1

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 derivative of $R(t)R(t)^\mathrm{T} $ which gives us a skew symmetric matrix $\dot{R}(t)R(t)^\mathrm{T} = -R(t)\dot{R}{}^\mathrm{T}(t) =: \phi(t)$, where $$\phi (t) = \left[ \begin{matrix} 0 & -\phi_3 & \phi_2 \\ \phi_3 & 0 & -\phi_1\\ -\phi_2 & \phi_1 & 0 \end{matrix} \right]. $$

The derivative of $R(t)$ is then given by $$ \dot{R}(t) = \phi (t) R(t).$$

To derive the Lie algebra, later, it gives the 1st order expansion of $R(t)$ $$ R(t) \approx R(t_0) + \dot{R}(t_0)\ (t-t_0) = I + t\ \phi (t_0) ,$$ where $t_0 = 0$ and $R(t_0) = I$. By given an assumption that $\phi (t_0) = \phi_0$ around $t_0$, it gets $$ \dot{R}(t) = \phi (t_0) R(t) = \phi_0 R(t).$$ This is the one I cannot get, i.e. the last equation is my question. Any suggestion would be appreciated.

Cosmas Zachos
  • 62,595
MakeItGreatAgain
  • 23

2 Answers2

2

$\phi$ can be thought of as the tangent vector $\text{d}/\text{d}t$ of $R(t)$. For any $R(t)$, we can use the exponential map to obtain $$R(t_0 + t) = R(t_0) + t\frac{\text{d}}{\text{d}t} R(t_0) + \frac{1}{2!}\left(t \frac{\text{d}}{\text{d}t}\right)^2 R(t_0) + \frac{1}{3!}\left(t \frac{\text{d}}{\text{d}t}\right)^3 R(t_0) + \ldots \\ = \sum^\infty_{n = 0} \frac{1}{n!} \left(t \frac{\text{d}}{\text{d}t}\right)^n = e^{t \phi } R |_{t_0}$$

By definition $R(t_0 = 0) = I$, we have $$R(t) = e^{t\phi}$$

and we can differentiate the matrix exponential to obtain $$\dot{R}(t) = \phi e^{t\phi} = \phi_0 R(t)$$

Vincent Thacker
  • 7,926
1

Use your $t_0 = 0$ so $R(0) = I$, $$\dot{R}(t)R(t)^\mathrm{T} =: \phi(t)= \phi(0)+ O(t)\leadsto \\ \dot{R}(t)= (\phi_0+ O(t) ) R(t)= (\phi_0+ O(t) )( I+ O(t))= \phi_0+ O(t) . $$
The order t terms vanish at 0, so $\dot{R}(0)=\phi_0$, antisymmetric, alright.

Cosmas Zachos
  • 62,595
  • Thanks as well, Cosmas. – MakeItGreatAgain Jun 17 '21 at 06:34
  • Cosmas, I have a little more to ask. It is simply about the symbol in my question you edited. Is that OK to use $$\phi(t) = \left[ \begin{matrix} x & x & x \ x & x & x \ x & x& x \end{matrix} \right]$$ instead of $\phi(t)^{\wedge}$ representing a skew symmetric matrix of vector $\phi(t)$. I want them in my assignment report. – MakeItGreatAgain Jun 17 '21 at 06:42
  • Cosmas, I have a little more to ask. It is simply about the symbol in my question you edited. Is that OK to use $$\phi(t) = \left [ – MakeItGreatAgain Jun 17 '21 at 06:42
  • No, the object you defined is manifestly antisymmetric, as you demonstrate. – Cosmas Zachos Jun 17 '21 at 10:28