{"id":29,"date":"2024-08-16T16:35:55","date_gmt":"2024-08-16T16:35:55","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/robotics\/chapter\/chapter-3\/"},"modified":"2026-03-16T14:22:45","modified_gmt":"2026-03-16T14:22:45","slug":"chapter-3","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/robotics\/chapter\/chapter-3\/","title":{"raw":"Forward Kinematics for Serial Manipulators","rendered":"Forward Kinematics for Serial Manipulators"},"content":{"raw":"[latexpage]\n

1) Basics of Forward Kinematics<\/h1>\n

Theory<\/h2>\nForward kinematics maps a robot's joint positions to its end effector pose.\u00a0 It describes a manipulator\u2019s motion irrespective of force and torque.\n\n[caption id=\"attachment_26\" align=\"aligncenter\" width=\"391\"]\"forward Figure 3.1: Forward Kinematics[\/caption]\n\nFor simplicity, assume each joint in the chain has 1 DoF (or is two joints, etc.).\u00a0 For serial manipulators, place a coordinate frame on each joint, and number the joints in order from base to tip.\u00a0 The robot's base is considered link 0, so the first joint (joint 1) drives link 1, joint 2 drives link 2, etc.\u00a0 The Z axis of each joint should indicate the axis of motion.\u00a0 An example robot (ABB IRB 6700<\/a>) is shown below, with coordinate frames attached to each joint.\n\n[caption id=\"attachment_26\" align=\"aligncenter\" width=\"422\"]\"Standard Figure 3.2: Standard 6-Axis Industrial Robot with Coordinate Frames[\/caption]\n\nFor a detailed introduction to forward kinematics, watch the Forward Kinematics Introduction video below.\n\nVideo: Forward Kinematics Intro<\/a> (click link for closed caption version)\n\nThe original version of this chapter contained H5P content. You may want to remove or replace this element.\n\nThere are three common methods of solving for a robot's forward kinematics equations.\u00a0 These are:\n
    \n \t
  1. Basic Method<\/li>\n \t
  2. Denavit-Hartenberg (DH) Method<\/li>\n \t
  3. Product of Exponentials (PE) Method<\/li>\n<\/ol>\nThe basic method is used for simple robots, generally those that have three or fewer joints.\u00a0 The DH and PE methods are used for more complicated robots.\u00a0 Each section of this chapter will cover a different method of forward kinematics, with theory and examples.\n\nThe procedure for the basic method of forward kinematics is as follows:\n
      \n \t
    1. Put one coordinate frame on each link<\/li>\n \t
    2. Find the link positions $o_1^0$ \u2026 $o_n^0$<\/li>\n \t
    3. Find the end effector frame orientation $R_n^0$<\/li>\n \t
    4. Make the homogeneous transformation $T_n^0 = \\begin{bmatrix} R_n^0 & o_n^0 \\\\ 0_{1x3} & 1 \\end{bmatrix} $<\/li>\n<\/ol>\n

      Examples<\/h2>\nFor a detailed explanation and example of forward kinematics for a two-link RR robot, see the video below.\n\nVideo: Forward Kinematics - Basic Method<\/a> (click link for closed caption version)\n\nThe original version of this chapter contained H5P content. You may want to remove or replace this element.\n

      Quick Quiz<\/h2>\nThe original version of this chapter contained H5P content. You may want to remove or replace this element.\n

      2) Denavit-Hartenberg (DH) Method<\/h1>\n

      Theory<\/h2>\nThe Denavit-Hartenberg (DH) method of forward kinematics uses a current-frame approach. Each transformation from one coordinate frame to another is represented by the product of four basic transformations:\n
        \n \t
      1. Rotation around current Z by angle $\\theta$ (\"joint angle\")<\/li>\n \t
      2. Translation along current Z by distance\u00a0d<\/em> (\"link offset\")<\/li>\n \t
      3. Translation along current X by distance\u00a0a<\/em> (\"link length\")<\/li>\n \t
      4. Rotation around current X by angle $\\alpha$ (\"link twist\")<\/li>\n<\/ol>\n[latex]\n\n$T_i^{i-1} = [Rot_{z,\\theta_i}] [Trans_{z,d_i}] [Trans_{x,a_i}] [Rot_{x,\\alpha_i}]\u00a0 \\\\ \\\\ $\n\n$T_i^{i-1} = \\left\\[ \\begin{bmatrix} c_{\\theta i} & -s_{\\theta i}c_{\\alpha i} & s_{\\theta i}s_{\\alpha i} & a_i c_{\\theta i} \\\\\ns_{\\theta i} & c_{\\theta i}c_{\\alpha i} & -c_{\\theta i}s_{\\alpha i} & a_i s_{\\theta i} \\\\\n0 & s_{\\alpha i} & c_{\\alpha i} & d_i\\\\\n0 & 0 & 0 & 1 \\end{bmatrix} =\n\\begin{bmatrix} R_i^{i-1} & o_i^{i-1} \\\\ 0_{1x3} & 1 \\end{bmatrix} \\right\\]$\n\n[\/latex]\n\nIn layman's terms, this transformation tells you the orientation and position coordinates of the robot's current joint location relative to the previous one.\u00a0 This is illustrated in the following diagram:\n\n[caption id=\"attachment_26\" align=\"aligncenter\" width=\"393\"]\"DH Figure 3.3: DH Frame Diagram[\/caption]\n\nwhere the parameter assignments are:\n