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[AMR] Some layout updates for smaller font sizes

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janishutz
2026-07-29 11:10:17 +02:00
parent 4a2c984ee6
commit e08ee018a9
6 changed files with 55 additions and 54 deletions
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electives/amr/parts/01_kinematics/00_intro.tex
+46 -46
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@@ -7,25 +7,25 @@ $_{\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),
$\mat{R}_y(\theta)$ (Pitch),
$\mat{R}_x(\varphi)$ (Roll)\\
$\begin{bmatrix}
c(\psi) & -s(\psi) & 0 \\
s(\psi) & c(\psi) & 0 \\
0 & 0 & 1
\end{bmatrix};
\begin{bmatrix}
c(\theta) & 0 & s(\theta) \\
0 & 1 & 0 \\
-s(\theta) & 0 & c(\theta) \\
\end{bmatrix};
\begin{bmatrix}
1 & 0 & 0 \\
0 & c(\varphi) & -s(\varphi) \\
0 & s(\varphi) & c(\varphi)
\end{bmatrix}$
$\begin{bmatrix}
c(\psi) & -s(\psi) & 0 \\
s(\psi) & c(\psi) & 0 \\
0 & 0 & 1
\end{bmatrix};
\begin{bmatrix}
c(\theta) & 0 & s(\theta) \\
0 & 1 & 0 \\
-s(\theta) & 0 & c(\theta) \\
\end{bmatrix};
\begin{bmatrix}
1 & 0 & 0 \\
0 & c(\varphi) & -s(\varphi) \\
0 & s(\varphi) & c(\varphi)
\end{bmatrix}$
}
\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
@@ -36,17 +36,17 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb
\shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
$\begin{smallmatrix}
\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
\theta = \arcsin(-R_{31})\\
\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
\end{smallmatrix}$
{\scriptsize
$[\vec{n}]^\times = \begin{bmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
\end{bmatrix}$
}
\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}$
$[\vec{n}]^\times = \begin{bmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
\end{bmatrix}$
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\\
@@ -58,31 +58,31 @@ To quat: $\vec{q} = [\vec{n}, \alpha]$
$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$).
$\vec{q} = \begin{bmatrix}
\vec{v}(\vec{q}), a(\vec{q})
\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
\vec{v}(\vec{q}), a(\vec{q})
\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
\bi{Mult} {\scriptsize
$\vec{q} \otimes \vec{p} = \begin{bmatrix}
a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
\end{bmatrix}$
$\vec{q} \otimes \vec{p} = \begin{bmatrix}
a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
\end{bmatrix}$
}
\bi{To Rot Mat} {\scriptsize
$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
}
\shortdefinition[Transf. M]
{\scriptsize
$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
$\mat{T}_{AB} = \begin{bmatrix}
\mat{R}_{AB} & {_A}\vec{t}_B \\
\mat{0}_{1\times 3} & 1
\end{bmatrix};
\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
\begin{bmatrix}
\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
\mat{0}_{1 \times 3} & 1
\end{bmatrix}$
$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
$\mat{T}_{AB} = \begin{bmatrix}
\mat{R}_{AB} & {_A}\vec{t}_B \\
\mat{0}_{1\times 3} & 1
\end{bmatrix};
\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
\begin{bmatrix}
\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
\mat{0}_{1 \times 3} & 1
\end{bmatrix}$
}