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I am using a robotics simulator named V-Rep. In V-Rep Euler angles $\alpha$, $\beta$ and $\gamma$ describe a rotation composed by three elemental rotations: $$Q=R_x(\alpha) R_y(\beta) R_z(\gamma)$$

where $R_x$, $R_y$ and $R_z$ represent elemental rotations about axes $x$, $y$ and $z$ respectively of the absolute reference frame. If the angular velocity of an object, whose orientation with respect to the reference frame is described by the Euler angles above, is $\omega=[\omega_x,\omega_y,\omega_z]$ with respect to the reference frame, then am I right to say that $\dot \alpha=\omega_x$, $\dot \beta=\omega_y$ and $\dot \gamma=\omega_z$ ?

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No, Euler angles are a poor choice for representing the orientation. Apart from that not all orientations can be represented it complicates the calculations.

What you instead could do is to use quaternion representation of orientation or use a rotation matrix. That would make the maths more easy, the relation between angular velocity and orientation is $\omega = {d\over dt}\operatorname{vec}(q)$, where $\operatorname{vec}(q)$ is the vector part of the quaternion.

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