Section 3.3 Direction Cosines and Angular Velocity
If the orientation of referential \(\cB\) relative to \(\cA\) is defined in terms of the direction cosines \(c_{ij} = \bha_i \cdot \bhb_j\) between basis \((\bha_1 , \bha_2 , \bha_3 )\) of \(\cA\) and basis \((\bhb_1 , \bhb_2 , \bhb_3 )\) of \(\cB\text{,}\) then it should be possible to relate the components of angular velocity \(\bom_{\cB / \cA}\) on either basis to the time-derivatives of these quantities.
First we resolve \(\bom_{\cB / \cA}\) on the bases of \(\cA\) and \(\cB\) in the following way
\begin{equation*}
\bom_{\cB / \cA}
= \Omega_1 \bha_1 + \Omega_2 \bha_2 + \Omega_3 \bha_3
= \omega_1 \bhb_1 + \omega_2 \bhb_2 + \omega_3 \bhb_3
\end{equation*}
We then determine \(\dot{c}_{ij}\) by differentiating \(\bha_i \cdot \bhb_j\) relative to referential \(\cA\text{:}\)
\begin{align*}
\dot{c}_{ij} \amp = \bha_i \cdot\left({d \bhb_j \over dt} \right)_{\cA} \\
\amp = (c_{i1} \bhb_1 + c_{i2} \bhb_2 + c_{i3} \bhb_3) \cdot (\bom_{\cB / \cA} \times \bhb_j) \\
\amp = \bom_{\cB / \cA} \cdot \left(c_{i1} \bhb_j\times \bhb_1 + c_{i2} \bhb_j \times \bhb_2 + c_{i3} \bhb_j \times \bhb_3 \right)
\end{align*}
where we have used the property
\(({\bf u} \times {\bf v})\cdot {\bf w} = {\bf u}
\cdot ( {\bf v} \times {\bf w})\) of the
triple scalar product (see
Definition A.1.10). Then letting
\(j=1,2,3\) we find:
Theorem 3.3.1. Angular Velocity vs direction cosines.
If \(c_{ij} = \bha_i \cdot \bhb_j\) denotes the \(ij\)th direction cosines between basis \((\bha_1 , \bha_2 , \bha_3 )\) of \(\cA\) and basis \((\bhb_1 , \bhb_2 , \bhb_3 )\) of \(\cB\text{,}\) then
\begin{equation}
\begin{array}{l}
\dot{c}_{i1} = c_{i2} \omega_3 - c_{i3} \omega_2 \\
\dot{c}_{i2} = c_{i3} \omega_1 - c_{i1} \omega_3 \\
\dot{c}_{i3} = c_{i1} \omega_2 - c_{i2} \omega_1
\end{array}
\qquad (i=1,2,3)\tag{3.3.1}
\end{equation}
with \(\omega_i = \bom_{\cB / \cA}\cdot \bhb_i\text{.}\)
Alternatively, we can express \(\dot{c}_{ij}\) in terms of the components \((\Om_1 , \Om_2 , \Om_3)\text{:}\)
\begin{equation}
\begin{array}{l}
\dot{c}_{1j} = c_{3j} \Omega_2 - c_{2j} \Omega_3 \\
\dot{c}_{2j} = c_{1j} \Omega_3 - c_{3j} \Omega_1 \\
\dot{c}_{3j} = c_{2j} \Omega_1 - c_{1j} \Omega_2
\end{array}
\qquad (j=1,2,3)\tag{3.3.2}
\end{equation}
In practical applications such as spacecraft attitude dynamics, the time evolution of
\(\bom_{\cB/\cA}\) is obtained as a vector resolved into the “body basis”, that is, in terms of components
\((\om_1 , \om_2 , \om_3)\text{:}\) the orientation of
\(\cB\) relative to
\(\cA\) is then obtained by integration of the system of equations
(3.3.1).
