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.