Add table with matlab files
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@ -70,6 +70,19 @@ Once the system is well damped, a feedback position controller is applied, and t
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Conclusion remarks are given in Section ref:sec:conclusion.
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#+name: tab:section_matlab_code
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#+caption: Report sections and corresponding Matlab files
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#+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
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#+attr_latex: :center t :booktabs t
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| *Sections* | *Matlab File* |
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|-------------------------------------------+-------------------------------------|
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| Section ref:sec:micro_station_model | =uniaxial_1_micro_station_model.m= |
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| Section ref:sec:nano_station_model | =uniaxial_2_nano_hexapod_model.m= |
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| Section ref:sec:uniaxial_disturbances | =uniaxial_3_disturbances.m= |
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| Section ref:sec:uniaxial_noise_budgeting | =uniaxial_4_dynamic_noise_budget.m= |
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| Section ref:sec:uniaxial_active_damping | =uniaxial_5_active_damping.m= |
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| Section ref:sec:uniaxial_position_control | =uniaxial_6_hac_lac.m= |
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#+begin_src latex :file uniaxial_overview_model_sections.pdf
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\begin{tikzpicture}
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% ====================
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@ -211,7 +224,6 @@ Conclusion remarks are given in Section ref:sec:conclusion.
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#+RESULTS:
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[[file:figs/uniaxial_overview_model_sections.png]]
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* Micro Station Model
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:PROPERTIES:
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:HEADER-ARGS:matlab+: :tangle matlab/uniaxial_1_micro_station_model.m
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Binary file not shown.
@ -1,7 +1,25 @@
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% Created 2023-02-17 Fri 11:26
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% Created 2023-02-20 Mon 10:57
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12, parskip=full]{scrreprt}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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\usepackage{grffile}
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\usepackage{longtable}
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\usepackage{wrapfig}
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\usepackage{rotating}
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\usepackage[normalem]{ulem}
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\usepackage{amsmath}
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\usepackage{textcomp}
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\usepackage{amssymb}
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\usepackage{capt-of}
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\usepackage{hyperref}
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\usepackage[most]{tcolorbox}
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\usepackage{bm}
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\usepackage{booktabs}
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\usepackage{tabularx}
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\usepackage{array}
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\usepackage{siunitx}
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\input{preamble.tex}
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\author{Dehaeze Thomas}
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\date{\today}
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@ -41,14 +59,31 @@ Once the system is well damped, a feedback position controller is applied, and t
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Conclusion remarks are given in Section \ref{sec:conclusion}.
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\begin{table}[htbp]
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\caption{\label{tab:section_matlab_code}Report sections and corresponding Matlab files}
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\centering
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\begin{tabularx}{0.5\linewidth}{lX}
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\toprule
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\textbf{Sections} & \textbf{Matlab File}\\
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\midrule
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Section \ref{sec:micro_station_model} & \texttt{uniaxial\_1\_micro\_station\_model.m}\\
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Section \ref{sec:nano_station_model} & \texttt{uniaxial\_2\_nano\_hexapod\_model.m}\\
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Section \ref{sec:uniaxial_disturbances} & \texttt{uniaxial\_3\_disturbances.m}\\
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Section \ref{sec:uniaxial_noise_budgeting} & \texttt{uniaxial\_4\_dynamic\_noise\_budget.m}\\
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Section \ref{sec:uniaxial_active_damping} & \texttt{uniaxial\_5\_active\_damping.m}\\
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Section \ref{sec:uniaxial_position_control} & \texttt{uniaxial\_6\_hac\_lac.m}\\
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\bottomrule
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\end{tabularx}
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\end{table}
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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:org4e07986}
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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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@ -61,6 +96,7 @@ From the measured frequency response functions (FRF), the model can be tuned to
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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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\label{sec:org525a6c0}
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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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@ -95,6 +131,7 @@ load('meas_microstation_frf.mat');
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\end{figure}
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\section{Uniaxial Model}
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\label{sec:orge223677}
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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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@ -137,6 +174,7 @@ 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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\label{sec:orgfa10aaa}
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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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@ -149,6 +187,7 @@ More accurate models will be used later on.
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\end{figure}
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\chapter{Nano-Hexapod Model}
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\label{sec:org5ef4103}
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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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@ -170,6 +209,7 @@ The effect of having resonances between the sample's point of interest and the n
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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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\label{sec:orge14d0af}
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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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@ -190,6 +230,7 @@ ms = 10; % [kg]
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\end{minted}
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\section{Obtained Dynamics}
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\label{sec:orgddb8cf5}
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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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@ -207,6 +248,7 @@ For further analysis, 9 configurations are considered: three nano-hexapod stiffn
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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:org2036233}
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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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@ -224,6 +266,7 @@ The geophone on the floor is used to measured the floor motion \(x_f\) while the
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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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\label{sec:org931b2ea}
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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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@ -284,6 +327,7 @@ The amplitude spectral density \(\Gamma_{x_f}\) of the measured displacement \(x
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\end{figure}
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\section{Stage Vibration}
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\label{sec:orga7b3779}
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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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@ -339,9 +383,11 @@ The vibrations induced by the \(T_y\) stage are not considered here as:
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\end{itemize}
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\chapter{Open-Loop Dynamic Noise Budgeting}
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\label{sec:org11bfa6e}
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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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\label{sec:org469c500}
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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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@ -369,6 +415,7 @@ From Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}
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\end{figure}
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\section{Open-Loop Dynamic Noise Budgeting}
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\label{sec:org99b18ec}
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Now, the power spectral density of the disturbances is taken into account to estimate the residual motion \(d\) in each case.
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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.
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@ -391,6 +438,7 @@ The conclusion is that the sample's mass has little effect on the cumulative amp
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\end{figure}
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\section{Conclusion}
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\label{sec:org6160aae}
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\begin{important}
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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:
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\begin{itemize}
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@ -405,6 +453,7 @@ This gives a first advantage to having a soft nano-hexapod.
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\end{important}
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\chapter{Active Damping}
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\label{sec:orgc25f9e5}
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\label{sec:uniaxial_active_damping}
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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).
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@ -421,6 +470,7 @@ These active damping techniques are compared in terms of:
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\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}
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\end{figure}
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\section{Active Damping Strategies}
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\label{sec:org224d0f3}
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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).
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@ -470,6 +520,7 @@ This is usually refers to as ``\emph{sky hook damper}''.
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\end{figure}
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\section{Plant Dynamics for Active Damping}
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\label{sec:orga11bdaf}
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The plant dynamics for all three active damping techniques are shown in Figure \ref{fig:uniaxial_plant_active_damping_techniques}.
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All have \textbf{alternating poles and zeros} meaning that the phase is bounded to \(\pm 90\,\text{deg}\) which makes the controller very robust.
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@ -485,6 +536,7 @@ Therefore, it is expected that the micro-station dynamics might impact the achie
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\end{figure}
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\section{Achievable Damping - Root Locus}
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\label{sec:org682230b}
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The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
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They are shown in Figure \ref{fig:uniaxial_root_locus_damping_techniques}.
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@ -506,6 +558,7 @@ This can be explained by the fact that above the suspension mode of the soft nan
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\end{figure}
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\section{Change of sensitivity to disturbances}
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\label{sec:org240dd20}
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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}.
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The comparison is done with the nano-hexapod having a stiffness \(k_n = 1\,N/\mu m\).
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@ -525,6 +578,7 @@ Conclusions from Figure \ref{fig:uniaxial_sensitivity_dist_active_damping} are:
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\end{figure}
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\section{Noise Budgeting after Active Damping}
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\label{sec:org59aa5fa}
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Cumulative Amplitude Spectrum of the distance \(d\) with all three active damping techniques are compared in Figure \ref{fig:uniaxial_cas_active_damping}.
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All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
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@ -537,6 +591,7 @@ Compare to the open-loop case, the active damping helps to lower the vibrations
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\end{figure}
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\section{Obtained Damped Plant}
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\label{sec:org534205b}
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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}.
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All three active damping techniques yield similar damped plants.
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@ -554,6 +609,7 @@ The damped plants are shown in Figure \ref{fig:uniaxial_damped_plant_change_samp
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\end{figure}
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\section{Robustness to change of payload's mass}
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\label{sec:org1194b7f}
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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).
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The closed-loop poles are shown by the squares for a specific gain.
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@ -566,6 +622,7 @@ We can see that having heavier samples yields larger damping for IFF and smaller
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\end{figure}
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\section{Conclusion}
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\label{sec:orgd9c89ae}
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\begin{important}
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Conclusions for Active Damping:
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@ -598,6 +655,7 @@ Conclusions for Active Damping:
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\end{table}
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\chapter{Position Feedback Controller}
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\label{sec:orgb8f8fbb}
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\label{sec:uniaxial_position_control}
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The High Authority Control - Low Authority Control (HAC-LAC) architecture is shown in Figure \ref{fig:uniaxial_hac_lac_architecture}.
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It corresponds to a \emph{two step} control strategy:
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@ -625,6 +683,7 @@ Combined with the uniaxial model, it is shown in Figure \ref{fig:uniaxial_hac_la
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\caption{\label{fig:uniaxial_hac_lac}High Authority Control - Low Authority Control (HAC-LAC)}
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\end{figure}
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\section{Damped Plant Dynamics}
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\label{sec:orge78415e}
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As was shown in Section \ref{sec:uniaxial_active_damping}, all three proposed active damping techniques yield similar damping plants.
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Therefore, \emph{Integral Force Feedback} will be used in this section to study the HAC-LAC performances.
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@ -637,6 +696,7 @@ The obtained damped plants for the three nano-hexapod stiffnesses are shown in F
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\end{figure}
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\section{Position Feedback Controller}
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\label{sec:orgbfebc9e}
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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.
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The required feedback bandwidth was approximately determined un Section \ref{sec:uniaxial_noise_budgeting}:
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@ -683,6 +743,7 @@ The goal is just to have a first estimation of the attainable performances.
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\end{figure}
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\section{Closed-Loop Noise Budgeting}
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\label{sec:org6f05561}
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The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
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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).
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As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
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@ -704,6 +765,7 @@ Obtained root mean square values of the distance \(d\) are better for the soft n
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\end{figure}
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\chapter{Conclusion}
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\label{sec:org1cd0a09}
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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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