From 4070eaadc5735c69e46405cdfb7c9af3b048a2d3 Mon Sep 17 00:00:00 2001 From: Dimitri Lozeve Date: Fri, 13 Nov 2020 18:41:23 +0100 Subject: [PATCH] Add references and improve text --- bib/bibliography.bib | 58 ++++++++++++++++++++++++++++++++++++++++++++ posts/lie-theory.org | 56 +++++++++++++++++++++++++++++++++++++++--- 2 files changed, 111 insertions(+), 3 deletions(-) diff --git a/bib/bibliography.bib b/bib/bibliography.bib index a0d1878..ac3038a 100644 --- a/bib/bibliography.bib +++ b/bib/bibliography.bib @@ -386,3 +386,61 @@ file = {/home/dimitri/Nextcloud/Zotero/storage/WBUCWRPK/Fong and Spivak - 2018 - Seven Sketches in Compositionality An Invitation .pdf;/home/dimitri/Nextcloud/Zotero/storage/MT7MPULY/1803.html} } +@article{sola2018_micro_lie_theor_state_estim_robot, + author = {Sol{\`a}, Joan and Deray, Jeremie and Atchuthan, + Dinesh}, + title = {A Micro Lie Theory for State Estimation in Robotics}, + journal = {CoRR}, + year = {2018}, + url = {http://arxiv.org/abs/1812.01537v7}, + abstract = {A Lie group is an old mathematical abstract object + dating back to the XIX century, when mathematician + Sophus Lie laid the foundations of the theory of + continuous transformation groups. As it often + happens, its usage has spread over diverse areas of + science and technology many years later. In + robotics, we are recently experiencing an important + trend in its usage, at least in the fields of + estimation, and particularly in motion estimation + for navigation. Yet for a vast majority of + roboticians, Lie groups are highly abstract + constructions and therefore difficult to understand + and to use. This may be due to the fact that most of + the literature on Lie theory is written by and for + mathematicians and physicists, who might be more + used than us to the deep abstractions this theory + deals with. In estimation for robotics it is often + not necessary to exploit the full capacity of the + theory, and therefore an effort of selection of + materials is required. In this paper, we will walk + through the most basic principles of the Lie theory, + with the aim of conveying clear and useful ideas, + and leave a significant corpus of the Lie theory + behind. Even with this mutilation, the material + included here has proven to be extremely useful in + modern estimation algorithms for robotics, + especially in the fields of SLAM, visual odometry, + and the like. Alongside this micro Lie theory, we + provide a chapter with a few application examples, + and a vast reference of formulas for the major Lie + groups used in robotics, including most jacobian + matrices and the way to easily manipulate them. We + also present a new C++ template-only library + implementing all the functionality described here.}, + archivePrefix ={arXiv}, + eprint = {1812.01537}, + primaryClass = {cs.RO}, +} + +@book{stillwell2008_naive_lie_theor, + author = {John Stillwell}, + title = {Naive Lie Theory}, + year = 2008, + publisher = {Springer New York}, + url = {https://doi.org/10.1007/978-0-387-78214-0}, + DATE_ADDED = {Sat Oct 31 17:40:56 2020}, + doi = {10.1007/978-0-387-78214-0}, + pages = {nil}, + series = {Undergraduate Texts in Mathematics}, +} + diff --git a/posts/lie-theory.org b/posts/lie-theory.org index c7080a5..cd3b7d0 100644 --- a/posts/lie-theory.org +++ b/posts/lie-theory.org @@ -112,12 +112,27 @@ higher dimension) on which we can compute derivatives at every point. This means that there is a tangent hyperplane at each point, which is a nice vector space where our derivatives will live. -And /that's all!/ What we have defined so far is a /Lie +You can think of the manifold as a tablecloth that has a weird shape, +all kinds of curvatures, but no edges or spikes. The idea here is that +we can define an /atlas/, i.e. a local approximation of the manifold +as a plane. The name is telling: they're called atlases because they +play the exact same role as maps. The Earth is not flat, it is a +sphere with all kinds of deformations (mountains, canyons, oceans), +but we can have maps that represent a small area with a very good +precision. Similarly, atlases are the vector spaces that provide the +best linear approximation of a small region around a point on the +manifold. + +So we know what a group and a differential manifold are. As it turns +out, that's all we need to know! What we have defined so far is a /Lie group/[fn:lie], i.e. a group that is also a differentiable -manifold. To take the example of rotation matrices: +manifold. The tangent vector space at the identity element is called +the /Lie algebra/. + +To take the example of rotation matrices: - We can combine them (i.e. by matrix multiplication): they form a group. -- if we have a function $R : \mathbb{R} \rightarrow +- If we have a function $R : \mathbb{R} \rightarrow \mathrm{GL}_3(\mathbb{R})$ defining a trajectory (e.g. the successive attitudes of a object in space), we can find derivatives of this trajectory! They would represent instantaneous orientation @@ -130,6 +145,41 @@ general theory for differential equations. * Interesting properties of Lie groups +For a complete overview of Lie theory, there are a lot of interesting +material that you can find online.[fn:princeton_companion] I +especially recommend the tutorial by +cite:sola2018_micro_lie_theor_state_estim_robot: just enough maths to +understand what is going on, but without losing track of the +applications. There is also a [[https://www.youtube.com/watch?v=QR1p0Rabuww][video tutorial]] for the [[https://www.iros2020.org/][IROS2020]] +conference[fn::More specifically for the workshop on [[https://sites.google.com/view/iros2020-geometric-methods/][Bringing +geometric methods to robot learning, optimization and control]].]. For a +more complete treatment, cite:stillwell2008_naive_lie_theor is +great.[fn::{-} John Stillwell is one of the best textbook writers. All +his books are extremely clear and a pleasure to read. You generally +read a book because you're interested in learning the topic; you begin +learning a topic just because Stillwell wrote a book on it.] + +[fn:princeton_companion] {-} There is also a chapter on Lie theory in +the amazing /Princeton Companion to Mathematics/ +[[citep:gowersPrincetonCompanionMathematics2010][::, §II.48]]. + + +The Lie group is a set of elements. At every element, there is a +tangent vector space. The tangent vector space at the identity is +called the Lie algebra, and as we will see, it plays a special role. + +TODO + * Applications +Lie theory is useful because it gives strong theoretical guarantees +whenever we need to linearize something. If you have a system evolving +on a complex geometric structure (for example, the space of rotations, +which is definitely not linear), but you need to use a linear +operation (if you need uncertainties, or you have differential +equations), you have to approximate somehow. + +Using the Lie structure of the underlying space, you immediately get a +principled way of defining derivatives, random variables, and so on. + * References