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---
title: Representations of 3D Motion
categories: lectures 3D mathematics
geometry: margin=2cm
fontsize: 12pt
---
# Representation of Rotations
Consider what happens when an object rotates continuously over time,
i.e. the rotational matrix is a function $R(t)$ of time.
## The derivative
1. Rotation is represented by an orthogonal matrix $R$
$$R(t)\cdot R^T(t)=I$$
2. Implicit derivation
$$\dot R(t)\cdot R^T(t)+R(t)\cdot\dot R^T(t)=I$$
3. by transposing the product and moving one term across, we have
$$\dot R(t)\cdot R^T(t) = -(\dot R(t)\cdot R^T(t))^T$$
4. This is a skew-symmetric matrix, hence
$$\exists \vec{\omega}\in\mathbb{R}^3, \text{s.t.}
\dot R(t)\cdot R^T(t) = \hat\omega(t)$$
5. Multiply by $R(t)$ to get
$$\dot R(t) = \hat\omega(t)\cdot R(t)$$
6. If $R(t_0)=I$ as an initial condition, then $\dot R(t)=\hat\omega(t)$
Note $so(3)$ is the space of all skew-symmetric matrices.
7. First Order approximation
$$ R(t_0+dt)\approx I + \hat\omega(t_0(dt$$
$$ R(t_0+dt)\approx I + \hat\omega(t_0)dt$$
## The differential equation
Let $x(t)$ be a point rotated over time.
Assume that $\omega$ is constant.
1. **ODE:**
$$\dot x(t) = \hat\omega x(t), \quad x(t)\in\mathbb{R}^3$$
2. Solution:
$$x(t) = e^{\hat\omega t} x(0)$$
3. where
$$e^{\hat\omega t} = I + \sum_{i=1}^\infty \frac{(\hat\omega )^i}{i!}$$
4. The rotational matrix $$R(t)=e^{\hat\omega t}$$
signifies a rotation around the axis $\omega$ by $t$ radians.
$$\exp : \mathrm{so}(3)\to\mathrm{SO}(3); \hat\omega\mapsto e^{\hat\omega}$$
This is a map from a Lie algebra to a Lie group.
For any $R$, such an $\hat\omega$ can be found,
not necessarily unique.
Rotation is obviously periodic. A rotation by $2\pi$ is back to start.
**Note** Only three degrees of freedom.
## Logarithm
*Theorem 2.8* page 27 in the textbook
$$
R =
\begin{bmatrix}
r_{11} & r_{12} & r_{13} \\
r_{21} & r_{22} & r_{23} \\
r_{31} & r_{32} & r_{33}
\end{bmatrix}
= \exp(\hat\omega)
$$
where
$$
\DeclareMathOperator{\tr}{trace}
||\omega|| = \cos^{-1}\big(\frac{\tr(R)-1}2\big)
$$
and
$$
\frac{\omega}{||\omega||} = \frac1{2\sin(||\omega||)}
\begin{bmatrix} r_{32}-r_{23}\\ r_{13}-r_{31}\\ r_{21}-r_{12} \end{bmatrix}
$$