part filt chapter finished, immpf nearly finished
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toni
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\section{Recursive State Estimation}
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\section{Standard Particle Filtering}
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\label{sec:rse}
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Wie immer. bisschen umschreiben halt.
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In this section, we present two common localisation schemes based on particle filtering using two different transition models for propagating new states and an identical evaluation step for udpating the weights.
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As mentioned before, most smoothing methods require a preceding filtering.
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Similar to our previous works, we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem.
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Therefore, a Bayes filter that satisfies the Markov property is used to calculate the posterior:
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As mentioned before, we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem.
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Recursive filters, like the aforementioned particle filter, use all observations $\mObsVec_{t}$ until the current time $t$ for computing an estimation of the hidden state $\mStateVec_{t}$.
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In a Bayesian setting, this can be formalized as the computation of the posterior distribution:
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%
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\begin{equation}
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\arraycolsep=1.2pt
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@@ -19,45 +18,163 @@ Therefore, a Bayes filter that satisfies the Markov property is used to calculat
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\label{equ:bayesInt}
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\end{equation}
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%
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Here, the previous observation $\mObsVec_{t-1}$ is included into the state transition \cite{Ebner-15}.
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For approximating eq. \eqref{equ:bayesInt} by means of MC methods, the transition is used as proposal distribution, also known as CONDENSATION algorithm \cite{isard1998smoothing}.
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This algorithm also performs a resampling step to handle the phenomenon of weight degeneracy.
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Here, the previous observation $\mObsVec_{t-1}$ is included into the state transition \cite{Ebner-15}.
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For approximating $p(\mStateVec_{t} \mid \mObsVec_{1:t})$ with a particle filter, a sample set of $N$ independet random variables, $\vec{X}^i_{t} \sim p(\mStateVec_t \mid \mObsVec_{1:t})$ for $i = 1,...,N$, is used.
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Due to importance sampling, a weight $W^i_t$ is assigned to each sample $\vec{X}^i_{t}$.
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A particle set of the filter is then given by $\{W^i_{1:t}, \vec{X}^i_{1:t} \}_{i=1}^N$.
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The transition is used as proposal distribution, also known as CONDENSATION algorithm \cite{Isard98:CCD}.
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This algorithm also performs a traditional resampling step.
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In the context of indoor localisation, the hidden state $\mStateVec$ is defined as follows:
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For indoor localisation we define the hidden state $\mStateVec$ as follows:
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\begin{equation}
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\mStateVec = (x, y, z, \mStateHeading, \mStatePressure),\enskip
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x, y, z, \mStateHeading, \mStatePressure \in \R \enspace,
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x, y, z, \mStateHeading, \mStatePressure \in \R \enspace.
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\end{equation}
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%
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where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the relative atmospheric pressure prediction in hectopascal (hPa). Further, the observation is given by
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where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the relative atmospheric pressure prediction in hectopascal (hPa).
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A particle is therefore a weighted representation of one possible system state $\mStateVec$. All relevant sensor measurements are incorporated into the observation $\mObsVec$ given by
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%
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\begin{equation}
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\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsSteps, \mObsPressure, \mObsActivity) \enspace,
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\mObsVec = (\mObsHeading, \mObsSteps, \mRssiVec_\text{wifi}, \mObsPressure) \enspace,
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\end{equation}
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%
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covering all relevant sensor measurements.
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Here, $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{}) and \docIBeacon{}s, respectively.
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$\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number of steps detected for the pedestrian.
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$\mObsPressure$ is the relative barometric pressure with respect to a fixed reference.
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Finally, $\mObsActivity$ contains the activity currently estimated for the pedestrian, which is one of:
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unknown, standing, walking, walking up the stairs or walking down the stairs.
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Here, $\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number of steps detected for the pedestrian.
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Further, $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{}).
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Finally, $\mObsPressure$ is the relative barometric pressure with respect to a fixed reference.
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The probability density of the state evaluation is given by
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We assume a statistical independence of all sensors. The probability density of the state evaluation is then given by the following components:
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%
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\begin{equation}
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%\begin{split}
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p(\vec{o}_t \mid \vec{q}_t) =
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p(\vec{o}_t \mid \vec{q}_t)_\text{baro}
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\,p(\vec{o}_t \mid \vec{q}_t)_\text{ib}
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\,p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}
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\enspace
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%\end{split}
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\label{eq:evalBayes}
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\end{equation}
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%
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and therefore similar to \cite{Ebner-16}.
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Here, we assume a statistical independence of all sensors and every single component refers to a probabilistic sensor model.
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The barometer information is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$, whereby absolute position information
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is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{ib}$ for \docIBeacon{}s and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
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The current pressure value is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$ and absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
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\subsection{Evaluation}
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%Barometer
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First, the smartphone’s barometer is used to infer the likeliness of the current $z$-location.
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Due to noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several sensor readings and the sensor's uncertainty $\sigma_\text{baro}$.
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This average serves as relative base for all future measurements and is carried out while the pedestrian chooses his destination \cite{Fetzer2016OMC}.
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The evaluation step for time $t$ is given by
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%
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\begin{equation}
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p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace.
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\label{eq:baroEval}
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\end{equation}
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%
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Here, every predicted relative pressure $\mState_t^{\mStatePressure}$ is compared with the observed one $\mObs_t^{\mObsPressure}$ using a normal distribution.
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The state's relative pressure prediction $\mStatePressure$ is estimated within each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ by tracking every height-change ($z$-axis):
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%
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\begin{equation}
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\mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b
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,\enskip
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\Delta z = \mState_{t-1}^{z} - \mState_{t}^z
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,\enskip
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b \in \R
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\enspace ,
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\label{eq:baroTransition}
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\end{equation}
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%
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Here, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$.
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%WI-FI
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Signal strength models are a well-established and popular method for estimating a pedestrian's position in indoor environments.
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We are using the wall attenuation factor model based on Friis transmission equation to predict an \docAP{}’s (\docAPshort{}) signal strength at an arbitrary position $\mStateVec_t$ \cite{Ebner-15}.
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Here, the positions of detected \docAP{}s (\docAPshort{}) are known beforehand.
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The main advantage of this approach is that no time-consuming initial calibration phase and updates in case of infrastructural changes are needed.
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Using the 3D distance $d$ and the number of floors $\Delta f$ between the transmitter and the state-in-question, it can be described by
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%
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\begin{equation}
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P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace ,
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\label{eq:waf}
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\end{equation}
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%
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where $\mTXP$ contains the AP’s signal strength at $\mMdlDist_0$ and $\mPLE$ models the signal’s depletion with growing distance.
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The attenuation per floor is described by $\mWAF$.
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To reduce the system's setup time, we use the same values for all \docAP{}s at the cost of accuracy.
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By assuming statistical independence, the overall probability can be determined using
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%
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\begin{equation}
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\mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} =
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\prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace .
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\label{eq:wifiTotal}
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\end{equation}
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The uncertainty of the measurements is given by $\Delta{f_{i}}$. More details on this approach and possible extensions can be found in \cite{Ebner-15} and \cite{Ebner-17}.
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\subsection{Transition}
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As mentioned before, we are searching for a solution to satisfy both, sample diversity and focus.
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In the following, two very different transition models, each providing one of this abilities, are presented.
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The first transition model is based upon random walks on a graph $G=(V,E)$ with vertices $\mVertexA \in V$ and undirected edges $\mEdgeAB \in E$, which is generated from the buildings floorplan \cite{Ebner-16}.
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Starting at the vertex of the position $\fPos{\mStateVec_{t-1}} = (x, y,z)^T$ a new particle is sampled by walking along adjacent nodes into a given walking-direction $\gHead$ until a distance $\gDist$ is reached \cite{Ebner-15}.
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During the random walk, each edge has its own probability $p(\mEdgeAB)$ which depends on the edge’s direction $\angle \mEdgeAB$ and the pedestrian’s current heading $\gHead$.
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The to-be-walked edge is thus drawn according to their resemblance:
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%
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\begin{equation}
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p(\mEdgeAB)_\text{head} = p(\mEdgeAB \mid \gHead) = \mathcal{N} (\angle \mEdgeAB \mid \gHead, \sigma_\text{head}^2)
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\enspace .
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\label{eq:transHeading}
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\end{equation}
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%
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While the distribution \refeq{eq:transHeading} does not integrate to $1.0$ due to circularity of angular data, in our case, the normal distribution can be assumed as sufficient for small enough $\sigma^2$.
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To provide $\gHead$ and $\gDist$, steps and turns are detected using the smartphone's IMU, implemented as described in \cite{Ebner-15}.
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The number of steps detected since the last transition is used to estimate the to-be-walked distance $\gDist$ by assuming a fixed step-size with some deviation:
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%
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\begin{equation}
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\gDist = \mObs_{t-1}^{\mObsSteps} \cdot \mStepSize + \mathcal{N}(0, \sigma_{\gDist}^2)
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\enspace .
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\end{equation}
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%
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Turn-Detection supplies the magnitude of the detected heading change by integrating over the gyroscope's change since the last transition.
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Together with some deviation and the state's previous heading, the magnitude is used to estimate the current state's heading:
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%
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\begin{equation}
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\gHead = \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_{t-1}^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
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\end{equation}
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%
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All this promises a very focused propagation for new particles and draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.
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Additionally, the graph-based approach offers plenty of scope for further extension as can be seen in \cite{Ebner2016OPN}.
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The second transition model is very simple and thus often used in scenarios with little or rough information provided by sensors.
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Especially in cases without any regular statements about the pedestrian's movement.
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This continuous model just moves into a random direction, ignoring the graph and thus any floorplan knowledge:
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%
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\begin{equation}
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\begin{split}
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\mProb(\mStateVec_{t} \mid \mStateVec_{t-1}) &=
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\mathcal{N}\left(
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\fPos{\mStateVec_{t}}
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\mid{}
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\fPos{\mStateVec_{t-1}},
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\mat{\Sigma}_{\text{move}}
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\right),\\
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\mat{\Sigma}_{\text{move}} &=
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\begin{pmatrix}
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\sigma_{\text{move}} & 0 & 0\\
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0 & \sigma_{\text{move}} & 0\\
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0 & 0 & \sigma_{\text{floor}}\\
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\end{pmatrix}
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\end{split}
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\label{eq:simpleTrans}
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\end{equation}
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%
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The only restriction made, is that newly drawn particles need to be somewhere in between the graphs boundaries and therefore have a valid vertex for $\fPos{\mStateVec_{t}}$.
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If the particle does not satisfy this condition, the position of nearest available vertex is chosen instead.
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This ensures that a particle resides always on a valid vertex $\mVertexA$, what will be of importance for the upcoming IMMPF.
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