Angular Velocities of CoM in Global Coordinate

Question

The block diagram of the robot is shown as above. Here, $\phi$ is the angle between $X_v$ and $X_0$ (Yaw angle of the robot), and $\psi$ is the angle between $Z_b$ and $Z_v$ (pitch angle of the robot's CoM).

Angular velocities of CoM in global coordinate $X_0, Y_0, Z_0$ are given as

$\Omega_g = [-\dot \phi sin\psi \ \ \dot \psi \ \ \dot \phi cos\psi]$

Whereas, according to my understanding it should be simply

$\Omega_g = [0 \ \ \ \dot \psi \ \ \dot \phi]$

I am pretty sure I am wrong, but I could not figure out how did the authors come up with $\Omega_g = [-\dot \phi sin\psi \ \ \dot \psi \ \ \dot \phi cos\psi]$.

Any help would be really appreciated.

Answer 1

After some search, I am only able to find the geometric solution which makes sense to me. As $\dot \phi$ is the instantaneous angular velocity about $Z_v$ axis, then it has two components on $Z_b$ and $X_b$ axes. Note that $\dot \phi sin \psi$ component is on the negative $X_b$ axis. Finally I get:

$\Omega_g= [-\dot \phi sin\psi \ \dot \psi \ \dot\phi cos\psi]$

as given in the paper.

P.S: If you have a mathematical solution please contribute here. Also, if my solution is wrong let me know.