Rigid Body Motions

October 26, 2022

Reference

Rotation Matrix

The special orthogonal group: $SO(3)$

The special orghogonal group $SO(3)$ , also known as the group of rotation matrices, is the set of all $3 \times 3$ real matrices $R$ that satisfy (i) $RR^T = I$ and (ii) $\det{R} = 1$

$SO(n)$ is called groups because they satisfy the properties required of a mathematical group. More specifically, the $SO(n)$ groups are also called matrix Lie groups as the elements of the group form a differentiable manifold (?).

Definition of a group

A group consists of a set of elements and an operation on two elements such that, for all $A, B$ in the group:

closure: $AB$ is also in the group
associativity: $(AB)C = A(BC)$
identity element existence: There exists an element $I$ in the group (the identity matrix for $SO(n)$ ) such that $AI=IA=A$
inverse element existence: There exists an element $A^{-1}$ in the group such that $AA^{-1} = A^{-1}A = I$

Definition of Lie group

A Lie group is a (non-empty) subset $G$ of $\mathbb{R}^N$ that fulfills:

$G$ is a group
$G$ is a manifold in $\mathbb{R}^N$
Both the group product operation and its inverse are smooth functions

Reference: https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf (P.40)

What is Manifold??

An $N$ -dimensional manifold $M$ is a topological space where every point $p \in M$ is endowed with local Euclidean structure.

Intuitively, this means that in an infinitely small vicinity of a point $p$ , the space looks "flat", or a $\mathbb{R}^2$ Euclidean space.

Reference https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf (P.40)

Exponential Coordinate Representation of Rotation (3.2.3; P.77)

The exponential coordinates parameterize a rotation matrix in terms of:

a rotation axis (as a unit vector $\hat{\omega} \in \mathbb{R}^3$ )
an angle of rotation $\theta \in \mathbb{R}$ about the axis

A rotation with this is represented as $\hat{\omega}\theta \in \mathbb{R}^3$ .

There are two views to explain the connection with a rotation matrix $R$ :

If a frame was rotated by $\theta$ about $\hat{\omega}$ , its final orientation (relative to the original frame) would be expressed by $R$
If a frame followed angular velocity $\hat{\omega}$ for $\theta$ units of time, its final orientation (relative to the original frame) would be expressed by $R$

Let's think about rotating a vector $p(0)$ by $\theta$ about $\hat{\omega}$ to $p(\theta)$ .
$p(t)$ forms the path traced by the tip of the vector.

We can see it as $p(0)$ rotates at a constant rate (1 [rad/s]) from time $t=0$ to $t=\theta$ . Then, its velocity is given by:

\dot{p} = \hat{\omega} \times p

Then we use 3 $\times$ 3 skew-symmetric matrix to transform it into a familiar form

skew-symmetric matrix

Given a vector $x = [x_1, x_2, x_3]^T \in \mathbb{R}^3$ ,

[x] \vcentcolon= \begin{bmatrix}0 & {\color{green} -x_3} & {\color{blue} x_2}\\ {\color{green} x_3} & 0 & {\color{purple} -x_1}\\ {\color{blue} -x_2} & {\color{purple} x_1} & 0 \end{bmatrix}

It's called skew-symmetric because $[x] = -[x]^T$ .

The set of all 3 $\times$ 3 skew-symmetric matrices is called $so(3)$ , and is also called Lie algebra of the Lie group $SO(3)$ .

\dot{p} = [\hat{\omega}] p

This is a linear differential equation ( $\dot{x} = Ax$ ), whose solution is given by:

p(t) = e^{[\hat{\omega}]t}p(0)

Summary

\hat{\omega}\theta \in \mathbb{R}^3 \longleftrightarrow [\hat{\omega}]\theta \in so(3) \longleftrightarrow e^{[\hat{\omega}\theta]} = R \in SO(3)

Rodrigues' formula

e^{[\hat{\omega}\theta]} = I + \sin\theta[\hat{\omega}] + (1 - \cos\theta)[\hat{\omega}] \in SO(3)