[AMR] Some fixes

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

@@ -11,6 +11,12 @@
 \renewcommand{\subsectionnumbering}{section}
 \renewcommand{\numberingpreset}{off}
 \renewcommand{\definitionShortNamingEN}{Def}
+\renewcommand{\remarkShortNamingEN}{Rem}
+\renewcommand{\lemmaShortNamingEN}{Lem}
+\renewcommand{\theoremShortNamingEN}{Thm}
+
+\renewcommand{\descriptorNameDisplay}[1]{\textbf{#1}}
+
 \fboxsep 1pt
 \fboxrule 0.1pt
 
@@ -41,7 +47,8 @@
 \section{Introduction}
 \input{parts/00_basics/00_probability.tex}
 \input{parts/00_basics/01_measurement-models.tex}
-\input{parts/00_basics/02_trigonometry.tex}
+% TODO: Add this (especially law of cosines)
+% \input{parts/00_basics/02_trigonometry.tex}
 
 \section{Locomotion \& Kinematics}
 \input{parts/01_kinematics/00_intro.tex}

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

@@ -1,27 +1,32 @@
 \subsection{Positioning}
-\shortdefinition[Position Vector]
+\shortdefinition[Pos Vec.]
 $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{ForestGreen}\fbox{W}}\,_{\color{red}\fbox{B}}$,
-{\color{blue} Original Frame}, {\color{red} End point}, {\color{ForestGreen} Target Frame},
+{\color{blue} Relative to Frame}, {\color{red} P. in other Frame}, {\color{ForestGreen} Start P. of vec in first F.},
 \hl{$\sin = s$, $\cos = c$}
 
 \shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors
 
-\shortdefinition[Rot. Mat.] $\mat{R}_{z}(\psi) = \begin{bmatrix}
+\shortdefinition[Rot. Mat.]
-        c(\psi) & -s(\psi) & 0 \\
+{\scriptsize
-        s(\psi) & c(\psi)  & 0 \\
+    $\mat{R}_{z}(\psi)$ (Yaw),
-        0       & 0        & 1
+    $\mat{R}_y(\theta)$ (Pitch),
-    \end{bmatrix}$\\
+    $\mat{R}_x(\varphi)$ (Roll)\\
-$\mat{R}_y(\theta) = \begin{bmatrix}
+    $\begin{bmatrix}
-        c(\theta)  & 0 & s(\theta) \\
+            c(\psi) & -s(\psi) & 0 \\
-        0        & 1 & 0       \\
+            s(\psi) & c(\psi)  & 0 \\
-        -s(\theta) & 0 & c(\theta) \\
+            0       & 0        & 1
-    \end{bmatrix};
+        \end{bmatrix};
-    \mat{R}_x(\varphi)
+        \begin{bmatrix}
-    \begin{bmatrix}
+            c(\theta)  & 0 & s(\theta) \\
-        1 & 0       & 0        \\
+            0          & 1 & 0         \\
-        0 & c(\varphi) & -s(\varphi) \\
+            -s(\theta) & 0 & c(\theta) \\
-        0 & s(\varphi) & c(\varphi)
+        \end{bmatrix};
-    \end{bmatrix}$
+        \begin{bmatrix}
+            1 & 0          & 0           \\
+            0 & c(\varphi) & -s(\varphi) \\
+            0 & s(\varphi) & c(\varphi)
+        \end{bmatrix}$
+}
 
 \shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
 
@@ -29,25 +34,54 @@ $\mat{R}_y(\theta) = \begin{bmatrix}
 
 \shortremark Cols of $\mat{R}_{WB}$ are basis vec. of Frame $\underset{\rightarrow}{\cF}{_B}$ in $\underset{\rightarrow}{\cF}{_W}$
 
-\shortdefinition[Euler Angles] Yaw ($z$), Pitch ($y$), Roll ($x$), mult. rotation matrices, e.g.
+\shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
-$\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$, \hl{bound.}.
+$\begin{smallmatrix}
-$\qquad [\vec{n}]^\times = \vec{n} \vec{x}^\top$ (matrix from vec + arg $\vec{x}$)
+        \psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
+        \theta = \arcsin(-R_{31})\\
+        \varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
+    \end{smallmatrix}$
+{\scriptsize
+$[\vec{n}]^\times = \begin{bmatrix}
+        0    & -a_3 & a_2  \\
+        a_3  & 0    & -a_1 \\
+        -a_2 & a_1  & 0
+    \end{bmatrix}$
+}
 
 \shortdefinition[Rot. Vec]
-$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal)\\
+$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
 $\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$
 
-\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
-$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$)
-% TODO: Finish this
 
-\shortdefinition[Transf. M] $\mat{T}_{AB} = \begin{bmatrix}
+\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
-        \mat{R}_{AB}        & {_A}\vec{t}_B \\
+$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$).
-        \mat{0}_{1\times 3} & 1
+
-    \end{bmatrix}$\\
+$\vec{q} = \begin{bmatrix}
-$\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
+        \vec{v}(\vec{q}), a(\vec{q})
-    \begin{bmatrix}
+    \end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
-        \mat{R}_{AB}^\top    & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
+
-        \mat{0}_{1 \times 3} & 1
+\bi{Mult} {\scriptsize
-    \end{bmatrix}$
+    $\vec{q} \otimes \vec{p} = \begin{bmatrix}
-$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$
+            a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
+            a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
+        \end{bmatrix}$
+}
+
+\bi{To Rot Mat} {\scriptsize
+    $R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
+}
+
+
+\shortdefinition[Transf. M]
+{\scriptsize
+    $\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
+    $\mat{T}_{AB} = \begin{bmatrix}
+            \mat{R}_{AB}        & {_A}\vec{t}_B \\
+            \mat{0}_{1\times 3} & 1
+        \end{bmatrix};
+        \mat{T}_{BA} = \mat{T}_{AB}^{-1} =
+        \begin{bmatrix}
+            \mat{R}_{AB}^\top    & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
+            \mat{0}_{1 \times 3} & 1
+        \end{bmatrix}$
+}

electives/amr/parts/01_kinematics/01_forward.tex

@@ -1,8 +1,12 @@
 \subsection{Forward Kinematics (FK)}
-$T_{WB_n}(\theta) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
+$\mat{T}_{WB_n}(\theta) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
-For 2R system: ${_W}\vec{t}_{WE} = \begin{bmatrix}
+For 2R system:
-    L_1 \cos(\theta_1) + L_2 \cos(\theta_1 + \theta_2)\\
+${_W}\vec{t}_{WE} =$
-    L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2)
+{\scriptsize
-\end{bmatrix}$\\
+$ \begin{bmatrix}
+        L_1 \cos(\theta_1) + L_2 \cos(\theta_1 + \theta_2) \\
+        L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2)
+    \end{bmatrix}$\\
+}
 With workspace (pos) $W$ for $\theta_1, \theta_2 \in [-\pi, \pi]$
 % TODO: Example? (w02s42 possibly)