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Engineering, Technology & Applied Science Research Vol. 11, No. 4, 2021, 7446-7451 7446 
 

www.etasr.com Duc et al.: A Study on the Response of the Rehabilitation Lower Device using Sliding Mode Controller 

 

A Study on the Response of the Rehabilitation Lower 

Device using Sliding Mode Controller  
 

Dao Minh Duc 

Faculty of Engineering and Technology 
Pham Van Dong University 

Quang Ngai, Vietnam 

dmduc@pdu.edu.vn  

Tran Xuan Tuy 

Faculty of Mechanical Engineering 
Da Nang University of Science and Technology 

Da Nang, Vietnam 

tranxuantuy@yahoo.fr 

Pham Dang Phuoc 

Faculty of Engineering and Technology 

Pham Van Dong University 

Quang Ngai, Vietnam 
pdphuoc@pdu.edu.vn 

 

 

Abstract-This paper presents the application of the sliding mode 

controller to simulate the control of a 3 Degree of Freedom (DoF) 
robot model that supports lower extremity rehabilitation 

exercises for stroke patients. The 3 DoF robot model has an 

established dynamic differential equation for each joint. This 

paper studied the response of links of the research model with a 

sliding mode controller proposed as the control solution. Matlab 

software was used in simulations of ankle, knee, and hip joints' 

drive for 2 cases, without and with load. Compared with other 

studies, the research results show that the sliding controller 

results in slight angular errors when applied to the research 

model, while the torque of the joints remains low. This result is 

the basis for calculating and selecting parameters for the actuator 
when manufacturing the actual equipment. 

Keywords-sliding mode controller; ankle; knee; hip; 
rehabilitation 

I. INTRODUCTION  

According to the World Health Organization (WHO), 
worldwide, each year 15 million people suffer a stroke, of 
which 5 million die and 5 million become permanently 
disabled [1]. Vietnam has a population of more than 90 million 
people and the respective number of newly infected people is 
225,000 with 540,720 surviving and 117,900 people dying 
annually [2]. A cerebrovascular accident will leave severe 
consequences to some patients. The survivors of strokes 
commonly are not able to perform daily activities such as 
dressing, eating, and bathing. In recent years, different 
researches have been done to develop robotic devices for 
rehabilitation, especially for the neurorehabilitation of post-
stroke patients [3]. Rehabilitation after stroke exercise for 
patients has become an urgent major social issue. Currently, in 
hospitals and rehabilitation centers, the demand for automatic 
rehabilitation equipment has increased significantly. Robotic 
rehabilitation has some benefits over conventional manual 
rehabilitation, such as:  

• Robots can efficiently perform repetitive therapeutic 
exercises. 

• Employing different sensors in robotic systems enables 
either accurate exertion of torques/forces or precise 
quantitative evaluation of the patient's recovery during the 
rehabilitation program. 

• Rehabilitation strategies using robotic systems provide 
simultaneous and coordinated actuation of the patient's 
joints. 

In this study a controller is designed for an exoskeleton 
rehabilitation robot, performing different passive, assistive, and 
resistive therapeutic exercises [4]. Research and manufacturing 
of equipment to support rehabilitation exercises for stroke 
patients have achieved certain achievements [5-6]. To 
successfully research and manufacture assistive devices for 
stroke patients, calculating and simulating the device's response 
is the first thing before manufacturing. Thanks to the 
development of computers over the years, simulation has 
become an indispensable tool for synthesizing control laws in 
mechanical systems, allowing the verification of the behavior 
of control laws on a virtual model of an automatic system 
called simulation model. 

Currently, the PID controller is widely applied [7], 
especially in robot control, to support rehabilitation for stroke 
patients with the advantages of simple and easy to control 
structure [8]. The disadvantage is that it is difficult to find the 
Kp, Ki, Kd coefficients when the load changes. In addition, 
modern control theory is increasingly widely applied [9]. The 
use of a fuzzy controller to control the device has resulted [10]. 
Authors in [11] used a combination of fuzzy and PID 
controllers to control their research equipment. The advantage 
of this controller is that it optimizes the search for control 
parameters and can automatically adjust the control parameters 
of other controllers when combined with a PID controller. 

Corresponding author: Dao Minh Duc



Engineering, Technology & Applied Science Research Vol. 11, No. 4, 2021, 7446-7451 7447 
 

www.etasr.com Duc et al.: A Study on the Response of the Rehabilitation Lower Device using Sliding Mode Controller 

 

However, for more accuracy, it needs more fuzzy grades, 
which increases the rules, the lower speed, and the long run 
time of the system exponentially. In this paper, we introduce 
the sliding mode controller for the research model. This 
controller has the advantage of being less affected by load 
changes. However, it is necessary to overcome the chattering 
phenomenon of the controller. 

II. DYNAMIC EQUATIONS OF THE RESEARCH MODEL 

In this study, a drive model of lower extremity joints was 
established with 3 Degrees of Freedom (DoFs) for hip, knee, 
and ankle joints flexion/extension exercises. The range of 
motion for hip, knee, and ankle joint was 0-90

0
, -135

0
-0

0
, 40

0
-

120
0
 respectively [12]. The mechanism that drives the joints is 

a linear actuator with the advantage of large load capacity and 
self-braking to protect the patient during the exercise. This 
paper introduces a research model to contribute to the creation 
of lower limb rehabilitation equipment. The hip, knee, and 
ankle joints are reduced to a system with only 3 DoFs, and the 
dynamic computational model for the research model is 
presented in Figure 1. In the model, link 1 is for the hip joint, 
link 2 is for the knee joint, and link 3 is for the ankle joint. 
Setting up the dynamic equation for each link is an essential 
factor to calculate and simulate the motion of the joints, from 
which it is possible to calculate the design of the mechanical 
structure and the controller so that it is suitable for research 
models. 

 

y
1

y
2

y
3

x
1

x
2

x
3

z
1

z
2

z
3

θ 
1

θ 
2

θ 
3

Linear actuator 

drive hip joint

Linear actuator 

drive knee joint

Linear actuator 

drive ankle joint

 
Fig. 1.  The research model. 

y

x

θ 
1

θ 
2

θ 
3

C 
1

C 
2

C 
3

O 
 

Fig. 2.  Dynamic mathematical model. 

To establish the dynamic equation of the device [13-14], we 
conduct a dynamic analysis of the joints. The diagram of the 
dynamic analysis of the research model is shown in Figure 2. 
The rotation angles of the joints are shown with the rotation 

angles of the hip joint is θ1, the knee joint is θ2, and the ankle 
joint is θ3. To simplify the establishment of kinematic 
equations of joints, we reduce the mass of each link to the 
centroid of each link. 

The dynamic equation of the model is described as follows: 

( ) ( ) ( )n n n n n n n n
1

n 2

3

D q H , Gθ θ θ θ τ

θ
θ θ

θ

+ + =

 
 =  
  

&&&

    (1) 

Equations (2)-(4) are the dynamic equations of each joint. 
The dynamic equation of link 1: 

( )
( )

( )

( )

( )

2

1 1 1

2 2

2 1 2 1 2 2

2 2 2
1 1

1 2 3 1 2 2

3

1 3 2 3 2 3 3

2 3

2

2 2 1 2 2 2

2 2

2 3 1 2 2

3

1 3 2 3 2 3 3

2 cos

2 cos

2 cos 2 cos

cos

cos

cos 2 cos

C C

C

C C

C C

C

C C

m L I

m L R L R

L L R L L
m

L R L R

I I

m R L R I

L R L L
m

L R L R

θ

τ θθ

θ θ θ

θ

θ

θ θ θ

 +
 
 + + +
 

=   + + + +
 +   + +  
 
+ + 

+ + +

+  + +
+  + + + 

&&

( )
( )

( )
( )

( )
( ) ( )

2

3

2 1 2 3 1 2 2 2

2

2 1 3 2 3

2

3 1 3 2 3 3 2 3 3 3

2 1 2 3 1 2 2

1 2

3 1 3 2 3

2 1 3 3 1 3 2 3

3 2 3 3

sin

sin

sin sin

2 2 sin

2 sin

sin

sin

C

C

C C

C

C

C C

C

I

m L R m L L

m L R

m L R m L R

m L R m L L

m L R

m L R m L R

m L R

θ

θ
θ

θ θ

θ θ θ θ

θ
θ θ

θ θ

θ θ

θ

 
 
 
 
  

 +
− 

+ +  

− + +  

 +
− 

+ +  

 + + 
− 

+ 

&&

&

&

& &

&

( )

( )
( )

( )

2 3

3 1 3 2 3 3 2 3 3 1 3

1 1 1

2 1 1 2 1 2

1 1 2 1 2

3

3 1 2 3

2 sin 2 sin

cos

cos cos

cos cos
.

cos

C C

C

C

C

m L R m L R

m gR

m g L R

L L
m g

R

θ θ

θ θ θ θ θ

θ

θ θ θ

θ θ θ

θ θ θ

− + +  
+

+  + +  

 + +
+  

+ + +  

&

& &

    (2) 

The dynamic equation of link 2: 

( )

( )

2

2 2 1 2 2 2

2 2

2 3 1 2 2

2 3 1 3 2 3 1

2 3 3

3

cos

cos

cos

cos

C C

C

C

C

m R L R I

L R L L

m L R

L R

I

θ

θ

τ θ θ θ

θ

 + +
 
  + +
  

= + + +  
  +  
 +
  

&&
    (3) 



Engineering, Technology & Applied Science Research Vol. 11, No. 4, 2021, 7446-7451 7448 
 

www.etasr.com Duc et al.: A Study on the Response of the Rehabilitation Lower Device using Sliding Mode Controller 

 

( )
[ ]
( )

( )
[ ]

( )

2 2

2 2 3

2 2 2 3

22 3 3

3

2

3 3 2 3 3 3 3

2

3 1 3 3 3

2 1 2 3 1 2 2

1 2

3 1 3 2 3

3 2 3 3 2 3

3 1 3 2 3

3 2 3

2 cos

cos

sin

sin

sin

2 sin

sin

2 sin

C

C

C

C C

C

C

C

C

C

C

L R
m R I m

L R

I

m R L R I

m L R

m L R m L L

m L R

m L R

m L R

m L R

θθ

θ θ

θ θ

θ
θ θ

θ θ

θ θ θ

θ θ

  +
+ +  

+ +  
 + 

 + + + 

−

 +
− 
+ +  

−

+
−
+

&&

&&

&

& &

& &

( )
( )
( )

1 3

3

2 2 1 2

2 1 2

3

3 1 2 3

cos

cos
.

cos

C

C

m gR

L
m g

R

θ θ
θ

θ θ

θ θ

θ θ θ

 
 
 

 + 
  

+  + + 
+   

+ +    

& &

 

The dynamic equation of link 3: 

( )

( )

[ ]
( )

[ ]
( )

2

3 1 3 2 3

3

3 12 3 3

3

2

3 3 2 3 3 3 2

2

3 3 3 3

2

3 1 3 3 3

3 1 3 2 3 1 2

3 2 3 3 2 3

3 1 3 2 3 3 2 3 3

cos

cos

cos

sin

sin

sin

sin sin

C C

C

C C

C

C

C

C

C C

R L R
m

L R

I

m R L R I

m R I

m L R

m L R

m L R

m L R m L R

θ θ
τ θθ

θ θ

θ

θ θ

θ θ θ θ

θ θ θ

θ θ θ

  + + +
   =   
 + 

 + + + 

 + + 

−

− +  

−

− + + 

&&

&&

&&

&

& &

& &

( ){ }
1 3

3 3 3 2 1
. . .cos

C
m g R

θ θ

θ θ θ



+ + +

& &

    (4) 

The general form as (1) is rewritten as follows: 

( ) ( ) ( )n n n nD C , Gθ θ θ θ θ τ+ + =&& &     (5) 
where L1, L2, L3 are the lengths, C1, C2, C3 the centers, R1, R2, 
R3 the lengths of the centers, Dn the inertial matrix, Hn the 
matrix of centrifugal and Coriolis forces, and Gn the 
gravitational force matrix of the links 1, 2, and 3 respectively. 

[ ]

[ ]

T

n 1 2 3

T

n 1 2 3

T

n 1 2 3

T

n 1 2 3

θ θ θ θ

θ θ θ θ

θ θ θ θ

τ τ τ τ

=

 =  

 =  

=

& & & &

&& && && &&

 

[ ]

11 12 13

n 21 22 23

31 32 33

11 12 13

n 21 22 23

31 32 33

T

n 1 2 3

D D D

D D D D

D D D

C C C

C C C C

C C C

G G G G

 
 =  
  

 
 =  
  

=

 

III. SLIDING MODE CONTROLLER FOR THE RESEARCHED 
MODEL 

The sliding mode controller is generally utilized in industry 
and biomedical robots [15-16]. The controller can be planned 
regardless of the boundaries of the exploration model. 
Consequently, the sliding mode controller priorities include 
robustness, responsiveness, insensitiveness to parameter 
changes and noise, a simple physical model, and easy 
programming for the software driver. The block diagram that 
controls the lower extremity rehabilitation device based on the 
sliding mode controller is shown in Figure 3. 

 

+

-

Setpoint Sliding 

surface 

function

Sliding 

mode 

controller

Actuator

Rehabilitation 

lower

Feedback

Error

 
Fig. 3.  Sliding mode controller block diagram. 

Tracking error: 

( ) ( ) ( )de t t tθ θ= −     
(6) 

where θd is the setpoint and θ is a reference. 

Velocity and acceleration tracking error: 

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

d

d

e t t t

e t t t

θ θ

θ θ

= −

= −

& &&

&& &&&&

    (7) 

The dynamic equation (5) can be written as: 

( ) ( ) ( ){ }1D q C , Gθ τ θ θ θ−= − −&& &
    
(8) 

Sliding surface choice function: 

( ) ( ) ( )s t e t e tλ= + &     (9) 

where λ is a positive constant. 

Derivative sliding surface: 

( ) ( ) ( )s t e t e tλ= +& & &&     (10) 

The control signal is the torque at the joints, according to 
[14] the control signal consists of two components which are: 

( )
( )

eq

N

u when s ,t 0
u

u when s ,t 0

θ
θ

 =
= 

≠
 



Engineering, Technology & Applied Science Research Vol. 11, No. 4, 2021, 7446-7451 7449 
 

www.etasr.com Duc et al.: A Study on the Response of the Rehabilitation Lower Device using Sliding Mode Controller 

 

We can calculate the control signal from the differential 
condition s(t)=0 and from (10) and (8). 

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

( ) ( ) ( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ) ( ) ( ){ }

d

1

d

1

d eq

s t e t e t 0

s t e t t t

s t e t t D C , G

s t e t t D u C , G

λ

λ θ θ

λ θ θ τ θ θ θ

λ θ θ θ θ θ

−

−

= + =

= + −

= + − − −

= + − − −

& & &&

&& &&& &

&& && &

&& && &

 

( ) ( ) ( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ) ( ) ( )

1

d eq

1 1

eq d

s t e t t D u C , G 0

u e t t D C , D G

λ θ θ θ θ θ

λ θ θ θ θ θ θ

−

− −

= + − − − =

= + + +

&& && &

&& &&

 

with 
N

u K sgn( s ).=
 
The sliding mode control for each joint 

is shown as: 

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

1

N eq d

1

u u u e t t D C ,

D G Ksgn( s )

λ θ θ θ θ

θ θ

−

−

= + = + +

+ +

&& &&
    (11) 

We use the Lyapunov method to check the stability of the 
sliding mode controller. The Lyapunov function associated 
with the research model is: 

21
V ( s ) s

2

V ( s ) ss

V ( s ) -K sgn( s )

=

=

=

& &

&

 

For the controller to be stable we must choose a positive K 
to satisfy the stability condition: 

V ( s ) K sgn( s ) 0= − <&  

IV. SIMULATION IN MATLAB  

We utilized Matlab to simulate the response of the research 
model when using the sliding mode controller. The parameters 
of the research model are shown in Table I. Simulations for the 
research model were conducted for 2 cases, namely without 
and with a torque of load acting on the joints of the research 
model. The position of the initial joints of the research model 
is: 

0
1

0
2

0
3

0

90

90

θ

θ

θ

 =


= −


=

 

The setpoint angle for the joint simulates the range of 
motion of the human lower limb joints.  

0 0
1 _ setpo int

0 0

2 _ setpo int

0 0
3 _ setpo int

45 sin( 2 * * 0 ,1* t ) 45

135 135
s in( 2 * * 0 ,1* t )

2 2

40 s in( 2 * * 0,1* t ) 80

θ π

θ π

θ π

 = +



= − −

 = +


 

The simulation results of the case without load are shown in 
Figures 4-6 and with load in Figures 7-9. 

TABLE I.  SIMULATION PARAMETERS OF THE RESEARCH MODEL 

 
Simulator parameters 

Description Value Parameter 

1 Mass of link 1 5 m1 (kg) 

2 Mass of link 2 5 m2 (kg) 

3 Mass of link 3 1 m3 (kg) 

4 Length of link 1 0.4 L1 (m) 

5 Length of link 2 0.45 L2 (m) 

6 Length of link 3 0.2 L3 (m) 

7 Length center of link 1 0.2 Rc1 (m) 

8 Length center of link 2 0.225 Rc2 (m) 

9 Length center of link 3 0.1 Rc3 (m) 

10 Moment of inertia of link 1 0.2667 I1 (kg.m
2
) 

11 Moment of inertia of link 2 0.3375 I2 (kg.m
2
) 

12 Moment of inertia of link 3 0.0133 I3 (kg.m
2
) 

13 Cofficient K 40  

14 Cofficient λ 20  
15 Load Torque on link 1 50 (N.m) 

16 Load Torque on link 2 50 (N.m) 

17 Load Torque on link 3 10 (N.m) 

 

 
Fig. 4.  Response of link 1without load. 

 
Fig. 5.  Response of link 2 without load. 

Figures 4-6 present the response of the research model's 
hip, knee, and ankle joints when using the sliding mode 
controller without load. The hip joint results are: settling time = 
1s, maximum torque = 150Nm, and overshoot = 0. The knee 
joint results are: settling time = 0.5s, maximum torque = 
30Nm, and overshoot = 0. The ankle joint results are: settling 
time = 0.5s, maximum torque = 6Nm, and overshoot = 0. 



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Fig. 6.  Response of link 3 without load. 

The sliding mode controller results for the case with load 
are shown in Figures 7-9. The hip joint results are: settling time 
= 1s, maximum torque = 80Nm, and overshoot = 0. The knee 
joint results are: settling time = 0.5s, maximum torque = 
31Nm, and overshoot = 0. The results of the ankle joint are: the 
settling time = 0.5s, maximum torque = 6Nm, and overshoot = 
0. 

 

 

Fig. 7.  Response of link 1 with load. 

 
Fig. 8.  Response of link 2 with load. 

 
Fig. 9.  Response of link 3 with load. 

V. DISCUSSION 

In this study, a sliding controller for a lower extremity 
rehabilitation exercise aid was designed and its response was 
simulated in Matlab. The simulation results of the device in the 
flexion/extension movements of the hip, knee, and ankle joints 
are shown in Figures 4-9. The hip joint results are shown in 
Figures 4 and 7, which show that this is the joint that bears the 
most significant internal load due to the influence of the weight 
of the structure and the external force, thus creating the most 
torque. For the knee joint, the results are shown in Figures 5 
and 8. The joint can only bear the internal load of the 
mechanism and the external force, so the maximum torque is 
31N.m. Finally, the results of the ankle joint are shown in 
Figures 6 and 9. This is the joint that only bears the internal 
load of the mechanism and the minor external force, so the 
torque is minimal. In addition, in order to assess the adequacy 
of the proposed controller, the comparison of the trajectory 
tracking of the proposed controller with several controllers 
studied in [17-19] was conducted. The trajectory error tracking 
of the study compared with the works in [17-19] is similar, 
however, the response time in this study was better (1s instead 
of 2s). In addition, the above studies have not considered the 
influence of external [17] or small [18-19] loads on the 
research model and have not analyzed the action moments of 
each joint. Therefore, this study demonstrates that the sliding 
controller can control the lower extremity rehabilitation device 
for patients with fast response time and slight error. The 
advantage of this study is that we have created a controller that 
is not affected by external forces, which makes the device 
response stable, which is essential in the rehabilitation setting 
for patients. The main limitation of the current study is that it 
only simulates the response, so it is not possible to conclude 
that this is the actual response when the controller is applied to 
real devices. The development direction of this research is to 
check the correctness of the controller when applied on real 
devices. 

VI. CONCLUSION 

A sliding controller that controls a 3 DoF robot model that 
supports lower extremity rehabilitation was presented in this 
paper. The dynamic model of the research model was studied. 
It is proposed to apply the sliding mode controller to study the 



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dynamic response of the links of the research model. The study 
results have been verified in Matlab simulations of the response 
of the links of the research model. The proposed sliding mode 
controller for the research model has shown a good response 
with small error tracking and torque compared to other studies. 
This result is the basis for manufacturing an actual device to 
support lower extremity rehabilitation for stroke patients using 
a linear actuator. 

ACKNOWLEDGMENT 

This work is funded by the Pham Van Dong University, 
Vietnam, project 13/2017/KHCN. The authors are grateful for 
the financial support. 

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