[AMR] Improve kinematics summary

electives/amr/parts/01_kinematics/06_wheeled-robot.tex

@@ -9,7 +9,7 @@
 \begin{itemize}
     \item \textit{Driving straight} all $\vec{v}$ equal
     \item \textit{Turning} Wheel axis must intersect the \bi{Instant Centre of Rotation} (ICR),
-          speeds: $v_i \div R_i = \Omega$ ($R_i$ dist. wheel-ICR, $\Omega$, vehicle body rotation rate)
+          speeds: $v_i \div R_i = \Omega$ ($R_i$ = dist. wheel-ICR; $\Omega$: vehicle body rotation rate)
 \end{itemize}
 
 \bi{Maneuverability}
@@ -19,7 +19,8 @@
     \item Deg. of Maneuverability: $\delta_M = \delta_m + \delta_s$
 \end{itemize}
 
-\bi{Wheel Configurations}
+\newpage
+\shade{ForestGreen}{Wheel Configurations}
 
 \includegraphics[width=1\columnwidth]{assets/wheel-config.png}
 
@@ -32,27 +33,27 @@
     \end{tabular}
 \end{scriptsize}
 
-\bi{Differential Drive Kinematics}
+\shortdefinition[Differential Drive Kinematics]
 
 \bi{State vec} $\vec{x} = [x_1, x_2, \theta]^\top$,
-\bi{Inputs} $\vec{u} = [\omega_l, \omega_r]^\top$, $r_r$ radius of right wheel, $w$ width of robot
+\bi{Inputs} $\vec{u} = [\omega_l, \omega_r]^\top$, radius of right (left) wheel $r_r$ ($r_l$), $w$ width of robot
 
 \bi{Gen. eq. of Motion} $\dot{x}_1 = v\cos(\theta)$, $\dot{x}_2 = v\sin(\theta)$, $\dot{\theta} = \Omega$,
 with $v = 0.5\cdot(\omega_l r_l + \omega_r + r_r)$, $\Omega = \frac{\omega_r r_r - \omega_l r_l}{w}$
 
 % TODO: Consider adding wheel constraints (planar) here as well (from W05 slides)
 \textit{Straight}: $v = \omega_l r_l = \omega_r r_r$, $\Omega = 0$, $D = v\Delta t$.\\
-$\vec{b}_s = \begin{bmatrix}
+$\vec{b}_s = {\scriptsize \begin{bmatrix}
         D \cos(\theta) \\
         D \sin(\theta) \\
         0
-    \end{bmatrix}$
-$\vec{b}_t = \begin{bmatrix}
-    R(\sin(\Delta \theta + \theta) - \sin(\theta))\\
-    -R(\cos(\Delta \theta + \theta) - \cos(\theta))\\
-    \Delta \theta
-\end{bmatrix}$
+    \end{bmatrix}}
+    \quad \vec{b}_t = {\scriptsize \begin{bmatrix}
+        R(\sin(\Delta \theta + \theta) - \sin(\theta))  \\
+        -R(\cos(\Delta \theta + \theta) - \cos(\theta)) \\
+        \Delta \theta
+    \end{bmatrix}}$
 
 \textit{Turning}: $\Omega = (\omega_l r_l) / R_l\! =\! (\omega_r r_r) / R_r$, $R\! =\! v / \Omega$, $\Delta \theta\! =\! \Omega \Delta t$
 
-\textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} b_i$ with $i \in \{s, t\}$. ($\int \ldots \dx \Delta t$)
+\textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} b_i$ with $i \in \{s, t\}$, respectively