International Journal of Control, Automation, and Systems (2012) 10(3):567-573 DOI 10.1007/s12555-012-0313-9 ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555 Efficient Closed-Form Solution of Inverse Kinematics for a Specific Six-DOF Arm Thanhtam Ho, Chul-Goo Kang*, and Sangyoon Lee Abstract: Inverse kinematics solutions for multi-DOF arms can be classified as analytical or numeri- cal. In general, analytical solutions are preferable to numerical solutions because analytical ones yield complete solutions and are computationally fast and reliable. However, analytical closed-form solu- tions for inverse kinematics of 6-DOF arms rarely exist for real-time control purposes of fast moving arms. In this paper, we propose a fast inverse kinematics algorithm with a closed-form solution for a specific 6-DOF arm. The proposed algorithm is verified using simulation modules developed by us for demonstrations. Keywords: Closed-form solution, inverse kinematics, robotic arm, simulation. 1. INTRODUCTION Inverse kinematics plays a key role in robotics and computer animations. Given the pose (position and orientation) of the end-effector, inverse kinematics prob- lems correspond to computing joint variables for that pose. Inverse kinematics solutions for 6-dergee-of- freedom (DOF) arms may be characterized as analytical or numerical [1]. Analytical solutions can be further sub- divided into geometry-based closed-form solutions [2] and algebraic-elimination-based solutions [3]. In general, closed-form solutions can only be obtained for 6 or less than 6-DOF systems with a specific structure. Solutions based on algebraic elimination express joint variables as solutions to a system of multivariable polynomial equa- tions, or express a single joint variable as the solution to a very high degree polynomial and determine the other joint variables using closed-form computations. In contrast to the analytical solutions, numerical ap- proaches iteratively converge to a solution based on an initial guess [4]. In numerical approaches, computation time to converge may vary, and thus numerical solutions are not appropriate to fast real-time control applications even if they are applicable to ill-posed systems. In gener- al, analytical solutions are preferable to numerical ones for real-time control applications because analytical ones yield all solutions (completeness) and are computational- ly fast and reliable. In some cases, partly closed-form and partly numerical solutions have been tried [5]. The existence of the closed-form solution depends on the kinematic structure of the arm. Pieper [2] showed that the 6-DOF manipulator with a spherical wrist has a closed-form solution. Many researchers have obtained closed-form solutions for inverse kinematics of 6-DOF manipulators including Lee et al. [6], Kang [7] and oth- ers [8-10] for 6-DOF PUMA robots, and Schilling [11] for a 6-DOF Intelledex 660T robot. However, these solu- tions are ones for industrial manipulators that are differ- ent in configuration from the human-like arm shown in Fig. 1. Fast and efficient closed-form solutions for the 6- DOF arm such as Fig. 1 rarely exist for the real-time control purposes of fast moving arms. Among earlier works on anthropomorphic arms, Tola- ni et al. [1] considered the inverse kinematics of 7-DOF anthropomorphic limbs, and they simplified the arm structure to two segments including the upper arm and the forearm while the shoulder blade is neglected. The simplification makes the inverse kinematics solvable but the solution is not able to apply to many robotic arms where shoulder blades are included. Also, Asfour et al. [12] derived a closed-form solution of the inverse kine- matics for a 7-DOF arm of a humanoid robot ARMAR. After selecting a specific value of the z axis of the elbow, they determined joint angles by matrix equations using the decomposition approach. This paper presents an efficient (i.e., fast and reliable) closed-form solution of inverse kinematics for a 6-DOF arm similar to the human arm structure. Previously for the commercial arm by Robot and Design Co., Ltd. shown in Fig. 1, a pseudo-inverse numerical solution has been used for inverse kinematics of the arm, and the av- erage converging time was reported to be about 1 ms. This is a big computational burden for real-time control of the fast moving arm, and, consequently, the sampling frequency of the feedback control system becomes quite low to control such a robot arm where high precision and fast response are the most primary requirements. Diffe- © ICROS, KIEE and Springer 2012 __________ Manuscript received September 4, 2010; revised August 9, 2011; accepted January 4, 2012. Recommended by Editorial Board member Shinsuk Park under the direction of Editor-in- Chief Jae-Bok Song. This work was supported by Konkuk University in 2010. Thanhtam Ho and Sangyoon Lee are with the Department of Mechanical Design and Production Engineering, Konkuk Univer- sity, Gwangjin-gu, Seoul 143-701, Korea (e-mails: thanhtam.h @gmail.com, slee@konkuk.ac.kr). Chul-Goo Kang is with the Department of Mechanical Engi- neering, Konkuk University, Gwangjin-gu, Seoul 143-701, Korea (e-mail: cgkang@konkuk.ac.kr). * Corresponding author.