Euler Angles - Why Post Multiplication

[phiby]

Normally with column vectors, we premultiply rotation matrices if the angles are with respect to fixed axis.

Why then do we post multiply if the angles are Euler Angles, angles with respect to the mobile axis?

 

[chiro]

Normally with column vectors, we premultiply rotation matrices if the angles are with respect to fixed axis.

Why then do we post multiply if the angles are Euler Angles, angles with respect to the mobile axis?

Hey phiby.

Could you give an example (in terms of the rotation matrices)?

The answer I would give is basically going to do with using the order of composition to see what happens in what order and then look at each composition uniquely and then the compositions in the order they are in.

This is also the basic way you can understand what a matrix is actually doing in the way that finding the inverse relates to finding a series of matrices that you premultiply by which gives a series of compositions that represent the 'row operations' you were taught when you did the first linear algebra course (i.e. Gaussian elimination).

The nature of these row-reduction matrices have the same kind of interpretation as compositions of general matrices and based on these you can follow step by step what the matrix is actually doing for each scale and translation.

 

[phiby]

Hey phiby.

Could you give an example (in terms of the rotation matrices)?

You have a Fixed Frame with Axis XYZ & Mobile Frame with Axis xyz.

Both the frames have the same origin. To describe the mobile frame, you could use 3 rotations about Fixed axis as t1 about X, t2 about Y, t3 about Z?
Now, the rotation of the mobile axis is R_Z(t3) * R_Y(t2) * R_X(t1)

or through zyx euler angles, i.e. t1 about z, t2 about y' and t1 about x".
(y' is the mobile y-axis after t1 rotation about z, x" is the mobile x-axis after the first 2 rotations)

This works out to be the same as the first matrix.

 

[phiby]

Anybody have any idea about it.
Basically, you have 2 frames who share the same origin, but axes are oriented differently. You want to describe the position and orientation of the 2nd frame with respect to the first.

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 rotate frame 2 by t1 about X - R_x(t1)
- you rotate frame 2 by t2 about Y - R_y(t2)
- you rotate frame 2 by t3 about Z - R_z(t2)

So new orientation of frame 2 is given by

R_z(t3) * R_y(t2) * R_x(t1)

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

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
R_z(t3) * R_y(t2) * R_x(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.

Basically, this is used in Industrial Robot Kinematics. Should I not be posting this in the math sub forum - is some other sub-forum the right place for this question?

 

[jadnohra]

phiby, one way to explain this is using linear algebra, here is how: http://jadnohra-tech.blogspot.de/2012_06_01_archive.html#7338113974578597231

Cheers.