Maximal Coordinates

The $i$-th body in a mechanism with $N$ bodies has state:

\[z^{(i)} = (x^{(i)}, v^{(i)}, q^{(i)}, \omega^{(i)}) \in \mathbf{R}^3 \times \mathbf{R}^3 \times \mathbf{H} \times \mathbf{R}^3,\]

represented in maximal coordinates, where $\mathbf{H}$ is the space of unit quaternions.

• $x$: position in world frame
• $v$: linear velocity in the world frame
• $q$: orientation represented as a unit quaternion
• $\omega$: angular velocity in the body frame

The mechanism state:

\[z = (z^{(1)}, \dots, z^{(N)}).\]

is the concatentation of all body states.