[AMR] Fix typos, improve space usage some more

2026-09-10 13:05:24 +02:00 · 2026-08-01 11:31:44 +02:00
parent 55030a0d0a
commit c80452f383
8 changed files with 15 additions and 14 deletions
@@ -25,7 +25,7 @@
 \renewcommand{\vec}[1]{\bm{#1}}
 \newcommand{\mat}[1]{\bm{#1}}

-\setupCheatSheet{Autonomous Mobile Robots}
+\setupCheatSheet[0.5cm]{Autonomous Mobile Robots}


 \begin{document}
@@ -2,8 +2,7 @@
 \shortdefinition[Rule of cosines] $c^2 = a^2 + b^2 - 2ab \cos(\gamma)$

 % TODO: Add convenient results (such as cos2 + sin2 = 1)
-\shortdefinition[Orthogonal vec] $v^\top w = 0$
+\shortdefinition[Orthogonal vec] $v^\top w = 0$;
+\shortdefinition[Vec Line Eq] $\vec{p} + \lambda \vec{d}$

 \shortdefinition[Determinant] $ad - bc$ for mat $[a, b; c, d]$
-
-\shortdefinition[Vec Line Eq] $\vec{p} + \lambda \vec{d}$
@@ -1,6 +1,6 @@
 \subsection{Rigid Body Dynamics}
 \label{sec:rigid-body-dynamics}
-\shortdefinition[Newton II] For fin. body w/ mass $m$ and inertia mat. $I$, with force $\vec{F}$ and torque $\vec{T}$ on \bi{Centre of Mass} (CoM), expressed in body frame:
+\shortdefinition[Newton II] For fin. body w/ mass $m$ and inertia mat. $I$, with force $\vec{F}$ and torque $\vec{T}$ on \bi{Centre of Mass} (CoM), in body frame:
 \begin{align*}
    {_B}\vec{F} & = \sum {_B}\vec{F}_i = m({_B} \vec{\dot{v}}_{CoM}) + m_B \vec{\omega} \times {_B}\vec{v}_{CoM}           \\
    {_B}\vec{T} & = \sum {_B}\vec{T}_i = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
@@ -6,10 +6,12 @@

 \bi{Wheel constraints} $v_i = \omega_i r_i$ ($r_i$ constraints)
 \begin{itemize}
-    \item \textit{Driving straight} all $\vec{v}$ equal
+    \item \textit{Driving straight} all $\vec{v}$ equal (ICR: R.Cent.)
    \item \textit{Turning} Wheel axis must intersect the \bi{Instant Centre of Rotation} (ICR) of vehicle,
          speeds: $v_i \div R_i = \Omega$ ($R_i$ = dist. wheel-ICR; $\Omega$: vehicle rotation rate (around ICR))
 \end{itemize}
+To compute ICR, use $v_i \div R_i = \Omega$ and similarity.
+
 Below: $\alpha$, $l$ pos in frame, $\beta$ rot at that pos ($z$-ax).
 To compute $\vec{c}$ in $\vec{c} \cdot {_B}\vec{v}_{WB} = \omega$
 (For multiple wheels, construct mat. from this)
@@ -1,4 +1,4 @@
-\subsection{Kalman Filtear (KF)}
+\subsection{Kalman Filter (KF)}
 Bayes Filter for Gauss. dist of R.V. \& linear meas. model.
 Initial state $\vec{x}_0 \sim \cN(\hat{\vec{x}}, \mat{P}_0)$, $\mat{P}_0$ previous covariance;

@@ -10,8 +10,8 @@ $\vec{x}_k = \mat{F}\vec{x}_{k - 1} + \mat{G}\vec{u}_k + \mat{L}\vec{w}_k$ with
 \end{itemize}
 \bi{Update} Lin. meas.: $\tilde{\vec{z}}_k = \mat{H}\vec{x}_k + \vec{v}_k$ with $\vec{v_k} \sim \cN(\vec{0}, \mat{R}_k)$:
 \begin{itemize}
-    \item \bi{Meas. residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \mat{H} \hat{\vec{x}}_{k | k - 1}$
-    \item \bi{Resid. Cov}: $\mat{S}_k = \mat{H}\mat{P}_{k | k - 1} \mat{H}^\top \mat{R}_k$
+    \item \bi{Measurement residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \mat{H} \hat{\vec{x}}_{k | k - 1}$
+    \item \bi{Residual Covariance}: $\mat{S}_k = \mat{H}\mat{P}_{k | k - 1} \mat{H}^\top \mat{R}_k$
    \item \bi{Kalman gain}: $\mat{K}_k = \mat{P}_{k | k - 1} \mat{H}^\top \mat{S}_k^{-1}$
    \item \bi{Updated mean}: $\hat{\vec{x}}_{k | k} = \hat{\vec{x}}_{k | k - 1} + \mat{K}_k \vec{y}_k$
    \item \bi{Updated Cov.}: $\mat{P}_{k | k} = (\mat{I} - \mat{K}_k \mat{H}) \mat{P}_{k | k - 1}$
@@ -1,14 +1,14 @@
-\subsection{Extended Kalman Filater (EKF)}
+\subsection{Extended Kalman Filter (EKF)}
 Non-l. state trans. model $\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$ as above:
 \begin{itemize}
    \item \bi{Mean}: $\hat{\vec{x}}_{k | k - 1} = \vec{f}(\hat{\vec{x}}_{k - 1 | k - 1}, \vec{u}_k)$
-    \item \bi{Cov.}: $\mat{P}_{k | k - 1} = \mat{F}_k \mat{P}_{k - 1 | k - 1} \mat{F}_k^\top + \mat{L}_k \mat{Q}_k \mat{L}_k^\top$\\
+    \item \bi{Covariance}: $\mat{P}_{k | k - 1} = \mat{F}_k \mat{P}_{k - 1 | k - 1} \mat{F}_k^\top + \mat{L}_k \mat{Q}_k \mat{L}_k^\top$\\
        With $\mat{F}_k$ linearisation $\frac{\partial \vec{f}}{\partial \vec{x}}$ and $\mat{L}_k$ lin. $\frac{\partial \vec{f}}{\partial \vec{w}}$
 \end{itemize}
-\bi{Update} N-Lin. meas.: $\tilde{\vec{z}}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$:
+\bi{Update} Non-Linear meas.: $\tilde{\vec{z}}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$:
 \begin{itemize}
    \item \bi{Meas. residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \vec{h}(\hat{\vec{x}}_{k | k - 1})$
 \end{itemize}
-Difference to above: $\mat{H}$ becomes $\mat{H}_k$, and $\mat{H}^\top$ is $\mat{H}_k^\top$
+Difference to KF: $\mat{H}$ becomes $\mat{H}_k$, and $\mat{H}^\top$ is $\mat{H}_k^\top$

 % TODO: Consider adding examples
@@ -34,7 +34,7 @@ Extension to RRT* to make path better:
        \State $x_{\min} \gets$ \Call{neighbours}{Graph, $x_f$, $R$}
        \State $c_{\min} \gets$ \Call{cost}{$x_n$} + \Call{edgeCost}{$x_n, x_f$}
        \For{each $x_{\text{near}}$ in $X_\text{near}$}
-            \If{\Call{edgeCollisionFree}{$x_\text{near}, x_f$} and \Call{cost}{$x_\text{near}$} + \Call{edgeCost}{$x_\text{near}, x_f$} $< c_{\min}$}
+            \If{\Call{edgeCollisionFree}{$x_\text{near}, x_f$} and\\ \Call{cost}{$x_\text{near}$} + \Call{edgeCost}{$x_\text{near}, x_f$} $< c_{\min}$}
                \State \Call{insertEdge}{\texttt{Graph}, $x_f, x_{\text{near}}$}
                \State $x_\text{parent} \gets$ \Call{parent}{\texttt{Graph}, $x_{\text{near}}$}
                \State \Call{removeEdge}{\texttt{Graph}, $x_\text{parent}, x_{\text{near}}$}