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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018 575
Adaptive Tracking Control of an Exoskeleton
Robot With Uncertain Dynamics Based
on Estimated Time-Delay Control
Brahim Brahmi , Maarouf Saad , Senior Member, IEEE, Cristobal Ochoa-Luna ,
Mohammad Habibur Rahman, Member, IEEE, and Abdelkrim Brahmi
Abstract—In this paper, we present a backstepping ap-
proach integrated with time-delay estimation to provide an
accurate estimation of unknown dynamics and to compen-
sate for external bounded disturbances. The control was
implemented to perform passive rehabilitation movements
with a 7-DOF exoskeleton robot named ETS-Motion Assis-
tive Robotic-Exoskeleton for Superior Extremity. The un-
known dynamics and external bounded disturbances can
affect the robotic system in the form of input saturation,
time-delay errors, friction forces, backlash, and different
upper-limb’s mass of each subject. The output of the time-
delay estimator is coupled directly to the control input of the
proposed adaptive tracking control through a feed-forward
loop. In this case, the control system ensures a highly ac-
curate tracking of the desired trajectory, while being robust
to the uncertainties and unforeseen external forces, and
flexible with variation of parameters. Due to the proposed
strategy, the designed control approach does not require
accurate knowledge of the dynamic parameters of the ex-
oskeleton robot to achieve the desired performance. The
stability of the exoskeleton robot and the convergence of its
state errors are established and proved based on Lyapunov–
Krasovskii functional theory. Experimental results and a
comparative study are presented to validate the advantages
of the proposed strategy.
Index Terms—Backstepping control, rehabilitation
robots, time-delay control, time-delay error (TDR).
I. INTRODUCTION
NEUROLOGICAL diseases have become a growing chal-
lenge and a difficult reality for the medical and scientific
community, as confirmed by the statistics of the World Health
Organization. Each year, at least 15 million people worldwide
suffer neurological diseases, such as stroke [1]. From them, six
Manuscript received October 4, 2017; accepted January 19, 2018.
Date of publication February 20, 2018; date of current version April 16,
2018. Recommended by Technical Editor Y.-J. Pan. (Corresponding
author: Brahim Brahmi.)
B. Brahmi, M. Saad, C. Ochoa-Luna, and A. Brahmi are with
the Department of Electrical Engineering, ´
Ecole de technologie
sup´
erieure, Montreal, QC H3C 1K3, Canada (e-mail: brahim.brahmi.
1@ens.etsmtl.ca; Maarouf.saad@etsmtl.ca; cristobal.ochoa-luna.1@
ens.etsmtl.ca; cc-Abdelkrim.Brahmi@etsmtl.ca).
M. H. Rahman is with the Department of Mechanical Engineer-
ing, University of Wisconsin-Milwaukee, Milwaukee, WI 53211 USA
(e-mail: rahmanmh@uwm.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMECH.2018.2808235
million die and five million more are living with a persistent
disability [1]. At all times, human activities of everyday life,
such as balance of legs and arms to move, walk, stand, sit or
even relax, take objects, eat, brushing teeth, or cleaning face,
depend on a healthy nervous system. Damage caused by nerve
cells causes a disruption of communication between the nervous
system and the rest of the body, resulting in the inability of the
nervous system to function adequately. This condition causes
symptoms that can include numbness, pain, balance disorders,
etc. Recently, the use of robotic devices known as rehabilitation
robots in physiotherapy assistive domains has demonstrated a
high potential in preventing the worsening of the subject’s dis-
ability, improving its functional movements, ensuring its return
to normal life, and helping the subject in daily living activities,
such as self-care skills and pick-and-place exercises [2]–[5]. The
robot’s significance is due to its attractive characteristics such
as its ability to provide intensive rehabilitation and its easiness
to design a physical therapy activity fitted to the needs of the
subject [6].
The control of these kinds of robots presents additional com-
plexity over the control of conventional robotic manipulators
due to their complex mechanical structure designed for human
use, the type of desired tasks, and the sensibility of the in-
teraction with a great diversity of human wearers [7]–[9]. To
address these problems, different control strategies have been
developed for rehabilitation robots. Among them, a simple PID
control implemented in [10]; a nonlinear modified computed
torque control that requires a good knowledge of dynamic pa-
rameters, implemented in [11] and [12]. A robust sliding-mode
control with exponential reaching law was proposed in [13] to
improve the performance of the robot and to limit the chattering
problem generated by the high-frequency activity of the control
signal. As well, a force controller was proposed in [14] for an
exoskeleton that permits to the wearer to achieve motor tasks
based on muscle activity data. We can also mention a back-
stepping control combined with human inverse kinematics [15]
to provide a humanlike motion. Nevertheless, in the previous
cited papers, the control scheme is the named model-based con-
troller, in which the control loop requires a dynamic model of
the exoskeleton.
In reality, the dynamics of these types of robots is typi-
cally uncertain due to their complex and sensitive structure.
1083-4435 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
576 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018
Particularly, when the number of degrees of freedom (DOFs) of
the robot increases, it is not straightforward to find the accurate
parameters of the exoskeleton robot; e.g., the parameters’ vector
of the robot can be greater than 100 if the number of DOFs of the
robot is greater than 4 [16]. Usually, these robots operate under
input saturation constraints to ensure the safety to the robot’s
user. This latter is one of the most serious nonlinearities of the
robotic system [17]. The effect of the constraint appears when
the exoskeleton actuators are unable to provide the required en-
ergy to perform a rehabilitation motion, which causes tracking
errors. Additionally, a repeating motion can cause the fatigue of
the motors [18], [19], which provokes many problems, such as
a dead zone [20], friction forces, etc. Moreover, the synergy be-
tween the subject and the exoskeleton robot forces the system to
operate under unforeseen external forces as the payload caused
by the attached upper limb of the subject.
Hence, the estimation of the uncertainties of the nonlinear
dynamic parameters of these robots is one of the most challeng-
ing problems in the control of high-DOFs robotics exoskeleton
systems. Numerous control strategies have been developed to
approximate the dynamic parameters. Within these approaches,
the linear parameterization of the dynamic equation of motion
is used in order to obtain the regressor matrix [21], [22]. This
matrix is required in the design of the updated control law.
However, it is not straightforward to find the parameters of the
exoskeleton robot, if it has a high number of DOFs (more than
four). Additionally, the integral action of the designed updated
law can cause the instability of the robot system in the presence
of disturbances, even if it is small. Recently, the approach by
fuzzy logic and neural networks presented a significant solution
due to their attractive characteristics of the robot’s unknown
nonlinear dynamics estimation with minimum feedback from
the robotic system [17], [23]–[25]. However, these strategies
demand a heavy computation cost, which makes their imple-
mentation very hard. Adaptive learning control is a robust ap-
proach designed to adapt the time-varying uncertainties and dis-
turbances in order to reject them from the rigid body robot [16],
[26]. In this approach, the authors have proposed that the robot
performs the same exercises over a fixed period, making this
approach limited to repetitive tasks only and actuators fatigue.
However, in this paper, a time-delay estimation (TDE) approach
[27]–[30] is used to estimate the unknown uncertain parameters
of the exoskeleton’s dynamic. The TDE has been implemented
in many robotic systems with consistently good performances
[29], [31]–[33]. The choice of use of TDE is that one can easily
estimate the unknown dynamics with a simple control scheme.
In addition, TDE is one of the strategies that is not affected
by the size of the estimated parameters (high DOFs). It can
only be employed in time-delayed knowledge from the previ-
ous state response of the system and its control input to provide
an accurate estimation of unknown dynamics. However, due
to noisy measurements and nonlinearity of signals along the
sampling time, a time-delay error (TDR) exists, which would
deteriorate the robustness and the accuracy of the exoskeleton
robot.
The cited constraints (mainly dynamic uncertainties, joint
conjunction, friction forces, backlash, mass changing, and
TDR), that limit the functionality of the exoskeleton robots,
motivate us to design a controller able to provide a highly ac-
curate tracking of the desired trajectory, to be robust to the
uncertainties and unforeseen external forces, and to be flexible
with the parameters’ variation. In this paper, we present a new
adaptive backstepping controller based on TDE, applied on an
upper-limb exoskeleton robot. The backstepping is employed to
estimate the unknown uncertainties, unforeseen disturbances,
and compensate for the TDR and unexpected disturbances. Fur-
thermore, the theoretical development is complemented by its
implementation on an exoskeleton robot. The output of the
time-delay estimator is added directly to the control input of
the proposed adaptive tracking control via feed-forward loops,
which makes the control system more powerful and faster to
estimate and compensate for unknown dynamics and external
disturbances.
Summarizing, through the proposed strategy, the control ap-
proach does not require any accurate knowledge of the dynamic
parameters of the exoskeleton robot to reach the desired per-
formance. This controller is designed to be robust and more
flexible to deal with the dynamic uncertainties taking into con-
sideration the TDR, and to be more robust to the parameter
variations. Moreover, the proposed strategy is not restricted by
the repetitive task or periodic desired trajectory. The stabil-
ity of the exoskeleton system and the convergence of its er-
rors are formulated and demonstrated based on the Lyapunov–
Krasovskii functional theory. The validation of the control plat-
form is done by creating a rehabilitation session performed
with healthy human subjects. A comparative study is provided
and is compared against the conventional approach to show
the advantages, feasibility, and the robustness of the proposed
approach.
The remainder of the paper is organized as follows. The
exoskeleton robot, its kinematics and dynamics are presented
in the following section. The control scheme is described in
Section III. Experimental and comparison results are shown in
Section IV. Finally, the conclusion and future work are presented
in Section V.
II. CHARACTERIZATION OF SYSTEM REHABILITATION
A. Exoskeleton Robot Development
The developed exoskeleton robot ETS-Motion Assistive
Robotic-Exoskeleton for Superior Extremity (ETS-MARSE) is
a redundant robot consisting of 7 DOFs, as shown in Fig. 1.It
was created to provide physical therapy and assisted motion to
the injured upper limb. The idea of the designed exoskeleton is
basically extracted from the anatomy of the upper limb of the
human, to be ergonomic for the user along with the physical
therapy session. The shoulder part consists of three joints; the
elbow part comprises one joint, and the wrist part consists of
three joints. Each part is responsible for performing a variety
of upper-limb motions, as shown in Table II. The design of the
ETS-MARSE has special features compared with the existing
exoskeleton robots. Among them, it has a comparatively low
weight, an excellent power/weight ratio, can be easily fitted or
removed, and is capable of adequately compensating for gravity.
BRAHMI et al.: ADAPTIVE TRACKING CONTROL OF AN EXOSKELETON ROBOT WITH UNCERTAIN DYNAMICS 577
Fig. 1. Reference frames of ETS-MARSE.
A new power carrying mechanism was included for support-
ing the shoulder joint internal/external rotation and for forearm
pronation/supination. This robot can be used with a wide range
of subjects, due to the length of its adjustable links. This ex-
oskeleton can perform passive (completely support and perform
the motion on the subjects’ upper limb) and active assistive mo-
tion (respond to force, electromyography, and /or be compliant
with the subject to accompany and assist him/her in the in-
tended motion). All special characteristics of the ETS-MARSE
and comparison with similar existing exoskeleton robots are
summarized in [34]–[37].
B. Kinematics of ETS-MARSE Robot
The transformation from Cartesian space to joint space is
done by a nonlinear function named the Jacobian matrix. In
order to maneuver the exoskeleton in Cartesian space, we used
the inverse Jacobian matrix method, since the control is executed
in the joint space. Due to the redundant nature of the ETS-
MARSE robot, the inverse kinematics can be achieved using
the pseudoinverse of the Jacobian, which can be expressed as
follows:
˙
θ=JTJJT−1˙xd(1)
where ˙xdis the desired Cartesian velocity, ˙
θis the calculated
joint velocity, and Jis the Jacobian matrix of the robot.
The modified Denavit–Hartenberg (DH) parameters [38] are
given in Table I. These parameters are obtained from the refer-
ence frames shown in Fig. 1 and are used to obtain the homo-
geneous transformation matrices.
The workspace of the designed robot is given in Table II.
Further detailed information of the parameters and design of
ETS- MARSE can be found in [11].
TABLE I
MODIFIED DH PARAMETERS
Joint (i)αi−1ai−1diθi
100dsθ1
2−π/2 0 0 θ2
3π/2 0 deθ3
4−π/2 0 0 θ4
5π/2 0 dwθ5
6−π/2 0 0 θ6−π/2
7−π/2 0 0 θ7
TABLE II
WORKSPACE ETS-MARSE
Joints Motion Workspace
1 Shoulder joint horizontal flexion/extension 0°/140°
2 Shoulder joint vertical flexion/extension 140°/0°
3 Shoulder joint internal/external rotation −85°/75°
4 Elbow joint flexion/extension 120°/0°
5 Forearm joint pronation/supination −85°/85°
6 Wrist joint ulnar/radial deviation −30°/20°
7 Wrist joint flexion/extension −50°/60°
C. Dynamics of the ETS-MARSE Robot
The dynamic equation of the ETS-MARSE is expressed in
joint space as follow:
M(θ)¨
θ+Cθ, ˙
θ˙
θ+G(θ)+Fθ, ˙
θ+fdis =τ(2)
where θ∈R7,˙
θ∈R7, and ¨
θ∈R7are, respectively, the joints
position, velocity, and acceleration vectors, M(θ)∈R7×7,
C(θ, ˙
θ)˙
θ∈R7, and G(θ)∈R7are, respectively, the symmetric
positive definite inertia matrix, the Coriolis and centrifugal vec-
tor, and the gravitational vector considering both the user’s arm
and the exoskeleton arm. τ∈R7is the torque vector, fdis ∈R7
is the external disturbances vector, and F(θ, ˙
θ)∈R7is the fric-
tion vector. Without loss of generality, the matrices of dynamic
model (2) can be written as follows:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
M(θ)=M0(θ)+ΔM(θ)
Cθ, ˙
θ=C0θ, ˙
θ+ΔCθ, ˙
θ
G(θ)=G0(θ)+ΔG(θ)
(3)
where M0(θ),C
0(θ, ˙
θ), and G0(θ)are, respectively, the known
parts of the inertia matrix, the Coriolis/centrifugal matrix, and
the gravity vector. ΔM(θ),ΔC(θ, ˙
θ), and ΔG(θ)are the un-
certain parts.
Let us introduce a new variable such that η1=θand η2=˙
θ;
hence, the dynamic model expressed in (2) can be rewritten as
follows:
˙η1=η2
˙η2=U(t)−f(t)−H(t)(4)
where, U(t)=U(η1),H(t)=H(η1,η
2,˙η2), and f(t)=
f(η1,η
2). This notation is employed to facilitate handling the
control methodology with
578 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018
1) U(t) = M−1
0(θ)τ(t);
2) H(t) = −M−1
0(θ)(ΔM(θ)¨
θ+ΔC(θ, ˙
θ)˙
θ+ΔG(θ)+
F(θ, ˙
θ)+fdis); and
3) f(t) = −M−1
0(θ)(C0(θ, ˙
θ)˙
θ+G0(θ)).
D. Problem Formulation
From (2) and (4), it is difficult to obtain H(t)due to the
uncertainties of the dynamic model of the exoskeleton robot and
the aforementioned unknown external effects. Consequently, to
solve this dilemma, the proposition of this paper is to obtain a
control input able to force the measured trajectory ηd∈R7to
track the desired trajectory even if the exoskeleton robot is under
the effect of uncertain and unforeseen external disturbances. In
this paper, the aforementioned desired trajectories correspond
to those of passive rehabilitation protocol investigated under
adaptive control. Before stating the control methodology, the
properties and the assumptions that are used in this paper are
given as follows.
Property 1: (see [39]): The known part of inertia matrix
M0(θ)is symmetric and positive definite for all θ∈Rnand
satisfying γmin(M0(θ))I7×7≤M0(θ)≤γmax(M0(θ))I7×7,
where γmin and γmax are minimum and maximum eigenval-
ues, respectively, of the known inertia matrix M0(θ)and I7×7is
the 7 ×7 identity matrix.
Assumption 1: The joint position and joint velocity are mea-
sured.
Assumption 2: The function H(t)is bounded and is a glob-
ally Lipschitz function.
Assumption 3: The desired trajectory is bounded.
Assumption 4: The external disturbance fdis is supposed to
be continuous, has finite energy, and satisfies fdis ≤ϑ, with an
unknown positive disturbance boundary ϑ.
Assumption 5: The variation of the unknown parameters
model H(t)in time is continuous with known delay td. It can
be expressed as
H(t)=H(t−td)+ε(td)(5)
where ε(td)∈R7is the TDR vector and tdis a very-small
time-delay constant.
III. ADAPTIVE CONTROL DESIGN
In this section, the design of a tracking control that can es-
timate the uncertainties and unexpected disturbances and de-
crease its effects to achieve the desired tracking performance is
described. Let us assume first that H(t)is known. Let us define
the position error and velocity error as follows:
e1=η1−ηd(6)
e2=η2−ξ(7)
where ηd∈R7and η1∈R7are the desired trajectory and mea-
sured trajectory, respectively. ξis a virtual control input to e1.
Step 1: The time derivative of (6) is given by
˙e1=η2−˙ηd=e2+ξ−˙ηd.(8)
Consider the first Lyapunov function candidate as follow:
V1=1
2eT
1e1.(9)
Taking time derivative of V1as follows:
˙
V1=eT
1˙e1=eT
1(e2+ξ−˙ηd).(10)
Let ξ(t)= ˙ηd−k1e1, with k1∈R7×7being a diagonal
positive-definite matrix. Equation (10) becomes
˙
V1=−eT
1k1e1+eT
1e2.(11)
The first term of (11) is stabilizing and the second term will
be addressed in the next step.
Step 2: Differentiating (7), using (4), with respect to time yields
˙e2=U(t)−f(t)−H(t)−˙
ξ(12)
where ˙
ξ(t)=¨ηd−k1˙e1, Therefore, the proposed model-based
control can be given as follows:
U(t)=−k2e2−e1+f(t)+H(t)+ ˙
ξ
with τ=M0(θ)U(t).(13)
Consider the second Lyapunov function candidate as
V2=V1+1
2eT
2e2.(14)
The time derivative of V2is given by
˙
V2=˙
V1+eT
2˙e2
=˙
V1+eT
2U(t)−f(t)−H(t)−˙
ξ.(15)
Considering the model-based control law (13), the above
equation can be written as follows
˙
V2=−eT
1k1e1−eT
2k2e2.(16)
Having the following condition: k1>0 and k2>0, this im-
plies that ˙
V2≤0, which means that the robot system is stable.
Practically, as established, all dynamics parameters of the
exoskeleton robot are not easy obtained due to the uncertainties
and their variation during the robot’s performance. Since H(t)
is uncertain, it might influence the control proposition. For now
on, we will consider H(t)uncertain. In such case, the model-
based control law (13) is not feasible. To overcome this problem,
a new adaptive time-delay controller is proposed as
U(t)=−k2e2−e1+f(t)+ ˆ
H(t)+ ˙
ξ
with τ=M0(θ)U(t)(17)
where ˆ
H(t)is the estimated value of H(t)obtained by the TDE
approach [27]. However, due to noisy measurements and nonlin-
earity of signals along the sampling time, a TDR, ε(td), exists.
This TDR would deteriorate the robustness and the accuracy of
the robot. To overcome this problem, we proposed the following
estimator:
ˆ
H(t)=ˆ
H(t−td)+ε(td)−k3e2
ˆ
H(t)=0∀t∈[−t, 0];k3=kI7×7(18)
BRAHMI et al.: ADAPTIVE TRACKING CONTROL OF AN EXOSKELETON ROBOT WITH UNCERTAIN DYNAMICS 579
where kis a positive scalar constant and I7×7is the 7 ×7 iden-
tity matrix. The proof of (18) is given in Appendix A. So, if
Assumptions 2 and 5 are verified, ˆ
H(t−td)can be calculated
using (4) such that [27]
ˆ
H(t−td)=U(t−td)−f(t−td)−˙η2(t−td)(19)
where tdis a very small time-delay constant. Practically,
the smallest constant that can be achieved in real time is
the sampling period. According to the Lipschitz condition
(Assumptions 2 and 5), ε(td)can be calculated as follows:
ε(td)=H(t)−H(t−td)≤td(20)
where >0 is the Lipschitz constant. To facilitate the proof of
stability, we can write d
dt t
t−td
˜
HT(w)˜
H(w)dw as follows:
d
dt t
t−td
˜
HT(w)˜
H(w)dw =˜
HT(t)˜
H(t)
−˜
HT(t−td)˜
H(t−td)(21)
where ˜
H(w)is the estimation error of the uncertainties that will
be defined latter. Additionally
1
2k˜
HT(t)˜
H(t)−1
2k˜
HT(t−td)˜
H(t−td)= ˜
HT(t)e2
−e2TkT
3
2e2(22)
where kT
3=k3. The details of (21) and (22) are given in
Appendix A.
Theorem 1: Consider the exoskeleton robot system (4) that
satisfies Assumptions 2–5, with the proposed adaptive TDE
(17). If the previous conditions are verified with a bounded
initial condition, the robot system (4) is stable and its errors e1
and e2, and ˜
Hare bounded.
Proof: Consider the following Lyapunov function candidate:
V3=1
2eT
1e1+1
2eT
2e2+1
2kt
t−td
˜
HT(w)˜
H(w)dw (23)
with ˜
H(t)=H(t)−ˆ
H(t). The derivative of the proposed
Lyapunov function with respect to time, using (21), is
obtained as
˙
V3=−eT
1k1e1+eT
1e2+eT
2U(t)−f(t)−H(t)−˙
ξ
+1
2k˜
HT(t)˜
H(t)−1
2k˜
HT(t−td)˜
H(t−td).(24)
Substituting the adaptive time-delay contol input (17) into
(24) and using (22), we find
˙
V3=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t)+ ˜
HT(t)e2−e2TkT
3
2e2.
(25)
Finally, we obtain
˙
V3=−eT
1k1e1−eT
2k2+kT
3
2e2.(26)
Since k1>0, k2>0,and k3=kT
3>0, this implies that
˙
V3≤0, which means that the robot system is stable.
Remark 1: We observe that ˙
V3is seminegative in the inter-
val [t−td,t], which means that V3is stable, but outside this
interval, the stability of V3cannot be ensured. To guarantee
the stability of V3in the interval [0,∞), we use the functional
Lyapunov–Krasovskii theorem [40].
To guarantee the asymptotic stability of the delayed system
in the interval[0,∞), we propose the following Lyapunov–
Krasovskii function: Eq. (27) as shown at the bottom of this
page, with
V2=1
2eT
1e1+1
2eT
2e2.(28)
The derivative of the proposed Lyapunov–Krasovskii func-
tion with respect to time is obtained as follows:
˙
V4≤−eT
1k1e1−eT
2k2e2−1
2
n
i=1
k3ie2
2i.(29)
It is clear from (29) that ˙
V4≤0, where all gains k1,k
2,
and k3iare positive. The proof of the stability is detailed in
Appendix B.
The structure of the control scheme is shown in Fig. 2.
IV. EXPERIMENTAL AND COMPARATIVE STUDY
A. Experiment Setup
Implementation was carried out on the ETS-MARSE sys-
tem described in the following. The system consists of three
V4=V2+
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
n
i=1
1
2k3it
0˜
H2
i(w)dw, t ∈[0,t
d1);
1
2k31 t
t−td1
˜
H2
i(w)dw +n
i=2
1
2k3it
0˜
H2
i(w)dw;t∈[td1,t
d2);
.
.
.
n−1
i=1
1
2k3it
t−tdi
˜
H2
i(w)dw +1
2k3nt
0˜
H2
n(w)dw;t∈[tdn−1,t
dn);
n
i=1
1
2k3it
t−tdi
˜
H2
i(w)dw, t ∈[tdn,∞);
(27)
580 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018
Fig. 2. General schematic of proposed control.
TABLE III
PHYSICAL PARAMETERS OF ETS-MARSE
Joints Mass (kg) Centre of mass (m) Link length (m)
1 3.475 0.0984 0.145
2 3.737 0.1959 0
30 0 0.25
4 2.066 0.163 0
5 0 0 0.267
6 0.779 0.121 0
7 0.496 0.0622 0
processing units; the first is a PC, where the top-level commands
are sent to the robot using LabVIEW interface, i.e., the control
scheme selection, joint or Cartesian space trajectory, gain ad-
justments, etc. This PC also receives the data after the robot task
is executed to analyze its performance. The other two processing
units are part of a National Instruments PXI platform. First is
an NI-PXI 8081 controller card with an Intel Core Duo proces-
sor; in this card, the main operating system of the robot and the
top-level control scheme are executed, in our case, the integral
backstepping based controller as well as the estimation based
on TDE approach, at a sampling time of 500 μs. The inverse
kinematics algorithm also runs inside this control loop. Finally,
at input–output level, an NI PXI-7813R remote input–output
card with a field programmable gate array (FPGA) executes
the low-level control; i.e., a PI current control loop (sampling
time of 50 μs) to maintain the current of the motors required by
the main controller. Also, in this FPGA, the position feedback
via Hall-sensors (joint position), and basic input–output tasks
are executed. Force sensor feedback is important to accurately
control the movement of the exoskeleton. A high linearity six-
axis force sensor [NANO17-R-1.8-M2-M1PCI, ATI industrial
automation]was chosen to obtain accurate real-time force mea-
surements for the ETS-MARSE. This sensor is mounted on the
tip of the robot. The joints of the ETS-MARSE are powered
by brushless dc motors (Maxon EC-45, EC-90) combined with
harmonic drives [gear ratio 120:1 for motor-1, motor-2, and
motor-4, and gear ratio 100:1 for motor-3 and motors (5–7)].
The physical parameters of ETS-MARSE relative to the base
reference frame are given in Table III. The diagram of the ar-
chitecture of the ETS-MARSE with a healthy subject is shown
in Fig. 3.
An experimental session was created to validate the proposed
control strategy. The physical therapy tasks are performed by
Fig. 3. General schematic of experiment architecture (the subject wear
the ETS-MARSE robot).
TABLE IV
CONTROLLER PARAMETERS
Gains Values
k1i15
k2i150
k3i0.12
two different healthy subjects (mean age: 27 ±4.6 years; mean
height: 170 ±8.75 cm; mean weight: 75 ±18 kg). Each subject
participated in a full session, under the supervision of a therapist
and a control engineer. The role of the therapist is to define the
range of motion of each subject and to determine a suitable ex-
ercise. During the session, the subject was ergonomically seated
in a chair (height of the chair is adjustable according to the size
of each subject) as shown in Fig. 3. The experimental session
was conceived in two scenarios. In the first scenario, each sub-
ject performed a basic joint physical therapy task consisting of
elbow joint flexion/extension and forearm supination/pronation
simultaneously, using the designed control. In this scenario, the
subjects repeat the same task with a conventional approach [41]
to show the feasibility of the proposed controller. In the second
scenario, the proposed controller was tested using a Cartesian
task described in the following section. The objective of this
task is to show the tracking of the 7 DOFs of the robot in a
Cartesian task. In this part, the subject also repeated the same
Cartesian trajectory using the conventional controller. It is im-
portant to mention that the initial position of the robot is with
the elbow joint position at 90°for all experiments, and the exter-
nal disturbances here are concordantly represented by different
physiological conditions of the subjects, the varying mass of the
upper limb with each subject and the TDR. The control gains
were chosen arbitrarily as shown in Table IV.
Remark 2: The conventional controller [41] is characterized
by its complex implementation due to the complex regressor
dynamic matrix, while the robot has a high degree-of-freedom
(7-DOFs).
B. Joint Space Tests
The experimental results of the proposed controller are il-
lustrated in Fig. 4. This exercise was performed with subject-1
(age: 30 years; height: 177 cm; weight: 75 kg). In this case, the
speed of motion is constant (48°/s) for the two joints. We can
easily see in Fig. 4 that for the movement of all joints, the desired
BRAHMI et al.: ADAPTIVE TRACKING CONTROL OF AN EXOSKELETON ROBOT WITH UNCERTAIN DYNAMICS 581
Fig. 4. Elbow and forearm motion, trajectory tracking in joint space
using the proposed controller. Experiment was conducted with subject-1
(age: 30 years; height: 177 cm; weight: 75 kg).
Fig. 5. Elbow and forearm motion, trajectory tracking in joint space us-
ing the conventional approach. Experiment was conducted with subject-1
(age: 30 years; height: 177 cm; weight: 75 kg).
trajectory, represented by the red line, practically overlaps the
measured trajectory, represented by the solid blue line. It is clear
from the plots in Fig. 4 that the proposed controller provides an
excellent performance, where the controller has the potential to
maintain stability of the system along the designed trajectory
with a position error (second column of Fig. 4) less than 1.5°for
elbow joint and less than 2.0°for forearm joint. The last column
of Fig. 4 shows the control input, which is clearly smooth and
without any chattering effect.
The test result of the conventional controller is given in Fig. 5.
The same subject (subject-1) repeated the task. The conventional
controller has good tracking performance, where the controller
preserves the stability of the robotic system.
However, the proposed controller exhibits a better perfor-
mance than the conventional controller does. That appears at
the level of tracking errors where the proposed strategy pro-
vides tracking errors smaller than the tracking errors given by
the conventional approach, especially for the forearm joint.
Fig. 6. (a) Reaching movement exercise, Cartesian trajectory track-
ing in three-dimensional (3-D) space using the proposed controller.
(b) Cartesian trajectory tracking error along X-axis, Y-axis, and Z-
axis. Experiments was conducted with subject-2 (age: 28 years; height:
173 cm; weight: 72 kg).
Fig. 7. Tracking performance of ETS-MARSE in joint space corre-
sponds to Cartesian tasks using the proposed controller.
C. Cartesian Space Tests
The experimental results with the ETS-MARSE robot in the
Cartesian space performed by subject-2 (age: 28 years; height:
173 cm; weight: 72 kg) using the designed controller are shown
in Figs. 6–8; as shown in Figs. 6(a) and 7, the desired trajectory
(red line) nearly overlapped with the measured trajectory (green
line). It can be noted that these results are very satisfactory.
Fig. 6(b) presents the Cartesian errors as functions of time.
582 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018
Fig. 8. Control input the proposed controller.
Fig. 9. (a) Reaching movement exercise, Cartesian trajectory tracking
in 3-D space using the conventional controller. (b) Cartesian trajectory
tracking error along X-axis, Y-axis, and Z-axis. Experiments was con-
ducted with subject-2 (age: 28 years; height: 173 cm; weight: 72 kg).
From this figure, it is obvious that the Cartesian errors are getting
smaller along the desired trajectory. Fig. 8 shows that the control
inputs are bounded without any noticeable control chattering.
So, these results confirm that the control strategy is able to
achieve the desired robot’s performance even if the nonlinear
dynamics of the exoskeleton robot is uncertain.
Fig. 9 presents the workspace performance of the robot in the
Cartesian space (red is the desired trajectory, green is the real
trajectory) performed by the same subject-2 using the conven-
tional controller.
Fig. 10. Tracking performance of ETS-MARSE in joint space corre-
sponds to Cartesian tasks using the conventional controller.
Fig. 11. Control input of the conventional controller.
In fact, we note from Figs. 9 and 10 that the proposed con-
troller shows a good performance where the Cartesian error [see
Fig. 9(b)]becomes smaller with time. However, the proposed
strategy presents an excellent performance compared with the
performance presented by the conventional approach where the
tacking errors of the proposed approach seen in Fig. 6(b) are
smaller than the tracking errors provided by the classical ap-
proach [see Fig. 9(b)].
Additionally, the control input (see Fig. 11) presented by
the conventional approach is noisier than the control input (see
Fig. 8) presented by the proposed approach. These noises may
damage the motors of the robot. From the comparison of the
two experimental scenarios, we can conclude that the proposed
strategy provides a high level of precision and robustness against
the nonlinear dynamic uncertainties and unknown disturbances.
V. CONCLUSION
In this paper, we have proposed an adaptive control of ex-
oskeleton robots with an unknown parameters model based
BRAHMI et al.: ADAPTIVE TRACKING CONTROL OF AN EXOSKELETON ROBOT WITH UNCERTAIN DYNAMICS 583
on a backstepping controller. A new time-delay approach is
proposed to estimate the uncertain part of the exoskeleton robot
and bounded external disturbances where the TDR, friction
forces, backlash, and different upper-limb’s mass of each sub-
ject are taken into consideration to improve the robot perfor-
mance. The main advantage of the proposed adaptive control
law is that accurate estimation of robots’ dynamic model is not
needed. The output of the time-delay estimator is added directly
to the control input through a feed-forward loop, whereby the
control scheme provides a highly accurate tracking of the de-
sired performance, robust to the uncertainties and unexpected
bounded external forces, and flexible with variations in param-
eters. The stability analysis of the proposed control technique
with a Lyapunov–Krasovskii function was presented. The ro-
bustness of the proposed controller was realized by maneuvering
the ETS-MARSE to provide both joint-based and end-effector-
based rehabilitation exercise to the different subjects. The ex-
perimental results demonstrate the excellent performance of the
proposed controller compared with the conventional controller.
In future work, we seek to overcome the limitations of this ap-
proach, in particular, the value of delayed acceleration for the
controller, where the estimation of this variable may deteriorate
the accuracy of the controller.
APPENDIX
A. Proof of (18), (21), and (22)
In such case, considering
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Hi(t)∈H(t)
ˆ
Hi(t)∈ˆ
H(t)
˜
Hi(t)∈˜
H(t)
g(t)∈R
(30)
where i∈{1,...,7},g(t)∈R7; it will be determined later.
Now, let us propose the following equation:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Hi(t)=Hi(t−td)+εi(td)
˜
Hi(t)=Hi(t)−ˆ
Hi(t)
ˆ
Hi(t)= ˆ
Hi(t−td)+εi(td)+g(t).
(31)
Then d
dt ∫t
t−td˜
HT(w)˜
H(w)dw =−2˜
HT(t)g(t)−gT(t)g(t)
Proof: d
dt t
t−td
˜
HT(w)˜
H(w)dw is ˜
HT(t)˜
H(t)−˜
HT(t−
td)˜
H(t−td). Considering ˆ
Hi(t)= ˆ
Hi(t−td)+εi(td)+
g(t):
˜
H2
i(t−td)=Hi(t−td)−ˆ
Hi(t−td)
×Hi(t−td)−ˆ
Hi(t−td)
=Hi(t)−ˆ
Hi(t)+gT(t)Hi(t)−ˆ
Hi(t)+g(t)
=˜
H2
i(t)+2˜
Hi(t)g(t)+g2(t)(32)
Thus, we can obtain
˜
H2
i(t)−˜
H2
i(t−td)=−2˜
Hi(t)g(t)−g2(t).(33)
Let us now define g(t)function as
g(t)=−k3ie2i(t)(34)
where e2i∈R7. Substituting (34) in (35), we obtain
˜
H2
i(t)−˜
H2
i(t−td)=2˜
Hi(t)k3ie2i−(k3ie2i)2.(35)
Multiplying both sides of (35) by 1
2k3ito obtain
1
2k3i
˜
H2
i(t)−1
2k3i
˜
H2
i(t−td)= ˜
Hi(t)e2i−k3i
2e2
2i.(36)
B. Proof Stability of Functional Lyapunov–Krasovskii
Equation (27)
The proposed adaptive time-delay control law (17) makes
the system converge asymptotically stable. Where e1→0 and
e2→0ast→∞. The proof is formed in three stages. Stage 1
proves the boundedness of V4(t)in the interval [0,t
dn). Stage 2
proves the negativeness of V4and asymptotically convergence
of tracking errors e1and e2in the interval [tdn,∞).
Stage 1: Boundedness of V4(t)in the interval [0,t
dn).
The derivative of (27) V4(t)with respect to time for t∈
[0,t
d1)is given by
˙
V4(t)= ˙
V2(t)+
n
i=1
1
2k3i
˜
H2
i(t).(37)
Differentiating (28) ˙
V2(t)with respect to time is given such
that
˙
V2(t)=−eT
1k1˙e1+eT
1e2+eT
2˙e2.(38)
Substituting (8) and (12) into (38) and applying the adaptive
control law (17) to find
˙
V2(t)=−eT
1k1˙e1+eT
1e2+eT
2−k2e2−e1+ˆ
H(t)−H(t)
=−eT
1k1e1+eT
1e2+eT
2−k2e2−e1−H(t)−ˆ
H(t)
.
(39)
One obtains
˙
V2(t)=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t).(40)
For ∈[0,t
d1),wehavefrom(18): ˆ
H(t)=−k3ie2i. Hence
˜
H2
i(t)=
Hi(t)−ˆ
Hi(t)2
=H2
i(t)−2Hi(t)ˆ
Hi(t)+ ˆ
H2
i(t)
=H2
i(t)−2ˆ
Hi(t)˜
Hi(t)+ ˆ
Hi(t)+ˆ
H2
i(t)
=H2
i(t)+2k3i˜
Hi(t)e2i−(k3ie2i)2.(41)
584 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018
Therefore, substituting (40) and (41) in (37) the Lyapunov
function ˙
V4(t)is rewritten as follows:
˙
V4(t)= −eT
1k1e1+eT
1e2−eT
2k2e2−eT
2˜
H(t)
+
n
i=1
1
2k3i
H2
i(t)+
n
i=1
˜
Hi(t)e2i−
n
i=1
1
2k3i
(k3ie2i)2.
=−eT
1k1e1−eT
2k2e2−
n
i=1
k3i
2e2
2i+
n
i=1
1
k3i
H2
i(t)
(42)
According to Assumption 2, Hi(t),i=1,...,nare bounded.
This implies ˙
V4(t)is bounded. Hence, ˙
V4(t)is bounded in time
interval [0,t
d1).
For t∈[tdn−1,t
dn), conforming to the derivative of ˙
V4(t)
with respect to time can be writing by
˙
V4=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t)+ 1
2k3n
˜
H2
n(t)
+
n−1
i=1
1
2k3i
(˜
H2
i(t)−˜
H2
i(t−tdn)
=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t)1
2k3n
H2
n(t)
+˜
Hn(t)e2n−1
2k3ne2
2n+
n−1
i=1
˜
Hi(t)e2i
−1
2
n−1
i=1
k3ie2
2i=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t)
+1
2k3n
H2
n(t)+
n
i=1
˜
Hi(t)e2i−1
2
n
i=1
k3ie2
2i
=−eT
1k1e1−eT
2k2e2−1
2
n
i=1
k3ie2
2i+1
2k3n
H2
n(t).(43)
It is clear from (43) that ˙
V4(t)is bounded for t∈[tdn−1,t
dn)
because Hi(t)is bounded. That means ˙
V4(t)is bounded in time
interval [tdn−1,t
dn).
Stage 2: The negativeness of ˙
V4(t)in the interval [tdn,∞).
The derivative of ˙
V4(t)with respect to time for t∈[tdn,∞)
is given by
˙
V4=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t)
+
n
i=1
1
2k3i˜
H2
i(t)−˜
H2
i(t−tdi).(44)
We have from (35): ˜
H2
i(t)−˜
H2
i(t−tdi)=2k3i˜
Hi(t)e2i−
k2
3ie2
2i. So, (44) can be rewritten as follows:
˙
V4=−eT
1k1e1−eT
2k2e2−eT
2˜
H(t)+
n
i=1
˜
Hi(t)e2i
−1
2
n
i=1
k3ie2
2i=−eT
1k1e1−eT
2k2e2−1
2
n
i=1
k3ie2
2i.
(45)
It is clear from (45) that ˙
V4(t)≤0fort∈[tdn ,∞), where
all gains k1,k
2,and k3iare positive, which means that the
robot system is stable. Hence, the signals e1,e2and ˜
H(t)are
bounded.
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Brahim Brahmi was born in Algeria. He re-
ceived the B.Eng. degree in automation con-
trol from the Department of Electronic and Au-
tomatic Engineering, University of Science and
Technology, Oran, Algeria, in 2011, the Master’s
degree in computer and control system engi-
neering from Lviv Polytechnic National Univer-
sity, Lviv, Ukraine, in 2014. He is currently work-
ing toward the Ph.D. degree in robotics at the
Department of Electrical Engineering, ´
Ecole de
technologie sup´
erieure, Montreal, QC, Canada.
His research interests include nonlinear control and adaptive control
applied to robotics, exoskeleton robots, intelligent systems, and biologi-
cal control.
Maarouf Saad (SM’05) received the B.Sc. and
M.Sc. degrees in electrical engineering from
´
Ecole Polytechnique of Montreal, Montreal, QC,
Canada, in 1982 and 1984, respectively, and
the Ph.D. degree in electrical engineering from
McGill University, Montreal, QC, Canada, in
1988.
In 1987, he joined ´
Ecole de technologie
sup´
erieure, Montreal, QC, Canada, where he
is currently teaching control theory and robotics
courses. His research interests include nonlin-
ear control and optimization applied to robotics and flight control system,
rehabilitation robotics, power systems, and distributed generation.
Cristobal Ochoa-Luna received the B.Eng.
degree in electronic and computer engineer-
ing and the M.Sc. degree in electronics engi-
neering from the Universidad de las Am´
ericas
Puebla, San Andr´
es Cholula, Mexico, in 2003
and 2005, respectively, and the Ph.D. in en-
gineering (robotics) from ´
Ecole de technolo-
gie sup´
erieure, Universit´
eduQu
´
ebec, Montreal,
QC, Canada, in 2016.
He is currently a Postdoctoral Fellow with
McGill University, Montreal. His research inter-
ests include robotics, control, and industrial automation.
Mohammad Habibur Rahman (M’10) received
the B.Sc. degree in mechanical engineering
from Khulna University of Engineering & Tech-
nology, Khulna, Bangladesh, in 2001, the Mas-
ter of Engineering degree in biorobotics from
Saga University, Saga, Japan, in 2005, and the
Ph.D. degree in engineering (biorobotics) from
´
Ecole de technologie sup´
erieure, Universit´
edu
Qu´
ebec, Montreal, QC, Canada, in 2011.
He was a Postdoctoral Research Fellow with
the School of Physical & Occupational Therapy,
McGill University, from 2012 to 2014. He also served as the Faculty
Member with the Department of Mechanical Engineering, Khulna Uni-
versity of Engineering & Technology, Bangladesh, from 2001 to 2011. He
is currently an Assistant Professor with the Department of Mechanical
Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI, USA.
His research interests include biorobotics, exoskeleton robot, intelligent
systems and control, mobile robotics, nonlinear control, artificial intelli-
gence, fuzzy systems, and control.
Abdelkrim Brahmi received the B.Sc. and
M.Sc. degrees in electrical engineering from
the University of Sciences and Technologies of
Oran, Algiers, Algeria, in 1997 and 2009, respec-
tively, and the Ph.D. degree in electrical engi-
neering from ´
Ecole de technologie sup´
erieure,
Universit´
eduQu
´
ebec, Montreal, QC, Canada,
in 2018.
His current research interests include nonlin-
ear control and adaptive control applied to coor-
dinated robotic systems.
