94 lines
5.3 KiB
TeX
94 lines
5.3 KiB
TeX
\section{Sensors}
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\label{sec:sensors}
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\subsection{Barometer}
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\label{sec:sensBaro}
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As stated by \cite{Muralidharan14-BPS}, ambient pressure readings are highly influenced by environmental conditions
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like the weather, time-of-day and others. Thus, relative pressure readings are preferred over absolute ones.
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However, due to noisy sensors, more than one reading is required to estimate the relative base.
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Harnessing the usual setup time of a navigation-system (route calculation, user checking the route, etc.)
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we use the average of all barometer readings during this timeframe as estimated base $\overline{\mObsPressure}$.
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Moreover, it is often necessary to omit some initial sensor readings, as the smartphone's sensor needs some time
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to settle. Besides, we use the setup timeframe to estimate the sensor's uncertainty $\sigma_\text{baro}$ for later
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use within the evaluation step. Fig. \ref{fig:baroSetupError} depicts actual sensor readings including aforementioned
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error conditions.
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\begin{figure}
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\include{gfx/baro/baro_setup_issue}
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\caption{Sometimes the smartphone's barometer (here: Motorola Nexus 6) provides erroneous pressure readings
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during the first seconds. Those need to be omitted before $\sigma_\text{baro}$ and
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$\overline{\mObsPressure}$ are estimated.}
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\label{fig:baroSetupError}
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\end{figure}
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During each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$, we need a corresponding, relative
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pressure prediction $\mStatePressure$ which is adjusted according to the resulting $z$-change:
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\begin{equation}
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\mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b
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,\enskip
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\Delta z = \mState_{t-1}^{z} - \mState_{t}^z
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,\enskip
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b \in \R
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\enspace .
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\label{eq:baroTransition}
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\end{equation}
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In \refeq{eq:baroTransition}, $b$ denotes the usual pressure change in $\frac{\text{hPa}}{\text{m}}$.
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The evaluation, following the transition, compares the predicted relative pressure with the observed
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one using a normal distribution with the previously estimated $\sigma_\text{baro}$:
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\begin{equation}
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p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace.
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\label{eq:baroEval}
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\end{equation}
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\subsection{Wi-Fi \& iBeacons}
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Absolute location hints are provided by the smartphone's \docWIFI{} and \docIBeacon{} component,
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measuring the signal-strengths of nearby transmitters. As the positions of both \docAP{}s and \docIBeacon{}s
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are known beforehand, we compare each measurement with its corresponding signal strength prediction using
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the wall-attenuation-factor model \cite{Ebner-15}. This prediction depends on the 3D distance $d$ from the
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\docAPshort{} and the number of floors $\Delta f$ between the \docAPshort{} and the state-in-question:
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\begin{equation}
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P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace ,
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\end{equation}
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Assuming statistical independence of all \docAPshort{}s,
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the probability to measure their predictions at a given location is:
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\begin{equation}
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\mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} =
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\prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace .
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\label{eq:wifiTotal}
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\end{equation}
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For the \docWIFI{} component we thus need three parameters per \docAPshort{}: $\mTXP$ measured at a distance
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$\mMdlDist_0$ (usually \SI{1}{\meter}), the path-loss exponent $\mPLE$ describing the environment
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and $\mWAF$ denoting the attenuation per floor.
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To reduce complexity and system setup time, we use the same values for all \docAP{}s at the cost of accuracy.
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While, $\mTXP$ is best determined using averaged measurements at a single location,
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a good estimation of $\mPLE$ and $\mWAF$ requires several measurements and numerical optimization
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\cite{PathLossPredictionModelsForIndoor}. $\mPLE$ and $\mWAF$ are thus chosen empirically.
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For the \docIBeacon{} component we also use \refeq{eq:wifiTotal} but $\mTXP$ is transmitted by each beacon.
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Due to the short-range coverage the model parameters require less consideration of the senders ambient conditions
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(e.g. walls). Therefore, a smaller $\mPLE$ can be chosen to model the signal strength prediction for \docIBeacon{}s.
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\subsection{Step- \& Turn-Detection}
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A big disadvantage of using the state transition as proposal distribution is the high possibility of sample
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impoverishment due to a small measurement noise. This happens since accurate observations result in high peaks
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of the evaluation density and therefore the proposal density is not able to sample outside that peak \cite{Isard98:CCD}.
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Additionally, erroneous or delayed measurements from absolute positioning sensors like \docWIFI{} may lead to misplaced turns.
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This causes a downvoting of the posterior distribution in areas where the heading deviates.
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Therefore, we incorporate the pedestrian's heading $\mObsHeading$, as well as the number of steps $\mObsSteps$, directly into the state transition
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$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$, which leads to a directed sampling instead of a random one.
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Steps and turns are detected using the smartphone's IMU and are implemented as described in \cite{Ebner-15}.
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