I have some troubles on differentiating a time-varying rotational matrix $R(t) \in SO(3)$. If I use a axis-angle representation with the Rodrigues formula we have:
$R(t) = \exp(\,A(t) \theta(t)\,) = I + A(t)\sin(\theta(t)) + A^2(t)(1-\cos(\theta(t))) $
where $\vec{a}(t)$ is a time-varying unit vector ($A(t) \in so(3)$ is its matrix representation), and $\theta(t)$ is the time-varying angle. I would like to compute $ \frac{d}{dt}( R(t) )$. Any hints? Thanks a lot!
Use the usual rules for derivative, keeping in mind that multiplication of matrices is not commutative. $$ \begin{split} \frac{d}{dt} I &=0 \\ \frac{d}{dt} A\sin \theta &= A'\sin \theta +A( \cos \theta)\theta' \\ \frac{d}{dt} A^2(1-\cos \theta ) &= (AA'+A'A)(1-\cos \theta )+A^2 (\sin \theta )\theta' \end{split} $$