Tex changes

tex/chapters/4_ftmloc.tex

@@ -1,7 +1,7 @@
 \section{Position Estimation}
 \label{sec:position}
 
-Bei Indoor Lokalisierung geht es darum eine Position zu ermitteln. Hierfür nutzen wir unterschiedliche Verfahren. namley... usw. 
+%Bei Indoor Lokalisierung geht es darum eine Position zu ermitteln. Hierfür nutzen wir unterschiedliche Verfahren. namley... usw. 
 
 After measuring several distances to different anchor points one can calculate his current position.
 %TODO Alles mit 2D, weil halt
@@ -12,26 +12,43 @@ After measuring several distances to different anchor points one can calculate h
 %Typische Nachteile: Wenn Schnittpunkt nicht analytisch exakt bestimmt werden können
 %FTM Nachteil: Häufig fallen die Messungen aus? Was tun? Alte Werte statisch halten? Keine est berechnen?
 %Conceptually, multilateration determines the position by analytically intersecting at least $3$ circles for a 2-dimensional position, or at least $4$ spheres in case of a 3-dimensional coordinate system.
-
-Each distance measurement $d_i$ constrains the position estimate $\hat{\mPosVec}$ to a circle, where the center of the circle is the known position $\mPosVec_i = (x,y)^T$ of AP $i$.
+In two dimensions each distance measurement $d_i$ constrains the position estimate $\hat{\mPosVec}$ to a circle with radius equals $d_i$, where the center of the circle is the known position $\mPosVec_i = (x,y)^T$ of AP $i$.
 Formally the distance is the euclidean distance between the known position and the estimate
 \begin{equation}
 d_i = \| \mPosVec_i - \hat{\mPosVec} \|   \text{.}
 \end{equation}
-In two dimensions three ideal distances form a system of linear equations which can be uniquely solved to obtain the position.
-Given more than three distances no solution can be found which stratifies all the constraints as the linear system is overdetermined.
+Three ideal distances form a system of linear equations which can be uniquely solved to obtain the position.
+Given more than three distances no solution can be found which satisfies all the constraints as the linear system is overdetermined.
 Additionally, in the presence of noise and inaccurate measurements an exact analytical solution is not possible.
 In this case an approximative solution $\mPosVec^*$ can be found by using a least squares approach which minimizes the quadratic error between the measured distance and the actual distance at a given point
 \begin{equation}
   \mPosVec^* = \argmin_{\hat{\mPosVec}} \sum_{i}^{} \left(  \| \mPosVec_i - \hat{\mPosVec} \| - d_i \right)^2  \text{.}
+\label{eq:leastSquare}
 \end{equation}
 
+This forms a classical non linear least squared optimization problem which can be solved with a numerical optimization method like \docGaussNetwon or \docLevenbergMarq.
+The position $\hat{\mPosVec}$ which minimizes the error provides an approximative estimate $\mPosVec^*$.
+In contrast to an analytical solution this results not in an ideal point intersection but an area where the error of \eqref{eq:leastSquare} is acceptable small.
 
-% each distance defines the position on a circle
-% intersecting the circles gives the position
-% at least 3 circles
-% only works for perfect measurments
+A good starting value for the optimization can be obtained by computing the pseudo inverse of the matrix. % TODO formeln
+Note that the linearization of \ref{?} is obtained by subtracting one equation from the remaining equations which introduces a potentially significant offset in the estimate.
+This offset depends on the particular choice of equation to subtract.
+%TODO local minima vs global?
+%TODO was noch?
 
+Another factor which influences the accuracy is the geometry of the setup where the measurements were taken.
+The accuracy of multilateration estimate depends on the position of the APs and the position of the smartphone relatively to each other.
+Therefore, it is important to consider the actual AP locations for localization which might differ from the AP locations which results the best signal coverage.
+And the walkable area where the localization system should be used.
+
+Best localization results are archived when the distance circles of the APs intersect in a near orthogonal angle.
+Localization performance degrades with wider intersection angles.
+The optimal setup in a simple scenario given four APs is to place them at the corners of a square.
+This yields best localization performance inside of the square.
+
+Usually non-optimal AP locations need to be chosen due to environmental constrains like building structure and signal coverage.
+These geometrical considerations can be founded on geometric dilution of precision (GDOP), which is a indicator which specifies the localization error based on the sender-receiver geometry.
+%DOP ganz nett aber signalstärken spielt auch eine Rolle
 
 
 \subsection{Probabilistic}

tex/misc/keywords.tex

@@ -43,5 +43,7 @@
 \newcommand{\docsRSSI}{RSSI}
 
 \newcommand{\docDSimplex}{downhill-simplex}
+\newcommand{\docGaussNetwon}{Gauss–Newton algorithm\xspace}
+\newcommand{\docLevenbergMarq}{Levenberg–Marquardt algorithm\xspace}
 
 \DeclareMathOperator{\atan}{atan2}