Clarification on SO3.to_euler order definition: Does '321' correspond to intrinsic XYZ?

[barikata1984]

Hello,

First of all, I really appreciate your work. I loved STARS Lab's liegroups but it has not been mainteined for a while. Thanks for developing an updated version.

Then, I have a question regarding the rotation order definition in SO3.to_euler.

I observed that SO3.to_euler(C, order="321") returns Euler angles that match the Intrinsic XYZ sequence (Roll $\to$ Pitch $\to$ Yaw) when compared with scipy.spatial.transform.Rotation.as_euler('xyz').

I believe that "321" usually implies rotating around the 3rd axis (Z), then the 2nd axis (Y), then the 1st axis (X), which corresponds to Intrinsic ZYX.

It appears that the implementation might be interpreting "321" (or "123") differently, possibly referring to the order of axes in the returned array rather than the sequence of rotations.

Could you please clarify if order="321" is indeed intended to implement the XYZ rotation sequence, and conversely, if order="123" corresponds to ZYX?

Thank you.

[arturodcb]

Hello,

Thank you for your comments, they are very welcome!

You are right, the order "321" in pymlg corresponds to the Roll -> Pitch -> Yaw rotation sequence. The DCM that encodes the orientation of frame a relative to frame b from the sequence Roll Pitch Yaw is

$ C_ab = C_1(\theta_1) C_2(\theta_2) C_3(\theta_3) $.

The reason it was named "321"
is because the DCM that encodes the orientation of frame b relative to frame a from the sequence Roll Pitch Yaw is

$ C_ba = C_3(-\theta_3) C_2(-\theta_2) C_1(-\theta_1) $.

Therefore, the naming comes from the order from left to right in which the elementary DCMs are multiplied to produce C_ba.

This is based from State Estimation for Robotics book by Prof. Barfoot. In the second edition, this equation is found in page 243.

I agree that the naming is confusing and we should just change it to "123" to refer to RPY, to avoid confusions and stay consistent with the rest of the literature.

Best regards,

Arturo.