Technology and Innovation for Sustainable Development International Conference (TISD2010) Faculty of Engineering, Khon Kaen University, Thailand, 4-6 March 2010 A Simple Approach to the Solution of Inverse Kinematics of Robot Manipulator and Simulation Kriangkrai Waiyagan 1 Kritsada Nakthewan 2 Supapan Chaiprapat 3 1 Department of Agro Industrial Management Technology, Faculty of Agro-Industry, Prince of Songkla University, Songkhla 90112 E-mail: kriangkrai.w@psu.ac.th 2,3 Department of Industrial Engineering, Faculty of Engineering, Prince of Songkla University, Songkhla 90112 E-mail: kn_tee@hotmail.com supapan.s@psu.ac.th Abstract Robot manipulators can be operated by two ways including teaching mode and automatic mode. The teaching mode requires an operator to input initial position and angle of each link for obtaining a destination. The automatic mode calculates angle of each link to move from an initial position to a destination. This paper proposed a simple method to operate robot manipulators in the automatic mode. The objective is focused on finding the solution of inverse kinematics of a robot. There are several manners in which forward kinematics and inverse kinematics are developed for robot manipulators. For example, Euler's equations describe the rotation of a rigid body about the axes of a moving referenced coordinate system, The Denavit-Hartenberg representation becomes the standard way of representing robot’s motions. However, inverse kinematics equations formulated by beginners based on the two methods previously mentioned are prone to human error. The main reason is the difficulty to achieve correct equations without understanding on the real motion of manipulators. Therefore, a simple way to solve the inverse kinematics based on the principles of transformation geometry is demonstrated step-by-step using a case study. Then, solutions are computed by numerical methods using a spreadsheet application (Microsoft Excel). The given solutions are simulated in 3-Dimensional space to verify positions and to prevent collisions before operating the robot. 3D robotic motion simulation software is not necessary. Available Compute-Aided Design (CAD) applications can be applied for checking and displaying results of movement. The benefits of proposed concept are: (1) fast and easy inverse kinematics solutions (2) corresponding motions of manipulators are based on basic transformation geometry. Keywords: Robot manipulator, Forward kinematics, Inverse kinematics, Euler's equations, The Denavit-Hartenberg representation 1. Introduction Robot manipulators operated by teaching and playback mode are suitable for fixed positioning applications. This method is not comfortable for variable positioning applications because an operator has to teach that how can a robot go to a new position. The automatic mode runs a robot to move to a new position without teaching based on forward and inverse kinematics. The forward kinematics is computation of final destination when knowing the starting point and angle of links. The solution is unique and easy. The inverse kinematics is computation about how to arrive to a destination. The solution is not unique and not easy due to additional constraints like dynamic limitations and obstacles [1]. The Denavit-Hartenberg representation becomes the standard way of representing robot’s motions and multi-axis machine tool’s motions as shown in [2], [3], [4] and [5]. However, The Denavit-Hartenberg method is difficult to understand and produce. There are alternative methods for solving inverse kinematics problems such as the principles of transformation geometry [6] as shown in Fig. 1. Referring to Fig. 1, the formulae for ( , ) P x y    are as follows: x = rcos( +)   = rcos cos rsin sin      = xcos ysin    (1) y = rsin( +)   = rsin cos rcos sin     + = xsin ycos   + (2) Figure 1. Rotation of a point P to P about the origin