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-\renewcommand{\vec}[1]{\textbf{#1}} -\newcommand{\mat}[1]{\textbf{#1}} -\noverticalspacing + +% Extra commands +\renewcommand{\vec}[1]{\bm{#1}} +\newcommand{\mat}[1]{\bm{#1}} \setupCheatSheet{Autonomous Mobile Robots} -\begin{document} -\startDocument +\begin{document} + +\vspace*{0mm} + +\begin{center} + \begin{Large} + \textbf{Autonomous Mobile Robots} + \end{Large} + + \textsc{Cheat Sheet by {\color{MidnightBlue}\href{https://janishutz.com}{Janis Hutz}}} + + \textsc{ETHZ, FS2026} +\end{center} + + +\section{Introduction} +\input{parts/00_basics/00_probability.tex} +\input{parts/00_basics/01_measurement-models.tex} \section{Locomotion \& Kinematics} \input{parts/01_kinematics/00_intro.tex} diff --git a/electives/others/amr/parts/00_basics/00_probability.tex b/electives/others/amr/parts/00_basics/00_probability.tex index 0ab609a..fe88645 100644 --- a/electives/others/amr/parts/00_basics/00_probability.tex +++ b/electives/others/amr/parts/00_basics/00_probability.tex @@ -1 +1,20 @@ \subsection{Probability} +\shortdefinition[Sum rule] $P(X) = \sum P(X, Y) = \sum P(X \cap Y)$ + +\shortdefinition[Prod] $P(X, Y) = P(X | Y) P(Y) = P(Y | X) P(X)$ + +\shorttheorem[Bayes] $\displaystyle P(Y_i | X) = \frac{P(X | Y_i) P(Y_i)}{\sum_{j = 1}^n P(X | Y_j) P(Y_j)}$ + +\shortdefinition[Cont. Var] Sums become integrals\\ +e.g. $\sum_{X} P(X) = 1$ becomes $\int p(x) \dx = 1$ + +\shortdefinition[Indep.] $x, y$ indep. iff $p(x, y) = p(x) p(y)$ + +\shortdefinition[Cond. Indep.] iff $p(x, y | z) = p(x|z) p(y|z)$ + +\shortdefinition $E[\vec{x}] = \int_{-\8}^{\8} \vec{x} p(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$ + +\shortdefinition $\text{Cov}[x] = E[\vec{x} \vec{x}^\top] - E[\vec{x}]E[\vec{x}]^\top = \mat{\Sigma}$ + +\shortdefinition[Gauss. Dist.] $\vec{x} \sim \cN(\vec{\mu}, \mat{\Sigma})$ ($\vec{\mu}$ mean, $\mat{\Sigma}$ cov.),\\ +PDF: $p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k |\mat{\Sigma}|}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$ diff --git a/electives/others/amr/parts/00_basics/01_measurement-models.tex b/electives/others/amr/parts/00_basics/01_measurement-models.tex new file mode 100644 index 0000000..3118abb --- /dev/null +++ b/electives/others/amr/parts/00_basics/01_measurement-models.tex @@ -0,0 +1,3 @@ +\subsection{Measurement models} +$\vec{z} = \vec{b}_C + s\mat{M} {_S}\vec{\omega} + \vec{b} + \vec{n} + \vec{o}$: +$\vec{b}_C$ const bias, $\vec{b}$ time bias, $\mat{M}$ missal., $\vec{n} \sim \cN(\vec{0}, \mat{R})$ noise, ${_S}\omega$ corr. meas., $\vec{o}$ other infl. diff --git a/electives/others/amr/parts/01_kinematics/00_intro.tex b/electives/others/amr/parts/01_kinematics/00_intro.tex index 19c431c..df988ac 100644 --- a/electives/others/amr/parts/01_kinematics/00_intro.tex +++ b/electives/others/amr/parts/01_kinematics/00_intro.tex @@ -30,4 +30,24 @@ $\mat{R}_y(\theta) = \begin{bmatrix} \shortremark Cols of $\mat{R}_{WB}$ are basis vec. of Frame $\underset{\rightarrow}{\cF}{_B}$ in $\underset{\rightarrow}{\cF}{_W}$ \shortdefinition[Euler Angles] Yaw ($z$), Pitch ($y$), Roll ($x$), mult. rotation matrices, e.g. -$\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$, \hl{bound.} +$\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$, \hl{bound.}. +$\qquad [\vec{n}]^\times = \vec{n} \vec{x}^\top$ (matrix from vec + arg $\vec{x}$) + +\shortdefinition[Rot. Vec] +$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal)\\ +$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$ + +\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\ +$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$) +% TODO: Finish this + +\shortdefinition[Transf. M] $\mat{T}_{AB} = \begin{bmatrix} + \mat{R}_{AB} & {_A}\vec{t}_B \\ + \mat{0}_{1\times 3} & 1 + \end{bmatrix}$\\ +$\mat{T}_{BA} = \mat{T}_{AB}^{-1} = + \begin{bmatrix} + \mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\ + \mat{0}_{1 \times 3} & 1 + \end{bmatrix}$ +$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$

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