Lie Groups

This repo houses an implementation-focused introduction to Lie Groups for roboticists. All material is written in Python and presented in Jupyter Notebook format.

For the best experience, start reading online with nbviewer or interacting with the code using binder by clicking the appropriate badge above. Alternatively, follow the Getting Started instructions below to run the notebook on your local machine.

Overview

The purpose of this repo is to present theory, tools, and examples of Lie groups and algebras for the practitioner working with the control, estimation, and optimization of rigid bodies.

To facilitate this, each chapter will present a section on using the group element to illustrate kinematics, solving differential equations, control, and estimation. We take the approach of starting with the unit circle (S^1) and working through more complicated spaces: SE(2), SO(3), SE(3), and SL(3).

Motivation

Many interesting problems in control and estimation exist which can be solved by using "hacks" like wrapping Euler angles, normalizing quaternions at every step, etc. Alternatively, Lie theory can be applied to implement algorithms that do their work on the manifold (i.e., the surface of all 3D rotation matrices, SO(3)).

In particular, vision-based control and estimation is an active research area that employs the use of Lie groups and Lie algebras to leverage computationally efficient representations of rigid body transformations. Many state of the art algorithms in visual-inertial odometry, SLAM, and robust control use these concepts.

Getting Started

To get started improving and learning with these notebooks, make sure you have Jupyter installed:

$ pip install jupyter --user

After cloning this repo and navigating into its directory on your machine, kick off the Jupyter notebook server with:

$ jupyter notebook

The server will start and a new browser tab will open.

For more information on Python/pip, see here.

You will also need ffmpeg installed to render matplotlib animations as inline videos. On Ubuntu, this is accomplished with

sudo apt install ffmpeg

For other operating systems, refer to the documentation here or here.

Handling Merge Conflicts: nbdime

Because Jupyter notebook are difficult to parse as raw text, the nbdime tool was created to help graphically manage merge conflicts. Install using pip with

$ pip install -U nbdime

and make sure to integrate with git using nbdime config-git --enable --global. This will allow git diff to use nbdime's command-line diff interface for any *.ipynb files. Alternatively, you can use nbdiff-web to compare Jupyter notebooks graphically.

See the nbdime docs to read more about git integration and using nbdime as the merge tool.