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Published in IET Control Theory and ApplicationsReceived on 3rd June 2009Revised on 22nd October 2009doi: 10.1049/iet-cta.2009.0269

ISSN 1751-8644

Concept and Takagi–Sugeno descriptortracking controller design of a closed muscularchain lower-limb rehabilitation deviceL. Seddiki1,2 K. Guelton1 J. Zaytoon1

1CReSTIC, EA3804, Universite de Reims Champagne-Ardenne, Moulin de la House BP1039, 51687 Reims Cedex 2, France2LIASD, EA 4383, Universite Paris 8, 2 rue de la liberte, 93200 Saint Denis Cedex, FranceE-mail: kevin.guelton@univ-reims.fr

Abstract: The authors consider the kinematic concepts of a new lower-limb rehabilitation device in closedmuscular chain. The proposed control structure is based on a trajectory generator and a continuous non-lineartracking controller. The human efforts applied to this device are considered as external disturbances to thesystem’s dynamics and as inputs to the trajectory generator and allow safe voluntary control of the system bythe user. A H1 control structure based on a Takagi–Sugeno descriptor model is proposed to track the desiredtrajectories and to attenuate external disturbances. Stability conditions are given in terms of linear matrixinequalities using a fuzzy Lyapunov function. Finally, the simulation results of the proposed control structurefor the new rehabilitation device during isokinetic movements illustrate the efficiency of the proposed approach.

T

1 IntroductionTechnical assistance to functional rehabilitation has attractedgreat interest recently (see, e.g. [1, 2] and references therein).Many rehabilitation devices have been designed. Forinstance, Lokomat is a device for gait rehabilitation ofpatients suffering from neuromuscular trauma [3]. Anotherexample is an open muscular chain (OMC) lower-limbrehabilitation device developed by our research centre andnamed Multi-Iso [4].

The design of a new device must start from specific needsexpressed by clinicians in rehabilitation. For a specificmuscular complex, several rehabilitation techniques may beconsidered. Here, we are interested in knee rehabilitation,and our interest naturally extends to the global lower-limbmusculoskeletal complex. For muscular deficiency, tworehabilitation techniques can be used: the first is namedOMC, which is characterised by strengthening an isolatedmuscle group; the second is named closed muscular chain(CMC), which is characterised by recruiting both theagonist and antagonist muscle groups that contribute to themovement [5, 6]. These two techniques are complementaryand are used in various stages of rehabilitation protocols [7].

Control Theory Appl., 2010, Vol. 4, Iss. 8, pp. 1407–1420i: 10.1049/iet-cta.2009.0269

For technical reasons, it is difficult to design a device that isable to reproduce these two techniques simultaneously. Theonly existing such device is named MotionMaker, but it isa very complex and expensive robotic structure [8].Consequently, it can be used in the context of researchstudies but is not appropriate for common rehabilitationcentres. OMC devices are commonly used in clinicalcontexts for lower-limb rehabilitation with isokineticdevices such as Cybex, Biodex or Multi-Iso [9]. Morerecently, CMC rehabilitation has received particularattention because of its ability to stabilise the targeted jointduring exercises similar to those used in daily life (such aswalking, sitting or standing up). For instance, Moflex,Contrex LP or Erigo devices are CMC rehabilitationdevices [10–12]. Previous studies in our research centre ledto the development of Multi-Iso [4]. To cover a largergroup of rehabilitation protocols, our aim is to design anisokinetic CMC rehabilitation apparatus and robust controlstructure to ensure the safety of the users. Rehabilitationdevices used for knee rehabilitation are controlled byclassical control laws. Although the results obtained withthese controllers are satisfying in terms of rehabilitationspecifications, they are restrictive in terms of controlperformance, mainly because they do not theoretically

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guarantee good behaviour in the entire state space and do notensure the rejection of external disturbances such as patientefforts. Thus, previous works have proposed employing anon-linear control scheme based on switching controllers[4] or, more recently, a coupled trajectory generator with aTakagi–Sugeno (T–S) fuzzy controller [13, 14]. However,these studies were only concerned with stability analysis,and the tracking performance of the rehabilitation deviceswas not guaranteed. Consequently, safe behaviour was notguaranteed, because the controllers could not reject externaldisturbances or uncontrolled users’ efforts with the device.

We present the concept of a new CMC rehabilitationdevice, called Sys-Reeduc, and we propose a convenienttracking controller design that ensures safe behaviour forthe users. In a preliminary study [15], the stabilityperformance in simulation was obtained through majormodelling simplifications (including linearisation anddecoupling) that make it difficult to apply the synthesisedcontrol law to the real system. In this paper, afterpresenting the kinematic concepts of the new rehabilitationdevice, a non-linear dynamic model is derived via the well-known Lagrange equations. Then, a tracking control plant,based on a trajectory generator added to a non-linear H1

tracking control law, is proposed. A description of thetrajectory generator is proposed for isokinetic cyclicrehabilitation protocols, which are the most commonlyused protocols in lower-limb rehabilitation [16].Afterwards, the design of a non-linear control law ensuringrobust trajectory tracking is investigated.

Among non-linear controllers, T–S fuzzy model-basedapproaches have become popular, because the modelprovides universal approximators of non-linear systems [17,18]. Therefore in the past few decades, T–S fuzzy controlhas been the subject of many theoretical studies (see, e.g.[18–20]). Moreover, this modelling approach has beensuccessfully employed for practical applications (see, e.g.[21, 22]). The major interest of such approaches is thatthey allow extending some linear control techniques tonon-linear systems. In most cases, quadratic Lyapunovfunctions are employed to derive controller designs basedon linear matrix inequalities (LMIs). Nevertheless, theseapproaches are conservative, because they require theexistence of a common Lyapunov matrix for the whole setof LMIs. More recently, non-quadratic fuzzy Lyapunovapproaches have been proposed to reduce the conservatismof LMI conditions [23–26]. Nevertheless, the majordrawback of such approaches is that they require knowledgeof the membership function time derivative’s lower bounds,which are difficult to obtain in practice. Complementary tothese approaches, tracking controller designs have beenconsidered [27, 28].

In the present study, a tracking controller design isproposed based on T–S fuzzy descriptor modelling of therehabilitation device [29, 30]. Indeed, although state-spacedescriptors can be used to model algebraic systems such as

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singular systems [31–33], they are convenient for dealingwith mechanical systems with time-varying inertia [21, 22].Moreover, LMI-based descriptor stability conditionsprovide relaxed quadratic fuzzy Lyapunov approacheswithout requiring knowledge of the membership functionderivative [34–36].

Note that despite numerous works dealing with stability andstabilisation of T–S descriptor systems, to the best of theauthors’ knowledge, there are no available results, since thefirst quadratic result in [29], dealing with robust T–Sdescriptor tracking control problems. In this paper, for thecontrol law synthesis, a H1 criterion is employed to guaranteethe attenuation of uncontrolled human disturbances. Thus, arelaxed quadratic fuzzy Lyapunov-based tracking controllerdesign methodology is provided, based on LMI conditionsfor the considered class of perturbed non-linear descriptors.Finally, the simulation results of isokinetic movements willillustrate the efficiency of the proposed control approach.

2 Concept of Sys-Reeduc2.1 Definitions, advantages anddrawbacks of OMC and CMC rehabilitationtechniques

Exercises in OMCs are defined by the contraction of theagonist muscle group that allows the movement of theconsidered segment. In this rehabilitation mode, for a lowerlimb, the foot is considered ‘free’, see Fig. 1. This approach isefficient and allows fast recovery when strengthening isolatedmuscle groups. However, the major drawback of OMC isthat it causes constraints localised on the anterior cruciateligament and on the patellar tendon that can generatesignificant pain during intensive use [5, 37]. Moreover, thistechnique may result in joint imbalance because ofasymmetrical stretching of the targeted joint complex.

Exercises in CMC are defined by simultaneous contractionof both the agonist and antagonist muscles for a particularmovement. In most cases, these exercises are possible whenthe foot is in contact with a support, see Fig. 2. Thus, allthe muscles in a group contribute to the joint’s actuation.

Figure 1 Exercise in OMC

IET Control Theory Appl., 2010, Vol. 4, Iss. 8, pp. 1407–1420doi: 10.1049/iet-cta.2009.0269

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Consequently, the musculoskeletal stretching is applied in asymmetrical way and contributes to increasing the jointstability. Thus, patellar pains are reduced during CMCrehabilitation compared with OMC rehabilitation [6, 38].Note that the benefit of CMC could be longer-lasting thanthat of OMC.

Finally, OMC and CMC rehabilitation techniques arecomplementary and together allow patients to reach specificrehabilitation goals relating to different pathologies. Forinstance, during long-term rehabilitation protocols, one canimagine proposing CMC exercises for stabilising the wholejoint complex and alleviating constraints generated by theOMC in the first step. In the second step, when the patient’sclinical state makes it possible, rehabilitation in OMC couldbe proposed for a specific muscular strengthening while thejoint balance achieved in CMC is maintained.

2.2 Kinematic concepts of Sys-Reeduc

Our objective now is to design an isokinetic CMCrehabilitation apparatus. By analogy with robotic systems, themechanical complex {lower limb, rehabilitation device} mustconstitute a closed kinematic chain (CKC). For each lowerlimb, a CKC, shown in Fig. 3, is proposed and is composedof the links C1, C2, C3, C4 and C5, which are the apparatusbase, thigh, leg, foot and mobile foot’s support, respectively.The joint between links i and j is denoted as Li,j.

According to the CKC depicted in Fig. 3, the design of anew device consists in defining the nature of the joints L1, 2,L2, 3, L3, 4, L4, 5 and L5, 1. In that way, we assume that thehuman body consists of a set of rigid polyarticulatedsegments. In the sagittal plane, the whole kinematicstructure, presented in Fig. 4, is supposed to be composedof perfect joints such that

Figure 2 Exercise in CMC

Figure 3 Kinematic chain of a lower-limb CMC rehabilitationdevice

T Control Theory Appl., 2010, Vol. 4, Iss. 8, pp. 1407–1420i: 10.1049/iet-cta.2009.0269

† L1, 2 is the joint between the base and the thigh. Thepatient sits on a chair fixed to the base of the system.The trunk and the pelvis are also assumed to be fixed tothe base. Then, L1, 2 is supposed to be a spherical jointcorresponding to the coxo-femoral joint (hip).

† L2,3 corresponds to the knee, which is the main joint wewish to rehabilitate. Its complex kinematics were thesubject of a previous study [39] and will not be detailed inthis paper. To simplify the specification of the Sys-Reeduckinematic concept, the knee will be represented as a hingejoint around the main lower-limb flexion–extension axis.

† The joints L3,4 (ankle) and L4,5 (mobile support rotations)can be reduced to a unique joint, L3,5. Indeed, L3,4

corresponds to the ankle plantar flexion/dorsal flexion Lfd

added with the ankle internal external rotation Lie. Moreover,L4,5 corresponds to the rotation q2(t) of the mobile support(which is collinear with Lie) added with the degree ofinclination of the mobile support in the sagittal plane (whichis centred with Lfd in the sagittal plane). Therefore themobile support mechanism is designed such that L3,4 andL4,5 are concentric (the clinicians have to line up the anklewith the support mobile rotations using a dedicated screw anda set of wedges). Thus, L3,5 can be resumed to two hingejoints, Lie and Lfd. Note that the adduction/abduction of theankle is omitted, because it has little influence on kneerehabilitation. Such mechanical design issues are introducedto reduce the constraints on the feet and, consequently,reduce ankle pains during rehabilitation exercises.

† The joint L5,1, between the foot’s mobile support and thebase, is realised by a prismatic joint along the axis defined bythe lower limb in complete extension.

Because the mechanical structure of Sys-Reeduc issymmetrical, Fig. 4 represents only the right lower limb. Tosimplify the presentation of the rehabilitation mode, onlythe right lower limb will be considered in the sequel.Moreover, according to the kinematics defined in Fig. 4, theinternal–external ankle rotation Lie (along the �zs-axis anddenoted by the variable q2(t)) and the translation L5,1 (alongthe �x-axis and denoted by the variable q1(t)) allow achievingcomplexes knee rehabilitation protocols. Indeed, even if the

Figure 4 Kinematical concept of Sys-Reeduc in the sagittalplane

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knee motion can be seen as planar for Sys-Reeduc controlpurpose since the main movement is flexion–extension inthe sagittal plane, when dealing with rehabilitationprotocols, one has to consider the complex kinematics of theknee [39]. Therefore controlling the knee flexion–extensionvia q1(t) as well as the ankle internal–external rotation viaq2(t) allows stimulating the knee in its six internal degrees offreedom. The motorisation of the joint L5,1 makes itpossible to rehabilitate the flexor muscular complex andlimits the constraints applied to the cruciate ligaments. Thefoot’s mobile support rotations allow the knee internal–external rotation Lie and the plantar flexion/dorsal flexion(free or forced by a mechanical brake) leading to a selectiveand precise muscular constraint. Then, rehabilitation ormuscular strengthening can be practiced in a specific wayduring a desired movement. For instance, lower-limbmovement with the foot in external rotation and inextension helps the stretching of medial hamstrings.

The efforts applied by the user to the device that mayinfluence the dynamic behaviour of Sys-Reeduc aredepicted in Fig. 5. Hence, fp/x(t) and Cp/zs(t) are,respectively, the force along the �x-axis and the torque along�zs applied by the user. These forces and torques aremeasured by six axes SENSIX sensors (www.sensix.fr)included in the mobile support under each foot.

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

† In Q1.1 and Q1.3, fp/x(t) is applied in the device’smovement direction, and the contractile mode is thereforecalled ‘concentric’.

† In Q1.2 and Q1.4, fp/x(t) is applied in opposition to thedevice’s movement direction, and the contractile mode istherefore called ‘eccentric’.

Figure 5 External efforts which influence Sys-Reeduc’sdynamics

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Fortunately, the observation depicted in Fig. 7 can also bemade relative to the internal–external rotation movement ofSys-Reeduc. In that case, the four-quadrant diagram ischaracterised by

† Q2.1 and Q2.3, where Cp/zs(t) is applied in the device’smovement direction, and the contractile mode is thereforecalled ‘concentric’;

† Q2.2 and Q2.4, where Cp/zs(t) is applied in opposition tothe device’s movement direction, and the contractile modeis therefore called ‘eccentric’.

3 Control structure design forSys-Reeduc3.1 Generic control structure

A generic control structure for rehabilitation devices ispresented in Fig. 8. This two-level structure is based on thefollowing components:

† A trajectory generator, which provides the desiredtrajectory xd(t) to be tracked by the continuous system statex(t). This trajectory generator, detailed in the next section,is based on the use of a state machine [4, 40] and allowsvoluntary movements activated from the measurement ofthe effort f(t) applied by the patient to the device.

Figure 7 External efforts which influence Sys-Reeduc’sinternal–external rotation

Figure 6 External efforts which influence Sys-Reeduc’sflexion–extension

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† A closed-loop rehabilitation device plant, which is used toguarantee the tracking of the desired trajectory, xd(t), by thedynamic system consisting of the device and the user. Toensure safe behaviour, this closed-loop system must ensurethe tracking error convergence of the system regardless ofthe patient’s efforts on the device. To achieve this goal,f(t) is considered as an external disturbance to beattenuated or, in the better case, to be rejected. Note that,without the trajectory generator, the closed-looprehabilitation device plant by itself does not allow thevoluntary control of the system by the patient.

3.2 Trajectory generator design

A rehabilitation exercise can be viewed as a succession of nelementary movement phases denoted as fi, for i ¼ 1, . . . , n.Hence, a discrete state machine can be used to representthese exercises, as depicted in Fig. 9. In this case, thedevice’s current position q1(t), q2(t) and the external effortf(t), consisting of fp/x(t) and Cp/zs(t), can be considered asinputs. The trajectory generator provides as outputs thedesired state, denoted as xd(t), consisting of the desiredpositions q1d(t) q2d(t)

[ ]and velocities q1d(t) q2d(t)

[ ],

to be tracked by the device.

Therefore it is necessary to define the whole movementcharacteristics for each elementary phase and the conditionsto ensure the validity of transitions between two phases.Elementary phases can be discriminated into two differentmajor classes: passive and active phases. A passive phase ischaracterised by an absence of voluntary control by the useron the elementary movement. In that case, independent ofthe user’s efforts, the device achieves the goal assigned tothe phase. For instance, a forced stop in the currentposition as well as any movements that do not depend onthe applied effort can be considered as passive phases.

Figure 8 Generic control structure for rehabilitation devices

Figure 9 Trajectory generator for Sys-Reeduc

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In an active phase, the device’s movements must be actuatedby the user. In the following, isokinetic movements areconsidered because they are commonly used in lower-limbrehabilitation protocols. These movements are traditionallyused as a standard for performance tests to evaluate musclestrengthening and applied forces during rehabilitation [16].Therefore isokinetic movements are the first wishes ofclinicians when developing a new rehabilitation device. Formore details on other rehabilitation protocols specification interms of elementary movements, the readers can refer to [4].Thus, during an isokinetic elementary phase, the velocitiesv1 and v2 are constant and the desired trajectory to betracked by the device can be written as

q1d = a1v1

q2d = a2v2

q1d = a1v1t + q1d(ti)q2d = a2v2t + q2d(ti)

⎧⎪⎪⎨⎪⎪⎩ (1)

where ti is the initial activation time of the ith phase and ak¼1

or 2 are parameters required to set the contractile mode duringthe movement (ak ¼ 1 for a concentric exercise and ak ¼ 21for an eccentric one, see Figs. 6 and 7).

To define a cyclic isokinetic exercise, the discrete statemachine, presented in Fig. 9 for flexion–extension, can bedefined. The transition between phases depends on theforce applied by the user on the mobile support as well asits current position. At the beginning of a training session(Start), f0 is activated. In that state, the desiredrehabilitation trajectory remains constant in its currentposition (the device is expected to be immobile). Then, thetransition from f0 to f1 takes place if the user force fp/x(t)is greater than the force threshold s1 . 0 (set by clinicians)or if the current position of the mobile support does not reachthe upper limit of the operational space �q1. When f1 isactivated, the transition to f0 occurs if fp/x(t) goes below thethreshold s1 or if the current position of the mobile supportreaches the upper limit of the operational space. The transitionfrom f0 to f2 occurs if fp/x(t) is lower than the thresholds2 , 0 or if the current position of the support mobile doesnot reach the lower limit of the operational space q

1. Finally,

the transition back to f0 occurs if the effort fp/x(t) is greaterthan the threshold 2s1 or if the current position of thesupport mobile reaches the lower limit of the operational space.

To illustrate the desired trajectory generation during acyclic isokinetic exercise described by this state machine, asimulation was performed with the parameters s1 ¼ 80 N,s2 ¼ 280 N, �q1 = 1.2 m, q

1= 0.1 m and v1 ¼ 2 m s21.

Fig. 10 shows the results obtained by fp/x(t) ¼ 500 sin(6t).Note that the desired trajectory generation for internal–external rotation is not presented in this section, because itrelies on the same principle as for flexion–extension. Withthe trajectory generator defined, the goal is now to ensurethe tracking of these trajectories by the dynamic system(consisting of the device and the user). Therefore the

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Figure 10 Desired trajectory generated during isokinetic exercises

following section will present the proposed non-lineartracking controller design methodology.

3.3 Continuous control levelof Sys-Reeduc

The goal is now to provide a convenient controller designmethodology that ensures the tracking of the desiredtrajectories provided by the trajectory generator. Note that,in that case, the external efforts consisting of the forces andtorques applied by the user must be rejected or attenuatedby the device’s dynamics. In that context, a H1-basedcontroller design can be employed. Moreover, in view ofthe non-linear dynamics of Sys-Reeduc, a Takagi–Sugenotracking controller design will be proposed.

3.4 Dynamic modelling 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/hermovement cannot be artificially controlled. This justifies the useof a H1-based design to attenuate these external disturbances tothe device. Thus, to synthesise the control law, we consider thatthe device is to be controlled via the motors’ torques while thepatient applies external effort. CM1(t) and CM2(t) are themotors’ torques allowing the movements associated with thedegrees of freedom q1(t) and q2(t), respectively.

Recall that w(t) = fp/x(t) Cp/zs(t)

[ ]Tdenotes the vector

of the efforts applied by the user on the device. Themechanical parameters used to model the Sys-Reeducdynamics are defined in Table 1. These parameters wereobtained from the design of the mechanical elementsconstituting Sys-Reeduc using the Catia software tools.Then, the dynamic model can be obtained using the

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well-known Lagrangian equations given by

d

dt

∂Ec

∂qi

( )−

∂(Ec − Ep)

∂qi

= Gi (2)

where qi and Gi are, respectively, the coordinates and thegeneralised efforts associated with the ith degree of freedom,and Ec and Ep are, respectively, the kinetic and potential energy.

The mobile support’s centre of gravity coordinates can bewritten in the frame (o, �x, �y, �z) as

OG/x = q1(t) + b cos (a) + b cos(q2(t)) (3)

OG/y = h − b sin (a) + d cos(q2(t)) (4)

OG/z = l sin(q2(t)) (5)

with b ¼ l sin(a) and d ¼ l cos(a).

Figure 11 Mechanical principle from the rehabilitation device

IET Control Theory Appl., 2010, Vol. 4, Iss. 8, pp. 1407–1420doi: 10.1049/iet-cta.2009.0269

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Thus, its velocity can be written in the frame (o, �x, �y, �z) as

VG/x = q1(t) − bq2(t) sin(q2(t)) (6)

VG/y = −d q2(t) sin(q2(t)) (7)

VG/z = l q2(t) cos(q2(t)) (8)

Therefore the kinetic energy can be expressed as

EC = 1

2(M + m)q2

1(t)

+ 1

2m[l 2q2(t) − 2bq1(t)q2(t) sin (q2(t))] + 1

2J q2

2(t) (9)

and the potential energy as

EP = mgh − mgb sin(a) + mgd cos(q2(t)) (10)

where g ¼ 9.81 m s22 is the gravitational constant.

Finally, substituting (9) and (10) in (2) and consideringthe generalised efforts G1(t) ¼ CM1(t) 2 fp/x(t) andG2(t) ¼ CM2(t) 2 Cp/zs(t), the dynamic model of Sys-Reeducis expressed by the following motion equation

M(q)q(t) + C(q, q)q(t) + G(q)q(t) = Ru(t) + Sw(t) (11)

Table 1 Numerical parameters of the dynamical model ofthe rehabilitation device Sys-Reeduc

Parameters Designation Values

M mobile support mass (intranslation)

14 kg

m mobile support mass (inrotation)

4 kg

J mobile support inertia alongthe �zs-axis (in rotation)

0.26 kg m2

a ray of the pulley-belt carryingout the translation along the

�x-axis

0.025 m

l distance between the rotationaxis and the gravity centre of

the support

0.05 m

a angle between the basehorizontal axis and the

rotation axis of the mobilesupport

208

b distance between the ankleaxis and the rotation axis of

the support

0.01 m

h height between the base andthe rotation axis of the mobile

support

0.6 m

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where

M(q) =M + m −ml sin(a) sin(q2(t))

−ml sin(a) sin(q2(t)) ml2 + J

[ ]

is the inertia matrix

C(q, q) = 0 −ml sin(a)q2(t) cos(q2(t))0 0

[ ]

is the Coriolis matrix

G(q) = 0 00 −mgl cos(a) sinc (q2(t))

[ ]

is the gravitational effect

R = 1/a 00 1

[ ]

is the input matrix

S = 1 00 1

[ ]

is the disturbance matrix. q(t) = q1(t) q2(t)[ ]T

denotes thegeneralised coordinates, and u(t) = CM1(t) CM2(t)

[ ]Tis

the input vector.

3.5 T–S fuzzy descriptor modellingof Sys-Reeduc

We now propose a descriptor-based controller design. State-space descriptors can be used to model algebraic systems suchas singular systems [31–33]. Moreover, these are aconvenient way to deal with mechanical systems withtime-varying inertia and reduce the computationalcomplexity of T–S-based LMI problems [21, 22]. Byconsidering the following state vector of the system,x(t) = q1(t) q2(t) q1(t) q2(t)

[ ]T, (11) can be rewritten

as the following state-space descriptor

E(x(t))x(t) = A(x(t))x(t) + Bu(t) + Hw(t) (12)

where

E(x(t)) =I 0

0 M(q)

[ ], A(x(t)) =

0 I

−G(q) −C(q, q)

[ ],

B =0

R

[ ]and H =

0

S

[ ]

The goal is now to rewrite (12) as a T–S fuzzy descriptor of

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the form

∑e

k=1

vk(z(t))Ekx(t) =∑r

i=1

hi(z(t))Aix(t) + Bu(t) + Hw(t)

(13)

where e and r are the numbers of fuzzy sets for the left andright-hand sides of the state (13), respectively; z(t) is thepremise vector depending on the state variables; vk(z(t)) ≥ 0for k ¼ 1, . . . , e and hi(z(t)) ≥ 0 for i ¼ 1, . . . , r are themembership functions that satisfy the convex sum property,that is, Sk¼1

e vk(z(t)) ¼ 1 and Si¼1r hi(z(t)) ¼ 1; Ek, Ai, Bi and

Hi are constant matrices, each defining e × r linear timeinvariant (LTI) descriptors that compose the T–S fuzzydescriptor.

The sector non-linearity approach [18] is a convenient wayto rewrite a non-linear descriptor such as (12) as one of itsT–S representations (13). Using this approach, theobtained T–S fuzzy model exactly matches the non-linearmodel on a compact set of the state space. Hence, from(12), one can consider the non-linear functions, included inE(x(t)) and A(x(t)), given by

v(q2(t)) = sin(q2(t)) [ [−1, 1] (14)

m1(q2(t), q2(t)) = q2(t) cos(q2(t)) [ [−�q2, �q2] (15)

m2(q2(t)) = sin(q2(t))

q2(t)[ [4, 1] (16)

where4 ¼ 20.2172 and �q2 ; 100 tr min21 ; 10.47 rad s21

is the maximal speed of the internal–external rotation set fromproduct design specification.

To obtain each LTI model and the membership functionscomposing a T–S model matching (12), the following sectornon-linearity transformations are employed

v(·) = (1)v(·) + 1

2︸���︷︷���︸v1(q2)

+(−1)1 − v(·)

2︸���︷︷���︸v2(q2)

(17)

m1(·) = (�q2)m1(·) + �q2

2�q2︸����︷︷����︸w11(q2,q2)

+(−�q2)�q2 − m1(·)

2�q2︸����︷︷����︸w12(q2,q2)

(18)

m2(·) = (1)m2(·) −4

1 −4︸����︷︷����︸w21(q2)

+(4)1 − m2(·)

1 −4︸����︷︷����︸w22(q2)

(19)

From (17), (18) and (19), the membership functions are

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given by

v1(q2) = v(q2) + 1

2(20)

v2(q2(t)) = 1 − v(q2(t))

2(21)

h1(·) = w11(·)w21(·) =(m1(·) + �q2)(m2(·) −4)

2�q2(1 −4)(22)

h2(·) = w11(·)w22(·) =(m1(·) + �q2)(1 − m2(·))

2�q2(1 −4)(23)

h3(·) = w12(·)w21(·) =(�q2 − m1(·))(m2(·) −4)

2�q2(1 −4)(24)

h4(·) = w12(·)w22(·) =(�q2 − m1(·))(1 − m2(·))

2�q2(1 −4)(25)

and the LTI matrices by

E1 =

1 0 0 0

0 1 0 0

0 0 M + m −ml sin a

0 0 −ml sin a ml 2 + J

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦,

E2 =

1 0 0 0

0 1 0 0

0 0 M + m ml sin a

0 0 ml sin a ml 2 + J

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

A1 =

0 0 1 0

0 0 0 1

0 0 0 ml �q2 sin a

0 mgl cosa 0 0

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦,

A2 =

0 0 1 0

0 0 0 1

0 0 0 m �q2 sin a

0 mgl 4 cos a 0 0

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

A3 =

0 0 1 0

0 0 0 1

0 0 0 −ml �q2 sin a

0 mgl cos a 0 0

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦ and

A4 =

0 0 1 0

0 0 0 1

0 0 0 −ml �q2 sin a

0 mgl 4 cos a 0 0

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

Consequently, considering z(t) = q2 q2

[ ]T, a T–S fuzzy

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model of Sys-Reeduc can be expressed as

∑2

k=1

vk(q2)Ekx(t) =∑4

i=1

hi(q2, q2)Aix(t) + Bu(t) + Hw(t)

(26)

3.6 LMI-based tracking controller designfor T–S descriptors

With the T–S descriptor model of Sys-Reeduc from the previoussection, we now provide a convenient controller designmethodology ensuring the tracking of desired trajectoriesprovided by the trajectory generator depicted in Section 3.2.

In the sequel, to simplify the mathematical expressions, wewill consider the notations Xh ¼ Sr

i¼1hi(z(t))Xi,

Yhv =∑e

k=1

∑r

i=1

vk(z(t))hi(z(t))Yik

Zhhv =∑e

k=1

∑r

i=1

∑r

j=1

vk(z(t))hi(z(t))hj(z(t))Zijk

and so on. Moreover, when there is no ambiguity, the time twill be omitted.

For the sake of generality, the following LMI results willbe developed for the general class of T–S fuzzy descriptorsdescribed by (13) and rewritten with the above-definednotations as

Evx(t) = Ahx(t) + Bhu(t) + Hhw(t) (27)

where Ah [ Rn×n, Br [ Rn×m, Hh [ Rn×p, x(t) [ Rn,u(t) [ Rm and f(t) [ Rp.

Consider the tracking control plant given by Fig. 12. Thisplant contains a reference model that is required to write theclosed-loop dynamics given by the state-space representation

xr(t) = Arxr(t) + Brxd(t) (28)

with Ar [ Rn×n a Hurwitz matrix, Br [ Rn×n, xr(t) [ Rn

the reference state vector and xd(t) [ Rm the desiredtrajectory to be tracked.

Note that (28) allows setting the dynamics of the trackingcontrol trajectory. A convenient way to choose it will beproposed in the simulation results section.

Figure 12 T–S trajectory tracking control plant

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The goal is now to write the closed-loop dynamics of thetracking control plant. Consider the parallel distributedcompensation (PDC) control law [41] defined by

u(t) = −KhvZ−111 (x(t) − xr(t)) (29)

where Khv and Z11 . 0 are the gain matrices to besynthesised.

As is classically the case for descriptors [29], by consideringthe extended state vector x∗(t) = xT(t) xT(t)

[ ]T, (27) can

be rewritten, for instance, as

E∗x∗(t) = A∗hvx∗(t) + B∗

h (t)u(t) + H ∗h w(t) (30)

where

E∗ =I 0

0 0

[ ], A∗

hv =0 I

Ah −Ev

[ ], B∗

h =0

Bh

[ ]and

H ∗h =

0

Hh

[ ]

In the same way, let us consider the extended reference statevector x∗r (t) = xT

r (t) xTr (t)

[ ]T. Equation (28) can be

rewritten as

E∗x∗r (t) = A∗r x∗r (t) + B∗

r xd(t) (31)

where A∗r = 0 I

Ar −I

[ ]and B∗

r = 0Br

[ ].

Let us define e∗(t) ¼ x∗(t) 2 xr∗(t) as the tracking trajectory

error. We can write

E∗e∗(t) = E∗x∗(t) − E∗x∗r (t) (32)

and (29) can be rewritten as

u(t) = −K ∗hve∗(t) (33)

where K ∗hv = KhvZ−1

11 0[ ]

.

Combining (30)–(33), the closed-loop dynamics can beexpressed by

E ˙e(t) = Ahhve(t) + H hf(t) (34)

with

e(t) =e∗(t)

x∗r (t)

[ ], f(t) =

w(t)

xd(t)

[ ], E =

E∗ 0

0 E∗

[ ]

Ahhv�h =(A∗

hv − B∗h K ∗

hv) A∗hv − A∗

r

0 A∗r

[ ]and

H h =H ∗

h −B∗r

0 B∗r

[ ]

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Now, the aim is to synthesise the matrices Khv and Z11 thatstabilise the closed-loop system (34) as well as to ensure theattenuation of the external disturbances f(t) with respect tothe trajectory-tracking error e(t) [27, 28]. Thus, weconsider the following H1 criterion

∫tf

t0

eT(t)e(t) dt ≤ h2

∫tf

t0

fT

(t)f(t) dt (35)

where h is the attenuation level to be minimised.

The main theoretical result is summarised in the followingtheorem.

Theorem 1: If the matrices Kjk, Z11 ¼ Z11T . 0,

Z41 ¼ Z41T . 0, Z13ij, Z14ij, Z23ij, Z24ij, Z33ij, Z34ij, Z43ij

and Z44ij and the scalar d exist such that the followingLMI conditions are satisfied for all i ¼ 1, . . . , r, j ¼ 1, . . . ,r and k ¼ 1, . . . , e under the minimisation problem min

R+d

Y(1,1)ij (Y(2,1)

ij )T ZT11 (Y(4,1)

ijk )T (Y(5,1)ij )T 0 0

Y(2,1)ij Y

(2,2)ij ZT

11 (Y(4,2)ijk )T (Y(5,2)

ij )T 0 0

Z11 Z11 −I 0 0 0 0

Y(4,1)ijk Y

(4,2)ijk 0 Y

(4,4)ijk (Y(5,4)

ijk )T Hi −Br

Y(5,1)ij Y

(5,2)ij 0 Y

(5,4)ijk Y

(5,5)ij 0 Br

0 0 0 H Ti 0 −dI 0

0 0 0 −BTr BT

r 0 −dI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, 0 (36)

where Yij(1,1) ¼ Z13ij

T + Z13ij, Yij(2,1) ¼ Z33ij + Z23ij

T , Yij(2,2) ¼

Z43ijT + Z43ij, Y

(4,1)ijk = ZT

14ij + 2AiZ11 − EkZ13ij− BiKjk−EkZ33ij − ArZ

T11 + Z33ij , Y

(4,2)ijk = ZT

34ij + AiZ11 − EkZ23ij−BiKjk + AiZ41 − EkZ43ij − ArZ41 + Z43ij , Y

(4,4)ijk = −ZT

14ij

ETk − EkZ14ij − ZT

34ijETk − EkZ34ij + ZT

34ij + Z34ij , Y(5,1)ij =

ArZT11 − Z33ij + ZT

24ij , Y(5,2)ijk = ZT

44ij + ArZ41 − Z43ijY(5,4)ijk =

−Z34ij − ZT24ijE

Tk − ZT

44ijETk + ZT

44ij and Y(5,5)ij = −ZT

44ij−Z44ij , then the asymptotic stability of the closed-loop fuzzysystem (34) is ensured and the H1 tracking controlperformance (35) is guaranteed with an externaldisturbance attenuation level h ¼

pd.

Proof: Let us consider the candidate Lyapunov function

V (e(t)) = eT(t)EZ−1

hhve(t) . 0 (37)

with Zhhv a non-singular C1 matrix satisfying EZ−1

hhv =

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Z−T

hhvE . 0 and defined as

Zhhv =

Z11 0 Z21hh 0Z13hh Z14hh Z23hh Z24hh

Z31hh 0 Z41 0Z33hh Z34hh Z43hh Z44hh

⎡⎢⎢⎣

⎤⎥⎥⎦ (38)

where Z11T ¼ Z11 . 0, Z21hh

T ¼ Z31hh . 0 and Z41T ¼ Z41 . 0.

Let us consider Q = diag I 0 0 0[ ]

, then the H1

criterion (35) can be rewritten as

∫tf

t0

eT(t)Qe(t) dt ≤ h2

∫tf

t0

fT

(t)f(t) dt (39)

Thus, the closed-loop descriptor (34) is stable and ensuresthe H1 performance h if

V (e(t)) + eT(t)Qe(t) − h2f

T(t)f(t) , 0 (40)

Considering (34) and (37), (40) is verified if

AT

hhvZ−1

hhv + Z−T

hhvAhhv + Q Z−T

hhvH h

HTh Z

−1

hhv −h2I

[ ], 0 (41)

By pre- and post-multiplying (41) by ZT

hhv 00 I

[ ]and

Zhhv 00 I

[ ], respectively, we obtain

(ZT

hhvAT

hhv + AhhvZhhv + ZT

hhvQZhhv) H h

HTh −h2I

[ ], 0 (42)

After developing (42) in its extended form with the matricesdefined in (34), (38) and (39), we apply the Schurcomplement [42] and the proof is completed. A

4 Simulation resultsThe theoretical results in the previous section ensure thetracking performance of a general class of T–S fuzzydescriptors. In this part, our goal is to show, by simulation,the efficiency of the proposed tracking controller designmethodology in the dynamic model of Sys-Reeduc given in(11). Following the tracking control plant depicted inFig. 12 and considering the parameters required to solvethe LMI problem (36), we must first choose an adequatereference model of the form (28). This reference model canbe used to set the dynamics of the trajectory trackinginputs, and its influence on the LMI solution cannot beneglected. In the Sys-Reeduc application, the goal is toforce the state x(t) to track the reference state xr(t).According to (28), the reference model contains an input

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vector xd(t), which must be close to xr(t). In this study, thereference model is chosen such that each transfer betweenan input variable xdp(s) and its associated reference statevariable xrp(s), for p ¼ 1, . . . , n, corresponds to a low-passfilter given in the frequency domain by

xrp(s)

xdp(s)= 1

1 + ks(43)

where k is the time constant and s the Laplace variable.

According to (43), a state-space reference model (28) canbe written by considering Ar ¼ 2(1/k)eye(4,4) andBr ¼ (1/k)eye(4,4). Obviously, the smaller the k is, thecloser the reference state xr(t) dynamics will be to thedesired reference xr(t). Indeed, decreasing the time constantleads to downgrade the H1 performance as the LMIproblem is compelled. To illustrate this phenomenon, theminimal attenuation level h has been evaluated, using theMATLAB LMI control toolbox [43], for several timeconstants k, and the results are presented in Table 2.

In the present application, a good compromise could bek ¼ 0.1 s, leading to h ¼ 3.08. The results computed fromthe LMI conditions (36) lead to the design of the PDCtracking control law given by

u(t) = −∑2

k=1

∑4

i=1

vk(q2(t))hi(q2(t), q2(t))KikZ−111 (x(t) − xr(t))

(44)

Table 2 Evolution of the attenuation rate regarding k

k 1/400 1/100 1/50 1/30 1/20 1/10

h 19.98 9.97 7.03 5.43 4.41 3.08

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Where

K11 = −31.6302 −0.0230 627.1351 0.09221.6806 −11.3922 −33.3268 206.1104

[ ]

K12 =−31.6302 −0.1708 627.1347 2.9790

−1.6804 −11.3930 33.3112 206.1174

[ ]

K21 =−31.6302 −0.0231 627.1350 0.0945

1.6799 −11.2670 −33.3143 204.9511

[ ]

K31 =−31.6302 0.1470 627.1351 −2.6469

1.6800 −11.3904 −33.3157 206.0744

[ ]

K32 =−31.6302 −0.0010 627.1349 0.2438

−1.6792 −11.3888 33.2885 206.0361

[ ]

K41 =−31.6302 0.1469 627.1351 −2.6450

1.6800 −11.2694 −33.3158 205.0098

[ ]

K42 =−31.6302 −0.0008 627.1349 0.2384

−1.6793 −11.2681 33.2885 204.9780

[ ]

and

Z11 =

−0.3926 0 −1.5155 00 −0.3907 0 −1.5116

−1.5155 0 19.4130 −0.00010 −1.5116 −0.0001 19.2394

⎡⎢⎢⎣

⎤⎥⎥⎦

Figs. 13 and 14 illustrate the tracking performance of theclosed-loop rehabilitation device plant. The simulations arerealised for a lower-limb isokinetic extension with velocityq1 = 1 m s21 between t ¼ 1 s and t ¼ 2 s from the initialposition q1(0) ¼ 0.1 m and for a sinusoidal internal–external rotation q2 ¼ (p/4)cos(2t). The efforts fp/x(t) andCp/zs(t) have been set as non-physiological externaldisturbances such that they outperform common usercapabilities. This setting is used to evaluate the robustnessof the designed non-linear tracking controller (44)

Figure 13 Tracking trajectory for translation of the mobile support

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Figure 14 Tracking trajectory of rotation of the mobile support

Figure 15 Simulation of the whole Sys-Reeduc control structure

independent of the trajectory generator. These externaldisturbances are given by

fp/x(t) = 500 sin(6t) + 250 sin(30t) + 125 sin(60t)

+ rand(t) (45)

and

Cp/zs(t) = 20 sin (5t) (46)

Note that in (45), the function rand(t) is used to simulate anoise measurement in the force applied by the patient. Thisfunction is set with a Gaussian disturbed signal of varianceequal to 1000 and a sample time equal to 0.01 s.

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These simulation results illustrate the efficiency of theperformance in trajectory tracking obtained by using theproposed control law. Indeed, Figs. 13 and 14 show thatthe H1 control law (44) successfully attenuates the externaldisturbances and compensates them via the input signal. Inthese simulations, the trajectory generator is not taken intoaccount. Fig. 15 illustrates the performance of the wholecontrol structure design for Sys-Reeduc, depicted in Fig. 8,including the isokinetic trajectory generator proposed inSection 3.2. This simulation shows that the trajectorygeneration is successfully realised while the trackingperformance of the device is ensured.

5 ConclusionsAfter discussing the advantages and drawbacks of the OMCand the CMC rehabilitation techniques, the concept of a new

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CMC lower-limb rehabilitation device has been presented. Theproposed control structure consists of a trajectory generator anda continuous level ensuring the tracking control stability of theclosed-loop mechanical system. For the sake of generality, theprinciple of the trajectory generator was described as aparameterised discrete-state machine. LMI-based trackingcontroller design methodology was developed for a class ofT–S fuzzy descriptors suitable to represent the non-lineardynamics of Sys-Reeduc. Moreover, a H1 criterion has beenemployed to attenuate the user’s efforts that are considered,for the closed-loop continuous dynamics, as externaldisturbances. Note that attenuating these disturbancesprovides a safe behaviour to the user of the rehabilitationdevice. Indeed, even if the user–rehabilitation deviceinteraction is attenuated by the continuous level, the trajectorygenerator allows voluntary movements to be performed on thebasis of the measurement of this interaction. One of theinterests of such a control structure is the possibility ofparameterising a particular trajectory generator that, forinstance, motivates the user to provide an appropriate effortduring rehabilitation movements. Finally, the proposedcontroller synthesis has been validated in simulation on Sys-Reeduc and has shown the efficiency of the proposed controlplant. Ongoing realisation of the first prototype of Sys-Reeduc will enable us to develop and validate newrehabilitation protocols and to evaluate accurately the benefitsof CMC against OMC rehabilitation.

6 AcknowledgmentThis work was supported by the French Ministry of Researchand the ‘Region Champagne-Ardenne’ within the CPERSYS-REEDUC and the CPER MOSYP. The authors wouldlike to thank Dr. Mathieu Boucher (SENSIX), Dr. SebastienLeteneur (La Rougeville Rehabilitation Center) and Ms. MegAmbettes for their valuable comments within this study.

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