Tham khảo tài liệu 'crc press - robotics and automation handbook episode 1 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Determine d and a Determine a D-H Parameters Table Axis Number Twist angle ai Link length ai Link offset di Joint angle 0 1 a1 a1 d1 Ỡ1 2 a2 a2 d2 q 2 Ị Ị n a n an dn q n FIGURE Continued . Algorithm for Determining the Homogenous Transformation Matrix A n After determining the D-H parameters the next step is to derive the homogeneous transformation matrix from one frame to the next. We will denote the Ui ỡi matrix as the rotation matrix about the Zi-1 axis while the Vi ai matrix as the rotation matrix denoting the rotation about the Xi axis. To perform coordinate transformation from reference frame Fi to reference frame Fi_1 we need to derive the rotation matrix and the displacement vector. The rotation matrix from Fi to Fi _1 is given by Ri-1 UiVi Copyright 2005 by CRC Press LLC where Ui Or is given by cosỌ sin ỡj 0 Ui O sin Or cosừỵ 0 0 0 1 and Vi ai is given by 1 0 0 V a 0 cos ai sin ai 0 sin ar cos ai The homogeneous transformation matrix transforming coordinates from F to Fi 1 is given by Ar 1 A F1 A1 where A11 is the screw displacement about the axis xi through an angle ai and distance ữị. Similarly Áị71 is the screw displacement about the axis zi through an angle Or and distance di. The expression for A and Ai is given by A 1 Ua l3x3 0 0 di .0 0 0 1 ữi A i Vi i3x3 0 0 0 0 0 1 Consequently A 1 Ui 0 0 di Vi ai 0 0 UiVi 0 UiSi 1 .0 0 0 1 0 0 0 1 Expanding the various terms in Equation we get c Oi s Oi 0 1 0 0 c Oi s Orc ar s Oi s ai Ui Vi s Oi c Or 0 0 c ai s i s Oi c Orc ar c ỡrs ar 0 0 1 0 s a c ar 0 s ai c ar c Oi sOi 0 ai 0 c 9iữi Uisi s Oi c Oi 0 0 0 s 9ịữi 0 0 1 0 di di where si ai 0 di Copyright 2005 by CRC Press LLC FIGURE Transformation from the endeffector frame to the base frame. Performing the composition from the nth frame to the base frame see Figure we get A n A 1 A2 . An-1 R 0 d n 0 1 Based on Equations and we can write R0 Ui VM V2 UnVn and d 0 UiS 1 Ui V1U2S2
