82 lines
3.2 KiB
TeX
82 lines
3.2 KiB
TeX
\section{Indoor Positioning System}
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\label{sec:system}
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Our smartphone-based indoor localization system estimates the current location and heading
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using recursive density estimation.
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A graph based movement model provides the transition,
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%$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$
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while the smartphone's accelerometer, gyroscope, magnetometer provide the observations
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for the following evaluation step to infer the hidden state, namely the pedestrian's location and heading
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\cite{Ebner-16, Fetzer-16}.
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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}} \enspace,
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\end{array}
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\label{eq:recursiveDensity}
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\end{equation}
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The hidden state $\mStateVec$ is given by
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\begin{equation}
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\mStateVec = (x, y, z, \mStateHeading),\enskip
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x, y, z, \mStateHeading \in \R \enspace,
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\end{equation}
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%
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where $x, y, z$ represent the pedestrian's position in 3D space
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and $\mStateHeading$ his current (absolute) heading.
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%
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The corresponding observation vector is defined as
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%
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\begin{equation}
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\mObsVec = (\mRssiVecWiFi{}, \mObsSteps, \mObsHeadingRel, \mObsHeadingAbs, \mObsGPS) \enspace.
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\end{equation}
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%
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$\mRssiVecWiFi$ contains the measurements of all nearby \docAP{}s (\docAPshort{}s),
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$\mObsSteps$ describes the number of steps detected since the last filter-step,
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$\mObsHeadingRel$ the (relative) angular change since the last filter-step,
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$\mObsHeadingAbs$ the current, vague absolute heading and
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$\mObsGPS = ( \mObsGPSlat, \mObsGPSlon )$ the current location (if available) given by the GPS.
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Assuming statistical independence, the state evaluation's density can be written as
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%
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\begin{equation}
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%\begin{split}
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p(\vec{o}_t \mid \vec{q}_t) =
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p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}\enskip
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p(\vec{o}_t \mid \vec{q}_t)_\text{gps}
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\enspace.
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\label{eq:evalDensity}
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\end{equation}
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%
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The remaining observations,
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namely: detected steps, relative- and absolute heading are
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used within the transition model, where potential movements
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$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ are sampled
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based on those sensor values.
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As this work focuses on \docWIFI{} optimization, not all parts of
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the localization system are discussed in detail.
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For missing explanations please refer to \cite{Ebner-16}.
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%
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Since then, absolute heading and GPS have been added as additional sensors
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to further enhance the localization by comparing the sensor values
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using some distribution.
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\todo{neues resampling?}
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\todo{ueberleitung}
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\todo{
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die absolute positionierung kommt aus dem wlant,
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dafür braucht man entweder viele fingerprints oder ein modell
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}
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As GPS will only work outdoors, e.g. when moving from one building into another,
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the system's absolute position indoors is solely provided by the \docWIFI{} component.
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Therefore its crucial for this component to provide location estimations
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that are as accurate as possible, while ensuring fast setup and
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maintenance times.
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