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Suppose we have an evolving system describing a rigid body by euler angles with respect to the bodies center. Suppose we do a constant offset parameter transformation for the euler angles, in other words suppose we re-define the center of the coordinate system with respect to a different set of euler angles. Does the body fixed frame angular velocity vector get changed during this process?
Since it's body fi xed frame I would like to say it does not get changed, can someone explain?
Two coordinate systems that not parallel to each other are attached to a rigid body
the transformation matrix between system "1" and inertial system is described by the three Euler angles
$$S_1=S_1[\alpha(t)~,\beta(t)~,\gamma(t)]$$
the transformation matrix between system "2" and inertial system is $$S_2=S_s^T\,S_1\,S_s$$
where $~S_s$ is a static transformation that make at $t=0$ the two coordinates system parallel to each other , you can describe the static coordinate system also by a three Euler angles they are time independent .
the angular velocity $~\vec\omega=[\omega_x'~,\omega_y'~,\omega_z']~$ can obtained from this equation
$$\begin{bmatrix} 0 & -\omega_z' & \omega_y' \\ \omega_z' & 0 & -\omega_x' \\ -\omega_y' & 0 & \omega_x' \\ \end{bmatrix}=S_2^T\,\dot{S_2}$$
where the components of the angular velocity $~\vec \omega~$ are given in body fixed coordinate system "2".
