Robotic Systems · Lecture 3

Frames, rigid transforms, and SO(3)

Simeon Nedelchev

Frame convention

p^W=R_{WB}p^B+p_{WB}

$R_{WB}$ maps body components to world components.

T_{WB}=\begin{bmatrix}R_{WB}&p_{WB}\\0&1\end{bmatrix}

T_{WC}=T_{WB}T_{BC}

R^\top R=I,\qquad\det R=1

Proper rotations preserve lengths and orientation.

\dot R=[\omega^W]_\times R=R[\omega^B]_\times

The multiplication side depends on the velocity frame.

Translation $(1,2,0)$ and a quarter-turn about z.

Local point $(1,0,0)$ becomes $(1,3,0)$ .

Local direction becomes $(0,1,0)$ .

Frame, point of application, and component order.