) -ζ ∞ ) X M e -ωt (ψ 0 -ψ ∞ , ζ 0 -ζ ∞ ) X ∀t 0.

Remark :

The equilibrium state is some constant state determined by y r , w o and w c . Though it is unknown a priori, the state of the closed-loop system is bounded because of the asymptotic stability property of the equilibrium. Moreover the output is always regulated to the set-point independently of the unknown disturbances. It is the virtue of the integral controller that allows to suppress the static error and hence achieves output regulation. A more general situation is explained in the paper [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF].

Remark :

The solution considered in Theorem 1 is a classical solution in the sense of Li and Yu [START_REF] Li | Boundary value problems for quasi linear hyperbolic equations[END_REF]. However the topology used here is the topology induced by the Hilbert H 2 norm instead of the C 1 norm. Moreover, we have only local exponential stability of the equilibrium of the closed-loop system. For initial compatible data outside some neighborhood of the equilibrium, the classical solution to the Cauchy problem (2) may not be extended on the whole positive time axis.

The proof of Theorem 1 is given in the next section. Our proof is based on the construction of an appropriate Lyapunov functional. The direct Lyapunov approach allows us to consider the tuning parameter k I relatively bigger than the classical method (see [START_REF] Davison | Multivariable tuning regulators: the feedforward and robust control of a general servomechanism problem[END_REF], [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF] and [START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF]). The proposed upper bound on k I is computed directly from the given system. This may be the advantage of our approach with respect to that of the literature [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF], [START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF], [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF].

III. PROOF OF THE MAIN RESULT

To prove Theorem 1 we consider the following transformation :

φ(x, t) = ψ(x, t) -ψ ∞ , ξ(t) = ζ(t) -ζ ∞ (6)
where ψ ∞ and ζ ∞ are defined in [START_REF] Bounit | Regulation of an irrigation channel through the semigroup approach[END_REF]. Then we obtain a perturbation free nonlinear closed-loop system as follows :

φ t (x, t) = -F (φ(x, t))φ x (x, t) (7) ξ(t) = φ(L, t) (8) φ(0, t) = -k I ξ(t) (9) φ(x, 0) = φ 0 (x) = ψ 0 (x) -ψ ∞ (10) ξ(0) = ξ 0 = ζ 0 -ζ ∞ ( 11
)
where φ t (x, t) denotes the time partial derivative of φ(x, t), and we have defined

F (φ) = F (φ + ψ ∞ ).
In the new coordinates, the output is written as

y(t) = φ(L, t) + y r .
Hence the output regulation to y r is achieved if

lim t→∞ |φ(L, t)| = 0.
To guarantee the output regulation of the disturbed nonlinear system (2), we design the integral controller so as to ensure local asymptotic stabilization to the origin of the equivalent system ( 7)- [START_REF] Davison | Multivariable tuning regulators: the feedforward and robust control of a general servomechanism problem[END_REF].

In the following, the integral stabilization problem of the equivalent system is considered first for the linearized case in Section III-A and then for the nonlinear case in Section III-B. Finally the complete proof of Theorem 1 is presented in Section III-C.

A. Linear hyperbolic system

The purpose of this Section is to study stability property of the origin for the nonlinear hyperbolic system with an integral controller on the boundary as described in ( 7)- [START_REF] Davison | Multivariable tuning regulators: the feedforward and robust control of a general servomechanism problem[END_REF]. To begin with, we consider the particular case where the system is linear, i.e., F does not depend on φ. This is the case if for instance the considered system is obtained by the tangent linearization of the nonlinear system around the equilibrium state. In this subsection, we consider the following linear system :

φ t = -rφ x , r > 0 (12) ξ = φ(L, t), φ(0, t) = -k I ξ(t) (13) φ(x, 0) = φ 0 (x), ξ(0) = ξ 0 . (14)
To the system ( 12)-( 14) is associated the state space Z which is the Hilbert space Z = L 2 (0, L)×R equipped with the scalar product

(φ a , ξ a ), (φ b , ξ b ) Z = L 0 φ a (x)φ b (x)dx + ξ a ξ b ∀ ((φ a , ξ a ), (φ b , ξ b )) ∈ Z 2 .
and we denote by • Z its associated norm. The first stability result is obtained by employing the Laplace transform approach.

Proposition 1: The closed-loop linear system ( 12)-( 14) is exponentially stable in Z w.r.t. L 2 norm if and only if k I ∈ 0, rπ 2L .

The proof of this result can be obtained from [4, p.444, Chapter 13] or from [START_REF] Hale | Introduction to functional differential equations[END_REF]Appendix Theorem A.5]. For the reader's convenience a simple proof is given in Appendix A.

By frequency-domain analysis it is possible to establish some necessary and sufficient conditions on the parameter k I for asymptotic stability of the equilibrium to the linear closedloop system (12)- [START_REF] Georges | Infinite-dimensional nonlinear predictive controller design for open channel hydraulic systems[END_REF]. However the approach is no longer applicable when dealing with a general nonlinear system. This is the reason why we introduce a Lyapunov functional for the linear system which allows us to tackle the nonlinear hyperbolic system in the following section.

The Lyapunov functional candidate V : Z → R has the following form :

V (φ, ξ) = L 0 φ 2 (x)e -µx + q 1 ξφ(x)e -µx 2 dx + q 2 ξ 2 (15)
where µ > 0 and

q i > 0 ∀ i = 1, 2. Consider the function Π : [0, 2] → R such that Π(z) = z(2 -z)e -z/2 . We have Π(2 - √ 2) 0.3395 that is the maximum value of Π(z) in [0, 2].
Given T > 0 and a function φ : (0, L) × (0, T ) → R, we use the notation φ(t) := φ(•, t) when there is no ambiguity. Assume that the initial condition is smooth enough so that the solution of ( 12)-( 14) is continuously differentiable with respect to time t and space x. Then, by differentiating V (φ(t), ξ(t)) with time along the solution and by using integration by parts we get

V (φ(t), ξ(t)) = -re -µL φ 2 (L, t) -k I r(q 1 -k I )ξ 2 (t) -µr L 0 e -µx φ 2 (x, t)dx + 2q 2 -q 1 re -µL 2 ξ(t)φ(L, t) - µq 1 r 2 ξ(t) L 0 e -µx 2 φ(x, t)dx + q 1 φ(L, t) L 0 e -µx 2 φ(x, t)dx. ( 16
) Lemma 1: Let k * I = r 2L Π(2 - √ 2). Take k I ∈ (0, k * I ) and µ ∈ (0, (2 - √ 2)
/L] such that r 2L Π(µL) > k I . Let q 1 = 2k I and let q 2 = rk I e -µL/2 . Then there exist positive constants M 1 and α > 0 such that

M -1 ||(φ, ξ)|| 2 Z V (φ, ξ) M ||(φ, ξ)|| 2 Z ∀ (φ, ξ) ∈ Z, (17)
and for every smooth compatible

(φ 0 , ξ 0 ) ∈ Z V (φ(t), ξ(t)) -αV (φ(t), ξ(t))- re -µL 2 φ 2 (L, t). (18)
Proof : Rewrite V (φ, ξ) as follows

V (φ, ξ) = L 0 φ(x)e -µx/2 ξ √ L P φ(x)e -µx/2 ξ √ L dx where P = 1 √ Lq1 2 √ Lq1 2 q 2 .
We claim that the matrix P is positive definite. Indeed, we have det(P ) =

Lk I rΠ(µL) 2L -k I + r 2L e -µL/2 2 -µL(2 -µL) .
Since r 2L Π(µL) > k I and µL < 2, it is easy to see that det(P ) rk I 2 e -µL 2 . Hence there is some real constant M 1 such that the inequality (17) holds.

By substituting the given q 1 and q 2 into (16) we have the following

V (φ(t), ξ(t)) = -re -µL φ 2 (L, t) -µr L 0 e -µx φ 2 (x, t)dx -k 2 I rξ 2 (t) -µrk I ξ(t)
L 0 e -µx 2 φ(x, t)dx

+ 2k I φ(L, t) L 0 e -µx 2 φ(x, t)dx. ( 19
)
By using the Young and Cauchy-Schwarz inequalities we get

2k I φ(L, t) L 0 e -µx 2 φ(x, t)dx re -µL 2 φ 2 (L, t) + 2Lk 2 I e µL r L 0 e -µx φ 2 (x, t)dx (20)
and

µrk I ξ(t) L 0 e -µx 2 φ(x, t)dx rk 2 I 2 ξ 2 (t) + rµ 2 L 2 L 0 e -µx φ 2 (x, t)dx. (21)
Substituting ( 20) and ( 21) into ( 19) leads us to the following inequality

V (φ(t), ξ(t)) - re -µL 2 φ 2 (L, t) - k 2 I r 2 ξ 2 (t) - r 2L Π(µL) + k I r 2L Π(µL) -k I J φ,ξ (22)
where

J φ,ξ = 2L r e µL L 0 e -µx φ 2 (x, t)dx.
By the choice of µ, we have r 2L Π(µL) -k I > 0. It follows from ( 22) that there exists a positive real number

M 1 such that V (φ(t), ξ(t)) -M 1 ξ 2 (t) + L 0 e -µx φ 2 (x, t)dx - re -µL 2 φ 2 (L, t). (23)
The required inequality [START_REF] Krstic | Backstepping boundary control for firstorder hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF] is true by [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF] and [START_REF] Kato | The Cauchy problem for quasi-linear symmetric hyperbolic systems[END_REF].

2

Remark :

It can be noticed that the set of parameter k I which makes the Lyapunov functional decreasing along solutions is smaller than the set of parameter obtained from Proposition 1. Hence, in the linear context our Lyapunov approach is conservative. However the Lyapunov functional allows us to deal with nonlinear systems as it will be shown in the next Section.

B. Nonlinear system

In this section, we consider the problem for the nonlinear system ( 7)- [START_REF] Davison | Multivariable tuning regulators: the feedforward and robust control of a general servomechanism problem[END_REF] with F (0) = r > 0. By the designed integral controller the nonlinear closed-loop system (7)-( 11) is written as follows

φ t + F (φ)φ x = 0 ξ = φ(L, t) φ(0, t) = -k I ξ φ(x, 0) = φ 0 (x), ξ(0) = ξ 0 . (24)
Let us set :

s(x, t) = φ x (x, t) , p(x, t) = φ xx (x, t).
By successive derivatives and compatibility conditions we find that the dynamics of s(x, t) and p(x, t) are governed by the following PDE, respectively,

s t + F (φ)s x = -F (φ) s 2 F (φ(0, t))s(0, t) = k I φ(L, t) s(x, 0) = φ 0 (x) (25)
and

p t + F (φ)p x = -3F (φ) s p -F (φ) s 3 F 2 (φ(0, t))p(0, t) = k I F (φ(L, t))s(L, t) -2k I F (φ(0, t))φ(L, t)s(0, t) p(x, 0) = φ 0 (x). ( 26
)
Now we use the idea presented in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] to extend the Lyapunov functional from the linear system (in L 2 norm) to the nonlinear system (in H 2 norm). Therefore local asymptotic stability of the equilibrium state and the set-point output regulation will be proved for the nonlinear closed-loop system [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF]. To do that, we consider the Lyapunov functional candidate S :

X → R + such that S(φ, ξ) = V (φ, ξ) + q 3 V 1 (φ x ) + q 4 V 1 (φ xx ) (27)
where V (φ, ξ) is defined in ( 15) with q 1 and q 2 given in Lemma 1 and

V 1 (φ x ) = L 0 e -µx φ 2 x (x)dx (28)
with the real positive constants q 3 and q 4 to be determined later.

For the moment we assume that all the required regularity is satisfied and carry out formal computations.

Lemma 2: The time derivative of V (φ(t), ξ(t)) along each regular trajectory of the nonlinear system ( 24) is written as follows

V (φ(t), ξ(t)) = -re -µL φ 2 (L, t) -k 2 I rξ 2 (t) -µr L 0 e -µx φ 2 (x, t)dx -µrk I ξ(t) L 0 e -µx 2 φ(x, t)dx+2k I φ(L, t) L 0 e -µx 2 φ(x, t)dx -φ 3 (L, t)F 1 (φ(L, t))e -µL + φ 3 (0, t)F 1 (φ(0, t)) + L 0 e -µx [F (0) + φ(x, t)F 2 (φ(x, t))] φ x (x, t)φ 2 (x, t)dx -µ L 0 e -µx F 1 (φ(x, t))φ 3 (x, t)dx -2k I ξ(t) L 0 e -µx/2 F 1 (φ(x, t))φ(x, t)φ x (x, t)dx (29)
where

F (z) = F (0) + F 1 (z)z F (z) = F (0) + F 2 (z)z (30)
with F 1 (z) = 1 0 F (λz)dλ and F 2 (z) = 1 0 F (λz)dλ. Proof : By differentiating V (φ(t), ξ(t)) along each regular trajectory of ( 24) the following identity holds true

V (φ(t), ξ(t)) = - L 0 2e -µx φ(x, t)F (φ(x, t))φ x (x, t)dx -q 1 L 0 e -µx/2 F (φ(x, t))φ x (x, t)ξ(t)dx + q 1 L 0 e -µx/2 φ x (x, t)dxφ(L, t) + 2q 2 ξ(t)φ(L, t).
By integration by parts and by using the boundary condition [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF] and the parameters q 1 and q 2 given in Lemma 1 as well as the relations [START_REF] Xu | A robust PI controller for infinite dimensional systems[END_REF] we prove the required identity [START_REF] Trinh | PI regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF]. 2

Similarly we may prove the following lemmas. Lemma 3: With the same notations as in Lemma 2, the time derivative of V 1 (φ x (t)) along every regular trajectory of the nonlinear system ( 24)-( 25) is written as follows

V1 (φ x (t)) = -re -µL s 2 (L, t) + r -1 k 2 I φ 2 (L, t) -rµ L 0 e -µx s 2 (x, t)dx -k 2 I F 3 (φ(0, t))φ(0, t)φ 2 (L, t) -e -µL F 1 (φ(L, t))φ(L, t)s 2 (L, t) - L 0 [(F (φ(x, t))) x + µF 1 (φ(x, t))φ(x, t)] e -µx s 2 (x, t)dx (31)
where

F 3 (z) = 1 0 F (λz) F 2 (λz) dλ.
Lemma 4: With the same notations as in Lemma 2, the time derivative of V 1 (φ xx (t)) along each regular trajectory of the nonlinear system ( 24)-( 26) is written as follows

V1 (φ xx (t)) = -e -µL F (φ(L, t))p 2 (L, t) + k 2 I F 2 (φ(L, t)) F 3 (φ(0, t)) s 2 (L, t) -µ L 0 e -µx F (φ(x, t))p 2 (x, t)dx + 4k 2 I (F (φ(0, t)) 2 F 3 (φ(0, t)) φ 2 (L, t)s 2 (0, t) - 4k 3 I F (φ(L, t))F (φ(0, t)) F 4 (φ(0, t)) s(L, t)φ 2 (L, t) -5 L 0 e -µx F (φ(x, t))s(x, t)p 2 (x, t)dx -2 L 0 e -µx F (φ(x, t))s 3 (x, t)p(x, t)dx. ( 32
) Let T > 0. For each function (φ, ξ) ∈ C([0, T ]; C 1 [0, L] × R) we define (φ, φ x , ξ) T,∞ = sup t∈[0,T ] |ξ(t)| + + sup x ∈ [0, L] t ∈ [0, T ] |φ(x, t)| + sup x ∈ [0, L] t ∈ [0, T ] |φ x (x, t)|.
By combining results of Lemma 1-4 the following theorem is obtained.

Theorem 2: Let the parameters k I , µ, q 1 and q 2 be determined as in Lemma 1. Then there are positive real constants q 3 , q 4 , δ and β such that, for each function

(φ, ξ) ∈ C([0, T ]; C 3 [0, L] × R) ∩ C 1 ([0, T ]; C 2 [0, L] × R) satisfying
the PDE ( 24)-( 26) and the condition (φ, φ x , ξ) T,∞ < δ, the following differential inequality holds true

Ṡ(φ(t), ξ(t)) -βS(φ(t), ξ(t)) ∀ t ∈ [0, T ]. (33)
Moreover there exists a positive constant K 1 such that

K -1 (φ, ξ) 2 X S(φ, ξ) K (φ, ξ) 2 X ∀ (φ, ξ) ∈ X. (34)
Proof : Without loss of generality we assume that δ 1. For the sake of simplicity we write C T = (φ, φ x , ξ) T,∞ . By Lemma 2, Lemma 1 and the Cauchy-Schwarz inequality there exists a positive constant K 1 > 0 such that

V (φ(t), ξ(t)) -αV (φ(t), ξ(t)) -(r/2)e -µL φ 2 (L, t)+ + K 1 C T L 0 e -µx φ 2 (x, t)dx + ξ 2 (t) + φ 2 (L, t) . (35)
Similarly, by Lemma 3 there exists a positive constant K 2 > 0 such that

V1 (φ x (t)) -(re -µL -K 2 C T )s 2 (L, t) + r -1 k 2 I φ 2 (L, t) -rµ L 0 e -µx s 2 (x, t)dx + K 2 C T φ 2 (L, t) + L 0 e -µx s 2 (x, t)dx . (36)
Similarly, by Lemma 4 there exists a positive constant

K 3 > 0 such that V1 (φ xx (t)) -(re -µL -K 3 C T )p 2 (L, t) + (r -2 k 2 I + K 3 C T )s 2 (L, t) + K 3 C T φ 2 (L, t) + K 3 C T L 0 e -µx s 2 (x, t)dx -(rµ -K 3 C T ) L 0 e -µx p 2 (x, t)dx. (37)
As C T can be made as small as we like with δ, adding the inequalities (35)-(37) and taking δ, q 3 and q 4 sufficiently small lead us directly to the following differential relation

Ṡ(φ(t), ξ(t)) - α 2 V (φ(t), ξ(t)) - q 3 rµ 2 L 0 e -µx s 2 (x, t)dx - q 4 rµ 2 L 0 e -µx p 2 (x, t)dx. (38)
Therefore the theorem is proved by using ( 17), ( 27) and (38). 2

C. Proof of Theorem 1

With Theorem 2, we are now ready to prove the main result of the paper. Proof of Theorem 1 : We first prove the local existence of a unique solution to the closed-loop system ( 7)-( 11) for each compatible initial state (φ 0 , ξ 0 ) in H 2 (0, L) × R. The closedloop control system (7)-( 11) is governed by the following PDE coupled with an ODE through the boundary as follows:

φ t = -F (φ)φ x , ξ = φ(L, t) φ(0, t) = -k I ξ, (φ(x, 0), ξ(0)) = (φ 0 (x), ξ 0 ). (39) Recall that X = H 2 (0, L) × R is equipped with the norm (f, z) 2 X = f 2 H 2 + z 2 .
Assume that the initial condition (φ 0 , ξ 0 ) is in B X (0, δ), δ > 0 and satisfies the C 0 and C 1 compatibility conditions as in [START_REF] Perrollaz | Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law[END_REF] and [START_REF] Pohjolainen | Robust multivariable PI-controllers for infinite dimensional systems[END_REF].

By using the Theorem 1.2 and the Propositions 1.3-1.5 in [28, pp.362-365], or [17, Theorem II] we deduce the existence of a unique solution to (39) for some δ > 0 and T > 0 :

(φ, ξ) ∈ C([0, T ]; H 2 (0, L) × R) ∩ C 1 ([0, T ]; H 1 (0, L) × R).
The reader is referred to [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and [2, Appendix B] for a rigorous proof to the initial boundary case. Now we prove local exponential stability of the null state to (39). Notice that each compatible w 0 = (φ 0 , ξ 0 ) ∈ X admits a sequence of w 0,n = (φ 0,n , ξ 0,n ) ∈ H 4 (0, L) × R satisfying the C k compatibility condition, k = 0, 1, 2, 3, such that lim n→∞ w 0,n -w 0 X = 0 (cf. [7, p.130]). Hence it is sufficient for us to prove the exponential stability for w 0 ∈ H 4 (0, L) × R. As the solution depends continuously on the initial condition (see [17, Theorem III]), then the exponential decay of solution from compatible w 0 ∈ X is proved by taking the limit.

Indeed, take a compatible w 0 ∈ (H 4 (0, L) × R) ∩ B X (0, δ). As stated above the system (39) has a unique solution w(t) in H 4 (0, L) × R (cf. [START_REF] Taylor | Partial Differential Equation III Nonlinear Equation[END_REF]) such that

w ∈ C([0, T ]; H 4 (0, L) × R) ∩ C 1 ([0, T ]; H 3 (0, L) × R)
where w(t) = (φ(t), ξ(t)). By the continuous embedding (cf. [7, p.167

]) H n (0, L) → C n-1 [0, L] ∀ n 1 integer, we have the solution w ∈ C([0, T ]; C 3 [0, L] × R) ∩ C 1 ([0, T ]; C 2 [0, L] × R).
Let w 0 X < δ 1 for some δ 1 > 0. We choose δ 1 > 0 sufficiently small such that w 0 X < Kδ 1 implies (φ, φ x , ξ) T,∞ < δ with smaller T if necessary (cf. [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF]Theorem 2.2 ,p.46]). Notice that K and δ are defined in our Theorem 2. Then direct application of Theorem 2 allows us to get w(T ) X < Kδ 1 . Since the system is autonomous, the same argument can be used on the time interval [T, 2T ]. By successive iterations we obtain the differential inequality [START_REF] Xu | Multivariable boundary PI control and regulation of a fluid flow system[END_REF] satisfied for all t 0. By ( 33)-( 34) we find positive constants M and ω such that w(t) X M e -ωt w 0 X ∀ w 0 ∈ B X (0, δ 1 ).

The regulation effect is automatically guaranteed, since w ∈ C([0, ∞); H 2 (0, L) × R). Hence the proof of Theorem 1 is complete. 2

IV. NUMERICAL SIMULATIONS

The performance of the integral controller on the linearized system has been studied through numerical simulations and discussed in the paper [START_REF] Trinh | PI regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF]. Here the simulations are done on the nonlinear system to validate the theoretical results of Theorem 1. These simulations are realized by using the discretization method with the Preissmann scheme [START_REF] Lyn | Stability of a general Preissmann scheme[END_REF]. The details of the method have been presented in our paper [START_REF] Trinh | PI regulation control of a fluid flow model governed by hyperbolic partial differential equations[END_REF] with the following parameters : N, θ, ∆t and ∆x. Note that N is the number of discretized space intervals, ∆t and ∆x are the time discretization step and the space discretization step, respectively, and the weight parameter θ ∈ [0.5, 1] is made use of to compute the value of a function from its neighbor values. In this section, the parameters take the following values: L = 50m, N = 100, θ = 0.55, and ∆t ∆x = 0.5. Moreover, to simulate the nonlinear closed-loop system (2), the following flux function and numerical values are applied: F (ψ) = ψ 2 + 3, k I = 0.05, and y r = 0.5. The constant perturbations on the output and on the control are given by w o = 0.1 and w c = 0.05, respectively.

Figure 1 shows asymptotic stability of the nonlinear closedloop system and illustrates the evolution of the state ψ(x, t).

V. CONCLUSIONS

We have considered the design of stabilizing integral controllers for the nonlinear systems described by scalar hyperbolic PDE. First we have proposed an interval of integral gain for stabilization and then proved exponential stability of the linearized system controlled by the designed controller. Moreover, for the linearized system we have been able to establish a necessary and sufficient condition for the integral gain to get exponential stability of the controlled system in the L 2 norm. Then we have proved local exponential stability of the nonlinear controlled system by the same integral controller in the H 2 norm. Both of two main proofs have used Lyapunov techniques with the Lyapunov functions in the quadratic form. The regulation of the output to the set-point is automatically guaranteed from the local exponential stability of the closedloop system in H 2 norm. Numerical simulations for the nonlinear closed-loop system have been carried out to validate the performance of the controlled system. In the future, the work is to extend the design of stabilizing PI controllers for networks of scalar systems governed by nonlinear hyperbolic PDE.

APPENDIX

A. Proof of Proposition 1

A necessary and sufficient condition for exponential stability of the system ( 12)-( 14) is that all the poles of the transfer function have negative real part (see [START_REF] Xu | Exponential stability of the heat exchanger equation[END_REF], [START_REF] Bellman | Differential-difference equations[END_REF]Chapter 13], or [START_REF] Hale | Introduction to functional differential equations[END_REF]Appendix Theorem 3.5]). To formulate the transfer function, we set v(t) as the new control input with y(t) as the output : φ(0, t) = -k I ξ(t) + v(t), y(t) = φ(L, t).

(40)

By taking the Laplace transform in ( 12)-( 14), we obtain:

s φ + r φx = 0 (41)

s ξ = φ(L, s), (42)
φ(0, s) = -k I ξ(s) + v(s) , ŷ(s) = φ(L, s)

From (41) we have the solution φ(x, s) = φ(0, s)e -sr -1 x . Combining it with (42) and (43), we obtain: The proposition is proved if we show that the equation ( 46) has all the solutions µ in the left-half complex plane e(µ) < 0 if and only if α ∈ (0, π 2 ). Let set µ = σ + iη, where σ, η ∈ R. Then (46) is rewritten as follows :

(σ + iη)e σ+iη + α = 0

By separating the real part and the imaginary part, we obtain:

-e σ (σcos(η) -ηsin(η)) = α (47) ηcos(η) + σsin(η) = 0 (48)

We consider the following two cases.

• If sin(η) = 0, by (48), η cos(η) = 0 implies η = 0. From (47), we have α = -σe σ . The last equation has no solution σ 0 whatever is α > 0. Hence each solution σ is negative if and only if α ∈ (0, π/2). • If sin(η) = 0, from (48), In addition, lim η→2kπ + H(η) = 0 ∀ k ∈ N and k > 0, and lim η→0 H(η) = e -1 . By (49) and η = γ + 2kπ,we have σ < 0 if and only if γ ∈ (0, π 2 ). Obviously σ 0 if γ ∈ [ π 2 , π). Therefore σ < 0 if and only if the equation H(η)-α = 0 has all its solutions in ∪ ∞ k=0 (2kπ, 2kπ+ π 2 ). Since α needs to be in one interval including 0, we have α ∈ (0, π 2 ). From the two cases, the proposition is proved.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Evolution of ψ(x, t)

Fig. 3 :Fig. 4 :

 34 Fig. 3: Evolution of error |y(t) -y r |

ŷ

  (s) = φ(L, s) = φ(0, s)e -sLr -1 = e -sLr -1 (-k I ξ(s) + v(s)) = e -sLr -1 -1 ŷ(s) = e -sLr -1 v(s)Therefore we get the transfer function as follows:G(s) = ŷ(s) v(s) = s k I + se sLr -1The poles of transfer function are solutions of the following equation :k I + se sLr -1 = 0(44)We set µ = sLr -1 and α = k I Lr -1 .(45)Note that α > 0. The characteristic equation now becomes α + µe µ = 0 (46)

π 2 H

 2 Because H(η) is a pair function, we only need to consider the case where η > 0. Thus α > 0 if and only if sin(η) > 0. As η > 0 and sin(η) > 0, we set η = γ + 2kπ, where γ ∈ (0, π) and k ∈ N. Now considering the function H(η), we have :∂H(η) ∂η = e -ηcos(η) sin(η) sin 2 (η) -ηsin(2η) + η 2 sin 3 (η)One can easily check that sin 2 (η) -ηsin(2η) + η 2 0 for all η > 0. Therefore, ∂H(η)∂η 0. Hence, on each interval (2kπ, 2kπ + π), the function H(η) is continuous and monotonic increasing. Moreover we have lim η→(2k+1)π -H(η) = +∞, lim η→2kπ+ (η) = 2kπ + π 2 .

ACKNOWLEDGMENT

The authors would like to thank the editors and the anonymous reviewers for their useful suggestions.


J-P Merlet

Direct kinematics of CDPR with extra cable orientation sensors: the 2 and 3 cables case with perfect measurement and sagging cables

Direct kinematics (DK) of cable-driven parallel robots (CDPR) based only on cable lengths measurements is a complex issue even with ideal cables and consequently even harder for more realistic cable models such as sagging cable. A natural way to simplify the DK solving is to add sensors. We consider here sensors that give a partial or complete measurement of the cable direction at the anchor points and/or measure the orientation of the platform of CDPR with 2 or 3 cables and we assume that the measurements are exact. We provide a solving procedure and maximal number of DK solutions for an extensive combination of sensors for CDPR with sagging cables. We show that at least two measurements are necessary for the planar 2 cables case while six are necessary for the spatial 3 cables case. For spatial CDPR with n cables we prove that at least 2n additional sensors will be required to get a closed-form solution of the DK.

I. INTRODUCTION

We consider cable-driven parallel robot (CDPR) with 3 cables whose output points on the base will be denoted by A i with anchor point B i on the platform (figure 1). The Fig. 1. A CDPR with sagging cables winch system assume that the cable is not submitted to any deformation and is able to control and measure the length at rest L 0 of the cable. But the elasticity and own mass of the cable induce a sagging effect on the cable that modify its shape and length. We will assume here that the sagging of the cable is such that the whole cable lies in a vertical plane P that includes A, B (see figure 5). A sagging cable model, called the Irvine model [START_REF] Irvine | Cable Structures[END_REF], may be defined in P. In this plane the coordinates of A, B are assumed to be respectively (0,0), (x b ≥ 0, y b ) while F z , F x denote the vertical and horizontal forces exerted on the cable at point B. With this notation the 1 HEPHAISTOS project, Université Côte d'Azur, Inria, France Jean-Pierre.Merlet@inria.fr coordinates of B are related to the forces F x , F z by:

x b = F x ( L 0 EA 0 + sinh -1 (F z ) -sinh -1 ((F z -µgL0 Fx ) µg ) (1)
z b = F 2 x + F 2 z -F 2 x + (F z -µgL 0 ) 2 µg + F z L 0 EA 0 - µgL 2 0 2EA 0 ( 2
)
where E is the Young modulus of the cable material, µ its linear density and A 0 the surface of the cable cross-section.

The direct kinematic (DK) problem amounts to find all the possible pose(s) of the platform being given the L 0 . Although this may seen to be mostly a theoretical problem, DK solving is important also in practice for a better understanding of singularity and workspace and also for providing an initializing solution for the real time DK that is then less problematic [START_REF] Merlet | On the real-time calculation of the forward kinematics of suspended cable-driven parallel robots[END_REF], [START_REF] Pott | An algorithm for real-time forward kinematics of cabledriven parallel robots[END_REF]. Although relatively well mastered for parallel robots with rigid legs, DK is still an open issue for CDPR. Even if we assume ideal cable (with no elasticity and no deformation of the cable due to its own mass) the DK problem leads to a larger number of equations than in the rigid leg case [START_REF] Bruckman | Parallel manipulators, New Developments[END_REF] and consequently to solving problems [START_REF] Abbasnejad | Real solutions of the direct geometrico-static problem of underconstrained cable-driven parallel robot with 3 cables: a numerical investigation[END_REF], [START_REF]Direct geometrico-static problem of underconstrained cabledriven parallel robots with n cables[END_REF], [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF], [START_REF] Carricato | Direct geometrico-static problem of under-constrained cable-driven parallel robots with three cables[END_REF], [START_REF] Carricato | Direct geometrico-static analysis of under-constrained cable-driven parallel robots with 4 cables[END_REF], [START_REF] Carricato | Stability analysis of underconstrained cable-driven parallel robots[END_REF], [START_REF] Pott | An algorithm for real-time forward kinematics of cabledriven parallel robots[END_REF]. The DK problem with the full Irvine model has been addressed in [START_REF] Merlet | The forward kinematics of cable-driven parallel robots with sagging cables[END_REF] where it has been shown that the DK always amount to solve a square system of equations whatever the number of cables is. A solving algorithm has also been presented in [START_REF] Merlet | The forward kinematics of cable-driven parallel robots with sagging cables[END_REF] but is computer intensive. A more efficient solving scheme has then been presented in [START_REF]A generic numerical continuation scheme for solving the direct kinematics of cable-driven parallel robot with deformable cables[END_REF] but it is still quite intensive. A major problem with the Irvine model is that the non algebraic nature of equations (1,2) prohibits to use elimination methods that have been quite successful for parallel robots with rigid legs to reduce the DK to the solving an univariate polynomial. Another issue is that the DK has usually several solutions while the real CDPR is always in a given pose among all the possible solutions. The later use of the real-time DK then imposes to visually determine what is the current pose of the platform among all the DK solutions.

An intuitive approach to avoid or reduce this non-unicity problem and to speed up the solving time of the DK is to add sensors that provide additional information on the cable beside the cable lengths, as already proposed for classical parallel robot [START_REF] Bonev | A closed-form solution to the direct kinematics of nearly general parallel manipulators with optimally located three linear extra sensors[END_REF], [START_REF] Han | New resolution scheme of the forward kinematics of parallel manipulators using extra sensor data[END_REF], [START_REF] Merlet | Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data[END_REF], [START_REF] Parenti-Castelli | Real-time computation of the actual posture of the general geometry 6-6 fully parallel mechanism using only two extra rotary sensors[END_REF]. A natural candidate will be to measure the cable tensions as they play an important role in the DK solving. Unfortunately force measurement are usually noisy and measuring these tensions on a moving platform submitted to various mechanical noises appears to be difficult [START_REF] Krauss | System identification and cable force control for a cable-driven parallel robot with industrial servo drives[END_REF], [START_REF] Miermeister | Auto calibration method for cable-driven parallel robot using force sensors[END_REF]. Although several attempts have been made of integrating force sensing in CDPR, none of them have presented clear result about the reliability of the measurement.

In this paper we are considering another measurement possibility which consists in getting complete or partial information on the cable direction at the anchor points A and B. The measurement at point A are (figure 2):

• the angle θ V between the x axis and the vertical plane that includes the cable • the angle θ H between the horizontal direction and the cable tangent at A We introduce a mobile frame attached to the platform whose center is G, the center of mass of the platform, and vectors x r , y r , z r . Let u be the unit vector of the cable tangent at B and u p its projection in the x r , y r plane. The measured angles (figure 3) may be Realizing such measurement has already been considered: for example our CDPR MARIONET-Assist uses a simple rotating guide at A whose rotation is measured by a potentiometer in order to obtain the measurement of θ V while our CDPR MARIONET-VR is instrumented with a more sophisticated cable guiding system which allows for the measurement of both θ V and θ H (figure 4). Our first trials with such a simple system have shown that the accuracy is poor as soon as the cable tension is low. We are therefore considering a non contact system: the idea is to have a range meter mounted on a rotary head in front of the A point (figure 5). The head rotates around the x axis until the range meter detects the cable at point M . The measurement of the distance d between A ′ and M and of the rotation angle β allows one to calculate the coordinates of M . The Fig. 4. On the left the rotation guide of MARIONET-Assist which allows for the measurement of θ V . On the right the system used on MARIONET-VR for the measurement of both θ V and θ H coordinates of the points A and M allow us to determine the cable plane and therefore the θ V angle, while the angle between AM and the x axis provides the θ H angle. This non contact sensing method should provide a better accuracy than the mechanical guides of figure 4. To the best of our knowledge no similar system has been proposed for the platform. For measuring theses angles we may also consider a vision system as proposed in [START_REF] Andreff | Image-based visual servoing of a Gough-Stewart parallel manipulator using leg observations[END_REF].

We may also consider having a 3D accelerometrer on the platform and assume that the CDPR motion is sufficiently slow to neglect the acceleration due to the motion. In that case the accelerometer will provide 2 orientation angles of the platform. Such a sensor will be called an IMU and will be counted as 1 sensor.

II. FUNDAMENTAL EQUATIONS

In this section a superscript j will denote an unknown of cable j. The components of u the mobile frame are

u m = cos(α V ) cos(α H ), cos(α V ) sin(α H ), sin(α H ). If R
is the rotation matrix between the reference frame and the mobile frame, R being expressed as functions of 3 angles ψ, θ, φ, then u in the reference frame is Ru m . The vector u should be perpendicular to the normal to P. A possible way to express this constraint is to consider the rotation matrix 1 corresponding to a rotation around the vertical of angle θ V so that the vector R 1 Ru m has 0 as second component. This constraint may be written as This constraint is a function of α V , α H , θ V , ψ, θ, φ. Let u v be the 2D vectors whose components are the first and third component of R 1 Ru m . In the plane frame the unit vector of the tangent to the cable at B is expressed as

R 1 Ru m .(0, 1, 0) T = 0 (3)
(u v (1), u v (2)). If we define τ B = F 2 x + F 2 z the components of this vector are also (F x /τ B , F z /τ B ). This allows to define a new constraint F z /F x = u v (2)/u v (1) (4)
This constraint is a function of F x , F z , α V , α H , θ V and of ψ, θ, φ. At A the unit tangent vector in the cable plane is

F x /τ A , (F Z -µgL 0 )/τ with τ A = F 2 x + (F z -µgL 0 ) 2 so that we have (F Z -µgL 0 )/F x = tan(θ H ) (5)
This constraint only involve θ H , F x , F z . We may also have to use the mechanical equilibrium of the platform, assuming that it is submitted only to gravity. The force equilibrium may be written as R 1 T (F x , 0, F z ) T = (0, 0, mg) where m is the platform mass, which leads to

j=3 j=1 -cos(θ j V )F j x = 0 j=3 j=1 sin(θ j V )F j x = 0 j=3 j=1 -F j z -mg = 0 (6)
These 3 constraints are evidently only function of the θ V , F x , F Z . The moment equations may be written as

j=3 j=1 GB j × (R 1 T (F x , 0, F z ) T ) = 0 (7)
which are functions of θ V , F x , F Z and of ψ, θ, φ. We may also use the fact that if the coordinates x b , z b have been determined, then the components of vector OB in the reference may be obtained as

OB = OA + R T 1 (x b , 0, z b ) T (8)
provided that the 3 angles θ V are known. Our objective is now to investigate various sensor placements and determine how the DK solutions may be obtained through only algebraic manipulation. In this paper we will assume that all sensor measurements are exact: we are aware that it is not realistic but this paper is a preliminary step for investigating which sensor placement may be used to determine theoretical solutions while uncertainties on the measurement will be studied next. It must be noted that the non algebraic Irvine equations (1,2) may be used only to determine first the x b , z b (provided that the F z , F z have been calculated) and then the coordinates of the B (provided that θ V been determined). Therefore our objective is not use these 6 equations in the DK solving unless the F x , F z , θ V have been obtained. now analyze how many sensors are required. If we assume that only the θ V , θ H are measured we have as constraints the 3 equations ( 5) and the 3 force equilibrium (6) with a total of 12 unknowns. For getting a system with a finite number of solutions we must therefore have 6 measurements. If we add the moment equilibrium [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] we have nine constraints but we add 3 unknowns (ψ, θ, φ) for a total of 15 unknowns. Here again we will need 6 measurements to end up with a square system.

If we measure only the α V , α H we have as unknowns 6 α V , α H , the F x , F z and ψ, θ, φ, θ V (because any constraints dealing with α V , α H use these 6 later variables) for a total of 18 unknowns. The constraints are the 6 force and moment equilibrium (6, 7), the 3 equations (3) and the 3 equations (4) for a total of 12 equations. Therefore we need at least 6 measurements to get a square system.

More generally assume that we are measuring n 1 θ V , n 2 θ H , n 3 α V , n 4 α H (on the same B than the measured α V ), n 5 α H (on a B that has no α V measurement) and n 6 measurement on the ψ, θ, φ. Assuming that n 3 or n 5 are greater than 0 we have as unknowns: 6 F x , F z , 3n 1 θ V , 3n 6 ψ, θ, φ, |n 4 -n 3 | α H or α V (on the same B) and n 5 α H for a total of 12 + |n 4 -n 3 | + n 5 -n 1 -n 6 . In terms of equations we have the 6 equilibrium, Sup(n 3 , n 4 ) equations (3), Sup(n 3 , n 4 ) equations ( 4) and n 2 equations (5) for a total of 6 + 2Sup(n 3 , n 4 ) + n 2 + n 5 . Hence to get a square system imposes to have 6 ≤ n 2 + 2Sup(n 3 , n 4 ) + n 1 + n 6 -|n 4 -n 3 |. This formula allows on to show that at least 6 measurements are necessary to solve the DK in analytic form.

III. PLANAR CASE, 2 CABLES

If we consider a planar robot with 2 cables we have only θ H and α H sensors and the equations of the previous section are simpler. If θ denote the orientation of the platform the mechanical equilibrium may written as

F 1 x + F 2 x = 0 F 1 z + F 2 z = mg (9) i=2 i=1 -GB i (1)F i z + GB i (2)F i x = 0 (10)
the later equation being a function of θ and of the F x , F z . As for the sensors we have

tan(θ + α H ) = F z F x (11) tan(θ H ) = (F Z -µgL 0 )/F x (12)
For solving the DK in symbolic form we cannot use the Irvine equations (1, 2) unless we have determined the F x , F z . Hence if we use the two first equations of (10), then we have 4 unknowns and 2 constraints, while if we use the 3 equations ( 10), then we have 5 unknowns. Hence it is necessary to have at least 2 sensor measurements to solve the DK in analytic form.

If we measure 2 θ H , then the 2 first equations of (10) and the 2 equations of (12) constitutes a system of 4 linear equations in the F x , F z . Solving this system and reporting the result in the Irvine equations leads to the location of B 1 , B 2 . In turn this leads to 2 solutions for the DK, one with G above B 1 B 2 , which is unstable, and one with G below B 1 B 2 , which is stable.

If we measure 2 α H we may obtain the F x , F z by solving the linear system (9), [START_REF] Merlet | The forward kinematics of cable-driven parallel robots with sagging cables[END_REF] which are linear in the F x , F z . These unknowns are obtained as function of θ and reporting their value in [START_REF] Carricato | Stability analysis of underconstrained cable-driven parallel robots[END_REF] leads to a function of θ, cos θ which is transformed into a 6th order algebraic equation using the tangent half-angle substitution. Hence there will be at most 6 stable DK solutions.

If we measure one θ H and one α H we may obtain the F x , F z by solving the linear system ( 9),( 11), [START_REF]A generic numerical continuation scheme for solving the direct kinematics of cable-driven parallel robot with deformable cables[END_REF] which are linear in the F x , F z . These unknowns are obtained as function of θ and reporting their value in [START_REF] Carricato | Stability analysis of underconstrained cable-driven parallel robots[END_REF] leads to a function of sin θ, cos θ which is transformed into a 4th order algebraic equation using the tangent half-angle substitution. Hence there will be at most 4 stable DK solutions.

IV. SPATIAL CASE, 3 CABLES

A. DK with 6 sensors: 3-θ V θ H

In this section we assume that all A anchor points have both θ V , θ H sensors. If we consider the constraints (5, 6) we get a system of 6 equations in the 6 F x , F z which is linear in these unknowns. Solving for these unknowns allows one to calculate the coordinates x b , z b for each cable. As the 3 angles θ V are known we may then use equation ( 8) to determine the coordinates of the three B in the reference frame and therefore the pose of the platform. In summary this placement allows to determine a single DK solutions with the effort of solving a linear system.

B. DK with 6 sensors: 2-θ V θ H , 1 θ V and IMU

In this section we assume that the anchor points A 1 , A 2 have both θ V , θ H sensors while A 3 has only a θ V sensor. An accelerometer is located on the platform. The 2 first equations of (5), the three equations ( 6) and the equations of ( 7) have as unknowns ψ, θ, φ and the F x , F z while being linear in these later unknowns. We select the first five equations and the first equation of the moment equilibrium [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] to get a linear system of 6 equations in the F x , F z . After solving this system the second and third equations of ( 7) are only function of ψ, θ, φ. As the accelerometer provides 2 of these unknowns these 2 equations are just function of the sine and cosine of the remaining unknown angle. If we use the Euler's angles the second equation factors out in 2 terms that are linear in the cosine, sine of any of the angle ψ, θ, φ. The first equation is linear in sin(ψ), cos(ψ) and therefore the 2 equations constitute a linear system in these 2 unknowns. Hence we will obtain at most 2 DK solutions by solving 2 linear systems.

C. DK with 5 sensors: 2-θ V θ H and IMU

In this section we assume that the anchor points A 1 , A 2 have both θ V , θ H sensors while an accelerometer is located on the platform. The 2 first equations of ( 5), the 6 equilibrium equations ( 6) and ( 7) have as unknowns ψ, θ, φ and the F x , F z while being linear in these later unknowns. We select the first five equations and the first equation of ( 7) to get a linear system of 6 equations in the F x , F z . After solving this system the second and third equations of ( 7) are only functions of ψ, θ, φ, θ 3

V . As the accelerometer provides 2 of these unknowns these 2 equations are just functions of the sine and cosine of the remaining unknown angle. If we use the Euler's angles the second equation factors out in 2 terms. The first factor is linear in the cosine, sine of any of the angle ψ, θ, φ and does not involve θ 3

V . The first equation is linear in sin(ψ), cos(ψ) and therefore the first factor and this equation constitute a linear system in these 2 unknowns. After solving this system the constraint sin 2 ψ + cos 2 ψ -1 involves only the sine and cosine of θ 3 V . By using the half-angle tangent substitution this equation becomes a 4th order polynomial in T = tan(θ 3 V /2) and hence we will obtain at most 4 DK solutions.

The second factor is linear in sin(θ 3 V ) and after solving for this variable it appears that the first equation may be written as cos θ 3

V F (ψ, θ, φ). As cos θ 3 V = 0 implies also sin θ 3 V = 0 only the equation F has to be considered. This equation is of degree 4 in T 1 = tan(ψ/2), T 2 = tan(θ/2) and T 3 = tan(φ/2). Whatever is the chosen variable we get for each root a special case where the mechanical equilibrium is not dependent upon the value of

θ 3 V while F 1 x , F 2 x , F 1 z , F 2 z , F 3 z
have constant values (meaning that B 1 , B 2 have a fixed position) and F 3

x is obtained as a/ cos θ 3 V where a is a constant. Using he Irvine equations (1,2) we get x 3 b , z 3 b as a function of cos θ 3

V and using equation ( 8) we obtain the coordinates of B 3 as a function of θ 3 V . At the same time B 3 must lie on a circle centered in a point located on the line going through B 1 , B 2 , the circle being perpendicular to this line. The center and the radius of this circle may easily be calculated from the known distances between (B 1 , B 3 ), (B 1 , B 2 ) and (B 2 , B 3 ). As B 3 must belong to this circle it induces 2 constraint equations on θ 3

V and unfortunately there is no way to determine the maximum number of solutions of this system as it is not algebraic. However we have found numerical examples with 2 solutions.

In this section we assume that the B 1 , B 2 anchor points have both α V , α H sensors and that an accelerometer is located on the platform. The unknowns are therefore the 6 F x , F z , the 3 θ V that are used to calculate the x b , z b and the angles ψ, θ, φ, two of which will be provided by the IMU for a total of 10 unknowns. In terms of constraint we have the 6 equilibrium constraints [START_REF]Direct geometrico-static problem of underconstrained cabledriven parallel robots with n cables[END_REF][START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF], the two equations in [START_REF] Pott | An algorithm for real-time forward kinematics of cabledriven parallel robots[END_REF] and in (4) involving the measured α V , α H for a total of 10 constraints. The 3 force equations ( 6), the two equations ( 4) and the first moment equation of [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] are linear in the F x , F z and are used to find these variables. We assume that θ is not measured by the IMU (but the process will the same whatever is the not measured angle). The 2 equations (3) are linear in the unknowns sin θ, cos θ and are used to determine these variables with the additional constraint sin 2 θ+cos 2 θ-1 = 0 which is now a function of θ 1 V , θ 2 V . The two remaining equations of the moment equilibrium [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] are now function of the 3 θ V . Hence we have now 3 equations in the 3 θ V that are converted into algebraic form using the tangent halfangle substitution. Successive resultant in T 1 , T 3 leads to a single polynomial in T 2 which factors out in 2 polynomial of degree 4 and one polynomial of degree 16. Each root of these polynomials leads to a single value of the 3 θ V and of θ which in turn leads to a single value for the F x , F z allowing to calculate first the x b , z b of all cables and using (8) a single location for all the B. Hence the DK may have up to 24 solutions.

V. USING A SIMPLER CABLE MODEL: THE PARABOLA

CASE

It must be noted that Irvine has proposed as simplified cable model where the shape of the cable is a parabola. This model leads to a single solution for the inverse kinematic [START_REF] Gouttefarde | Simplified static analysis of large-dimension parallel cable-driven robots[END_REF], while they are multiple solutions with the full model [START_REF] Merlet | On the inverse kinematics of cable-driven parallel robots with up to 6 sagging cables[END_REF]. From the DK view point the unknowns are the 3 coordinates (x i , y i , z i ) of the B i and the F x , F z for a total of 15 unknowns as the θ V may be expressed as functions of the x i , y i . The constraint are the 6 equilibrium equations (6, 7) and the 3 equations (4) (which are identical to the equations ( 5)) which amount to:

F z = µgL 0 2 + F x z b x b (13)
where x b , z b are the coordinates of B in the cable plane, that are derived from the (x i , y i , z i ) Three other equations are obtained by writing that the distance d ij between the pair of points B i , B j is a known constant, leading to quadratic equations in the x i , y i , z i . The final 3 constraints are that the L 0 may be expressed as non linear, non algebraic equations (involving the logarithm function) of the unknowns, that therefore cannot be used to have an algebraic system. Hence we end up with a square system of 15 equations, 12 of them being algebraic. A benefit of this approach is that we theoretically need only 3 measurements to get algebraic system that may provide solution(s) in closed-form. For that purpose we consider the force equilibrium equations ( 6) and the 3 equations ( 13) that constitutes a linear system in the F x , F z . We may solve this system in these variables but the presence of z b /x b in equation ( 13) leads to relatively large expression for the equations ( 7) and our trials have shown that even with 5 additional sensors the elimination process leads to an univariate polynomial of high degree.

VI. EXTENSION FOR A CDPR WITH n CABLES

If we consider a spatial CDPR with n cables the unknowns are the 2n F x , F z , the n θ V and the 6 parameters of the pose for a total of 3n + 6 unknowns. The constraints are the 2n Irvine equations (1, 2), the n constraints (3) and the 6 equations of the mechanical equilibrium (6, 7) for a total of 3n + 6 constraints. Hence if we want to get rid of the non algebraic equations the measurements should provide 2n data. The simplest case is to measure all θ V , θ H : in that case the same reasoning than in section IV-A allows one to show that the DK will have a single solution.

VII. CONCLUSION

This investigation on additional sensors for simplifying the direct kinematics of CDPR with sagging cable has shown that the best sensor arrangement is to use ground-based sensors that measure the θ V , θ H angles for all cables. This arrangement has the advantages of a simple implementation and of providing usually a single DK solution. We have also shown that for a CDPR with n cables at least 2n additional measurement are required for getting the DK solutions in closed-form and that the θ V , θ H arrangement on all cables will provide a single DK solution whatever n is.

Having sensors on the platform is still a possibility but leads to a more complex solving process that may be simplified if we have an estimate of the platform orientation with the drawback of leading to multiple DK solutions. Accelerometer may provide this information but their accuracy in practice is doubtful.

We have proceeded under the assumption that all measurements are exact, an assumption that is clearly not true in practice. Taking into account the effective accuracy of the orientation sensors and its influence on the DK solving is another issue that has to be investigated.

Another issue is that adding sensors beside the cable lengths provides sensor redundancy. To reduce the effect of uncertainties on the DK solution we may consider the DK as an optimization problem. However using the deterministic approach to provide an initial guess for the optimization may make sense.

At this stage it is uncertain if adding orientation sensors on the cable will provide a reliable solution for the DK problem and may lead to an improvement of the accuracy of the CDPR. Possibly direct estimation of the platform pose using vision, optical markers or telemeter may provide a better accuracy even for large CDPR.

Fig. 2 .

 2 Fig. 2. Orientation sensors at A may provide the value of θ V and/or θ H

•Fig. 3 .

 3 Fig.3. Orientation sensors at B may provide the value of α V and/or α H

Fig. 5 .

 5 Fig.5. A new sensor arrangement for measuring the θ V , θ H angles. A range meter is mounted on a rotary head located at A ′ rotating around the x axis. The range meter is rotated until it detects the cable and the system provides the distance d between A ′ and the cable and the rotation angle β

D. DK with 6 sensors: 3-α V α H

In this section we assume that the anchor points B have both α V , α H sensors. The 3 equations of the force equilibrium [START_REF]Direct geometrico-static problem of underconstrained cabledriven parallel robots with n cables[END_REF] and the equations (4) are linear in the F x , F z . Solving this system leads to 6 equations, the 3 moment equilibrium [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] and the 3 equations (3), whose unknowns are the 3 θ V and ψ, θ, φ. Hence this system (which may be transformed into an algebraic form) has in general a finite number of solution but we have been unable to reduce this system to an univariate polynomial.

E. DK with 5 sensors: 3-θ V , 1 θ H and IMU

In this section we assume that the A anchor points have all θ V sensors while A 1 has also a θ H sensor and an accelerometer is located on the platform. We consider the system constituted of the 3 equations of the force equilibrium ( 6), the 2 first equations of the moment equilibrium [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] and the first equation of (5). This system has as unknowns ψ, θ, φ and the F x , F z and is linear in the 6 later variables. Solving this system and reporting the result in the last equation of the moment equilibrium [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] leads to an expression that factors out into 2 components, both of which are linear in the sine, cosine of any angle ψ, θ, φ considered independently. As two of these variables are provided by the accelerometer we end up with two systems that can be written as U cos β + V sin β + W = 0, β being any angle in the set ψ, θ, φ, which admits two solutions in β. Therefore we get up to 4 solutions in the unknown angle. For each of the solution we get the position of the B of each cable in the cable plane and equation [START_REF] Carricato | Direct geometrico-static problem of under-constrained cable-driven parallel robots with three cables[END_REF] allows to calculate them in the reference frame: hence there may up to 4 solutions of the DK that are obtained by solving two quadratic polynomials.

F. DK with 5 sensors: 3-θ H , 1 θ V and IMU

In this section we assume that the A anchor points have all θ H sensors while A 1 has also a θ V sensor. An accelerometer is located on the platform. We consider the system constituted of the 3 equations of the force equilibrium (6) and the 3 equations (4). This system of 6 equations has as unknowns θ 2

V , θ 3 V , and the F x , F z and is linear in the 6 later variables. Solving this system and reporting the result in the 3 moment equations ( 7) lead to a system having ψ, θ, φ, θ 2

V , θ 3 V as unknowns. Any of the 3 equations of the moment equilibrium is linear in sin θ 2 V , cos θ 2 V . We consider any two pair of these equations to solve in this variables and report the result in the remaining equation of [START_REF] Berti | Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis[END_REF] and in the constraint equation sin 2 θ 2 V + cos 2 θ 2 V -1 = 0. This leads to 2 equations in the unknown θ 3

V , ψ, θ, φ. Using the half-angle tangent substitution on θ 3

V and calculating the resultant of the 2 equations in T 3 leads to a single equation in ψ, θ, φ. As the IMU provides two of these angles we have therefore an univariate equation and using the half-angle substitution on any of the angle leads to a polynomial of degree 32. Each of the root of this polynomial leads to a single value for θ 2 V , θ 3

V and for the F x , F z . This allows one to calculate the x b , z b for each cable and using equation ( 8) a single position for the B. Hence the DK may have up to 32 solutions.