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