Kinematic modeling and verification for a SCARA robot

The kinematic model of robots is to describe the nonlinear relationship between the position and orientation of the end-effector and the displacement of each joint, which is an important content of robot calibration. The coordinate systems of a SCARA robot are established according to DH method, through which the structural parameters are determined. Then, the kinematic equations between the working space and the joint space are deduced by homogeneous transformation principle. The graphic and calculating model of the SCARA robot are established in Matlab. Through the comparison between graphic and calculating model, the kinematic model is verified. Introduction SCARA robot is a kind of special cylindrical coordinate type of industrial robots, which relies on two rotary joints for rapid positioning within the horizontal plane as well as a linear joint and a rotary joint for rotary and stretchy movement within the Z direction. The structure of the SCARA robot studied in this paper is shown in Fig. 1. It’s movement is smooth and reliable in the horizontal direction, and it has a greater stiffness in the vertical direction [1]. In order to improve the accuracy of the robot, kinematic parameters calibration must be done first. Furthermore, kinematic modeling is the theoretical foundation of calibration. Through a function relationship between the end-effector position, orientation matrix and the joint variables, we can determine the parameters of the adjacent coordinate system to obtain a homogeneous transformation relationship. Finally, we can get the kinematic model of the robot using multiplication[2]. The purpose of the establishment of the kinematic model is to find the original error of each parameter transfer coefficient for the results influence, as well as synthetic relationship between the various errors. SCARA robot movement usually needs to be high-speed and high-precision. Therefore, the analysis of the robot kinematics must be careful, accurate and efficient[3]. Due to the DH model has clear physical meaning and easy programming, this paper established the kinematic model with DH model method, and Matlab simulation was used to verify the model. Coordinate systems of the robot based on DH model Denavit-Hertenberg method, referred to as DH method, is now the most commonly used method of robot kinematic model, which uses a homogeneous transformation matrix to describe the relative position between two adjacent links[4]. The reference coordinate system on the base should be established first in this paper. Then, we established four joint coordinate systems of No.1 to 4. The reference coordinate system is stationary in the working process of the SCARA robot, and others coordinate systems move with the corresponding joints. In addition, the kinematic equation is the product of the coordinate transformation[5]. Method of establishing the coordinate system of each joint are as follows: (1) Determining the zi axis, the zi axis is along the axis direction of the joint zi+1; (2) Determining the origin oi, the oi is on common perpendicular of zi and zi-1 axis, intersection of the common perpendicular and axis zi is the origin oi ; 3rd International Conference on Materials Engineering, Manufacturing Technology and Control (ICMEMTC 2016) © 2016. The authors Published by Atlantis Press 918 (3) Determining the xi axis, the xi axis is along the direction of the common perpendicular of zi and zi-1 axis and leaving the zi-1 axis; (4) Determining the yi axis with the right hand rule. The coordinate systems of the SCARA robot established with DH method are shown as Fig. 2. Fig. 1 The structure of the SCARA robot Fig. 2 The coordinate systems of the SCARA robot According to the principle of DH method, there are four groups structural parameters of the SCARA robot[6]. They are, respectively, linkage length di, joint length ai, torsion angle αi and joint angle θi. These parameters are defined as follows: (1) di is the translation between the axis of xi-1 and xi, along zi direction is positive; (2) ai-1 is the translation between the axis of zi-1 and zi, along xi-1 direction is positive; (3) αi-1 is the rotation between the axis of zi-1 and zi, about xi-1 counter-clockwise; (4) θi is the rotation between the axis of xi and xi-1, about zi counter-clockwise; The DH parameters of the SCARA robot are shown in Table 1, where θi and d3 is variable; L1 = 250mm; L2 = 350mm; d1 = 300mm. Table 1. The DH parameters of SCARA robot Joints di(mm) ai-1(mm) αi-1 (°) θi (°) 1 d1 L1 0 θ1 2 0 L2 0 θ2 3 -d3 0 180 0 4 0 0 0 θ4 Kinematic equations of the robot According to the coordinate systems and homogeneous transformation method, the homogeneous transformation matrix of the SCARA robot can be determined from (1): 1, 1 ( , ) (0,0, ) ( ,0,0) ( , ) cos sin cos sin sin cos sin cos cos cos sin sin . 0 sin cos 0 0 0 1 i i i i i i i i i i i i i i i i i i i i i i i i i T Rot z Trans d Trans a Rot x a a d                                   (1) We obtained pose transformation matrix between two adjacent coordinates through the DH parameters of table 1, they are shown in (2) and (3): Joint 2