From 1ecbd0c5c08a3dbd2618da8fb6380bd97df53fea Mon Sep 17 00:00:00 2001 From: toni Date: Tue, 22 May 2018 17:45:26 +0200 Subject: [PATCH] started sample impoverishment section --- tex/chapters/eval.tex | 2 +- tex/chapters/misc.tex | 68 ++++++++++++++++++++++++++++++++++++++---- tex/misc/functions.tex | 3 ++ 3 files changed, 66 insertions(+), 7 deletions(-) diff --git a/tex/chapters/eval.tex b/tex/chapters/eval.tex index ed63756..5128fa8 100644 --- a/tex/chapters/eval.tex +++ b/tex/chapters/eval.tex @@ -133,7 +133,7 @@ A corresponding activity is chosen by \Delta \bar\omega = \bar\omega_\text{l} - \bar\omega_\text{s} \end{equation} \noindent and $\bar\omega$ provides the arithmetic mean of the respective windows and thus represents a moving average. -We set $t_\text{acc} = $ \SI{0.015}{m/s$^2$} and $t_\text{baro} = $ \SI{0.042}{m/s$^2$}. +We set $t_\text{acc} = $ \SI{0.015}{\meter/\square\second} and $t_\text{baro} = $ \SI{0.042}{\meter/\square\second}. For both involved sensors we suggest to set the size of $\vec{\omega}_\text{s}$ between \SI{0.3}{\second} and \SI{0.6}{\second}. Recognizing if the pedestrian is standing or walking requires less prior data, then climbing stairs. Therefore, $\vec{\omega}_\text{l, acc}$ is recommended between \SI{1}{\second} and \SI{2}{\second}, while $\vec{\omega}_\text{l, baro}$ between \SI{3}{\second} and \SI{5}{\second}. diff --git a/tex/chapters/misc.tex b/tex/chapters/misc.tex index 36b52ba..6004a04 100644 --- a/tex/chapters/misc.tex +++ b/tex/chapters/misc.tex @@ -1,4 +1,64 @@ -\section{Misc} +\section{Particle Filtering} + +As described earlier, we use a CONDENSATION particle filter to implement the recursive state estimator described in section \ref{sec:rse}. +A set of particles is defined by $\{\vec{X}^i_{t}, w^i_{t} \}_{i=1}^N$, where $\mParticleVec^{i}_{t}$ is sampled based on the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$. +The weight $w_t^i$ is obtained by the probability density of the state evaluation $p(\mObsVec_{t} \mid \mStateVec_{t})$. +A particle set approximates the posterior as follows: + +\begin{equation} +p(\mStateVec_{t} \mid \mObsVec_{1:t}) \approx \sum^N_{i=1} w^i_t \delta_{\vec{X}^i_{t}}(\vec{q}_{t}) \enspace, +\label{eg:monteEstimation} +\end{equation} + +\noindent where $\delta_{x_0}(x)$ denotes the Dirac delta mass located at $x_0$. +As one can imagine, after a few iterations with continuously reweighting particles, the weight will concentrate on a few particles only. +To handle this phenomenon of weight degeneracy, a resampling procedure is performed after every filter step \cite{robotics}. + +\subsection{Sample Impoverishment} +As we have extensively discussed in \cite{Fetzer-17}, besides sample degeneracy, particle filters (and nearly all of its modifications) continue to suffer from another notorious problem: sample impoverishment. +It refers to a situation, in which the filter is unable to sample enough particles into proper regions of the building, caused by a high concentration of misplaced particles. +%Such situations are strongly influenced by the resampling step and most of all by restrictive transition models. +As described in section \ref{sec:relatedWork}, sample impoverishment is often a problem of environmental restrictions and system dynamics. +An example using the so far presented approach can be seen in fig. \ref{fig:multimodalPath}. +Due to uncertain measurements, the posterior distribution of the particle filter is captured within a room. +Between time $t-1$ and $t$, the resampling step abandons all particles on the corridor and drawing new particles outside the room is not possible due to the restricted transition. +At this point, standard filtering methods are not able to recover. +% +\begin{figure}[t] + \centering + \def\svgwidth{0.75\columnwidth} + \input{gfx/multimodalPath.eps_tex} + \caption[An example of the occurrence of sample impoverishment.]{ + An example of the occurrence of sample impoverishment enhanced by a restrictive transition model that prevents sampling through walls. At time $t-1$ the approximated position (green line) drifts apart from the ground truth (black line) due to uncertain measurements. The posterior distribution is then captured within the room and not able to recover by itself \cite{Fetzer-17}. } + \label{fig:multimodalPath} +\end{figure} +% + +%todo: umschreiben weng +The simplest solution to handle sample impoverishment is by drawing a handful new particles randomly in the building. +For this, we add a slight chance of \SI{0.01}{\percent} to the resampling step, so that every particle can be chosen for repositioning instead of the standard procedure. +A new position for those particles is then drawn uniformely from the underlying mesh. +It is obvious that this leads to a higher uncertainty and possibly a multimodal posterior distribution. +Additionally, very uncertain absolute measurements, like attenuated Wi-Fi signals, can cause unpredictable jumps to such a newly drawn position, which would otherwise not be possible. +Especially, methods using relative measurements like pedestrian dead reckoning are losing their importance. +Nevertheless, this method is very easy to implement and we expect that the system should be able to recover from nearly every situation regardless of the cause. + +A second method we suggest within this paper is a simplified version of our approach presented in \cite{Fetzer-17}. +Here, we used an additional, very simple particle filter to monitor if our primary (localisation) filter suffers from sample impoverishment. +If that is true, both filters are combined by exchanging particles among each other. +This allows the primary filter to recover, while retaining prior knowledge. +However, we believe that such a combination of two independent filters is not necessary for most scenarios and thus the resulting overhead can be avoided. + +%neue methode: + + +we sample 10k samples uniformaly within the whole building.. + +\begin{itemize} + \item zufällig einen partikel streuen + \item partikel bekommen eine kleine chance durch wände zu laufen + \item KLD zwischen wifi und aktuellen particeln des filters. +\end{itemize} \subsection{State Estimation} @@ -12,11 +72,7 @@ -\subsection{Sample Impoverishment} -\begin{itemize} - \item einfache methode um das zu beheben. - \item falls ichs schaff, wifi method über die frank und ich mal gesprochen haben. -\end{itemize} + %fürs verständnis, diesen satz hier nicht vergessen. diff --git a/tex/misc/functions.tex b/tex/misc/functions.tex index 12db569..84dfb32 100644 --- a/tex/misc/functions.tex +++ b/tex/misc/functions.tex @@ -24,6 +24,9 @@ \newcommand{\mObsVec}{\vec{o}} % observation vector variable \newcommand{\mObsWifi}{\vec{o}_{\text{wifi}}} % wifi observation +\newcommand{\mParticle}{X} +\newcommand{\mParticleVec}{\vec{X}} + \newcommand{\mProb}{p} % char for probability \newcommand{\mMovingAvgWithSize}[1]{\ensuremath{\text{avg}_{#1}}}