Introductory Mathematics for Engineering Applications

Trigonometry in Engineering

CHAPTER 3

3.1 INTRODUCTION

In this chapter, the direct (forward) and inverse (reverse) kinematics of one-link and two-link planar robots are considered to explain the trigonometric functions and their identities. Kinematics is the branch of mechanics that studies the motion of an object. The direct or forward kinematics is the static geometric problem of determining the position and orientation of the end-effector (hand) of the robot from the knowledge of the joint displacement. In general, the joint displacement can be linear or rotational (angular). But in this chapter, only rotational motion is considered. Furthermore, it is assumed that the planar robot is wristless (i.e., it has no end-effector or hand) and that only the position but not the orientation of the tip of the robot can be changed.

Going in the other direction, the inverse or reverse kinematics is the problem of determining all possible joint variables (angles) that lead to the given Cartesian position and orientation of the end-effector. Since no end-effector is considered in this chapter, the inverse kinematics will determine the joint angle(s) from the Cartesian position of the tip.

3.2 ONE-LINK PLANAR ROBOT

Consider a one-link planar robot of length l (Fig. 3.1) that is being rotated in the x-y plane by a motor mounted at the center of the table, which is also the location of the robot's joint. The robot has a position sensor installed at the joint that gives the value of ...