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Mechatronics 000 (2015) 1–10

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Mechatronics

journal homepage: www.elsevier.com/locate/mechatronics

Non-singular terminal sliding mode controller: Application to an

actuated exoskeleton

T. Madani a,∗, B. Daachib, K. Djouania,c

aLISSI Lab., University of Paris-Est Creteil, Creteil, France

bLIASD Lab., University of Paris 8, Saint Denis, France & CNRS-AIST JRL, UMI 3218/CRT, Tsukuba, Japan

cF’SATIE/TUT, Pretoria, South Africa

article info

Article history:

Received8July2014

Accepted 26 October 2015

Available online xxx

Keywords:

Nonlinear control

Non-singular terminal sliding mode

Finite-time convergence

Exoskeleton

abstract

This paper presents a robust controller of an active orthosis used for rehabilitation purposes. The system is

composed of the orthosis worn by the shank and has a complex dynamical model. No prior knowledge is con-

sidered on the dynamical model and the ﬂexion/extension movements considered are of sinusoidal form and

are generally defined by the doctor. The used non-singular terminal sliding mode technique permits to have

a finite time convergence. The experimental results have been conducted online on an appropriate dummy

and then on three healthy subjects. A comparison of performances obtained by the proposed approach with

those obtained by a conventional controller has also been realized. Several situations have been considered

to test the robustness and it has been concluded with the effectiveness of the developed controller.

1. Introduction

Currently, exoskeletons/orthoses represent a considerable part of

robotics research. These wearable robots are used in various fields

such as rehabilitation, assistance and can sometimes replace com-

pletely the upper or lower limbs of human. In addition, exoskele-

tons can also be used to improve comfort and help on various daily

tasks (gardening, carry heavy loads, climb stairs, walk longer, etc.).

Some of main problems considered for this type of robot are related

to control, identification of dynamic parameters or behavior and to

observe some physical phenomena that can improve exoskeletons

performance. Now, the challenge is to improve the cognitive abilities

of exoskeletons to enable them to learn, adapt and make decisions

based on their own mistakes in the same way than humans. In liter–

ature, it can be found several exoskeletons/orthoses developed and

used for various applications. The University of Agriculture and Tech-

nology of Tokyo has designed an exoskeleton to help the wearer to

perform agricultural work considered difficult and tough [7].Another

kind of exoskeleton namely “Hercules” has been made to improve the

performance of soldiers.1The University of Berkeley has recently de-

veloped a lower limb exoskeleton called BLEEX that allows the holder

to carry heavy loads [12]. Furthermore, Refs. [4,9] constitute a good

state of the art on the exoskeletons and their applications.

∗Corresponding author. Tel.: +33 141807335; fax: +33 141807320.

E-mail addresses: tarek.madani@u-pec.fr (T. Madani), bd@ai.univ-paris8.fr

(B. Daachi), djouani@u-pec.fr (K. Djouani).

1http://www.army-technology.com/features/featurefrench-hercule-robotic-

exoskeleton.

To allow the exoskeleton to meet the needs of the wearer, it is nec-

essary to apply an appropriate control law. The complexity of the sys-

tem dynamics consisting of the exoskeleton and its wearer associated

with external perturbations make the traditional controllers ineffi-

cient. This complexity has led researchers to propose a variety of ap-

propriate controllers. Some control schemes are based on the prelim-

inary step of identifying the dynamic parameters of the set consisting

of the exoskeleton and its wearer. Other approaches are adaptive and

are dedicated to generic exoskeletons designed to be worn by humans

of different morphologies. An example of such controllers is based on

neural networks [10,25]. The universal approximation of neural net-

works [5,21] is one of their advantages. However, neural approaches

require generally oﬄine learning step in order to avoid undesirable

behavior of the exoskeleton during the initialization step. In [1],the

authors use a dynamical model of the upper limb exoskeleton to am-

plify the human power. In that human-robot cooperation, it is rarely

possible to know the exact dynamical model and consequently the

scope of this approach is reduced. Several works on nonlinear control

of exoskeletons [8,11] can be also found.

Sliding mode controllers having the advantage of robustness

against external disturbances and model uncertainties, have been

widely applied to the robotic systems [19,22,23]. The classic sliding

mode controllers use a linear sliding surface and the convergence of

the system’s states is asymptotic. Terminal Sliding Mode (TSM) tech-

nique based on a non-linear sliding surface is proposed to ensure, in

finite time, the stability of the closed loop system [16,18,20].TheFast

Terminal Sliding Mode (FTSM) surface has been introduced [6,13,26]

to further reduce the finite-settling-time. The Non-singular Terminal

http://dx.doi.org/10.1016/j.mechatronics.2015.10.012

Please cite this article as: T. Madani et al., Non-singular terminal sliding mode controller: Application to an actuated exoskeleton, Mechatronics

(2015), http://dx.doi.org/10.1016/j.mechatronics.2015.10.012

2T. Madani et al. / Mechatronics 000 (2015) 1–10

ARTICLE IN PRESS

JID: MECH [m5G;November 25, 2015;20:1]

Fig. 1. The dummy and human wearing an actuated knee joint orthosis (EICoSI).

Sliding Mode (NTSM) has been proposed [2,14,15] to overcome the

classical singularity problem of the conventional TSM. Furthermore,

a boundary layer augmented sliding mode controller is proposed in

[3] for trajectory tracking of a robotic orthosis. Even if the obtained

results are good, the stability of the system in closed loop is not

ensured.

In this article, it is proposed to construct and experiment a NTSM

controller for a powered knee orthosis. The goal of the proposed con-

troller is to show the effectiveness of a sliding technique that ensures

the convergence toward the sliding surface and avoiding any singu-

larity problem. Surely it can be found other versions of this technique

in the literature but as given in the article, it is unique with a par–

ticular way to compute the finite time convergence. In fact, a new

optimization strategy at the vicinity of the singularity is proposed.

This allows to compute the minimum reaching time to the sliding

surface. Furthermore, this technique requires no prior knowledge on

the dynamic model. Also, the advantage of this method over other

methods is its robustness against parameter uncertainties and exter–

nal disturbances. The system considered in this paper, consisting of

the orthosis and knee of the wearer, has a complex dynamic that is

difficult to mobilize by equations. For safety reasons, it is very impor–

tant to control it by techniques including a very short transient toler–

ated by the standards set by the doctor. This is precisely what offers

the NTSM technique that is used in this work. As this technique is al-

most never used in the field of control of exoskeletons, the presented

work proposes a control taking into account the state of the carrier

of the exoskeleton that can be active or completely passive (no mus-

cular effort developed). So the proposed controller is firstly tested on

an appropriate dummy and is only after eliminating all the risks that

application can be made. Final validation is performed on a healthy

subject.

The paper is organized as follows: in Section 2, the exoskeleton

used for the experiments and formulate the considered problem are

described. Section 3 deal with the analysis and controller proposed

stability. The experimental results and their analysis are given in

Section 4. Finally, Section 5 is dedicated for the conclusion and future

work.

Fig. 2. Electrical architecture of the joint orthosis actuator.

2. Active orthosis system

The EICoSI (Exoskeleton Intelligently Communicating and Sensi-

tive to Intention) is used to carry out the experimental validations

of the proposed controller. EICoSI is an active knee-joint orthosis

developed in LISSI Laboratory (Laboratoire Images, Signaux et Sys-

tèmes Intelligents) of University Paris-Est Créteil (UPEC), France. Two

jointed segments compose the considered actuated orthosis, upper

and lower. The mechanical part and the actuator are placed on the

upper part of the orthosis. The torque generated by the orthosis re-

alizes ﬂexion/extension movements of the lower part, formed by the

dummy shank and the lower part of the orthosis. Moreover, the knee

joint is constrained by a range of motion between 0 and 2π

3for safety

reason. In Fig. 1, a dummy and a human wearing the used actuated

knee joint orthosis are presented.

2.1. Electrical part

The joint of the orthosis is actuated by a brushless DC motor

(BLDC). A power supply and an adequate electrical system are used

to provide the regulation for the current in the motor. A mechanical

Please cite this article as: T. Madani et al., Non-singular terminal sliding mode controller: Application to an actuated exoskeleton, Mechatronics

(2015), http://dx.doi.org/10.1016/j.mechatronics.2015.10.012

T. Madani et al. / Mechatronics 000 (2015) 1–10 3

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Fig. 3. Position of the joint orthosis.

transmission is used to increase the orthosis applied torque. Fig. 2

shows the schematic diagram of the used electromechanical system.

Assumption 1. The time constant of the electrical system is negligi-

ble compared to the mechanical time constant.

According to regulation system characteristics of the BLDC motor

and the Assumption 1, the following equation can be written:

τa=cmi(1)

where iis the electrical current of the BLDC motor, τais the applied

torque and cmis a positive constant.

2.2. Mechanical part

The mechanical structure scheme of the considered active ortho-

2.3. Dynamic model

The following dynamic model of the whole system is obtained us-

ing (1) and (3):

¨

θ=f(θ,˙

θ,t)+ϕu(t)(4)

with u=iis the electrical current control input, ϕ=cm

Jand the func-

tions f(θ,˙

θ,t)∈Ris given by:

f(θ,˙

θ,t)=1

J[mgl cos (θ) −kv˙

θ−kssgn(˙

θ)+τk(t)+τd(t)](5)

According to Assumptions 3 and 4, it is obvious to write:

1

λ≤ϕ

|f(θ,˙

θ,t)|≤a0+a1|cos (θ)|+a2|˙

θ|P(θ,˙

θ) (6)

where a0,a1,a2and λare adequate positive constants.

One can synthesize the control law forcing the states {θ,˙

θ}of a

orthosis to follow the desired trajectory by using the non-singular

terminal sliding mode technique.

3. Sliding mode controller design

Our objective is to design a sliding mode control law for the actu-

ated orthosis. The controller generates the control signal u(t)ensur–

ing that the real position θ(t) tracks the desired one θd(t) in finite-

time. This can be done in two main steps:

•Select the NTSM switching manifold so that the system in slid-

ing mode guarantees the convergence to the equilibrium point in

finite-time (settling time).

•Determine the control law that guarantees the reachability of the

sliding manifold and the appearance of the sliding mode in finite-

time (reaching time).

The following lemmas are used to prove the finite time stability

of the proposed controller [24]. Their proofs are given respectively in

Sections A.1 and A.2 of Appendix A.

Lemma 1. Let V(t)be a C1(continuously differentiable) scalar positive-