

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

 
Al-Khwarizmi 

Engineering 

Journal 

Al-Khwarizmi Engineering Journal 

           ISSN (printed): 1818 – 1171, ISSN (online): 2312 – 0789 

Vol. 20, No. 1, March, (2024), P P. 63- 75 
 

Adaptive Robust Tracking Control of Robotic Manipulator based on 

SMC and Fuzzy Control Strategy 
  

Ali Hussien Mary*     Ahmad Al-Talabi**      Tolgay Kara***                                 

Dina Saadi Muneam****           Mohammad Yahya Almuhanna*****  

Laith Awda Kadhim Mayyahi****** 

*,****,***** Department of Mechatronics  Engineering/Al-khwarizmi College of Engineering/ University of Baghdad/ 

Baghdad/ Iraq 

**Department of Medical Instrumentation Techniques Engineering/ College of Engineering and Information 

Technology / AlShaab University/ Baghdad/Iraq 

*** Gaziantep University/ Türkiye 

******Carleton University / Faculty of Engineering and Design/ Ottawa/ Canada 

                            Corresponding Author: *Email: Alimary76@kecbu.uobaghdad.edu.iq 

**Email: ahmad.altalabi@alshaab.edu.iq 

***Email: kara@gantep.edu.tr 

 ****Email: deena@kecbu.uobaghdad.edu.iq 

******Email: Mohammad.Yahya@kecbu.uobaghdad.edu.iq 

******Email: Laith.mayyahi@carleton.ca 

 
)Received 22 August 2023; Accepted 12 November  2023 Published 1 March 2024( 

002.11.4https://doi.org/10.22153/kej.202 
 

Abstract 

 
In recent years, robotic systems have been widely used in different applications, and this has motivated researchers to 

develop different control methods.  A model-free, intelligent, robust control method for a nonlinear robotic manipulator 

system is proposed in this work. This paper presents a novel solution for the major drawbacks of the sliding mode control 

scheme, which are chattering. Prior knowledge is needed about the dynamic model of the controlled system and the upper 

bound of uncertainty. In this paper, a fuzzy-like PD controller with SMC (FLPDSM) is proposed. The fuzzy-like PD 

controller was designed according to fuzzy rules and membership functions based on the nominal model of the robot 

manipulator. A robust control term was added to the control signal to compensate for the system uncertainty, and external 

disturbances are compensated by adding an auxiliary robust term to the SMC control law. Two methods for designing 

robust control terms are proposed. The first proposed method assumes that the upper bound of system uncertainty is 

known although it cannot be exactly determined due to external disturbances and uncertainty. Hence, a second method 

was proposed that assumes this bound to be unknown, and an adaptive gain based on Lyapunov theory was used to derive 

the adaptation law. The Lyapunov second method was used to ensure the stability of the closed loop system. Performance 

tests on the proposed methods were implemented through simulation studies for the two-link robotic manipulator, and the 
test results were compared with the standard SMC to verify the effectiveness of the proposed method. A good trajectory 

tracking with a high robustness against parameter variations and external disturbances was observed under the presented 

control scheme. 

 
Keywords: SMC, Robotic Systems, Fuzzy Control, Trajectory Tracking. 

 
1.  Introduction 
 

Robotic manipulators are used successfully in 
many particular applications, and especially in 

industrial factories. Accuracy and precision are 
important features that encourage the use of robotic 
manipulators in plants that aim to enhance their 
products and manufacturing processes [1]. There 

mailto:Alimary76@kecbu.uobaghdad.edu.iq
mailto:ahmad.altalabi@alshaab.edu.iq
mailto:Yahya@kecbu.uobaghdad.edu.iq
mailto:Laith.mayyahi@carleton.ca
https://doi.org/10.22153/kej.2024.11.002


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    64 

are many challenges that make tracking accuracy 
in robotic manipulators difficult, including high 
nonlinearity, system uncertainties, and strong 
coupling between adjacent joints, hence, several 
control schemes have been proposed to solve these 
problems by designing a stable and robust 
controller. Because of its simplicity in structure 

and the relatively easy tuning of parameters, the 
Proportional Integral Derivative (PID) controller is 
applied in different control system [2]. Fuzzy logic 
was used to schedul of the PID controller to control 
the hybrid robot manipulator [3]. Fuzzy type 2 had 
been proposed for the control of the 2dof robotic 
manipulator with Grey wolf optimization used for 
tuning the parameters [4]. The particle swarm 

optimization method was combined with the PID 
controller to stabilize the humanoid robot [5] . An 
adaptive backstepping control with a simple 
adaptive estimated Lyapunov theorem has been 
used for the control of the robotic manipulator [6]. 
Many advanced control schemes, including 
adaptive and artificial intelligence methods, have 

been used to tune the parameters of the PID. The 
number of degrees of freedom and system 
uncertainties of the robotic manipulator have 
significant effects on control performance. The 
sliding mode control (SMC) represents an efficient 
control scheme for nonlinear systems, that is 
applied successfully in many mechanical systems 

and robotic manipulators. Chattering is the major 
disadvantage of SMC and different control 
schemes have been proposed to eliminate the 
chattering. Among the solutions to the chattering 
problem in the SMC is the use of saturation 
approximation functions instead of the 
discontinues function, and low-pass filtering [7-
12]. Recently, the fuzzy logic technique has been 

widely applied to approximate the signum 
discontinues term [13-18]. Implementing the SMC 
control law requires the upper bound of the 
uncertainties, and the external disturbance must be 
known [19]. The values of these bounds are very 
important in the selection of the switching gain. To 
ensure stability, the switching gain must be greater 

than the upper bound of uncertainty, which is 
unknown, and assuming large values for the upper 
bound may be the reason for chattering. This paper 
presents two robust control schemes based on 
fuzzy control and SMC. The important features of 
the proposed controller in this paper can be 
summarized as follows: i) the upper bound of 

uncertainty is not required, ii) the proposed 
controller is model-free, iii) Lyapunov theory is 
used to avoid the overestimation of the switching 
gain, construct an adaptation law for the switching 

gain, and also guarantee the stability of the 
controlled system.  

 
2. Robotic Manipulator Dynamic 
 

The dynamic equations of n rigid-link robotic 
manipulator system based on the Lagrange-Euler 
equations of motion are: 

𝑀(𝑞)�̈� + 𝐶(𝑞, �̇�)�̇� + 𝐹(�̇�) + 𝐺(𝑞) + 𝜏𝑑 =  … (1) 

𝑀(𝑞) = 𝑀𝑜(𝑞) + ∆𝑀(𝑞)                     …(2) 
𝐶(𝑞, �̇�) = 𝐶𝑜(𝑞, �̇�) + ∆𝐶(𝑞, �̇�)                     …(3) 

𝐹(�̇�) = 𝐹𝑜(�̇�) + ∆𝐹(�̇�)              …(4) 

Where q = [𝑞1, 𝑞2, ⋯ , 𝑞𝑛]𝑇   ∈ Rn  is the joint 

angular position vector, q̇  = [�̇�1, �̇�2, ⋯ , �̇�𝑛]𝑇   ∈
Rn is the joint angular velocity vector, M(q) ∈
Rnx n  denotes the inertia matrix, C(q, q̇) ∈ Rnx n 

represents the centrifugal-Coriolis matrix,F(q̇) ∈
Rn is the friction torque vector, G(q) ∈ Rn denotes 

the gravity term, τd ∈ Rn  is the external 

disturbance vector, and τ = [𝜏1, 𝜏2, ⋯ , 𝜏𝑛]𝑇 is the 

torque vector. Mo(q), Co(q, q̇) and Fo(q̇) refer to 
the nominal model of the robotic manipulator, and 

∆M(q),∆C(q, q̇) and ∆F(q̇) refer to the uncertainty 
in the dynamic model of the robotic manipulator. 
The proposed control method assumes the 
following [ 20]: 
Assumption 1: Boundedness of the inertia matrix 
‖𝑀(𝑞)‖ ≤ 𝑘1                                …(5) 

where 𝑘1 is a positive scalar. 
Assumption 2: Boundedness of the centrifugal 
matrix 
‖𝐶(𝑞, �̇�)‖ ≤ 𝑘2          …(6) 

where 𝑘2 is a positive scalar. 
Assumption 3: Boundedness of the friction vector 
‖𝐹(�̇�)‖ ≤ k3‖q̇‖ + F0                     …(7)  

where 𝑘3 and 𝐹0 are positive scalars.  
Assumption 4: Boundedness of the gravity vector 
‖𝐺(𝑞)‖ ≤ 𝑘4                             …(8) 

where 𝑘4 is a positive scalar. 
Assumption 5: The model in (1) is linearly 
parameterized, so it can be represented by the 
following expression:  

Y∅ = M(q)q̈r + C(q, q̇)q̇r + G(q) + F(q̇)    …(9) 

�̇�𝑟 = �̇�𝑑 + 𝛾(𝑞𝑑 − 𝑞) …(10) 
where Y = Y(q, q̇, q̇r, q̈r) ∈ Rn×p is a matrix that 

contains a known nonlinear function, ∅ ∈ Rp is a 

vector that contains unknown parameters, and 𝛄 is 
a positive diagonal matrix. 
Assumption 6: The desired trajectories and their 

derivativesqd(𝑡),q̇d(𝑡), and q̈d(𝑡)are bounded as 
follows: 
|𝑞𝑑(𝑡)| ≤ 𝑀𝑑1 , |�̇�𝑑(𝑡)| ≤ 𝑀𝑑2 , |�̈�𝑑(𝑡)| ≤
𝑀𝑑3, …(11) 

with 𝑀𝑑1, 𝑀𝑑2, and 𝑀𝑑3being positive constants.  


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    65 

3. Sliding Mode Control 

 
The objective of SMC is to make trajectory 

states 𝐪track the desired trajectory𝐪𝐝. The first step 

in the SMC is designing the sliding surface. For a 

second-order system, the sliding surface is given 

by: 

𝑠(𝑡) = 𝛾 𝑒(𝑡) + �̇�(𝑡) …(12) 

𝑒(𝑡) = 𝑞 − 𝑞𝑑 …(13) 

𝑒(𝑡) = [𝑒1(𝑡) 𝑒2(𝑡)  ⋯ 𝑒𝑛(𝑡)]𝑇 …(14) 

where 𝐞(𝑡)is  the error signal that represents the 
difference between the desired trajectory and the 
actual trajectory. An equivalent control term in the 

conventional SMC is calculated by setting  �̇�(𝑡) =
0 , and this will determine the control effort 
required to achieve a good performance without 
considering the external disturbance and system 
uncertainties. 

ṡ(𝑡) = γė(𝑡) + ë(𝑡) …(15) 

= γė(𝑡) + q̈ − q̈d …(16) 

q̈ = M−1(q)[τ − C(q, q̇)q̇ − F(q̇) − G(q) − τd] …(17) 

where 𝛄is a diagonal matrix. 

In the equivalent control term, only the known part 

of the dynamic model of the controlled system is 

taken into account, which yields: 

�̇�(𝑡) = 𝛾 �̇�(𝑡) + 𝑀𝑜
−1(𝑞)[𝜏 − 𝐶𝑜(𝑞, �̇�)�̇� −

𝐹𝑜(�̇�) − 𝐺𝑜(𝑞)] …(18) 

�̇�(𝑡) = 0 …(19) 

γ ė(t) + Mo
−1(q)[τeq − Co(q, q̇)q̇ − Fo(q̇) −

Go(q)] = 0 …(20) 

𝜏𝑒𝑞 = 𝐶𝑜(𝑞, �̇�)�̇� − 𝐹𝑜(�̇�) − 𝐺𝑜(𝑞) +

𝑀𝑜(𝑞)𝛾 �̇�(𝑡) …(21) 

However, the equivalent control term is not 
sufficient to achieve good performance in practical 
applications due to many challenges like parameter 
variations and external disturbances. A control law 
is presented to compensate for these uncertainties. 
The overall SMC control signal is: 

τ = τ𝑒𝑞 + τ𝑟 …(22) 

τ𝑟 = 𝑘 𝑠𝑔𝑛(s) =

[𝑘1𝑠𝑔𝑛(𝑠1)  ⋯ 𝑘𝑛𝑠𝑔𝑛(𝑠𝑛)]𝑇 …(23) 

𝑠𝑔𝑛(𝑠𝑖) = {
 1         𝑖𝑓 𝑠𝑖 > 0
−1     𝑖𝑓 𝑠𝑖 < 0

 …(24) 

where 𝛕𝒓 is the robust term and the value of k is 
large and  must be greater than the upper bound of 

uncertainty. 
There are many challenges to implementing the 

control law of the conventional SMC. As shown in 
(21), the exact dynamic mode of the robotic system 
and the upper bound of uncertainty must be known 
to select the gain of the robust term. Moreover, the 
sign function causes the chattering phenomena that 
may cause damage to the actuators. 

 
4. Proposed FLPDSM Design 
 

This section discusses in detail the two terms of 
the proposed control method. Figure 1 displays the 

block diagram for the proposed control scheme. 
The proposed control law is: 

τ = τ𝐹 + τ𝑟  …(25) 

where 𝛕𝑭  is the output of the fuzzy-like PD 
controller that was designed based on the nominal 

model of the robotic system, and 𝛕𝒓refers to the 
robust control signal that overcomes the 
uncertainties and external disturbances that were 
not included in the fuzzy-like PD controller’s 
design. Two methods are proposed for the robust 

control term𝛕𝒓 : In the first method, the upper 
bound of the dynamic model of the robotic system 
is assumed to be known, while in the second 
method, this bound is assumed to be unknown. 

 
4.1  Fuzzy-Like PD Controller 

 
A fuzzy-like PD controller is proposed to 

achieve good performance with fuzzy rules, and 
was designed without considering parameter 
variations, and external disturbances. Only the 

nominal model of the robot manipulator is 
considered. The inputs to the fuzzy controller are 
the normalized error and the derivative of the error. 

The output of the fuzzy controller is τ𝐹 .  
τ𝐹 = kf𝑢𝑓 …(26) 

𝑢𝑓(𝑡) =  𝐹𝑃𝐷(𝑒(𝑡), �̇�(𝑡))  …(27) 

 
Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    66 

 
Fig. 1. Basic decentralized control scheme for two link robotic manipulator. 

 
where 𝒌𝒇  is a positive diagonal matrix that 

refers to the output scaling factor, and 

𝐹𝑃𝐷(𝒆(𝑡), �̇�(𝑡))denotes the fuzzy logic decision 

system. The membership functions for the input 

variables 𝒆(𝑡) and �̇�(𝑡)  and the output  
variable𝒖𝒇(𝑡)  are shown in figure 2. Five fuzzy 

functions, defined as Negative Big (NB), Negative 
Small (NS), Zero (Z), Positive Small (PS), and 
Positive Big (PB), are used as membership 

functions. Table 1 lists the rules used in this 
controller. The intersection minimum has been 
used for the fuzzification process, while the center 
average operations were used for defuzzification. 

 
Fig. 2. Membership functions for the fuzzy controller 

 
Table 1, 

 Fuzzy controller rules 

𝒆(𝒕) 

�̇�(𝒕) 
NB NS Z PS PB 

NB NB NB NB NS Z 

NS NB NB NS Z PS 

Z NB NS Z PS PB 

PS NS Z PS PB PB 

PB Z PS PB PB PB 

4.2 Robust Auxiliary Controller 
 

Any robust control scheme must take into 
account parameter variations and external 
disturbances in order to provide a controller that is 
robust to unpredictable variations. In this section, 
two methods are presented to select a suitable 
auxiliary controller. In the first proposed method, 

the upper bound of uncertainty is assumed to be 
known, while in the second proposed method, it 
assumed to be unknown. 
 

4.2.1 Proposed I: Constant Switching Gain  
  

In this method, the switching gain, whose value 
is determined based on Lyapunov theory, is kept 
constant, as shown in figure 3. 

𝛕𝒓 = 𝒌 𝑠𝑔𝑛 (𝐬) =
[𝑘1𝑠𝑔𝑛(𝑠1)  ⋯ 𝑘𝑛𝑠𝑔𝑛(𝑠𝑛)]𝑇 …(28) 
The proposed control law can be written as: 

τ = 𝑘𝑓𝑢𝑓 + 𝑘 𝑠𝑔𝑛(s) …(29) 

Theorem 1 
Considering the nonlinear robotic system in (1), 
and the proposed robust fuzzy control method in 

(29), the closed loop system will be asymptotically 
stable with approximately zero error signals if the 
controller parameters are selected as follows: 

‖k‖ > ‖Y∅ + 𝑘𝑓‖ …(30) 

𝑘𝑓 > 0 …(31) 

Proof. 

Let 𝑉  be the candidate Lyapunov function used to 
verify the stability. 

𝑉 =
1

2
s𝑇Ms …(32) 

�̇� = s𝑇Mṡ +
1

2
s𝑇Ṁs = s𝑇Mṡ + s𝑇Cs …(33) 

�̇� = s𝑇[M(q)q̈r + C(q, q̇)q̇r + G(q) + F(q̇) −
τ] …(34) 

�̇� = s𝑇[Y∅ − τ] …(35) 

Gc 1 

- 
𝑞1𝑑  +  

𝜏2 𝑞2 

𝜏1 

𝑞1 
Robotic 

Manipulator 

𝑞2𝑑 
Gc 2 

+ - 

-1 -0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

d
e
g
re

e
 o

f 
m

e
m

b
e
rs

h
ip

 
NB NS Z PS PB

𝑒(𝑡)  , �̇�(𝑡) , 𝑢𝑓  


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    67 

�̇� = s𝑇[Y∅ − τF − τr] …(36) 

�̇� = s𝑇[Y∅ − 𝑘𝑓𝑢𝑓 − τ𝑟] …(37) 

The outputs of the fuzzy controller are normalized 
between [-1, 1], then  

‖uf‖ ≤ 1 …(38) 

s𝑇𝑘𝑓uf ≤ ‖s‖‖kf‖ …(39) 

�̇� ≤ ‖s‖[‖Y∅‖ − ‖k‖ + ‖kf‖] …(40) 
 

Fig. 3. Block diagram for the proposed scheme I (constant robust gain). 

 
�̇� ≤ ‖s‖[‖Y∅‖ + ‖kf‖ − ‖k‖] …(41) 

If  k is selected based on the following condition: 
‖k‖ > ‖Y∅‖ + ‖kf‖ …(42) 
Then  

�̇� ≤ 0 …(43) 
thus, the closed loop system is asymptotically 
stable. 

Remark: 
The problem is that the upper bound of the 

robotic system dynamic (𝐘∅) is not exactly known 
and is related to the upper bound of uncertainty. 

Selecting a larger value for  𝐤  will cause 
chattering, and a smaller value may make the 
system unstable. A second method was proposed to 
overcome this problem.  

 
4.2.2 Proposed II: Adaptive gain  
 

To avoid the problem of the unknown upper 

bound of the dynamic model of the robotic system, 
Lyapunov’s theorem is applied to design an 
adaptation law for the gain of the robust term to 
estimate the upper bound of the robot dynamic. As 
a result, there is no need to know the upper bound 
of the dynamic model, which is related to the upper 

bound of system uncertainty and external 
disturbance, which cannot be easily determined in 
practical applications. Figure 4 shows the second 
proposed method. 
Let 

𝜌 = Y∅ − 𝑘𝑓𝑢𝑓 …(44) 

𝝆cannot be determined exactly because it’s based 
on dynamic of the robot manipulator. 
The proposed robust control is: 

τ𝑟 = 𝜌 ...(45) 

Where �̂�  represents estimation of 𝝆 . Then the 
estimation error can be defined as: 

�̃� = 𝜌 − 𝜌                                                     … (46) 
The proposed control law is: 

τ = 𝑘𝑓uf + 𝜌 ...(47) 

Theorem 2 If the control law in (46) is used for the 
nonlinear robotic manipulator in (1), then the 
controlled system will be asymptotically stable 

with zero error signals if the parameters �̂�  are 
adjusted by the following adaptation law: 

�̇̃� = −𝐿𝑇 −1
𝑠 …(48) 

𝜌(𝑡) = ∫ �̇̃�(𝑡)
𝑡

𝑡−1
𝑑𝑡 + 𝜌(𝑡 − 1) …(49) 

 
where 𝐋 ∈ R𝑛×𝑛 is the adaptation rate. 

 
Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    68 

 
Fig. 4. Block diagram for the proposed scheme II (adaptive robust gain). 

 
Proof. 

Let 

Let 𝑉be the candidate Lyapunov function. 

𝑉 =
1

2
s𝑇Ms +

1

2
�̃�𝑇L�̃� …(50) 

�̇� = s𝑇Mṡ +
1

2
s𝑇Ṁs + �̇̃�𝑇L�̃� …(51) 

= s𝑇Mṡ + s𝑇Cs + �̇̃�L�̃� …(52) 

s𝑇[M(q)q̈r + C(q, q̇)q̇r + G(q) + F(q̇) − τ] +
�̇̃�𝑇L�̃� …(53) 

�̇� = s𝑇[Y∅ − 𝑘𝑓uf − τr] + �̇̃�𝑇L�̃� …(54) 

�̇� = s𝑇[𝜌 − 𝜌] + �̇̃�𝑇L�̃� …(55) 

�̇� = s𝑇[�̃�] + �̇̃�𝑇L�̃� …(56) 

�̇� = [s𝑇 + �̇̃�𝑇L]�̃� …(57) 

If  �̇̃� = −𝐿𝑇 −1
s …(58) 

Then  

�̇� = 0 …(59) 
As a result, the adaptation law for the control 

parameter  𝜌(𝑡)will be as follows: 

𝜌(𝑡) = ∫ �̇̃�(𝑡)
𝑡

𝑡−1
𝑑𝑡 + 𝜌(𝑡 − 1) …(60) 

Thus, the controlled system with the proposed 
method is asymptotically stable with zero tracking 
error. 
 
 
5. Simulation Results 
 

In this section, a two-link rigid planar robotic 
manipulator system is used to illustrate the 
robustness and effectiveness of the presented 
method. Figure 5 shows the schematic diagram of 

the two-link planar robotic manipulator, with the 
dynamic model expressed as follows: 

[
𝜏1

𝜏2
] = [

𝑀11 𝑀12

𝑀12 𝑀22
] [

�̈�1

�̈�2
] +

[
−𝑏�̇�2 −𝑏�̇�1 − 𝑏�̇�2

−𝑏�̇�1 0
] [

�̇�1

�̇�2
] + [

𝑣1�̇�1

𝑣2�̇�2
] +

[
𝑝1𝑠𝑔𝑛(�̇�1)
𝑝2𝑠𝑔𝑛(�̇�2)

] + [
𝑔1

𝑔2
] …(61) 

with 

𝑀11 = 𝐼1 + 𝐼2 + 𝑚1𝐿𝑐1
2

+ 𝑚2(𝐿𝑐2
2 + 𝐿1

2

+ 2𝐿1𝐿𝑐2𝑐𝑜𝑠(𝑞2))𝑀12

= 𝐼2

+ 𝑚2(𝐿𝑐2
2 + 𝐿1𝐿𝑐2cos (𝑞2)) 

𝑀22 = 𝐼2 + 𝑚2𝐿𝑐2
2,  

𝑏 = 𝑚2𝐿1𝐿𝑐2sin (𝑞2), 
𝑔1 = 𝑚1𝐿𝑐1𝑔𝑐𝑜𝑠(𝑞1) + 𝑚2𝑔(𝐿𝑐2𝑐𝑜𝑠(𝑞1 + 𝑞2) +
𝐿1𝑐𝑜𝑠(𝑞1)), 

𝑔2 = 𝑚2𝑔𝐿𝑐2𝑐𝑜𝑠(𝑞1 + 𝑞2). 
where 𝑞1 and 𝑞2  are angular positions, 𝜏1  and 𝜏2 

are torques, 𝐿1 and 𝐿2 are lengths, 𝑚1 and 𝑚2 are 
masses, 𝐼1and𝐼2  are lengthwise centroid inertia, 

𝐿𝑐1  and 𝐿𝑐2 are distances from the joint to  the 

center of gravity, 𝑣1  and 𝑣2 are coefficients of 

viscous friction, and  𝑝1 and 𝑝2 are coefficients of 
dynamic friction of  Link1 and Link2, respectively. 
The parameters of the two-link robotic manipulator 
used in the current simulation study are listed in 
table 2. In order to prove its effectiveness, the 
proposed control scheme is compared with the 


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    69 

standard SMC. Table 3 lists the values of the 
parameters of the proposed controllers and the 
SMC that are used in the simulation.  The integral 

time absolute error (𝐼𝑇𝐴𝐸) performance index is 
used for the comparison, which was used to  
numerically evaluate the performance of the 
tracking error. 

𝐼𝑇𝐴𝐸 = ∫ 𝑡|𝑒(𝑡)|𝑑𝑡
𝑡𝑓

0
     …(62) 

 
Fig. 5. Schematic diagram of two linkobotic system. 

 
Table 2, 
Robotic Manipulator parameters 

 
5.1 Robustness test: Model Uncertainties 
 

The robustness and effectiveness of the 
presented control methods are examined in the 
presence of the model of uncertainties and 

compared with the SMC. The system uncertainty 
includes variations in the mass, static, and dynamic 
coefficients of friction of Link1 as well as Link2. 
These parameters are increased by 15% of their 
nominal values. The desired trajectory used in this 
simulation is given as: 

𝑞1𝑑(𝑡) = −0.1 + cos (2𝜋𝑡) …(63) 

𝑞2𝑑(𝑡) = 0.5 + 𝑠𝑖𝑛 (2𝜋𝑡) …(64) 
The ITAE values for the proposed method I, 

proposed method II, and SMC are listed in table 4. 
To illustrate the comparison, variations in the 
ITAE for Link1 and Link2 are shown in figure 6. 

This comparison indicates that the ITAE for the 
proposed method II is less than for the other 
methods, whereas the ITAE for the proposed 
method I and SMC are approximately equal which 
means that the proposed method II is more robust 
than the SMC and the proposed method I. Figures 
7 and 8 show the tracking position, tracking error, 

and input torque for Link1 and Link2, respectively. 
These figures clearly indicate that the proposed 
control methods have very good tracking 
performance and smaller position tracking errors in 
Link1 and Link2. However, with respect to the 
chattering problem, figures 7 (c) and 8 (c) show 
that the control signals of proposed methods I and 

II are significantly smoother than that of the SMC. 
 
 
Table 3, 

 The parameters of proposed and SMC controllers 

 
x 

y 

I,1m

1L 1 

,2m

 2I

2L 

q

 
2q 

mp 

c2I 

c1I 

1L 

2L 

Parameter    Link1 Link2 

Mass (kg) 
1.0 1.0 

Length (m) 1.0 1.0 

Viscous friction coefficient  0.1 0.1 

Ddynamic friction coefficient  0.1 0.1 

Lengthwise centroid inertia (kg 

m2) 

0.5 0.5 

Distance from joint to center of 

gravity (m) 

0.2 0.2 

Control Method  Law Parameter Link1 link2 

Proposed II 

τ = 𝑘𝑓 𝑢𝑓 + �̂� 

�̇̂� = 𝐿−1𝑠𝑇 

𝑠(𝑡) = 𝛾 𝑒(𝑡) + �̇�(𝑡) 

𝑘𝑓  250 250 

𝐿 1 1 

𝛾 5 5 

Proposed I 

τ = 𝑘𝑓 𝑢𝑓 + 𝑘 𝑠𝑎𝑡(𝑠, ∅) 

 
𝑘𝑓  250 250 

𝑘 300 300 

∅ 0.05 0.05 

SMC 

𝑢 = 𝑀0(𝑞)�̈�𝑟 + 𝑁0(𝑞)�̇�𝑟 + 𝐺0 (𝑞) + 𝐻0(𝑞)
+ 𝑘1𝑠𝑎𝑡(𝑠, ∅) 

�̇�𝑟(𝑞) = �̇�𝑑 − 𝛾(𝑞 − 𝑞𝑑 ) 

𝑠(𝑡) = 𝛾𝑒(𝑡) + �̇�(𝑡) 

𝑘1 400 400 

∅ 0.05 0.05 

𝛾 5 5 


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    70 

 
Fig. 6. ITAE variations for model uncertainties 

 
Table 4, 
Performance index ITAE values for model 

uncertainties 

 
Fig. 7. Angular position (a), tracking error (b), and input torque (c) of Link1 under model uncertainties 

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

proposed
II

Proposed I SMC

Link1

Link2

 Proposed II Proposed I SMC 

Link1  0.0483 0.0648 0.0645 

Link2 0.0552 0.0551 0.0556 


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    71 

5.2 Robustness Test: Disturbance Rejection 
 

This section discusses the robustness of the 
proposed controller in the case of an external 
disturbance when applied to the controller output 

for Link1 and Link2. The disturbance signals 𝑑1(𝑡) 

and 𝑑2(𝑡) that were applied respectively, on Link1 
and Link2 are: 

𝑑1(𝑡) = 9𝑠𝑖𝑛(5𝑡), 𝑑2(𝑡) = 9sin (7𝑡) …(65) 
The desired trajectory used in this simulation is 
given as: 

𝑞1𝑑(𝑡) =  𝑞2𝑑(𝑡) = sin(2𝜋𝑡) … (66) 
The ITAE values are listed in table 5. The ITAE 
variations for all methods are also shown in figure 

9, which indicates the clear superiority of proposed 
method II. The simulation results for this case are 
shown in figures 10 and 11. The results obtained 
show a fast response of the proposed and SMC 
methods with good tracking performance. The 
results also clearly indicate the superiority of the 
proposed method II in comparison with the SMC 

and the proposed method I. Moreover, the 
adaptation technique that was used to estimate the 
upper bound of the dynamic model eliminates the 
chattering. Therefore, the control signal of the 
proposed method II is very smooth. 
 

Fig. 8. Angular position (a), tracking error (b), and input torque (c) of Link2 under model uncertainties 


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    72 

 
Fig. 9.  ITAE variations for adding external 

disturbance  

Table 5, 
Performance index ITAE values for disturbance 

rejection. 

 
Fig. 10. Angular position (a), tracking error (b), and input torque (c) of link1 subjected to external disturbance 

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

proposed
II

Proposed I SMC

Link1

Link2

 
0 0.5 1 1.5 2 2.5 3 3.5 4
-1.5

-1

-0.5

0

0.5

1

1.5

p
o
s
it
io

n
(r

a
d
)

 
ideal position

Proposed II

SMC

Proposed I

0 0.5 1 1.5 2 2.5 3 3.5 4
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

e
rr

o
r(

ra
d
)

 
Proposed II

SMC

Proposed I

0 0.5 1 1.5 2 2.5 3 3.5 4
-200

-100

0

100

200

300

tr
o
q
u
e
 (

N
m

)

time(s)

 
Proposed II

SMC

Proposed I

 Proposed II Proposed I SMC 

Link1  0.0493 0.0655 0.0653 

Link2 0.0590 0.0613 0.0633 


Ali Hussien Mary                                      Al-Khwarizmi Engineering Journal, Vol. 20, No. 1, P.P. 63- 75 (2024) 

 
    73 

 
Fig. 11. Angular position (a), tracking error (b), and input torque (c) of link2 subjected to external disturbance. 

 
6. Conclusion 
 

This paper proposes an adaptive fuzzy robust 
control system for robotic manipulators. Two 
robust controllers are proposed for the two cases in 
which the upper bound of the dynamic model is 

known, and the upper bound is unkown. The 
proposed intelligent, robust, and model-free 
control scheme based on FLC can be applied 
successfully in practical applications due to its 
simplicity in structure. It combines the robustness 
of the SMC and an intelligent adaptation of the 
FLC. The Lyapunov theorem is used to approve the 
stability of the controlled system with the proposed 

control method and estimate the upper bound of the 
dynamic model. The simulation results show the 
effectiveness of the proposed control methods and 
indicate the superiority of the proposed method in 
the response to model uncertainty and external 
disturbance. 

 
References 

 
[1] Jasim, H. H., Mary, A. H., & Ahmed, M. S. 

(2021). Robust Computed Torque Control for 
Uncertain Robotic Manipulators. Al-
Khwarizmi Engineering Journal, 17(3). 

[2] Kara, T., & Mary, A. H. (2018). Robust 
trajectory tracking control of robotic 
manipulators based on model-free PID-SMC 
approach. Journal of Engineering Research, 
6(3). 

[3] Han, J., Shan, X., Liu, H., Xiao, J., & Huang, T. 
(2023). Fuzzy gain scheduling PID control of a 
hybrid robot based on dynamic characteristics. 
Mechanism and Machine Theory, 184, 105283. 

[4] Gaidhane, Prashant J., et al. "Design of 
Gaidhane, P. J., Nigam, M. J., Kumar, A., & 
Pradhan, P. M. (2019). Design of interval type-

2 fuzzy precompensated PID controller applied 


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to two-DOF robotic manipulator with variable 
payload. ISA transactions, 89, 169-185. 

[5] Kashyap, A. K., & Parhi, D. R. (2021). Particle 
Swarm Optimization aided PID gait controller 
design for a humanoid Mai, T., & Tran, H. 
(2023). An adaptive robust backstepping 
improved control scheme for mobile 

manipulators robot. ISA transactions, 137, 446-
456.  

[6] Tseng, M. L., & Chen, M. S. (2010). Chattering 
reduction of sliding mode control by low‐pass 
filtering the control signal. Asian Journal of 
control, 12(3), 392-398. 

[7] Feng, Y., Han, F., & Yu, X. (2014). Chattering 
free full-order sliding-mode control. 

Automatica, 50(4), 1310-1314. 
[8] Chaudhary, K. S., & Kumar, N. (2023). 

Fractional order fast terminal sliding mode 
control scheme for tracking control of robot 
manipulators. ISA transactions.  

[9] Hasan, S. K., & Dhingra, A. K. (2022). 
Development of a sliding mode controller 

with chattering suppressor for human lower 
extremity exoskeleton robot. Results in 
Control and Optimization, 7, 100123.  

[10] Kara, T., & Mary, A. H. (2017). Adaptive PD-
SMC for nonlinear robotic manipulator 
tracking control. Studies in Informatics and 
Control, 26(1), 49-58.  

[11] Sun, Z., Hu, S., Xie, H., Li, H., Zheng, J., & 
Chen, B. (2023). Fuzzy adaptive recursive 
terminal sliding mode control for an 
agricultural omnidirectional mobile robot. 
Computers and Electrical Engineering, 105, 
108529.  

[12] Wu, G., Zhang, X., Zhu, L., Lin, Z., & Liu, J. 
(2021). Fuzzy sliding mode variable structure 

control of a high-speed parallel PnP robot. 
Mechanism and Machine Theory, 162, 
104349. 

[13] Zaare, S., & Soltanpour, M. R. (2022). 
Adaptive fuzzy global coupled nonsingular 
fast terminal sliding mode control of n-rigid-
link elastic-joint robot manipulators in 
presence of uncertainties. Mechanical 
Systems and Signal Processing, 163, 108165. 

[14] Wu, X., & Huang, Y. (2022). Adaptive 

fractional-order non-singular terminal sliding 
mode control based on fuzzy wavelet neural 
networks for omnidirectional mobile robot 
manipulator. ISA transactions, 121, 258-267. 

[15] Lian, R. J. (2013). Enhanced adaptive grey-
prediction self-organizing fuzzy sliding-mode 
controller for robotic systems. Information 
Sciences, 236, 186-204. 

[16] Sadati, N., & Talasaz, A. (2006). Variable 
structure control with adaptive fuzzy sliding 
surface. Journal of Vibration and Control, 
12(11), 1251-1270. 

[17] Li, P., Ma, J., & Zheng, Z. (2016). Robust 
adaptive sliding mode control for uncertain 
nonlinear MIMO system with guaranteed 

steady state tracking error bounds. Journal of 
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[18] Mary, A. H., & Kara, T. (2016). Robust 
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 (2024) 63-75، صفحة 1، العدد20المجلد  جلة الخوارزمي الهندسيةمعلي حسين مري                                                               

 
    75 

 
واستراتيجية التحكم  التحكم المتين لذراع االنسان االلي بناء على طريقة الوضع المنزلق 

 الغامض

 
 ****دينا سعدي منعم           تولكاي كارار***         احمد  عبد عطية**         مري*علي حسين     

 ليث عودة كاظم******                         محمد يحيى ادريس*****
 العراق /جامعة بغداد /كلية الهندسة الخوارزمي /قسم هندسة الميكاترونكس *****،****،*

 العراق /جامعة الشعب /كلية الهندسة وتكنولوجيا المعلومات /الطبية** قسم تقنيات األجهزة 
 تركيا / ***جامعة غازي عنتاب كلية الهندسة

 كندا /جامعة  كارلتون   ******

  kecbu.uobaghdad.edu.iq76Alimary@ :البريد االلكتروني*

  ahmad.altalabi@alshaab.edu.iq :االلكتروني* البريد *

  kara@gantep.edu.tr :* البريد االلكتروني**

  deena@kecbu.uobaghdad.edu.iq :البريد االلكتروني ****

  .Yahya@kecbu.uobaghdad.edu.iqMohammad:البريد االلكتروني ******

  Laith.mayyahi@carleton.ca:البريد االلكتروني ******

 
 الخالصة
 

م المختلفة. في السنوات األخيرة، تم استخدام األنظمة الروبوتية على نطاق واسع في تطبيقات مختلفة، وهذا ما حفز الباحثون على تطوير أساليب التحك

الرئيسة لنظام التحكم في  يُقترح في هذا العمل طريقة تحكم ذكية وقوية وخالية من النماذج لنظام مناور آلي غير خطي. تقدم هذه الورقة حالً جديدًا للعيوب

تم اقتراح وحدة تحكم  الوضع المنزلق، هنالك حاجة إلى معرفة مسبقة حول النموذج الديناميكي للنظام المتحكم فيه والحد األعلى من عدم اليقين. في هذا البحث،

PD  معSMC (FLPDSM) وتم تصميم وحدة التحكم .PD م . وتمت إضافة مصطلح تحكم متين إلى إشارة اعتمادا على المعلومات المتوافرة عن النظا

. وتم اقتراح SMCالتحكم للتعويض عن عدم اليقين في النظام، ويتم تعويض االضطرابات الخارجية عن طريق إضافة مصطلح قوي مساعد إلى قانون التحكم 

عدم اليقين في النظام معروف على الرغم من أنه ال يمكن تحديده بدقة طريقتين لتصميم شروط التحكم القوية. تفترض الطريقة األولى المقترحة أن الحد األعلى ل

على نظرية  بسبب االضطراب الخارجي وعدم اليقين. ومن ثم تم اقتراح طريقة ثانية تفترض أن هذا الحد غير معروف، وتم استخدام الكسب التكيفي القائم

الثانية لضمان استقرار نظام الحلقة المغلقة. تم تنفيذ اختبارات األداء على الطرق المقترحة من ليابونوف الشتقاق قانون التكيف. تم استخدام طريقة البونوف 

القياسي للتحقق من فعالية الطريقة المقترحة. وقد لوحظ تتبع مسار  SMCخالل دراسات المحاكاة للمناول اآللي ثنائي الوصلة، وتمت مقارنة نتائج االختبار مع 

 ختالفات المعلمات واالضطرابات الخارجية في إطار مخطط التحكم المقدم.جيد بمتانة عالية ضد ا

 
mailto:Alimary76@kecbu.uobaghdad.edu.iq
mailto:ahmad.altalabi@alshaab.edu.iq
mailto:kara@gantep.edu.tr
mailto:deena@kecbu.uobaghdad.edu.iq
mailto:Yahya@kecbu.uobaghdad.edu.iq

	1.  Introduction
	2. Robotic Manipulator Dynamic
	3. Sliding Mode Control
	4. Proposed FLPDSM Design
	4.1  Fuzzy-Like PD Controller

	5. Simulation Results
	5.1 Robustness test: Model Uncertainties

	6. Conclusion