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This is an open access article under the CC BY license : 

 
 
 

 
 

   
Al-Khwarizmi 
Engineering 

Journal 
 

Al-Khwarizmi Engineering Journal, Vol. 17, No. 3, Sptember, (2021) 
P. P. 22-  28  

 

 
Robust Computed Torque Control for Uncertain Robotic 

Manipulators 
 

 Maryam S. Ahmed*           Ali Hussien Mary**           
    Hisham H. Jasim*** 

*,**,***Department of Mechatronics/ Al-Khwarizmi College of Engineering/ University of Baghdad 
*Email: Maryamsadeq97@gmail.com 

**Email: Alimary76@kecbu.uobaghdad.edu.iq 
***Email: mschisham@gmail.com 

 
(Received 27 July 2021; Revised 20 August 2021; Accepted 1 September 2021) 

https://doi.org/10.22153/kej.2021.09.002 

 
 

ABSTRACT 
 

This paper presents a robust control method for the trajectory control of the robotic manipulator. The standard 
Computed Torque Control (CTC) is an important method in the robotic control systems but its not robust to system 
uncertainty and external disturbance. The proposed method overcome the system uncertainty and external disturbance 
problems. In this paper, a robustification term has been added to the standard CTC. The stability of the proposed control 
method is approved by the Lyapunov stability theorem.  The performance of the presented controller is tested by 
MATLAB-Simulink environment and is compared with different control methods to illustrate its robustness and 
performance.  

 
Keywords: Computed Torque Control, Linear Matrix Inequality, Robotic Control, Robotic Manipulato. 
 
 
1. Introduction 

 
Today, the robotics systems enter into many 

applications such as, industrial applications, 
manufacturing, medical fields, and space 
application [1] . Precise positioning is an important 
desired feature of the robot manipulator. The robot 
manipulator can be considered as a nonlinear 
system that suffering from high nonlinearity and 
external disturbance. The main problem is how to 
control the robot taking into account the system 
uncertainty and external disturbance. However, in 
recent years, significant and rapid progress had 
been made in the field of control, to solve the 
control problem. Different control strategies have 
been proposed. PID control proposed by many 
researchers because Due to its simplicity and ease 

of implementation. Abhishek and Dayal presented 
PID controller optimized by Particle Swarm 
Optimization (PSO) to stabilize the gait humanoid 
robot [2]. Ignacio proposed design an adaptive PID 
controller based reinforcement learning [3].   
Structured and unstructured uncertainties and 
external interference, it is difficult to obtain an 
accurate dynamics model of the robotic arm. 
Therefore, many control schemes for unknown 
dynamic models of robot manipulators have been 
proposed. A robust control strategy is an important 
method that can handle the system uncertainties 
and disturbances. Sliding mode control (SMC) is 
an effect robust control method that applied 
successfully in control different robotic 
manipulators.in [4], SMC is combined with fuzzy 
logic technique to control robot manipulator in task 
space. Mary and Kara presented robust control 



 Maryam S. Ahmed                                                 Al-Khwarizmi Engineering Journal, Vol. 17, No. 3, P.P. 22- 28 (2021) 

 

23 
 

method for control 2 links robotic arm by using 
Linear Matrix Inequality (LMI) to tune the control 
gains [5]. In (Wu and Huang , 2021), fraction 
fractional calculus utilized to design hybrid 
controller with the sliding mode control for 
trajectory tracking of mobile robot manipulator. In 
(Chen et al. , 2019), radial basis function neural 
network used with SMC for control n links robot 
arm taking into account the actuator dynamic. 
However, the Chattering is the important drawback 
in SMC [8,9]. CTC is an efficient control used in 
robotic control problem. CTC method require 
knowing the dynamic of robotic arm and in 
practice, determine the accurate dynamic of robotic 
arm may be difficult [10,11,12]. Thus, this paper 
proposed a new method that can improve the 
robustness of the standard CTC against system 
uncertainty and external disturbance. 
 
 
2. Dynamic Model of the Robotic 

Manipulator   
 

The dynamic model of the robotic arm system 
can be written as follows: 
𝑀 𝑞 𝑞 𝐶 𝑞, 𝑞 𝑞 𝐹 𝑞 𝐺 𝑞 𝜏 𝜏…(1) 
where 𝑞, 𝑞, 𝑞 ∈ 𝑅  refer respectively to the angular 
position, velocity, and acceleration vectors of 
robotic manipulator joints, 𝑀 𝑞 ∈ 𝑅  refers to 
the inertia matrix, 𝐶 𝑞, 𝑞 ∈ 𝑅  denotes a 
centripetal and Coriolis vector force, 𝐹 𝑞 ∈ 𝑅  is 
the friction torque vector, 𝐺 𝑞 ∈ 𝑅  represents 
loading of the gravity, 𝜏 ∈ 𝑅  is the external 
disturbance, and 𝜏 ∈ 𝑅  refers to the  joints torque 
vector. In particular applications, getting the 
accurate dynamic of the robotic manipulator are 
not easy due to model uncertainties and external 
disturbance. Therefore, the model uncertainty can 
be included in the dynamic model as follows: 
𝑀 𝑞 𝑀 𝑞 ∆𝑀 𝑞                                 …(2) 
𝐶 𝑞, 𝑞 𝐶 𝑞, 𝑞 ∆𝐶 𝑞, 𝑞                        …(3) 
𝐹 𝑞 𝐹 𝑞 ∆𝐹 𝑞                                     …(4) 
𝐺 𝑞 𝐺 𝑞 ∆𝐺 𝑞                                     …(5) 
where 𝑀 𝑞 , 𝐶 𝑞, 𝑞 , 𝐹 𝑞 , and 𝐺 𝑞  
represent the nominal model of the robotic system 
and it can be known, whereas ∆𝑀 𝑞 , ∆𝐶 𝑞, 𝑞 , 
∆𝐹 𝑞  and ∆𝐺 𝑞  denote the uncertainty part of 
the robotic manipulator dynamic model and it 
cannot be determined exactly. In this paper, we 
assume the following: 
Assumption 1. The desired trajectories 𝑞  with 
their first and second derivatives 𝑞  and 𝑞  are 
continuous and bounded as follows: 
 
|𝑞 𝑡 | 𝑀 , |𝑞 𝑡 | 𝑀 , |𝑞 𝑡 | 𝑀 ,                  

                                                                        …(6) 
with 𝑀 , 𝑀 , and 𝑀  being positive 
constants[5]. 
 
  
3. Computed Torque Control 
 

Computed torque control is an important 
control method that applied successfully in robotic 
system control when there is no model uncertainty 
Then, the dynamic model in (1) becomes  
 𝑀 𝑞 𝑞 𝐶 𝑞, 𝑞 𝑞 𝐹 𝑞 𝐺 𝑞 𝜏  …(7) 
The standard CTC is 
𝜏 𝑀 𝑞 𝑞 𝑘  𝑒 𝑘 𝑒 𝐶 𝑞, 𝑞 𝑞
𝐹 𝑞 𝐺 𝑞                                                   …(8) 
𝑒 𝑞 𝑞                                                         …(9) 
where 𝑒  is the tracking error, 𝑘   and 𝑘  are the 
proportional and derivative control gain matrices. 
By substituting (8) in (7), 
𝑞 𝑘  𝑒 𝑘 𝑒 0                                     …(10) 
It is obvious that the roots of (10) will lie on the left 
half plane if the control gain matrices 𝑘   and 𝑘  
are positive, which implies that the actual 
trajectory can track desired trajectory and error 
signal will converge to zero.  
 
 
4. Proposed Robust Ctc Design 
 

A robust  CTC controller will be presented to 
improve the performance of the CTC. The 
proposed control law is 
𝜏 𝑢 𝑢                                                 …(11) 
𝑢 𝑀 𝑞 𝑞 𝑘  𝑒 𝑘 𝑒  

𝐶 𝑞, 𝑞 𝑞 𝐹 𝑞 𝐺 𝑞                              …(12) 
where  
𝑢  is a standard computed torque that defined in 
(8). 
𝑢  is a robust control term that can handle the 
system uncertainties and external disturbance. 
 
Design of robust compensator controller 
  

To solve the problem robustness of the standard 
CTC, the following control is proposed:  
𝑢 𝑘 𝑠𝑖𝑔𝑛 𝛾e t e t                           …(13) 
Where  
𝑘  is the gain of the robust term,  𝛾  is a positive 
constant.  
Sign is the sign function. 
 
Theorem 1. Consider the robotic manipulator 
dynamic model (1) with the proposed control law 
in (2), the closed loop controlled system will be 



 Maryam S. Ahmed                                                 Al-Khwarizmi Engineering Journal, Vol. 17, No. 3, P.P. 22- 28 (2021) 

 

24 
 

stable and the tracking error and its derivative will 
converge to zero. 
 
Proof. The Lyapunov function candidate selected 
as follows:  
𝑆 t 𝛾e t e t     …(14) 
𝑞 𝑞 𝛾 𝑞 𝑞                                    …(15) 
By simple calculations it can be obtained the 
following  
𝑆 𝑡 𝑞 𝑞                                               …(16) 
𝑆 𝑡 𝑞 𝑞                                               …(17) 
𝑉 𝑡 𝑆 𝑀𝑆          …(18) 

  𝑉 𝑡 𝑆 𝑀𝑆 𝑆 𝑀𝑆                             …(19) 

𝑆 𝑀𝑆 𝑆 𝐶𝑆                                           ...(20) 
𝑆 𝑀 𝑞 𝑞 𝐶 𝑞 𝑞                     …(21) 
𝑆 𝑀𝑞 𝐶𝑞 𝑀𝑞 𝐶𝑞                       …(22) 
𝑆 𝑀𝑞 𝐶𝑞 𝐹 𝐺 𝜏                      …(23) 

Sub (11) in  (23)  
𝑉 𝑡 𝑆 𝑀𝑞 𝐶𝑞 𝐹 𝐺 𝑢 𝑢   
                                                                     …(24) 
Sub (12) and (13) in (24) 
 𝑆 𝑀𝑞 𝐶𝑞 𝐹 𝐺 𝑀 𝑞 𝑞
𝑘  𝑒 𝑘 𝑒 𝐶 𝑞, 𝑞 𝑞 𝐹 𝑞 𝐺 𝑞
𝑘 𝑠𝑖𝑔𝑛 S                                                     …(25) 
Let  
𝜌 𝑡 𝑀𝑞 𝐶𝑞 𝐹 𝐺 𝑀 𝑞 𝑞
𝐶 𝑞, 𝑞 𝑞 𝐹 𝑞 𝐺 𝑞                           …(26) 

𝑆 𝜌 𝑡 𝑀 𝑞 𝑘  𝑒 𝑘 𝑒 𝑞
𝑘 𝑠𝑖𝑔𝑛 S  

                                                                       ...(27) 
 𝑉 𝑡 𝑆 𝜌 𝑡 𝑘 𝑀 𝑞 𝑒 𝑘 𝑀 𝑞 𝑒 t
𝑘 𝑠𝑖𝑔𝑛 S      …(28) 
𝑉 𝑡 𝑆 𝜌 𝑡 𝑀 𝑞 𝑘 𝑘 𝑘 𝑒 𝑒 t
𝑘 𝑠𝑖𝑔𝑛 S    …(29) 
If parameters 𝑘  and 𝑘  selected as follows 
𝑘 𝑘 𝛾   …(30) 
Sub (30) in (29), yields:  
 𝑉 𝑡 𝑆 𝜌 𝑡 𝑘 𝑀 𝑆 𝑘 𝑠𝑖𝑔𝑛 S   …(31) 

𝑆 𝑘 𝑀 𝑆 𝑆 𝜌 𝑡 𝑘     …(32) 
‖𝑀 𝑞 ‖‖𝑘 ‖‖𝑆‖ ‖𝑘‖‖𝑆‖ ‖𝜌 𝑡 ‖‖𝑆‖                                                                             

…(33) 
If  𝑘  selected large enough greater than  𝜌 𝑡 , then     
𝑉 𝑡 ‖𝑀 𝑞 ‖‖𝑘 ‖‖𝑆‖   …(34) 
which implies  that: 
𝑉 𝑡 0   …(35) 
As a result the stability of the controlled system in 
(1) with the proposed control law in (11) is 
guaranteed. 
 
 
 

5. Simulation Results 
 

In this section, the performance and robustness 
of the proposed control method is tested by 
applying it on the two link robotic arm. Moreover, 
the proposed method is compared with the standard 
CTC method. Additionally, integral absolute value 
error (𝐼𝐴𝐸) performance index is used to examine 
the tracking error performances in this comparison 
that can be expressed as follows: 

𝐼𝐴𝐸 |𝑒 𝑡 |𝑑𝑡  …(36) 
 The dynamic model of two link robotic 
manipulator is [21]: 

𝜏
𝜏

𝐴 𝐴
𝐴 𝐴

𝑞
𝑞

2𝑏𝑞 𝑧𝑞
𝑏𝑞 0

𝑞
𝑞

𝑣 𝑞
𝑣 𝑞

𝑝 𝑠𝑔𝑛 𝑞
𝑝 𝑠𝑔𝑛 𝑞

𝐺
𝐺

   …(37) 

with 
𝐴 𝑚 𝑙 𝑚 𝑙 𝑙 2𝑙 𝑙 𝑐𝑜𝑠 𝑞     
𝐴 𝑚 𝑙 𝑙 𝑐𝑜𝑠 𝑞 𝑙     
𝐴 𝑚 𝑙     
𝑏 2𝑚 𝑙 𝑙 𝑠𝑖𝑛 𝑞    
𝑧 𝑚 𝑙 𝑙 𝑠𝑖𝑛 𝑞    
𝐺 𝑚 𝐿 𝑔𝑐𝑜𝑠 𝑞 𝑚 𝑔 𝐿 𝑐𝑜𝑠 𝑞 𝑞
𝐿 𝑐𝑜𝑠 𝑞   
𝐺 𝑚 𝑙 𝑔 𝑐𝑜𝑠 𝑞 𝑞    
where 𝑞 and 𝑞  are angular positions, 𝜏 and 𝜏  are 
torques, 𝐿  and 𝐿  are lengths, 𝑚  and 𝑚  are 
masses, 𝑣 , and 𝑣  are coefficients of viscous 
friction, and 𝑝  and 𝑝  are coefficients of dynamic 
friction of Link1and Link2, respectively. The 
parameters of the robotic manipulator are selected 
as: 𝑚 10 𝑘𝑔 , 𝑚 2 𝑘𝑔 ,  𝑙 1.1 m, 𝑙
0.8 m,  𝑃 𝑃 30 ,and  𝑉 𝑉 10 . The 
desired trajectory is 𝑞 𝑡 𝑞   𝑞  , where 
q sin 2πt  …(38) 
𝑞 sin 2πt                                                   …(39) 

The controller gains selected as follows: 𝑘
3, 𝑘 5, 𝑘 15.The robustness of the proposed 
method and standard CTC are checked by varying 
the parameters of the robotic system by 15% of 
their nominal. Moreover, a sin 𝑡 disturbance 
signal is applied. At second 3, Angular position and 
error in this position of robotic manipulators are 
shown in Figures 1 and 2 for the proposed and CTC 
methods. These figures indicated faster response of 
the proposed method. The proposed method needs 
approximately 0.06 seconds until tracking error 
converges to zero. Tables 1 and 2 list the 𝐼𝐴𝐸 for 
proposed control scheme and CTC methods for 
Link1 and Link2 respectively. 



 Maryam S. Ahmed                                                 Al-Khwarizmi Engineering Journal, Vol. 17, No. 3, P.P. 22- 28 (2021) 

 

25 
 

 
 
Fig. 1. Results for system uncertainty.  
 
 

These indices indicate that the performance of 
the prosed control method is better than standard 
CTC method in reduction tracking error.  Finally, 
it should be noted that all simulation results 
illustrate good robustness of proposed method 

against system uncertainty and external 
disturbance. 

 
Table 1, 
Performance of controllers under system 
uncertainty 

 Proposed  CTC 
Link 1 0.3464 2.1859 
Link 2 0.0419 1.7772 

 
Table 2 
Performance of controllers for disturbance rejection   

 Proposed  CTC 
Link 1 0.4679 2.7099 
Link 2 0.0485 2.2542 

 
 
6. Conclusion 

 
In this paper, a robust control method proposed 

for control a robotic manipulator. Robotic 
manipulator is non-linear system, especially with 
the inability to represent the system perfectly due 
to the change in the parameters and the external 
disturbance. Although, CTC is good control 
method, but it’s not robust to system uncertainties. 
This paper improves CTC method by adding a new 
robust term 
 
 
 

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 Maryam S. Ahmed                                                 Al-Khwarizmi Engineering Journal, Vol. 17, No. 3, P.P. 22- 28 (2021) 

 

26 
 

 
 
Fig. 2. Results for disturbance rejection.   
 
 

Moreover, Lyapunov theorem stability has been 
used to approve stability of the proposed control 
method. It was concluded from the simulation 
results that there was a significant improvement in 

the performance of the proposed control in 
response to external disturbance and uncertainties 
in parameters of robotic manipulator. In future 
work, reinforcement learning can be used to select 
the optimal gains for the proposed controller 
parameters. Moreover, the proposed algorithm can 
be tested in the real robotic manipulator.   

 
 

7. References  
 
[1] A. Mary, T. Kara and A. Miry, “Inverse 

kinematics solution for robotic manipulators 
based on fuzzy logic and PD control”, Al-
Sadiq International Conference on 
Multidisciplinary in IT and Communication 
Techniques Science and Applications(AIC-
MITCSA), vol.pp.160-165, 2016. 

[2] A. Kashyap and D. Parhi, “Particle Swarm 
Optimization aided PID gait controller design 
for a humanoid robot”, ISA Transaction, vol. 
114, pp.306-330, 2020.  

[3] I. Carlucho and G. Acosta, “An adaptive deep 
reinforcement learning approach for MIMO 
PID control”, ISA Transactions, vol.102, 
pp.280-249, 2020. 

[4] [4] X. Yin, L. Pan and S. Cai, “Robust 
adaptive fuzzy sliding mode trajectory 
tracking control for serial robotic 
manipulators”, Robotics and Computer-
Integrated Manufacturing, vol. 72, pp. 
101884, 2021. 

[5] A. Mary and T. Kara,” Robust Proportional 
Control for Trajectory Tracking of a 
Nonlinear Robotic Manipulator: LMI 
Optimization Approach”, Arabian Journal for 
Science and Engineering, vol. 41, pp. 5027-
5036, 2016. 

[6] X. Wu and Y. Huang, “Adaptive fractional-
order non-singular terminal sliding mode 
control based on fuzzy wavelet neural 
networks for omnidirectional mobile robot 
manipulator”, ISA Transactions, vol. 115, 
2021. 

[7] Z. Chen, X.Yang and X. Liu, “RBFNN-based 
non-singular fast terminal sliding mode 
control for robotic manipulators including 
actuator dynamics”, Neurocomputing,  vol. 
362, pp. 72-92, 2019. 

[8] T.  Kara and A. Mary,” Robust trajectory 
tracking control of robotic manipulators based 
on model-free PID-SMC approach” Journal of 
Engineering Research, vol. 6, pp. 170-188, 
2018. 

[9] T.  Kara and A. Mary,” Adaptive PD-SMC for                     
Nonlinear Robotic Manipulator Tracking 

0 1 2 3 4 5
-1.5

-1

-0.5

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time(s)

P
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c
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lin

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ideal position

Proposed

CTC

0 1 2 3 4 5
-10

-5

0

5

10

time(s)

S
p
e
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d
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c
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g
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f 
lin

k
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ideal speed

Proposed

CTC

0 1 2 3 4 5
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-5

0

5

10

time(s)

S
p
e
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d
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c
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g
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lin

k
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ideal speed

Proposed

CTC

0 1 2 3 4 5
-3

-2

-1

0

1

time(s)

P
o
s
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o
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c
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f 
lin

k
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ideal position

Proposed

CTC



 Maryam S. Ahmed                                                 Al-Khwarizmi Engineering Journal, Vol. 17, No. 3, P.P. 22- 28 (2021) 

 

27 
 

Control” Studies in Informatics and Control, 
vol. 26, pp. 49-58, 2017. 

[10] Th. Costa, F. Lara-Molina, A. Junior and E. 
Taketa,”Robust H∞ Computed torque Control 
for Manipulators”, IEEE Latin America 
Transactions, vol. 16, pp. 398-407, 2018. 

[11] M. Alwan, Hassan and R. Zaid Hikmat,” 
Motion Control of Three Links Robot 
Manipulator (Open Chain) with Spherical 
Wrist”, Al-Khwarizmi Engineering Journal 
15(2) ,13-23. 2019. 

[12] C. Chen, C. Zhang, T. Hu, H. Ni1, and W. 
Luo,“Model-assisted extended state observer-
based computed torque control for trajectory 
tracking of uncertain robotic manipulator 
systems”, International Journal of Advanced 
Robotic Systems, vol. 15, pp. 1-12, 2018. 

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



 ) 2021( 22- 28 صفحة ، 3العدد ، 17المجلد الهندسية الخوارزمي جلةم                                   مريم صادق احمد                          

 

28 
 

 
 
 

  التحكم المتين في متحكم  عزم  دوران   ذراع االنسان االلي
  

***هشام  حسن  جاسم            **علي حسين مري              احمد* مريم صادق
 *،**،***قسم هندسة الميكاترونكس/ كلية الهندسة الخوارزمي/ جامعة بغداد

Maryamsadeq97@gmail.com :البريد  االلكتروني* 
Alimary76@kecbu.uobaghdad.edu.iq  :البريد االلكتروني** 

mschisham@gmail.com  :البريد االلكتروني*** 

 
 
 

  الصةالخ
  

ثل نظام يدرس هذا البحث السيطرة على ذراع روبوت حيث تم إقتراح بناء منظومة سيطرة باالعتماد على طرق السيطرة الذكية. إذ إن ذراع الروبوت تم
 uncertaintyال خطيا وخاصة مع عدم القدرة على تمثيل النظام بشكل مثالي بسبب ازعاجات الحمل واإلخطاء التي تحصل عند نمذجة النظام وهو مايسمى 

system  وهذا من خالل تحسين تحكم عزم الدورانCTCاستخدمت نظرية, Lyapunov   للحصول على استقرارية النظام. و تم اختبار أداء المتحكم المقترح
مقترحة ومتانتها والحصول  متحكم عزم الدوران المحسوب التقليدي, حيث توضح نتيجة المحاكاة قوة الطريقة الومقارنتها مع    matlab - simulink بواسطه 

على تتبع مسار جيد وفق االداء الطلوب.