result section but also related stuff

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paper/main.bib

+@INPROCEEDINGS{7528057, 
+author={D. Örn and M. Szilassy and B. Dil and F. Gustafsson}, 
+booktitle={2016 19th International Conference on Information Fusion (FUSION)}, 
+title={A novel multi-step algorithm for low-energy positioning using GPS}, 
+year={2016}, 
+volume={}, 
+number={}, 
+pages={1469-1476}, 
+month={July},}
+
 @book{Liggins:2008,
  author = {Liggins, Martin E. and Llinas, James and Hall, David L.},
  title = {Multisensor Data Fusion},

paper/main.tex

@@ -47,7 +47,7 @@
 \usepackage{pgfplotstable}
 \usepackage{ifthen}
 \pgfplotsset{compat=newest}
-\usetikzlibrary{shapes,arrows}
+\usetikzlibrary{shapes,arrows,spy}
 \usetikzlibrary{calc,patterns,decorations.pathmorphing,decorations.markings}
 \usetikzlibrary{positioning,automata}
 \usetikzlibrary{pgfplots.groupplots}

paper/sections/01-intro.tex

 Global Positioning System (GPS) receivers are well known to be
 power-hungry with respect to the power consumed by a small electronic
-device~\cite{
+device~\cite{7528057,
   bib:microsoft-leap, bib:enloc-smartphones,
   bib:virtualGPS, bib:accuracy-adaptation,
   bib:feasibility-duty-cycling, bib:traffic-delay,

paper/sections/02-related-work.tex

@@ -7,17 +7,17 @@ i.e., work optimizing the behavior of the sensor, (ii) work that try
 to reduce the usage of the GPS sensor -- i.e., work that tries to
 sample less frequently or only when needed.
 
-The first class includes results
-like~\cite{bib:computation-offloading, bib:selective-tracking,
+The first class includes results like~\cite{7528057,
+bib:computation-offloading, bib:selective-tracking,
 bib:microsoft-leap, bib:sparse-fourier}. The authors of
 \cite{bib:computation-offloading} aim at outsourcing the device
 computation (once the data has been received) to some server, using a
 network connnection. \cite{bib:selective-tracking} improves the GPS
 receiver power-efficiency selecting only a subset of visible
 satellites to be tracked. Other works aim at improving the speed of
-the signal acquisition either using information from previous
+the signal acquisition, either using information from previous
 acquisitions~\cite{bib:microsoft-leap}, or using different algorithms
-for the decoding of the signal~\cite{bib:sparse-fourier}.
+for the decoding of the signal~\cite{bib:sparse-fourier, 7528057}.
 
 The second class includes several attempts to build an
 \emph{adaptation} layer that controls the usage of positioning

paper/sections/06-results.tex

@@ -44,7 +44,7 @@ GPS module. GPS sensors in smartphones implement an
 \emph{Assisted-GPS} function, that allows the retrieval of the
 ephemeris data from the internet instead of listening to the
 satellites\footnote{The model presented in this paper can be adapted
-  to reprent this allowing for an external input that possibly
+  to represent this allowing for an external input that possibly
   triggers the transition \texttt{get\_ephemeris} before the delay
   that instead represents the action of listening to the
   satellites. The modeling of how much time the device requires for
@@ -190,92 +190,206 @@ data (provided that it kept in memory the ones acquired at the start
 up and that those were still valid). To capture this
 \emph{phenomenon}, we would need to describe separately the
 acquisition of the signal and ephemeris data of the different
-satellites, together with their visibiliy. Apparently, this would
+satellites, together with their visibility. Apparently, this would
 increase the complexity of the model and decrease it usability. An
 extension of the model to include also this phenomenon would not be
 very difficult to obtain. It is enough to have parallel state machines
 similar to the one shown in Figure~\ref{fig:cyberdynamics}, that
 independently capture the tracking of individual satellites.
 
-
 \subsection{Tracking the Trade-Off between Performance and Power
   Consumption}
 \label{sec:res:tradeoff}
 
-Here we use real traces, recorded from a GPS receiver and an IMU
-sensor that includes accelerometer and gyroscope. We recorded data in
-two different conditions: a car and a bicycle ride. We recorded traces
-with continuous GPS sampling and simulated different GPS sensor
-dynamics on top of that, to compare different sampling policies. We
-show what the tracking would have been when the sensor fusion
-algorithm was live, compared to the continuous sampling of the GPS. We
-then use simulations to further analyze the trade-off between power
-(and therefore battery) consumption and performance (positioning
-accuracy).
-
-\begin{figure}[t]
+\begin{figure*}[t]
+\begin{minipage}{0.95\columnwidth}
 \begin{tikzpicture}
 \begin{axis}[%
- height = 0.9\columnwidth,
+ height = 0.8\textwidth,
  grid style = {black!30, dashed},
  grid = major,
- width = 0.9\columnwidth,
+ width = 0.95\textwidth,
  scaled x ticks = false,
  scaled y ticks = false,
  ylabel style = {align=center},
  xtick = {55000, 55500, 56000},
+ legend pos = {north west},
+ xlabel = {East [m]},
+ ylabel = {North [m]},
 ]
+\pgfplotsset{filter discard warning=false}
 \pgfkeys{/pgf/number format/.cd,1000 sep={}}
 \addplot[thick, blue, solid]
  table[x index = {0}, y index = {1}, col sep=comma]
  {data/exp_biketrace.csv};
+\addlegendentry{GPS trace}
+\addplot[thick, red, dotted]
+ table[x index = {0}, y index = {1}, col sep=comma, each nth point=10]
+ {data/exp_biketracePsat.csv};
+\addlegendentry{Sensor Fusion $th = 0.1$}
+\addplot[thick, black!70!green, dashed]
+ table[x index = {0}, y index = {1}, col sep=comma, each nth point=10]
+ {data/exp_biketraceP25.csv};
+\addlegendentry{Sensor Fusion $th = 2.5$}
 \end{axis}
 \end{tikzpicture}
-\caption{X.}
+\caption{Trace tracked when cycling. GPS measurements and sensor fusion algorithm with different thresholds.}
 \label{fig:cycling-trace}
-\end{figure}
-
-
-
-\begin{figure}[t]
-\begin{center}
-\includegraphics[height=0.70\columnwidth, width=0.90\columnwidth]{images/cycling_trace.png}
-\caption{Tracked trace when cycling. The sampled sensor fusion algorithm is compared to the one using all the avaiable GPS measurements.
-\label{fig:cycling-trace}
-}
-\end{center}
-\end{figure} 
-
-\begin{figure}[t]
-\begin{center}
-\includegraphics[height=0.70\columnwidth, width=0.90\columnwidth]{images/car_trace.png}
-\caption{Tracked trace when driving a car. The sampled sensor fusion algorithm is compared to the one using all the avaiable GPS measurements.
+\end{minipage}
+\hspace{1mm}
+\begin{minipage}{0.95\columnwidth}
+\vspace{-2mm}
+\begin{tikzpicture}[spy using outlines={rounded rectangle, width=3.5cm, height=3.5cm, magnification=2.5, connect spies}]
+\begin{axis}[%
+ height = 0.8\textwidth,
+ grid style = {black!30, dashed},
+ grid = major,
+ width = 0.95\textwidth,
+ scaled x ticks = false,
+ scaled y ticks = false,
+ ylabel style = {align=center},
+ legend pos = {north west},
+ xlabel = {East [m]},
+ ylabel = {North [m]},
+ xmax = -200,
+ xmin = -300,
+ ymax = 400,
+ ymin = 200,
+]
+\pgfplotsset{filter discard warning=false}
+\pgfkeys{/pgf/number format/.cd,1000 sep={}}
+\addplot[thick, blue, solid]
+ table[x index = {0}, y index = {1}, col sep=comma]
+ {data/exp_cartrace.csv};
+\addlegendentry{GPS trace}
+\addplot[thick, red, dotted]
+ table[x index = {0}, y index = {1}, col sep=comma, each nth point=10]
+ {data/exp_cartracePsat.csv};
+\addlegendentry{Sensor Fusion $th = 0.1$}
+\addplot[thick, black!70!green, dashed]
+ table[x index = {0}, y index = {1}, col sep=comma, each nth point=10]
+ {data/exp_cartraceP41.csv};
+\addlegendentry{Sensor Fusion $th = 4.1$}
+\coordinate (spypoint) at (axis cs:-240,250);
+\coordinate (magnifyglass) at (axis cs:-200,310);
+\spy [blue] on (spypoint) in node[fill=white] at (magnifyglass);
+\end{axis}
+\end{tikzpicture}
+\caption{Trace tracked when driving. GPS measurements and sensor fusion algorithm with different thresholds.}
 \label{fig:car-trace}
-}
-\end{center}
-\end{figure} 
-
-Figures~\ref{fig:car-trace} and~\ref{fig:cycling-trace} show part of the traces generates respectively for the tracking of the car and the bike. In general the cycling trace exposes more complex dynamics that are tracked with more difficulty by the sensor fusion algorithm. Specifically we can see how the lower sampling of the GPS can still guarantee some form of tracking, whether this is acceptable or not it will be dependant on the specific application\footnote{This is also a basic implementation of a sensor fusion algorithm, better performances in terms of tracking could be achieved with more advanced versions of it and better quality IMU sensors. In any case this is not part of the discussion here.}.
-%\footnote{The reader might argue that the sensor fusion algorithm for the cycling data is even less performing than the pure GPS track. This is due to the fact that the algorithm here implemented is quite simple and for instance doesn't bound in any way the possible movements of the bike. We chosed instead this implementation since it provides more expressive figures, the same considerations would hold.}
-From the same simulations we show also plots of the signal used for the triggering of the GPS (i.e. the sum of the trace of $P$) toghether with the state of the GPS antenna. Those are shown in figure~\ref{fig:cycling-control} for the cycling trace and~\ref{fig:car-control} for the car trace. Those figures show how the sampling is able to keep bounded the variance of the estimation while reducing the time that the antenna spends being turned on.
+\end{minipage}
+\end{figure*}
+\begin{figure*}
+\hspace{2mm}
+\begin{minipage}{0.95\columnwidth}
+\begin{tikzpicture}
+\begin{axis}[%
+ height = 0.6\textwidth,
+ grid style = {black!30, dashed},
+ grid = major,
+ width = 0.95\textwidth,
+ scaled x ticks = false,
+ scaled y ticks = false,
+ ylabel style = {align=center},
+ legend style={at={(1.3,1.1)},anchor=south},
+ legend columns=2,
+ xlabel = {Time [s]},
+ ylabel = {Sum of Trace \\ Turn On Signal},
+ xmin=500,xmax=3500,
+]
+\pgfplotsset{filter discard warning=false}
+\pgfkeys{/pgf/number format/.cd,1000 sep={}}
+\addplot[thick, red, solid]
+ table[x index = {0}, y index = {1}, col sep=comma]
+ {data/exp_biketraceCtl.csv};
+\addlegendentry{Sensor Fusion $th=0.1$, Sampling On}
+\addplot[thick, black!70!green, solid]
+ table[x index = {0}, y index = {3}, col sep=comma]
+ {data/exp_biketraceCtl.csv};
+\addlegendentry{Sensor Fusion $th=4.1$, Sampling On}
+\addplot[thick, red, dashed]
+ table[x index = {0}, y index = {2}, col sep=comma]
+ {data/exp_biketraceCtl.csv};
+\addlegendentry{Sensor Fusion $th = 0.1$, $\sum Tr(P)$}
+\addplot[thick, black!70!green, dashed]
+ table[x index = {0}, y index = {4}, col sep=comma]
+ {data/exp_biketraceCtl.csv};
+\addlegendentry{Sensor Fusion $th = 4.1$, $\sum Tr(P)$}
+\end{axis}
+\end{tikzpicture}
+\caption{Control Signal and Sampling when cycling.}
+\label{fig:bike-trace-ctl}
+\end{minipage}
+\hspace{1mm}
+\begin{minipage}{0.95\columnwidth}
+  \vspace{1.1cm}
+\begin{tikzpicture}
+\begin{axis}[%
+ height = 0.6\textwidth,
+ grid style = {black!30, dashed},
+ grid = major,
+ width = 0.95\textwidth,
+ scaled x ticks = false,
+ scaled y ticks = false,
+ ylabel style = {align=center},
+ xlabel = {Time [s]},
+ ylabel = {Sum of Trace \\ Turn On Signal},
+ xmin=500,xmax=3500,
+]
+\pgfplotsset{filter discard warning=false}
+\pgfkeys{/pgf/number format/.cd,1000 sep={}}
+\addplot[thick, black!70!green, solid]
+ table[x index = {0}, y index = {1}, col sep=comma]
+ {data/exp_cartraceCtl.csv};
+\addplot[thick, black!70!green, dashed]
+ table[x index = {0}, y index = {2}, col sep=comma]
+ {data/exp_cartraceCtl.csv};
+\addplot[thick, red, solid]
+ table[x index = {0}, y index = {3}, col sep=comma]
+ {data/exp_cartraceCtl.csv};
+\addplot[thick, red, dashed]
+ table[x index = {0}, y index = {4}, col sep=comma]
+ {data/exp_cartraceCtl.csv};
+\end{axis}
+\end{tikzpicture}
+\caption{Control Signal and Sampling when driving.}
+\label{fig:car-trace-ctl}
+\end{minipage}
+\end{figure*}
 
-\begin{figure}[t]
- \begin{center}
-  \includegraphics[height=0.70\columnwidth, width=0.90\columnwidth]{images/cycling_control.png}
-  \caption{Control signal and control action for the triggering of the GPS sensor in the cycling trace.
-           \label{fig:cycling-control}
-          }
- \end{center}
-\end{figure} 
+Here we use real traces, recorded from a GPS receiver and an IMU
+sensor that includes accelerometer and gyroscope. We recorded data in
+two different conditions: a car and a bicycle ride. We recorded traces
+with continuous GPS sampling and simulated different GPS sensor
+dynamics on top of that, to compare different sampling policies. We
+show what the tracking would have been when the sensor fusion
+algorithm was live, compared to the continuous sampling of the GPS. We
+then use simulations to further analyze the trade-off between power
+(and therefore battery) consumption and performance (positioning
+accuracy).
 
-\begin{figure}[t]
- \begin{center}
-  \includegraphics[height=0.70\columnwidth, width=0.90\columnwidth]{images/car_control.png}
-  \caption{Control signal and control action for the triggering of the GPS sensor in the car trace.
-           \label{fig:car-control}
-          }
- \end{center}
-\end{figure} 
+Figures~\ref{fig:cycling-trace} and~\ref{fig:car-trace} respectively
+shows traces for the tracking of the bike and the car. In each
+figure, the GPS trace is represented using solid blue lines, while
+two different executions of the sensor fusion algorithm (with
+different values of the threshold $th$) are shown in red dotted lines
+and green dashed lines. The red dotted lines correspond to the use of
+a threshold $th=0.1$, which in turn means leaving the GPS receiver
+always on, but using the sensor fusion algorithm to incorporate also
+the IMU data. In general, the cycling trace exposes more complex
+dynamics, that are harder to track for the sensor fusion algorithm.
+Low frequency GPS sampling can still guarantee some form of tracking,
+depending on the precision needed for the given application. In the
+car trace, the movements of the GPS receiver are reduced and the IMU
+sensors are able to much better allow for low frequency sampling of
+the GPS signal. Notice that our implementation of the sensor fusion
+algorithm is very basic and more advanced versions could improve the
+tracking performance also in the biking case. For the same
+simulations, we also show in Figures~\ref{fig:bike-trace-ctl}
+and~\ref{fig:car-trace-ctl} the signal used for the GPS triggering
+(i.e., the sum of the trace of $P$) and the state of the GPS antenna.
+The figures show how the sampling strategy is able to keep the
+variance of the estimation bounded, while reducing the on time.
 
 Finally we run a large number \todo{specify} of simulations with different triggering thresholds of $P$ \todo{specify}. The idea is that higher values of P  will guarantee less power consumption at the price of lareger errors in the estimation and vice versa. In the simulations the acquisition time of the satellites signals is modeled as a random variable uniformly distributed. Moreover, while in the simulations with the cycling data the number of visible satellites is constant, in the ones with the car data it randomly changes\footnote{Specifically when the number of visible satellites is a given number there is some probability at every time step that t increases or decreases by one, always bounded betwen 3 and 6 in any case.}. The overall error of a tracking trace is here defined as the root-mean-square of the distance between the trace and the pure GPS signal.
 
@@ -313,23 +427,3 @@ Finally figure~\ref{fig:car-trade-off} shows the simulations performed using the
           }
  \end{center}
 \end{figure}
-
-
-%Power-accuracy trade-off: Montecarlo simulations. Characteristics:
-%\begin{itemize}
-%\item We generate 10000 traces, 60 minutes long.
-%\item For each point in each trace, we randomly extract from
-%  probability distributions the visibility of satellites. We also
-%  randomize the time to fetch signals.
-%\item Figure comparing clouds of points with only GPS and GPS+IMU in
-%  the axis \emph{accuracy} (sum of distances from the ideal GPS trace)
-%  and \emph{power consumption} (due to antenna).
-%\end{itemize}
-
-%To show how the proposed model joined with the sensor fusion algorithm allows to capture the trade-off betwen accuracy and power consumption we run simulations for different threshold values for triggering the sampling of the GPS. Coherently with the intuition, we can see that higher values for the threshold allow to save power at the cost of lower precision in the positioning. The accuracy is measured as the distance from the positioning that uses the GPS continuously.
-
-%In these simulations the GPS model exposes a random acquisition time for the fetching of the ephemeris data, drawed for a uniform distribution between \textcolor{red}{2 and 12 milliseconds}. Also the number of satellites is randomized as in the previous simulations.
-
-
-
-