Rotational Kinematics¶
Here, Three topics are described.
- Definition of angular velocity
- Derivative of Rotation Matrix
- Relationship between Angular Velocity and Euler Rate
Definition of angular velocity¶
Let frame \(\vec{B}\) rotate with respect to frame \(\vec{I}\).
The angular velocity of frame \(\vec{B}\) respect to frame \(\vec{I}\) is denoted by \(\vec{\omega}_{BI} = -\vec{\omega}_{IB}\)
Starts with a basic theorem, without demonstration,
Derivative of Rotation Matrix¶
Take derivative of both side,
or, from
derive
Relationship between Angular Velocity and Euler Rate¶
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PASSIVE or ACTIVE does not affects relationship between Angular Velocity and Euler Rate.
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However, Intrinsic or Extrinsic affects relationship between Angular Velocity and Euler Rate.
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Here we derive relationship between Angular Velocity and Euler Rate of generalized euler angle \(\alpha \rightarrow \beta \rightarrow \gamma\) with arbitrary axis of rotation.
Case of Intrinsic Euler Angle¶
\(R = R_{\gamma \leftarrow \beta \leftarrow \alpha \leftarrow I} = R_{\gamma}R_{\beta}R_{\alpha}\) (PASSIVE)
In case of principal axis rotation,
and using,
Case of Extrinsic Euler Angle¶
\(R = R_{I \rightarrow \alpha \rightarrow \beta \rightarrow \gamma} = R_{\alpha}R_{\beta}R_{\gamma}\) (PASSIVE)