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-%\begin{itemize}
-%\item Introduci come case study. cite the applications from the introduction (drones, small trackers for sport or animals..) and explain the commonalities (no interenet connection, lightweight and small battery, availability of other sensors..). To maximize battery duration different information sources with different effect on the battery and different quality can be merged. The most common one is perhaps GPS+IMU.
-%\item Spiega cosa vuol dire sensor fusion. Using IMU in short time scales and GPS in longer ones. So we do GPS sampling, but we want to guarantee availability of GPS and do it power efficiently.
-%\item Spiega con quali sensori fondi (anche caratteristiche tipo deve consumare meno batteria, altrimenti grazie al cavolo) IMU, which measures do we get from it, how much power it consumes. cite the book~\cite{bib:gps-imu}.
-%\item Spiega come lo fai (equazioni). rifrasa dal tast del corso AS1
-%\end{itemize}
-
-
-It is now introduced one possible implementation and use of the presented model. This will be used to discuss some of the different analysis that can be done thanks to this abstarction. The chosen case study is an implementation of \emph{GPS-IMU sensor fusion}. Sensor fusion refers to a set of techniques for merging measurements that come from different sensors with the aim of enhancing the advantages of the specific sensors while compensating for the disadvantages.
-
-Inertial measurement units (IMUs) are sensors that usually include an accelerometer and a gyroscope. The former measures the accelerations along the three axis while the latter measures the angular velocities of the orientation changes. This kind of sensors can be considered in different ways complementary to the GPS. First they provide a relative measurement (while the position estimaiton of the GPS is absolute). Secondly they are characterized by low power consumption (\todo{quantify and reference this}), since they dont have to rely on any external device providing a reference no antenna for communication must be powered. Third, for the same reason, they are continuously available, not having to rely on the communication with any external device. The usage of only inertial sensor for navigation is called \emph{Inertial Navigation System} (INS).
-
-The main weakness of INS is that, given the differential nature of the sensors, their measure must be integrated. This exposes the estimation to significant low frequency errors introduced by the biases. By rule of thumb, those are usually considered to be growing cubically in time. For this reason inertial sensor are considered only reliable on a short time scale. The idea of GPS-IMU sensor fusion is to use the inertial sensor unit to provide continuous and low-power positioning, while the GPS, not being affected by biases can be sampled at lower rate to compensate for the low frequency erors.
-
-In this framework, the presented model for the GPS sensor can capture the delays in the position measure of the GPS and its power consumption. This allows rigorous study on what is the optimal (in terms of trade-off betwen power consumption and accuracy) sampling method for the GPS sensor.
+The proposed model is general and can be used for many different
+purposes.  The abstraction offered by the model devised in this paper
+allows for analysis and synthesis of control strategies. This section
+introduces its use in the context of sensor fusion. Sensor
+fusion~\todo{ref} refers to a set of techniques for merging
+measurements that come from different sensors with the aim of
+enhancing the advantages of the specific sensors while compensating
+for the disadvantages. For our specific case, we would like to limit
+battery consumption without penalizing accuracy, using Inertial
+Measurement Units (IMUs) in conjunction with the GPS receiver.
+
+IMUs are sensors that usually include an accelerometer and a
+gyroscope. The accelerometer measures the acceleration along the three
+axis while the gyroscope measures the angular velocity of the
+orientation changes. IMUs can be considered complementary to the
+GPS. In fact, they provide a relative measurement (while the GPS
+position is absolute), and are characterized by low power
+consumption~\todo{ref}, since they do not consume the extra power
+needed to use an antenna. Their measurements are also continuously
+available, since they do not rely on other devices. A navigation
+system only based on IMUs is referred to as an Inertial Navigation
+System (INS).
+
+The main weakness of an INS is that, given the differential nature of
+the sensors, their measurements must be integrated. This exposes the
+estimation to significant low frequency errors introduced by the
+biases. As a rule of thumb, those errors are usually considered grow
+cubically in time. IMUs are therefore considered reliable only for
+short time intervals. The idea of \emph{GPS-IMU} sensor fusion
+algorithm is to exploit the IMUs to provide continuous low-power
+positioning. At the same time, the GPS can be sampled at lower rate to
+compensate for the low frequency erors.
+
+The GPS model devised in this paper captures the delays introduced by
+the GPS receiver and its power consumption. This allows us to
+rigorously study an optimal sampling method for the GPS sensor, with
+respect to a given accuracy and power consumption target.
 
 \subsection{The Sensor Fusion Algorithm}
 In this section the actual implementation of the sensor fusion algorithm is discussed. The proposed algorithm integrates the inertial measurements in order to obtain the trajectory profile.When GPS positioning is triggered an \emph{extended Kalman filter} is implemented to estimate and compensate the IMU biases.