- The problem
I am having three coordinate systems $(X,Y,Z)$, $(x,y,z)$ and $(\hat x, \hat y, \hat z)$ in $\mathbb R^3$, which can be mapped into each other by Euler rotations. I will refer to them in short as the $X$, $x$ and $\hat x$ system, respectively.
• I have given the Euler angles $(\Omega,i,\omega)$ which map the $x$ into the $X$ system.
• I have given the Euler angles $(\phi,\theta,\psi)$ which map the $\hat x$ into the $X$ system.
• I need to determine the Euler angles $(\alpha,\beta,\gamma)$ which map the $\hat x$ into the $x$ system.
- Definitions
First, I define an Euler rotation by the matrix $R(\Omega,i,\omega):=R_3(\Omega)\circ R_2(i)\circ R_1(\omega)$ with elementary rotations
$R_1(\omega):=\begin{bmatrix} \cos\omega & -\sin\omega & 0 \\ \sin\omega & \cos\omega & 0 \\ 0 & 0 & 1 \end{bmatrix}$,$\;\;$ $R_2(i):=\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos i & -\sin i \\ 0 & \sin i & \cos(i) \end{bmatrix}$,$\;\;$ $R_3(\Omega):=\begin{bmatrix} \cos\Omega & -\sin\Omega & 0 \\ \sin\Omega & \cos\Omega & 0 \\ 0 & 0 & 1 \end{bmatrix}$.
A reverse rotation can then be written by $R^{-1}(\Omega,i,\omega)= R_1(-\omega)\circ R_2(-i)\circ R_3(-\Omega)$.
- Solution attempt
Then, I thought that the rotation $R(\alpha, \beta, \gamma)$, which maps the $\hat x$ into the $x$ system should be composable by
$R(\alpha, \beta, \gamma)=R^{-1}(\Omega,i,\omega)\circ R(\phi,\theta,\psi)$,
correct? I then performed the corresponding matrix multiplication, and equated the resulting matrix to the rotation matrix for $R(\alpha, \beta, \gamma)$. My idea was to read off the angles $(\alpha, \beta, \gamma)$ from those equations. However it is not obvious to me how to read them off, or obtain them from these equations. The expressions are too long to post them here.
Do I have an error? Is there a simpler way to do this then the path I have chosen?