# Rigid Body Motions

# Reference

- Modern Robotics
- Modern Robotics (YouTube)
- https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf

# Rotation Matrix

SO(3)

The special orthogonal group: The

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

*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

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

## Definition of Lie group

A Lie group is a (non-empty) subset

is a groupG is a manifold inG \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 *local* Euclidean structure.

Intuitively, this means that in an infinitely small vicinity of a point

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
about the axis\theta \in \mathbb{R}

A rotation with this is represented as

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

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

Let's think about rotating a vector

We can see it as

Then we use 3 **skew-symmetric** matrix to transform it into a familiar form

## skew-symmetric matrix

Given a vector

It's called skew-symmetric because

The set of all 3 *Lie algebra* of the Lie group

This is a linear differential equation (

## Summary

**Rodrigues' formula**