To write the explicit forms of the frametoframe transformation matrices, we use the following TaitBryan angles (angles of roll, pitch, and yaw) to describe the orientation of one frame relative to another

ϕ_{ lb }, θ_{ lb }, and ψ_{ lb }: the roll (around the x ^{ b } axis), pitch (around the y ^{ b } axis), and yaw (around the z ^{ b } axis) angles of the body relative to the lab frame, respectively;

ϕ_{ bw }, θ_{ bw }, and ψ_{ bw }: the roll (around the x ^{ w } axis), pitch (around the y ^{ w } axis), and yaw (around the z ^{ w } axis) angles of a wing relative to the body, respectively.
We then define \(\vec{{\alpha }}_{lb} = {[{\phi }_{lb},{\theta }_{lb},{\psi }_{lb}]}^{T}\) and \(\vec{{\alpha }}_{bw} = {[{\phi }_{bw},{\theta }_{bw},{\psi }_{bw}]}^{T}\).
Applying the rotations in the order of yaw, pitch, and roll, we have
$$\begin{array}{rcl} {R}_{lb}& =& {R}_{lb}^{roll}{R}_{ lb}^{pitch}{R}_{ lb}^{yaw}, \\ {R}_{bw}& =& {R}_{bw}^{roll}{R}_{ bw}^{pitch}{R}_{ bw}^{yaw}, \\ {R}_{lw}& =& {R}_{bw}{R}_{lb} \end{array}$$
(34)
The labtobody transformation matrices are
$$\begin{array}{rcl} & & {R}_{lb}^{roll} = \left (\begin{array}{ccc} 1& 0 & 0\\ 0 & \cos {\phi }_{lb } &\sin {\phi }_{lb} \\ 0& \sin {\phi }_{lb}&\cos {\phi }_{lb} \end{array} \right ),\quad {R}_{lb}^{pitch} = \left (\begin{array}{ccc} \cos {\theta }_{lb}&0& \sin {\theta }_{lb}\\ 0 &1 & 0 \\ \sin {\theta }_{lb}&0& \cos {\theta }_{lb} \end{array} \right ), \\ & & {R}_{lb}^{yaw} = \left (\begin{array}{ccc} \cos {\psi }_{lb} &\sin {\psi }_{lb}&0\\ \sin {\psi }_{ lb}&\cos {\psi }_{lb}&0\\ 0 & 0 &1 \end{array} \right ). \end{array}$$
(35)
We can obtain bodytowing transformation matrices similarly. Every transformation matrix is orthogonal, so its inverse is its transpose.
The angular velocity
\(\vec{{\Omega }}_{B}\) in the body frame is
$$\begin{array}{rcl} & & \vec{{\Omega }}_{B}^{b} = {K}_{ B}\dot{\vec{{\alpha }}}_{lb},\end{array}$$
(36)
where
$$\begin{array}{rcl} & & {K}_{B} = \left (\begin{array}{ccc} 1& 0 & \sin {\theta }_{lb} \\ 0& \cos {\phi }_{lb} &\sin {\phi }_{lb}\cos {\theta }_{lb} \\ 0& \sin {\phi }_{lb}&\cos {\phi }_{lb}\cos {\theta }_{lb} \end{array} \right ).\end{array}$$
(37)
Similarly the angular velocity
\(\vec{{\Pi }}_{W}\) in the wing frame can be written as
\(\vec{{\Pi }}_{W}^{w} = {K}_{W}\dot{\vec{{\alpha }}}_{bw}\). If the TaitBryan angles of the wing relative to the body are prescribed, then
\(\dot{\vec{{\Pi }}}_{W}^{w} =\dot{ {K}}_{W}\dot{\vec{{\alpha }}}_{bw} + {K}_{W}\ddot{\vec{{\alpha }}}_{bw}\) is known.