A Mathematical Review

A.4 Rotational velocity

Figure A.8: Frame B rotating with respect to frame A.

Given two 3D coordinate frames A and B, the rotational, or angular, velocity of B with respect to A is given by a 3D vector \boldsymbol{\omega}_{BA} (Figure A.8). \boldsymbol{\omega}_{BA} is related to the derivative of {\bf R}_{BA} by

\dot{\bf R}_{BA}=[{}^{A}\boldsymbol{\omega}_{BA}]{\bf R}_{BA}={\bf R}_{BA}[{}^%
{B}\boldsymbol{\omega}_{BA}] (A.21)

where {}^{A}\boldsymbol{\omega}_{BA} and {}^{B}\boldsymbol{\omega}_{BA} indicate \boldsymbol{\omega}_{BA} with respect to frames A and B and [\boldsymbol{\omega}] denotes the 3\times 3 cross product matrix

\end{matrix}\right). (A.22)

If we consider instead the velocity of A with respect to B, it is straightforward to show that

\boldsymbol{\omega}_{AB}=-\boldsymbol{\omega}_{BA}. (A.23)