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[AMR] Fix errors pointed out to me

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janishutz
2026-08-09 08:30:17 +02:00
parent cc5f111813
commit a8b3bd7b20
24 changed files with 50 additions and 47 deletions
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electives/amr/parts/01_kinematics/00_intro.tex
+3 -3
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@@ -38,7 +38,7 @@ $\begin{bmatrix}
$\begin{smallmatrix}
\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
\theta = \arcsin(-R_{31})\\
\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
\varphi = \arcsin\left( R_{32} \div \sqrt{1 - R_{31}^2} \right)\\
\end{smallmatrix}$
$[\vec{n}]^\times = \begin{bmatrix}
0 & -n_3 & n_2 \\
@@ -51,7 +51,7 @@ For pitch axis $\theta = \pm 90\deg$, \bi{Gimbal Lock}, Jacobian \bi{singular}.
\shortdefinition[Angle-Axis]
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); To \bi{rotation matrix}:\\
$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
To \bi{quaternion}: $\vec{q} = [\vec{n}, \alpha]$
To \bi{quaternion}: $\vec{q} = [\vec{n} \sin(\alpha \div 2), \cos(\alpha \div 2)]$
\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
@@ -63,7 +63,7 @@ $\vec{q} = \begin{bmatrix}
\bi{Mult} {\scriptsize
$\vec{q} \otimes \vec{p} = \begin{bmatrix}
a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p})\vec{v}(\vec{q}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
\end{bmatrix}$
}