diff --git a/paper/main.tex b/paper/main.tex
index 8f403f76896bb5d38c22dce0d2b49a6c71c2c6f7..1e37ab9346ade7cbeeac8ffe4ec5b1909dfc0f51 100644
--- a/paper/main.tex
+++ b/paper/main.tex
@@ -39,6 +39,9 @@
 \usepackage{amsfonts}
 \usepackage{listings}
 
+%amsmath
+\DeclareMathOperator{\Tr}{Tr}
+
 % TiKz stuff
 \usepackage{tikz,xstring,siunitx,pgfplots,psfrag}
 \usepackage{pgfplotstable}
diff --git a/paper/sections/05-control.tex b/paper/sections/05-control.tex
index d8a85f904d28a0473dfaef8fa82fbdbbb07e9785..ea53d85b2184022f7a2223d21c94e3d22f073ad6 100644
--- a/paper/sections/05-control.tex
+++ b/paper/sections/05-control.tex
@@ -44,7 +44,7 @@ aims at keeping the ephemeris data updated, and sampling the GPS
 sensor according to the uncertainty of the position estimation (in the
 Kalman filter). We use the trace of the covariance matrix $P$, which
 represents the estimation variance of the position. When the trace
-overcomes a predefined threshold, we consider the position too
+overcomes a predefined threshold $th$, we consider the position too
 uncertain and request the GPS intervention. This is formally encoded
 in the state machine shown in Figure~\ref{fig:controller}.
 
@@ -76,7 +76,7 @@ font=\footnotesize]
 \draw [arr, bend left]  (gp)  to node [near start, left, yshift=2mm, align=right] {\texttt{sensor in position\_available} \\ \textcolor{red}{\texttt{turn\_off}}} (wst);
 \draw [arr, bend right] (gp)  to node [below, align=center, yshift=-2mm] {\texttt{lost\_visibility} \\ \textbf{\texttt{or}} \texttt{ephemeris\_about\_to\_expire}} (re);
 %arrows from 4
-\draw [arr, bend left]  (wst) to node [right, align=left, xshift=2mm, yshift=-3mm, at start] {\texttt{estimation} \\ \texttt{triggering} \\ \textcolor{red}{\texttt{turn\_on}}} (gp); 
+\draw [arr, bend left]  (wst) to node [right, align=left, xshift=2mm, yshift=-3mm, at start] {\texttt{$\sum(\Tr(P)>th)$} \\ \texttt{\textcolor{red}{\texttt{turn\_on}}}} (gp); 
 \draw [arr, bend right] (wst) to node [left, at start, xshift=-2mm, yshift=3mm, align=right] {\texttt{ephemeris\_about\_to\_expire} \\ \textcolor{red}{\texttt{turn\_on}}} (re);
 \end{tikzpicture}
 \caption{State Machine of the Sampling Strategy Controller.}
@@ -85,21 +85,6 @@ font=\footnotesize]
 %-------------------------------------------
 
 
-The logical controller sends a \textcolor{red}{\texttt{turn\_on}}
-signal when the system is starting, to collect the ephemeris data
-(State \textcircled{\scriptsize 2} in
-Figure~\ref{fig:controller}). Then, once the ephemeris data are
-available, it starts cycling between states \textcircled{\scriptsize
-  3} and \textcircled{\scriptsize 4}, alternatively triggering the
-\textcolor{red}{\texttt{turn\_off}} and
-\textcolor{red}{\texttt{turn\_on}} signals. For readability, and
-consistently with the sensor model shown in
-Figure~\ref{fig:cyberDynamics}, the states in which the antenna is
-turned on are filled in green.
+The logical controller sends a \textcolor{red}{\texttt{turn\_on}} signal when the system is starting, to collect the ephemeris data (State \textcircled{\scriptsize 2} in Figure~\ref{fig:controller}). Then, once the ephemeris data are available (which is defined by the very same transition of the sensor model), it starts cycling between states \textcircled{\scriptsize 3} and \textcircled{\scriptsize 4}, alternatively triggering the \textcolor{red}{\texttt{turn\_off}} and \textcolor{red}{\texttt{turn\_on}} signals. For readability, and consistently with the sensor model shown in Figure~\ref{fig:cyberDynamics}, the states in which the antenna is turned on are filled in green.
 
-When the ephemeris data are about to expire, or the sensor loses
-visibility of the tracked satellites, the controller goes back to
-State \textcircled{\scriptsize 2} and keeps the antenna on, to refresh
-the ephemeris data. If the ephemeris data are valid (and about to
-expire) the sensor can actually still be sampled, represented by
-taking the transition \texttt{sensor in position\_avaialable}.
+When the ephemeris data are about to expire (intuitively defined as $time>expiry\_time\_ephemeris-60$), or the sensor loses visibility of the tracked satellites, the controller goes back to State \textcircled{\scriptsize 2} and keeps the antenna on, to refresh the ephemeris data. If the ephemeris data are valid (and about to expire) the sensor can actually still be sampled, represented by taking the transition \texttt{sensor in position\_avaialable}.
diff --git a/paper/sections/06-results.tex b/paper/sections/06-results.tex
index fe94fdafe416a8b7b5477a5cc3644f13e9d5956e..3ed9fd495ba5c2808bf47d4763f2d9679c3fa331 100644
--- a/paper/sections/06-results.tex
+++ b/paper/sections/06-results.tex
@@ -13,11 +13,11 @@
 %you want to distinguish, you can have a finite state machine for each
 %satellite.
 
-In this section we will discuss two implementations of the model specified above: one in the modeling language Modelica\footnote{www.modelica.org} and a second one in the Matlab language. The purpose of the first one is to show how the model captures the relevant dynamics of a GPS sensor and the object-oriented nature of the Modelica language makes it ready to use for other applications. The second implementation is instead used to present how the model can be combined with the sensor fusion algorithm discussed in section~\ref{sec:fusion} to evaluate the possible accuracy-over-battery-consumption trade-offs. The two implementations also correspond to the two parts in which this section is organized.
+In this section we will discuss two implementations of the model specified above: one in the modeling language Modelica\footnote{www.modelica.org} and a second one in the Matlab language. The purpose of the first one is to show how the model captures the relevant dynamics of a GPS sensor and the object-oriented nature of the Modelica language makes the it ready to use for other applications. The second is instead used to present how the model can be combined with the sensor fusion algorithm discussed in section~\ref{sec:fusion} to evaluate the possible accuracy-over-battery-consumption trade-offs. This section is organized in two parts, each of them presenting simulations of each of the models.
 
 \subsection{GPS sensor dynamics}
 \label{sec:res:gps}
-The phenomena we will show in this section are: the TTFF, the loss of ephemeris data and the loss of visibility.
+The phenomena we want to show in this section are: the TTFF, the loss of ephemeris data and the loss of visibility.
 
 Figure~\ref{fig:control1} shows the command signal used to show the time that passes at the start up beofore the position bcomes available. As we can see, first the sensor is kept turned on for one minute and then it is sampled at regular intervals. The resulting position availability of the sensor is then shown in figure~\ref{fig:position1}. 
 
@@ -39,11 +39,9 @@ Figure~\ref{fig:control1} shows the command signal used to show the time that pa
  \end{center}
 \end{figure}
 
-Some previous works cited in section~\ref{sec:related} discuss the TTFF but none of them does it by looking in detail at the technology of the GPS~\cite{bib:entracked-datadriven-modeling}~\cite{bib:feasibility-duty-cycling}~\cite{bib:accuracy-adaptation}. The performances are evaluated for smartphones looking at how much time it takes for an application to get a position measure after the API request \textcolor{red}{??}. An important remark is that GPS sensors in smartphones implement the so called Assisted-GPS that allows the retrieval of the ephemeris data from the internet instead of listening to the satellites. The model presented in this paper can be adapted to reprent this allowing for an external input that triggers the transition \texttt{get\_ephemeris}, possibly before the delay that represent the action of listening instead to the satellites.
+Some previous works cited in section~\ref{sec:related} discuss the TTFF but none of them does it by looking in detail at the technology of the GPS~\cite{bib:entracked-datadriven-modeling}~\cite{bib:feasibility-duty-cycling}~\cite{bib:accuracy-adaptation}. The performances are evaluated for smartphones looking at how much time it takes for an application to get a position measure after the API request \textcolor{red}{(can we say API?)}. An important remark is that GPS sensors in smartphones implement the so called Assisted-GPS 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 triggers the transition \texttt{get\_ephemeris} before the delay that instead represents the action of listening to the satellites. Anyhow the modeling of how much time the device requires for fetching these informations from the internet and how much power this procedure can take is non trivial because of the many different (and often difficult to predict) involved components.}.
 
-Up to the limited extent to which they are comparable -- due to the fact that are used Assisted-GPS sensors and the extra software layers included in the experiments -- the results are coherent with what discussed here. Our model on the other side achieves more generality not being dependent on the specific implementation on the given device. Moreover it allows to look directly at what are the theoretical performances we should expect from a GPS sensor without the overhead that is introduced by the operative system of a smartphone.
-
----------------------------------------------------
+Up to the limited extent to which they are comparable -- due to the fact that are used Assisted-GPS sensors and the extra software layers included in the experiments -- the results presented in these works are coherent with what discussed here. Our model on the other side achieves more generality not being dependent on the specific implementation on the given device. Moreover it allows to look directly at what are the theoretical performances we should expect from a GPS sensor without the overhead that is introduced by the operative system of a smartphone.
 
 Next we want to show how our model captures the expiration of the ephemeris data and the loss of visibility or the satellites. Figure~\ref{fig:control2} and~\ref{fig:position2} show a duty cycling of the sensor for quickly acquiring position -- i.e. a series of warm starts. At time $t = 1861 sec$ the ephemeris data expire making the duty cycling ineffective. Only after a prolonged period of time of the sensor being turned on, during which the device updates the ephemeris data, the position becomes available again as well as the possibility of performing warm starts.
 
@@ -65,7 +63,7 @@ Next we want to show how our model captures the expiration of the ephemeris data
  \end{center}
 \end{figure}
 
-Last simulation of this section will show how the model captures the loss of visibility of the satellites. This is done with the variable \texttt{visible\_satellites} which as defined in section~\ref{sec:gps} is an input of the system. The number of visible  satellites (shown in figure`\ref{fig:control3}) is initially 5 for 100 seconds, allowing for the acquisition of the ephemeris data. At time $t = 100 sec$ two satellites disappear from the visible sky. Since three is not a sufficient number for the device to position itself the position stops being available, this is shon in figure~\ref{fig:position3}. Then at time $t = 200 sec$ one new satellite appears in the visible sky and after the device acquires its ephemeris data the position becomes visible again.
+Last simulation of this part of the section shows how the model captures the loss of visibility of the satellites. This is done with the variable \texttt{visible\_satellites} which as defined in section~\ref{sec:gps} is an input of the system. The number of visible  satellites (shown in figure`\ref{fig:control3}) is initially 5 for 100 seconds, allowing for the acquisition of the ephemeris data. At time $t = 100 sec$ two satellites disappear from the visible sky. Since three is not a sufficient number for the device to position itself the position stops being available, this is shown in figure~\ref{fig:position3}. Then at time $t = 200 sec$ one new satellite appears in the visible sky and after the device acquires its ephemeris data the position becomes visible again.
 
 \begin{figure}[h]
  \begin{center}
@@ -85,7 +83,7 @@ Last simulation of this section will show how the model captures the loss of vis
  \end{center}
 \end{figure} 
 
-This last simulation points out also one limitation of the proposed model. In fact theoretically if the satellite that appears again after the loss of visibility was one of the two that previously disappeared the device would not need to re-acquire the ephemeris data, having still in memory the ones acquired at the start up and being those still valid. But to capture this we would have to capture separately the acquisition of the signal and ephemeris data of the different satellites, together with their visibiliy. This apparently would increase the complexity of the model and decrease it usability\footnote{An extension of the model to include also this phenomenon would't be very difficult. It would be sufficient to have parallel state machines similar to the one shown in figure~\ref{fig:cyberdynamics} that independently capture the tracking of the different satellites. Some more adjustments would of course be needede but the basic ideas would still hold.}.
+This last simulation points out also one limitation of the proposed model. In fact theoretically if the satellite that appears again after the loss of visibility was one of the two that previously disappeared the device would not need to re-acquire the ephemeris data, having still in memory the ones acquired at the start up and being those still valid. But to capture this we would have to describe separately the acquisition of the signal and ephemeris data of the different satellites, together with their visibiliy. This apparently would increase the complexity of the model and decrease it usability\footnote{An extension of the model to include also this phenomenon would't be very difficult. It would be sufficient to have parallel state machines similar to the one shown in figure~\ref{fig:cyberdynamics} that independently capture the tracking of the different satellites. Some more adjustments would of course be needede but the basic idea would still hold.}.
 ------------------------------------------------
 
 \subsection{Power Consumption Accuracy Trade Off}