# Angular Velocities of CoM in Global Coordinate

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.

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