MEV


 

 

Journal of Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 

Journal of Mechatronics, Electrical Power, 

and Vehicular Technology 
 

e-ISSN: 2088-6985  

p-ISSN: 2087-3379  

 

 

www.mevjournal.com 
 

 

 

doi: https://dx.doi.org/10.14203/j.mev.2018.v9.65-72 

2088-6985 / 2087-3379 ©2018 Research Centre for Electrical Power and Mechatronics - Indonesian Institute of Sciences (RCEPM LIPI).  

This is an open access article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/).  

Accreditation Number: (LIPI) 633/AU/P2MI-LIPI/03/2015 and (RISTEKDIKTI) 1/E/KPT/2015. 

Study on the transient response of lower limb rehabilitation actuator 

using the pneumatic cylinder 
 

Dao Minh Duc a, *, Pham Dang Phuoc a, Tran Xuan Tuy b, Le Thi Thuy Tram c 
a Faculty of Engineering Technology, Pham Van Dong University 

509 Phan Dinh Phung Street, Quang Ngai City, Viet Nam 
b Faculty of Mechanical, University of Science and Technology 

41 Nguyen Luong Bang Street, Lien Chieu, Da Nang City, Viet Nam 
c Department of Electrical and Electronics, Technological Colleges Quang Nam 

224 Huynh Thuc Khang, Tam Ky City, Viet Nam 

 

Received 23 January 2018; received in revised form 2 December 2018; accepted 12 December 2018 

Published online 30 December 2018 

 

 

Abstract 

A lower limb rehabilitation device was designed using the compressed air cylinder in order to answer the particular request 

in Viet Nam. This paper is presenting the results of a study of the device response. Dynamic equation of the actuator and 

equations of the proportional valve have been established. The relationship between the input signal and the output signal of the 

actuator was derived. Inventor® software was used to design the mechanical structure of the device. Matlab® software was used 

to calculate the parameters values of the PID controller by simulating the response of the actuator. The results show that the 

response time of both knee drive and hip drive mechanisms are 8 seconds while the overshoot of both knee drive and hip drive 

mechanisms are 1%. Moreover, the starting torque of the knee drive mechanism is 17 Nm, and the starting torque of the hip 

drive mechanism is 35 Nm. The simulation results show that the PID controller gives a fast response time and a low overshoot. 

©2018 Research Centre for Electrical Power and Mechatronics - Indonesian Institute of Sciences. This is an open access 

article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/).  

Keywords: lower limb rehabilitation; hip and knee joint; pneumatic cylinder; PID controller. 

 

 

I. Introduction 

At present, the mechanical engineering industry 

contributes greatly to the development of society. The 

products of the mechanical industry are diversified and 

rich in various fields. In the biomedical field, there are 

also many products that help the patients and 

physicians in treating the disease [1]. In the treatment 

of herniated disc patients with spinal stretch bed, many 

authors are interested in research, design, manufacture 

and have achieved certain success [2][3][4].  

Some studies have applied robots to support 

rehabilitation for patients with underlying limb disease, 

helping patients to shorten treatment and helping 

doctors to monitor the course of treatment for patients 

[5]. However, in the treatment of patients with 

hemiplegia due to a cerebrovascular accident in Viet 

Nam, at present, there is no automatic device that 

supports the treatment of equipment for purely 

mechanical treatment. Overseas studies have also been 

available for the treatment of patients and have also 

been successful [6][7]. 

Today, compressed air is widely used in industry as 

well as in social life. The main advantages of 

compressed air include large capacity, low cost, and 

ability to be used in harsh environments. These 

advantages can make the actuator extremely useful in 

the applications of rehabilitation techniques for stroke 

or knee pain patients [8][9][10]. The use of compressed 

air to drive the lower limb rehabilitation device has also 

been investigated. Previous research had used 

compressed air to control the actuators and produce 

very promising results [11][12][13]. 

The request for equipment to support the treatment 

of patients with hemiplegia caused by accident has been 

approved by the researchers. This paper has studied the 

response of an exercise support structure, which 

evaluates the feasibility of the structure before practical 

modeling is expected to contribute to patient treatment 

 

 

* Corresponding Author. Tel: +84 905 423 314 

E-mail address: dmduc@pdu.edu.vn 

https://dx.doi.org/10.14203/j.mev.2018.v9.65-72
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D.M. Duc et al. / Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 

66 

support. This paper’s aim is studying the use of 

pneumatic cylinders to drive rehabilitation equipment 

for the hip and knee joint. The advantage of this 

solution is to build passive exercises for patients with 

levels 1 to 3 and active impedance exercises for patients 

with muscle levels of 3 to 5. Combining two active and 

passive exercises will help the patient shorten the joints 

rehabilitation time. 

II. Materials and methods 

A. The lower limb research model 

Analysis of the lower limb joints including hip 

joints, knee joints, and ankle joints shows the total 

degrees of freedom of the lower limb are 9, as shown in 

Figure 1 [14]. However, this paper introduces the 

degrees of hip and knee arthritis that are commonly 

used in rehabilitation exercises in hospitals and centers 

nationwide, as illustrated in Figure 2 and Figure 3. Hip 

and knee joints are reduced to a two-degree system, and 

the model is shown in Figure 4. 

B. Model of lower limb rehabilitation actuator 

The functional lower limb rehabilitation actuator 

model is illustrated in Figure 5. Its structure consists of 

two stages with two degrees of freedom around the z 

axis. Stitches are driven by pneumatic cylinders, 

controlling the cylinders through pneumatic servo 

valves. 

C. The dynamic equations of the actuator 

The dynamic equation of the actuator is expressed 

by equation (1) and (2). 

     D q q H q,q G q     (1) 

n cc ms
       (2) 

where τn is the torque caused by the hip, and thigh 
muscles τcc and τms are the friction torque components 
at the robotic stage.  

Because torque components have negligible effects, 

both components can be ignored and only observing the 

torque generated by the mechanism. Inertial matrix, 

centrifugal force and Coriolis matrix, and gravity force 

matrix are given as follow.  

1) Inertial matrix 𝐷(𝑞): 

2 2 2

11 1 1 2 1 2 2 1 2 2

1 1
D m l m l l m l l C

3 3


 
    

 
 

2 2

12 21 2 2 2 1 2 1 2 2

1
D D m l m l m l l C

3
     

2

22 2 2

1
D m l

3
  

 

Figure 1. The degrees of freedom of the lower limb 

 

 

Figure 2. Hip and knee joints in extension state 

 

Figure 3. Hip and knee joints in the contraction state 

 

 

 

Figure 4. Lower limb model 

 

 

Figure 5. Model of rehabilitation lower limb actuator 



D.M. Duc et al. / Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 
67 

2) Centrifugal force and Coriolis matrix: 

2

2 1 2 2 2 2 1 2 1 2 2

2

2 1 2 1 2

1
m l l S m l l S

2
H

1
m l l S

2

    

 

 
  

  
 
  

 

3) Gravity force matrix: 

 1 1 1 1 2 1 1 2 2 1 1
1 1

G m gl C m gl C m gl C
2 2

      
 

 2 2 2 1 2
1

G m gl C
2

  
 

From the above matrix D, H, G, the dynamical 

equation (1) becomes: 

2 2 2

1 1 2 1 2 1 2 2

1

2 2 2

2 1 2 2 1 2 2

2 2

2 1 2 1 2 2

1

22

2 2

1 1
m l m l l l l C

3 3

1
m l l m l l C

3

1
m l l l l C

3

1
l m

3












  
    

        
   

 

 
        

  



2

2 1 2 2 2 2 1 2 1 2 2

2

2 1 2 1 2

1
m l l S m l l S

2

1
m l l S

2

    

 

 
  

  
 
  

 

 

 

1 1 1 2 1 1 2 2 1 1

2 2 1 2

1 1
m gl C m gl C m gl C

2 2

1
m gl C

2

   

 

 
   

  
 


  

 (3) 

D. Design of controller for the actuator 

Figure 6 shows a model of control mechanism at 

each joint. At each torque joint, the piston's impact 

force in the cylinder creates the following torque 

equation: 

1

2

F .r.sin


 


 
  
 

  (4) 

where F is the force generated by the pressure in the 

cylinders affecting the piston area, r is the rotational 

radius of the link. 

Force equation in the cylinder is given by (5). 

F p.A  (5) 

where A is the area of the piston. The valve pressure 

dependent gas pressure is given by (6). 

v
p K .u  (6) 

From equations (3), (4), (5), and estimating point θ, 

the equation between torque and voltage can be found 

as follow [15]. 

1

v

2

K .r.A. .u


 


 
  
 

 (7) 

PID controller was used to examine and evaluate the 

response of the structure as shown in Figure 7. 

Furthermore, equation (1) has been transformed into 

the equation (8). 

 
   

 
1 1

q H q,q G q
D q D q

        (8) 

The PID controller is described as follow: 

t

p I D

0

u K e K edt K e    (9) 

In controlling the actuator, it is necessary to provide 

the valve opening and closing pressure, from which the 

pneumatic pressure acts on the piston to produce the 

torque that controls rotates the rotation at each joint. 

The rotation angle of the stages are as follows: 

1 1d 1

2 2d 2

e

e

 

 

 


 
 

where θ1d, θ2d are the angles set values of joints 1 and 
2. Additionally, denoted control voltage can be found 

as: 
 

Figure 6. Model of control at each joint 

 

Figure 7. Control chart for the actuator 



D.M. Duc et al. / Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 

68 

1

2

u
u

u

 
  
 

 (10) 

Equation (11) can be found by substituting equation 

(9) into equation (10): 

 

 

1 1 1

2 2 2

t

1 P 1d 1 I 1 D 1

0

t

2 P 2d 2 I 2 D 2

0

u K K e dt K

u K K e dt K

  

  


   





   







 (11) 

The define state variable as follows : 

t

1 1 1 1d 1

0

t

2 2 2 2 d 2

0

x e dt x

x e dt x

 

 


   





   







 (12) 

Equation (13) can be made by substituting equation 

(12) into equation (10): 

1 1 1

2 2 2

1 P 1 I 1 D 1

2 P 2 I 2 D 2

u K x K x K

u K x K x K





   


  

  (13) 

Moreover, substituting equation (12) into equation 

(6) can emerge equation (14): 

 

 
1 1 1

2 2 2

v 1 1 1 P 1 I 1 D 1

v 2 2 2 P 2 I 2 D 2

K .r .A . . K x K x K

K .r .A . . K x K x K

 


 

  
 
    

  (14) 

Combining equations (8) and (14) will generate this 

following equation 

 
   

 

 

 
1 1 1

2 2 2

1

2

v 1 1 1 P 1 I 1 D 1

v 2 2 2 P 2 I 2 D 2

1
H q,q G g

D q

K .r .A . . K x K x K1
 

D q K .r .A . . K x K x K





 

 

 
      

 

  
 
    

 

III. Results and discussions 

A. Device 3D model 

After analyzing the dynamics of the lower limb 

rehabilitation actuator, Inventor® software was used to 

design mechanical structure for the device. A 3D 

drawing of the device is shown in Figure 8. The device 

supports the patient to practice both feet. The patient is 

able to sit and lie down for training. At the extension 

and Flexion exercise, the limit angle of the knee is from 

0° to 135°. For the hip joint, when the patient is in the 

sitting position, the limit angle is 0° to 60° in the 

extension exercise. When the patient is lying down, the 

limit angle is 0° to 110° at the extension and Flexion 

exercise. 

B. Simulation results 

This study was using Simulink® tool in Matlab® to 

simulate the dynamics of the actuator. The parameter 

values are selected as follows.  

𝐴1 =  0.00524 (𝑚
2);  

𝐴2 =  0.002826 (𝑚
2);  

𝑙1 = 0.5 (𝑚);  

𝑙2 = 0.45 (𝑚);  

𝑝 = 4.105 (𝑁/𝑚2); 

𝑚1 = 13 (𝑘𝑔);  𝑚2 = 10 (𝑘𝑔);  

𝐾𝑣 = 0,0283 (𝑁/𝑚
2. 𝑉);  

01

0

02

0

q

2






 
            ;

1d

d

2 d

4
q

4




 

 
  

    
   
     

𝐾𝑃 = 50;  

𝐾𝐼 = 15;  

𝐾𝐷 = 5; 

 

 

Figure 8. 3D model of the device 



D.M. Duc et al. / Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 
69 

Figures 9 to Figure 14 show simulated responses of 

the actuator. Analysis and discussion of the simulation 

results are described as follows. In Flexion exercises, 

the initial values and set point were set using this 

configuration: the knee joint initial angle was 90°, and 

the angle set point was 45°, while the hip initial angle 

was 0° and angle set point was 45°. The following 

results are obtained: 

1) The response time of the knee drive mechanism is 
8 seconds, and the response time of the hip drive 

mechanism is 8 seconds. 

2) Overshoot of the knee drive mechanism is 1%, 
and overshoot of the hip drive mechanism is 1%.  

3) The starting torque of the knee drive mechanism 
is 17 Nm, and the starting torque of the hip drive 

mechanism is 35 Nm. 

 

Figure 9. Torque response of joint 1 

 

 

Figure 10. Torque response of joint 2 

 

 

Figure 11. Angle response of joint 1 



D.M. Duc et al. / Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 

70 

From the simulation results of the actuator at 

Flexion exercise for knee and hip, the error is zero and 

the overshoot is low. However, the response time is 

long (approximately 6 to 8 seconds), and the starting 

torque is quite large. These results are due to the 

actuator and the human load are quite large. Therefore, 

the high level of torque is very much required to drive 

the actuator. From this result, the application of 

pneumatic cylinder can be used for practical 

experiments. 

 

Figure 12. Angle response of joint 2 

 

 

Figure 13. Error angle response of joint 1 

 

 

Figure 14. Error angle response of joint 2 



D.M. Duc et al. / Mechatronics, Electrical Power, and Vehicular Technology 9 (2018) 65–72 

 
71 

IV. Conclusion 

In this paper, a dynamic equation for the actuators 

of a lower limb rehabilitation device was derived, and 

their responses were simulated using Matlab®. In a 

Flexion exercise simulation, initial values, and set 

points were set as: the knee joint initial angle was 90° 

and its angle setpoint was 45°, while the hip initial 

angle was 0° and its angle set point was 45°. The results 

show that the response time for both knee drive and hip 

drive mechanisms are 8 seconds while the overshoot of 

both knee drive and hip drive mechanisms are 1%. 

Moreover, the starting torque of the knee drive 

mechanism is 17 Nm, and the starting torque of the hip 

drive mechanism is 35 Nm. The simulation results 

show that the response is consistent throughout the 

training of the patient. In the next step, it would be best 

to examine the empirical model in order to evaluate the 

results obtained in theoretical simulation. 

Acknowledgement 

The authors would like to thank the Faculty of the 

College of Technology, the Pham Van Dong University 

for cooperation and help. 

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