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89 lines
3.0 KiB
TeX
89 lines
3.0 KiB
TeX
\subsection{Positioning}
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\shortdefinition[Pos Vec.]
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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} Relative to Frame}, {\color{red} P. in other Frame}, {\color{ForestGreen} Start P. of vec in first F.},
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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.]
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{\scriptsize
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$\mat{R}_{z}(\psi)$ (Yaw),
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$\mat{R}_y(\theta)$ (Pitch),
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$\mat{R}_x(\varphi)$ (Roll)\\
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$\begin{bmatrix}
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c(\psi) & -s(\psi) & 0 \\
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s(\psi) & c(\psi) & 0 \\
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0 & 0 & 1
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\end{bmatrix};
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\begin{bmatrix}
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c(\theta) & 0 & s(\theta) \\
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0 & 1 & 0 \\
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-s(\theta) & 0 & c(\theta) \\
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\end{bmatrix};
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\begin{bmatrix}
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1 & 0 & 0 \\
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0 & c(\varphi) & -s(\varphi) \\
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0 & s(\varphi) & c(\varphi)
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\end{bmatrix}$
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}
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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 Ang] (Tait-Brian) $\mat{R}_{EB} = \mat{R}_z(\psi) \cdot \mat{R}_y(\theta) \cdot \mat{R}_x(\varphi)$.
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$\begin{smallmatrix}
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\psi = \arcsin\left( R_{21} \div \sqrt{1 - R_{31}^2} \right)\\
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\theta = \arcsin(-R_{31})\\
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\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
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\end{smallmatrix}$
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{\scriptsize
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$[\vec{n}]^\times = \begin{bmatrix}
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0 & -a_3 & a_2 \\
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a_3 & 0 & -a_1 \\
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-a_2 & a_1 & 0
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\end{bmatrix}$
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}
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\shortdefinition[Angle-Axis]
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$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
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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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To quat: $\vec{q} = [\vec{n}, \alpha]$
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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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$\vec{q} = \begin{bmatrix}
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\vec{v}(\vec{q}), a(\vec{q})
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\end{bmatrix}^\top$, $a(\vec{q}) = q_w$, Add like $\C$ (by ``groups'')
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\bi{Mult} {\scriptsize
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$\vec{q} \otimes \vec{p} = \begin{bmatrix}
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a(\vec{q}) \vec{v}(\vec{p}) + a(\vec{p}) + \vec{v}(\vec{q}) \times \vec{v}(\vec{p}) \\
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a(\vec{q}) a(\vec{p}) - \vec{v}(\vec{q})^\top \vec{v}(\vec{p})
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\end{bmatrix}$
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}
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\bi{To Rot Mat} {\scriptsize
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$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
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}
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\shortdefinition[Transf. M]
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{\scriptsize
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$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
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$\mat{T}_{AB} = \begin{bmatrix}
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\mat{R}_{AB} & {_A}\vec{t}_B \\
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\mat{0}_{1\times 3} & 1
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\end{bmatrix};
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\mat{T}_{BA} = \mat{T}_{AB}^{-1} =
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\begin{bmatrix}
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\mat{R}_{AB}^\top & -\mat{R}_{AB}^\top {_A}\vec{t}_B \\
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\mat{0}_{1 \times 3} & 1
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\end{bmatrix}$
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}
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