Angular Velocity
Definition
The angular velocity vector, \(\boldsymbol{\omega}^{\gamma}_{\beta \alpha}\), is the rate of rotation of the frame \(F_\alpha\) axes with respect to the frame \(F_\beta\) axes, resolved about the frame \(F_\gamma\) axes. The rotation is within the plane perpendicular to the angular rate vector, and the angular rate vector direction follows the right-hand rule.
| Properties of Angular Velocity | |
|---|---|
| Reverse Angular Rate | \(\boldsymbol{\omega}^{\gamma}_{\beta \alpha} = -\boldsymbol{\omega}^{\gamma}_{\alpha \beta}\) |
| Addition | \(\boldsymbol{\omega}^{\gamma}_{\beta \alpha} = \boldsymbol{\omega}^{\gamma}_{\beta \delta} + \boldsymbol{\omega}^{\gamma}_{\delta \alpha}\) |
| Resolving Axes Transform | \(\boldsymbol{\omega}^{\gamma}_{\beta \alpha} = \mathbf{R}^{\gamma}_{\delta} \boldsymbol{\omega}^{\delta}_{\beta \alpha}\) |
| Skew Symmetric Form | \(\boldsymbol{\Omega}^{\gamma}_{\beta \alpha} = \left[ \boldsymbol{\omega}^{\gamma}_{\beta \alpha} \right]_\times = \left[ \begin{array}{ccc} 0 & -\omega^{\gamma}_{\beta \alpha, 3} & \omega^{\gamma}_{\beta \alpha, 2} \\ \omega^{\gamma}_{\beta \alpha, 3} & 0 & -\omega^{\gamma}_{\beta \alpha, 1} \\ -\omega^{\gamma}_{\beta \alpha, 2} & \omega^{\gamma}_{\beta \alpha, 1} & 0 \\ \end{array} \right]\) |
| Skew Form Axes Transformation | \(\boldsymbol{\Omega}^{\gamma}_{\beta \alpha} = \mathbf{R}^{\gamma}_{\delta} \boldsymbol{\Omega}^{\delta}_{\beta \alpha} \mathbf{R}^{\delta}_{\gamma}\) |
References
- Groves, P., Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Second Edition, pp. 44-45