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janishutz
2026-06-19 10:36:07 +02:00
parent 816b4b5942
commit cf41a61e6b
5 changed files with 63 additions and 57 deletions
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electives/amr/parts/01_kinematics/00_intro.tex
+48 -47
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@@ -8,24 +8,24 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb
\shortdefinition[Rot. Mat.]
{\scriptsize
$\mat{R}_{z}(\psi)$ (Yaw),
$\mat{R}_y(\theta)$ (Pitch),
$\mat{R}_x(\varphi)$ (Roll)\\
$\begin{bmatrix}
c(\psi) & -s(\psi) & 0 \\
s(\psi) & c(\psi) & 0 \\
0 & 0 & 1
\end{bmatrix};
\begin{bmatrix}
c(\theta) & 0 & s(\theta) \\
0 & 1 & 0 \\
-s(\theta) & 0 & c(\theta) \\
\end{bmatrix};
\begin{bmatrix}
1 & 0 & 0 \\
0 & c(\varphi) & -s(\varphi) \\
0 & s(\varphi) & c(\varphi)
\end{bmatrix}$
$\mat{R}_{z}(\psi)$ (Yaw),
$\mat{R}_y(\theta)$ (Pitch),
$\mat{R}_x(\varphi)$ (Roll)\\
$\begin{bmatrix}
c(\psi) & -s(\psi) & 0 \\
s(\psi) & c(\psi) & 0 \\
0 & 0 & 1
\end{bmatrix};
\begin{bmatrix}
c(\theta) & 0 & s(\theta) \\
0 & 1 & 0 \\
-s(\theta) & 0 & c(\theta) \\
\end{bmatrix};
\begin{bmatrix}
1 & 0 & 0 \\
0 & c(\varphi) & -s(\varphi) \\
0 & s(\varphi) & c(\varphi)
\end{bmatrix}$
}
\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
@@ -36,52 +36,53 @@ $_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fb
\shortdefinition[Euler Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
$\begin{smallmatrix}
\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
\theta = \arcsin(-R_{31})\\
\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
\end{smallmatrix}$
\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
$[\vec{n}]^\times = \begin{bmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
\end{bmatrix}$
}
$[\vec{n}]^\times = \begin{bmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
\end{bmatrix}$
}
\shortdefinition[Rot. Vec]
\shortdefinition[Angle-Axis]
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$
$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
To quat: $\vec{q} = [\vec{n}, \alpha]$
\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 $ki$).
$\vec{q} = \begin{bmatrix}
\vec{v}(\vec{q}), a(\vec{q})
\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, 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\\
$\mat{T}_{AB} = \begin{bmatrix}
\mat{R}_{AB} & {_A}\vec{t}_B \\
\mat{0}_{1\times 3} & 1
\end{bmatrix};
\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
\begin{bmatrix}
\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
\mat{0}_{1 \times 3} & 1
\end{bmatrix}$
}
electives/amr/parts/01_kinematics/01_forward.tex
+1 -1
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@@ -1,5 +1,5 @@
\subsection{Forward Kinematics (FK)}
$\mat{T}_{WB_n}(\theta) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.\\
For 2R system:
${_W}\vec{t}_{WE} =$
{\scriptsize