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@default_files=('nass-rotating-3dof-model.tex'); @default_files=('nass-rotating-3dof-model.tex');
# PDF-generating modes are: # PDF-generating modes are:
# 1: pdflatex, as specified by $pdflatex variable (still largely in use) # 1: pdflatex, as specified by $pdflatex variable (still largely in use)
# 2: postscript conversion, as specified by the $ps2pdf variable (useless) # 2: postscript conversion, as specified by the $ps2pdf variable (useless)
# 3: dvi conversion, as specified by the $dvipdf variable (useless) # 3: dvi conversion, as specified by the $dvipdf variable (useless)
# 4: lualatex, as specified by the $lualatex variable (best) # 4: lualatex, as specified by the $lualatex variable (best)
# 5: xelatex, as specified by the $xelatex variable (second best) # 5: xelatex, as specified by the $xelatex variable (second best)
$pdf_mode = 1; $pdf_mode = 4;
# Treat undefined references and citations as well as multiply defined references as # Treat undefined references and citations as well as multiply defined references as
# ERRORS instead of WARNINGS. # ERRORS instead of WARNINGS.
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$warnings_as_errors = 0; $warnings_as_errors = 0;
# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853 # Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853
$show_time = 1; $show_time = 0;
# Default is 5; we seem to need more owed to the complexity of the document. # Default is 5; we seem to need more owed to the complexity of the document.
# Actual documents probably don't need this many since they won't use all features, # Actual documents probably don't need this many since they won't use all features,
# plus won't be compiling from cold each time. # plus won't be compiling from cold each time.
$max_repeat=7; $max_repeat=10;
# --shell-escape option (execution of code outside of latex) is required for the # --shell-escape option (execution of code outside of latex) is required for the
#'svg' package. #'svg' package.
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set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S"); set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S");
# Use default pdf viewer # Use default pdf viewer
$pdf_previewer = 'zathura'; $pdf_update_method = 1;
$pdf_previewer = "zathura %O %S";
# option 2 is same as 1 (run biber when necessary), but also deletes the # option 2 is same as 1 (run biber when necessary), but also deletes the
# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original # regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original

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org-ref-acronyms-before-parsing)) org-ref-acronyms-before-parsing))
#+END_SRC #+END_SRC
* Notes :noexport:
Prefix is =rotating=
* Introduction :ignore: * Introduction :ignore:
An important aspect of the Nano Active Stabilization System (NASS) is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. An important aspect of the Nano Active Stabilization System (NASS) is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not.

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nass-rotating-3dof-model.tex
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% Created 2023-07-03 Mon 17:38 % Created 2024-03-21 Thu 18:28
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -18,7 +18,7 @@
pdftitle={NASS - Effect of rotation}, pdftitle={NASS - Effect of rotation},
pdfkeywords={}, pdfkeywords={},
pdfsubject={}, pdfsubject={},
pdfcreator={Emacs 28.2 (Org mode 9.5.2)}, pdfcreator={Emacs 29.2 (Org mode 9.7)},
pdflang={English}} pdflang={English}}
\usepackage{biblatex} \usepackage{biblatex}
@ -28,7 +28,6 @@
\tableofcontents \tableofcontents
\clearpage \clearpage
An important aspect of the Nano Active Stabilization System (NASS) is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. An important aspect of the Nano Active Stabilization System (NASS) is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not.
Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performances. Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performances.
@ -65,9 +64,9 @@ Section \ref{sec:rotating_nass} & \texttt{rotating\_6\_nass.m}\\
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\end{table} \end{table}
\chapter{System Description and Analysis} \chapter{System Description and Analysis}
\label{sec:rotating_system_description} \label{sec:rotating_system_description}
The studied system consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure \ref{fig:rotating_3dof_model_schematic}). The studied system consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure \ref{fig:rotating_3dof_model_schematic}).
The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\s}\). The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\s}\).
@ -83,7 +82,6 @@ The position of the payload is represented by \((d_u, d_v, 0)\) expressed in the
\includegraphics[scale=1]{figs/rotating_3dof_model_schematic.png} \includegraphics[scale=1]{figs/rotating_3dof_model_schematic.png}
\caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system} \caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system}
\end{figure} \end{figure}
\section{Equations of motion} \section{Equations of motion}
To obtain the equations of motion for the system represented in Figure \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equations are used: To obtain the equations of motion for the system represented in Figure \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equations are used:
\begin{equation} \begin{equation}
@ -117,7 +115,6 @@ The uniform rotation of the system induces \textbf{two gyroscopic effects} as sh
\end{itemize} \end{itemize}
One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two uncoupled one degree of freedom mass-spring-damper systems. One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two uncoupled one degree of freedom mass-spring-damper systems.
\section{Transfer Functions in the Laplace domain} \section{Transfer Functions in the Laplace domain}
To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation \eqref{eq:Gd_mimo_tf} is obtained. To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation \eqref{eq:Gd_mimo_tf} is obtained.
Its elements are shown in equation \eqref{eq:Gd_indiv_el}. Its elements are shown in equation \eqref{eq:Gd_indiv_el}.
@ -149,7 +146,6 @@ The elements of transfer function matrix \(\mathbf{G}_d\) are now described by e
\mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
\end{align} \end{align}
\end{subequations} \end{subequations}
\section{System Poles: Campbell Diagram} \section{System Poles: Campbell Diagram}
The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation \eqref{eq:poles}. The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation \eqref{eq:poles}.
@ -178,7 +174,6 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo
\includegraphics[scale=1]{figs/rotating_campbell_diagram.png} \includegraphics[scale=1]{figs/rotating_campbell_diagram.png}
\caption{\label{fig:rotating_campbell_diagram}Campbell diagram - Real and Imaginary parts of the poles as a function of the rotating velocity} \caption{\label{fig:rotating_campbell_diagram}Campbell diagram - Real and Imaginary parts of the poles as a function of the rotating velocity}
\end{figure} \end{figure}
\section{System Dynamics: Effect of rotation} \section{System Dynamics: Effect of rotation}
The system dynamics from actuator forces \([F_u, F_v]\) to the relative motion \([d_u, d_v]\) is identified for several rotating velocities. The system dynamics from actuator forces \([F_u, F_v]\) to the relative motion \([d_u, d_v]\) is identified for several rotating velocities.
@ -192,9 +187,9 @@ For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p
\includegraphics[scale=1]{figs/rotating_direct_coupling_bode_plot.png} \includegraphics[scale=1]{figs/rotating_direct_coupling_bode_plot.png}
\caption{\label{fig:rotating_direct_coupling_bode_plot}Bode plot of the direct and coupling terms for several rotating velocities} \caption{\label{fig:rotating_direct_coupling_bode_plot}Bode plot of the direct and coupling terms for several rotating velocities}
\end{figure} \end{figure}
\chapter{Integral Force Feedback} \chapter{Integral Force Feedback}
\label{sec:rotating_iff_pure_int} \label{sec:rotating_iff_pure_int}
In order to further decrease the residual vibrations, active damping can be used for reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat}. In order to further decrease the residual vibrations, active damping can be used for reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat}.
Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. \par Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. \par
@ -210,7 +205,6 @@ Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to deri
However, none of these study have been applied to a rotating system. However, none of these study have been applied to a rotating system.
In this section, Integral Force Feedback strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alters the system dynamics and that IFF cannot be applied as is. In this section, Integral Force Feedback strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alters the system dynamics and that IFF cannot be applied as is.
\section{System and Equations of motion} \section{System and Equations of motion}
In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure \ref{fig:rotating_3dof_model_schematic_iff}). In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure \ref{fig:rotating_3dof_model_schematic_iff}).
Two identical controllers \(K_F\) are then used to feedback each of the sensed force to its associated actuator: Two identical controllers \(K_F\) are then used to feedback each of the sensed force to its associated actuator:
@ -275,7 +269,6 @@ The low frequency gain of \(\mathbf{G}_f\) increases with the rotational speed \
This can be explained as follows: a constant actuator force \(F_u\) induces a small displacement of the mass \(d_u = \frac{F_u}{k - m\Omega^2}\) (Hooke's law taking into account the negative stiffness induced by the rotation). This can be explained as follows: a constant actuator force \(F_u\) induces a small displacement of the mass \(d_u = \frac{F_u}{k - m\Omega^2}\) (Hooke's law taking into account the negative stiffness induced by the rotation).
This small displacement then increases the centrifugal force \(m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u\) which is then measured by the force sensors. This small displacement then increases the centrifugal force \(m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u\) which is then measured by the force sensors.
\section{Effect of the rotation speed on the IFF plant dynamics} \section{Effect of the rotation speed on the IFF plant dynamics}
The transfer functions from actuator forces \([F_u,\ F_v]\) to the measured force sensors \([f_u,\ f_v]\) are identified for several rotating velocities and shown in Figure \ref{fig:rotating_iff_bode_plot_effect_rot}. The transfer functions from actuator forces \([F_u,\ F_v]\) to the measured force sensors \([f_u,\ f_v]\) are identified for several rotating velocities and shown in Figure \ref{fig:rotating_iff_bode_plot_effect_rot}.
@ -292,7 +285,6 @@ A pair of (minimum phase) complex conjugate zeros appears between the two comple
\includegraphics[scale=1]{figs/rotating_iff_bode_plot_effect_rot.png} \includegraphics[scale=1]{figs/rotating_iff_bode_plot_effect_rot.png}
\caption{\label{fig:rotating_iff_bode_plot_effect_rot}Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation} \caption{\label{fig:rotating_iff_bode_plot_effect_rot}Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation}
\end{figure} \end{figure}
\section{Decentralized Integral Force Feedback} \section{Decentralized Integral Force Feedback}
The control diagram for decentralized Integral Force Feedback is shown in Figure \ref{fig:rotating_iff_diagram}. The control diagram for decentralized Integral Force Feedback is shown in Figure \ref{fig:rotating_iff_diagram}.
@ -327,9 +319,9 @@ The control system is thus canceling the spring forces which makes the suspended
\includegraphics[scale=1]{figs/rotating_root_locus_iff_pure_int.png} \includegraphics[scale=1]{figs/rotating_root_locus_iff_pure_int.png}
\caption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown.} \caption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown.}
\end{figure} \end{figure}
\chapter{Integral Force Feedback with an High Pass Filter} \chapter{Integral Force Feedback with an High Pass Filter}
\label{sec:rotating_iff_pseudo_int} \label{sec:rotating_iff_pseudo_int}
As was explained in the previous section, the instability of the IFF controller applied on the rotating system comes in part from the high gain at low frequency caused by the pure integrators. As was explained in the previous section, the instability of the IFF controller applied on the rotating system comes in part from the high gain at low frequency caused by the pure integrators.
In order to limit the low frequency controller gain, an High Pass Filter (HPF) can be added to the controller as shown in equation \eqref{eq:iff_lhf}. In order to limit the low frequency controller gain, an High Pass Filter (HPF) can be added to the controller as shown in equation \eqref{eq:iff_lhf}.
@ -343,7 +335,6 @@ This is equivalent as to slightly \textbf{shifting the controller pole to the le
This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}. This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}.
This is however not why this high pass filter is added here. This is however not why this high pass filter is added here.
\section{Modified Integral Force Feedback Controller} \section{Modified Integral Force Feedback Controller}
The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used: The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
\begin{equation} \begin{equation}
@ -388,7 +379,6 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro
\includegraphics[scale=1]{figs/rotating_iff_root_locus_hpf.png} \includegraphics[scale=1]{figs/rotating_iff_root_locus_hpf.png}
\caption{\label{fig:rotating_iff_root_locus_hpf}Root Locus for the initial IFF and the modified IFF} \caption{\label{fig:rotating_iff_root_locus_hpf}Root Locus for the initial IFF and the modified IFF}
\end{figure} \end{figure}
\section{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)} \section{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)}
Two parameters can be tuned for the modified controller in equation \eqref{eq:iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\). Two parameters can be tuned for the modified controller in equation \eqref{eq:iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\).
The optimal values of \(\omega_i\) and \(g\) are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized. The optimal values of \(\omega_i\) and \(g\) are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
@ -417,7 +407,6 @@ Three regions can be observed:
\includegraphics[scale=1]{figs/rotating_iff_hpf_optimal_gain.png} \includegraphics[scale=1]{figs/rotating_iff_hpf_optimal_gain.png}
\caption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio \(\xi_\text{cl}\) as a function of \(\omega_i/\omega_0\). Corresponding control gain \(g_\text{opt}\) and \(g_\text{max}\) are also shown} \caption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio \(\xi_\text{cl}\) as a function of \(\omega_i/\omega_0\). Corresponding control gain \(g_\text{opt}\) and \(g_\text{max}\) are also shown}
\end{figure} \end{figure}
\section{Obtained Damped Plant} \section{Obtained Damped Plant}
Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compute the damped plant. Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compute the damped plant.
@ -450,9 +439,9 @@ The same trade-off can be seen between achievable damping and loss of compliance
\includegraphics[scale=1]{figs/rotating_iff_hpf_effect_wi_compliance.png} \includegraphics[scale=1]{figs/rotating_iff_hpf_effect_wi_compliance.png}
\caption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of \(\omega_i\) on the obtained compliance} \caption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of \(\omega_i\) on the obtained compliance}
\end{figure} \end{figure}
\chapter{IFF with a stiffness in parallel with the force sensor} \chapter{IFF with a stiffness in parallel with the force sensor}
\label{sec:rotating_iff_parallel_stiffness} \label{sec:rotating_iff_parallel_stiffness}
In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects. In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects.
Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} where \(k_a\) is the stiffness of the actuator and \(k_p\) the added stiffness in parallel with the actuator and force sensor. Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} where \(k_a\) is the stiffness of the actuator and \(k_p\) the added stiffness in parallel with the actuator and force sensor.
@ -462,7 +451,6 @@ Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_sche
\includegraphics[scale=1]{figs/rotating_3dof_model_schematic_iff_parallel_springs.png} \includegraphics[scale=1]{figs/rotating_3dof_model_schematic_iff_parallel_springs.png}
\caption{\label{fig:rotating_3dof_model_schematic_iff_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors (shown in red)} \caption{\label{fig:rotating_3dof_model_schematic_iff_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors (shown in red)}
\end{figure} \end{figure}
\section{Equations} \section{Equations}
The forces measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} are described by Eq. \eqref{eq:measured_force_kp}. The forces measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} are described by Eq. \eqref{eq:measured_force_kp}.
@ -509,7 +497,6 @@ The two real zeros \(z_r\) in Eq. \eqref{eq:iff_zero_real} that were inducing a
\begin{important} \begin{important}
Thus, if the added \textbf{parallel stiffness} \(k_p\) is \textbf{higher than the negative stiffness induced by centrifugal forces} \(m \Omega^2\), the dynamics from actuator to its collocated force sensor will show minimum phase behavior. Thus, if the added \textbf{parallel stiffness} \(k_p\) is \textbf{higher than the negative stiffness induced by centrifugal forces} \(m \Omega^2\), the dynamics from actuator to its collocated force sensor will show minimum phase behavior.
\end{important} \end{important}
\section{Effect of the parallel stiffness on the IFF plant} \section{Effect of the parallel stiffness on the IFF plant}
The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified in three different cases: The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified in three different cases:
\begin{itemize} \begin{itemize}
@ -535,7 +522,6 @@ It is shown that if the added stiffness is higher than the maximum negative stif
\includegraphics[scale=1]{figs/rotating_iff_kp_root_locus.png} \includegraphics[scale=1]{figs/rotating_iff_kp_root_locus.png}
\caption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring} \caption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring}
\end{figure} \end{figure}
\section{Effect of \(k_p\) on the attainable damping} \section{Effect of \(k_p\) on the attainable damping}
Even though the parallel stiffness \(k_p\) has no impact on the open-loop poles (as the overall stiffness \(k\) is kept constant), it has a large impact on the transmission zeros. Even though the parallel stiffness \(k_p\) has no impact on the open-loop poles (as the overall stiffness \(k\) is kept constant), it has a large impact on the transmission zeros.
@ -561,7 +547,6 @@ This is confirmed by the Figure \ref{fig:rotating_iff_kp_optimal_gain} where the
\includegraphics[scale=1]{figs/rotating_iff_kp_optimal_gain.png} \includegraphics[scale=1]{figs/rotating_iff_kp_optimal_gain.png}
\caption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio \(\xi_\text{cl}\) as a function of the parallel stiffness \(k_p\). Corresponding control gain \(g_\text{opt}\) is also shown. Values for \(k_p < m\Omega^2\) are not shown as the system is unstable.} \caption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio \(\xi_\text{cl}\) as a function of the parallel stiffness \(k_p\). Corresponding control gain \(g_\text{opt}\) is also shown. Values for \(k_p < m\Omega^2\) are not shown as the system is unstable.}
\end{figure} \end{figure}
\section{Damped plant} \section{Damped plant}
Let's choose a parallel stiffness equal to \(k_p = 2 m \Omega^2\) and compute the damped plant. Let's choose a parallel stiffness equal to \(k_p = 2 m \Omega^2\) and compute the damped plant.
The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure \ref{fig:rotating_iff_kp_damped_plant}. The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure \ref{fig:rotating_iff_kp_damped_plant}.
@ -602,9 +587,9 @@ The added high pass filter gives almost the same damping properties while giving
\includegraphics[scale=1]{figs/rotating_iff_kp_added_hpf_damped_plant.png} \includegraphics[scale=1]{figs/rotating_iff_kp_added_hpf_damped_plant.png}
\caption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with IFF - Transfer function from \(F_u\) to \(d_u\)} \caption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with IFF - Transfer function from \(F_u\) to \(d_u\)}
\end{figure} \end{figure}
\chapter{Relative Damping Control} \chapter{Relative Damping Control}
\label{sec:rotating_relative_damp_control} \label{sec:rotating_relative_damp_control}
In order to apply a ``relative damping control strategy'', relative motion sensors are added in parallel with the actuators as shown in Figure \ref{fig:rotating_3dof_model_schematic_rdc}. In order to apply a ``relative damping control strategy'', relative motion sensors are added in parallel with the actuators as shown in Figure \ref{fig:rotating_3dof_model_schematic_rdc}.
Two controllers \(K_d\) are used to fed back the relative motion to the actuator. Two controllers \(K_d\) are used to fed back the relative motion to the actuator.
@ -624,7 +609,6 @@ K_d(s) = \frac{s}{s + \omega_d}
\includegraphics[scale=1]{figs/rotating_3dof_model_schematic_rdc.png} \includegraphics[scale=1]{figs/rotating_3dof_model_schematic_rdc.png}
\caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensor and decentralized ``relative damping control'' applied.} \caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensor and decentralized ``relative damping control'' applied.}
\end{figure} \end{figure}
\section{Equations of motion} \section{Equations of motion}
Let's note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators: Let's note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators:
\begin{equation} \begin{equation}
@ -651,7 +635,6 @@ Which are between the two pairs of complex conjugate poles at:
\end{align} \end{align}
Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functions for Relative Damping Control have \textbf{alternating complex conjugate poles and zeros}. Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functions for Relative Damping Control have \textbf{alternating complex conjugate poles and zeros}.
\section{Decentralized Relative Damping Control} \section{Decentralized Relative Damping Control}
The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) is identified and shown in Figure \ref{fig:rotating_rdc_plant_effect_rot} for several rotating velocities. The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) is identified and shown in Figure \ref{fig:rotating_rdc_plant_effect_rot} for several rotating velocities.
@ -669,7 +652,6 @@ The closed-loop system is unconditionally stable and the poles can be damped as
\includegraphics[scale=1]{figs/rotating_rdc_root_locus.png} \includegraphics[scale=1]{figs/rotating_rdc_root_locus.png}
\caption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control} \caption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control}
\end{figure} \end{figure}
\section{Damped Plant} \section{Damped Plant}
Let's select a reasonable ``Relative Damping Control'' gain, and compute the closed-loop damped system. Let's select a reasonable ``Relative Damping Control'' gain, and compute the closed-loop damped system.
The open-loop and damped plants are compared in Figure \ref{fig:rotating_rdc_damped_plant}. The open-loop and damped plants are compared in Figure \ref{fig:rotating_rdc_damped_plant}.
@ -682,14 +664,13 @@ It does not increase the low frequency coupling as compared to Integral Force Fe
\includegraphics[scale=1]{figs/rotating_rdc_damped_plant.png} \includegraphics[scale=1]{figs/rotating_rdc_damped_plant.png}
\caption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control} \caption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control}
\end{figure} \end{figure}
\chapter{Comparison of Active Damping Techniques} \chapter{Comparison of Active Damping Techniques}
\label{sec:rotating_comp_act_damp} \label{sec:rotating_comp_act_damp}
These two proposed IFF modifications as well as relative damping control are now compared in terms of added damping and closed-loop behavior. These two proposed IFF modifications as well as relative damping control are now compared in terms of added damping and closed-loop behavior.
For the following comparisons, the cut-off frequency for the added HPF is set to \(\omega_i = 0.1 \omega_0\) and the stiffness of the parallel springs is set to \(k_p = 5 m \Omega^2\) (corresponding to \(\alpha = 0.05\)). For the following comparisons, the cut-off frequency for the added HPF is set to \(\omega_i = 0.1 \omega_0\) and the stiffness of the parallel springs is set to \(k_p = 5 m \Omega^2\) (corresponding to \(\alpha = 0.05\)).
These values are chosen based on previous discussion about optimal parameters. These values are chosen based on previous discussion about optimal parameters.
\section{Root Locus} \section{Root Locus}
Figure \ref{fig:rotating_comp_techniques_root_locus} shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control. Figure \ref{fig:rotating_comp_techniques_root_locus} shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control.
@ -706,7 +687,6 @@ It is interesting to note that the maximum added damping is very similar for bot
\includegraphics[scale=1]{figs/rotating_comp_techniques_root_locus.png} \includegraphics[scale=1]{figs/rotating_comp_techniques_root_locus.png}
\caption{\label{fig:rotating_comp_techniques_root_locus}Comparison of active damping techniques for rotating platform - Root Locus} \caption{\label{fig:rotating_comp_techniques_root_locus}Comparison of active damping techniques for rotating platform - Root Locus}
\end{figure} \end{figure}
\section{Obtained Damped Plant} \section{Obtained Damped Plant}
The actively damped plants are computed for the three techniques and compared in Figure \ref{fig:rotating_comp_techniques_dampled_plants}. The actively damped plants are computed for the three techniques and compared in Figure \ref{fig:rotating_comp_techniques_dampled_plants}.
@ -720,8 +700,6 @@ Integral Force Feedback strategy is adding some coupling at low frequency which
\includegraphics[scale=1]{figs/rotating_comp_techniques_dampled_plants.png} \includegraphics[scale=1]{figs/rotating_comp_techniques_dampled_plants.png}
\caption{\label{fig:rotating_comp_techniques_dampled_plants}Comparison of the damped plants obtained with the three active damping techniques} \caption{\label{fig:rotating_comp_techniques_dampled_plants}Comparison of the damped plants obtained with the three active damping techniques}
\end{figure} \end{figure}
\section{Transmissibility And Compliance} \section{Transmissibility And Compliance}
The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance. The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance.
@ -744,7 +722,6 @@ This is very well known characteristics of these common active damping technique
\includegraphics[scale=1]{figs/rotating_comp_techniques_transmissibility_compliance.png} \includegraphics[scale=1]{figs/rotating_comp_techniques_transmissibility_compliance.png}
\caption{\label{fig:rotating_comp_techniques_transmissibility_compliance}Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques} \caption{\label{fig:rotating_comp_techniques_transmissibility_compliance}Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques}
\end{figure} \end{figure}
\chapter{Rotating Nano-Hexapod} \chapter{Rotating Nano-Hexapod}
\label{sec:rotating_nano_hexapod} \label{sec:rotating_nano_hexapod}
The current analysis is now applied on a model representing the rotating nano-hexapod. The current analysis is now applied on a model representing the rotating nano-hexapod.
@ -771,7 +748,6 @@ It is shown that the rotation has the largest effect on the soft nano-hexapod:
\includegraphics[scale=1]{figs/rotating_nano_hexapod_dynamics.png} \includegraphics[scale=1]{figs/rotating_nano_hexapod_dynamics.png}
\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics - Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity, and shaded lines are coupling terms at maximum rotating velocity} \caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics - Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity, and shaded lines are coupling terms at maximum rotating velocity}
\end{figure} \end{figure}
\section{Optimal IFF with High Pass Filter} \section{Optimal IFF with High Pass Filter}
Let's apply Integral Force Feedback with an added High Pass Filter to the three nano-hexapods. Let's apply Integral Force Feedback with an added High Pass Filter to the three nano-hexapods.
@ -816,7 +792,6 @@ The Root Locus for all three nano-hexapods are shown in Figure \ref{fig:rotating
\includegraphics[scale=1]{figs/rotating_root_locus_iff_hpf_nass.png} \includegraphics[scale=1]{figs/rotating_root_locus_iff_hpf_nass.png}
\caption{\label{fig:rotating_root_locus_iff_hpf_nass}Root Locus for modified IFF with high pass filter. Optimal \(\omega_i\) is used. The three nano-hexapod stiffnesses are compared. The grey line indicates the minimum damping obtained with the optimal chosen control parameters.} \caption{\label{fig:rotating_root_locus_iff_hpf_nass}Root Locus for modified IFF with high pass filter. Optimal \(\omega_i\) is used. The three nano-hexapod stiffnesses are compared. The grey line indicates the minimum damping obtained with the optimal chosen control parameters.}
\end{figure} \end{figure}
\section{Optimal IFF with Parallel Stiffness} \section{Optimal IFF with Parallel Stiffness}
For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness to have unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness to have unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness).
In order to keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased: In order to keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased:
@ -875,7 +850,6 @@ Similarly to what was found with the IFF and added High Pass Filter:
\includegraphics[scale=1]{figs/rotating_root_locus_iff_kp_nass.png} \includegraphics[scale=1]{figs/rotating_root_locus_iff_kp_nass.png}
\caption{\label{fig:rotating_root_locus_iff_kp_nass}Root Locus for optimal parameters (IFF + \(k_p\) strategy) - Comparison of attainable damping with the three nano-hexapod stiffnesses} \caption{\label{fig:rotating_root_locus_iff_kp_nass}Root Locus for optimal parameters (IFF + \(k_p\) strategy) - Comparison of attainable damping with the three nano-hexapod stiffnesses}
\end{figure} \end{figure}
\section{Optimal Relative Motion Control} \section{Optimal Relative Motion Control}
For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is shown in Figure \ref{fig:rotating_rdc_optimal_gain}. For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is shown in Figure \ref{fig:rotating_rdc_optimal_gain}.
@ -899,7 +873,6 @@ It can be easily applied on the nano-hexapod with and without rotation without m
\includegraphics[scale=1]{figs/rotating_root_locus_rdc_nass.png} \includegraphics[scale=1]{figs/rotating_root_locus_rdc_nass.png}
\caption{\label{fig:rotating_root_locus_rdc_nass}Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods} \caption{\label{fig:rotating_root_locus_rdc_nass}Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods}
\end{figure} \end{figure}
\section{Comparison of the obtained damped plants} \section{Comparison of the obtained damped plants}
Let's now compare the obtained damped plants for the three active damping techniques applied on the three nano-hexapod stiffnesses (Figure \ref{fig:rotating_nass_damped_plant_comp}). Let's now compare the obtained damped plants for the three active damping techniques applied on the three nano-hexapod stiffnesses (Figure \ref{fig:rotating_nass_damped_plant_comp}).
@ -917,7 +890,6 @@ Similarly to what was concluded in previous analysis:
\includegraphics[scale=1]{figs/rotating_nass_damped_plant_comp.png} \includegraphics[scale=1]{figs/rotating_nass_damped_plant_comp.png}
\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with \(k_p\) and RDC) applied on the three nano-hexapod stiffnesses. \(\Omega = 60\,\text{rmp}\) and \(m_n + m_s = \SI{16}{\kg}\).} \caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with \(k_p\) and RDC) applied on the three nano-hexapod stiffnesses. \(\Omega = 60\,\text{rmp}\) and \(m_n + m_s = \SI{16}{\kg}\).}
\end{figure} \end{figure}
\chapter{Nano-Active-Stabilization-System with rotation} \chapter{Nano-Active-Stabilization-System with rotation}
\label{sec:rotating_nass} \label{sec:rotating_nass}
Up until now, the model used consisted of an infinitely stiff vertical rotating stage with a X-Y suspended stage. Up until now, the model used consisted of an infinitely stiff vertical rotating stage with a X-Y suspended stage.
@ -943,7 +915,6 @@ A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the pay
\includegraphics[scale=1]{figs/rotating_nass_model.png} \includegraphics[scale=1]{figs/rotating_nass_model.png}
\caption{\label{fig:rotating_nass_model}3D view of the Nano-Active-Stabilization-System model.} \caption{\label{fig:rotating_nass_model}3D view of the Nano-Active-Stabilization-System model.}
\end{figure} \end{figure}
\section{System dynamics} \section{System dynamics}
The dynamics of the undamped and damped plants are identified. The dynamics of the undamped and damped plants are identified.
@ -976,7 +947,6 @@ To confirm that the coupling is smaller when the stiffness of the nano-hexapod i
\includegraphics[scale=1]{figs/rotating_nass_plant_coupling_comp.png} \includegraphics[scale=1]{figs/rotating_nass_plant_coupling_comp.png}
\caption{\label{fig:rotating_nass_plant_coupling_comp}Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses} \caption{\label{fig:rotating_nass_plant_coupling_comp}Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses}
\end{figure} \end{figure}
\section{Effect of disturbances} \section{Effect of disturbances}
The effect of three disturbances are considered: The effect of three disturbances are considered:
@ -1018,7 +988,6 @@ Conclusions are similar than with the uniaxial (non-rotating) model:
\includegraphics[scale=1]{figs/rotating_nass_effect_direct_forces.png} \includegraphics[scale=1]{figs/rotating_nass_effect_direct_forces.png}
\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses} \caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses}
\end{figure} \end{figure}
\chapter{Conclusion} \chapter{Conclusion}
In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a spindle model (Section \ref{sec:rotating_system_description}). In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a spindle model (Section \ref{sec:rotating_system_description}).
@ -1039,6 +1008,5 @@ While having very different implementations, both proposed modifications were fo
Then, this study has been applied to a rotating system that corresponds to the nano-hexapod parameters (Section \ref{sec:rotating_nano_hexapod}). Then, this study has been applied to a rotating system that corresponds to the nano-hexapod parameters (Section \ref{sec:rotating_nano_hexapod}).
To be closer to the real system dynamics, the limited compliance of the micro-station has been taken into account. To be closer to the real system dynamics, the limited compliance of the micro-station has been taken into account.
Results show that the two proposed IFF modifications can be applied for the NASS even in the presence of spindle rotation. Results show that the two proposed IFF modifications can be applied for the NASS even in the presence of spindle rotation.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]
\end{document} \end{document}

7
preamble.tex
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@ -1,6 +1,9 @@
\usepackage{float} \usepackage{float}
\usepackage{caption,tabularx,booktabs}
\usepackage{caption,tabularx,booktabs}
\usepackage{bm}
\usepackage{xpatch} % Recommanded for biblatex
\usepackage[ % use biblatex for bibliography \usepackage[ % use biblatex for bibliography
backend=biber, % use biber backend (bibtex replacement) or bibtex backend=biber, % use biber backend (bibtex replacement) or bibtex
style=ieee, % bib style style=ieee, % bib style
@ -19,7 +22,7 @@
mincitenames=1 % mincitenames=1 %
]{biblatex} ]{biblatex}
\usepackage{fontawesome} \setlength\bibitemsep{1.1\itemsep}
\usepackage{caption} \usepackage{caption}
\usepackage{subcaption} \usepackage{subcaption}