Mechatronics 23 (2013) 1141–1149

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Mechatronics

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Nested saturation based control of an actuated knee joint orthosis q

0957-4158/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mechatronics.2013.09.007

q This work lies within the scope of the project EICOSI (Exoskeleton IntelligentlyCommunicating and Sensitive to Intention), sponsored by the regional council ofIle-De-France.⇑ Corresponding author. Tel.: +33 141807318.

E-mail addresses: hala.rifai@u-pec.fr (H. Rifaï), samer.mohammed@u-pec.fr (S.Mohammed), walid.hassani@u-pec.fr (W. Hassani), amirat@u-pec.fr (Y. Amirat).

Hala Rifaï, Samer Mohammed ⇑, Walid Hassani, Yacine AmiratLISSI, EA 3956 – University of Paris-Est Créteil, 122 rue Paul Armangot, 94400 Vitry-Sur-Seine, France

a r t i c l e i n f o

Article history:Received 10 July 2012Accepted 12 September 2013Available online 9 October 2013

Keywords:Bounded controlAsymptotic stabilityKnee joint orthosis

a b s t r a c t

Wearable robots have opened a new horizon for assistance and rehabilitation of dependent/elderly per-sons. The present study deals with the control of an actuated lower limb orthosis at the knee joint level.The dynamics of the shank–foot–orthosis system are expressed through a nonlinear second order modeltaking into account viscous, inertial and gravitational properties. Shank–foot–orthosis system parametersare identified experimentally. Since the underlying dynamic model is nonlinear, a robust control strategyis needed to guarantee an accurate and precise movement generation. The proposed control strategyensures, at the same time, the stability of the closed-loop system. A bounded control torque is appliedto guarantee the asymptotic stability of the shank–foot–orthosis. The generated control respects thephysical constraints imposed by the system. The effectiveness of the proposed control strategy is shownin real-time in terms of stability, position tracking performances and robustness with respect to identi-fication errors and external disturbances.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the last decade, rehabilitation and assistance of dependentpeople have drawn particular attention due to the considerablyincreasing rate of disabled and elderly. This fact has encouragedthe development of wearable robots, designed to fit the shapeand function of the human limbs [1,2]. They are equipped withsensors and actuators and are mainly dedicated for [2,3]: (i) theaugmentation of the wearer’s performance in terms of energy sav-ing, joint strength and endurance [4,5], (ii) the assistance of peoplesuffering from physical weaknesses to help them achieving tasks ofdaily living, and (iii) the physical and neurological rehabilitation ofjoints or muscles in order to allow patients regain control of theirlimbs. Some orthoses of lower limbs have been used in combina-tion with treadmills for gait rehabilitation such as the Lokomat[6] and LOPES [7].

Within the third aforecited category, passive rehabilitation ismainly dedicated for people having been subject to injuries or sur-gical operations of articular tissues (ligaments, cartilage or ten-dons) as well as for skeletal muscles reinforcement [8]. It can bealso performed after total knee arthroplasty and is beneficial forlimb movements rehabilitation of spinal cord injured patients[9,10]. This kind of rehabilitation can be done by means of actuated

orthoses performing continuous motions of the user’s joint. Thestrategy consists of moving the joint without any active contrac-tion of the muscles by tracking a predefined trajectory for a periodof time, allowing automation of therapist actions, control themovements and increase the duration of intervention. Even thoughthe continuous passive motion is debatable, it presents benefits forthe patient in short-term rehabilitation [11]. Among the advanta-ges of the passive rehabilitation, one can cite the stiffness reduc-tion, the fast increase of the joint range of motion, the musclesstrengthening, the stimulation of new cartilage production, the in-crease of synovial fluid movement and the joint blood clearing[12].

Proportional Integral Derivative (PID) controllers have beenwidely used for exoskeletons having multiple degrees of freedom.This control has been associated to gravity, friction or coriolisforces balancing in order to control the leg’s trajectory, stabilizethe body at the zero moment point and ensure a gait rehabilitation[13–19]. Exoskeletons or orthoses having only one degree of free-dom have been developed mainly to reduce the dimensions,weight and electronic equipments (sensors, actuators, processors,battery, etc.) when assistance or rehabilitation are only neededfor this degree of freedom. Some works concerning the knee jointare presented in the following. In [20], an orthosis is used to reducethe joint stiffness of patients having been subject to total kneearthroplasty surgery and to increase the flexion and extensionrange of motion. In [21], a robotic knee orthosis is designed to as-sist stroke patients in untethered functional training. Strengthen-ing muscles has also been considered in [22–28]. The controller,for the last two orthoses, acts as an amplifier of the human torque

1142 H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149

when no desired trajectory and therefore no position controller isdefined. In [29,30], impedance-based PID and Linear Quadratic(LQ) controllers are used respectively to track the desired orienta-tion. Note that the inherent compliance of the actuation/transmis-sion system [31,32] has also been used to ensure an impedancecontrol allowing high-impedance robot-in-charge and low imped-ance patient-in-charge tasks [33]. One should emphasize that noneof the aforementioned works did considered the physical con-straints of the orthosis, a crucial point for avoiding the saturationof the actuator which may induce instability of the system andendanger the wearer.

Other safety criteria have been considered in the literature, re-lated to the alignment of the human–exoskeleton joint axes [34,35]and the comfort of the user by monitoring the human–exoskeletonphysical interaction [36].

The present paper deals with the control of the EICOSI (Exoskel-eton Intelligently COmmunicating and Sensitive to Intention)orthosis having one degree of freedom at the knee joint level.EICOSI orthosis has a simple structure, easy to don and doff, whichmakes it very practical to use by knee-joint impaired people. Theintended applications of this study concern chiefly the passiverehabilitation, at low frequencies, of patients having undergone asurgery of the knee joint or suffering from a knee osteoarthritis,patellar pain, knee dysplasia or knee ligament deficiency. Besides,the orthosis use increases the neuromuscular force (the quadricepsand hamstrings in this case) as has been proved in [22,37]. For thispurpose, flexion and extension movements of the knee joint areperformed by means of the orthosis’ actuator. The control is com-puted using nested saturation functions of the state feedback, i.e.the angular position and velocity, with a gravity torque balance.The bounded input allows to (i) take into consideration the pro-vided power limitation, (ii) avoid the saturation of the actuator,(iii) prevent problems related to nonlinearities and resulting inthe orthosis’ instability, and (iv) enhance the wearer’s security.To the authors best knowledge, a control law that satisfies theaforecited criteria has not been considered for controlling orthoses.The considered control laws in the literature are mainly based onclassical Proportional, Integral, Derivative actions with no care ofthe actuator’s bounds. Moreover, the asymptotic stability of theshank–foot–orthosis system, subjected to the proposed boundedcontrol law, has been proved theoretically by means of a Lyapunovanalysis. Unlike linear control laws, the proposed control strategyallows to avoid the saturation of the actuator and guarantees thestability of the closed loop system even in presence of external dis-turbances. Robustness tests have been conducted with respect toparameters identification errors. A part of this paper has been pre-sented in [38], notably a brief description of the identificationstrategy, the control approach and experimental tests. In this ver-sion, a more realistic model of the shank–foot–orthosis is pro-posed, the control law and consequently the stability analysis areslightly modified. Theoretical proof of the control law’s robustnesswith respect to a knee blockade is presented. Additional experi-ments and robustness tests are also performed.

The paper is structured as follows. The orthosis model as well asits parameters identification are presented in Section 2. Thebounded control torque is proposed in Section 3 and the system’sstability is proved. Real-time experiments and robustness testsare presented in Section 4. Finally, conclusions are addressed inSection 5 and future works are introduced.

Fig. 1. Human leg embodying the orthosis: fixed and shank frames.

2. Shank–foot–orthosis model and parameters identification

Closed-loop control of a powered orthosis actuating the kneejoint of a given subject constitutes a prerequisite step towardsany upward mobility such as standing up, walking, and climbing

stairs. The biomechanical model chosen in this study consists oftwo segments representing the shank and the thigh connected toeach other by one degree of freedom revolute joint and coupledwith the embodied orthosis. Knee joint flexion and extension arestudied for a person in a sitting position, with the thigh supportedand the shank–foot swinging. Within the passive rehabilitation,these movements are ensured by the powered orthosis. Denoteby h the angle between the shank and the full extension position,as shown in Fig. 1. h = 0 rad corresponds to the full extension ofthe knee-orthosis system. h = �2.1 rad corresponds to maximumflexion and hr denotes the rest position. The orthosis torque repre-sents the input of the biomechanical model while h is the corre-sponding output. This model has been particularly chosen for itseasiness of implementation as well as for the limited number ofdegrees of freedom. This model allows at the same time the repro-duction of the knee joint movements for flexion and extension inthe sagittal plane.

The system considered in the present study includes the ortho-sis as well as the wearer shank–foot. In the following, the shankand foot will be considered as one system in rotation about theknee joint, that will be referred by shank. Recall that the move-ments of flexion and extension of the shank will be studied for awearer in a sitting position for passive rehabilitation purposes.

2.1. Shank–orthosis modeling

The design of the orthosis fits the geometry of the human leg. Inthe project EICOSI, the orthosis is composed of two segments re-lated along a rotational axis. A segment is attached to the thighwhile the other is attached to the shank by means of braces (Figs. 1and 5). The orthosis is fixed to the human leg such that they havethe same rotational axis at the knee joint. The orthosis and the legare therefore assumed coupled. Previous works dealing with mis-alignment problems have considered chiefly exoskeletons withmultiple degrees of freedom [33,31]. In this study, this problemhas been considerably reduced by adjusting manually the orthosisto every wearer’s morphology using adaptable straps as shown inFig. 5. Special cares have been taken during experiments in orderto avoid reaching of the full knee joint flexion which considerablyreduce the joints misalignment.

Denote by Fð~xf ;~yf ;~zf Þ a fixed frame in the space and bySð~xs;~ys;~zsÞ a frame attached to the shank at the knee joint definedsuch that the directions of~yf and~ys coincide (Fig. 1). The knee, andtherefore the orthosis, are in rotation about the pitch axis~ys of theangle h. The system having a unique degree of freedom, its angularvelocity _h is equal to the derivative of the rotational position. In thefollowing, the subscripts s and o refer to the shank and orthosis

H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149 1143

respectively. Deriving the shank–orthosis Lagrangian, the system’sdynamics can be written as:

J€h ¼ �sg cos h� Kðh� hrÞ � Asign _h� B _hþ s; ð1Þ

with J = Js + Jo the system’s inertia, sg = (msls + molo)g the system’sgravity torque in full extension position of the thigh, A = As + Ao

the system’s solid friction coefficient, B = Bs + Bo the system’s vis-cous friction coefficient, K = Ks + Ko is the stiffness coefficient, hr isthe angle of the shank relative to the thigh at the rest positionand s the control torque applied by the actuator to drive the ortho-sis and consequently the shank to a desired orientation.

2.2. Parameters identification

The shank–orthosis system is composed of two components.The identification of its parameters amounts from the separateidentification of each of its components. The rest position hr, whichrefers to the knee joint angle of the shank–orthosis, has been mea-sured for the wearer by means of the orthosis incremental encoderand has been used in the identification process. Muscular activitiesof both quadriceps and hamstrings, supposed to be null at the restposition, have been monitored through the EMG recordings.

2.2.1. Shank’s parametersThe choice of a given wearer is subject to three exclusion crite-

ria: (i) complete flexion and extension of the knee joint, (ii) nospasticity or contracture during the flexion and extension, and(iii) no prior knee joint pathology. The mass of the shank ms andthe position of its center of gravity ls are determined using theregression equations of Winter, knowing the height and weightof the subject [39]. The other parameters are identified using thepassive pendulum test [40]. The shank angle relative to the hori-zontal axis is extracted from the kinematic data measured bymeans of an electrogoniometer attached to the leg at the knee le-vel. An EMG signal analysis of the quadriceps muscles is performedto identify and reject any undesired voluntary/reflex muscle con-tractions. In case of a patient suffering of muscle spasticity charac-terized by stiff or rigid muscles with exaggerated deep tendonreflexes, an increase in the stiffness coefficient will be noticed dur-ing the identification process [41].

(1), applied to the shank, can be written as:

�msgls cos h ¼ Js€hþ Ksðh� hrÞ þ Assign _hþ Bs

_h; ð2Þ

with msgls; h; _h; €h; hr known and Js, As, Bs, Ks the parameters to beidentified.

2.2.2. Orthosis’ parametersAn excitation sequence describing the desired trajectory of the

angle h is applied to the orthosis [42]. The angular velocity andacceleration, _h and €h, are computed by derivation of the excitationtrajectory. During the movement, the torque developed by theactuator is computed proportionally to the current measurementsdelivered by the current sensor. (1), applied to the orthosis, can berewritten as:

s ¼ Jo€hþmoglo cos hþ Koðh� hrÞ þ Aosign _hþ Bo

_h; ð3Þ

with s; h; _h; €h; hr known and Jo, mog lo, Ao, Bo, Ko the parameters to beidentified.

The identified parameters of the shank and orthosis are ob-tained using a nonlinear least square optimization of Eqs. (2) and(3) [43]. Numerical values will be given in Section 4.

3. Exoskeleton control

The control law should take into consideration two criteria re-lated principally to the safety of the mechanism since it is in directrelation with the human body. On one hand, the solid friction hasnot been balanced in the control torque because its quick varia-tions, induced by unpredictable movements can cause wear load-ing of the actuator and engender unacceptable behavior of theorthosis. On the other hand, the control torque has been boundedto reduce the power consumption, a necessary condition for anysustainable wearable robot solution, and to ensure the safety ofthe wearer since the saturation of the actuator can lead to undesir-able closed loop behaviors resulting in orthosis’ instability.

Bounded control has been treated in the literature for systemsfalling in the framework of manipulators [44–46] subject of inter-est in the present study since the orthosis falls within the sameframework. For these applications, the gravity balance should betaken into consideration. In the present work, the control torqueis based on nested saturations [47] to allow separate tuning of pro-portional and derivative actions in order to obtain maximal benefitof the system’s physical constraints without saturating the actua-tor. The control law is simple to compute and implement; there-fore, it is adapted for real-time applications such as the orthosis.Moreover, it ensures the asymptotic stability of the shank–orthosissystem.

The control torque and stability analysis are presented in thefollowing:

Proposition 1. Consider the knee-joint shank–orthosis modeldescribed by (1) with h and _h the knee joint angle and angularvelocity, respectively. Denote by hd the desired orientation and~h ¼ h� hd the orientation error. hr stands for the shank angle relativethe horizontal at the rest position. The control torque boundedbetween ��s and defined by:

s ¼ �satN1 ½k1_hþ satN2 ðk2

~hÞ� þ sg cos hþ Kðhd � hrÞ; ð4Þ

asymptotically stabilizes (1) at ðh; _hÞ ¼ ðhd; 0Þ with a domain of attrac-tion equal to ð�p;pÞ � R. k1 and k2 are positive scalar parameters andsatNi

ð�Þ; i 2 f1; 2g, are classical saturation functions with Ni the satura-tion bounds chosen such that N1 > 2N2 > 2A. The saturation bound ofthe control torque is �s ¼ N1 þ sg þ K p

2.

Proof. Consider firstly that k2j~hj > N2 and k1j _hj > N1 � N2 > N2.Consider the Lyapunov function V positive definite and radiallyunbounded:

V ¼ 12

J _h2 þ 12

K~h2: ð5Þ

Based on the system’s model (1) and the control torque (4), thederivative of the Lyapunov function V is given by:

_V ¼ _h �B _h� Asign _h� K~h� satN1 ½k1_hþ satN2 ðk2

~h�� �

þ K~h _h;

¼ �B _h2 � A _hsign _h� _hsatN1 ½k1_hþ satN2 ðk2

~h�:

Since jk1_hj > N2, then jk1

_hþ satN2 ðk2~hÞj > 0. Therefore, _h and

k1_hþ satN2 ðk2

~hÞ are of the same sign. The Lyapunov functionbecomes:

_V ¼ �B _h2 � A _hsign _h� j _hjN1: ð6Þ

The Lyapunov function V is decreasing and j _hj consequently. _henters the set X1 : f _h; ~h : k1j _hj < N1 � N2; k2j~hj > N2g. InX1; k1j _hj þ N2 < N1. satN1 ð�Þ operates then in the linear region:satN1 ½k1

_hþ satN2 ðk2~hÞ� ¼ k1

_hþ satN2 ðk2~hÞ, and the control torque

(4) becomes:

Lower segment

Uppersegment

Pulley -Revolute joint

Ball screw

BrushlessDC motor

Tractioncable

Gear motor

Mechanicalsupport

Belt transmission

Angular abutment

Incrementalencoder

Fig. 2. The structure of the orthosis EICOSI.

1144 H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149

s ¼ �k1_h� satN2 ðk2

~hÞ þ sg cos hþ Kðhd � hrÞ: ð7Þ

In X1, define the Lyapunov function W as:

W ¼ 12

Jð _hþ k~hÞ2 þZ ~h

0satN2 ðk2xÞdxþ 1

2k Bþ k1 � Jkþ K

k

� �~h2; ð8Þ

with 0 < k < Bþk1J and j ¼ N2 � A > 0; satN2 ð�Þ locally Lipschitz,

satN2 ð0Þ ¼ 0 and zsatN2 ðzÞ > 0 for all z – 0.The derivative of the Lyapunov function within the trajectories

of the system is given by:

_W ¼ ð _hþ k~hÞðJ€hþ Jk _hÞ þ _hsatN2 ðk2~hÞ þ k Bþ k1 � Jkþ K

k

� �~h _h;

¼ ð _hþ k~hÞ½�ðBþ k1 � JkÞ _h� Asign _h� K~h� satN2 ðk2~hÞ�

þ _hsatN2 ðk2~hÞ þ k Bþ k1 � Jkþ K

k

� �~h _h:

Since k2j~hj > N2, then jsatN2 ðk2~hÞj ¼ N2. The derivative of the

Lyapunov function W becomes:

_W ¼ �ðBþ k1 � JkÞ _h2 � A _hsign _h� kK~h2 � k~hsatN2 ðk2~hÞ � Ak~hsign _h;

6 �ðBþ k1 � JkÞ _h2 � A _hsign _h� kK~h2 � kj~hjN2 þ Akj~hj;6 �ðBþ k1 � JkÞ _h2 � A _hsign _h� kK~h2 � kj~hjj:

W is then decreasing. _h; ~h enter the set X2 defined byX2 : f _h; ~h : k1j _hj < N1 � N2; k2j~hj < N2g. In X2; satN2 ð�Þ operates inthe linear region: satN2 ðk2

~hÞ ¼ k2~h, and the control torque (7)

becomes:

s ¼ �k1_h� k2

~hþ sg cos hþ Kðhd � hrÞ: ð9Þ

Define in X2 the Lyapunov function L as:

L ¼ 12

J _h2 þ 12ðK þ k2Þ~h2: ð10Þ

Replacing (9) in (1), the derivative of the Lyapunov function L isgiven by:

_L ¼ �ðBþ k1Þ _h2 � A _hsign _h 6 0: ð11Þ

The Lyapunov function L is then decreasing till the angularvelocity reaches the origin _h � 0. In order to complete the proof,the LaSalle Invariance Principle is invoked. All the trajectories con-verge to the largest invariant set �X3 inX3 ¼ fð~h; _hÞ : _L ¼ 0g ¼ fð~h; _hÞ : _h ¼ 0g. To remain in this set, onemust ensure that J€h ¼ �ðK þ k2Þ~h ¼ 0 with K + k2 > 0. Therefore,to remain in the set �X3, one should satisfy ~h ¼ 0 which means thath = hd. Therefore, ðh; _hÞ ¼ ðhd;0Þ is an asymptotically stable point ofthe closed-loop system with a domain of attraction equal toð�p;pÞ � R. h

Fig. 3. Block diagram of the actuated orthosis in closed loop.

4. Experimentation and robustness tests

The identification process as well as the control law are per-formed in real-time using the EICOSI orthosis. A brief descriptionof the actuated orthosis is given in the following.

4.1. Description of the orthosis EICOSI

The mechanical structure of the orthosis is composed of twosegments coupled with the thigh and shank respectively usingsome straps; the rotation axis is fixed at the knee joint level. Themaximal rotation angle and angular velocity are respectively of�2.1 rad and 2.1 rad/s. The actuation system of the orthosis EICOSIis shown in Fig. 2. It comprises a cylinder based on a ball screw,having a high efficiency coefficient of 0.9, that actuates a pulleyby means of a traction cable. The cable has a breaking strength of

4690 N. The pulley is fixed to the orthosis at the knee level. A belttransmission system 1:1 is used to connect the cylinder to a gear-motor, mounted in parallel and characterized by a reduction ratioof 14:1. A brushless DC motor (BLDC) is chosen to drive the wholesystem for its relatively high torque and smoothly running at lowspeeds. The motor delivers a maximal torque of 129 � 10�3 N mat nominal voltage. The maximal torque that can be delivered bythe actuation system is �s ¼ 54 N m which corresponds to the kneejoint peak torque determined in [48] (0.32 ± 0.15 N m/(kg m)). Theorthosis is also equipped with an incremental encoder that mea-sures the rotation angle with a resolution of 1000 counts/turn.The control torque is computed using a controller board(dSPACE-DS1103) equipped with an IBM processor (PowerPC604th) running at 400 MHz. The controller takes the measurementof the angle delivered by the EICOSI’s sensor and the angular veloc-ity obtained by a simple derivation as well as the desired angle andvelocity. The controller board delivers the pulse width modulation(PWM) level to control the actuator’s velocity. The control loopruns at 1 kHz, fixed due to current and position sensors constraints.The block diagram is presented in Fig. 3.

4.2. Experimental results

The experiments are conducted on a healthy subject being35 years old, weighing 97 kg and measuring 1.82 m. The subject

Table 2Shank parameters identification.

Parameter Value ± s.d.

Inertia (Js) 1.5383 ± 0.7052 kg m2

Stiffness (Ks) 0.45 ± 0.5476 N m rad�1

Solid friction coefficient (As) 0.3975 ± 1.3265 N mViscous friction coefficient (Bs) 4.7528±2.4557 N m s rad�1,Gravity torque (sgs

) 17.4576 N m

0 1 2 3 4 5 6 7−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Fig. 4. The cross-validation of the orthosis parameters identification: the torquecomputed using the inverse dynamics of the orthosis in red dashed line and themeasured one in blue continuous line. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149 1145

is able to ensure a complete flexion and extension of the knee jointwith no spasticity or contracture. He has been sufficiently in-formed of the experiment’s protocol and the benefit of the study.Abutments are placed at the full extension position (0 rad) andthe full flexion (�2.1 rad) to guarantee the security of the wearer.Moreover, all the precautions have been taken such that the exper-iment does not affect the health of the subject. Privacy and confi-dentiality have been also respected. Two electromyogramelectrodes have been placed at the thigh of the subject to detectthe activity of the Rectus Femoris (RF) muscle (quadriceps group)and the Biceps Femoris (BF) long head muscle (hamstrings group)detecting respectively voluntary extension and flexionmovements.

The identification process of the shank’s and the orthosis’parameters is described in Section 2. The identified parameters:inertia, stiffness, solid and viscous coefficients, gravity torque aswell as their standard deviations are shown in Tables 1 and 2.The orthosis’ identified parameters are verified using the cross-val-idation process. It consists of considering a trajectory, computingthe inverse dynamics of the orthosis then comparing the resultingtorque to the one measured when applying this trajectory to theorthosis. Fig. 4 shows a good convergence of the computed andmeasured torques. One should note that the trajectory used for val-idation is different from the one used for identification. Notice thatthe identified orthosis stiffness is null; it has been added into theshank–orthosis model only to harmonize the expression of theparameters as the sum of the orthosis and the shank ones.

The control torque (4) is applied to the orthosis. The saturationbounds are chosen such that the control respects the limitation ofthe supplied power and the maximal torque delivered by the actu-ator: N1 = 34 and N2 ¼ N1

2:1. The scaling control parameters are setexperimentally to: k1 = 15 and k2 = 340 such that good perfor-mance is obtained. The knee joint angle at rest position: hr =�1.29 rad, has been measured by means of the encoder.

Three experiments are performed to test the efficiency of thecontrol law.

The first one considers flexion and extension movements of theknee, actions that are often performed during the passive rehabil-itation process. The desired trajectory is defined by successive stepfunctions. Fig. 5 shows snapshots of the shank–orthosis system’strajectory. The angle, angular velocity and control torque are plot-ted in Fig. 6. Experiments show a good convergence of the anglewith relatively short response time (about 0.7 s). The steady stateerror at the extension position of the shank is of 8 � 10�4 radand at the flexion position of 4 � 10�3 rad. The control torque re-mains within the saturation bounds avoiding the nonlinearitiesof the actuator. The angular velocity is not high, avoiding the wear-er to endure high velocities that may be harmful for disabled peo-ple. When the shank reaches the desired position, the angularvelocity is null. If the desired position reaches the full extensiondefined by angle h = 0 rad, the control torque will have the valueof sg. Note that the RF and BF muscle activities remain null duringthis test.

If the saturation functions have not been present in the controllaw, i.e. (9) is applied to the shank–orthosis system, the motor will

Table 1Orthosis parameters identification.

Parameter Value ± s.d.

Inertia (Jo) 0.0117 ± 3.5238 kg m2

Stiffness (Ko) 0 N m rad�1

Solid friction coefficient (Ao) 0.3525 ± 0.2491 N mViscous friction coefficient (Bo) 0.6928 ± 0.3811 N m s rad�1

Gravity torque (sgo) 0.2424 ± 0.5518 N m

reach its saturation bounds. Fig. 7 shows that the control torque,computed using (9), exceeds the saturation bounds of the actuatorduring the knee joint extension. Therefore, this torque cannot bedelivered by the actuator because the latter is saturated. The actu-ator performs then in the nonlinear regime and the convergence tothe desired trajectory cannot be guaranteed. The steady state errorduring the knee joint extension reaches 0.26 rad and 5 � 10�3 radduring flexion. The difference between the steady state errors dur-ing extension and flexion is due to the saturation of the actuatorduring extension and its performance in linear regime duringflexion.

The second experiment is performed to determine the band-width of the system subject to the control law. For this purpose,a sine function with increasing frequencies over the time is appliedas a desired trajectory. The bode diagram of the closed loop systemwith hd as input and h as output is estimated as the ratio of thepower spectral density of the current and desired knee joint angles.The bode diagram is presented in Fig. 8. The cutoff frequency mea-sured at �3 dB is equal to 4.97 rad/s and the phase to �0.79 rad,which limits the closed loop system’s bandwidth to [0,0.79] Hz.This bandwidth is perfectly adaptable for passive rehabilitation.

In the third experiment, the control torque (4), applied on theerror dynamics, is tested for its tracking performance. A sine refer-ence trajectory is applied having frequencies of 0.4 Hz in the timeinterval [0,18.5] s and 0.6 Hz in the time interval [18.5,40] s. Notethat the frequencies belong to the system’s closed loop bandwidth.The angle, angular velocity and control torque are presented inFig. 9. The experiments show a good tracking of the angle andangular velocity for the two considered frequencies, besidesacceptable values of angular velocity and torque, guaranteeingthe safety of the subject. The control reaches greater values forthe frequency of 0.6 Hz because the angular velocity has a higheramplitude. The root mean square error of the angle in the wholetime interval is evaluated to 3 � 10�2 rad. As in the previousexperiment, the RF and BF muscle activities are null (passiverehabilitation).

Fig. 5. Successive positions of the shank during flexion–extension: (a) shows the rest position, (b) presents the shank during extension, (c) shows the full extension position,and (d) the shank during flexion phase.

0 10 20 30 40 50 60−1.5

−1−0.5

0

0 10 20 30 40 50 60−1

0

1

0 10 20 30 40 50 60−10

01020

0 10 20 30 40 50 60−100

0

100

0 10 20 30 40 50 60−100

0

100

Fig. 6. Flexions and extensions: the knee-joint angle, angular velocity, controltorque, RF and BF muscle activities. The current values are plotted in blue, thedesired values in dashed red lines. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

0 5 10 15 20 25 30 35 40 45 50 55−1.5

−1−0.5

0

0 5 10 15 20 25 30 35 40 45 50 55−1

0

1

0 5 10 15 20 25 30 35 40 45 50 55−200

0

200

0 5 10 15 20 25 30 35 40 45 50 55−100

0

100

0 5 10 15 20 25 30 35 40 45 50 55−100

0

100

Fig. 7. Flexions and extensions with removed saturation functions: the knee-jointangle, angular velocity, control torque, RF and BF muscle activities. The currentvalues are plotted in blue, the desired values in dashed red lines. The saturationbounds of the actuator are plotted with dashed green line. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

1146 H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149

4.3. Robustness with respect to parameters identification

The control law does not depend on the system’s inertia, solidand viscous friction coefficients. However, the saturation boundof the scaled orientation error should always satisfy N2 > A to guar-antee the stability, condition that is always valid because the solidfriction coefficient’s value is smaller than the chosen bound N2. Thegravity torque compensation as well as the stiffness parameter Khave a major influence on the control. An error in the estimation

of these parameters does not induce instability but it may createa static error, i.e. the shank does not follow exactly the desired tra-jectory including flexion and extension. In order to show therobustness of the controller with respect to sg and K, a load of2 kg has been fixed to the subject’s leg to change its identifiedparameters. Fig. 10 shows a good performance for the step

10−1 100 101−40

−30

−20

−10

0

10

10−1 100 101−2

−1.5

−1

−0.5

0

0.5

Fig. 8. The bode diagram of the estimated transfer function of the closed loopsystem.

0 5 10 15 20 25 30 35 40−1.5

−1−0.5

0

0 5 10 15 20 25 30 35 40−2

02

0 5 10 15 20 25 30 35 40−10

0102030

0 5 10 15 20 25 30 35 40−100

0

100

0 5 10 15 20 25 30 35 40−100

0

100

Fig. 9. Sinusoidal trajectory: the knee-joint angle, angular velocity and controltorque. The current values are plotted in blue, the desired values in dashed red lines.(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

0 10 20 30 40 50−1.5

−1−0.5

0

0 10 20 30 40 50−1

0

1

0 10 20 30 40 50−10

0102030

0 10 20 30 40 50−100

0

100

0 10 20 30 40 50−100

0

100

Fig. 10. Flexions and extensions in presence of an added load: the knee-joint angle,angular velocity, control torque and muscle activities. The current values areplotted in blue, the desired values in dashed red lines. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

0 5 10 15 20 25−1.5

−1−0.5

0

0 5 10 15 20 25−2

0

2

0 5 10 15 20 250

2040

0 5 10 15 20 25−100

0

100

0 5 10 15 20 25−100

0

100

Fig. 11. Sinusoidal trajectory in presence of an added load: the knee-joint angle,angular velocity, control torque and muscle activities. The current values areplotted in blue, the desired values in dashed red lines. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149 1147

function, the steady state error is equal to 2 � 10�2 rad at exten-sion and 10�3 rad at flexion of the knee joint. Fig. 11 shows alsosatisfactory tracking of the sine desired trajectory. In this case,the root mean square is equal to 4 � 10�2 rad. Note that the errorinduced in the parameters identification by using the load is verylow. Actually, in both experiments, the control torques reach great-er values relative to the nominal cases, in order to balance the dis-turbing effect of the added load and ensure convergence andstability with satisfactory performances.

4.4. Robustness with respect to muscles co-contraction

A co-contraction, i.e. simultaneous contraction of the quadri-ceps extensor muscular group and hamstring flexor muscles, sim-ulates a spasm and consequently a blockade, or a joint locking, atthe knee level. This is, in a some way, equivalent to increasingthe knee stiffness. In this study, the co-contraction effect is mod-eled as a bounded external torque d generated by the muscles ofthe wearer and acting in the opposite direction of the desired

movement for a short time. One main property of the control lawis to maintain the system’s stability during the co-contraction thenmake it regain the intended position whenever the unpredictablemovement of the knee joint disappears.

Proposition 2. The dynamics of the system subject to the boundedexternal torque d, jdj 6 D, is given by:

J€h ¼ �sg cos h� Asign _h� B _h� Kðh� hrÞ þ sþ d: ð12Þ

Applying the control law defined in (4) with k2j~hj > N2 andk1j _hj > N1 � N2 > N2 ensures stability of the system atð _h; ~hÞ ¼ ð0; d

Kþk2Þ and at ð _h; ~hÞ ¼ ð0;0Þ when the disturbance vanishes.

0 5 10 15 20 25 30−1.5

−1−0.5

0

0 5 10 15 20 25 30−2

0

2

0 5 10 15 20 25 30−20

02040

0 5 10 15 20 25 30−200

0

200

0 5 10 15 20 25 30−200

0

200

Fig. 13. Sinusoidal trajectory in presence of muscle co-contraction: the knee-jointangle, angular velocity, control torque, RF and BF muscle activities. The currentvalues are plotted in blue, the desired values in dashed red lines. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

1148 H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149

Proof. A sketch of the stability analysis is stated in the following.Consider the Lyapunov function V defined in (5). Its derivativerespects:

_V ¼ �A _hsign _h� B _h2 � _hsatN1 ½k1_hþ satN2 ðk2

~hÞ� þ _hd;

¼ �Aj _hj � B _h2 � N1j _hj þ _hd;

6 �B _h2 � ðN1 þ A� jdjÞj _hj:

If jdj < N1 + A, one can ensure the decrease of the Lyapunovfunction V and j _hj consequently until the set X1 is reached andthe control law s becomes (7). Consider then the Lyapunovfunction W defined by (8). The derivative of W is given by:

_W ¼ �A _hsign _h� ðBþ k1 � JkÞ _h2 � kK~h2 � k~hsatN2 ðk2~hÞ

þ d _h� Ak~hsign _hþ kd~h;

6 �ðBþ k1 � JkÞ _h2 � kK~h2 � ðA� jdjÞj _hj � kðN2 � A� jdjÞj~hj:

If jdj < A and jdj < N2 � A, one can ensure the decrease of the Lyapu-nov function W and consequently j _hj and j~hj until the set X2 isreached and the control law becomes (9). Consider finally theLyapunov function L defined by (10). Its derivative is:

_L ¼ �Aj _hj � ðBþ k1Þ _h2 � ðk2 þ KÞ~h _hþ d _hþ ðk2 þ KÞ~h _h;

6 �ðBþ k1Þ _h2 � ðA� jdjÞj _hj:

If jdj < A, one can ensure the decrease of the Lyapunov function Land consequently _h converges to the origin _h � 0. Therefore, €h � 0.From (12), ~h ¼ d

Kþk2. As a conclusion, if the shank–orthosis system

is subject to a bounded co-contraction torque of amplitude d, suchthat jdj < A, one can ensure the stability of the system atð _h; ~hÞ ¼ 0; d

Kþk2

� �. When the effect of the disturbance torque van-

ishes: d = 0, the stability at the origin ð _h; ~hÞ ¼ ð0;0Þ is reached. h

The experiments consist of generating a co-contraction of theRF and BF muscles, by the wearer, for a short period of time in or-der to highlight the robustness of the control law. The results arepresented in Figs. 12 and 13 in both cases: flexion/extension andsinusoidal trajectory, respectively. The muscle co-contractionsare detectable on the curves of the muscular activities. One can no-tice that the effect of the disturbance is more detectable for thestep trajectory than the sine one. Notice also that the control

0 10 20 30 40 50 60−1.5

−1−0.5

0

0 10 20 30 40 50 60−1

0

1

0 10 20 30 40 50 60−20

0

20

0 10 20 30 40 50 60−200−100

0100

0 10 20 30 40 50 60−100

0

100

Fig. 12. Flexions and extensions in presence of muscle co-contraction: the knee-joint angle, angular velocity, control torque, RF and BF muscle activities. The currentvalues are plotted in blue, the desired values in dashed red lines. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

torque acts in both cases to reject the disturbance and regain thedesired trajectory. After the disturbance has vanished, the steadystate error in the case of the step desired trajectory is evaluatedto 15 � 10�4 rad for extension and 3 � 10�3 for flexion movements.The root mean square computed within the sine desired trajectoryis equal to 6 � 10�2 rad. The error in both cases remains low whichemphasizes the robustness of the proposed control law.

5. Discussion

Orthoses have created a new hope for disabled people and el-derly because they improve and make easier their daily activities.However, these robots should respect many constraints in orderto make their use secure and comfortable at the same time. Theelectromechanical structure should be simple and safe with anintuitive interface adapted to each wearer. The control of theorthosis should be robust while ensuring the stability of the wear-er-orthosis system. Besides, it should detect and react to all torquesthat may act on the system.

Therefore, in the present work, the EICOSI orthosis, having asimple structure and easy to don and doff, is adopted. A boundedcontrol torque, based on nested saturations, is then proposed. Thiscontrol respects the power limitation and avoids the saturation ofthe actuator. It spares thus all nonlinearities associated to hyster-esis cycles, guarantees the linear movement of the motor and thesecurity of the wearer as a sequel. In the phases where the controllaw is not bounded, it is equivalent to a PD controller which iswidely used for orthoses’ applications thanks to its simplicity asmentioned earlier. When ensuring a movement, the control lawasymptotically stabilizes the system. Besides, it is easy to imple-ment, it is independent of the system’s inertia and friction modeland is robust with respect to modeling and identification errors.Moreover, it is robust with respect to external disturbances, as aknee blockade, allowing to guarantee the stability of the wearerin case of an unpredictable movement.

The EICOSI orthosis has been tested in a passive rehabilitationcontext where the wearer is not supposed to deliver any voluntaryeffort. The cutoff frequency of the shank–foot–orthosis, subject tothe control torque, is equal to 0.79 Hz allowing to ensure continu-ous motions for the passive rehabilitation. The robustness of the

H. Rifaï et al. / Mechatronics 23 (2013) 1141–1149 1149

control torque with respect to identification errors, induced by fix-ing a load of 2 kg on the wearer’s leg, has shown satisfactory per-formances. A co-contraction of agonist/antagonist muscles,quadriceps and hamstrings respectively, engenders a knee jointblockade that is modeled as an external disturbance. If this distur-bance has a great amplitude, it may cause the orthosis movementto differ from the desired one as has been shown during therobustness tests section. During all robustness tests, the controllaw reacts to allow the shank–foot–orthosis system regain the de-sired trajectory.

Future works will consider the assistance context. The intentionand effort of the wearer will be taken into account and integratedin the loop. The choice of the signals to be exploited and treated todetermine the desired movement of the subject represents a majorchallenge. Future works will consider the contribution of the wear-er in the closed control loop by means of the EMG signalmeasurements.

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