Phase 1 abgeschlossen: Positionen werden erkannt.
Positionen aus den Merkern heraus erkennbar. Viele Bilder gleichzeitig verarbeitbar.
This commit is contained in:
chk
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documentation/Phase1.aux
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\relax
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\@writefile{toc}{\contentsline {section}{\numberline {1}Ziel}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {2}Koordinatensysteme}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {3}solvePnP}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {4}Kamerapose}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {5}Markerposition in Weltkoordinaten}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {6}Markerrotation in Weltkoordinaten}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {7}Gewichtung der Beobachtungen}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {8}Gewichtete Positionsfusion}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {9}Rotationsfusion}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {10}Qualitätsmetriken}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {11}Rigid-Body Erweiterung}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {12}Spätere Erweiterungen}{4}{}\protected@file@percent }
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\gdef \@abspage@last{4}
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documentation/Phase1.pdf
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documentation/Phase1.pdf
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248
documentation/Phase1.tex
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248
documentation/Phase1.tex
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\documentclass[a4paper,11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{geometry}
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\geometry{margin=2.5cm}
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\title{Mathematische Beschreibung von \texttt{3\_fuse\_markers\_world.py}}
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\author{}
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\date{}
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\begin{document}
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\maketitle
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\section{Ziel}
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Das Script fusioniert mehrere ArUco-Detektionen aus mehreren Kameras zu einem gemeinsamen Weltmodell.
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Gegeben sind:
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\begin{itemize}
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\item Kameraposen im Weltkoordinatensystem
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\item 2D Marker-Detektionen pro Kamera
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\item Kameraintrinsics
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\item Markergröße
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\end{itemize}
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Gesucht sind:
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\begin{itemize}
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\item Weltposition aller Marker
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\item Markerorientierungen
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\item Qualitätsmetriken
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\end{itemize}
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\section{Koordinatensysteme}
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Verwendete Systeme:
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\begin{itemize}
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\item Weltkoordinatensystem $W$
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\item Kamerakoordinatensystem $C$
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\item Markerkoordinatensystem $M$
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\end{itemize}
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\section{solvePnP}
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Für jeden Marker wird mittels OpenCV solvePnP berechnet:
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\[
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X_C = R_{CM} X_M + t_{CM}
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\]
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Dabei gilt:
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\begin{itemize}
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\item $R_{CM}$ = Rotation Marker $\rightarrow$ Kamera
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\item $t_{CM}$ = Translation Marker $\rightarrow$ Kamera
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\end{itemize}
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\section{Kamerapose}
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Aus der vorher berechneten Kamerapose:
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\[
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X_W = R_{WC} X_C + t_{WC}
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\]
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mit:
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\[
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t_{WC} = C_W
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\]
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(Kameraposition in Weltkoordinaten)
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\section{Markerposition in Weltkoordinaten}
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Markerzentrum:
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\[
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p_M =
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\begin{bmatrix}
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0 \\
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0 \\
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0
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\end{bmatrix}
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\]
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Markerposition im Kamerasystem:
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\[
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p_C = R_{CM} p_M + t_{CM}
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\]
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Da $p_M = 0$:
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\[
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p_C = t_{CM}
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\]
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Transformation ins Weltkoordinatensystem:
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\[
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p_W = R_{WC} p_C + t_{WC}
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\]
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Somit:
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\[
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p_W = R_{WC} t_{CM} + t_{WC}
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\]
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\section{Markerrotation in Weltkoordinaten}
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Die Markerrotation relativ zur Kamera:
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\[
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R_{CM}
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\]
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Die Kamerarotation relativ zur Welt:
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\[
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R_{WC}
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\]
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Markerrotation relativ zur Welt:
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\[
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R_{WM} = R_{WC} R_{CM}
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\]
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Dies ist die zentrale Rotationsgleichung des Scripts.
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\section{Gewichtung der Beobachtungen}
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Mehrere Kameras können denselben Marker beobachten.
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Für jede Beobachtung wird ein Gewicht berechnet:
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\[
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w_i =
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w_{\text{confidence}}
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\cdot
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w_{\text{area}}
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\cdot
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w_{\text{view}}
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\cdot
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w_{\text{reprojection}}
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\]
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Typische Faktoren:
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\begin{itemize}
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\item Marker Confidence
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\item Markergröße in Pixel
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\item Sichtwinkel
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\item Distanz zum Bildrand
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\item Reprojektionsfehler
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\end{itemize}
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\section{Gewichtete Positionsfusion}
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Die endgültige Markerposition:
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\[
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p =
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\frac{
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\sum_i w_i p_i
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}{
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\sum_i w_i
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}
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\]
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Dies entspricht einem gewichteten Mittelwert.
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\section{Rotationsfusion}
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Rotationen werden gesammelt:
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\[
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R_1, R_2, ..., R_n
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\]
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Eine einfache erste Näherung:
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\begin{itemize}
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\item Eulerwinkel mitteln
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\item oder Quaternionen mitteln
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\end{itemize}
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Später empfohlen:
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\begin{itemize}
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\item SVD-basierte Rotationsmittelung
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\item Lie-Group Mittelung auf $SO(3)$
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\end{itemize}
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\section{Qualitätsmetriken}
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Das Script berechnet:
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\begin{itemize}
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\item Anzahl beobachtender Kameras
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\item Positionsstreuung
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\item Reprojektionsfehler
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\item Gesamtgewicht
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\item Sichtwinkel
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\end{itemize}
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Beispiel:
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\[
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\sigma =
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\sqrt{
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\frac{1}{N}
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\sum_i ||p_i - \bar{p}||^2
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}
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\]
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\section{Rigid-Body Erweiterung}
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Später können Marker über bekannte Relativpositionen gekoppelt werden.
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Beispiel:
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\[
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p_{M2} = p_{M1} + R_{Body} \Delta p
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\]
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Dadurch können Marker rekonstruiert werden, selbst wenn sie nicht direkt sichtbar sind.
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\section{Spätere Erweiterungen}
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Geplant:
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\begin{itemize}
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\item Bundle Adjustment
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\item Kinematic Constraints
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\item Joint Solver
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\item Graph Optimization
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\item Temporal Tracking
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\end{itemize}
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\end{document}
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28
documentation/Phase1_Phase2.aux
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documentation/Phase1_Phase2.aux
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\relax
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||||
\@writefile{toc}{\contentsline {section}{\numberline {1}Ziel}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {2}Ausgangslage}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {3}Grundidee}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {4}Vorteil}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {5}Erwartete Verbesserungen}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Stabilität}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Konsistenz}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Multi-Camera-Verkettung}{3}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {6}Benötigte Daten}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Bereits vorhanden}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.1}Absolute Marker}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.2}Relative Markerpositionen}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.3}Body-Zuordnung}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {6.1.4}Gelenke}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {7}Noch fehlende Daten}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Marker-Orientierung relativ zum Body}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {8}Geplante Solver-Strategie}{5}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {8.1}Phase 2A --- Rigid Body Fit}{5}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {8.2}Phase 2B --- Joint Constraints}{5}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {8.3}Phase 2C --- Global Optimization}{5}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {9}Wichtige Architekturentscheidung}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {10}Geplante Datenstruktur}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {10.1}Weltpose eines Körpers}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {10.2}Relative Markerdefinition}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {11}Phase 1 Reminder}{6}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {12}Zielbild}{7}{}\protected@file@percent }
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\gdef \@abspage@last{7}
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BIN
documentation/Phase1_Phase2.pdf
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documentation/Phase1_Phase2.pdf
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documentation/Phase1_Phase2.tex
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documentation/Phase1_Phase2.tex
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\documentclass[a4paper,11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{geometry}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{listings}
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\usepackage{xcolor}
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\title{Phase 2 --- Kinematic-Constrained Multi-Camera Solver}
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\author{}
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\date{}
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\begin{document}
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\maketitle
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\section{Ziel}
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Nach Phase 1 existiert:
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\begin{itemize}
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\item ein gemeinsames Weltkoordinatensystem
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\item Kameraposen aller Kameras
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\item bekannte absolute Markerpositionen
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\item fusionierte Beobachtungen mehrerer Kameras
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\end{itemize}
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Phase 2 erweitert das System um:
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\begin{itemize}
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\item Rigid-Body Constraints
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\item mechanische Zusammenhänge
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\item Gelenke
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\item relative Marker-Geometrien (\texttt{relPos})
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\item Stabilisierung bei wenigen sichtbaren Markern
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\end{itemize}
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\section{Ausgangslage}
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Aktuell wird jeder Marker unabhängig behandelt.
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Das ist suboptimal, weil:
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|
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\begin{itemize}
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||||
\item viele Marker nur kurz sichtbar sind
|
||||
\item oft nur 1--2 Marker eines Bauteils sichtbar sind
|
||||
\item solvePnP bei wenigen Markern instabil wird
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||||
\item Markerrauschen direkt in die Weltkoordinaten eingeht
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||||
\end{itemize}
|
||||
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Mechanisch sind die Marker jedoch nicht unabhängig.
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||||
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||||
Mehrere Marker gehören jeweils zu:
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||||
|
||||
\begin{itemize}
|
||||
\item Arm1
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||||
\item Arm2
|
||||
\item Joint1
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||||
\item Base
|
||||
\item Finger1
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||||
\item Finger2
|
||||
\end{itemize}
|
||||
|
||||
und bilden jeweils starre Körper (Rigid Bodies).
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||||
|
||||
\section{Grundidee}
|
||||
|
||||
Statt einzelne Marker zu lösen:
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||||
|
||||
\begin{verbatim}
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||||
Marker -> Welt
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||||
\end{verbatim}
|
||||
|
||||
wird gelöst:
|
||||
|
||||
\begin{verbatim}
|
||||
RigidBody -> Welt
|
||||
\end{verbatim}
|
||||
|
||||
und daraus:
|
||||
|
||||
\begin{verbatim}
|
||||
Marker = RigidBody * relTransform
|
||||
\end{verbatim}
|
||||
|
||||
\section{Vorteil}
|
||||
|
||||
Schon ein einzelner sichtbarer Marker kann:
|
||||
|
||||
\begin{itemize}
|
||||
\item einen ganzen Körper stabilisieren
|
||||
\item andere unsichtbare Marker indirekt bestimmen
|
||||
\end{itemize}
|
||||
|
||||
Beispiel:
|
||||
|
||||
Wenn Marker 198 sichtbar ist
|
||||
und Marker 229 relativ dazu bekannt ist,
|
||||
dann kann Marker 229 geschätzt werden,
|
||||
auch wenn er aktuell unsichtbar ist.
|
||||
|
||||
\section{Erwartete Verbesserungen}
|
||||
|
||||
\subsection{Stabilität}
|
||||
|
||||
Deutlich stabilere Pose-Schätzung bei:
|
||||
|
||||
\begin{itemize}
|
||||
\item Motion Blur
|
||||
\item wenigen sichtbaren Markern
|
||||
\item schlechten Blickwinkeln
|
||||
\item Teilverdeckungen
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Konsistenz}
|
||||
|
||||
Marker eines Bauteils bleiben:
|
||||
|
||||
\begin{itemize}
|
||||
\item geometrisch korrekt
|
||||
\item starr
|
||||
\item ohne unrealistische Verzerrungen
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Multi-Camera-Verkettung}
|
||||
|
||||
Kameras können indirekt gekoppelt werden.
|
||||
|
||||
Beispiel:
|
||||
|
||||
Cam1 sieht:
|
||||
\begin{itemize}
|
||||
\item Marker 1,2,3
|
||||
\end{itemize}
|
||||
|
||||
Cam2 sieht:
|
||||
\begin{itemize}
|
||||
\item Marker 3,198
|
||||
\end{itemize}
|
||||
|
||||
Cam3 sieht:
|
||||
\begin{itemize}
|
||||
\item Marker 198,229
|
||||
\end{itemize}
|
||||
|
||||
Dadurch wird:
|
||||
\begin{itemize}
|
||||
\item Arm1 relativ zur Welt bestimmbar
|
||||
\item obwohl keine einzelne Kamera alles sieht
|
||||
\end{itemize}
|
||||
|
||||
\section{Benötigte Daten}
|
||||
|
||||
\subsection{Bereits vorhanden}
|
||||
|
||||
\subsubsection{Absolute Marker}
|
||||
|
||||
\begin{verbatim}
|
||||
"position":[x,y,z]
|
||||
\end{verbatim}
|
||||
|
||||
für Board-Marker.
|
||||
|
||||
\subsubsection{Relative Markerpositionen}
|
||||
|
||||
\begin{verbatim}
|
||||
"relPos":[x,y,z]
|
||||
\end{verbatim}
|
||||
|
||||
für Marker auf einem Rigid Body.
|
||||
|
||||
\subsubsection{Body-Zuordnung}
|
||||
|
||||
\begin{verbatim}
|
||||
"on":"Arm1"
|
||||
\end{verbatim}
|
||||
|
||||
\subsubsection{Gelenke}
|
||||
|
||||
\begin{verbatim}
|
||||
"type":"revolute"
|
||||
"axis":[1,0,0]
|
||||
\end{verbatim}
|
||||
|
||||
\section{Noch fehlende Daten}
|
||||
|
||||
\subsection{Marker-Orientierung relativ zum Body}
|
||||
|
||||
Aktuell existiert nur:
|
||||
|
||||
\begin{verbatim}
|
||||
"relPos"
|
||||
\end{verbatim}
|
||||
|
||||
Empfohlen wird zusätzlich:
|
||||
|
||||
\begin{verbatim}
|
||||
"relRot":[rx,ry,rz]
|
||||
\end{verbatim}
|
||||
|
||||
oder alternativ:
|
||||
|
||||
\begin{verbatim}
|
||||
"normal":[x,y,z]
|
||||
"up":[x,y,z]
|
||||
\end{verbatim}
|
||||
|
||||
Denn Marker besitzen nicht nur Position,
|
||||
sondern auch Orientierung.
|
||||
|
||||
Das verbessert spätere Pose-Fits deutlich.
|
||||
|
||||
\section{Geplante Solver-Strategie}
|
||||
|
||||
\subsection{Phase 2A --- Rigid Body Fit}
|
||||
|
||||
Zunächst:
|
||||
|
||||
\begin{itemize}
|
||||
\item pro Element einen starren Körper fitten
|
||||
\item noch keine Gelenkoptimierung
|
||||
\end{itemize}
|
||||
|
||||
Beispiel:
|
||||
|
||||
\begin{verbatim}
|
||||
T_world_arm1
|
||||
\end{verbatim}
|
||||
|
||||
wird geschätzt.
|
||||
|
||||
Alle Marker von Arm1 folgen daraus automatisch.
|
||||
|
||||
\subsection{Phase 2B --- Joint Constraints}
|
||||
|
||||
Danach:
|
||||
|
||||
\begin{itemize}
|
||||
\item Gelenkachsen erzwingen
|
||||
\item Rotationen einschränken
|
||||
\item mechanische Grenzen verwenden
|
||||
\end{itemize}
|
||||
|
||||
Beispiel:
|
||||
|
||||
\begin{verbatim}
|
||||
Arm2 darf sich nur um Y drehen
|
||||
\end{verbatim}
|
||||
|
||||
\subsection{Phase 2C --- Global Optimization}
|
||||
|
||||
Später optional:
|
||||
|
||||
\begin{itemize}
|
||||
\item vollständiges Bundle Adjustment
|
||||
\item gleichzeitige Optimierung aller:
|
||||
\begin{itemize}
|
||||
\item Kameras
|
||||
\item Marker
|
||||
\item Bodies
|
||||
\item Gelenkwinkel
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
|
||||
\section{Wichtige Architekturentscheidung}
|
||||
|
||||
NICHT:
|
||||
|
||||
\begin{verbatim}
|
||||
Marker einzeln lösen
|
||||
\end{verbatim}
|
||||
|
||||
SONDERN:
|
||||
|
||||
\begin{verbatim}
|
||||
Bodies + Constraints lösen
|
||||
\end{verbatim}
|
||||
|
||||
Marker werden damit Beobachtungen,
|
||||
nicht mehr primäre Zustände.
|
||||
|
||||
\section{Geplante Datenstruktur}
|
||||
|
||||
\subsection{Weltpose eines Körpers}
|
||||
|
||||
\begin{verbatim}
|
||||
{
|
||||
"body":"Arm1",
|
||||
"worldPose":{
|
||||
"position":[x,y,z],
|
||||
"rotationMatrix":[...]
|
||||
}
|
||||
}
|
||||
\end{verbatim}
|
||||
|
||||
\subsection{Relative Markerdefinition}
|
||||
|
||||
\begin{verbatim}
|
||||
{
|
||||
"id":198,
|
||||
"on":"Arm1",
|
||||
"relPos":[x,y,z],
|
||||
"relRot":[rx,ry,rz]
|
||||
}
|
||||
\end{verbatim}
|
||||
|
||||
\section{Phase 1 Reminder}
|
||||
|
||||
Phase 1 bleibt weiterhin wichtig:
|
||||
|
||||
\begin{itemize}
|
||||
\item alle Kameras finden
|
||||
\item alle Detection-Dateien laden
|
||||
\item gemeinsame Marker fusionieren
|
||||
\item Weltkoordinaten berechnen
|
||||
\item Qualitätsmetriken speichern
|
||||
\item auch schlechte / unvollständige Beobachtungen abspeichern
|
||||
\end{itemize}
|
||||
|
||||
Die Ergebnisse von Phase 1 dienen als Eingang für Phase 2.
|
||||
|
||||
\section{Zielbild}
|
||||
|
||||
Langfristig entsteht:
|
||||
|
||||
\begin{itemize}
|
||||
\item ein globales Robotermodell
|
||||
\item mit mehreren Kameras
|
||||
\item mehreren Rigid Bodies
|
||||
\item Gelenken
|
||||
\item Unsicherheiten
|
||||
\item Qualitätsmetriken
|
||||
\item temporaler Stabilisierung
|
||||
\end{itemize}
|
||||
|
||||
basierend auf:
|
||||
|
||||
\begin{itemize}
|
||||
\item ArUco-Beobachtungen
|
||||
\item Mechanik
|
||||
\item Kinematik
|
||||
\item Multi-View-Geometrie
|
||||
\end{itemize}
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user