How can I convert a 4x4 transformation matrix from one coordinate system to another using Python?

Question

I am trying to convert a transformation matrix from one coordinate system to another. The first coordinate system looks like this in a pybullet simulation:

I'm assuming this coordinate system would be:

X = Forward
Y = Away From Camera
Z = Up

Though I'm not sure about the orientation of the XY plane.

The second coordinate system looks like this:

I'm assuming this coordinate system would be:

X = Forward
Y= Up
Z = Toward The Camera

Though I'm not sure about the orientation of the XZ plane.

The coordinate conversion should then be something like this:

(X,Y,Z)->(X,-Z,Y)

This is the following code I wrote to achieve the transformation:

def transform_matrix(self,transformation_matrix):
    #X maps to X (1,0,0)
    #Y maps to -Z (0,0,-1)
    #Z maps to -Y (0,-1,0)
        

    C = np.matrix([
        [1, 0, 0, 0],
        [0, 0, -1,0],
        [0, -1, 0, 0],
        [0, 0, 0, 1]])

    C_prime = np.transpose(C)

    return C @ transformation_matrix @ C_prime

Which I derived from here

This code, however, isn't working. I'm not sure if it's because the code itself is incorrect, or if my mapping is incorrect. Any help would be appreciated!

@PabloC do you mean making the y column equal to the -z column? I've tried that as well. — Sergio, May 26 '23 at 16:28
Why does the comment say Y maps to -Z? In the pictures both Y and Z go up? — Joooeey, May 26 '23 at 17:40
I took the same logic that was specified in the link I provided, but Y maps to -Z because in the original coordinate system, Y goes away from the camera; whereas, Z goes towards the camera — Sergio, May 26 '23 at 17:57

score 1 · Answer 1 · answered May 27 '23 at 03:42

So @JamesParrott was correct in that you do have a mathematical error. The concept that you are missing is not that you are mapping coordinates simply applying a sign change or such. I think we need more context here. Are you applying a rotation matrix (as link indicated)

Also the link you provided is a discussion around a 4D rotation matrix (x,y,z,w) - I am guessing your application would be a 3x3 Matrix.

Furthermore, I am gonna link here - don't kill this post, haha. https://en.wikipedia.org/wiki/Rotation_matrix

That link is on Rotation Matrices - a standard mathematical concept you can find any number of links on.

If you are simply wanting to map the space (X,Y,Z) to (X, -Z, -Y), then your matrix should be [[1 0 0], [0, 0, -1], [0, -1, 0]]

To understand why this works, do the following matrix multiplication (I recommend using Wolfram Alpha for simple understandings like this)

score 0 · Answer 2 · answered May 26 '23 at 17:33

0

Is there a simple sign error? In the link, the middle two non-zero entries are +1 and -1. Your matrix has both equal to -1.

4x4 matrix from the link

answered May 26 '23 at 17:33

JamesParrott

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Sorry, I should have been more clear with the link. The only portion I derived from the link is the equation M_{new} = C * M * C^-1, which yields the transformation matrix in the new coordinate space. My coordinate systems are not the same from the link, however. – Sergio May 26 '23 at 17:44
Ah sorry. Which error are you getting, and what transformation_matrix are you calling it with? – JamesParrott May 26 '23 at 18:19
So some answers I need help with are, is my transform_matrix function mathematically correct in terms of changing a transformation matrix from one coordinate system to another? Additionally, how should I interpret the pictures of the coordinate systems I provided? I don't know how to tell what is left, right, forward, or back. – Sergio May 26 '23 at 18:34
The equation ought to be C^-1 M C, but it should make no difference in your case as your C matrix satisfies C = C' = C^-1 https://en.wikipedia.org/wiki/Change_of_basis#Endomorphisms – JamesParrott May 26 '23 at 19:03

How can I convert a 4x4 transformation matrix from one coordinate system to another using Python?

2 Answers2