Robot Kinematics
2D Rotation
Rotation matrix from frame B to frame A:
$${}^AR_B=\begin{bmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{bmatrix}, ,, {}^AR_B\in SO(2)$$
$SO(2)$ is 2D Special Orthogonal Group, which has these properties:
- $det(R)=+1$
- $R\cdot R^T=I$
2D Rigid Motion
A point $\overset\rightharpoonup p$’s coordinate transformation from frame B to frame A:
$${}^A\overset\rightharpoonup p={}^AR_B\cdot{}^B\overset\rightharpoonup p+{}^A\overset\rightharpoonup t_B$$