[AMR] Improve kinematics summary

electives/amr/parts/01_kinematics/02_inverse.tex

@@ -4,7 +4,7 @@
 \end{wrapfigure}
 \bi{Option}: Solve Forward Kinematics for angles.\\
 \bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,\\
-$\theta_1 = \phi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
+$\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
 
 (Positive for {\color{ForestGreen} Elbow Down}, Negative for {\color{red} Elbow Up})
 

electives/amr/parts/01_kinematics/03_temporal-models.tex

@@ -1,14 +1,10 @@
 \subsection{Temporal Models}
-For \bi{Cont-time n.-lin. system of ODE} $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t))$, with measurements $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.\\
+Model \bi{robot dyn} as \bi{Cont-time n.-lin. system of ODE} $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t), \vec{w}(t))$, with meas. $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.\\
-Need linearised (around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium}):\\
+Linearize around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium}:\\
 $\delta \vec{\dot{x}}(t) = \vec{f}_C(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F}_C \delta \vec{x}(t) + \mat{G}_C \delta \vec{u}(t) + \mat{L}_C \vec{w}(t)$\\
 $\delta \vec{z}(t) = \mat{H} \delta \vec{x}(t) + \vec{v}(t)$.
-Herein, $\mat{H}$ is measurements, $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process noise, $\vec{v}$ measurement noise, both zero-mean \bi{Gaussian White Noise Process}.
+$\mat{H}$ is meas., $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process noise, $\vec{v}$ measurement noise,
-
+both zero-mean \bi{Gaussian White Noise Process}.
-For \bi{n-lin. cont-time system}:
-$\vec{\dot{x}}(t) = \vec{f}_C(\vec{x}(t), \vec{u}(t), \vec{w}(t))$\\
-$\vec{z}(t) = \vec{h}(\vec{x}(t)) = \vec{v})(t)$,
-linearization is the same
 
 To \bi{discretize}, integrate from $t_{k - 1}$ to $t_k$:\\
 $\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$
@@ -17,7 +13,7 @@ $\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$,
 $\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;
 $\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k$
 
-\bi{Trapezoidal num. int} 
+\shortdefinition[Trapezoidal num. int]
 $\Delta \vec{x}_1 = \Delta t \vec{f}_C (\vec{x}_{k - 1}, \vec{u}_{k - 1}, t_{k - 1})$\\
 $\Delta \vec{x}_2 = \Delta t \vec{f}_C (\vec{x}_{k - 1} + \Delta \vec{x}_1, \vec{u}_{k}, t_{k})$, then:\\
 $\vec{x}_k = \vec{x}_{k - 1} + 0.5 \cdot (\Delta \vec{x}_1 + \Delta \vec{x}_2)$

electives/amr/parts/01_kinematics/04_rigid-body-imu-kinematics.tex

@@ -3,9 +3,9 @@
 \begin{wrapfigure}[5]{r}{0.3\columnwidth}
     \includegraphics[width=0.3\columnwidth]{assets/rigid-body-6d.png}
 \end{wrapfigure}
-\bi{Velocity} ${_I}\vec{v}_{IB} = \diff{t} ({_I}\vec{t}_B)$
+\bi{Velocity} ${_I}\vec{v}_{IB} = {_I}\dot{\vec{t}}_B$
 
-\bi{Rot. Velocity} ${_I}\vec{\omega}_{IB} = \diff{t} (\alpha)\; {_I}\vec{t}$
+\bi{Rot. Velocity} ${_I}\vec{\omega}_{IB} = \dot{\alpha} \cdot {_I}\vec{n}$
 
 \bi{Velocity point $P$} ${_B}\vec{v}_{IP} = {_B}\vec{v}_{IB} + {_B}\vec{\omega}_{IB} \times {_B}\vec{t}_{P}$
 
@@ -15,35 +15,45 @@
           $\mat{\dot{R}}_{IB} = [{_I} \omega_{IB}]^\times \mat{R}_{IB}$
     \item For right pertubing
           $\mat{\dot{R}}_{IB} = \mat{R}_{IB} [{_I} \omega_{IB}]^\times$
-    \item Constant angular velocity ($\exp{[\Delta \alpha]^\times} = \delta \mat{R}(\Delta \alpha)$)\\
+    \item Constant angular velocity ($\exp{[\Delta \vec{\alpha}]^\times} = \delta \mat{R}(\Delta \vec{\alpha})$)\\
-          $\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \alpha]^\times} \mat{R}_{IB}(t)$
+          $\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \vec{\alpha}]^\times} \mat{R}_{IB}(t) \quad \vec{\alpha} = {_I}\vec{\omega}_{IB} \delta t$
 \end{itemize}
 
 \bi{Quaternions}
 \begin{itemize}
     \item For left pertubing
-          $\displaystyle \vec{\dot{q}}_{IB} = \frac{1}{2} \begin{bmatrix}
+          $\vec{\dot{q}}_{IB} =
-                  {_I}\vec{\omega}_{IB} \\
+              \frac{1}{2}
-                  0
+              {\scriptsize
-              \end{bmatrix}
+                  \begin{bmatrix}
+                      {_I}\vec{\omega}_{IB} \\
+                      0
+                  \end{bmatrix}
+              }
               \otimes \vec{q}_{IB}$
     \item For right pertubing
-          $\displaystyle \vec{\dot{q}}_{IB} = \frac{1}{2} \vec{q}_{IB} \otimes \begin{bmatrix}
+          $\vec{\dot{q}}_{IB} =
-                  {_B}\vec{\omega}_{IB} \\
+              \frac{1}{2} \vec{q}_{IB} \otimes
-                  0
+              {\scriptsize
-              \end{bmatrix}$
+                  \begin{bmatrix}
+                      {_B}\vec{\omega}_{IB} \\
+                      0
+                  \end{bmatrix}
+              }$
 \end{itemize}
 
 \bi{IMU} (Outputs {\color{blue} ${_S}\vec{\tilde{a}}$} (accel.), {\color{red} ${_S}\vec{\tilde{\omega}}$} (rot. accel.))\\
-${_W}\vec{\dot{t}}_S = {_W} \vec{v}$,
+${_W}\vec{\dot{t}}_S = {_W} \vec{v}$;
-$\displaystyle \vec{\dot{q}}_{WS} = \frac{1}{2} \vec{q}_{WS} \otimes
+$\quad \displaystyle \vec{\dot{q}}_{WS} = \frac{1}{2} \vec{q}_{WS} \otimes
-    \begin{bmatrix}
+    {\scriptsize
-        {\color{red}{_S}\vec{\tilde{\omega}}} {\color{gray} + \vec{w}_g - \vec{b}_g} \\
+        \begin{bmatrix}
-        0
+            {\color{red}{_S}\vec{\tilde{\omega}}} {\color{gray} + \vec{w}_g - \vec{b}_g} \\
-    \end{bmatrix}$
+            0
+        \end{bmatrix}
+    }$
 
-    ${_W}\vec{\dot{v}} = \mat{R}_{WS}\; ({\color{blue}{_S}\vec{\tilde{a}}} {\color{gray} + \vec{w}_a - \vec{b}_a}) + {_W}\vec{g}$
+${_W}\vec{\dot{v}} = \mat{R}_{WS}\; ({\color{blue}{_S}\vec{\tilde{a}}} {\color{gray} + \vec{w}_a - \vec{b}_a}) + {_W}\vec{g}$
-    where {\color{gray} gray parts} only IRL (in theor. models, leave out), with $\vec{\dot{b}}_g = \vec{w}_{b_g}$ and $\vec{\dot{b}}_a = \vec{w}_{b_a}$
+where {\color{gray} gray parts} only IRL (in theor. models, leave out), with $\vec{\dot{b}}_g = \vec{w}_{b_g}$ and $\vec{\dot{b}}_a = \vec{w}_{b_a}$
 
 \bi{IMU Sensor Model}: $\vec{\tilde{z}} = \vec{b}_C + s\mat{M}\vec{z} + \vec{b} + \vec{n} + \vec{o}$
 where bias $\vec{b}$ and scale $s$ often modelled time-varying $\dot{b}(t) = \sigma_C n(t)$.

electives/amr/parts/01_kinematics/05_rigid-body-dynamics.tex

@@ -6,4 +6,4 @@
     {_B}\vec{T} &= \sum {_B}\vec{T}_i = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
 \end{align*}
 
-${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. world frame
+${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame

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}$
+    \end{bmatrix}}
-$\vec{b}_t = \begin{bmatrix}
+    \quad \vec{b}_t = {\scriptsize \begin{bmatrix}
-    R(\sin(\Delta \theta + \theta) - \sin(\theta))\\
+        R(\sin(\Delta \theta + \theta) - \sin(\theta))  \\
-    -R(\cos(\Delta \theta + \theta) - \cos(\theta))\\
+        -R(\cos(\Delta \theta + \theta) - \cos(\theta)) \\
-    \Delta \theta
+        \Delta \theta
-\end{bmatrix}$
+    \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