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,

\[ \dot{\vec{I}}^T = 0, \ \ \dot{\vec{B}}^T = \vec{\omega}_{BI} \times \vec{B}^T \]

Derivative of Rotation Matrix

\[B^T R_{BI} = I^T\]

Take derivative of both side,

\[\vec{\omega}_{BI} \times \vec{B}^TR_{BI} + \vec{B}^T \dot{R}_{BI} = \vec{B}^T \vec{\omega}_{BI}^B \times \vec{B}^TR_{BI} + \vec{B}^T \dot{R}_{BI} = 0\]
\[\therefore \ \ \dot{R}_{BI}= R_{BI} [\vec{\omega}_{BI}^B]^\times\]

or, from

\[I^T R_{IB} = B^T\]

derive

\[\therefore \ \ \dot{R}_{IB}= - [\vec{\omega}_{BI}^B]^\times R_{IB}\]

Relationship between Angular Velocity and Euler Rate

  1. PASSIVE or ACTIVE does not affects relationship between Angular Velocity and Euler Rate.

  2. However, Intrinsic or Extrinsic affects relationship between Angular Velocity and Euler Rate.

  3. 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,

\[\therefore \ \ -\dot{R_i}R_i^T= [1_i]^\times{\dot{\theta}}_i\]

and using,

\[[Rv]^\times=Rv^\times R^T\]
\[ \dot{R}=\dot{R}_\gamma R_\beta R_\alpha + R_\gamma \dot{R}_\beta R_\alpha + R_\gamma R_\beta \dot{R}_\alpha = - [\omega]^\times (R_\gamma R_\beta R_\alpha) \]
\[ \therefore \ \ \omega = [R_\gamma R_\beta 1_1 \ \ R_\gamma 1_2 \ \ 1_3] \begin{bmatrix} \dot{\alpha} \\ \dot{\beta} \\ \dot{\gamma} \end{bmatrix} \]

Case of Extrinsic Euler Angle

\(R = R_{I \rightarrow \alpha \rightarrow \beta \rightarrow \gamma} = R_{\alpha}R_{\beta}R_{\gamma}\) (PASSIVE)

\[ \therefore \ \ \omega = [1_1 \ \ R_\alpha 1_2 \ \ R_\alpha R_\beta 1_1] \begin{bmatrix} \dot{\alpha} \\ \dot{\beta} \\ \dot{\gamma} \end{bmatrix} \]