[AMR] Some layout updates for smaller font sizes

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

@@ -7,11 +7,11 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb
 \shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors
 
 \shortdefinition[Rot. Mat.]
+$\mat{R}_{z}(\psi)$ (Yaw),
+$\mat{R}_y(\theta)$ (Pitch),
+$\mat{R}_x(\varphi)$ (Roll)\\
 {\scriptsize
-	$\mat{R}_{z}(\psi)$ (Yaw),
+$\begin{bmatrix}
-	$\mat{R}_y(\theta)$ (Pitch),
-	$\mat{R}_x(\varphi)$ (Roll)\\
-	$\begin{bmatrix}
         c(\psi) & -s(\psi) & 0 \\
         s(\psi) & c(\psi)  & 0 \\
         0       & 0        & 1
@@ -40,13 +40,13 @@ $\begin{smallmatrix}
         \theta = \arcsin(-R_{31})\\
         \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
     \end{smallmatrix}$
-{\scriptsize
+$[\vec{n}]^\times = \begin{bmatrix}
-		$[\vec{n}]^\times = \begin{bmatrix}
         0    & -a_3 & a_2  \\
         a_3  & 0    & -a_1 \\
         -a_2 & a_1  & 0
     \end{bmatrix}$
-	}
+
+For pitch axis $= \pm 90\deg$, Gimbal Lock, Jacobian singular.
 
 \shortdefinition[Angle-Axis]
 $\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\

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

@@ -1,12 +1,11 @@
 \subsection{Inverse Kinematics (IK)}
-\begin{wrapfigure}[10]{r}{0.4\columnwidth}
+\begin{wrapfigure}[10]{r}{0.35\columnwidth}
-    \includegraphics[width=0.4\columnwidth]{assets/inverse-kinematics.png}
+    \includegraphics[width=0.35\columnwidth]{assets/inverse-kinematics.png}
 \end{wrapfigure}
 \bi{Option}: Solve Forward Kinematics for angles.\\
-\bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,\\
+\bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,
 $\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
-
+(Positive for {\color{ForestGreen} Elbow Down}, Neg. for {\color{red} Elbow Up})
-(Positive for {\color{ForestGreen} Elbow Down}, Negative for {\color{red} Elbow Up})
 
 \bi{Extension to 6R}:
 1. Waist: spherical coords (2 sol.)\\

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

@@ -7,7 +7,7 @@ $\mat{H}$ is meas., $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process
 both zero-mean \bi{Gaussian White Noise Process}.
 
 To \bi{discretize}, integrate from $t_{k - 1}$ to $t_k$:\\
-$\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$
+$\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$;
 $\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$,
 \bi{linearised}:\\
 $\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;

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

@@ -2,8 +2,10 @@
 \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:
 \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{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}
+    {_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. inert. frame
+
+    {\scriptsize Legged robots don't need to maintain stability!}