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---
title: Angular Motion
categories: repetition lecture
---
# 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
# Continuous Motion
The exponential form can be found via implicit derivation of
$R(t)R^T(t)=I$ and solution of the resulting ODE.
# Angular Motion
+ $\dot R(t)R^T(t) =\hat\omega(t)$ has to be skew-symmetric
# Formulæ
**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 $\omega$.