Procrustes with dcm2angle: sign error?

Hi!

I have an issue with my script and even when I simplify the numbers, it still looks funny...here's the simplifed form, question still applies.

I am using two arrays which create squares:

original = [ 1 0 1 1 1 1 0 1 1 0 0 1];

rotated = [ 1 0 1 1 0 0 0 0 0 0 0 1];

I use the following commands:

[D, Z, TRANSFORM] = procrustes(rotated, original);

[yawR, pitchR, rollR] = dcm2angle(TRANSFORM.T);

yaw = rad2deg(yawR);

pitch = rad2deg(pitchR);

roll = rad2deg(rollR);

In this case, I correctly find the roll to be -90deg. Right hand rule!

If I swap the two, I would expect to find the roll to be +90 deg. It still comes up as roll = -90deg. Why does it calculate -90deg for roll in either configuration? I see that the original plane could be translated "above" the rotated plane, allowing for a -90 deg rotation, but that is not the objective: I have ensured that the points on the original correspond to the points in the rotated square. That is to say, if I were to leave the adjacent points in place (the first point both the original and rotated are the same and I desire the rotation to be around this point), the shortest roll should be in the positive 90 deg direction, which would bring the far edges of the squares together.

I can add the plots if it is helpful. Understanding this should help me identify the problem in my alignment! Thank you.

Answer 1

[AJ von Alt]

The function procrustes allows reflection by default. If you want T to belong to SO(3) you must manually set the 'reflection' flag to false.

You can see that this is happening because the DCM for the transformation from rotated to original has a determinant of -1. (Recall that a 3x3 matrix is a member of SO(3) only if it is orthogonal and has a determinant of 1.)

[D, Z, Tr2d] = procrustes(original,rotated);
det(Tr2d.T)

ans =

      -1
roll =

    -90

Turning off scaling and reflection fixes this problem.

[D, Z, Tr2oSO3] = procrustes(original ,rotated,'reflection',0,'scaling',0);

det(Tr2oSO3.T)

ans =

       1
  roll =

      90