tdehaeze/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update Figures, paper compiling fine

Browse Source
This commit is contained in:
Thomas Dehaeze
2020-06-23 19:34:34 +02:00
parent 664c2bb4dd
commit e1eae66184
32 changed files with 9388 additions and 5136 deletions
Download Patch File Download Diff File Expand all files Collapse all files

13
paper/paper.org
View File

@@ -20,6 +20,9 @@
#+LATEX_HEADER: \usepackage{algorithmic, graphicx, textcomp}
#+LATEX_HEADER: \usepackage{xcolor, import, hyperref}
#+LATEX_HEADER: \usepackage[USenglish]{babel}
#+LATEX_HEADER_EXTRA: \usepackage{tikz}
#+LATEX_HEADER_EXTRA: \usetikzlibrary{shapes.misc}
#+LATEX_HEADER: \setcounter{footnote}{1}
#+LATEX_HEADER: \input{config.tex}
@@ -56,6 +59,10 @@
*** Establish the importance of the research topic :ignore:
# Active Damping + Rotating System
Controller Poles are shown by black crosses (
\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
).
*** Applications of active damping :ignore:
# Link to previous paper / tomography
@@ -122,7 +129,7 @@ The Lagrangian is the kinetic energy minus the potential energy:
L = T - V
\end{equation}
From the Lagrange's equations of the second kind eqref:eq:lagrange_second_kind, the equation of motion eqref:eq:eom_mixed is obtained ($q_1 = u$, $q_2 = v$).
From the Lagrange's equations of the second kind, the equation of motion is obtained ($q_1 = u$, $q_2 = v$).
\begin{equation}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
\end{equation}
@@ -319,7 +326,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
#+name: fig:root_locus_modified_iff
#+caption: Figure caption
#+attr_latex: :scale 1
[[file:figs/root_locus_modified_iff_bis.pdf]]
[[file:figs/root_locus_modified_iff_ter.pdf]]
** Optimal Cut-Off Frequency
@@ -354,7 +361,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
#+name: fig:root_locus_iff_kp_bis
#+caption: Figure caption
#+attr_latex: :scale 1
[[file:figs/root_locus_iff_kp_bis.pdf]]
[[file:figs/root_locus_iff_kp_ter.pdf]]
#+name: fig:root_locus_opt_gain_iff_kp
#+caption: Figure caption

BIN
paper/paper.pdf
View File

Binary file not shown.

61
paper/paper.tex
View File

@@ -1,4 +1,4 @@
% Created 2020-06-22 lun. 17:38
% Created 2020-06-23 mar. 19:34
% Intended LaTeX compiler: pdflatex
\documentclass{ISMA_USD2020}
\usepackage[utf8]{inputenc}
@@ -32,8 +32,17 @@
\affil[2] {BEAMS Department\NewLineAffil Free University of Brussels, Belgium \NewAffil}
\affil[3] {European Synchrotron Radiation Facility \NewLineAffil Grenoble, France e-mail: \textbf{thomas.dehaeze@esrf.fr}}
\bibliographystyle{IEEEtran}
\usepackage{tikz}
\usetikzlibrary{shapes.misc}
\date{}
\title{Active Damping of Rotating Positioning Platforms}
\hypersetup{
pdfauthor={},
pdftitle={Active Damping of Rotating Positioning Platforms},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 27.0.91 (Org mode 9.4)},
pdflang={English}}
\begin{document}
\maketitle
@@ -43,14 +52,17 @@
}
\section{Introduction}
\label{sec:org67e0a4e}
\label{sec:org977317c}
\label{sec:introduction}
Controller Poles are shown by black crosses (
\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
).
\cite{dehaeze18_sampl_stabil_for_tomog_exper}
\section{System Under Study}
\label{sec:org85bcde2}
\label{sec:org042e800}
\subsection{Rotating Positioning Platform}
\label{sec:org4959a5e}
\label{sec:org489e4b9}
Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
\begin{itemize}
@@ -75,7 +87,7 @@ Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
\end{figure}
\subsection{Equation of Motion}
\label{sec:orgdb109d9}
\label{sec:orgb1836d5}
The system has two degrees of freedom and is thus fully described by the generalized coordinates \(u\) and \(v\).
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\) (neglecting the rotational energy):
@@ -96,7 +108,7 @@ The Lagrangian is the kinetic energy minus the potential energy:
L = T - V
\end{equation}
From the Lagrange's equations of the second kind \eqref{eq:lagrange_second_kind}, the equation of motion \eqref{eq:eom_mixed} is obtained (\(q_1 = u\), \(q_2 = v\)).
From the Lagrange's equations of the second kind, the equation of motion is obtained (\(q_1 = u\), \(q_2 = v\)).
\begin{equation}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
\end{equation}
@@ -120,7 +132,7 @@ Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to \tex
\subsection{Transfer Functions in the Laplace domain}
\label{sec:orgfcd3def}
\label{sec:orgb1002ed}
\begin{subequations}
\begin{align}
@@ -159,7 +171,7 @@ With:
\subsection{Constant Rotating Speed}
\label{sec:org81c7074}
\label{sec:orga4faf60}
To simplify, let's consider a constant rotating speed \(\dot{\theta} = \Omega\) and thus \(\ddot{\theta} = 0\).
\begin{equation}
@@ -214,9 +226,9 @@ The magnitude of the coupling terms are increasing with the rotation speed.
\end{figure}
\section{Integral Force Feedback}
\label{sec:orgc6c1b99}
\label{sec:orgaf500b0}
\subsection{Control Schematic}
\label{sec:orgb93b297}
\label{sec:orgbd9f859}
Force Sensors are added in series with the actuators as shown in Figure \ref{fig:system_iff}.
@@ -227,7 +239,7 @@ Force Sensors are added in series with the actuators as shown in Figure \ref{fig
\end{figure}
\subsection{Equations}
\label{sec:org4072ea4}
\label{sec:org48206d5}
The sensed forces are equal to:
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
@@ -260,7 +272,7 @@ Which then gives:
\subsection{Plant Dynamics}
\label{sec:org0250ac0}
\label{sec:orgec8431d}
\begin{figure}[htbp]
\centering
@@ -269,19 +281,18 @@ Which then gives:
\end{figure}
\subsection{Physical Interpretation}
\label{sec:orgb2d79d2}
\label{sec:org159680e}
At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
\section{Integral Force Feedback with High Pass Filters}
\label{sec:orgabf7a6a}
\label{sec:org694707d}
\subsection{Modification of the Control Low}
\label{sec:org4766bd6}
\label{sec:org931fb10}
\subsection{Close Loop Analysis}
\label{sec:org4c639fd}
\label{sec:org9de0aa7}
\begin{figure}[htbp]
\centering
@@ -291,12 +302,12 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_modified_iff_bis.pdf}
\includegraphics[scale=1]{figs/root_locus_modified_iff_ter.pdf}
\caption{\label{fig:root_locus_modified_iff}Figure caption}
\end{figure}
\subsection{Optimal Cut-Off Frequency}
\label{sec:orge829a45}
\label{sec:org9808de1}
\begin{figure}[htbp]
\centering
@@ -312,7 +323,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\section{Integral Force Feedback with Parallel Springs}
\label{sec:orgd96ea25}
\label{sec:orgd4915d5}
\begin{figure}[htbp]
\centering
@@ -334,7 +345,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_iff_kp_bis.pdf}
\includegraphics[scale=1]{figs/root_locus_iff_kp_ter.pdf}
\caption{\label{fig:root_locus_iff_kp_bis}Figure caption}
\end{figure}
@@ -351,7 +362,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\section{Direct Velocity Feedback}
\label{sec:org027d051}
\label{sec:orgb0a5870}
\begin{figure}[htbp]
\centering
@@ -360,7 +371,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\section{Comparison of the Proposed Active Damping Techniques}
\label{sec:org1eaa959}
\label{sec:org6097c1d}
\begin{figure}[htbp]
\centering
@@ -381,12 +392,12 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\section{Conclusion}
\label{sec:org1b2b4ae}
\label{sec:org1624a6b}
\label{sec:conclusion}
\section*{Acknowledgment}
\label{sec:org2ae16a5}
\label{sec:org1b29790}
\bibliography{ref.bib}
\end{document}