[AMR] Catch up with summary

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: $p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k |\mat{\Sigma}|}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$

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.

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] $\mat{T}_{AB} = \begin{bmatrix}
+        \mat{R}_{AB}        & {_A}\vec{t}_B \\
+        \mat{0}_{1\times 3} & 1
+    \end{bmatrix}$\\
+$\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
+    \begin{bmatrix}
+        \mat{R}_{AB}^\top    & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
+        \mat{0}_{1 \times 3} & 1
+    \end{bmatrix}$
+$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$