# Angular velocity and transformation of Euler Angles

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".

• So do you just apply the transformation to the angular velocity vector? Jun 5, 2021 at 19:30
• yes , this is how you obtain the angular velocity matrix with the transmission matrix and the time derivative of the transformation matrix.
– Eli
Jun 5, 2021 at 20:43