Transform velocities from one frame to an other within a rigid body I come from non-physics background but just came to face the following problem. I have a rigid body with two attached frames of reference A and A'.
I know:


*

*the rotation and translation between A and A'

*the instantaneous translational velocities V$_x$, V$_y$, V$_z$ along the axes of A

*the instantaneous angular velocities $\omega_x$, $\omega_y$, $\omega_z$ around the axes of A


Based on the assumption that A and A' are attached to the same rigid body I want to compute:


*

*the instantaneous translational velocities V$_x$', V$_y$', V$_z$' along the axes of A'

*the instantaneous angular velocities $\omega_x$', $\omega_y$', $\omega_z$' around the axes of A'




The context is that I have a robot and I know it's instantaneous velocities in the frame of reference of it's sensor, but would like to know the instantaneous velocities at some other point of the rigidly attached body.
 A: I think that what you want to achieve is described in the following lecture: Robotics, Geometry and Control - Rigid body motion and geometry by Ravi Banavar.
You know the homogeneous transformation matrix that transforms the coordinate of a point in the frame A to the coordinate of the same point in the frame A' (using the same notation as in the lecture):
$ \bar{q}_{A'} = \begin{pmatrix}
q_{A'}\\ 
1
\end{pmatrix} = \begin{pmatrix}
R_{A'A} & p_{A'A} \\ 
0 & 1
\end{pmatrix} \begin{pmatrix}
q_{A} \\ 
1
\end{pmatrix} = \bar{g}_{A'A} \bar{q}_{A} $
with $ R $ the rotation matrix and $ p $ the translation vector.
The translational and angular velocities in the frame A' should be:
$ \begin{pmatrix}
v_{A'} \\ 
\omega_{A'}
\end{pmatrix} = \begin{pmatrix}
R_{A'A} & \hat{p}_{A'A}R_{A'A} \\ 
0 & R_{A'A}
\end{pmatrix} \begin{pmatrix}
v_{A} \\ 
\omega_{A}
\end{pmatrix} $
With:
$ a = \begin{pmatrix}
a_1 & a_2 & a_3
\end{pmatrix}^T $
and the skew-symmetric matrix: 
$ \hat{a} = \begin{pmatrix}
0 & -a_3 & a_2 \\ 
a_3 & 0 & -a_1 \\ 
-a_2 & a_1 & 0
\end{pmatrix} $
