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51 lines
2.2 KiB
TeX
51 lines
2.2 KiB
TeX
\newpage
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\subsection{Rigid body \& IMU kinematics}
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\begin{wrapfigure}[5]{r}{0.3\columnwidth}
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\includegraphics[width=0.3\columnwidth]{assets/rigid-body-6d.png}
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\end{wrapfigure}
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\bi{Velocity} ${_I}\vec{v}_{IB} = \diff{t} ({_I}\vec{t}_B)$
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\bi{Rot. Velocity} ${_I}\vec{\omega}_{IB} = \diff{t} (\alpha)\; {_I}\vec{t}$
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\bi{Velocity point $P$} ${_B}\vec{v}_{IP} = {_B}\vec{v}_{IB} + {_B}\vec{\omega}_{IB} \times {_B}\vec{t}_{P}$
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\bi{Rotation Matrices}
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\begin{itemize}
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\item For left pertubing\\
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$\mat{\dot{R}}_{IB} = [{_I} \omega_{IB}]^\times \mat{R}_{IB}$
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\item For right pertubing
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$\mat{\dot{R}}_{IB} = \mat{R}_{IB} [{_I} \omega_{IB}]^\times$
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\item Constant angular velocity ($\exp{[\Delta \alpha]^\times} = \delta \mat{R}(\Delta \alpha)$)\\
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$\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \alpha]^\times} \mat{R}_{IB}(t)$
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\end{itemize}
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\bi{Quaternions}
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\begin{itemize}
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\item For left pertubing
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$\displaystyle \vec{\dot{q}}_{IB} = \frac{1}{2} \begin{bmatrix}
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{_I}\vec{\omega}_{IB} \\
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0
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\end{bmatrix}
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\otimes \vec{q}_{IB}$
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\item For right pertubing
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$\displaystyle \vec{\dot{q}}_{IB} = \frac{1}{2} \vec{q}_{IB} \otimes \begin{bmatrix}
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{_B}\vec{\omega}_{IB} \\
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0
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\end{bmatrix}$
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\end{itemize}
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\bi{IMU} (Outputs {\color{blue} ${_S}\vec{\tilde{a}}$} (accel.), {\color{red} ${_S}\vec{\tilde{\omega}}$} (rot. accel.))\\
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${_W}\vec{\dot{t}}_S = {_W} \vec{v}$,
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$\displaystyle \vec{\dot{q}}_{WS} = \frac{1}{2} \vec{q}_{WS} \otimes
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\begin{bmatrix}
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{\color{red}{_S}\vec{\tilde{\omega}}} {\color{gray} + \vec{w}_g - \vec{b}_g} \\
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0
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\end{bmatrix}$
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${_W}\vec{\dot{v}} = \mat{R}_{WS}\; ({\color{blue}{_S}\vec{\tilde{a}}} {\color{gray} + \vec{w}_a - \vec{b}_a}) + {_W}\vec{g}$
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where {\color{gray} gray parts} only IRL (in theor. models, leave out), with $\vec{\dot{b}}_g = \vec{w}_{b_g}$ and $\vec{\dot{b}}_a = \vec{w}_{b_a}$
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\bi{IMU Sensor Model}: $\vec{\tilde{z}} = \vec{b}_C + s\mat{M}\vec{z} + \vec{b} + \vec{n} + \vec{o}$
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where bias $\vec{b}$ and scale $s$ often modled as time-varying state $\dot{b}(t) = \sigma_C n(t)$.
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$\vec{b}_C$ const. calib. and $\vec{n}$ the model.
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