360 lines
No EOL
12 KiB
TeX
360 lines
No EOL
12 KiB
TeX
\documentclass[aspectratio=169]{beamer} \usepackage[utf8]{inputenc}
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\usepackage[french]{babel}
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\usepackage{minted}
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\usepackage{xcolor}
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\usepackage{tikz}
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\usepackage{appendixnumberbeamer}
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\definecolor{boxcolor}{HTML}{23373B}
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\usetikzlibrary{shapes,arrows}
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\tikzstyle{block} = [draw, fill=boxcolor!30, rectangle, minimum height=3em, minimum width=6em]
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\tikzstyle{sum} = [draw, fill=boxcolor!30, circle, node distance=1cm]
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\tikzstyle{input} = [coordinate]
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\tikzstyle{output} = [coordinate]
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\tikzstyle{pinstyle} = [pin edge={to-,thin,black}]
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\usepackage{amsmath,amsmath,amsfonts,amssymb}
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\usetheme{metropolis}
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\setbeamertemplate{footline}[frame number]
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\newcommand*\intervalleEntier[2]{\intervalle{\llbracket}{#1}{#2}{\rrbracket}}
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\newcommand*\intervalle[4]{\left#1 #2 \, ; #3 \right#4}
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\newcommand*\floor[1]{\left \lfloor{#1}\right \rfloor}
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\title{\includegraphics[width=8cm]{LogoCS.png}\\Asservissement rigide de drones}
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\author{Hugo \bsc{Levy--Falk} et Joanne \bsc{Steiner}}\institute{CentraleSupélec}
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\date{Soutenance de projet}
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\begin{document}
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\begin{frame}
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\titlepage
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\end{frame}
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\section{Introduction}
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\begin{frame}{Objectifs du projet}
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Asservissement plus "rigide" d'un drone
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\begin{itemize}
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\item<2-> Stabilité
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\item<3-> Rapidité
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\end{itemize}
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\end{frame}
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\begin{frame}{Livrables}
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\begin{itemize}
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\item Une bibliothèque d'automatique;
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\item Un exemple d'asservissement en simple boucle.
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\end{itemize}
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\end{frame}
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\section{Utilisation de la bibliothèque}
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\begin{frame}{Pourquoi ?}
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\begin{itemize}
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\item Faire de l'automatique avec ROS;
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\item Utilisable dans des cas plus généraux que l'asservissement d'un drone.
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\end{itemize}
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\end{frame}
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\begin{frame}{Fonctionnement}
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\begin{itemize}
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\item Script Python permettant de créer des n\oe uds;
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\item Fichiers de configuration pour gérer les paramètres dynamiquement;
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\item Fichiers launch pour relier les n\oe uds entre eux.
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\end{itemize}
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\begin{center}
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\includegraphics[width=0.3\linewidth]{images/presentation/brique.jpg}
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\end{center}
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\end{frame}
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\begin{frame}{Différents n\oe uds}
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\begin{itemize}
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\item On a les n\oe uds suivants :
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\begin{itemize}
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\item \mintinline{python}|ProportionalNode|;
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\item \mintinline{python}|SaturateNode|;
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\item \mintinline{python}|DerivativeNode|;
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\item \mintinline{python}|DifferenciateNode|;
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\item \mintinline{python}|SumNode|;
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\item \mintinline{python}|InputNode|;
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\item \mintinline{python}|RateNode|.
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\end{itemize}
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\item Calcul de la vitesse du n\oe ud \mintinline{python}|DerivativeNode| grâce à un filtre de \bsc{Savitzky-Golay}.
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\end{itemize}
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\end{frame}
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\begin{frame}{Performances de \bsc{Savitzky-Golay}}
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\begin{figure}
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\centering
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\includegraphics[height=0.7\textheight]{images/realisation/mesure_vitesse_big_quadra.eps}
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\caption{Filtre de \bsc{Savitsky-Golay}}
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\end{figure}
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\end{frame}
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\begin{frame}[fragile]{Utilisation du script}
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\begin{minted}{bash}
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python control_compute.py derivative
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python control_compute.py proportional
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python control_compute.py sum 3
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...
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\end{minted}
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\end{frame}
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\begin{frame}{Utilisation du script}
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\includegraphics[width=\linewidth]{images/realisation/control_launch.png}
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\end{frame}
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\section{Interface avec le drone}
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\begin{frame}{Rappels}
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\begin{itemize}
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\item Objectif : Asservir le drone;
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\item Livrable : Une bibliothèque pour faire de l'automatique.
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\end{itemize}
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\end{frame}
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\begin{frame}{Détection des cibles}
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\begin{itemize}
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\item Idée du script : Détecter les zones d'une couleur choisie;
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\item Possibilité de modifier dynamiquement la couleur.
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\end{itemize}
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\end{frame}
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\begin{frame}{Détection du bleu}
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\begin{figure}
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\centering
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\includegraphics[height=0.7\textheight]{images/utilisation/targets.png}
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\caption{Cibles bleues}
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\end{figure}
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\end{frame}
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\begin{frame}{Détection du jaune}
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\begin{figure}
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\centering
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\includegraphics[height=0.7\textheight]{images/utilisation/targets_yellow.png}
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\caption{Cibles jaunes}
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\end{figure}
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\end{frame}
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\begin{frame}{Interface avec le contrôleur}
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\centering
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\includegraphics[width=\linewidth]{images/presentation/schema_mesure.png}
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\end{frame}
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\begin{frame}{Simple boucle : Principe}
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\centering
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\begin{tikzpicture}[auto, node distance=2cm,>=latex']
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% We start by placing the blocks
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\node [input, name=input] {};
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\node [sum, right of=input, node distance=3cm] (sum) {};
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\node [block, right of=sum] (controller) {$C(p)$};
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\node [block, right of=controller,
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node distance=3cm] (system) {Drone};
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% We draw an edge between the controller and system block to
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% calculate the coordinate u. We need it to place the measurement block.
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\node [output, right of=system, node distance=5cm] (output) {};
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\node [block, below of=system] (measure) {Mesure};
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% Once the nodes are placed, connecting them is easy.
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\draw [draw,->] (input) -- node[pos=0.99] {$+$} (sum);
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\draw [->] (sum) -- node {$\epsilon$} (controller);
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\draw [->] (controller) -- (system);
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\draw [->] (system) -- node [name=y] {position }(output);
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\draw [->] (y) |- (measure);
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\draw [->] (measure) -| node[pos=0.99] {$-$}
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node [near end] {mesure de la position} (sum);
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\end{tikzpicture}
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\end{frame}
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\begin{frame}{Axes du drone}
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\includegraphics[height=\textheight]{images/modelisation/drone_axes.png}
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\end{frame}
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\begin{frame}{Diagramme de bode selon l'axe z et l'angle z}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=0.5\linewidth]{images/modelisation/bode_z.eps}
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\caption{Diagramme de Bode simplifié}
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$C(p)=K_p+K_i/p + K_d p$
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\end{figure}
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\end{frame}
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\begin{frame}{Diagramme de Bode selon les axes x et y}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=0.5\linewidth]{images/modelisation/bode_u.eps}
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\caption{Diagramme de Bode simplifié}
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$C(p)=K_p+K_d p$
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\end{figure}
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\end{frame}
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\begin{frame}{Réglage de la simple boucle}
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\begin{itemize}
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\item Axe z et angle z : Méthode de \bsc{Ziegler-Nichols} en oscillations entretenues, puis ajustements;
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\item Axe x et y : Méthode empirique car \bsc{Ziegler-Nichols} trop instable;
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\item Amélioration éventuelle avec une double boucle
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\end{itemize}
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\end{frame}
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\section{Conclusion}
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\begin{frame}{Conclusion}
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\begin{itemize}
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\item Bibliothèque fonctionnelle;
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\item Asservissement fonctionnelle extensible à d'autres applications;
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\item Extension éventuelle à une double boucle.
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\end{itemize}
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\includegraphics[height=0.5\textheight]{images/presentation/marvin.jpg}
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\end{frame}
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\appendix
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\begin{frame}{Diagramme UML}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/realisation/uml.png}
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\caption{Diagramme UML}
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\label{fig:uml}
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\end{figure}
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\end{frame}
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\begin{frame}{Savitzky-Golay}
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\begin{columns}[T]
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\begin{column}{.48\textwidth}
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$((x_i,y_i))_{i\in\intervalleEntier{1}{n}}$ les $n$ points de la fenêtre
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Polynôme $a_0 + a_1 X + \dots + a_k X^k$
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\begin{equation*}
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\left \|
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\begin{pmatrix}
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y_1 \\
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y_2 \\
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\vdots \\
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y_n
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\end{pmatrix}
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-
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\begin{pmatrix}
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1 & x_1 & x_1 ^2 & \cdots & x_1 ^ k \\
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1 & x_2 & x_2 ^2 & \cdots & x_2 ^ k \\
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\vdots & \vdots & \vdots & & \vdots \\
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1 & x_n & x_n ^2 & \cdots & x_n ^ k
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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a_0 \\
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a_1 \\
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\vdots \\
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a_k
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\end{pmatrix}
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\right \| ^2
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\label{eq:moindrecarres_raw}
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\end{equation*}
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\end{column}
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\begin{column}{.48\textwidth}
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En supposant les $x_i$ centrés en 0 et espacés régulièrement d'un pas $h$, et $k$ impair,
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Savitsky-Golay}
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\begin{eqnarray*}
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L((y_k),(a_k)) &=&
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\left \|
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\underbrace{
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\begin{pmatrix}
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y_1 \\
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y_2 \\
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\vdots \\
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y_n
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\end{pmatrix}}_{=Y}
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-
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\underbrace{
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\begin{pmatrix}
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1 & -\floor{\frac{n}{2}} & \left(-\floor{\frac{n}{2}}\right) ^2 & \cdots & \left(-\floor{\frac{n}{2}}\right) ^ k \\
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1 & \left(-\floor{\frac{n-1}{2}}\right) & \left(-\floor{\frac{n-1}{2}}\right) ^2 & \cdots & \left(-\floor{\frac{n-1}{2}}\right) ^ k \\
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\vdots & \vdots & \vdots & \vdots & \vdots \\
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1 & -1 & \left(-1\right) ^2 & \cdots & \left(-1\right) ^ k \\
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1 & 0 & 0 & \cdots & 0 \\
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1 & \left(1\right) & \left(1\right) ^2 & \cdots & \left(1\right) ^ k \\
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\vdots & \vdots & \vdots & \vdots & \vdots \\
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1 & \left(\floor{\frac{n-1}{2}}\right) & \left(\floor{\frac{n-1}{2}}\right) ^2 & \cdots & \left(\floor{\frac{n-1}{2}}\right) ^ k \\
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1 & \left(\floor{\frac{n}{2}}\right) & \left(\floor{\frac{n}{2}}\right) ^2 & \cdots & \left(\floor{\frac{n}{2}}\right) ^ k
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\end{pmatrix}}_{=M}
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\cdot
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\underbrace{
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\begin{pmatrix}
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a'_0 \\
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a'_1 \\
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\vdots \\
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a'_k
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\end{pmatrix}}_{=A}
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\right \| ^2 \\
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\label{eq:moindrecarres}
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\end{eqnarray*}
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\end{frame}
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\begin{frame}{Savitsky-Golay}
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\begin{eqnarray}
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\frac{\partial L}{\partial (A)}(A) &=& -2 \cdot M^\intercal \cdot (Y - M \cdot A)\\
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A &=& \underbrace{(M^\intercal\cdot M)^{-1} \cdot M^\intercal}_{=B} \cdot Y
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\end{eqnarray}
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Dérivée d'ordre $d$, on convolue le signal par le filtre RIF $[b_1, \dots b_n ]$, avec
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\begin{equation}
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\forall i \in \intervalleEntier{0}{n}, b_i = B_{d,n-i}
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\end{equation}
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\end{frame}
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\begin{frame}{Méthode de \bsc{Ziegler-Nichols}}
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\centering
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\begin{tabular}{|c|c|}
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\hline
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Type de contrôle & Paramètres de réglage \\
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\hline
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P.I.D & $K_p=0.6 K_u$ $K_i=\frac{T_u}{2}$ $K_d=\frac{T_u}{8}$ \\
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\hline
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P.I.D peu de dépassement & $K_p=0.33 K_u$ $K_i=\frac{T_u}{2}$ $K_d=\frac{T_u}{3}$ \\
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\hline
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P.I.D aucun dépassement & $K_p=0.2 K_u$ $K_i=\frac{T_u}{2}$ $K_d=\frac{T_u}{3}$ \\
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\hline
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\end{tabular}
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\end{frame}
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\begin{frame}{Double boucle}
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\begin{tikzpicture}[auto, node distance=2cm,>=latex',scale=0.7, every node/.style={scale=0.7}]
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% We start by placing the blocks
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\node [input, name=input] {};
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\node [sum, right of=input, node distance=3cm] (sum) {};
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\node [block, right of=sum] (controller) {$C_p(p)$};
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\node [sum, right of=controller, node distance=3cm] (sum_speed) {};
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\node [block, right of=sum_speed] (controller_speed) {$C_v(p)$};
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\node [block, right of=controller_speed,
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node distance=3cm] (system) {Drone};
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\node [block, right of=system,
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node distance=3cm] (integrate) {$\frac{1}{p}$};
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\node [block, below of=controller_speed] (measurements_speed) {Mesure};
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% We draw an edge between the controller and system block to
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% calculate the coordinate u. We need it to place the measurement block.
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\draw [->] (controller) -- node[pos=0.99] {$+$}
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node [near end] {$u$} (sum_speed);
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\node [output, right of=system, node distance=5cm] (output) {};
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\node [block, below of=measurements_speed] (measurements) {Mesure};
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% Once the nodes are placed, connecting them is easy.
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\draw [draw,->] (input) -- node {position attendue +} (sum);
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\draw [->] (sum) -- node {$\epsilon_p$} (controller);
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\draw [->] (sum_speed) -- node {$\epsilon_v$} (controller_speed);
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\draw [->] (controller_speed) -- node [name=x] {} (system);
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\draw [->] (system) -- node [name=v] {} (integrate);
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\draw [->] (measurements_speed) -| node[pos=0.99] {$-$}
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node [near end] {mesure de la vitesse} (sum_speed);
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\draw [->] (integrate) -- node [name=y] {position }(output);
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\draw [->] (v) |- (measurements_speed);
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\draw [->] (y) |- (measurements);
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\draw [->] (measurements) -| node[pos=0.99] {$-$}
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node [near end] {mesure de la position} (sum);
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\end{tikzpicture}
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\end{frame}
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\end{document} |