Merge branch 'master' of https://git.frank-ebner.de/FHWS/FtmPrologic

tex/chapters/4_ftmloc.tex

@@ -12,6 +12,7 @@ After measuring several distances to different anchor points one can calculate h
 %TODO Alles mit 2D, weil halt
 
 \subsection{Multilateration}
+\label{sec:multilateration}
 %Ganz kurz erläutern was Multilateration eigentlich ist. in 2D min 3 aps und in 3D min. 4D. Aber grundsätzlich gilt: viel hilft viel.
 %Es ist uns klar, dass Trilat nichts taugt aber ist halt der einfachste Schritt
 %Typische Nachteile: Wenn Schnittpunkt nicht analytisch exakt bestimmt werden können
@@ -84,18 +85,37 @@ However, GDOP still gives a good first impression of the theoretical suitability
 %Vorteil: Nicht ideale Schnittpunkte sind kein Problem, weil die Dichte sowas abbilden kann
 %FTM Vorteil: Fehlen von Messungen kann probabilistisch erfasst werden indem Streuung der NV größer wird / Oder es enstehen einfach mehrere Hypthesen über die Position
 Probabilistic frameworks allow to natively incorporate faulty or inaccurate measurements into the position model.
-While multilateration produces a single point as position estimate the probabilistic framework allows to describe the estimated position as probability density function.
-This approach has the advantage to model inaccurate measurments as variances and quantify them.
+While multilateration produces a single point as position estimate a probabilistic framework allows to describe the estimated position in terms of probability density functions.
+This approach has the advantage to model inaccurate measurements as variances and quantify them.
 
 Assuming that FTM measurements follow a Gaussian distribution, the measured distance $d_i$ between the sender and AP $i$ is described by $\mathcal{N}(d^*_i, \sigma^2_i)$ where $d^*_i=\|\mPosVec_i - \mPosVec \|$ is the actual distance between the AP position $\mPosVec_i$ and a given position $\mPosVec$. $\sigma^2_i$ is the variance of the measurement.
-Visualized on a two dimensional plane this produces a ring shaped density function where the cross section of the ring is the well known Gaussian curve.
-
-Multiple distance measurements can be combined by a joint density function, if the individual distance measurements to the APs are statistical independent.
-The probability to observe a distance measurement $d$ at position $\mPosVec$ is given as
+Hence, the probability to observe a distance $d_i$ measured to the AP $i$ at position $\mPosVec$ is given by the density function:
 \begin{equation}
-  p(d | \mPosVec ) = \prod_i \frac{1}{\sqrt{2\pi\sigma^2_i}} \exp \left( - \frac{(d - d^*_i)^2}{2\sigma^2_i} \right)
+\label{eq:distDensity}
+  p(d_i | \mPosVec ) = \frac{1}{\sqrt{2\pi\sigma^2_i}} \exp \left( - \frac{(d_i - d^*_i)^2}{2\sigma^2_i} \right)
 \end{equation}
 
+Visualized on a two dimensional plane this produces a ring shaped density function where the cross section of the ring is the well known Gaussian curve.
+Multiple distance measurements can be combined by a joint density function, if the individual distance measurements to the individual APs are statistical independent.
+The areas where the individual density functions overlap have higher likelihood to observe the given distance.
+Thus, the geometrical considerations of \autoref{sec:multilateration} still apply.
+
+% + Dynamic:
+In order to estimate the position of a moving pedestrian at each discrete timestep $\Delta t$ the most likely position given the observed distances is computed.
+Note that multiple distance measurements can be available if the measurement rate is higher than the update rate $\Delta t$.
+It is also possible that no measurements are available because the sight to a particular AP is blocked.
+Hence, the distance measurements for a given AP $i$ are given with the vector $\vec{d}_i$ of length equal to the number of successful measurements.
+
+The probability to observe the given distances $\vec{d}_{1:i} =(\vec{d}_1, \dots, \vec{d}_i)$ to each AP at the position $\mPosVec$ is given by the joint probability density function
+\begin{equation}
+  p(\vec{d}_{1:i} | \mPosVec ) = \prod_i p(\vec{d}_i | \mPosVec ) = \prod_i\prod_{m=1}^{M_i} p(d_{i,m} | \mPosVec )
+\end{equation}
+where $M_i$ is the number of successful measurements to AP $i$.
+For each AP its measurements form a joint distribution by itself.
+This allows to integrate several measurements per update time step.
+An alternative approach could be to compute the mean measurement over all $M_i$ distances in $\vec{d}_i$ and use the density in \eqref{eq:distDensity} directly.
+
+
 \subsection{Particle Filter}
 
 %Particle Filter Introduction