[Electives] Restructure

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

@@ -0,0 +1,53 @@
+\subsection{Positioning}
+\shortdefinition[Position Vector]
+$_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{ForestGreen}\fbox{W}}\,_{\color{red}\fbox{B}}$,
+{\color{blue} Original Frame}, {\color{red} End point}, {\color{ForestGreen} Target Frame},
+\hl{$\sin = s$, $\cos = c$}
+
+\shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors
+
+\shortdefinition[Rot. Mat.] $\mat{R}_{z}(\psi) = \begin{bmatrix}
+        c(\psi) & -s(\psi) & 0 \\
+        s(\psi) & c(\psi)  & 0 \\
+        0       & 0        & 1
+    \end{bmatrix}$\\
+$\mat{R}_y(\theta) = \begin{bmatrix}
+        c(\theta)  & 0 & s(\theta) \\
+        0        & 1 & 0       \\
+        -s(\theta) & 0 & c(\theta) \\
+    \end{bmatrix};
+    \mat{R}_x(\varphi)
+    \begin{bmatrix}
+        1 & 0       & 0        \\
+        0 & c(\varphi) & -s(\varphi) \\
+        0 & s(\varphi) & c(\varphi)
+    \end{bmatrix}$
+
+\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
+
+\shortlemma $\mat{R}_{BW} = \mat{R}_{WB}^{-1} = \mat{R}_{WB}^\top$, $\det(\mat{R}_{WB}) = 1$ (orth.)
+
+\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.
+$\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$, \hl{bound.}.
+$\qquad [\vec{n}]^\times = \vec{n} \vec{x}^\top$ (matrix from vec + arg $\vec{x}$)
+
+\shortdefinition[Rot. Vec]
+$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal)\\
+$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$
+
+\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] $\mat{T}_{AB} = \begin{bmatrix}
+        \mat{R}_{AB}        & {_A}\vec{t}_B \\
+        \mat{0}_{1\times 3} & 1
+    \end{bmatrix}$\\
+$\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
+    \begin{bmatrix}
+        \mat{R}_{AB}^\top    & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
+        \mat{0}_{1 \times 3} & 1
+    \end{bmatrix}$
+$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$