80 lines
2.7 KiB
TeX
80 lines
2.7 KiB
TeX
\newcommand{\mPosVec}{\vec{\mPos}} % position vector
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\newcommand{\mRssiVec}{\vec{s}} % client signal strength measurements
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\newcommand{\mState}{q} % state variable
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\newcommand{\mStateVec}{\vec{q}} % state vector variable
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\newcommand{\mObs}{o} % observation variable
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\newcommand{\mObsVec}{\vec{o}} % observation vector variable
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\newcommand{\mObsWifi}{\vec{o}_{\text{wifi}}} % wifi observation
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\newcommand{\mPressure}{\rho}
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\newcommand{\mObsPressure}{\mPressure_\text{rel}} % symbol for observation pressure
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\newcommand{\mStatePressure}{\hat{\mPressure}_\text{rel}} % symbol for state pressure
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\newcommand{\mHeading}{\theta}
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\newcommand{\mObsHeading}{\Delta\mHeading} % symbol used for the observation heading
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\newcommand{\mStateHeading}{\mHeading} % symbol used for the state heading
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\newcommand{\mSteps}{n_\text{steps}}
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\newcommand{\mObsSteps}{\mSteps}
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\newcommand{\mActivity}{\Omega}
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\newcommand{\mObsActivity}{\mActivity}
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\newcommand{\R}{\mathbb{R}}
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%\newcommand{\N}{\mathbb{N}}
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\begin{frame}[fragile]
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\frametitle{System}
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%\includegraphics[width=4cm]{gfx/map1}
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\small
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\begin{minipage}{0.49\textwidth}
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Unbekannter Zustand
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\begin{equation*}
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\mStateVec = (\underbrace{x, y, z}_{\text{Position}}, \underbrace{\mStateHeading}_{\text{Richtung}}),\enskip
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x, y, z, \mStateHeading \in \R
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\end{equation*}
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\vspace{1mm}
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\end{minipage}
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%
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\begin{minipage}{0.49\textwidth}
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Smartphone Sensordaten
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\begin{equation*}
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\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{beacon}, \mObsSteps, \mObsHeading, \mStateHeading, \mObsActivity, ) \enspace .
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\end{equation*}
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\vspace{1mm}
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\end{minipage}
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Sensorfusion über rekursive Dichteschätzung
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\begin{equation*}
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\arraycolsep=1.2pt
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%\begin{array}{ll}
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p(\mStateVec_{t} \mid \mObsVec_{1:t}) \propto
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\underbrace{p(\mObsVec_{t} \mid \mStateVec_{t})}_{\text{evaluation}}
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\int \underbrace{p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})}_{\text{transition}}
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\underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}}
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%\end{array}
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%\label{equ:bayesInt}
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\end{equation*}
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%\vspace{1mm}
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%
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\begin{equation*}
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\begin{aligned}
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p(\vec{o}_t \mid \vec{q}_t) &=
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\,p(\mRssiVec_\text{wifi} \mid \vec{q}_t)%_\text{wifi}
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\,p(\mRssiVec_\text{beacon} \mid \vec{q}_t)%_\text{beacon}
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\,p(\mStateHeading \mid \vec{q}_t)%_\text{kompass}
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\,p(\mObsActivity \mid \vec{q}_t)%_\text{activity}
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\footnote{\tiny Annahme: Sensoren sind statistisch unabhängig}
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\\
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p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1}) &= \text{Bewegungsmodell mit Zusatzwissen}
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\end{aligned}
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\end{equation*}
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\end{frame}
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