SO(3)

Also known as the Special Orthogonal Group. This is also a Lie Group.

Definition: Special Orthogonal group in 3D

$$\text{SO}(3)=R\in {\mathbb{R}}^{3\times 3}:R^{T}R=I,\det (R)=1$$

Group operation: Matrix multiplication

$$R_1,R_2\in \text{SO}(3)⟹R_1{R}_2\in \text{SO}(3)$$

Identity: $I_{3\times 3}$

The Lie Algebra $\mathfrak{so}(3)$

Definition: Skew-symmetric 3×3 matrices

$$\mathfrak{so}(3)={\omega }^{\wedge }\in {\mathbb{R}}^{3\times 3}:({\omega }^{\wedge }{)}^T=-{\omega }^{\wedge }$$

“Hat” operator ${}^{\wedge }:{\mathbb{R}}^3\to \mathfrak{so}(3)$: Maps vectors to skew-symmetric matrices

$$\omega =\left[\begin{array}{c}{\omega }_1\\ {\omega }_2\\ {\omega }_3\end{array}\right]\stackrel{{}^{\wedge }}{\to }{\omega }^{\wedge }=\left[\begin{array}{ccc}0& -{\omega }_3& {\omega }_2\\ {\omega }_3& 0& -{\omega }_1\\ -{\omega }_2& {\omega }_1& 0\end{array}\right]$$

“Vee” operator ${}^{\vee }:\mathfrak{so}(3)\to {\mathbb{R}}^3$: Inverse of hat (extracts the vector)

Physical meaning: $\omega$ is the angular velocity vector (axis-angle representation)

• Direction of $\omega$: axis of rotation
• Magnitude $|\omega |$: angle of rotation (in radians)

Lie bracket: $[{\omega }_1^{\wedge },{\omega }_2^{\wedge }]={\omega }_1^{\wedge }{\omega }_2^{\wedge }-{\omega }_2^{\wedge }{\omega }_1^{\wedge }=({\omega }_1\times {\omega }_2{)}^{\wedge }$

Exponential Map: $\mathfrak{so}$(3) → SO(3)

Rodrigues’ Formula:

$$\exp ({\omega }^{\wedge })=I+\frac{\sin (\theta )}{\theta }{\omega }^{\wedge }+\frac{1-\cos (\theta )}{{\theta }^2}({\omega }^{\wedge }{)}^2$$

where $\theta =|\omega |$ is the rotation angle.

Special cases:

• If $\theta =0$: $\exp (0)=I$
• Small angles: $\exp ({\omega }^{\wedge })\approx I+{\omega }^{\wedge }$ (first-order approximation)

Alternative form (axis-angle):

\exp(\omega^\wedge) = I + \sin(\theta)\hat{u}^\wedge + (1-\cos(\theta))(\hat{u}^\wedge)^2 $$ where $\hat{u} = \omega/\theta$ is the unit axis. ## Logarithm Map: SO(3) → so(3)

\log(R) = \frac{\theta}{2\sin(\theta)}(R - R^T)

where $\theta = \arccos\left(\frac{\text{trace}(R) - 1}{2}\right)$ **Extract the vector**:

\omega = \log(R)^\vee = \frac{\theta}{2\sin(\theta)}\begin{bmatrix} R_{32} - R_{23} \ R_{13} - R_{31} \ R_{21} - R_{12} \end{bmatrix}

**Special case**: If $\theta = 0$ (R = I), then $\omega = 0$ #worldModeling