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index c7bbff3..ada60a1 100644
--- a/figs/inkscape/introduction_id31_station_detector.svg
+++ b/figs/inkscape/introduction_id31_station_detector.svg
@@ -23,13 +23,13 @@
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+ inkscape:zoom="0.78673956"
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diff --git a/figs/introduction_endstation_id16b.pdf b/figs/introduction_endstation_id16b.pdf
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index a5807ee..4cc686e 100644
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diff --git a/figs/test_joints_profilometer_setup.jpg b/figs/test_joints_profilometer_setup.jpg
index b00e85e..abdcc40 100644
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diff --git a/figs/test_joints_received_zoom.jpg b/figs/test_joints_received_zoom.jpg
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diff --git a/phd-thesis.org b/phd-thesis.org
index 4b6ae51..9efe35d 100644
--- a/phd-thesis.org
+++ b/phd-thesis.org
@@ -20,8 +20,6 @@
#+BIND: org-latex-bib-compiler "biber"
-#+TODO: TODO(t) MAKE(m) REVIEW(r) COPY(c) | DONE(d)
-
#+BIND: org-latex-image-default-width ""
#+LATEX_HEADER: \input{config.tex}
@@ -2110,8 +2108,6 @@ Therefore, it is important to take special care when designing sampling environm
:END:
<>
-# TODO - Make a table summarizing the findings
-
In this study, a uniaxial model of the nano-active-stabilization-system was tuned from both dynamical measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model) and from disturbances measurements (Section\nbsp{}ref:sec:uniaxial_disturbances).
Three active damping techniques can be used to critically damp the active platform resonances (Section\nbsp{}ref:sec:uniaxial_active_damping).
@@ -10024,8 +10020,8 @@ Lastly, the design needed to permit the mounting of an encoder parallel to the s
#+end_figure
The flexible joints, shown in Figure\nbsp{}ref:fig:detail_design_flexible_joint, were manufactured using wire-cut acrfull:edm.
-First, the part's inherent fragility, stemming from its $0.25\,\text{mm}$ neck dimension, makes it susceptible to damage from cutting forces typical in classical machining.
-Furthermore, wire-cut acrshort:edm allows for the very tight machining tolerances critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure\nbsp{}ref:fig:detail_design_flexible_joint) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties.
+First, the part being quite fragile, stemming from its $0.25\,\text{mm}$ neck dimension, is easier to machine using wire-cut acrshort:edm thanks to the very small cutting forces compared to classical machining.
+Furthermore, wire-cut acrshort:edm allows for tight machining tolerances of complex shapes.
The material chosen for the flexible joints is a stainless steel designated /X5CrNiCuNb16-4/ (alternatively known as F16Ph).
This selection was based on its high specified yield strength (exceeding $1\,\text{GPa}$ after appropriate heat treatment) and its high fatigue resistance.
diff --git a/phd-thesis.pdf b/phd-thesis.pdf
index 2ac22f8..54ece65 100644
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diff --git a/phd-thesis.tex b/phd-thesis.tex
index 82e7bec..77c850d 100644
--- a/phd-thesis.tex
+++ b/phd-thesis.tex
@@ -1,4 +1,4 @@
-% Created 2025-04-22 Tue 23:52
+% Created 2025-04-23 Wed 23:54
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -44,31 +44,18 @@
\newacronym{rdc}{RDC}{Relative Damping Control}
\newacronym{rga}{RGA}{Relative Gain Array}
\newacronym{rms}{RMS}{Root Mean Square}
+\newacronym{rpm}{RPM}{Rotations Per Minute}
\newacronym{rp}{RP}{Robust Performance}
\newacronym{rs}{RS}{Robust Stability}
\newacronym{siso}{SISO}{Single Input Single Output}
\newacronym{sps}{SPS}{Samples per Second}
\newacronym{svd}{SVD}{Singular Value Decomposition}
\newacronym{vc}{VC}{Voice Coil}
-\newglossaryentry{ms}{name=\ensuremath{m_s},description={{Mass of the sample}}}
-\newglossaryentry{mn}{name=\ensuremath{m_n},description={{Mass of the nano-hexapod}}}
-\newglossaryentry{mh}{name=\ensuremath{m_h},description={{Mass of the positioning hexapod}}}
-\newglossaryentry{mt}{name=\ensuremath{m_t},description={{Mass of the micro-station stages}}}
-\newglossaryentry{mg}{name=\ensuremath{m_g},description={{Mass of the granite}}}
-\newglossaryentry{xf}{name=\ensuremath{x_f},description={{Floor motion}}}
-\newglossaryentry{ft}{name=\ensuremath{f_t},description={{Disturbance force of the micro-station}}}
-\newglossaryentry{fs}{name=\ensuremath{f_s},description={{Direct forces applied on the sample}}}
-\newglossaryentry{d}{name=\ensuremath{d},description={{Measured motion between the nano-hexapod and the granite}}}
-\newglossaryentry{fn}{name=\ensuremath{f_n},description={{Force sensor on the nano-hexapod}}}
-\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
-\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
-\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
-\newglossaryentry{casx}{name=\ensuremath{\Gamma_{x}},description={{Cumulative Amplitude Spectrum of signal $x$}}}
\input{config_extra.tex}
\addbibresource{ref.bib}
\addbibresource{phd-thesis.bib}
\author{Dehaeze Thomas}
-\date{2025-04-22}
+\date{2025-04-23}
\title{Nano Active Stabilization of samples for tomography experiments: A mechatronic design approach}
\subtitle{PhD Thesis}
\hypersetup{
@@ -128,7 +115,7 @@ Prof. Gérard Scorletti\newline
Dr. Olivier Mathon\newline
European Synchrotron Radiation Facility (Grenoble, France)
\chapter*{Abstract}
-The \(4^{\text{th}}\) generation synchrotron light sources has yielded X-ray beams with a 100-fold increase in brightness and sub-micron focusing capabilities, offering unprecedented scientific opportunities while requiring end-stations with enhanced sample positioning accuracy.
+The \(4^{\text{th}}\) generation synchrotron light sources have yielded X-ray beams with a 100-fold increase in brightness and sub-micron focusing capabilities, offering unprecedented scientific opportunities while requiring end-stations with enhanced sample positioning accuracy.
At the European Synchrotron (ESRF), the ID31 beamline features an end-station for positioning samples along complex trajectories.
However, its micrometer-range accuracy, limited by thermal drifts and mechanical vibrations, prevents maintaining the point of interest on the focused beam during experiments.
@@ -215,18 +202,18 @@ The research presented in this manuscript has been possible thanks to the Fonds
\dominitoc
\tableofcontents
-\clearpage
-\listoftables
+% \clearpage
+% \listoftables
-\clearpage
-\listoffigures
+% \clearpage
+% \listoffigures
\chapter{Introduction}
\label{chap:introduction}
\begingroup
\def\clearpage{\par}
\section{Context of this thesis}
\endgroup
-Synchrotron radiation facilities, are particle accelerators where electrons are accelerated to near the speed of light.
+Synchrotron radiation facilities are particle accelerators where electrons are accelerated to near the speed of light.
As these electrons traverse magnetic fields, typically generated by insertion devices or bending magnets, they produce exceptionally bright light known as synchrotron light.
This intense electromagnetic radiation, particularly in the X-ray spectrum, is subsequently used for the detailed study of matter.
Approximately 70 synchrotron light sources are operational worldwide, some of which are indicated in Figure~\ref{fig:introduction_synchrotrons}.
@@ -244,7 +231,7 @@ The \acrfull{esrf}, shown in Figure~\ref{fig:introduction_esrf_picture}, is a jo
The \acrshort{esrf} started user operations in 1994 as the world's first third-generation synchrotron.
Its accelerator complex, schematically depicted in Figure~\ref{fig:introduction_esrf_schematic}, includes a linear accelerator where electrons are initially generated and accelerated, a booster synchrotron to further accelerate the electrons, and an 844-meter circumference storage ring where electrons are maintained in a stable orbit.
-Synchrotron light are emitted in more than 40 beamlines surrounding the storage ring, each having specialized experimental stations.
+Synchrotron light is emitted in more than 40 beamlines surrounding the storage ring, each having specialized experimental stations.
These beamlines host diverse instrumentation that enables a wide spectrum of scientific investigations, including structural biology, materials science, and study of matter under extreme conditions.
\begin{figure}[h!tbp]
@@ -252,7 +239,7 @@ These beamlines host diverse instrumentation that enables a wide spectrum of sci
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/introduction_esrf_schematic.png}
\end{center}
-\subcaption{\label{fig:introduction_esrf_schematic} Schematic of the ESRF. The linear accelerator is shown in blue, the booster synchrotron in purple and the storage ring in green. There are over 40 beamlines, the ID31 beamline is highlighted in red}
+\subcaption{\label{fig:introduction_esrf_schematic} Schematic of the ESRF. The linear accelerator is shown in blue, the booster synchrotron in purple and the storage ring in green. There are over 40 beamlines. The ID31 beamline is highlighted in red}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
@@ -260,7 +247,7 @@ These beamlines host diverse instrumentation that enables a wide spectrum of sci
\end{center}
\subcaption{\label{fig:introduction_esrf_picture} European Synchrotron Radiation Facility}
\end{subfigure}
-\caption{\label{fig:instroduction_esrf}Schematic (\subref{fig:introduction_esrf_schematic}) and picture (\subref{fig:introduction_esrf_picture}) of the European Synchrotron Radiation Facility, situated in Grenoble, France}
+\caption{\label{fig:instroduction_esrf}Schematic (\subref{fig:introduction_esrf_schematic}) and picture (\subref{fig:introduction_esrf_picture}) of the European Synchrotron Radiation Facility, situated in Grenoble, France.}
\end{figure}
In August 2020, following an extensive 20-month upgrade period, the \acrshort{esrf} inaugurated its \acrfull{ebs}, establishing it as the world's premier fourth-generation synchrotron~\cite{raimondi21_commis_hybrid_multib_achrom_lattic}.
@@ -279,7 +266,7 @@ While this enhanced beam quality presents unprecedented scientific opportunities
Each beamline begins with a ``white'' beam generated by the insertion device.
This beam carries substantial power, typically exceeding kilowatts, and is generally unsuitable for direct application to samples.
-Instead, the beam passes through a series of optical elements—including absorbers, mirrors, slits, and monochromators—that filter and shape the X-rays to the desired specifications.
+The goal of the beamline is therefore to filter and shape the X-rays to the desired specifications using a series of optical elements such as absorbers, mirrors, slits, and monochromators.
These components are housed in multiple Optical Hutches, as depicted in Figure~\ref{fig:introduction_id31_oh}.
\begin{figure}[h!tbp]
@@ -297,7 +284,7 @@ These components are housed in multiple Optical Hutches, as depicted in Figure~\
\end{center}
\subcaption{\label{fig:introduction_id31_oh2}OH2}
\end{subfigure}
-\caption{\label{fig:introduction_id31_oh}Schematic of the two ID31 optical hutches: OH1 (\subref{fig:introduction_id31_oh1}) and OH2 (\subref{fig:introduction_id31_oh2}). Distance from the source (the insertion device) is indicated in meters.}
+\caption{\label{fig:introduction_id31_oh}Schematic of the two ID31 optical hutches: OH1 (\subref{fig:introduction_id31_oh1}) and OH2 (\subref{fig:introduction_id31_oh2}). Distance from the source (i.e. from the insertion device) is indicated in meters.}
\end{figure}
Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure~\ref{fig:introduction_id31_cad}), where, for experiments pertinent to this work, focusing optics are used.
@@ -323,7 +310,7 @@ Each stage serves distinct positioning functions; for example, the positioning h
\end{center}
\subcaption{\label{fig:introduction_micro_station_dof} Micro-Station}
\end{subfigure}
-\caption{\label{fig:introduction_micro_station}3D view of the ID31 Experimal Hutch (\subref{fig:introduction_id31_cad}). There are typically four main elements: the focusing optics in yellow, the sample stage in green, the sample itself in purple and the detector in blue. All these elements are fixed to the same granite. 3D view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}).}
+\caption{\label{fig:introduction_micro_station}3D view of the ID31 Experimental Hutch (\subref{fig:introduction_id31_cad}). There are typically four main elements: the focusing optics in yellow, the sample stage in green, the sample itself in purple and the detector in blue. All these elements are fixed to the same granite. 3D view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}).}
\end{figure}
The ``stacked-stages'' configuration of the micro-station provides high mobility, enabling diverse scientific experiments and imaging techniques.
@@ -345,7 +332,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/introduction_tomography_schematic.png}
\end{center}
-\subcaption{\label{fig:introduction_tomography_schematic} Experimental setup}
+\subcaption{\label{fig:introduction_tomography_schematic} Typical experimental setup for tomography experiment}
\end{subfigure}
\begin{subfigure}{0.34\textwidth}
\begin{center}
@@ -353,7 +340,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
\end{center}
\subcaption{\label{fig:introduction_tomography_results} Obtained image \cite{schoeppler17_shapin_highl_regul_glass_archit}}
\end{subfigure}
-\caption{\label{fig:introduction_tomography}Exemple of a tomography experiment. The sample is rotated and images are taken at several angles (\subref{fig:introduction_tomography_schematic}). Example of one 3D image obtained after tomography (\subref{fig:introduction_tomography_results}).}
+\caption{\label{fig:introduction_tomography}Example of a tomography experiment. The sample is rotated and images are taken at several angles (\subref{fig:introduction_tomography_schematic}). Example of one 3D image obtained using tomography (\subref{fig:introduction_tomography_results}).}
\end{figure}
\begin{figure}[h!tbp]
@@ -361,7 +348,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/introduction_scanning_schematic.png}
\end{center}
-\subcaption{\label{fig:introduction_scanning_schematic} Experimental setup}
+\subcaption{\label{fig:introduction_scanning_schematic} Typical experimental setup for a scanning experiment}
\end{subfigure}
\begin{subfigure}{0.34\textwidth}
\begin{center}
@@ -369,7 +356,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
\end{center}
\subcaption{\label{fig:introduction_scanning_results} Obtained image \cite{sanchez-cano17_synch_x_ray_fluor_nanop}}
\end{subfigure}
-\caption{\label{fig:introduction_scanning}Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of \(20\,\text{nm}\) (\subref{fig:introduction_scanning_results}).}
+\caption{\label{fig:introduction_scanning}Example of a scanning experiment. The sample is scanned in the YZ plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of \(20\,\text{nm}\) (\subref{fig:introduction_scanning_results}).}
\end{figure}
\subsubsection*{Need of Accurate Positioning End-Stations with High Dynamics}
Continuous progress in both synchrotron source technology and X-ray optics have led to the availability of smaller, more intense, and more stable X-ray beams.
@@ -391,7 +378,7 @@ The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source s
\caption{\label{fig:introduction_beam_3rd_4th_gen}View of the ESRF X-ray beam before the EBS upgrade (\subref{fig:introduction_beam_3rd_gen}) and after the EBS upgrade (\subref{fig:introduction_beam_4th_gen}). The brilliance is increased, whereas the horizontal size and emittance are reduced.}
\end{figure}
-Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of \acrshort{esrf}, where typical spot sizes were on the order of \(10\,\mu\text{m}\) \cite{riekel89_microf_works_at_esrf}.
+Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of \acrshort{esrf}, where typical spot sizes were on the order of \(10\,\upmu\text{m}\) \cite{riekel89_microf_works_at_esrf}.
Various technologies, including zone plates, Kirkpatrick-Baez mirrors, and compound refractive lenses, have been developed and refined, each presenting unique advantages and limitations~\cite{barrett16_reflec_optic_hard_x_ray}.
The historical reduction in achievable spot sizes is represented in Figure~\ref{fig:introduction_moore_law_focus}.
Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines.
@@ -399,7 +386,7 @@ Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Ha
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp,scale=0.9]{figs/introduction_moore_law_focus.png}
-\caption{\label{fig:introduction_moore_law_focus}Evolution of the measured spot size for different hard X-ray focusing elements. Adapated from~\cite{barrett24_x_optic_accel_based_light_sourc}}
+\caption{\label{fig:introduction_moore_law_focus}Evolution of the measured spot size for different hard X-ray focusing elements. Adapted from \cite{barrett24_x_optic_accel_based_light_sourc}.}
\end{figure}
The increased brilliance introduces challenges related to radiation damage, particularly at high-energy beamlines like ID31.
@@ -421,7 +408,7 @@ While effective for mitigating radiation damage, this sequential process can be
\end{center}
\subcaption{\label{fig:introduction_scan_fly} Fly scan}
\end{subfigure}
-\caption{\label{fig:introduction_scan_mode}Two acquisition modes. In step-by-step mode (\subref{fig:introduction_scan_step}), the motor moves at the wanted imaged position, the detector acquisition is started, the shutter is openned briefly to have the wanted exposition, the detector acquisition is stopped, and the motor can move to a new position. In \emph{fly-scan} mode (\subref{fig:introduction_scan_fly}), the shutter is openned while the sample is in motion, and the detector is acquired only at the wanted positions, on the \emph{fly}.}
+\caption{\label{fig:introduction_scan_mode}Two acquisition modes. In step-by-step mode (\subref{fig:introduction_scan_step}), the motor moves to the desired imaged position, the detector acquisition is started, the shutter is opened briefly to have the wanted exposure, the detector acquisition is stopped, and the motor can move to a new position. In \emph{fly-scan} mode (\subref{fig:introduction_scan_fly}), the shutter is opened while the sample is in motion, and the detector acquires data only at the desired positions while in motion (``on the fly'').}
\end{figure}
An alternative, more efficient approach is the ``fly-scan'' or ``continuous-scan'' methodology~\cite{xu23_high_nsls_ii}, depicted in Figure~\ref{fig:introduction_scan_fly}.
@@ -459,7 +446,7 @@ Similarly, the overall dynamic performance (stiffness, resonant frequencies) is
\end{center}
\subcaption{\label{fig:introduction_parallel_kinematics} Parallel Kinematics}
\end{subfigure}
-\caption{\label{fig:introduction_kinematics}Two positioning platforms with \(D_x/D_y/R_z\) degrees of freedom. One is using serial kinematics (\subref{fig:introduction_serial_kinematics}), while the other uses parallel kinematics (\subref{fig:introduction_parallel_kinematics})}
+\caption{\label{fig:introduction_kinematics}Two positioning platforms with \(D_x/D_y/R_z\) degrees of freedom. One is using serial kinematics (\subref{fig:introduction_serial_kinematics}), while the other uses parallel kinematics (\subref{fig:introduction_parallel_kinematics}).}
\end{figure}
Conversely, parallel kinematic architectures (Figure~\ref{fig:introduction_parallel_kinematics}) involve the coordinated motion of multiple actuators to achieve the desired end-effector motion.
@@ -484,7 +471,7 @@ However, when a large number of DoFs are required, the cumulative errors and lim
\end{center}
\subcaption{\label{fig:introduction_endstation_id11}ID11 end-station \cite{wright20_new_oppor_at_mater_scien}}
\end{subfigure}
-\caption{\label{fig:introduction_passive_stations}Example of two nano end-stations without online metrology}
+\caption{\label{fig:introduction_passive_stations}Example of two nano end-stations lacking online metrology for measuring the sample's position.}
\end{figure}
The concept of using an external metrology to measure and potentially correct for positioning errors is increasing used for nano-positioning end-stations.
@@ -511,7 +498,7 @@ The PtiNAMi microscope at DESY P06 (Figure~\ref{fig:introduction_stages_schroer}
\end{center}
\subcaption{\label{fig:introduction_stages_schroer} DESY P06 - PtiNAMi microscope \cite{schroer17_ptynam}}
\end{subfigure}
-\caption{\label{fig:introduction_metrology_stations}Two examples of end-station with integrated online metrology}
+\caption{\label{fig:introduction_metrology_stations}Two examples of end-station with integrated online metrology.}
\end{figure}
For applications requiring active compensation of measured errors, particularly with nano-beams, feedback control loops are implemented.
@@ -537,7 +524,7 @@ A more comprehensive review of actively controlled end-stations is provided in S
\end{center}
\subcaption{\label{fig:introduction_stages_nazaretski} NSLS-II HXN - Microscope. 1 and 2 are focusing optics, 3 is the sample location, 4 the sample stage and 5 the interferometers \cite{nazaretski17_desig_perfor_x_ray_scann}}
\end{subfigure}
-\caption{\label{fig:introduction_active_stations}Example of two end-stations with real-time position feedback based on an online metrology}
+\caption{\label{fig:introduction_active_stations}Example of two end-stations with real-time position feedback based on an online metrology.}
\end{figure}
For tomography experiments, correcting spindle guiding errors is critical.
@@ -546,19 +533,19 @@ In most reported cases, only translation errors are actively corrected.
Payload capacities for these high-precision systems are usually limited, typically handling calibrated samples on the micron scale, although capacities up to 500g have been reported~\cite{nazaretski22_new_kirkp_baez_based_scann,kelly22_delta_robot_long_travel_nano}.
The system developed in this thesis aims for payload capabilities approximately 100 times heavier (up to \(50\,\text{kg}\)) than previous stations with similar positioning requirements.
-End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few \acrshortpl{dof} with strokes around \(100\,\mu\text{m}\).
+End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few \acrshortpl{dof} with strokes around \(100\,\upmu\text{m}\).
Recently, \acrfull{vc} actuators were used to increase the stroke up to \(3\,\text{mm}\) \cite{kelly22_delta_robot_long_travel_nano,geraldes23_sapot_carnaub_sirius_lnls}
An alternative strategy involves a ``long stroke-short stroke'' architecture, illustrated conceptually in Figure~\ref{fig:introduction_two_stage_schematic}.
In this configuration, a high-accuracy, high-bandwidth short-stroke stage is mounted on top of a less precise long-stroke stage.
The short-stroke stage actively compensates for errors based on metrology feedback, while the long-stroke stage performs the larger movements.
-This approach allows combining extended travel with high precision and good dynamical response, but is often implemented for only one or a few DoFs, as seen in Figures~\ref{fig:introduction_two_stage_schematic} and~\ref{fig:introduction_two_stage_control_h_bridge}.
+This approach allows the combination of extended travel with high precision and good dynamical response, but is often implemented for only one or a few DoFs, as seen in Figures~\ref{fig:introduction_two_stage_schematic} and~\ref{fig:introduction_two_stage_control_h_bridge}.
\begin{figure}[h!tbp]
\begin{subfigure}{0.64\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/introduction_two_stage_schematic.png}
\end{center}
-\subcaption{\label{fig:introduction_two_stage_schematic} Typical Long Stroke - Short Stroke control architecture}
+\subcaption{\label{fig:introduction_two_stage_schematic} Typical Long Stroke-Short Stroke control architecture}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
@@ -566,7 +553,7 @@ This approach allows combining extended travel with high precision and good dyna
\end{center}
\subcaption{\label{fig:introduction_two_stage_control_h_bridge} Schematic of the "H-bridge" \cite{schmidt20_desig_high_perfor_mechat_third_revis_edition}}
\end{subfigure}
-\caption{\label{fig:introduction_two_stage_example}Schematic of a typical Long stroke - Short stroke control architecture (\subref{fig:introduction_two_stage_schematic}). A 3DoF long stroke - short stroke is shown in (\subref{fig:introduction_two_stage_control_h_bridge}) where \(y_1\), \(y_2\) and \(x\) are 3-phase linear motors and short stroke actuators are voice coils.}
+\caption{\label{fig:introduction_two_stage_example}Schematic of a typical Long stroke-Short stroke control architecture (\subref{fig:introduction_two_stage_schematic}). A 3-DoFs long stroke-short stroke is shown in (\subref{fig:introduction_two_stage_control_h_bridge}) where \(y_1\), \(y_2\) and \(x\) are 3-phase linear motors and short stroke actuators are voice coils.}
\end{figure}
\section{Challenge definition}
The advent of fourth-generation light sources, coupled with advancements in focusing optics and detector technology, imposes stringent new requirements on sample positioning systems.
@@ -578,25 +565,25 @@ Given the high frame rates of modern detectors, these specified positioning erro
These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to \(50\,\text{kg}\).
-The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately \(\SI{10}{\mu\m}\) and \(\SI{10}{\mu\rad}\) due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations.
+The existing micro-station, despite being composed of high-performance stages, has a positioning accuracy limited to approximately \(\SI{10}{\micro\m}\) and \(\SI{10}{\micro\rad}\) due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations.
The primary objective of this project is therefore defined as enhancing the positioning accuracy and stability of the ID31 micro-station by roughly two orders of magnitude, to fully leverage the capabilities offered by the ESRF-EBS source and modern detectors, without compromising its existing mobility and payload capacity.
\paragraph{The Nano Active Stabilization System Concept}
To address these challenges, the concept of a \acrfull{nass} is proposed.
-As schematically illustrated in Figure~\ref{fig:introduction_nass_concept_schematic}, the \acrshort{nass} comprises four principal components integrated with the existing micro-station (yellow): a 5-DoF online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple).
+As schematically illustrated in Figure~\ref{fig:introduction_nass_concept_schematic}, the \acrshort{nass} comprises four principal components integrated with the existing micro-station (yellow): a 5-DoFs online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple).
This system essentially functions as a high-performance, multi-axis vibration isolation and error correction platform situated between the micro-station and the sample.
It actively compensates for positioning errors measured by the external metrology system.
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp]{figs/introduction_nass_concept_schematic.png}
-\caption{\label{fig:introduction_nass_concept_schematic}The Nano Active Stabilization System concept}
+\caption{\label{fig:introduction_nass_concept_schematic}The Nano Active Stabilization System concept.}
\end{figure}
\paragraph{Online Metrology system}
The performance of the \acrshort{nass} is fundamentally reliant on the accuracy and bandwidth of its online metrology system, as the active control is based directly on these measurements.
-This metrology system must fulfill several criteria: measure the sample position in 5 DoF (excluding rotation about the vertical Z-axis); possess a measurement range compatible with the micro-station's extensive mobility and continuous spindle rotation; achieve an accuracy compatible with the sub-100 nm positioning target; and offer high bandwidth for real-time control.
+This metrology system must fulfill several criteria: measure the sample position in 5-DoFs (excluding rotation about the vertical Z-axis); possess a measurement range compatible with the micro-station's extensive mobility and continuous spindle rotation; achieve an accuracy compatible with the sub-100 nm positioning target; and offer high bandwidth for real-time control.
\begin{figure}[htbp]
\centering
@@ -609,7 +596,7 @@ Fiber interferometers target both surfaces.
A tracking system maintains perpendicularity between the interferometer beams and the mirror, such that Abbe errors are eliminated.
Interferometers pointing at the spherical surface provides translation measurement, while the ones pointing at the flat bottom surface yield tilt angles.
The development of this complex metrology system constitutes a significant mechatronic project in itself and is currently ongoing; it is not further detailed within this thesis.
-For the work presented herein, the metrology system is assumed to provide accurate, high-bandwidth 5-DoF position measurements.
+For the work presented herein, the metrology system is assumed to provide accurate, high-bandwidth 5-DoFs position measurements.
\paragraph{Active Stabilization Platform Design}
The active stabilization platform, positioned between the micro-station top plate and the sample, must satisfy several demanding requirements.
@@ -618,7 +605,7 @@ It must possess excellent dynamic properties to enable high-bandwidth control ca
Consequently, it must be free from backlash and play, and its active components (e.g., actuators) should introduce minimal vibrations.
Critically, it must accommodate payloads up to \(50\,\text{kg}\).
-A suitable candidate architecture for this platform is the Stewart platform (also known as ``hexapod''), a parallel kinematic mechanism capable of 6-DoF motion.
+A suitable candidate architecture for this platform is the Stewart platform (also known as ``hexapod''), a parallel kinematic mechanism capable of 6-DoFs motion.
Stewart platforms are widely employed in positioning and vibration isolation applications due to their inherent stiffness and potential for high precision.
Various designs exist, differing in geometry, actuation technology, sensing methods, and control strategies, as exemplified in Figure~\ref{fig:introduction_stewart_platform_piezo}.
A central challenge addressed in this thesis is the optimal mechatronic design of such an active platform tailored to the specific requirements of the NASS.
@@ -666,7 +653,7 @@ The goal is to rigorously evaluate different concepts, predict performance limit
Key challenges within this approach include developing appropriate design methodologies, creating accurate models capable of comparing different concepts quantitatively, and converging on a final design that achieves the target performance levels.
\section{Original Contributions}
This thesis presents several original contributions aimed at addressing the challenges inherent in the design, control, and implementation of the Nano Active Stabilization System, primarily within the fields of Control Theory, Mechatronic Design, and Experimental Validation.
-\paragraph{6DoF vibration control of a rotating platform}
+\paragraph{6-DoFs vibration control of a rotating platform}
Traditional long-stroke/short-stroke architectures typically operate in one or two degrees of freedom.
This work extends the concept to six degrees of freedom, with the active platform designed not only to correct rotational errors but to simultaneously compensate for errors originating from all underlying micro-station stages.
@@ -707,10 +694,10 @@ The integration of such filters into feedback control architectures can also lea
\paragraph{Experimental validation of the Nano Active Stabilization System}
The conclusion of this work involved the experimental implementation and validation of the complete NASS on the ID31 beamline.
-Experimental results, presented in Section~\ref{sec:test_id31}, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native \(\approx 10\,\mu\text{m}\) level down to the target \(\approx 100\,\text{nm}\) range during representative scientific experiments.
+Experimental results, presented in Section~\ref{sec:test_id31}, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native \(\approx 10\,\upmu\text{m}\) level down to the target \(\approx 100\,\text{nm}\) range during representative scientific experiments.
Crucially, robustness to variations in sample mass and diverse experimental conditions was verified.
The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal use of the advanced capabilities of the ESRF-EBS beam and associated detectors.
-To the author's knowledge, this represents the first demonstration of such a 5-DoF active stabilization platform being used to enhance the accuracy of a complex positioning system to this level.
+To the author's knowledge, this represents the first demonstration of such a 5-DoFs active stabilization platform being used to enhance the accuracy of a complex positioning system to this level.
\section{Outline}
This is divided into three chapters, each corresponding to a distinct phase of this methodology: Conceptual Design, Detailed Design, and Experimental Validation.
While the chapters follow this logical progression, care has been taken to structure each chapter such that its constitutive sections may also be consulted independently based on the reader's specific interests.
@@ -718,7 +705,7 @@ While the chapters follow this logical progression, care has been taken to struc
The conceptual design phase, detailed in Chapter~\ref{chap:concept}, followed a methodical progression from simplified uniaxial models to more complex multi-body representations.
Initial uniaxial analysis (Section~\ref{sec:uniaxial}) provided fundamental insights, particularly regarding the influence of active platform stiffness on performance.
-The introduction of rotation in a 3-DoF model (Section~\ref{sec:rotating}) allowed investigation of gyroscopic effects, revealing challenges for softer platform designs.
+The introduction of rotation in a 3-DoFs model (Section~\ref{sec:rotating}) allowed investigation of gyroscopic effects, revealing challenges for softer platform designs.
Experimental modal analysis of the existing micro-station (Section~\ref{sec:modal}) confirmed its complex dynamics but supported a rigid-body assumption for the different stages, justifying the development of a detailed multi-body model.
This model, tuned against experimental data and incorporating measured disturbances, was validated through simulation (Section~\ref{sec:ustation}).
The Stewart platform architecture was selected for the active stage, and its kinematics, dynamics, and control were analyzed (Section~\ref{sec:nhexa}).
@@ -751,7 +738,7 @@ The conceptual design of the Nano Active Stabilization System (NASS) follows a m
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp,width=\linewidth]{figs/chapter1_overview.png}
-\caption{\label{fig:chapter1_overview}Overview of the conceptual design development. The approach evolves from simplified analytical models to a multi-body model tuned from experimental modal analysis. It is concluded by closed-loop simulations of tomography experiments, validating the conceptual design.}
+\caption{\label{fig:chapter1_overview}Overview of the conceptual design development. The approach evolves from simplified analytical models to a multi-body model tuned from experimental modal analysis. Closed-loop simulations of tomography experiments are used to validate the concept.}
\end{figure}
The design process begins with a uniaxial model, presented in Section~\ref{sec:uniaxial}, which provides initial insights into fundamental challenges associated with this complex system.
@@ -783,7 +770,7 @@ The confidence gained through this systematic investigation provides a solid fou
\section{Uni-axial Model}
\label{sec:uniaxial}
In this report, a uniaxial model of the \acrfull{nass} is developed and used to obtain a first idea of the challenges involved in this complex system.
-Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions.
+Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, leading to similar conclusions.
To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section~\ref{sec:uniaxial_micro_station_model}).
Then, a model of the active platform is added on top of the micro-station.
@@ -803,7 +790,7 @@ Two key effects that may limit that positioning performances are then considered
\subsection{Micro Station Model}
\label{sec:uniaxial_micro_station_model}
In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station.
-\subsubsection{Measured dynamics}
+\paragraph{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 positioning hexapod's top platform (force \(F_{h}\)).
The vertical inertial motion of the granite \(x_{g}\) and the top platform of the positioning hexapod \(x_{h}\) are measured using geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 1\,\text{Hz}\)}.
@@ -826,7 +813,7 @@ Due to the poor coherence at low frequencies, these \acrlongpl{frf} will only be
\end{subfigure}
\caption{\label{fig:micro_station_uniaxial_model}Schematic of the Micro-Station measurement setup and uniaxial model.}
\end{figure}
-\subsubsection{Uniaxial Model}
+\paragraph{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 \acrshortpl{dof}.
A mass-spring-damper system represents the granite (with mass \(m_g\), stiffness \(k_g\) and damping \(c_g\)).
@@ -838,22 +825,23 @@ The damping coefficients were tuned to match the damping identified from the mea
The parameters obtained are summarized in Table~\ref{tab:uniaxial_ustation_parameters}.
\begin{table}[htbp]
-\caption{\label{tab:uniaxial_ustation_parameters}Physical parameters used for the micro-station uniaxial model}
\centering
\begin{tabularx}{0.6\linewidth}{Xccc}
\toprule
\textbf{Stage} & \textbf{Mass} & \textbf{Stiffness} & \textbf{Damping}\\
\midrule
-Hexapod & \(m_h = 15\,\text{kg}\) & \(k_h = 61\,\text{N}/\mu\text{m}\) & \(c_h = 3\,\frac{\text{kN}}{\text{m/s}}\)\\
-\(T_y\), \(R_y\), \(R_z\) & \(m_t = 1200\,\text{kg}\) & \(k_t = 520\,\text{N}/\mu\text{m}\) & \(c_t = 80\,\frac{\text{kN}}{\text{m/s}}\)\\
-Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,\text{N}/\mu\text{m}\) & \(c_g = 250\,\frac{\text{kN}}{\text{m/s}}\)\\
+Hexapod & \(m_h = 15\,\text{kg}\) & \(k_h = 61\,\text{N}/\upmu\text{m}\) & \(c_h = 3\,\frac{\text{kN}}{\text{m/s}}\)\\
+\(T_y\), \(R_y\), \(R_z\) & \(m_t = 1200\,\text{kg}\) & \(k_t = 520\,\text{N}/\upmu\text{m}\) & \(c_t = 80\,\frac{\text{kN}}{\text{m/s}}\)\\
+Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,\text{N}/\upmu\text{m}\) & \(c_g = 250\,\frac{\text{kN}}{\text{m/s}}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:uniaxial_ustation_parameters}Physical parameters used for the micro-station uniaxial model.}
+
\end{table}
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.
-\subsubsection{Comparison of model and measurements}
+\paragraph{Comparison of Model and Measurements}
The transfer functions from the forces injected by the hammers to the measured inertial motion of the positioning 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 \acrshortpl{dof}, only three modes with frequencies at \(70\,\text{Hz}\), \(140\,\text{Hz}\) and \(320\,\text{Hz}\) are modeled.
@@ -864,12 +852,12 @@ More accurate models will be used later on.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_comp_frf_meas_model.png}
-\caption{\label{fig:uniaxial_comp_frf_meas_model}Comparison of the measured FRF and identified ones from the uniaxial model}
+\caption{\label{fig:uniaxial_comp_frf_meas_model}Comparison of the measured FRF and the uniaxial model dynamics.}
\end{figure}
\subsection{Active Platform Model}
\label{sec:uniaxial_nano_station_model}
A model of the active platform and sample is now added on top of the uniaxial model of the micro-station (Figure~\ref{fig:uniaxial_model_micro_station_nass}).
-Disturbances (shown in red) are \gls{fs} the direct forces applied to the sample (for example cable forces), \gls{ft} representing the vibrations induced when scanning the different stages and \gls{xf} the floor motion.
+Disturbances (shown in red) are \(f_s\) the direct forces applied to the sample (for example cable forces), \(f_t\) representing the vibrations induced when scanning the different stages and \(x_f\) the floor motion.
The control signal is the force applied by the active platform \(f\) and the measurement is the relative motion between the sample and the granite \(d\).
The sample is here considered as a rigid body and rigidly fixed to the active platform.
The effect of resonances between the sample's \acrshort{poi} and the active platform actuator will be considered in Section~\ref{sec:uniaxial_payload_dynamics}.
@@ -877,7 +865,7 @@ The effect of resonances between the sample's \acrshort{poi} and the active plat
\begin{figure}[htbp]
\begin{subfigure}{0.39\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_model_micro_station_nass.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_model_micro_station_nass.png}
\end{center}
\subcaption{\label{fig:uniaxial_model_micro_station_nass}Uniaxial mass-spring-damper model of the NASS}
\end{subfigure}
@@ -885,21 +873,21 @@ The effect of resonances between the sample's \acrshort{poi} and the active plat
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_plant_first_params.png}
\end{center}
-\subcaption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces $f$ to measured displacement $d$ by the metrology}
+\subcaption{\label{fig:uniaxial_plant_first_params}Plant Dynamics}
\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 active platform 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}).}
+\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 active platform in blue, the sample in green and disturbances in red. Transfer function from \(f\) to \(d\) (\subref{fig:uniaxial_plant_first_params}).}
\end{figure}
-\subsubsection{Active Platform Parameters}
+\paragraph{Active Platform Parameters}
The active platform 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 \(50\,\text{kg}\).
+Its mass \(m_n\) 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 \(m_s\) that can vary from \(1\,\text{kg}\) up to \(50\,\text{kg}\).
-As a first example, the active platform stiffness of is set at \(k_n = 10\,\text{N}/\mu\text{m}\) and the sample mass is chosen at \(m_s = 10\,\text{kg}\).
-\subsubsection{Obtained Dynamic Response}
+As a first example, the active platform stiffness of is set at \(k_n = 10\,\text{N}/\upmu\text{m}\) and the sample mass is chosen at \(m_s = 10\,\text{kg}\).
+\paragraph{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}.
+The \emph{plant} (i.e., the transfer function from actuator force \(f\) to displacement \(d\)) is shown in Figure~\ref{fig:uniaxial_plant_first_params}.
-For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~\ref{fig:uniaxial_model_micro_station_nass} will be considered: three active platform stiffnesses (\(k_n = 0.01\,\text{N}/\mu\text{m}\), \(k_n = 1\,\text{N}/\mu\text{m}\) and \(k_n = 100\,\text{N}/\mu\text{m}\)) combined with three sample's masses (\(m_s = 1\,\text{kg}\), \(m_s = 25\,\text{kg}\) and \(m_s = 50\,\text{kg}\)).
+For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~\ref{fig:uniaxial_model_micro_station_nass} will be considered: three active platform stiffnesses (\(k_n = 0.01\,\text{N}/\upmu\text{m}\), \(k_n = 1\,\text{N}/\upmu\text{m}\) and \(k_n = 100\,\text{N}/\upmu\text{m}\)) combined with three sample's masses (\(m_s = 1\,\text{kg}\), \(m_s = 25\,\text{kg}\) and \(m_s = 50\,\text{kg}\)).
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@@ -920,9 +908,9 @@ For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_xf}Floor motion}
\end{subfigure}
-\caption{\label{fig:uniaxial_sensitivity_dist_first_params}Sensitivity of the relative motion \(d\) to disturbances: \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_first_params_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_first_params_fs})}
+\caption{\label{fig:uniaxial_sensitivity_dist_first_params}Sensitivity of the relative motion \(d\) to the following disturbances: \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_first_params_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_first_params_xf}).}
\end{figure}
-\subsection{Disturbance Identification}
+\subsection{Identification of Disturbances}
\label{sec:uniaxial_disturbances}
To quantify disturbances (red signals in Figure~\ref{fig:uniaxial_model_micro_station_nass}), three geophones\footnote{Mark Product L-22D geophones are used with a sensitivity of \(88\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 2\,\text{Hz}\)} are used.
One is located on the floor, another one on the granite, and the last one on the positioning hexapod's top platform (see Figure~\ref{fig:uniaxial_ustation_meas_disturbances}).
@@ -941,9 +929,9 @@ The geophone located on the floor was used to measure the floor motion \(x_f\) w
\end{center}
\subcaption{\label{fig:uniaxial_ustation_dynamical_id_setup}Geophones used to measure vibrations induced by $T_y$ and $R_z$ scans}
\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})}
+\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}
-\subsubsection{Ground Motion}
+\paragraph{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 \(2.4\,\text{nV}/\sqrt{\text{Hz}}\)} with a gain of \(60\,\text{dB}\) before going to the \acrfull{adc}.
This is done to improve the signal-to-noise ratio.
@@ -960,7 +948,7 @@ G_{geo}(s) = \frac{V_{x_f}}{x_f}(s) = T_{g} \cdot s \cdot \frac{s^2}{s^2 + 2 \xi
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/uniaxial_geophone_meas_chain.png}
-\caption{\label{fig:uniaxial_geophone_meas_chain}Measurement setup for one geophone. The inertial displacement \(x\) is converted to a voltage \(V\) by the geophone. This voltage is amplified by a factor \(g_0 = 60\,\text{dB}\) using a low-noise voltage amplifier. It is then converted to a digital value \(\hat{V}_x\) using a 16bit ADC.}
+\caption{\label{fig:uniaxial_geophone_meas_chain}Measurement setup for one geophone. The inertial displacement \(x_f\) is converted to a voltage \(V_{x_f}\) by the geophone. This voltage is amplified by a factor \(g_0 = 60\,\text{dB}\) using a low-noise voltage amplifier. It is then converted to a digital value \(\hat{V}_{x_f}\) using a 16bit ADC.}
\end{figure}
The \acrfull{asd} of the floor motion \(\Gamma_{x_f}\) can be computed from the \acrlong{asd} of measured voltage \(\Gamma_{\hat{V}_{x_f}}\) using~\eqref{eq:uniaxial_asd_floor_motion}.
@@ -983,11 +971,11 @@ The estimated \acrshort{asd} \(\Gamma_{x_f}\) of the floor motion \(x_f\) is sho
\end{center}
\subcaption{\label{fig:uniaxial_asd_disturbance_force}Estimated ASD of $f_t$}
\end{subfigure}
-\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})}
+\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}
-\subsubsection{Stage Vibration}
+\paragraph{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 positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm.
+The vertical relative velocity between the top platform of the positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6 \acrfull{rpm}.
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.
The amplitude spectral density of the relative motion with and without the Spindle rotation are compared in Figure~\ref{fig:uniaxial_asd_vibration_spindle_rotation}.
@@ -997,7 +985,7 @@ The sharp peak observed at \(24\,\text{Hz}\) is believed to be induced by electr
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_asd_vibration_spindle_rotation.png}
-\caption{\label{fig:uniaxial_asd_vibration_spindle_rotation}Amplitude Spectral Density \(\Gamma_{R_z}\) of the relative motion measured between the granite and the positioning hexapod's top platform during Spindle rotating}
+\caption{\label{fig:uniaxial_asd_vibration_spindle_rotation}Amplitude Spectral Density \(\Gamma_{R_z}\) of the relative motion measured between the granite and the positioning hexapod's top platform during continuous Spindle rotation.}
\end{figure}
To compute the equivalent disturbance force \(f_t\) (Figure~\ref{fig:uniaxial_model_micro_station}) that induces such motion, the transfer function \(G_{f_t}(s)\) from \(f_t\) to the relative motion between the positioning hexapod's top platform and the granite \((x_{h} - x_{g})\) is extracted from the model.
@@ -1011,18 +999,17 @@ The amplitude spectral density \(\Gamma_{f_{t}}\) of the disturbance force is th
Now that a model of the \acrshort{nass} has been obtained (see section~\ref{sec:uniaxial_nano_station_model}) and that the disturbances have been estimated (see section~\ref{sec:uniaxial_disturbances}), it is possible to perform an \emph{open-loop dynamic noise budgeting}.
To perform such noise budgeting, the disturbances need to be modeled by their spectral densities (done in section~\ref{sec:uniaxial_disturbances}).
-Then, the transfer functions from disturbances to the performance metric (here the distance \(d\)) are computed (Section~\ref{ssec:uniaxial_noise_budget_sensitivity}).
+Then, the transfer functions from disturbances to the performance metric (here the distance \(d\)) are computed.
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}).
-\subsubsection{Sensitivity to disturbances}
-\label{ssec:uniaxial_noise_budget_sensitivity}
+This is very useful to identify what is limiting the performance of the system, and to compare the achievable performance with different system parameters.
+\paragraph{Sensitivity to Disturbances}
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.
This is done for two extreme sample masses \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and three active platform stiffnesses:
\begin{itemize}
-\item \(k_n = 0.01\,\text{N}/\mu\text{m}\) that represents a voice coil actuator with soft flexible guiding
-\item \(k_n = 1\,\text{N}/\mu\text{m}\) that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator
-\item \(k_n = 100\,\text{N}/\mu\text{m}\) that represents a stiff piezoelectric stack actuator
+\item \(k_n = 0.01\,\text{N}/\upmu\text{m}\) that represents a voice coil actuator with soft flexible guiding
+\item \(k_n = 1\,\text{N}/\upmu\text{m}\) that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator
+\item \(k_n = 100\,\text{N}/\upmu\text{m}\) that represents a stiff piezoelectric stack actuator
\end{itemize}
The obtained sensitivity to disturbances for the three active platform stiffnesses are shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses} for the sample mass \(m_s = 1\,\text{kg}\) (the same conclusions can be drawn with \(m_s = 50\,\text{kg}\)):
@@ -1051,10 +1038,9 @@ The obtained sensitivity to disturbances for the three active platform stiffness
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf}Floor motion}
\end{subfigure}
-\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})}
+\caption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}Sensitivity of \(d\) to disturbances for three different active platform 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_xf}).}
\end{figure}
-\subsubsection{Open-Loop Dynamic Noise Budgeting}
-\label{ssec:uniaxial_noise_budget_result}
+\paragraph{Open-Loop Dynamic Noise Budgeting}
Now, the amplitude spectral densities of the disturbances are considered to estimate the residual motion \(d\) for each active platform and sample configuration.
The \acrfull{cas} 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 active platform stiffnesses.
It is shown that the effect of floor motion is much less than that of stage vibrations, except for the soft active platform below \(5\,\text{Hz}\).
@@ -1077,17 +1063,17 @@ The conclusion is that the sample mass has little effect on the cumulative ampli
\end{subfigure}
\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 active platform stiffnesses is shown in (\subref{fig:uniaxial_cas_d_disturbances_payload_masses}). The control objective of having a residual error of \(20\,\text{nm RMS}\) is shown by the horizontal black dashed line.}
\end{figure}
-\subsubsection{Conclusion}
+\paragraph{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 \(100\,\text{nm RMS}\) for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)), \(200\,\text{nm RMS}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)) and \(1\,\mu\text{m}\,\text{RMS}\) for the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)).
+This residual vibration of \(d\) is found to be in the order of \(100\,\text{nm RMS}\) for the stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)), \(200\,\text{nm RMS}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\upmu\text{m}\)) and \(1\,\upmu\text{m}\,\text{RMS}\) for the soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)).
From this analysis, it may be concluded that the stiffer the active platform the better.
However, what is more important is the \emph{closed-loop} residual vibration of \(d\) (i.e., while the feedback controller is used).
The goal is to obtain a closed-loop residual vibration \(\epsilon_d \approx 20\,\text{nm RMS}\) (represented by an horizontal dashed black line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}).
The bandwidth of the feedback controller leading to a closed-loop residual vibration of \(20\,\text{nm RMS}\) can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}.
-A closed loop bandwidth of \(\approx 10\,\text{Hz}\) is found for the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)), \(\approx 50\,\text{Hz}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)), and \(\approx 100\,\text{Hz}\) for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)).
+A closed loop bandwidth of \(\approx 10\,\text{Hz}\) is found for the soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)), \(\approx 50\,\text{Hz}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\upmu\text{m}\)), and \(\approx 100\,\text{Hz}\) for the stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)).
Therefore, while the \emph{open-loop} vibration is the lowest for the stiff active platform, it requires the largest feedback bandwidth to meet the specifications.
The advantage of the soft active platform can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}.
@@ -1095,31 +1081,29 @@ The advantage of the soft active platform can be explained by its natural isolat
\label{sec:uniaxial_active_damping}
In this section, three active damping techniques are applied to the active platform (see Figure~\ref{fig:uniaxial_active_damping_strategies}): Integral Force Feedback (IFF)~\cite{preumont91_activ}, Relative Damping Control (RDC)~\cite[Chapter 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition} and Direct Velocity Feedback (DVF)~\cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}.
-These damping strategies are first described (Section~\ref{ssec:uniaxial_active_damping_strategies}) and are then compared in terms of achievable damping of the active platform mode (Section~\ref{ssec:uniaxial_active_damping_achievable_damping}), reduction of the effect of disturbances (i.e., \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\) (Sections~\ref{ssec:uniaxial_active_damping_sensitivity_disturbances}).
+These damping strategies are first described and are then compared in terms of achievable damping of the active platform mode, reduction of the effect of disturbances (i.e., \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\).
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_strategies_iff.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_strategies_iff.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_strategies_iff}IFF}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_strategies_rdc.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_strategies_rdc.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_strategies_rdc}RDC}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_strategies_dvf.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_strategies_dvf.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_strategies_dvf}DVF}
\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}
+\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}
-\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}.
@@ -1132,17 +1116,17 @@ The mechanical equivalent of this \acrshort{iff} strategy is a dashpot in series
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_iff_schematic.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_iff_schematic.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_iff_schematic}Integral Force Feedback}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_iff_equiv.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_iff_equiv.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_iff_equiv}Equivalent mechanical representation}
\end{subfigure}
-\caption{\label{fig:uniaxial_active_damping_iff}Integral Force Feedback (\subref{fig:uniaxial_active_damping_iff_schematic}) is equivalent to a damper in series with the actuator stiffness (\subref{fig:uniaxial_active_damping_iff_equiv})}
+\caption{\label{fig:uniaxial_active_damping_iff}Integral Force Feedback (\subref{fig:uniaxial_active_damping_iff_schematic}) is equivalent to a damper in series with the actuator stiffness (\subref{fig:uniaxial_active_damping_iff_equiv}).}
\end{figure}
\paragraph{Relative Damping Control (RDC)}
For the Relative Damping Control strategy, a relative motion sensor that measures the motion of the actuator is used (see Figure~\ref{fig:uniaxial_active_damping_rdc_schematic}) and a ``derivative'' feedback controller is used~\eqref{eq:uniaxial_rdc_controller}.
@@ -1156,17 +1140,17 @@ The mechanical equivalent of \acrshort{rdc} is a dashpot in parallel with the ac
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_rdc_schematic.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_rdc_schematic.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_rdc_schematic}Relative motion control}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_rdc_equiv.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_rdc_equiv.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_rdc_equiv}Equivalent mechanical representation}
\end{subfigure}
-\caption{\label{fig:uniaxial_active_damping_rdc}Relative Damping Control (\subref{fig:uniaxial_active_damping_rdc_schematic}) is equivalent to a damper in parallel with the actuator (\subref{fig:uniaxial_active_damping_rdc_equiv})}
+\caption{\label{fig:uniaxial_active_damping_rdc}Relative Damping Control (\subref{fig:uniaxial_active_damping_rdc_schematic}) is equivalent to a damper in parallel with the actuator (\subref{fig:uniaxial_active_damping_rdc_equiv}).}
\end{figure}
\paragraph{Direct Velocity Feedback (DVF)}
Finally, the direct velocity feedback strategy consists of using an inertial sensor (usually a geophone) that measures the ``absolute'' velocity of the body fixed on top of the actuator (see Figure~\ref{fig:uniaxial_active_damping_dvf_schematic}).
@@ -1182,20 +1166,19 @@ This is usually referred to as ``\emph{sky hook damper}''.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_dvf_schematic.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_dvf_schematic.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_dvf_schematic}Direct velocity feedback}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_active_damping_dvf_equiv.png}
+\includegraphics[scale=1,scale=0.9]{figs/uniaxial_active_damping_dvf_equiv.png}
\end{center}
\subcaption{\label{fig:uniaxial_active_damping_dvf_equiv}Equivalent mechanical representation}
\end{subfigure}
-\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})}
+\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}
-\subsubsection{Plant Dynamics for Active Damping}
-\label{ssec:uniaxial_active_damping_plants}
+\paragraph{Plant Dynamics for Active Damping}
The plant dynamics for all three active damping techniques are shown in Figure~\ref{fig:uniaxial_plant_active_damping_techniques}.
All have \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.
@@ -1225,11 +1208,9 @@ Therefore, it is expected that the micro-station dynamics might impact the achie
\end{center}
\subcaption{\label{fig:uniaxial_plant_active_damping_techniques_dvf}DVF}
\end{subfigure}
-\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 active platform stiffnesses (\(k_n = 0.01\,\text{N}/\mu\text{m}\) in blue, \(k_n = 1\,\text{N}/\mu\text{m}\) in red and \(k_n = 100\,\text{N}/\mu\text{m}\) in yellow) and three sample's masses (\(m_s = 1\,\text{kg}\): solid curves, \(m_s = 25\,\text{kg}\): dot-dashed curves, and \(m_s = 50\,\text{kg}\): dashed curves).}
+\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}) and DVF (\subref{fig:uniaxial_plant_active_damping_techniques_dvf})). Three active platform stiffnesses (\(k_n = 0.01\,\text{N}/\upmu\text{m}\) in blue, \(k_n = 1\,\text{N}/\upmu\text{m}\) in red and \(k_n = 100\,\text{N}/\upmu\text{m}\) in yellow) and three sample's masses (\(m_s = 1\,\text{kg}\): solid curves, \(m_s = 25\,\text{kg}\): dot-dashed curves, and \(m_s = 50\,\text{kg}\): dashed curves) are considered in each case.}
\end{figure}
-\subsubsection{Achievable Damping and Damped Plants}
-\label{ssec:uniaxial_active_damping_achievable_damping}
-
+\paragraph{Achievable Damping and Damped Plants}
To compare the added damping using the three considered active damping strategies, the root locus plot is used.
Indeed, the damping ratio \(\xi\) of a pole in the complex plane can be estimated from the angle \(\phi\) it makes with the imaginary axis~\eqref{eq:uniaxial_damping_ratio_angle}.
Increasing the angle with the imaginary axis therefore means that more damping is added to the considered resonance.
@@ -1239,7 +1220,7 @@ This is illustrated in Figure~\ref{fig:uniaxial_root_locus_damping_techniques_mi
\xi = \sin(\phi)
\end{equation}
-The Root Locus for the three active platform stiffnesses and the three active damping techniques are shown in Figure~\ref{fig:uniaxial_root_locus_damping_techniques}.
+The root locus for the three active platform stiffnesses and the three active damping techniques are shown in Figure~\ref{fig:uniaxial_root_locus_damping_techniques}.
All three active damping approaches can lead to \emph{critical damping} of the active platform suspension mode (angle \(\phi\) can be increased up to 90 degrees).
There is even some damping authority on micro-station modes in the following cases:
\begin{description}
@@ -1255,27 +1236,27 @@ The micro-station and the active platform masses are connected through a large d
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_soft.png}
\end{center}
-\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_mid.png}
\end{center}
-\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_stiff.png}
\end{center}
-\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed in the suspension mode of the active platform.}
+\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The root loci are zoomed on the suspension mode of the active platform.}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png}
-\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques applied with the soft active platform. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the positioning hexapod.}
+\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root locus for the three damping techniques applied with the soft active platform. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the positioning hexapod.}
\end{figure}
The transfer functions from the plant input \(f\) to the relative displacement \(d\) while active damping is implemented are shown in Figure~\ref{fig:uniaxial_damped_plant_three_active_damping_techniques}.
@@ -1286,28 +1267,26 @@ All three active damping techniques yielded similar damped plants.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png}
\end{center}
-\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_md.png}
\end{center}
-\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_pz.png}
\end{center}
-\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques}Obtained damped transfer function from \(f\) to \(d\) for the three damping techniques.}
+\caption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques}Obtained damped transfer functions from \(f\) to \(d\) for the three damping techniques.}
\end{figure}
-\subsubsection{Sensitivity to disturbances and Noise Budgeting}
-\label{ssec:uniaxial_active_damping_sensitivity_disturbances}
-
+\paragraph{Sensitivity to Disturbances and Noise Budgeting}
Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped.
The sensitivity to disturbances (direct forces \(f_s\), stage vibrations \(f_t\) and floor motion \(x_f\)) for all three active damping techniques are compared in Figure~\ref{fig:uniaxial_sensitivity_dist_active_damping}.
-The comparison is done with the active platform having a stiffness \(k_n = 1\,\text{N}/\mu\text{m}\).
+The comparison is done with the active platform having a stiffness \(k_n = 1\,\text{N}/\upmu\text{m}\).
Several conclusions can be drawn by comparing the obtained sensitivity transfer functions:
\begin{itemize}
@@ -1337,7 +1316,7 @@ This is because the equivalent damper in parallel with the actuator (see Figure~
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_xf}Floor motion}
\end{subfigure}
-\caption{\label{fig:uniaxial_sensitivity_dist_active_damping}Change of sensitivity to disturbance with all three active damping strategies. \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_active_damping_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs})}
+\caption{\label{fig:uniaxial_sensitivity_dist_active_damping}Change of sensitivity to disturbances for all three active damping strategies. Considered disturbances are \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_active_damping_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_active_damping_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_active_damping_xf}).}
\end{figure}
From the amplitude spectral density of the disturbances (computed in Section~\ref{sec:uniaxial_disturbances}) and the sensitivity to disturbances estimated using the three active damping strategies, a noise budget can be calculated.
@@ -1349,39 +1328,37 @@ All three active damping methods give similar results.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_soft.png}
\end{center}
-\subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_mid.png}
\end{center}
-\subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_stiff.png}
\end{center}
-\subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_cas_active_damping}Comparison of the cumulative amplitude spectrum (CAS) of the distance \(d\) for all three active damping techniques (\acrshort{ol} in black, IFF in blue, RDC in red and DVF in yellow).}
+\caption{\label{fig:uniaxial_cas_active_damping}Comparison of the \acrlong{cas} of the distance \(d\) for all three active damping techniques.}
\end{figure}
-\subsubsection{Conclusion}
-
+\paragraph{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.
+Equivalent mechanical representations were derived which are helpful for understanding the specific effects of each strategy.
+The plant dynamics were then compared and were found to all have alternating poles and zeros, which helps in the design of the active damping controller.
However, this property is not guaranteed for \acrshort{dvf}.
-The achievable damping of the active platform suspension mode can be made as large as possible for all three active damping techniques (Section~\ref{ssec:uniaxial_active_damping_achievable_damping}).
+The achievable damping of the active platform suspension mode can be made as large as possible for all three active damping techniques.
Even some damping can be applied to some micro-station modes in specific cases.
The obtained damped plants were found to be similar.
-The damping strategies were then compared in terms of disturbance reduction in Section~\ref{ssec:uniaxial_active_damping_sensitivity_disturbances}.
+The damping strategies were then compared in terms of disturbance reduction.
The comparison between the three active damping strategies is summarized in Table~\ref{tab:comp_active_damping}.
It is difficult to conclude on the best active damping strategy for the \acrfull{nass} yet.
-Which one will be used will be determined by the use of more accurate models and will depend on which is the easiest to implement in practice
+The one used will be determined by the use of more accurate models and will depend on which is easiest to implement in practice
\begin{table}[htbp]
-\caption{\label{tab:comp_active_damping}Comparison of active damping strategies}
\centering
\begin{tabularx}{0.9\linewidth}{Xccc}
\toprule
@@ -1398,10 +1375,12 @@ Which one will be used will be determined by the use of more accurate models and
\(x_f\) \textbf{Disturbance} & \(\nearrow\) at low frequency & \(\searrow\) near resonance & \(\nearrow\) at low frequency\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:comp_active_damping}Comparison of active damping strategies for the NASS.}
+
\end{table}
\subsection{Position Feedback Controller}
\label{sec:uniaxial_position_control}
-The \gls{haclac} architecture is shown in Figure~\ref{fig:uniaxial_hac_lac_architecture}.
+The \acrfull{haclac} architecture is shown in Figure~\ref{fig:uniaxial_hac_lac_architecture}.
This corresponds to a \emph{two step} control strategy:
\begin{itemize}
\item First, an active damping controller \(\bm{K}_{\textsc{LAC}}\) is implemented (see Section~\ref{sec:uniaxial_active_damping}).
@@ -1417,23 +1396,22 @@ This control architecture applied to the uniaxial model is shown in Figure~\ref{
\begin{figure}[htbp]
\begin{subfigure}{0.54\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_hac_lac_architecture.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_lac_architecture.png}
\end{center}
\subcaption{\label{fig:uniaxial_hac_lac_architecture}Typical HAC-LAC Architecture}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_hac_lac_model.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_lac_model.png}
\end{center}
\subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-IFF strategy}
\end{subfigure}
-\caption{\label{fig:uniaxial_hac_lac}\acrfull{haclac}}
+\caption{\label{fig:uniaxial_hac_lac}\acrfull{haclac}.}
\end{figure}
-\subsubsection{Damped Plant Dynamics}
-\label{ssec:uniaxial_position_control_damped_dynamics}
+\paragraph{Damped Plant Dynamics}
The damped plants obtained for the three active platform stiffnesses are shown in Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses}.
-For \(k_n = 0.01\,\text{N}/\mu\text{m}\) and \(k_n = 1\,\text{N}/\mu\text{m}\), 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}).
-However, this is not the case for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)) where two modes can be seen (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}).
+For \(k_n = 0.01\,\text{N}/\upmu\text{m}\) and \(k_n = 1\,\text{N}/\upmu\text{m}\), 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}).
+However, this is not the case for the stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)) where two modes can be seen (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}).
This is due to the interaction between the micro-station (modeled modes at \(70\,\text{Hz}\), \(140\,\text{Hz}\) and \(320\,\text{Hz}\)) and the active platform.
This effect will be further explained in Section~\ref{sec:uniaxial_support_compliance}.
@@ -1442,38 +1420,37 @@ This effect will be further explained in Section~\ref{sec:uniaxial_support_compl
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_soft.png}
\end{center}
-\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_mid.png}
\end{center}
-\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_stiff.png}
\end{center}
-\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_hac_iff_damped_plants_masses}Obtained damped plant using Integral Force Feedback for three sample masses}
+\caption{\label{fig:uniaxial_hac_iff_damped_plants_masses}Obtained damped plant using Integral Force Feedback for three sample masses.}
\end{figure}
-\subsubsection{Position Feedback Controller}
-\label{ssec:uniaxial_position_control_design}
+\paragraph{Position Feedback Controller}
The objective is to design high-authority feedback controllers for the three active platforms.
This controller must be robust to the change of sample's mass (from \(1\,\text{kg}\) up to \(50\,\text{kg}\)).
The required feedback bandwidths were estimated in Section~\ref{sec:uniaxial_noise_budgeting}:
\begin{itemize}
-\item \(f_b \approx 10\,\text{Hz}\) for the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)).
+\item \(f_b \approx 10\,\text{Hz}\) for the soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)).
Near this frequency, the plants (shown in Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_soft}) are equivalent to a mass line (i.e., slope of \(-40\,\text{dB/dec}\) and a phase of -180 degrees).
The gain of this mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,\text{kg}\) up to \(65\,\text{kg}\)).
This means that the designed controller will need to have \emph{large gain margins} to be robust to the change of sample's mass.
-\item \(\approx 50\,\text{Hz}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)).
+\item \(\approx 50\,\text{Hz}\) for the relatively stiff active platform (\(k_n = 1\,\text{N}/\upmu\text{m}\)).
Similar to the soft active platform, the plants near the crossover frequency are equivalent to a mass line (Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_mid}).
It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the active platform suspension mode.
-\item \(\approx 100\,\text{Hz}\) for the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)).
+\item \(\approx 100\,\text{Hz}\) for the stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)).
Contrary to the two first active platform stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure~\ref{fig:uniaxial_hac_iff_damped_plants_masses_stiff}).
The micro-station is not stiff enough to have a clear stiffness line at this frequency.
Therefore, there is both a change of phase and gain depending on the sample mass.
@@ -1503,7 +1480,6 @@ K_{\text{stiff}}(s) &= g \cdot
\end{subequations}
\begin{table}[htbp]
-\caption{\label{tab:uniaxial_feedback_controller_parameters}Parameters used for the position feedback controllers}
\centering
\begin{tabularx}{0.75\linewidth}{Xccc}
\toprule
@@ -1515,6 +1491,8 @@ K_{\text{stiff}}(s) &= g \cdot
\textbf{LPF} & \(\omega_l = 200\,\text{Hz}\) & \(\omega_l = 300\,\text{Hz}\) & \(\omega_l = 500\,\text{Hz}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:uniaxial_feedback_controller_parameters}Parameters used for the position feedback controllers.}
+
\end{table}
The loop gains corresponding to the designed \acrlongpl{hac} for the three active platform are shown in Figure~\ref{fig:uniaxial_loop_gain_hac}.
@@ -1531,21 +1509,21 @@ The goal is to have a first estimation of the attainable performance.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_vc.png}
\end{center}
-\subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_md.png}
\end{center}
-\subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_pz.png}
\end{center}
-\subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_nyquist_hac}Nyquist Plot for the high authority controller. The minimum modulus margin is illustrated by a black circle.}
+\caption{\label{fig:uniaxial_nyquist_hac}Nyquist Plot for the High Authority Controllers. The modulus margin is illustrated by the black circles.}
\end{figure}
\begin{figure}[htbp]
@@ -1553,24 +1531,23 @@ The goal is to have a first estimation of the attainable performance.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_vc.png}
\end{center}
-\subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_md.png}
\end{center}
-\subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_pz.png}
\end{center}
-\subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_loop_gain_hac}Loop gains for the High Authority Controllers}
+\caption{\label{fig:uniaxial_loop_gain_hac}Loop gains for the High Authority Controllers.}
\end{figure}
-\subsubsection{Closed-Loop Noise Budgeting}
-\label{ssec:uniaxial_position_control_cl_noise_budget}
+\paragraph{Closed-Loop Noise Budgeting}
The \acrlong{hac} are then implemented and the closed-loop sensitivities to disturbances are computed.
These are compared with the open-loop and damped plants cases in Figure~\ref{fig:uniaxial_sensitivity_dist_hac_lac} for just one configuration (moderately stiff active platform with \(25\,\text{kg}\) sample's mass).
@@ -1595,7 +1572,7 @@ As expected, the sensitivity to disturbances decreased in the controller bandwid
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_xf}Floor motion}
\end{subfigure}
-\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with \acrshort{lac} and with \acrshort{haclac}. An active platform with \(k_n = 1\,\text{N}/\mu\text{m}\) and a sample mass of \(25\,\text{kg}\) is used. \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), \(f_t\) disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs})}
+\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with \acrshort{lac} and with \acrshort{haclac}. An active platform with \(k_n = 1\,\text{N}/\upmu\text{m}\) and a sample mass of \(25\,\text{kg}\) are used. Disturbances are: \(f_s\) the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), \(f_t\) the disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and \(x_f\) the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_xf}).}
\end{figure}
The cumulative amplitude spectrum of the motion \(d\) is computed for all active platform configurations, all sample masses and in the \acrfull{ol}, damped (IFF) and position controlled (HAC-IFF) cases.
@@ -1607,38 +1584,38 @@ Obtained root mean square values of the distance \(d\) are better for the soft a
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_soft.png}
\end{center}
-\subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_mid.png}
\end{center}
-\subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_stiff.png}
\end{center}
-\subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectrum for all three active platform stiffnesses - Comparison of OL, IFF and \acrshort{haclac} cases}
+\caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectra for all three active platform stiffnesses in OL, with IFF and with \acrshort{haclac}.}
\end{figure}
-\subsubsection{Conclusion}
+\paragraph{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 \(\approx 20\,\text{nm RMS}\) 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.
In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass.
The attainable vibration control performances were estimated for the three active platform stiffnesses and were found to be close to the required values.
-However, the stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass.
+However, the stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)) requires the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass.
A slight advantage can be given to the soft active platform as it requires less feedback bandwidth while providing better stability results.
-\subsection{Effect of limited micro-station compliance}
+\subsection{Effect of Limited Support Compliance}
\label{sec:uniaxial_support_compliance}
In this section, the impact of the compliance of the support (i.e., the micro-station) on the dynamics of the plant to control is studied.
This is a critical point because the dynamics of the micro-station is complex, depends on the considered direction (see measurements in Figure~\ref{fig:uniaxial_comp_frf_meas_model}) and may vary with position and time.
It would be much better to have a plant dynamics that is not impacted by the micro-station.
-Therefore, the objective of this section is to obtain some guidance for the design of a active platform that will not be impacted by the complex micro-station dynamics.
+Therefore, the objective of this section is to obtain some guidance for the design of an active platform that will not be impacted by the complex micro-station dynamics.
To study this, two models are used (Figure~\ref{fig:uniaxial_support_compliance_models}).
The first one consists of the active platform directly fixed on top of the granite, thus neglecting any support compliance (Figure~\ref{fig:uniaxial_support_compliance_nano_hexapod_only}).
The second one consists of the active platform fixed on top of the micro-station having some limited compliance (Figure~\ref{fig:uniaxial_support_compliance_test_system})
@@ -1656,9 +1633,9 @@ The second one consists of the active platform fixed on top of the micro-station
\end{center}
\subcaption{\label{fig:uniaxial_support_compliance_test_system}Active platform fixed on top of the Micro-Station}
\end{subfigure}
-\caption{\label{fig:uniaxial_support_compliance_models}Models used to study the effect of limited support compliance}
+\caption{\label{fig:uniaxial_support_compliance_models}Models used to study the effect of limited support compliance.}
\end{figure}
-\subsubsection{Neglected support compliance}
+\paragraph{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 active platform mass (including the payload) is set at \(20\,\text{kg}\) and three active platform stiffnesses are considered, such that their resonance frequencies are at \(\omega_{n} = 10\,\text{Hz}\), \(\omega_{n} = 70\,\text{Hz}\) and \(\omega_{n} = 400\,\text{Hz}\).
@@ -1684,9 +1661,9 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_stiff}$\omega_{n} \gg \omega_{\mu}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_effect_support_compliance_neglected}Obtained transfer functions from \(F\) to \(L^{\prime}\) when neglecting support compliance}
+\caption{\label{fig:uniaxial_effect_support_compliance_neglected}Obtained transfer functions from \(F\) to \(L^{\prime}\) when neglecting support compliance.}
\end{figure}
-\subsubsection{Effect of support compliance on \(L/F\)}
+\paragraph{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}).
@@ -1695,6 +1672,7 @@ The transfer functions from \(F\) to \(L\) (i.e., control of the relative motion
When the relative displacement of the active platform \(L\) is controlled (dynamics shown in Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics}), having a stiff active platform (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_stiff}).
This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages.
In such a case, the control of the piezoelectric stage using its integrated metrology (typically capacitive sensors) is quite simple as the plant is not much affected by the dynamics of the support on which it is fixed.
+
If a soft active platform is used, the support dynamics appears in the dynamics between \(F\) and \(L\) (see Figure~\ref{fig:uniaxial_effect_support_compliance_dynamics_soft}) which will impact the control robustness and performance.
\begin{figure}[htbp]
@@ -1716,9 +1694,9 @@ If a soft active platform is used, the support dynamics appears in the dynamics
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_stiff}$\omega_{n} \gg \omega_{\mu}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_effect_support_compliance_dynamics}Effect of the support compliance on the transfer functions from \(F\) to \(L\)}
+\caption{\label{fig:uniaxial_effect_support_compliance_dynamics}Effect of the support compliance on the transfer functions from \(F\) to \(L\).}
\end{figure}
-\subsubsection{Effect of support compliance on \(d/F\)}
+\paragraph{Effect of support compliance on \(d/F\)}
When the motion to be controlled is the relative displacement \(d\) between the granite and the active platform'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'' active platform (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}).
@@ -1743,9 +1721,9 @@ Conversely, if a ``stiff'' active platform is used, the support dynamics appears
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_stiff}$\omega_{n} \gg \omega_{\mu}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_effect_support_compliance_dynamics_d}Effect of the support compliance on the transfer functions from \(F\) to \(d\)}
+\caption{\label{fig:uniaxial_effect_support_compliance_dynamics_d}Effect of the support compliance on the transfer functions from \(F\) to \(d\).}
\end{figure}
-\subsubsection{Conclusion}
+\paragraph{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 active platform) with respect to \(\omega_\mu\) (modes of the support), the interaction between the support and the active platform dynamics can drastically change (observations made are summarized in Table~\ref{tab:uniaxial_effect_compliance}).
@@ -1754,7 +1732,6 @@ For the \acrfull{nass}, having the suspension mode of the active platform at low
Note that the observations made in this section are also affected by the ratio between the support mass \(m_{\mu}\) and the active platform mass \(m_n\) (the effect is more pronounced when the ratio \(m_n/m_{\mu}\) increases).
\begin{table}[htbp]
-\caption{\label{tab:uniaxial_effect_compliance}Impact of the support dynamics on the plant dynamics}
\centering
\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
@@ -1764,12 +1741,14 @@ Note that the observations made in this section are also affected by the ratio b
\(L/F\) & large & large & small\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:uniaxial_effect_compliance}Impact of the support dynamics on the plant dynamics.}
+
\end{table}
\subsection{Effect of Payload Dynamics}
\label{sec:uniaxial_payload_dynamics}
Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic}).
-However, such a sample may present internal dynamics, and its fixation to the active platform may have limited stiffness.
+However, such a sample may present internal dynamics, and its mounting on the active platform may have limited stiffness.
To study the effect of the sample dynamics, the models shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_schematic} are used.
\begin{figure}[htbp]
@@ -1783,18 +1762,17 @@ To study the effect of the sample dynamics, the models shown in Figure~\ref{fig:
\begin{center}
\includegraphics[scale=1,scale=1]{figs/uniaxial_paylaod_dynamics_schematic.png}
\end{center}
-\subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Payload with some flexibility}
+\subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Flexible payload}
\end{subfigure}
-\caption{\label{fig:uniaxial_payload_dynamics_models}Models used to study the effect of payload dynamics}
+\caption{\label{fig:uniaxial_payload_dynamics_models}Models used to study the effect of payload dynamics.}
\end{figure}
-\subsubsection{Impact on plant dynamics}
-\label{ssec:uniaxial_payload_dynamics_effect_dynamics}
+\paragraph{Impact on Plant Dynamics}
To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic} is used.
Then ``flexible'' payload whose model is shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_schematic} are considered.
The resonances of the payload are set at \(\omega_s = 20\,\text{Hz}\) and at \(\omega_s = 200\,\text{Hz}\) while its mass is either \(m_s = 1\,\text{kg}\) or \(m_s = 50\,\text{kg}\).
-The transfer functions from the active platform force \(f\) to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_soft_nano_hexapod}.
+The transfer functions from the active platform force \(f\) to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_soft_nano_hexapod}.
It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern).
The frequency of the anti-resonance corresponds to the ``free'' resonance of the sample \(\omega_s = \sqrt{k_s/m_s}\).
The flexibility of the sample also changes the high frequency gain (the mass line is shifted from \(\frac{1}{(m_n + m_s)s^2}\) to \(\frac{1}{m_ns^2}\)).
@@ -1804,18 +1782,18 @@ The flexibility of the sample also changes the high frequency gain (the mass lin
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png}
\end{center}
-\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$}
+\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_heavy.png}
\end{center}
-\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$}
+\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,\text{N}/\upmu\text{m}$, $m_s = 50\,\text{kg}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod}Effect of the payload dynamics on the soft active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy})}
+\caption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod}Effect of the payload dynamics on the soft active platform with light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}).}
\end{figure}
-The same transfer functions are now compared when using a stiff active platform (\(k_n = 100\,\text{N}/\mu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}.
+The same transfer functions are now compared when using a stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}.
In this case, the sample's resonance \(\omega_s\) is smaller than the active platform resonance \(\omega_n\).
This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to \(\omega_s\)).
Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}).
@@ -1825,18 +1803,17 @@ Even though the added sample's flexibility still shifts the high frequency mass
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png}
\end{center}
-\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$}
+\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_heavy.png}
\end{center}
-\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$}
+\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,\text{N}/\upmu\text{m}$, $m_s = 50\,\text{kg}$}
\end{subfigure}
-\caption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}Effect of the payload dynamics on the stiff active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy})}
+\caption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}Effect of the payload dynamics on the stiff active platform with 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}
-\subsubsection{Impact on close loop performances}
-\label{ssec:uniaxial_payload_dynamics_effect_stability}
+\paragraph{Impact on Close Loop Performances}
Having a flexibility between the measured position (i.e., the top platform of the active platform) and the \acrshort{poi} to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration).
@@ -1846,8 +1823,8 @@ In this case, the measured (i.e., controlled) distance \(d\) is no longer equal
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/uniaxial_sample_flexibility_control.png}
-\caption{\label{fig:uniaxial_sample_flexibility_control}Uniaxial model considering some flexibility between the active platform top platform and the sample. In this case, the measured and controlled distance \(d\) is different from the distance \(y\) which is the real performance index}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sample_flexibility_control.png}
+\caption{\label{fig:uniaxial_sample_flexibility_control}Uniaxial model considering some flexibility between the active platform top platform and the sample. In this case, the measured and controlled distance \(d\) is different from the distance \(y\) which is the real performance index.}
\end{figure}
The system dynamics is computed and IFF is applied using the same gains as those used in Section~\ref{sec:uniaxial_active_damping}.
@@ -1874,9 +1851,9 @@ What happens is that above \(\omega_s\), even though the motion \(d\) can be con
\end{center}
\subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_y}Cumulative Amplitude Spectrum of $y$}
\end{subfigure}
-\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 active platform stiffness of \(100\,\text{N}/\mu\text{m}\) are used for the simulations.}
+\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 an active platform stiffness of \(100\,\text{N}/\upmu\text{m}\) are used for the simulations.}
\end{figure}
-\subsubsection{Conclusion}
+\paragraph{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 active platform \(\omega_n\).
@@ -1886,7 +1863,7 @@ Such additional dynamics can induce stability issues depending on their position
The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload.
This is why high-bandwidth soft positioning stages are usually restricted to constant and calibrated payloads (CD-player, lithography machines, isolation system for gravitational wave detectors, \ldots{}), whereas stiff positioning systems are usually used when the control must be robust to a change of payload's mass (stiff piezo nano-positioning stages for instance).
-Having some flexibility between the measurement point and the \acrshort{poi} (i.e., the sample point to be position on the x-ray) also degrades the position stability as shown in Section~\ref{ssec:uniaxial_payload_dynamics_effect_stability}.
+Having some flexibility between the measurement point and the \acrshort{poi} (i.e., the sample point to be position on the x-ray) also degrades the position stability.
Therefore, it is important to take special care when designing sampling environments, especially if a soft active platform is used.
\subsection*{Conclusion}
\label{sec:uniaxial_conclusion}
@@ -1899,14 +1876,14 @@ However, this model does not allow the determination of which one is most suited
Position feedback controllers have been developed for three considered active platform stiffnesses (Section~\ref{sec:uniaxial_position_control}).
These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances.
Having a soft active platform makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section~\ref{sec:uniaxial_support_compliance}) and requires less position feedback bandwidth to fulfill the requirements.
-The moderately stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)) is requiring a higher feedback bandwidth, but still gives acceptable results.
+The moderately stiff active platform (\(k_n = 1\,\text{N}/\upmu\text{m}\)) requires a higher feedback bandwidth, but still gives acceptable results.
However, the stiff active platform is the most complex to control and gives the worst positioning performance.
\section{Effect of Rotation}
\label{sec:rotating}
An important aspect of the \acrfull{nass} is that the active platform continuously rotates around a vertical axis, whereas the external metrology is not.
Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance.
To study these effects, a model of a rotating suspended platform is first presented (Section~\ref{sec:rotating_system_description})
-This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects in play.
+This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects at play.
\acrfull{iff} is then applied to the rotating platform, and it is shown that the unconditional stability of \acrshort{iff} is lost due to the gyroscopic effects induced by the rotation (Section~\ref{sec:rotating_iff_pure_int}).
Two modifications of the Integral Force Feedback are then proposed.
@@ -1921,7 +1898,7 @@ This study of adapting \acrshort{iff} for the damping of rotating platforms has
It is then shown that \acrfull{rdc} is less affected by gyroscopic effects (Section~\ref{sec:rotating_relative_damp_control}).
Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, damped plant and closed-loop compliance and transmissibility (Section~\ref{sec:rotating_comp_act_damp}).
-The previous analysis was applied to three considered active platform stiffnesses (\(k_n = 0.01\,\text{N}/\mu\text{m}\), \(k_n = 1\,\text{N}/\mu\text{m}\) and \(k_n = 100\,\text{N}/\mu\text{m}\)) and the optimal active damping controller was obtained in each case (Section~\ref{sec:rotating_nano_hexapod}).
+The previous analysis was applied to three considered active platform stiffnesses (\(k_n = 0.01\,\text{N}/\upmu\text{m}\), \(k_n = 1\,\text{N}/\upmu\text{m}\) and \(k_n = 100\,\text{N}/\upmu\text{m}\)) and the optimal active damping controller was obtained in each case (Section~\ref{sec:rotating_nano_hexapod}).
Up until this section, the study was performed on a very simplistic model that only captures the rotation aspect, and the model parameters were not tuned to correspond to the NASS.
In the last section (Section~\ref{sec:rotating_nass}), a model of the micro-station is added below the active platform with a rotating spindle and parameters tuned to match the NASS dynamics.
The goal is to determine whether the rotation imposes performance limitation on the NASS.
@@ -1939,9 +1916,9 @@ After the dynamics of this system is studied, the objective will be to dampen th
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic.png}
-\caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system}
+\caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied 2-DoFs translation stage on top of a rotation stage.}
\end{figure}
-\subsubsection{Equations of motion and transfer functions}
+\paragraph{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}.
@@ -2000,7 +1977,7 @@ The elements of the transfer function matrix \(\bm{G}_d\) are described by equat
\bm{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
\end{align}
\end{subequations}
-\subsubsection{System Poles: Campbell Diagram}
+\paragraph{System Poles: Campbell Diagram}
The poles of \(\bm{G}_d\) are the complex solutions \(p\) of equation~\eqref{eq:rotating_poles} (i.e. the roots of its denominator).
\begin{equation}\label{eq:rotating_poles}
@@ -2034,9 +2011,9 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo
\end{center}
\subcaption{\label{fig:rotating_campbell_diagram_imag}Imaginary part}
\end{subfigure}
-\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 \(\Omega\).}
+\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}
-\subsubsection{System Dynamics: Effect of rotation}
+\paragraph{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 \(\bm{G}_d\) 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 \(\Omega\).
@@ -2056,13 +2033,13 @@ For \(\Omega > \omega_0\), the low-frequency pair of complex conjugate poles \(p
\end{center}
\subcaption{\label{fig:rotating_bode_plot_coupling}Coupling terms: $d_u/F_v$, $d_v/F_u$}
\end{subfigure}
-\caption{\label{fig:rotating_bode_plot}Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities}
+\caption{\label{fig:rotating_bode_plot}Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_coupling}) terms for several rotating velocities.}
\end{figure}
\subsection{Integral Force Feedback}
\label{sec:rotating_iff_pure_int}
The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation.
-As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances~\cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller.
+As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances~\cite{collette11_review_activ_vibrat_isolat_strat} and for making the plant easier to control for the high authority controller.
Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF)~\cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF)~\cite{preumont91_activ} and Direct Velocity Feedback (DVF)~\cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}.
In~\cite{preumont91_activ}, the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system.
@@ -2076,7 +2053,7 @@ 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.
-\subsubsection{System and Equations of motion}
+\paragraph{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.
@@ -2087,7 +2064,7 @@ Two identical controllers \(K_F\) described by~\eqref{eq:rotating_iff_controller
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_3dof_model_schematic_iff.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_iff.png}
\end{center}
\subcaption{\label{fig:rotating_3dof_model_schematic_iff}System with added Force Sensor in series with the actuators}
\end{subfigure}
@@ -2147,7 +2124,7 @@ This small displacement then increases the centrifugal force \(m\Omega^2d_u = \f
0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2}
\end{bmatrix}
\end{equation}
-\subsubsection{Effect of rotation speed on IFF plant dynamics}
+\paragraph{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}
@@ -2168,11 +2145,11 @@ A pair of (minimum phase) complex conjugate zeros appears between the two comple
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_root_locus_iff_pure_int.png}
\end{center}
-\subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus}
+\subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root locus}
\end{subfigure}
-\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})}
+\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}
-\subsubsection{Decentralized Integral Force Feedback}
+\paragraph{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}.
@@ -2183,11 +2160,11 @@ The decentralized \acrshort{iff} controller \(\bm{K}_F\) corresponds to a diagon
\end{aligned}
\end{equation}
-To determine how the \acrshort{iff} controller affects the poles of the closed-loop system, a Root Locus plot (Figure~\ref{fig:rotating_root_locus_iff_pure_int}) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain \(g\) varies from \(0\) to \(\infty\) for the two controllers \(K_{F}\) simultaneously.
+To determine how the \acrshort{iff} controller affects the poles of the closed-loop system, a Root locus plot (Figure~\ref{fig:rotating_root_locus_iff_pure_int}) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain \(g\) varies from \(0\) to \(\infty\) for the two controllers \(K_{F}\) simultaneously.
As explained in~\cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr}, the closed-loop poles start at the open-loop poles (shown by crosses) for \(g = 0\) and coincide with the transmission zeros (shown by circles) as \(g \to \infty\).
Whereas collocated IFF is usually associated with unconditional stability~\cite{preumont91_activ}, this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null.
-This can be seen in the Root Locus plot (Figure~\ref{fig:rotating_root_locus_iff_pure_int}) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability.
+This can be seen in the Root locus plot (Figure~\ref{fig:rotating_root_locus_iff_pure_int}) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability.
Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure~\ref{fig:rotating_iff_bode_plot_effect_rot}).
The control system is thus cancels the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability.
\subsection{Integral Force Feedback with a High-Pass Filter}
@@ -2202,12 +2179,12 @@ This is however not the reason why this \acrlong{hpf} is added here.
\begin{equation}\label{eq:rotating_iff_lhf}
\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}
-\subsubsection{Modified Integral Force Feedback Controller}
+\paragraph{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.
-The Root Locus plots for the decentralized \acrshort{iff} with and without the \acrshort{hpf} are displayed in Figure~\ref{fig:rotating_iff_root_locus_hpf_large}.
+The Root locus plots for the decentralized \acrshort{iff} with and without the \acrshort{hpf} are displayed in Figure~\ref{fig:rotating_iff_root_locus_hpf_large}.
With the added \acrshort{hpf}, the poles of the closed-loop system are shown to be stable up to some value of the gain \(g_\text{max}\) given by equation~\eqref{eq:rotating_gmax_iff_hpf}.
It is interesting to note that \(g_{\text{max}}\) also corresponds to the controller gain at which the low-frequency loop gain reaches one (for instance the gain \(g\) can be increased by a factor \(5\) in Figure~\ref{fig:rotating_iff_modified_loop_gain} before the system becomes unstable).
@@ -2226,15 +2203,15 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_root_locus_hpf_large.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus}
+\subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root locus}
\end{subfigure}
-\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})}
+\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}
-\subsubsection{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)}
+\paragraph{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.
-To visualize how \(\omega_i\) does affect the attainable damping, the Root Locus plots for several \(\omega_i\) are displayed in Figure~\ref{fig:rotating_root_locus_iff_modified_effect_wi}.
+To visualize how \(\omega_i\) does affect the attainable damping, the Root locus plots for several \(\omega_i\) are displayed in Figure~\ref{fig:rotating_root_locus_iff_modified_effect_wi}.
It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes (see Root locus in Figure~\ref{fig:rotating_root_locus_iff_modified_effect_wi}), the control gain \(g\) may be limited to small values due to equation~\eqref{eq:rotating_gmax_iff_hpf}.
To study this trade-off, the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) is computed as a function of \(\omega_i/\omega_0\).
The gain \(g_{\text{opt}}\) at which this maximum damping is obtained is also displayed and compared with the gain \(g_{\text{max}}\) at which the system becomes unstable (Figure~\ref{fig:rotating_iff_hpf_optimal_gain}).
@@ -2247,17 +2224,17 @@ For larger values of \(\omega_i\), the attainable damping ratio decreases as a f
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_root_locus_iff_modified_effect_wi.png}
\end{center}
-\subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus}
+\subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root locus}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_optimal_gain.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown}
+\subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio as a function of $\omega_i/\omega_0$. Maximum and optical control gains are also shown}
\end{subfigure}
-\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})}
+\caption{\label{fig:rotating_iff_modified_effect_wi}Root loci 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 (\subref{fig:rotating_iff_hpf_optimal_gain}).}
\end{figure}
-\subsubsection{Obtained Damped Plant}
+\paragraph{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 \(\omega_i\).
@@ -2276,9 +2253,9 @@ The same trade-off can be seen between achievable damping and loss of compliance
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of $\omega_i$ on the compliance}
\end{subfigure}
-\caption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi}Effect of \(\omega_i\) on the damped plant coupling}
+\caption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi}Effect of \(\omega_i\) on the damped plant coupling (\subref{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}) and on the compliance (\subref{fig:rotating_iff_hpf_effect_wi_compliance}).}
\end{figure}
-\subsection{IFF with a stiffness in parallel with the force sensor}
+\subsection{IFF with a Stiffness in Parallel with the Force Sensor}
\label{sec:rotating_iff_parallel_stiffness}
In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects.
@@ -2287,9 +2264,9 @@ Such springs are schematically shown in Figure~\ref{fig:rotating_3dof_model_sche
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_iff_parallel_springs.png}
-\caption{\label{fig:rotating_3dof_model_schematic_iff_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors (shown in red)}
+\caption{\label{fig:rotating_3dof_model_schematic_iff_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors (shown in red).}
\end{figure}
-\subsubsection{Equations}
+\paragraph{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}
@@ -2327,13 +2304,13 @@ Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{ne
\begin{equation}\label{eq:rotating_kp_cond_cc_zeros}
\boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2}
\end{equation}
-\subsubsection{Effect of parallel stiffness on the IFF plant}
+\paragraph{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 \(k_p > m \Omega^2\).
In that case, the system shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case.
-Figure~\ref{fig:rotating_iff_kp_root_locus} shows the Root Locus plots for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\) when \(K_F\) is a pure integrator, as shown in Eq.~\eqref{eq:rotating_Kf_pure_int}.
+Figure~\ref{fig:rotating_iff_kp_root_locus} shows the Root locus plots for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\) when \(K_F\) is a pure integrator, as shown in Eq.~\eqref{eq:rotating_Kf_pure_int}.
It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the unconditional stability of \acrshort{iff} is recovered.
\begin{figure}[htbp]
@@ -2341,20 +2318,20 @@ It is shown that if the added stiffness is higher than the maximum negative stif
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_effect_kp.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$}
+\subcaption{\label{fig:rotating_iff_effect_kp}Bode plots of $f_u/F_u$ without parallel spring (blue), with parallel spring $k_p < m \Omega^2$ (red) and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ (yellow)}
\end{subfigure}
\begin{subfigure}{0.44\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_root_locus.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring}
+\subcaption{\label{fig:rotating_iff_kp_root_locus}Root locus for IFF without parallel spring, with soft parallel spring and with stiff parallel spring}
\end{subfigure}
-\caption{\label{fig:rotating_iff_plant_effect_kp}Effect of parallel stiffness on the IFF plant}
+\caption{\label{fig:rotating_iff_plant_effect_kp}Effect of parallel stiffness on the IFF plant (\subref{fig:rotating_iff_effect_kp}) and on the control stability (\subref{fig:rotating_iff_kp_root_locus}).}
\end{figure}
-\subsubsection{Effect of \(k_p\) on the attainable damping}
+\paragraph{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}.
+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}.
The frequencies of the transmission zeros of the system increase with an increase in the parallel stiffness \(k_p\) (thus getting closer to the poles), and the associated attainable damping is reduced.
Therefore, even though the parallel stiffness \(k_p\) should be larger than \(m \Omega^2\) for stability reasons, it should not be taken too large as this would limit the attainable damping.
This is confirmed by the Figure~\ref{fig:rotating_iff_kp_optimal_gain} where the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) and the associated optimal control gain \(g_\text{opt}\) are computed as a function of the parallel stiffness.
@@ -2364,17 +2341,17 @@ This is confirmed by the Figure~\ref{fig:rotating_iff_kp_optimal_gain} where the
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_root_locus_effect_kp.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$}
+\subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root locus: Effect of parallel stiffness, $\Omega = 0.1 \omega_0$}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_optimal_gain.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown because the system is unstable.}
+\subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown}
\end{subfigure}
-\caption{\label{fig:rotating_iff_optimal_kp}Effect of parallel stiffness on the IFF plant}
+\caption{\label{fig:rotating_iff_optimal_kp}Effect of the parallel stiffness on the achievable damping with IFF.}
\end{figure}
-\subsubsection{Damped plant}
+\paragraph{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.
@@ -2395,26 +2372,26 @@ Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compare the obtained damped p
The added \acrshort{hpf} gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior.
\begin{figure}[htbp]
-\begin{subfigure}{0.34\linewidth}
+\begin{subfigure}{0.44\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_added_hpf_effect_damping.png}
\end{center}
\subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$}
\end{subfigure}
-\begin{subfigure}{0.65\linewidth}
+\begin{subfigure}{0.54\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_added_hpf_damped_plant.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF}
+\subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Effect of the added HPF on the damped plant}
\end{subfigure}
-\caption{\label{fig:rotating_iff_optimal_hpf}Effect of high-pass filter cut-off frequency on the obtained damping}
+\caption{\label{fig:rotating_iff_optimal_hpf}Effect of high-pass filter cut-off frequency on the obtained damping (\subref{fig:rotating_iff_kp_added_hpf_effect_damping}) and on the damped plant (\subref{fig:rotating_iff_kp_added_hpf_damped_plant}).}
\end{figure}
\subsection{Relative Damping Control}
\label{sec:rotating_relative_damp_control}
-To apply a ``Relative Damping Control'' strategy, relative motion sensors are added in parallel with the actuators as shown in Figure~\ref{fig:rotating_3dof_model_schematic_rdc}.
+To apply a \acrfull{rdc} strategy, relative motion sensors are added in parallel with the actuators as shown in Figure~\ref{fig:rotating_3dof_model_schematic_rdc}.
Two controllers \(K_d\) are used to feed back the relative motion to the actuator.
-These controllers are in principle pure derivators (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high-pass filter~\eqref{eq:rotating_rdc_controller}.
+These controllers have in principle pure derivative action (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high-pass filter~\eqref{eq:rotating_rdc_controller}.
\begin{equation}\label{eq:rotating_rdc_controller}
K_d(s) = g \cdot \frac{s}{s + \omega_d}
@@ -2423,9 +2400,9 @@ K_d(s) = g \cdot \frac{s}{s + \omega_d}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_rdc.png}
-\caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensor and decentralized ``relative damping control'' applied.}
+\caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensors and decentralized \acrfull{rdc} applied.}
\end{figure}
-\subsubsection{Equations of motion}
+\paragraph{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}.
@@ -2446,7 +2423,7 @@ Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functi
\begin{equation}\label{eq:rotating_rdc_zeros_poles}
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}
-\subsubsection{Decentralized Relative Damping Control}
+\paragraph{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}).
@@ -2462,7 +2439,7 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_rdc_root_locus.png}
\end{center}
-\subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control}
+\subcaption{\label{fig:rotating_rdc_root_locus}Root locus for Relative Damping Control}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\begin{center}
@@ -2470,7 +2447,7 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
\end{center}
\subcaption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control}
\end{subfigure}
-\caption{\label{fig:rotating_rdc_result}Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant})}
+\caption{\label{fig:rotating_rdc_result}Relative Damping Control. Root locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant}).}
\end{figure}
\subsection{Comparison of Active Damping Techniques}
\label{sec:rotating_comp_act_damp}
@@ -2478,8 +2455,8 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
These two proposed IFF modifications and relative damping control are compared in terms of added damping and closed-loop behavior.
For the following comparisons, the cut-off frequency for the added HPF is set to \(\omega_i = 0.1 \omega_0\) and the stiffness of the parallel springs is set to \(k_p = 5 m \Omega^2\) (corresponding to \(\alpha = 0.05\)).
These values are chosen one the basis of previous discussions about optimal parameters.
-\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.
+\paragraph{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.
@@ -2492,21 +2469,21 @@ It is interesting to note that the maximum added damping is very similar for bot
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_root_locus.png}
\end{center}
-\subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus}
+\subcaption{\label{fig:rotating_comp_techniques_root_locus}Root locus}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_dampled_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_damped_plants.png}
\end{center}
-\subcaption{\label{fig:rotating_comp_techniques_dampled_plants}Damped plants}
+\subcaption{\label{fig:rotating_comp_techniques_damped_plants}Damped plants}
\end{subfigure}
-\caption{\label{fig:rotating_comp_techniques}Comparison of active damping techniques for rotating platform}
+\caption{\label{fig:rotating_comp_techniques}Comparison of active damping techniques for rotating platform.}
\end{figure}
-\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}.
+\paragraph{Obtained Damped Plant}
+The actively damped plants are computed for the three techniques and compared in Figure~\ref{fig:rotating_comp_techniques_damped_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.
-\subsubsection{Transmissibility And Compliance}
+\paragraph{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.
@@ -2531,14 +2508,14 @@ This is very well known characteristics of these common active damping technique
\end{center}
\subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance}
\end{subfigure}
-\caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques}
+\caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques.}
\end{figure}
\subsection{Rotating Active Platform}
\label{sec:rotating_nano_hexapod}
The previous analysis is now applied to a model representing a rotating active platform.
-Three active platform 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 \(k_n = \SI{100}{\N\per\mu\m}\).
+Three active platform stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\micro\m}\), \(k_n = \SI{1}{\N\per\micro\m}\) and \(k_n = \SI{100}{\N\per\micro\m}\).
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.
-\subsubsection{Nano-Active-Stabilization-System - Plant Dynamics}
+\paragraph{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 active platform's actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\).
The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \approx \SI{0.6}{\newton\per\mm}\).
@@ -2552,30 +2529,30 @@ The coupling (or interaction) in a \acrshort{mimo} \(2 \times 2\) system can be
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_md.png}
\end{center}
-\subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the active platform 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}
+\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the active platform 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}
-\subsubsection{Optimal IFF with a High-Pass Filter}
+\paragraph{Optimal IFF with a High-Pass Filter}
Integral Force Feedback with an added \acrlong{hpf} is applied to the three active platforms.
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:
\begin{itemize}
-\item for \(k_n = \SI{0.01}{\N\per\mu\m}\) (Figure~\ref{fig:rotating_iff_hpf_nass_optimal_gain}): \(\omega_i\) is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable.
+\item for \(k_n = \SI{0.01}{\N\per\micro\m}\) (Figure~\ref{fig:rotating_iff_hpf_nass_optimal_gain}): \(\omega_i\) is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable.
This is done to have some control robustness.
-\item for \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\) (Figure~\ref{fig:rotating_iff_hpf_nass_optimal_gain_md} and \ref{fig:rotating_iff_hpf_nass_optimal_gain_pz}): the largest \(\omega_i\) is chosen such that the obtained damping is \(\SI{95}{\percent}\) of the maximum achievable damping.
+\item for \(k_n = \SI{1}{\N\per\micro\m}\) and \(k_n = \SI{100}{\N\per\micro\m}\) (Figure~\ref{fig:rotating_iff_hpf_nass_optimal_gain_md} and \ref{fig:rotating_iff_hpf_nass_optimal_gain_pz}): the largest \(\omega_i\) is chosen such that the obtained damping is \(\SI{95}{\percent}\) of the maximum achievable damping.
Large \(\omega_i\) is chosen here to limit the loss of compliance and the increase of coupling at low-frequency as shown in Section~\ref{sec:rotating_iff_pseudo_int}.
\end{itemize}
The obtained IFF parameters and the achievable damping are visually shown by large dots in Figure~\ref{fig:rotating_iff_hpf_nass_optimal_gain} and are summarized in Table~\ref{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}.
@@ -2585,37 +2562,38 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_md.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\caption{\label{fig:rotating_iff_hpf_nass_optimal_gain}For each value of \(\omega_i\), the maximum damping ratio \(\xi\) is computed (blue), and the corresponding controller gain is shown (in red). The chosen controller parameters used for further analysis are indicated by the large dots.}
\end{figure}
\begin{table}[htbp]
-\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping of the two modes \(\xi\) is also shown.}
\centering
\begin{tabularx}{0.3\linewidth}{Xccc}
\toprule
\(k_n\) & \(\omega_i\) & \(g\) & \(\xi_\text{opt}\)\\
\midrule
-\(0.01\,\text{N}/\mu\text{m}\) & 7.3 & 51 & 0.45\\
-\(1\,\text{N}/\mu\text{m}\) & 39 & 427 & 0.93\\
-\(100\,\text{N}/\mu\text{m}\) & 500 & 3775 & 0.94\\
+\(0.01\,\text{N}/\upmu\text{m}\) & 7.3 & 51 & 0.45\\
+\(1\,\text{N}/\upmu\text{m}\) & 39 & 427 & 0.93\\
+\(100\,\text{N}/\upmu\text{m}\) & 500 & 3775 & 0.94\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping \(\xi_{\text{opt}}\) of the two modes is also shown.}
+
\end{table}
-\subsubsection{Optimal IFF with Parallel Stiffness}
+\paragraph{Optimal IFF with Parallel Stiffness}
For each considered active platform 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 active platform 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 active platform 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 (\(\omega_i = \omega_0/10\)).
@@ -2626,13 +2604,13 @@ For the two stiff options, the achievable damping decreases when the parallel st
Such behavior can be explained by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero~\cite[chapt 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
This distance is larger for stiff active platform because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff active platforms.
-Let's choose \(k_p = 1\,\text{N/mm}\), \(k_p = 0.01\,\text{N}/\mu\text{m}\) and \(k_p = 1\,\text{N}/\mu\text{m}\) for the three considered active platforms.
+Let's choose \(k_p = 1\,\text{N/mm}\), \(k_p = 0.01\,\text{N}/\upmu\text{m}\) and \(k_p = 1\,\text{N}/\upmu\text{m}\) for the three considered active platforms.
The corresponding optimal controller gains and achievable damping are summarized in Table~\ref{tab:rotating_iff_kp_opt_iff_kp_params_nass}.
\begin{minipage}[b]{0.49\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_nass_optimal_gain.png}
-\captionof{figure}{\label{fig:rotating_iff_kp_nass_optimal_gain}Maximum damping \(\xi\) as a function of the parallel stiffness \(k_p\)}
+\captionof{figure}{\label{fig:rotating_iff_kp_nass_optimal_gain}Maximum damping \(\xi\) as a function of the parallel stiffness \(k_p\).}
\end{center}
\end{minipage}
\hfill
@@ -2643,21 +2621,22 @@ The corresponding optimal controller gains and achievable damping are summarized
\toprule
\(k_n\) & \(k_p\) & \(g\) & \(\xi_{\text{opt}}\)\\
\midrule
-\(0.01\,\text{N}/\mu\text{m}\) & \(1\,\text{N/mm}\) & 47.9 & 0.44\\
-\(1\,\text{N}/\mu\text{m}\) & \(0.01\,\text{N}/\mu\text{m}\) & 465.57 & 0.97\\
-\(100\,\text{N}/\mu\text{m}\) & \(1\,\text{N}/\mu\text{m}\) & 4624.25 & 0.99\\
+\(0.01\,\text{N}/\upmu\text{m}\) & \(1\,\text{N/mm}\) & 48 & 0.44\\
+\(1\,\text{N}/\upmu\text{m}\) & \(0.01\,\text{N}/\upmu\text{m}\) & 465 & 0.97\\
+\(100\,\text{N}/\upmu\text{m}\) & \(1\,\text{N}/\upmu\text{m}\) & 4624 & 0.99\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses}
\end{minipage}
-\subsubsection{Optimal Relative Motion Control}
+\paragraph{Optimal Relative Motion Control}
For each considered active platform 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}).
\begin{minipage}[b]{0.49\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_rdc_optimal_gain.png}
-\captionof{figure}{\label{fig:rotating_rdc_optimal_gain}Maximum damping \(\xi\) as a function of the RDC gain \(g\)}
+\captionof{figure}{\label{fig:rotating_rdc_optimal_gain}Maximum damping \(\xi\) as a function of the RDC gain \(g\).}
\end{center}
\end{minipage}
\hfill
@@ -2668,14 +2647,15 @@ The gain is chosen such that 99\% of modal damping is obtained (obtained gains a
\toprule
\(k_n\) & \(g\) & \(\xi_{\text{opt}}\)\\
\midrule
-\(0.01\,\text{N}/\mu\text{m}\) & 1600 & 0.99\\
-\(1\,\text{N}/\mu\text{m}\) & 8200 & 0.99\\
-\(100\,\text{N}/\mu\text{m}\) & 80000 & 0.99\\
+\(0.01\,\text{N}/\upmu\text{m}\) & 1600 & 0.99\\
+\(1\,\text{N}/\upmu\text{m}\) & 8200 & 0.99\\
+\(100\,\text{N}/\upmu\text{m}\) & 80000 & 0.99\\
\bottomrule
-\end{tabularx}}
-\captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC}
+\end{tabularx}
+}
+\captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the acrlong:rdc}
\end{minipage}
-\subsubsection{Comparison of the obtained damped plants}
+\paragraph{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:
@@ -2690,35 +2670,35 @@ Similar to what was concluded in the previous analysis:
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_md.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. For this analysis the rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).}
+\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. Rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).}
\end{figure}
-\subsection{Nano-Active-Stabilization-System with rotation}
+\subsection{Nano Active Stabilization System with Rotation}
\label{sec:rotating_nass}
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.
-\subsubsection{Nano Active Stabilization System model}
-To have a more realistic dynamics model of the NASS, the 2-DoF active platform (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):
+\paragraph{Nano Active Stabilization System Model}
+To have a more realistic dynamics model of the NASS, the 2-DoFs active platform (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
-\item a 2-DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\))
-\item a 2-DoF \(T_y\) stage (\(k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}\), \(m_t = \SI{600}{\kg}\))
+\item a 2-DoFs granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\micro\m}\), \(m_g = \SI{2500}{\kg}\))
+\item a 2-DoFs \(T_y\) stage (\(k_{t,x} = k_{t,y} = \SI{520}{\N\per\micro\m}\), \(m_t = \SI{600}{\kg}\))
\item a spindle (vertical rotation) stage whose rotation is imposed (\(m_s = \SI{600}{\kg}\))
-\item a 2-DoF positioning hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}\), \(m_h = \SI{15}{\kg}\))
+\item a 2-DoFs positioning hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\micro\m}\), \(m_h = \SI{15}{\kg}\))
\end{itemize}
A payload is rigidly fixed to the active platform and the \(x,y\) motion of the payload is measured with respect to the granite.
@@ -2728,7 +2708,7 @@ A payload is rigidly fixed to the active platform and the \(x,y\) motion of the
\includegraphics[scale=1,scale=0.7]{figs/rotating_nass_model.png}
\caption{\label{fig:rotating_nass_model}3D view of the Nano-Active-Stabilization-System model.}
\end{figure}
-\subsubsection{System dynamics}
+\paragraph{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.
@@ -2745,23 +2725,23 @@ It can be observed that:
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_md.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from active platform actuator to measured motion by the external metrology}
+\caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from active platform actuator to measured motion by the external metrology.}
\end{figure}
-\subsubsection{Effect of disturbances}
+\paragraph{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.
@@ -2784,19 +2764,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_md.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\caption{\label{fig:rotating_nass_effect_floor_motion}Effect of floor motion \(x_{f,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency.}
\end{figure}
@@ -2806,21 +2786,21 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_md.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
-\caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations \(f_{t,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc})}
+\caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations \(f_{t,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc}).}
\end{figure}
\begin{figure}[htbp]
@@ -2828,19 +2808,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_vc.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_md.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_pz.png}
\end{center}
-\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,\text{N}/\mu\text{m}$}
+\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,\text{N}/\upmu\text{m}$}
\end{subfigure}
\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three active platform stiffnesses. Integral Force Feedback degrades this compliance at low-frequency.}
\end{figure}
@@ -2862,12 +2842,12 @@ While having very different implementations, both proposed modifications were fo
This study has been applied to a rotating platform that corresponds to the active platform parameters.
As for the uniaxial model, three active platform stiffnesses values were considered.
-The dynamics of the soft active platform (\(k_n = 0.01\,\text{N}/\mu\text{m}\)) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects).
+The dynamics of the soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects).
In addition, the attainable damping ratio of the soft active platform when using \acrshort{iff} is limited by gyroscopic effects.
To be closer to the \acrlong{nass} dynamics, the limited compliance of the micro-station has been considered.
Results are similar to those of the uniaxial model except that come complexity is added for the soft active platform due to the spindle's rotation.
-For the moderately stiff active platform (\(k_n = 1\,\text{N}/\mu\text{m}\)), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model.
+For the moderately stiff active platform (\(k_n = 1\,\text{N}/\upmu\text{m}\)), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model.
\section{Micro Station - Modal Analysis}
\label{sec:modal}
To further improve the accuracy of the performance predictions, a model that better represents the micro-station dynamics is required.
@@ -2888,7 +2868,7 @@ This modal model can then be used to tune the spatial model (i.e. the multi-body
\end{figure}
The measurement setup used to obtain the response model is described in Section~\ref{sec:modal_meas_setup}.
-This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals.
+This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planning, and a first analysis of the obtained signals.
In Section~\ref{sec:modal_frf_processing}, the obtained \acrshortpl{frf} between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed.
These measurements are projected at the \acrfull{com} of each considered solid body to facilitate the further use of the results.
@@ -2929,7 +2909,7 @@ These accelerometers were glued to the micro-station using a thin layer of wax f
\end{center}
\subcaption{\label{fig:modal_oros}OROS acquisition system}
\end{subfigure}
-\caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis}
+\caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis.}
\end{figure}
Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,\text{mV/N}\) and measurement range of \(2\,\text{kN}\)} (figure~\ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled manner.
@@ -2937,7 +2917,7 @@ Tests were conducted to determine the most suitable hammer tip (ranging from a m
The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved.
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.
-\subsubsection{Structure Preparation and Test Planing}
+\subsubsection{Structure Preparation and Test Planning}
\label{ssec:modal_test_preparation}
To obtain meaningful results, the modal analysis of the micro-station is performed \emph{in-situ}.
@@ -2949,7 +2929,7 @@ The top part representing the active stabilization stage was disassembled as the
To perform the modal analysis from the measured responses, the \(n \times n\) \acrshort{frf} matrix \(\bm{H}\) needs to be measured, where \(n\) is the considered number of \acrshortpl{dof}.
The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the \acrshort{frf} from a force \(F_k\) applied at \acrfull{dof} \(k\) to the displacement of the structure \(X_j\) at \acrshort{dof} \(j\).
-Measuring this \acrshort{frf} matrix is time consuming as it requires to make \(n \times n\) measurements.
+Measuring this \acrshort{frf} matrix is time consuming as it requires making \(n \times n\) measurements.
However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\bm{H}\) \cite[chapt. 5.2]{ewins00_modal}.
Therefore, a minimum set of \(n\) \acrshortpl{frf} is required.
This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
@@ -2970,7 +2950,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs
\begin{minipage}[b]{0.63\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_location_accelerometers.png}
-\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers}
+\captionof{figure}{\label{fig:modal_location_accelerometers}Position of the accelerometers.}
\end{center}
\end{minipage}
\hfill
@@ -3005,7 +2985,8 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs
(3) Hexapod & 64 & 64 & -270\\
(4) Hexapod & 64 & -64 & -270\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
\end{minipage}
@@ -3022,7 +3003,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs
\end{center}
\subcaption{\label{fig:modal_accelerometers_hexapod} Positioning Hexapod}
\end{subfigure}
-\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages}
+\caption{\label{fig:modal_accelerometer_pictures}Accelerometers fixed on the micro-station stages.}
\end{figure}
\subsubsection{Hammer Impacts}
\label{ssec:modal_hammer_impacts}
@@ -3051,7 +3032,7 @@ The impacts were performed in three directions, as shown in figures~\ref{fig:mod
\end{center}
\subcaption{\label{fig:modal_impact_z} $Z$ impact}
\end{subfigure}
-\caption{\label{fig:modal_hammer_impacts}The three hammer impacts used for the modal analysis}
+\caption{\label{fig:modal_hammer_impacts}The three hammer impacts used for the modal analysis.}
\end{figure}
\subsubsection{Force and Response signals}
\label{ssec:modal_measured_signals}
@@ -3078,7 +3059,7 @@ Similar results were obtained for all measured \acrshortpl{frf}.
\end{center}
\subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
\end{subfigure}
-\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer 1 in the \(x\) direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the \(z\) direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
+\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer 1 in the \(x\) direction (blue) and of the force sensor at the Hammer tip (red) for an impact in the \(z\) direction (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force}).}
\end{figure}
The \acrshort{frf} from the applied force to the measured acceleration is then computed and shown Figure~\ref{fig:modal_frf_acc_force}.
@@ -3098,7 +3079,7 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
\end{center}
\subcaption{\label{fig:modal_coh_acc_force} Coherence}
\end{subfigure}
-\caption{\label{fig:modal_frf_coh_acc_force}Computed frequency response function from the applied force \(F_{z}\) to the measured response \(X_{1,x}\) (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})}
+\caption{\label{fig:modal_frf_coh_acc_force}Computed \acrshort{frf} from the applied force \(F_{z}\) to the measured response \(X_{1,x}\) (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force}).}
\end{figure}
\subsection{Frequency Analysis}
\label{sec:modal_frf_processing}
@@ -3130,7 +3111,7 @@ 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 \acrlong{com}.
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}.
-\subsubsection{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).
@@ -3139,7 +3120,7 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 \acrsho
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_local_to_global_coordinates.png}
-\caption{\label{fig:modal_local_to_global_coordinates}Schematic of the measured motions of a solid body}
+\caption{\label{fig:modal_local_to_global_coordinates}Schematic of the measured motion of a solid body at 4 distinct locations.}
\end{figure}
The motion of the rigid body of figure~\ref{fig:modal_local_to_global_coordinates} can be described by its displacement \(\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]\) and (small) rotations \([\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]\) with respect to the reference frame \(\{O\}\).
@@ -3186,7 +3167,6 @@ From the 3D model, the position of the \acrlong{com} of each solid body is compu
The position of each accelerometer with respect to the \acrlong{com} of the corresponding solid body can easily be determined.
\begin{table}[htbp]
-\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
\centering
\begin{tabularx}{0.45\linewidth}{Xccc}
\toprule
@@ -3200,6 +3180,8 @@ Spindle & \(0\) & \(0\) & \(-580\,\text{mm}\)\\
Positioning Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the \acrlong{poi}.}
+
\end{table}
Using~\eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the \acrlong{com} of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\bm{H}\).
@@ -3217,7 +3199,7 @@ Using~\eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\bm{H}_\text
\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
\end{bmatrix}
\end{equation}
-\subsubsection{Verification of solid body assumption}
+\subsubsection{Verification of the 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 \(\bm{H}_{\text{CoM}}\)), 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.
@@ -3232,7 +3214,7 @@ This also validates the reduction in the number of \acrshortpl{dof} from 69 (23
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/modal_comp_acc_solid_body_frf.png}
-\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the positioning hexapod are shown. Input is a hammer force applied on the positioning hexapod in the \(x\) direction}
+\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses with responses reconstructed from the solid body response. Accelerometers 1 to 4, corresponding to the positioning hexapod, are shown. Input is a hammer force applied on the positioning hexapod in the \(x\) direction.}
\end{figure}
\subsection{Modal Analysis}
\label{sec:modal_analysis}
@@ -3246,7 +3228,7 @@ 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 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}).
-\subsubsection{Number of modes determination}
+\subsubsection{Determination of the Number of Modes}
\label{ssec:modal_number_of_modes}
The \acrshort{mif} is 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 \(p=3\)).
@@ -3269,7 +3251,7 @@ The obtained natural frequencies and associated modal damping are summarized in
\begin{minipage}[b]{0.65\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/modal_indication_function.png}
-\captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function}
+\captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function.}
\end{center}
\end{minipage}
\hfill
@@ -3297,10 +3279,11 @@ Mode & Frequency & Damping\\
15 & \(150.5\,\text{Hz}\) & \(2.4\,\%\)\\
16 & \(165.4\,\text{Hz}\) & \(1.4\,\%\)\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
\end{minipage}
-\subsubsection{Modal parameter extraction}
+\subsubsection{Modal Parameter Extraction}
\label{ssec:modal_parameter_extraction}
Generally, modal identification is using curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
@@ -3330,7 +3313,7 @@ From the obtained modal parameters, the mode shapes are computed and can be disp
\end{center}
\subcaption{\label{fig:modal_mode13_animation}$13^{th}$ mode at $124.2\,\text{Hz}$: lateral hexapod resonance}
\end{subfigure}
-\caption{\label{fig:modal_mode_animations}Three obtained mode shape animations}
+\caption{\label{fig:modal_mode_animations}Three obtained mode shape animations.}
\end{figure}
These animations are useful for visually obtaining a better understanding of the system's dynamic behavior.
@@ -3342,7 +3325,7 @@ The levelers were then better adjusted.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.6\linewidth]{figs/modal_airlock_picture.jpg}
-\caption{\label{fig:modal_airloc}AirLoc used for the granite (2120-KSKC)}
+\caption{\label{fig:modal_airloc}AirLoc used for the granite (2120-KSKC).}
\end{figure}
The modal parameter extraction is made using a proprietary software\footnote{NVGate software from OROS company.}.
@@ -3357,7 +3340,7 @@ The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation~\eqref{
\begin{equation}\label{eq:modal_eigenvalues}
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}
-\subsubsection{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 \(\bm{H}_{\text{syn}}\) is first synthesized from the modal parameters.
@@ -3427,9 +3410,9 @@ However, the measurements are useful for tuning the parameters of the micro-stat
From the start of this work, it became increasingly clear that an accurate micro-station model was necessary.
First, during the uniaxial study, it became clear that the micro-station dynamics affects the active platform dynamics.
-Then, using the 3-DoF rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled.
+Then, using the 3-DoFs rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled.
Finally, a modal analysis of the micro-station showed how complex the dynamics of the station is.
-The modal analysis also confirm that each stage behaves as a rigid body in the frequency range of interest.
+The modal analysis also confirms that each stage behaves as a rigid body in the frequency range of interest.
Therefore, a multi-body model is a good candidate to accurately represent the micro-station dynamics.
In this report, the development of such a multi-body model is presented.
@@ -3456,7 +3439,7 @@ Such a stacked architecture allows high mobility, but the overall stiffness is r
\end{figure}
There are different ways of modeling the stage dynamics in a multi-body model.
-The one chosen in this work consists of modeling each stage by two solid bodies connected by one 6-DoF joint.
+The one chosen in this work consists of modeling each stage by two solid bodies connected by one 6-DoFs joint.
The stiffness and damping properties of the joint
s can be tuned separately for each DoF.
@@ -3485,14 +3468,14 @@ To precisely control the \(R_y\) angle, a stepper motor and two optical encoders
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/ustation_ty_stage.png}
-\captionof{figure}{\label{fig:ustation_ty_stage}Translation Stage}
+\captionof{figure}{\label{fig:ustation_ty_stage}Translation Stage.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/ustation_ry_stage.png}
-\captionof{figure}{\label{fig:ustation_ry_stage}Tilt Stage}
+\captionof{figure}{\label{fig:ustation_ry_stage}Tilt Stage.}
\end{center}
\end{minipage}
\paragraph{Spindle}
@@ -3512,14 +3495,14 @@ It can also be used to precisely position the \acrfull{poi} vertically with resp
\begin{minipage}[t]{0.49\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/ustation_rz_stage.png}
-\captionof{figure}{\label{fig:ustation_rz_stage}Rotation Stage (Spindle)}
+\captionof{figure}{\label{fig:ustation_rz_stage}Rotation Stage (Spindle).}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/ustation_hexapod_stage.png}
-\captionof{figure}{\label{fig:ustation_hexapod_stage}Positioning Hexapod}
+\captionof{figure}{\label{fig:ustation_hexapod_stage}Positioning Hexapod.}
\end{center}
\end{minipage}
\subsubsection{Mathematical description of a rigid body motion}
@@ -3529,7 +3512,7 @@ In this section, mathematical tools\footnote{The tools presented here are largel
First, the tools to describe the pose of a solid body (i.e. it's position and orientation) are introduced.
The motion induced by a positioning stage is described by transformation matrices.
Finally, the motions of all stacked stages are combined, and the sample's motion is computed from each stage motion.
-\paragraph{Spatial motion representation}
+\paragraph{Spatial Representation of Motion}
The \emph{pose} of a solid body relative to a specific frame can be described by six independent parameters.
Three parameters are typically used to describe its position, and three other parameters describe its orientation.
@@ -3661,7 +3644,7 @@ Frame \(\{A\}\) represents the initial location, frame \(\{B\}\) is an intermedi
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/ustation_combined_transformation.png}
-\caption{\label{fig:ustation_combined_transformation}Motion of a rigid body represented at three locations by frame \(\{A\}\), \(\{B\}\) and \(\{C\}\)}
+\caption{\label{fig:ustation_combined_transformation}Motion of a rigid body represented at three locations by frame \(\{A\}\), \(\{B\}\) and \(\{C\}\).}
\end{figure}
Furthermore, suppose the position vector of a point \(P\) of the rigid body is given in the final location, that is \({}^CP\) is given, and the position of this point is to be found in the fixed frame \(\{A\}\), that is \({}^AP\).
@@ -3699,11 +3682,11 @@ Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/ustation_stage_motion.png}
-\caption{\label{fig:ustation_stage_motion}Example of the motion induced by the tilt-stage \(R_y\). ``Rest'' position in shown in blue while a arbitrary position in shown in red. Parasitic motions are here magnified for clarity.}
+\caption{\label{fig:ustation_stage_motion}Example of the motion induced by the tilt-stage \(R_y\). Initial position is shown in blue while an arbitrary position is shown in red. Parasitic motions are here magnified for clarity.}
\end{figure}
The motion induced by a positioning stage can be described by a homogeneous transformation matrix from frame \(\{A\}\) to frame \(\{B\}\) as explain in Section~\ref{ssec:ustation_kinematics}.
-As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in~\eqref{eq:ustation_translation_stage_errors}.
+As any motion stage induces parasitic motion in all 6-DoFs, the transformation matrix representing its induced motion can be written as in~\eqref{eq:ustation_translation_stage_errors}.
\begin{equation}\label{eq:ustation_translation_stage_errors}
{}^A\bm{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) =
@@ -3766,8 +3749,8 @@ The inertia of the solid bodies and the stiffness properties of the guiding mech
The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section~\ref{ssec:ustation_model_comp_dynamics}).
-As the dynamics of the active platform 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}).
+As the dynamics of the active platform is impacted by the micro-station compliance (see Section \ref{sec:uniaxial_support_compliance}), the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station.
+To do so, the 6-DoFs compliance of the micro-station is measured and then compared with the 6-DoFs compliance extracted from the multi-body model (Section~\ref{ssec:ustation_model_compliance}).
\subsubsection{Multi-Body Model}
\label{ssec:ustation_model_simscape}
@@ -3782,47 +3765,48 @@ External forces can be used to model disturbances, and ``sensors'' can be used t
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/ustation_simscape_stage_example.png}
-\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model software (Simulink - Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included}
+\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model software (Simulink - Simscape). It is composed of two solid bodies connected by a 6-DoFs joint. One joint DoF (here the tilt angle) can be ``controlled'', the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included.}
\end{figure}
Therefore, the micro-station is modeled by several solid bodies connected by joints.
-A typical stage (here the tilt-stage) is modeled as shown in Figure~\ref{fig:ustation_simscape_stage_example} where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoF joint.
-One DoF of the 6-DoF joint is ``imposed'' by a setpoint (i.e. modeled as infinitely stiff), while the other 5 are each modeled by a spring and damper.
+A typical stage (here the tilt-stage) is modeled as shown in Figure~\ref{fig:ustation_simscape_stage_example} where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoFs joint.
+One DoF of the 6-DoFs joint is ``imposed'' by a setpoint (i.e. modeled as infinitely stiff), while the other 5 are each modeled by a spring and damper.
Additional forces can be used to model disturbances induced by the stage motion.
The obtained 3D representation of the multi-body model is shown in Figure~\ref{fig:ustation_simscape_model}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/ustation_simscape_model.jpg}
-\caption{\label{fig:ustation_simscape_model}3D view of the micro-station multi-body model}
+\caption{\label{fig:ustation_simscape_model}3D view of the micro-station multi-body model.}
\end{figure}
The ground is modeled by a solid body connected to the ``world frame'' through a joint only allowing 3 translations.
-The granite was then connected to the ground using a 6-DoF joint.
-The translation stage is connected to the granite by a 6-DoF joint, but the \(D_y\) motion is imposed.
-Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoF joint, with 1 imposed DoF each time.
-Finally, the positioning hexapod has 6-DoF.
+The granite was then connected to the ground using a 6-DoFs joint.
+The translation stage is connected to the granite by a 6-DoFs joint, but the \(D_y\) motion is imposed.
+Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoFs joint, with 1 imposed DoF each time.
+Finally, the positioning hexapod has 6-DoFs.
The total number of ``free'' \acrshortpl{dof} is 27, so the model has 54 states.
The springs and dampers values were first estimated from the joint/stage specifications and were later fined-tuned based on the measurements.
The spring values are summarized in Table~\ref{tab:ustation_6dof_stiffness_values}.
\begin{table}[htbp]
-\caption{\label{tab:ustation_6dof_stiffness_values}Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by ``-''. The frames in which the 6-DoF joints are defined are indicated in figures found in Section~\ref{ssec:ustation_stages}}
\centering
\begin{tabularx}{0.9\linewidth}{Xcccccc}
\toprule
\textbf{Stage} & \(D_x\) & \(D_y\) & \(D_z\) & \(R_x\) & \(R_y\) & \(R_z\)\\
\midrule
-Granite & \(5\,\text{kN}/\mu\text{m}\) & \(5\,\text{kN}/\mu\text{m}\) & \(5\,\text{kN}/\mu\text{m}\) & \(25\,\text{Nm}/\mu\text{rad}\) & \(25\,\text{Nm}/\mu\text{rad}\) & \(10\,\text{Nm}/\mu\text{rad}\)\\
-Translation & \(200\,\text{N}/\mu\text{m}\) & - & \(200\,\text{N}/\mu\text{m}\) & \(60\,\text{Nm}/\mu\text{rad}\) & \(90\,\text{Nm}/\mu\text{rad}\) & \(60\,\text{Nm}/\mu\text{rad}\)\\
-Tilt & \(380\,\text{N}/\mu\text{m}\) & \(400\,\text{N}/\mu\text{m}\) & \(380\,\text{N}/\mu\text{m}\) & \(120\,\text{Nm}/\mu\text{rad}\) & - & \(120\,\text{Nm}/\mu\text{rad}\)\\
-Spindle & \(700\,\text{N}/\mu\text{m}\) & \(700\,\text{N}/\mu\text{m}\) & \(2\,\text{kN}/\mu\text{m}\) & \(10\,\text{Nm}/\mu\text{rad}\) & \(10\,\text{Nm}/\mu\text{rad}\) & -\\
-Hexapod & \(10\,\text{N}/\mu\text{m}\) & \(10\,\text{N}/\mu\text{m}\) & \(100\,\text{N}/\mu\text{m}\) & \(1.5\,\text{Nm/rad}\) & \(1.5\,\text{Nm/rad}\) & \(0.27\,\text{Nm/rad}\)\\
+Granite & \(5\,\text{kN}/\upmu\text{m}\) & \(5\,\text{kN}/\upmu\text{m}\) & \(5\,\text{kN}/\upmu\text{m}\) & \(25\,\text{Nm}/\upmu\text{rad}\) & \(25\,\text{Nm}/\upmu\text{rad}\) & \(10\,\text{Nm}/\upmu\text{rad}\)\\
+Translation & \(200\,\text{N}/\upmu\text{m}\) & - & \(200\,\text{N}/\upmu\text{m}\) & \(60\,\text{Nm}/\upmu\text{rad}\) & \(90\,\text{Nm}/\upmu\text{rad}\) & \(60\,\text{Nm}/\upmu\text{rad}\)\\
+Tilt & \(380\,\text{N}/\upmu\text{m}\) & \(400\,\text{N}/\upmu\text{m}\) & \(380\,\text{N}/\upmu\text{m}\) & \(120\,\text{Nm}/\upmu\text{rad}\) & - & \(120\,\text{Nm}/\upmu\text{rad}\)\\
+Spindle & \(700\,\text{N}/\upmu\text{m}\) & \(700\,\text{N}/\upmu\text{m}\) & \(2\,\text{kN}/\upmu\text{m}\) & \(10\,\text{Nm}/\upmu\text{rad}\) & \(10\,\text{Nm}/\upmu\text{rad}\) & -\\
+Hexapod & \(10\,\text{N}/\upmu\text{m}\) & \(10\,\text{N}/\upmu\text{m}\) & \(100\,\text{N}/\upmu\text{m}\) & \(1.5\,\text{Nm/rad}\) & \(1.5\,\text{Nm/rad}\) & \(0.27\,\text{Nm/rad}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:ustation_6dof_stiffness_values}Summary of the stage stiffnesses. The constrained degrees-of-freedom are indicated by ``-''. The frames in which the 6-DoFs joints are defined are indicated in figures found in Section \ref{ssec:ustation_stages}.}
+
\end{table}
-\subsubsection{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.
@@ -3852,9 +3836,9 @@ Tuning the numerous model parameters to better match the measurements is a highl
\end{center}
\subcaption{\label{fig:ustation_comp_com_response_ry_z}Tilt, $z$ response}
\end{subfigure}
-\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.}
+\caption{\label{fig:ustation_comp_com_response}FRFs from a hammer impact to the stage acceleration, both expressed at its CoM. The measured FRFs are compared with the multi-body model. Different directions are computed for different stages.}
\end{figure}
-\subsubsection{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.
@@ -3870,7 +3854,7 @@ For each impact position, 10 impacts were performed to average and improve the d
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/ustation_compliance_meas.png}
-\caption{\label{fig:ustation_compliance_meas}Schematic of the measurement setup used to estimate the compliance of the micro-station. The top platform of the positioning hexapod is shown with four 3-axis accelerometers (shown in red) are on top. 10 hammer impacts are performed at different locations (shown in blue).}
+\caption{\label{fig:ustation_compliance_meas}Schematic of the measurement setup used to estimate the compliance of the micro-station. Four 3-axis accelerometers (shown in red) are fix on top of the positioning hexapod platform. 10 hammer impacts are performed at different locations (shown in blue).}
\end{figure}
To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]\) to the acceleration expressed in the \(\{\mathcal{X}\}\) frame \(a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]\), a Jacobian matrix \(\bm{J}_a\) is written based on the positions and orientations of the accelerometers~\eqref{eq:ustation_compliance_acc_jacobian}.
@@ -3941,7 +3925,7 @@ Considering the complexity of the micro-station compliance dynamics, the model c
\end{center}
\subcaption{\label{fig:ustation_frf_compliance_Rxyz_model}Compliance in rotation}
\end{subfigure}
-\caption{\label{fig:ustation_frf_compliance_model}Compliance of the micro-station expressed in frame \(\{\mathcal{X}\}\). The measured FRFs are display by translucent lines, while the FRFs extracted from the multi-body models are shown by opaque lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed.}
+\caption{\label{fig:ustation_frf_compliance_model}Compliance of the micro-station expressed in frame \(\{\mathcal{X}\}\). The measured FRFs are displayed by translucent lines, while the FRFs extracted from the multi-body models are shown by opaque lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed.}
\end{figure}
\subsection{Estimation of Disturbances}
\label{sec:ustation_disturbances}
@@ -3956,7 +3940,7 @@ 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}.
-\subsubsection{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.
@@ -3972,17 +3956,17 @@ The obtained ground motion displacement is shown in Figure~\ref{fig:ustation_gro
\begin{minipage}[b]{0.54\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/ustation_ground_disturbance.png}
-\captionof{figure}{\label{fig:ustation_ground_disturbance}Measured ground motion}
+\captionof{figure}{\label{fig:ustation_ground_disturbance}Measured ground motion.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.44\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.92\linewidth]{figs/ustation_geophone_picture.jpg}
-\captionof{figure}{\label{fig:ustation_geophone_picture}(3D) L-4C geophone}
+\captionof{figure}{\label{fig:ustation_geophone_picture}(3D) L-4C geophone.}
\end{center}
\end{minipage}
-\paragraph{Ty Stage}
+\paragraph{Translation Stage}
To measure the positioning errors of the translation stage, the setup shown in Figure~\ref{fig:ustation_errors_ty_setup} is used.
A special optical element (called a ``straightness interferometer''\footnote{The special optics (straightness interferometer and reflector) are manufactured by Agilent (10774A).}) is fixed on top of the micro-station, while a laser source\footnote{Laser source is manufactured by Agilent (5519b).} and a straightness reflector are fixed on the ground.
@@ -3991,7 +3975,7 @@ A similar setup was used to measure the horizontal deviation (i.e. in the \(x\)
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/ustation_errors_ty_setup.png}
-\caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the straightness (vertical deviation) of the translation stage}
+\caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the straightness (vertical deviation) of the translation stage.}
\end{figure}
Six scans were performed between \(-4.5\,\text{mm}\) and \(4.5\,\text{mm}\).
@@ -4012,9 +3996,9 @@ Similar result is obtained for the \(x\) lateral direction.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_dy_vertical_remove_mean.png}
\end{center}
-\subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing linear fit}
+\subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing a linear fit}
\end{subfigure}
-\caption{\label{fig:ustation_errors_dy}Measurement of the vertical error of the translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).}
+\caption{\label{fig:ustation_errors_dy}Measurement of the straightness (vertical error) of the translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).}
\end{figure}
\paragraph{Spindle}
@@ -4028,7 +4012,7 @@ From the 5 measured displacements \([d_1,\,d_2,\,d_3,\,d_4,\,d_5]\), the transla
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_rz_meas_lion.jpg}
\end{center}
-\subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoF metrology}
+\subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoFs metrology}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
@@ -4036,10 +4020,10 @@ From the 5 measured displacements \([d_1,\,d_2,\,d_3,\,d_4,\,d_5]\), the transla
\end{center}
\subcaption{\label{fig:ustation_rz_meas_lion_zoom}Zoom on the metrology system}
\end{subfigure}
-\caption{\label{fig:ustation_rz_meas_lion_setup}Experimental setup used to estimate the errors induced by the Spindle rotation (\subref{fig:ustation_rz_meas_lion}). The motion of the two reference spheres is measured using 5 capacitive sensors (\subref{fig:ustation_rz_meas_lion_zoom})}
+\caption{\label{fig:ustation_rz_meas_lion_setup}Experimental setup used to estimate the errors induced by the Spindle rotation (\subref{fig:ustation_rz_meas_lion}). The motion of the two reference spheres is measured using 5 capacitive sensors (\subref{fig:ustation_rz_meas_lion_zoom}).}
\end{figure}
-A measurement was performed during a constant rotational velocity of the spindle of 60rpm and during 10 turns.
+A measurement was performed during a constant rotational velocity of the spindle of \(60\,\text{rpm}\) and during 10 turns.
The obtained results are shown in Figure~\ref{fig:ustation_errors_spindle}.
A large fraction of the radial (Figure~\ref{fig:ustation_errors_spindle_radial}) and tilt (Figure~\ref{fig:ustation_errors_spindle_tilt}) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle.
This is displayed by the dashed circle.
@@ -4066,9 +4050,9 @@ The vertical motion induced by scanning the spindle is in the order of \(\pm 30\
\end{center}
\subcaption{\label{fig:ustation_errors_spindle_tilt}Tilt errors}
\end{subfigure}
-\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.}
+\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}) errors during a Spindle rotation at \(60\,\text{rpm}\). The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis.}
\end{figure}
-\subsubsection{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.
@@ -4096,7 +4080,7 @@ The obtained transfer functions are shown in Figure~\ref{fig:ustation_model_sens
\end{subfigure}
\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}
-\subsubsection{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.
@@ -4159,9 +4143,9 @@ Second, a constant velocity scans with the translation stage was performed and a
\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.
+To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of \(60\,\text{rpm}\).
Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section~\ref{sec:ustation_disturbances}.
-A radial offset of \(\approx 1\,\mu\text{m}\) between the \acrfull{poi} and the spindle's rotation axis is introduced to represent what is experimentally observed.
+A radial offset of \(\approx 1\,\upmu\text{m}\) between the \acrfull{poi} and the spindle's rotation axis is introduced to represent what is experimentally observed.
During the 10 second simulation (i.e. 10 spindle turns), the position of the \acrshort{poi} with respect to the granite was recorded.
Results are shown in Figure~\ref{fig:ustation_errors_model_spindle}.
A good correlation with the measurements is observed both for radial errors (Figure~\ref{fig:ustation_errors_model_spindle_radial}) and axial errors (Figure~\ref{fig:ustation_errors_model_spindle_axial}).
@@ -4179,9 +4163,9 @@ A good correlation with the measurements is observed both for radial errors (Fig
\end{center}
\subcaption{\label{fig:ustation_errors_model_spindle_axial}Axial error}
\end{subfigure}
-\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}).}
+\caption{\label{fig:ustation_errors_model_spindle}Simulation results for a tomography experiment at constant velocity of \(60\,\text{rpm}\). 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}
-\subsubsection{Scans with the translation stage}
+\subsubsection{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.
@@ -4219,7 +4203,7 @@ The control of the Stewart platform introduces additional complexity due to its
Section~\ref{sec:nhexa_control} explores how the \acrfull{haclac} strategy, previously validated on the uniaxial model, can be adapted to address the coupled dynamics of the Stewart platform.
This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space).
Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts.
-\subsection{Active Vibration Platforms}
+\subsection{Review of Active Vibration Platforms}
\label{sec:nhexa_platform_review}
The conceptual phase started with the use of simplified models, such as uniaxial and three-degree-of-freedom rotating systems.
These models were chosen for their ease of analysis, and despite their simplicity, the principles derived from them usually apply to more complex systems.
@@ -4240,7 +4224,7 @@ To overcome this limitation, external metrology systems have been implemented to
A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems.
In many cases, sample position control is limited to translational \acrshortpl{dof}.
-At NSLS-II, for instance, a system capable of \(100\,\mu\text{m}\) stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure~\ref{fig:nhexa_stages_nazaretski}).
+At NSLS-II, for instance, a system capable of \(100\,\upmu\text{m}\) stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure~\ref{fig:nhexa_stages_nazaretski}).
Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately \(100\,\text{Hz}\) (Figure~\ref{fig:nhexa_stages_sapoti}).
\begin{figure}[h!tbp]
@@ -4248,21 +4232,21 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/nhexa_stages_nazaretski.png}
\end{center}
-\subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope}
+\subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope \cite{nazaretski15_pushin_limit}}
\end{subfigure}
\begin{subfigure}{0.60\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/nhexa_stages_sapoti.png}
\end{center}
-\subcaption{\label{fig:nhexa_stages_sapoti} SAPOTI sample stage}
+\subcaption{\label{fig:nhexa_stages_sapoti} SAPOTI sample stage \cite{geraldes23_sapot_carnaub_sirius_lnls}}
\end{subfigure}
-\caption{\label{fig:nhexa_stages_translations}Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope~\cite{nazaretski15_pushin_limit} at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline~\cite{geraldes23_sapot_carnaub_sirius_lnls} at Sirius facility (\subref{fig:nhexa_stages_sapoti})}
+\caption{\label{fig:nhexa_stages_translations}Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline at Sirius facility (\subref{fig:nhexa_stages_sapoti}).}
\end{figure}
The integration of \(R_z\) rotational capability, which is necessary for tomography experiments, introduces additional complexity.
At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section~\ref{sec:nhexa_stewart_platform}) using piezoelectric actuators has been positioned below the spindle (Figure~\ref{fig:nhexa_stages_villar}).
-While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to \(50\,\mu\text{m}\) due to the inherent constraints of piezoelectric actuators.
-In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering \(100\,\mu\text{m}\) stroke (Figure~\ref{fig:nhexa_stages_schroer}).
+While this configuration enables the correction of spindle motion errors through 5-DoFs control based on capacitive sensor measurements, the stroke is limited to \(50\,\upmu\text{m}\) due to the inherent constraints of piezoelectric actuators.
+In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering \(100\,\upmu\text{m}\) stroke (Figure~\ref{fig:nhexa_stages_schroer}).
However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration.
\begin{figure}[h!tbp]
@@ -4270,22 +4254,21 @@ However, attempts to implement real-time feedback using YZ external metrology pr
\begin{center}
\includegraphics[scale=1,height=5.5cm]{figs/nhexa_stages_villar.png}
\end{center}
-\subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station}
+\subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station \cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}}
\end{subfigure}
\begin{subfigure}{0.40\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/nhexa_stages_schroer.png}
\end{center}
-\subcaption{\label{fig:nhexa_stages_schroer} PtyNAMi microscope}
+\subcaption{\label{fig:nhexa_stages_schroer} PtyNAMi microscope \cite{schropp20_ptynam,schroer17_ptynam}}
\end{subfigure}
-\caption{\label{fig:nhexa_stages_spindle}Example of two sample stages for tomography experiments. ID16a endstation~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml} at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope~\cite{schropp20_ptynam,schroer17_ptynam} at PETRA III (\subref{fig:nhexa_stages_schroer})}
+\caption{\label{fig:nhexa_stages_spindle}Example of two sample stages for tomography experiments. ID16a endstation at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope at PETRA III (\subref{fig:nhexa_stages_schroer}).}
\end{figure}
Table~\ref{tab:nhexa_sample_stages} provides an overview of existing end-stations that incorporate feedback loops based on online metrology for sample positioning.
Although direct performance comparisons between these systems are challenging due to their varying experimental requirements, scanning velocities, and specific use cases, several distinctive characteristics of the NASS can be identified.
\begin{table}[!ht]
-\caption{\label{tab:nhexa_sample_stages}End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are ommited or indicated between parentheses. The specifications for the NASS are indicated in the last row.}
\centering
\begin{tabularx}{0.8\linewidth}{ccccc}
\toprule
@@ -4297,7 +4280,7 @@ Sample & light & Interferometers & 3 PID, n/a & APS\\
Sample & light & Capacitive sensors & \(\approx 10\,\text{Hz}\) & ESRF\\
Spindle & \(R_z: \pm 90\,\text{deg}\) & \(D_{xyz},\ R_{xy}\) & & ID16a\\
\textbf{Hexapod (piezo)} & \(D_{xyz}: 0.05\,\text{mm}\) & & & ~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}\\
- & \(R_{xy}: 500\,\mu\text{rad}\) & & & \\
+ & \(R_{xy}: 500\,\upmu\text{rad}\) & & & \\
\midrule
Sample & light & Interferometers & n/a & PETRA III\\
\textbf{XYZ stage (piezo)} & \(D_{xyz}: 0.1\,\text{mm}\) & \(D_{yz}\) & & P06\\
@@ -4331,6 +4314,8 @@ Tilt-Stage & \(R_y: \pm 3\,\text{deg}\) & & & \\
Translation Stage & \(D_y: \pm 10\,\text{mm}\) & & & \\
\bottomrule
\end{tabularx}
+\caption{\label{tab:nhexa_sample_stages}End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are omitted or indicated between parentheses. The specifications for the NASS are indicated in the last row.}
+
\end{table}
The first key distinction of the NASS is in the continuous rotation of the active vibration platform.
@@ -4353,10 +4338,10 @@ The primary control requirements focus on \([D_y,\ D_z,\ R_y]\) motions; however
\label{ssec:nhexa_active_platforms}
The choice of the active platform architecture for the NASS requires careful consideration of several critical specifications.
-The platform must provide control over five \acrshortpl{dof} (\(D_x\), \(D_y\), \(D_z\), \(R_x\), and \(R_y\)), with strokes exceeding \(100\,\mu\text{m}\) to correct for micro-station positioning errors, while fitting within a cylindrical envelope of 300 mm diameter and 95 mm height.
+The platform must provide control over five \acrshortpl{dof} (\(D_x\), \(D_y\), \(D_z\), \(R_x\), and \(R_y\)), with strokes exceeding \(100\,\upmu\text{m}\) to correct for micro-station positioning errors, while fitting within a cylindrical envelope of \(300\,\text{mm}\) diameter and \(95\,\text{mm}\) height.
It must accommodate payloads up to \(50\,\text{kg}\) while maintaining high dynamical performance.
For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to \(100\,\text{Hz}\).
-However, achieving such resonance frequencies with a \(50\,\text{kg}\) payload would require unrealistic stiffness values of approximately \(2000\,\text{N}/\mu\text{m}\).
+However, achieving such resonance frequencies with a \(50\,\text{kg}\) payload would require unrealistic stiffness values of approximately \(2000\,\text{N}/\upmu\text{m}\).
This limitation necessitates alternative control approaches, and the High \acrfull{haclac} strategy is proposed to address this challenge.
To this purpose, the design includes force sensors for active damping.
Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities.
@@ -4378,15 +4363,15 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/nhexa_serial_architecture_kenton.png}
\end{center}
-\subcaption{\label{fig:nhexa_serial_architecture_kenton} Serial positioning stage}
+\subcaption{\label{fig:nhexa_serial_architecture_kenton} XYZ Serial positioning stage \cite{kenton12_desig_contr_three_axis_serial}}
\end{subfigure}
\begin{subfigure}{0.55\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/nhexa_parallel_architecture_shen.png}
\end{center}
-\subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoF stage}
+\subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoFs stage \cite{shen19_dynam_analy_flexur_nanop_stage}}
\end{subfigure}
-\caption{\label{fig:nhexa_serial_parallel_examples}Examples of an XYZ serial positioning stage~\cite{kenton12_desig_contr_three_axis_serial} (\subref{fig:nhexa_serial_architecture_kenton}) and of a 5-DoF hybrid (parallel/serial) positioning platform~\cite{shen19_dynam_analy_flexur_nanop_stage} (\subref{fig:nhexa_parallel_architecture_shen}).}
+\caption{\label{fig:nhexa_serial_parallel_examples}Examples of a serial positioning stage (\subref{fig:nhexa_serial_architecture_kenton}) and of a hybrid (parallel/serial) positioning platform (\subref{fig:nhexa_parallel_architecture_shen}).}
\end{figure}
After evaluating the different options, the Stewart platform architecture was selected for several reasons.
@@ -4400,17 +4385,17 @@ Furthermore, the successful implementation of Integral Force Feedback (IFF) cont
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/nhexa_stewart_piezo_furutani.png}
\end{center}
-\subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning}
+\subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning \cite{furutani04_nanom_cuttin_machin_using_stewar}}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.9\linewidth]{figs/nhexa_stewart_vc_preumont.png}
\end{center}
-\subcaption{\label{fig:nhexa_stewart_vc_preumont} Stewart platform for vibration isolation}
+\subcaption{\label{fig:nhexa_stewart_vc_preumont} Stewart platform for vibration isolation \cite{preumont07_six_axis_singl_stage_activ,preumont18_vibrat_contr_activ_struc_fourt_edition}}
\end{subfigure}
-\caption{\label{fig:nhexa_stewart_examples}Two examples of Stewart platform. A Stewart platform based on piezoelectric stack actuators and used for nano-positioning is shown in (\subref{fig:nhexa_stewart_piezo_furutani})~\cite{furutani04_nanom_cuttin_machin_using_stewar}. A Stewart platform based on voice coil actuators and used for vibration isolation is shown in (\subref{fig:nhexa_stewart_vc_preumont})~\cite{preumont07_six_axis_singl_stage_activ,preumont18_vibrat_contr_activ_struc_fourt_edition}}
+\caption{\label{fig:nhexa_stewart_examples}Two examples of Stewart platforms. (\subref{fig:nhexa_stewart_piezo_furutani}) Stewart platform based on piezoelectric actuators and used for nano-positioning. (\subref{fig:nhexa_stewart_vc_preumont}) Stewart platform based on voice coil actuators and used for vibration isolation.}
\end{figure}
-\subsection{The Stewart platform}
+\subsection{The Stewart Platform}
\label{sec:nhexa_stewart_platform}
The Stewart platform, first introduced by Stewart in 1965~\cite{stewart65_platf_with_six_degrees_freed} for flight simulation applications, represents a significant milestone in parallel manipulator design.
This mechanical architecture has evolved far beyond its original purpose, and has been applied across diverse field, from precision positioning systems to robotic surgery.
@@ -4426,7 +4411,7 @@ Second, its compact design compared to serial manipulators makes it ideal for in
Third, the good dynamical properties should enable high-bandwidth positioning control.
While Stewart platforms excel in precision and stiffness, they typically exhibit a relatively limited workspace compared to serial manipulators.
-However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of \(10\,\mu\text{m}\).
+However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of \(10\,\upmu\text{m}\).
This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications.
The analysis encompasses the platform's kinematic relationships (Section~\ref{ssec:nhexa_stewart_platform_kinematics}), the use of the Jacobian matrix (Section~\ref{ssec:nhexa_stewart_platform_jacobian}), static behavior (Section~\ref{ssec:nhexa_stewart_platform_static}), and dynamic characteristics (Section~\ref{ssec:nhexa_stewart_platform_dynamics}).
@@ -4435,13 +4420,13 @@ These theoretical foundations form the basis for subsequent design decisions and
\label{ssec:nhexa_stewart_platform_architecture}
The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure~\ref{fig:nhexa_stewart_architecture}).
-Each strut is modelled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints.
+Each strut is modeled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints.
The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical)}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_architecture.png}
-\caption{\label{fig:nhexa_stewart_architecture}Schematical representation of the Stewart platform architecture.}
+\caption{\label{fig:nhexa_stewart_architecture}Schematic representation of the Stewart platform architecture.}
\end{figure}
To facilitate the rigorous analysis of the Stewart platform, four reference frames were defined:
@@ -4464,7 +4449,7 @@ This is summarized in Figure~\ref{fig:nhexa_stewart_notations}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_notations.png}
-\caption{\label{fig:nhexa_stewart_notations}Frame and key notations for the Stewart platform}
+\caption{\label{fig:nhexa_stewart_notations}Typical defined frames for the Stewart platform and key notations.}
\end{figure}
\subsubsection{Kinematic Analysis}
\label{ssec:nhexa_stewart_platform_kinematics}
@@ -4482,7 +4467,7 @@ This equation links the pose\footnote{The \emph{pose} represents the position an
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_loop_closure.png}
-\caption{\label{fig:nhexa_stewart_loop_closure}Notations to compute the kinematic loop closure}
+\caption{\label{fig:nhexa_stewart_loop_closure}Geometrical representation of the loop closure.}
\end{figure}
\paragraph{Inverse Kinematics}
@@ -4530,7 +4515,7 @@ By multiplying both sides by \({}^A\hat{\bm{s}}_i\),~\eqref{eq:nhexa_loop_closur
{}^A\hat{\bm{s}}_i {}^A\bm{v}_p + \underbrace{{}^A\hat{\bm{s}}_i ({}^A\bm{\omega} \times {}^A\bm{b}_i)}_{=({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) {}^A\bm{\omega}} = \dot{l}_i + \underbrace{{}^A\hat{s}_i l_i \left( {}^A\bm{\omega}_i \times {}^A\hat{\bm{s}}_i \right)}_{=0}
\end{equation}
-Equation~\eqref{eq:nhexa_loop_closure_velocity_bis} can be rearranged in matrix form to obtain~\eqref{eq:nhexa_jacobian_velocities}, with \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^{\intercal}\) the vector of strut velocities, and \(\dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\ {}^A\bm{\omega}]^{\intercal}\) the vector of platform velocity and angular velocity.
+Equation~\eqref{eq:nhexa_loop_closure_velocity_bis} can be rearranged in matrix form to obtain~\eqref{eq:nhexa_jacobian_velocities}, with \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^{\intercal}\) the vector of strut velocities, and \(\dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\ {}^A\bm{\omega}]^{\intercal}\) the vector of platform velocity and angular velocity.
\begin{equation}\label{eq:nhexa_jacobian_velocities}
\boxed{\dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}}}
@@ -4551,7 +4536,7 @@ The matrix \(\bm{J}\) is called the Jacobian matrix and is defined by~\eqref{eq:
Therefore, the Jacobian matrix \(\bm{J}\) links the rate of change of the strut length to the velocity and angular velocity of the top platform with respect to the fixed base through a set of linear equations.
However, \(\bm{J}\) needs to be recomputed for every Stewart platform pose because it depends on the actual pose of the manipulator.
-\paragraph{Approximate solution to the Forward and Inverse Kinematic problems}
+\paragraph{Approximate Solution to the Forward and Inverse Kinematic Problems}
For small displacements \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^{\intercal}\) around an operating point \(\bm{\mathcal{X}}_0\) (for which the Jacobian was computed), the associated joint displacement \(\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^{\intercal}\) can be computed using the Jacobian~\eqref{eq:nhexa_inverse_kinematics_approximate}.
@@ -4567,22 +4552,22 @@ Similarly, for small joint displacements \(\delta\bm{\mathcal{L}}\), it is possi
These two relations solve the forward and inverse kinematic problems for small displacement in a \emph{approximate} way.
While this approximation offers limited value for inverse kinematics, which can be solved analytically, it proves particularly useful for the forward kinematic problem where exact analytical solutions are difficult to obtain.
-\paragraph{Range validity of the approximate inverse kinematics}
+\paragraph{Range Validity of the Approximate Inverse Kinematics}
The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple analysis.
For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation~\eqref{eq:nhexa_inverse_kinematics}.
These strut lengths are then used with the Jacobian to estimate the platform pose~\eqref{eq:nhexa_forward_kinematics_approximate}, from which the error between the estimated and true poses can be calculated, both in terms of position \(\epsilon_D\) and orientation \(\epsilon_R\).
-For motion strokes from \(1\,\mu\text{m}\) to \(10\,\text{mm}\), the errors are estimated for all direction of motion, and the worst case errors are shown in Figure~\ref{fig:nhexa_forward_kinematics_approximate_errors}.
-The results demonstrate that for displacements up to approximately \(1\,\%\) of the hexapod's size (which corresponds to \(100\,\mu\text{m}\) as the size of the Stewart platform is here \(\approx 100\,\text{mm}\)), the Jacobian approximation provides excellent accuracy.
+For motion strokes from \(1\,\upmu\text{m}\) to \(10\,\text{mm}\), the errors are estimated for all direction of motion, and the worst case errors are shown in Figure~\ref{fig:nhexa_forward_kinematics_approximate_errors}.
+The results demonstrate that for displacements up to approximately \(1\,\%\) of the hexapod's size (which corresponds to \(100\,\upmu\text{m}\) as the size of the Stewart platform is here \(\approx 100\,\text{mm}\)), the Jacobian approximation provides excellent accuracy.
-Since the maximum required stroke of the active platform (\(\approx 100\,\mu\text{m}\)) is three orders of magnitude smaller than its overall size (\(\approx 100\,\text{mm}\)), the Jacobian matrix can be considered constant throughout the workspace.
+Since the maximum required stroke of the active platform (\(\approx 100\,\upmu\text{m}\)) is three orders of magnitude smaller than its overall size (\(\approx 100\,\text{mm}\)), the Jacobian matrix can be considered constant throughout the workspace.
It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/nhexa_forward_kinematics_approximate_errors.png}
-\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,\text{mm}\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.}
+\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,\text{mm}\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true position and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.}
\end{figure}
\paragraph{Static Forces}
@@ -4711,7 +4696,7 @@ This will be performed in the next section using a multi-body model.
All these characteristics (maneuverability, stiffness, dynamics, etc.) are fundamentally determined by the platform's geometry.
While a reasonable geometric configuration will be used to validate the NASS during the conceptual phase, the optimization of these geometric parameters will be explored during the detailed design phase.
-\subsection{Multi-Body Model}
+\subsection{Multi-Body Model of Stewart Platforms}
\label{sec:nhexa_model}
The dynamic modeling of Stewart platforms has traditionally relied on analytical approaches.
However, these analytical models become increasingly complex when the dynamical behaviors of struts and joints must be captured.
@@ -4737,7 +4722,7 @@ From these parameters, key kinematic properties can be derived: the strut orient
\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_model_def.png}
-\captionof{figure}{\label{fig:nhexa_stewart_model_def}Geometry of the stewart platform}
+\captionof{figure}{\label{fig:nhexa_stewart_model_def}Geometrical parameters of the Stewart platform.}
\end{center}
\end{minipage}
\hfill
@@ -4763,7 +4748,8 @@ From these parameters, key kinematic properties can be derived: the strut orient
\({}^M\bm{b}_5\) & \(-78\) & \(78\) & \(-20\)\\
\({}^M\bm{b}_6\) & \(-106\) & \(28\) & \(-20\)\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]}
\end{minipage}
\paragraph{Inertia of Plates}
@@ -4784,7 +4770,7 @@ These joints are considered massless and exhibit no stiffness along their degree
The actuator model comprises several key elements (Figure~\ref{fig:nhexa_actuator_model}).
At its core, each actuator is modeled as a prismatic joint with internal stiffness \(k_a\) and damping \(c_a\), driven by a force source \(f\).
-Similarly to what was found using the rotating 3-DoF model, a parallel stiffness \(k_p\) is added in parallel with the force sensor to ensure stability when considering spindle rotation effects.
+Similarly to what was found using the rotating 3-DoFs model, a parallel stiffness \(k_p\) is added in parallel with the force sensor to ensure stability when considering spindle rotation effects.
Each actuator is equipped with two sensors: a force sensor providing measurements \(f_n\) and a relative motion sensor that measures the strut length \(l_i\).
The actuator parameters used in the conceptual phase are listed in Table~\ref{tab:nhexa_actuator_parameters}.
@@ -4794,7 +4780,7 @@ This modular approach to actuator modeling allows for future refinements as the
\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/nhexa_actuator_model.png}
-\captionof{figure}{\label{fig:nhexa_actuator_model}Model of the active platform actuators}
+\captionof{figure}{\label{fig:nhexa_actuator_model}Model of the active platform actuators.}
\end{center}
\end{minipage}
\hfill
@@ -4805,14 +4791,15 @@ This modular approach to actuator modeling allows for future refinements as the
\toprule
& Value\\
\midrule
-\(k_a\) & \(1\,\text{N}/\mu\text{m}\)\\
+\(k_a\) & \(1\,\text{N}/\upmu\text{m}\)\\
\(c_a\) & \(50\,\text{Ns}/\text{m}\)\\
-\(k_p\) & \(0.05\,\text{N}/\mu\text{m}\)\\
+\(k_p\) & \(0.05\,\text{N}/\upmu\text{m}\)\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters}
\end{minipage}
-\subsubsection{Validation of the multi-body model}
+\subsubsection{Validation of the Multi-body Model}
\label{ssec:nhexa_model_validation}
The developed multi-body model of the Stewart platform is represented schematically in Figure~\ref{fig:nhexa_stewart_model_input_outputs}, highlighting the key inputs and outputs: actuator forces \(\bm{f}\), force sensor measurements \(\bm{f}_n\), and relative displacement measurements \(\bm{\mathcal{L}}\).
@@ -4829,12 +4816,12 @@ A three-dimensional visualization of the model is presented in Figure~\ref{fig:n
\begin{minipage}[b]{0.35\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.8\linewidth]{figs/nhexa_simscape_screenshot.jpg}
-\captionof{figure}{\label{fig:nhexa_simscape_screenshot}3D representation of the multi-body model}
+\captionof{figure}{\label{fig:nhexa_simscape_screenshot}3D representation of the multi-body model.}
\end{center}
\end{minipage}
The validation of the multi-body model was performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section~\ref{ssec:nhexa_stewart_platform_dynamics}.
-This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness \(k_a = 1\,\text{N}/\mu\text{m}\) and damping \(c_a = 10\,\text{N}/({\text{m}/\text{s}})\).
+This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness \(k_a = 1\,\text{N}/\upmu\text{m}\) and damping \(c_a = 10\,\text{N}/({\text{m}/\text{s}})\).
The geometric parameters remain as specified in Table~\ref{tab:nhexa_actuator_parameters}.
While the moving platform itself is considered massless, a \(10\,\text{kg}\) cylindrical payload is mounted on top with a radius of \(r = 110\,\text{mm}\) and a height \(h = 300\,\text{mm}\).
@@ -4859,7 +4846,7 @@ The close agreement between both approaches across the frequency spectrum valida
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/nhexa_comp_multi_body_analytical.png}
-\caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model}
+\caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model.}
\end{figure}
\subsubsection{Active Platform Dynamics}
\label{ssec:nhexa_model_dynamics}
@@ -4895,7 +4882,7 @@ The inclusion of parallel stiffness introduces an additional complex conjugate z
\end{center}
\subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{n}$}
\end{subfigure}
-\caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed using the active platform multi-body model}
+\caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed using the multi-body model.}
\end{figure}
\subsubsection{Conclusion}
@@ -4940,7 +4927,7 @@ In the context of the active platform, two distinct control strategies were exam
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_decentralized_control.png}
-\caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.}
+\caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown.}
\end{figure}
\subsubsection{Choice of the Control Space}
\label{ssec:nhexa_control_space}
@@ -4971,9 +4958,9 @@ Furthermore, at low frequencies, the plant exhibits good decoupling between the
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_control_cartesian.png}
\end{center}
-\subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-\intercal}$ is used to project force and torques on each strut}
+\subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-\intercal}$ is used to project forces and torques on each strut}
\end{subfigure}
-\caption{\label{fig:nhexa_control_frame}Two control strategies}
+\caption{\label{fig:nhexa_control_frame}Two control strategies using the Jacobian matrix.}
\end{figure}
\paragraph{Control in Cartesian Space}
@@ -5003,7 +4990,7 @@ More sophisticated control strategies will be explored during the detailed desig
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame}
\end{subfigure}
-\caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed using the active platform multi-body model}
+\caption{\label{fig:nhexa_plant_frame}Bode plots of plants corresponding to the two control strategies shown in Figure \ref{fig:nhexa_control_frame}.}
\end{figure}
\subsubsection{Active Damping with Decentralized IFF}
\label{ssec:nhexa_control_iff}
@@ -5015,7 +5002,7 @@ The corresponding block diagram of the control loop is shown in Figure~\ref{fig:
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/nhexa_decentralized_iff_schematic.png}
-\caption{\label{fig:nhexa_decentralized_iff_schematic}Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs \(\bm{f}^{\prime}\)}
+\caption{\label{fig:nhexa_decentralized_iff_schematic}Schematic of the implemented decentralized IFF controller. The damped plant has input \(\bm{f}^{\prime}\).}
\end{figure}
\begin{equation}\label{eq:nhexa_kiff}
@@ -5029,11 +5016,11 @@ The corresponding block diagram of the control loop is shown in Figure~\ref{fig:
In this section, the stiffness in parallel with the force sensor was omitted since the Stewart platform is not subjected to rotation.
The effect of this parallel stiffness is examined in the next section when the platform is integrated into the complete NASS.
-Root Locus analysis, shown in Figure~\ref{fig:nhexa_decentralized_iff_root_locus}, reveals the evolution of the closed-loop poles as the controller gain \(g\) varies from \(0\) to \(\infty\).
+Root locus analysis, shown in Figure~\ref{fig:nhexa_decentralized_iff_root_locus}, reveals the evolution of the closed-loop poles as the controller gain \(g\) varies from \(0\) to \(\infty\).
A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability~\cite{preumont08_trans_zeros_struc_contr_with}.
This property is particularly valuable because the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone.
-The bode plot of an individual loop gain (i.e. the loop gain of \(K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)\)), presented in Figure~\ref{fig:nhexa_decentralized_iff_loop_gain}, exhibits the typical characteristics of integral force feedback of having a phase bounded between \(-90^o\) and \(+90^o\).
+The bode plot of an individual loop gain (i.e. the loop gain of \(K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)\)), presented in Figure~\ref{fig:nhexa_decentralized_iff_loop_gain}, exhibits the typical characteristics of integral force feedback of having a phase bounded between \(\SI{-90}{\degree}\) and \(\SI{+90}{\degree}\).
The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes.
This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability.
@@ -5048,11 +5035,11 @@ This high gain, combined with the bounded phase, enables effective damping of th
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_iff_root_locus.png}
\end{center}
-\subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus}
+\subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root locus}
\end{subfigure}
-\caption{\label{fig:nhexa_decentralized_iff_results}Decentralized IFF}
+\caption{\label{fig:nhexa_decentralized_iff_results}Decentralized IFF. Loop Gain for an individual controller (\subref{fig:nhexa_decentralized_iff_loop_gain}) and root locus (\subref{fig:nhexa_decentralized_iff_root_locus}). Black crosses are indicating the closed-loop poles for the chosen controller gain.}
\end{figure}
-\subsubsection{MIMO High-Authority Control - Low-Authority Control}
+\subsubsection{High Authority Control / Low Authority Control}
\label{ssec:nhexa_control_hac_lac}
The design of the High Authority Control positioning loop is now examined.
@@ -5065,7 +5052,7 @@ A diagonal High Authority Controller \(\bm{K}_{\text{HAC}}\) then processes thes
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/nhexa_hac_iff_schematic.png}
-\caption{\label{fig:nhexa_hac_iff_schematic}HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts}
+\caption{\label{fig:nhexa_hac_iff_schematic}HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts.}
\end{figure}
The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions.
@@ -5120,7 +5107,7 @@ Additionally, the distance of the loci from the \(-1\) point provides informatio
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Characteristic Loci}
\end{subfigure}
-\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection performances. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop.}
+\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection level. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop.}
\end{figure}
\subsubsection{Conclusion}
@@ -5153,7 +5140,7 @@ Although the coupled dynamics of the system suggest the potential benefit of adv
This approach combines decentralized Integral Force Feedback for active damping with High Authority Control for positioning, which was implemented in the strut space to leverage the natural decoupling observed at low frequencies.
This study establishes the theoretical framework necessary for the subsequent development and validation of the NASS.
-\section{Validation of the Concept}
+\section{Validation of the NASS Concept}
\label{sec:nass}
The previous chapters have established crucial foundational elements for the development of the Nano Active Stabilization System (NASS).
The uniaxial model study demonstrated that very stiff active platform configurations should be avoided due to their high coupling with the micro-station dynamics.
@@ -5164,11 +5151,11 @@ Furthermore, a multi-body model of the active platform was created, that can the
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp,width=0.8\linewidth]{figs/nass_simscape_model.jpg}
-\caption{\label{fig:nass_simscape_model}3D view of the NASS multi-body model}
+\caption{\label{fig:nass_simscape_model}3D view of the NASS multi-body model.}
\end{figure}
Building upon these foundations, this chapter presents the validation of the NASS concept.
-The investigation begins with the previously established active platform model with actuator stiffness \(k_a = 1\,\text{N}/\mu\text{m}\).
+The investigation begins with the previously established active platform model with actuator stiffness \(k_a = 1\,\text{N}/\upmu\text{m}\).
A thorough examination of the control kinematics is presented in Section~\ref{sec:nass_kinematics}, detailing how both external metrology and active platform internal sensors are used in the control architecture.
The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section~\ref{sec:nass_active_damping}), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section~\ref{sec:nass_hac}).
@@ -5184,7 +5171,7 @@ This section focuses on the components of the ``Instrumentation and Real-Time Co
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp]{figs/nass_concept_schematic.png}
-\caption{\label{fig:nass_concept_schematic}Schematic of the Nano Active Stabilization System}
+\caption{\label{fig:nass_concept_schematic}Schematic of the Nano Active Stabilization System.}
\end{figure}
As established in the previous section on Stewart platforms, the proposed control strategy combines Decentralized Integral Force Feedback with a High Authority Controller performed in the frame of the struts.
@@ -5239,7 +5226,7 @@ Using these reference signals, the desired sample position relative to the fixed
\end{bmatrix}
\end{align}
\end{equation}
-\subsubsection{Computation of the sample's pose error}
+\subsubsection{Computation of the Sample's Pose Error}
\label{ssec:nass_sample_pose_error}
The external metrology system measures the sample position relative to the fixed granite.
@@ -5260,7 +5247,7 @@ The measured sample pose is represented by the homogeneous transformation matrix
0 & 0 & 0 & 1
\end{array} \right]
\end{equation}
-\subsubsection{Position error in the frame of the struts}
+\subsubsection{Position Error in the Frame of the Struts}
\label{ssec:nass_error_struts}
The homogeneous transformation formalism enables straightforward computation of the sample position error.
@@ -5311,9 +5298,9 @@ Then, the high authority controller uses the computed errors in the frame of the
\subsection{Decentralized Active Damping}
\label{sec:nass_active_damping}
Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the \acrshort{haclac} strategy.
-The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3DoF rotating model.
+The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3-DoFs rotating model.
The objective here is to design a decentralized IFF controller that provides good damping of the active platform modes across payload masses ranging from \(1\) to \(50\,\text{kg}\) and rotational velocity up to \(360\,\text{deg/s}\).
-The payloads used for validation have a cylindrical shape with 250 mm height and with masses of \(1\,\text{kg}\), \(25\,\text{kg}\), and \(50\,\text{kg}\).
+The payloads used for validation have a cylindrical shape with \(250\,\text{mm}\) height and with masses of \(1\,\text{kg}\), \(25\,\text{kg}\), and \(50\,\text{kg}\).
\subsubsection{IFF Plant}
\label{ssec:nass_active_damping_plant}
@@ -5338,10 +5325,10 @@ Although both cases show significant coupling around the resonances, stability i
\end{center}
\subcaption{\label{fig:nass_iff_plant_kp}with parallel stiffness}
\end{subfigure}
-\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and a payload mass of \(25\,\text{kg}\). The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp})}
+\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness in parallel with the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and a payload mass of \(25\,\text{kg}\). The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}).}
\end{figure}
-The effect of rotation, as shown in Figure~\ref{fig:nass_iff_plant_effect_rotation}, is negligible as the actuator stiffness (\(k_a = 1\,\text{N}/\mu\text{m}\)) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model).
+The effect of rotation, as shown in Figure~\ref{fig:nass_iff_plant_effect_rotation}, is negligible as the actuator stiffness (\(k_a = 1\,\text{N}/\upmu\text{m}\)) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3-DoFs rotating model).
Figure~\ref{fig:nass_iff_plant_effect_payload} illustrate the effect of payload mass on the plant dynamics.
The poles and zeros shift in frequency as the payload mass varies.
@@ -5360,12 +5347,12 @@ However, their alternating pattern is preserved, which ensures the phase remains
\end{center}
\subcaption{\label{fig:nass_iff_plant_effect_payload}Effect of payload mass}
\end{subfigure}
-\caption{\label{fig:nass_iff_plant_effect_rotation_payload}Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload})}
+\caption{\label{fig:nass_iff_plant_effect_rotation_payload}Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload}).}
\end{figure}
\subsubsection{Controller Design}
\label{ssec:nass_active_damping_control}
-The previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation.
+The previous analysis using the 3-DoFs rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation.
This finding was also confirmed with the multi-body model of the NASS: the system was unstable when using pure integrators and without parallel stiffness.
This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors.
@@ -5387,7 +5374,7 @@ The overall gain was then increased to obtain a large loop gain around the reson
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp,scale=0.8]{figs/nass_iff_loop_gain.png}
-\caption{\label{fig:nass_iff_loop_gain}Loop gain for the decentralized IFF: \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)}
+\caption{\label{fig:nass_iff_loop_gain}Loop gain for the decentralized IFF: \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\).}
\end{figure}
To verify stability, the root loci for the three payload configurations were computed, as shown in Figure~\ref{fig:nass_iff_root_locus}.
@@ -5412,7 +5399,7 @@ The results demonstrate that the closed-loop poles remain within the left-half p
\end{center}
\subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$}
\end{subfigure}
-\caption{\label{fig:nass_iff_root_locus}Root Loci for Decentralized IFF for three payload masses. The closed-loop poles are shown by the black crosses.}
+\caption{\label{fig:nass_iff_root_locus}Root loci for decentralized IFF for three payload masses. The closed-loop poles are shown by the black crosses.}
\end{figure}
\subsection{Centralized Active Vibration Control}
\label{sec:nass_hac}
@@ -5456,7 +5443,7 @@ This also validates the developed control strategy.
\end{center}
\subcaption{\label{fig:nass_undamped_plant_effect_mass}Effect of payload's mass}
\end{subfigure}
-\caption{\label{fig:nass_undamped_plant_effect}Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})}
+\caption{\label{fig:nass_undamped_plant_effect}Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass}).}
\end{figure}
The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs \(\bm{f}^\prime\) of the damped plant to the strut errors \(\bm{\epsilon}_{\mathcal{L}}\) were extracted from this model.
@@ -5483,7 +5470,7 @@ For the undamped plants (shown in blue), achieving robust control with bandwidth
\end{center}
\subcaption{\label{fig:nass_hac_plants}Effect of IFF on the set of plants to control}
\end{subfigure}
-\caption{\label{fig:nass_hac_plant}Effect of Decentralized Integral Force Feedback on the positioning plant for a \(1\,\text{kg}\) sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}).}
+\caption{\label{fig:nass_hac_plant}Effect of decentralized Integral Force Feedback on the positioning plant for a \(1\,\text{kg}\) sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). Direct terms are shown by solid lines while coupling terms are shown by shaded lines. The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}).}
\end{figure}
The coupling between the active platform and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions.
@@ -5496,20 +5483,20 @@ This result confirms effective dynamic decoupling between the active platform an
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp,scale=0.8]{figs/nass_effect_ustation_compliance.png}
-\caption{\label{fig:nass_effect_ustation_compliance}Effect of the micro-station limited compliance on the plant dynamics}
+\caption{\label{fig:nass_effect_ustation_compliance}Effect of the micro-station limited compliance on the plant dynamics.}
\end{figure}
\subsubsection{Effect of Active Platform Stiffness on System Dynamics}
\label{ssec:nass_hac_stiffness}
-The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models.
-These models suggest that a moderate stiffness of approximately \(1\,\text{N}/\mu\text{m}\) would provide better performance than either very stiff or very soft configurations.
+The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3-DoFs) models.
+These models suggest that a moderate stiffness of approximately \(1\,\text{N}/\upmu\text{m}\) would provide better performance than either very stiff or very soft configurations.
-For the stiff active platform analysis, a system with an actuator stiffness of \(100\,\text{N}/\mu\text{m}\) was simulated with a \(25\,\text{kg}\) payload.
+For the stiff active platform analysis, a system with an actuator stiffness of \(100\,\text{N}/\upmu\text{m}\) was simulated with a \(25\,\text{kg}\) payload.
The transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station.
As shown in Figure~\ref{fig:nass_stiff_nano_hexapod_coupling_ustation}, significant coupling was observed between the active platform and micro-station dynamics.
This coupling introduces complex behavior that is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model.
-The soft active platform configuration was evaluated using a stiffness of \(0.01\,\text{N}/\mu\text{m}\) with a \(25\,\text{kg}\) payload.
+The soft active platform configuration was evaluated using a stiffness of \(0.01\,\text{N}/\upmu\text{m}\) with a \(25\,\text{kg}\) payload.
The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s.
Figure~\ref{fig:nass_soft_nano_hexapod_effect_Wz} demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies.
The current approach of controlling the position in the strut frame is inadequate for soft active platforms; but even shifting control to a frame matching the payload's \acrlong{com} would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects.
@@ -5519,17 +5506,17 @@ The current approach of controlling the position in the strut frame is inadequat
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/nass_stiff_nano_hexapod_coupling_ustation.png}
\end{center}
-\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\mu\text{m}$ - Coupling with the micro-station}
+\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\upmu\text{m}$ - Coupling with the micro-station}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/nass_soft_nano_hexapod_effect_Wz.png}
\end{center}
-\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,\text{N}/\mu\text{m}$ - Effect of Spindle rotation}
+\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,\text{N}/\upmu\text{m}$ - Effect of Spindle rotation}
\end{subfigure}
-\caption{\label{fig:nass_soft_stiff_hexapod}Coupling between a stiff active platform (\(k_a = 100\,\text{N}/\mu\text{m}\)) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance (\(k_a = 0.01\,\text{N}/\mu\text{m}\)) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz})}
+\caption{\label{fig:nass_soft_stiff_hexapod}Coupling between a stiff active platform (\(k_a = 100\,\text{N}/\upmu\text{m}\)) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a soft (\(k_a = 0.01\,\text{N}/\upmu\text{m}\)) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}).}
\end{figure}
-\subsubsection{Controller design}
+\subsubsection{Controller Design}
\label{ssec:nass_hac_controller}
A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure~\ref{fig:nass_hac_plants}), and achievement of sufficient bandwidth (targeted at \(10\,\text{Hz}\)) for high performance operation.
@@ -5556,9 +5543,9 @@ Second, the characteristic loci analysis presented in Figure~\ref{fig:nass_hac_l
\end{center}
\subcaption{\label{fig:nass_hac_loci}Characteristic Loci}
\end{subfigure}
-\caption{\label{fig:nass_hac_controller}High Authority Controller - ``Diagonal Loop Gain'' (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci})}
+\caption{\label{fig:nass_hac_controller}High Authority Controller - ``Diagonal Loop Gain'' (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci}).}
\end{figure}
-\subsubsection{Tomography experiment}
+\subsubsection{Tomography Experiment}
\label{ssec:nass_hac_tomography}
The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with a particular focus on tomography scanning because of its demanding performance requirements.
@@ -5592,8 +5579,8 @@ The robustness of the NASS to payload mass variation was evaluated through addit
As illustrated in Figure~\ref{fig:nass_tomography_hac_iff}, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis.
While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions.
-It should be noted that the maximum rotational velocity of 360deg/s is primarily intended for lightweight payload applications.
-For higher mass configurations, rotational velocities are expected to be below 36deg/s.
+It should be noted that the maximum rotational velocity of \(360\,\text{deg/s}\) is primarily intended for lightweight payload applications.
+For higher mass configurations, rotational velocities are expected to be below \(36\,\text{deg/s}\).
\begin{figure}[h!tbp]
\begin{subfigure}{0.33\textwidth}
@@ -5614,7 +5601,7 @@ For higher mass configurations, rotational velocities are expected to be below 3
\end{center}
\subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,\text{kg}$}
\end{subfigure}
-\caption{\label{fig:nass_tomography_hac_iff}Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse}
+\caption{\label{fig:nass_tomography_hac_iff}Simulation of tomography experiments at \(360\,\text{deg/s}\). Beam size is indicated by the dashed black ellipse.}
\end{figure}
\subsection*{Conclusion}
\label{sec:nass_conclusion}
@@ -5626,7 +5613,7 @@ The control strategy implements a High Authority Control - Low Authority Control
The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping under various operating conditions.
The addition of parallel springs to the force sensors has been shown to ensure stability during spindle rotation.
The centralized High Authority Controller, operating in the frame of the struts for simplicity, has successfully achieved the desired performance objectives of maintaining a bandwidth of \(10\,\text{Hz}\) while maintaining robustness against payload mass variations.
-This investigation has confirmed that the moderate actuator stiffness of \(1\,\text{N}/\mu\text{m}\) represents an adequate choice for the active platform, as both very stiff and very compliant configurations introduce significant performance limitations.
+This investigation has confirmed that the moderate actuator stiffness of \(1\,\text{N}/\upmu\text{m}\) represents an adequate choice for the active platform, as both very stiff and very compliant configurations introduce significant performance limitations.
Simulations of tomography experiments have been performed, with positioning accuracy requirements defined by the expected minimum beam dimensions of \(200\,\text{nm}\) by \(100\,\text{nm}\).
The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples.
@@ -5639,7 +5626,7 @@ Through a systematic progression from simplified to increasingly complex models,
Using the simple uniaxial model revealed that a very stiff stabilization stage was unsuitable due to its strong coupling with the complex micro-station dynamics.
Conversely, the three-degree-of-freedom rotating model demonstrated that very soft stabilization stage designs are equally problematic due to the gyroscopic effects induced by spindle rotation.
-A moderate stiffness of approximately \(1\,\text{N}/\mu\text{m}\) was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection.
+A moderate stiffness of approximately \(1\,\text{N}/\upmu\text{m}\) was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection.
The multi-body modeling approach proved essential for capturing the complex dynamics of both the micro-station and the active platform.
This model was tuned based on extensive modal analysis and vibration measurements.
@@ -5656,7 +5643,7 @@ As anticipated by the control analysis, some performance degradation was observe
\minitoc
\subsubsection*{Abstract}
Following the validation of the Nano Active Stabilization System concept in the previous chapter through simulated tomography experiments, this chapter addresses the refinement of the preliminary conceptual model into an optimized implementation.
-The initial validation used a active platform with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise.
+The initial validation used an active platform with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise.
This detailed design phase aims to optimize each component while ensuring none will limit the system's overall performance.
This chapter begins by determining the optimal geometric configuration for the active platform (Section~\ref{sec:detail_kinematics}).
@@ -5675,9 +5662,9 @@ Third, the optimization of controllers for decoupled plants is discussed, introd
Section~\ref{sec:detail_instrumentation} focuses on instrumentation selection using a dynamic error budgeting approach to establish maximum acceptable noise specifications for each component.
The selected instrumentation is then experimentally characterized to verify compliance with these specifications, ensuring that the combined effect of all noise sources remains within acceptable limits.
-The chapter concludes with a concise presentation of the obtained optimized active platform design, called the ``nano-hexapod'', in Section~\ref{sec:detail_design}, summarizing how the various optimizations contribute to a system that balances the competing requirements of precision positioning, vibration isolation, and practical implementation constraints.
+The chapter concludes with a concise presentation of the obtained optimized active platform design, called the ``nano-hexapod'' (Section~\ref{sec:detail_design}).
With the detailed design completed and components procured, the project advances to the experimental validation phase, which will be addressed in the subsequent chapter.
-\section{Optimal Geometry}
+\section{Optimal Active Platform Geometry}
\label{sec:detail_kinematics}
The performance of a Stewart platform depends on its geometric configuration, especially the orientation of its struts and the positioning of its joints.
During the conceptual design phase of the active platform, a preliminary geometry was selected based on general principles without detailed optimization.
@@ -5689,7 +5676,7 @@ The chapter begins with a comprehensive review of existing Stewart platform desi
Section~\ref{sec:detail_kinematics_geometry} develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process.
Section~\ref{sec:detail_kinematics_cubic} examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the NASS applications.
Finally, Section~\ref{sec:detail_kinematics_nano_hexapod} presents the optimized active platform geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS.
-\subsection{Review of Stewart platforms}
+\subsection{Review of Stewart Platforms}
\label{sec:detail_kinematics_stewart_review}
The first parallel platform similar to the Stewart platform was built in 1954 by Gough~\cite{gough62_univer_tyre_test_machin}, for a tyre test machine (shown in Figure~\ref{fig:detail_geometry_gough_paper}).
@@ -5709,19 +5696,19 @@ Since then, the Stewart platform (sometimes referred to as the Stewart-Gough pla
\end{center}
\subcaption{\label{fig:detail_geometry_stewart_flight_simulator}Flight simulator proposed by Stewart \cite{stewart65_platf_with_six_degrees_freed}}
\end{subfigure}
-\caption{\label{fig:detail_geometry_stewart_origins}Two of the earliest developments of Stewart platforms}
+\caption{\label{fig:detail_geometry_stewart_origins}Two of the earliest developments of Stewart platforms.}
\end{figure}
-As explained in the conceptual phase, Stewart platforms comprise the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
+As explained in Section~\ref{sec:nhexa_stewart_platform}, Stewart platforms comprise the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
The specific geometry (i.e., position of joints and orientation of the struts) can be selected based on the application requirements, resulting in numerous designs throughout the literature.
This discussion focuses primarily on Stewart platforms designed for nano-positioning and vibration control, which necessitates the use of flexible joints.
-The implementation of these flexible joints, will be discussed when designing the active platform flexible joints.
+The implementation of these flexible joints, will be discussed when designing the active platform flexible joints in Section~\ref{sec:detail_fem_joint}.
Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics~\cite{merlet06_paral_robot}.
In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators.
Voice coil actuators, providing stroke ranges from \(0.5\,\text{mm}\) to \(10\,\text{mm}\), are commonly implemented in cubic architectures (as illustrated in Figures~\ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) and are mainly used for vibration isolation~\cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax,thayer98_stewar,mcinroy99_dynam,preumont07_six_axis_singl_stage_activ}.
-For applications requiring short stroke (typically smaller than \(500\,\mu\text{m}\)), piezoelectric actuators present an interesting alternative, as shown in~\cite{agrawal04_algor_activ_vibrat_isolat_spacec,furutani04_nanom_cuttin_machin_using_stewar,yang19_dynam_model_decoup_contr_flexib}.
+For applications requiring short stroke (typically smaller than \(500\,\upmu\text{m}\)), piezoelectric actuators present an interesting alternative, as shown in~\cite{agrawal04_algor_activ_vibrat_isolat_spacec,furutani04_nanom_cuttin_machin_using_stewar,yang19_dynam_model_decoup_contr_flexib}.
Examples of piezoelectric-actuated Stewart platforms are presented in Figures~\ref{fig:detail_kinematics_ulb_pz}, \ref{fig:detail_kinematics_uqp} and \ref{fig:detail_kinematics_yang19}.
Although less frequently encountered, magnetostrictive actuators have been successfully implemented in~\cite{zhang11_six_dof} (Figure~\ref{fig:detail_kinematics_zhang11}).
@@ -5752,7 +5739,7 @@ Although less frequently encountered, magnetostrictive actuators have been succe
\end{center}
\subcaption{\label{fig:detail_kinematics_uqp}Naval Postgraduate School - USA \cite{agrawal04_algor_activ_vibrat_isolat_spacec}}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_stewart_examples_cubic}Some examples of developped Stewart platform with Cubic geometry}
+\caption{\label{fig:detail_kinematics_stewart_examples_cubic}Some examples of developed Stewart platform with Cubic geometry.}
\end{figure}
The sensors integrated in these platforms are selected based on specific control requirements, as different sensors offer distinct advantages and limitations~\cite{hauge04_sensor_contr_space_based_six}.
@@ -5800,11 +5787,11 @@ The influence of strut orientation and joint positioning on Stewart platform pro
\end{center}
\subcaption{\label{fig:detail_kinematics_naves}University of Twente - Netherlands \cite{naves21_desig_optim_large_strok_flexur_mechan}}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_stewart_examples_non_cubic}Some examples of developped Stewart platform with non-cubic geometry}
+\caption{\label{fig:detail_kinematics_stewart_examples_non_cubic}Some examples of developed Stewart platform with non-cubic geometry.}
\end{figure}
-\subsection{Effect of geometry on Stewart platform properties}
+\subsection{Kinematic Study of Stewart Platforms}
\label{sec:detail_kinematics_geometry}
-As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator.
+As was demonstrated in Section~\ref{sec:nhexa_stewart_platform}, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator.
It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications.
A useful analytical tool for this study is the Jacobian matrix, which depends on \(\bm{b}_i\) (joints' position with respect to the top platform) and \(\hat{\bm{s}}_i\) (struts' orientation).
@@ -5833,7 +5820,7 @@ The analysis is significantly simplified when considering small motions, as the
Therefore, the mobility of the Stewart platform (defined as the set of achievable \([\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]\)) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix).
More specifically, the XYZ mobility only depends on the \(\hat{\bm{s}}_i\) (orientation of struts), while the mobility in rotation also depends on \(\bm{b}_i\) (position of top joints).
-\paragraph{Mobility in translation}
+\paragraph{Mobility in Translation}
For simplicity, only translations are first considered (i.e., the Stewart platform is considered to have fixed orientation).
In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions.
@@ -5864,7 +5851,7 @@ The vertically oriented struts configuration leads to greater stroke in the hori
Conversely, horizontal oriented struts configuration provides more stroke in the vertical direction.
It may seem counterintuitive that less stroke is available in the direction of the struts.
-This phenomenon occurs because the struts form a lever mechanism that amplifies the motion.
+This phenomenon occurs because the struts form a lever arm mechanism that amplifies the motion.
The amplification factor increases when the struts have a high angle with the direction of motion and equals one (i.e. is minimal) when aligned with the direction of motion.
\begin{figure}[htbp]
@@ -5886,9 +5873,9 @@ The amplification factor increases when the struts have a high angle with the di
\end{center}
\subcaption{\label{fig:detail_kinematics_mobility_translation_strut_orientation}Translational mobility}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_stewart_mobility_translation_examples}Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).}
+\caption{\label{fig:detail_kinematics_stewart_mobility_translation_examples}Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometries are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented horizontally (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).}
\end{figure}
-\paragraph{Mobility in rotation}
+\paragraph{Mobility in Rotation}
As shown by equation~\eqref{eq:detail_kinematics_jacobian}, the rotational mobility depends both on the orientation of the struts and on the location of the top joints.
Similarly to the translational case, to increase the rotational mobility in one direction, it is advantageous to have the struts more perpendicular to the rotational direction.
@@ -5918,9 +5905,9 @@ Having struts further apart decreases the ``lever arm'' and therefore reduces th
\end{center}
\subcaption{\label{fig:detail_kinematics_mobility_angle_strut_distance}Rotational mobility}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_stewart_mobility_rotation_examples}Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).}
+\caption{\label{fig:detail_kinematics_stewart_mobility_rotation_examples}Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometries are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further apart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).}
\end{figure}
-\paragraph{Combined translations and rotations}
+\paragraph{Combined Translations and Rotations}
It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex.
For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke.
@@ -5970,7 +5957,7 @@ Having a diagonal stiffness matrix \(\bm{K}\) can be beneficial for control purp
This property depends on both the geometry and the chosen \(\{A\}\) frame.
For specific geometry and choice of \(\{A\}\) frame, it is possible to achieve a diagonal \(K\) matrix.
This is discussed in Section~\ref{ssec:detail_kinematics_cubic_static}.
-\subsubsection{Dynamical properties}
+\subsubsection{Dynamical Properties}
\label{ssec:detail_kinematics_geometry_dynamics}
The dynamical equations (both in the Cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints).
@@ -5998,7 +5985,6 @@ These results could have been easily deduced based on mechanical principles, but
These trade-offs provide important guidelines when choosing the Stewart platform geometry.
\begin{table}[htbp]
-\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness and mobility}
\centering
\begin{tabularx}{0.65\linewidth}{Xcc}
\toprule
@@ -6015,6 +6001,8 @@ Vertical rotation mobility & \(\nearrow\) & \(\searrow\)\\
Horizontal rotation mobility & \(\searrow\) & \(\searrow\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness and mobility.}
+
\end{table}
\subsection{The Cubic Architecture}
\label{sec:detail_kinematics_cubic}
@@ -6038,7 +6026,7 @@ It is also possible to implement designs with strut lengths smaller than the cub
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_architecture_examples}Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).}
+\caption{\label{fig:detail_kinematics_cubic_architecture_examples}Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges' length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).}
\end{figure}
Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption~\cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}: simplified kinematics relationships and dynamical analysis~\cite{geng94_six_degree_of_freed_activ}; uniform stiffness in all directions~\cite{hanieh03_activ_stewar}; uniform mobility~\cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}; and minimization of the cross coupling between actuators and sensors in different struts~\cite{preumont07_six_axis_singl_stage_activ}.
@@ -6051,7 +6039,7 @@ Given that the cubic architecture imposes strict geometric constraints, alternat
The ultimate objective is to determine the suitability of the cubic architecture for the active platform.
\subsubsection{Static Properties}
\label{ssec:detail_kinematics_cubic_static}
-\paragraph{Stiffness matrix for the Cubic architecture}
+\paragraph{Stiffness Matrix for the Cubic Architecture}
Consider the cubic architecture shown in Figure~\ref{fig:detail_kinematics_cubic_schematic_full}.
The unit vectors corresponding to the edges of the cube are described by equation~\eqref{eq:detail_kinematics_cubic_s}.
@@ -6078,7 +6066,7 @@ The unit vectors corresponding to the edges of the cube are described by equatio
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})}
+\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Cubic architecture. Struts are represented in blue. The cube's center is indicated by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic}).}
\end{figure}
Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation~\eqref{eq:detail_kinematics_cubic_vertices}.
@@ -6113,7 +6101,7 @@ In that case, the location of the top joints can be expressed by equation~\eqref
The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)).
This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
This specific location where the stiffness matrix is diagonal is referred to as the \acrfull{cok}, analogous to the \acrfull{com} where the mass matrix is diagonal.
-\paragraph{Effect of having frame \(\{B\}\) off-centered}
+\paragraph{Effect of Having Frame \(\{B\}\) Off-centered}
When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
@@ -6163,7 +6151,7 @@ Furthermore, an inverse relationship exists between the cube's dimension and rot
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_mobility_rotations}Mobility in rotation}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_mobility}Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations})}
+\caption{\label{fig:detail_kinematics_cubic_mobility}Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations}).}
\end{figure}
\subsubsection{Dynamical Decoupling}
\label{ssec:detail_kinematics_cubic_dynamic}
@@ -6173,9 +6161,9 @@ When relative motion sensors are integrated in each strut (measuring \(\bm{\math
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
-\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame}
+\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame.}
\end{figure}
-\paragraph{Low frequency and High frequency coupling}
+\paragraph{Low Frequency and High Frequency Coupling}
As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation~\eqref{eq:detail_kinematics_transfer_function_cart}.
At low frequency, the behavior of the platform depends on the stiffness matrix~\eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
@@ -6197,7 +6185,7 @@ To achieve a diagonal mass matrix, the \acrlong{com} of the mobile components mu
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.5\linewidth]{figs/detail_kinematics_cubic_payload.png}
-\caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload}
+\caption{\label{fig:detail_kinematics_cubic_payload}Cubic Stewart platform with cylindrical payload located on the top platform.}
\end{figure}
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure~\ref{fig:detail_kinematics_cubic_payload}).
@@ -6218,9 +6206,9 @@ Conversely, when positioned at the \acrlong{cok}, coupling occurred at high freq
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).}
+\caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).}
\end{figure}
-\paragraph{Payload's CoM at the cube's center}
+\paragraph{Payload's CoM at the Cube's Center}
An effective strategy for improving dynamical performances involves aligning the cube's center (\acrlong{cok}) with the \acrlong{com} of the moving components~\cite{li01_simul_fault_vibrat_isolat_point}.
This can be achieved by positioning the payload below the top platform, such that the \acrlong{com} of the moving body coincides with the cube's center (Figure~\ref{fig:detail_kinematics_cubic_centered_payload}).
@@ -6242,7 +6230,7 @@ If a design similar to Figure~\ref{fig:detail_kinematics_cubic_centered_payload}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})}
+\caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok}).}
\end{figure}
\paragraph{Conclusion}
@@ -6270,7 +6258,7 @@ The second uses a non-cubic Stewart platform shown in Figure~\ref{fig:detail_kin
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.5\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
-\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture}
+\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture.}
\end{figure}
\paragraph{Relative Displacement Sensors}
@@ -6294,7 +6282,7 @@ The resonance frequencies differ between the two cases because the more vertical
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})}
+\caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL}).}
\end{figure}
\paragraph{Force Sensors}
@@ -6315,7 +6303,7 @@ The system demonstrates good decoupling at high frequency in both cases, with no
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})}
+\caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn}).}
\end{figure}
\paragraph{Conclusion}
@@ -6333,7 +6321,7 @@ Three key parameters define the geometry of the cubic Stewart platform: \(H\), t
Depending on the cube's size \(H_c\) in relation to \(H\) and \(H_{CoM}\), different designs emerge.
In the following examples, \(H = 100\,\text{mm}\) and \(H_{CoM} = 20\,\text{mm}\).
-\paragraph{Small cube}
+\paragraph{Small Cube}
When the cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}\) \eqref{eq:detail_kinematics_cube_small}, the resulting design is shown in Figure~\ref{fig:detail_kinematics_cubic_above_small}.
@@ -6341,7 +6329,7 @@ When the cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}
H_c < 2 H_{CoM}
\end{equation}
-This configuration is similar to that described in~\cite{furutani04_nanom_cuttin_machin_using_stewar}, although they do not explicitly identify it as a cubic configuration.
+This configuration is similar to that described in~\cite{furutani04_nanom_cuttin_machin_using_stewar} (Figure~\ref{fig:nhexa_stewart_piezo_furutani}, page~\pageref{fig:nhexa_stewart_piezo_furutani}), although they do not explicitly identify it as a cubic configuration.
Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other.
This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness.
@@ -6365,9 +6353,9 @@ This approach yields a compact architecture, but the small cube size may result
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_above_small_top}Top view}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.}
+\caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of \(40\,\text{mm}\) is used.}
\end{figure}
-\paragraph{Medium sized cube}
+\paragraph{Medium Sized Cube}
Increasing the cube's size such that~\eqref{eq:detail_kinematics_cube_medium} is verified produces an architecture with intersecting struts (Figure~\ref{fig:detail_kinematics_cubic_above_medium}).
@@ -6396,9 +6384,9 @@ This configuration resembles the design proposed in~\cite{yang19_dynam_model_dec
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_above_medium_top}Top view}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.}
+\caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of \(140\,\text{mm}\) is used.}
\end{figure}
-\paragraph{Large cube}
+\paragraph{Large Cube}
When the cube's height exceeds twice the sum of the platform height and CoM height~\eqref{eq:detail_kinematics_cube_large}, the architecture shown in Figure~\ref{fig:detail_kinematics_cubic_above_large} is obtained.
@@ -6425,9 +6413,9 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_above_large_top}Top view}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.}
+\caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of \(240\,\text{mm}\) is used.}
\end{figure}
-\paragraph{Platform size}
+\paragraph{Platform Size}
For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom joints \(\bm{a}_i\)) and are positioned on a circle with radius \(R_{b_i}\) (resp. \(R_{a_i}\)) described by Equation~\eqref{eq:detail_kinematics_cube_joints}.
@@ -6459,7 +6447,7 @@ However, this arrangement presents practical challenges, as the cube's center is
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's \acrlong{com} with the \acrlong{cok}, resulting in beneficial decoupling properties in the Cartesian frame.
-\subsection{Active Platform for the NASS}
+\subsection{Kinematics of the Active Platform}
\label{sec:detail_kinematics_nano_hexapod}
Based on previous analysis, this section aims to determine the active platform optimal geometry.
For the NASS, the chosen reference frames \(\{A\}\) and \(\{B\}\) coincide with the sample's \acrshort{poi}, which is positioned \(150\,\text{mm}\) above the top platform.
@@ -6469,8 +6457,8 @@ This is the location where precise control of the sample's position is required,
The design of the active platform must satisfy several constraints.
The device should fit within a cylinder with radius of \(120\,\text{mm}\) and height of \(95\,\text{mm}\).
-Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of \(\pm 50\,\mu\text{m}\).
-At any position, the system should be capable of performing \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\).
+Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of \(\pm 50\,\upmu\text{m}\).
+At any position, the system should be capable of performing \(R_x\) and \(R_y\) rotations of \(\pm 50\,\upmu \text{rad}\).
Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics.
In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts.
\subsubsection{Obtained Geometry}
@@ -6489,7 +6477,7 @@ Regarding dynamical properties, particularly for control in the frame of the str
Consequently, the geometry was selected according to practical constraints.
The height between the two plates is maximized and set at \(95\,\text{mm}\).
Both platforms take the maximum available size, with joints offset by \(15\,\text{mm}\) from the plate surfaces and positioned along circles with radii of \(120\,\text{mm}\) for the fixed joints and \(110\,\text{mm}\) for the mobile joints.
-The positioning angles, as shown in Figure~\ref{fig:detail_kinematics_nano_hexapod_top}, are \([255,\ 285,\ 15,\ 45,\ 135,\ 165]\) degrees for the top joints and \([220,\ 320,\ 340,\ 80,\ 100,\ 200]\) degrees for the bottom joints.
+The positioning angles, as shown in Figure~\ref{fig:detail_kinematics_nano_hexapod_top}, are [255, 285, 15, 45, 135, 165] degrees for the top joints and [220, 320, 340, 80, 100, 200] degrees for the bottom joints.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@@ -6504,28 +6492,29 @@ The positioning angles, as shown in Figure~\ref{fig:detail_kinematics_nano_hexap
\end{center}
\subcaption{\label{fig:detail_kinematics_nano_hexapod_top}Top view}
\end{subfigure}
-\caption{\label{fig:detail_kinematics_nano_hexapod}Obtained architecture for the active platform}
+\caption{\label{fig:detail_kinematics_nano_hexapod}Obtained architecture for the active platform.}
\end{figure}
The resulting geometry is illustrated in Figure~\ref{fig:detail_kinematics_nano_hexapod}.
While minor refinements may occur during detailed mechanical design to address manufacturing and assembly considerations, the fundamental geometry will remain consistent with this configuration.
-This geometry serves as the foundation for estimating required actuator stroke (Section~\ref{ssec:detail_kinematics_nano_hexapod_actuator_stroke}), determining flexible joint stroke requirements (Section~\ref{ssec:detail_kinematics_nano_hexapod_joint_stroke}), performing noise budgeting for instrumentation selection, and developing control strategies.
+This geometry serves as the foundation for estimating the required actuator stroke (Section~\ref{ssec:detail_kinematics_nano_hexapod_actuator_stroke}), flexible joint stroke (Section~\ref{ssec:detail_kinematics_nano_hexapod_joint_stroke}) and to perform noise budgeting for instrumentation selection (Section~\ref{sec:detail_instrumentation}).
+
Implementing a cubic architecture as proposed in Section~\ref{ssec:detail_kinematics_cubic_design} was considered.
However, positioning the cube's center \(150\,\text{mm}\) above the top platform would have resulted in platform dimensions exceeding the maximum available size.
Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the active platform, ensuring that its \acrlong{com} coincides with the cube's center.
Given the impracticality of consistently aligning the \acrlong{com} with the cube's center, the cubic architecture was deemed unsuitable for the NASS.
-\subsubsection{Required Actuator stroke}
+\subsubsection{Required Actuator Stroke}
\label{ssec:detail_kinematics_nano_hexapod_actuator_stroke}
With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined.
-The required mobility parameters include combined translations in the XYZ directions of \(\pm 50\,\mu\text{m}\) (essentially a cubic workspace).
-Additionally, at any point within this workspace, combined \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\), with \(R_z\) maintained at 0, should be possible.
+The required mobility parameters include combined translations in the XYZ directions of \(\pm 50\,\upmu\text{m}\) (essentially a cubic workspace).
+Additionally, at any point within this workspace, combined \(R_x\) and \(R_y\) rotations of \(\pm 50\,\upmu\text{rad}\), with \(R_z\) maintained at 0, should be possible.
-Calculations based on the selected geometry indicate that an actuator stroke of \(\pm 94\,\mu\text{m}\) is required to achieve the desired mobility.
+Calculations based on the selected geometry indicate that an actuator stroke of \(\pm 94\,\upmu\text{m}\) is required to achieve the desired mobility.
This specification will be used during the actuator selection process in Section~\ref{sec:detail_fem_actuator}.
-Figure~\ref{fig:detail_kinematics_nano_hexapod_mobility} illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the active platform with an actuator stroke of \(\pm 94\,\mu\text{m}\).
+Figure~\ref{fig:detail_kinematics_nano_hexapod_mobility} illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the active platform with an actuator stroke of \(\pm 94\,\upmu\text{m}\).
The diagram confirms that the required workspace fits within the system's capabilities.
\begin{figure}[htbp]
@@ -6533,7 +6522,7 @@ The diagram confirms that the required workspace fits within the system's capabi
\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_nano_hexapod_mobility.png}
\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Specified translation mobility of the active platform (grey cube) and computed Mobility (red volume).}
\end{figure}
-\subsubsection{Required Joint angular stroke}
+\subsubsection{Required Joint Angular Stroke}
\label{ssec:detail_kinematics_nano_hexapod_joint_stroke}
With the active platform geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined.
@@ -6557,7 +6546,7 @@ Modified cubic architectures with the cube's center positioned above the top pla
For the active platform design, a key challenge was addressing the wide range of potential payloads (1 to \(50\,\text{kg}\)), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios.
This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies.
-\section{Component Optimization}
+\section{Hybrid Modelling for Component Optimization}
\label{sec:detail_fem}
Addressing the need for both detailed component optimization and efficient system-level simulation—especially considering the limitations of full \acrshort{fem} for real-time control—a hybrid modeling approach was used.
This combines \acrfull{fea} with multi-body dynamics, employing reduced-order flexible bodies.
@@ -6565,11 +6554,11 @@ This combines \acrfull{fea} with multi-body dynamics, employing reduced-order fl
The theoretical foundations and implementation are presented in Section~\ref{sec:detail_fem_super_element}, where experimental validation was performed using an Amplified Piezoelectric Actuator.
The framework was then applied to optimize two critical active platform elements: the actuators (Section~\ref{sec:detail_fem_actuator}) and the flexible joints (Section~\ref{sec:detail_fem_joint}).
Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated.
-\subsection{Reduced order flexible bodies}
+\subsection{Reduced Order Flexible Bodies}
\label{sec:detail_fem_super_element}
Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models.
These components are traditionally analyzed using \acrshort{fea} software.
-However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models~\cite{hatch00_vibrat_matlab_ansys}.
+However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be integrated into multi-body models~\cite{hatch00_vibrat_matlab_ansys}.
This combined multibody-FEA modeling approach presents significant advantages, as it enables the accurate FE modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system~\cite{rankers98_machin}.
The investigation of this hybrid modeling approach is structured in three sections.
@@ -6591,7 +6580,7 @@ Subsequently, interface frames are defined at locations where the multi-body mod
These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames.
Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method~\cite{craig68_coupl_subst_dynam_analy} (also known as the ``fixed-interface method''), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics.
-This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF.
+This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100-DoFs.
The number of \acrshortpl{dof} in the reduced model is determined by~\eqref{eq:detail_fem_model_order} where \(n\) represents the number of defined frames and \(p\) denotes the number of additional modes to be modeled.
The outcome of this procedure is an \(m \times m\) set of reduced mass and stiffness matrices, \(m\) being the total retained number of \acrshortpl{dof}, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior.
@@ -6611,7 +6600,7 @@ The specific design of the \acrshort{apa} allows for the simultaneous modeling o
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa95ml_picture.png}
-\captionof{figure}{\label{fig:detail_fem_apa95ml_picture}Picture of the APA95ML}
+\captionof{figure}{\label{fig:detail_fem_apa95ml_picture}Picture of the APA95ML.}
\end{center}
\end{minipage}
\hfill
@@ -6622,11 +6611,12 @@ The specific design of the \acrshort{apa} allows for the simultaneous modeling o
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
-Nominal Stroke & \(100\,\mu\text{m}\)\\
+Nominal Stroke & \(100\,\upmu\text{m}\)\\
Blocked force & \(2100\,\text{N}\)\\
-Stiffness & \(21\,\text{N}/\mu\text{m}\)\\
+Stiffness & \(21\,\text{N}/\upmu\text{m}\)\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
@@ -6635,7 +6625,6 @@ The development of the \acrfull{fem} for the APA95ML required the knowledge of t
The finite element mesh, shown in Figure~\ref{fig:detail_fem_apa95ml_mesh}, was then generated.
\begin{table}[htbp]
-\caption{\label{tab:detail_fem_material_properties}Material properties used for FEA. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density}
\centering
\begin{tabularx}{0.55\linewidth}{Xccc}
\toprule
@@ -6645,6 +6634,8 @@ Stainless Steel & \(190\,\text{GPa}\) & \(0.31\) & \(7800\,\text{kg}/\text{m}^3\
Piezoelectric Ceramics (PZT) & \(49.5\,\text{GPa}\) & \(0.31\) & \(7800\,\text{kg}/\text{m}^3\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_fem_material_properties}Material properties used for FEA. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density.}
+
\end{table}
The definition of interface frames constitutes a critical aspect of the model preparation.
@@ -6666,7 +6657,7 @@ The modal reduction procedure was then executed, yielding the reduced mass and s
\end{center}
\subcaption{\label{fig:detail_fem_apa_model_schematic} }
\end{subfigure}
-\caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}).}
+\caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interfaces with the multi-body model are shown in (\subref{fig:detail_fem_apa_model_schematic}).}
\end{figure}
\paragraph{Super Element in the Multi-Body Model}
@@ -6698,28 +6689,28 @@ Unfortunately, it is difficult to know exactly which material is used for the pi
Yet, based on the available properties of the stacks in the data-sheet (summarized in Table~\ref{tab:detail_fem_stack_parameters}), the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
\begin{table}[htbp]
-\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters}
\centering
\begin{tabularx}{0.3\linewidth}{Xc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
-Nominal Stroke & \(20\,\mu\text{m}\)\\
+Nominal Stroke & \(20\,\upmu\text{m}\)\\
Blocked force & \(4700\,\text{N}\)\\
-Stiffness & \(235\,\text{N}/\mu\text{m}\)\\
+Stiffness & \(235\,\text{N}/\upmu\text{m}\)\\
Voltage Range & \(-20/150\,\text{V}\)\\
-Capacitance & \(4.4\,\mu\text{F}\)\\
+Capacitance & \(4.4\,\upmu\text{F}\)\\
Length & \(20\,\text{mm}\)\\
Stack Area & \(10\times 10\,\text{mm}^2\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters.}
+
\end{table}
The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table~\ref{tab:detail_fem_piezo_properties}.
-From these parameters, \(g_s = 5.1\,\text{V}/\mu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained.
+From these parameters, \(g_s = 5.1\,\text{V}/\upmu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained.
\begin{table}[htbp]
-\caption{\label{tab:detail_fem_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities}
\centering
\begin{tabularx}{0.8\linewidth}{ccX}
\toprule
@@ -6734,6 +6725,8 @@ From these parameters, \(g_s = 5.1\,\text{V}/\mu\text{m}\) and \(g_a = 26\,\text
\(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_fem_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuator sensitivities.}
+
\end{table}
\paragraph{Identification of the APA Characteristics}
@@ -6741,22 +6734,22 @@ Initial validation of the \acrlong{fem} and its integration as a reduced-order f
The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure~\ref{fig:detail_fem_apa95ml_compliance}), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames.
The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML.
-A value of \(23\,\text{N}/\mu\text{m}\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,\text{N}/\mu\text{m}\).
+A value of \(23\,\text{N}/\upmu\text{m}\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,\text{N}/\upmu\text{m}\).
The multi-body model predicted a resonant frequency under block-free conditions of \(\approx 2\,\text{kHz}\) (Figure~\ref{fig:detail_fem_apa95ml_compliance}), which is in agreement with the nominal specification.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_compliance.png}
-\caption{\label{fig:detail_fem_apa95ml_compliance}Estimated compliance of the APA95ML}
+\caption{\label{fig:detail_fem_apa95ml_compliance}Estimated axial compliance of the APA95ML.}
\end{figure}
In order to estimate the stroke of the APA95ML, the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, was first determined.
This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of \(1.5\) was derived.
-The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) relative to their length (here equal to \(20\,\text{mm}\)), produce an individual nominal stroke of \(20\,\mu\text{m}\) (see data-sheet of the piezoelectric stacks on Table~\ref{tab:detail_fem_stack_parameters}, page~\pageref{tab:detail_fem_stack_parameters}).
-As three stacks are used, the horizontal displacement is \(60\,\mu\text{m}\).
-Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\mu\text{m}\) which falls within the manufacturer-specified range of \(80\,\mu\text{m}\) and \(120\,\mu\text{m}\).
+The piezoelectric stacks, exhibiting a typical strain response of \(0.1\,\%\) relative to their length (here equal to \(20\,\text{mm}\)), produce an individual nominal stroke of \(20\,\upmu\text{m}\) (see data-sheet of the piezoelectric stacks on Table~\ref{tab:detail_fem_stack_parameters}, page~\pageref{tab:detail_fem_stack_parameters}).
+As three stacks are used, the horizontal displacement is \(60\,\upmu\text{m}\).
+Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\upmu\text{m}\) which falls within the manufacturer-specified range of \(80\,\upmu\text{m}\) and \(120\,\upmu\text{m}\).
The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include \acrshort{fem} into multi-body model.
\subsubsection{Experimental Validation}
@@ -6765,16 +6758,16 @@ Further validation of the reduced-order flexible body methodology was undertaken
The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
The test bench illustrated in Figure~\ref{fig:detail_fem_apa95ml_bench_schematic} was used, which consists of a \(5.7\,\text{kg}\) granite suspended on top of the APA95ML.
-The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement \(y\).
+The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measure its vertical displacement \(y\).
A \acrfull{dac} was used to generate the control signal \(u\), which was subsequently conditioned through a voltage amplifier with a gain of \(20\), ultimately yielding the effective voltage \(V_a\) across the two piezoelectric stacks.
Measurement of the sensor stack voltage \(V_s\) was performed using an \acrshort{adc}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=\linewidth]{figs/detail_fem_apa95ml_bench_schematic.png}
-\caption{\label{fig:detail_fem_apa95ml_bench_schematic}Test bench used to validate ``reduced order solid bodies'' using an APA95ML.}
+\caption{\label{fig:detail_fem_apa95ml_bench_schematic}Test bench used to validate the presented modeling strategy.}
\end{figure}
-\paragraph{Comparison of the dynamics}
+\paragraph{Comparison of the Dynamics}
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
The identification procedure required careful choice of the excitation signal~\cite[, chap. 5]{pintelon12_system_ident}.
@@ -6804,7 +6797,7 @@ Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:detail_fem_apa95ml_comp_plant}Comparison of the measured frequency response functions and the finite element model of the APA95ML. Both for the dynamics from \(V_a\) to \(y\) (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})}
+\caption{\label{fig:detail_fem_apa95ml_comp_plant}Comparison of the measured frequency response functions and the finite element model of the APA95ML. Both for the dynamics from \(V_a\) to \(y\) (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa95ml_comp_plant_sensor}).}
\end{figure}
\paragraph{Integral Force Feedback with APA}
@@ -6828,7 +6821,7 @@ The close agreement between experimental measurements and theoretical prediction
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_iff_root_locus.png}
\end{center}
-\subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot}
+\subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root locus plot}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
@@ -6836,7 +6829,7 @@ The close agreement between experimental measurements and theoretical prediction
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants}
\end{subfigure}
-\caption{\label{fig:detail_fem_apa95ml_iff_results}Results using Integral Force Feedback with the APA95ML. Closed-loop poles as a function of the controller gain \(g\) are predicted by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.}
+\caption{\label{fig:detail_fem_apa95ml_iff_results}Results using Integral Force Feedback with the APA95ML. Closed-loop poles as a function of the controller gain \(g\) are predicted by the root locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.}
\end{figure}
\subsubsection{Conclusion}
@@ -6853,13 +6846,13 @@ The actuator selection process was driven by several critical requirements deriv
A primary consideration is the actuator stiffness, which significantly impacts system dynamics through multiple mechanisms.
The spindle rotation induces gyroscopic effects that modify plant dynamics and increase coupling, necessitating sufficient stiffness.
Conversely, the actuator stiffness must be carefully limited to ensure the active platform's suspension modes remain below the problematic modes of the micro-station to limit the coupling between the two structures.
-These competing requirements suggest an optimal stiffness of approximately \(1\,\text{N}/\mu\text{m}\).
+These competing requirements suggest an optimal stiffness of approximately \(1\,\text{N}/\upmu\text{m}\).
Additional specifications arise from the control strategy and physical constraints.
The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator.
-The system's geometric constraints limit the actuator height to 50mm, given the active platform's maximum height of 95mm and the presence of flexible joints at each strut extremity.
+The system's geometric constraints limit the actuator height to \(50\,\text{mm}\), given the active platform's maximum height of \(95\,\text{mm}\) and the presence of flexible joints at each strut extremity.
Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility.
-An actuator stroke of \(\approx 200\,\mu\text{m}\) is therefore required.
+An actuator stroke of \(\approx 200\,\upmu\text{m}\) is therefore required.
Three actuator technologies were evaluated (examples of such actuators are shown in Figure~\ref{fig:detail_fem_actuator_pictures}): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity.
@@ -6886,13 +6879,13 @@ Variable reluctance actuators were not considered despite their superior efficie
\caption{\label{fig:detail_fem_actuator_pictures}Example of actuators considered for the active platform. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).}
\end{figure}
-Voice coil actuators (shown in Figure~\ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness (\(\approx 1\,\text{N}/\mu\text{m}\)), would require forces in the order of \(200\,\text{N}\) to achieve the specified \(200\,\mu\text{m}\) displacement.
+Voice coil actuators (shown in Figure~\ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness (\(\approx 1\,\text{N}/\upmu\text{m}\)), would require forces in the order of \(200\,\text{N}\) to achieve the specified \(200\,\upmu\text{m}\) displacement.
While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability.
Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions.
Conventional piezoelectric stack actuators (shown in Figure~\ref{fig:detail_fem_piezo_picture}) present two significant limitations for the current application.
-Their stroke is inherently limited to approximately \(0.1\,\%\) of their length, meaning that even with the maximum allowable height of \(50\,\text{mm}\), the achievable stroke would only be \(50\,\mu\text{m}\), insufficient for the application.
-Additionally, their extremely high stiffness, typically around \(100\,\text{N}/\mu\text{m}\), exceeds the desired specifications by two orders of magnitude.
+Their stroke is inherently limited to approximately \(0.1\,\%\) of their length, meaning that even with the maximum allowable height of \(50\,\text{mm}\), the achievable stroke would only be \(50\,\upmu\text{m}\), insufficient for the application.
+Additionally, their extremely high stiffness, typically around \(100\,\text{N}/\upmu\text{m}\), exceeds the desired specifications by two orders of magnitude.
Amplified Piezoelectric Actuators emerged as the optimal solution by addressing these limitations through a specific mechanical design.
The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile.
@@ -6905,19 +6898,20 @@ This selection was further reinforced by previous experience with \acrshortpl{ap
The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses.
\begin{table}[htbp]
-\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the active platform}
\centering
\begin{tabularx}{0.9\linewidth}{Xccccc}
\toprule
\textbf{Specification} & APA150M & \textbf{APA300ML} & APA400MML & FPA-0500E-P & FPA-0300E-S\\
\midrule
-Stroke \(> 200\,\mu\text{m}\) & 187 & 304 & 368 & 432 & 240\\
-Stiffness \(\approx 1\,\text{N}/\mu\text{m}\) & 0.7 & 1.8 & 0.55 & 0.87 & 0.58\\
-Resolution \(< 2\,\text{nm}\) & 2 & 3 & 4 & & \\
+Stroke \(> 200\,\upmu\text{m}\) & 187 & 304 & 368 & 432 & 240\\
+Stiffness \(\approx 1\,\text{N}/\upmu\text{m}\) & 0.7 & 1.8 & 0.55 & 0.87 & 0.58\\
+Resolution \(< 2\,\text{nm}\) & 2 & 3 & 4 & n/a & n/a\\
Blocked Force \(> 100\,\text{N}\) & 127 & 546 & 201 & 376 & 139\\
Height \(< 50\,\text{mm}\) & 22 & 30 & 24 & 27 & 16\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the active platform.}
+
\end{table}
\subsubsection{APA300ML - Reduced Order Flexible Body}
\label{ssec:detail_fem_actuator_apa300ml}
@@ -6945,21 +6939,21 @@ While this high order provides excellent accuracy for validation purposes, it pr
\end{figure}
The sensor and actuator ``constants'' (\(g_s\) and \(g_a\)) derived in Section~\ref{ssec:detail_fem_super_element_example} for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks.
-\subsubsection{Simpler 2DoF Model of the APA300ML}
+\subsubsection{Simpler 2-DoFs Model of the APA300ML}
\label{ssec:detail_fem_actuator_apa300ml_2dof}
To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from~\cite{souleille18_concep_activ_mount_space_applic}, was developed.
This model, illustrated in Figure~\ref{fig:detail_fem_apa_2dof_model}, comprises three components.
The mechanical shell is characterized by its axial stiffness \(k_1\) and damping \(c_1\).
-The actuator is modelled with stiffness \(k_a\) and damping \(c_a\), incorporating a force source \(f\).
+The actuator is modeled with stiffness \(k_a\) and damping \(c_a\), incorporating a force source \(f\).
This force is related to the applied voltage \(V_a\) through the actuator constant \(g_a\).
The sensor stack is modeled with stiffness \(k_e\) and damping \(c_e\), with its deformation \(d_L\) being converted to the output voltage \(V_s\) through the sensor sensitivity \(g_s\).
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_apa_2dof_model.png}
-\caption{\label{fig:detail_fem_apa_2dof_model}Schematic of the 2DoF model of the Amplified Piezoelectric Actuator}
+\caption{\label{fig:detail_fem_apa_2dof_model}Schematic of the 2-DoFs model of the Amplified Piezoelectric Actuator.}
\end{figure}
While providing computational efficiency, this simplified model has inherent limitations.
@@ -6970,7 +6964,7 @@ Nevertheless, the model's primary advantage lies in its simplicity, adding only
The model requires tuning of 8 parameters (\(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\), and \(g_a\)) to match the dynamics extracted from the \acrshort{fea}.
The shell parameters \(k_1\) and \(c_1\) were determined first through analysis of the zero in the \(V_a\) to \(V_s\) transfer function.
-The physical interpretation of this zero can be understood through Root Locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros.
+The physical interpretation of this zero can be understood through root locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros.
The open-loop zero therefore corresponds to the poles of the system with a theoretical infinite-gain controller that ensures zero force in the sensor stack.
This condition effectively represents the dynamics of an \acrshort{apa} without the force sensor stack (i.e. an \acrshort{apa} with only the shell).
This physical interpretation enables straightforward parameter tuning: \(k_1\) determines the frequency of the zero, while \(c_1\) defines its damping characteristic.
@@ -6983,22 +6977,23 @@ The resulting parameters, listed in Table~\ref{tab:detail_fem_apa300ml_2dof_para
While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range.
\begin{table}[htbp]
-\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
\centering
\begin{tabularx}{0.25\linewidth}{cc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
-\(k_1\) & \(0.30\,\text{N}/\mu\text{m}\)\\
-\(k_e\) & \(4.3\,\text{N}/\mu\text{m}\)\\
-\(k_a\) & \(2.15\,\text{N}/\mu\text{m}\)\\
+\(k_1\) & \(0.30\,\text{N}/\upmu\text{m}\)\\
+\(k_e\) & \(4.3\,\text{N}/\upmu\text{m}\)\\
+\(k_a\) & \(2.15\,\text{N}/\upmu\text{m}\)\\
\(c_1\) & \(18\,\text{Ns/m}\)\\
\(c_e\) & \(0.7\,\text{Ns/m}\)\\
\(c_a\) & \(0.35\,\text{Ns/m}\)\\
\(g_a\) & \(2.7\,\text{N}/V\)\\
-\(g_s\) & \(0.53\,\text{V}/\mu\text{m}\)\\
+\(g_s\) & \(0.53\,\text{V}/\upmu\text{m}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2-DoFs APA300ML model.}
+
\end{table}
\begin{figure}[htbp]
@@ -7014,7 +7009,7 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}from $V_a$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from \(V_a\) to \(d_i\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor})}
+\caption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2-DoFs model. Both for the dynamics from \(V_a\) to \(d_i\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}).}
\end{figure}
\subsubsection{Electrical characteristics of the APA}
\label{ssec:detail_fem_actuator_apa300ml_electrical}
@@ -7029,7 +7024,7 @@ The developed models of the \acrshort{apa} do not represent such behavior, but a
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_effect_electrical_boundaries.png}
-\caption{\label{fig:detail_fem_apa95ml_effect_electrical_boundaries}Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency}
+\caption{\label{fig:detail_fem_apa95ml_effect_electrical_boundaries}Effect of the electrical boundaries of the force sensor stack on the APA95ML resonance frequency.}
\end{figure}
However, the electrical characteristics of the \acrshort{apa} remain crucial for instrumentation design.
@@ -7038,16 +7033,16 @@ These aspects will be addressed in the instrumentation chapter.
\subsubsection{Validation with the Active Platform}
\label{ssec:detail_fem_actuator_apa300ml_validation}
-The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with \acrshort{apa} modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full \acrshort{fem} implementation.
+The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with \acrshort{apa} modeled as flexible bodies, and to validate the simplified 2-DoFs model through comparative analysis with the full \acrshort{fem} implementation.
The dynamics predicted using the flexible body model align well with the design requirements established during the conceptual phase.
The dynamics from \(\bm{u}\) to \(\bm{V}_s\) exhibits the desired alternating pole-zero pattern (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}), a critical characteristic for implementing robust decentralized Integral Force Feedback.
Additionally, the model predicts no problematic high-frequency modes in the dynamics from \(\bm{u}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}), maintaining consistency with earlier conceptual simulations.
These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator.
-Comparative analysis between the high-order \acrshort{fem} implementation and the simplified 2DoF model (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_plants}) demonstrates remarkable agreement in the frequency range of interest.
+Comparative analysis between the high-order \acrshort{fem} implementation and the simplified 2-DoFs model (Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_plants}) demonstrates remarkable agreement in the frequency range of interest.
This validates the use of the simplified model for time-domain simulations.
-The reduction in model order is substantial: while the \acrshort{fem} implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete active platform.
+The reduction in model order is substantial: while the \acrshort{fem} implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2-DoFs model requires only 24 states for the complete active platform.
These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the active platform.
@@ -7064,7 +7059,7 @@ These results validate both the selection of the APA300ML and the effectiveness
\end{center}
\subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
-\caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a active platform having actuators modelled a 2DoF system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).}
+\caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between an active platform having the actuators modeled with FEM and an active platform having actuators modeled as 2-DoFs system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).}
\end{figure}
\subsection{Flexible Joint Design}
\label{sec:detail_fem_joint}
@@ -7092,7 +7087,7 @@ For design simplicity and component standardization, identical joints are employ
\end{center}
\subcaption{\label{fig:detail_fem_joints_wire}}
\end{subfigure}
-\caption{\label{fig:detail_fem_joints_examples}Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical ``universal'' flexible joint used in~\cite{preumont07_six_axis_singl_stage_activ}. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in~\cite{yang19_dynam_model_decoup_contr_flexib}. (\subref{fig:detail_fem_joints_wire}) ``Thin'' flexible joints having ``notch curves'' are also used~\cite{du14_piezo_actuat_high_precis_flexib}.}
+\caption{\label{fig:detail_fem_joints_examples}Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical ``universal'' flexible joint used in \cite{preumont07_six_axis_singl_stage_activ}. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitly specified as done in \cite{yang19_dynam_model_decoup_contr_flexib}. (\subref{fig:detail_fem_joints_wire}) ``Thin'' flexible joints having ``notch curves'' \cite{du14_piezo_actuat_high_precis_flexib}.}
\end{figure}
While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other \acrshortpl{dof}, practical implementations exhibit parasitic stiffness that can impact control performance~\cite{mcinroy02_model_desig_flexur_joint_stewar}.
@@ -7106,7 +7101,7 @@ The validation process, detailed in Section~\ref{ssec:detail_fem_joint_validatio
The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction~\cite{mcinroy02_model_desig_flexur_joint_stewar} and can affect system dynamics.
-To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of \(1\,\text{N}/\mu\text{m}\)) without parallel stiffness to the force sensors.
+To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1-DoF actuators (stiffness of \(1\,\text{N}/\upmu\text{m}\)) without parallel stiffness to the force sensors.
Flexible joint bending stiffness was varied from 0 (ideal case) to \(500\,\text{Nm}/\text{rad}\).
Analysis of the plant dynamics reveals two significant effects.
@@ -7119,7 +7114,7 @@ For the force sensor plant, bending stiffness introduces complex conjugate zeros
This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure~\ref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}).
However, this time the parallel stiffness does not comes from the considered strut, but from the bending stiffness of the flexible joints of the other five struts.
This characteristic impacts the achievable damping using decentralized Integral Force Feedback~\cite{preumont07_six_axis_singl_stage_activ}.
-This is confirmed by the Root Locus plot in Figure~\ref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}.
+This is confirmed by the root locus plot in Figure~\ref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}.
This effect becomes less significant when using the selected APA300ML actuators (Figure~\ref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness.
A parallel analysis of torsional stiffness revealed similar effects, though these proved less critical for system performance.
@@ -7137,7 +7132,7 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
-\caption{\label{fig:detail_fem_joints_bending_stiffness_plants}Effect of bending stiffness of the flexible joints on the plant dynamics. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_bending_stiffness_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_bending_stiffness_iff_plant})}
+\caption{\label{fig:detail_fem_joints_bending_stiffness_plants}Effect of bending stiffness of the flexible joints on the plant dynamics. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_bending_stiffness_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_bending_stiffness_iff_plant}).}
\end{figure}
\begin{figure}[h!tbp]
@@ -7145,7 +7140,7 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_bending_stiffness_iff_locus_1dof.png}
\end{center}
-\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1DoF actuators}
+\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1-DoF actuators}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
@@ -7153,7 +7148,7 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}APA300ML actuators}
\end{subfigure}
-\caption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus}Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. When having an actuator modelled as 1DoF without parallel stiffness to the force sensor (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2DoF model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml})}
+\caption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus}Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. For 1-DoF actuators (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2-DoFs model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}).}
\end{figure}
\subsubsection{Axial Stiffness}
\label{ssec:detail_fem_joint_axial}
@@ -7163,7 +7158,7 @@ As explained in~\cite[, chapter 6]{preumont18_vibrat_contr_activ_struc_fourt_edi
Therefore, determining the minimum acceptable axial stiffness that maintains active platform performance becomes crucial.
The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects.
-A parametric study was conducted by varying the axial stiffness from \(1\,\text{N}/\mu\text{m}\) (matching actuator stiffness) to \(1000\,\text{N}/\mu\text{m}\) (approximating rigid behavior).
+A parametric study was conducted by varying the axial stiffness from \(1\,\text{N}/\upmu\text{m}\) (matching actuator stiffness) to \(1000\,\text{N}/\upmu\text{m}\) (approximating rigid behavior).
The resulting dynamics (Figure~\ref{fig:detail_fem_joints_axial_stiffness_plants}) reveal distinct effects on system dynamics.
The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both \acrshortpl{frf} (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_plant}) and root locus analysis (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_locus}).
@@ -7176,7 +7171,7 @@ First, the system exhibits strong coupling between control channels, making dece
Second, control authority diminishes significantly near the resonant frequencies.
These effects fundamentally limit achievable control bandwidth, making high axial stiffness essential for system performance.
-Based on this analysis, an axial stiffness specification of \(100\,\text{N}/\mu\text{m}\) was established for the active platform joints.
+Based on this analysis, an axial stiffness specification of \(100\,\text{N}/\upmu\text{m}\) was established for the active platform joints.
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
@@ -7191,7 +7186,7 @@ Based on this analysis, an axial stiffness specification of \(100\,\text{N}/\mu\
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
-\caption{\label{fig:detail_fem_joints_axial_stiffness_plants}Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant})}
+\caption{\label{fig:detail_fem_joints_axial_stiffness_plants}Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant}).}
\end{figure}
\begin{figure}[h!tbp]
@@ -7199,7 +7194,7 @@ Based on this analysis, an axial stiffness specification of \(100\,\text{N}/\mu\
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_axial_stiffness_iff_locus.png}
\end{center}
-\subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root Locus}
+\subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root locus}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
@@ -7207,9 +7202,9 @@ Based on this analysis, an axial stiffness specification of \(100\,\text{N}/\mu\
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}RGA number}
\end{subfigure}
-\caption{\label{fig:detail_fem_joints_axial_stiffness_iff_results}Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant})}
+\caption{\label{fig:detail_fem_joints_axial_stiffness_iff_results}Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}).}
\end{figure}
-\subsubsection{Specifications and Design flexible joints}
+\subsubsection{Specifications and Design of Flexible Joints}
\label{ssec:detail_fem_joint_specs}
The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics.
@@ -7217,19 +7212,20 @@ Critical specifications include sufficient bending stroke to ensure long-term op
Based on the dynamic analysis presented in previous sections, quantitative specifications were established and are summarized in Table~\ref{tab:detail_fem_joints_specs}.
\begin{table}[htbp]
-\caption{\label{tab:detail_fem_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
\centering
\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
& \textbf{Specification} & \textbf{FEM}\\
\midrule
-Axial Stiffness \(k_a\) & \(> 100\,\text{N}/\mu\text{m}\) & 94\\
-Shear Stiffness \(k_s\) & \(> 1\,\text{N}/\mu\text{m}\) & 13\\
+Axial Stiffness \(k_a\) & \(> 100\,\text{N}/\upmu\text{m}\) & 94\\
+Shear Stiffness \(k_s\) & \(> 1\,\text{N}/\upmu\text{m}\) & 13\\
Bending Stiffness \(k_f\) & \(< 100\,\text{Nm}/\text{rad}\) & 5\\
Torsion Stiffness \(k_t\) & \(< 500\,\text{Nm}/\text{rad}\) & 260\\
Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_fem_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model.}
+
\end{table}
Among various possible flexible joint architectures, the design shown in Figure~\ref{fig:detail_fem_joints_design} was selected for three key advantages.
@@ -7239,7 +7235,7 @@ Third, the architecture inherently provides high axial stiffness while maintaini
The joint geometry was optimized through parametric \acrshort{fea}.
The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits.
-The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through \acrshort{fea} and summarized in Table~\ref{tab:detail_fem_joints_specs}.
+The final design, featuring a neck dimension of \(0.25\,\text{mm}\), achieves mechanical properties closely matching the target specifications, as verified through \acrshort{fea} and summarized in Table~\ref{tab:detail_fem_joints_specs}.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@@ -7266,7 +7262,7 @@ The computed transfer functions from actuator forces to both force sensor measur
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_fem_joints_frames.png}
-\caption{\label{fig:detail_fem_joints_frames}Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames \(\{F\}\) and \(\{M\}\) both located at the center of the rotation.}
+\caption{\label{fig:detail_fem_joints_frames}Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames \(\{F\}\) and \(\{M\}\) both located at the center of rotation.}
\end{figure}
While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order.
@@ -7290,7 +7286,7 @@ While additional \acrshortpl{dof} could potentially capture more dynamic feature
\end{center}
\subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
-\caption{\label{fig:detail_fem_joints_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a active platform having bottom joint modelled by bending stiffness \(k_f\) and axial stiffness \(k_a\) and top joints modelled by bending stiffness \(k_f\), torsion stiffness \(k_t\) and axial stiffness \(k_a\). Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}).}
+\caption{\label{fig:detail_fem_joints_fem_vs_perfect_plants}Comparison of the dynamics obtained between an active platform including joints modeled with FEM and an active platform having 2-DoFs bottom joints and 3-DoFs top joints. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}).}
\end{figure}
\subsection*{Conclusion}
\label{sec:detail_fem_conclusion}
@@ -7354,7 +7350,7 @@ From the literature, three principal approaches for combining sensors have been
\end{center}
\subcaption{\label{fig:detail_control_sensor_arch_sensor_fusion}Sensor Fusion}
\end{subfigure}
-\caption{\label{fig:detail_control_control_multiple_sensors}Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control})}
+\caption{\label{fig:detail_control_control_multiple_sensors}Different control architectures combining multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}), Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}) and Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}).}
\end{figure}
The \acrshort{haclac} approach employs a dual-loop control strategy in which two control loops are using different sensors for distinct purposes (Figure~\ref{fig:detail_control_sensor_arch_hac_lac}).
@@ -7467,13 +7463,13 @@ The sensor dynamics estimate \(\hat{G}_i(s)\) may be a simple gain or a more com
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_sensor_model.png}
\end{center}
-\subcaption{\label{fig:detail_control_sensor_model}Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$}
+\subcaption{\label{fig:detail_control_sensor_model}Model with noise $n_i$ and acrshort:lti transfer function $G_i(s)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_sensor_model_calibrated.png}
\end{center}
-\subcaption{\label{fig:detail_control_sensor_model_calibrated}Normalized sensors using the inverse of an estimate $\hat{G}}
+\subcaption{\label{fig:detail_control_sensor_model_calibrated}Normalized sensor using the inverse of an estimate $\hat{G}}
\end{subfigure}
\caption{\label{fig:detail_control_sensor_models}Sensor models with and without normalization.}
\end{figure}
@@ -7531,7 +7527,7 @@ Therefore, by appropriately shaping the norm of the complementary filters, the n
\paragraph{Sensor Fusion Robustness}
In practical systems, sensor normalization is rarely perfect, and condition~\eqref{eq:detail_control_sensor_perfect_dynamics} is not fully satisfied.
-To analyze such imperfections, a multiplicative input uncertainty is incorporated into the sensor dynamics (Figure~\ref{fig:detail_control_sensor_model_uncertainty}).
+To analyze such imperfections, a multiplicative input uncertainty is included into the sensor dynamics (Figure~\ref{fig:detail_control_sensor_model_uncertainty}).
The nominal model is the estimated model used for normalization \(\hat{G}_i(s)\), \(\Delta_i(s)\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(w_i(s)\) is a weighting transfer function representing the magnitude of uncertainty.
Since the nominal sensor dynamics is taken as the normalized filter, the normalized sensor model can be further simplified as shown in Figure~\ref{fig:detail_control_sensor_model_uncertainty_simplified}.
@@ -7549,10 +7545,10 @@ Since the nominal sensor dynamics is taken as the normalized filter, the normali
\end{center}
\subcaption{\label{fig:detail_control_sensor_model_uncertainty_simplified}Simplified normalized sensor model}
\end{subfigure}
-\caption{\label{fig:detail_control_sensor_models_uncertainty}Sensor models with dynamical uncertainty}
+\caption{\label{fig:detail_control_sensor_models_uncertainty}Sensor models with dynamical uncertainty.}
\end{figure}
-The sensor fusion architecture incorporating sensor models with dynamical uncertainty is illustrated in Figure~\ref{fig:detail_control_sensor_fusion_dynamic_uncertainty}.
+The sensor fusion architecture including sensor models with dynamical uncertainty is illustrated in Figure~\ref{fig:detail_control_sensor_fusion_dynamic_uncertainty}.
The super sensor dynamics~\eqref{eq:detail_control_sensor_super_sensor_dyn_uncertainty} is no longer unity but depends on the sensor dynamical uncertainty weights \(w_i(s)\) and the complementary filters \(H_i(s)\).
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on \(1\) with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (Figure~\ref{fig:detail_control_sensor_uncertainty_set_super_sensor}).
@@ -7571,19 +7567,19 @@ The dynamical uncertainty of the super sensor can be graphically represented in
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_sensor_uncertainty_set_super_sensor.png}
\end{center}
-\subcaption{\label{fig:detail_control_sensor_uncertainty_set_super_sensor}Uncertainty regions}
+\subcaption{\label{fig:detail_control_sensor_uncertainty_set_super_sensor}Uncertainty region}
\end{subfigure}
-\caption{\label{fig:detail_control_sensor_uncertainty}Sensor fusion architecture with sensor dynamics uncertainty (\subref{fig:detail_control_sensor_fusion_dynamic_uncertainty}). Uncertainty region (\subref{fig:detail_control_sensor_uncertainty_set_super_sensor}) of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency \(\omega\) is here omitted.}
+\caption{\label{fig:detail_control_sensor_uncertainty}Sensor fusion architecture with sensor dynamics uncertainty (\subref{fig:detail_control_sensor_fusion_dynamic_uncertainty}). Uncertainty region (\subref{fig:detail_control_sensor_uncertainty_set_super_sensor}) of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The uncertainty region is function of frequency, which is omitted.}
\end{figure}
-The super sensor dynamical uncertainty, and consequently the robustness of the fusion, clearly depends on the complementary filters' norm.
+The super sensor dynamical uncertainty, and consequently the robustness of the fusion clearly depends on the complementary filters' norm.
As it is generally desired to limit the dynamical uncertainty of the super sensor, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
\subsubsection{Complementary Filters Shaping}
\label{ssec:detail_control_sensor_hinf_method}
As established in Section~\ref{ssec:detail_control_sensor_fusion_requirements}, the super sensor's noise characteristics and robustness are directly dependent on the complementary filters' norm.
A synthesis method enabling precise shaping of these norms would therefore offer substantial practical benefits.
This section develops such an approach by formulating the design objective as a standard \(\mathcal{H}_\infty\) optimization problem.
-The methodology for designing appropriate weighting functions (which specify desired complementary filter shape during synthesis) is examined in detail, and the efficacy of the proposed method is validated with a simple example.
+The methodology for designing appropriate weighting functions (which specify desired complementary filter shape during synthesis) is examined in detail, and the efficiency of the proposed method is validated with a simple example.
\paragraph{Synthesis Objective}
The primary objective is to shape the norms of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring they maintain their complementary property as defined in~\eqref{eq:detail_control_sensor_comp_filter}.
@@ -7618,7 +7614,7 @@ The synthesis objective can be expressed as a standard \(\mathcal{H}_\infty\) op
\end{center}
\subcaption{\label{fig:detail_control_sensor_h_infinity_robust_fusion_fb}Generalized plant with the synthesized filter}
\end{subfigure}
-\caption{\label{fig:detail_control_sensor_h_infinity_robust_fusion}Architecture for the \(\mathcal{H}_\infty\text{-synthesis}\) of complementary filters}
+\caption{\label{fig:detail_control_sensor_h_infinity_robust_fusion}Architecture for the \(\mathcal{H}_\infty\text{-synthesis}\) of complementary filters.}
\end{figure}
Applying standard \(\mathcal{H}_\infty\text{-synthesis}\) to the generalized plant \(P(s)\) is equivalent to finding a stable filter \(H_2(s)\) that, based on input \(v\), generates an output signal \(u\) such that the \(\mathcal{H}_\infty\) norm of the system shown in Figure~\ref{fig:detail_control_sensor_h_infinity_robust_fusion_fb} from \(w\) to \([z_1, \ z_2]\) does not exceed unity, as expressed in~\eqref{eq:detail_control_sensor_hinf_syn_obj}.
@@ -7659,7 +7655,7 @@ The typical magnitude response of a weighting function generated using~\eqref{eq
\begin{minipage}[]{0.45\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_control_sensor_weight_formula.png}
-\captionof{figure}{\label{fig:detail_control_sensor_weight_formula}Magnitude of a weighting function generated using~\eqref{eq:detail_control_sensor_weight_formula}, \(G_0 = 10^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
+\captionof{figure}{\label{fig:detail_control_sensor_weight_formula}Magnitude of a weighting function generated using \eqref{eq:detail_control_sensor_weight_formula}, \(G_0 = 10^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
\end{center}
\end{minipage}
\hfill
@@ -7672,7 +7668,7 @@ The typical magnitude response of a weighting function generated using~\eqref{eq
}\right)^n
\end{equation}
\end{minipage}
-\paragraph{Validation of the proposed synthesis method}
+\paragraph{Validation of the Proposed Synthesis Method}
The proposed methodology for designing complementary filters is now applied to a simple example.
Consider the design of two complementary filters \(H_1(s)\) and \(H_2(s)\) with the following requirements:
@@ -7692,7 +7688,7 @@ The inverse magnitudes of the designed weighting functions, which represent the
\footnotesize\sf
\begin{tabularx}{0.7\linewidth}{ccc}
\toprule
-Parameter & \(W_1(s)\) & \(W_2(s)\)\\
+ & \(W_1(s)\) & \(W_2(s)\)\\
\midrule
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_{\infty}\) & \(1000\) & \(0.1\)\\
@@ -7701,6 +7697,7 @@ Parameter & \(W_1(s)\) & \(W_2(s)\)\\
\(n\) & \(2\) & \(3\)\\
\bottomrule
\end{tabularx}
+
\end{center}
\captionof{table}{\label{tab:detail_control_sensor_weights_params}Parameters for \(W_1(s)\) and \(W_2(s)\)}
\end{minipage}
@@ -7708,7 +7705,7 @@ Parameter & \(W_1(s)\) & \(W_2(s)\)\\
\begin{minipage}[b]{0.52\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_control_sensor_hinf_filters_results.png}
-\captionof{figure}{\label{fig:detail_control_sensor_hinf_filters_results}Weights and obtained filters}
+\captionof{figure}{\label{fig:detail_control_sensor_hinf_filters_results}Weights and obtained filters.}
\end{center}
\end{minipage}
@@ -7717,10 +7714,10 @@ This yields the filter \(H_2(s)\) that minimizes the \(\mathcal{H}_\infty\) norm
The resulting \(\mathcal{H}_\infty\) norm is found to be close to unity, indicating successful synthesis: the norms of the complementary filters remain below the specified upper bounds.
This is confirmed by the Bode plots of the obtained complementary filters in Figure~\ref{fig:detail_control_sensor_hinf_filters_results}.
This straightforward example demonstrates that the proposed methodology for shaping complementary filters is both simple and effective.
-\subsubsection{Synthesis of a set of three complementary filters}
+\subsubsection{Synthesis of a set of Three Complementary Filters}
\label{ssec:detail_control_sensor_hinf_three_comp_filters}
-Certain applications necessitate the fusion of more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,carreira15_compl_filter_desig_three_frequen_bands}.
+Some applications require the fusion of more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,carreira15_compl_filter_desig_three_frequen_bands}.
At LIGO, for example, a super sensor is formed by merging three distinct sensors: a \acrshort{lvdt}, a seismometer, and a geophone~\cite{matichard15_seism_isolat_advan_ligo}.
For merging \(n>2\) sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure~\ref{fig:detail_control_sensor_fusion_three}.
@@ -7743,7 +7740,7 @@ This section presents a generalization of the proposed complementary filter synt
\end{center}
\subcaption{\label{fig:detail_control_sensor_fusion_three_parallel}Parallel fusion}
\end{subfigure}
-\caption{\label{fig:detail_control_sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged}
+\caption{\label{fig:detail_control_sensor_fusion_three}Sensor fusion architectures when more than two sensors are to be merged.}
\end{figure}
The synthesis objective is to compute a set of \(n\) stable transfer functions \([H_1(s),\ H_2(s),\ \dots,\ H_n(s)]\) that satisfy conditions~\eqref{eq:detail_control_sensor_hinf_cond_compl_gen} and \eqref{eq:detail_control_sensor_hinf_cond_perf_gen}.
@@ -7808,7 +7805,7 @@ Consider the generalized plant \(P_3(s)\) shown in Figure~\ref{fig:detail_contro
\end{center}
\subcaption{\label{fig:detail_control_sensor_three_complementary_filters_results}Weights and obtained filters}
\end{subfigure}
-\caption{\label{fig:detail_control_sensor_comp_filter_three_hinf}Architecture for the \(\mathcal{H}_\infty\text{-synthesis}\) of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results})}
+\caption{\label{fig:detail_control_sensor_comp_filter_three_hinf}Architecture for the \(\mathcal{H}_\infty\text{-synthesis}\) of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results}).}
\end{figure}
Standard \(\mathcal{H}_\infty\text{-synthesis}\) is performed on the generalized plant \(P_3(s)\).
@@ -7829,7 +7826,7 @@ Consequently, typical sensor fusion objectives can be effectively translated int
For the NASS, the \acrshort{haclac} strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune.
Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensors and external metrology) compares to the HAC-IFF approach in terms of performance and robustness.
-\subsection{Decoupling}
+\subsection{Decoupling Strategies for Parallel Manipulators}
\label{sec:detail_control_decoupling}
The control of parallel manipulators (and any \acrshort{mimo} system in general) typically involves a two-step approach: first decoupling the plant dynamics (using various strategies discussed in this section), followed by the application of \acrshort{siso} control for the decoupled plant (discussed in section~\ref{sec:detail_control_cf}).
@@ -7862,12 +7859,12 @@ A simplified parallel manipulator model is introduced in Section~\ref{ssec:detai
The decentralized plant (transfer functions from actuators to sensors integrated in the struts) is examined in Section~\ref{ssec:detail_control_decoupling_decentralized}.
Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section~\ref{ssec:detail_control_decoupling_jacobian}), modal decoupling (Section~\ref{ssec:detail_control_decoupling_modal}), and Singular Value Decomposition (SVD) decoupling (Section~\ref{ssec:detail_control_decoupling_svd}).
Finally, a comparative analysis with concluding observations is provided in Section~\ref{ssec:detail_control_decoupling_comp}.
-\subsubsection{Test Model}
+\subsubsection{3-DoFs Test Model}
\label{ssec:detail_control_decoupling_model}
-Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis.
+Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate the analysis.
The system illustrated in Figure~\ref{fig:detail_control_decoupling_model_test} is used for this purpose.
-It possesses three \acrshortpl{dof} and incorporates three parallel struts.
+It has three \acrshortpl{dof} and incorporates three parallel struts.
Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform.
Two reference frames are defined within this model: frame \(\{M\}\) with origin \(O_M\) at the \acrlong{com} of the solid body, and frame \(\{K\}\) with origin \(O_K\) at the \acrlong{cok} of the parallel manipulator.
@@ -7875,7 +7872,7 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin
\begin{minipage}[b]{0.60\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test.png}
-\captionof{figure}{\label{fig:detail_control_decoupling_model_test}Model used to compare decoupling strategies}
+\captionof{figure}{\label{fig:detail_control_decoupling_model_test}Model used to compare decoupling strategies.}
\end{center}
\end{minipage}
\hfill
@@ -7888,12 +7885,13 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin
\midrule
\(l_a\) & & \(0.5\,\text{m}\)\\
\(h_a\) & & \(0.2\,\text{m}\)\\
-\(k\) & Actuator stiffness & \(10\,\text{N}/\mu\text{m}\)\\
+\(k\) & Actuator stiffness & \(10\,\text{N}/\upmu\text{m}\)\\
\(c\) & Actuator damping & \(200\,\text{Ns/m}\)\\
\(m\) & Payload mass & \(40\,\text{kg}\)\\
\(I\) & Payload \(R_z\) inertia & \(5\,\text{kgm}^2\)\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
\end{minipage}
@@ -7949,7 +7947,7 @@ The matrices representing the payload inertia, actuator stiffness, and damping a
\end{equation}
The parameters employed for the subsequent analysis are summarized in Table~\ref{tab:detail_control_decoupling_test_model_params}, which includes values for geometric parameters (\(l_a\), \(h_a\)), mechanical properties (actuator stiffness \(k\) and damping \(c\)), and inertial characteristics (payload mass \(m\) and rotational inertia \(I\)).
-\subsubsection{Control in the frame of the struts}
+\subsubsection{Control in the Frame of the Struts}
\label{ssec:detail_control_decoupling_decentralized}
The dynamics in the frame of the struts are first examined.
@@ -7998,7 +7996,7 @@ The resulting plant (Figure~\ref{fig:detail_control_jacobian_decoupling_arch}) h
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_control_jacobian.png}
-\caption{\label{fig:detail_control_jacobian_decoupling_arch}Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{O\}}\) to \(\bm{\mathcal{X}}_{\{O\}}\)}
+\caption{\label{fig:detail_control_jacobian_decoupling_arch}Block diagram of the decoupling the plant in a frame \(\{O\}\) using Jacobian matrix \(\bm{J}_{\{O\}}\)}
\end{figure}
The transfer function from \(\bm{\mathcal{F}}_{\{O\}\) to \(\bm{\mathcal{X}}_{\{O\}}\), denoted \(\bm{G}_{\{O\}}(s)\) can be computed using~\eqref{eq:detail_control_decoupling_plant_jacobian}.
@@ -8009,7 +8007,7 @@ The transfer function from \(\bm{\mathcal{F}}_{\{O\}\) to \(\bm{\mathcal{X}}_{\{
The frame \(\{O\}\) can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice.
Two natural reference frames are particularly relevant: the \acrlong{com} and the \acrlong{cok}.
-\paragraph{Center Of Mass}
+\paragraph{Center of Mass}
When the decoupling frame is located at the \acrlong{com} (frame \(\{M\}\) in Figure~\ref{fig:detail_control_decoupling_model_test}), the Jacobian matrix and its inverse are expressed as in~\eqref{eq:detail_control_decoupling_jacobian_CoM_inverse}.
@@ -8064,7 +8062,7 @@ This phenomenon is illustrated in Figure~\ref{fig:detail_control_decoupling_mode
\end{subfigure}
\caption{\label{fig:detail_control_jacobian_decoupling_plant_CoM_results}Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}).}
\end{figure}
-\paragraph{Center Of Stiffness}
+\paragraph{Center of Stiffness}
When the decoupling frame is located at the \acrlong{cok}, the Jacobian matrix and its inverse are expressed as in~\eqref{eq:detail_control_decoupling_jacobian_CoK_inverse}.
@@ -8147,7 +8145,7 @@ The resulting decoupled system features diagonal elements each representing seco
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_modal.png}
-\caption{\label{fig:detail_control_decoupling_modal}Modal Decoupling Architecture}
+\caption{\label{fig:detail_control_decoupling_modal}Modal Decoupling Architecture.}
\end{figure}
Modal decoupling was then applied to the test model.
@@ -8193,7 +8191,7 @@ Each of these diagonal elements corresponds to a specific mode, as shown in Figu
\end{center}
\subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes}
\end{subfigure}
-\caption{\label{fig:detail_control_decoupling_modal_plant_modes}Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal})}
+\caption{\label{fig:detail_control_decoupling_modal_plant_modes}Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}). Decoupled elements can be used to invidiually address the modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal}).}
\end{figure}
\subsubsection{SVD Decoupling}
\label{ssec:detail_control_decoupling_svd}
@@ -8228,14 +8226,14 @@ These singular input and output matrices are then applied to decouple the system
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_decoupling_svd.png}
-\caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{\text{SVD}}\) using the Singular Value Decomposition}
+\caption{\label{fig:detail_control_decoupling_svd}Decoupled plant \(\bm{G}_{\text{SVD}}\) using the Singular Value Decomposition.}
\end{figure}
Implementation of SVD decoupling requires access to the system's \acrshort{frf}, at least in the vicinity of the desired decoupling frequency.
This information can be obtained either experimentally or derived from a model.
While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency.
Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping.
-\paragraph{Example}
+\paragraph{Test on the 3-DoFs model}
Plant decoupling using the Singular Value Decomposition was then applied on the test model.
A decoupling frequency of \(\SI{100}{Hz}\) was used.
@@ -8272,7 +8270,7 @@ Additionally, the diagonal terms manifest as second-order dynamic systems, facil
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_svd_plant.png}
-\caption{\label{fig:detail_control_decoupling_svd_plant}Plant dynamics \(\bm{G}_{\text{SVD}}(s)\) obtained after decoupling using Singular Value Decomposition}
+\caption{\label{fig:detail_control_decoupling_svd_plant}Plant dynamics \(\bm{G}_{\text{SVD}}(s)\) obtained after decoupling using Singular Value Decomposition.}
\end{figure}
As it was surprising to obtain such a good decoupling at all frequencies, a variant system with identical dynamics but different sensor configurations was examined.
@@ -8293,13 +8291,13 @@ Notably, the coupling demonstrates local minima near the decoupling frequency, c
\end{center}
\subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant}
\end{subfigure}
-\caption{\label{fig:detail_control_svd_decoupling_not_symmetrical}Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).}
+\caption{\label{fig:detail_control_svd_decoupling_not_symmetrical}SVD decoupling applied on the system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).}
\end{figure}
The exceptional performance of SVD decoupling on the plant with collocated sensors warrants further investigation.
This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure~\ref{fig:detail_control_decoupling_coupled_plant_bode}.
The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics~\cite{hovd97_svd_contr_contr}.
-\subsubsection{Comparison of decoupling strategies}
+\subsubsection{Comparison of Decoupling Strategies}
\label{ssec:detail_control_decoupling_comp}
While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table~\ref{tab:detail_control_decoupling_strategies_comp}.
@@ -8324,7 +8322,6 @@ Modal decoupling offers good decoupling across all frequencies, though its effec
SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point.
\begin{table}[htbp]
-\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies}
\centering
\scriptsize
\begin{tabularx}{\linewidth}{lXXX}
@@ -8347,9 +8344,11 @@ SVD decoupling can be implemented using measured data without requiring a model,
\midrule
\textbf{Pros} & Retain physical meaning of inputs / outputs. Controller acts on a meaningfully ``frame'' & Ability to target specific modes. Simple \(2^{nd}\) order diagonal plants & Good Decoupling near the crossover. Very General and requires no model\\
\midrule
-\textbf{Cons} & Good decoupling at all frequency can only be obtained for specific mechanical architecture & Relies on the accuracy of equation of motions. Robustness to unmodelled dynamics may be poor & Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor\\
+\textbf{Cons} & Good decoupling at all frequency can only be obtained for specific mechanical architecture & Relies on the accuracy of equation of motions. Robustness to unmodeled dynamics may be poor & Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies.}
+
\end{table}
\subsection{Closed-Loop Shaping using Complementary Filters}
\label{sec:detail_control_cf}
@@ -8399,12 +8398,12 @@ In this arrangement, the physical plant is controlled at low frequencies, while
\end{center}
\subcaption{\label{fig:detail_control_cf_arch_eq}Equivalent Architecture}
\end{subfigure}
-\caption{\label{fig:detail_control_cf_arch_and_eq}Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}). An equivalent architecture is shown in (\subref{fig:detail_control_cf_arch_eq}). The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).}
+\caption{\label{fig:detail_control_cf_arch_and_eq}Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}) and equivalent architecture (\subref{fig:detail_control_cf_arch_eq}). Signals are the reference input \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).}
\end{figure}
Although the control architecture shown in Figure~\ref{fig:detail_control_cf_arch} appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing \(k\), \(G\), and \(H_H\), all numerically implemented).
Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure~\ref{fig:detail_control_cf_arch_eq}.
-\paragraph{Asymptotic behavior}
+\paragraph{Asymptotic Behavior}
When considering the extreme case of very high values for \(k\), the effective controller \(K(s)\) converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in~\eqref{eq:detail_control_cf_high_k}.
@@ -8421,7 +8420,7 @@ The dynamics of this closed-loop system are described by equations~\eqref{eq:det
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_cf_arch_class.png}
-\caption{\label{fig:detail_control_cf_arch_class}Equivalent classical feedback control architecture}
+\caption{\label{fig:detail_control_cf_arch_class}Equivalent classical feedback control architecture.}
\end{figure}
\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf}
@@ -8442,7 +8441,7 @@ At frequencies where the model accurately represents the physical plant (\(G^{-1
The sensitivity transfer function equals the high-pass filter \(S = \frac{y}{dy} = H_H\), and the complementary sensitivity transfer function equals the low-pass filter \(T = \frac{y}{n} = H_L\).
Hence, when the plant model closely approximates the actual dynamics, the closed-loop transfer functions converge to the designed complementary filters, allowing direct translation of performance requirements into the design of the complementary.
-\subsubsection{Translating the performance requirements into the shape of the complementary filters}
+\subsubsection{Translating the Performance Requirements into the Shape of the Complementary Filters}
\label{ssec:detail_control_cf_trans_perf}
Performance specifications in a feedback system can usually be expressed as upper bounds on the magnitudes of closed-loop transfer functions such as the sensitivity and complementary sensitivity transfer functions~\cite{bibel92_guidel_h}.
The design of a controller \(K(s)\) to obtain the desired shape of these closed-loop transfer functions is known as closed-loop shaping.
@@ -8507,7 +8506,7 @@ The set of possible plants \(\Pi_i\) is described by~\eqref{eq:detail_control_cf
\end{center}
\subcaption{\label{fig:detail_control_cf_nyquist_uncertainty}Nyquist plot - Effect of multiplicative uncertainty}
\end{subfigure}
-\caption{\label{fig:detail_control_cf_input_uncertainty_nyquist}Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty})}
+\caption{\label{fig:detail_control_cf_input_uncertainty_nyquist}Input multiplicative uncertainty used to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is illustrated on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty}).}
\end{figure}
When considering input multiplicative uncertainty, \acrfull{rs} can be derived graphically from the Nyquist plot (illustrated in Figure~\ref{fig:detail_control_cf_nyquist_uncertainty}), yielding to~\eqref{eq:detail_control_cf_robust_stability_graphically}, as demonstrated in~\cite[, chapt. 7.5.1]{skogestad07_multiv_feedb_contr}.
@@ -8538,7 +8537,7 @@ Transforming this condition into constraints on the complementary filters yields
The \acrfull{rp} condition effectively combines both nominal performance~\eqref{eq:detail_control_cf_nominal_performance} and robust stability conditions~\eqref{eq:detail_control_cf_condition_robust_stability}.
If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2~\cite[, chapt. 7.6]{skogestad07_multiv_feedb_contr}.
Therefore, for \acrshort{siso} systems, ensuring robust stability and nominal performance is typically sufficient.
-\subsubsection{Complementary filter design}
+\subsubsection{Complementary Filter Design}
\label{ssec:detail_control_cf_analytical_complementary_filters}
As proposed in Section~\ref{sec:detail_control_sensor}, complementary filters can be shaped using standard \(\mathcal{H}_{\infty}\text{-synthesis}\) techniques.
@@ -8554,22 +8553,13 @@ For some applications, first-order complementary filters as shown in Equation~\e
\end{align}
\end{subequations}
-These filters can be transformed into the digital domain using the Bilinear transformation, resulting in the digital filter representations shown in Equation~\eqref{eq:detail_control_cf_1st_order_z}.
-
-\begin{subequations}\label{eq:detail_control_cf_1st_order_z}
- \begin{align}
- H_L(z^{-1}) &= \frac{T_s \omega_0 + T_s \omega_0 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} \\
- H_H(z^{-1}) &= \frac{2 - 2 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}}
- \end{align}
-\end{subequations}
-
A significant advantage of using analytical formulas for complementary filters is that key parameters such as \(\omega_0\) can be tuned in real-time, as illustrated in Figure~\ref{fig:detail_control_cf_arch_tunable_params}.
This real-time tunability allows rapid testing of different control bandwidths to evaluate performance and robustness characteristics.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_cf_arch_tunable_params.png}
-\caption{\label{fig:detail_control_cf_arch_tunable_params}Implemented digital complementary filters with parameter \(\omega_0\) that can be changed in real-time}
+\caption{\label{fig:detail_control_cf_arch_tunable_params}Implemented digital complementary filters with parameter \(\omega_0\) that can be changed in real-time.}
\end{figure}
For many practical applications, first order complementary filters are not sufficient.
@@ -8627,7 +8617,7 @@ The positioning stage itself is characterized by stiffness \(k\), internal dampi
The model of the plant \(G(s)\) from actuator force \(F\) to displacement \(y\) is described by Equation~\eqref{eq:detail_control_cf_test_plant_tf}.
\begin{equation}\label{eq:detail_control_cf_test_plant_tf}
- G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = 1\si{\N/\mu\m},\ c = 10^2\si{\N\per(\m\per\s)}
+ G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = \SI{1}{\N/\micro\m},\ c = \SI{100}{\N\per(\m\per\s)}
\end{equation}
The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics and payload dynamics.
@@ -8652,9 +8642,9 @@ Figure~\ref{fig:detail_control_cf_bode_plot_mech_sys} illustrates both the nomin
\end{center}
\subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set}
\end{subfigure}
-\caption{\label{fig:detail_control_cf_test_model_plant}Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function \(G(s)\) from \(F\) to \(y\) and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}).}
+\caption{\label{fig:detail_control_cf_test_model_plant}Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of \(G(s) = y/F\) and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}).}
\end{figure}
-\paragraph{Requirements and choice of complementary filters}
+\paragraph{Requirements and Choice of Complementary Filters}
As discussed in Section~\ref{ssec:detail_control_cf_trans_perf}, nominal performance requirements can be expressed as upper bounds on the shape of the complementary filters.
For this example, the requirements are:
@@ -8687,9 +8677,9 @@ There magnitudes are displayed in Figure~\ref{fig:detail_control_cf_specs_S_T},
\end{center}
\subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$}
\end{subfigure}
-\caption{\label{fig:detail_control_cf_specs_S_T_obtained_filters}Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}).}
+\caption{\label{fig:detail_control_cf_specs_S_T_obtained_filters}Performance requirements are compared with the complementary filters in (\subref{fig:detail_control_cf_specs_S_T}). The bode plot of the obtained controller is shown in (\subref{fig:detail_control_cf_bode_Kfb}).}
\end{figure}
-\paragraph{Controller analysis}
+\paragraph{Controller Analysis}
The controller to be implemented takes the form \(K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)\), where \(\tilde{G}^{-1}(s)\) represents the plant inverse, which must be both stable and proper.
To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation~\eqref{eq:detail_control_cf_test_plant_inverse}.
@@ -8705,7 +8695,7 @@ The loop gain reveals several important characteristics:
\item A notch at the plant resonance frequency (arising from the plant inverse)
\item A lead component near the control bandwidth of approximately \(20\,\text{Hz}\), enhancing stability margins
\end{itemize}
-\paragraph{Robustness and Performance analysis}
+\paragraph{Robustness and Performance Analysis}
Robust stability is assessed using the Nyquist plot shown in Figure~\ref{fig:detail_control_cf_nyquist_robustness}.
Even when considering all possible plants within the uncertainty set, the Nyquist plot remains sufficiently distant from the critical point \((-1,0)\), indicating robust stability with adequate margins.
@@ -8718,15 +8708,15 @@ It is shown that the sensitivity transfer function achieves the desired \(+2\) s
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_nyquist_robustness.png}
\end{center}
-\subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability}
+\subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust stability}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_robust_perf.png}
\end{center}
-\subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and Robust performance}
+\subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and robust performance}
\end{subfigure}
-\caption{\label{fig:detail_control_cf_simulation_results}Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf})}
+\caption{\label{fig:detail_control_cf_simulation_results}Validation of robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf}).}
\end{figure}
\subsubsection{Conclusion}
@@ -8767,7 +8757,7 @@ Figure~\ref{fig:detail_instrumentation_plant} illustrates the control diagram wi
The selection process follows a three-stage methodology.
First, dynamic error budgeting is performed in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting} to establish maximum acceptable noise specifications for each instrumentation component (\acrshort{adc}, \acrshort{dac}, and voltage amplifier).
-This analysis is based on the multi-body model with a 2DoF \acrshort{apa} model, focusing particularly on the vertical direction due to its more stringent requirements.
+This analysis is based on the multi-body model with a 2-DoFs \acrshort{apa} model, focusing particularly on the vertical direction due to its more stringent requirements.
From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived.
Section~\ref{sec:detail_instrumentation_choice} then presents the selection of appropriate components based on these noise specifications and additional requirements.
@@ -8786,12 +8776,12 @@ The measured noise characteristics are then incorporated into the multi-body mod
The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (\acrshort{adc}, \acrshort{dac}, and voltage amplifier) that would result in acceptable vibration levels in the system.
The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section~\ref{ssec:detail_instrumentation_cl_sensitivity}).
-This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs.
+This analysis is conducted using the multi-body model with a 2-DoFs Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs.
Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints.
From these transfer functions, the maximum acceptable \acrfull{asd} of the noise sources is derived (Section~\ref{ssec:detail_instrumentation_max_noise_specs}).
Since the voltage amplifier gain affects the amplification of \acrshort{dac} noise, an assumption of an amplifier gain of 20 was made.
-\subsubsection{Closed-Loop Sensitivity to Instrumentation Disturbances}
+\subsubsection{Closed-Loop Sensitivity to Instrumentation Noise}
\label{ssec:detail_instrumentation_cl_sensitivity}
Several key noise sources are considered in the analysis (Figure~\ref{fig:detail_instrumentation_plant}).
@@ -8804,9 +8794,9 @@ The transfer functions from these three noise sources (for one strut) to the ver
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_noise_sensitivities.png}
-\caption{\label{fig:detail_instrumentation_noise_sensitivities}Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy.}
+\caption{\label{fig:detail_instrumentation_noise_sensitivities}Transfer function from noise sources to vertical error, in closed-loop with the implemented HAC-LAC strategy.}
\end{figure}
-\subsubsection{Estimation of maximum instrumentation noise}
+\subsubsection{Estimation of Maximum Acceptable Instrumentation Noise}
\label{ssec:detail_instrumentation_max_noise_specs}
The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of \(100\,\text{nm}\), which corresponds to a maximum allowed vibration of \(15\,\text{nm RMS}\).
@@ -8823,11 +8813,11 @@ In order to derive specifications in terms of noise spectral density for each in
The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications.
-Based on this analysis, the obtained maximum noise levels are as follows: \acrshort{dac} maximum output noise \acrshort{asd} is established at \(14\,\mu\text{V}/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise \acrshort{asd} at \(280\,\mu\text{V}/\sqrt{\text{Hz}}\), and \acrshort{adc} maximum measurement noise \acrshort{asd} at \(11\,\mu\text{V}/\sqrt{\text{Hz}}\).
+Based on this analysis, the obtained maximum noise levels are as follows: \acrshort{dac} maximum output noise \acrshort{asd} is established at \(14\,\upmu\text{V}/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise \acrshort{asd} at \(280\,\upmu\text{V}/\sqrt{\text{Hz}}\), and \acrshort{adc} maximum measurement noise \acrshort{asd} at \(11\,\upmu\text{V}/\sqrt{\text{Hz}}\).
In terms of RMS noise, these translate to less than \(1\,\text{mV RMS}\) for the \acrshort{dac}, less than \(20\,\text{mV RMS}\) for the voltage amplifier, and less than \(0.8\,\text{mV RMS}\) for the \acrshort{adc}.
If the Amplitude Spectral Density of the noise of the \acrshort{adc}, \acrshort{dac}, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below \(15\,\text{nm RMS}\).
-\subsection{Choice of Instrumentation}
+\subsection{Selection of Instrumentation}
\label{sec:detail_instrumentation_choice}
\subsubsection{Piezoelectric Voltage Amplifier}
Several characteristics of piezoelectric voltage amplifiers must be considered for this application.
@@ -8849,11 +8839,11 @@ When combined with the piezoelectric load (represented as a capacitance \(C_p\))
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_amp_output_impedance.png}
-\caption{\label{fig:detail_instrumentation_amp_output_impedance}Electrical model of a voltage amplifier with output impedance \(R_0\) connected to a piezoelectric stack with capacitance \(C_p\)}
+\caption{\label{fig:detail_instrumentation_amp_output_impedance}Electrical model of an amplifier with output impedance \(R_0\) connected to a piezoelectric stack with capacitance \(C_p\).}
\end{figure}
Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases.
-For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to \(C_p = 8.8\,\mu\text{F}\).
+For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to \(C_p = 8.8\,\upmu\text{F}\).
If a small signal bandwidth of \(f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}\) is desired, the voltage amplifier output impedance should be less than \(R_0 = 3.6\,\Omega\).
\paragraph{Large signal Bandwidth}
@@ -8879,7 +8869,7 @@ As established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgetin
It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure~\ref{fig:detail_instrumentation_amp_output_impedance}.
Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement~\cite{spengen20_high_voltag_amplif}.
-For this application, the output noise must remain below \(20\,\text{mV RMS}\) with a load of \(8.8\,\mu\text{F}\) and a bandwidth exceeding \(5\,\text{kHz}\).
+For this application, the output noise must remain below \(20\,\text{mV RMS}\) with a load of \(8.8\,\upmu\text{F}\) and a bandwidth exceeding \(5\,\text{kHz}\).
\paragraph{Choice of voltage amplifier}
The specifications are summarized in Table~\ref{tab:detail_instrumentation_amp_choice}.
@@ -8896,7 +8886,6 @@ Note that for the WMA-200, the manufacturer proposed to remove the \(50\,\Omega\
The PD200 from PiezoDrive was ultimately selected because it meets all the requirements and is accompanied by clear documentation, particularly regarding noise characteristics and bandwidth specifications.
\begin{table}[htbp]
-\caption{\label{tab:detail_instrumentation_amp_choice}Specifications for the Voltage amplifier and considered commercial products}
\centering
\begin{tabularx}{0.8\linewidth}{Xcccc}
\toprule
@@ -8907,14 +8896,16 @@ Input Voltage Range: \(\pm 10\,\text{V}\) & \(\pm 10\,\text{V}\) & \(\pm8.75\,\
Output Voltage Range: \(-20/150\,\text{V}\) & \(-50/150\,\text{V}\) & \(\pm 175\,\text{V}\) & \(-20/150\,\text{V}\) & \(-30/130\,\text{V}\)\\
Gain \(>15\) & 20 & 20 & 20 & 10\\
Output Current \(> 300\,\text{mA}\) & \(900\,\text{mA}\) & \(150\,\text{mA}\) & \(360\,\text{mA}\) & \(215\,\text{mA}\)\\
-Slew Rate \(> 34\,\text{V/ms}\) & \(150\,\text{V}/\mu\text{s}\) & \(80\,\text{V}/\mu\text{s}\) & n/a & n/a\\
+Slew Rate \(> 34\,\text{V/ms}\) & \(150\,\text{V}/\upmu\text{s}\) & \(80\,\text{V}/\upmu\text{s}\) & n/a & n/a\\
Output noise \(< 20\,\text{mV RMS}\) & \(0.7\,\text{mV}\) & \(0.05\,\text{mV}\) & \(3.4\,\text{mV}\) & \(0.6\,\text{mV}\)\\
-(10uF load) & (\(10\,\mu\text{F}\) load) & (\(10\,\mu\text{F}\) load) & (n/a) & (n/a)\\
+(10uF load) & (\(10\,\upmu\text{F}\) load) & (\(10\,\upmu\text{F}\) load) & (n/a) & (n/a)\\
Small Signal Bandwidth \(> 5\,\text{kHz}\) & \(6.4\,\text{kHz}\) & \(300\,\text{Hz}\) & \(30\,\text{kHz}\) & n/a\\
-(\(10\,\mu\text{F}\) load) & (\(10\,\mu\text{F}\) load) & (\(10\,\mu\text{F}\) load) & (unloaded) & (n/a)\\
+(\(10\,\upmu\text{F}\) load) & (\(10\,\upmu\text{F}\) load) & (\(10\,\upmu\text{F}\) load) & (unloaded) & (n/a)\\
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_instrumentation_amp_choice}Specifications for the voltage amplifier and considered commercial products.}
+
\end{table}
\subsubsection{ADC and DAC}
Analog-to-digital converters and digital-to-analog converters play key roles in the system, serving as the interface between the digital RT controller and the analog physical plant.
@@ -8937,10 +8928,10 @@ Sigma-Delta \acrshortpl{adc} can provide excellent noise characteristics, high b
Typically, the latency can reach 20 times the sampling period~\cite[, chapt. 8.4]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
Consequently, while Sigma-Delta \acrshortpl{adc} are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor.
-For real-time control applications, \acrfull{sar} remain the predominant choice due to their single-sample latency characteristics.
+For real-time control applications, successive-approximation ADC remain the predominant choice due to their single-sample latency characteristics.
\paragraph{ADC Noise}
-Based on the dynamic error budget established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise \acrshort{asd} should not exceed \(11\,\mu V/\sqrt{\text{Hz}}\).
+Based on the dynamic error budget established in Section~\ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise \acrshort{asd} should not exceed \(11\,\upmu V/\sqrt{\text{Hz}}\).
\acrshortpl{adc} are subject to various noise sources.
Quantization noise, which results from the discrete nature of digital representation, is one of these sources.
@@ -8954,7 +8945,7 @@ Since the integral of this probability density function \(p(e)\) equals one, its
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_adc_quantization.png}
-\caption{\label{fig:detail_instrumentation_adc_quantization}Probability density function \(p(e)\) of the ADC quantization error \(e\)}
+\caption{\label{fig:detail_instrumentation_adc_quantization}Probability density function \(p(e)\) of the ADC quantization error \(e\).}
\end{figure}
The variance (or time-average power) of the quantization noise is expressed by~\eqref{eq:detail_instrumentation_quant_power}.
@@ -8979,10 +8970,10 @@ From a specified noise amplitude spectral density \(\Gamma_{\text{max}}\), the m
n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right)
\end{equation}
-With a sampling frequency \(F_s = 10\,\text{kHz}\), an input range \(\Delta V = 20\,\text{V}\) and a maximum allowed \acrshort{asd} \(\Gamma_{\text{max}} = 11\,\mu\text{V}/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is readily achievable with commercial \acrshortpl{adc}.
+With a sampling frequency \(F_s = 10\,\text{kHz}\), an input range \(\Delta V = 20\,\text{V}\) and a maximum allowed \acrshort{asd} \(\Gamma_{\text{max}} = 11\,\upmu\text{V}/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is readily achievable with commercial \acrshortpl{adc}.
\paragraph{DAC Output voltage noise}
-Similar to the \acrshort{adc} requirements, the \acrshort{dac} output voltage noise \acrshort{asd} should not exceed \(14\,\mu\text{V}/\sqrt{\text{Hz}}\).
+Similar to the \acrshort{adc} requirements, the \acrshort{dac} output voltage noise \acrshort{asd} should not exceed \(14\,\upmu\text{V}/\sqrt{\text{Hz}}\).
This specification corresponds to a \(\pm 10\,\text{V}\) \acrshort{dac} with 13-bit resolution, which is easily attainable with current technology.
\paragraph{Choice of the ADC and DAC Board}
@@ -8990,13 +8981,13 @@ Based on the preceding analysis, the selection of suitable \acrshort{adc} and \a
For optimal synchronicity, a Speedgoat-integrated solution was chosen.
The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, \(\pm 10\,\text{V}\) range, maximum sampling rate of 200kSPS (\acrlong{sps}), simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity.
-The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, \(\pm 10\,\text{V}\) range, conversion time of \(10\,\mu s\), and simultaneous update capability.
+The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, \(\pm 10\,\text{V}\) range, conversion time of \(10\,\upmu s\), and simultaneous update capability.
Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements.
This will be experimentally verified in Section~\ref{sec:detail_instrumentation_characterization}.
\subsubsection{Relative Displacement Sensors}
-The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below \(6\,\text{nm RMS}\) (derived from the \(15\,\text{nm RMS}\) vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding \(100\,\mu\text{m}\).
+The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below \(6\,\text{nm RMS}\) (derived from the \(15\,\text{nm RMS}\) vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding \(100\,\upmu\text{m}\).
Several sensor technologies are capable of meeting these requirements~\cite{fleming13_review_nanom_resol_posit_sensor}.
These include optical encoders (Figure~\ref{fig:detail_instrumentation_sensor_encoder}), capacitive sensors (Figure~\ref{fig:detail_instrumentation_sensor_capacitive}), and eddy current sensors (Figure~\ref{fig:detail_instrumentation_sensor_eddy_current}), each with their own advantages and implementation considerations.
@@ -9020,7 +9011,7 @@ These include optical encoders (Figure~\ref{fig:detail_instrumentation_sensor_en
\end{center}
\subcaption{\label{fig:detail_instrumentation_sensor_capacitive}Capacitive Sensor}
\end{subfigure}
-\caption{\label{fig:detail_instrumentation_sensor_examples}Relative motion sensors considered for measuring the active platform strut motion}
+\caption{\label{fig:detail_instrumentation_sensor_examples}Relative motion sensors considered for measuring the active platform strut motion.}
\end{figure}
From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the \acrshort{apa}, as illustrated in Figure~\ref{fig:detail_instrumentation_capacitive_implementation}.
@@ -9039,7 +9030,7 @@ In contrast, optical encoders are bigger and they must be offset from the strut'
\end{center}
\subcaption{\label{fig:detail_instrumentation_capacitive_implementation}Capacitive Sensor}
\end{subfigure}
-\caption{\label{fig:detail_instrumentation_sensor_implementation}Implementation of relative displacement sensor to measure the motion of the APA}
+\caption{\label{fig:detail_instrumentation_sensor_implementation}Implementation of relative displacement sensors to measure the motion of the APA.}
\end{figure}
A significant consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
@@ -9050,7 +9041,6 @@ Based on this criterion, an optical encoder with digital output was selected, wh
The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table~\ref{tab:detail_instrumentation_sensor_specs}, alongside alternative options that were considered.
\begin{table}[htbp]
-\caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products}
\centering
\begin{tabularx}{0.65\linewidth}{Xccc}
\toprule
@@ -9059,11 +9049,13 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic
Technology & Digital Encoder & Capacitive & Eddy Current\\
Bandwidth \(> 5\,\text{kHz}\) & \(> 500\,\text{kHz}\) & \(10\,\text{kHz}\) & \(20\,\text{kHz}\)\\
Noise \(< 6\,\text{nm RMS}\) & \(1.6\,\text{nm RMS}\) & \(4\,\text{nm RMS}\) & \(15\,\text{nm RMS}\)\\
-Range \(> 100\,\mu\text{m}\) & Ruler length & \(250\,\mu \text{m}\) & \(500\,\mu \text{m}\)\\
+Range \(> 100\,\upmu\text{m}\) & Ruler length & \(250\,\upmu \text{m}\) & \(500\,\upmu \text{m}\)\\
In line measurement & & \(\times\) & \(\times\)\\
Digital Output & \(\times\) & & \\
\bottomrule
\end{tabularx}
+\caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products.}
+
\end{table}
\subsection{Characterization of Instrumentation}
\label{sec:detail_instrumentation_characterization}
@@ -9072,7 +9064,7 @@ Digital Output & \(\times\) & & \\
The measurement of \acrshort{adc} noise was performed by short-circuiting its input with a \(50\,\Omega\) resistor and recording the digital values at a sampling rate of \(10\,\text{kHz}\).
The amplitude spectral density of the recorded values was computed and is presented in Figure~\ref{fig:detail_instrumentation_adc_noise_measured}.
-The \acrshort{adc} noise exhibits characteristics of white noise with an amplitude spectral density of \(5.6\,\mu\text{V}/\sqrt{\text{Hz}}\) (equivalent to \(0.4\,\text{mV RMS}\)), which satisfies the established specifications.
+The \acrshort{adc} noise exhibits characteristics of white noise with an amplitude spectral density of \(5.6\,\upmu\text{V}/\sqrt{\text{Hz}}\) (equivalent to \(0.4\,\text{mV RMS}\)), which satisfies the established specifications.
All \acrshort{adc} channels demonstrated similar performance, so only one channel's noise profile is shown.
If necessary, oversampling can be applied to further reduce the noise~\cite{lab13_improv_adc}.
@@ -9083,9 +9075,9 @@ This approach is effective because the noise approximates white noise and its am
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_adc_noise_measured.png}
-\caption{\label{fig:detail_instrumentation_adc_noise_measured}Measured ADC noise (IO318)}
+\caption{\label{fig:detail_instrumentation_adc_noise_measured}Measured ADC noise (IO318).}
\end{figure}
-\paragraph{Reading of piezoelectric force sensor}
+\paragraph{Reading of Piezoelectric Force Sensor}
To further validate the ADC's capability to effectively measure voltage generated by a piezoelectric stack, a test was conducted using the APA95ML.
The setup is illustrated in Figure~\ref{fig:detail_instrumentation_force_sensor_adc_setup}, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor.
@@ -9094,7 +9086,7 @@ The voltage amplifier employed in this setup has a gain of 20.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_force_sensor_adc_setup.png}
-\caption{\label{fig:detail_instrumentation_force_sensor_adc_setup}Schematic of the setup to validate the use of the ADC for reading the force sensor volage}
+\caption{\label{fig:detail_instrumentation_force_sensor_adc_setup}Schematic of the setup to validate the use of the ADC for reading the force sensor voltage.}
\end{figure}
Step signals with an amplitude of \(1\,\text{V}\) were generated using the \acrshort{dac}, and the \acrshort{adc} signal was recorded.
@@ -9108,7 +9100,7 @@ The charge generated by the piezoelectric effect across the stack's capacitance
Consequently, the transfer function from the generated voltage \(V_p\) to the measured voltage \(V_{\text{ADC}}\) is a first-order high-pass filter with the time constant \(\tau\).
An exponential curve was fitted to the experimental data, yielding a time constant \(\tau = 6.5\,\text{s}\).
-With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu\text{F}\), the internal impedance of the Speedgoat \acrshort{adc} was calculated as \(R_i = \tau/C_p = 1.5\,M\Omega\), which closely aligns with the specified value of \(1\,M\Omega\) found in the datasheet.
+With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\upmu\text{F}\), the internal impedance of the Speedgoat \acrshort{adc} was calculated as \(R_i = \tau/C_p = 1.5\,M\Omega\), which closely aligns with the specified value of \(1\,M\Omega\) found in the datasheet.
\begin{figure}[htbp]
\begin{subfigure}{0.61\textwidth}
@@ -9123,12 +9115,12 @@ With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu\te
\end{center}
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor}Measured Signals}
\end{subfigure}
-\caption{\label{fig:detail_instrumentation_force_sensor}Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from~\cite{reza06_piezoel_trans_vibrat_contr_dampin}. Measured voltage \(V_s\) while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}).}
+\caption{\label{fig:detail_instrumentation_force_sensor}Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from \cite{reza06_piezoel_trans_vibrat_contr_dampin}. Measured voltage \(V_s\) while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}).}
\end{figure}
The constant voltage offset can be explained by the input bias current \(i_n\) of the \acrshort{adc}, represented in Figure~\ref{fig:detail_instrumentation_force_sensor_adc}.
At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the \acrshort{adc}, and therefore the input bias current \(i_n\) passing through the internal resistance \(R_i\) produces a constant voltage offset \(V_{\text{off}} = R_i \cdot i_n\).
-The input bias current \(i_n\) is estimated from \(i_n = V_{\text{off}}/R_i = 1.5\mu A\).
+The input bias current \(i_n\) is estimated from \(i_n = V_{\text{off}}/R_i = 1.5\,\upmu\text{A}\).
In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor \(R_p\) can be added in parallel to the force sensor, as illustrated in Figure~\ref{fig:detail_instrumentation_force_sensor_adc_R}.
This modification produces two beneficial effects: a reduction of input voltage offset through the relationship \(V_{\text{off}} = (R_p R_i)/(R_p + R_i) i_n\), and an increase in the high pass corner frequency \(f_c\) according to the equations \(\tau = 1/(2\pi f_c) = (R_i R_p)/(R_i + R_p) C_p\).
@@ -9155,7 +9147,7 @@ These results validate both the model of the \acrshort{adc} and the effectivenes
\end{figure}
\subsubsection{Instrumentation Amplifier}
-Because the \acrshort{adc} noise may be too low to measure the noise of other instruments (anything below \(5.6\,\mu\text{V}/\sqrt{\text{Hz}}\) cannot be distinguished from the noise of the \acrshort{adc} itself), a low noise instrumentation amplifier was employed.
+Because the \acrshort{adc} noise may be too low to measure the noise of other instruments (anything below \(5.6\,\upmu\text{V}/\sqrt{\text{Hz}}\) cannot be distinguished from the noise of the \acrshort{adc} itself), a low noise instrumentation amplifier was employed.
A Femto DLPVA-101-B-S amplifier with adjustable gains from \(20\,text{dB}\) up to \(80\,text{dB}\) was selected for this purpose.
The first step was to characterize the input\footnote{For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain.} noise of the amplifier.
@@ -9169,14 +9161,14 @@ The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and th
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_instrumentation_femto_meas_setup.png}
-\captionof{figure}{\label{fig:detail_instrumentation_femto_meas_setup}Measurement of the instrumentation amplifier input voltage noise}
+\captionof{figure}{\label{fig:detail_instrumentation_femto_meas_setup}Measurement of the instrumentation amplifier input voltage noise.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_femto_input_noise.png}
-\captionof{figure}{\label{fig:detail_instrumentation_femto_input_noise}Obtained ASD of the instrumentation amplifier input voltage noise}
+\captionof{figure}{\label{fig:detail_instrumentation_femto_input_noise}Obtained ASD of the instrumentation amplifier input voltage noise.}
\end{center}
\end{minipage}
\subsubsection{Digital to Analog Converters}
@@ -9187,7 +9179,7 @@ The \acrshort{dac} was configured to output a constant voltage (zero in this cas
The Amplitude Spectral Density \(\Gamma_{n_{da}}(\omega)\) of the measured signal was computed, and verification was performed to confirm that the contributions of \acrshort{adc} noise and amplifier noise were negligible in the measurement.
The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure~\ref{fig:detail_instrumentation_dac_output_noise}.
-The noise profile is predominantly white with an \acrshort{asd} of \(0.6\,\mu\text{V}/\sqrt{\text{Hz}}\).
+The noise profile is predominantly white with an \acrshort{asd} of \(0.6\,\upmu\text{V}/\sqrt{\text{Hz}}\).
Minor \(50\,\text{Hz}\) noise is present, along with some low frequency \(1/f\) noise, but these are not expected to pose issues as they are well within specifications.
It should be noted that all \acrshort{dac} channels demonstrated similar performance, so only one channel measurement is presented.
@@ -9216,7 +9208,7 @@ The observed \acrshort{frf} corresponds to exactly one sample delay, which align
\end{center}
\subcaption{\label{fig:detail_instrumentation_dac_adc_tf}Transfer function from DAC to ADC}
\end{subfigure}
-\caption{\label{fig:detail_instrumentation_dac}Measurement of the output voltage noise of the ADC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a ``1-sample'' delay.}
+\caption{\label{fig:detail_instrumentation_dac}Measurement of the output voltage noise of the DAC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a ``1-sample'' delay.}
\end{figure}
\subsubsection{Piezoelectric Voltage Amplifier}
\paragraph{Output Voltage Noise}
@@ -9228,7 +9220,7 @@ Two piezoelectric stacks from the APA95ML were connected to the PD200 output to
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_pd200_setup.png}
-\caption{\label{fig:detail_instrumentation_pd200_setup}Setup used to measured the output voltage noise of the PD200 voltage amplifier. A gain \(G_a = 1000\) was used for the instrumentation amplifier.}
+\caption{\label{fig:detail_instrumentation_pd200_setup}Setup used to measure the output voltage noise of the PD200 voltage amplifier. A gain \(G_a = 1000\) was used for the instrumentation amplifier.}
\end{figure}
The Amplitude Spectral Density \(\Gamma_{n}(\omega)\) of the signal measured by the \acrshort{adc} was computed.
@@ -9248,7 +9240,7 @@ While the exact cause of these peaks is not fully understood, their amplitudes r
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_pd200_noise.png}
-\caption{\label{fig:detail_instrumentation_pd200_noise}Measured output voltage noise of the PD200 amplifiers}
+\caption{\label{fig:detail_instrumentation_pd200_noise}Measured output voltage noise of the PD200 amplifiers.}
\end{figure}
\paragraph{Small Signal Bandwidth}
@@ -9265,7 +9257,7 @@ The identified dynamics shown in Figure~\ref{fig:detail_instrumentation_pd200_tf
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_pd200_tf.png}
-\caption{\label{fig:detail_instrumentation_pd200_tf}Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier}
+\caption{\label{fig:detail_instrumentation_pd200_tf}Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier.}
\end{figure}
\subsubsection{Linear Encoders}
@@ -9281,19 +9273,19 @@ The noise profile exhibits characteristics of white noise with an amplitude of a
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_vionic_bench.jpg}
-\captionof{figure}{\label{fig:detail_instrumentation_vionic_bench}Test bench used to measured the encoder noise}
+\captionof{figure}{\label{fig:detail_instrumentation_vionic_bench}Test bench used to measure the encoder noise.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_vionic_asd.png}
-\captionof{figure}{\label{fig:detail_instrumentation_vionic_asd}Measured encoder noise ASD}
+\captionof{figure}{\label{fig:detail_instrumentation_vionic_asd}Measured encoder noise ASD.}
\end{center}
\end{minipage}
-\subsubsection{Noise budgeting from measured instrumentation noise}
+\subsubsection{Noise Budgeting from Measured Instrumentation Noise}
-After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier.
+After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model.
The vertical motion induced by the noise sources, specifically the \acrshort{adc} noise, \acrshort{dac} noise, and voltage amplifier noise, is presented in Figure~\ref{fig:detail_instrumentation_cl_noise_budget}.
The total motion induced by all noise sources combined is approximately \(1.5\,\text{nm RMS}\), which remains well within the specified limit of \(15\,\text{nm RMS}\).
This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application.
@@ -9301,7 +9293,7 @@ This confirms that the selected instrumentation, with its measured noise charact
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_cl_noise_budget.png}
-\caption{\label{fig:detail_instrumentation_cl_noise_budget}Closed-loop noise budgeting using measured noise of instrumentation}
+\caption{\label{fig:detail_instrumentation_cl_noise_budget}Closed-loop noise budgeting using measured noise of instrumentation.}
\end{figure}
\subsection*{Conclusion}
\label{sec:detail_instrumentation_conclusion}
@@ -9314,28 +9306,28 @@ Based on these specifications, appropriate instrumentation components were selec
The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions.
Despite these challenges, suitable components were identified that theoretically met all requirements.
-The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized.
+The selected instrumentation was procured and thoroughly characterized.
Initial measurements of the \acrshort{adc} system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit.
-All components were found to meet or exceed their respective specifications. The \acrshort{adc} demonstrated noise levels of \(5.6\,\mu\text{V}/\sqrt{\text{Hz}}\) (versus the \(11\,\mu\text{V}/\sqrt{\text{Hz}}\) specification), the \acrshort{dac} showed \(0.6\,\mu\text{V}/\sqrt{\text{Hz}}\) (versus \(14\,\mu\text{V}/\sqrt{\text{Hz}}\) required), the voltage amplifiers exhibited noise well below the \(280\,\mu\text{V}/\sqrt{\text{Hz}}\) limit, and the encoders achieved \(1\,\text{nm RMS}\) noise (versus the \(6\,\text{nm RMS}\) specification).
+All components were found to meet or exceed their respective specifications. The \acrshort{adc} demonstrated noise levels of \(5.6\,\upmu\text{V}/\sqrt{\text{Hz}}\) (versus the \(11\,\upmu\text{V}/\sqrt{\text{Hz}}\) specification), the \acrshort{dac} showed \(0.6\,\upmu\text{V}/\sqrt{\text{Hz}}\) (versus \(14\,\upmu\text{V}/\sqrt{\text{Hz}}\) required), the voltage amplifiers exhibited noise well below the \(280\,\upmu\text{V}/\sqrt{\text{Hz}}\) limit, and the encoders achieved \(1\,\text{nm RMS}\) noise (versus the \(6\,\text{nm RMS}\) specification).
Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance.
The combined effect of all noise sources was estimated to induce vertical sample vibrations of only \(1.5\,\text{nm RMS}\), which is substantially below the \(15\,\text{nm RMS}\) requirement.
This rigorous methodology spanning requirement formulation, component selection, and experimental characterization validates the instrumentation's ability to fulfill the nano active stabilization system's demanding performance specifications.
-\section{Obtained Design}
+\section{Obtained Design: the ``Nano-Hexapod''}
\label{sec:detail_design}
The detailed mechanical design of the active platform (also referred to as the ``nano-hexapod''), depicted in Figure~\ref{fig:detail_design_nano_hexapod_elements}, is presented in this section.
Several primary objectives guided the mechanical design.
First, to ensure a well-defined Jacobian matrix used in the control architecture, accurate positioning of the top flexible joint rotation points and correct orientation of the struts were required.
Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of \(120\,\text{mm}\) and a height of \(95\,\text{mm}\).
Thirdly, because performance predicted by the multi-body model was fulfilling the requirements, the final design was intended to approximate the behavior of this ``idealized'' active platform as closely as possible.
-This objective implies that the frequencies of (un-modelled) flexible modes potentially detrimental to control performance needed to be maximized.
+This objective implies that the frequencies of (un-modeled) flexible modes potentially detrimental to control performance needed to be maximized.
Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_nano_hexapod_elements.png}
-\caption{\label{fig:detail_design_nano_hexapod_elements}Obtained mechanical design of the Active platform, called the ``nano-hexapod''}
+\caption{\label{fig:detail_design_nano_hexapod_elements}Obtained mechanical design of the active platform, called the ``nano-hexapod''.}
\end{figure}
\subsection{Mechanical Design}
\label{sec:detail_design_mechanics}
@@ -9347,7 +9339,7 @@ Due to the limited angular stroke of the flexible joints, it was critical that t
To facilitate this alignment, cylindrical washers (Figure~\ref{fig:detail_design_strut_without_enc}) were integrated into the design to compensate for potential deviations from perfect flatness between the two \acrshort{apa} interface planes (Figure~\ref{fig:detail_design_apa}).
Furthermore, a dedicated mounting bench was developed to enable precise alignment of each strut, even when accounting for typical machining inaccuracies.
The mounting procedure is described in Section~\ref{sec:test_struts_mounting}.
-Lastly, the design needed to permit the fixation of an encoder parallel to the strut axis, as shown in Figure~\ref{fig:detail_design_strut_with_enc}.
+Lastly, the design needed to permit the mounting of an encoder parallel to the strut axis, as shown in Figure~\ref{fig:detail_design_strut_with_enc}.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -9362,12 +9354,12 @@ Lastly, the design needed to permit the fixation of an encoder parallel to the s
\end{center}
\subcaption{\label{fig:detail_design_strut_with_enc}With the mounted encoder}
\end{subfigure}
-\caption{\label{fig:detail_design_strut}Design of the Nano-Hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration.}
+\caption{\label{fig:detail_design_strut}Design of the nano-hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration.}
\end{figure}
The flexible joints, shown in Figure~\ref{fig:detail_design_flexible_joint}, were manufactured using wire-cut \acrfull{edm}.
-First, the part's inherent fragility, stemming from its \(0.25\,\text{mm}\) neck dimension, makes it susceptible to damage from cutting forces typical in classical machining.
-Furthermore, wire-cut \acrshort{edm} allows for the very tight machining tolerances critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure~\ref{fig:detail_design_flexible_joint}) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties.
+First, the part being quite fragile, stemming from its \(0.25\,\text{mm}\) neck dimension, is easier to machine using wire-cut \acrshort{edm} thanks to the very small cutting forces compared to classical machining.
+Furthermore, wire-cut \acrshort{edm} allows for tight machining tolerances of complex shapes.
The material chosen for the flexible joints is a stainless steel designated \emph{X5CrNiCuNb16-4} (alternatively known as F16Ph).
This selection was based on its high specified yield strength (exceeding \(1\,\text{GPa}\) after appropriate heat treatment) and its high fatigue resistance.
@@ -9407,7 +9399,7 @@ Although topology optimization methods were considered, the implemented ribbed d
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/detail_design_top_plate.png}
-\caption{\label{fig:detail_design_top_plate}The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is detailed later.}
+\caption{\label{fig:detail_design_top_plate}The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is later detailed.}
\end{figure}
The interfaces for the joints on the plates incorporate V-grooves (red planes in Figure~\ref{fig:detail_design_top_plate}).
@@ -9434,7 +9426,7 @@ High machining accuracy for these features is essential to ensure that the flexi
\end{center}
\subcaption{\label{fig:detail_design_location_bot_flex}Bottom Positioning}
\end{subfigure}
-\caption{\label{fig:detail_design_fixation_flexible_joints_platform}Fixation of the flexible points to the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joint is in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joint is just oriented (\subref{fig:detail_design_location_bot_flex}).}
+\caption{\label{fig:detail_design_fixation_flexible_joints_platform}Clamping of the flexible points on the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joints are in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joints are just oriented (\subref{fig:detail_design_location_bot_flex}).}
\end{figure}
Furthermore, the flat interface surface of each top flexible joint is designed to be in direct contact with the top platform surface, as shown in Figure~\ref{fig:detail_design_location_top_flexible_joints}.
@@ -9474,11 +9466,11 @@ Finally, the FEA indicated that flexible modes of the top plate itself begin to
\end{center}
\subcaption{\label{fig:detail_design_fem_plate_mode}Top plate mode}
\end{subfigure}
-\caption{\label{fig:detail_design_fem_nano_hexapod}Measurement of strut flexible modes. First six modes are ``suspension'' modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from \(205\,\text{Hz}\) to \(420\,\text{Hz}\) (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at \(650\,\text{Hz}\) (\subref{fig:detail_design_fem_plate_mode})}
+\caption{\label{fig:detail_design_fem_nano_hexapod}Finite Element Model of the nano-hexapod. The first six modes are ``suspension'' modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from \(205\,\text{Hz}\) to \(420\,\text{Hz}\) (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at \(650\,\text{Hz}\) (\subref{fig:detail_design_fem_plate_mode}).}
\end{figure}
\paragraph{Alternative Encoder Placement}
-In anticipation of potential issues arising from the local modes of the struts affecting encoder measurements, an alternative fixation strategy for the encoders was designed.
+In anticipation of potential issues arising from the local modes of the struts affecting encoder measurements, an alternative mounting strategy for the encoders was designed.
In this configuration, the encoders are fixed directly to the top and bottom plates instead of the struts, as illustrated in Figure~\ref{fig:detail_design_enc_plates_design}.
\begin{figure}[htbp]
@@ -9492,9 +9484,9 @@ In this configuration, the encoders are fixed directly to the top and bottom pla
\begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_design_encoders_plates.jpg}
\end{center}
-\subcaption{\label{fig:detail_design_encoders_plates}Zoom on encoder fixation}
+\subcaption{\label{fig:detail_design_encoders_plates}Zoom on encoder mounting}
\end{subfigure}
-\caption{\label{fig:detail_design_enc_plates_design}Alternative way of using the encoders: they are fixed directly to the plates.}
+\caption{\label{fig:detail_design_enc_plates_design}Alternative location of the encoders: fixed to the plates.}
\end{figure}
Dedicated supports, machined from aluminum, were designed for this purpose.
@@ -9527,18 +9519,18 @@ In these models, the top and bottom plates were represented as rigid bodies, wit
Several levels of detail were considered for modeling the flexible joints within the multi-body model.
Models with two \acrshortpl{dof} incorporating only bending stiffnesses, models with three \acrshortpl{dof} adding torsional stiffness, and models with four \acrshortpl{dof} further adding axial stiffness were evaluated.
-The multi-body representation corresponding to the 4DoF configuration is shown in Figure~\ref{fig:detail_design_simscape_model_flexible_joint}.
+The multi-body representation corresponding to the 4-DoFs configuration is shown in Figure~\ref{fig:detail_design_simscape_model_flexible_joint}.
This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from \acrshort{fea} of the joint component.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/detail_design_simscape_model_flexible_joint.png}
-\caption{\label{fig:detail_design_simscape_model_flexible_joint}4DoF multi-body model of the flexible joints}
+\caption{\label{fig:detail_design_simscape_model_flexible_joint}4-DoFs multi-body model of the flexible joints. Axial, bending and torsional stiffnesses are modeled.}
\end{figure}
\paragraph{Amplified Piezoelectric Actuators}
The \acrlongpl{apa} were incorporated into the multi-body model following the methodology detailed in Section~\ref{sec:detail_fem_actuator}.
-Two distinct representations of the \acrshort{apa} can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex ``Reduced Order Flexible Body'' model derived from a \acrshort{fem}.
+Two distinct representations of the \acrshort{apa} can be used within the simulation: a simplified 2-DoFs model capturing the axial behavior, or a more complex ``Reduced Order Flexible Body'' model derived from a \acrshort{fem}.
\paragraph{Encoders}
In earlier modeling stages, the relative displacement sensors (encoders) were implemented as a direct measurement of the relative distance between the joint connection points \(\bm{a}_i\) and \(\bm{b}_i\).
@@ -9547,8 +9539,8 @@ Therefore, a more sophisticated model of the optical encoder was necessary.
The optical encoders operate based on the interaction between an encoder head and a graduated scale or ruler.
The optical encoder head contains a light source that illuminates the ruler.
-A reference frame \(\{E\}\) fixed to the scale, represents the the light position on the scale, as illustrated in Figure~\ref{fig:detail_design_simscape_encoder_model}.
-The ruler features a precise grating pattern (in this case, with a \(20\,\mu\text{m}\) pitch), and its position is associated with the reference frame \(\{R\}\).
+A reference frame \(\{E\}\) fixed to the scale, represents the light position on the scale, as illustrated in Figure~\ref{fig:detail_design_simscape_encoder_model}.
+The ruler features a precise grating pattern (in this case, with a \(20\,\upmu\text{m}\) pitch), and its position is associated with the reference frame \(\{R\}\).
The displacement measured by the encoder corresponds to the relative position of the encoder frame \(\{E\}\) (specifically, the point where the light interacts with the scale) with respect to the ruler frame \(\{R\}\), projected along the measurement direction defined by the scale.
An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure~\ref{fig:detail_design_simscape_encoder_disp}, can induce a measured displacement.
@@ -9566,9 +9558,9 @@ An important consequence of this measurement principle is that a relative rotati
\end{center}
\subcaption{\label{fig:detail_design_simscape_encoder_disp}Rotation of the encoder head}
\end{subfigure}
-\caption{\label{fig:detail_design_simscape_encoder_model}Representation of the encoder model in the multi-body model. Measurement \(d_i\) corresponds to the \(x\) position of the encoder frame \(\{E\}\) expresssed in the ruller frame \(\{R\}\) (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}).}
+\caption{\label{fig:detail_design_simscape_encoder_model}Representation of the encoder multi-body model. Measurement \(d_i\) corresponds to the \(x\) position of the encoder frame \(\{E\}\) expresssed in the ruller frame \(\{R\}\) (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}).}
\end{figure}
-\paragraph{Validation of the designed active platform}
+\paragraph{Validation of the Designed Active Platform}
The refined multi-body model of the nano-hexapod was integrated into the multi-body micro-station model.
Dynamical analysis was performed, confirming that the platform's behavior closely approximates the dynamics of the ``idealized'' model used during the conceptual design phase.
@@ -9631,9 +9623,9 @@ The \acrshort{haclac} control architecture is implemented and tested under vario
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp,width=\linewidth]{figs/chapter3_overview.png}
-\caption{\label{fig:chapter3_overview}Overview of the Experimental validation phase. The actuators and flexible joints and individual tested and then integrated into the struts. The Nano-hexapod is then mounted and the complete system is validated on the ID31 beamline.}
+\caption{\label{fig:chapter3_overview}Overview of the experimental validation phase. The actuators and flexible joints and individual tested and then integrated into the struts. The Nano-hexapod is then mounted and the complete system is validated on the ID31 beamline.}
\end{figure}
-\section{Amplified Piezoelectric Actuator}
+\section{Amplified Piezoelectric Actuators}
\label{sec:test_apa}
In this chapter, the goal is to ensure that the received APA300ML (shown in Figure~\ref{fig:test_apa_received}) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics.
@@ -9654,9 +9646,9 @@ This more complex model also captures well capture the axial dynamics of the APA
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_received.jpg}
-\caption{\label{fig:test_apa_received}Picture of 5 out of the 7 received APA300ML}
+\caption{\label{fig:test_apa_received}5 of the 7 received APA300ML.}
\end{figure}
-\subsection{First Basic Measurements}
+\subsection{Static Measurements}
\label{sec:test_apa_basic_meas}
Before measuring the dynamical characteristics of the APA300ML, simple measurements are performed.
@@ -9668,14 +9660,14 @@ Finally, in Section~\ref{ssec:test_apa_spurious_resonances}, the flexible modes
\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.
-As shown in Figure~\ref{fig:test_apa_flatness_setup}, the \acrshort{apa} is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu\text{m}\)} is used to measure the height of four points on each of the APA300ML interfaces.
-From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.
+As shown in Figure~\ref{fig:test_apa_flatness_setup}, the \acrshort{apa} is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\upmu\text{m}\)} is used to measure the height of four points on each of the APA300ML interfaces.
+From the XYZ coordinates of the measured eight points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.
The measured flatness values, summarized in Table~\ref{tab:test_apa_flatness_meas}, are within the specifications.
\begin{minipage}[b]{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_flatness_setup.png}
-\captionof{figure}{\label{fig:test_apa_flatness_setup}Measurement setup for flatness estimation}
+\captionof{figure}{\label{fig:test_apa_flatness_setup}Measurement setup for flatness estimation.}
\end{center}
\end{minipage}
\hfill
@@ -9684,7 +9676,7 @@ The measured flatness values, summarized in Table~\ref{tab:test_apa_flatness_mea
{\footnotesize\sf
\begin{tabularx}{0.5\linewidth}{Xc}
\toprule
- & \textbf{Flatness} \([\mu\text{m}]\)\\
+ & \textbf{Flatness} \([\upmu\text{m}]\)\\
\midrule
APA 1 & 8.9\\
APA 2 & 3.1\\
@@ -9694,17 +9686,18 @@ APA 5 & 1.9\\
APA 6 & 7.1\\
APA 7 & 18.7\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
\end{minipage}
\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 \(18\,\mu\text{F}\) and \(26\,\mu\text{F}\) with a nominal capacitance of \(20\,\mu\text{F}\).
+From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\upmu\text{F}\) and \(26\,\upmu\text{F}\) with a nominal capacitance of \(20\,\upmu\text{F}\).
The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter\footnote{LCR-819 from Gwinstek, with a specified accuracy of \(0.05\%\). The measured frequency is set at \(1\,\text{kHz}\)} shown in Figure~\ref{fig:test_apa_lcr_meter}.
The two stacks used as the actuator and the stack used as the force sensor were measured separately.
-The measured capacitance values are summarized in Table~\ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu\text{F}\).
+The measured capacitance values are summarized in Table~\ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \upmu\text{F}\).
However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance.
This may indicate a manufacturing defect.
@@ -9714,7 +9707,7 @@ This may be because the manufacturer measures the capacitance with large signals
\begin{minipage}[b]{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_apa_lcr_meter.jpg}
-\captionof{figure}{\label{fig:test_apa_lcr_meter}Used LCR meter}
+\captionof{figure}{\label{fig:test_apa_lcr_meter}Used LCR meter.}
\end{center}
\end{minipage}
\hfill
@@ -9733,28 +9726,29 @@ APA 5 & 4.90 & 9.66\\
APA 6 & 4.99 & 9.91\\
APA 7 & 4.85 & 9.85\\
\bottomrule
-\end{tabularx}}
-\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu\text{F}$}
+\end{tabularx}
+}
+\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\upmu\text{F}$}
\end{minipage}
\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 \acrshort{apa} is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu\text{m}\)} is located on the other side as shown in Figure~\ref{fig:test_apa_stroke_bench}.
+To compare the stroke of the APA300ML with the datasheet specifications, one side of the \acrshort{apa} is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\upmu\text{m}\)} is located on the other side as shown in Figure~\ref{fig:test_apa_stroke_bench}.
The voltage across the two actuator stacks is varied from \(-20\,\text{V}\) to \(150\,\text{V}\) using a DAC\footnote{The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of \(\pm 10\,\text{V}\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,\text{V/V}\)}.
Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure~\ref{fig:test_apa_stroke_voltage}).
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_stroke_bench.jpg}
-\caption{\label{fig:test_apa_stroke_bench}Bench to measure the APA stroke}
+\includegraphics[scale=1,width=0.5\linewidth]{figs/test_apa_stroke_bench.jpg}
+\caption{\label{fig:test_apa_stroke_bench}Test bench to measure the APA stroke.}
\end{figure}
The measured \acrshort{apa} displacement is shown as a function of the applied voltage in Figure~\ref{fig:test_apa_stroke_hysteresis}.
Typical hysteresis curves for piezoelectric stack actuators can be observed.
-The measured stroke is approximately \(250\,\mu\text{m}\) when using only two of the three stacks.
-This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu\text{m}\), therefore \(\approx 200\,\mu\text{m}\) if only two stacks are used).
-For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\mu\text{m}\).
+The measured stroke is approximately \(250\,\upmu\text{m}\) when using only two of the three stacks.
+This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\upmu\text{m}\), therefore \(\approx 200\,\upmu\text{m}\) if only two stacks are used).
+For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\upmu\text{m}\).
It is clear from Figure~\ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared with the other units.
This confirms the abnormal electrical measurements made in Section~\ref{ssec:test_apa_electrical_measurements}.
@@ -9774,7 +9768,7 @@ From now on, only the six remaining amplified piezoelectric actuators that behav
\end{center}
\subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
\end{subfigure}
-\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})}
+\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}
\subsubsection{Flexible Mode Measurement}
\label{ssec:test_apa_spurious_resonances}
@@ -9789,23 +9783,23 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
\begin{figure}[htbp]
\begin{subfigure}{0.35\textwidth}
\begin{center}
-\includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_1.png}
+\includegraphics[scale=1,height=4cm]{figs/test_apa_mode_shapes_1.png}
\end{center}
\subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode ($268\,\text{Hz}$)}
\end{subfigure}
\begin{subfigure}{0.27\textwidth}
\begin{center}
-\includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_2.png}
+\includegraphics[scale=1,height=4cm]{figs/test_apa_mode_shapes_2.png}
\end{center}
\subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode ($399\,\text{Hz}$)}
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\begin{center}
-\includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_3.png}
+\includegraphics[scale=1,height=4cm]{figs/test_apa_mode_shapes_3.png}
\end{center}
\subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode ($706\,\text{Hz}$)}
\end{subfigure}
-\caption{\label{fig:test_apa_mode_shapes}First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model}
+\caption{\label{fig:test_apa_mode_shapes}First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model.}
\end{figure}
\begin{figure}[htbp]
@@ -9821,7 +9815,7 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
\end{center}
\subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
\end{subfigure}
-\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure the flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
+\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure the flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the hammer impact point is at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the hammer impact point is located at the back of the top measurement point.}
\end{figure}
The measured \acrshortpl{frf} computed from the experimental setups of figures~\ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure~\ref{fig:test_apa_meas_freq_compare}.
@@ -9834,9 +9828,9 @@ Another explanation is the shape difference between the manufactured APA300ML an
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_freq_compare.png}
-\caption{\label{fig:test_apa_meas_freq_compare}Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
+\caption{\label{fig:test_apa_meas_freq_compare}Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\).}
\end{figure}
-\subsection{Dynamical measurements}
+\subsection{Dynamical Measurements}
\label{sec:test_apa_dynamics}
After the measurements on the \acrshort{apa} were performed in Section~\ref{sec:test_apa_basic_meas}, a new test bench was used to better characterize the dynamics of the APA300ML.
This test bench, depicted in Figure~\ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a \(5\,\text{kg}\) granite block that is vertically guided by an air bearing.
@@ -9846,46 +9840,19 @@ An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,\text{nm}\)} is used t
\begin{figure}[htbp]
\begin{subfigure}{0.3\textwidth}
\begin{center}
-\includegraphics[scale=1,height=8cm]{figs/test_apa_bench_picture.jpg}
+\includegraphics[scale=1,height=7cm]{figs/test_apa_bench_picture.jpg}
\end{center}
\subcaption{\label{fig:test_apa_bench_picture}Picture of the test bench}
\end{subfigure}
\begin{subfigure}{0.69\textwidth}
\begin{center}
-\includegraphics[scale=1,height=8cm]{figs/test_apa_bench_picture_encoder.jpg}
+\includegraphics[scale=1,height=7cm]{figs/test_apa_schematic.png}
\end{center}
-\subcaption{\label{fig:test_apa_bench_picture_encoder}Zoom on the APA with the encoder}
+\subcaption{\label{fig:test_apa_schematic}Schematic of the test bench}
\end{subfigure}
-\caption{\label{fig:test_bench_apa}Schematic of the test bench used to estimate the dynamics of the APA300ML}
+\caption{\label{fig:test_bench_apa}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}
-
-The bench is schematically shown in Figure~\ref{fig:test_apa_schematic} with the associated signals.
-It will be first used to estimate the hysteresis from the piezoelectric stack (Section~\ref{ssec:test_apa_hysteresis}) as well as the axial stiffness of the APA300ML (Section~\ref{ssec:test_apa_stiffness}).
-The \acrshortpl{frf} from the \acrshort{dac} voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section~\ref{ssec:test_apa_meas_dynamics}.
-The presence of a non-minimum phase zero found on the transfer function from \(u\) to \(V_s\) is investigated in Section~\ref{ssec:test_apa_non_minimum_phase}.
-To limit the low-frequency gain of the transfer function from \(u\) to \(V_s\), a resistor is added across the force sensor stack (Section~\ref{ssec:test_apa_resistance_sensor_stack}).
-Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section~\ref{ssec:test_apa_iff_locus}.
-
-\begin{figure}[htbp]
-\centering
-\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}
-\subsubsection{Hysteresis}
-\label{ssec:test_apa_hysteresis}
-
-Because the payload is vertically guided without friction, the hysteresis of the \acrshort{apa} can be estimated from the motion of the payload.
-A quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,\text{V}\) (halfway between \(-20\,\text{V}\) and \(150\,\text{V}\)) and with an amplitude varying from \(4\,\text{V}\) up to \(80\,\text{V}\) is generated using the \acrshort{dac}.
-For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage in Figure~\ref{fig:test_apa_meas_hysteresis}.
-This is the typical behavior expected from a \acrfull{pzt} stack actuator, where the hysteresis increases as a function of the applied voltage amplitude~\cite[chap. 1.4]{fleming14_desig_model_contr_nanop_system}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_hysteresis.png}
-\caption{\label{fig:test_apa_meas_hysteresis}Displacement as a function of applied voltage for multiple excitation amplitudes}
-\end{figure}
-\subsubsection{Axial stiffness}
-\label{ssec:test_apa_stiffness}
+\paragraph{Axial stiffness}
To estimate the stiffness of the \acrshort{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.
The \acrshort{apa} stiffness can then be estimated from equation~\eqref{eq:test_apa_stiffness}, with \(g \approx 9.8\,\text{m}/\text{s}^2\) the acceleration of gravity.
@@ -9899,12 +9866,12 @@ It can be seen that there are some drifts in the measured displacement (probably
These two effects induce some uncertainties in the measured stiffness.
The stiffnesses are computed for all \acrshortpl{apa} from the two displacements \(d_1\) and \(d_2\) (see Figure~\ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\).
-These estimated stiffnesses are summarized in Table~\ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,\text{N}/\mu\text{m}\).
+These estimated stiffnesses are summarized in Table~\ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,\text{N}/\upmu\text{m}\).
\begin{minipage}[b]{0.57\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_stiffness_time.png}
-\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding (at \(t \approx 3\,\text{s}\)) and removing (at \(t \approx 13\,\text{s}\)) the mass}
+\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Displacement when adding and removing the payload.}
\end{center}
\end{minipage}
\hfill
@@ -9922,17 +9889,18 @@ APA & \(k_1\) & \(k_2\)\\
6 & 1.7 & 1.92\\
8 & 1.73 & 1.98\\
\bottomrule
-\end{tabularx}}
-\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\mu\text{m}$}
+\end{tabularx}
+}
+\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\upmu\text{m}$}
\end{minipage}
-The stiffness can also be computed using equation~\eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 95\,\text{Hz}\) (estimated by the dynamical measurements shown in section~\ref{ssec:test_apa_meas_dynamics}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).
+The stiffness can also be computed using equation~\eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 95\,\text{Hz}\) (estimated from the dynamical measurements shown in Figure~\ref{fig:test_apa_frf_dynamics}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).
\begin{equation} \label{eq:test_apa_res_freq}
\omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
\end{equation}
-The obtained stiffness is \(k \approx 2\,\text{N}/\mu\text{m}\) which is close to the values found in the documentation and using the ``static deflection'' method.
+The obtained stiffness is \(k \approx 2\,\text{N}/\upmu\text{m}\) which is close to the values found in the documentation and using the ``static deflection'' method.
It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack~\cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}.
@@ -9942,16 +9910,15 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the `
\item \(k_{\text{sc}}\): piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance)
\end{itemize}
-The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,\text{N}/\mu\text{m}\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,\text{N}/\mu\text{m}\).
-\subsubsection{Dynamics}
-\label{ssec:test_apa_meas_dynamics}
+The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,\text{N}/\upmu\text{m}\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,\text{N}/\upmu\text{m}\).
+\paragraph{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.
First, the dynamics from \(u\) to \(d_e\) for the six APA300ML are compared in Figure~\ref{fig:test_apa_frf_encoder}.
The obtained \acrshortpl{frf} are similar to those of a (second order) mass-spring-damper system with:
\begin{itemize}
-\item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu\text{m}/V\).
+\item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\upmu\text{m/V}\).
The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the \acrshort{apa}
\item A lightly damped resonance at \(95\,\text{Hz}\)
\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the \acrshort{apa} support.
@@ -9963,13 +9930,13 @@ The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\)
A lightly damped resonance (pole) is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance (zero) at \(41\,\text{Hz}\).
No additional resonances are present up to at least \(2\,\text{kHz}\) indicating that Integral Force Feedback can be applied without stability issues from high-frequency flexible modes.
The zero at \(41\,\text{Hz}\) seems to be non-minimum phase (the phase \emph{decreases} by 180 degrees whereas it should have \emph{increased} by 180 degrees for a minimum phase zero).
-This is investigated in Section~\ref{ssec:test_apa_non_minimum_phase}.
+This is investigated further investigated.
-As illustrated by the Root Locus plot, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases.
+As illustrated by the root locus plot, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases.
The significance of this behavior varies with the type of sensor used, as explained in~\cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
Considering the transfer function from \(u\) to \(V_s\), if a controller with a very high gain is applied such that the sensor stack voltage \(V_s\) is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell.
-From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,\text{N}/\mu\text{m}\) (which is close to what is found using a \acrshort{fem}).
+From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,\text{N}/\upmu\text{m}\) (which is close to what is found using a \acrshort{fem}).
All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure~\ref{fig:test_apa_frf_encoder} and at the force sensor in Figure~\ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
@@ -9986,10 +9953,9 @@ All the identified dynamics of the six APA300ML (both when looking at the encode
\end{center}
\subcaption{\label{fig:test_apa_frf_force}FRF from $u$ to $V_s$}
\end{subfigure}
-\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}
+\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}
-\subsubsection{Non Minimum Phase Zero?}
-\label{ssec:test_apa_non_minimum_phase}
+\paragraph{Non Minimum Phase Zero?}
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}).
It was initially thought that this non-minimum phase behavior was an artifact arising from the measurement.
@@ -9999,7 +9965,7 @@ The coherence (Figure~\ref{fig:test_apa_non_minimum_phase_coherence}) is good ev
Such non-minimum phase zero when using load cells has also been observed on other mechanical systems~\cite{spanos95_soft_activ_vibrat_isolat,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}.
It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear.
-However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure~\ref{fig:test_apa_iff_root_locus}) should not be unstable, except for very large controller gains that will never be applied in practice.
+However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the root locus plot in Figure~\ref{fig:test_apa_iff_root_locus}) should not be unstable, except for very large controller gains that will never be applied in practice.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -10014,12 +9980,11 @@ However, this is not so important here because the zero is lightly damped (i.e.
\end{center}
\subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
\end{subfigure}
-\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.}
+\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}) shows a non-minimum phase behavior.}
\end{figure}
-\subsubsection{Effect of the resistor on the IFF Plant}
-\label{ssec:test_apa_resistance_sensor_stack}
+\paragraph{Effect of the resistor on the IFF Plant}
-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 \(\approx 5\,\mu\text{F}\)).
+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 \(\approx 5\,\upmu\text{F}\)).
As explained before, this is done to limit the voltage offset due to the input bias current of the \acrshort{adc} as well as to limit the low frequency gain.
@@ -10029,10 +9994,9 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_apa_effect_resistance.png}
-\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}
+\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}
-\subsubsection{Integral Force Feedback}
-\label{ssec:test_apa_iff_locus}
+\paragraph{Integral Force Feedback}
To implement the Integral Force Feedback strategy, the measured \acrshort{frf} 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}.
The parameters were manually tuned, and the obtained values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\).
@@ -10046,7 +10010,7 @@ A comparison between the identified plant and the manually tuned transfer functi
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_apa_iff_plant_comp_manual_fit.png}
-\caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\mu s\) is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.}
+\caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\upmu\text{s}\) is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.}
\end{figure}
The implemented Integral Force Feedback Controller transfer function is shown in equation~\eqref{eq:test_apa_Kiff_formula}.
@@ -10062,7 +10026,7 @@ The transfer function from the ``damped'' plant input \(u\prime\) to the encoder
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_iff_schematic.png}
-\caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\)}
+\caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\).}
\end{figure}
The identified dynamics were then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see~\cite{gustavsen99_ration_approx_frequen_domain_respon}}.
@@ -10085,11 +10049,11 @@ The two obtained root loci are compared in Figure~\ref{fig:test_apa_iff_root_loc
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_apa_iff_root_locus.png}
\end{center}
-\subcaption{\label{fig:test_apa_iff_root_locus}Root Locus plot using the plant model (black) and poles of the identified damped plants (color crosses)}
+\subcaption{\label{fig:test_apa_iff_root_locus}Root locus plot using the plant model (black) and poles of the identified damped plants (color crosses)}
\end{subfigure}
-\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}}
+\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and root locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}.}
\end{figure}
-\subsection{APA300ML - 2 degrees-of-freedom Model}
+\subsection{Two degrees-of-freedom Model}
\label{sec:test_apa_model_2dof}
In this section, a multi-body model (Figure~\ref{fig:test_apa_bench_model}) of the measurement bench is used to tune the two degrees-of-freedom model of the \acrshort{apa} using the measured \acrshortpl{frf}.
@@ -10100,7 +10064,7 @@ After the model is presented, the procedure for tuning the model is described, a
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_bench_model.png}
-\caption{\label{fig:test_apa_bench_model}Screenshot of the multi-body model}
+\caption{\label{fig:test_apa_bench_model}Screenshot of the multi-body model.}
\end{figure}
\paragraph{Two degrees-of-freedom APA Model}
@@ -10125,23 +10089,23 @@ Such a simple model has some limitations:
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_2dof_model.png}
-\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees-of-freedom model of the APA300ML, adapted from~\cite{souleille18_concep_activ_mount_space_applic}}
+\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees-of-freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}.}
\end{figure}
9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure~\ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section~\ref{sec:test_apa_dynamics}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_2dof_model_simscape.png}
-\caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees-of-freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
+\caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees-of-freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\).}
\end{figure}
First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
Both methods lead to an estimated mass of \(m = 5.7\,\text{kg}\).
-Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,\text{N}/\mu\text{m}\) in Section~\ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure~\ref{fig:test_apa_frf_force}.
+Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,\text{N}/\upmu\text{m}\) in Section~\ref{sec:test_apa_dynamics} from the frequency of the anti-resonance seen on Figure~\ref{fig:test_apa_frf_force}.
Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(5\,\text{Ns/m}\).
-Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not completely correct as it was shown in Section~\ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}.
+Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not completely correct as electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}.
Therefore, we have \(k_e = 2 k_a\) and \(c_e = 2 c_a\) as the actuator stack is composed of two stacks in series.
In this case, the total stiffness of the \acrshort{apa} model is described by~\eqref{eq:test_apa_2dof_stiffness}.
@@ -10149,10 +10113,10 @@ In this case, the total stiffness of the \acrshort{apa} model is described by~\e
k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a
\end{equation}
-Knowing from~\eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{\text{tot}} = 2\,\text{N}/\mu\text{m}\), we get from~\eqref{eq:test_apa_2dof_stiffness} that \(k_a = 2.5\,\text{N}/\mu\text{m}\) and \(k_e = 5\,\text{N}/\mu\text{m}\).
+Knowing from~\eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{\text{tot}} = 2\,\text{N}/\upmu\text{m}\), we get from~\eqref{eq:test_apa_2dof_stiffness} that \(k_a = 2.5\,\text{N}/\upmu\text{m}\) and \(k_e = 5\,\text{N}/\upmu\text{m}\).
\begin{equation}\label{eq:test_apa_tot_stiffness}
-\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,\text{N}/\mu\text{m} \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s
+\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,\text{N}/\upmu\text{m} \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s
\end{equation}
Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance.
@@ -10163,23 +10127,24 @@ In the last step, \(g_s\) and \(g_a\) can be tuned to match the gain of the iden
The obtained parameters of the model shown in Figure~\ref{fig:test_apa_2dof_model_simscape} are summarized in Table~\ref{tab:test_apa_2dof_parameters}.
\begin{table}[htbp]
-\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
\centering
\begin{tabularx}{0.25\linewidth}{cc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
\(m\) & \(5.7\,\text{kg}\)\\
-\(k_1\) & \(0.38\,\text{N}/\mu\text{m}\)\\
-\(k_e\) & \(5.0\,\text{N}/\mu\text{m}\)\\
-\(k_a\) & \(2.5\,\text{N}/\mu\text{m}\)\\
+\(k_1\) & \(0.38\,\text{N}/\upmu\text{m}\)\\
+\(k_e\) & \(5.0\,\text{N}/\upmu\text{m}\)\\
+\(k_a\) & \(2.5\,\text{N}/\upmu\text{m}\)\\
\(c_1\) & \(5\,\text{Ns/m}\)\\
\(c_e\) & \(100\,\text{Ns/m}\)\\
\(c_a\) & \(50\,\text{Ns/m}\)\\
\(g_a\) & \(-2.58\,\text{N/V}\)\\
-\(g_s\) & \(0.46\,\text{V}/\mu\text{m}\)\\
+\(g_s\) & \(0.46\,\text{V}/\upmu\text{m}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2-DoFs APA300ML model.}
+
\end{table}
The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table~\ref{tab:test_apa_2dof_parameters}) from the multi-body model.
This is compared with the experimental data in Figure~\ref{fig:test_apa_2dof_comp_frf}.
@@ -10199,9 +10164,9 @@ This indicates that this model represents well the axial dynamics of the APA300M
\end{center}
\subcaption{\label{fig:test_apa_2dof_comp_frf_force}from $u$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
+\caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2-DoFs model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force}).}
\end{figure}
-\subsection{APA300ML - Super Element}
+\subsection{Reduced Order Flexible Model}
\label{sec:test_apa_model_flexible}
In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
@@ -10212,18 +10177,18 @@ Several \emph{remote points} are defined in the \acrshort{fem} (here illustrated
For the APA300ML \emph{super element}, 5 \emph{remote points} are defined.
Two \emph{remote points} (\texttt{1} and \texttt{2}) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used to connect the APA300ML with other mechanical elements.
Two \emph{remote points} (\texttt{3} and \texttt{4}) are located across two piezoelectric stacks and are used to apply internal forces representing the actuator stacks.
-Finally, two \emph{remote points} (\texttt{4} and \texttt{5}) are located across the third piezoelectric stack, and will be used to measured the strain of the sensor stack.
+Finally, two \emph{remote points} (\texttt{4} and \texttt{5}) are located across the third piezoelectric stack, and will be used to measure the strain of the sensor stack.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_simscape.png}
\caption{\label{fig:test_apa_super_element_simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Multi-Body model with included ``Reduced Order Flexible Solid'' on the right (here in Simulink-Simscape software).}
\end{figure}
-\paragraph{Identification of the Actuator and Sensor constants}
+\paragraph{Identification of the Actuator and Sensor ``Constants''}
Once the APA300ML \emph{super element} is included in the multi-body model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be extracted.
The gains \(g_a\) and \(g_s\) are then tuned such that the gains of the transfer functions match the identified ones.
-By doing so, \(g_s = 4.9\,\text{V}/\mu\text{m}\) and \(g_a = 23.2\,\text{N/V}\) are obtained.
+By doing so, \(g_s = 4.9\,\text{V}/\upmu\text{m}\) and \(g_a = 23.2\,\text{N/V}\) are obtained.
To ensure that the sensitivities \(g_a\) and \(g_s\) are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material.
@@ -10240,10 +10205,9 @@ Unfortunately, the manufacturer of the stack was not willing to share the piezoe
However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table~\ref{tab:test_apa_piezo_properties}.
-From these parameters, \(g_s = 5.1\,\text{V}/\mu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained, which are close to the constants identified using the experimentally identified transfer functions.
+From these parameters, \(g_s = 5.1\,\text{V}/\upmu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained, which are close to the constants identified using the experimentally identified transfer functions.
\begin{table}[htbp]
-\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities}
\centering
\begin{tabularx}{0.8\linewidth}{ccX}
\toprule
@@ -10258,8 +10222,10 @@ From these parameters, \(g_s = 5.1\,\text{V}/\mu\text{m}\) and \(g_a = 26\,\text
\(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuator sensitivities.}
+
\end{table}
-\paragraph{Comparison of the obtained dynamics}
+\paragraph{Comparison of the Obtained Dynamics}
The obtained dynamics using the \emph{super element} with the tuned ``sensor sensitivity'' and ``actuator sensitivity'' are compared with the experimentally identified \acrshortpl{frf} in Figure~\ref{fig:test_apa_super_element_comp_frf}.
A good match between the model and the experimental results was observed.
@@ -10280,7 +10246,7 @@ Using this simple test bench, it can be concluded that the \emph{super element}
\end{center}
\subcaption{\label{fig:test_apa_super_element_comp_frf_force}from $u$ to $V_s$}
\end{subfigure}
-\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})}
+\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}
\label{sec:test_apa_conclusion}
@@ -10312,19 +10278,20 @@ Deviations from these ideal properties will impact the dynamics of the Nano-Hexa
During the detailed design phase, specifications in terms of stiffness and stroke were determined and are summarized in Table~\ref{tab:test_joints_specs}.
\begin{table}[htbp]
-\caption{\label{tab:test_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model}
\centering
\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
& \textbf{Specification} & \textbf{FEM}\\
\midrule
-Axial Stiffness & \(> 100\,\text{N}/\mu\text{m}\) & 94\\
-Shear Stiffness & \(> 1\,\text{N}/\mu\text{m}\) & 13\\
+Axial Stiffness & \(> 100\,\text{N}/\upmu\text{m}\) & 94\\
+Shear Stiffness & \(> 1\,\text{N}/\upmu\text{m}\) & 13\\
Bending Stiffness & \(< 100\,\text{Nm}/\text{rad}\) & 5\\
Torsion Stiffness & \(< 500\,\text{Nm}/\text{rad}\) & 260\\
Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model.}
+
\end{table}
After optimization using a \acrshort{fem}, the geometry shown in Figure~\ref{fig:test_joints_schematic} has been obtained and the corresponding flexible joint characteristics are summarized in Table~\ref{tab:test_joints_specs}.
@@ -10356,7 +10323,7 @@ The rotation axes are represented by the dashed lines that intersect
\end{center}
\subcaption{\label{fig:test_joints_xz_plane}XZ plane}
\end{subfigure}
-\caption{\label{fig:test_joints_schematic}Geometry of the optimized flexible joints}
+\caption{\label{fig:test_joints_schematic}Geometry of the optimized flexible joints.}
\end{figure}
Sixteen flexible joints have been ordered (shown in Figure~\ref{fig:test_joints_received}) such that some selection can be made for the twelve that will be used on the nano-hexapod.
@@ -10374,7 +10341,7 @@ Sixteen flexible joints have been ordered (shown in Figure~\ref{fig:test_joints_
\end{center}
\subcaption{\label{fig:test_joints_received_zoom}Zoom on one flexible joint}
\end{subfigure}
-\caption{\label{fig:test_joints_picture}Pictures of the received 16 flexible joints}
+\caption{\label{fig:test_joints_picture}Pictures of the received 16 flexible joints.}
\end{figure}
In this document, the received flexible joints are characterized to ensure that they fulfill the requirements and such that they can well be modeled.
@@ -10390,11 +10357,11 @@ Results are shown in Section~\ref{sec:test_joints_bending_stiffness_meas}
Two dimensions are critical for the bending stiffness of the flexible joints.
These dimensions can be measured using a profilometer.
-The dimensions of the flexible joint in the Y-Z plane will contribute to the X-bending stiffness, whereas the dimensions in the X-Z plane will contribute to the Y-bending stiffness.
+The dimensions of the flexible joint in the YZ plane will contribute to the X-bending stiffness, whereas the dimensions in the X-Z plane will contribute to the Y-bending stiffness.
The setup used to measure the dimensions of the ``X'' flexible beam is shown in Figure~\ref{fig:test_joints_profilometer_setup}.
What is typically observed is shown in Figure~\ref{fig:test_joints_profilometer_image}.
-It is then possible to estimate the dimension of the flexible beam with an accuracy of \(\approx 5\,\mu\text{m}\),
+It is then possible to estimate the dimension of the flexible beam with an accuracy of \(\approx 5\,\upmu\text{m}\),
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -10409,43 +10376,42 @@ It is then possible to estimate the dimension of the flexible beam with an accur
\end{center}
\subcaption{\label{fig:test_joints_profilometer_image}Picture of the gap}
\end{subfigure}
-\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})}
+\caption{\label{fig:test_joints_profilometer}Setup to measure the dimensions of the flexible ``neck'' corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and an image is obtained with which the ``neck'' size can be estimated (\subref{fig:test_joints_profilometer_image}).}
\end{figure}
\subsubsection{Measurement Results}
-The specified flexible beam thickness (gap) is \(250\,\mu\text{m}\).
+The specified flexible beam thickness (gap) is \(250\,\upmu\text{m}\).
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.
A histogram of the measured beam thicknesses is shown in Figure~\ref{fig:test_joints_size_hist}.
-The measured thickness is less than the specified value of \(250\,\mu\text{m}\), but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment.
-
+The measured thickness is less than the specified value of \(250\,\upmu\text{m}\), but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment.
However, what is more important than the true value of the thickness is the consistency between all flexible joints.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_joints_size_hist.png}
-\caption{\label{fig:test_joints_size_hist}Histogram for the (16x2) measured beams' thicknesses}
+\caption{\label{fig:test_joints_size_hist}Histogram for the measured beams' thicknesses.}
\end{figure}
-\subsubsection{Bad flexible joints}
+\subsubsection{Defects in 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}).
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=6cm]{figs/test_joints_bad_shape.jpg}
+\includegraphics[scale=1,height=5cm]{figs/test_joints_bad_shape.jpg}
\end{center}
\subcaption{\label{fig:test_joints_bad_shape}Non-Symmetrical shape}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=6cm]{figs/test_joints_bad_chips.jpg}
+\includegraphics[scale=1,height=5cm]{figs/test_joints_bad_chips.jpg}
\end{center}
\subcaption{\label{fig:test_joints_bad_chips}"Chips" stuck in the air gap}
\end{subfigure}
-\caption{\label{fig:test_joints_bad}Example of two flexible joints that were considered unsatisfactory after visual inspection}
+\caption{\label{fig:test_joints_bad}Example of two flexible joints that were considered unsatisfactory after visual inspection.}
\end{figure}
-\subsection{Compliance Measurement Test Bench}
+\subsection{Characterization Test Bench}
\label{sec:test_joints_test_bench_desc}
The most important characteristic of the flexible joint to be measured is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
@@ -10455,9 +10421,9 @@ 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}
-\subsubsection{Measurement principle}
+\subsubsection{Measurement Principle}
\label{ssec:test_joints_meas_principle}
-\paragraph{Torque and Rotation measurement}
+\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.
In this case, the equivalent applied torque can be estimated from equation~\eqref{eq:test_joints_force_torque_distance}.
Note that the application point of the force should be sufficiently far from the rotation axis such that the resulting bending motion is much larger than the displacement due to shear.
@@ -10490,9 +10456,9 @@ The deflection of the joint \(d_x\) is measured using a displacement sensor.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_joints_bench_working_principle.png}
-\caption{\label{fig:test_joints_bench_working_principle}Working principle of the test bench used to estimate the bending stiffness \(k_{R_y}\) of the flexible joints by measuring \(F_x\), \(d_x\) and \(h\)}
+\caption{\label{fig:test_joints_bench_working_principle}Test bench used to estimate the bending stiffness \(k_{R_y}\) of the flexible joints by measuring \(F_x\), \(d_x\) and \(h\).}
\end{figure}
-\paragraph{Required external applied force}
+\paragraph{Required External Applied Force}
The bending stiffness is foreseen to be \(k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,\text{mrad}\).
The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 22.5\,\text{mm}\) (see Figure~\ref{fig:test_joints_bench_working_principle}).
@@ -10508,7 +10474,7 @@ The measurement range of the force sensor should then be higher than \(5.5\,\tex
\begin{equation}\label{eq:test_joints_max_force}
F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} \approx 5.5\,\text{N}
\end{equation}
-\paragraph{Required actuator stroke and sensors range}
+\paragraph{Required Actuator Stroke and Sensors Range}
The flexible joint is designed to allow a bending motion of \(\pm 25\,\text{mrad}\).
The corresponding stroke at the location of the force sensor is given by~\eqref{eq:test_joints_max_stroke}.
To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least \(0.5\,\text{mm}\).
@@ -10517,12 +10483,12 @@ Similarly, the measurement range of the displacement sensor should also be highe
\begin{equation}\label{eq:test_joints_max_stroke}
d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,\text{mm}
\end{equation}
-\paragraph{Force and Displacement measurements}
+\paragraph{Force and Displacement Measurements}
To determine the applied force, a load cell will be used in series with the mechanism that applied the force.
The measured deflection of the flexible joint will be indirectly estimated from the displacement of the force sensor itself (see Section~\ref{ssec:test_joints_test_bench}).
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})
-\subsubsection{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.
@@ -10556,8 +10522,8 @@ The estimated bending stiffness \(k_{\text{est}}\) then depends on the shear sti
k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_s h^2}}_{\epsilon_{s}} \Bigl)
\end{equation}
-With an estimated shear stiffness \(k_s = 13\,\text{N}/\mu\text{m}\) from the \acrshort{fem} and an height \(h=25\,\text{mm}\), the estimation errors of the bending stiffness due to shear is \(\epsilon_s < 0.1\,\%\)
-\paragraph{Effect of load cell limited stiffness}
+With an estimated shear stiffness \(k_s = 13\,\text{N}/\upmu\text{m}\) from the \acrshort{fem} and an height \(h=25\,\text{mm}\), the estimation errors of the bending stiffness due to shear is \(\epsilon_s < 0.1\,\%\)
+\paragraph{Effect of Load Cell Limited Stiffness}
As explained in the previous section, because the measurement of the flexible joint deflection is indirectly performed with the encoder, errors will be made if the load cell experiences some compression.
Suppose the load cell has an internal stiffness \(k_f\), the same reasoning that was made for the effect of shear can be applied here.
@@ -10567,8 +10533,8 @@ The estimation error of the bending stiffness due to the limited stiffness of th
k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_F h^2}}_{\epsilon_f} \Bigl)
\end{equation}
-With an estimated load cell stiffness of \(k_f \approx 1\,\text{N}/\mu\text{m}\) (from the documentation), the errors due to the load cell limited stiffness is around \(\epsilon_f = 1\,\%\).
-\paragraph{Estimation error due to height estimation error}
+With an estimated load cell stiffness of \(k_f \approx 1\,\text{N}/\upmu\text{m}\) (from the documentation), the errors due to the load cell limited stiffness is around \(\epsilon_f = 1\,\%\).
+\paragraph{Height Estimation Error}
Now consider an error \(\delta h\) in the estimation of the height \(h\) as described by~\eqref{eq:test_joints_est_h_error}.
\begin{equation}\label{eq:test_joints_est_h_error}
@@ -10582,7 +10548,7 @@ The computed bending stiffness will be~\eqref{eq:test_joints_stiffness_height_er
\end{equation}
The height estimation is foreseen to be accurate to within \(|\delta h| < 0.4\,\text{mm}\) which corresponds to a stiffness error \(\epsilon_h < 3.5\,\%\).
-\paragraph{Estimation error due to force and displacement sensors accuracy}
+\paragraph{Force and Displacement Sensors Accuracy}
An optical encoder is used to measure the displacement (see Section~\ref{ssec:test_joints_test_bench}) whose maximum non-linearity is \(40\,\text{nm}\).
As the measured displacement is foreseen to be \(0.5\,\text{mm}\), the error \(\epsilon_d\) due to the encoder non-linearity is negligible \(\epsilon_d < 0.01\,\%\).
@@ -10594,7 +10560,6 @@ The most important source of error is the estimation error of the distance betwe
An overall accuracy of \(\approx 5\,\%\) can be expected with this measurement bench, which should be sufficient for an estimation of the bending stiffness of the flexible joints.
\begin{table}[htbp]
-\caption{\label{tab:test_joints_error_budget}Summary of the error budget for estimating the bending stiffness}
\centering
\begin{tabularx}{0.35\linewidth}{Xc}
\toprule
@@ -10607,6 +10572,8 @@ Displacement sensor & \(\epsilon_d < 0.01\,\%\)\\
Force sensor & \(\epsilon_F < 1\,\%\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_joints_error_budget}Summary of the error budget for the estimation of the bending stiffness.}
+
\end{table}
\subsubsection{Mechanical Design}
\label{ssec:test_joints_test_bench}
@@ -10622,7 +10589,7 @@ Instead of measuring the displacement directly at the tip of the flexible joint
To do so, an encoder\footnote{Resolute\texttrademark{} encoder with \(1\,\text{nm}\) resolution and \(\pm 40\,\text{nm}\) maximum non-linearity} is used, which measures the motion of a ruler.
This ruler is fixed to the translation stage in line (i.e. at the same height) with the application point to reduce Abbe errors (see Figure~\ref{fig:test_joints_bench_overview}).
-The flexible joint can be rotated by \(90^o\) in order to measure the bending stiffness in the two directions.
+The flexible joint can be rotated by \(\SI{90}{\degree}\) in order to measure the bending stiffness in the two directions.
The obtained design of the measurement bench is shown in Figure~\ref{fig:test_joints_bench_overview} while a zoom on the flexible joint with the associated important quantities is shown in Figure~\ref{fig:test_joints_bench_side}.
\begin{figure}[htbp]
@@ -10638,7 +10605,7 @@ The obtained design of the measurement bench is shown in Figure~\ref{fig:test_jo
\end{center}
\subcaption{\label{fig:test_joints_bench_side} Zoom}
\end{subfigure}
-\caption{\label{fig:test_joints_bench}3D view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side})}
+\caption{\label{fig:test_joints_bench}3D view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side}).}
\end{figure}
\subsection{Bending Stiffness Measurement}
\label{sec:test_joints_bending_stiffness_meas}
@@ -10658,7 +10625,7 @@ A closer view of the force sensor tip is shown in Figure~\ref{fig:test_joints_pi
\end{center}
\subcaption{\label{fig:test_joints_picture_bench_zoom}Zoom on the tip}
\end{subfigure}
-\caption{\label{fig:test_joints_picture_bench}Manufactured test bench for compliance measurement of the flexible joints}
+\caption{\label{fig:test_joints_picture_bench}Manufactured test bench for compliance measurement of the flexible joints.}
\end{figure}
\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 \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range}.
@@ -10682,14 +10649,14 @@ However, the estimated non-linearity is bellow \(0.2\,\%\) for forces between \(
\end{center}
\subcaption{\label{fig:test_joints_force_sensor_calib_fit}Measured two forces}
\end{subfigure}
-\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})}
+\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}
\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 \(k_F \approx 1\,\text{N}/\mu\text{m}\)) is mounted in front of the force sensor, as shown in Figure~\ref{fig:test_joints_meas_force_sensor_stiffness_picture}.
+To do so, a stiff element (much stiffer than the estimated \(k_F \approx 1\,\text{N}/\upmu\text{m}\)) 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.
The measured displacement as a function of the measured force is shown in Figure~\ref{fig:test_joints_force_sensor_stiffness_fit}.
-The load cell stiffness can then be estimated by computing a linear fit and is found to be \(k_F \approx 0.68\,\text{N}/\mu\text{m}\).
+The load cell stiffness can then be estimated by computing a linear fit and is found to be \(k_F \approx 0.68\,\text{N}/\upmu\text{m}\).
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -10706,7 +10673,7 @@ The load cell stiffness can then be estimated by computing a linear fit and is f
\end{subfigure}
\caption{\label{fig:test_joints_meas_force_sensor_stiffness}Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}), and results are shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}).}
\end{figure}
-\subsubsection{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}.
@@ -10731,7 +10698,7 @@ The bending stroke can also be estimated as shown in Figure~\ref{fig:test_joints
\end{subfigure}
\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}
-\subsubsection{Measured flexible joint stiffness}
+\subsubsection{Measured Flexible Joints' Stiffnesses}
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.
@@ -10753,7 +10720,7 @@ Most of the bending stiffnesses are between \(4.6\,\text{Nm/rad}\) and \(5.0\,\t
\end{center}
\subcaption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness in the x and y directions}
\end{subfigure}
-\caption{\label{fig:test_joints_meas_bending_results}Result of measured \(k_{R_x}\) and \(k_{R_y}\) stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}).}
+\caption{\label{fig:test_joints_meas_bending_results}Measured \(k_{R_x}\) and \(k_{R_y}\) stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}).}
\end{figure}
\subsection*{Conclusion}
\label{sec:test_joints_conclusion}
@@ -10774,7 +10741,7 @@ The Nano-Hexapod struts (shown in Figure~\ref{fig:test_struts_picture_strut}) ar
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_struts_picture_strut.jpg}
-\caption{\label{fig:test_struts_picture_strut}One strut including two flexible joints, an amplified piezoelectric actuator and an encoder}
+\caption{\label{fig:test_struts_picture_strut}One strut including two flexible joints, an amplified piezoelectric actuator and an encoder.}
\end{figure}
After the strut elements have been individually characterized (see previous sections), the struts are assembled.
@@ -10791,7 +10758,7 @@ The strut models were then compared with the measured dynamics (Section~\ref{sec
The model dynamics from the \acrshort{dac} voltage to the axial motion of the strut (measured by an interferometer) and to the force sensor voltage well match the experimental results.
However, this is not the case for the dynamics from \acrshort{dac} voltage to the encoder displacement.
It is found that the complex dynamics is due to a misalignment between the flexible joints and the \acrshort{apa}.
-\subsection{Mounting Procedure}
+\subsection{Assembly Procedure}
\label{sec:test_struts_mounting}
A mounting bench was developed to ensure:
@@ -10805,8 +10772,8 @@ This is important not to loose to much angular stroke during their mounting into
The mounting bench is shown in Figure~\ref{fig:test_struts_mounting_bench_first_concept}.
It consists of a ``main frame'' (Figure~\ref{fig:test_struts_mounting_step_0}) precisely machined to ensure both correct strut length and strut coaxiality.
-The coaxiality is ensured by good flatness (specified at \(20\,\mu\text{m}\)) between surfaces A and B and between surfaces C and D.
-Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\mu\text{m}\)} (see Figure~\ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements.
+The coaxiality is ensured by good flatness (specified at \(20\,\upmu\text{m}\)) between surfaces A and B and between surfaces C and D.
+Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\upmu\text{m}\)} (see Figure~\ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements.
The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes.
\begin{figure}[htbp]
@@ -10822,7 +10789,7 @@ The strut length (defined by the distance between the rotation points of the two
\end{center}
\subcaption{\label{fig:test_struts_mounting_overview}Exploded view}
\end{subfigure}
-\caption{\label{fig:test_struts_mounting}Strut mounting bench}
+\caption{\label{fig:test_struts_mounting}Strut mounting bench.}
\end{figure}
\begin{figure}[htbp]
@@ -10838,7 +10805,7 @@ The strut length (defined by the distance between the rotation points of the two
\end{center}
\subcaption{\label{fig:test_struts_check_dimensions_bench}Dimensional check}
\end{subfigure}
-\caption{\label{fig:test_struts_mounting_base_part}Main element of the mounting bench for the struts that ensure good coaxiality of the two flexible joints and correct struts length.}
+\caption{\label{fig:test_struts_mounting_base_part}Part that ensures good coaxiality of the two flexible joints and correct struts length.}
\end{figure}
The flexible joints were not directly fixed to the mounting bench but were fixed to a cylindrical ``sleeve'' shown in Figures~\ref{fig:test_struts_cylindrical_mounting_part_top} and \ref{fig:test_struts_cylindrical_mounting_part_bot}.
@@ -10850,13 +10817,13 @@ These ``sleeves'' have one dowel groove (that are fitted to the dowel holes show
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_cylindrical_mounting_part_top.jpg}
\end{center}
-\subcaption{\label{fig:test_struts_cylindrical_mounting_part_top}Cylindral Interface (Top)}
+\subcaption{\label{fig:test_struts_cylindrical_mounting_part_top}Cylindrical Interface (Top)}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
\includegraphics[scale=1,height=4.5cm]{figs/test_struts_cylindrical_mounting_part_bot.jpg}
\end{center}
-\subcaption{\label{fig:test_struts_cylindrical_mounting_part_bot}Cylindrlcal Interface (Bottom)}
+\subcaption{\label{fig:test_struts_cylindrical_mounting_part_bot}Cylindrical Interface (Bottom)}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
@@ -10864,7 +10831,7 @@ These ``sleeves'' have one dowel groove (that are fitted to the dowel holes show
\end{center}
\subcaption{\label{fig:test_struts_mounting_joints}Mounted flexible joints}
\end{subfigure}
-\caption{\label{fig:test_struts_cylindrical_mounting}Preparation of the flexible joints by fixing them in their cylindrical ``sleeve''}
+\caption{\label{fig:test_struts_cylindrical_mounting}Preparation of the flexible joints by fixing them in their cylindrical ``sleeves''.}
\end{figure}
The ``sleeves'' were mounted to the main element as shown in Figure~\ref{fig:test_struts_mounting_step_0}.
@@ -10884,13 +10851,13 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_1.jpg}
\end{center}
-\subcaption{\label{fig:test_struts_mounting_step_1}Step 1}
+\subcaption{\label{fig:test_struts_mounting_step_1}Fix the flexible joints}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_2.jpg}
\end{center}
-\subcaption{\label{fig:test_struts_mounting_step_2}Step 2}
+\subcaption{\label{fig:test_struts_mounting_step_2}Mount the APA with the cylindrical washers}
\end{subfigure}
\bigskip
@@ -10898,17 +10865,17 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_3.jpg}
\end{center}
-\subcaption{\label{fig:test_struts_mounting_step_3}Step 3}
+\subcaption{\label{fig:test_struts_mounting_step_3}Mount and align the encoder}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_mounting_step_4.jpg}
\end{center}
-\subcaption{\label{fig:test_struts_mounting_step_4}Step 4}
+\subcaption{\label{fig:test_struts_mounting_step_4}Obtained mounted strut}
\end{subfigure}
\caption{\label{fig:test_struts_mounting_steps}Steps for mounting the struts.}
\end{figure}
-\subsection{Measurement of flexible modes}
+\subsection{Measurement of Flexible Modes}
\label{sec:test_struts_flexible_modes}
A Finite Element Model\footnote{Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called \emph{17-4 PH}. Encoder and ruler support material is aluminium.} of the struts is developed and is used to estimate the flexible modes.
@@ -10935,7 +10902,7 @@ The mode shapes are displayed in Figure~\ref{fig:test_struts_mode_shapes}: an ``
\end{center}
\subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode ($400\,\text{Hz}$)}
\end{subfigure}
-\caption{\label{fig:test_struts_mode_shapes}Spurious resonances of the struts estimated from a Finite Element Model}
+\caption{\label{fig:test_struts_mode_shapes}Flexible modes of the struts estimated from a Finite Element Model.}
\end{figure}
To experimentally measure these mode shapes, a Laser vibrometer\footnote{OFV-3001 controller and OFV512 sensor head from Polytec} was used.
@@ -10965,7 +10932,7 @@ These tests were performed with and without the encoder being fixed to the strut
\end{center}
\subcaption{\label{fig:test_struts_meas_z_torsion}Z-torsion mode}
\end{subfigure}
-\caption{\label{fig:test_struts_meas_modes}Measurement of strut flexible modes}
+\caption{\label{fig:test_struts_meas_modes}Measurement of the flexible modes of the struts using a laser vibrometer.}
\end{figure}
The obtained \acrshortpl{frf} for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure~\ref{fig:test_struts_spur_res_frf_no_enc} when the encoder is not fixed to the strut and in Figure~\ref{fig:test_struts_spur_res_frf_enc} when the encoder is fixed to the strut.
@@ -10983,7 +10950,7 @@ The obtained \acrshortpl{frf} for the three configurations (X-bending, Y-bending
\end{center}
\subcaption{\label{fig:test_struts_spur_res_frf_enc}with the encoder}
\end{subfigure}
-\caption{\label{fig:test_struts_spur_res_frf}Measured frequency response functions without the encoder~\ref{fig:test_struts_spur_res_frf} and with the encoder~\ref{fig:test_struts_spur_res_frf_enc}}
+\caption{\label{fig:test_struts_spur_res_frf}Measured frequency response functions without the encoder (\subref{fig:test_struts_spur_res_frf_no_enc}) and with the encoder (\subref{fig:test_struts_spur_res_frf_enc}).}
\end{figure}
Table~\ref{tab:test_struts_spur_mode_freqs} summarizes the measured resonance frequencies and the computed ones using the \acrfull{fem}.
@@ -10992,7 +10959,6 @@ In addition, the computed resonance frequencies from the \acrshort{fem} are very
This validates the quality of the \acrshort{fem}.
\begin{table}[htbp]
-\caption{\label{tab:test_struts_spur_mode_freqs}Measured frequency of the flexible modes of the strut}
\centering
\begin{tabularx}{0.7\linewidth}{Xccc}
\toprule
@@ -11003,8 +10969,10 @@ Y-Bending & \(285\,\text{Hz}\) & \(293\,\text{Hz}\) & \(337\,\text{Hz}\)\\
Z-Torsion & \(400\,\text{Hz}\) & \(381\,\text{Hz}\) & \(398\,\text{Hz}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_struts_spur_mode_freqs}Estimated and measured frequencies of the flexible modes of the struts.}
+
\end{table}
-\subsection{Dynamical measurements}
+\subsection{Dynamical Measurements}
\label{sec:test_struts_dynamical_meas}
In order to measure the dynamics of the strut, the test bench used to measure the APA300ML dynamics is being used again.
@@ -11015,13 +10983,13 @@ A fiber interferometer\footnote{Two fiber intereferometers were used: an IDS3010
\begin{figure}[htbp]
\begin{subfigure}{0.3\textwidth}
\begin{center}
-\includegraphics[scale=1,height=210px]{figs/test_struts_bench_leg_overview.jpg}
+\includegraphics[scale=1,height=6cm]{figs/test_struts_bench_leg_overview.jpg}
\end{center}
\subcaption{\label{fig:test_struts_bench_leg_overview}Overview Picture}
\end{subfigure}
\begin{subfigure}{0.66\textwidth}
\begin{center}
-\includegraphics[scale=1,height=210px]{figs/test_struts_bench_schematic.png}
+\includegraphics[scale=1,height=6cm]{figs/test_struts_bench_schematic.png}
\end{center}
\subcaption{\label{fig:test_struts_bench_schematic}Schematic}
\end{subfigure}
@@ -11031,7 +10999,7 @@ 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}.
-\subsubsection{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}).
@@ -11049,7 +11017,7 @@ System identification was performed without the encoder being fixed to the strut
\end{center}
\subcaption{\label{fig:test_struts_bench_leg_front}Strut without encoder}
\end{subfigure}
-\caption{\label{fig:test_struts_bench_leg_with_without_enc}Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})}
+\caption{\label{fig:test_struts_bench_leg_with_without_enc}Strut fixed to the test bench with clamped flexible joints. The encoder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front}).}
\end{figure}
The obtained \acrshortpl{frf} are compared in Figure~\ref{fig:test_struts_effect_encoder}.
@@ -11077,9 +11045,9 @@ This means that the encoder should have little effect on the effectiveness of th
\end{center}
\subcaption{\label{fig:test_struts_comp_enc_int}$u$ to $d_e$, $d_a$}
\end{subfigure}
-\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})}
+\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}
-\subsubsection{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}.
@@ -11101,21 +11069,21 @@ A very good match can be observed between the struts.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_interf_plants.png}
\end{center}
-\subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$}
+\subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$ (interferometer)}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_iff_plants.png}
\end{center}
-\subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$}
+\subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$ (force sensor)}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_enc_plants.png}
\end{center}
-\subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$}
+\subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$ (encoder)}
\end{subfigure}
-\caption{\label{fig:test_struts_comp_plants}Comparison of the measured plants}
+\caption{\label{fig:test_struts_comp_plants}Comparison of the measured dynamics for five of the struts..}
\end{figure}
The same comparison is made for the transfer function from \(u\) to \(d_e\) (encoder output) in Figure~\ref{fig:test_struts_comp_enc_plants}.
@@ -11136,20 +11104,20 @@ The struts were then disassembled and reassemble a second time to optimize align
\begin{figure}[htbp]
\centering
\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}
+\caption{\label{fig:test_struts_simscape_model}Multi-body model of the strut fixed to the bench.}
\end{figure}
-\subsubsection{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}.
These two models of the APA300ML were tuned to best match the measured \acrshortpl{frf} of the \acrshort{apa} alone.
-The flexible joints were modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness).
+The flexible joints were modeled with the 4-DoFs model (axial stiffness, two bending stiffnesses and one torsion stiffness).
These two models are compared using the measured \acrshortpl{frf} in Figure~\ref{fig:test_struts_comp_frf_flexible_model}.
The model dynamics from DAC voltage \(u\) to the axial motion of the strut \(d_a\) (Figure~\ref{fig:test_struts_comp_frf_flexible_model_int}) and from DAC voltage \(u\) to the force sensor voltage \(V_s\) (Figure~\ref{fig:test_struts_comp_frf_flexible_model_iff}) are well matching the experimental identification.
However, the transfer function from \(u\) to encoder displacement \(d_e\) are not well matching for both models.
-For the 2DoF model, this is normal because the resonances affecting the dynamics are not modelled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis).
+For the 2-DoFs model, this is normal because the resonances affecting the dynamics are not modeled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis).
For the flexible model, it will be shown in the next section that by adding some misalignment between the flexible joints and the APA300ML, this model can better represent the observed dynamics.
\begin{figure}[htbp]
@@ -11157,23 +11125,23 @@ For the flexible model, it will be shown in the next section that by adding some
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_frf_flexible_model_int.png}
\end{center}
-\subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$}
+\subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$ (interferometer)}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_frf_flexible_model_enc.png}
\end{center}
-\subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$}
+\subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$ (encoder)}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_frf_flexible_model_iff.png}
\end{center}
-\subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$}
+\subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$ (force sensor)}
\end{subfigure}
-\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).}
+\caption{\label{fig:test_struts_comp_frf_flexible_model}Comparison of the measured dynamics of the struts (black) with dynamics extracted from the multi-body model using the 2-DoFs APA model (blue), and using the reduced order flexible model of the APA300ML model (red).}
\end{figure}
-\subsubsection{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.
@@ -11184,7 +11152,7 @@ In this case, the ``x-bending'' mode at \(200\,\text{Hz}\) (see Figure~\ref{fig:
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_struts_misalign_schematic.png}
-\caption{\label{fig:test_struts_misalign_schematic}Mis-alignement between the joints and the APA}
+\caption{\label{fig:test_struts_misalign_schematic}Misalignment between the joints and the APA.}
\end{figure}
To verify this assumption, the dynamics from the output DAC voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed using the flexible \acrshort{apa} model for several misalignments in the \(y\) direction.
@@ -11220,9 +11188,9 @@ This similarity suggests that the identified differences in dynamics are caused
\end{center}
\subcaption{\label{fig:test_struts_effect_misalignment_x}Misalignment along $x$}
\end{subfigure}
-\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})}
+\caption{\label{fig:test_struts_effect_misalignment}Effect of a misalignment between the flexible joints and the APA300ML in the \(y\) (\subref{fig:test_struts_effect_misalignment_y}) and in the \(x\) direction (\subref{fig:test_struts_effect_misalignment_x}).}
\end{figure}
-\subsubsection{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 \acrshort{apa} with respect to the flexible joints in the \(y\) directions were not used (not received at the time).
@@ -11231,7 +11199,7 @@ Therefore, large \(y\) misalignments are expected.
To estimate the misalignments between the two flexible joints and the \acrshort{apa}:
\begin{itemize}
\item the struts were fixed horizontally on the mounting bench, as shown in Figure~\ref{fig:test_struts_mounting_step_3} but without the encoder
-\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu\text{m}\)}, the height difference between the flexible joints surface and the \acrshort{apa} shell surface was measured for both the top and bottom joints and for both sides
+\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\upmu\text{m}\)}, the height difference between the flexible joints surface and the \acrshort{apa} shell surface was measured for both the top and bottom joints and for both sides
\item as the thickness of the flexible joint is \(21\,\text{mm}\) and the thickness of the \acrshort{apa} shell is \(20\,\text{mm}\), \(0.5\,\text{mm}\) of height difference should be measured if the two are perfectly aligned
\end{itemize}
@@ -11241,7 +11209,6 @@ To check the validity of the measurement, it can be verified that the sum of the
Thickness differences for all the struts were found to be between \(0.94\,\text{mm}\) and \(1.00\,\text{mm}\) which indicate low errors compared to the misalignments found in Table~\ref{tab:test_struts_meas_y_misalignment}.
\begin{table}[htbp]
-\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment at the top and bottom of the APA. Measurements are in \(\text{mm}\)}
\centering
\begin{tabularx}{0.2\linewidth}{Xcc}
\toprule
@@ -11254,6 +11221,8 @@ Thickness differences for all the struts were found to be between \(0.94\,\text{
5 & 0.15 & 0.02\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment for each strut. Measurements are in \(\text{mm}\).}
+
\end{table}
By using the measured \(y\) misalignment in the model with the flexible \acrshort{apa} model, the model dynamics from \(u\) to \(d_e\) is closer to the measured dynamics, as shown in Figure~\ref{fig:test_struts_comp_dy_tuned_model_frf_enc}.
@@ -11268,19 +11237,18 @@ With a better alignment, the amplitude of the spurious resonances is expected to
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_dy_tuned_model_frf_enc.png}
-\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}
+\caption{\label{fig:test_struts_comp_dy_tuned_model_frf_enc}Comparison of the frequency response functions from DAC voltage \(u\) to measure displacement \(d_e\) by the encoders for three struts. The measured dynamics is shown in blue, the dynamics extracted from the model with the \(y\) misalignment estimated from measurements is shown in red, and the dynamics extracted from the model when both the \(x\) and \(y\) misalignments are tuned is shown in yellow.}
\end{figure}
-\subsubsection{Proper struts alignment}
+\subsubsection{Better Struts Alignment}
\label{sec:test_struts_meas_all_aligned_struts}
After receiving the positioning pins, the struts were mounted again with the positioning pins.
This should improve the alignment of the \acrshort{apa} with the two flexible joints.
The alignment is then estimated using a length gauge, as described in the previous sections.
-Measured \(y\) alignments are summarized in Table~\ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\mu\text{m}\) for all the struts, which is much better than before (see Table~\ref{tab:test_struts_meas_y_misalignment}).
+Measured \(y\) alignments are summarized in Table~\ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\upmu\text{m}\) for all the struts, which is much better than before (see Table~\ref{tab:test_struts_meas_y_misalignment}).
\begin{table}[htbp]
-\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in \(\text{mm}\).}
\centering
\begin{tabularx}{0.25\linewidth}{Xcc}
\toprule
@@ -11294,6 +11262,8 @@ Measured \(y\) alignments are summarized in Table~\ref{tab:test_struts_meas_y_mi
6 & -0.005 & 0.055\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment after realigning the struts using dowel pins. Measurements are in \(\text{mm}\).}
+
\end{table}
The dynamics of the re-aligned struts were then measured on the same test bench (Figure~\ref{fig:test_struts_bench_leg}).
@@ -11307,7 +11277,7 @@ Therefore, fixing the encoders to the nano-hexapod plates instead may be an inte
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_enc_frf_realign.png}
-\caption{\label{fig:test_struts_comp_enc_frf_realign}Comparison of the dynamics from \(u\) to \(d_e\) before and after proper alignment using the dowel pins}
+\caption{\label{fig:test_struts_comp_enc_frf_realign}Comparison of the dynamics from \(u\) to \(d_e\) before and after proper alignment using the dowel pins.}
\end{figure}
\subsection*{Conclusion}
\label{sec:test_struts_conclusion}
@@ -11329,7 +11299,7 @@ To identify the dynamics of the nano-hexapod, a special suspended table was deve
The Nano-Hexapod was then mounted on top of the suspended table such that its dynamics is not affected by complex dynamics except from the suspension modes of the table that can be well characterized and modeled (Section~\ref{sec:test_nhexa_table}).
The obtained nano-hexapod dynamics is analyzed in Section~\ref{sec:test_nhexa_dynamics}, and compared with the multi-body model in Section~\ref{sec:test_nhexa_model}.
-\subsection{Nano-Hexapod Assembly Procedure}
+\subsection{Assembly Procedure}
\label{sec:test_nhexa_mounting}
The assembly of the nano-hexapod is critical for both avoiding additional stress in the flexible joints (that would result in a loss of stroke) and for precisely determining the Jacobian matrix.
The goal was to fix the six struts to the two nano-hexapod plates (shown in Figure~\ref{fig:test_nhexa_nano_hexapod_plates}) while the two plates were parallel and aligned vertically so that all the flexible joints did not experience any stress.
@@ -11348,11 +11318,11 @@ To do so, a precisely machined mounting tool (Figure~\ref{fig:test_nhexa_center_
\end{center}
\subcaption{\label{fig:test_nhexa_center_part_hexapod_mounting}Mounting tool}
\end{subfigure}
-\caption{\label{fig:test_nhexa_received_parts}Nano-Hexapod plates (\subref{fig:test_nhexa_nano_hexapod_plates}) and mounting tool used to position the two plates during assembly (\subref{fig:test_nhexa_center_part_hexapod_mounting})}
+\caption{\label{fig:test_nhexa_received_parts}Nano-Hexapod plates (\subref{fig:test_nhexa_nano_hexapod_plates}) and mounting tool used to position the two plates during assembly (\subref{fig:test_nhexa_center_part_hexapod_mounting}).}
\end{figure}
-The mechanical tolerances of the received plates were checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\mu\text{m}\)} (Figure~\ref{fig:test_nhexa_plates_tolerances}) and were found to comply with the requirements\footnote{Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.}.
-The same was done for the mounting tool\footnote{The height dimension is better than \(40\,\mu\text{m}\). The diameter fitting of 182g6 and 24g6 with the two plates is verified.}.
+The mechanical tolerances of the received plates were checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\upmu\text{m}\)} (Figure~\ref{fig:test_nhexa_plates_tolerances}) and were found to comply with the requirements\footnote{Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.}.
+The same was done for the mounting tool\footnote{The height dimension is better than \(40\,\upmu\text{m}\). The diameter fitting of 182g6 and 24g6 with the two plates is verified.}.
The two plates were then fixed to the mounting tool, as shown in Figure~\ref{fig:test_nhexa_mounting_tool_hexapod_top_view}.
The main goal of this ``mounting tool'' is to position the flexible joint interfaces (the ``V'' shapes) of both plates so that a cylinder can rest on the 4 flat interfaces at the same time.
@@ -11367,31 +11337,32 @@ The main goal of this ``mounting tool'' is to position the flexible joint interf
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_mounting_tool_hexapod_top_view.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_mounting_tool_hexapod_top_view}Wanted coaxiality between interfaces}
+\subcaption{\label{fig:test_nhexa_mounting_tool_hexapod_top_view}Wanted coaxiality between strut interfaces}
\end{subfigure}
-\caption{\label{fig:test_nhexa_dimensional_check}A FARO arm is used to dimensionally check the received parts (\subref{fig:test_nhexa_plates_tolerances}) and after mounting the two plates with the mounting part (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view})}
+\caption{\label{fig:test_nhexa_dimensional_check}A FARO arm is used to dimensionally check the plates (\subref{fig:test_nhexa_plates_tolerances}) and to verify coaxiality of the strut interfaces (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view}).}
\end{figure}
The quality of the positioning can be estimated by measuring the ``straightness'' of the top and bottom ``V'' interfaces.
This corresponds to the diameter of the smallest cylinder which contains all points along the measured axis.
This was again done using the FARO arm, and the results for all six struts are summarized in Table~\ref{tab:measured_straightness}.
-The straightness was found to be better than \(15\,\mu\text{m}\) for all struts\footnote{As the accuracy of the FARO arm is \(\pm 13\,\mu\text{m}\), the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.}, which is sufficiently good to not induce significant stress of the flexible joint during assembly.
+The straightness was found to be better than \(15\,\upmu\text{m}\) for all struts\footnote{As the accuracy of the FARO arm is \(\pm 13\,\upmu\text{m}\), the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.}, which is sufficiently good to not induce significant stress of the flexible joint during assembly.
\begin{table}[htbp]
-\caption{\label{tab:measured_straightness}Measured straightness between the two ``V'' shapes for the six struts. These measurements were performed twice for each strut.}
\centering
\begin{tabularx}{0.25\linewidth}{Xcc}
\toprule
\textbf{Strut} & \textbf{Meas 1} & \textbf{Meas 2}\\
\midrule
-1 & \(7\,\mu\text{m}\) & \(3\, \mu\text{m}\)\\
-2 & \(11\, \mu\text{m}\) & \(11\, \mu\text{m}\)\\
-3 & \(15\, \mu\text{m}\) & \(14\, \mu\text{m}\)\\
-4 & \(6\, \mu\text{m}\) & \(6\, \mu\text{m}\)\\
-5 & \(7\, \mu\text{m}\) & \(5\, \mu\text{m}\)\\
-6 & \(6\, \mu\text{m}\) & \(7\, \mu\text{m}\)\\
+1 & \(7\,\upmu\text{m}\) & \(3\, \upmu\text{m}\)\\
+2 & \(11\, \upmu\text{m}\) & \(11\, \upmu\text{m}\)\\
+3 & \(15\, \upmu\text{m}\) & \(14\, \upmu\text{m}\)\\
+4 & \(6\, \upmu\text{m}\) & \(6\, \upmu\text{m}\)\\
+5 & \(7\, \upmu\text{m}\) & \(5\, \upmu\text{m}\)\\
+6 & \(6\, \upmu\text{m}\) & \(7\, \upmu\text{m}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:measured_straightness}Measured straightness between the V grooves for the six struts. Measurements were performed twice for each strut.}
+
\end{table}
The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure~\ref{fig:test_nhexa_mount_encoder}), and the encoder heads were aligned to maximize the received contrast.
@@ -11409,7 +11380,7 @@ The encoder rulers and heads were then fixed to the top and bottom plates, respe
\end{center}
\subcaption{\label{fig:test_nhexa_mount_encoder_heads}Encoder heads}
\end{subfigure}
-\caption{\label{fig:test_nhexa_mount_encoder}Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate (\subref{fig:test_nhexa_mount_encoder_rulers}) while encoders heads are fixed to the bottom plate (\subref{fig:test_nhexa_mount_encoder_heads})}
+\caption{\label{fig:test_nhexa_mount_encoder}Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate (\subref{fig:test_nhexa_mount_encoder_rulers}) while encoders heads are fixed to the bottom plate (\subref{fig:test_nhexa_mount_encoder_heads}).}
\end{figure}
The six struts were then fixed to the bottom and top plates one by one.
@@ -11421,7 +11392,7 @@ After mounting all six struts, the mounting tool (Figure~\ref{fig:test_nhexa_cen
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_mounted_hexapod.jpg}
-\caption{\label{fig:test_nhexa_nano_hexapod_mounted}Mounted Nano-Hexapod}
+\caption{\label{fig:test_nhexa_nano_hexapod_mounted}Mounted Nano-Hexapod.}
\end{figure}
\subsection{Suspended Table}
\label{sec:test_nhexa_table}
@@ -11443,7 +11414,7 @@ Finally, the multi-body model representing the suspended table was tuned to matc
\label{ssec:test_nhexa_table_setup}
The design of the suspended table is quite straightforward.
-First, an optical table with high frequency flexible mode was selected\footnote{The 450 mm x 450 mm x 60 mm Nexus B4545A from Thorlabs.}.
+First, an optical table with high frequency flexible mode was selected\footnote{The \(450\,\text{mm} \times 450\,\text{mm} \times 60\,\text{mm}\) Nexus B4545A from Thorlabs.}.
Then, four springs\footnote{``SZ8005 20 x 044'' from Steinel. The spring rate is specified at \(17.8\,\text{N/mm}\)} were selected with low spring rate such that the suspension modes are below \(10\,\text{Hz}\).
Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure~\ref{fig:test_nhexa_suspended_table_cad}).
@@ -11452,7 +11423,7 @@ Finally, some interface elements were designed, and mechanical lateral mechanica
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_suspended_table_cad.jpg}
\caption{\label{fig:test_nhexa_suspended_table_cad}3D View of the vibration table. The purple cylinders are representing the soft springs.}
\end{figure}
-\subsubsection{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 \(1\,\text{V/g}\), measurement range is \(\pm 5\,\text{g}\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} were fixed to the breadboard.
@@ -11460,10 +11431,10 @@ Using an instrumented hammer, the first 9 modes could be identified and are summ
The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below \(10\,\text{Hz}\).
The next modes are the flexible modes of the breadboard as shown in Figure~\ref{fig:test_nhexa_table_flexible_modes}, and are located above \(700\,\text{Hz}\).
-\begin{minipage}[t]{0.45\textwidth}
+\begin{minipage}[b]{0.45\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.99\linewidth]{figs/test_nhexa_suspended_table.jpg}
-\captionof{figure}{\label{fig:test_nhexa_suspended_table}Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top}
+\captionof{figure}{\label{fig:test_nhexa_suspended_table}Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top.}
\end{center}
\end{minipage}
\hfill
@@ -11483,7 +11454,8 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{
8 & \(989\,\text{Hz}\) & Complex mode\\
9 & \(1025\,\text{Hz}\) & Complex mode\\
\bottomrule
-\end{tabularx}}
+\end{tabularx}
+}
\captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table}
\end{minipage}
@@ -11506,9 +11478,9 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{
\end{center}
\subcaption{\label{fig:test_nhexa_table_flexible_mode_3}Flexible mode at $1025\,\text{Hz}$}
\end{subfigure}
-\caption{\label{fig:test_nhexa_table_flexible_modes}Three identified flexible modes of the suspended table}
+\caption{\label{fig:test_nhexa_table_flexible_modes}Three identified flexible modes of the suspended table.}
\end{figure}
-\subsubsection{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.
@@ -11521,7 +11493,6 @@ The stiffness of the springs in the horizontal plane is set at \(0.5\,\text{N/mm
The obtained suspension modes of the multi-body model are compared with the measured modes in Table~\ref{tab:test_nhexa_suspended_table_simscape_modes}.
\begin{table}[htbp]
-\caption{\label{tab:test_nhexa_suspended_table_simscape_modes}Comparison of suspension modes of the multi-body model and the measured ones}
\centering
\begin{tabularx}{0.5\linewidth}{Xcccc}
\toprule
@@ -11531,8 +11502,10 @@ Multi-body & \(1.3\,\text{Hz}\) & \(1.8\,\text{Hz}\) & \(6.8\,\text{Hz}\) & \(9.
Experimental & \(1.3\,\text{Hz}\) & \(2.0\,\text{Hz}\) & \(6.9\,\text{Hz}\) & \(9.5\,\text{Hz}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_nhexa_suspended_table_simscape_modes}Comparison of suspension modes of the multi-body model and the measured ones.}
+
\end{table}
-\subsection{Nano-Hexapod Measured Dynamics}
+\subsection{Measured Active Platform Dynamics}
\label{sec:test_nhexa_dynamics}
The Nano-Hexapod was then mounted on top of the suspended table, as shown in Figure~\ref{fig:test_nhexa_hexa_suspended_table}.
@@ -11541,7 +11514,7 @@ All instrumentation (Speedgoat with \acrshort{adc}, DAC, piezoelectric voltage a
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_hexa_suspended_table.jpg}
-\caption{\label{fig:test_nhexa_hexa_suspended_table}Mounted Nano-Hexapod on top of the suspended table}
+\caption{\label{fig:test_nhexa_hexa_suspended_table}Mounted Nano-Hexapod on top of the suspended table.}
\end{figure}
A modal analysis of the nano-hexapod is first performed in Section~\ref{ssec:test_nhexa_enc_struts_modal_analysis}.
@@ -11554,9 +11527,9 @@ The effect of the payload mass on the dynamics is discussed in Section~\ref{ssec
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_nhexa_nano_hexapod_signals.png}
-\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the studied system. The command signal generated by the speedgoat is \(\bm{u}\), and the measured dignals are \(\bm{d}_{e}\) and \(\bm{V}_s\). Units are indicated in square brackets.}
+\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the studied system. The command signal is \(\bm{u}\), and the measured signals are \(\bm{d}_{e}\) and \(\bm{V}_s\).}
\end{figure}
-\subsubsection{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.
@@ -11565,7 +11538,7 @@ Five 3-axis accelerometers were fixed on the top platform of the nano-hexapod (F
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_modal_analysis.jpg}
-\caption{\label{fig:test_nhexa_modal_analysis}Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis}
+\caption{\label{fig:test_nhexa_modal_analysis}Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis.}
\end{figure}
Between \(100\,\text{Hz}\) and \(200\,\text{Hz}\), 6 suspension modes (i.e. rigid body modes of the top platform) were identified.
@@ -11573,7 +11546,6 @@ At around \(700\,\text{Hz}\), two flexible modes of the top plate were observed
These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}.
\begin{table}[htbp]
-\caption{\label{tab:test_nhexa_hexa_modal_modes_list}Description of the identified modes of the Nano-Hexapod}
\centering
\begin{tabularx}{0.6\linewidth}{ccX}
\toprule
@@ -11589,6 +11561,8 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}.
8 & \(709\,\text{Hz}\) & Second flexible mode of the top platform\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_nhexa_hexa_modal_modes_list}Description of the identified modes of the Nano-Hexapod.}
+
\end{table}
\begin{figure}[htbp]
@@ -11604,9 +11578,9 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}.
\end{center}
\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at $709\,\text{Hz}$}
\end{subfigure}
-\caption{\label{fig:test_nhexa_hexa_flexible_modes}Two identified flexible modes of the top plate of the Nano-Hexapod}
+\caption{\label{fig:test_nhexa_hexa_flexible_modes}Two identified flexible modes of the top plate of the Nano-Hexapod.}
\end{figure}
-\subsubsection{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.
@@ -11630,7 +11604,7 @@ This would not have occurred if the encoders were fixed to the struts.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_de.png}
-\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\bm{u}\) to \(\bm{d}_e\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.}
+\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRFs from \(\bm{u}\) to \(\bm{d}_e\). The 6 direct terms are the colored lines, and the 30 coupling terms are the gray lines.}
\end{figure}
Similarly, the \(6 \times 6\) \acrshort{frf} matrix from \(\bm{u}\) to \(\bm{V}_s\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs}.
@@ -11641,9 +11615,9 @@ The first flexible mode of the struts as \(235\,\text{Hz}\) has large amplitude,
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_Vs.png}
-\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\bm{u}\) to \(\bm{V}_s\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.}
+\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF from \(\bm{u}\) to \(\bm{V}_s\). The 6 direct terms are the colored lines, and the 30 coupling terms are the gray lines.}
\end{figure}
-\subsubsection{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.
@@ -11654,7 +11628,7 @@ These three cylindrical masses on top of the nano-hexapod are shown in Figure~\r
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_table_mass_3.jpg}
-\caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,\text{kg}\)}
+\caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,\text{kg}\).}
\end{figure}
The obtained \acrshortpl{frf} from actuator signal \(u_i\) to the associated encoder \(d_{ei}\) for the four payload conditions (no mass, \(13\,\text{kg}\), \(26\,\text{kg}\) and \(39\,\text{kg}\)) are shown in Figure~\ref{fig:test_nhexa_identified_frf_de_masses}.
@@ -11677,17 +11651,17 @@ For all tested payloads, the measured \acrshort{frf} always have alternating pol
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_de_masses.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$}
+\subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$ (encoder)}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_Vs_masses.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$}
+\subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$ (force sensor)}
\end{subfigure}
\caption{\label{fig:test_nhexa_identified_frf_masses}Measured Frequency Response Functions from \(u_i\) to \(d_{ei}\) (\subref{fig:test_nhexa_identified_frf_de_masses}) and from \(u_i\) to \(V_{si}\) (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown.}
\end{figure}
-\subsection{Nano-Hexapod Model Dynamics}
+\subsection{Model Dynamics}
\label{sec:test_nhexa_model}
In this section, the dynamics measured in Section~\ref{sec:test_nhexa_dynamics} is compared with those estimated from the multi-body model.
@@ -11696,7 +11670,7 @@ The nano-hexapod multi-body model was therefore added on top of the vibration ta
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_hexa_simscape.png}
-\caption{\label{fig:test_nhexa_hexa_simscape}3D representation of the multi-body model with the nano-hexapod on top of the suspended table. Three mass ``layers'' are here added}
+\caption{\label{fig:test_nhexa_hexa_simscape}Multi-body model of the nano-hexapod on top of the suspended table. Three mass ``layers'' are here added.}
\end{figure}
The model should exhibit certain characteristics that are verified in this section.
@@ -11704,10 +11678,10 @@ First, it should match the measured system dynamics from actuators to sensors pr
Both the ``direct'' terms (Section~\ref{ssec:test_nhexa_comp_model}) and ``coupling'' terms (Section~\ref{ssec:test_nhexa_comp_model_coupling}) of the multi-body model are compared with the measured dynamics.
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}.
-\subsubsection{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 \acrshort{apa}, and rigid top and bottom plates.
+The multi-body model of the nano-hexapod was first configured with 4-DoFs flexible joints, 2-DoFs \acrshort{apa}, and rigid top and bottom plates.
The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the \acrshort{fem}.
The parameters of the \acrshort{apa} model were determined from the test bench of the \acrshort{apa}.
The \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(\bm{d}_e\) and from \(\bm{u}\) to \(\bm{V}_s\) are then extracted from the multi-body model.
@@ -11715,7 +11689,7 @@ The \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(\bm{d}_e\) an
First, is it evaluated how well the models matches the ``direct'' terms of the measured \acrshort{frf} matrix.
To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured \acrshort{frf} in Figure~\ref{fig:test_nhexa_comp_simscape_diag}.
It can be seen that the 4 suspension modes of the nano-hexapod (at \(122\,\text{Hz}\), \(143\,\text{Hz}\), \(165\,\text{Hz}\) and \(191\,\text{Hz}\)) are well modeled.
-The three resonances that were attributed to ``internal'' flexible modes of the struts (at \(237\,\text{Hz}\), \(349\,\text{Hz}\) and \(395\,\text{Hz}\)) cannot be seen in the model, which is reasonable because the \acrshortpl{apa} are here modeled as a simple uniaxial 2-DoF system.
+The three resonances that were attributed to ``internal'' flexible modes of the struts (at \(237\,\text{Hz}\), \(349\,\text{Hz}\) and \(395\,\text{Hz}\)) cannot be seen in the model, which is reasonable because the \acrshortpl{apa} are here modeled as a simple uniaxial 2-DoFs system.
At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies.
\begin{figure}[htbp]
@@ -11723,17 +11697,17 @@ At higher frequencies, no resonances can be observed in the model, as the top pl
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_diag.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$}
+\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$ (encoder)}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_Vs_diag.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$}
+\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$ (force sensor)}
\end{subfigure}
-\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})}
+\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}
-\subsubsection{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 \acrshort{mimo} systems.
@@ -11744,7 +11718,7 @@ Similar results are observed for all other coupling terms and for the transfer f
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_all.png}
-\caption{\label{fig:test_nhexa_comp_simscape_de_all}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 2-DoF mass-spring-damper system.}
+\caption{\label{fig:test_nhexa_comp_simscape_de_all}Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\). The APA are here modeled with a 2-DoFs mass-spring-damper system. No payload us used.}
\end{figure}
The APA300ML was then modeled with a \emph{super-element} extracted from a FE-software.
@@ -11756,9 +11730,9 @@ Therefore, if the modes of the struts are to be modeled, the \emph{super-element
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_all_flex.png}
-\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''.}
+\caption{\label{fig:test_nhexa_comp_simscape_de_all_flex}Comparison of the measured (in blue) and modeled (in red) FRFs 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''. No payload us used.}
\end{figure}
-\subsubsection{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.
@@ -11775,15 +11749,15 @@ However, as decentralized IFF will be applied, the damping is actively brought,
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_diag_masses.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$}
+\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$ (encoder)}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_Vs_diag_masses.png}
\end{center}
-\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$}
+\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$ (force sensor)}
\end{subfigure}
-\caption{\label{fig:test_nhexa_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 \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})}
+\caption{\label{fig:test_nhexa_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 \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag}).}
\end{figure}
In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from \(u_1\) to \(d_{e1}\) to \(d_{e6}\) are compared in the case of the \(39\,\text{kg}\) payload in Figure~\ref{fig:test_nhexa_comp_simscape_de_all_high_mass}.
@@ -11793,7 +11767,7 @@ Therefore, the model effectively represents the system coupling for different pa
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_all_high_mass.png}
-\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 \(d_{e6}\)}
+\caption{\label{fig:test_nhexa_comp_simscape_de_all_high_mass}Comparison of the measured (in blue) and modeled (in red) FRF from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\). \(39\,\text{kg}\) payload is used.}
\end{figure}
\subsection*{Conclusion}
\label{sec:test_nhexa_conclusion}
@@ -11826,9 +11800,9 @@ If a model of the nano-hexapod was developed in one time, it would be difficult
To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure~\ref{fig:test_id31_micro_station_nano_hexapod}.
This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions.
-Initially, the project planned to develop a long-stroke (\(\approx 1 \, \text{cm}^3\)) 5-DoF metrology system to measure the sample position relative to the granite base.
+Initially, the project planned to develop a long-stroke (\(\approx 1 \, \text{cm}^3\)) 5-DoFs metrology system to measure the sample position relative to the granite base.
However, the complexity of this development prevented its completion before the experimental testing phase on ID31.
-To validate the nano-hexapod and its associated control architecture, an alternative short-stroke (\(\approx 100\,\mu\text{m}^3\)) metrology system was developed, which is presented in Section~\ref{sec:test_id31_metrology}.
+To validate the nano-hexapod and its associated control architecture, an alternative short-stroke (\(\approx 100\,\upmu\text{m}^3\)) metrology system was developed, which is presented in Section~\ref{sec:test_id31_metrology}.
Then, several key aspects of the system validation are examined.
Section~\ref{sec:test_id31_open_loop_plant} analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities.
@@ -11854,16 +11828,16 @@ These include tomography scans at various speeds and with different payload mass
\end{center}
\subcaption{\label{fig:test_id31_fixed_nano_hexapod}Nano-hexapod fixed on top of the micro-station}
\end{subfigure}
-\caption{\label{fig:test_id31_micro_station_nano_hexapod}Picture of the micro-station without the nano-hexapod (\subref{fig:test_id31_micro_station_cables}) and with the nano-hexapod (\subref{fig:test_id31_fixed_nano_hexapod})}
+\caption{\label{fig:test_id31_micro_station_nano_hexapod}Picture of the micro-station without the nano-hexapod (\subref{fig:test_id31_micro_station_cables}) and with the nano-hexapod (\subref{fig:test_id31_fixed_nano_hexapod}).}
\end{figure}
\subsection{Short Stroke Metrology System}
\label{sec:test_id31_metrology}
The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite.
-As a long-stroke (\(\approx 1 \,\text{cm}^3\)) metrology system was not yet developed, a stroke stroke (\(\approx 100\,\mu\text{m}^3\)) was used instead to validate the nano-hexapod control.
+As a long-stroke (\(\approx 1 \,\text{cm}^3\)) metrology system was not yet developed, a stroke stroke (\(\approx 100\,\upmu\text{m}^3\)) was used instead to validate the nano-hexapod control.
The first considered option was to use the ``Spindle error analyzer'' shown in Figure~\ref{fig:test_id31_lion}.
This system comprises 5 capacitive sensors facing two reference spheres.
-However, as the gap between the capacitive sensors and the spheres is very small\footnote{Depending on the measuring range, gap can range from \(\approx 1\,\mu\text{m}\) to \(\approx 100\,\mu\text{m}\).}, the risk of damaging the spheres and the capacitive sensors is too high.
+However, as the gap between the capacitive sensors and the spheres is very small\footnote{Depending on the measuring range, gap can range from \(\approx 1\,\upmu\text{m}\) to \(\approx 100\,\upmu\text{m}\).}, the risk of damaging the spheres and the capacitive sensors is too high.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@@ -11884,7 +11858,7 @@ However, as the gap between the capacitive sensors and the spheres is very small
\end{center}
\subcaption{\label{fig:test_id31_interf_head}Interferometer head}
\end{subfigure}
-\caption{\label{fig:test_id31_short_stroke_metrology}Short stroke metrology system used to measure the sample position with respect to the granite in 5DoF. The system is based on a ``Spindle error analyzer'' (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). The interferometer heads are shown in (\subref{fig:test_id31_interf_head})}
+\caption{\label{fig:test_id31_short_stroke_metrology}Short stroke metrology system used to measure the sample position with respect to the granite in 5-DoFs. The system is based on a ``Spindle error analyzer'' (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). One interferometer head is shown in (\subref{fig:test_id31_interf_head}).}
\end{figure}
Instead of using capacitive sensors, 5 fibered interferometers were used in a similar manner (Figure~\ref{fig:test_id31_interf}).
@@ -11935,7 +11909,7 @@ The five equations~\eqref{eq:test_id31_metrology_kinematics} can be written in m
d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
\end{bmatrix}
\end{equation}
-\subsubsection{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 must be well aligned with the rotation axis of the spindle.
@@ -11945,10 +11919,10 @@ To not damage the sensitive sphere surface, the probes are instead positioned on
The probes are first fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis.
The probes are then fixed to the top (adjustable) cylinder, and the same alignment is performed.
-With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around \(10\,\mu\text{m}\).
+With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around \(10\,\upmu\text{m}\).
The accuracy was probably limited by the poor coaxiality between the cylinders and the spheres.
However, this first alignment should be sufficient to position the two sphere within the acceptance range of the interferometers.
-\subsubsection{Tip-Tilt adjustment of the interferometers}
+\subsubsection{Tip-Tilt Adjustment of the Interferometers}
\label{ssec:test_id31_metrology_alignment}
The short-stroke metrology system was placed on top of the main granite using granite blocs (Figure~\ref{fig:test_id31_short_stroke_metrology_overview}).
@@ -11957,7 +11931,7 @@ Granite is used for its good mechanical and thermal stability.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.8\linewidth]{figs/test_id31_short_stroke_metrology_overview.jpg}
-\caption{\label{fig:test_id31_short_stroke_metrology_overview}Granite gantry used to fix the short-stroke metrology system}
+\caption{\label{fig:test_id31_short_stroke_metrology_overview}Granite gantry used to fix the short-stroke metrology system.}
\end{figure}
The interferometer beams must be placed with respect to the two reference spheres as close as possible to the ideal case shown in Figure~\ref{fig:test_id31_metrology_kinematics}.
@@ -11971,7 +11945,7 @@ This allows them to be individually oriented so that they all point to the spher
This is achieved by maximizing the intensity of the reflected light of each interferometer.
After the alignment procedure, the top interferometer should coincide with the spindle axis, and the lateral interferometers should all be in the horizontal plane and intersect the centers of the spheres.
-\subsubsection{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 interferometer orientations (Section~\ref{ssec:test_id31_metrology_alignment}), the spindle can perform complete rotations while still having interference for all five interferometers.
@@ -11983,7 +11957,7 @@ From the errors, the motion of the positioning hexapod to better align the spher
Then, the spindle is scanned again, and new alignment errors are recorded.
This iterative process is first performed for angular errors (Figure~\ref{fig:test_id31_metrology_align_rx_ry}) and then for lateral errors (Figure~\ref{fig:test_id31_metrology_align_dx_dy}).
-The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\) in \(R_x\) and \(R_y\) orientations, \(\pm 1\,\mu\text{m}\) in \(D_x\) and \(D_y\) directions, and less than \(0.1\,\mu\text{m}\) vertically.
+The remaining errors after alignment are in the order of \(\pm5\,\upmu\text{rad}\) in \(R_x\) and \(R_y\) orientations, \(\pm 1\,\upmu\text{m}\) in \(D_x\) and \(D_y\) directions, and less than \(0.1\,\upmu\text{m}\) vertically.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -12000,7 +11974,7 @@ The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\)
\end{subfigure}
\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 full spindle rotation. Between two rotations, the positioning hexapod is adjusted to better align the two spheres with the rotation axis.}
\end{figure}
-\subsubsection{Estimated measurement volume}
+\subsubsection{Estimated Measurement Volume}
\label{ssec:test_id31_metrology_acceptance}
Because the interferometers point to spheres and not flat surfaces, the lateral acceptance is limited.
@@ -12010,7 +11984,6 @@ Results are summarized in Table~\ref{tab:test_id31_metrology_acceptance}.
The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(\pm0.5\,\text{mm}\), which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS.
\begin{table}[htbp]
-\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
\centering
\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
@@ -12023,8 +11996,10 @@ The obtained lateral acceptance for pure displacements in any direction is estim
\(d_5\) (z) & \(1.33\,\text{mm}\) & \(1.06\,\text{mm}\) & \(>2\,\text{mm}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
+
\end{table}
-\subsubsection{Estimated measurement errors}
+\subsubsection{Estimated Measurement Errors}
\label{ssec:test_id31_metrology_errors}
When using the NASS, the accuracy of the sample positioning is determined by the accuracy of the external metrology.
@@ -12038,11 +12013,11 @@ As the top interferometer points to a sphere and not to a plane, lateral motion
Then, the reference spheres have some deviations relative to an ideal sphere \footnote{The roundness of the spheres is specified at \(50\,\text{nm}\).}.
These sphere are originally intended for use with capacitive sensors that integrate shape errors over large surfaces.
-When using interferometers, the size of the ``light spot'' on the sphere surface is a circle with a diameter approximately equal to \(50\,\mu\text{m}\), and therefore the measurement is more sensitive to shape errors with small features.
+When using interferometers, the size of the ``light spot'' on the sphere surface is a circle with a diameter approximately equal to \(50\,\upmu\text{m}\), and therefore the measurement is more sensitive to shape errors with small features.
As the light from the interferometer travels through air (as opposed to being in vacuum), the measured distance is sensitive to any variation in the refractive index of the air.
Therefore, any variation in air temperature, pressure or humidity will induce measurement errors.
-For instance, for a measurement length of \(40\,\text{mm}\), a temperature variation of \(0.1\,{}^oC\) (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of \(\approx 4\,\text{nm}\).
+For instance, for a measurement length of \(40\,\text{mm}\), a temperature variation of \(\SI{0.1}{\degree}\) (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of \(\approx 4\,\text{nm}\).
Interferometers are also affected by noise~\cite{watchi18_review_compac_inter}.
The effect of noise on the translation and rotation measurements is estimated in Figure~\ref{fig:test_id31_interf_noise}.
@@ -12081,7 +12056,7 @@ Voltages generated by the force sensor piezoelectric stacks \(\bm{V}_s = [V_{s1}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.9]{figs/test_id31_block_schematic_plant.png}
-\caption{\label{fig:test_id31_block_schematic_plant}Schematic of the NASS plant}
+\caption{\label{fig:test_id31_block_schematic_plant}Schematic of the NASS plant.}
\end{figure}
\subsubsection{Open-Loop Plant Identification}
\label{ssec:test_id31_open_loop_plant_first_id}
@@ -12110,7 +12085,7 @@ This issue was later solved.
\end{center}
\subcaption{\label{fig:test_id31_first_id_iff}Force Sensors}
\end{subfigure}
-\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 displayed with shaded lines.}
+\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 for force sensors (\subref{fig:test_id31_first_id_iff}). Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are displayed with shaded lines.}
\end{figure}
\subsubsection{Better Angular Alignment}
\label{ssec:test_id31_open_loop_plant_rz_alignment}
@@ -12137,7 +12112,7 @@ Results shown in Figure~\ref{fig:test_id31_Rz_align_correct} are indeed indicati
\end{center}
\subcaption{\label{fig:test_id31_Rz_align_correct}After alignment}
\end{subfigure}
-\caption{\label{fig:test_id31_Rz_align_error_and_correct}Measurement of the Nano-Hexapod axes in the frame of the external metrology. Before alignment (\subref{fig:test_id31_Rz_align_error}) and after alignment (\subref{fig:test_id31_Rz_align_correct}).}
+\caption{\label{fig:test_id31_Rz_align_error_and_correct}Measurement of nano-hexapod's axes in the frame of the external metrology. Before (\subref{fig:test_id31_Rz_align_error}) and after alignment (\subref{fig:test_id31_Rz_align_correct}).}
\end{figure}
The dynamics of the plant was identified again after fine alignment and compared with the model dynamics in Figure~\ref{fig:test_id31_first_id_int_better_rz_align}.
@@ -12149,7 +12124,7 @@ The flexible modes of the top platform can be passively damped, whereas the mode
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_first_id_int_better_rz_align.png}
-\caption{\label{fig:test_id31_first_id_int_better_rz_align}Decrease of the coupling with better Rz alignment}
+\caption{\label{fig:test_id31_first_id_int_better_rz_align}Decrease of the coupling after better \(R_z\) alignment.}
\end{figure}
\subsubsection{Effect of Payload Mass}
\label{ssec:test_id31_open_loop_plant_mass}
@@ -12186,7 +12161,7 @@ It is interesting to note that the anti-resonances in the force sensor plant now
\end{center}
\subcaption{\label{fig:test_id31_picture_mass_m3}$m=39\,\text{kg}$}
\end{subfigure}
-\caption{\label{fig:test_id31_picture_masses}The four tested payload conditions: (\subref{fig:test_id31_picture_mass_m0}) no payload, (\subref{fig:test_id31_picture_mass_m1}) \(13\,\text{kg}\) payload, (\subref{fig:test_id31_picture_mass_m2}) \(26\,\text{kg}\) payload, (\subref{fig:test_id31_picture_mass_m3}) \(39\,\text{kg}\) payload.}
+\caption{\label{fig:test_id31_picture_masses}Four tested payload conditions: (\subref{fig:test_id31_picture_mass_m0}) no payload, (\subref{fig:test_id31_picture_mass_m1}) \(13\,\text{kg}\) payload, (\subref{fig:test_id31_picture_mass_m2}) \(26\,\text{kg}\) payload, (\subref{fig:test_id31_picture_mass_m3}) \(39\,\text{kg}\) payload.}
\end{figure}
\begin{figure}[htbp]
@@ -12194,15 +12169,15 @@ It is interesting to note that the anti-resonances in the force sensor plant now
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_comp_simscape_int_diag_masses.png}
\end{center}
-\subcaption{\label{fig:test_id31_comp_simscape_int_diag_masses}from $u$ to $\epsilon\mathcal{L}$}
+\subcaption{\label{fig:test_id31_comp_simscape_int_diag_masses}from $u$ to $\epsilon\mathcal{L}$ (strut error)}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_comp_simscape_iff_diag_masses.png}
\end{center}
-\subcaption{\label{fig:test_id31_comp_simscape_iff_diag_masses}from $u$ to $V_s$}
+\subcaption{\label{fig:test_id31_comp_simscape_iff_diag_masses}from $u$ to $V_s$ (force sensor)}
\end{subfigure}
-\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})}
+\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}
\subsubsection{Effect of Spindle Rotation}
\label{ssec:test_id31_open_loop_plant_rotation}
@@ -12252,7 +12227,7 @@ The decentralized Integral Force Feedback is implemented as shown in the block d
\begin{figure}[htbp]
\centering
\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 for every diagonal term \(K_{\text{IFF}}\).}
+\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.}
\end{figure}
\subsubsection{IFF Plant}
\label{ssec:test_id31_iff_plant}
@@ -12268,7 +12243,7 @@ This confirms that the multi-body model can be used to tune the IFF controller.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_comp_simscape_Vs.png}
-\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 \(V_{s6}\)}
+\caption{\label{fig:test_id31_comp_simscape_Vs}Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal \(u_1\) to the six force sensor voltages \(V_{s1}\) to \(V_{s6}\).}
\end{figure}
\subsubsection{IFF Controller}
\label{ssec:test_id31_iff_controller}
@@ -12295,9 +12270,9 @@ It can be seen that the loop-gain is larger than \(1\) around the suspension mod
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_Kiff_loop_gain.png}
\end{center}
-\subcaption{\label{fig:test_id31_Kiff_loop_gain}Decentralized Loop gains}
+\subcaption{\label{fig:test_id31_Kiff_loop_gain}Decentralized loop gains}
\end{subfigure}
-\caption{\label{fig:test_id31_Kiff}Bode plot of the decentralized IFF controller (\subref{fig:test_id31_Kiff_bode_plot}). The decentralized controller \(K_{\text{IFF}}\) multiplied by the identified dynamics from \(u_1\) to \(V_{s1}\) for all payloads are shown in (\subref{fig:test_id31_Kiff_loop_gain})}
+\caption{\label{fig:test_id31_Kiff}Bode plot of the decentralized IFF controller (\subref{fig:test_id31_Kiff_bode_plot}). The decentralized controller \(K_{\text{IFF}}\) multiplied by the identified dynamics from \(u_1\) to \(V_{s1}\) for all payloads are shown in (\subref{fig:test_id31_Kiff_loop_gain}).}
\end{figure}
To estimate the added damping, a root-locus plot was computed using the multi-body model (Figure~\ref{fig:test_id31_iff_root_locus}).
@@ -12332,7 +12307,7 @@ However, in this study, it was chosen to implement a ``fixed'' (i.e. non-adaptiv
\end{center}
\subcaption{\label{fig:test_id31_iff_root_locus_m3}$m = 39\,\text{kg}$}
\end{subfigure}
-\caption{\label{fig:test_id31_iff_root_locus}Root Locus plots for the designed decentralized IFF controller, computed using the multy-body model. Black crosses indicate the closed-loop poles for the choosen value of the gain.}
+\caption{\label{fig:test_id31_iff_root_locus}Root loci for the decentralized IFF controller, computed using the multi-body model. Black crosses indicate the closed-loop poles for the chosen value of the gain.}
\end{figure}
\subsubsection{Damped Plant}
\label{ssec:test_id31_iff_perf}
@@ -12357,7 +12332,7 @@ The obtained \acrshortpl{frf} are compared with the model in Figure~\ref{fig:tes
\end{center}
\subcaption{\label{fig:test_id31_hac_plant_effect_mass}Comparison of model and experimental results}
\end{subfigure}
-\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 measured damped 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}
+\caption{\label{fig:test_id31_hac_plant_effect_mass_comp_model}Comparison of the open-loop plant and the damped plant with decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped 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 simplicity.}
\end{figure}
\subsubsection*{Conclusion}
The implementation of a decentralized Integral Force Feedback controller was successfully demonstrated.
@@ -12365,7 +12340,7 @@ Using the multi-body model, the controller was designed and optimized to ensure
The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0 to \(39\,\text{kg}\)).
Although higher gains could achieve better damping performance for lighter payloads, the chosen fixed-gain configuration represents a robust compromise that maintains stability and performance under all operating conditions.
The good correlation between the modeled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction.
-\subsection{High Authority Control in the frame of the struts}
+\subsection{High Authority Control in the Frame of the Struts}
\label{sec:test_id31_hac}
In this section, a High-Authority-Controller is developed to actively stabilize the sample position.
The corresponding control architecture is shown in Figure~\ref{fig:test_id31_iff_hac_schematic}.
@@ -12383,7 +12358,7 @@ K_{\text{HAC}} & & 0 \\
\begin{figure}[htbp]
\centering
\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 \(K_{\text{HAC}}\).}
+\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.}
\end{figure}
\subsubsection{Damped Plant}
\label{ssec:test_id31_iff_hac_plant}
@@ -12394,7 +12369,7 @@ Considering the complexity of the system's dynamics, the model can be considered
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_comp_simscape_hac.png}
-\caption{\label{fig:test_id31_comp_simscape_hac}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal (\(u_1^\prime\)) of the damped plant to the estimated errors (\(\epsilon_{\mathcal{L}_i}\)) in the frame of the six struts by the external metrology}
+\caption{\label{fig:test_id31_comp_simscape_hac}Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal (\(u_1^\prime\)) of the damped plant to the estimated errors (\(\epsilon_{\mathcal{L}_i}\)) in the frame of the six struts by the external metrology.}
\end{figure}
The challenge here is to tune a high authority controller such that it is robust to the change in dynamics due to different payloads being used.
@@ -12433,7 +12408,7 @@ This design choice, while beneficial for system simplicity, introduces inherent
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_rga_number.png}
-\caption{\label{fig:test_id31_hac_rga_number}RGA-number for the damped plants - Comparison of all the payload conditions}
+\caption{\label{fig:test_id31_hac_rga_number}RGA-number for the damped plants - Comparison of all the payload conditions.}
\end{figure}
\subsubsection{Robust Controller Design}
\label{ssec:test_id31_iff_hac_controller}
@@ -12449,7 +12424,7 @@ K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot
The obtained ``decentralized'' loop-gains (i.e. the diagonal element of the controller times the diagonal terms of the plant) are shown in Figure~\ref{fig:test_id31_hac_loop_gain}.
The closed-loop stability was verified by computing the characteristic Loci (Figure~\ref{fig:test_id31_hac_characteristic_loci}).
-However, small stability margins were observed for the highest mass, indicating that some multivariable effects are in play.
+However, small stability margins were observed for the highest mass, indicating that some multivariable effects are at play.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -12464,12 +12439,12 @@ However, small stability margins were observed for the highest mass, indicating
\end{center}
\subcaption{\label{fig:test_id31_hac_characteristic_loci}Characteristic Loci}
\end{subfigure}
-\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})}
+\caption{\label{fig:test_id31_hac_loop_gain_loci}``Decentralized loop-gains'' (\subref{fig:test_id31_hac_loop_gain}) and characteristic loci (\subref{fig:test_id31_hac_characteristic_loci}) for the robust high authority controller.}
\end{figure}
-\subsubsection{Performance estimation with simulation of Tomography scans}
+\subsubsection{Performance - Tomography Scans}
\label{ssec:test_id31_iff_hac_perf}
-To estimate the performances that can be expected with this \acrshort{haclac} 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.}.
+To estimate the performances that can be expected with this \acrshort{haclac} architecture and the designed controller, simulations of tomography experiments were performed\footnote{Note that the eccentricity of the PoI with respect to the Spindle rotation axis has been tuned based on measurements.}.
The rotational velocity was set to \(180\,\text{deg/s}\), and no payload was added on top of the nano-hexapod.
An open-loop simulation and a closed-loop simulation were performed and compared in Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}.
The obtained closed-loop positioning accuracy was found to comply with the requirements as it succeeded to keep the \acrshort{poi} on the beam (Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}).
@@ -12489,7 +12464,7 @@ The obtained closed-loop positioning accuracy was found to comply with the requi
\end{subfigure}
\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}
-\subsubsection{Robustness estimation with simulation of Tomography scans}
+\subsubsection{Robustness - Tomography Scans}
\label{ssec:test_id31_iff_hac_robustness}
To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure~\ref{fig:test_id31_picture_masses} (i.e. \(0\,\text{kg}\), \(13\,\text{kg}\), \(26\,\text{kg}\) and \(39\,\text{kg}\)).
@@ -12502,7 +12477,7 @@ However, it was decided that this controller should be tested experimentally and
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_tomography_Wz36_simulation.png}
-\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)}
+\caption{\label{fig:test_id31_hac_tomography_Wz36_simulation}Positioning errors in the YZ plane during closed-loop simulations of tomography experiments.}
\end{figure}
\subsubsection*{Conclusion}
In this section, a High-Authority-Controller was developed to actively stabilize the sample position.
@@ -12517,7 +12492,7 @@ The closed-loop system remained stable under all tested payload conditions (0 to
With no payload at \(180\,\text{deg/s}\), the NASS successfully maintained the sample \acrshort{poi} in the beam, which fulfilled the specifications.
At \(6\,\text{deg/s}\), although the positioning errors increased with the payload mass (particularly in the lateral direction), the system remained stable.
These results demonstrate both the effectiveness and limitations of implementing control in the frame of the struts.
-\subsection{Validation with Scientific experiments}
+\subsection{Validation with Scientific Experiments}
\label{sec:test_id31_experiments}
In this section, the goal is to evaluate the performance of the NASS and validate its use to perform typical scientific experiments.
However, the online metrology prototype (presented in Section~\ref{sec:test_id31_metrology}) does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam.
@@ -12537,26 +12512,27 @@ Higher performance controllers using complementary filters are investigated in S
For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to \(200\,\text{nm}\times 100\,\text{nm}\).
In this case, the goal is to keep the sample's \acrshort{poi} in the beam, and therefore the \(D_y\) and \(D_z\) positioning errors should be less than \(200\,\text{nm}\) and \(100\,\text{nm}\) peak-to-peak, respectively.
-The \(R_y\) error should be less than \(1.7\,\mu\text{rad}\) peak-to-peak.
+The \(R_y\) error should be less than \(1.7\,\upmu\text{rad}\) peak-to-peak.
In terms of RMS errors, this corresponds to \(30\,\text{nm}\) in \(D_y\), \(15\,\text{nm}\) in \(D_z\) and \(250\,\text{nrad}\) in \(R_y\) (a summary of the specifications is given in Table~\ref{tab:test_id31_experiments_specifications}).
Results obtained for all experiments are summarized and compared to the specifications in Section~\ref{ssec:test_id31_scans_conclusion}.
\begin{table}[htbp]
-\caption{\label{tab:test_id31_experiments_specifications}Specifications for the Nano-Active-Stabilization-System}
\centering
\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
& \(D_y\) & \(D_z\) & \(R_y\)\\
\midrule
-peak 2 peak & \(200\,\text{nm}\) & \(100\,\text{nm}\) & \(1.7\,\mu\text{rad}\)\\
+peak 2 peak & \(200\,\text{nm}\) & \(100\,\text{nm}\) & \(1.7\,\upmu\text{rad}\)\\
RMS & \(30\,\text{nm}\) & \(15\,\text{nm}\) & \(250\,\text{nrad}\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_id31_experiments_specifications}Positioning specifications for the Nano-Active-Stabilization-System.}
+
\end{table}
\subsubsection{Tomography Scans}
\label{ssec:test_id31_scans_tomography}
-\paragraph{Slow Tomography scans}
+\paragraph{Slow Tomography Scans}
First, tomography scans were performed with a rotational velocity of \(6\,\text{deg/s}\) for all considered payload masses (shown in Figure~\ref{fig:test_id31_picture_masses}).
Each experimental sequence consisted of two complete spindle rotations: an initial open-loop rotation followed by a closed-loop rotation.
@@ -12572,7 +12548,7 @@ While this approach likely underestimates actual open-loop errors, as perfect al
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit.png}
\end{center}
-\subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in $(x,y)$ plane}
+\subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in XY plane}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
@@ -12580,19 +12556,19 @@ While this approach likely underestimates actual open-loop errors, as perfect al
\end{center}
\subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}Removed eccentricity}
\end{subfigure}
-\caption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff}Tomography experiment with a rotation velocity of \(6\,\text{deg/s}\), and payload mass of \(26\,\text{kg}\). Errors in the \((x,y)\) plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}).}
+\caption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff}Tomography experiment with a rotation velocity of \(6\,\text{deg/s}\), and payload mass of \(26\,\text{kg}\). Errors in the XY plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}).}
\end{figure}
-The residual motion (i.e. after compensating for eccentricity) in the \(Y-Z\) is compared against the minimum beam size, as illustrated in Figure~\ref{fig:test_id31_tomo_Wz36_results}.
+The residual motion (i.e. after compensating for eccentricity) in the YZ is compared against the minimum beam size, as illustrated in Figure~\ref{fig:test_id31_tomo_Wz36_results}.
Results are indicating the NASS succeeds in keeping the sample's \acrshort{poi} on the beam, except for the highest mass of \(39\,\text{kg}\) for which the lateral motion is a bit too high.
These experimental findings are consistent with the predictions from the tomography simulations presented in Section~\ref{ssec:test_id31_iff_hac_robustness}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_Wz36_results.png}
-\caption{\label{fig:test_id31_tomo_Wz36_results}Measured errors in the \(Y-Z\) plane during tomography experiments at \(6\,\text{deg/s}\) for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data.}
+\caption{\label{fig:test_id31_tomo_Wz36_results}Measured errors in the YZ plane during tomography experiments at \(6\,\text{deg/s}\) for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data.}
\end{figure}
-\paragraph{Fast Tomography scans}
+\paragraph{Fast Tomography Scans}
A tomography experiment was then performed with the highest rotational velocity of the Spindle: \(180\,\text{deg/s}\)\footnote{The highest rotational velocity of \(360\,\text{deg/s}\) could not be tested due to an issue in the Spindle's controller.}.
The trajectory of the \acrshort{poi} during the fast tomography scan is shown in Figure~\ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}.
@@ -12622,27 +12598,28 @@ For this specific measurement, an enhanced high authority controller (discussed
Figure~\ref{fig:test_id31_hac_cas_cl} presents the cumulative amplitude spectra of the position errors for all three cases.
The results reveal two distinct control contributions: the decentralized IFF effectively attenuates vibrations near the nano-hexapod suspension modes (an achievement not possible with HAC alone), while the high authority controller suppresses low-frequency vibrations primarily arising from Spindle guiding errors.
-Notably, the spectral patterns in Figure~\ref{fig:test_id31_hac_cas_cl} closely resemble the cumulative amplitude spectra computed in the project's early stages.
-This experiment also illustrates that when needed, performance can be enhanced by designing controllers for specific experimental conditions rather than relying solely on robust controllers that can accommodate all payload ranges.
+Notably, the spectral patterns in Figure~\ref{fig:test_id31_hac_cas_cl} closely resemble the cumulative amplitude spectra computed in the project's early stages (Figure~\ref{fig:uniaxial_cas_hac_lac_mid} in page\nbps{}\pageref{fig:uniaxial_cas_hac_lac_mid}).
+
+This experiment also illustrates that when needed, performance can be enhanced by designing controllers for specific experimental conditions rather than relying solely on robust controllers able to accommodate all payloads.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_cas_cl_dy.png}
\end{center}
-\subcaption{\label{fig:test_id31_hac_cas_cl_dy} $D_y$}
+\subcaption{\label{fig:test_id31_hac_cas_cl_dy}Lateral ($D_y$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_cas_cl_dz.png}
\end{center}
-\subcaption{\label{fig:test_id31_hac_cas_cl_dz} $D_z$}
+\subcaption{\label{fig:test_id31_hac_cas_cl_dz}Vertical ($D_z$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_cas_cl_ry.png}
\end{center}
-\subcaption{\label{fig:test_id31_hac_cas_cl_ry} $R_y$}
+\subcaption{\label{fig:test_id31_hac_cas_cl_ry}Tilt ($R_y$)}
\end{subfigure}
\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. The RMS values are indicated in the legend.}
\end{figure}
@@ -12650,7 +12627,7 @@ This experiment also illustrates that when needed, performance can be enhanced b
\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.
-In this experiment, a \(R_y\) scan was executed at a rotational velocity of \(100\,\mu \text{rad/s}\), and the closed-loop positioning errors were monitored (Figure~\ref{fig:test_id31_reflectivity}).
+In this experiment, a \(R_y\) scan was executed at a rotational velocity of \(100\,\upmu \text{rad/s}\), and the closed-loop positioning errors were monitored (Figure~\ref{fig:test_id31_reflectivity}).
The results confirmed that the NASS successfully maintained the \acrshort{poi} within the specified beam parameters throughout the scanning process.
\begin{figure}[htbp]
@@ -12658,35 +12635,35 @@ The results confirmed that the NASS successfully maintained the \acrshort{poi} w
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_reflectivity_dy.png}
\end{center}
-\subcaption{\label{fig:test_id31_reflectivity_dy}$D_y$}
+\subcaption{\label{fig:test_id31_reflectivity_dy}Lateral ($D_y$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_reflectivity_dz.png}
\end{center}
-\subcaption{\label{fig:test_id31_reflectivity_dz}$D_z$}
+\subcaption{\label{fig:test_id31_reflectivity_dz}Vertical ($D_z$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_reflectivity_ry.png}
\end{center}
-\subcaption{\label{fig:test_id31_reflectivity_ry}$R_y$}
+\subcaption{\label{fig:test_id31_reflectivity_ry}Tilt ($R_y$)}
\end{subfigure}
-\caption{\label{fig:test_id31_reflectivity}Reflectivity scan (\(R_y\)) with a rotational velocity of \(100\,\mu \text{rad}/s\).}
+\caption{\label{fig:test_id31_reflectivity}Reflectivity scan (\(R_y\)) with a rotational velocity of \(100\,\upmu \text{rad}/s\).}
\end{figure}
\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.
-\paragraph{Step by Step \(D_z\) motion}
+\paragraph{Step by Step \(D_z\) Motion}
The vertical step motion was performed exclusively with the nano-hexapod.
-Testing was conducted across step sizes ranging from \(10\,\text{nm}\) to \(1\,\mu\text{m}\).
+Testing was conducted across step sizes ranging from \(10\,\text{nm}\) to \(1\,\upmu\text{m}\).
Results are presented in Figure~\ref{fig:test_id31_dz_mim_steps}.
The system successfully resolved \(10\,\text{nm}\) steps (red curve in Figure~\ref{fig:test_id31_dz_mim_10nm_steps}) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements.
In step-by-step scanning procedures, the settling time is a critical parameter as it significantly affects the total experiment duration.
-The system achieved a response time of approximately \(70\,\text{ms}\) to reach the target position (within \(\pm 20\,\text{nm}\)), as demonstrated by the \(1\,\mu\text{m}\) step response in Figure~\ref{fig:test_id31_dz_mim_1000nm_steps}.
+The system achieved a response time of approximately \(70\,\text{ms}\) to reach the target position (within \(\pm 20\,\text{nm}\)), as demonstrated by the \(1\,\upmu\text{m}\) step response in Figure~\ref{fig:test_id31_dz_mim_1000nm_steps}.
The settling duration typically decreases for smaller step sizes.
\begin{figure}[htbp]
@@ -12706,40 +12683,40 @@ The settling duration typically decreases for smaller step sizes.
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_1000nm_steps.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu\text{m}$ step}
+\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\upmu\text{m}$ step}
\end{subfigure}
-\caption{\label{fig:test_id31_dz_mim_steps}Vertical steps performed with the nano-hexapod. \(10\,\text{nm}\) steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of \(50\,\text{ms}\). \(100\,\text{nm}\) steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of \(\pm 20\,\text{nm}\) is \(\approx 70\,\text{ms}\) as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a \(1\,\mu\text{m}\) step.}
+\caption{\label{fig:test_id31_dz_mim_steps}Vertical steps performed with the nano-hexapod. \(10\,\text{nm}\) steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of \(50\,\text{ms}\). \(100\,\text{nm}\) steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of \(\pm 20\,\text{nm}\) is \(\approx 70\,\text{ms}\) as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a \(1\,\upmu\text{m}\) step.}
\end{figure}
-\paragraph{Continuous \(D_z\) motion: Dirty Layer Scans}
+\paragraph{Continuous \(D_z\) Motion: Dirty Layer Scans}
For these and subsequent experiments, the NASS performs ``ramp scans'' (constant velocity scans).
To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies.
-Initial testing at \(10\,\mu\text{m/s}\) demonstrated positioning errors well within specifications (indicated by dashed lines in Figure~\ref{fig:test_id31_dz_scan_10ums}).
+Initial testing at \(10\,\upmu\text{m/s}\) demonstrated positioning errors well within specifications (indicated by dashed lines in Figure~\ref{fig:test_id31_dz_scan_10ums}).
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_10ums_dy.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$}
+\subcaption{\label{fig:test_id31_dz_scan_10ums_dy}Vertical ($D_y$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_10ums_dz.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_10ums_dz}$D_z$}
+\subcaption{\label{fig:test_id31_dz_scan_10ums_dz}Horizontal ($D_z$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_10ums_ry.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$}
+\subcaption{\label{fig:test_id31_dz_scan_10ums_ry}Tilt ($R_y$)}
\end{subfigure}
-\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan at a velocity of \(10\,\mu \text{m/s}\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry})}
+\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan at a velocity of \(10\,\upmu \text{m/s}\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}).}
\end{figure}
-A subsequent scan at \(100\,\mu\text{m/s}\) - the maximum velocity for high-precision \(D_z\) scans\footnote{Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and \(100\,\text{nm}\) ``resolution'' (equal to the vertical beam size).} - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure~\ref{fig:test_id31_dz_scan_100ums}).
+A subsequent scan at \(100\,\upmu\text{m/s}\) - the maximum velocity for high-precision \(D_z\) scans\footnote{Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and \(100\,\text{nm}\) ``resolution'' (equal to the vertical beam size).} - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure~\ref{fig:test_id31_dz_scan_100ums}).
Since detectors typically operate only during the constant velocity phase, these transient deviations do not compromise the measurement quality.
However, performance during acceleration phases could be enhanced through the implementation of feedforward control.
@@ -12748,35 +12725,35 @@ However, performance during acceleration phases could be enhanced through the im
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_100ums_dy.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$}
+\subcaption{\label{fig:test_id31_dz_scan_100ums_dy}Lateral ($D_y$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_100ums_dz.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_100ums_dz}$D_z$}
+\subcaption{\label{fig:test_id31_dz_scan_100ums_dz}Vertical ($D_z$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_100ums_ry.png}
\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$}
+\subcaption{\label{fig:test_id31_dz_scan_100ums_ry}Tilt ($R_y$)}
\end{subfigure}
-\caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan at a velocity of \(100\,\mu\text{m/s}\). \(D_z\) 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})}
+\caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan at a velocity of \(100\,\upmu\text{m/s}\). \(D_z\) 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}
\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.
+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.
Within the Speedgoat, the system computes the positioning error by comparing the measured \(D_y\) sample position against the received setpoint, and the Nano-Hexapod compensates for positioning errors introduced during \(T_y\) stage scanning.
-The scanning range is constrained \(\pm 100\,\mu\text{m}\) due to the limited acceptance of the metrology system.
-\paragraph{Slow scan}
+The scanning range is constrained \(\pm 100\,\upmu\text{m}\) due to the limited acceptance of the metrology system.
+\paragraph{Slow Scan}
-Initial testing were made with a scanning velocity of \(10\,\mu\text{m/s}\), which is typical for these experiments.
+Initial testing were made with a scanning velocity of \(10\,\upmu\text{m/s}\), which is typical for these experiments.
Figure~\ref{fig:test_id31_dy_10ums} compares the positioning errors between open-loop (without NASS) and closed-loop operation.
In the scanning direction, open-loop measurements reveal periodic errors (Figure~\ref{fig:test_id31_dy_10ums_dy}) attributable to the \(T_y\) stage's stepper motor.
These micro-stepping errors, which are inherent to stepper motor operation, occur 200 times per motor rotation with approximately \(1\,\text{mrad}\) angular error amplitude.
-Given the \(T_y\) stage's lead screw pitch of \(2\,\text{mm}\), these errors manifest as \(10\,\mu\text{m}\) periodic oscillations with \(\approx 300\,\text{nm}\) amplitude, which can indeed be seen in the open-loop measurements (Figure~\ref{fig:test_id31_dy_10ums_dy}).
+Given the \(T_y\) stage's lead screw pitch of \(2\,\text{mm}\), these errors manifest as \(10\,\upmu\text{m}\) periodic oscillations with \(\approx 300\,\text{nm}\) amplitude, which can indeed be seen in the open-loop measurements (Figure~\ref{fig:test_id31_dy_10ums_dy}).
In the vertical direction (Figure~\ref{fig:test_id31_dy_10ums_dz}), open-loop errors likely stem from metrology measurement error because the top interferometer points at a spherical target surface (see Figure~\ref{fig:test_id31_xy_map_sphere}).
Under closed-loop control, positioning errors remain within specifications in all directions.
@@ -12786,25 +12763,25 @@ Under closed-loop control, positioning errors remain within specifications in al
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_10ums_dy.png}
\end{center}
-\subcaption{\label{fig:test_id31_dy_10ums_dy} $D_y$}
+\subcaption{\label{fig:test_id31_dy_10ums_dy}Lateral ($D_y$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_10ums_dz.png}
\end{center}
-\subcaption{\label{fig:test_id31_dy_10ums_dz} $D_z$}
+\subcaption{\label{fig:test_id31_dy_10ums_dz}Vertical ($D_z$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_10ums_ry.png}
\end{center}
-\subcaption{\label{fig:test_id31_dy_10ums_ry} $R_y$}
+\subcaption{\label{fig:test_id31_dy_10ums_ry}Tilt ($R_y$)}
\end{subfigure}
-\caption{\label{fig:test_id31_dy_10ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(10\,\mu\text{m/s}\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_10ums_dy}).}
+\caption{\label{fig:test_id31_dy_10ums}Open-loop (in blue) and closed-loop (i.e. using the NASS, in red) during a \(10\,\upmu\text{m/s}\) scan with the \(T_y\) stage.}
\end{figure}
\paragraph{Fast Scan}
-The system performance was evaluated at an increased scanning velocity of \(100\,\mu\text{m/s}\), and the results are presented in Figure~\ref{fig:test_id31_dy_100ums}.
+The system performance was evaluated at an increased scanning velocity of \(100\,\upmu\text{m/s}\), and the results are presented in Figure~\ref{fig:test_id31_dy_100ums}.
At this velocity, the micro-stepping errors generate \(10\,\text{Hz}\) vibrations, which are further amplified by micro-station resonances.
These vibrations exceeded the NASS feedback controller bandwidth, resulting in limited attenuation under closed-loop control.
This limitation exemplifies why stepper motors are suboptimal for ``long-stroke/short-stroke'' systems requiring precise scanning performance~\cite{dehaeze22_fastj_uhv}.
@@ -12819,34 +12796,34 @@ For applications requiring small \(D_y\) scans, the nano-hexapod can be used exc
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_100ums_dy.png}
\end{center}
-\subcaption{\label{fig:test_id31_dy_100ums_dy} $D_y$}
+\subcaption{\label{fig:test_id31_dy_100ums_dy}Lateral ($D_y$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_100ums_dz.png}
\end{center}
-\subcaption{\label{fig:test_id31_dy_100ums_dz} $D_z$}
+\subcaption{\label{fig:test_id31_dy_100ums_dz}Vertical ($D_z$)}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_100ums_ry.png}
\end{center}
-\subcaption{\label{fig:test_id31_dy_100ums_ry} $R_y$}
+\subcaption{\label{fig:test_id31_dy_100ums_ry}Tilt ($R_y$)}
\end{subfigure}
-\caption{\label{fig:test_id31_dy_100ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(100\,\mu\text{m/s}\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_100ums_dy}).}
+\caption{\label{fig:test_id31_dy_100ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(100\,\upmu\text{m/s}\) scan with the \(T_y\) stage.}
\end{figure}
\subsubsection{Diffraction Tomography}
\label{ssec:test_id31_scans_diffraction_tomo}
In diffraction tomography experiments, the micro-station performs 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 performs the lateral scanning motion.
-To avoid high-frequency vibrations typically induced by the stepper motor, the \(T_y\) stage was not used, which constrained the scanning range to approximately \(\pm 100\,\mu\text{m/s}\).
+To avoid high-frequency vibrations typically induced by the stepper motor, the \(T_y\) stage was not used, which constrained the scanning range to approximately \(\pm 100\,\upmu\text{m/s}\).
The system performance was evaluated at three lateral scanning velocities: \(0.1\,\text{mm/s}\), \(0.5\,\text{mm/s}\), and \(1\,\text{mm/s}\). Figure~\ref{fig:test_id31_diffraction_tomo_setpoint} presents both the \(D_y\) position setpoints and the corresponding measured \(D_y\) positions for all tested velocities.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/test_id31_diffraction_tomo_setpoint.png}
-\caption{\label{fig:test_id31_diffraction_tomo_setpoint}Dy motion for several configured velocities}
+\caption{\label{fig:test_id31_diffraction_tomo_setpoint}Lateral (\(D_y\)) motion for several configured velocities.}
\end{figure}
The positioning errors measured along \(D_y\), \(D_z\), and \(R_y\) directions are displayed in Figure~\ref{fig:test_id31_diffraction_tomo}.
@@ -12874,12 +12851,12 @@ Alternatively, a feedforward controller could improve the lateral positioning ac
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_ry} $R_y$}
\end{subfigure}
-\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 \(6\,\text{deg/s}\)).}
+\caption{\label{fig:test_id31_diffraction_tomo}Diffraction tomography scans (combined \(R_z\) and \(D_y\) motions) at several \(D_y\) velocities, \(\Omega_z = 6\,\text{deg/s}\).}
\end{figure}
-\subsubsection{Feedback control using Complementary Filters}
+\subsubsection{Feedback Control using Complementary Filters}
\label{ssec:test_id31_cf_control}
-A control architecture based on complementary filters to shape the closed-loop transfer functions was proposed during the detail design phase.
+A control architecture based on complementary filters to shape the closed-loop transfer functions was proposed during the detail design phase (Section~\ref{sec:detail_control_cf}).
Experimental validation of this architecture using the NASS is presented herein.
Given that performance requirements are specified in the Cartesian frame, decoupling of the plant within this frame was achieved using Jacobian matrices.
@@ -12890,12 +12867,13 @@ A schematic of the proposed control architecture is illustrated in Figure~\ref{f
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_id31_cf_control.png}
-\caption{\label{fig:test_id31_cf_control}Control architecture in the Cartesian frame. Only the controller corresponding to the \(D_z\) direction is shown. \(H_L\) and \(H_H\) are complementary filters.}
+\caption{\label{fig:test_id31_cf_control}Control architecture using complementary filters proposed in Section~\ref{sec:detail_control_cf}, here adapted for the NASS. Jacobian matrices are used to have the control in the Cartesian frame. Only the \(D_z\) controller is shown. \(H_L\) and \(H_H\) are complementary filters.}
\end{figure}
Implementation of this control architecture necessitates a plant model, which must subsequently be inverted.
-This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF \acrshort{apa} model, such that the model order stays relatively low.
-Proposed analytical formulas for complementary filters having \(40\,\text{dB/dec}\) were used during this experimental validation.
+This plant model was derived from the multi-body model incorporating the previously detailed 2-DoFs \acrshort{apa} (Section~\ref{sec:test_apa_model_2dof}) model and 4-DoFs flexible joints, such that the model order stays relatively low.
+Analytical formulas for complementary filters having \(40\,\text{dB/dec}\) slopes, proposed in Section~\ref{ssec:detail_control_cf_analytical_complementary_filters}, were used during this experimental validation.
+
An initial experimental validation was conducted under no-payload conditions, with control applied solely to the \(D_y\), \(D_z\), and \(R_y\) directions.
Increased control bandwidth was achieved for the \(D_z\) and \(R_y\) directions through appropriate tuning of the parameter \(\omega_0\).
The experimentally measured closed-loop sensitivity transfer functions corresponding to these three controlled directions are presented in Figure~\ref{fig:test_id31_cf_control_dy_dz_diff}.
@@ -12911,7 +12889,7 @@ It also shows that the parameter \(\alpha\) provides a mechanism for managing th
\begin{center}
\includegraphics[scale=1,scale=0.9]{figs/test_id31_cf_control_dy_dz_diff.png}
\end{center}
-\subcaption{\label{fig:test_id31_cf_control_dy_dz_diff}Chose of bandwidth using $\omega_0$, $m = 0\,\text{kg}$}
+\subcaption{\label{fig:test_id31_cf_control_dy_dz_diff}Choice of bandwidth using $\omega_0$, $m = 0\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
@@ -12941,7 +12919,7 @@ For higher values of \(\omega_0\), the system became unstable in the vertical di
\end{center}
\subcaption{\label{fig:test_id31_high_bandwidth_T}Complementary Sensitivity}
\end{subfigure}
-\caption{\label{fig:test_id31_high_bandwidth}Measured Closed-Loop Sensitivity (\subref{fig:test_id31_high_bandwidth_S}) and Complementary Sensitivity (\subref{fig:test_id31_high_bandwidth_T}) transfer functions for the highest test bandwidth \(\omega_0 = 2\pi\cdot 60\,\text{rad/s}\).}
+\caption{\label{fig:test_id31_high_bandwidth}Measured Closed-Loop Sensitivity (\subref{fig:test_id31_high_bandwidth_S}) and Complementary Sensitivity (\subref{fig:test_id31_high_bandwidth_T}) transfer functions for the highest tested parameter \(\omega_0 = 2\pi\cdot 60\,\text{rad/s}\).}
\end{figure}
\subsubsection*{Conclusion}
\label{ssec:test_id31_scans_conclusion}
@@ -12953,9 +12931,9 @@ For tomography experiments, the NASS successfully maintained good positioning ac
The \acrshort{haclac} control architecture proved particularly effective, with the decentralized IFF providing damping of nano-hexapod suspension modes, while the high authority controller addressed low-frequency disturbances.
The vertical scanning capabilities were validated in both step-by-step and continuous motion modes.
-The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to \(100\,\mu\text{m/s}\).
+The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to \(100\,\upmu\text{m/s}\).
-For lateral scanning, the system performed well at moderate speeds (\(10\,\mu\text{m/s}\)) but showed limitations at higher velocities (\(100\,\mu\text{m/s}\)) due to stepper motor-induced vibrations in the \(T_y\) stage.
+For lateral scanning, the system performed well at moderate speeds (\(10\,\upmu\text{m/s}\)) but showed limitations at higher velocities (\(100\,\upmu\text{m/s}\)) due to stepper motor-induced vibrations in the \(T_y\) stage.
The most challenging test case - diffraction tomography combining rotation and lateral scanning - demonstrated the system's ability to maintain vertical and angular stability while highlighting some limitations in lateral positioning during rapid accelerations.
These limitations could be addressed through feedforward control or alternative detector triggering strategies.
@@ -12964,7 +12942,6 @@ Overall, the experimental results validate the effectiveness of the developed co
The identified limitations, primarily related to high-speed lateral scanning and heavy payload handling, provide clear directions for future improvements.
\begin{table}[htbp]
-\caption{\label{tab:test_id31_experiments_results_summary}Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated by bold number.}
\centering
\begin{tabularx}{0.85\linewidth}{Xccc}
\toprule
@@ -12978,13 +12955,13 @@ Tomography (\(6\,\text{deg/s}\), \(39\,\text{kg}\)) & \(297 \Rightarrow \bm{53}\
Tomography (\(180\,\text{deg/s}\)) & \(143 \Rightarrow \bm{38}\) & \(24 \Rightarrow 11\) & \(252 \Rightarrow 130\)\\
Tomography (\(180\,\text{deg/s}\), custom HAC) & \(143 \Rightarrow 29\) & \(24 \Rightarrow 5\) & \(252 \Rightarrow 142\)\\
\midrule
-Reflectivity (\(100\,\mu\text{rad}/s\)) & \(28\) & \(6\) & \(118\)\\
+Reflectivity (\(100\,\upmu\text{rad}/s\)) & \(28\) & \(6\) & \(118\)\\
\midrule
-\(D_z\) scan (\(10\,\mu\text{m/s}\)) & \(25\) & \(5\) & \(108\)\\
-\(D_z\) scan (\(100\,\mu\text{m/s}\)) & \(\bm{35}\) & \(9\) & \(132\)\\
+\(D_z\) scan (\(10\,\upmu\text{m/s}\)) & \(25\) & \(5\) & \(108\)\\
+\(D_z\) scan (\(100\,\upmu\text{m/s}\)) & \(\bm{35}\) & \(9\) & \(132\)\\
\midrule
-Lateral Scan (\(10\,\mu\text{m/s}\)) & \(585 \Rightarrow 21\) & \(155 \Rightarrow 10\) & \(6300 \Rightarrow 60\)\\
-Lateral Scan (\(100\,\mu\text{m/s}\)) & \(1063 \Rightarrow \bm{732}\) & \(167 \Rightarrow \bm{20}\) & \(6445 \Rightarrow \bm{356}\)\\
+Lateral Scan (\(10\,\upmu\text{m/s}\)) & \(585 \Rightarrow 21\) & \(155 \Rightarrow 10\) & \(6300 \Rightarrow 60\)\\
+Lateral Scan (\(100\,\upmu\text{m/s}\)) & \(1063 \Rightarrow \bm{732}\) & \(167 \Rightarrow \bm{20}\) & \(6445 \Rightarrow \bm{356}\)\\
\midrule
Diffraction tomography (\(6\,\text{deg/s}\), \(0.1\,\text{mm/s}\)) & \(\bm{36}\) & \(7\) & \(113\)\\
Diffraction tomography (\(6\,\text{deg/s}\), \(0.5\,\text{mm/s}\)) & \(29\) & \(8\) & \(81\)\\
@@ -12993,6 +12970,8 @@ Diffraction tomography (\(6\,\text{deg/s}\), \(1\,\text{mm/s}\)) & \(\bm{53}\) &
\textbf{Specifications} & \(30\) & \(15\) & \(250\)\\
\bottomrule
\end{tabularx}
+\caption{\label{tab:test_id31_experiments_results_summary}Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated in bold.}
+
\end{table}
\subsection*{Conclusion}
\label{ssec:test_id31_conclusion}
@@ -13000,7 +12979,7 @@ Diffraction tomography (\(6\,\text{deg/s}\), \(1\,\text{mm/s}\)) & \(\bm{53}\) &
This chapter presented a comprehensive experimental validation of the Nano Active Stabilization System (NASS) on the ID31 beamline, demonstrating its capability to maintain precise sample positioning during various experimental scenarios.
The implementation and testing followed a systematic approach, beginning with the development of a short-stroke metrology system to measure the sample position, followed by the successful implementation of a \acrshort{haclac} control architecture, and concluding in extensive performance validation across diverse experimental conditions.
-The short-stroke metrology system, while designed as a temporary solution, proved effective in providing high-bandwidth and low-noise 5-DoF position measurements.
+The short-stroke metrology system, while designed as a temporary solution, proved effective in providing high-bandwidth and low-noise 5-DoFs position measurements.
The careful alignment of the fibered interferometers targeting the two reference spheres ensured reliable measurements throughout the testing campaign.
The implementation of the control architecture validated the theoretical framework developed earlier in this project.
@@ -13138,7 +13117,7 @@ One possible configuration, illustrated in Figure~\ref{fig:conclusion_nass_archi
\begin{figure}[htbp]
\centering
\includegraphics[h!tbp]{figs/conclusion_nass_architecture.png}
-\caption{\label{fig:conclusion_nass_architecture}Proposed alternative configuration for an end-station including the Nano Active Stabilization System}
+\caption{\label{fig:conclusion_nass_architecture}Proposed alternative configuration for an end-station including the Nano Active Stabilization System.}
\end{figure}
With this architecture, the online metrology could be divided into two systems, as proposed by~\cite{geraldes23_sapot_carnaub_sirius_lnls}: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by~\cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}.
@@ -13149,7 +13128,7 @@ Stages based on voice coils, offering nano-positioning capabilities with \(3\,\t
Magnetic levitation also emerges as a particularly interesting technology to be explored, especially for microscopy~\cite{fahmy22_magnet_xy_theta_x,heyman23_levcub} and tomography~\cite{dyck15_magnet_levit_six_degree_freed_rotar_table,fahmy22_magnet_xy_theta_x} end-stations.
Two notable designs illustrating these capabilities are shown in Figure~\ref{fig:conclusion_maglev}.
-Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately \(1\,\text{cm}^3\), is depicted in Figure~\ref{fig:conclusion_maglev_heyman23}, while a 6DoF stage featuring infinite rotation, is shown in Figure~\ref{fig:conclusion_maglev_dyck15}.
+Specifically, a compact 6-DoFs stage known as LevCube, providing a mobility of approximately \(1\,\text{cm}^3\), is depicted in Figure~\ref{fig:conclusion_maglev_heyman23}, while a 6-DoFs stage featuring infinite rotation, is shown in Figure~\ref{fig:conclusion_maglev_dyck15}.
However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature.
\begin{figure}[htbp]
@@ -13165,7 +13144,7 @@ However, implementations of such magnetic levitation stages on synchrotron beaml
\end{center}
\subcaption{\label{fig:conclusion_maglev_dyck15}Stage with infinite $R_z$ rotation \cite{dyck15_magnet_levit_six_degree_freed_rotar_table}}
\end{subfigure}
-\caption{\label{fig:conclusion_maglev}Example of magnetic levitation stages. LevCube allowing for 6DoF control of the position with \(\approx 1\,\text{cm}^3\) mobility (\subref{fig:conclusion_maglev_heyman23}). Magnetic levitation stage with infinite \(R_z\) rotation mobility (\subref{fig:conclusion_maglev_dyck15})}
+\caption{\label{fig:conclusion_maglev}Example of magnetic levitation stages. LevCube allowing for 6-DoFs control of the position with \(\approx 1\,\text{cm}^3\) mobility (\subref{fig:conclusion_maglev_heyman23}). Magnetic levitation stage with infinite \(R_z\) rotation mobility (\subref{fig:conclusion_maglev_dyck15}).}
\end{figure}
\paragraph{Extending the design methodology to complete beamlines}
@@ -13175,6 +13154,7 @@ However, the frequency content of these performance metrics (such as beam stabil
Therefore, adopting a design approach using dynamic error budgets, cascading from overall beamline requirements down to individual component specifications, is considered a potentially valuable direction for future investigation.
\printbibliography[heading=bibintoc,title={Bibliography}]
\chapter*{List of Publications}
+\addcontentsline{toc}{chapter}{List of Publications} % Put the list of publications in the ToC
\begin{refsection}[ref.bib]
\renewcommand{\clearpage}{} % Désactive \clearpage temporairement
% List all papers even if not cited
@@ -13186,5 +13166,5 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro
% Proceedings
\printbibliography[keyword={publication},heading={subbibliography},title={In Proceedings},env=mypubs,type={inproceedings}]
\end{refsection}
-\printglossaries
+\printglossary[type=\acronymtype]
\end{document}