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 $J_{q}(\theta)$?



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