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54 lines
2.2 KiB
TeX
54 lines
2.2 KiB
TeX
\subsection{Positioning}
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\shortdefinition[Position Vector]
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$_{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{red}\fbox{B}} = \, _{\color{blue}\fbox{W}}\,\vec{t}\,_{\color{ForestGreen}\fbox{W}}\,_{\color{red}\fbox{B}}$,
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{\color{blue} Original Frame}, {\color{red} End point}, {\color{ForestGreen} Target Frame},
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\hl{$\sin = s$, $\cos = c$}
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\shortdefinition[State vector] $x_R$: $x$, $v$ of rob in $W$, pos of sensors
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\shortdefinition[Rot. Mat.] $\mat{R}_{z} = \begin{bmatrix}
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c(\psi) & -s(\psi) & 0 \\
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s(\psi) & c(\psi) & 0 \\
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0 & 0 & 1
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\end{bmatrix}$\\
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$\mat{R}_y(\theta) = \begin{bmatrix}
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c(\psi) & 0 & s(\psi) \\
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0 & 1 & 0 \\
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-s(\psi) & 0 & c(\psi) \\
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\end{bmatrix}
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\mat{R}_x(\varphi)
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\begin{bmatrix}
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1 & 0 & 0 \\
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0 & c(\psi) & -s(\psi) \\
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0 & s(\psi) & c(\psi)
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\end{bmatrix}$
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\shortremark Application: ${_W} \vec{a} = \mat{R}_{WB} {_B} \vec{a}$
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\shortlemma $\mat{R}_{BW} = \mat{R}_{WB}^{-1} = \mat{R}_{WB}^\top$, $\det(\mat{R}_{WB}) = 1$ (orth.)
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\shortremark Cols of $\mat{R}_{WB}$ are basis vec. of Frame $\underset{\rightarrow}{\cF}{_B}$ in $\underset{\rightarrow}{\cF}{_W}$
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\shortdefinition[Euler Angles] Yaw ($z$), Pitch ($y$), Roll ($x$), mult. rotation matrices, e.g.
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$\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$, \hl{bound.}.
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$\qquad [\vec{n}]^\times = \vec{n} \vec{x}^\top$ (matrix from vec + arg $\vec{x}$)
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\shortdefinition[Rot. Vec]
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$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal)\\
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$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$
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\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
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$i^2 = j^2 = k^2 = -1$, ($ij = -ji = k$, same for $jk$ and $ki$)
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% TODO: Finish this
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\shortdefinition[Transf. M] $\mat{T}_{AB} = \begin{bmatrix}
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\mat{R}_{AB} & {_A}\vec{t}_B \\
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\mat{0}_{1\times 3} & 1
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\end{bmatrix}$\\
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$\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
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\begin{bmatrix}
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\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
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\mat{0}_{1 \times 3} & 1
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\end{bmatrix}$
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$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$
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