\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

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Angular velocity tensor and vectors

Mathematica note is available here.

Angular velocity tensor

The tensor $\bs{\Omega}_t$ is called a (spatial) angular velocity tensor. In the following we will show that \(\bs{\Omega}\) is a skew-symmetric tensor.

\[\begin{align} \bs{R}(t)\bs{R}^{\textsf{T}}(t)&=\bs{I}\\ \dot{\bs{R}}(t)\bs{R}^{\textsf{T}}(t)+\bs{R}(t)\dot{\bs{R}}^{\textsf{T}}(t) &=0\\ \dot{\bs{R}}(t)\bs{R}^{\textsf{T}}(t)&=-\bs{R}(t)\dot{\bs{R}}^{\textsf{T}}(t)\\ \dot{\bs{R}}(t)\bs{R}^{\textsf{T}}(t)&=-\left(\dot{\bs{R}}(t)\bs{R}(t)\right)^{\textsf{T}} \end{align}\]

Angular velocity vector

Since \(\bs{\Omega}\) is skew, we can define the axial vector \(\bs{\omega}\) corresponding to \(\bs{\Omega}\) as follows:

\[\begin{align} \omega_i=-\frac{1}{2}\epsilon_{ijk}\Omega_{jk} \end{align}\]