[AMR] Various fixes, error propagation, more explanations

electives/amr/autonomous-mobile-robots-cheatsheet.tex

@@ -49,6 +49,7 @@
 \input{parts/00_basics/00_probability.tex}
 \input{parts/00_basics/01_measurement-models.tex}
 \input{parts/00_basics/02_trigonometry.tex}
+\input{parts/00_basics/03_error-propagation.tex}
 
 \section{Locomotion \& Kinematics}
 \input{parts/01_kinematics/00_intro.tex}

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

@@ -2,5 +2,5 @@
 $\vec{z} = \vec{b}_C + s\mat{M} {_S}\vec{\omega} + \vec{b} + \vec{n} + \vec{o}$:
 $\vec{b}_C$ 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
+\hl{Finding}: Is in $W$-frame: may need $\mat{T}_{BW}$ or $\mat{R}_{BW}$.
+Also see Sec.~\ref{sec:sensors}

electives/amr/parts/00_basics/03_error-propagation.tex

@@ -0,0 +1,3 @@
+\subsection{Error Propagation}
+For functions $\vec{f}(\vec{x}) = \mat{A}\vec{x}$, the \bi{linear error propagation}
+is given by $\Sigma^f = A \Sigma^x \A^\top$, with $\Sigma^x$ the uncertanty of $\vec{x}$ (covariance mat.)

electives/amr/parts/02_Sensors-Actuators/00_intro.tex

@@ -1,5 +1,7 @@
-\bi{Meas. Model}: $\vec{z} = \vec{h}(\vec{x}) + \vec{v} + \vec{o}$, with $\vec{h}(\vec{x})$ deterministic mean, 
-$\vec{v}$ zero-mean noise, $\vec{o}$ unmodelled effects, $\vec{x}$ true state
+\label{sec:sensors}
+\bi{Meas. Model}: $\vec{z} = \vec{h}(\vec{x}) + \vec{v} + \vec{o}$, with $\vec{h}(\vec{x})$ determ. mean,
+$\vec{v}$ zero-mean noise, $\vec{o}$ unmodelled effects, $\vec{x}$ true state.
+$\vec{h}(\vec{x})$ describes how to compute $x$ from known values.
 
 \bi{Motor encoders} Typ. 64-2048 incrm. per rev; Estim. rot
 

electives/amr/parts/02_Sensors-Actuators/02_actuators.tex

@@ -13,5 +13,5 @@
 
 (Induced V, Faraday) $U_i = k_i \omega$
 
-(Mech. pow. eq. el. pow)\\
+(Mech. pow. $=$ electric power)\\
 $U_i I_a = k_i \omega I_a = T_\omega = k_T I_a \omega \Rightarrow k_i = k_T =: k$

electives/amr/parts/02_Sensors-Actuators/03_cameras.tex

@@ -7,12 +7,10 @@ $\begin{bmatrix}
     \begin{bmatrix}
         x & y
     \end{bmatrix}^\top$
-with $f$ the distance to the lens and $z$ the full distance
+with $f$ the distance to the lens and $z$ the distance from object
 
-\newpage
 $u = c_u + f \cdot x'$ and $v = c_v + f \cdot y'$ where $x' = t_x \div t_z$ and $y' = t_y \div t_z$
-where $u, v$ are the pixel $x, y$ coords, $\vec{c} = [c_u, c_v]^\top$ is optical centre of cam in pixel coords, $f$ scale factor,
-% and $\vec{{_C}\vec{t}_P} = [t_x, t_y, t_z]^\top$
+where $u, v$ are the pixel $x, y$ coords, $\vec{c} = [c_u, c_v]^\top$ is optical centre of cam in pixel coords, $f$ scale factor.
 
 The full proj:
 $\vec{u} =
@@ -34,7 +32,7 @@ $\vec{u} =
     }
     = \mat{K}\; {_C}\vec{t}_P$
 
-If p. in diff frame ${_W} \vec{t}_P$, then $\vec{u} = \mat{K}[\mat{R}_{CW}\; {_C}\vec{t}_{CW}] {_W}\vec{t}_P$
+If p. in diff frame, e.g. $W$-frame then $\vec{u} = \mat{K}[\mat{R}_{CW}\; {_C}\vec{t}_{CW}] {_W}\vec{t}_P$
 
 \subsubsection{Pinhole Camera Projection with distortion}
 \shortdefinition Model: $\vec{u} = \vec{k}(\vec{d}(\vec{p}({_C}\vec{t}_P)))$, with ($\vec{c}$ as above):

electives/amr/parts/03_multi-sensor-estimation/02_nonlinear-least-squares.tex

@@ -2,6 +2,7 @@
 Find $\vec{x}^* = \text{argmax}\; \P(\vec{x} | \vec{z}) = \argmin{}(-\log(\P(\vec{x}|\vec{z})))$
 
 \bi{Gauss-Newton} % TODO: Do we really need these? If so, use from NumCS (much simpler notation)
+% TODO: Error propagation laws
 
 \bi{Levenberg-Marquardt}