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!

Answer 1

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

Answer 2

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)