90 lines
5.7 KiB
TeX
90 lines
5.7 KiB
TeX
\section{Position Estimation}
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\label{sec:position}
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%Bei Indoor Lokalisierung geht es darum eine Position zu ermitteln. Hierfür nutzen wir unterschiedliche Verfahren. namley... usw.
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After measuring several distances to different anchor points one can calculate his current position.
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%TODO Alles mit 2D, weil halt
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\subsection{Multilateration}
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%Ganz kurz erläutern was Multilateration eigentlich ist. in 2D min 3 aps und in 3D min. 4D. Aber grundsätzlich gilt: viel hilft viel.
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%Es ist uns klar, dass Trilat nichts taugt aber ist halt der einfachste Schritt
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%Typische Nachteile: Wenn Schnittpunkt nicht analytisch exakt bestimmt werden können
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%FTM Nachteil: Häufig fallen die Messungen aus? Was tun? Alte Werte statisch halten? Keine est berechnen?
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%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.
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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 $i$.
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Formally the distance is the euclidean distance between the known position and the estimate
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\begin{equation}
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d_i = \| \mPosVec_i - \hat{\mPosVec} \| \text{.}
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\end{equation}
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Three ideal distances form a system of linear equations which can be uniquely solved to obtain the position.
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Given more than three distances no solution can be found which satisfies all the constraints as the linear system is overdetermined.
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Additionally, in the presence of noise and inaccurate measurements an exact analytical solution is not possible.
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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
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\begin{equation}
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\mPosVec^* = \argmin_{\hat{\mPosVec}} \sum_{i}^{} \left( \| \mPosVec_i - \hat{\mPosVec} \| - d_i \right)^2 \text{.}
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\label{eq:leastSquare}
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\end{equation}
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This forms a classical non linear least squared optimization problem which can be solved with a numerical optimization method like \docGaussNetwon or \docLevenbergMarq.
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The position $\hat{\mPosVec}$ which minimizes the error provides an approximative estimate $\mPosVec^*$.
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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.
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A good starting value for the optimization can be obtained by computing the pseudo inverse of the matrix. % TODO formeln
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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.
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This offset depends on the particular choice of equation to subtract.
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%TODO local minima vs global?
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%TODO was noch?
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Another factor which influences the localization accuracy is the geometry of the setup where the measurements were taken.
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The accuracy of multilateration estimate depends on the position of the APs and the position of the smartphone relatively to each other.
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Therefore, it is important to consider the actual AP locations for improved localization accuracy and the walkable area where the localization system is used.
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However, the best geometrical setup for localization is not necessarily the best setup for signal coverage.
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Best localization results are archived when the distance circles of the APs intersect in a near orthogonal angle.
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Localization performance degrades with wider intersection angles.
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The optimal setup in a simple scenario given four APs is to place them at the corners of a square.
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This yields best localization performance inside of the square.
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Usually non-optimal AP locations need to be chosen due to environmental constrains like building structure and signal coverage.
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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.
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%DOP ganz nett aber signalstärken spielt auch eine Rolle
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\begin{equation}
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A =
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\begin{pmatrix}
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\frac{(x_1 - x)}{R_1} & \frac{(y_1 - y)}{R_1} & \frac{(z_1 - z)}{R_1} \\[0.3em]
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\frac{(x_2 - x)}{R_2} & \frac{(y_2 - y)}{R_2} & \frac{(z_2 - z)}{R_2} \\[0.3em]
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\vdots & \vdots & \vdots \\[0.3em]
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\frac{(x_i - x)}{R_i} & \frac{(y_i - y)}{R_i} & \frac{(z_i - z)}{R_i} \\
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\end{pmatrix}
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\end{equation}
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where $R_i=\sqrt{(x_i-x)^2+(y_i-y)^2+(z_i-z)^2}$
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\begin{equation}
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Q = (A^TA)^{-1}
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\end{equation}
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\begin{equation}
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\text{GDOP} = \sqrt{\text{trace}(Q)}
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\end{equation}
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\subsection{Probabilistic}
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%Dichte aus Messungen erzeugen.
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%Distanzern werden mit Normalverteilung gewichtet
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%Vorteil: Nicht ideale Schnittpunkte sind kein Problem, weil die Dichte sowas abbilden kann
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%FTM Vorteil: Fehlen von Messungen kann probabilistisch erfasst werden indem Streuung der NV größer wird / Oder es enstehen einfach mehrere Hypthesen über die Position
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\subsection{Particle Filter}
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%Warum auch noch PF? Weil... die meisten lokalisierungs systeme diesen als nicht-linearen schätzer benutzen. er ist vielseitig und kann einfach mit anderen sensoren kombiniert werden. ist das gängigste sensor fusionsverfahren. die dichte wird in samples repräsentiert und ist damit nur eine approximatino der wahrsch. dichte.. dadurch ganz ander repräsentation als probabilistic teil. außerdem ist es ein FILTER, hat also vergangenheit mit drin.
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%Ganz schnell nochmal den PF einführen mit standard formel. ein größerer absatz.
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%Einfaches Bewegungsmodell mit NV random
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%ftm bzw. rssi einfach über normalverteilung in die evaluation rein.
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