Coordinate Transformations¶

Overview¶

Coordinate transformations are fundamental to robotics, enabling representation of positions and orientations in different reference frames.

Orthogonal matrix $R$ where $R^T R = I$ and $\det(R) = 1$

Properties: - 9 parameters, 6 constraints → 3 DOF - Intuitive for composition - Memory intensive for transmission

Sequential rotations about principal axes.

ZYZ Convention: $$R = R_z(\alpha) R_y(\beta) R_z(\gamma)$$

Common issues: Gimbal lock at $\beta = \pm 90°$

\[R = e^{[\omega]_\times \theta}\]

Where $[\omega]_\times$ is the skew-symmetric matrix of $\omega$.

4-parameter representation: $q = [w, x, y, z]$ or $q = [cos(\theta/2), \sin(\theta/2)\omega]$

Advantages: - No gimbal lock - Efficient interpolation (Slerp) - Minimal storage

Combining rotation and translation:

\[T = \begin{bmatrix} R & d \\ 0 & 1 \end{bmatrix}\]

Where $R$ is 3x3 rotation matrix and $d$ is 3x1 translation vector.

\[T_{AC} = T_{AB} \cdot T_{BC}\]

\[T^{-1} = \begin{bmatrix} R^T & -R^T d \\ 0 & 1 \end{bmatrix}\]

\[v_A = R_{AB} \cdot v_B\]