# so(3)

The space of skew-symmetric matrices in 3D Euclidean space. They satisfy the constraint that $$\{[\textbf{x}] = -[\textbf{x}]^T\}$$ where $$\textbf{x}\in \mathbb{R}^3$$ and $$[\textbf{x}]=\begin{bmatrix}0 & -x_3 & x_2 \\ x_3 & 0 & -x_1 \\ -x_2 & x_1 & 0 \end{bmatrix}$$.

In robotics, they are used to represent angular velocities. Any such angular velocity can be represented by a unit vector $$\hat{\omega}$$ pointing in the direction of the rotation axis, and speed of rotation $$\dot{\theta}$$ around it. Denote the angular velocity in frame $$\{s\}$$ as $$\omega_s = \dot{\theta}\hat{\omega}$$. As frame $$\{b\}$$ rotates, its x-axis traces out a circle whose linear velocity is in a direction tangent to this circle, which can be computed as $$\dot{\hat{x_b}}= \omega_s \times \hat{x}_b = [\omega_s]\hat{x}_b$$. Similarly, $$\dot{\hat{y_b}}= [\omega_s]\hat{y}_b$$ and $$\dot{\hat{z_b}}=[\omega_s]\hat{z}_b$$. This results in the matrix representation of angular velocity as $$\dot{R}_{sb} = [\dot{\hat{x}}_b \; \dot{\hat{x}}_b \; \dot{\hat{x}}_b] = [\omega_s]R_{sb}$$.

## References

Lynch, Kevin M., and Frank C. Park. 2017. Modern Robotics: Mechanics, Planning, and Control. 1st ed. USA: Cambridge University Press.

Created: 2022-03-13 Sun 21:45

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