I am working on an application of CV, in which a way to calculate the derivative of rotation matrix is involved.
$$R$$ is the rotation matrix and $$R \in SO(3)$$ Also, $R$ is changing with $t$ giving $R(t)$.
$R(t)R(t)^\mathrm{T} = I$ is known. We calculate the derivate of $R(t)R(t)^\mathrm{T} $ which gives us a skew symmetric matrix $\dot{R(t)}R(t)^\mathrm{T}$ and $\dot{R(t)}R(t)^\mathrm{T} = \phi(t) \hat{}$, where $$\phi (t) \hat{ } = \left[ \begin{matrix} 0 & -\phi_3 & \phi_2 \\ \phi_3 & 0 & -\phi_1\\ -\phi_2 & \phi_1 & 0 \end{matrix} \right]. $$ Maybe $\phi \hat{}$ is more suitable than $\phi (t) \hat{}$ in the one above.
The derivative of $R(t)$ is then given by $$ \dot{R(t)} = \phi (t) \hat{} R(t).$$
To derive the Lie algebra, later, it gives the 1st order expansion of $R(t)$ $$ R(t) \approx R(t_0) + \dot{R(t_0)}(t-t_0) = I + \phi (t_0) \hat{} (t) ,$$ where $t_0 = 0$ and $R(t_0) = I$. By given an assumption that $\phi (t_0) = \phi_0$ around $t_0$, it gets $$ \dot{R(t)} = \phi (t_0) \hat{} R(t)$ = \phi_0 \hat{} R(t).$$ This is the one (problem) I cannot get. Any suggestion would be appreciated.