For a robot manipulator with joint angles $\theta$, the well-known relationship is
$$\omega = J_\omega(\theta)\dot{\theta}.$$
What if the angular velocity is represented as the time derivative of quaternions $\dot{q}$? What is the relationship between $\omega$ and $\dot{q}$?
We can also write the Jacobians w.r.t to the quaternions, i.e.,
$$\dot{q} = J_{q}(\theta)\dot{\theta},$$
what is the relationship between $J_\omega(\theta)$ and $J_{q}(\theta)$?
