720 lines
40 KiB
TeX
720 lines
40 KiB
TeX
% Created 2023-02-17 Fri 11:26
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12, parskip=full]{scrreprt}
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\input{preamble.tex}
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\author{Dehaeze Thomas}
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\date{\today}
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\title{NASS - Uniaxial Model}
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\hypersetup{
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pdfauthor={Dehaeze Thomas},
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pdftitle={NASS - Uniaxial Model},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 28.2 (Org mode 9.5.2)},
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pdflang={English}}
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\usepackage{biblatex}
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\begin{document}
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\maketitle
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\tableofcontents
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\clearpage
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In this report, a uniaxial model of the Nano Active Stabilization System (NASS) is developed and used to have a first idea of the challenges involved in this complex system.
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Note that in this document, only the vertical direction is considered (which is the most stiff), but other directions were considered as well and yields similar conclusions.
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The model is schematically shown in Figure \ref{fig:uniaxial_overview_model_sections} where the colors are representing the studied parts in different sections.
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In order to have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section \ref{sec:micro_station_model}).
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Then, a model of the nano-hexapod is added on top of the micro-station.
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With added sample and sensors, this gives a uniaxial dynamical model of the NASS that will be used for further analysis (Section \ref{sec:nano_station_model}).
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The disturbances affecting the position accuracy are identified experimentally (Section \ref{sec:uniaxial_disturbances}) and included in the model for dynamical noise budgeting (Section \ref{sec:uniaxial_noise_budgeting}).
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In all the following analysis, there nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate stiffness.
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Three sample masses are also considered to verify the robustness of the applied control strategies to a change of sample.
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Three active damping techniques are then applied on the nano-hexapod.
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This helps to reduce the effect of disturbances as well as render the system easier to control afterwards (Section \ref{sec:uniaxial_active_damping}).
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Once the system is well damped, a feedback position controller is applied, and the obtained performances are compared (Section \ref{sec:uniaxial_position_control}).
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Conclusion remarks are given in Section \ref{sec:conclusion}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_overview_model_sections.png}
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\caption{\label{fig:uniaxial_overview_model_sections}Uniaxial Micro-Station model in blue (Section \ref{sec:micro_station_model}), Nano-Hexapod and sample models in red (Section \ref{sec:nano_station_model}), Disturbances in yellow (Section \ref{sec:uniaxial_disturbances}), Active Damping in green (Section \ref{sec:uniaxial_active_damping}) and Position control in purple (Section \ref{sec:uniaxial_position_control})}
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\end{figure}
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\chapter{Micro Station Model}
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\label{sec:micro_station_model}
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In this section, a uni-axial model of the micro-station is tuned in order to match measurements made on the micro-station
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The measurement setup is shown in Figure \ref{fig:micro_station_first_meas_dynamics} where several geophones are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station.
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From the measured frequency response functions (FRF), the model can be tuned to approximate the uniaxial dynamics of the micro-station.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=\linewidth]{figs/micro_station_first_meas_dynamics.jpg}
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\caption{\label{fig:micro_station_first_meas_dynamics}Experimental Setup for the first dynamical measurements on the Micro-Station. Geophones are fixed to the micro-station, and the granite as well as the micro-hexapod's top platform are impact with an instrumented hammer}
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\end{figure}
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\section{Measured dynamics}
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The measurement setup is schematically shown in Figure \ref{fig:micro_station_meas_dynamics_schematic} where:
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\begin{itemize}
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\item Two hammer hits are performed, one on the Granite (force \(F_g\)), and one on the micro-hexapod's top platform (force \(F_h\))
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\item The inertial motion of the granite \(x_g\) and the micro-hexapod's top platform \(x_h\) are measured using geophones.
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\end{itemize}
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From the forces applied by the instrumented hammer and the responses of the geophones, the following frequency response functions can be computed:
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\begin{itemize}
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\item from \(F_h\) to \(d_h\) (i.e. the compliance of the micro-station)
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\item from \(F_g\) to \(d_h\) (or from \(F_h\) to \(d_g\))
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\item from \(F_g\) to \(d_g\)
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\end{itemize}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/micro_station_meas_dynamics_schematic.png}
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\caption{\label{fig:micro_station_meas_dynamics_schematic}Measurement setup - Schematic}
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\end{figure}
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Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure \ref{fig:uniaxial_measured_frf_vertical}).
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\begin{minted}[]{matlab}
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%% Load measured FRF
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load('meas_microstation_frf.mat');
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\end{minted}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_measured_frf_vertical.png}
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\caption{\label{fig:uniaxial_measured_frf_vertical}Measured Frequency Response Functions in the vertical direction}
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\end{figure}
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\section{Uniaxial Model}
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The uni-axial model of the micro-station is shown in Figure \ref{fig:uniaxial_comp_frf_meas_model}, with:
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\begin{itemize}
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\item Disturbances:
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\begin{itemize}
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\item \(x_f\): Floor motion
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\item \(f_t\): Stage vibrations
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\end{itemize}
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\item Hammer impacts: \(F_h\) and \(F_g\).
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\item Geophones: \(x_h\) and \(x_g\)
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\end{itemize}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_model_micro_station.png}
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\caption{\label{fig:uniaxial_model_micro_station}Uniaxial model of the micro-station}
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\end{figure}
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Masses are estimated from the CAD.
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\begin{minted}[]{matlab}
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%% Parameters - Mass
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mh = 15; % Micro Hexapod [kg]
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mt = 1200; % Ty + Ry + Rz [kg]
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mg = 2500; % Granite [kg]
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\end{minted}
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And stiffnesses from the data-sheet of stage manufacturers.
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\begin{minted}[]{matlab}
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%% Parameters - Stiffnesses
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kh = 6.11e+07; % [N/m]
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kt = 5.19e+08; % [N/m]
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kg = 9.50e+08; % [N/m]
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\end{minted}
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The damping coefficients are tuned to match the identified damping from the measurements.
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\begin{minted}[]{matlab}
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%% Parameters - damping
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ch = 2*0.05*sqrt(kh*mh); % [N/(m/s)]
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ct = 2*0.05*sqrt(kt*mt); % [N/(m/s)]
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cg = 2*0.08*sqrt(kg*mg); % [N/(m/s)]
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\end{minted}
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\section{Comparison of the model and measurements}
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The comparison between the measurements and the model is done in Figure \ref{fig:uniaxial_comp_frf_meas_model}.
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As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation.
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More accurate models will be used later on.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_comp_frf_meas_model.png}
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\caption{\label{fig:uniaxial_comp_frf_meas_model}Comparison of the measured FRF and identified ones from the uni-axial model}
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\end{figure}
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\chapter{Nano-Hexapod Model}
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\label{sec:nano_station_model}
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A model of the nano-hexapod and sample is now added on top of the uni-axial model of the micro-station (Figure \ref{fig:uniaxial_model_micro_station-nass}).
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Disturbances (shown in red) are:
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\begin{itemize}
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\item \(f_s\): direct forces applied to the sample (for instance cable forces)
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\item \(f_t\): disturbances coming from the imperfect stage scanning performances
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\item \(x_f\): floor motion
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\end{itemize}
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The control signal is the force applied by the nano-hexapod \(f\) and the measurement is the relative motion between the sample and the granite \(d\).
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The sample is here considered as a rigid body and rigidly fixed to the nano-hexapod.
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The effect of having resonances between the sample's point of interest and the nano-hexapod actuator will be considered in further analysis.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_model_micro_station-nass.png}
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\caption{\label{fig:uniaxial_model_micro_station-nass}Uni-axial model of the micro-station with added nano-hexapod (represented in blue) and sample (represented in green)}
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\end{figure}
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\section{Nano-Hexapod Parameters}
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The parameters for the nano-hexapod and sample are:
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\begin{itemize}
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\item \(m_s\) the sample mass that can vary from 1kg up to 50kg
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\item \(m_n\) the nano-hexapod mass which is set to 15kg
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\item \(k_n\) the nano-hexapod stiffness, which can vary depending on the chosen architecture/technology
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\end{itemize}
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As a first example, let's choose a nano-hexapod stiffness of \(10\,N/\mu m\) and a sample mass of 10kg.
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\begin{minted}[]{matlab}
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%% Nano-Hexapod Parameters
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mn = 15; % [kg]
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kn = 1e7; % [N/m]
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cn = 2*0.01*sqrt(mn*kn); % [N/(m/s)]
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%% Sample Mass
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ms = 10; % [kg]
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\end{minted}
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\section{Obtained Dynamics}
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The sensitivity to disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) are shown in Figure \ref{fig:uniaxial_sensitivity_dist_first_params}.
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The \emph{plant} (i.e. the transfer function from actuator force \(f\) to measured displacement \(d\)) is shown in Figure \ref{fig:uniaxial_plant_first_params}.
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For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) combined with three sample's masses (\(m_s = 1\,kg\), \(m_s = 25\,kg\) and \(m_s = 50\,kg\)).
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_sensitivity_dist_first_params.png}
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\caption{\label{fig:uniaxial_sensitivity_dist_first_params}Sensitivity to disturbances}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_plant_first_params.png}
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\caption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces to measured displacement by the metrology}
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\end{figure}
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\chapter{Disturbance Identification}
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\label{sec:uniaxial_disturbances}
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In order to measure disturbances, two geophones are used, on located on the floor and on on the micro-hexapod's top platform (see Figure \ref{fig:micro_station_meas_disturbances}).
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The geophone on the floor is used to measured the floor motion \(x_f\) while the geophone on the micro-hexapod is used to measure vibrations introduced by scanning of the \(T_y\) stage and \(R_z\) stage.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/micro_station_meas_disturbances.png}
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\caption{\label{fig:micro_station_meas_disturbances}Disturbance measurement setup - Schematic}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,width=0.6\linewidth]{figs/micro_station_dynamical_id_setup.jpg}
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\caption{\label{fig:micro_station_dynamical_id_setup}Two geophones are used to measure the micro-station vibrations induced by the scanning of the \(T_y\) and \(R_z\) stages}
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\end{figure}
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\section{Ground Motion}
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The geophone fixed to the floor to measure the floor motion.
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\begin{minted}[]{matlab}
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%% Load floor motion data
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% t: time in [s]
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% V: measured voltage genrated by the geophone and amplified by a 60dB gain voltage amplifier [V]
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load('ground_motion_measurement.mat', 't', 'V');
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\end{minted}
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The voltage generated by each geophone is amplified using a voltage amplifier (gain of 60dB) before going to the ADC.
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The sensitivity of the geophone as well as the gain of the voltage amplifier are then taken into account to reconstruct the floor displacement.
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\begin{minted}[]{matlab}
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%% Sensitivity of the geophone
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S0 = 88; % Sensitivity [V/(m/s)]
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f0 = 2; % Cut-off frequency [Hz]
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S = S0*(s/2/pi/f0)/(1+s/2/pi/f0); % Geophone's transfer function [V/(m/s)]
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%% Gain of the voltage amplifier
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G0_db = 60; % [dB]
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G0 = 10^(G0_db/20); % [abs]
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%% Transfer function from measured voltage to displacement
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G_geo = 1/S/G0/s; % [m/V]
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\end{minted}
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The PSD \(S_{V_f}\) of the measured voltage \(V_f\) is computed.
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\begin{minted}[]{matlab}
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%% Compute measured voltage PSD
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Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
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win = hanning(ceil(2*Fs)); % Hanning window
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[psd_V, f] = pwelch(V, win, [], [], Fs); % [V^2/Hz]
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\end{minted}
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The PSD of the corresponding displacement can be computed as follows:
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\begin{equation}
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S_{x_f}(\omega) = \frac{S_{V_f}(\omega)}{|S_{\text{geo}}(j\omega)| \cdot G_{\text{amp}} \cdot \omega}
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\end{equation}
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with:
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\begin{itemize}
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\item \(S_{\text{geo}}\) the sensitivity of the Geophone in \([Vs/m]\)
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\item \(G_{\text{amp}}\) the gain of the voltage amplifier
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\item \(\omega\) is here to integrate and have the displacement instead of the velocity
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\end{itemize}
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\begin{minted}[]{matlab}
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%% Ground Motion ASD
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psd_xf = psd_V.*abs(squeeze(freqresp(G_geo, f, 'Hz'))).^2; % [m^2/Hz]
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\end{minted}
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The amplitude spectral density \(\Gamma_{x_f}\) of the measured displacement \(x_f\) is shown in Figure \ref{fig:asd_floor_motion_id31}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/asd_floor_motion_id31.png}
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\caption{\label{fig:asd_floor_motion_id31}Measured Amplitude Spectral Density of the Floor motion on ID31}
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\end{figure}
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\section{Stage Vibration}
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During Spindle rotation (here at 6rpm), the granite velocity and micro-hexapod's top platform velocity are measured with the geophones.
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\begin{minted}[]{matlab}
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%% Measured velocity of granite and hexapod during spindle rotation
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% t: time in [s]
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% vg: measured granite velocity [m/s]
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% vg: measured micro-hexapod's top platform velocity [m/s]
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load('meas_spindle_on.mat', 't', 'vg', 'vh');
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spindle_off = load('meas_spindle_off.mat', 't', 'vg', 'vh'); % No Rotation
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\end{minted}
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The Power Spectral Density of the relative velocity between the hexapod and the granite is computed.
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\begin{minted}[]{matlab}
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%% Compute Power Spectral Density of the relative velocity between granite and hexapod during spindle rotation
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Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
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win = hanning(ceil(2*Fs)); % Hanning window
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[psd_vft, f] = pwelch(vh-vg, win, [], [], Fs); % [(m/s)^2/Hz]
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[psd_off, ~] = pwelch(spindle_off.vh-spindle_off.vg, win, [], [], Fs); % [(m/s)^2/Hz]
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\end{minted}
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It is then integrated to obtain the Amplitude Spectral Density of the relative motion which is compared with a non-rotating case (Figure \ref{fig:asd_vibration_spindle_rotation}).
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It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/asd_vibration_spindle_rotation.png}
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\caption{\label{fig:asd_vibration_spindle_rotation}Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating}
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\end{figure}
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In order to compute the equivalent disturbance force \(f_t\) that induces such motion, the transfer function from \(f_t\) to the relative motion of the hexapod's top platform and the granite is extracted from the model.
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The power spectral density \(\Gamma_{f_{t}}\) of the disturbance force can be computed as follows:
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\begin{equation}
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\Gamma_{f_{t}}(\omega) = \frac{\Gamma_{v_{t}}(\omega)}{|G_{\text{model}}(j\omega)|^2}
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\end{equation}
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with:
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\begin{itemize}
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\item \(\Gamma_{v_{t}}\) the measured power spectral density of the relative motion between the micro-hexapod's top platform and the granite during the spindle's rotation
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\item \(G_{\text{model}}\) the transfer function (extracted from the uniaxial model) from \(f_t\) to the relative motion between the micro-hexapod's top platform and the granite
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\end{itemize}
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The obtained amplitude spectral density of the disturbance force \(f_t\) is shown in Figure \ref{fig:asd_disturbance_force}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/asd_disturbance_force.png}
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\caption{\label{fig:asd_disturbance_force}Estimated disturbance force ft from measurement and uniaxial model}
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\end{figure}
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The vibrations induced by the \(T_y\) stage are not considered here as:
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\begin{itemize}
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\item the induced vibrations have less amplitude than the vibrations induced by the \(R_z\) stage
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\item it can be scanned at lower velocities if the induced vibrations are an issue
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\end{itemize}
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\chapter{Open-Loop Dynamic Noise Budgeting}
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\label{sec:uniaxial_noise_budgeting}
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Now that we have a model of the NASS and an estimation of the power spectral density of the disturbances, it is possible to perform an \emph{open-loop dynamic noise budgeting}.
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\section{Sensitivity to disturbances}
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From the Uni-axial model, the transfer function from the disturbances (\(f_s\), \(x_f\) and \(f_t\)) to the displacement \(d\) are computed.
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This is done for \textbf{two extreme sample masses} \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and \textbf{three nano-hexapod stiffnesses}:
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\begin{itemize}
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\item \(k_n = 0.01\,N/\mu m\) that could represent a voice coil actuator with soft flexible guiding
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\item \(k_n = 1\,N/\mu m\) that could represent a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator
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\item \(k_n = 100\,N/\mu m\) that could represent a stiff piezoelectric stack actuator
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\end{itemize}
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The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses} for the light sample (same conclusions can be drawn with the heavy one).
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\begin{important}
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From Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}, following can be concluded for the \textbf{soft nano-hexapod}:
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\begin{itemize}
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\item It is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to the lower stiffness
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\item Between the suspension mode of the nano-hexapod (here at 5Hz) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out.
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\item Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is almost \(1\).
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\end{itemize}
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\end{important}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses.png}
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\caption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}Sensitivity to disturbances for three different nano-hexpod stiffnesses}
|
|
\end{figure}
|
|
|
|
\section{Open-Loop Dynamic Noise Budgeting}
|
|
Now, the power spectral density of the disturbances is taken into account to estimate the residual motion \(d\) in each case.
|
|
|
|
The Cumulative Amplitude Spectrum of the relative motion \(d\) due to both the floor motion \(x_f\) and the stage vibrations \(f_t\) are shown in Figure \ref{fig:uniaxial_cas_d_disturbances_stiffnesses} for the three nano-hexapod stiffnesses.
|
|
|
|
It is shown that the effect of the floor motion is much less than the stage vibrations, except for the soft nano-hexapod below 5Hz.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_cas_d_disturbances_stiffnesses.png}
|
|
\caption{\label{fig:uniaxial_cas_d_disturbances_stiffnesses}Cumulative Amplitude Spectrum of the relative motion d, due to both the floor motion and the stage vibrations (light sample)}
|
|
\end{figure}
|
|
|
|
The total cumulative amplitude spectrum for the three nano-hexapod stiffnesses and for the two sample's masses are shown in Figure \ref{fig:uniaxial_cas_d_disturbances_payload_masses}.
|
|
The conclusion is that the sample's mass has little effect on the cumulative amplitude spectrum of the relative motion \(d\).
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_cas_d_disturbances_payload_masses.png}
|
|
\caption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Cumulative Amplitude Spectrum of the relative motion d due to all disturbances, for two sample masses}
|
|
\end{figure}
|
|
|
|
\section{Conclusion}
|
|
\begin{important}
|
|
The conclusion is that in order to have a closed-loop residual vibration \(d \approx 20\,nm\text{ rms}\), if a simple feedback controller is used, the required closed-loop bandwidth would be:
|
|
\begin{itemize}
|
|
\item \(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\))
|
|
\item \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\))
|
|
\item \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\))
|
|
\end{itemize}
|
|
|
|
This can be explain by the fact that above the suspension mode of the nano-hexapod, the stage vibrations are filtered out (see Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}).
|
|
|
|
This gives a first advantage to having a soft nano-hexapod.
|
|
\end{important}
|
|
|
|
\chapter{Active Damping}
|
|
\label{sec:uniaxial_active_damping}
|
|
In this section, three active damping are applied on the nano-hexapod (see Figure \ref{fig:uniaxial_active_damping_strategies}): Integral Force Feedback (IFF), Relative Damping Control (RDC) and Direct Velocity Feedback (DVF).
|
|
|
|
These active damping techniques are compared in terms of:
|
|
\begin{itemize}
|
|
\item Reduction of the effect of disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\)
|
|
\item Achievable damping
|
|
\item Robustness to a change of sample's mass
|
|
\end{itemize}
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_active_damping_strategies.png}
|
|
\caption{\label{fig:uniaxial_active_damping_strategies}Three active damping strategies: Integral Force Feedback (IFF) using a force sensor, Relative Damping Control (RDC) using a relative displacement sensor, and Direct Velocity Feedback (DVF) using a geophone}
|
|
\end{figure}
|
|
\section{Active Damping Strategies}
|
|
|
|
The Integral Force Feedback strategy consists of using a force sensor in series with the actuator (see Figure \ref{fig:uniaxial_active_damping_iff_equiv}, left).
|
|
|
|
The control strategy consists of integrating the measured force and feeding it back to the actuator:
|
|
\begin{equation}
|
|
K_{\text{IFF}}(s) = \frac{g}{s}
|
|
\end{equation}
|
|
|
|
The mechanical equivalent of this strategy is to add a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain \(k/g\) (see Figure \ref{fig:uniaxial_active_damping_iff_equiv}, right).
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_active_damping_iff_equiv.png}
|
|
\caption{\label{fig:uniaxial_active_damping_iff_equiv}Integral Force Feedback is equivalent as to add a damper in series with the stiffness (the initial damping is here neglected for simplicity)}
|
|
\end{figure}
|
|
|
|
|
|
For the Relative Damping Control strategy, a relative motion sensor that measures the motion of the actuator is used (see Figure \ref{fig:uniaxial_active_damping_rdc_equiv}, left).
|
|
|
|
The derivative of this relative motion is used for the feedback signal:
|
|
\begin{equation}
|
|
K_{\text{RDC}}(s) = - g \cdot s
|
|
\end{equation}
|
|
|
|
The mechanical equivalent is to add a dashpot in parallel with the actuator with a damping coefficient equal to the controller gain \(g\) (see Figure \ref{fig:uniaxial_active_damping_rdc_equiv}, right).
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_active_damping_rdc_equiv.png}
|
|
\caption{\label{fig:uniaxial_active_damping_rdc_equiv}Relative Damping Control is equivalent as adding a damper in parallel with the actuator/relative motion sensor}
|
|
\end{figure}
|
|
|
|
Finally, the Direct Velocity Feedback strategy consists of using an inertial sensor (usually a geophone), that measured the ``absolute'' velocity of the body fixed on top of the actuator (se Figure \ref{fig:uniaxial_active_damping_dvf_equiv}, left).
|
|
|
|
The measured velocity is then fed back to the actuator:
|
|
\begin{equation}
|
|
K_{\text{DVF}}(s) = - g
|
|
\end{equation}
|
|
|
|
This is equivalent as to fix a dashpot (with a damping coefficient equal to the controller gain \(g\)) between the body (one which the inertial sensor is fixed) and an inertial reference frame (see Figure \ref{fig:uniaxial_active_damping_dvf_equiv}, right).
|
|
This is usually refers to as ``\emph{sky hook damper}''.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_active_damping_dvf_equiv.png}
|
|
\caption{\label{fig:uniaxial_active_damping_dvf_equiv}Direct velocity Feedback using an inertial sensor is equivalent to a ``sky hook damper''}
|
|
\end{figure}
|
|
|
|
\section{Plant Dynamics for Active Damping}
|
|
The plant dynamics for all three active damping techniques are shown in Figure \ref{fig:uniaxial_plant_active_damping_techniques}.
|
|
All have \textbf{alternating poles and zeros} meaning that the phase is bounded to \(\pm 90\,\text{deg}\) which makes the controller very robust.
|
|
|
|
When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure \ref{fig:uniaxial_plant_active_damping_techniques}), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
|
|
|
|
For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure \ref{fig:uniaxial_plant_active_damping_techniques}.
|
|
Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_plant_active_damping_techniques.png}
|
|
\caption{\label{fig:uniaxial_plant_active_damping_techniques}Plant dynamics for the three active damping techniques (IFF: right, RDC: middle, DVF: left), for three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\) in blue, \(k_n = 1\,N/\mu m\) in red and \(k_n = 100\,N/\mu m\) in yellow) and three sample's masses (\(m_s = 1\,kg\): solid curves, \(m_s = 25\,kg\): dot-dashed curves, and \(m_s = 50\,kg\): dashed curves).}
|
|
\end{figure}
|
|
|
|
\section{Achievable Damping - Root Locus}
|
|
The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
|
|
They are shown in Figure \ref{fig:uniaxial_root_locus_damping_techniques}.
|
|
|
|
All three active damping approach can lead to \textbf{critical damping} of the nano-hexapod suspension mode.
|
|
|
|
There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques}, right) and with RDC for a soft nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}).
|
|
This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_root_locus_damping_techniques.png}
|
|
\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.}
|
|
\end{figure}
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png}
|
|
\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.}
|
|
\end{figure}
|
|
|
|
\section{Change of sensitivity to disturbances}
|
|
The sensitivity to disturbances (direct forces \(f_s\), stage vibrations \(f_t\) and floor motion \(x_f\)) for all three active damping techniques are compared in Figure \ref{fig:uniaxial_sensitivity_dist_active_damping}.
|
|
The comparison is done with the nano-hexapod having a stiffness \(k_n = 1\,N/\mu m\).
|
|
|
|
\begin{important}
|
|
Conclusions from Figure \ref{fig:uniaxial_sensitivity_dist_active_damping} are:
|
|
\begin{itemize}
|
|
\item IFF degrades the sensitivity to direct forces on the sample (i.e. the compliance) below the resonance of the nano-hexapod
|
|
\item RDC degrades the sensitivity to stage vibrations around the nano-hexapod's resonance as compared to the other two methods
|
|
\item both IFF and DVF degrades the sensitivity to floor motion below the resonance of the nano-hexapod
|
|
\end{itemize}
|
|
\end{important}
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_sensitivity_dist_active_damping.png}
|
|
\caption{\label{fig:uniaxial_sensitivity_dist_active_damping}Change of sensitivity to disturbance with all three active damping strategies}
|
|
\end{figure}
|
|
|
|
\section{Noise Budgeting after Active Damping}
|
|
Cumulative Amplitude Spectrum of the distance \(d\) with all three active damping techniques are compared in Figure \ref{fig:uniaxial_cas_active_damping}.
|
|
All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
|
|
|
|
Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_cas_active_damping.png}
|
|
\caption{\label{fig:uniaxial_cas_active_damping}Comparison of the cumulative amplitude spectrum (CAS) of the distance \(d\) for all three active damping techniques (OL in black, IFF in blue, RDC in red and DVF in yellow).}
|
|
\end{figure}
|
|
|
|
\section{Obtained Damped Plant}
|
|
The transfer functions from the plant input \(f\) to the relative displacement \(d\) while the active damping is implemented are shown in Figure \ref{fig:uniaxial_damped_plant_three_active_damping_techniques}.
|
|
All three active damping techniques yield similar damped plants.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_damped_plant_three_active_damping_techniques.png}
|
|
\caption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques}Obtained damped transfer function from f to d for the three damping techniques}
|
|
\end{figure}
|
|
|
|
The damped plants are shown in Figure \ref{fig:uniaxial_damped_plant_change_sample_mass} for all three techniques, with the three considered nano-hexapod stiffnesses and sample's masses.
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_damped_plant_change_sample_mass.png}
|
|
\caption{\label{fig:uniaxial_damped_plant_change_sample_mass}Damped plant \(d/f\) - Robustness to change of sample's mass for all three active damping techniques. Grey curves are the open-loop (i.e. undamped) plants.}
|
|
\end{figure}
|
|
|
|
\section{Robustness to change of payload's mass}
|
|
The Root Locus for the three damping techniques are shown in Figure \ref{fig:uniaxial_active_damping_robustness_mass_root_locus} for three sample's mass (1kg, 25kg and 50kg).
|
|
The closed-loop poles are shown by the squares for a specific gain.
|
|
|
|
We can see that having heavier samples yields larger damping for IFF and smaller damping for RDC and DVF.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_active_damping_robustness_mass_root_locus.png}
|
|
\caption{\label{fig:uniaxial_active_damping_robustness_mass_root_locus}Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses}
|
|
\end{figure}
|
|
|
|
\section{Conclusion}
|
|
|
|
\begin{important}
|
|
Conclusions for Active Damping:
|
|
\begin{itemize}
|
|
\item All three active damping techniques yields good damping (Figure \ref{fig:uniaxial_root_locus_damping_techniques}) and similar remaining vibrations (Figure \ref{fig:uniaxial_cas_active_damping})
|
|
\item The obtained damped plants (Figure \ref{fig:uniaxial_damped_plant_change_sample_mass}) are equivalent for the three active damping techniques
|
|
\item Which one to be used will be determined with the use of more accurate models and will depend on which is the easiest to implement in practice
|
|
\end{itemize}
|
|
\end{important}
|
|
|
|
\begin{table}[htbp]
|
|
\caption{\label{tab:comp_active_damping}Comparison of active damping strategies}
|
|
\centering
|
|
\scriptsize
|
|
\begin{tabularx}{\linewidth}{Xccc}
|
|
\toprule
|
|
& \textbf{IFF} & \textbf{RDC} & \textbf{DVF}\\
|
|
\midrule
|
|
\textbf{Sensor} & Force sensor & Relative motion sensor & Inertial sensor\\
|
|
\midrule
|
|
\textbf{Damping} & Up to critical & Up to critical & Up to Critical\\
|
|
\midrule
|
|
\textbf{Robustness} & Requires collocation & Requires collocation & Impacted by geophone resonances\\
|
|
\midrule
|
|
\(f_s\) \textbf{Disturbance} & \(\nearrow\) at low frequency & \(\searrow\) near resonance & \(\searrow\) near resonance\\
|
|
\(f_t\) \textbf{Disturbance} & \(\searrow\) near resonance & \(\nearrow\) near resonance & \(\searrow\) near resonance\\
|
|
\(x_f\) \textbf{Disturbance} & \(\nearrow\) at low frequency & \(\searrow\) near resonance & \(\nearrow\) at low frequency\\
|
|
\bottomrule
|
|
\end{tabularx}
|
|
\end{table}
|
|
|
|
\chapter{Position Feedback Controller}
|
|
\label{sec:uniaxial_position_control}
|
|
The High Authority Control - Low Authority Control (HAC-LAC) architecture is shown in Figure \ref{fig:uniaxial_hac_lac_architecture}.
|
|
It corresponds to a \emph{two step} control strategy:
|
|
\begin{itemize}
|
|
\item First, an active damping controller \(\bm{K}_{\textsc{LAC}}\) is implemented (see Section \ref{sec:uniaxial_active_damping}).
|
|
It allows to reduce the vibration level, and it also makes the damped plant (transfer function from \(u^{\prime}\) to \(y\)) easier to control than the undamped plant (transfer function from \(u\) to \(y\)).
|
|
\item Then, a position controller \(\bm{K}_{\textsc{HAC}}\) is implemented.
|
|
\end{itemize}
|
|
|
|
Combined with the uniaxial model, it is shown in Figure \ref{fig:uniaxial_hac_lac_model}.
|
|
|
|
\begin{figure}
|
|
\begin{subfigure}{0.54\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,width=1.0\linewidth]{figs/uniaxial_hac_lac_architecture.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:uniaxial_hac_lac_architecture}Typical HAC-LAC Architecture}
|
|
\end{subfigure}
|
|
\begin{subfigure}{0.45\textwidth}
|
|
\begin{center}
|
|
\includegraphics[scale=1,scale=1]{figs/uniaxial_hac_lac_model.png}
|
|
\end{center}
|
|
\subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-LAC strategy}
|
|
\end{subfigure}
|
|
\caption{\label{fig:uniaxial_hac_lac}High Authority Control - Low Authority Control (HAC-LAC)}
|
|
\end{figure}
|
|
\section{Damped Plant Dynamics}
|
|
As was shown in Section \ref{sec:uniaxial_active_damping}, all three proposed active damping techniques yield similar damping plants.
|
|
Therefore, \emph{Integral Force Feedback} will be used in this section to study the HAC-LAC performances.
|
|
|
|
The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:uniaxial_hac_iff_damped_plants_masses}.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_hac_iff_damped_plants_masses.png}
|
|
\caption{\label{fig:uniaxial_hac_iff_damped_plants_masses}Obtained damped plant using Integral Force Feedback for three sample's masses}
|
|
\end{figure}
|
|
|
|
\section{Position Feedback Controller}
|
|
The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample's mass.
|
|
|
|
The required feedback bandwidth was approximately determined un Section \ref{sec:uniaxial_noise_budgeting}:
|
|
\begin{itemize}
|
|
\item \(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)).
|
|
Near this frequency, the plants are equivalent to a mass line.
|
|
The gain of the mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,kg\) up to \(65\,kg\)).
|
|
This mean that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
|
|
\item \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)).
|
|
Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line.
|
|
It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.
|
|
\item \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)).
|
|
Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the wanted crossover frequency.
|
|
The micro-station is not stiff enough to have a clear stiffness line at this frequency.
|
|
Therefore, there are both a change of phase and gain depending on the sample's mass.
|
|
This makes the robust design of the controller a little bit more complicated.
|
|
\end{itemize}
|
|
|
|
|
|
Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure \ref{fig:uniaxial_nyquist_hac}).
|
|
These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
|
|
The loop gains for the three nano-hexapod are shown in Figure \ref{fig:uniaxial_loop_gain_hac}.
|
|
We can see that:
|
|
\begin{itemize}
|
|
\item for the soft and moderately stiff nano-hexapod, the crossover frequency varies a lot with the sample mass.
|
|
This is due to the fact that the crossover frequency corresponds to the mass line of the plant.
|
|
\item for the stiff nano-hexapod, the obtained crossover frequency is not at high as what was estimated necessary.
|
|
The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.
|
|
\end{itemize}
|
|
|
|
Note that these controller were quickly tuned by hand and not designed using any optimization methods.
|
|
The goal is just to have a first estimation of the attainable performances.
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_nyquist_hac.png}
|
|
\caption{\label{fig:uniaxial_nyquist_hac}Nyquist Plot - Hight Authority Controller for all three nano-hexapod stiffnesses (soft one in blue, moderately stiff in red and very stiff in yellow) and all sample masses (corresponding to the three curves of each color)}
|
|
\end{figure}
|
|
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[scale=1]{figs/uniaxial_loop_gain_hac.png}
|
|
\caption{\label{fig:uniaxial_loop_gain_hac}Loop Gain - High Authority Controller for all three nano-hexapod stiffnesses (soft one in blue, moderately stiff in red and very stiff in yellow) and all sample masses (corresponding to the three curves of each color)}
|
|
\end{figure}
|
|
|
|
\section{Closed-Loop Noise Budgeting}
|
|
The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
|
|
These are compared with the open-loop and damped plants cases in Figure \ref{fig:uniaxial_sensitivity_dist_hac_lac} for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass).
|
|
As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_sensitivity_dist_hac_lac.png}
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\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with LAC and with HAC-LAC}
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\end{figure}
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The cumulative amplitude spectrum of the motion \(d\) is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases.
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The results are shown in Figure \ref{fig:uniaxial_cas_hac_lac}.
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Obtained root mean square values of the distance \(d\) are better for the soft nano-hexapod (\(\approx 25\,nm\) to \(\approx 35\,nm\) depending on the sample's mass) than for the stiffer nano-hexapod (from \(\approx 30\,nm\) to \(\approx 70\,nm\)).
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/uniaxial_cas_hac_lac.png}
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\caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and HAC-LAC cases}
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\end{figure}
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\chapter{Conclusion}
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\label{sec:conclusion}
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In this study, a uniaxial model of the nano-active-stabilization-system has been tuned both from dynamical measurements (Section \ref{sec:micro_station_model}) and from disturbances measurements (Section \ref{sec:uniaxial_disturbances}).
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It has been shown that three active damping techniques can be used to critically damp the nano-hexapod resonances (Section \ref{sec:uniaxial_active_damping}).
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However, this model does not allows to determine which one is most suited to this application.
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Finally, position feedback controllers have been developed for three considered nano-hexapod stiffnesses.
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These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances.
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It has been found that having a soft nano-hexapod makes the plant dynamics easier to control (because decoupled from the micro-station dynamics) and requires less position feedback bandwidth to fulfill the requirements.
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The moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)) is requiring a bit more position feedback bandwidth, but it still seems to give acceptable results.
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However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performances.
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\end{document}
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