Given the 6-dimensional vector of the exponential coordinates of the homogeneous transformation: $S\theta$, where $S$ is the screw axis consisting of the pair $(\omega, v)$ and both of them are $3$ dimensional vectors, $\theta$ is the angle followed by the transformation around that screw axis. Also for a screw axis, either $v$ or $\omega$ is a unit vector.
How to separate the screw axis $S$ and the angle of rotation $\theta$ from the given exponential coordinates 6-vector $S\theta$? Note that the unit vector is not given. it may be $v$ or $\omega$.
Is the following the correct algorithm?
\begin{align*} \theta &= \|S\theta\| &\text{#which is the magnitude of $S\theta$ $6$-vector.} \\ S &= \frac{S\theta}{\|S\theta\|} &\text{#the screw axis is the $S\theta$ $6$-vector divided by its magnitude.} \end{align*}
Thanks a lot in advance.
S_\theta
in place ofS\theta
. $\endgroup$