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16. Appendix A: Kinematics of a Two Link Revolute Manipulator

16.1 A.1 Forward Kinematics:
The manipulator to be considered is shown in the figure below. The robot base is assumed to be at the origin, and space is assigned cartesian coordinates as shown. The links have lengths of ‘l1’ and ‘l2’. The joints may be rotated angles ‘θ1’ and ‘θ2’ about their axes. The location of the end effector in space will be assigned the coordinates ‘Px’, and ‘Py’.

Figure 16.1 Figure A.1: A Typical Two Link Manipulator

For the kinematic transformation, the location of the end effector is calculated using the joint coordinates. There are many sophisticated techniques for finding the kinematics relationships, but here the coordinates may be found quite simply with some basic trigonometry.

Figure 16.2 Figure A.2: Inverse Kinematics for a Two Link Manipulator

16.2 A.2 Inverse Kinematics:
The inverse kinematics transformation calculates the robot joint coordinates required to obtain the given Px, Py coordinates for the end effector. These equations are more involved to derive, and may be derived with several sophisticated methods. The derivation can also be done with simple trigonometry, as shown below.

The reader should note that these equations provide two solutions, this is because the arm can reach any point in space with the elbow either up or down. The solution is also singular when Px and Py are both equal zero (at the origin), or outside the reach of the manipulator.

Figure 16.3 Figure A.3: Derivation of Inverse Kinematics for a Two Link Manipulator (With Trigonometric Methods)