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.
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.
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.
2.1 Mathematical Model of 2-DOF Robot Arm
The dynamics of a robot arm is explicitly derived based on the Lagrange-Euler formulation to elucidate the problems involved in dynamic modelling. Figure 1 shows the schematic diagram of two degree of freedom (DOF) of the robot arm with the robot arm link1 and link 2, joint displacement are and ,link lengths are and , , represent the masses of each link and and are torque for the link 1and 2 respectively. In the model the following assumptions are been made:
§ The actuators dynamics (motor and gear boxes) is not taken into account.
§ The effect of friction forces is assumed to be negligible
§ The mass of each link is assumed to be concentrated at the end of each link.