From 9e9b6882cd4ffa1ec49ba1934029bb410e38be0a Mon Sep 17 00:00:00 2001 From: FrankE Date: Fri, 5 Feb 2016 20:20:52 +0100 Subject: [PATCH] some new TeX aspects --- tex/bare_conf.tex | 1 + tex/chapters/sensors.tex | 49 ++++++++++++++++++++++++++++++++++++++++ tex/chapters/system.tex | 29 +++++++++++++++++++----- tex/misc/functions.tex | 7 +++++- 4 files changed, 79 insertions(+), 7 deletions(-) diff --git a/tex/bare_conf.tex b/tex/bare_conf.tex index a45fbc2..9fc1fba 100644 --- a/tex/bare_conf.tex +++ b/tex/bare_conf.tex @@ -95,6 +95,7 @@ \newcommand{\percent}{\%} \newcommand{\decibel}{dB} \newcommand{\dB}{dB} +\newcommand{\hpa}{hPa} \newcommand{\degree}{\ensuremath{^{\circ}}} % missing math operators diff --git a/tex/chapters/sensors.tex b/tex/chapters/sensors.tex index 4566b57..af28e53 100644 --- a/tex/chapters/sensors.tex +++ b/tex/chapters/sensors.tex @@ -8,7 +8,56 @@ route calculation, user checking the route) we use the average of all barometer readings during this timeframe as realtive base $\overline{\mPressure}$. + During each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$, the predicted pressure $\mStatePressure$ is + adjusted according to the resulting $z$-change, if any: + + \begin{equation} + \mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot \SI{0.105}{\hpa} + ,\enskip + \Delta z = \mState_{t-1}^{z} - \mState_{t}^z + . + \end{equation} + \subsection{Wi-Fi \& iBeacons} + For additional absolute location hints, we use the Smartphones Wi-Fi and iBeacon sensor to measure the signal-strengths + of nearby transmitters. As the positions of both \docAP{}s and and \docIBeacon{}s are known beforehand, we compare + each measurement with its corresponding signal strength prediction which is defined by the 3D distance $d$ + and the number of floors $\Delta f$ between the \docAPshort{} and the particle + + \begin{equation} + P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF, + \end{equation} + + and calculate the resulting probability as described in \cite{ipin2015}: + + \begin{equation} + \mProb(\mObsVec \mid \mStateVec)_\text{wifi} = + \prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2). + \label{eq:wifiTotal} + \end{equation} + + For the \docWIFI{} component we thus need two parameters per \docAPshort{}: $\mTXP$ measured at a distance + $\mMdlDist_0$ (usually \SI{1}{\meter}) and the path-loss exponent $\mPLE$ describing the environment. + To reduce complexity and system setup time, we use the same values for all \docAP{}s at the cost of accuracy. + While, $\mTXP$ is best determined using averaged measurements at a single location, + a good estimation of $\mPLE$ requires several measurements and numerical optimization \cite{etwas_aus_der_MA}. + $\mPLE$ is thus chosen empirically. + + For the \docIBeacon{} component we also use \refeq{eq:wifiTotal} but $\mTXP$ is transmitted by each beacon. + Again, $\mPLE$ is determined emprically. \todo{faellt hier meist kleiner aus, weil ja kuerzere reichweite etc} + \subsection{Step- \& Turn-Detection} + + To prevent degradation within the particle-filter \cite{??} due to downvoting of particles with increased + heading deviation, we incorporate the turn-detection as control-data directly into the transition + $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$. + \cite{thrun?}\cite{lukas2014?} to get a more directed sampling instead of a truly random one. + + + + \commentByFrank{todo: wie wird die unsicherheit in der transition eingebracht, sigma, ..} + + + diff --git a/tex/chapters/system.tex b/tex/chapters/system.tex index be81303..0064215 100644 --- a/tex/chapters/system.tex +++ b/tex/chapters/system.tex @@ -1,7 +1,24 @@ -\section{Indoor Localisation System} +\section{Recursive Density Estimation} -\subsection{Overview} - -\subsection{Transition} - -\subsection{Evaluation} + \commentByFrank{particle-filter wie bei lukas mit $\vec{o}_{t}$ in transition und $\vec{q}_{t-1}$ in eval??} + \commentByFrank{brauchen wir in der observation ueberhaupt noch $q_{t-1}$??} + \commentByFrank{das ist die basis fuer unser system} + + + \begin{equation} + p(\mStateVec_{t} \mid \langle \mObsVec \rangle_{t}) = \\ + p(\mObsVec_{t} \mid \mStateVec_{t}) + \int + p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t}) + p(\mStateVec_{t-1} \mid \langle \mObsVec \rangle_{t-1} + d\mStateVec_{t-1} + \end{equation} + + \begin{equation} + \mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsPressure) + \end{equation} + + \begin{equation} + \mStateVec = (x, y, z, \mObsHeading, \mStatePressure),\enskip + x,y,z,\mStatePressure \in \R + \end{equation} diff --git a/tex/misc/functions.tex b/tex/misc/functions.tex index 5d6aca0..92dfbb8 100644 --- a/tex/misc/functions.tex +++ b/tex/misc/functions.tex @@ -28,7 +28,12 @@ \newcommand{\mMovingAvgWithSize}[1]{\ensuremath{\text{avg}_{#1}}} -\newcommand{\mPressure}{\rho} % symbol for pressure readings +\newcommand{\mPressure}{\rho} +\newcommand{\mObsPressure}{\mPressure_\text{rel}} % symbol for observation pressure +\newcommand{\mStatePressure}{\hat{\mPressure}_\text{rel}} % symbol for state pressure +\newcommand{\mHeading}{\theta} +\newcommand{\mObsHeading}{\Delta\mHeading} % symbol used for the observation heading +\newcommand{\mStateHeading}{\mHeading} % symbol used for the state heading %\newcommand{\docIBeacon}{iBeacon}