Robot Modeling and Control, 2nd Edition

CHAPTER 5INVERSE KINEMATICS

In Chapter 3 we showed how to determine the end effector’s position and orientation in terms of the joint variables. This chapter is concerned with the inverse problem, that of finding the joint variables in terms of the end effector’s position and orientation. This is the problem of inverse kinematics, and it is, in general, more difficult than the forward kinematics problem.

We begin by formulating the general inverse kinematics problem. Following this, we describe the principle of kinematic decoupling and how it can be used to simplify the inverse kinematics of most modern manipulators that are equipped with spherical wrists. Using kinematic decoupling, we can consider the position and orientation problems independently. We describe a geometric approach for solving the positioning problem, while we exploit the Euler angle parameterization to solve the orientation problem.

We also discuss numerical solution of the inverse kinematics using methods based on both the Jacobian inverse and the Jacobian transpose. The Jacobian inverse method is similar to a Newton–Raphson search whereas the Jacobian transpose method is a gradient search method.

5.1 The General Inverse Kinematics Problem

The general problem of inverse kinematics can be stated as follows. Given a 4 × 4 homogeneous transformation

(5.1)

find a solution, or possibly multiple solutions, ...