Rotation matrix defined by two Euler angles

Question

I need to find the expressions for $\hat{e_{CX}}$, $\hat{e_{CY}}$ and $\hat{e_{CZ}}$ accordingly to that picture. How can I write a proper axis transformation so I can obtain $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ in terms of $\hat{e_{CX}}$, $\hat{e_{CY}}$ and $\hat{e_{CZ}}$?

Answer 1

What you want for the local to global rotation matrix is a sequence of rotations about the z and the y axis:

$$R=\rm{Rot}_Z(\phi_0) \rm{Rot}_Y(\theta_0) $$ $$ R = \begin{vmatrix} \cos \phi_0 \cos \theta_0 & -\sin \phi_0 & \cos \phi_0 \sin \theta_0 \\ \sin \phi_0 \cos \theta_0 & \cos \phi_0 & \sin \phi_0 \sin \theta_0 \\ -\sin\theta_0 & 0 & \cos \theta_0 \end{vmatrix}$$

Each column of $R$ contains the coordinates of local axes $\hat{e}_{cx}$, $\hat{e}_{cy}$ and $\hat{e}_{cz}$.

The inverse transformation is $R^\top$.