\item what are the limits that the sensor dynamics poses. (i)initial delay, (ii) delay when position is requested unless always on, (iii) periodical drain of battery for updating ephemeris data, (iv) you can se you have lost visibility only if you are turned on ,(v) the best you can do is turn off the antenna (for this reaosn as soon as we get the position we assume you want to turn off the antenna)
\item what are the phenomena we have to account for in regular ``working'' conditions. cycling on two different periods. plot of how this looks like?
\item what are the possible disturbances we have to deal with. expiration of ephemeris data and loss of visibility. examples?
\item how do we do this? state machine of the controller`/ref{fig:controller}. three kind of transitions: some are observed events, some are control actions, some are both
\end{itemize}
%\begin{itemize}
%\item Title: Analysis and sampling strategy
%\item what are the limits that the sensor dynamics poses. (i)initial delay, (ii) delay when position is requested unless always on, (iii) periodical drain of battery for updating ephemeris data, (iv) you can se you have lost visibility only if you are turned on ,(v) the best you can do is turn off the antenna (for this reaosn as soon as we get the position we assume you want to turn off the antenna)
%\item what are the phenomena we have to account for in regular ``working'' conditions. cycling on two different periods. plot of how this looks like?
%\item what are the possible disturbances we have to deal with. expiration of ephemeris data and loss of visibility. the latter is not observable when the antenna is turned off. examples?
%\item how do we do this? state machine of the controller`/ref{fig:controller}. three kind of transitions: some are observed events, some are control actions, some are both
%\end{itemize}
\textcolor{red}{Title: Analysis and sampling strategy}
\begin{figure*}[t]
Given the sensor fusion algorithm and the sensor model we can now describe which dynamical limitations the sensor imposes and therefore how it can be sampled. In this section we will discuss the general features that characterize an effective sampling strategy. Complementary in the next section a simulation evaluation of the available trade-offs will be performed.
\subsection{The dynamics}
The model discussed in section~\ref{sec:gps} points out two dynamics that characterize the sensor: the availability of the ephemeris data and the availability of ranging data. The two are caracterized by very different time scales, both in terms of acquisition time and validity.
The ephemeris data live in the time scale of minutes requiring betwen 30 to 59 seconds to be aquired and being valid for 30 minutes. This poses two main constraints. First at the startup a delay equivalent to the acquisition time of the ephemeris data will be present -- this is the so called \emph{Time To First Fix} TTFF. Second during regular working conditions every 30 minutes the ephemeris data must be updated, requiring the antenna to be turned on for enough time and affecting the battery drain of the sensor.
The ranging data are instead caracterized by a time scale of the order of milliseconds. They require from 2 to 10 milliseconds to be acquired and they have istantaneous validitiy since they are the ones used to compute the present position. This means that there is a lower bound the sampling period of the sensor under which the sampling will be equivalent to keep the sensor always turned on. Also for some applications with real time constraints the varying delay for the acquisition could be critical.
Another important consequence of the sampling policy is the observability of the event \texttt{lost\_visibility}. This is in fact observable only when the antenna is on and is listening to the visible satellites. When a satellite disappears, if the antenna is turned off the device wont observe this and at the next sampling it will have to acquire new ephemeris data before being available of providing new positioning (assuming that enough satellites are visible).
\subsection{Sampling Strategy}
Given these considerations we designed a simple sampling stategy that tries to keep the ephemeris data updated and samples the GPS sensor according to the uncertainty of the state estimation of the Kalman filter. To do this we use the trace of the covariance matrix $P$ which represents the estimation variance of the position. When this quantity overcomes a defined threshold the position is requested to the sensor. This is formally encoded in the state machine represented in figure~\ref{fig:controller}.
\textbf{The controller}: The logical controller sends a \texttt{turn\_on} signal at the start up in order to collect the ephemeris data (state 2). Then once those are available it starts cycling between states 3 and 4 through alternatively triggering the \texttt{turn\_off}, \texttt{turn\_on} signals. For readability, in the same way as we did in the sensor model, the states in which the antenna is turned on are filled with green.
When the ephemeris data are about to expire or the sensor loses visibility of the satellites it was tracking, the controller goes back to state 2 and keeps the antenna on in order to update the ephemeris data. In the case that the ephemeris data are only about to expire the sensor can actually still be sampled, represented by taking the transition \texttt{sensor in position\_avaialable}. The previous ephemeris data are in fact still valid and the ranging data come along with the fetching of the satellites signals.
