Revision 16737ddd47409afcb85ddd15f9f233d3c8dcd976 (click the page title to view the current version)
Changes from 16737ddd47409afcb85ddd15f9f233d3c8dcd976 to current
---
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.
So far we have thought of motion as a discrete event.
An object had some position and orientation, and then suddenly,
by application of a transformation $(R,T)$ is has a different
position and orientation.
When objects move in the real world, motion is continuous.
Position and orientation are functions of time, and the object cannot
skip in space/time. It has to pass through every intermediate state.
How can we describe this continuous motion, particularly the
the rotational matrix as a function $R(t)$ of time?
Well, we have to go back to first-year *calculus*, because calculus
is the study of continuous behaviour.
## The derivative
1. Rotation is represented by an orthogonal matrix $R$
$$R(t)\cdot R^T(t)=I$$
2. Implicit derivation
$$\dot R(t)\cdot R^T(t)+R(t)\cdot\dot R^T(t)=0$$
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
$$ R(t_0+dt)\approx I + \hat\omega(t_0)dt$$
-->
## 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.
$$
\begin{align}
\exp : \mathrm{so}(3)\to\mathrm{SO}(3)
\exp : \mathrm{so}(3)&\to\mathrm{SO}(3)
\\
\hat\omega\mapsto e^{\hat\omega}
\hat\omega& \mapsto e^{\hat\omega}
\end{align}$$
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.
+ $\hat\omega$ is the axis of rotation
+ **Note** Only three degrees of freedom; since $\hat\omega$ is a 3D vector
+ a scalar factor can be applied to $t$ (change of unit) or
to $\hat\omega$
+ useful to normalise $\hat\omega$ to unit norm
+ Rotation is obviously periodic.
+ A rotation by $2\pi$ is back to start.
## 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
$$
\frac{\omega}{||\omega||} = \frac1{2\sin(||\omega||)}
\begin{bmatrix} r_{32}-r_{23}\\ r_{13}-r_{31}\\ r_{21}-r_{12} \end{bmatrix}
$$