i was working on something and i'm wondering if the following is true
Can we decompose any given rotation matrix of angle $\theta$ and along $u$, $A_{u}({\theta})$ in terms of the rotations $M_x(\theta_1),M_y(\theta_2),M_z(\theta_3)$.
This feels like it should be possible to just compose the $3$ elementary rotations with different angles to get any rotation matrix $A_{u}({\theta})$. I know that the rotations are a group for the composition of applications. I feel like i'm lacking knowledge about rotations and that this is obvious. Would those 3 elementary rotation matrix be like a base of a vector space in which any rotation matrix is a "linear combination" of the elementary ones except that it's for the composition and not addition.
Any idea on how to proceed ?
Thank you.
Best Answer
Yes, it's possible, but the solution might not be unique. Step 1 is to write out the rotation matrix in terms of the axis and angle, like here. Now write the matrix obtained by multiplying the rotations around the three axis. What you get is the rotation matrix according to Tait–Bryan XYZ convention. You can then identify the corresponding coefficients, to get the angles. Let's assume that $c2=\cos\theta_2$ is positive ($0\le\theta_2\le \pi/2$). Then from the table you immediately get $$\theta_1=\mathrm{atan2}(R_{33},-R_{23}) \\\theta_2=\arcsin(R_{13})\\\theta_3=\mathrm{atan2}(R_{11},-R_{12})$$