[AMR] More notes

electives/amr/parts/00_basics/00_probability.tex

@@ -1,20 +1,22 @@
 \subsection{Probability}
-\shortdefinition[Sum rule] $P(X) = \sum P(X, Y) = \sum P(X \cap Y)$
+\shortdefinition[Sum rule] $\P(X) = \sum \P(X, Y) = \sum \P(X \cap Y)$
 
-\shortdefinition[Prod] $P(X, Y) = P(X | Y) P(Y) = P(Y | X) P(X)$
+\shortdefinition[Prod] $\P(X \cap Y) = \P(X | Y) \P(Y) = \P(Y | X) P(X)$
 
-\shorttheorem[Bayes] $\displaystyle P(Y_i | X) = \frac{P(X | Y_i) P(Y_i)}{\sum_{j = 1}^n P(X | Y_j) P(Y_j)}$
+\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 $\int p(x) \dx = 1$
+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[Indep.] $x, y$ indep. iff $\P(\cX \cap \cY) = \P(\cX) \P(\cY)$
 
-\shortdefinition[Cond. Indep.] iff $p(x, y | z) = p(x|z) p(y|z)$
+\shortdefinition[Cond. Indep.] iff $\P(\cX \cap \cY | \cZ) = \P(\cX | \cY) \P(\cY | \cZ)$
 
-\shortdefinition $E[\vec{x}] = \int_{-\8}^{\8} \vec{x} p(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$
+\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 $\text{Cov}[x] = \E[\vec{x} \vec{x}^\top] - \E[\vec{x}]\E[\vec{x}]^\top = \mat{\Sigma}$
 
 \shortdefinition[Gauss. Dist.] $\vec{x} \sim \cN(\vec{\mu}, \mat{\Sigma})$ ($\vec{\mu}$ mean, $\mat{\Sigma}$ cov.),\\
-PDF: $p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k |\mat{\Sigma}|}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$
+PDF: $f(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k \det(\mat{\Sigma})}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$
+
+$\Sigma^{-1}$ for $\Sigma$ diagonal, inverse of diag els (e.g. $\sigma^{-1}$)

electives/amr/parts/00_basics/01_measurement-models.tex

@@ -1,3 +1,6 @@
 \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.
+
+\hl{Finding}: Is in $W$-frame, so may need $\mat{T}_{BW}$ or $\mat{R}_{BW}$.
+Other model: $\vec{z} = \vec{h}(\vec{x}) + \vec{v}$, $\vec{h}(\vec{x})$ is pos of rob. dep. model

electives/amr/parts/01_kinematics/00_intro.tex

@@ -48,9 +48,10 @@ $[\vec{n}]^\times = \begin{bmatrix}
 			\end{bmatrix}$
 	}
 
-\shortdefinition[Rot. Vec]
+\shortdefinition[Angle-Axis]
 $\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
-$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$
+$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
+To quat: $\vec{q} = [\vec{n}, \alpha]$
 
 
 \shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\

electives/amr/parts/01_kinematics/01_forward.tex

@@ -1,5 +1,5 @@
 \subsection{Forward Kinematics (FK)}
-$\mat{T}_{WB_n}(\theta) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
+$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
 For 2R system:
 ${_W}\vec{t}_{WE} =$
 {\scriptsize