Horst Schulte
email: horst.schulte@boschrexroth.de

Kevin Guelton
email: kevin.guelton@univ-reims.fr

Descriptor modelling toward control of two links pneumatic robot manipulator: a T-S multimodel approach

Keywords: Descriptor systems, Control oriented models, Robotic manipulators, Fluid Power System

This paper presents the first steps towards a robust model-based controller design for two-link manipulators using a Takagi Sugeno multimodel descriptor form. Due to the circumstances that this form is more similar to a given original nonlinear equation as a Takagi Sugeno multimodel, it allows to reduce the conservatism of the controller design by using common matrix structures. The model-based control law here is equivalent to the well-known parallel distributed compensation scheme. The challenge of the investigated modelling and control problem in this case is the highly nonlinear dynamics of the dual-actuator drive powered by air-pressure that interacts with the dynamics of the robot manipulator.

Introduction

With the growing interest in modelling and control of systems with complex dynamics, nonlinear control theory becomes popular and several applications have been proposed, for instance in robotics [START_REF] Hu | Fuzzy supervisory sliding-mode and neural-network control for robotic manipulators[END_REF][START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF][START_REF] Schulte | Modelling and simulation of two-link robot manipulators based on Takagi-Sugeno fuzzy descriptor systems[END_REF]. For several years Takagi Sugeno (T-S) systems [START_REF] Takagi | Fuzzy identification of systems and its application to modelling in control[END_REF] are widely used in a context of modelling for control design [START_REF] Sala | Perspectives of fuzzy systems and control[END_REF][START_REF] Tanaka | Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs[END_REF][START_REF] Wang | An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues[END_REF][START_REF] Lian | LMI-Based Sensorless Control of Permanent-Magnet Synchronous Motors[END_REF][START_REF] Precup | Fuzzy Controllers With Maximum Sensitivity for Servosystems[END_REF][START_REF] Mansouri | Output feedback LMI tracking control conditions with H? criterion for uncertain and disturbed TS models[END_REF]. Indeed, it is well known that T-S multimodels is an universal approximator of continuous nonlinear systems by using, for instance, the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis[END_REF] or a tensor product approach [START_REF] Petres | Trajectory Tracking by TP Model Transformation: Case Study of a Benchmark Problem[END_REF], [START_REF] Schulte | Approximate Modeling of a Class of Nonlinear Oscillators using Takagi-Sugeno Fuzzy Systems and Its Application to Control Design[END_REF]. Thus, by analogy to switched systems [START_REF] Mansouri | Robust pole placement controller design in LMI region for uncertain and disturbed switched systems[END_REF], a T-S mutlimodel can be view as a set of local LTI systems blended together by smooth switching functions. Also, linear descriptor systems, which differ from state-space representations, have generated a great deal of interest in control system design. Descriptor systems describe a wider class of systems including physical models and nondynamic constraints [START_REF] Luenberger | Dynamic Equations in Descriptor Form[END_REF]. The interest of choosing a descriptor system rather than a classical state space representation, in particular for mechanical systems, is based on: 1. Reducing the number of rules → reducing conservatism, see [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown inputs observer in the descriptor form[END_REF]. [START_REF] Mansouri | Output feedback LMI tracking control conditions with H? criterion for uncertain and disturbed TS models[END_REF]. Using the natural model structure due to the inertia matrix. Then, the use of a descriptor representation is justified to model a robot if the inertia matrix contains nonlinear terms, i.e. if the structure of the considered robot is geometrically variable, that is the case of the SCARA robot. There exist a large number of papers on the stability analysis of T-S multimodels based on the state-space representation and also some applications in the control design process e.g. [START_REF] Messoussi | Four-Wheel Steering Vehicle Control using Takagi-Sugeno Fuzzy Models[END_REF][START_REF] Schulte | Fuzzy state feedback gain scheduling control of servo-pneumatic actuators[END_REF][START_REF] Sun | Neuro-Fuzzy Dynamic-Inversion-Based Adaptive Control for Robotic Manipulators-Discrete Time Case[END_REF]. But in contrast, stability analysis of T-S multimodels based on descriptor representation as weighted combinations of linear descriptor systems (from now called T-S descriptor systems) have not been investigated until [START_REF] Taniguchi | Fuzzy Descriptor Systems and Nonlinear Model Following Control[END_REF] and recently applied in biomechanics [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown inputs observer in the descriptor form[END_REF] and in robotics [START_REF] Schulte | Modelling and simulation of two-link robot manipulators based on Takagi-Sugeno fuzzy descriptor systems[END_REF]. The purpose of this work is to propose a robust multimodel controller design based on a first derived descriptor form of a two-link robot manipulator powered by pneumatic actuators as a reasonable example of a highly nonlinear complex system. This paper is organized as follows: First, for a class of serial manipulators, the so-called SCARA-type robot manipulators [START_REF] Sciavicco | Modelling and Control of Robot Manipulators[END_REF], a nonlinear statespace model is formulated by the Newton-Euler equation of the manipulator arm coupled with a reduced model of the actuators which contains two bounded uncertainties. Then the state-space model combined with dynamic constraints is transformed into a T-S descriptor form [START_REF] Taniguchi | Fuzzy Descriptor Systems and Nonlinear Model Following Control[END_REF]. It is shown by simulation results that this proposed model is capable to represent the coupled nonlinearities. Finally, in a prospective way, a global controller is designed based on the previously derived model as a robust parallel distributed compensation (PDC) [START_REF] Sala | Perspectives of fuzzy systems and control[END_REF].

Physical System Modelling

In the following the dynamic model of a SCARA-type robot manipulator is presented. The robot arm is powered by an actuator mechanism based on a tangential feed as the main drive and an electric direct-drive as the actuator of the second axis presented in Fig. 1. The main drive consists of two rodless cylinders with pistons, two deflection rollers and a metal belt as a frictionlocked connection between the rollers and the pistons (also called Belt-motor [START_REF] Schillings | Servopneumatische Antriebssysteme und Handhabungsgeräte[END_REF]). The forces on the pistons caused by opposed pressure differences in Actuator 1 and 2 are transmitted by the belt to the deflection rollers, where the linear motion of the pistons is converted to angular motion of the robot link 1. We assume that the normal strain of the belt is negligibly small (infinitely stiff) and the tangent stiffness is pure elastic.

Mechanical model of the manipulator dynamics

Consider the two-link planar arm shown in Fig. 1 that represents a reduced mechanical system of the manipulator. The robot arm has two vertical revolute joints, both joint axes are orthogonal to the xy-plane. Here, the masses of the actuator cylinders and the mass of link 1 are combined to m sp . Let m e be the mass of the rotor of the electric drive, and m t the mass of the tool and vertical drive in the tool center point (TCP, see Fig. 1), whereby the mass of link 2 is small in comparison to m t . The distance of the centers of mass of m sp and m e to the Joint-axis 1 is defined by l 1 /2, and l 1 as the length of link 1. The distance of the center of mass of m t to the joint-axis 2 is defined by l 2 as the length of link 2. The equation of motion for this mechanical system can be written in the known matrix form [START_REF] Sciavicco | Modelling and Control of Robot Manipulators[END_REF] which represents the joint space dynamics model:

H(q) q + h(q, q) = τ (1)
where q = [ ψ ϕ ] T is the vector of generalized coordinates with ψ as the angle of link 1 from the x-axis, and with ϕ as the angle between link 1 and link 2. The inertia matrix

H(q) = H 11 (q) H 12 (q) H 21 (q) H 22 (2)
depends on the current arm configuration with:

H 11 (q) = I L1 + m sp l 1 2 2 + m e l 2 1 + I L2 +m t l 2 1 + l 2 2 + 2 l 1 l 2 cos ϕ , H 12 (q) = H 21 (q) = I L2 + m t l 2 (l 2 + l 1 cos ϕ) , H 22 = I L2 + m t l 2 2 .
Note that, for this application, the assumption holds (the links are made of carbon-fiber-reinforced plastic) that the moments of inertias relative to the centers of mass of link 1 and link 2 as I L1 and I L2 can be neglected against the other terms, such as (m e l 2 1 ). So we set in the following considerations

I L1 = I L2 = 0. The actuation torques in (2) are represented by τ = [ τ 1 τ 2 ] T . Finally, the vector h(q, q) = h 1 (q, q) h 2 (q, q) (3)
in (1) represents the centrifugal effects and Coriolis effects with h 1 (q, q) = m t l 1 l 2 sin ϕ (-φ2 -2 ψ φ) and h 2 (q, q) = m t l 1 l 2 sin ϕ ψ2 .

P I 2 P II 2 p S p R p R . m 2 1 m . x v 3 m m . .

Dual-Actuator model

The physical description of the main drive (see Fig. 1) is based on the continuity and enthalpy balance equation of the gas flow through the actuator and the constitutive relations of the air. Whereby the most important aspect of the actuator dynamics that interact with the robot arm is described by the pressure evolution of each chamber. Assuming isentropic behavior of the gas flow, the pressure evolution in the chambers of actuator j (j = 1, 2 , see Fig. 1) is modeled by the equations:

ṗI j (V 0I + A K (x K -x K0 )) = κ R g T I ṁ2 -ṁ1 -p I A K ẋK (4a)
and

ṗII j (V 0II -A K (x K -x K0 )) = κ -R g T II ṁ4 -ṁ3 + p II A K ẋK . ( 4b
)
where p I , p II [N/m 2 ] as the pressure in actuator chambers I, II, x K [m] as the actuator piston displacement and ṁn [kg/s] as the mass flow over the servovalve control edges n = 1, 2, 3, 4 (see Fig. 2).

The model parameters are: x K0 [m] as the start position of the actuator piston, V 0I,0II [m 3 ] as the start volume of the actuator chamber I, II , A K [m 3 ] as the actuator piston area, R g as the gas constant [ J kg•K ], and κ [-] as the adiabatic exponent of the gas. The two variable volume chambers in each actuator cylinder are connected to a four-way servovalve. The configuration for one actuator cylinder in combination with the servovalve is shown in Fig. 2. In this case the servovalve controls the air-mass-flow ṁ1 , ..., ṁ4 by adjusting four orifice areas between the constant pressure supply p s , the exhaust pressure p R , and the chamber pressures p I,II . The four orifice areas are continuously controlled by the servovalve voltage u v . The equations for the mass flows are:

ṁ1 = -α D1 A 1 (x v ) ψ p R p II p I 2 R g T II (5a) ṁ2 = α D2 A 2 (x v ) ψ p II p S p S 2 R g T S (5b) ṁ3 = -α D3 A 3 (x v ) ψ p I p S p S 2 R g T S (5c) ṁ4 = α D4 A 4 (x v ) ψ p R p I p I 2 R g T I (5d)
with the servovalve displacement x v and the orifice areas of the control edges for n = 2, 4 :

A n (x v ) = x v π d n for x v ≥ x ün 0 for x v < x ün (6a)
and for n = 1, 3 :

A n (x v ) = x v π d n for x v ≤ -x ün 0 for x v > -x ün , ( 6b
)
and the flow function:

ψ p a p b =      ψ 0 1 - pa p b -p krit 1-p krit 2 for pa p b ≥ p krit ψ 0 for pa p b < p krit . ( 6c
)
The not yet defined model parameters are:

α

Reduced physical model in T-S multimodel descriptor form

Reduced actuator model in descriptor form

The detailed physical description of the servo-pneumatic actuators is reduced by simplification of the mass flow relations [START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown inputs observer in the descriptor form[END_REF] and by linearization of the switching functions for zero valve overlap (x ün = 0 for n = 1, ..., 4). The influence of the model reductions on the differential equation of the chamber pressures p I , p II is condensed in two bounded uncertainties as ∆B and ∆a. The remaining nonlinear terms are transfered into T-S multimodel relations using a sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis[END_REF]. It leads to the following descriptor form:

2 j=1 ν P j (x K ) E P j ẋP = 8 i=1
h P i ( ẋK , p I , p II ) A P i x P + B P i x v + ∆B x v + ∆a [START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF] with the state vector

x P = [ p I , p II ] T (8a)
and

E P j = V 0I +A K ( * 1 -x K0 ) κ 0 0 V 0II -A K ( * 1 -x K0 ) κ , * 1 ∈ {x K , xK } (8b )
A P i = A K * 2 0 0 A K * 2 , * 2 ∈ { ẋK , xK } , ( 8c
)
B P i = R g T I α D π d ( ã02 + b02 * 3 ) -R g T II α D π d ( ã03 + b03 * 4 ) , * 3 ∈ {p I , pI } , * 4 ∈ {p II , pII } . ( 8d
)
and the uncertainty terms

∆a = ∆p I ∆p II , ( 8e
) ∆B = R g T I α D π d ∆ã 2 -R g T II α D π d ∆ã 3 . (8f)

Model of the manipulator dynamics in T-S multimodel descriptor form

The equations ( 1),( 2) can be directly written in the following "exact" T-S multimodel descriptor form

2 j=1 ν R j (ϕ) E R j ẋR = 4 i=1 h R i (ϕ, φ, ψ) A R i x R + B R τ 1 τ 2 (9)
with

x R = ψ , ϕ , ψ , φ T , ( 10a
) E R j =   I 2×2 0 2×2 0 2×2 a + 2 b * 5 c + b * 5 c + b * 5 c   , * 5 ∈ {f 1 , f1 } , ( 10b
) A R i =   0 2×2 I 2×2 0 2×2 2 b * 6 b * 6 -b * 7 0   , * 6 ∈ {g 1 , ḡ1 } , * 7 ∈ {g 2 , ḡ2 } , (10c)
and

B R = 0 2×2 I 2×2 , ( 10d
)
whereby

f 1 = cos ϕ , g 1 = φ sin ϕ , g 2 = ψ sin ϕ , a := m sp l 1 2 2 + m e l 2 1 + m t ( l 2 1 + l 2 2 ) , b := m t l 1 l 2 , c := m t l 2 2 .
Based on the assumptions that, first, the piston area of the actuators are equal A K 1 = A K 2 =: A K , and second, the mass flow control of each actuator caused by opposed pressure differences p L j = p I jp II j , j = 1, 2 (see Fig. 1) the magnitudes are

p L 1 = p L 2 =: p L , (11)
the driving torque of joint 1 can be written as:

τ 1 = p L 1 A K 1 d r 2 + p L 2 A K 2 d r 2 = p L A K d r , ( 12
)
with d r as the roller diameter. In this consideration it is assumed that the electric actuator of joint 2, see Fig. 1, behaves as an ideal torque-controlled generator, which gives a proportional relation between the torque τ 2 and the control voltage u m established by the motor constant k m .

τ 2 = k m u m (13)

Complete dynamic model in descriptor form

The complete dynamic model is based on a combination of the actuator model in descriptor form [START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF] and the robot arm dynamics in descriptor form [START_REF] Luenberger | Dynamic Equations in Descriptor Form[END_REF]. It was derived by using the following algebraic constraints between the translational motion of the actuator piston and the rotational motion of the deflection roller:

x K = ψ d r 2 , ẋK = ψ d r 2 . ( 14
)
With the assumption that the piston areas are A K = A K 1 = A K 2 and the mass flow control of each actuator is caused by opposed pressure difference in both actuators the chamber pressures are:

p I = p I 1 = p I 2 , p II = p II 1 = p II 2 .
The driving torque in joint 1 is

τ 1 = ( p I 1 -p II 1 ) A K 1 d r 2 + ( p I 2 -p II 2 ) A K 2 d r 2 = ( p I -p II ) A K d r . ( 15
)
So the complete dynamic model can be presented as:

4 k=1 νk ( ψ, ϕ ) E k ẋ = 32 l=1 h l (ϕ, φ, ψ, ψ, p I , p II ) [ A l x + ( B l + ∆B ) u ] + ∆ã (16) with E k ∈ R 6×6 , A l ∈ R 6×6 , B l ∈ R 6×2 , ∆ã ∈ R 6×1 , the input vector u = [ x v , τ 1 ] T , ( 17a
)
and the state vector

x = ψ , ϕ , ψ , φ , p I , p II T . ( 17b
)
The left hand side weighting functions νk in [START_REF] Schulte | Modelling and simulation of two-link robot manipulators based on Takagi-Sugeno fuzzy descriptor systems[END_REF] are obtained from all combinations of ν P j in [START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF] and ν R i in (9) for i and j = 1, 2 and the right hand side weighting functions h l in ( 16) from all combinations of h P j in [START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF] and h R i in (9) for i = 1, ..., 4 and j = 1, ..., 8. In detail the matrices are:

E 1 = E R 1 0 4×2 0 2×4 E P 1 , E 2 = E R 2 0 4×2 0 2×4 E P 1 , ( 18a
)
E 3 = E R 1 0 4×2 0 2×4 E P 2 , E 4 = E R 2 0 4×2 0 2×4 E P 2 , ( 18b
)
A l =       0 2×2 I 2×2 0 2×2 0 2×2 2 b * 6 b * 6 -b * 7 0 A K d r -A K d r 0 0 0 2 × 2 0 2 × 2 A K * 8 0 0 A K * 8       , ( 18c
)
with * 8 ∈ { ẋK , xK } where ẋK = ψ dr 2 , xK = ψ dr 2 and

B l =         0 0 0 0 0 1 0 0 b 5 ( * 3 ) 0 b 6 ( * 4 ) 0         , ∆B =   0 4×2 ∆b 5 0 ∆b 6 0   (18d) with b 5 ( * 3 ) = R g T I α D π d ( ã02 + b02 * 3 ) , b 6 ( * 4 ) = -R g T II α D π d ( ã03 + b03 * 4 ) , ∆b 5 = R g T I α D π d ∆ã 2 ∆b 6 = -R g T II α D π d ∆ã 3 and finally ∆ã =   0 4×1 ∆p I ∆p II   . (18e)

Experimental validation of the actuator model

The proposed actuator model in descriptor form ( 7) is validated by comparsion of the simulated pressure evolution with experimental data measured in each chamber of the dual-actuator, see Fig. 2. For the simulation purpose the nominal model [START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF] with ∆B = 0 and ∆a = 0 was transferred into the Matlab/Simulink framework using the S-function application programming interface (API). At each integration step n a system of linear descriptor equations of the form:

E P (n) ẋP (n) = A P (n) x P (n) + B P (n) x v

Model-based controller design concept

The basic idea behind the investigated controller concept is that the model base of the feedback loop controllers will be increased so a less accurate knowledge of the system model is required in the feedforward compensation. In this paper we consider the special case of a structure without a common feedforward compensation as an appropriate starting point.

By defining x = [ x T ẋT ], the descriptor system can be rewritten as:

Ẽ ẋ = 32 l=1 4 k=1 h l νk Ãlk x + ( Bl + ∆ B ) u + ∆ ã ( 19
)
with h l = h l (ϕ, φ, ψ, ψ, p I , p II ) and νk = νk ( ψ, ϕ ) where:

Ẽ = I 0 0 0 , Ãlk = 0 I A l -E k , Bl = 0 B l , ∆ B = 0 ∆B , ∆ ã = 0 ∆ã .
We propose a modified parallel distributed compensation scheme (PDC):

u = 32 l=1 4 k=1 h l νk F lk ẋ (20)
with F lk = [ F lk 0 ] to stabilize the multimodel descriptor system [START_REF] Takagi | Fuzzy identification of systems and its application to modelling in control[END_REF]. The controller design problem here is to determine the local feedback gains using the local models { Ãlk , Bl , ∆ B, ∆ ã} that globally stabilize the multimodel descriptor system with uncertainties by [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs[END_REF]. We currently solve this problem using a novel LMI (Linear Matrix Inequalities) condition derived in [START_REF] Bouarar | Stabilization of uncertain Takagi-Sugeno descriptors: a fuzzy Lyapunov approach[END_REF].

Simulation results

We will now consider the results of the simulation with the global decriptor model of the plant [START_REF] Takagi | Fuzzy identification of systems and its application to modelling in control[END_REF], that is operated in closed loop. For the model validation purpose a fixed linear state-space controller for the whole operating space is used. It is worth to note that this controller is not able to handle the dominate nonlinearities in the plant. The validation process is exemplified by a reference trajectory shown in Fig. 5 that defines the desired tool center position (TCP), see Fig. 1, in the xy-plane of the working space. The corresponding joint angles in Fig. 4 have been calculated by the well-known inverse kinematics of a two-link planar arm [START_REF] Sciavicco | Modelling and Control of Robot Manipulators[END_REF]. The simulation model of the dual actuators is based on the descriptor form [START_REF] Khayati | A Multistage Position/Force Control for Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization, and Multiobjective Observer/Controller Synthesis Using LMI Formalism[END_REF], but in the first step without uncertainties. The influence of the air-pressure evolution in the actuator cylinders that causes a weakly damped dynamics is recognized in Fig. 5 where the desired trajectory distinctly differs from the actual path. The large tracking error at the corners of the rectangle is an indication that the linear controller is not able to handle the dynamics of the dual-actuator pneumatic drive that interacts with the dynamics of the robot arm. This fact is also a motivation for a nonlinear control design, for instance based on [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs[END_REF].

Conclusion

In this paper the possibilities of the use of a T-S descriptor form to describe the dynamics of a two-link manipulators with dual-actuators were analysed. A complete manipulator model using the combination of the actuator models in descriptor form and the robot arm dynamics in descriptor form was derived for a model-based control concept. We must insist on the fact that this paper is a first step dealing with T-S descriptor modelization and a further paper will present the whole T-S multimodel control study. We think this is an appropriate starting point for a robot control concept that systematically take into consideration the influnce of model reduction by structured uncertainties.

Figure 1 :

 1 Figure 1: Two-link manipulator with a dual servo-pneumatic actuator as main drive (ψ) and an electric direct-drive (ϕ)

Figure 2 :

 2 Figure 2: One cylinder of the dual-actuator with the accessory servovalve

  Dn as the flow coefficient of the control edge n = 1, ..., 4 , d n as the effective diameter of the control edge n [m], R g as the gas constant [ J kg•K ], κ as the adiabatic exponent of the gas [-], p krit as the critical pressure ratio [-], ψ 0 as the maximum value of the flow function ψ, and x ün as the valve overlap of the control edge n [m].

Figure 3 :

 3 Figure 3: Difference pressure evolution p L = p I -p II in one cylinder, see (Fig.2)

Figure 4 :

 4 Figure 4: Time history of the reference joint positions ψ d (t), ϕ d (t) and simulated joint positions ψ(t), ϕ(t) relate to Fig. 5.

Figure 5 :

 5 Figure 5: The reference (desired) rectangle trajectory of the TCP, see Fig. 1, and the actual simulated TCP-position. Starting position: x = 1.2 m, y = -0.5 m

supported by the GIS 3SGS within the project COSMOS.


Lynda Seddiki

Kevin Guelton
email: kevin.guelton@univ-reims.fr

Janan Zaytoon

Concept and T-S tracking controller design of a CMC lower-limb rehabilitation device

Keywords: Closed muscular chain, Descriptors, Disturbances attenuation, Linear Matrix Inequalities (LMIs), Rehabilitation device, Takagi-Sugeno, Tracking controller

We consider the kinematic concepts of a new lower-limb rehabilitation device in closed muscular chain. The proposed control structure is based on a trajectory generator and a continuous nonlinear tracking controller. The human efforts applied to this device are considered as external disturbances to the system's dynamics and as inputs to the trajectory generator and allow safe voluntary control of the system by the user. A H ∞ control structure based on a Takagi-Sugeno descriptor model is proposed to track the desired trajectories and to attenuate external disturbances. Stability conditions are given in terms of Linear Matrix Inequalities (LMIs) using a fuzzy Lyapunov function. Finally, simulation results of the proposed control structure for with new rehabilitation device during isokinetic movements illustrate the efficiency of the proposed approach.

INTRODUCTION

Technical assistance to functional rehabilitation has attracted great interest recently; see, for example, [START_REF] Dallaway | Rehabilitation robotics in europe[END_REF][START_REF] Riener | control of robots for rehabilitation[END_REF] and references therein. Many rehabilitation devices have been designed. For instance, Lokomat is a device for gait rehabilitation of patients suffering from neuromuscular trauma [START_REF] Jezernik | Automatic gait-pattern adaptation algorithms for rehabilitation with a 4-dof robotics orthosis[END_REF]. Another example is an open muscular chain lower-limb rehabilitation device developed by our research centre and named Multi-Iso [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF].

The design of a new device must start from specific needs expressed by clinicians in rehabilitation. For a specific muscular complex, several rehabilitation techniques may be considered. We are interested here in knee rehabilitation, and our interest naturally extends to the global lower limb musculoskeletal complex. For muscular deficiency, two rehabilitation techniques can be used: the first is named open muscular chain (OMC), which is characterised by strengthening an isolated muscle group; the second is named closed muscular chain (CMC), which is characterised by recruiting both the agonist and antagonist muscle groups that contribute to the movement [START_REF] Lutz | Comparaison of tibio-femoral joint forces during open kinetic chain and closed kinetic chain exercices[END_REF][START_REF] Steinkamp | Biomechanical consideration in patellofemoral joint rehabilitation[END_REF]. These two techniques are complementary and are used in various stages of rehabilitation protocols [START_REF] Mickkelsen | Closed kinetic chain alone compared to combined open and closed kinetic chain exercices for quadricpes strenthening after anterior cruciate ligament reconstruction with respect to return to sports: a prespective matched follow-up study[END_REF].

For technical reasons, it is difficult to design a device that is able to reproduce these two techniques simultaneously. The only existing such device is named MotionMaker, but it is a very complex and expensive robotic structure [START_REF] Metrailler | Improvement of rehabilitation possibilities with the MotionMaker TM[END_REF]. Consequently, it can be used in the context of research studies but is not appropriate for common rehabilitation centres. OMC devices are commonly used in clinical contexts for lower-limb rehabilitation with isokinetic devices such as Cybex, Biodex or Multi-Iso [START_REF] Anaes | Les appareils d'isocinétisme en évaluation et en rééducation musculaire: intérêt et utilisation[END_REF]. More recently, CMC rehabilitation has received particular attention because of its ability to stabilise the targeted joint during exercises similar to those used in daily life (such as walking, sitting or standing up). For instance, Moflex, Contrex LP or Erigo devices are CMC rehabilitation devices [START_REF] Patterson | A comparison of different tests to assess lower extremity left/right strength imbalances[END_REF][START_REF] Petermann | Efficacy of linear isocinetic versus dynamic auxotonic muscle training for the shoulder-A prospective, randomised, controlled study in men[END_REF][START_REF] Colombo | Novel tilt table with integrated robotic stepping mechanism: Design principles and clinical application[END_REF]. Previous studies in our research centre led to the development of Multi-Iso [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF]. To cover a larger group of rehabilitation protocols, our aim is to design an isokinetic CMC rehabilitation apparatus and robust control structure to ensure the safety of the users. Rehabilitation devices used for knee rehabilitation are controlled by classical control laws. Although the results obtained with these controllers are satisfying in terms of rehabilitation specifications, they are restrictive in terms of control performance, mainly because they do not theoretically guarantee good behaviour in the entire state space and do not ensure the rejection of external disturbances such as patient efforts.

Thus, previous works have proposed employing a nonlinear control scheme based on switching controllers [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF] or, more recently, a coupled trajectory generator with a Takagi-Sugeno (T-S) fuzzy controller [START_REF] Seddiki | H-infinity Takagi-Sugeno fuzzy control of a lower limbs rehabilitation device[END_REF][START_REF] Seddiki | Modélisation T-S et commande H ∞ d'une machine de rééducation des members inférieurs[END_REF]. However, these studies were only concerned with stability analysis, and the tracking performance of the rehabilitation devices was not guaranteed. Consequently, safe behaviour was not guaranteed, because the controllers could not reject external disturbances or uncontrolled users' efforts with the device.

We present the concept of a new CMC rehabilitation device, called Sys-Reeduc, and we propose a convenient tracking controller design that ensures safe behaviour for the users. In a preliminary study [START_REF] Seddiki | Concept and simulation of "Sys-Rééduc": Closed muscular chain lower limbs rehabilitation device[END_REF], the stability performance in simulation was obtained through major modeling simplifications (including linearisation and decoupling) that make it difficult to apply the synthesised control law to the real system. In this paper, after presenting the kinematic concepts of the new rehabilitation device, a nonlinear dynamic model is derived via the well-known Lagrange equations. Then, a tracking control plant, based on a trajectory generator added to a nonlinear H ∞ tracking control law, is proposed. A description of the trajectory generator is proposed for isokinetic cyclic rehabilitation protocols, which are the most commonly used protocols in lower limb rehabilitation. Afterward, the design of a nonlinear control law ensuring robust trajectory tracking is investigated.

Among nonlinear controllers, T-S fuzzy model-based approaches have become popular,

because the model provides universal approximators of nonlinear systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF][START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF]. Therefore, in the past few decades, T-S fuzzy control has been the subject of many theoretical studies; see, for example, [START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF][START_REF] Sala | Perspectives of fuzzy systems and control[END_REF][START_REF] Zerar | Linear Fractional Transformation Based Hinfinity Output Stabilization for Takagi-Sugeno Fuzzy Models[END_REF]. Moreover, this modeling approach has been successfully employed for practical applications; see, for example, [START_REF] Guelton | An Alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknow-input observer in the descriptor form[END_REF][START_REF] Schulte | Descriptor modelling towards control of a two link pneumatic robot manipulator: A T-S multimodel approach[END_REF]. The major interest of such approaches is that they allow extending some linear control techniques to nonlinear systems. In most cases, quadratic Lyapunov functions are employed to derive controller designs based on linear matrix inequalities (LMIs). Nevertheless, these approaches are conservative, because they require the existence of a common Lyapunov matrix for the whole set of LMIs. More recently, non-quadratic fuzzy Lyapunov approaches have been proposed to reduce the conservatism of LMI conditions [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Guerra | LMI based relaxed nonquadratic stabilizations for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Feng | A Survey on Analysis and Design of Model-Based Fuzzy Control Systems[END_REF][START_REF] Rhee | A new Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. Nevertheless, the major drawback of such approaches is that they require knowledge of the membership function time derivative's lower bounds, which are difficult to obtain in practice. Complementary to these approaches, tracking controller designs have been considered [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF][START_REF] Mansouri | Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models[END_REF].

In the present study, a tracking controller design is proposed based on T-S fuzzy descriptor modeling of the rehabilitation device [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF][START_REF] Bouarar | Stabilization of uncertain Takagi-Sugeno descriptors: A fuzzy Lyapunov approach[END_REF]. Indeed, although state space descriptors can be used to model algebraic systems such as singular systems [START_REF] Luenberger | Dynamic equation in descriptor form[END_REF][START_REF] Dai | Singular control systems[END_REF][START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF], they are convenient for dealing with mechanical systems with time-varying inertia [START_REF] Guelton | An Alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknow-input observer in the descriptor form[END_REF][START_REF] Schulte | Descriptor modelling towards control of a two link pneumatic robot manipulator: A T-S multimodel approach[END_REF]. Moreover, LMI-based descriptor stability conditions provide relaxed quadratic fuzzy Lyapunov approaches without requiring knowledge of the membership function derivative [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Guerra | A way to improve results for the stabilisation of continuous time fuzzy descriptor models[END_REF][START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF].

Note that, despite numerous works dealing with stability and stabilisation of T-S descriptor systems, to the best of the authors' knowledge, there are no available results, since the first quadratic result in [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF], dealing with robust T-S descriptor tracking control problems. In this paper, for the control law synthesis, a H ∞ criterion is employed to guarantee the attenuation of uncontrolled human disturbances. Thus, a relaxed quadratic fuzzy Lyapunov-based tracking controller design methodology is provided, based on linear matrix inequality conditions for the considered class of perturbed nonlinear descriptors. Finally, simulation results of isokinetic movements will illustrate the efficiency of the proposed control approach.

CONCEPT OF SYS-REEDUC

Definitions, advantages and drawbacks of OMC and CMC rehabilitation techniques

Exercises in open muscular chains are defined by the contraction of the agonist muscle group that allows the movement of the considered segment. In this rehabilitation mode, for a lower limb, the foot is considered "free", see Fig. 1. This approach is efficient and allows fast recovery when strengthening isolated muscle groups. However, the major drawback of OMC is that it causes constraints localised on the anterior cruciate ligament and on the patellar tendon that can generate significant pain during intensive use [START_REF] Lutz | Comparaison of tibio-femoral joint forces during open kinetic chain and closed kinetic chain exercices[END_REF][START_REF] Bynum | Open versus closed chain kinetic exercices after anterior cruplate ligament reconstruction', A prospective randomized study[END_REF]. Moreover, this technique may result in joint imbalance because of asymmetrical stretching of the targeted joint complex.

Exercises in CMC are defined by simultaneous contraction of both the agonist and antagonist muscles for a particular movement. In most cases, these exercises are possible when the foot is in contact with a support, see Fig. 2. Thus, all the muscles in a group contribute to the joint's actuation. Consequently, the musculoskeletal stretching is applied in a symmetrical way and contributes to increasing the joint stability. Thus, patellar pains are reduced during CMC rehabilitation compared with OMC rehabilitation [START_REF] Steinkamp | Biomechanical consideration in patellofemoral joint rehabilitation[END_REF][START_REF] O'connor | Can muscle co-knee ligaments after injury or repair[END_REF]. Note that the benefit of CMC could be longer-lasting than that of OMC.

Finally, OMC and CMC rehabilitation techniques are complementary and together allow patients to reach specific rehabilitation goals relating to different pathologies. For instance, during long-term rehabilitation protocols, one can imagine proposing CMC exercises for stabilising the whole joint complex and alleviating constraints generated by the OMC in a first step. In a second step, when the patient's clinical state makes it possible, rehabilitation in OMC could be proposed for a specific muscular strengthening while the joint balance achieved in CMC is maintained.

Kinematic concepts of Sys-Reeduc

Our objective is now to design an isokinetic CMC rehabilitation apparatus. By analogy with robotic systems, the mechanical complex {lower limb, rehabilitation device} must constitute a closed kinematic chain (CKC). For each lower limb, a CKC, shown in Fig. 3, is proposed and is composed of the links 1 C , 2 C , 3 C , 4 C and 5 C , which are respectively the apparatus base, thigh, leg, foot and mobile foot's support. The joint between links i and j is denoted ,

i j L .
According to the CKC depicted in Fig. 3, the design of a new device consists of defining the nature of the joints 1,2 L , 2,3 L , 3,4 L , 4,5 L and 5,1 L . In that way, we assume that the human body consists of a set of rigid polyarticulated segments. In the sagittal plane, the whole kinematic structure, presented in Fig. 4, is supposed to be composed of perfect joints such that:

-

1,2
L is the joint between the base and the thigh. The patient sits on a chair fixed to the base of the system. The trunk and the pelvis are also assumed to be fixed to the base. Then, 1,2 L is supposed to be a spherical joint corresponding to the coxo-femoral joint (hip).

-2,3

L corresponds to the knee, which is the main joint we wish to rehabilitate. Its complex kinematics were the subject of a previous study [START_REF] Seddiki | A 6 degrees of freedom kinematical model of the knee for the design of a new rehabilitation device[END_REF] and will not be detailed in this paper.

To simplify the specification of the Sys-Reeduc kinematic concept, the knee will be represented as a hinge joint around the main lower limb flexion-extension axis.

-The joints 3,4 L and 4,5 L can be reduced to a unique joint, 3,5 L . This requires that the ankle axes and mobile support hinge joint, in the sagittal plane, must be perfectly aligned with those of the foot mobile support. This constraint allows removing the foot from the kinematic chain. Then, 3,5 L consists of two hinge joints. The first joint is the plantar flexion/dorsal flexion ( fd L ankle axis, which is collinear with the main lower limb flexion-extension axis). The second joint is the ankle internal-external rotation ( ie L ankle rotation in the tibia axis). Note that the adduction/abduction of the ankle was voluntarily omitted, because it has little influence on knee rehabilitation.

-The joint 5,1 L , between the foot's mobile support and the base, is realised by a prismatic joint along the axis defined by the lower limb in complete extension.

Because the mechanical structure of Sys-Reeduc is symmetrical, Fig. 4 represents only the right lower limb. To simplify the presentation of the rehabilitation mode, only the right lower limb will be considered in the sequel. Moreover, according to the kinematics defined in Fig. 4, the internal-external ankle rotation ie L (along the s z axis and denoted by the variable ( )

2 q t )
and the translation 5,1

L (along the x axis and denoted by the variable ( )

1
q t ) are sufficient to stimulate the knee in its six degrees of freedom [START_REF] Seddiki | A 6 degrees of freedom kinematical model of the knee for the design of a new rehabilitation device[END_REF]. These two axes will therefore be motorised, and the plantar flexion/dorsal flexion will remain passive (free or forced by a mechanical brake). In terms of rehabilitation, the motorisation of the joint 5,1 L brings about a movement of the lower limb along the medial-lateral axis (flexion-extension movement).

This makes it possible to rehabilitate the flexor muscular complex and limits the constraints applied to the cruciate ligaments. Moreover, the foot's mobile support rotations allow the knee internal-external rotation and the plantar flexion/dorsal flexion. These lead to a selective and precise muscular constraint. Then, rehabilitation or muscular strengthening can be practiced in a specific way during a desired movement. For instance, lower limb movement with the foot in external rotation and in extension helps the stretching of medial hamstrings.

The efforts applied by the user to the device that may influence the dynamic behaviour of Sys-Reeduc are depicted in Fig. 5. Hence, ( )

/ p x f t and ( ) / s p z C
t are respectively the force along the x -axis and the torque along s z applied by the user.

Then, by considering the kinematics of Sys-Reeduc and the effort supplied by the user, we can define the type of muscle contraction occurring during a particular exercise. Indeed, as depicted in Fig. 6 for flexion-extension, we can define four quadrants in the effort-speed plan for the characterisation of the muscular exercise, as follows.

-

In 1.1 Q and 1.3 Q , ( ) / p x f
t is applied in the device's movement direction, and the contractile mode is therefore called "concentric".

-

In 1.2 Q and 1.4 Q , ( ) / p x f
t is applied in opposition to the device's movement direction, and the contractile mode is therefore called "eccentric".

Fortunately, the observation depicted in Fig. 7 can also be made relative to the internalexternal rotation movement of Sys-Reeduc. In that case, the four-quadrant diagram is characterised by: -

/ s p z C
t is applied in opposition to the device's movement direction, and the contractile mode is therefore called "eccentric".

CONTROL STRUCTURE DESIGN FOR SYS-REEDUC

Generic control structure

A generic control structure for rehabilitation devices is presented in Fig. 8. This two-level structure is based on the following components.

-A trajectory generator, which provides the desired trajectory ( )

d
x t to be tracked by the continuous system state ( )

x t . This trajectory generator, detailed in the next section, is based on the use of a state machine [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF][START_REF] Zaytoon | Formal specifications of sequential control for training machines for the lower limbs[END_REF] and allows voluntary movements activated from the measurement of the effort ( ) t ϕ applied by the patient to the device.

-A closed-loop rehabilitation device plant, which is used to guarantee the tracking of the desired trajectory, ( )

d
x t , by the dynamic system consisting of the device and the user. To ensure safe behaviour, this closed-loop system must ensure the tracking error convergence of the system regardless of the patient's efforts on the device. To achieve this goal, ( ) t ϕ is considered as an external disturbance to be attenuated or, in the better case, to be rejected. Note that, without the trajectory generator, the closed-loop rehabilitation device plant by itself does not allow the voluntary control of the system by the patient.

Trajectory generator design

A rehabilitation exercise can be viewed as a succession of n elementary movement phases

denoted i φ for 1,..., i n =
. Hence, a discrete state machine can be used to represent these exercises, as depicted in Fig. 9. In this case, the device's current position ( )

1
q t , ( )

2
q t and the external effort ( ) t ϕ , consisting of ( )

/ p x f t and
( )

/ s p z C
t , can be considered as inputs. The trajectory generator provides as outputs the desired state, denoted ( )

d
x t , consisting of the desired positions ( ) ( )

1 2 d d q t q t ⎡ ⎤ ⎣
⎦ and velocities ( ) ( )

1 2 d d q t q t ⎡ ⎤ ⎣
⎦ , to be tracked by the device.

Therefore, it is necessary to define the whole movement characteristics for each elementary phase and the conditions to ensure the validity of transitions between two phases. Elementary phases can be discriminated into two different major classes: passive and active phases.

A passive phase is characterised by an absence of voluntary control by the user on the elementary movement. In that case, independently of the user's efforts, the device achieves the goal assigned to the phase. For instance, a forced stop in the current position as well as any movements that do not depend on the applied effort can be considered as passive phases.

In an active phase, the device's movements must be actuated by the user. In the following, isokinetic movements are considered, because they are commonly used in lower limb rehabilitation protocols. Thus, during an isokinetic elementary phase, the velocities 1 v and 2 v are constant and the desired trajectory to be tracked by the device can be written as:

( ) ( ) 1 1 1 2 2 2 1 1 1 1 2 2 2 2 d d d d i d d i q v q v q vt q t q vt q t α α α α = ⎧ ⎪ = ⎪ ⎨ = + ⎪ ⎪ = + ⎩ , ( 1
)
where i t is the initial activation time of the ith phase and To define a cyclic isokinetic exercise, the discrete state machine, presented in Fig. 9 for flexion-extension, can be defined. The transition between phases depends on the force applied by the user on the mobile support as well as its current position.

At the beginning of a training session (Start), 0 φ is activated. In that state, the desired rehabilitation trajectory remains constant in its current position (the device is expected to be immobile). Then, the transition from 0

φ
( ) / p x f
t is lower than the threshold 2 0 s < and if the current position of the support mobile does not reach the lower limit of the operational space 1 q . Finally, the transition back to 0 φ occurs if the effort

( ) / p x f
t is greater than the threshold 1 s or if the current position of the support mobile reaches the lower limit of the operational space.

To illustrate the desired trajectory generation during a cyclic isokinetic exercise described by this state machine, a simulation was performed with the parameters 1 80

s = N, 2 80 s = -N, 1 1.2 q =
m, 1 0.1 q = m and 1 2 v = m.s -1 . Figure 10 shows the results obtained with

( ) / 500sin(6 ) p x f t t =
. Note that the desired trajectory generation for internal-external rotation is not presented in this section, because it relies on the same principle as for flexionextension. With the trajectory generator defined, the goal is now to ensure the tracking of these trajectories by the dynamic system (consisting of the device and the user). Therefore, the following section will present the proposed nonlinear tracking controller design methodology.

Continuous control level of Sys-Reeduc

The goal is now to provide a convenient controller design methodology that ensures the tracking of the desired trajectories provided by the trajectory generator. Note that, in that case, the external efforts, consisting of the forces and torques applied by the user, must be rejected or attenuated by the device's dynamics. In that context, a H ∞ -based controller design can be employed. Moreover, in view of the nonlinear dynamics of Sys-Reeduc, a Takagi-Sugeno tracking controller design will be proposed.

Dynamic modeling of Sys-Reeduc

The mechanical scheme of Sys-Reeduc is presented in Fig. 11. Note that the user is not included in this model because his/her movement cannot be artificially controlled. This justifies the use of a H ∞ -based design to attenuate these external disturbances to the device.

Thus, to synthesise the control law, we consider that the device is to be controlled via the motors' torques while the patient applies external effort.

( )

1 M C t and ( ) 2 M C
t are the motors' torques allowing the movements associated with the degrees of freedom ( )

1 q t and
( )

2 q t ,
respectively.

Recall that ( ) ( ) ( )

/ / s T p x p z t f t C t ϕ ⎡ ⎤ = ⎣
⎦ denotes the vector of the efforts applied by the user on the device. The mechanical parameters used to model the Sys-Reeduc dynamics are defined in Table 1. These parameters were obtained from the design of the mechanical elements constituting Sys-Reeduc using the Catia software tools. Then, the dynamic model can be obtained using the well-known Lagrangian equations given by: ( )

c p c i i i E E E d dt q q ∂ - ⎛ ⎞ ∂ - =Γ ⎜ ⎟ ∂ ∂ ⎝ ⎠ , (2)
where i q and i Γ are respectively the coordinates and the generalised efforts associated with the th i degree of freedom, and c E and p E are respectively the kinetic and the potential energy.

The mobile support's centre of gravity coordinates can be written in the frame ( ) , , , o x y z as:

( ) ( ) ( ) ( ) / 1 2 cos cos x OG q t b q t β α = + + , (3) ( ) ( ) ( ) / 2
sin cos

y OG h d q t β α = - + , (4) ( ) ( )
/ 2 sin z OG l q t = , (5) with (
)

sin b l α = and
( )

cos d l α = .
Thus, its velocity can be written in the frame ( ) , , , o x y z as:

( ) ( ) ( ) ( ) / 1 2 2 sin G x V q t bq t q t = - , (6) ( ) ( ) ( )
/ 2 2 sin G y V dq t q t = - , (7) ( ) ( ) ( )
/ 2 2 cos G z V lq t q t = . (8)
Therefore, the kinetic energy can be expressed as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2 1 2 1 2 2 2 1 1 2 s i n 2 2 C E M m q t l q t bq t q t q t Jq t ⎡ ⎤ = + + - + ⎣ ⎦ , (9)
and the potential energy as:

( ) ( ) ( ) 2 sin cos P E mgh mg mgd q t β α = - + , (10)
where 9.81 g = m.s -2 is the gravitational constant.

Finally, substituting ( 9) and ( 10) in (2) and considering the generalised efforts ( ) ( ) ( )

1 1 / M px t C t f t Γ = - and
( ) ( ) ( ) 2 2 / s M p z t C t C t Γ = -
, the dynamic model of Sys-Reeduc is expressed by the following motion equation:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
, q q t C q q q t G q q t Ru t S t

ϕ Μ + + = + , (11)
where

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 sin sin sin sin M m M ml q t q M m l q t M m l J α α ⎡ ⎤ + - + Μ =⎢ ⎥ - + + + ⎢ ⎥ ⎣ ⎦ is the inertia matrix, ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 s i n c o s , 0 0 M m l q t q t C q q α ⎡ ⎤ - + = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ is the Coriolis matrix, ( ) ( ) ( )
( )

2 0 0 0 cos sin c G q mgl q t α ⎡ ⎤ = ⎢ ⎥ - ⎣ ⎦ is the gravitational effect, 1/ 0 0 1 a R ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ is the input matrix, 1 0 0 1 S ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦
is the disturbance matrix, ( ) ( ) ( )

1 2 T q t q t q t = ⎡ ⎤ ⎣
⎦ denotes the generalised coordinates, and ( ) ( ) ( )

1 2 T M M u t C t C t = ⎡ ⎤ ⎣
⎦ is the input vector.

T-S fuzzy descriptor modeling of Sys-Reeduc

We now propose a descriptor-based controller design. State space descriptors can be used to model algebraic systems such as singular systems [START_REF] Luenberger | Dynamic equation in descriptor form[END_REF][START_REF] Dai | Singular control systems[END_REF][START_REF] Marx | Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis[END_REF]. Moreover, these are a convenient way to deal with mechanical systems with time-varying inertia and reduce the computational complexity of T-S-based LMI problems [START_REF] Guelton | An Alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknow-input observer in the descriptor form[END_REF][START_REF] Schulte | Descriptor modelling towards control of a two link pneumatic robot manipulator: A T-S multimodel approach[END_REF]. By considering the following state vector of the system, ( ) ( ) ( ) ( ) ( )

1 2 1 2

T

x t q t q t q t q t = ⎡ ⎤ ⎣ ⎦ , ( 11) can be rewritten as the following state space descriptor:

( ) ( ) ( ) ( ) ( ) ( ) ( )
( )

E x t x t A x t x t Bu t H t ϕ = + + , (12) where ( ) ( )
( )

0 0 I E x t q ⎡ ⎤ = ⎢ ⎥ Μ ⎣ ⎦ , ( ) ( ) ( ) ( ) 0 ,
I A x t G q C q q ⎡ ⎤ = ⎢ ⎥ - - ⎣ ⎦ , 0 B R ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ and 0 H S ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ .
The goal is now to rewrite (12) as a T-S fuzzy descriptor of the form:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 e r k k i i k i v z t E x t h z t A x t Bu t H t ϕ = = = + + ∑ ∑ , (13)
where: e and r are the numbers of fuzzy sets for the left and right hand sides of the state equation ( 13), respectively; ( ) z t is the premise vector depending on the state variables; The sector nonlinearity approach [START_REF] Tanaka | Fuzzy control systems design and analysis: a linear matrix inequality approach[END_REF] is a convenient way to rewrite a nonlinear descriptor such as [START_REF] Colombo | Novel tilt table with integrated robotic stepping mechanism: Design principles and clinical application[END_REF] as one of its T-S representations [START_REF] Seddiki | H-infinity Takagi-Sugeno fuzzy control of a lower limbs rehabilitation device[END_REF]. Using this approach, the obtained T-S fuzzy model exactly matches the nonlinear model on a compact set of the state space. Hence, from [START_REF] Colombo | Novel tilt table with integrated robotic stepping mechanism: Design principles and clinical application[END_REF], one can consider the nonlinear functions, included in ( ) ( )

E x t and
( ) ( )

A x t , given by:

( ) ( ) ( ) ( ) [ ] 2 2 sin 1,1 q t q t ω = ∈ - , (14) ( ) ( ) ( ) ( ) ( )
( )

1 2 2 2 2 2 2 , c o s , q t q t q t q t q q μ ⎡ ⎤ = ∈-⎣ ⎦ , (15) ( ) ( ) ( ) ( )
( ) [ ] 2 2 2 2 sin ,1 q t q t q t μ ϖ = ∈ , (16)
where 0.2172 ϖ =and 2 100 q ≡ tr.min -1 10.47 ≡ rad.s -1 is the maximal speed of the internalexternal rotation set from product design specification.

To obtain each LTI model and the membership functions composing a T-S model matching [START_REF] Colombo | Novel tilt table with integrated robotic stepping mechanism: Design principles and clinical application[END_REF], the following sector nonlinearity transformations are employed: w q q w q q q q q q q q μ μ μ

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 . 1 1 . . 1 1 2 2 v q v q ω ω ω + - = + - , (17) ( ) ( ) ( ) ( ) ( ) ( ) (
+ - = +- , (18) ( ) ( ) ( ) ( ) ( ) ( )
( )

21 2 22 2 2 2 2 . 1 . . 1 1 1 w q w q μ ϖ μ μ ϖ ϖ ϖ - - = + - - . (19)
From equations ( 17), ( 18) and ( 19), the membership functions are given by: ( ) ( )

2 1 2 1 2 q v q ω + = , (20) ( ) ( ) ( )
( )

2 2 2 1 2 q t v q t ω - = , (21)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 1 2 1 2 . . . 2 1 q h w w q μ μ ϖ ϖ + - = ⋅ ⋅ = - , (22) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 2 2 2 1 1 2 2 2 . 1 . 2 1 q h w w q μ μ ϖ + - ⋅ = ⋅ ⋅ = - , (23)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 3 1 2 2 1 2 . . 2 1 q h w w q μ μ ϖ ϖ - - ⋅ = ⋅ ⋅ = - , (24) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 1 2 4 1 2 2 2 2 . 1 . 2 1 q h w w q μ μ ϖ - - ⋅ = ⋅ ⋅ = - , (25)
and the LTI matrices by:

( ) ( ) ( ) 1 2 1 0 0 0 0 1 0 0 0 0 sin 0 0 sin E M m M m l M m l M m l J α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ + - + ⎢ ⎥ - + + + ⎣ ⎦ , ( ) ( ) ( ) 2 2
1 0 0 0

0 1 0 0 0 0 sin 0 0 sin E M m M m l M m l M m l J α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ + + ⎢ ⎥ + + + ⎣ ⎦ , ( )
1 2 0 0 1 0 0 0 0 1 0 0 0 sin 0 cos 0 0 A M m l q mgl α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ + ⎢ ⎥ ⎣ ⎦ , ( )
2 2 0 0 1 0 0 0 0 1 0 0 0 sin 0 cos 0 0 A M m l q mgl α ϖ α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ + ⎢ ⎥ ⎣ ⎦ , ( )
3 2 0 0 1 0 0 0 0 1 0 0 0 sin 0 cos 0 0 A M m l q mgl α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - + ⎢ ⎥ ⎣ ⎦ and
( )

4 2 0 0 1 0 0 0 0 1 0 0 0 sin 0 cos 0 0 A M m l q mgl α ϖ α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ - + ⎢ ⎥ ⎣ ⎦ . Consequently, considering ( ) [ ] 2 2 T z t q q =
, a T-S fuzzy model of Sys-Reeduc can be expressed as:

( ) ( ) ( ) ( ) ( ) ( ) 2 4 2 22 1 1 , k k i i k i v q E x t h q q A x t Bu t H t ϕ = = = + + ∑ ∑ . ( 26
)

LMI-based tracking controller design for T-S descriptors

With the T-S descriptor model of Sys-Reeduc from the previous section, we now provide a convenient controller design methodology ensuring the tracking of desired trajectories provided by the trajectory generator depicted in Section 3.2.

In the sequel, to simplify the mathematical expressions, we will consider the notations

( ) ( ) 1 r h i i i X h z t X = = ∑ , ( ) ( ) ( ) ( ) 1 1 e r hv k i ik k i Y v z t h z t Y = = = ∑∑ , ( )
( ) ( ) ( ) ( ) ( ) 1 1 1 e r r hhv k i j ijk k i j Z v z t h z t h z t Z = = = = ∑∑∑
, and so on.

As usual, a star (*) in an expression indicates a transposed quantity. Moreover, when there is no ambiguity, the time t will be omitted.

For the sake of generality, the following LMI results will be developed for the general class of T-S fuzzy descriptors described by ( 13) and rewritten with the above-defined notations as:

( ) ( ) ( ) ( ) v h h h E x t A x t B u t H t ϕ = + + , ( 27
)
where

n n h A × ∈ , n m r B × ∈ , n p h H × ∈ , ( ) n x t ∈ , ( ) m u t ∈ and ( ) p t ϕ ∈ .
Consider the tracking control plant given by Fig. 12. This plant contains a reference model that is required to write the closed loop dynamics given by the state space representation: Note that (28) allows setting the dynamics of the tracking control trajectory. A convenient way to choose it will be proposed in the simulation results section.

( ) ( ) ( ) r r r r d x t A x t B x t = + , (28)
The goal is now to write the closed loop dynamics of the tracking controller plant. Consider the Parallel Distributed Compensation (PDC) control law [START_REF] Wang | Parallel distributed compensation of nonlinear systems by Takagi-Sugeno[END_REF] defined by: ( ) ( ) ( ) ( )

1 11 hv r u t K Z x t x t - = - - , (29)
where hv K and 11 0 Z > are the gain matrices to be synthesised.

As is classically the case for descriptors [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF], by considering the extended state vector 27) can be rewritten, for instance, as:

( ) ( ) ( ) * T T T x t x t x t ⎡ ⎤ = ⎣ ⎦ , (
( ) ( ) ( ) ( ) ( ) * * * * * * hv h h E x t A x t B t u t H t ϕ = + + , (30)
where * 0 0 0

I E ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , * 0 hv h v I A A E ⎡ ⎤ = ⎢ ⎥ - ⎣ ⎦ , * 0 h h B B ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ and * 0 h h H H ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ .
In the same way, let us consider the extended reference state vector

( ) ( ) ( ) * T T T r r r x t x t x t ⎡ ⎤ = ⎣
⎦ . ( 28) can be rewritten as:

( ) ( ) ( ) * * * * * r r r r d E x t A x t B x t = + , ( 31
)
where * 0

r r I A A I ⎡ ⎤ = ⎢ ⎥ - ⎣ ⎦ and * 0 r r B B ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ .
Let us define ( ) ( ) ( )

* * * r e t x t x t = -
as the tracking trajectory error. We can write:

( ) ( ) ( ) * * * * * * r E e t E x t E x t = - , (32)
and ( 29) can be rewritten:

( ) ( ) * * hv u t K e t = - , (33)
where * 1 11

0 hv hv K K Z - ⎡ ⎤ = ⎣ ⎦ .
Combining ( 30), ( 31), ( 32) and ( 33), the closed loop dynamics can be expressed by:

( ) ( ) ( ) hhv h Ee t A e t H t φ = + , ( 34
) with ( ) ( ) ( ) * * r e t e t x t ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , ( ) ( ) ( ) d t t x t ϕ φ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , * * 0 0 E E E ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ , ( ) * * * * * * 0 hv h hv hv r hhv r A B K A A A A ⎡ ⎤ - - = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ and * * * 0 h r h r H B H B ⎡ ⎤ - = ⎢ ⎥ ⎣ ⎦ .
Now, the aim is to synthesise the matrices hv K and 11 Z that stabilise the closed loop system (34) as well as ensure the attenuation of the external disturbances ( )

t φ
with respect to the trajectory-tracking error ( ) e t [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF][START_REF] Mansouri | Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models[END_REF]. Thus, we consider the following H ∞ criterion:

( ) ( ) ( ) ( ) 0 0 2 f f t t T T t t
e t e t dt t t dt

η φ φ ≤ ∫ ∫ , ( 35
)
where η is the attenuation level to be minimised.

The main theoretical result is summarised in the following theorem.

Theorem: If the matrices jk K , 11 11 0

T Z Z = > , 41 41 0 T Z Z = > , 13ij Z , 14ij Z , 23ij Z , 24ij Z , 33ij Z , 34ij Z , 43ij Z
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1,1) (2,1) (2,2) 11 11 (4,1) (4,2) (4,4) (5,1) (5,2) (5,4) (5,5)
* * * * 0 0 * * * 0 0 0 0 0 0 0 0 * * * 0 0 * 0 0 0 0 0 0 0 0 0 ij ij ij ijk ijk ijk ij ij ijk ij T i T T r r Z Z I H I B B I δ δ ⎡ ⎤ ϒ ⎢ ⎥ ϒ ϒ ⎢ ⎥ ⎢ ⎥ - ⎢ ⎥ ≤ ϒ ϒ ϒ ⎢ ⎥ ⎢ ⎥ ϒ ϒ ϒ ϒ ⎢ ⎥ - ⎢ ⎥ ⎢ ⎥ - - ⎣ ⎦ , (36) where (1,1) 13 13
T ij ij ij Z Z ϒ = + , (2,1) 33 23
T ij ij ij Z Z ϒ = + , (2,2) 43 43
T ij ij ij Z Z ϒ = + , (4,1) 14 11 13 33 11 33 2
T T ijk ij i k ij i jk k ij r ij Z A Z E Z B K E Z A Z Z ϒ = + - - - - + , ( 4
T ijk ij i k ij i jk i k ij r ij Z A Z E Z B K A Z E Z A Z Z ϒ = + - - + - - + , ( 4
T T T T T ijk ij k k ij ij k k ij ij ij Z E E Z Z E E Z Z Z ϒ =- - - - + +
, (5,1) 11 [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF] 24

T T ij r ij ij A Z Z Z ϒ = - + , (5,2) 44 41 43
T ijk ij r ij Z A Z Z ϒ = + -
, (5,4) 34 24 44 44

T T T T T ijk ij ij k ij k ij Z Z E Z E Z ϒ =- - - +
and (5,5) 44 44

T ij ij ij Z Z ϒ = - - ,
then the asymptotic stability of the closed loop fuzzy system (34) is ensured and the H ∞ tracking control performance ( 35) is guaranteed with an external disturbance attenuation level η δ = .

Proof: Let us consider the candidate Lyapunov function:

( ) ( ) ( ) ( ) 1 0 T hhv V e t e t EZ e t - = > , (37)
with hhv Z a non-singular 1 C matrix satisfying

Z Z Z Z Z Z Z Z Z Z Z Z Z ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , (38)

Let us consider

[ ] 35) can be rewritten as:

0 0 0 Q diag I = , then the H ∞ criterion (
( ) ( ) ( ) ( ) 0 0 2 f f t t T T t t e t Qe t dt t t dt η φ φ ≤ ∫ ∫ . (39)
Thus, the closed-loop descriptor ( 34) is stable and ensures the H ∞ performance η if:

( ) ( ) ( ) ( ) ( ) ( ) 2 0 T T V e t e t Qe t t t η φ φ + - ≤ . (40)
Considering ( 34) and ( 37), ( 40) is verified if:

1 1 2 0 T T T hhv hhv hhv hhv hhv h T h hhv A Z Z A Q Z H H Z I η - - - - ⎡ ⎤ + + ≤ ⎢ ⎥ - ⎣ ⎦ . ( 41
)
By pre-and post-multiplying (41) by 0 0

T hhv Z I ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ and 0 0 hhv Z I ⎡ ⎤ ⎢ ⎥ ⎣ ⎦
, respectively, we obtain:

( ) 2 0 T T T hhv hhv hhv hhv hhv hhv h T h Z A A Z Z QZ H H I η ⎡ ⎤ + + ≤ ⎢ ⎥ ⎢ ⎥ - ⎣ ⎦ . (42)
After developing [START_REF] Gahinet | LMI control Toolbox[END_REF] in its extended form with the matrices defined in [START_REF] Guerra | A way to improve results for the stabilisation of continuous time fuzzy descriptor models[END_REF], [START_REF] Seddiki | A 6 degrees of freedom kinematical model of the knee for the design of a new rehabilitation device[END_REF] and (39), we apply the Schur complement [START_REF] Boyd | Linear matrix inequality in systems and control theory[END_REF] and the proof is completed. ■

SIMULATION RESULTS

The theoretical results in the previous section ensure the tracking performance of a general class of T-S fuzzy descriptors. In this part, our goal is to show, by simulation, the efficiency of the proposed tracking controller design methodology in the dynamic model of Sys-Reeduc given in [START_REF] Petermann | Efficacy of linear isocinetic versus dynamic auxotonic muscle training for the shoulder-A prospective, randomised, controlled study in men[END_REF]. Following the tracking control plant depicted in Fig. 12 and considering the parameters required to solve the LMI problem (36), we must first choose an adequate reference model of the form [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF]. This reference model can be used to set the dynamics of the trajectory tracking inputs, and its influence on the LMI solution cannot be neglected. In the Sys-Reeduc application, the goal is to force the state ( ) x t to track the reference state ( )

r x t .
According to [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF], the reference model contains an input vector ( ) ( )

1 1 rp dp x s x s s κ = + , ( 43
)
where κ is the time constant and s the Laplace variable.

According to (43), a state-space reference model can be written by considering x t dynamics will be to the desired reference ( )

( ) 1
r
x t . Indeed, decreasing the time constant leads to downgrade the H ∞ performance as the LMI problem is compelled. To illustrate this phenomenon, the minimal attenuation level η has been evaluated, using the Matlab LMI control toolbox [START_REF] Gahinet | LMI control Toolbox[END_REF], for several time constants κ , and the results are presented in Table 2.

In the present application, a good compromise could be 0.1 κ = s, leading to 3.08 η =

. The results computed from the LMI conditions (36) lead to the design of the PDC tracking control law given by: ( These simulation results illustrate the efficiency of the performance in trajectory tracking obtained by using the proposed control law. Indeed, Figures 13 and14 show that the H ∞ control law (44) successfully attenuates the external disturbances and compensates them via the input signal. In these simulations, the trajectory generator is not taken into account.

2 4 1 2 2 2 1 1 1 1 , k i i k r k i u t v q t h q t q t K Z x t x t - = = = - - ∑∑ , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Figure 15 illustrates the performance of the whole control structure design for Sys-Reeduc, depicted in Fig. 8, including the isokinetic trajectory generator proposed in Section 3.2. This simulation shows that the trajectory generation is successfully realised while the tracking performance of the device is ensured.

CONCLUSIONS

After discussing the advantages and drawbacks of the OMC and the CMC rehabilitation techniques, the concept of a new CMC lower-limb rehabilitation device has been presented.

The proposed control structure consists of a trajectory generator and a continuous level ensuring the tracking control stability of the closed loop mechanical system. For the sake of generality, the principle of the trajectory generator was described as a parameterised discretestate machine. LMI-based tracking controller design methodology was developed for a class of T-S fuzzy descriptors suitable to represent the nonlinear dynamics of Sys-Reeduc.

Moreover, a H ∞ criterion has been employed to attenuate the user's efforts that are considered, for the closed loop continuous dynamics, as external disturbances. Note that attenuating these disturbances provides a safe behaviour to the user of the rehabilitation device. Indeed, even if the user-rehabilitation device interaction is attenuated by the continuous level, the trajectory generator allows voluntary movements to be performed on the basis of the measurement of this interaction. One of the interests of such a control structure is the possibility of parameterising a particular trajectory generator that, for instance, motivates the user to provide an appropriate effort during rehabilitation movements. Finally, the proposed controller synthesis has been validated in simulation on Sys-Reeduc and has shown the efficiency of the proposed control plant. Ongoing realisation of the first prototype of Sys-Reeduc will enable us to develop and validate new rehabilitation protocols and to evaluate accurately the benefits of CMC versus OMC rehabilitation.

Figures and Tables: ) Ray of the pulley-belt carrying out the translation along the x axis 0.025 m l Distance between the rotation axis and the gravity centre of the support 0.05 m α Angle between the base horizontal axis and the rotation axis of the mobile support 20°

/

  α = are parameters required to set the contractile mode during the movement ( 1 k α = for a concentric exercise, and 1 k α =for an eccentric one, see Figs. 6 and 7).

H

  are constant matrices, each defining e r × linear time invariant (LTI) descriptors that compose the T-S fuzzy descriptor.

  the desired trajectory to be tracked.

  and 44ij Z and the scalar δ exist such that the following LMI conditions are

  the smaller κ is, the closer the reference state ( ) r

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Exercise in OMC .Fig. 2. Exercise in CMC .Fig. 3. Kinematic chain of a lower limbs CMC rehabilitation device.

Fig. 4 .

 4 Fig. 4. Kinematical concept of Sys-Reeduc in the sagittal plane.

Fig. 5 .

 5 Fig. 5. External efforts which influence Sys-Reeduc's dynamics.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. External efforts which influence Sys-Reeduc's flexion-extension.Fig. 7. External efforts which influence Sys-Reeduc's internal-external rotation.Fig. 8. Generic control structure for rehabilitation devices.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Trajectory generator for Sys-Reeduc Fig. 10. Desired trajectory generated during isokinetic exercises.

Fig. 11 .

 11 Fig. 11. Mechanical principle from the rehabilitation device.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. T-S Trajectory tracking control plant.Fig. 13. Tracking trajectory for translation of the mobile support.

Fig. 14 .

 14 Fig. 14. Tracking trajectory of rotation of the mobile support.

Fig. 15 .

 15 Fig. 15. Simulation of the whole Sys-Reeduc control structure.

Fig. 1 .Fig. 2 . 1 L 2 L 3 L 5 L 1 C 2 C 4 C 4 L 3 CCFig. 3 .Fig. 4 .

 1212351244334 Fig. 1. Exercise in OMC.

Fig. 5 . 4 QFig. 6 . 2 QFig. 7 .Fig. 8 .

 546278 Fig. 5. External efforts that influence Sys-Reeduc's dynamics.

(

  )

	1	.	2	1	. 2	2	2	2	2	2	1 2	.
					11 2 2 ,				12 2 2 ,

Table I :

 I Numerical parameters of the dynamical model of the rehabilitation device Sys-Reeduc

Table II :

 II Evolution of the attenuation rate regarding to κ

  Fig. 10. Desired trajectory generated during isokinetic exercises.Fig. 15. Simulation of the whole Sys-Reeduc control structure.

		500 500
	p/x (N) p/x (N)	0 0				Start (initial position)					f f	/ p x / p x
	f f										f f	/ p x / p x			Patient force Patient force
		Immobile mode 0 -500 0 -500 1.5 1.5	0.2 0.2		φ 0	0.4 0.4	0.6 0.6	0.8 0.8	1 1	1.2 1.2 Desired trajectory 1.4 1.4 Desired trajectory	1 φ 1.6 1.6	Extension mode 1.8 1.8	2 2
	q 1 (m) q 1 (m)	0.5 1 0.5 1													Translation support position Translation support position
		0 0	0 0							0.2 0.2			0.4 0.4	0.6 0.6	0.8 0.8	f	p x	2 t s AND q t 1 ( ) ≤ 1 1.2 1.4 1 1.2 1.4	>	1 q	1.6 1.6	1.8 1.8	2 2
		4 4
	dq 1 /dt (m/s) 1 /dt (m/s) dq	-2 0 2 -2 0 2 -4 -4	0 0	t t	0 0		(	f	/ p x 0.2 0.2	) t s OR q t q ( ) ( 2 1 1 ( ) > 0.4 0.6 0.4 0.6 ≥ 1 t 2 t 1 t 2 t	) 0.8 0.8	1 1 Time (s) Time (s)	1.2 1.2 3 t 4 t 3 t 4 t	Desired velocity Translation support velocity Desired velocity Translation support velocity 1.4 1.6 1.8 1.4 1.6 1.8 Flexion mode 2 φ	2 2
												Fig. 9. Trajectory generator for Sys-Reeduc
				ϕ ϕ	0 0					1 ϕ 1 ϕ	ϕ ϕ	0 0	ϕ ϕ	2 2	ϕ ϕ	0 0	1 ϕ 1 ϕ
			500 500
		p/x (N) p/x (N)	0 0										f f	/ p x / p x
		f f										f f	/ p x / p x		Effort-patient Effort-patient
			-500 -500	0 0				0.2 0.2		0.4 0.4	0.6 0.6	0.8 0.8	1 1	1.2 1.2	1.4 1.4	1.6 1.6	1.8 1.8	2 2
				1.5 1.5
		q 1d (t) (m) 1d (t) (m) q	0.5 1 1 0.5										Desired position Desired position
				0 0	0 0				0.2 0.2		0.4 0.4	0.6 0.6	0.8 0.8	1 1	1.2 1.2	1.4 1.4	1.6 1.6	1.8 1.8	2 2
				4 4
		dq 1d /dt (m/s) dq /dt (m/s) 1d	-2 0 2 -2 0 2										Desired velocity Desired velocity
				-4 -4	0 0				0.2 0.2		0.4 0.4	0.6 0.6	0.8 0.8	1 1	1.2 1.2	1.4 1.4	1.6 1.6	1.8 1.8	2 2
							t t	0 0						1 t 1 t	t t	2 2	Time (s) Time (s)	t t	3 3	t t	4 4

TABLE 2 EVOLUTION

 2 OF THE ATTENUATION RATE AS κ CHANGES κ

		1/400 1/100 1/50 1/30 1/20 1/10
	η	19.98	9.97	7.03 5.43 4.41 3.08

TABLE 1 NUMERICAL

 1 PARAMETERS OF THE DYNAMIC MODEL OF THE REHABILITATION DEVICE SYS-REEDUC

	Param.	Designation	Value
	M	Mobile support mass (in translation)	14 kg
	m	Mobile support mass (in rotation)	4 kg
	J	Mobile support inertia along the s z axis (in rotation)	0.26 kg.m 2
	a

ACKNOWLEDGEMENT

This work was supported by the French Ministry of Research and the "Région Champagne-Ardenne" within the CPER SYS-REEDUC and the CPER MOSYP.

Immobile mode Extension mode

Flexion mode

( )

( )

Fig. 11. Mechanical principle for the rehabilitation device.