My manipulator has a gyroscope attached to its end-effector (ee). The sensor provides angular velocity in the form of angular rates. Those rates are wrt fixed axis, so they effectively represent roll-pitch-yaw extrinsic sequence of angular rates. Let's call this angular velocity $\dot{\gamma}$. Now, $\dot{\gamma}$ is expressed with respect to the end-effector frame. For my control scheme I have to compute the joint velocity command, $\dot{q}$, as follow: $$\dot{q} = J_{A}^{-1}\dot{x}$$ where
- $J_{A}^{-1}$ is the inverse of the analytical Jacobian, which relates cartesian velocities with respect to the base link to joint velocities
- $\dot{x}$ is the cartesian velocity, whose angular component is not the angular velocity vector commonly denoted as $\omega$, but it's the time derivative (i.e. angle rates) of a minimal representation of orientation of the end-effector with respect to the base link. In this application, the angular component should be the above mentioned $\dot{\gamma}$
Here's the issue: $\dot{\gamma}$ is the angular rates expressed in the end-effector frame. Let's denote it as $^{ee}\dot{\gamma}$. For my computation, I need it to represent the angular rates wrt the base link of the manipulator (i.e. $^{base}\dot{\gamma}$).
How can I achieve this change of reference frame for the angular rates in order to use it with the analytical jacobian?
I'm also given $^{base}R_{ee}$, which is the matrix expressing the orientation of the end-effector frame wrt the base link frame. If $^{ee}\dot{\gamma}$ was an element of a vector space, it would be simple to change the frame of reference: $$^{base}\dot{\gamma} \text{ = } ^{base}R_{ee} \text{ } ^{ee}\dot{\gamma}$$ But since that's not the case, I don't think that multiplication is even feasible from a mathematical point of view.