# Euler Angles - Why Post Multiplication

1. Jun 2, 2012

### phiby

In robotics, you have a co-ordinate frame which is at the base & another which is at the next joint. You want to describe the position and orientation of the 2nd frame with respect to the first in terms of position and orientation. For the position, you use a translation matrix.

The orientation is described by a rotation matrix which rotates the 2nd frame from it's coincident position with the first frame and brings it to it's final position.

Let the 1st frame have axis X, Y, Z
Let 2nd frame have axis x, y, z

To describe frame 2 in terms or frame 1, you start with both frames very fully coincident at the beginning (i.e. even in orientation).

Then you have 2 methods.

1) Fixed Axis angles.
Then
- you rotate frame 2 by t1 about X - Rx(t1)
- you rotate frame 2 by t2 about Y - Ry(t2)
- you rotate frame 2 by t3 about Z - Rz(t2)

So new orientation of frame 2 is given by

Rz(t3) * Ry(t2) * Rx(t1)

(Obviously, you pre-multiply the 1st matrix by the 2nd. And the premultiply the result with the 3rd matrix)

2) Euler Angles
2nd way of describing it is by Euler angles - i.e. you rotate the 2nd frame about one of it's own axis (x or y or z), instead of (X, Y or Z)

- rotate frame 2 by t3 about x - (y becomes y' & z becomes z')
- rotate frame 2 by t2 about y' - (x becomes x' & z becomes z'')
- rotate frame 2 by t1 about z" - (x' becomes x'' & y becomes y")

Now this transform is described again by
Rz(t3) * Ry(t2) * Rx(t1)

I want to know how is this 2nd transform derived?

I know how the first one is derived because I know how to find the rotation matrix for rotating a point about about a fixed axis. And I know that if you are working with column vectors (for the point), you premultiply the 1st rotation matrix by the 2nd rotation matrix.

However, I am not able to grok how you write the transformation matrices for the 2nd case.