Kinematics

Abstract

One of the more useful applications of the matrix techniques that you have been learning is to the control of a robot. Most industrial robots consist of a chain of ‘revolute’ axes which we can think of waist, shoulder, elbow, wrist and so on. But the position of the ‘end effector’, the hand that does the work, is a somewhat complicated combination of the functions that depend on all these angles.

By now you should be familiar with the three-by-three matrices

$$\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & {\rm{c}} & -{\rm{s}}\\ 0 & {\rm{s}} & {\rm{c}}\end{array}\right]$$

$$\left[\begin{array}{ccc}{\rm{c}} & 0 & {\rm{s}}\\ 0 & 1 & 0\\ -{\rm{s}} & 0 & {\rm{c}}\end{array}\right]$$

and

$$\left[\begin{array}{ccc}{\rm{c}} & -{\rm{s}} & 0\\ {\rm{s}} & {\rm{c}} & 0\\ 0 & 0 & 1\end{array}\right]$$

that define rotations about the x, y and z axes respectively.

But now we need to combine translations with these rotations, so that we can ‘move down’ the parts of the robot, from shoulder to elbow, say.

We could simply add the displacement to our present coordinate, but we would really like something that can be applied using the standard computer matrix multiplication routine.

So we ‘fatten up’ the matrix to become four-by-four and add a fourth component, which is always 1, to our position vector to become (x, y, z, 1)′. Now

$$\left[\begin{array}{cccc}{\rm{c}} & 0 & {\rm{s}} & L\\ 0 & 1 & 0 & 0\\ -{\rm{s}} & 0 & {\rm{c}} & 0\\ 0 & 0 & 0 & 1\end{array}\right]$$

will represent a rotation about the y axis combined with a translation L in the x direction. But does the translation happen before or after the rotation when we are describing the kinematics of a robot arm?

This chapter tries to remind you of the theory, while the next will put it into practice.