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A preamble: I know to convert quaternion in Eulerian Angles, I have to know the rotation orders adopted in that situation (i.e ZYX, or 321, YZX, or 231).

Now, let's assume:

q = (q0, q1, q2, q3)

I could write the quaternion rate as an ODEs set:

dq/dt = M * q

where M is a 4x4 matrix.

The preamble is to be sure that the matrix M does not depend on the order of the rotations used, but instead that its construction is invariant to the adopted rotations order.

In fact M should be set up only with angular velocity on X, Y and Z axis for any conventional system of rotation.

Am I right?

Mat
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    It is not apparent what you wish to achieve. Could you extend the frame a little more, what is the given situation and what the desired output and its further purpose? In general it looks like you want to express some rigid-body dynamics in quaternions? – Lutz Lehmann May 29 '22 at 08:37
  • Yes, I would like to write the ode for quaternion rate. The quaternion rate is the results between the 4x4 matrix M and the actual quaternion values q. I don't remember the values inside the matrix. It should be based on the angular velocity, but I am not sure – Mat May 29 '22 at 16:21
  • The meaning of "the rotation about axis number $i$ by angle $\alpha_i$ for $i=1,2,3$" depends on a choice of ordering of the axes. How could the rate of change of the rotation NOT depend on the choice? I have always thought that point of using quaternions to describe the rotation is exactly to not think in terms of the $\alpha_i$s. – Jyrki Lahtonen Jun 17 '22 at 05:55
  • I guess the problem of my previous comment can be studied by treating the angular velocities as skew-symmetric matrix $\Omega$ like here. IIRC books on mechanics use those. – Jyrki Lahtonen Jun 17 '22 at 06:06

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