angle rates wrt another frame

Question

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.

Answer 1

It sounds like part of the problem here is what frame the RPY derivative, $\dot{\mathbf{x}}_{RPY}$, is in. Considering the fact that the RPY values, lets call them $\psi, \theta, \phi$ respectively, are used to parameterize the rotation matrix between the world frame and the end-effector frame - it is unlikely then that $\dot{\mathbf{x}}_{RPY}$ is defined in the end-effector frame. Consider

$$ ^{base}R_{ee}(\psi, \theta, \phi) = \ R_{z}(\psi) R_{y}(\theta) R_{x}(\phi) $$

So a change in these parameters is perhaps best not viewed as being in one frame vs another, but instead as changes in elementary parameters that effect the final orientation. That being said, if you use the analytical Jacobian you should not have to perform any change of bases for $\dot{\mathbf{x}}_{RPY}$ as these variables are the change in bases.

Now, for my personal schpeal on why the geometric Jacobian is better here. RPY is traditionally used for aircrafts and the like (the naming convention being a clear indicator) and they generally perform well in this arena. This is because it is highly unlikely that a passenger jet reaches the RPY singular configurations ($\theta = \pm \frac{\pi}{2}$) - as this would mean the plane is pointed "upwardly" or "downwardly" or perhaps "about-to-crashedly". So, for objects who are generally constrained to certain orientation ranges this should not be a problem. However, revolute-joint robots are not at all orientationally challenged - and as such find themselves in a wide range of orientations. In fact, it is within reason to assume that every orientation (i.e. all of $\text{SO}(3)$) is reachable for specific locations in a 6 DoF robot's workspace (up to self-collision and joint limits of course).

That being said, using a representation that contains singularites should be avoided if it can. Also note, the analytical Jacobian is not technically valid at these representational singularities (if you have access to Spong's robotics textbook check out section 4.8). We know how to convert RPY derivatives to angular velocity (here), the geometric Jacobian is required to compute the analytical Jacobian, and the analytical Jacobian can be undefined - angular velocity may be difficult to conceptualize at first but the math is better defined than its parameterized counterparts. As a final note, once the angular velocity has been computed from the RPY derivatives, this will be in the base frame - so no re-framing necessary.