163 lines
12 KiB
TeX
163 lines
12 KiB
TeX
\section{Experiments}
|
|
|
|
% allgemeine infos über pfade und gebäude. wo
|
|
% bild: mit pfaden drauf und eventl. wifi qualität in jeweiligen bereichen? (kann frank das)
|
|
\begin{figure}
|
|
\centering
|
|
\input{gfx/eval/paths.tex}
|
|
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
|
|
\label{fig:paths}
|
|
\end{figure}
|
|
%
|
|
All upcoming experiments were carried out on four floors (0 to 3) of a \SI{77}{m} x \SI{55}{m} sized faculty building.
|
|
It includes several staircases and elevators and has a ceiling height of about \SI{3}{m}.
|
|
Nevertheless, the grid was generated for the complete campus and thus outdoor areas like the courtyard are also walkable.
|
|
As Wi-Fi is attenuated by obstacles and walls, it does not provide a consistent quality over the complete building.
|
|
In fig. \ref{fig:paths} we illustrate the quality obtained by the wall attenuation factor model presented earlier.
|
|
Here, the intensity of red indicates a low coverage and thus a bad quality for localisation.
|
|
To obtain this information we interpolated the Wi-Fi quality factor given by all test walks using $l_{\text{max}} = \SI{-75}{\dBm}$ and $l_{\text{min}} = \SI{-90}{\dBm}$.
|
|
As mentioned before, we omit any time-consuming calibration processes and use the same values for all access-points. That would be $P_{0_{\text{wifi}}} = \SI{-46}{\dBm}, \mPLE_{\text{wifi}} = \SI{2.7}{}, \mWAF_{\text{wifi}} = \SI{8}{\dB}$.
|
|
The position of the access-points (about five per floor) is known beforehand.
|
|
Due to legal terms, we are not allowed to depict their positions and therefore omit this information within the figures.
|
|
|
|
% gewählte parameter (auch mal die optimieren wifi parameter testen)
|
|
We arranged three distinct walks (see also fig. \ref{fig:paths}).
|
|
The measurements for the walks were recorded using a Motorola Nexus 6 at 2.4 GHz band only.
|
|
The computation was done offline as described in algorithm \ref{fig:paths}.
|
|
For each walk we deployed $50$ MC runs using 5000 Particles for each mode.
|
|
Instead of an initial position and heading, all walks start with a uniform distribution (random position and heading) as prior.
|
|
For the filtering we used $\sigma_\text{wifi} = 8.0$ as uncertainties, both growing with each measurement's age.
|
|
While the pressure change was assumed to be \SI{0.105}{$\frac{\text{\hpa}}{\text{\meter}}$}, all other barometer-parameters are determined automatically.
|
|
The step size $\mStepSize$ for the transition was configured to be \SI{70}{\centimeter} with an allowed derivation of \SI{10}{\percent}.
|
|
The heading deviation was set to \SI{25}{\degree}.
|
|
The pedestrian's position (state) was estimated using the weighted arithmetic mean of
|
|
the particle set.
|
|
|
|
% wie für die kld gezogen? begründen warum wir nun keine parzenschätzung machen (weil ähnliche ergebnisse)
|
|
To calculate \eqref{equ:KLD} and thus the Kullback-Leibler divergence, we need to sample densities from both modes likewise.
|
|
The grid is suitable for this purpose.
|
|
However, sampling at any vertex $\mVertexA$ of the grid, given just a set of random variables (particles), is not the easiest task.
|
|
We need to estimate the posterior distribution given by the respective particle sets.
|
|
A common way is to deploy a kernel density estimation using a Gaussian distribution as kernel.
|
|
The density of a specific point $\hat\mStateVec_{t} = \fPos{\mVertexA}$ is then given by
|
|
%
|
|
\begin{equation}
|
|
p(\hat\mStateVec_{t} \mid m_t, \mObsVec_{1:t}) = \sum_{i=1}^{N_{m_t}} \mathcal{N}(d^i_{\text{KL}} \mid 0, \sigma_{\text{KL}})
|
|
\enspace ,
|
|
\end{equation}
|
|
%
|
|
while $d^i_{\text{KL}}$ is the euclidean distance between the considered point's $\hat\mStateVec_{t}$ and all particles $\fPos{\vec{X}_t^{i,m_t}}$ of the mode. The variance $\sigma_{\text{KL}}$ is set to \SI{1}{m}.
|
|
It is well known, that the computation of the kernel density estimation is rather slow, thus we also used a much simpler estimation by assuming a multivariate Gaussian distribution for both modes.
|
|
Here, the mean is given by weighted arithmetic mean of the particles and the variance is defined by the sample covariance matrix.
|
|
Calculating a meaningful $D_{\text{KL}}$, both estimation methods performed almost identical and therefore we used the multivariate Gaussian distribution for both modes with $\lambda = 0.03$ for the upcoming experimental discussion.
|
|
|
|
% ground truth
|
|
The ground truth is measured by recording a timestamp at marked spots on the walking route. When passing a marker, the pedestrian clicked a button on the smartphone application.
|
|
Between two consecutive points, a constant movement speed is assumed.
|
|
Thus, the ground truth might not be \SI{100}{\percent} accurate, but fair enough for error measurements.
|
|
The approximation error is then calculated by comparing the interpolated ground truth position with the current estimation \cite{Fetzer2016OMC}.
|
|
|
|
%error at the beginning always very high. about 44 meters. therefore the median is better value oder 75 quantil.
|
|
|
|
% zeigen das es stucken verhindert (eventl. hier eine andere aufnahme die mitten drinnen stecken bleibt)
|
|
% bild: stucken im raum + nicht mehr stucken im raum + kld mit anzeigen
|
|
\begin{figure}
|
|
\centering
|
|
\input{gfx/eval/path3.tex}
|
|
\input{gfx/eval/path3-kld.tex}
|
|
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
|
|
\label{fig:path3}
|
|
\end{figure}
|
|
%
|
|
|
|
At first, we discuss the results of path 3, starting at the left-hand side of the building.
|
|
Exemplary estimation results, using the modes standalone and combined within the IMMPF, can be seen in fig. \ref{fig:path3}.
|
|
As mentioned above, every run of a walk starts with a uniform distribution as prior.
|
|
Due to a low Wi-Fi coverage at the starting point, the pedestrian's position is falsely estimated into a room instead of the corridor.
|
|
All three filters are able to overcome this false detection.
|
|
However, the common particle filter (red) gets then indissoluble captured within a room, because of its restrictive behaviour and the aftereffects of the initial Wi-Fi readings.
|
|
It provides an \SI{75}{\percent}-quantil of $\tilde{x}_{0.75} = \SI{3884}{\centimeter}$ and got captured in \SI{100}{\percent} of all runs.
|
|
As expected and discussed earlier, the simple transition (blue) is less prone to bad observations and provides not so accurate, but very robust results of $\tilde{x}_{0.75}= \SI{809}{\centimeter}$ and a standard deviation over all results of $\bar{\sigma} = \SI{529}{\centimeter}$.
|
|
Looking at $D_{\text{KL}}$ over time confirms our assumption made in section \ref{sec:immpf}.
|
|
It is clearly visible, that the Kullback-Leibler divergence between both modes (purple) is a very good indicator to observe, if the dominant filter gets stuck or loses track.
|
|
Following, the IMMPF (green) results in a very natural and straight path estimation and a low $D_{\text{KL}}$ between modes and no sticking.
|
|
The benefits of mixing both filtering schemes within the scenario of path 3 are thus obvious.
|
|
The IMMPF filters with an error of $\tilde{x}_{0.75} = \SI{667}{\centimeter}$ and $\bar{\sigma} = \SI{558}{\centimeter}$.
|
|
|
|
% zeigen das schlechtes wi-fi (zu hohe diversity) behoben wird.
|
|
% bild: lauf auf der rechten seite des gebäudes zeige mit und ohne wifi faktor (schlechtes wifi einzeichnen)
|
|
\begin{figure}[b]
|
|
\centering
|
|
\input{gfx/eval/path2.tex}
|
|
\input{gfx/eval/path2-wifi-quality.tex}
|
|
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
|
|
\label{fig:path2}
|
|
\end{figure}
|
|
%
|
|
Next, we investigate the performance of our approach by considering the scenario in path 2.
|
|
Here, the overall Wi-Fi quality is rather low, especially in the zig-zag stairwell on the buildings back and the small entrance area at floor 1 (cf. fig. \ref{fig:paths}).
|
|
Path 2 starts in the second floor, walking town the centred stairs into the first floor, then making a right turn and walking the stairs down to zero floor, from there we walk back to second floor using the zig-zag stairwell and after finally crossing a room we are back at the start.
|
|
This is a very challenging scenario, at first the estimation got stuck on the first floor in a room's corner and after that the Wi-Fi is highly attenuated.
|
|
Looking at fig. \ref{fig:path2}, one can observe the impact of the Wi-Fi quality factor within the Markov transition matrix.
|
|
Without it, the position estimation (red) is drifting in the courtyard, missing the stairwell and producing high errors between \SI{80}{th} and \SI{130}{th} second.
|
|
As described before, the bad Wi-Fi readings are causing $D_{\text{KL}}$ to grow.
|
|
It follows that the accurate dominant filter draws new particles from the uncertain support and therefore worsen the position estimation.
|
|
In this scenario it is cold comfort that the system is able to recover thanks to its high diversity during situations with uncertain measurements.
|
|
Only by adding the Wi-Fi quality factor the system is able to improve the approximated path (green) and the overall estimation results from $\tilde{x}_{0.75} = \SI{1278}{\centimeter}$ with $\bar{\sigma} = \SI{948}{\centimeter}$ to $\tilde{x}_{0.75} = \SI{811}{\centimeter}$ with $\bar{\sigma} = \SI{340}{\centimeter}$.
|
|
However, this is far from perfect and in some cases ($\sim \SI{9}{\percent}$) the quality factor was not able to prevent the estimation to drift in the courtyard.
|
|
This solely happened when particles were sampled directly onto the courtyard while changing from first to zero floor.
|
|
Those particles then received a high weight due to the attenuated measurements, causing a weight degeneracy.
|
|
Adapting the bounds $l_{\text{max}}$ and $l_{\text{min}}$ of the quality factor or optimizing the access-points parameters can resolve this problem \cite{}.
|
|
|
|
% zeigen das immpf nicht viel schlechter als normaler pf (ohne stucken) ist.
|
|
% bild: er schafft es nicht die treppe rauf + er schafft es immpf + er schafft es normal filter
|
|
\begin{figure}[t]
|
|
\centering
|
|
\input{gfx/eval/path1.tex}
|
|
\input{gfx/eval/path1-time.tex}
|
|
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
|
|
\label{fig:path1}
|
|
\end{figure}
|
|
%
|
|
An exemplary result for path 1 is illustrated in fig. \ref{fig:path1}.
|
|
The path starts on the first floor and finishes on the third after walking two straight stairs.
|
|
Using the grid-based particle filter for localisation, we were able to provide an very accurate path (blue) in \SI{80}{\percent} of the MC runs providing $\tilde{x}_{0.75} = \SI{526}{\centimeter}$ with $\bar{\sigma} = \SI{316}{\centimeter}$.
|
|
Due to a lack of particles near the stairs, the other \SI{20}{\percent} failed to detect the first floor change (red).
|
|
Using our approach (green), we were able detect all floor changes and thus never lost track.
|
|
It performs with $\tilde{x}_{0.75} = \SI{544}{\centimeter}$ and $\bar{\sigma} = \SI{281}{\centimeter}$.
|
|
Those very similar estimation results confirm the efficiency of the mixing and how it is able to keep the accuracy while providing a higher robustness against failures.
|
|
|
|
|
|
% gegenüberstellung aller pfade und werte in tabelle
|
|
\definecolor{header}{rgb}{.8, .8, .8}
|
|
\begin{table}
|
|
\caption{Median error for all conducted walks.}
|
|
\label{tbl:err}
|
|
\centering
|
|
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
|
|
\hline
|
|
& \multicolumn{3}{c}{Path 1} & \multicolumn{3}{|c|}{Path 2} & \multicolumn{3}{|c|}{Path 3}\\
|
|
\hline
|
|
& $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{0.75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{0.75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{0.75}$ \\
|
|
\hline
|
|
PF_{\text{grid}}& $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ \\
|
|
\hline
|
|
PF_{\text{simple}}& $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ \\
|
|
\hline
|
|
IMMPF & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{table}
|
|
|
|
An overview of all localisation results can be seen in table \ref{tbl:err}.
|
|
Again, it should be noted that the localisation system used for the experiments is very basic and can be seen as a slimmed version of our previous works \cite{}.
|
|
Optimizing the Wi-Fi parameters and adding additional methods will improve the localisation results significantly.
|
|
More importantly, the here presented approach was able to recover in all situations and thus never got completely stuck within a demarcated area.
|
|
All results were similar or more accurate then the ones provided by the standalone filters, even when the localisation did not suffer from any problems.
|
|
|
|
%how the Markov transition matrix regulates the impact of the supporting filter in the right amount.
|
|
|
|
|
|
|
|
|