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Janis Hutz
2026-03-13 17:24:06 +01:00
parent d7d75c789c
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electives/others/amr/autonomous-mobile-robots-cheatsheet.pdf
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electives/others/amr/parts/01_kinematics/03_temporal-models.tex
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\subsection{Temporal Models}
Often use cont. time n.-lin. system of ODE $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t))$, with measurements $\vec{z}(t) = \vec{h}(\vec{x}(t))$.
Need linearised (around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium})
Often use cont. time n.-lin. system of ODE $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t))$, with measurements $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.
Need linearised (around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium}):\\
$\delta \vec{\dot{x}}(t) = \vec{f}_C(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F}_C \delta \vec{x}(t) + \mat{G}_C \delta \vec{u}(t) + \mat{L}_C \vec{w}(t)$\\
$\delta \vec{z}(t) = \mat{H} \delta \vec{x}(t) + \vec{v}(t)$.
Herein, $\mat{H}$ is measurements, $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process noise, $\vec{v}$ measurement noise