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\section{Transition}
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\label{sec:transition}
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\begin{figure}[t]
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\centering
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\begin{subfigure}{0.325\textwidth}
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\begin{figure}[t]
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\centering
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\includegraphics[width=5.1cm]{gfx/transition/museumMap.pdf}
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\caption{3D Floorplan}
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\label{fig:museumMap}
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\end{subfigure}
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\begin{subfigure}{0.325\textwidth}
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\centering
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\includegraphics[width=5.1cm]{gfx/transition/museumMapGrid.pdf}
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\caption{Navigation graph}
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\label{fig:museumMapGrid}
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\end{subfigure}
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\begin{subfigure}{0.325\textwidth}
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\centering
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\includegraphics[width=5.1cm]{gfx/transition/museumMapMesh.pdf}
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\caption{Navigation mesh}
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\label{fig:museumMapMesh}
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\end{subfigure}
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\caption{
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Floorplan and navigation data structures for the lower floors of the building.
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To reach every nook and cranny, the graph based approach (b) requires many nodes and edges.
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The depicted version uses a coarse node-spacing of \SI{90}{\centi\meter} (1700 nodes) and barely reaches all doors and stairs.
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A navigation mesh (c) requires only 320 triangles to tightly reach every corner within the building.
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}
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\label{fig:transition}
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\end{figure}
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\begin{subfigure}{0.325\textwidth}
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\centering
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\includegraphics[width=5.1cm]{gfx/transition/museumMap.pdf}
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\caption{3D Floorplan}
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\label{fig:museumMap}
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\end{subfigure}
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\begin{subfigure}{0.325\textwidth}
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\centering
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\includegraphics[width=5.1cm]{gfx/transition/museumMapGrid.pdf}
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\caption{Navigation graph}
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\label{fig:museumMapGrid}
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\end{subfigure}
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\begin{subfigure}{0.325\textwidth}
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\centering
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\includegraphics[width=5.1cm]{gfx/transition/museumMapMesh.pdf}
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\caption{Navigation mesh}
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\label{fig:museumMapMesh}
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\end{subfigure}
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\caption{
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Floorplan and transition data structures for the lower floors of the building.
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To reach every nook and cranny, the graph based approach (b) requires many nodes and edges.
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The depicted version uses a coarse node-spacing of \SI{90}{\centi\meter} (1700 nodes) and barely reaches all doors and stairs.
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A navigation mesh (c) requires only 320 triangles to tightly reach every corner within the building.
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}
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\label{fig:transition}
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\end{figure}
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Within previous works, we used a graph of equidistant nodes (see \reffig{fig:museumMapGrid})
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to model the buildings floorplan, representing the basis for the transition step \cite{Ebner-15, Ebner-16}.
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% in 15 und 16 haben wir stueckweise den graph eingefuhert
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%
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The graph equals a grid, where each node constitutes the center of a grid-cell.
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Cells, usually around \SI{30} x \SI{30}{\centi\meter} in size,
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are only placed in regions that are actually walkable and not intersected by any walls
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or other obstacles. After placement, each cell is connected with their, up to 8, potential
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neighbors in the plane, creating a walkable graph for each floor. The resulting graphs are
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hereafter connected via stairs or elevators, to form the final data structure
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for the whole building.
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This allowed for (semi-)random walks along the graph, by assigning probabilities to each edge,
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using prior knowledge provided by sensors, forming the transition probability
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$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ \cite{Ebner-16}.
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Due to the equidistant spacing, the resulting graph was rather rigid and
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only well-suited for rectangular buildings. For more contorted buildings, like many
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historic ones, the node-spacing needs to be small, to reliably reach every door, stair
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and corner of the building. Within \reffig{fig:museumMapGrid} we used a
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\SI{90}{\centi\meter} spacing, that is barely able to reach all places within
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the lower floors of the building, and failing to connect the upper floors reliably.
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While using smaller spacings remedies the problem, it requires huge amounts of memory:
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up to several hundred megabytes and millions of nodes and edges to model a single building.
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% musuem aus figure: 90cm grid : ca 2000 nodes, ca 6500 edges
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% museum aus figure: 30cm grid : ca 32k nodes und 120k edges
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% museum ganz, 20cm grid : ca 75k nodes, 280k edges
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Because of both, required memory amounts and inaccuracies of the graph-based
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model, we developed a new basis for the transition step, that is still able to answer
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$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$.
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The new foundation is provided by well-known navigation meshes \cite{navMesh1}
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where the walkable area is spanned by convex polygons, sharing
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their outline edges. Each polygon knows its adjacent
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neighbors, creating a walkable mesh.
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Using variable shaped/sized elements instead of rigid grid-cells
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provides both, higher accuracy for reaching every corner, and a reduced
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memory footprint as a single polygon is able to cover arbitrarily
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large regions. However, polygons impose several drawbacks on
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common operations used within the transition step, like checking whether
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a point is contained within some region. This is much more costly for polygons
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compared to grid-cells, which are axis-aligned rectangles.
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% museum aus figure: 305 3-ecke
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% museum ganz : 789 fuer alles
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%
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Such issues can be mitigated by using triangles instead of polygons, depicted within \reffig{fig:museumMapMesh}.
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Doing so, each element within the mesh has exactly three edges and a maximum of three neighbors.
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While this usually requires some additional memory, as more triangles are need compared to polygons,
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operations, such as aforementioned contains-check, can now easily be performed,
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\eg{} by using barycentric coordinates.
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\newcommand{\turnNoise}{\mathcal{T}}
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\newcommand{\stepSize}{\mathcal{S}}
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This data structure yields room for various strategies to be applied within the transition step.
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The most simple approach uses an average pedestrian step size together with the
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number of detected steps $\mObsSteps$ together and change in heading $\mObsHeading$
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gathered from sensor observations $\mObsVec_{t-1}$.
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Combined with previously estimated position $(x,y)^T$ and heading $\mStateHeading$
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%from $\mStateVec_{t-1}$
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, including uncertainties for step-size $\stepSize$
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and turn-angle $\turnNoise$,
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this directly defines new potential whereabouts
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$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$:
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%
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\begin{equation}
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\begin{aligned}
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x_t &=& \overbrace{x_{t-1}}^{\text{old pos.}}& & &+& \overbrace{\mObsSteps \cdot \stepSize}^{\text{distance}}& & &\cdot& \overbrace{\cos(\mStateHeading + \turnNoise)}^{\text{direction}}& & ,\enskip \turnNoise &\sim \mathcal{N}(\mObsHeading, \sigma_\text{turn}^2) \\
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y_t &=& y_{t-1}\phantom{.}& & &+& \mObsSteps \cdot \stepSize& & &\cdot& \sin(\mStateHeading + \turnNoise)& & ,\enskip \stepSize &\sim \mathcal{N}(\SI{70}{\centi\meter}, \sigma_\text{step}^2)
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\end{aligned}
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\end{equation}
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\noindent{}with
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\begin{equation*}
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\mObsSteps,\mObsHeading \in \mObsVec_{t-1}
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\enskip\enskip\enskip
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\text{and}
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\enskip\enskip\enskip
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x_{t-1},y_{t-1},\mStateHeading \in \mStateVec_{t-1}
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\enskip.
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\end{equation*}
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Whether the determined destination $(x_t, y_t)^T$ is actually reachable from the start $(x_{t-1}, y_{t-1})^T$ can be determined
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by checking if their corresponding triangles are connected with each other.
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If so, the corresponding $z_t$ can be interpolated using the barycentric coordinates of $(x_t, y_t)^T$
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within a 2D projection of the triangle the position belongs to and applying them to the original 3D triangle.
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If the destination is unreachable,
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\eg{} due to walls or other obstacles. Those occurrences demand for different handling strategies. Simply trying again might
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be a viable solution, as uncertainty induced by $\turnNoise$ and $\stepSize$ will yield a slightly different destination
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that might be reachable. Increasing $\sigma_\text{step}$ and $\sigma_\text{turn}$ for those cases might also be a viable choice.
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Likewise, just using some random position, omitting heading/steps might be viable as well.
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\commentByFrank{es gaebe noch ganz andere ansaetze etc. aber wir haben wohl nicht mehr genug platz :P}
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like
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max. 1 Seite
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toni