Angular Velocity Vector in the Body Frame

The angular velocity vector may be decomposed into components along each of the rotation axes used to construct the transformation matrix. If we transform those components to the body frame, then we can express the angular velocity vector in the body frame in terms of the Euler angles ( ). The component of along the first rotation axis, as viewed in the body frame, is

The component along the Y' axis is, in the body frame,

Finally, the component along the Z'' axis is simply

Hence, we have the angular velocity vector in the body frame,