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\bibcite{thesis:mueller}{{Christian Müller}(2012)}
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@ -133,46 +135,46 @@
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\bibcite{web:statista:laerm}{Statista(2010)}
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<use ext/MessgruppentabelleSued-2013-01-08.pdf> [32] [33 <C:/daten/source/colle
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ge/ss2013/bachelor thesis/thesis_ug/ext/overviewmap.pdf>] [34 <C:/daten/source/
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<use ext/MessgruppentabelleSued-2013-01-08.pdf> [34] [35 <C:/daten/source/colle
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ge/ss2013/bachelor thesis/thesis_ug/ext/overviewmap.pdf>] [36 <C:/daten/source/
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college/ss2013/bachelor thesis/thesis_ug/ext/MessgruppentabelleNord-2013-01-08.
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pdf>] [35 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/Messgru
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pdf>] [37 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/Messgru
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ppentabelleSued-2013-01-08.pdf>]
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] [39 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/KreuzungA3.
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pdf>] [40] [41 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/Kr
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euzungA5.pdf>] [42 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ex
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t/KreuzungA12.pdf>] [43] [44 <C:/daten/source/college/ss2013/bachelor thesis/th
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esis_ug/ext/KreuzungA28.pdf>] [45 <C:/daten/source/college/ss2013/bachelor thes
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is/thesis_ug/ext/KreuzungA29.pdf>] [46 <C:/daten/source/college/ss2013/bachelor
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thesis/thesis_ug/ext/KreuzungA46.pdf>] [47 <C:/daten/source/college/ss2013/bac
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] [41 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/KreuzungA3.
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pdf>] [42] [43 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/Kr
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euzungA5.pdf>] [44 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ex
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t/KreuzungA12.pdf>] [45] [46 <C:/daten/source/college/ss2013/bachelor thesis/th
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esis_ug/ext/KreuzungA28.pdf>] [47 <C:/daten/source/college/ss2013/bachelor thes
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is/thesis_ug/ext/KreuzungA29.pdf>] [48 <C:/daten/source/college/ss2013/bachelor
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thesis/thesis_ug/ext/KreuzungA46.pdf>] [49 <C:/daten/source/college/ss2013/bac
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helor thesis/thesis_ug/ext/KreuzungA59.pdf>]
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[48 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/KreuzungA104.
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] [50 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A3_Ueberset
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zungstabelle.pdf>] [51 <C:/daten/source/college/ss2013/bachelor thesis/thesis_u
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] [52 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A3_Ueberset
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zungstabelle.pdf>] [53 <C:/daten/source/college/ss2013/bachelor thesis/thesis_u
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g/ext/A4_Uebersetzungstabelle.pdf>]
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[52 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A5_Uebersetzu
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ngstabelle.pdf>] [53 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/
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ext/A12_Uebersetzungstabelle.pdf>] [54 <C:/daten/source/college/ss2013/bachelor
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thesis/thesis_ug/ext/A23_Uebersetzungstabelle.pdf>] [55 <C:/daten/source/colle
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ge/ss2013/bachelor thesis/thesis_ug/ext/A28_Uebersetzungstabelle.pdf>] [56 <C:/
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[54 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A5_Uebersetzu
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ngstabelle.pdf>] [55 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/
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||||
ext/A12_Uebersetzungstabelle.pdf>] [56 <C:/daten/source/college/ss2013/bachelor
|
||||
thesis/thesis_ug/ext/A23_Uebersetzungstabelle.pdf>] [57 <C:/daten/source/colle
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ge/ss2013/bachelor thesis/thesis_ug/ext/A28_Uebersetzungstabelle.pdf>] [58 <C:/
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daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A29_Uebersetzungstabe
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lle.pdf>] [57 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A46
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_Uebersetzungstabelle.pdf>] [58 <C:/daten/source/college/ss2013/bachelor thesis
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/thesis_ug/ext/A59_Uebersetzungstabelle.pdf>] [59 <C:/daten/source/college/ss20
|
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lle.pdf>] [59 <C:/daten/source/college/ss2013/bachelor thesis/thesis_ug/ext/A46
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_Uebersetzungstabelle.pdf>] [60 <C:/daten/source/college/ss2013/bachelor thesis
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/thesis_ug/ext/A59_Uebersetzungstabelle.pdf>] [61 <C:/daten/source/college/ss20
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13/bachelor thesis/thesis_ug/ext/A104_Uebersetzungstabelle.pdf>]
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@ -14,7 +14,7 @@
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\BOOKMARK [2][-]{subsection.4.7}{Abbiegewahrscheinlichkeiten}{section.4}% 14
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|
||||
\BOOKMARK [3][-]{subsubsection.6.4.2}{Berechnungsmatrizen}{subsection.6.4}% 36
|
||||
\BOOKMARK [1][-]{section.7}{Visualisierung}{}% 37
|
||||
\BOOKMARK [2][-]{subsection.7.1}{Visualisierung des Graphen mit Geoinformationen}{section.7}% 38
|
||||
\BOOKMARK [2][-]{subsection.7.2}{Visualisierung des JGraphT-Graphen}{section.7}% 39
|
||||
\BOOKMARK [1][-]{section.8}{Validierung}{}% 40
|
||||
\BOOKMARK [2][-]{subsection.8.1}{Testdatenmenge}{section.8}% 41
|
||||
\BOOKMARK [2][-]{subsection.8.2}{Verkehrsz\344hlung}{section.8}% 42
|
||||
\BOOKMARK [2][-]{subsection.8.3}{Validierung der Verkehrsaufkommensvorhersage}{section.8}% 43
|
||||
\BOOKMARK [1][-]{section.9}{Ausblick}{}% 44
|
||||
\BOOKMARK [1][-]{section.9}{Glossar}{}% 45
|
||||
\BOOKMARK [1][-]{section.9}{Abbildungsverzeichnis}{}% 46
|
||||
\BOOKMARK [1][-]{section.9}{Quellcodeverzeichnis}{}% 47
|
||||
\BOOKMARK [1][-]{section.13}{Anhang}{}% 48
|
||||
\BOOKMARK [1][-]{section.7}{Visualisierung}{}% 36
|
||||
\BOOKMARK [2][-]{subsection.7.1}{Visualisierung des Graphen mit Geoinformationen}{section.7}% 37
|
||||
\BOOKMARK [2][-]{subsection.7.2}{Visualisierung des JGraphT-Graphen}{section.7}% 38
|
||||
\BOOKMARK [1][-]{section.8}{Validierung}{}% 39
|
||||
\BOOKMARK [2][-]{subsection.8.1}{Testdatenmenge}{section.8}% 40
|
||||
\BOOKMARK [2][-]{subsection.8.2}{Verkehrsz\344hlung}{section.8}% 41
|
||||
\BOOKMARK [2][-]{subsection.8.3}{Validierung der Verkehrsaufkommensvorhersage}{section.8}% 42
|
||||
\BOOKMARK [1][-]{section.9}{Ausblick}{}% 43
|
||||
\BOOKMARK [1][-]{section.9}{Glossar}{}% 44
|
||||
\BOOKMARK [1][-]{section.9}{Abbildungsverzeichnis}{}% 45
|
||||
\BOOKMARK [1][-]{section.9}{Quellcodeverzeichnis}{}% 46
|
||||
\BOOKMARK [1][-]{section.13}{Anhang}{}% 47
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@ -50,7 +50,7 @@
|
||||
\newpage
|
||||
\section{Datenbasis und Grundlagen}{\input{tex/datengrund}}
|
||||
\newpage
|
||||
\section{Modell}{\input{tex/modell}}
|
||||
\section{Verkehrsmodell}{\input{tex/modell}}
|
||||
\newpage
|
||||
\section{L"osungsans"atze}{\input{tex/berechnung}}
|
||||
\newpage
|
||||
|
||||
@ -8,41 +8,40 @@
|
||||
\contentsline {subsection}{\numberline {4.3}Macro- und microskopische Modellierung von Verkehr}{8}{subsection.4.3}
|
||||
\contentsline {subsection}{\numberline {4.4}Induktionsschleifenwerte}{8}{subsection.4.4}
|
||||
\contentsline {subsubsection}{\numberline {4.4.1}CSV-Daten der Stadt Darmstadt}{8}{subsubsection.4.4.1}
|
||||
\contentsline {subsubsection}{\numberline {4.4.2}MYSQL-Daten des Projektes Verkehrsvisualisierung}{9}{subsubsection.4.4.2}
|
||||
\contentsline {subsection}{\numberline {4.5}Geographischer Ausschnitt der Daten}{10}{subsection.4.5}
|
||||
\contentsline {subsection}{\numberline {4.6}CAD-Zeichnungen der Kreuzungen}{10}{subsection.4.6}
|
||||
\contentsline {subsection}{\numberline {4.7}Abbiegewahrscheinlichkeiten}{11}{subsection.4.7}
|
||||
\contentsline {subsection}{\numberline {4.8}Das Zeitproblem}{11}{subsection.4.8}
|
||||
\contentsline {subsection}{\numberline {4.9}Das Abbiegeproblem}{12}{subsection.4.9}
|
||||
\contentsline {section}{\numberline {5}Modell}{13}{section.5}
|
||||
\contentsline {subsection}{\numberline {5.1}Modell der Ministadt}{13}{subsection.5.1}
|
||||
\contentsline {subsection}{\numberline {5.2}Modell als Graph}{15}{subsection.5.2}
|
||||
\contentsline {subsubsection}{\numberline {5.2.1}Kreuzungsgraph}{16}{subsubsection.5.2.1}
|
||||
\contentsline {subsubsection}{\numberline {5.2.2}Kreuzungs\active@dq \dq@prtct {u}bersicht}{17}{subsubsection.5.2.2}
|
||||
\contentsline {subsection}{\numberline {5.3}Modell als Matrix}{17}{subsection.5.3}
|
||||
\contentsline {subsubsection}{\numberline {5.3.1}Ausgangsmatrix}{18}{subsubsection.5.3.1}
|
||||
\contentsline {subsubsection}{\numberline {5.3.2}Eingangsmatrix}{18}{subsubsection.5.3.2}
|
||||
\contentsline {subsubsection}{\numberline {5.3.3}Sonderfall: Vallidierungssensor}{19}{subsubsection.5.3.3}
|
||||
\contentsline {subsection}{\numberline {5.4}Datenbankmodell}{19}{subsection.5.4}
|
||||
\contentsline {subsubsection}{\numberline {5.4.1}Datenbank Schema}{20}{subsubsection.5.4.1}
|
||||
\contentsline {section}{\numberline {6}L\active@dq \dq@prtct {o}sungsans\active@dq \dq@prtct {a}tze}{23}{section.6}
|
||||
\contentsline {subsection}{\numberline {6.1}L\active@dq \dq@prtct {o}sungsansatz: Hidden Markow Modell}{23}{subsection.6.1}
|
||||
\contentsline {subsection}{\numberline {6.2}L\active@dq \dq@prtct {o}sungsansatz: Wegfindungsalgorithmen}{23}{subsection.6.2}
|
||||
\contentsline {subsection}{\numberline {6.3}L\active@dq \dq@prtct {o}sungsansatz: Lineares Gleichungssystem}{23}{subsection.6.3}
|
||||
\contentsline {subsubsection}{\numberline {6.3.1}Grundlagen}{24}{subsubsection.6.3.1}
|
||||
\contentsline {subsubsection}{\numberline {6.3.2}Lineares Gleichungssystem einer Kreuzung}{24}{subsubsection.6.3.2}
|
||||
\contentsline {subsection}{\numberline {6.4}Lineares Gleichungssystem am Graph}{24}{subsection.6.4}
|
||||
\contentsline {subsubsection}{\numberline {6.4.1}Verkehrsfluss zwischen Kreuzungen}{25}{subsubsection.6.4.1}
|
||||
\contentsline {subsubsection}{\numberline {6.4.2}Berechnungsmatrizen}{25}{subsubsection.6.4.2}
|
||||
\contentsline {section}{\numberline {7}Visualisierung}{26}{section.7}
|
||||
\contentsline {subsection}{\numberline {7.1}Visualisierung des Graphen mit Geoinformationen}{26}{subsection.7.1}
|
||||
\contentsline {subsection}{\numberline {7.2}Visualisierung des JGraphT-Graphen}{26}{subsection.7.2}
|
||||
\contentsline {section}{\numberline {8}Validierung}{27}{section.8}
|
||||
\contentsline {subsection}{\numberline {8.1}Testdatenmenge}{27}{subsection.8.1}
|
||||
\contentsline {subsection}{\numberline {8.2}Verkehrsz\active@dq \dq@prtct {a}hlung}{27}{subsection.8.2}
|
||||
\contentsline {subsection}{\numberline {8.3}Validierung der Verkehrsaufkommensvorhersage}{27}{subsection.8.3}
|
||||
\contentsline {section}{\numberline {9}Ausblick}{28}{section.9}
|
||||
\contentsline {section}{\numberline {10}Glossar}{30}{section.9}
|
||||
\contentsline {section}{\numberline {11}Abbildungsverzeichnis}{31}{section.9}
|
||||
\contentsline {section}{\numberline {12}Quellcodeverzeichnis}{31}{section.9}
|
||||
\contentsline {section}{\numberline {13}Anhang}{32}{section.13}
|
||||
\contentsline {subsubsection}{\numberline {4.4.2}MYSQL-Daten des Projektes Verkehrsvisualisierung}{10}{subsubsection.4.4.2}
|
||||
\contentsline {subsection}{\numberline {4.5}Geographischer Ausschnitt der Daten}{11}{subsection.4.5}
|
||||
\contentsline {subsection}{\numberline {4.6}CAD-Zeichnungen der Kreuzungen}{11}{subsection.4.6}
|
||||
\contentsline {subsection}{\numberline {4.7}Abbiegewahrscheinlichkeiten}{12}{subsection.4.7}
|
||||
\contentsline {subsection}{\numberline {4.8}Das Zeitproblem}{12}{subsection.4.8}
|
||||
\contentsline {subsection}{\numberline {4.9}Das Abbiegeproblem}{13}{subsection.4.9}
|
||||
\contentsline {section}{\numberline {5}Verkehrsmodell}{14}{section.5}
|
||||
\contentsline {subsection}{\numberline {5.1}Modell der Ministadt}{14}{subsection.5.1}
|
||||
\contentsline {subsection}{\numberline {5.2}Modell als Graph}{16}{subsection.5.2}
|
||||
\contentsline {subsubsection}{\numberline {5.2.1}Kreuzungsgraph}{17}{subsubsection.5.2.1}
|
||||
\contentsline {subsubsection}{\numberline {5.2.2}Kreuzungs\active@dq \dq@prtct {u}bersicht}{18}{subsubsection.5.2.2}
|
||||
\contentsline {subsection}{\numberline {5.3}Modell als Matrix}{18}{subsection.5.3}
|
||||
\contentsline {subsubsection}{\numberline {5.3.1}Ausgangsmatrix}{19}{subsubsection.5.3.1}
|
||||
\contentsline {subsubsection}{\numberline {5.3.2}Eingangsmatrix}{19}{subsubsection.5.3.2}
|
||||
\contentsline {subsubsection}{\numberline {5.3.3}Sonderfall: Vallidierungssensor}{20}{subsubsection.5.3.3}
|
||||
\contentsline {subsection}{\numberline {5.4}Datenbankmodell}{20}{subsection.5.4}
|
||||
\contentsline {subsubsection}{\numberline {5.4.1}Datenbank Schema}{21}{subsubsection.5.4.1}
|
||||
\contentsline {section}{\numberline {6}L\active@dq \dq@prtct {o}sungsans\active@dq \dq@prtct {a}tze}{24}{section.6}
|
||||
\contentsline {subsection}{\numberline {6.1}L\active@dq \dq@prtct {o}sungsansatz: Hidden Markow Modell}{24}{subsection.6.1}
|
||||
\contentsline {subsection}{\numberline {6.2}L\active@dq \dq@prtct {o}sungsansatz: Wegfindungsalgorithmen}{24}{subsection.6.2}
|
||||
\contentsline {subsection}{\numberline {6.3}L\active@dq \dq@prtct {o}sungsansatz: Lineares Gleichungssystem}{24}{subsection.6.3}
|
||||
\contentsline {subsubsection}{\numberline {6.3.1}Grundlagen}{25}{subsubsection.6.3.1}
|
||||
\contentsline {subsubsection}{\numberline {6.3.2}Lineares Gleichungssystem einer Kreuzung}{25}{subsubsection.6.3.2}
|
||||
\contentsline {subsection}{\numberline {6.4}Lineares Gleichungssystem am Graph}{26}{subsection.6.4}
|
||||
\contentsline {subsubsection}{\numberline {6.4.1}Verkehrsfluss zwischen Kreuzungen}{27}{subsubsection.6.4.1}
|
||||
\contentsline {section}{\numberline {7}Visualisierung}{28}{section.7}
|
||||
\contentsline {subsection}{\numberline {7.1}Visualisierung des Graphen mit Geoinformationen}{28}{subsection.7.1}
|
||||
\contentsline {subsection}{\numberline {7.2}Visualisierung des JGraphT-Graphen}{28}{subsection.7.2}
|
||||
\contentsline {section}{\numberline {8}Validierung}{29}{section.8}
|
||||
\contentsline {subsection}{\numberline {8.1}Testdatenmenge}{29}{subsection.8.1}
|
||||
\contentsline {subsection}{\numberline {8.2}Verkehrsz\active@dq \dq@prtct {a}hlung}{29}{subsection.8.2}
|
||||
\contentsline {subsection}{\numberline {8.3}Validierung der Verkehrsaufkommensvorhersage}{29}{subsection.8.3}
|
||||
\contentsline {section}{\numberline {9}Ausblick}{30}{section.9}
|
||||
\contentsline {section}{\numberline {10}Glossar}{32}{section.9}
|
||||
\contentsline {section}{\numberline {11}Abbildungsverzeichnis}{33}{section.9}
|
||||
\contentsline {section}{\numberline {12}Quellcodeverzeichnis}{33}{section.9}
|
||||
\contentsline {section}{\numberline {13}Anhang}{34}{section.13}
|
||||
|
||||
@ -1,6 +1,6 @@
|
||||
\label{sec:berechnung}
|
||||
In diesem Kapitel werden einige Lösungsansätze für die Herausforderungen der Verkehrsflussberechnung anhand des in Kapitel \autoref{sec:modell} beschrieben Modells vorgestellt. Dabei konnten für Kreuzungen ein lineares Gleichungssystem entwickelt werden, welches sowohl die Verkehrswerte für die Ausgänge als auch für die Eingänge der entsprechenden Kreuzung berechnen kann. Anhand der Berechnungen innerhalb von Kreuzungen konnten Verkehrsflusswerte für zwischen den Kreuzungen bestimmt werden.\\ \\
|
||||
Es galt folgende Problem zu lösen:
|
||||
In diesem Kapitel werden drei Lösungsansätze für die Herausforderungen der Verkehrsflussberechnung anhand des in Kapitel \autoref{sec:modell} beschrieben Modells vorgestellt. Dabei konnten für Kreuzungen ein lineares Gleichungssystem entwickelt werden, welches sowohl die Verkehrswerte für die Ausgänge als auch für die Eingänge der entsprechenden Kreuzung berechnen kann. Anhand der Berechnungen innerhalb von Kreuzungen konnten Verkehrsflusswerte für zwischen den Kreuzungen bestimmt werden.\\ \\
|
||||
Es galt folgende Herausforderung zu lösen:
|
||||
\begin{enumerate}
|
||||
\item{Werte für virtuelle Sensoren}\label{problem:1}
|
||||
\item{Wie viele Autos verlassen die Kreuzung in Richtung Norden/Süden/Westen/Osten}\label{problem:3}
|
||||
@ -8,8 +8,8 @@ Es galt folgende Problem zu lösen:
|
||||
\item{Validierung von Sensorwerten, mithilfe von Validierungssensoren}\label{problem:2}
|
||||
\item{Verkehrswerte für Seitenstraßen ohne Sensoren}\label{problem:5}
|
||||
\end{enumerate}
|
||||
Es wurden im Rahmen dieser Arbeit mehrere Berechnungsansätze daraufhin überprüft, ob sie das gegeben eines der gegebenen Problem lösen kann. Die beschriebenen Ansätze sind 'Hidden Markow Modell', 'Wegfindungsalorithmen' wie A* und 'lineare Gleichungssystem'. \\ \\
|
||||
Das Ziel für \autoref{problem:1} virtuelle Sensoren Werte zu berechnen war das erste Ziel, welche zu erreichen galt. Da virtuelle Sensoren in dem entwickelten Verkehrsmodell ausschließlich Aus- und Eingänge modellieren würden damit ebenfalls die Herausforderung einen Verkehrswert für die jeweiligen Kreuzungsein- und -ausgänge zu berechnen gelöst werden.
|
||||
Es wurden im Rahmen dieser Arbeit mehrere Berechnungsansätze daraufhin überprüft, ob sie eines der gegebenen Problem lösen kann. Die beschriebenen Ansätze sind 'Hidden Markow Modell', 'Wegfindungsalorithmen' wie A* und 'lineares Gleichungssystem'. \\ \\
|
||||
Das Ziel für \ref{problem:1} virtuelle Sensoren Werte zu berechnen war das erste Ziel, welche zu erreichen galt. Da virtuelle Sensoren in dem entwickelten Verkehrsmodell ausschließlich Aus- und Eingänge modellieren würden damit ebenfalls die Herausforderung einen Verkehrswert für die jeweiligen Kreuzungsein- und -ausgänge zu berechnen gelöst werden.
|
||||
\subsection{L"osungsansatz: Hidden Markow Modell}{\label{sec:berechnung:hmm}
|
||||
Problem, nicht zyklischer Graph. Ein Verkehrsnetz hat viele Kreise und ein solches Modell ist aus diesen Gr"unden nicht sinnvoll.
|
||||
Als Ansatz kann man ein Markov-Modell auf einen zuf"allig ausgew"ahlten nicht zyklischen Graphen berechnen. Berechnet man nun viele solcher zuf"allig nicht zyklischen Graphen, und mittelt man die Werte f"ur die einzelnen unbekannten Sensoren,
|
||||
@ -25,7 +25,7 @@ Das Ziel für \autoref{problem:1} virtuelle Sensoren Werte zu berechnen war das
|
||||
\subsection{L"osungsansatz: Lineares Gleichungssystem}{\label{sec:berechnung:lgs}
|
||||
Gehen wir von einem Geregelten, Verkehrsordnung achtenden Verkehr aus, k"onnen wir f"ur jeden Ausgangsknoten durch addition derjenigen Sensorwerte, welche auf den Ausgang zeigen berechnen, wieviele Autos am Ausgangsknoten angekommen sein m"ussen. Das Problem dabei sind die Mischspuren, "uber welche der Verkehr in verschiedene Richtungen flie"sen kann. F"ur diese Sensoren werden Abbiegewahrscheinlichkeiten ben"otigt um zu berechnen, wie viel Prozent der Autos, welche "uber den Sensor Fahren an dem jeweiligen Ausgangsknoten angelangen. Der Sensorwert wird entsprechend mit der jeweiligen Abbiegewahrscheinlichkeit multipliziert und auf den Ausgang addiert.\\
|
||||
Da lineare Gleichungssysteme das Ergebnis dieser Arbeit sind ist ihnen ein eigenes Kapitel, um genauer darauf einzugehen.
|
||||
\subsubsection{Grundlagen}{
|
||||
\subsubsection{Grundlagen}{\label{sec:berechnung:lgs:grund}
|
||||
Ein lineares Gleichungssystem ist ein System linearer Gleichungen. Ein solches System hat n Unbekannte und m Gleichungen. Eine allgemeine Darstellungsform ist die folgende:\\
|
||||
\begin{equation}
|
||||
\begin{matrix}
|
||||
@ -64,10 +64,20 @@ Das Ziel für \autoref{problem:1} virtuelle Sensoren Werte zu berechnen war das
|
||||
Eine solche Gleichung kann mit einem beliebigem Lösungsverfahren wie z.b. dem Gaus-Algorithmus gelöst werden. Die Genauigkeit der Lösung hängt dabei von dem gewählten Verfahren ab.
|
||||
}
|
||||
|
||||
\subsubsection{Lineares Gleichungssystem einer Kreuzung}{
|
||||
Die Beziehungen zwischen Induktionsschleifensensoren und Kreuzungsein- und Ausgangängen kann durch eine lineares Gleichungssystem ausgedrückt werden. Der Wert für den Ausgang errechnet sich aus der Summe aller darauffolgenden, diesem Kreuzungseingang zugeordneten, Sensoren.
|
||||
\subsubsection{Lineares Gleichungssystem einer Kreuzung}{\label{sec:berechnung:lgs:xr}
|
||||
Die Beziehungen zwischen Induktionsschleifensensoren und Kreuzungsein- und Ausgangängen kann durch eine lineares Gleichungssystem ausgedrückt werden. Der Wert für den Ausgang errechnet sich aus der Summe aller darauffolgenden, diesem Kreuzungseingang zugeordneten, Sensoren. Ein Kreuzungsausgang errechnet sich aus den Werten derjenigen Sensoren, wessen Spur dem Verkehr erlauben diesen Kreuzungsausgang zu bedienen. Dabei muss zwischen Einspursensoren under Mischspursensoren unterschieden werden. Einspursensoren können direkt auf den Ausgang addiert werden, während für Mischspursensoren eine Abbiegewahrscheinlichkeit benötigt wird, die Angibt, wie viel Verkehr in die entsprechende Richtung fließt. Durch Multiplikation des Sensorwertes mit der Abbiegewahrscheinlichkeit erhält man den gesuchten Teil des Verkehrs und kann diesen auf den Ausgang addieren.
|
||||
Für den Ausgang der Kreuzung [] in richtung [] wäre dies folgende:
|
||||
[Gleichung nur Einspur]
|
||||
\begin{equation}
|
||||
Kreuzungausgang_x = \sum ES_{xi} + \sum MS_{xj}*ABW_{xj}
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
Kreuzungseingang_x = \sum S_{xi}
|
||||
\end{equation}
|
||||
$ES_x$ bezeichnet alle Einspursensoren welche auf den Kreuzungsausgang x zeigen.\\
|
||||
$MS_x$ bezeichnet alle Mehrspursensoren welche auf den Kreuzungsausgang x zeigen.\\
|
||||
$ABW_xi$ bezeichnet die Abbiegewahrscheinlichkeit des Mehrspursensors $MS_{xi}$ in Richtung Kreuzung x.\\
|
||||
$S_x$ bezeichnet alle Sensoren welche direkt nach einem Kreuzungseingang in der Straße verbaut sind und entsprechend eine Verbindung, gegen die Fließrichtung, zu dem Kreuzungseingang x hat.
|
||||
|
||||
Diese Berechnung erweist sich, da nur Einspursensoren in das Ergebnis einfließen, als trivial. Im Gegensatz zu Mischspursensoren, welche einen Wert für zwei oder mehr Ausgänge liefern. Ein Koeffizient für jede Richtung, in die ein Autofahrer, der den Sensor überfahren hat, bestimmt welcher Anteil des gemessen Verkehrswertes auf den jeweiligen Ausgang übertragen werden muss. Für diese Koeffizienten wurden die im Kapitel [Daten] vorgestellten Abbiegewahrscheinlichkeiten verwendet. Da alle Autos eine der Richtungen, welche ihnen die mit dem Sensor bestückte, Straße vorgibt, wählen müssen, muss die Summe aller Abbiegewahrscheinlichkeiten der verschiedenen Richtungen eins ergeben.\\ \\
|
||||
|
||||
@ -77,24 +87,8 @@ Das Ziel für \autoref{problem:1} virtuelle Sensoren Werte zu berechnen war das
|
||||
[Form]
|
||||
aufstellen, welches jedem Kreuzungsaus- sowie -eingang einen Wert zuweist.
|
||||
|
||||
Ausgang = A{K € e(K,Ausgang)| += k.count * e.abbiegewahrscheinlichkeit}
|
||||
Eingang = A{K € e(Eingang,K)| += k.count}
|
||||
|
||||
Zweirichtungeniterration vom Sensor weg.
|
||||
}
|
||||
\subsection{Lineares Gleichungssystem am Graph}{
|
||||
Ein Gleichungssystem der Form (oben beschrieben) kann als graph dargestellt werden. Die berechnung des System kann ebenfalls am Graphen passieren.
|
||||
|
||||
Ausgehend von einer direkten Verbindungen von Sensorknoten zu virtuellem Kreuzungsein- bzw. ausgang Knoten, kann um die Ausgänge zu berechnen der Graph entlangiterriert werden. Die Ausgänge berechnen sich dabei gegen die Flussrichtung.
|
||||
Die Eingänge berechnen sich mit der Flussrichtung.
|
||||
|
||||
Forward/Backward Iteration.
|
||||
}
|
||||
\subsubsection{Verkehrsfluss zwischen Kreuzungen}{
|
||||
|
||||
zwischen den Kreuzungen ist das Modell ungenau. Hier k"onnen wir die Verkehrsbelastung bestimmen. Und wir k"onnen Voraussagen, dass in der n"achsten Minute viele/wenige Autos an Kreuzung x von Kreuzung y ankommen werden
|
||||
}
|
||||
\subsubsection{Berechnungsmatrizen}{
|
||||
W"ahrend oben Verbindungsmatrizen beschrieben werden, welche ausschlie"slich modellieren, ob ein Knoten(virtuell oder Sensor) mit einem anderen verbunden ist,
|
||||
so kann man bei bekannten Abbiegewahrscheinlichkeiten f"ur die Kreuzung diese direkt in der Ausgangsmatrix verzeichnen:
|
||||
|
||||
@ -128,5 +122,17 @@ Das Ziel für \autoref{problem:1} virtuelle Sensoren Werte zu berechnen war das
|
||||
\end{Bmatrix}$
|
||||
|
||||
Diese Darstellung erwies sich als hilfreich zur Berechnung von Ausgangswerten f"ur eine Kreuzung. Siehe hierzu das Kapitel [Berechnung]
|
||||
}
|
||||
}
|
||||
\subsection{Lineares Gleichungssystem am Graph}{
|
||||
Ein Gleichungssystem der Form (oben beschrieben) kann als graph dargestellt werden. Die berechnung des System kann ebenfalls am Graphen passieren.
|
||||
|
||||
Ausgehend von einer direkten Verbindungen von Sensorknoten zu virtuellem Kreuzungsein- bzw. ausgang Knoten, kann um die Ausgänge zu berechnen der Graph entlangiterriert werden. Die Ausgänge berechnen sich dabei gegen die Flussrichtung.
|
||||
Die Eingänge berechnen sich mit der Flussrichtung.
|
||||
|
||||
Forward/Backward Iteration.
|
||||
}
|
||||
\subsubsection{Verkehrsfluss zwischen Kreuzungen}{
|
||||
|
||||
zwischen den Kreuzungen ist das Modell ungenau. Hier k"onnen wir die Verkehrsbelastung bestimmen. Und wir k"onnen Voraussagen, dass in der n"achsten Minute viele/wenige Autos an Kreuzung x von Kreuzung y ankommen werden
|
||||
}
|
||||
}
|
||||
@ -55,49 +55,95 @@ In diesem Kapitel werden die Grundlagen der Verkehrstechnik und -Modellierung be
|
||||
\item{Die Werte 'load' und 'count' für bis zu 64 Sensoren}
|
||||
\end{enumerate}
|
||||
Die Bezeichnung der Kreuzung richtet sich dabei nach der Kreuzungsbezeichnung des Verkehrsamtes Darmstadt und ist eindeutig. Eine übersicht über ganz Darmstadt mit eingezeichneten kreuzung ist im Anhang \ref{abb:4} zu finden. Ein Kreuzungsname in Darmstadt besteht dabei aus einem 'A' und einer eindeutigen Nummer. Die L"ange des Messintervalls ist in Darmstadt als eine Minute definiert.
|
||||
\begin{tabular}{lcr}
|
||||
Spalte 1 & Spalte 2 & Spalte 3 \\
|
||||
1 & 2 & 3 \\
|
||||
%\caption{CSV-Datei Ausschnitt}
|
||||
\end{tabular}
|
||||
\begin{figure}
|
||||
\begin{tabular}{|l|l|l|l|lllllllllllll|}
|
||||
Datum& Uhrzeit& Bez.& Intv.& D1Z& D1B& D2Z& D2B& D3Z& D3B& D4Z& D4B& D5Z& D5B& \dots& D64Z& D64B\\
|
||||
\hline
|
||||
08.08.13 & 00:19:00 & A 8 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 9 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 10 & 1 & 0 & 0 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 11 & 1 & 4 & 36 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 12 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 23 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 24 & 1 & 2 & 3 & 0 & 0 & 0 & 0 & 1 & 3 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 29 & 1 & 0 & 0 & 0 & 0 & & & & & & & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 30 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 31 & 1 & & & & & & & & & & & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 33 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 34 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 35 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 36 & 1 & 0 & 0 & 1 & 10 & 0 & 0 & 1 & 5 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 37 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 38 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 39 & 1 & 0 & 0 & 0 & 0 & & & & & & & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 40 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 46 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 59 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 61 & 1 & 0 & 0 & 0 & 0 & & & & & & & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 62 & 1 & & & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 64 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 10 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 65 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 69 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 70 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 72 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 73 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 74 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 75 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 18 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 76 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 82 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 84 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 87 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 90 & 1 & 0 & 25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
08.08.13 & 00:19:00 & A 94 & 1 & 0 & 0 & 1 & 10 & 1 & 10 & 0 & 0 & 0 & 0 & \dots & 0 & 0\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{CSV-Datei Ausschnitt}
|
||||
\end{figure}
|
||||
Sensorspalten, welche auf Z enden, bezeichnen den 'count'-Wert des jeweiligen Sensors, solche die auf B enden den 'load'-Wert. Jede Zeile in der CSV-Datei repr"asentiert dabei eine Kreuzung zu dem gegebenen Zeitpunkt. Für eine Kreuzung k"onnen dabei bis zu 64 Sensoren in der CSV bereitgestellt werden.
|
||||
Um die Zuordnung von CSV-Spalte zu dem ta"achlichen Sensornamen der CAD-Zeichnungen\gls{gls:cad} herzustellen, werden sog. "Ubersetzungstabellen ben"otigt. Diese ordnen einer CSV-Sensorspalte (1-64) einen kreuzungsspezifischen Sensornamen zu. In der Praxis wird eine solche "Ubersetzungstabelle durch einen Offset auf der CSV-Datei im Computer nachgebildet. Einige Übersetzungstabellen sowie die CAD-Zeichnung sind im Anhang zu finden [].
|
||||
}
|
||||
\subsubsection{MYSQL-Daten des Projektes Verkehrsvisualisierung}{
|
||||
Im Rahmen der Arbeit von \cite{thesis:mueller}, welche die Sensorwerte auf eine Karte projeziert, wurden die CSV-Datein bereits in eine MYSQL-Datenbank überführt.
|
||||
Die Sensordaten sind dort bereits mit Geoinformationen des OpenStreetMap Projektes verknüpft und geben Auskunft über Stra"sen- und Sensorpositionen.\\ \\
|
||||
Im Rahmen der Arbeit von \cite{thesis:mueller}, welche die Sensorwerte auf eine Karte projeziert, wurden die CSV-Dateien bereits in eine MYSQL-Datenbank überführt.
|
||||
Die Sensordaten sind dort bereits mit Geoinformationen des OpenStreetMap Projektes verknüpft und geben Auskunft über Kreuzung- und Sensorpositionen.\\ \\
|
||||
Die Daten verwendeten Daten befinden sich dort in drei Tabelle:
|
||||
\begin{enumerate}
|
||||
\item{jee\_crmodel\_CrossroadDim: In dieser Tabelle werden Kreuzungsname und -Position abgespeichert.}
|
||||
\item{jee\_crmodel\_SensorDim: In dieser Tabelle werden Sensorname und -Position, sowie die Kreuzung auf der der Sensor verbaut ist, gespeichert. Ein CSV-Offset identifiziert den Sensor bezüglich seiner Position in der CSV-Datei.}
|
||||
\item{jee\_trafficlight\_rawevents: In dieser Tabelle werden die eingelesenen CSV-Dateien der Stadt gespeichert. Dies umfasst neben den Werten 'load' und 'count' und Messdatum, die zugeh"orige Kreuzung. Ein Sensor kann anhand des gespeicherten CSV-Offsets mit jee\_crmodel\_SensorDim identifiziert werden.}
|
||||
\item{jee\_trafficlight\_rawevents: In dieser Tabelle werden die eingelesenen CSV-Dateien der Stadt gespeichert. Dies umfasst neben den Werten 'load' und 'count' und Messdatum, die zugeh"orige Kreuzung. Ein Sensor kann anhand des gespeicherten CSV-Offsets mit jee\_crmodel\_SensorDim identifiziert und lokalisiert werden.}
|
||||
\end{enumerate}
|
||||
Um die gespeicherte Sensorposition zu ermitteln wurde folgende SQL-Abfrage entwickelt, welche Kreuzungs- und Sensornamen, sowie dessen jeweiligen Latitude und Longitude zurückgibt. Die Auswahl ist dabei auf die zehn untersuchten Kreuzungen beschr"ankt. Mehr Informationen zu dem gew"ahlten Ausschnitt sind [] zu finden.
|
||||
Um die gespeicherte Sensorposition zu ermitteln wurde eine SQL-Abfrage\ref{lst:sql_sensorquery} entwickelt, welche Kreuzungs- und Sensornamen, sowie dessen jeweiligen Latitude und Longitude zurückgibt. Die Auswahl ist dabei auf die zehn untersuchten Kreuzungen beschr"ankt. Mehr Informationen zu dem gew"ahlten Ausschnitt sind [] zu finden.
|
||||
\begin{minipage}[t]{\dimexpr\textwidth-3\fboxsep-2\fboxrule-1em}
|
||||
\begin{lstlisting}[caption={[abc] Die Stra"senklasse ST}, label={listing1}, captionpos=bsec]
|
||||
SELECT * FROM (SELECT CD.REALNAME AS CR_NAME, SD.LAT as Sensor_LAT, SD.LON AS Sensor_Long, SD.REALNAME FROM
|
||||
`jee_crmodel_CrossroadDim` AS CD,
|
||||
`jee_crmodel_SensorDim` as SD WHERE CD.VALIDTO is null AND CD.REALNAME IN ('A 46','A 12','A 59','A104','A 23','A 28','A 29','A 3','A 4','A 5') AND SD.REALNAME LIKE 'D%' AND SD.CROSSROAD_ID = CD.ID) AS sensors
|
||||
\begin{lstlisting}[caption={[abc] Die Stra"senklasse ST}, label={lst:sql_sensorquery}, captionpos=bsec]
|
||||
SELECT CD.REALNAME AS CR_NAME,
|
||||
SD.LAT as Sensor_LAT,
|
||||
SD.LON AS Sensor_Long, SD.REALNAME
|
||||
FROM `jee_crmodel_CrossroadDim` AS CD,
|
||||
`jee_crmodel_SensorDim` as SD
|
||||
WHERE CD.VALIDTO is null
|
||||
AND CD.REALNAME IN ('A 3','A 4',...,'A 59','A104')
|
||||
AND SD.REALNAME LIKE 'D%'
|
||||
AND SD.CROSSROAD_ID = CD.ID
|
||||
AS sensors
|
||||
\end{lstlisting}
|
||||
\end{minipage}
|
||||
Eine weitere Abfrage dient dem Ermitteln der Sensorwerte über einen bestimmten Zeitraum. Zu Identifizierung des Sensors wird der CSV-Offset benutzt.
|
||||
\begin{minipage}[t]{\dimexpr\textwidth-3\fboxsep-2\fboxrule-1em}
|
||||
\begin{lstlisting}[caption={[abc] Die Stra"senklasse ST}, label={listing1}, captionpos=bsec]
|
||||
SELECT * FROM ( SELECT CD.REALNAME AS CR_NAME,
|
||||
SD.REALNAME, SD.CSVOFFSET,
|
||||
CD.VALIDFROM
|
||||
FROM `jee_crmodel_CrossroadDim` AS CD,
|
||||
jee_crmodel_SensorDim` as SD
|
||||
WHERE CD.VALIDTO is null
|
||||
AND CD.REALNAME IN
|
||||
('A 46','A 12','A 59','A104','A 23','A 28','A 29','A 3','A 4','A 5')
|
||||
AND SD.REALNAME LIKE 'D%' AND SD.CROSSROAD_ID = CD.ID) AS sensors,
|
||||
`jee_trafficlight_rawevents` AS raw
|
||||
WHERE sensors.CR_NAME=raw.CROSSROAD
|
||||
AND raw.DATETIME IN ('2013-05-20 15:01:00','2013-05-20 15:02:00','2013-05-20 15:03:00')
|
||||
AND sensors.CSVOFFSET=raw.CSVOFFSET
|
||||
\end{lstlisting}
|
||||
\end{minipage}
|
||||
\begin{minipage}[t]{\dimexpr\textwidth-3\fboxsep-2\fboxrule-1em}
|
||||
\begin{lstlisting}[caption={[abc] Die Stra"senklasse ST}, label={listing1}, captionpos=bsec]
|
||||
SELECT * FROM ( SELECT CD.REALNAME AS CR_NAME,
|
||||
SD.REALNAME, SD.CSVOFFSET,
|
||||
CD.VALIDFROM
|
||||
FROM `jee_crmodel_CrossroadDim` AS CD,
|
||||
jee_crmodel_SensorDim` as SD
|
||||
WHERE CD.VALIDTO is null
|
||||
AND CD.REALNAME IN('A 3','A 4','A 5',...,'A 59','A104')
|
||||
AND SD.REALNAME LIKE 'D%'
|
||||
AND SD.CROSSROAD_ID = CD.ID) AS sensors,
|
||||
`jee_trafficlight_rawevents` AS raw
|
||||
WHERE sensors.CR_NAME=raw.CROSSROAD
|
||||
AND raw.DATETIME IN ('2013-05-20 15:01:00','2013-05-20 15:02:00','2013-05-20 15:03:00')
|
||||
AND sensors.CSVOFFSET=raw.CSVOFFSET
|
||||
\end{lstlisting}
|
||||
\end{minipage}
|
||||
Zu beachten ist, dass die Zeitabfrage nach UTC-Zeit durchgeführt werden muss.
|
||||
Mithilfe dieser Abfragen konnten alle ben"otigten Informationen aus der Datenbank extrahiert werden. Wie diese abgespeichert wurde wird im Kapitel \autoref{sec:modell} n"aher behandelt.
|
||||
}
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user