Modeling and Control of 2-DOF Robot Arm

1 International Journal of Emerging Engineering Research and Technology Volume 6, Issue 11, 2018, PP 24-31. ISSN 2349-4395 (Print) & ISSN 2349-4409 (Online). Modeling and Control of 2-DOF Robot Arm Nasr M. Ghaleb1 and Ayman A. Aly1, 2. 1. Mechanical Engineering Department, College of Engineering, Taif University, PO Box 888, Taif, Saudi Arabia 2. Mechanical Engineering Department, Faculty of Engineering, Assiut University, PO Box 71516, Assiut, Egypt *Corresponding Author: Nasr M. Ghaleb, Mechanical Engineering Department, College of Engineering, Taif University, PO Box 888, Taif, Saudi Arabia ABSTRACT. This paper presents a Modeling , simulation and Control of a Two Degree of Freedom ( 2-DOF ) Robot Work is taken from the Final Year capstone project. First The Robot specifications , Robot Kinematics with Denavit-Hartenberg parameters (DH)for Forward kinematics and Inverse Kinematicsof 2- DOF Robot armwere presented.

2 Then The dynamics of the 2-DOF Robot arm was studied to derive the equations of motion based on Eular-Lagrange Equation of motion. A Control Design was performed using PID controller for the Modeling and Control Technique. The models has been done based on Matlab/Simulink software. Keywords: Robotics, 2-DOF Robot arm, Kinematic, Dynamic, PID Control and Modeling . INTRODUCTION most of industrial processes. Most Robot manipulators found in industrial operations are Robotics is defined practically as the study, controlled by PID algorithms independently at design and use of Robot systems for each joint . manufacturing and generally are used to perform highly repetitive, unsafe, hazardous, There are many Control techniques used for and unpleasant tasks. Robotics many different controlling the Robot arm.

3 Functions that used either in industry and The most familiar Control techniques are the manufacture or in complex, clatter and changing PID Control , adaptive Control , optimal Control environment such as pick and place, assembly, and robust Control . As the final goal is to design drilling, welding, machine tool load and unload and manufacturing real robots, it's helpful doing functions, painting, spraying, etc. or in A the simulation before the investigations with delivery in a hospital and Hotels, Discovering real robots, to enhance the final real Robot the space As a results of these different tasks performance and behavior. there are different Robot arm configuration such as rectangular, spherical, cylindrical, revolute Robot SPECIFICATION AND KINEMATICS. and prismatic jointed.

4 Robot Specification A pick and place Robot arm is used to ease A two degree of freedom Robot arm is described process of moving materials and supplying the in Figure(1). motion required in the manufacturing processes. The transfer process of the materials is usually which consists primarily of two links with the being accomplished, using man power and as following specifications in OXY coordinates: the transfer process is repeated for a period of 1 = 1 m is the length of the first link. time, it can cause injuries to the operator. the 2 = 1 m is the length of the second link. Robot arm preventing injuries and increasing the efficiency of the work, with reducing the human 1 = 1 kg is the mass of the first link. being errors that cost highly time and martial. 2 = 1 kg is the link of the second link.

5 The proportional-integral-derivative (PID) 1 = the rotation angel of the first link. Control has simple structure for its three gains. 2 = is the rotation angel of the second link. The Control performances are acceptable in the International Journal of Emerging Engineering Research and Technology V6 I11 2018 24. Modeling and Control of 2-DOF Robot Arm Robot Kinematics Forward Kinematics The Forward kinematics of a robotic arm is determined a group of parameters called Denavit-Hartenberg (DH) parameters which used for deriving the homogenous transformation matrices between the different frames assigned on the Robot arm structure. The DH parameters for a two degree of freedom robotic arm are defined as follows: Figure 1. Two degree of freedom Robot Arm Table 1. DH-parameters for the 2-DOF robotic arm Link ai i di.

6 1 L1 0 0 1. 2 L2 0 0 2. The homogenous transformation matrices for the 2-DOF robotic arm shown in Figure(1) are derived as follows: cos 1 sin 1 0 1 cos 1. sin 1 cos 1 0 1 sin 1. = (1). 0 0 1 0. 0 0 0 1. cos 2 sin 2 0 2 cos 2. sin 2 cos 2 0 2 sin 2 (2). =. 0 0 1 0. 0 0 0 1. Using the Eq. (1) and (2), the homogenous transformation matrix 02 can be derived as follows: cos( 1 + 2 ) sin( 1 + 2 ) 0 1 cos 1 + 2 cos( 1 + 2 ). sin( 1 + 2 ) cos( 1 + 2 ) 0 1 sin 1 + 2 sin( 1 + 2 ). = ( 3). 0 0 1 0. 0 0 0 1. Therefore, can be derived either algebraically or geometrically. The geometrical approach is . considered to be much easier for Robot arms of = (4) high degrees of freedom. In our Case, we solved 0 0 0 1 the inverse kinematics equations for the 2-DOF .. robotic arm shown in Figure(2) using the = (5) geometrical method.

7 From Eq. (5), the position coordinates of the manipulator end-effector is given by: = 1 cos 1 + 2 cos( 1 + 2 ) (6). = 1 sin 1 + 2 sin( 1 + 2 ) (7). And the end-effector's orientation matrix is defined by the first three rows and three columns of the transformation matrix in Eq.(3). Inverse Kinematics The inverse kinematics of a robotic arm is a solution of finding the Robot arm joint variables of given the position Cartesian coordinates of Figure 2. Two degree of freedom Robot Arm Inverse the end-effector. The mathematical equations Kinematic used to solve the inverse kinematics problem From Figure2, a mathematical equation for 25 International Journal of Emerging Engineering Research and Technology V6 I11 2018. Modeling and Control of 2-DOF Robot Arm solving the elbow joint angle 2 can be derived arm: Eular-Lagrange method and Newten-Eular using Pythagoras theorem as follows: method.

8 In this work , We used the Eular- Lagrange method which depends on calculating 2 + 2 = 1 2 + 2 2 + 2 1 2 cos 2 (8) the total Kineatic and Potential Energies of the 1 Robot arm to determine the Lagrangian ( ) of cos 2 = ( 2 + 2 1 2 2 2 ) (9). 2 1 2 the whole system in order to calculate the force or torque applied of each the sin 2 = 1 cos 2 2 (10) Lagrangian , we can solve the Euler- Lagrange equation which relies on the partial derivative of Therefore, kinetic and potential energy properties of sin 2 mechanical systems to compute the equations of 2 = (11). cos 2 motion and is defined as follow: For The joint variable 1 : . F= (21).. = ( 1 + 2 cos 2 ) cos 1 2 sin 1 sin 2 (12). where F is the external force acting on the = 2 sin 2 cos 1 + ( 1 + 2 cos 2 )sin 1 (13). generalized coordinate, represents the torque 1 + 2 cos 2 2 sin 2 applied to therobot and is the Lagrangian = (14).

9 2 sin 2 1 + 2 cos 2 equation of the motion, given in Eq.(22). 2 + 2 = ( 1 + 2 cos 2 )2 + ( 2 sin 2 )2 (15) , = , (22). 1 + 2 cos 2 To solve the Lagrangian Eq.(22),we need first to sin 1 = (16). 2 sin 2 calculate the kinetic Energy and potential Energy as follow: 2 sin 2. cos 1 = (17) 1. 1 + 2 cos 2 = 2. 2. sin 1 1 + 2 cos 2 2 sin 2 . sin 1 = = (18) = . 2 + 2. cos 1 1 + 2 cos 2 + 2 sin 2 The velocity is determined by taking the cos 1 = = (19) derivative of the position respect to time. so the 2 + 2. position in the end of the link is known using sin 1 1 + 2 cos 2 2 sin 2 . 1 = = (20) known familiar variables : cos 1 1 + 2 cos 2 2 sin 2 . 1 = 1 1. Robot DYNAMICS. 1 = 1 cos 1. The dynamic model of a Robot is concerned with the movement and the forces involved in the 2 = 1 1 + 2 sin.

10 ( 1 + 2 ) (23). Robot arm and establishes a mathematical 2 = 1 cos 1 + 2 cos( 1 + 2 ). relationship between the location of the Robot joint variables and the dimensional parameters Then substitute in the kinetic's energy equation: of the Robot . There are two methods for 1 1 1 1. = 1 1 2 + 1 1 2 + 2 2 2 + 2 2 2 (24). performing the dynamics equations of a Robot 2 2 2 2. After simplification, 1 2 1 2 1 2 2. = ( 1 + 2 ) 1 2 1 + 2 1 2 1 + 2 2 2 1 2 + 2 2 2 2 + 2 1 2 2 1 2 + 1 (25). 2 2 2. The Potential energy equation is defined as: = 1 1 cos 1 + 2 ( 1 cos 1 + 2 ( 1 + 2 )) (26). Substitute Eq.(25) and(26) in Eq.(21) to form the lagrangian equation as: 1 2 1 2 1 2 2. = ( 1 + 2 ) 1 2 1 + 2 1 2 1 + 2 2 2 1 2 + 2 2 2 2 + 2 1 2 2 1 2 + 1 . 2 2 2. 1 1 cos 1 + 2 1 cos 1 + 2 1 + 2 (27).