This is a bunch of tables that are useful for computing jacobians of the Lie Groups $SO(3)$, $\mathcal{S}^3$, $SE(3)$ or $\mathbb{D}\mathcal{S}^3$
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
Useful Tables Table 1: Group-Group Jacobians Expression Left Jacobian Right Jacobian $\frac{\partial}{\partial\x}\y\cdot\x$ $\Ad\left(\y\right)$ $I$ $\frac{\partial}{\partial\x}\x^{-1}\cdot\y$ $-\Ad\left(\x\right)^{-1}$ $-\Ad\left(\x^{-1}\cdot\y\right)^{-1}$ $\frac{\partial}{\partial\x}\y\cdot\x^{-1}$ $-\Ad\left(\y\cdot\x^{-1}\right)$ $-\Ad\left(\x\right)$ $\frac{\partial}{\partial\x}\x\cdot\y$ $I$ $\Ad\left(\y\right)^{-1}$ $$ \begin{align*} \x,\y & \in G \end{align*} $$ Table 2: Useful Jacobians for $\SO\left(3\right)$ Expression Left Jacobian Right Jacobian $\frac{\partial}{\partial R}R \cdot \v$ $-\skew{R\cdot\v}$ $-R\cdot\skew{\v}$ $\frac{\partial}{\partial\ R}R^{\top}\cdot\v$ $R^{\top}\skew{\v}$ $\skew{R^{\top}\cdot\v}$ $\frac{\partial}{\partial\w}\exp\left(\skew{\w}\right)$ $aI+b\skew{\w}+c\w\w^{\top}$ $aI-b\skew{\w}+c\w\w^{\top}$ $\frac{\partial}{\partial R}\log\left(R\right)$ $I-\frac{1}{2}\skew{\dd}+e\skew{\dd}^{2}$ $I+\frac{1}{2}\skew{\dd}+e\skew{\dd}^{2}$ $$ R\in\SO\left(3\right),\quad\v,\w\in\mathbb{R}^{3}\quad\dd=\log\left(R\right)\quad\theta=\norm{\dd} $$
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.
This is the second 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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\textcolor{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
This is the third 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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
This is the fourth 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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
This is the fifth 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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
This is the seventh 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.
$$ \def\v{\mathbf{v}} \def\u{\mathbf{u}} \def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}} \def\w{\boldsymbol{\omega}} \def\p{\mathbf{p}} \def\t{\mathbf{t}} \def\i{\mathbf{i}} \def\j{\mathbf{j}} \def\e{\mathbf{e}} \def\k{\mathbf{k}} \def\r{\mathbf{r}} \def\d{\mathbf{d}} \def\x{\mathbf{x}} \def\y{\mathbf{y}} \def\q{\mathbf{q}} \def\qq{\Gamma} \def\qr{\q_{r}} \def\qd{\q_{d}} \def\SO{\mathit{SO}} \def\SE{\mathit{SE}} \def\skew#1{\left\lfloor #1\right\rfloor _{\times}} \def\norm#1{\left\Vert #1\right\Vert } \def\grey#1{\color{gray}{#1}} \def\abs#1{\left|#1\right|} \def\S{\mathcal{S}} \def\dd{\boldsymbol{\delta}} \def\Ad{\textrm{Ad}} \def\SU{\mathit{SU}} \def\R{\mathbb{R}} \def\so{\mathfrak{so}} \def\se{\mathfrak{se}} \def\su{\mathfrak{su}} $$
I do a lot of scientific computing in C++, and often have a need for plotting the internals of my algorithms. I love the Eigen Matrix Library, but directly plotting the contents of an Eigen Matrix can be difficult. I’ve narrowed in on a workflow that I really like and want to share.
One example might be if I have a Matrix X which contains in each column the estimated state of the quadrotor at each time sample.