This is the sixth 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.

$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$

The Adjoint as the Jacobian of Group Action

Most people’s first introduction to calculus typically focuses on computing derivatives. For completeness, we’re going to start here and work our way to computing Jacobians of the Lie group action. The formula for computing the derivative of a scalar function $f$, is given as ¹

$$ \frac{\partial f}{\partial x}=\lim_{\varepsilon\to0}\frac{1}{\varepsilon}\left(f\left(x+\varepsilon\right)-f\left(x\right)\right). $$

In conversational English, this means, “How does the output of the function $f$ change if I bump the input a tiny bit?” This is very naturally adapted for a vector function $g\in\mathbb{R}^{m}\to\mathbb{R}^{n}$ as

$$ \begin{align*} \frac{\partial g}{\partial\x} & =\lim_{\varepsilon\to0}\frac{1}{\varepsilon}\begin{bmatrix}\dd_{1}\left(\x,\varepsilon\right) & \dd_{2}\left(\x,\varepsilon\right)\cdots\dd_{m}\left(\x,\varepsilon\right)\end{bmatrix}\in\mathbb{R}^{n\times m}\\ \dd_{i}\left(\x,\varepsilon\right) & =g\left(\x+\e_{i}\varepsilon\right)-g\left(\x\right) \end{align*} $$

Again, in conversational English, we’re saying that $g$ is a function that takes in $m$-vectors and produces $n$-vectors. We want to know what $g$ looks like along each of the input axes, so we’re going to make a matrix where each column is the derivative along each of these directions. This is known as the Jacobian matrix, and can also be interpreted as, “If I change the input in some way, what does it do to the output?” For example, if I go further along the $\e_{1}$-axis, would I expect the output to go up or down? This would be captured in the first column of the Jacobian. This relationship lets us look at Jacobians as a sort of mapping from input space to output space, and lets us quickly compute how changes in input should affect the output of the function $g$.

The concept of differentiation works quite naturally with Lie groups as well. For example, consider the case of some function $h\in\SE\left(3\right)\to\mathbb{R}^{n}.$ What we want is a matrix that tells us how the output of $h$ behaves as we tweak the argument of $h$ along the *generators *of $\se\left(3\right).$ To compute this, we use the fact that every member of the group can be described by the exponential of some linear combination of generators, as in ²

$$ \begin{align*} T & =\exp\left(\x^{\wedge}\right), & \x^{\wedge} & =\log\left(T\right). \end{align*} $$

When we write it this way, we can see how we might perturb $\x$ (and therefore, $T$) in the $J_{i}$ direction:

$$ \begin{align*} T_{i}^{+} & =\exp\left(\x^{\wedge}+J_{i}\varepsilon\right)\\ & \approx T\cdot\exp\left(J_{i}\varepsilon\right). & \grey{\left(\text{BCH}\vert\varepsilon\to0\right)} \end{align*} $$

Next, we need to think about how we might compute the difference between $T_{i}^{+}$ and the original $T$. Again, looking to the Lie Algebra, we should expect the difference $\dd_{i}\in\se\left(3\right)$ to be

$$ \begin{align} \exp\left(\dd_{i}\right) & =\exp\left(\x^{\wedge}\right)\cdot\exp\left(J_{i}\varepsilon\right)\cdot\exp\left(-\x^{\wedge}\right)\nonumber \\ \dd_{i} & =\log\left(T\cdot\exp\left(J_{i}\varepsilon\right)\cdot T^{-1}\right)^{\vee}.\label{eq:adjoint_start} \end{align} $$

If we make a matrix out of the six $\dd_{i}$ for $\se\left(3\right)$, one for each generator $J_{i},$ and take the limit as $\varepsilon\to0$, we’re going to get a $6\times6$ matrix that describes how the generators of the output of $T$ change as we perturb the input. To be specific, lets attach coordinate frames to our transform $T_{a}^{b}$. The resulting matrix will tell us how things change along the generators in the $b$ frame if we perturb along the generators the $a$ frame. This has a special name–the Adjoint–and we can think about it as the Jacobian of the group action by a member of the group.¹

We can come at the Adjoint in another way: because the Adjoint is the Jacobian of the group action of a Lie group, we can use it to map vectors from the “output” of a group action to the “input”, and vice-versa. In mathematical terms³

$$ T\cdot\exp\left(\dd\right)=\exp\left(\Ad_{T}\cdot\dd\right)\cdot T. $$

See how we have “moved” $\dd$ from the “input” of $T$ (the right side) to the “output” (the left side)? If we attach coordinate frames to $T$ this becomes even more clear:

$$ \begin{align} T_{a}^{b}\cdot\exp\left(\dd^{a}\right) & =\exp\left(\Ad\left(T_{a}^{b}\right)\cdot\dd^{a}\right)\cdot T_{a}^{b}.\nonumber \\ & =\exp\left(\dd^{b}\right)\cdot T_{a}^{b}.\label{eq:adjoint_show} \end{align} $$

The Adjoint transforms $\dd$ from $a$ to $b.$

Remember all that talk about left-invariance and right-invariance? This property of the Adjoint of the Lie group lets us swap back and forth however we need to. This is really helpful when computing Jacobians of complex expressions involving Lie Groups.

Next: Computing the Adjoint