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I have trouble understanding calculation sequences for RPY-transformation and Euler-transformation.
Let's assume that there are 3 (rotation/translation) instructions.
I will notate transformation matrix for instruction 1 as [1],
transformation matrix for instruction 2 as [2],
transformation matrix for instruction 3 as [3]

In this situation when using RPY-method, calculation order will be [3][2][1] -> [3][2 * 1] -> [3 * 2 * 1].
First multiply [2] and [1], and then multiply [3]. Right?

When using Euler-method, I am not sure what is correct order.

  1. [1][2][3] -> [1 * 2][3] -> [1 * 2 * 3]
    First multiply [1] and [2], and then multiply [3]
  2. [3][2][1] -> [3 * 2][1] -> [3 * 2 * 1]
    First multiply [3] and [2], and then multiply [1]

What is correct sequence when calculation Euler-method? Is it 1) or 2) ?
Also I think the resulting 4x4 matrix should be same whether I use RPY or Euler transformation. Is it correct?

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I was quite confused but now got an answer by myself.
Let's assume 3 instructions like below.
#1 rotation about world x axis(reference frame)
#2 rotation about world z axis(reference frame)
#3 rotation about world y axis(reference frame)

When using RPY convention, you process in order of #1 -> #2 -> #3.
Then your matrix will be like [3][2][1]
because in absolute frame, we use pre-multiplication. So new matrix goes to the left side of an existing matrix.

On the other hand, when using Euler convention, you process in order of #3 -> #2 -> #1.
In this case, matrix will also be like [3][2][1]
because in relative frame, we use post-multiplication. So new matrix goes to the right side of an existing matrix.

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