International audienceIn this paper, we have designed a sampled data control law for the single-link flexible joint robotic manipulator using two different approaches. First, we have designed the sampled-data sliding mode control (SMC) based on the continuous time system. In the second approach, we have obtained the approximate discrete model of the system and then designed discrete sliding mode control based on this approximate system. Simulation results have been obtained and a performance comparison has been presented for both techniques

Khan, Rameez

Malik, Fahad,

Raza, Abid

English

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Sampled-Data Sliding Mode Control Design of Single-
Link Flexible Joint Robotic Manipulator
Abid Raza, Rameez Khan, Fahad Malik
To cite this version:
Abid Raza, Rameez Khan, Fahad Malik. Sampled-Data Sliding Mode Control Design of Single- Link
Flexible Joint Robotic Manipulator. 52nd INTERNATIONAL SCIENTIFIC CONFERENCE ON
INFORMATION, COMMUNICATION AND ENERGY SYSTEMS AND TECHNOLOGIES (ICEST
2017), Jun 2017, Nis, Serbia. ￿hal-02372416￿
 75 
Sampled-Data Sliding Mode Control Design of Single-
Link Flexible Joint Robotic Manipulator  
Abid Raza1, Rameez Khan2, Fahad Mumtaz Malik3  
Abstract – In this paper, we have designed a sampled data 
control law for the single-link flexible joint robotic manipulator 
using two different approaches. First, we have designed the 
sampled-data sliding mode control (SMC) based on the 
continuous time system. In the second approach, we have 
obtained the approximate discrete model of the system and then 
designed discrete sliding mode control based on this approximate 
system. Simulation results have been obtained and a 
performance comparison has been presented for both 
techniques.  
 
Keywords – Sampled-data control, Sliding mode control, 
Robotic manipulator. 
I.  INTRODUCTION 
Robotic manipulators are widely being used in industry are 
highly nonlinear systems which often suffer from unmodeled 
dynamics and uncertainties. Recently control engineers have 
designed controllers for flexible joint robots but it is still very 
less as compared to the rigid robots. However, in many 
applications, the robotic manipulators have some flexible joints 
as well. At the same time, the flexibility of joints may limits 
the performance and robustness of the controllers designed for 
robots or may lead to instability [1]. Also, the flexibility of 
joints can be considered a step for the flexible robot link [2] so 
to by studying the flexible joint manipulator can lead us to the 
flexible link which helps to have link lighter in weight that 
ultimately results in fast robotic motions. If the flexibility of 
joint is modeled by the spring, dynamic model of flexible 
joints can be obtained. Some advantages of flexible joints 
robotic manipulators over rigid manipulators are less control 
effort, fast motion, light weight and smaller dimension [3]-[6]. 
To design sliding mode controller for the flexible joint 
manipulator, state variables of each joint, velocities, 
acceleration and jerks of the link must be known [5]. Since last 
two decades, dynamic modeling and non-linear control of 
robotic manipulators (flexible) is the area of research which 
received notable consideration. To design a control topology 
that is efficient and robust, the first and fundamental step is to 
develop an accurate dynamic model of a flexible joint robot. In 
industrial and space applications, we need the controller that 
can reduce the disturbance effects and can cope with modeling 
uncertainties. For flexible joints, a robust controller should be 
designed for the stabilization or tracking purpose that has 
negligible link vibration. 
 For this purpose, different techniques have been suggested 
for last two decades. For example,  linear quadratic regulation 
(LQR) control technique is widely used for such control [7], 
another technique which is vastly used is adaptive output 
feedback controller based on a back stepping [8]-[10]. In the 
case of nonlinear control, feedback linearization is the most 
renowned technique that is used for robotic manipulators [11].  
The integral control technique described in [12] and [13],  
robust control that uses PD control and H∞ control, PD fuzzy 
and optimal control and sliding mode control [14]. In [15], an 
adaptive second order terminal sliding mode controller for 
robotic manipulators is proposed.  
Almost all the research for control of robotic manipulators 
is carried out in the continuous time domain. However, due to 
the increasing availability of microprocessor hardware, control 
algorithms are now usually implemented in digital controllers. 
In this paper, we have focused on sampled data sliding mode 
control, as SMC is nonlinear control and is more robust as 
compared to most of the conventional control techniques. 
In this paper, we have focused on the stabilization of 
flexible joint robotic manipulator in discrete time domain. 
Sliding mode control is designed using sampled data control 
techniques and their comparison is carried out on the basis of 
simulation results. This paper is divided into five sections. In 
section 2 dynamic modeling of single link flexible joint robot 
is discussed. In section 3, sliding mode controller is designed 
that would stabilize the robotic manipulator. In section 4 
sampled data control techniques are discussed. Using these 
techniques SMC is designed for the system, simulation results 
are presented to show the comparison of sampled data control 
techniques. Section 5 discusses the conclusion on the basis of 
these simulation results. 
II. DYNAMIC-MODELLING OF FLEXIBLE JOINT 
SINGLE-LINK ROBOTIC MANIPULATOR 
In this work, we have focused upon the flexible joint single 
link robotic manipulator. The dynamic model of single-link 
robotic manipulator with the flexible joints is derived using the 
Euler-Lagrange equations and is given by [16], [17]  
 
 
Fig. 1 shows the model of single link flexible joint robotic 
manipulator [18].  are the link positions,  describes the 
link inertia whereas   is the actuator inertia, the nominal load 
is denoted by ,    represents the joint stiffness and input 
torque is denoted by . 
1Abid Raza is with the Department of Electrical Engineering at 
College of Electrical and Mechanical Engineering, National 
University of Science and Technology, Islamabad, Pakistan, E-mail: 
abid.raza@ceme.nust.edu.pk 
2Rameez Khan is with the Department of Electrical Engineering at 
College of Electrical and Mechanical Engineering, National 
University of Science and Technology, Islamabad, Pakistan 
3Fahad Mumtaz Malik is with the Department of Electrical 
Engineering at College of Electrical and Mechanical Engineering, 
National University of Science and Technology, Islamabad, Pakistan 
 76 
 
Fig. 1. Model of a single-link, flexible joint manipulator 
 
Using this dynamic model state space model of this robotic 
manipulator is derived. State space model of manipulator is 
given by 
 
 
 
 
where the constants are given by  
 and the input torque is . 
we have designed a sliding mode controller using this state 
space model for the stabilization of this system in next section. 
III. STABILIZATION USING SAMPLED DATA SMC 
BASED ON EMULATION 
Sampled-data control refers to the control of a continuous 
system with discrete controller. There are two main techniques 
for sampled data control [19] [20]. 
• Emulation design 
• Approximate discrete model based control design 
Emulation design is considered as a simpler method to 
design a controller for sampled-data systems. In terms of 
system stability and performance, it is also considered as 
inferior design technique as compared to other methods. In 
designing controller by this technique sampling is completely 
overlooked. The continuous controller is discretized, in next 
step implementation is done using sample and hold devices.  
Fig. 2 shows the block diagram of steps that are followed 
during the emulation design. In this technique of control 
design, the control law derived for the continuous time system 
is discretized. The classical sliding mode control suffers from 
chattering. Chattering is a severe problem especially in the 
mechanical system because it causes wear and tear in the 
system. There are different methods developed to reduce 
chattering. A  well-known  method  is  the  replacement of  the 
 
Fig. 2. Emulation design 
discontinuous signum function with the saturation function 
[17]. In this method, the real sliding is replaced with the sliding 
in a small vicinity of the discontinuous surface. The chattering 
issue can be overcome if we use saturation function instead of 
signum function. Consider the sliding surface 
                        (3) 
The following control law has been proposed for the 
stabilization of the single link flexible joint robotic 
manipulator. 
where 
            (4) 
 
0 2 4 6 8 10 12 14 16 18 20
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time(Seconds)
Stabilization
 
 
x1
x2
x3
x4
 
Fig. 3. States Stabilization using Sampled Data SMC based on 
Emulation 
 
In Fig. 3, state stabilization based on Emulation design is 
shown. These simulation results are generated by using 
MATLAB/SIMULINK. State is the position of the link and 
we can see this state stabilizes to the origin within 12 sec. All 
other states also stabilize to the origin within a certain time. 
This shows that we have designed efficient control law using 
emulation that stabilizes all the system states. 
IV. STABILIZATION USING SAMPLED DATA SMC 
BASED ON APPROXIMATE DISCRETE MODEL 
DESIGN  
In approximate discrete model based control design 
technique, the controller is designed by using the approximate 
discrete model of the plant in discrete time domain. For non-
linear systems, exact modeling of the system is almost 
impossible so approximate discrete model of the plant is used 
to design a controller for nonlinear systems. Then, this discrete 
controller is implemented using sample-and-hold devices. In 
this paper, we have discussed the controller design for sampled 
data control. 
 
Fig. 4.  SMC based on approximate discrete model 
 77 
Fig. 4 shows the block diagram of steps that are followed 
during the sampled data control using approximate discrete 
model. Using this technique SMC is designed in this section. 
The discretization is done using Euler forward difference 
method. The Euler model approximation is 
 
 State space model of manipulator (1) is 
 
 
 
 
Using the approximate discrete model of the manipulator 
we will design SMC that will stabilize the system. The 
approximate discrete model is given by 
 
 
 
 Consider the following sliding surface 
 
To stabilize it globally using Sliding mode control law  
 
where 
 
The constants are . 
 
Fig. 5. States Stabilization using sampled data SMC based on 
approximate discrete model 
 
In Fig. 5, state stabilization based on SMC (approximate 
discrete model) is shown. These simulations are generated by 
using MATLAB SIMULINK. State is the position of the 
link and we can see this state stabilizes to the origin within 4 
seconds. All other states also stabilize to origin approximately 
within four seconds. This shows that SMC designed using 
approximate discrete model is more efficient as compared to 
the previous technique and it also stabilizes all the system 
states. 
V. COMPARISON OF APPROXIMATE DISCRETE 
MODEL AND EMULATION DESIGN SMC  
In this section, a comparison of both sampled-data control 
techniques is presented. The simulation is done using 
MATLAB/SIMULINK for the implementation of the 
controller on robotic manipulator. The results of the 
comparison of both techniques are shown in Fig. 6. 
In simulation results, stabilization of Single link flexible 
joint robotic manipulator using SMC based on emulation 
design and SMC based on approximate discrete model are 
compared. It is obvious that stabilization of states using 
approximate discrete model is quicker than that of SMC based 
on emulation design. In the case of SMC based on emulation 
design, there are oscillations of states before settling down to 
the origin. Also, time taken to stabilize using emulation design 
is more than that of approximate discrete model design. 
Results of SMC based on approximate discrete model stabilize 
the states quickly which shows SMC based on approximate 
discrete model is better stabilization approach for single link 
robotic manipulator than that of SMC based on emulation 
design. 
VI. CONCLUSION  
In this paper, SMC feedback control for single-link flexible 
joint robotic manipulator system is discussed. Two approaches 
for the design of sampled data sliding mode control are studied 
and simulation results are presented. It is clear that the SMC 
based on approximate discrete model is better as it stabilizes 
the states in less time for the same parametric values of the 
system. Moreover, by using SMC based on approximate 
discrete model the region of attraction is increased as 
compared to emulation design. The main outcome of Sampled 
Data SMC is the robustness. It (SMC based on approximate 
discrete model) is more robust against disturbance to 
conventional techniques used for robotic manipulator control 
(feedback linearization).  SMC based on approximate discrete 
model has the insensitivity to parameter variations to a larger 
extent (greater region of attraction), fast dynamic responses 
and better disturbance rejection which is required for the 
smooth motions of the robotic manipulator.   
Now sampled data control is becoming popular than the 
continuous time. The sampled data controls that are derived 
and simulated can be further extended to sampled data 
observer. The states can be estimated with the help of an 
observer so we do not need to have sensors on all states. This 
work can also be extended for the n-link flexible joint robotic 
manipulator. Based on this work, a robust controller can also 
be designed for the tracking problem for the robotic 
manipulator. 
 78 
 
(a) 
0 2 4 6 8 10 12 14 16 18 20
-1.5
-1
-0.5
0
0.5
Time(Seconds)
Stabilization
 
 
SMC based on CT System
SMC based on ADTE System
 
(b) 
0 2 4 6 8 10 12 14 16 18 20
-3
-2
-1
0
1
Time(Seconds)
Stabilization
 
 
SMC based on CT System
SMC based on ADTE System
 
(c) 
0 2 4 6 8 10 12 14 16 18 20
-10
-5
0
5
10
Time(Seconds)
Stabilization
 
 
SMC based on CT System
SMC based on ADTE System
 
(d) 
Fig. 6. Estimated States Comparison of approximate discrete model, 
Emulation SMC: (a) State X1 stabilization, (b) State X2 stabilization, 
(c) State X3 stabilization, (d) State X4 stabilization 
 
 
 
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Sampled-Data Sliding Mode Control Design of Single- Link Flexible Joint Robotic Manipulator

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Sampled-Data Sliding Mode Control Design of Single- Link Flexible Joint Robotic Manipulator

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