diff --git a/config_extra.tex b/config_extra.tex index 10b7637..e00fb3b 100644 --- a/config_extra.tex +++ b/config_extra.tex @@ -17,6 +17,9 @@ \RequirePackage{setspace} \onehalfspacing +\usepackage{tocloft} +\setlength{\cftfignumwidth}{2.5em} % Adjust this value as needed + \usepackage{booktabs} \usepackage{multirow} \usepackage{tabularx} @@ -28,50 +31,12 @@ \widowpenalty = 10000 \displaywidowpenalty = 10000 -\usepackage{scrlayer-scrpage} - -\pagestyle{scrheadings} - -\renewcommand{\partformat}{\huge\partname~\thepart\autodot} -\renewcommand{\raggedpart}{\flushleft} - -\setkomafont{part}{\normalfont\huge\scshape} - -\setkomafont{sectioning}{\normalfont\scshape} -\setkomafont{descriptionlabel}{\normalfont\bfseries} - -\setkomafont{caption}{\small} -\setkomafont{captionlabel}{\usekomafont{caption}} - -\setcounter{secnumdepth}{\subsubsectionnumdepth} - -\makeatletter -\renewcommand*{\chapterformat}{ \mbox{\chapappifchapterprefix{\nobreakspace}{\color{BrickRed}\fontsize{40}{45}\selectfont\thechapter}\autodot\enskip}} -% Section with color -% \renewcommand\@seccntformat[1]{\color{BrickRed} {\csname the#1\endcsname}\hspace{0.3em}} -\renewcommand\@seccntformat[1]{{\csname the#1\endcsname}\hspace{0.3em}} -\makeatother - \renewcommand{\floatpagefraction}{.8}% \usepackage{etoolbox} \pretocmd{\section}{\clearpage}{}{} -\usepackage[ % - colorlinks=true, % - citecolor=BrickRed, % - linkcolor=BrickRed, % - urlcolor=BrickRed, % - unicode % - ]{hyperref} - -\usepackage{hypcap} - -\usepackage{bookmark} - -\bookmarksetup{depth=2} - \makeatletter \preto\Gin@extensions{png,} \DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf} @@ -198,6 +163,9 @@ \AtEndEnvironment{listing}{\vspace{-16pt}} +\let\OldTexttt\texttt +\renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}} + \usepackage{scrhack} \usepackage{float} @@ -207,5 +175,67 @@ \usepackage{soul} \sethlcolor{my-pale-grey} +% \usepackage[headsepline]{scrlayer-scrpage} +% \pagestyle{scrheadings} + +\usepackage[autooneside=false,headsepline]{scrlayer-scrpage} +% \pagestyle{scrheadings} +% \automark[section]{chapter} + +% Clear default header styles +\clearpairofpagestyles +\automark[section]{chapter} + +% Set the header content +\ihead{\headmark} % Chapter (or section on even pages) aligned to the left +\ohead{\pagemark} % Page number aligned to the right + +% Enable the page style +\pagestyle{scrheadings} + +\setkomafont{headsepline}{\color{black}} % Change color if desired +\ModifyLayer[addvoffset=\dp\strutbox]{headsepline} % Fine-tune position + +% \automark{section} +% \renewhead*{headings}{ +% \ifstr{\headmark}{}{}{% +% \headmark\hfill +% } +% } + +\renewcommand{\partformat}{\huge\partname~\thepart\autodot} +\renewcommand{\raggedpart}{\flushleft} + +\setkomafont{part}{\normalfont\huge\scshape} + +\setkomafont{sectioning}{\normalfont\scshape} +\setkomafont{descriptionlabel}{\normalfont\bfseries} + +\setkomafont{caption}{\small} +\setkomafont{captionlabel}{\usekomafont{caption}} + +\setcounter{secnumdepth}{\subsubsectionnumdepth} + +\makeatletter +\renewcommand*{\chapterformat}{ \mbox{\chapappifchapterprefix{\nobreakspace}{\color{BrickRed}\fontsize{40}{45}\selectfont\thechapter}\autodot\enskip}} +% Section with color +% \renewcommand\@seccntformat[1]{\color{BrickRed} {\csname the#1\endcsname}\hspace{0.3em}} +\renewcommand\@seccntformat[1]{{\csname the#1\endcsname}\hspace{0.3em}} +\makeatother + +\usepackage[ % + colorlinks=true, % + citecolor=BrickRed, % + linkcolor=BrickRed, % + urlcolor=BrickRed, % + unicode % + ]{hyperref} + +\usepackage{hypcap} + +\usepackage{bookmark} + +\bookmarksetup{depth=2} + \makeindex \makeglossaries diff --git a/phd-thesis.org b/phd-thesis.org index c3d93fb..cf305fa 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -55,7 +55,6 @@ ("\\paragraph{%s}" . "\\paragraph*{%s}") )) - ;; Remove automatic org heading labels (defun my-latex-filter-removeOrgAutoLabels (text backend info) "Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings." @@ -86,11 +85,6 @@ org-ref-acronyms-before-parsing)) #+END_SRC -* Useful snippets :noexport: - -- acronyms acrshort:nass acrshort:mimo acrshort:lti [[acrfull:siso][Single-Input Single-Output (SISO)]] -- glossary terms gls:ka, gls:phi. - * Glossary and Acronyms - Tables :ignore: #+name: glossary @@ -173,7 +167,6 @@ :UNNUMBERED: notoc :END: -\gls{phi} * Résumé :PROPERTIES: @@ -198,7 +191,6 @@ * Introduction # [[file:/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org][NASS - Introduction]] - * Conceptual Design Development \minitoc **** Abstract @@ -388,8 +380,6 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure re #+end_subfigure #+end_figure -**** Identification of all combination of stiffnesses / masses :noexport: - *** Disturbance Identification :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/uniaxial_3_disturbances.m @@ -734,8 +724,6 @@ Therefore, it is expected that the micro-station dynamics might impact the achie #+end_subfigure #+end_figure -**** Active Damping Controller Optimization and Damped plants :noexport: - **** Achievable Damping and Damped Plants <> @@ -1627,8 +1615,6 @@ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forc #+end_subfigure #+end_figure -**** Identify Generic Dynamics :noexport: - **** 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. Looking at the transfer function matrix $\mathbf{G}_d$ in equation eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. @@ -1953,8 +1939,6 @@ Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiff \boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2} \end{equation} -**** Identify plant with parallel stiffnesses :noexport: - **** Effect of parallel stiffness on the IFF plant The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified without parallel stiffness $k_p = 0$, with a small parallel stiffness $k_p < m \Omega^2$ and with a large parallel stiffness $k_p > m \Omega^2$. Bode plots of the obtained dynamics are shown in Figure ref:fig:rotating_iff_effect_kp. @@ -2129,8 +2113,6 @@ These two proposed IFF modifications and relative damping control are compared i 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 one the basis of previous discussions about optimal parameters. -**** Identify plants :noexport: - **** Root Locus Figure ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications and the relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. @@ -2203,8 +2185,6 @@ The previous analysis is now applied to a model representing a rotating nano-hex Three nano-hexapod stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) because this is the worst identified case scenario in terms of gyroscopic effects. -**** Identify NASS dynamics :noexport: - **** Nano-Active-Stabilization-System - Plant Dynamics For the NASS, the maximum rotating velocity is $\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}$ for a suspended mass on top of the nano-hexapod's actuators equal to $m_n + m_s = \SI{16}{\kilo\gram}$. The parallel stiffness corresponding to the centrifugal forces is $m \Omega^2 \approx \SI{0.6}{\newton\per\mm}$. @@ -2524,6 +2504,7 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: :PROPERTIES: :UNNUMBERED: t :END: +<> In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section ref:sec:rotating_system_description). Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int). @@ -3154,6 +3135,9 @@ This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the fre #+end_figure *** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: <> In this study, a modal analysis of the micro-station was performed. @@ -4752,6 +4736,9 @@ Using this simple test bench, it can be concluded that the /super element/ model #+end_figure *** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: <> In this study, the amplified piezoelectric actuators "APA300ML" have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase. @@ -6360,6 +6347,11 @@ Therefore, the model effectively represents the system coupling for different pa [[file:figs/test_nhexa_comp_simscape_de_all_high_mass.png]] *** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: +<> + The goal of this test bench was to obtain an accurate model of the nano-hexapod that could then be included on top of the micro-station model. The adopted strategy was to identify the nano-hexapod dynamics under conditions in which all factors that could have affected the nano-hexapod dynamics were considered. This was achieved by developing a suspended table with low frequency suspension modes that can be accurately modeled (Section ref:sec:test_nhexa_table). @@ -7630,6 +7622,7 @@ Moreover, the systematic approach to system development and validation, along wi #+begin_export latex \begin{refsection}[ref.bib] + \renewcommand{\clearpage}{} % Désactive \clearpage temporairement % List all papers even if not cited \nocite{*} % Sort by year @@ -7642,10 +7635,11 @@ Moreover, the systematic approach to system development and validation, along wi #+end_export * Glossary :ignore: +[[printglossaries:]] -#+latex: \printglossary[type=\acronymtype] -#+latex: \printglossary[type=\glossarytype] -#+latex: \printglossary +# #+latex: \printglossary[type=\acronymtype] +# #+latex: \printglossary[type=\glossarytype] +# #+latex: \printglossary * Footnotes diff --git a/phd-thesis.pdf b/phd-thesis.pdf index 2e16dc1..f84d1a7 100644 Binary files a/phd-thesis.pdf and b/phd-thesis.pdf differ diff --git a/phd-thesis.tex b/phd-thesis.tex index 0753319..c99f2c2 100644 --- a/phd-thesis.tex +++ b/phd-thesis.tex @@ -1,4 +1,4 @@ -% Created 2025-02-04 Tue 18:05 +% Created 2025-02-04 Tue 23:07 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -86,7 +86,7 @@ \newpage \chapter*{Abstract} -\gls{phi} + \chapter*{Résumé} @@ -100,10 +100,9 @@ \chapter{Introduction} - \chapter{Conceptual Design Development} \minitoc -\subsection*{Abstract} +\subsubsection*{Abstract} \begin{figure}[htbp] \centering \includegraphics[scale=1,width=\linewidth]{figs/chapter1_overview.png} @@ -149,7 +148,7 @@ From the measured frequency response functions (FRF), the model can be tuned to \includegraphics[scale=1,width=\linewidth]{figs/uniaxial_ustation_first_meas_dynamics.jpg} \caption{\label{fig:uniaxial_ustation_first_meas_dynamics}Experimental setup used for the first dynamical measurements on the Micro-Station. Geophones are fixed to different stages of the micro-station.} \end{figure} -\subsection{Measured dynamics} +\subsubsection{Measured dynamics} The measurement setup is schematically shown in Figure \ref{fig:uniaxial_ustation_meas_dynamics_schematic} where two vertical hammer hits are performed, one on the Granite (force \(F_{g}\)) and the other on the micro-hexapod's top platform (force \(F_{h}\)). The vertical inertial motion of the granite \(x_{g}\) and the top platform of the micro-hexapod \(x_{h}\) are measured using geophones. @@ -173,7 +172,7 @@ Due to the poor coherence at low frequencies, these frequency response functions \caption{\label{fig:micro_station_uniaxial_model}Schematic of the Micro-Station measurement setup and uniaxial model.} \end{figure} -\subsection{Uniaxial Model} +\subsubsection{Uniaxial Model} The uniaxial model of the micro-station is shown in Figure \ref{fig:uniaxial_model_micro_station}. It consists of a mass spring damper system with three degrees of freedom. A mass-spring-damper system represents the granite (with mass \(m_g\), stiffness \(k_g\) and damping \(c_g\)). @@ -202,7 +201,7 @@ Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,N/\mu m\) & \(c_g = 250\,\fra Two disturbances are considered which are shown in red: the floor motion \(x_f\) and the stage vibrations represented by \(f_t\). The hammer impacts \(F_{h}, F_{g}\) are shown in blue, whereas the measured inertial motions \(x_{h}, x_{g}\) are shown in black. -\subsection{Comparison of model and measurements} +\subsubsection{Comparison of model and measurements} The transfer functions from the forces injected by the hammers to the measured inertial motion of the micro-hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure \ref{fig:uniaxial_comp_frf_meas_model}. Because the uniaxial model has three degrees of freedom, only three modes with frequencies at \(70\,\text{Hz}\), \(140\,\text{Hz}\) and \(320\,\text{Hz}\) are modeled. @@ -239,14 +238,14 @@ The effect of resonances between the sample's point of interest and the nano-hex \end{subfigure} \caption{\label{fig:uniaxial_model_micro_station_nass_with_tf}Uniaxial model of the NASS \subref{fig:uniaxial_model_micro_station_nass} with the micro-station shown in black, the nano-hexapod represented in blue and the sample represented in green. Disturbances are shown in red. Extracted transfer function from \(f\) to \(d\) \subref{fig:uniaxial_plant_first_params}.} \end{figure} -\subsection{Nano-Hexapod Parameters} +\subsubsection{Nano-Hexapod Parameters} The nano-hexapod is represented by a mass spring damper system (shown in blue in Figure \ref{fig:uniaxial_model_micro_station_nass}). Its mass \gls{mn} is set to \(15\,\text{kg}\) while its stiffness \(k_n\) can vary depending on the chosen architecture/technology. The sample is represented by a mass \gls{ms} that can vary from \(1\,\text{kg}\) up to \(50\,\text{kg}\). As a first example, the nano-hexapod stiffness of is set at \(k_n = 10\,N/\mu m\) and the sample mass is chosen at \(m_s = 10\,\text{kg}\). -\subsection{Obtained Dynamic Response} +\subsubsection{Obtained Dynamic Response} The sensitivity to disturbances (i.e., the transfer functions from \(x_f,f_t,f_s\) to \(d\)) can be extracted from the uniaxial model of Figure \ref{fig:uniaxial_model_micro_station_nass} and are shown in Figure \ref{fig:uniaxial_sensitivity_dist_first_params}. 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}. @@ -295,7 +294,7 @@ The geophone located on the floor was used to measure the floor motion \(x_f\) w \end{subfigure} \caption{\label{fig:uniaxial_ustation_meas_disturbances_setup}Identification of the disturbances coming from the micro-station. The measurement schematic is shown in \subref{fig:uniaxial_ustation_meas_disturbances}. A picture of the setup is shown in \subref{fig:uniaxial_ustation_dynamical_id_setup}} \end{figure} -\subsection{Ground Motion} +\subsubsection{Ground Motion} To acquire the geophone signals, the measurement setup shown in Figure \ref{fig:uniaxial_geophone_meas_chain} is used. The voltage generated by the geophone is amplified using a low noise voltage amplifier\footnote{DLPVA-100-B from Femto with a voltage input noise is \(2.4\,nV/\sqrt{\text{Hz}}\)} with a gain of 60dB before going to the ADC. This is done to improve the signal-to-noise ratio. @@ -338,7 +337,7 @@ The estimated amplitude spectral density \(\Gamma_{x_f}\) of the floor motion \( \caption{\label{fig:uniaxial_asd_disturbance}Estimated amplitude spectral density of the floor motion \(x_f\) \subref{fig:uniaxial_asd_floor_motion_id31} and of the stage disturbances \(f_t\) \subref{fig:uniaxial_asd_disturbance_force}} \end{figure} -\subsection{Stage Vibration} +\subsubsection{Stage Vibration} To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure \ref{fig:uniaxial_ustation_dynamical_id_setup}. The vertical relative velocity between the top platform of the micro hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. The vibrations induced by the \(T_y\) stage are not considered here because they have less amplitude than the vibrations induced by the \(R_z\) stage and because the \(T_y\) stage can be scanned at lower velocities if the induced vibrations are found to be an issue. @@ -368,7 +367,7 @@ To perform such noise budgeting, the disturbances need to be modeled by their sp Then, the transfer functions from disturbances to the performance metric (here the distance \(d\)) are computed (Section \ref{ssec:uniaxial_noise_budget_sensitivity}). Finally, these two types of information are combined to estimate the corresponding spectral density of the performance metric. This is very useful to identify what is limiting the performance of the system, or the compare the achievable performance with different system parameters (Section \ref{ssec:uniaxial_noise_budget_result}). -\subsection{Sensitivity to disturbances} +\subsubsection{Sensitivity to disturbances} \label{ssec:uniaxial_noise_budget_sensitivity} From the uniaxial model of the \acrshort{nass} (Figure \ref{fig:uniaxial_model_micro_station_nass}), the transfer function from the disturbances (\(f_s\), \(x_f\) and \(f_t\)) to the displacement \(d\) are computed. @@ -408,7 +407,7 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses \caption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}Sensitivity of \(d\) to disturbances for three different nano-hexpod stiffnesses. \(f_s\) the direct forces applied on the sample \subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}, \(f_t\) disturbances from the micro-station stages \subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft} and \(x_f\) the floor motion \subref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}} \end{figure} -\subsection{Open-Loop Dynamic Noise Budgeting} +\subsubsection{Open-Loop Dynamic Noise Budgeting} \label{ssec:uniaxial_noise_budget_result} Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion \(d\) for each nano-hexapod and sample configuration. The Cumulative Amplitude Spectrum of the relative motion \(d\) due to both floor motion \(x_f\) and stage vibrations \(f_t\) are shown in Figure \ref{fig:uniaxial_cas_d_disturbances_stiffnesses} for the three nano-hexapod stiffnesses. @@ -433,7 +432,7 @@ The conclusion is that the sample mass has little effect on the cumulative ampli \caption{\label{fig:uniaxial_cas_d_disturbances}Cumulative Amplitude Spectrum of the relative motion \(d\). The effect of \(x_f\) and \(f_t\) are shown in \subref{fig:uniaxial_cas_d_disturbances_stiffnesses}. The effect of sample mass for the three hexapod stiffnesses is shown in \subref{fig:uniaxial_cas_d_disturbances_payload_masses}. The control objective of having a residual error of 20 nm RMS is shown by the horizontal black dashed line.} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} The open-loop residual vibrations of \(d\) can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure \ref{fig:uniaxial_cas_d_disturbances_payload_masses}. This residual vibration of \(d\) is found to be in the order of \(100\,nm\,\text{RMS}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)), \(200\,nm\,\text{RMS}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)) and \(1\,\mu m\,\text{RMS}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)). From this analysis, it may be concluded that the stiffer the nano-hexapod the better. @@ -473,7 +472,7 @@ These damping strategies are first described (Section \ref{ssec:uniaxial_active_ \end{subfigure} \caption{\label{fig:uniaxial_active_damping_strategies}Three active damping strategies. Integral Force Feedback \subref{fig:uniaxial_active_damping_strategies_iff} using a force sensor, Relative Damping Control \subref{fig:uniaxial_active_damping_strategies_rdc} using a relative displacement sensor, and Direct Velocity Feedback \subref{fig:uniaxial_active_damping_strategies_dvf} using a geophone} \end{figure} -\subsection{Active Damping Strategies} +\subsubsection{Active Damping Strategies} \label{ssec:uniaxial_active_damping_strategies} \paragraph{Integral Force Feedback (IFF)} The Integral Force Feedback strategy consists of using a force sensor in series with the actuator (see Figure \ref{fig:uniaxial_active_damping_iff_schematic}) and applying an ``integral'' feedback controller \eqref{eq:uniaxial_iff_controller}. @@ -552,7 +551,7 @@ This is usually referred to as ``\emph{sky hook damper}''. \caption{\label{fig:uniaxial_active_damping_dvf}Direct velocity Feedback \subref{fig:uniaxial_active_damping_dvf_schematic} is equivalent to a ``sky hook damper'' \subref{fig:uniaxial_active_damping_dvf_equiv}} \end{figure} -\subsection{Plant Dynamics for Active Damping} +\subsubsection{Plant Dynamics for Active Damping} \label{ssec:uniaxial_active_damping_plants} The plant dynamics for all three active damping techniques are shown in Figure \ref{fig:uniaxial_plant_active_damping_techniques}. All have \emph{alternating poles and zeros} meaning that the phase does not vary by more than \(180\,\text{deg}\) which makes the design of a \emph{robust} damping controller very easy. @@ -586,7 +585,7 @@ Therefore, it is expected that the micro-station dynamics might impact the achie \caption{\label{fig:uniaxial_plant_active_damping_techniques}Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), 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} -\subsection{Achievable Damping and Damped Plants} +\subsubsection{Achievable Damping and Damped Plants} \label{ssec:uniaxial_active_damping_achievable_damping} To compare the added damping using the three considered active damping strategies, the root locus plot is used. @@ -662,7 +661,7 @@ All three active damping techniques yielded similar damped plants. \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} -\subsection{Sensitivity to disturbances and Noise Budgeting} +\subsubsection{Sensitivity to disturbances and Noise Budgeting} \label{ssec:uniaxial_active_damping_sensitivity_disturbances} Reasonable gains are chosen for the three active damping strategies such that the nano-hexapod suspension mode is well damped. @@ -726,7 +725,7 @@ All three active damping methods give similar results. \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} -\subsection*{Conclusion} +\subsubsection*{Conclusion} Three active damping strategies have been studied for the \acrfull{nass}. Equivalent mechanical representations were derived in Section \ref{ssec:uniaxial_active_damping_strategies} which are helpful for understanding the specific effects of each strategy. The plant dynamics were then compared in Section \ref{ssec:uniaxial_active_damping_plants} and were found to all have alternating poles and zeros, which helps in the design of the active damping controller. @@ -792,7 +791,7 @@ This control architecture applied to the uniaxial model is shown in Figure \ref{ \end{subfigure} \caption{\label{fig:uniaxial_hac_lac}\acrfull{haclac}} \end{figure} -\subsection{Damped Plant Dynamics} +\subsubsection{Damped Plant Dynamics} \label{ssec:uniaxial_position_control_damped_dynamics} The damped plants obtained for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:uniaxial_hac_iff_damped_plants_masses}. For \(k_n = 0.01\,N/\mu m\) and \(k_n = 1\,N/\mu m\), the dynamics are quite simple and can be well approximated by a second-order plant (Figures \ref{fig:uniaxial_hac_iff_damped_plants_masses_soft} and \ref{fig:uniaxial_hac_iff_damped_plants_masses_mid}). @@ -822,7 +821,7 @@ This effect will be further explained in Section \ref{sec:uniaxial_support_compl \caption{\label{fig:uniaxial_hac_iff_damped_plants_masses}Obtained damped plant using Integral Force Feedback for three sample masses} \end{figure} -\subsection{Position Feedback Controller} +\subsubsection{Position Feedback Controller} \label{ssec:uniaxial_position_control_design} The objective is to design high-authority feedback controllers for the three nano-hexapods. @@ -935,7 +934,7 @@ The goal is to have a first estimation of the attainable performance. \caption{\label{fig:uniaxial_loop_gain_hac}Loop gain for the High Authority Controller} \end{figure} -\subsection{Closed-Loop Noise Budgeting} +\subsubsection{Closed-Loop Noise Budgeting} \label{ssec:uniaxial_position_control_cl_noise_budget} The high authority position feedback controllers are then implemented and the closed-loop sensitivities to disturbances are computed. @@ -990,7 +989,7 @@ Obtained root mean square values of the distance \(d\) are better for the soft n \caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and \acrshort{haclac} cases} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} On the basis of the open-loop noise budgeting made in Section \ref{sec:uniaxial_noise_budgeting}, the closed-loop bandwidth required to obtain a vibration level of \(\approx 20\,nm\,\text{RMS}\) was estimated. To achieve such bandwidth, the \acrshort{haclac} strategy was followed, which consists of first using an active damping controller (studied in Section \ref{sec:uniaxial_active_damping}) and then adding a high authority position feedback controller. @@ -1025,7 +1024,7 @@ The second one consists of the nano-hexapod fixed on top of the micro-station ha \end{subfigure} \caption{\label{fig:uniaxial_support_compliance_models}Models used to study the effect of limited support compliance} \end{figure} -\subsection{Neglected support compliance} +\subsubsection{Neglected support compliance} The limited compliance of the micro-station is first neglected and the uniaxial model shown in Figure \ref{fig:uniaxial_support_compliance_nano_hexapod_only} is used. The nano-hexapod mass (including the payload) is set at \(20\,\text{kg}\) and three hexapod stiffnesses are considered, such that their resonance frequencies are at \(\omega_{n} = 10\,\text{Hz}\), \(\omega_{n} = 70\,\text{Hz}\) and \(\omega_{n} = 400\,\text{Hz}\). @@ -1054,7 +1053,7 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev \caption{\label{fig:uniaxial_effect_support_compliance_neglected}Obtained transfer functions from \(F\) to \(L^{\prime}\) when neglecting support compliance} \end{figure} -\subsection{Effect of support compliance on \(L/F\)} +\subsubsection{Effect of support compliance on \(L/F\)} Some support compliance is now added and the model shown in Figure \ref{fig:uniaxial_support_compliance_test_system} is used. The parameters of the support (i.e., \(m_{\mu}\), \(c_{\mu}\) and \(k_{\mu}\)) are chosen to match the vertical mode at \(70\,\text{Hz}\) seen on the micro-station (Figure \ref{fig:uniaxial_comp_frf_meas_model}). @@ -1087,7 +1086,7 @@ If a soft nano-hexapod is used, the support dynamics appears in the dynamics bet \caption{\label{fig:uniaxial_effect_support_compliance_dynamics}Effect of the support compliance on the transfer functions from \(F\) to \(L\)} \end{figure} -\subsection{Effect of support compliance on \(d/F\)} +\subsubsection{Effect of support compliance on \(d/F\)} When the motion to be controlled is the relative displacement \(d\) between the granite and the nano-hexapod's top platform (which is the case for the \acrshort{nass}), the effect of the support compliance on the plant dynamics is opposite to that previously observed. Indeed, using a ``soft'' nano-hexapod (i.e., with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure \ref{fig:uniaxial_effect_support_compliance_dynamics_d_soft}). @@ -1115,7 +1114,7 @@ Conversely, if a ``stiff'' nano-hexapod is used, the support dynamics appears in \caption{\label{fig:uniaxial_effect_support_compliance_dynamics_d}Effect of the support compliance on the transfer functions from \(F\) to \(d\)} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} To study the impact of support compliance on plant dynamics, simple models shown in Figure \ref{fig:uniaxial_support_compliance_models} were used. Depending on the quantity to be controlled (\(L\) or \(d\) in Figure \ref{fig:uniaxial_support_compliance_test_system}) and on the relative location of \(\omega_\nu\) (suspension mode of the nano-hexapod) with respect to \(\omega_\mu\) (modes of the support), the interaction between the support and the nano-hexapod dynamics can drastically change (observations made are summarized in Table \ref{tab:uniaxial_effect_compliance}). @@ -1158,7 +1157,7 @@ To study the effect of the sample dynamics, the models shown in Figure \ref{fig: \caption{\label{fig:uniaxial_payload_dynamics_models}Models used to study the effect of payload dynamics} \end{figure} -\subsection{Impact on plant dynamics} +\subsubsection{Impact on plant dynamics} \label{ssec:uniaxial_payload_dynamics_effect_dynamics} To study the impact of the flexibility between the nano-hexapod and the payload, a first (reference) model with a rigid payload, as shown in Figure \ref{fig:uniaxial_paylaod_dynamics_rigid_schematic} is used. @@ -1207,7 +1206,7 @@ Even though the added sample's flexibility still shifts the high frequency mass \caption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample \subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}, and heavy sample \subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}} \end{figure} -\subsection{Impact on close loop performances} +\subsubsection{Impact on close loop performances} \label{ssec:uniaxial_payload_dynamics_effect_stability} Having a flexibility between the measured position (i.e., the top platform of the nano-hexapod) and the point-of-interest to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). @@ -1249,7 +1248,7 @@ What happens is that above \(\omega_s\), even though the motion \(d\) can be con \caption{\label{fig:uniaxial_sample_flexibility_noise_budget}Cumulative Amplitude Spectrum of the distances \(d\) and \(y\). The effect of the sample's flexibility does not affect much \(d\) but is detrimental to the stability of \(y\). A sample mass \(m_s = 1\,\text{kg}\) and a nano-hexapod stiffness of \(100\,N/\mu m\) are used for the simulations.} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} Payload dynamics is usually a major concern when designing a positioning system. In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample \(\omega_s\) and of the nano-hexapod \(\omega_n\). The larger the sample mass, the larger the effect (i.e., change of high frequency gain, appearance of additional resonances and anti-resonances). @@ -1321,7 +1320,7 @@ After the dynamics of this system is studied, the objective will be to dampen th \caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system} \end{figure} -\subsection{Equations of motion and transfer functions} +\subsubsection{Equations of motion and transfer functions} To obtain the equations of motion for the system represented in Figure \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equation \eqref{eq:rotating_lagrangian_equations} is used. \(L = T - V\) is the Lagrangian, \(T\) the kinetic coenergy, \(V\) the potential energy, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\). These terms are derived in \eqref{eq:rotating_energy_functions_lagrange}. @@ -1381,7 +1380,7 @@ The elements of the transfer function matrix \(\mathbf{G}_d\) are described by e \end{align} \end{subequations} -\subsection{System Poles: Campbell Diagram} +\subsubsection{System Poles: Campbell Diagram} The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation \eqref{eq:rotating_poles} (i.e. the roots of its denominator). \begin{equation}\label{eq:rotating_poles} @@ -1418,7 +1417,7 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo \caption{\label{fig:rotating_campbell_diagram}Campbell diagram - Real \subref{fig:rotating_campbell_diagram_real} and Imaginary \subref{fig:rotating_campbell_diagram_imag} parts of the poles as a function of the rotating velocity \(\Omega\).} \end{figure} -\subsection{System Dynamics: Effect of rotation} +\subsubsection{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. Looking at the transfer function matrix \(\mathbf{G}_d\) in equation \eqref{eq:rotating_Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. The bode plots of these two terms are shown in Figure \ref{fig:rotating_bode_plot} for several rotational speeds \(\Omega\). @@ -1459,7 +1458,7 @@ Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to deri However, none of these studies have been applied to rotating systems. In this section, the \acrshort{iff} strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alter the system dynamics and that IFF cannot be applied as is. -\subsection{System and Equations of motion} +\subsubsection{System and Equations of motion} 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 \(K_F\) described by \eqref{eq:rotating_iff_controller} are then used to feedback each of the sensed force to its associated actuator. @@ -1531,7 +1530,7 @@ This small displacement then increases the centrifugal force \(m\Omega^2d_u = \f \end{bmatrix} \end{equation} -\subsection{Effect of rotation speed on IFF plant dynamics} +\subsubsection{Effect of rotation speed on 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 are shown in Figure \ref{fig:rotating_iff_bode_plot_effect_rot}. As expected from the derived equations of motion: \begin{itemize} @@ -1557,7 +1556,7 @@ A pair of (minimum phase) complex conjugate zeros appears between the two comple \caption{\label{fig:rotating_iff_bode_plot_effect_rot}Effect of the rotation velocity on the bode plot of the direct terms \subref{fig:rotating_iff_bode_plot_effect_rot_direct} and on the IFF root locus \subref{fig:rotating_root_locus_iff_pure_int}} \end{figure} -\subsection{Decentralized Integral Force Feedback} +\subsubsection{Decentralized Integral Force Feedback} The control diagram for decentralized \acrshort{iff} is shown in Figure \ref{fig:rotating_iff_diagram}. The decentralized \acrshort{iff} controller \(\bm{K}_F\) corresponds to a diagonal controller with integrators \eqref{eq:rotating_Kf_pure_int}. @@ -1588,7 +1587,7 @@ This is however not the reason why this high-pass filter is added here. \boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} \end{equation} -\subsection{Modified Integral Force Feedback Controller} +\subsubsection{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 \eqref{eq:rotating_iff_lhf} where \(\omega_i\) characterize the frequency down to which the signal is integrated. The loop gains (\(K_F(s)\) times the direct dynamics \(f_u/F_u\)) with and without the added HPF are shown in Figure \ref{fig:rotating_iff_modified_loop_gain}. The effect of the added HPF limits the low-frequency gain to finite values as expected. @@ -1617,7 +1616,7 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro \caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high-pass filter (\(\Omega = 0.1\omega_0\)). The loop gain is shown in \subref{fig:rotating_iff_modified_loop_gain} with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). The root locus is shown in \subref{fig:rotating_iff_root_locus_hpf_large}} \end{figure} -\subsection{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)} +\subsubsection{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)} Two parameters can be tuned for the modified controller in equation \eqref{eq:rotating_iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\). The optimal values of \(\omega_i\) and \(g\) are considered here as the values for which the damping of all the closed-loop poles is simultaneously maximized. @@ -1645,7 +1644,7 @@ For larger values of \(\omega_i\), the attainable damping ratio decreases as a f \caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high-pass filter cut-off frequency \subref{fig:rotating_root_locus_iff_modified_effect_wi}. The achievable damping ratio decreases as \(\omega_i\) increases, as confirmed in \subref{fig:rotating_iff_hpf_optimal_gain}} \end{figure} -\subsection{Obtained Damped Plant} +\subsubsection{Obtained Damped Plant} To study how the parameter \(\omega_i\) affects the damped plant, the obtained damped plants for several \(\omega_i\) are compared in Figure \ref{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}. It can be seen that the low-frequency coupling increases as \(\omega_i\) increases. Therefore, there is a trade-off between achievable damping and added coupling when tuning \(\omega_i\). @@ -1678,7 +1677,7 @@ Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_sche \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} -\subsection{Equations} +\subsubsection{Equations} The forces measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} are described by \eqref{eq:rotating_measured_force_kp}. \begin{equation}\label{eq:rotating_measured_force_kp} @@ -1717,7 +1716,7 @@ Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{ne \boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2} \end{equation} -\subsection{Effect of parallel stiffness on the IFF plant} +\subsubsection{Effect of parallel stiffness on the IFF plant} The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified without parallel stiffness \(k_p = 0\), with a small parallel stiffness \(k_p < m \Omega^2\) and with a large parallel stiffness \(k_p > m \Omega^2\). Bode plots of the obtained dynamics are shown in Figure \ref{fig:rotating_iff_effect_kp}. The two real zeros for \(k_p < m \Omega^2\) are transformed into two complex conjugate zeros for \(k_p > m \Omega^2\). @@ -1742,7 +1741,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif \caption{\label{fig:rotating_iff_plant_effect_kp}Effect of parallel stiffness on the IFF plant} \end{figure} -\subsection{Effect of \(k_p\) on the attainable damping} +\subsubsection{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. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is expected to have some impact on the attainable damping. To study this effect, Root Locus plots for several parallel stiffnesses \(k_p > m \Omega^2\) are shown in Figure \ref{fig:rotating_iff_kp_root_locus_effect_kp}. @@ -1766,7 +1765,7 @@ This is confirmed by the Figure \ref{fig:rotating_iff_kp_optimal_gain} where the \caption{\label{fig:rotating_iff_optimal_kp}Effect of parallel stiffness on the IFF plant} \end{figure} -\subsection{Damped plant} +\subsubsection{Damped plant} The parallel stiffness are chosen to be \(k_p = 2 m \Omega^2\) and the damped plant is computed. The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure \ref{fig:rotating_iff_kp_added_hpf_damped_plant}. Even though the two resonances are well damped, the IFF changes the low-frequency behavior of the plant, which is usually not desired. @@ -1818,7 +1817,7 @@ K_d(s) = g \cdot \frac{s}{s + \omega_d} \caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensor and decentralized ``relative damping control'' applied.} \end{figure} -\subsection{Equations of motion} +\subsubsection{Equations of motion} Let's note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators \eqref{eq:rotating_rdc_plant_matrix}. The elements of \(\bm{G}_d\) were derived in Section \ref{sec:rotating_system_description} are shown in \eqref{eq:rotating_rdc_plant_elements}. @@ -1840,7 +1839,7 @@ Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functi z = \pm j \sqrt{\omega_0^2 - \omega^2}, \quad p_1 = \pm j (\omega_0 - \omega), \quad p_2 = \pm j (\omega_0 + \omega) \end{equation} -\subsection{Decentralized Relative Damping Control} +\subsubsection{Decentralized Relative Damping Control} The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) were identified for several rotating velocities in Section \ref{sec:rotating_system_description} and are shown in Figure \ref{fig:rotating_bode_plot} (page \pageref{fig:rotating_bode_plot}). To see if large damping can be added with Relative Damping Control, the root locus is computed (Figure \ref{fig:rotating_rdc_root_locus}). @@ -1873,7 +1872,7 @@ These two proposed IFF modifications and relative damping control are compared i 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 one the basis of previous discussions about optimal parameters. -\subsection{Root Locus} +\subsubsection{Root Locus} Figure \ref{fig:rotating_comp_techniques_root_locus} shows the Root Locus plots for the two proposed IFF modifications and the relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. This means that the closed-loop behavior of both systems will differ when large control gains are used. @@ -1898,12 +1897,12 @@ It is interesting to note that the maximum added damping is very similar for bot \caption{\label{fig:rotating_comp_techniques}Comparison of active damping techniques for rotating platform} \end{figure} -\subsection{Obtained Damped Plant} +\subsubsection{Obtained Damped Plant} The actively damped plants are computed for the three techniques and compared in Figure \ref{fig:rotating_comp_techniques_dampled_plants}. It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, the off-diagonal (coupling) terms are not. The \acrshort{iff} strategy is adding some coupling at low-frequency, which may negatively impact the positioning performance. -\subsection{Transmissibility And Compliance} +\subsubsection{Transmissibility And Compliance} The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance. The transmissibility is defined as the transfer function from the displacement of the rotating stage along \(\vec{i}_x\) to the displacement of the payload along the same direction. It is used to characterize the amount of vibration is transmitted through the suspended platform to the payload. @@ -1936,7 +1935,7 @@ This is very well known characteristics of these common active damping technique The previous analysis is now applied to a model representing a rotating nano-hexapod. Three nano-hexapod stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). Only the maximum rotating velocity is here considered (\(\Omega = \SI{60}{rpm}\)) with the light sample (\(m_s = \SI{1}{kg}\)) because this is the worst identified case scenario in terms of gyroscopic effects. -\subsection{Nano-Active-Stabilization-System - Plant Dynamics} +\subsubsection{Nano-Active-Stabilization-System - Plant Dynamics} For the NASS, the maximum rotating velocity is \(\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}\) for a suspended mass on top of the nano-hexapod's actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\). The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \approx \SI{0.6}{\newton\per\mm}\). @@ -1967,7 +1966,7 @@ The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually es \caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity} \end{figure} -\subsection{Optimal IFF with a High-Pass Filter} +\subsubsection{Optimal IFF with a High-Pass Filter} Integral Force Feedback with an added high-pass filter is applied to the three nano-hexapods. First, the parameters (\(\omega_i\) and \(g\)) of the IFF controller that yield the best simultaneous damping are determined from Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain}. The IFF parameters are chosen as follows: @@ -2016,7 +2015,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar \end{table} -\subsection{Optimal IFF with Parallel Stiffness} +\subsubsection{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 that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). To keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased (\(k_a = k_n - k_p\), with \(k_n\) the total nano-hexapod stiffness). A high-pass filter is also added to limit the low-frequency gain with a cut-off frequency \(\omega_i\) equal to one tenth of the system resonance (\(\omega_i = \omega_0/10\)). @@ -2053,7 +2052,7 @@ The corresponding optimal controller gains and achievable damping are summarized \end{center} \end{minipage} -\subsection{Optimal Relative Motion Control} +\subsubsection{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 computed (Figure \ref{fig:rotating_rdc_optimal_gain}). The gain is chosen such that 99\% of modal damping is obtained (obtained gains are summarized in Table \ref{tab:rotating_rdc_opt_params_nass}). @@ -2080,7 +2079,7 @@ The gain is chosen such that 99\% of modal damping is obtained (obtained gains a \end{center} \end{minipage} -\subsection{Comparison of the obtained damped plants} +\subsubsection{Comparison of the obtained damped plants} Now that the optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure \ref{fig:rotating_nass_damped_plant_comp}. Similar to what was concluded in the previous analysis: @@ -2117,7 +2116,7 @@ Similar to what was concluded in the previous analysis: Until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top. While quite simplistic, this allowed us to study the effects of rotation and the associated limitations when active damping is to be applied. In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. -\subsection{Nano Active Stabilization System model} +\subsubsection{Nano Active Stabilization System model} To have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modeled as shown in Figure \ref{fig:rotating_3dof_model_schematic}) is now located on top of a model of the micro-station including (see Figure \ref{fig:rotating_nass_model} for a 3D view): \begin{itemize} \item the floor whose motion is imposed @@ -2135,7 +2134,7 @@ A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the pay \caption{\label{fig:rotating_nass_model}3D view of the Nano-Active-Stabilization-System model.} \end{figure} -\subsection{System dynamics} +\subsubsection{System dynamics} The dynamics of the undamped and damped plants are identified using the optimal parameters found in Section \ref{sec:rotating_nano_hexapod}. The obtained dynamics are compared in Figure \ref{fig:rotating_nass_plant_comp_stiffness} in which the direct terms are shown by the solid curves and the coupling terms are shown by the shaded ones. @@ -2169,7 +2168,7 @@ It can be observed that: \caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology} \end{figure} -\subsection{Effect of disturbances} +\subsubsection{Effect of disturbances} The effect of three disturbances are considered (as for the uniaxial model), floor motion \([x_{f,x},\ x_{f,y}]\) (Figure \ref{fig:rotating_nass_effect_floor_motion}), micro-Station vibrations \([f_{t,x},\ f_{t,y}]\) (Figure \ref{fig:rotating_nass_effect_stage_vibration}) and direct forces applied on the sample \([f_{s,x},\ f_{s,y}]\) (Figure \ref{fig:rotating_nass_effect_direct_forces}). Note that only the transfer functions from the disturbances in the \(x\) direction to the relative position \(d_x\) between the sample and the granite in the \(x\) direction are displayed because the transfer functions in the \(y\) direction are the same due to the system symmetry. @@ -2254,6 +2253,8 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: \end{figure} \subsection*{Conclusion} +\label{sec:rotating_conclusion} + In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section \ref{sec:rotating_system_description}). Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms (Section \ref{sec:rotating_iff_pure_int}). Two modifications of the classical \acrshort{iff} control have been proposed to overcome this issue. @@ -2313,7 +2314,7 @@ In order to perform an experimental modal analysis, a suitable measurement setup This includes using appropriate instrumentation (presented in Section \ref{ssec:modal_instrumentation}) and properly preparing the structure to be measured (Section \ref{ssec:modal_test_preparation}). Then, the locations of the measured motions (Section \ref{ssec:modal_accelerometers}) and the locations of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully. The obtained force and acceleration signals are described in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is assessed. -\subsection{Used Instrumentation} +\subsubsection{Used Instrumentation} \label{ssec:modal_instrumentation} Three type of equipment are essential for a good modal analysis. @@ -2349,7 +2350,7 @@ The softer tip was found to give best results as it injects more energy in the l Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta ADC.} (figure \ref{fig:modal_oros}) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise. -\subsection{Structure Preparation and Test Planing} +\subsubsection{Structure Preparation and Test Planing} \label{ssec:modal_test_preparation} To obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}. @@ -2369,7 +2370,7 @@ This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode). In this modal analysis, it is chosen to measure the response of the structure at all considered \acrshort{dof}, and to excite the structure at one location in three directions in order to have some redundancy, and to ensure that all modes are properly identified. -\subsection{Location of the Accelerometers} +\subsubsection{Location of the Accelerometers} \label{ssec:modal_accelerometers} The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured. @@ -2439,7 +2440,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs \caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages} \end{figure} -\subsection{Hammer Impacts} +\subsubsection{Hammer Impacts} \label{ssec:modal_hammer_impacts} The selected location of the hammer impact corresponds to the location of accelerometer number \(11\) fixed to the translation stage. @@ -2469,7 +2470,7 @@ The impacts were performed in three directions, as shown in figures \ref{fig:mod \caption{\label{fig:modal_hammer_impacts}The three hammer impacts used for the modal analysis} \end{figure} -\subsection{Force and Response signals} +\subsubsection{Force and Response signals} \label{ssec:modal_measured_signals} The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure \ref{fig:modal_raw_meas}. @@ -2547,7 +2548,7 @@ The coordinate transformation from accelerometers \acrshort{dof} to the solid bo The \(69 \times 3 \times 801\) frequency response matrix is then reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to its center of mass. To validate this reduction of \acrshort{dof} and the solid body assumption, the frequency response function at the accelerometer location are ``reconstructed'' from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}. -\subsection{From accelerometer DOFs to solid body DOFs} +\subsubsection{From accelerometer DOFs to solid body DOFs} \label{ssec:modal_acc_to_solid_dof} Let us consider the schematic shown in Figure \ref{fig:modal_local_to_global_coordinates} where the motion of a solid body is measured at 4 distinct locations (in \(x\), \(y\) and \(z\) directions). @@ -2653,7 +2654,7 @@ Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\m \end{bmatrix} \end{equation} -\subsection{Verification of solid body assumption} +\subsubsection{Verification of solid body assumption} \label{ssec:modal_solid_body_assumption} From the response of one solid body expressed by its 6 \acrshortpl{dof} (i.e. from \(\mathbf{H}_{\text{CoM}}\)), and using equation \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any considered location. @@ -2683,7 +2684,7 @@ In section \ref{ssec:modal_parameter_extraction}, the modal parameter extraction The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes. To validate the quality of the modal model, the full \acrshort{frf} matrix is computed from the modal model and compared to the initial measured \acrshort{frf} (section \ref{ssec:modal_model_validity}). -\subsection{Number of modes determination} +\subsubsection{Number of modes determination} \label{ssec:modal_number_of_modes} The \acrshort{mif} is here applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)). @@ -2739,7 +2740,7 @@ Mode & Frequency & Damping\\ \end{scriptsize} \end{minipage} -\subsection{Modal parameter extraction} +\subsubsection{Modal parameter extraction} \label{ssec:modal_parameter_extraction} Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured \acrshort{frf} data. @@ -2797,7 +2798,7 @@ The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation \eqref{ s_r = \omega_r (-\xi_r + i \sqrt{1 - \xi_r^2}), \quad s_r^* = \omega_r (-\xi_r - i \sqrt{1 - \xi_r^2}) \end{equation} -\subsection{Verification of the modal model validity} +\subsubsection{Verification of the modal model validity} \label{ssec:modal_model_validity} To check the validity of the modal model, the complete \(n \times n\) \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) is first synthesized from the modal parameters. @@ -2850,7 +2851,7 @@ This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the f \caption{\label{fig:modal_comp_acc_frf_modal}Comparison of the measured FRF with the synthesized FRF from the modal model.} \end{figure} -\subsection{Conclusion} +\subsection*{Conclusion} \label{sec:modal_conclusion} In this study, a modal analysis of the micro-station was performed. @@ -2904,7 +2905,7 @@ The stiffness and damping properties of the joint s can be tuned separately for each DoF. The ``controlled'' DoF of each stage (for instance the \(D_y\) direction for the translation stage) is modeled as infinitely rigid (i.e. its motion is imposed by a ``setpoint'') while the other DoFs have limited stiffness to model the different micro-station modes. -\subsection{Motion Stages} +\subsubsection{Motion Stages} \label{ssec:ustation_stages} \paragraph{Translation Stage} @@ -2970,7 +2971,7 @@ It can also be used to precisely position the PoI vertically with respect to the \end{center} \end{minipage} -\subsection{Mathematical description of a rigid body motion} +\subsubsection{Mathematical description of a rigid body motion} \label{ssec:ustation_motion_description} In this section, mathematical tools\footnote{The tools presented here are largely taken from \cite{taghirad13_paral}.} that are used to describe the motion of positioning stages are introduced. @@ -3141,7 +3142,7 @@ Another key advantage of homogeneous transformation is the easy inverse transfor \end{array} \right] \end{equation} -\subsection{Micro-Station Kinematics} +\subsubsection{Micro-Station Kinematics} \label{ssec:ustation_kinematics} Each stage is described by two frames; one is attached to the fixed platform \(\{A\}\) while the other is fixed to the mobile platform \(\{B\}\). @@ -3222,7 +3223,7 @@ The obtained dynamics is then compared with the modal analysis performed on the As the dynamics of the nano-hexapod is impacted by the micro-station compliance, the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section \ref{ssec:ustation_model_compliance}). -\subsection{Multi-Body Model} +\subsubsection{Multi-Body Model} \label{ssec:ustation_model_simscape} By performing a modal analysis of the micro-station, it was verified that in the frequency range of interest, each stage behaved as a rigid body. @@ -3278,7 +3279,7 @@ Hexapod & \(10\,N/\mu m\) & \(10\,N/\mu m\) & \(100\,N/\mu m\) & \(1.5\,Nm/rad\) \end{table} -\subsection{Comparison with the measured dynamics} +\subsubsection{Comparison with the measured dynamics} \label{ssec:ustation_model_comp_dynamics} The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions. @@ -3311,7 +3312,7 @@ Tuning the numerous model parameters to better match the measurements is a highl \caption{\label{fig:ustation_comp_com_response}FRFs between the hammer impacts on the translation stage and the measured stage acceleration expressed at its CoM. Comparison of the measured and extracted FRFs from the multi-body model. Different directions are computed for different stages.} \end{figure} -\subsection{Micro-station compliance} +\subsubsection{Micro-station compliance} \label{ssec:ustation_model_compliance} As discussed in the previous section, the dynamics of the micro-station is complex, and tuning the multi-body model parameters to obtain a perfect match is difficult. @@ -3414,7 +3415,7 @@ Instead, the vibrations of the micro-station's top platform induced by the distu To estimate the equivalent disturbance force that induces such vibration, the transfer functions from disturbance sources (i.e. forces applied in the stages' joint) to the displacements of the micro-station's top platform with respect to the granite are extracted from the multi-body model (Section \ref{ssec:ustation_disturbances_sensitivity}). Finally, the obtained disturbance sources are compared in Section \ref{ssec:ustation_disturbances_results}. -\subsection{Disturbance measurements} +\subsubsection{Disturbance measurements} \label{ssec:ustation_disturbances_meas} In this section, ground motion is directly measured using geophones. Vibrations induced by scanning the translation stage and the spindle are also measured using dedicated setups. @@ -3529,7 +3530,7 @@ The vertical motion induced by scanning the spindle is in the order of \(\pm 30\ \caption{\label{fig:ustation_errors_spindle}Measurement of the radial \subref{fig:ustation_errors_spindle_radial}, axial \subref{fig:ustation_errors_spindle_axial} and tilt \subref{fig:ustation_errors_spindle_tilt} Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis.} \end{figure} -\subsection{Sensitivity to disturbances} +\subsubsection{Sensitivity to disturbances} \label{ssec:ustation_disturbances_sensitivity} To compute the disturbance source (i.e. forces) that induced the measured vibrations in Section \ref{ssec:ustation_disturbances_meas}, the transfer function from the disturbance sources to the stage vibration (i.e. the ``sensitivity to disturbances'') needs to be estimated. @@ -3558,7 +3559,7 @@ The obtained transfer functions are shown in Figure \ref{fig:ustation_model_sens \caption{\label{fig:ustation_model_sensitivity}Extracted transfer functions from disturbances to relative motion between the micro-station's top platform and the granite. The considered disturbances are the ground motion \subref{fig:ustation_model_sensitivity_ground_motion}, the translation stage vibrations \subref{fig:ustation_model_sensitivity_ty}, and the spindle vibrations \subref{fig:ustation_model_sensitivity_rz}.} \end{figure} -\subsection{Obtained disturbance sources} +\subsubsection{Obtained disturbance sources} \label{ssec:ustation_disturbances_results} From the measured effect of disturbances in Section \ref{ssec:ustation_disturbances_meas} and the sensitivity to disturbances extracted from the multi-body model in Section \ref{ssec:ustation_disturbances_sensitivity}, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated. @@ -3619,7 +3620,7 @@ To fully validate the micro-station multi-body model, two time-domain simulation First, a tomography experiment (i.e. a constant Spindle rotation) was performed and was compared with experimental measurements (Section \ref{sec:ustation_experiments_tomography}). Second, a constant velocity scans with the translation stage was performed and also compared with the experimental data (Section \ref{sec:ustation_experiments_ty_scans}). -\subsection{Tomography Experiment} +\subsubsection{Tomography Experiment} \label{sec:ustation_experiments_tomography} To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm. @@ -3645,7 +3646,7 @@ A good correlation with the measurements is observed both for radial errors (Fig \caption{\label{fig:ustation_errors_model_spindle}Simulation results for a tomography experiment at constant velocity of 60rpm. The comparison is made with measurements for both radial \subref{fig:ustation_errors_model_spindle_radial} and axial errors \subref{fig:ustation_errors_model_spindle_axial}.} \end{figure} -\subsection{Raster Scans with the translation stage} +\subsubsection{Raster Scans with the translation stage} \label{sec:ustation_experiments_ty_scans} A second experiment was performed in which the translation stage was scanned at constant velocity. @@ -3682,7 +3683,7 @@ After experimentally estimating the disturbances (Section \ref{sec:ustation_dist \chapter{Detailed Design} \minitoc -\subsection*{Abstract} +\subsubsection*{Abstract} \begin{figure}[htbp] \centering \includegraphics[scale=1,width=\linewidth]{figs/chapter2_overview.png} @@ -3707,7 +3708,7 @@ After experimentally estimating the disturbances (Section \ref{sec:ustation_dist \chapter{Experimental Validation} \minitoc -\subsection*{Abstract} +\subsubsection*{Abstract} \begin{figure}[htbp] \centering \includegraphics[scale=1,width=\linewidth]{figs/chapter3_overview.png} @@ -3745,7 +3746,7 @@ Then, the capacitance of the piezoelectric stacks is measured in Section \ref{ss The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}. Finally, in Section \ref{ssec:test_apa_spurious_resonances}, the flexible modes of the APA are measured and compared with a finite element model. -\subsection{Geometrical Measurements} +\subsubsection{Geometrical Measurements} \label{ssec:test_apa_geometrical_measurements} To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness. @@ -3780,7 +3781,7 @@ APA 7 & 18.7\\ \end{center} \end{minipage} -\subsection{Electrical Measurements} +\subsubsection{Electrical Measurements} \label{ssec:test_apa_electrical_measurements} From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu F\) and \(26\,\mu F\) with a nominal capacitance of \(20\,\mu F\). @@ -3821,7 +3822,7 @@ APA 7 & 4.85 & 9.85\\ \end{center} \end{minipage} -\subsection{Stroke and Hysteresis Measurement} +\subsubsection{Stroke and Hysteresis Measurement} \label{ssec:test_apa_stroke_measurements} To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}. @@ -3862,7 +3863,7 @@ From now on, only the six remaining amplified piezoelectric actuators that behav \caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML \subref{fig:test_apa_stroke_voltage}. Measured displacement as a function of applied voltage \subref{fig:test_apa_stroke_hysteresis}} \end{figure} -\subsection{Flexible Mode Measurement} +\subsubsection{Flexible Mode Measurement} \label{ssec:test_apa_spurious_resonances} In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. @@ -3958,7 +3959,7 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a \includegraphics[scale=1,scale=1]{figs/test_apa_schematic.png} \caption{\label{fig:test_apa_schematic}Schematic of the Test Bench used to measure the dynamics of the APA300ML. \(u\) is the output DAC voltage, \(V_a\) the output amplifier voltage (i.e. voltage applied across the actuator stacks), \(d_e\) the measured displacement by the encoder and \(V_s\) the measured voltage across the sensor stack.} \end{figure} -\subsection{Hysteresis} +\subsubsection{Hysteresis} \label{ssec:test_apa_hysteresis} Because the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload. @@ -3972,7 +3973,7 @@ This is the typical behavior expected from a PZT stack actuator, where the hyste \caption{\label{fig:test_apa_meas_hysteresis}Displacement as a function of applied voltage for multiple excitation amplitudes} \end{figure} -\subsection{Axial stiffness} +\subsubsection{Axial stiffness} \label{ssec:test_apa_stiffness} To estimate the stiffness of the APA, a weight with known mass \(m_a = 6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(\Delta d_e\) is measured using the encoder. @@ -4033,7 +4034,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the ` The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\). -\subsection{Dynamics} +\subsubsection{Dynamics} \label{ssec:test_apa_meas_dynamics} In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified. @@ -4079,7 +4080,7 @@ All the identified dynamics of the six APA300ML (both when looking at the encode \caption{\label{fig:test_apa_frf_dynamics}Measured frequency response function from generated voltage \(u\) to the encoder displacement \(d_e\) \subref{fig:test_apa_frf_encoder} and to the force sensor voltage \(V_s\) \subref{fig:test_apa_frf_force} for the six APA300ML} \end{figure} -\subsection{Non Minimum Phase Zero?} +\subsubsection{Non Minimum Phase Zero?} \label{ssec:test_apa_non_minimum_phase} It was surprising to observe a non-minimum phase zero on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}). @@ -4108,7 +4109,7 @@ However, this is not so important here because the zero is lightly damped (i.e. \caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found in the transfer function from \(u\) to \(V_s\). The coherence \subref{fig:test_apa_non_minimum_phase_coherence} is quite good around the anti-resonance frequency. The phase \subref{fig:test_apa_non_minimum_phase_zoom} shoes a non-minimum phase behavior.} \end{figure} -\subsection{Effect of the resistor on the IFF Plant} +\subsubsection{Effect of the resistor on the IFF Plant} \label{ssec:test_apa_resistance_sensor_stack} A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at \(\approx 5\,\mu F\)). @@ -4124,7 +4125,7 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil \caption{\label{fig:test_apa_effect_resistance}Transfer function from \(u\) to \(V_s\) with and without the resistor \(R\) in parallel with the piezoelectric stack used as the force sensor} \end{figure} -\subsection{Integral Force Feedback} +\subsubsection{Integral Force Feedback} \label{ssec:test_apa_iff_locus} To implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation \eqref{eq:test_apa_iff_manual_fit}. @@ -4383,7 +4384,7 @@ Using this simple test bench, it can be concluded that the \emph{super element} \caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) \subref{fig:test_apa_super_element_comp_frf_enc} and from \(u\) to \(V_s\) \subref{fig:test_apa_super_element_comp_frf_force}} \end{figure} -\subsection{Conclusion} +\subsection*{Conclusion} \label{sec:test_apa_conclusion} In this study, the amplified piezoelectric actuators ``APA300ML'' have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase. @@ -4490,7 +4491,7 @@ Results are shown in Section \ref{sec:test_joints_bending_stiffness_meas} \subsection{Dimensional Measurements} \label{sec:test_joints_flex_dim_meas} -\subsection{Measurement Bench} +\subsubsection{Measurement Bench} Two dimensions are critical for the bending stiffness of the flexible joints. These dimensions can be measured using a profilometer. @@ -4516,7 +4517,7 @@ It is then possible to estimate the dimension of the flexible beam with an accur \caption{\label{fig:test_joints_profilometer}Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer \subref{fig:test_joints_profilometer_setup} and a image is obtained with which the gap can be estimated \subref{fig:test_joints_profilometer_image}} \end{figure} -\subsection{Measurement Results} +\subsubsection{Measurement Results} The specified flexible beam thickness (gap) is \(250\,\mu m\). Four gaps are measured for each flexible joint (2 in the \(x\) direction and 2 in the \(y\) direction). The ``beam thickness'' is then estimated as the mean between the gaps measured on opposite sides. @@ -4532,7 +4533,7 @@ However, what is more important than the true value of the thickness is the cons \caption{\label{fig:test_joints_size_hist}Histogram for the (16x2) measured beams' thicknesses} \end{figure} -\subsection{Bad flexible joints} +\subsubsection{Bad flexible joints} Using this profilometer allowed to detect flexible joints with manufacturing defects such as non-symmetrical shapes (see Figure \ref{fig:test_joints_bad_shape}) or flexible joints with machining chips stuck in the gap (see Figure \ref{fig:test_joints_bad_chips}). @@ -4562,7 +4563,7 @@ The bending stiffness can then be computed from equation \eqref{eq:test_joints_b \begin{equation}\label{eq:test_joints_bending_stiffness} \boxed{k_{R_x} = \frac{T_x}{\theta_x}, \quad k_{R_y} = \frac{T_y}{\theta_y}} \end{equation} -\subsection{Measurement principle} +\subsubsection{Measurement principle} \label{ssec:test_joints_meas_principle} \paragraph{Torque and Rotation measurement} To apply torque \(T_{y}\) between the two mobile parts of the flexible joint, a known ``linear'' force \(F_{x}\) can be applied instead at a certain distance \(h\) with respect to the rotation point. @@ -4633,7 +4634,7 @@ The measured deflection of the flexible joint will be indirectly estimated from Indirectly measuring the deflection of the flexible joint induces some errors because of the limited stiffness between the force sensor and the displacement sensor. Such an effect will be estimated in the error budget (Section \ref{ssec:test_joints_error_budget}) -\subsection{Error budget} +\subsubsection{Error budget} \label{ssec:test_joints_error_budget} To estimate the accuracy of the measured bending stiffness that can be obtained using this measurement principle, an error budget is performed. @@ -4725,7 +4726,7 @@ Force sensor & \(\epsilon_F < 1\,\%\)\\ \end{table} -\subsection{Mechanical Design} +\subsubsection{Mechanical Design} \label{ssec:test_joints_test_bench} As explained in Section \ref{ssec:test_joints_meas_principle}, the flexible joint's bending stiffness is estimated by applying a known force to the flexible joint's tip and by measuring its deflection at the same point. @@ -4778,7 +4779,7 @@ A closer view of the force sensor tip is shown in Figure \ref{fig:test_joints_pi \end{subfigure} \caption{\label{fig:test_joints_picture_bench}Manufactured test bench for compliance measurement of the flexible joints} \end{figure} -\subsection{Load Cell Calibration} +\subsubsection{Load Cell Calibration} In order to estimate the measured errors of the load cell ``FC2231'', it is compared against another load cell\footnote{XFL212R-50N from TE Connectivity. The measurement range is \(50\,N\). The specified accuracy is \(1\,\%\) of the full range}. The two load cells are measured simultaneously while they are pushed against each other (see Figure \ref{fig:test_joints_force_sensor_calib_picture}). The contact between the two load cells is well defined as one has a spherical interface and the other has a flat surface. @@ -4803,7 +4804,7 @@ However, the estimated non-linearity is bellow \(0.2\,\%\) for forces between \( \caption{\label{fig:test_joints_force_sensor_calib}Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in \subref{fig:test_joints_force_sensor_calib_picture}. Comparison of the two measured forces and estimated non-linearity are shown in \subref{fig:test_joints_force_sensor_calib_fit}} \end{figure} -\subsection{Load Cell Stiffness} +\subsubsection{Load Cell Stiffness} The objective of this measurement is to estimate the stiffness \(k_F\) of the force sensor. To do so, a stiff element (much stiffer than the estimated \(k_F \approx 1\,N/\mu m\)) is mounted in front of the force sensor, as shown in Figure \ref{fig:test_joints_meas_force_sensor_stiffness_picture}. Then, the force sensor is pushed against this stiff element while the force sensor and the encoder displacement are measured. @@ -4826,7 +4827,7 @@ The load cell stiffness can then be estimated by computing a linear fit and is f \caption{\label{fig:test_joints_meas_force_sensor_stiffness}Estimation of the load cell stiffness. The measurement setup is shown in \subref{fig:test_joints_meas_force_sensor_stiffness_picture}. The measurement results are shown in \subref{fig:test_joints_force_sensor_stiffness_fit}.} \end{figure} -\subsection{Bending Stiffness estimation} +\subsubsection{Bending Stiffness estimation} The actual stiffness is now estimated by manually moving the translation stage from a start position where the force sensor is not yet in contact with the flexible joint to a position where the flexible joint is on its mechanical stop. The measured force and displacement as a function of time are shown in Figure \ref{fig:test_joints_meas_bend_time}. @@ -4852,7 +4853,7 @@ The bending stroke can also be estimated as shown in Figure \ref{fig:test_joints \caption{\label{fig:test_joints_meas_example}Results obtained on the first flexible joint. The measured force and displacement are shown in \subref{fig:test_joints_meas_bend_time}. The estimated angular displacement \(\theta_x\) as a function of the estimated applied torque \(T_{x}\) is shown in \subref{fig:test_joints_meas_F_d_lin_fit}. The bending stiffness \(k_{R_x}\) of the flexible joint can be estimated by computing a best linear fit (red dashed line).} \end{figure} -\subsection{Measured flexible joint stiffness} +\subsubsection{Measured flexible joint stiffness} The same measurement was performed for all the 16 flexible joints, both in the \(x\) and \(y\) directions. The measured angular motion as a function of the applied torque is shown in Figure \ref{fig:test_joints_meas_bending_all_raw_data} for the 16 flexible joints. @@ -5158,7 +5159,7 @@ A fiber interferometer\footnote{Two fiber intereferometers were used: an IDS3010 First, the effect of the encoder on the measured dynamics is investigated in Section \ref{ssec:test_struts_effect_encoder}. The dynamics observed by the encoder and interferometers are compared in Section \ref{ssec:test_struts_comp_enc_int}. Finally, all measured struts are compared in terms of dynamics in Section \ref{ssec:test_struts_comp_all_struts}. -\subsection{Effect of the Encoder on the measured dynamics} +\subsubsection{Effect of the Encoder on the measured dynamics} \label{ssec:test_struts_effect_encoder} System identification was performed without the encoder being fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) and with one encoder being fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}). @@ -5207,7 +5208,7 @@ This means that the encoder should have little effect on the effectiveness of th \caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) \subref{fig:test_struts_effect_encoder_int} and from \(u\) to \(V_s\) \subref{fig:test_struts_effect_encoder_iff}. Comparison of the observed dynamics by the encoder and interferometers \subref{fig:test_struts_comp_enc_int}} \end{figure} -\subsection{Comparison of the encoder and interferometer} +\subsubsection{Comparison of the encoder and interferometer} \label{ssec:test_struts_comp_enc_int} The dynamics measured by the encoder (i.e. \(d_e/u\)) and interferometers (i.e. \(d_a/u\)) are compared in Figure \ref{fig:test_struts_comp_enc_int}. @@ -5218,7 +5219,7 @@ These resonance frequencies match the frequencies of the flexible modes studied The good news is that these resonances are not impacting the axial motion of the strut (which is what is important for the hexapod positioning). However, these resonances make the use of an encoder fixed to the strut difficult from a control perspective. -\subsection{Comparison of all the Struts} +\subsubsection{Comparison of all the Struts} \label{ssec:test_struts_comp_all_struts} The dynamics of all the mounted struts (only 5 at the time of the experiment) were then measured on the same test bench. @@ -5268,7 +5269,7 @@ The struts were then disassembled and reassemble a second time to optimize align \includegraphics[scale=1,width=0.65\linewidth]{figs/test_struts_simscape_model.png} \caption{\label{fig:test_struts_simscape_model}Screenshot of the multi-body model of the strut fixed to the bench} \end{figure} -\subsection{Model dynamics} +\subsubsection{Model dynamics} \label{ssec:test_struts_comp_model} Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a \acrlong{fem}. @@ -5304,7 +5305,7 @@ For the flexible model, it will be shown in the next section that by adding some \caption{\label{fig:test_struts_comp_frf_flexible_model}Comparison of the measured frequency response functions, the multi-body model using the 2 DoF APA model, and using the ``flexible'' APA300ML model (Super-Element extracted from a Finite Element Model).} \end{figure} -\subsection{Effect of strut misalignment} +\subsubsection{Effect of strut misalignment} \label{ssec:test_struts_effect_misalignment} As shown in Figure \ref{fig:test_struts_comp_enc_plants}, the identified dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\) are very different from one strut to the other. @@ -5354,7 +5355,7 @@ This similarity suggests that the identified differences in dynamics are caused \caption{\label{fig:test_struts_effect_misalignment}Effect of a misalignment between the flexible joints and the APA300ML in the \(y\) direction \subref{fig:test_struts_effect_misalignment_y} and in the \(x\) direction \subref{fig:test_struts_effect_misalignment_x}} \end{figure} -\subsection{Measured strut misalignment} +\subsubsection{Measured strut misalignment} \label{ssec:test_struts_meas_misalignment} During the initial mounting of the struts, as presented in Section \ref{sec:test_struts_mounting}, the positioning pins that were used to position the APA with respect to the flexible joints in the \(y\) directions were not used (not received at the time). @@ -5404,7 +5405,7 @@ With a better alignment, the amplitude of the spurious resonances is expected to \caption{\label{fig:test_struts_comp_dy_tuned_model_frf_enc}Comparison of the frequency response functions from DAC voltage \(u\) to measured displacement \(d_e\) by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the \(y\) misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the \(x\) and \(y\) misalignments are tuned} \end{figure} -\subsection{Proper struts alignment} +\subsubsection{Proper struts alignment} \label{sec:test_struts_meas_all_aligned_struts} After receiving the positioning pins, the struts were mounted again with the positioning pins. @@ -5564,7 +5565,7 @@ After mounting all six struts, the mounting tool (Figure \ref{fig:test_nhexa_cen \subsection{Suspended Table} \label{sec:test_nhexa_table} -\subsection{Introduction} +\subsubsection{Introduction} When a dynamical system is fixed to a support (such as a granite or an optical table), its dynamics will couple to the support dynamics. This may results in additional modes appearing in the system dynamics, which are difficult to predict and model. @@ -5579,7 +5580,7 @@ The developed suspended table is described in Section \ref{ssec:test_nhexa_table The modal analysis of the table is done in \ref{ssec:test_nhexa_table_identification}. Finally, the multi-body model representing the suspended table was tuned to match the measured modes (Section \ref{ssec:test_nhexa_table_model}). -\subsection{Experimental Setup} +\subsubsection{Experimental Setup} \label{ssec:test_nhexa_table_setup} The design of the suspended table is quite straightforward. @@ -5593,7 +5594,7 @@ Finally, some interface elements were designed, and mechanical lateral mechanica \caption{\label{fig:test_nhexa_suspended_table_cad}CAD View of the vibration table. The purple cylinders are representing the soft springs.} \end{figure} -\subsection{Modal analysis of the suspended table} +\subsubsection{Modal analysis of the suspended table} \label{ssec:test_nhexa_table_identification} In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} were fixed to the breadboard. @@ -5653,7 +5654,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure \ref{ \caption{\label{fig:test_nhexa_table_flexible_modes}Three identified flexible modes of the suspended table} \end{figure} -\subsection{Multi-body Model of the suspended table} +\subsubsection{Multi-body Model of the suspended table} \label{ssec:test_nhexa_table_model} The multi-body model of the suspended table consists simply of two solid bodies connected by 4 springs. @@ -5703,7 +5704,7 @@ The effect of the payload mass on the dynamics is discussed in Section \ref{ssec \caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the studied system. The command signal generated by the speedgoat is \(\mathbf{u}\), and the measured dignals are \(\mathbf{d}_{e}\) and \(\mathbf{V}_s\). Units are indicated in square brackets.} \end{figure} -\subsection{Modal analysis} +\subsubsection{Modal analysis} \label{ssec:test_nhexa_enc_struts_modal_analysis} To facilitate the future analysis of the measured plant dynamics, a basic modal analysis of the nano-hexapod is performed. @@ -5755,7 +5756,7 @@ These modes are summarized in Table \ref{tab:test_nhexa_hexa_modal_modes_list}. \caption{\label{fig:test_nhexa_hexa_flexible_modes}Two identified flexible modes of the top plate of the Nano-Hexapod} \end{figure} -\subsection{Identification of the dynamics} +\subsubsection{Identification of the dynamics} \label{ssec:test_nhexa_identification} The dynamics of the nano-hexapod from the six command signals (\(u_1\) to \(u_6\)) to the six measured displacement by the encoders (\(d_{e1}\) to \(d_{e6}\)) and to the six force sensors (\(V_{s1}\) to \(V_{s6}\)) were identified by generating low-pass filtered white noise for each command signal, one by one. @@ -5793,7 +5794,7 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor \caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.} \end{figure} -\subsection{Effect of payload mass on the dynamics} +\subsubsection{Effect of payload mass on the dynamics} \label{ssec:test_nhexa_added_mass} One major challenge for controlling the NASS is the wanted robustness to a variation of payload mass; therefore, it is necessary to understand how the dynamics of the nano-hexapod changes with a change in payload mass. @@ -5855,7 +5856,7 @@ Both the ``direct'' terms (Section \ref{ssec:test_nhexa_comp_model}) and ``coupl Second, it should also represents how the system dynamics changes when a payload is fixed to the top platform. This is checked in Section \ref{ssec:test_nhexa_comp_model_masses}. -\subsection{Nano-Hexapod model dynamics} +\subsubsection{Nano-Hexapod model dynamics} \label{ssec:test_nhexa_comp_model} The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF APA, and rigid top and bottom plates. @@ -5885,7 +5886,7 @@ At higher frequencies, no resonances can be observed in the model, as the top pl \caption{\label{fig:test_nhexa_comp_simscape_diag}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(d_e\) \subref{fig:test_nhexa_comp_simscape_de_diag} and from \(u\) to \(V_s\) \subref{fig:test_nhexa_comp_simscape_Vs_diag}} \end{figure} -\subsection{Dynamical coupling} +\subsubsection{Dynamical coupling} \label{ssec:test_nhexa_comp_model_coupling} Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of MIMO systems. @@ -5911,7 +5912,7 @@ Therefore, if the modes of the struts are to be modeled, the \emph{super-element \caption{\label{fig:test_nhexa_comp_simscape_de_all_flex}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\). The APA are here modeled with a ``super-element''.} \end{figure} -\subsection{Effect of payload mass} +\subsubsection{Effect of payload mass} \label{ssec:test_nhexa_comp_model_masses} Another important characteristic of the model is that it should represents the dynamics of the system well for all considered payloads. @@ -5949,7 +5950,9 @@ Therefore, the model effectively represents the system coupling for different pa \caption{\label{fig:test_nhexa_comp_simscape_de_all_high_mass}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)} \end{figure} -\subsection{Conclusion} +\subsection*{Conclusion} +\label{sec:test_nhexa_conclusion} + The goal of this test bench was to obtain an accurate model of the nano-hexapod that could then be included on top of the micro-station model. The adopted strategy was to identify the nano-hexapod dynamics under conditions in which all factors that could have affected the nano-hexapod dynamics were considered. This was achieved by developing a suspended table with low frequency suspension modes that can be accurately modeled (Section \ref{sec:test_nhexa_table}). @@ -6047,7 +6050,7 @@ This way, the gap between the head and the reference sphere is much larger (here Nevertheless, the metrology system still has limited measurement range due to limited angular acceptance of the fibered interferometers. Indeed, when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and above some perpendicular displacement, the reflected light does not comes back into the fiber, and no interference is produced. -\subsection{Metrology Kinematics} +\subsubsection{Metrology Kinematics} \label{ssec:test_id31_metrology_kinematics} The developed short-stroke metrology system is schematically shown in Figure \ref{fig:test_id31_metrology_kinematics}. @@ -6089,7 +6092,7 @@ The five equations \eqref{eq:test_id31_metrology_kinematics} can be written in a \end{bmatrix} \end{equation} -\subsection{Rough alignment of the reference spheres} +\subsubsection{Rough alignment of the reference spheres} \label{ssec:test_id31_metrology_sphere_rought_alignment} The two reference spheres are aligned with the rotation axis of the spindle. @@ -6103,7 +6106,7 @@ With this setup, the alignment accuracy of both spheres with the spindle axis is The accuracy is probably limited due to the poor coaxiality between the cylinders and the spheres. However, this first alignment should permit to position the two sphere within the acceptance of the interferometers. -\subsection{Tip-Tilt adjustment of the interferometers} +\subsubsection{Tip-Tilt adjustment of the interferometers} \label{ssec:test_id31_metrology_alignment} The short-stroke metrology system is placed on top of the main granite using a gantry made of granite blocs (Figure \ref{fig:test_id31_short_stroke_metrology_overview}). @@ -6127,7 +6130,7 @@ This is done by maximizing the coupling efficiency of each interferometer. After the alignment procedure, the top interferometer should coincide with with spindle axis, and the lateral interferometers should all be in the horizontal plane and intersect the spheres' center. -\subsection{Fine Alignment of reference spheres using interferometers} +\subsubsection{Fine Alignment of reference spheres using interferometers} \label{ssec:test_id31_metrology_sphere_fine_alignment} Thanks to the first alignment of the two reference spheres with the spindle axis (Section \ref{ssec:test_id31_metrology_sphere_rought_alignment}) and to the fine adjustment of the interferometers orientations (Section \ref{ssec:test_id31_metrology_alignment}), the spindle can perform complete rotations while still having interference for all five interferometers. @@ -6157,7 +6160,7 @@ The remaining errors after alignment is in the order of \(\pm5\,\mu\text{rad}\) \caption{\label{fig:test_id31_metrology_align}Measured angular \subref{fig:test_id31_metrology_align_rx_ry} and lateral \subref{fig:test_id31_metrology_align_dx_dy} errors during a full spindle rotation. Between two rotations, the micro-hexapod is adjusted to better align the two spheres with the rotation axis.} \end{figure} -\subsection{Estimated measurement volume} +\subsubsection{Estimated measurement volume} \label{ssec:test_id31_metrology_acceptance} Because the interferometers are pointing to spheres and not flat surfaces, the lateral acceptance is limited. @@ -6183,7 +6186,7 @@ The obtained lateral acceptance for pure displacements in any direction is estim \end{table} -\subsection{Estimated measurement errors} +\subsubsection{Estimated measurement errors} \label{ssec:test_id31_metrology_errors} When using the NASS, the accuracy of the sample's positioning is determined by the accuracy of the external metrology. @@ -6243,7 +6246,7 @@ Voltages generated by the force sensor piezoelectric stacks \(\bm{V}_s = [V_{s1} \includegraphics[scale=1]{figs/test_id31_block_schematic_plant.png} \caption{\label{fig:test_id31_block_schematic_plant}Schematic of the NASS plant} \end{figure} -\subsection{Open-Loop Plant Identification} +\subsubsection{Open-Loop Plant Identification} \label{ssec:test_id31_open_loop_plant_first_id} The plant dynamics is first identified for a fixed spindle angle (at \(0\,\text{deg}\)) and without any payload. @@ -6273,7 +6276,7 @@ This issue was later solved. \caption{\label{fig:test_id31_first_id}Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology \subref{fig:test_id31_first_id_int} and force sensors \subref{fig:test_id31_first_id_iff}. Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are shown with shaded lines.} \end{figure} -\subsection{Better Angular Alignment} +\subsubsection{Better Angular Alignment} \label{ssec:test_id31_open_loop_plant_rz_alignment} One possible explanation of the increased coupling observed in Figure \ref{fig:test_id31_first_id_int} is the poor alignment between the external metrology axes (i.e. the interferometer supports) and the nano-hexapod axes. @@ -6313,7 +6316,7 @@ The flexible modes of the top platform can be passively damped while the modes o \caption{\label{fig:test_id31_first_id_int_better_rz_align}Decrease of the coupling with better Rz alignment} \end{figure} -\subsection{Effect of Payload Mass} +\subsubsection{Effect of Payload Mass} \label{ssec:test_id31_open_loop_plant_mass} In order to see how the system dynamics changes with the payload, open-loop identification was performed for four payload conditions that are shown in Figure \ref{fig:test_id31_picture_masses}. @@ -6367,7 +6370,7 @@ It is interesting to note that the anti-resonances in the force sensor plant are \caption{\label{fig:test_id31_comp_simscape_diag_masses}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(\epsilon\mathcal{L}\) \subref{fig:test_id31_comp_simscape_int_diag_masses} and from \(u\) to \(V_s\) \subref{fig:test_id31_comp_simscape_iff_diag_masses}} \end{figure} -\subsection{Effect of Spindle Rotation} +\subsubsection{Effect of Spindle Rotation} \label{ssec:test_id31_open_loop_plant_rotation} To verify that all the kinematics in Figure \ref{fig:test_id31_block_schematic_plant} are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at \(36\,\text{deg}/s\) and at \(180\,\text{deg}/s\). @@ -6394,7 +6397,7 @@ This also indicates that the metrology kinematics is correct and is working in r \caption{\label{fig:test_id31_effect_rotation}Effect of the spindle rotation on the plant dynamics from \(u\) to \(\epsilon\mathcal{L}\). Three rotational velocities are tested (\(0\,\text{deg}/s\), \(36\,\text{deg}/s\) and \(180\,\text{deg}/s\)). Both direct terms \subref{fig:test_id31_effect_rotation_direct} and coupling terms \subref{fig:test_id31_effect_rotation_coupling} are displayed.} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} The identified frequency response functions from command signals \(\bm{u}\) to the force sensors \(\bm{V}_s\) and to the estimated strut errors \(\bm{\epsilon\mathcal{L}}\) are well matching the developed multi-body model. Effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used. The spindle rotation has no visible effect on the measured dynamics, indicating that controllers should be robust to the spindle rotation. @@ -6421,7 +6424,7 @@ And it is implemented as shown in the block diagram of Figure \ref{fig:test_id31 \includegraphics[scale=1]{figs/test_id31_iff_schematic.png} \caption{\label{fig:test_id31_iff_block_diagram}Block diagram of the implemented decentralized IFF controller. The controller \(\bm{K}_{\text{IFF}}\) is a diagonal controller with the same elements on every diagonal term \(K_{\text{IFF}}\).} \end{figure} -\subsection{IFF Plant} +\subsubsection{IFF Plant} \label{ssec:test_id31_iff_plant} As the multi-body model is going to be used to evaluate the stability of the IFF controller and to optimize the achievable damping, it is first checked whether this model accurately represents the system dynamics. @@ -6438,7 +6441,7 @@ This confirms that the multi-body model can be used to tune the IFF controller. \caption{\label{fig:test_id31_comp_simscape_Vs}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six force sensor voltages \(V_{s1}\) to \(V_{s6}\)} \end{figure} -\subsection{IFF Controller} +\subsubsection{IFF Controller} \label{ssec:test_id31_iff_controller} A decentralized IFF controller was designed such that it adds damping to the suspension modes of the nano-hexapod for all considered payloads. @@ -6503,7 +6506,7 @@ However, in this study, it was chosen to implement a fix (i.e. non-adaptive) dec \caption{\label{fig:test_id31_iff_root_locus}Root Locus plots for the designed decentralized IFF controller and using the multy-body model. Black crosses indicate the closed-loop poles for the choosen value of the gain.} \end{figure} -\subsection{Damped Plant} +\subsubsection{Damped Plant} \label{ssec:test_id31_iff_perf} As the model is accurately modelling the system dynamics, it can be used to estimate the damped plant, i.e. the transfer functions from \(\bm{u}^\prime\) to \(\bm{\mathcal{L}}\). @@ -6529,7 +6532,7 @@ The obtained frequency response functions are compared with the model in Figure \caption{\label{fig:test_id31_hac_plant_effect_mass_comp_model}Comparison of the open-loop plants and the damped plant with Decentralized IFF, estimated from the multi-body model \subref{fig:test_id31_comp_ol_iff_plant_model}. Comparison of the measured damped plants and modeled plants for all considered payloads \subref{fig:test_id31_hac_plant_effect_mass}. Only ``direct'' terms (\(\epsilon\mathcal{L}_i/u_i^\prime\)) are displayed for simplificty} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} The implementation of a decentralized Integral Force Feedback controller has been successfully demonstrated. Using the multi-body model, the controller was designed and optimized to ensure stability across all payload conditions while providing significant damping of suspension modes. The experimental results validated the model predictions, showing a reduction of peak amplitudes by approximately a factor of 10 across the full payload range (0-39 kg). @@ -6556,7 +6559,7 @@ K_{\text{HAC}} & & 0 \\ \includegraphics[scale=1]{figs/test_id31_iff_hac_schematic.png} \caption{\label{fig:test_id31_iff_hac_schematic}Block diagram of the implemented HAC-IFF controllers. The controller \(\bm{K}_{\text{HAC}}\) is a diagonal controller with the same elements on every diagonal term \(K_{\text{HAC}}\).} \end{figure} -\subsection{Damped Plant} +\subsubsection{Damped Plant} \label{ssec:test_id31_iff_hac_plant} To verify whether the multi body model accurately represents the measured damped dynamics, both direct terms and coupling terms corresponding to the first actuator are compared in Figure \ref{fig:test_id31_comp_simscape_hac}. @@ -6579,7 +6582,7 @@ This is one of the key benefit of using the HAC-LAC strategy. \caption{\label{fig:test_id31_comp_all_undamped_damped_plants}Comparison of the (six) direct terms for all (four) payload conditions in the undamped case (in blue) and the damped case (i.e. with the decentralized IFF being implemented, in red).} \end{figure} -\subsection{Interaction Analysis} +\subsubsection{Interaction Analysis} \label{sec:test_id31_hac_interaction_analysis} As the control strategy here is to apply a diagonal control in the frame of the struts, it is important to determine the frequency at which multivariable effects become significant, as this represents a critical limitation of the control approach. @@ -6608,7 +6611,7 @@ This design choice, while beneficial for system simplicity, introduces inherent \caption{\label{fig:test_id31_hac_rga_number}RGA-number for the damped plants - Comparison of all the payload conditions} \end{figure} -\subsection{Robust Controller Design} +\subsubsection{Robust Controller Design} \label{ssec:test_id31_iff_hac_controller} A diagonal controller was designed to be robust to change of payloads, which means that every damped plants shown in Figure \ref{fig:test_id31_comp_all_undamped_damped_plants} should be considered during the controller design. @@ -6640,7 +6643,7 @@ However, small stability margins are observed for the highest mass, indicating t \caption{\label{fig:test_id31_hac_loop_gain_loci}Robust High Authority Controller. ``Decentralized loop-gains'' are shown in \subref{fig:test_id31_hac_loop_gain} and characteristic loci are shown in \subref{fig:test_id31_hac_characteristic_loci}} \end{figure} -\subsection{Performance estimation with simulation of Tomography scans} +\subsubsection{Performance estimation with simulation of Tomography scans} \label{ssec:test_id31_iff_hac_perf} To estimate the performances that can be expected with this HAC-LAC architecture and the designed controller, simulations of tomography experiments were performed\footnote{Note that the eccentricity of the ``point of interest'' with respect to the Spindle rotation axis has been tuned based on measurements.}. @@ -6664,7 +6667,7 @@ The obtained closed-loop positioning accuracy was found to comply with the requi \caption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}Position error of the sample in the XY \subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy} and YZ \subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz} planes during a simulation of a tomography experiment at \(180\,\text{deg/s}\). No payload is placed on top of the nano-hexapod.} \end{figure} -\subsection{Robustness estimation with simulation of Tomography scans} +\subsubsection{Robustness estimation with simulation of Tomography scans} \label{ssec:test_id31_iff_hac_robustness} To verify the robustness to the change of payload mass, four simulations of tomography experiments were performed with payloads as shown Figure \ref{fig:test_id31_picture_masses} (i.e. \(0\,kg\), \(13\,kg\), \(26\,kg\) and \(39\,kg\)). @@ -6680,7 +6683,7 @@ Yet it was decided that this controller will be tested experimentally, and impro \caption{\label{fig:test_id31_hac_tomography_Wz36_simulation}Positioning errors in the Y-Z plane during tomography experiments simulated using the multi-body model (in closed-loop)} \end{figure} -\subsection*{Conclusion} +\subsubsection*{Conclusion} In this section, a High-Authority-Controller was developed to actively stabilize the sample's position. The multi-body model was first validated by comparing it with measured frequency responses of the damped plant, showing good agreement for both direct terms and coupling terms. This validation confirmed that the model could be reliably used to tune the feedback controller and evaluate its performances. @@ -6732,7 +6735,7 @@ RMS & 30nm & 15nm & \(250\,\text{nrad}\)\\ \caption{\label{tab:test_id31_experiments_specifications}Specifications for the Nano-Active-Stabilization-System} \end{table} -\subsection{Tomography Scans} +\subsubsection{Tomography Scans} \label{ssec:test_id31_scans_tomography} \paragraph{Slow Tomography scans} @@ -6828,7 +6831,7 @@ This experiment also illustrates that when needed, performance can be enhanced b \caption{\label{fig:test_id31_hac_cas_cl}Cumulative Amplitude Spectrum for tomography experiments at \(180\,\text{deg}/s\). Open-Loop case, IFF, and HAC-LAC are compared. Specifications are indicated by black dashed lines. RSM values are indicated in the legend.} \end{figure} -\subsection{Reflectivity Scans} +\subsubsection{Reflectivity Scans} \label{ssec:test_id31_scans_reflectivity} X-ray reflectivity measurements involve scanning thin structures, particularly solid/liquid interfaces, through the beam by varying the \(R_y\) angle. @@ -6857,7 +6860,7 @@ The results confirm that the NASS successfully maintains the point of interest w \caption{\label{fig:test_id31_reflectivity}Reflectivity scan (\(R_y\)) with a rotational velocity of \(100\,\mu \text{rad}/s\).} \end{figure} -\subsection{Dirty Layer Scans} +\subsubsection{Dirty Layer Scans} \label{ssec:test_id31_scans_dz} In some cases, samples are composed of several atomic ``layers'' that are first aligned in the horizontal plane through \(R_x\) and \(R_y\) positioning, followed by vertical scanning with precise \(D_z\) motion. These vertical scans can be executed either continuously or in a step-by-step manner. @@ -6947,7 +6950,7 @@ Yet, performance during acceleration phases could potentially be enhanced throug \caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan with a velocity of \(100\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in \subref{fig:test_id31_dz_scan_100ums_dz}. Errors in \(D_y\) and \(R_y\) are respectively shown in \subref{fig:test_id31_dz_scan_100ums_dy} and \subref{fig:test_id31_dz_scan_100ums_ry}} \end{figure} -\subsection{Lateral Scans} +\subsubsection{Lateral Scans} \label{ssec:test_id31_scans_dy} Lateral scans are executed using the \(T_y\) stage. The stepper motor controller\footnote{The ``IcePAP'' \cite{janvier13_icepap} which is developed at the ESRF.} generates a setpoint that is transmitted to the Speedgoat. @@ -7020,7 +7023,7 @@ For applications requiring small \(D_y\) scans, the nano-hexapod can be used exc \caption{\label{fig:test_id31_dy_100ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(100\,\mu m/s\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in \subref{fig:test_id31_dy_100ums_dy}.} \end{figure} -\subsection{Diffraction Tomography} +\subsubsection{Diffraction Tomography} \label{ssec:test_id31_scans_diffraction_tomo} In diffraction tomography experiments, the micro-station executes combined motions: continuous rotation around the \(R_z\) axis while performing lateral scans along \(D_y\). For this validation, the spindle maintained a constant rotational velocity of \(6\,\text{deg/s}\) while the nano-hexapod executed the lateral scanning motion. @@ -7061,7 +7064,7 @@ Alternatively, developing a feedforward controller could improve lateral positio \caption{\label{fig:test_id31_diffraction_tomo}Diffraction tomography scans (combined \(R_z\) and \(D_y\) motions) at several \(D_y\) velocities (\(R_z\) rotational velocity is \(6\,\text{deg/s}\)).} \end{figure} -\subsection{Conclusion} +\subsubsection{Conclusion} \label{ssec:test_id31_scans_conclusion} A comprehensive series of experimental validations was conducted to evaluate the NASS performance across a wide range of typical scientific experiments. @@ -7150,6 +7153,7 @@ Moreover, the systematic approach to system development and validation, along wi \chapter*{List of Publications} \begin{refsection}[ref.bib] + \renewcommand{\clearpage}{} % Désactive \clearpage temporairement % List all papers even if not cited \nocite{*} % Sort by year @@ -7160,7 +7164,5 @@ Moreover, the systematic approach to system development and validation, along wi \printbibliography[keyword={publication},heading={subbibliography},title={In Proceedings},env=mypubs,type={inproceedings}] \end{refsection} -\printglossary[type=\acronymtype] -\printglossary[type=\glossarytype] -\printglossary +\printglossaries \end{document} diff --git a/setup.org b/setup.org index 6f0ac46..2b4c632 100644 --- a/setup.org +++ b/setup.org @@ -95,6 +95,12 @@ Do not modify itemize/enumerate environments by default \onehalfspacing #+end_src +** List of figures +#+begin_src latex +\usepackage{tocloft} +\setlength{\cftfignumwidth}{2.5em} % Adjust this value as needed +#+end_src + ** Tables #+begin_src latex \usepackage{booktabs} @@ -119,40 +125,6 @@ I reduce the size of tables so that longer tables can still fit into an A4 (redu \displaywidowpenalty = 10000 #+end_src -** Headers -#+begin_src latex -\usepackage{scrlayer-scrpage} - -\pagestyle{scrheadings} -#+end_src - -** Section/Figure format -#+begin_src latex -\renewcommand{\partformat}{\huge\partname~\thepart\autodot} -\renewcommand{\raggedpart}{\flushleft} - -\setkomafont{part}{\normalfont\huge\scshape} - -\setkomafont{sectioning}{\normalfont\scshape} -\setkomafont{descriptionlabel}{\normalfont\bfseries} - -\setkomafont{caption}{\small} -\setkomafont{captionlabel}{\usekomafont{caption}} - -\setcounter{secnumdepth}{\subsubsectionnumdepth} -#+end_src - -Improve chapter font colors and font size. -The following commands make chapter numbers BrickRed. -#+begin_src latex -\makeatletter -\renewcommand*{\chapterformat}{ \mbox{\chapappifchapterprefix{\nobreakspace}{\color{BrickRed}\fontsize{40}{45}\selectfont\thechapter}\autodot\enskip}} -% Section with color -% \renewcommand\@seccntformat[1]{\color{BrickRed} {\csname the#1\endcsname}\hspace{0.3em}} -\renewcommand\@seccntformat[1]{{\csname the#1\endcsname}\hspace{0.3em}} -\makeatother -#+end_src - ** Floating images configuration By default, if a figure consumes 60% of the page it will get its own float-page. To change that we have to adjust the value of the =floatpagefraction= derivative. @@ -169,28 +141,6 @@ See more information [[https://tex.stackexchange.com/questions/68516/avoid-that- \pretocmd{\section}{\clearpage}{}{} #+end_src -** Hyperref and Bookmarks -#+begin_src latex -\usepackage[ % - colorlinks=true, % - citecolor=BrickRed, % - linkcolor=BrickRed, % - urlcolor=BrickRed, % - unicode % - ]{hyperref} - -\usepackage{hypcap} -#+end_src - -The bookmark package implements a new bookmark (outline) organisation for package hyperref. -This lets us change the "tree-navigation" associated with the generated pdf and constrain the menu only to H:2. - -#+begin_src latex -\usepackage{bookmark} - -\bookmarksetup{depth=2} -#+end_src - ** Use pdf instead of png #+begin_src latex \makeatletter @@ -388,7 +338,7 @@ And reduce the distance between a minted listing and its caption. \AtEndEnvironment{listing}{\vspace{-16pt}} #+end_src -#+begin_src matlab +#+begin_src latex \let\OldTexttt\texttt \renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}} #+end_src @@ -426,6 +376,94 @@ Add the cover image as background to the first page. Only do so when outputting \sethlcolor{my-pale-grey} #+end_src +** Headers +#+begin_src latex +% \usepackage[headsepline]{scrlayer-scrpage} +% \pagestyle{scrheadings} +#+end_src + +Config to have the chapter name until there is a section, and then displays the sections (from the doc) +#+begin_src latex +\usepackage[autooneside=false,headsepline]{scrlayer-scrpage} +% \pagestyle{scrheadings} +% \automark[section]{chapter} +#+end_src + +#+begin_src latex +% Clear default header styles +\clearpairofpagestyles +\automark[section]{chapter} + +% Set the header content +\ihead{\headmark} % Chapter (or section on even pages) aligned to the left +\ohead{\pagemark} % Page number aligned to the right + +% Enable the page style +\pagestyle{scrheadings} + +\setkomafont{headsepline}{\color{black}} % Change color if desired +\ModifyLayer[addvoffset=\dp\strutbox]{headsepline} % Fine-tune position +#+end_src + +Kind of working solution: +#+begin_src latex +% \automark{section} +% \renewhead*{headings}{ +% \ifstr{\headmark}{}{}{% +% \headmark\hfill +% } +% } +#+end_src + +** Section/Figure format +#+begin_src latex +\renewcommand{\partformat}{\huge\partname~\thepart\autodot} +\renewcommand{\raggedpart}{\flushleft} + +\setkomafont{part}{\normalfont\huge\scshape} + +\setkomafont{sectioning}{\normalfont\scshape} +\setkomafont{descriptionlabel}{\normalfont\bfseries} + +\setkomafont{caption}{\small} +\setkomafont{captionlabel}{\usekomafont{caption}} + +\setcounter{secnumdepth}{\subsubsectionnumdepth} +#+end_src + +Improve chapter font colors and font size. +The following commands make chapter numbers BrickRed. +#+begin_src latex +\makeatletter +\renewcommand*{\chapterformat}{ \mbox{\chapappifchapterprefix{\nobreakspace}{\color{BrickRed}\fontsize{40}{45}\selectfont\thechapter}\autodot\enskip}} +% Section with color +% \renewcommand\@seccntformat[1]{\color{BrickRed} {\csname the#1\endcsname}\hspace{0.3em}} +\renewcommand\@seccntformat[1]{{\csname the#1\endcsname}\hspace{0.3em}} +\makeatother +#+end_src + +** Hyperref and Bookmarks +#+begin_src latex +\usepackage[ % + colorlinks=true, % + citecolor=BrickRed, % + linkcolor=BrickRed, % + urlcolor=BrickRed, % + unicode % + ]{hyperref} + +\usepackage{hypcap} +#+end_src + +The bookmark package implements a new bookmark (outline) organisation for package hyperref. +This lets us change the "tree-navigation" associated with the generated pdf and constrain the menu only to H:2. + +#+begin_src latex +\usepackage{bookmark} + +\bookmarksetup{depth=2} +#+end_src + ** Index and glossaries #+begin_src latex \makeindex