In of exponential formula; and the result under

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In of exponential formula; and the result under

In this
technology-driven economy, the demand for the robot is increasing rapidly and
its applications are widespread across all sectors. The study of robot arm
control has gained a lot of interest in manufacturing industry, military,
education, biomechanics, welding, automotive industry, pipeline monitoring,
space exploration and online trading (Mohammed, 2015; O zkan, 2016; Rajeev Agrawal,
Koushik Kabiraj, 2012; Salem, 2014; Virgala, 2014) due to the fact that it
works in unpredictable, dangerous, and hostile circumstances which human cannot
be reached. Recently, the robot arm is on increasing demand in health services
to administer drugs to patients and rehabilitate the disabled and aged people;
of which high accuracy and precision with zero-tolerance to error are of high
significance for efficient utilization. (Olanrewaju, Faieza, & Syakirah, 2013; Paper,
Wongphati, & Co, 2012; Virendra, Patidar, 2016).

A
robot arm is a kind of mechanical device, programmable, multi-functional
manipulator (Sanchez-Sanchez & Reyes-Cortes, 2010) designed with an intention
to interact with the environment in a safe manner. It is a mechanical device in
the sense that it has links and joint that provide stability and durability but
are redundant from a kinematic perspective since the forces involve in the
motion are not considered. The problems of high non-linearity in the coupling
reaction forces between joints, as result of coupling effect and inertia
loading(Craig, 2005; Munro, 2004; Virgala, 2014) are not well captured from
the kinematics perspective. However, in-depth understanding of dynamic modeling
is essential to address the controlling problem associated with the robot arm.

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Modeling,
simulation and control of robot arm had received tremendous attention in the
field of mechatronics over the past few decades and the quest for new
development of robot arm control still continues. In literature (Mohammed, 2015), kinematics model of a
4-DOF robot arm is addressed using both Denavit-Hartenberg (DH) method and
product of exponential formula; and the result under study has shown that both
approaches resulted in an identical solution. In the study (Gea & Kirchner, 2008),the impedance control is
implemented to control the interaction forces of a simulated 2 link planar arm;
a mathematical model of a robot is modelled, linearized and decoupled in order
to establish a  model-based controller.
Simmechanic is used as a simulation tool to model the mechanics of the robot which
permit the possibility to vary model-based control algorithms. The fundamental
and concepts of 5 DOF of educational robot arm study in (Mohammed Abu Qassem, Abuhadrous, & Elaydi, 2010) to promote the teaching of
the robot in higher institution of learning. To achieve this, a detailed
kinematic analysis of an ALSB robot arm was investigated and a graphical user
interface (GUI) platform was developed with Matlab programming language which
also includes on-line motional simulator of the robot arm to fascinate and encourage
experimental aspect of robot manipulator motion in real time among undergraduates
and graduates.

The
research work in (Virgala, 2014) centered on analyzing,
modeling and simulation of humanoid robot hand from the perspective of biology
focusing on bones and joints. A new method for the inverse kinematic model is
introduced using Matlab functions and dynamic model of humanoid hand is
established using model-based design with aid of Matlab/Simmechanics. The
conclusion of their work is that they established a model in Matlab which can
be used to control finger motion. The author in (Lafmejani & Zarabadipour, 2014) modeled, simulated and
controlled 3-DOF articulated robot manipulator by extracting the kinematic and
dynamic equations using Lagrange method and compared the derived analytical
model with a simulated model using Simmechanics toolbox. The model is further
linearized with feedback and a PID controller is implemented to track a
reference trajectory. It was concluded in the research work that robot
manipulator is difficult to control as result of complexity and nonlinearity
associated with the dynamic model.

Author  (Mahil & Al-durra, 2016), presented a
linearized mathematical model and control of 2-DOF robotic manipulator and
derived a mathematical model based on kinematic and dynamic equations using the
combination of  Denavit Hartenberg and Lagrangian
methods. In his work, two different control strategies were implemented to compare
the performance of the robot manipulator.

According
to (Salem, 2014), a robot arm  model and control issues based on Simulink for
educational purpose is presented. It established a comprehensive transfer
function for both the motor and the robot arm which provide an insight into the
dynamic behaviour of the robot arm. It later proposed a model for research and
education purposes; which is used to select and analyze the performance of the
system both in open and closed loop systems. Author (Razali et al., 2010), employed 2-DOF robot arm
for agricultural purposes such as planting and harvesting and computer
simulation based on visual basic is developed which enable the users to control
the way the robot moves and grab selected target according to real line
situation. Many authors (Mailah, Zain, Jahanabadi, & A, 2009; S. Manjaree,
2017; Salem, 2014) developed a model for the robot arm and controlled
the dynamic response of the robot arm using Simmechanics as a software tool.
However, detail essential functions of each block that describe the
mathematical model of the dynamic equations are not well captured with
Simmechanics.

The
accurate control of motion is a fundamental concern in the robot arm, where
placing an object in the specific desired location with the exact possible
amount of force and torque at the correct definite time is essential for
efficient system operation. In other words, control of the robot arm attempts
to shape the dynamic of the arm while achieving the constraints foisted by the
kinematics of the arm and this has been a key research area to increase robot
performance and to introduce new functionalities. In general, the control problem involves finding suitable mathematical
models that describe the dynamic behaviour of the physical robot arm for
designing the controller and identifying corresponding control strategies to realize
the expected system response and performance.  New strategies for controlling the robot arm
has been more recently introduced such as PID (David, I , Robles, 2012; Guler & Ozguler, 2012;
Lafmejani & Zarabadipour, 2014; Rajeev Agrawal, Koushik Kabiraj, 2012),Fuzzy logic and Fuzzy
pattern comparison technique (Bonkovic, Stipanicev, & Stula, 1999), Impedance control (Gea & Kirchner, 2008; Jezierski, Gmerek,
Jezierski, & Gmerek, 2013), LQR Hybrid control (Humberto et al., 2016), GA Based adaptive control (Vijay, 2014),neuro-fuzzy controller(Branch, 2012) and Neural networks  (Pajaziti & Cana, 2014).  The objective of this research is to
establish a mathematical model which represents the dynamic behaviour of the
robot and effectively control the joint angle of the robot arm within a
specified trajectory.

2.0          Methodology

The dynamics of 2-DOF robot arm was modelled using a set of
nonlinear, second-order, ordinary differential equations and to simulate the dynamics
accurately the Lagrangian and Lagrange-Euler was adopted.  The Euler’s formulation is chosen for its
simplicity, robustness (Amin, Rahim, & Low, 2014) energy based property (David, I , Robles, 2012),easy determination and
exploitation of  dynamic structural
property and minimal computational error as compared to Newton-Euler approach (Murray, 1994) to solve the derived
mathematical model. The formulation of the mathematical model is considered
crucial in the research because the control strategy is investigated based on
these derived dynamics equations, hence the model must be accurately predicted
to represent the dynamic behaviour of the robot arm. The control algorithm is
expanded on the derived mathematical model to control the movement of the robot
arm within the specified trajectory or workspace, hence, we further design a
PID controller and tuned the PID based on trial and error method to obtain
suitable controller parameters for proper controlling of the robot arm within
the specified trajectory. Simulation studies based on MATLAB and Simulink are
performed on the robot arm taken into the consideration the obtained PID
controller parameters and the obtained parameters are used to validate the mathematical
model in the joint space. The evaluation of the results obtained is presented
and discussed extensively concerning achievement as well as providing
recommendations for further work.

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