Angular Motion

Representations of the Rotation

  • rotation matrix \(R\)
  • rotation around a vector \(\omega\) with angle \(\theta\)
    • \(R\omega=1\omega\), i.e. \(\omega\) is an eigenvector
    • one is the only real eigenvalue
  • exponential form \(\exp\hat\omega\theta\)
    • Rodrigues’ formula

The exponential form can be found via implicit derivation of \(R(t)R^T(t)=I\) and solution of the resulting ODE.

  • \(\dot R(t)R^T(t) =\hat\omega(t)\) has to be skew-symmetric


Definition \[\exp(\hat\omega) = I + \sum_{i=1}^\infty \frac{(\hat\omega t)^i}{i!}.\]

Rodrigues \[\exp(\hat\omega) = I + \frac{\hat\omega}{||\omega||}\sin(||\omega||} + \frac{\hat\omega^2}{||\omega||^2}(1-\cos(||\omega||})\]

Angular Velocity

Derivation is easy from the exponential form. We have \[\mathbf{X}(t) = \exp(\hat\omega t)\mathbf{X}_0\] so \[\dot\mathbf{X}=\frac{\partial}{\parial t}\exp(\hat\omega t)\mathbf{X}_0 = \hat\omega\exp(\hat\omega t)\mathbf{X}_0=\hat\omega\mathbf{X}(t)\]

The angular velocity is the vector \(\omega\).