Euler Angles - Why Post Multiplication

[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.

 

[Achy47]

Thank you for your question regarding the transformation matrices for describing the position and orientation of a second coordinate frame with respect to the first coordinate frame in robotics. The two methods you mentioned, fixed axis angles and Euler angles, are both commonly used in robotics and each has its own advantages and disadvantages. Let me explain the derivation of the transformation matrices for the second method, Euler angles.

To understand the derivation, it is important to first understand the concept of rotation matrices. A rotation matrix represents a transformation that rotates a vector or a coordinate frame about a given axis. In a 3-dimensional space, there are three possible axes of rotation - X, Y and Z. A rotation matrix for each of these axes can be written as:

Rotation about X-axis:
[1 0 0]
[0 cos(t) -sin(t)]
[0 sin(t) cos(t)]

Rotation about Y-axis:
[cos(t) 0 sin(t)]
[0 1 0]
[-sin(t) 0 cos(t)]

Rotation about Z-axis:
[cos(t) -sin(t) 0]
[sin(t) cos(t) 0]
[0 0 1]

In the first method, fixed axis angles, the rotation matrix for the second coordinate frame is constructed by multiplying the individual rotation matrices for each axis in a specific order. This is because each rotation matrix represents a transformation about a fixed axis, and the order in which these transformations are applied matters. This method is straightforward and easy to understand, but it becomes complicated when dealing with multiple rotations and can lead to a phenomenon known as "gimbal lock".

In the second method, Euler angles, the rotation matrix is constructed by rotating the second coordinate frame about its own axes instead of the fixed axes of the first coordinate frame. This is similar to rotating a rigid object in space, where the object's orientation is described by its own axes. To understand the derivation of the transformation matrices for this method, we need to introduce the concept of rotation matrices about arbitrary axes.

A rotation about an arbitrary axis can be represented by a rotation matrix as follows:

[cos(t) + u^2(1-cos(t)) u*v(1-cos(t)) - w*sin(t) u*w(1-cos(t)) + v*sin(t)]
[u*v(1-cos(t)) + w*sin