Properties of SO(3) and so(3)

Properties of the “hat” notation:

• Equivalent to cross product: $v\times w=\hat{v}w$, $\forall v,w\in \mathbb{R}$
• $R\hat{\omega }{R}^T=\widehat{R\omega }$

Rotation is a rigid body transformation:

• $||Rq-Rp||=||q-p||,\forall q,p\in {\mathbb{R}}^3$
• $R(v\times w)=Rv\times Rw,\forall q,p\in {\mathbb{R}}^3$

Matrix exponential of so(3) and SO(3):

• Given $\hat{\omega }\in so(3)$, $\exp (\hat{\omega }\theta )=I+\theta \hat{\omega }+\frac{{\theta }^2}{2!}{\hat{\omega }}^2+\cdots \in SO(3)$
• Rodrigues’ formula: $e^{\hat{\omega }\theta }=I+\hat{\omega }\sin \theta +{\omega }^2(1-\cos \theta )$
• If $R(t)=e^{\hat{\omega }t}$, then $\dot{R}=\hat{\omega }{e}^{\hat{\omega }t}=\hat{\omega }R$
  • $⟹\hat{\omega }=\dot{R}{R}^T$