This is the first of a 8-part series of posts designed as a quick-start guide for students new to the field of robotics and estimation, specifically on the use of Lie groups to describe rotations and rigid body transformations. This is a web port of the full pdf document, which is hosted here.

Introduction

The topic of Lie groups is fundamental to much of modern physics, and therefore has a deep and rich history going back several decades. At a high level, Lie theory connects the symmetries of nature with differential equations. Rotations and rigid body transformations are two examples of such natural symmetries. The use of Lie group theory has recently become prevalent in state-of-the-art methods in robotics and state estimation, since it provides a natural way to connect the symmetries that often come up in robotics (rotations and rigid transformations) to differential equations.

Unfortunately, rigorous Lie theory also comes with a lot of vocabulary and subtlety that can discourage the uninitiated. Therefore, I’m are going to deliberately gloss over a lot of the nuance and subtlety that exists in the ideas of Lie Groups, and attempt to supply just enough information to give you the vocabulary and intuition necessary to understand and implement most of the state-of-the art robotics literature. Pretty much everything in this document comes from the following papers: Barfoot2019, Drummond2014, Ethan2019, Sola2019, and Schwichtenberg2015. These are all excellent resources that can take your understanding to the next level, and I would recommend any and all of them if you’re interested in being more rigorous in the theory.

One difference with this document when compared with others is the use of extra notation throughout. This notation is much more detailed than a lot of other literature on robotics or computer vision, but the explicit syntax clarifies what is going on, and makes it easier for someone like me (who likes to think of these ideas in terms of physical relationships, rather than abstract mathematical ideas) to visualize and understand what is going on. The extra notation has also allowed me to make distinctions between different fields of research that may have different conventions when it comes to transformations.

As many of my fellow graduate students and I used to say, “The two hardest problems in robotics are coordinate frames and naming things.” While this won’t help you much with the second problem, this will hopefully help dispel a lot of the confusion I encountered in the first.

Vectors

Let us first describe the notation used in the rest of the document. Succinctly, it is described as follows: $$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\textcolor{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$

$$ \begin{aligned} a & \quad\text{scalar quantity }a\in\R\\ \v & \quad\text{vector quantity }\v\\ \r_{a/b}^{c} & \quad\text{the vector }\r, \text{ representing some quantity (e.g. position) of point }a \text{ with respect to }\\ & \quad\text{point } b, \text{expressed in frame } c.\\ R_{a}^{b} & \quad\text{a transformation or rotation that executes a change of basis from frame } a \text{ to frame} b\\ \skew{\v} & \quad\text{The skew-symmetric matrix formed from }\v\end{aligned} $$

Vector Notation

If that’s all you need, then great! You can carry on to the next section. However, for people like me, let’s dig into this notation and (hopefully) let it sink in. Consider the illustration in Figure 1. We have some point $a$ and another point $b$. Let $\r_{a/b}$ be the vector which describes the position of $a$ with respect to $b$. Note that we could flip the vector around by negating the quantity, which means that for any arbitrary frame of reference

Although it may be sometimes hard to actually draw some quantities clearly (like the angular rate between two reference frames), in physics and engineering, we usually use vectors to represent some type of relationship between two reference frames. For example, the angular velocity of a rotating body with respect to the earth might be written as $\w_{b/E}$ or the position of some point $a$ with respect to another point $b$ $\r_{a/b}$. While it may be difficult to visualize, we could also consider the opposite case: the angular velocity of the earth with respect to the rotating body, $\w_{E/b}$ and position of $b$ with respect to $a$, $\r_{b/a}$. Conveniently, flipping the direction is just applying a negative.

$$ \w_{E/b}=-\w_{b/E}. \r_{a/b}=-\r_{b/a}. $$

In the abstract, a vector can exist anywhere in space and isn’t necessarily tied to a specific coordinate frame. However, we typically want to represent vectors somewhere in a computer’s memory or combine multiple vectors together with addition or subtraction. To do this, we have to represent the actual values of the vector according to some coordinate frame. This is also known as the vector’s basis.

There isn’t usually anything inherently special about the coordinate frame we choose. For example, in Figure 1, we could describe $\r$ in any coordinate frame we want. We could use frame $a$, $b$ or $c$, it doesn’t really matter, so long as we always do all operations between vectors represented in the same frame.

We will use the superscript notation to describe the frame of reference, as in

$$ \r_{a/b}^{c}, $$

which in words means “the position of point $a$ with respect to point $b$, expressed in frame $c$.” Any vector quantity we want to manipulate in robotics will have this concept of frame of reference, even if it is hard to visualize such as angular rate and acceleration.¹

Rotating Vectors

I’m going to jump ahead a little bit and introduce the concept of a rotation as a change of basis, (change in coordinate frame). Let us say that we have some $\r$ expressed in the $c$ frame, but we want to express it in the $b$ frame. How do we do that? Easy, we simply change the basis, as in

$$ \r_{a/b}^{b}=R_{c}^{b}\mathbf{r}_{a/b}^{c}. $$

I will get into the details of how this actually works a little later, but for now, just believe me that we can change this basis. The notation is such that you can cancel out the frames, as in

$$ \require{cancel} \r_{a/b}^{b}=R_{\cancel{c}}^{b}\mathbf{r}_{a/b}^{\cancel{c}}. $$

So what if I want to go all the way to frame $a$ but I only have rotations from $c\to b$ and $b\to a$? That’s easy, just compose the rotations, and everything cancels out.

$$ \r_{a/b}^{a}=R_\cancel{b}^a R_{\cancel{c}}^{\cancel{b}}\r_{a/b}^{\cancel{c}}. $$

Again, I’m deliberately skimming over most of the actual math. We’ll get into this a lot more later, but we first need to get the mechanics of working with this notation.

Composing Vectors

Let’s now take a minute to remember what defines a vector space. From Wikipedia:

A vector space is a collection of objects called vectors, which may be added together and multiplied (scaled) by numbers, called scalars.

All vector spaces abide by the rules shown in [Table 1](#tab:vector_rules}. If you’ve taken a course in linear algebra, these will not be new to you. However, when I first learned these rules, I had never worked with objects that were *not *in a vector space, so the rules seemed obvious and redundant. We like vector spaces because computers are well-suited for solving linear algebra problems. There are high-performance libraries such as BLAS (basic linear algebra subprograms) and methods for performing matrix decompositions (SVD, LU, QR, Cholesky) that allow us to solve complicated problems quickly and accurately.

It turns out that rotations and rigid-body transformations do not lie in a vector space. This means that we end up performing manipulations to get our problems into a vector space where we can use powerful linear algebra techniques, and the distinction between vector spaces and non-vector spaces will become very important.

Let us say we have three bodies, each with an associated coordinate frame $a$, $b$, and $c$, as shown in Figure 2. Let’s also say that we have only the orange vectors. (the vector from $b\to a$ and the vector from $c\to b$) but we want the green vector (the one that goes all the way from $c\to a$).

Well, this would be easy if they were all represented in the same coordinate frame

$$ \r_{a/c}=\r_{a/b}+\r_{b/c}, $$

but they aren’t. However, we just have to remember that vectors can only be added or subtracted if they are in the same frame, so we just rotate them into a common frame before doing the additions, like this:

$$ \r_{a/c}^{c}=R_{b}^{c}\left(\r_{b/c}^{b}+R_{a}^{b}\r_{a/b}^{a}\right). $$

From a theoretical perspective, the choice of coordinate frame doesn’t matter, however in practice there is often a choice of frame that makes things easier.

Skew-symmetric matrices

Before going much further, I also need to introduce skew-symmetric matrices, and the the skew-symmetric matrix operator ² $\skew{\v}$. It is defined as

$$ \skew{\v}=\left[\begin{array}{ccc} 0 & -\v_{z} & \v_{y}\\ \v_{z} & 0 & -\v_{x}\\ -\v_{y} & \v_{x} & 0 \end{array}\right],\quad\v=\begin{pmatrix}\v_{x}\\ \v_{y}\\ \v_{z} \end{pmatrix} $$ and is related to taking the cross-product between two vectors as

$$ \a\times\b=\skew{\a}\b. $$

The skew-symmetric matrix has some really interesting and helpful properties. The first is that the operator is anti-commutable, that is

$$ \begin{equation} \skew{\a}\b=-\skew{\b}\a.\label{eq:skew_trick_1} \end{equation} $$

The second is that rotation matrices can be moved in and out of the skew-symmetric operator with³

$$ \begin{equation} \skew{R\v}=R\skew{\v}R^{-1}.\label{eq:skew_trick_2} \end{equation} $$

Finally, you could probably see from inspection that the transpose of a skew-symmetric matrix is its negative

$$ \left\lfloor \v\right\rfloor _{\times}^{\top}=-\skew{\v}. $$

These are all super useful tricks that show up all the time. I’ll try to point out when I’m using one of these tricks, but I might miss some.

Next: The Matrix Exponential