From 35f695b0f6442d03eb31bbec5ceaa0887d910864 Mon Sep 17 00:00:00 2001
From: Thomas Dehaeze <dehaeze.thomas@gmail.com>
Date: Fri, 30 Jun 2023 20:01:22 +0200
Subject: [PATCH] Add two sections (payload dynamics and limited support
 compliance)

---
 .latexmkrc                                    |  111 +
 .../micro_station_meas_dynamics_schematic.pdf |  Bin 61865 -> 59046 bytes
 ...ial_effect_support_compliance_dynamics.pdf | 2644 ++++++++++++++
 ...ial_effect_support_compliance_dynamics.png |  Bin 0 -> 153254 bytes
 ...l_effect_support_compliance_dynamics_d.pdf | 2607 ++++++++++++++
 ...l_effect_support_compliance_dynamics_d.png |  Bin 0 -> 152245 bytes
 ...al_effect_support_compliance_neglected.pdf | 2529 ++++++++++++++
 ...al_effect_support_compliance_neglected.png |  Bin 0 -> 140524 bytes
 .../uniaxial_hac_iff_damped_plants_masses.pdf | 2103 ++++++-----
 .../uniaxial_hac_iff_damped_plants_masses.png |  Bin 235300 -> 291150 bytes
 figs/uniaxial_overview_model_sections.pdf     |  Bin 89711 -> 88182 bytes
 figs/uniaxial_overview_model_sections.png     |  Bin 38610 -> 43257 bytes
 figs/uniaxial_overview_model_sections.svg     |  553 +--
 ...axial_paylaod_dynamics_rigid_schematic.pdf |  Bin 0 -> 36842 bytes
 ...axial_paylaod_dynamics_rigid_schematic.png |  Bin 0 -> 10970 bytes
 ...axial_paylaod_dynamics_rigid_schematic.svg |  171 +
 figs/uniaxial_paylaod_dynamics_schematic.pdf  |  Bin 0 -> 41162 bytes
 figs/uniaxial_paylaod_dynamics_schematic.png  |  Bin 0 -> 15114 bytes
 figs/uniaxial_paylaod_dynamics_schematic.svg  |  206 ++
 ...iaxial_paylaod_dynamics_schematic_beam.pdf |  Bin 0 -> 41330 bytes
 ...iaxial_paylaod_dynamics_schematic_beam.png |  Bin 0 -> 16057 bytes
 ...iaxial_paylaod_dynamics_schematic_beam.svg |  232 ++
 ...ial_payload_dynamics_soft_nano_hexapod.pdf | 3061 +++++++++++++++++
 ...ial_payload_dynamics_soft_nano_hexapod.png |  Bin 0 -> 221869 bytes
 ...al_payload_dynamics_stiff_nano_hexapod.pdf | 2992 ++++++++++++++++
 ...al_payload_dynamics_stiff_nano_hexapod.png |  Bin 0 -> 197590 bytes
 figs/uniaxial_sample_flexibility_control.pdf  |  Bin 0 -> 84765 bytes
 figs/uniaxial_sample_flexibility_control.png  |  Bin 0 -> 39093 bytes
 figs/uniaxial_sample_flexibility_control.svg  |  495 +++
 ...xial_sample_flexibility_noise_budget_d.pdf | 1522 ++++++++
 ...xial_sample_flexibility_noise_budget_d.png |  Bin 0 -> 94387 bytes
 ...xial_sample_flexibility_noise_budget_y.pdf |  Bin 0 -> 96025 bytes
 ...xial_sample_flexibility_noise_budget_y.png |  Bin 0 -> 103087 bytes
 ...l_support_compliance_nano_hexapod_only.pdf |  Bin 0 -> 39889 bytes
 ...l_support_compliance_nano_hexapod_only.png |  Bin 0 -> 10278 bytes
 ...l_support_compliance_nano_hexapod_only.svg |  163 +
 ...niaxial_support_compliance_test_system.pdf |  Bin 0 -> 38371 bytes
 ...niaxial_support_compliance_test_system.png |  Bin 0 -> 17454 bytes
 ...niaxial_support_compliance_test_system.svg |  226 ++
 matlab/nass_uniaxial_model.slx                |  Bin 58238 -> 61033 bytes
 matlab/uniaxial_1_micro_station_model.m       |   62 +-
 matlab/uniaxial_2_nano_hexapod_model.m        |    6 +-
 matlab/uniaxial_3_disturbances.m              |    6 +-
 matlab/uniaxial_4_dynamic_noise_budget.m      |    4 +-
 matlab/uniaxial_5_active_damping.m            |   32 +-
 matlab/uniaxial_6_hac_lac.m                   |  148 +-
 matlab/uniaxial_7_support_compliance.m        |  245 ++
 matlab/uniaxial_8_payload_dynamics.m          |  479 +++
 nass-uniaxial-model.bib                       |   81 +
 nass-uniaxial-model.org                       | 1739 +++++++++-
 nass-uniaxial-model.pdf                       |  Bin 4354959 -> 5083342 bytes
 nass-uniaxial-model.tex                       |  579 ++--
 preamble.tex                                  |    5 +
 53 files changed, 21384 insertions(+), 1617 deletions(-)
 create mode 100644 .latexmkrc
 create mode 100644 figs/uniaxial_effect_support_compliance_dynamics.pdf
 create mode 100644 figs/uniaxial_effect_support_compliance_dynamics.png
 create mode 100644 figs/uniaxial_effect_support_compliance_dynamics_d.pdf
 create mode 100644 figs/uniaxial_effect_support_compliance_dynamics_d.png
 create mode 100644 figs/uniaxial_effect_support_compliance_neglected.pdf
 create mode 100644 figs/uniaxial_effect_support_compliance_neglected.png
 create mode 100644 figs/uniaxial_paylaod_dynamics_rigid_schematic.pdf
 create mode 100644 figs/uniaxial_paylaod_dynamics_rigid_schematic.png
 create mode 100644 figs/uniaxial_paylaod_dynamics_rigid_schematic.svg
 create mode 100644 figs/uniaxial_paylaod_dynamics_schematic.pdf
 create mode 100644 figs/uniaxial_paylaod_dynamics_schematic.png
 create mode 100644 figs/uniaxial_paylaod_dynamics_schematic.svg
 create mode 100644 figs/uniaxial_paylaod_dynamics_schematic_beam.pdf
 create mode 100644 figs/uniaxial_paylaod_dynamics_schematic_beam.png
 create mode 100644 figs/uniaxial_paylaod_dynamics_schematic_beam.svg
 create mode 100644 figs/uniaxial_payload_dynamics_soft_nano_hexapod.pdf
 create mode 100644 figs/uniaxial_payload_dynamics_soft_nano_hexapod.png
 create mode 100644 figs/uniaxial_payload_dynamics_stiff_nano_hexapod.pdf
 create mode 100644 figs/uniaxial_payload_dynamics_stiff_nano_hexapod.png
 create mode 100644 figs/uniaxial_sample_flexibility_control.pdf
 create mode 100644 figs/uniaxial_sample_flexibility_control.png
 create mode 100644 figs/uniaxial_sample_flexibility_control.svg
 create mode 100644 figs/uniaxial_sample_flexibility_noise_budget_d.pdf
 create mode 100644 figs/uniaxial_sample_flexibility_noise_budget_d.png
 create mode 100644 figs/uniaxial_sample_flexibility_noise_budget_y.pdf
 create mode 100644 figs/uniaxial_sample_flexibility_noise_budget_y.png
 create mode 100644 figs/uniaxial_support_compliance_nano_hexapod_only.pdf
 create mode 100644 figs/uniaxial_support_compliance_nano_hexapod_only.png
 create mode 100644 figs/uniaxial_support_compliance_nano_hexapod_only.svg
 create mode 100644 figs/uniaxial_support_compliance_test_system.pdf
 create mode 100644 figs/uniaxial_support_compliance_test_system.png
 create mode 100644 figs/uniaxial_support_compliance_test_system.svg
 create mode 100644 matlab/uniaxial_7_support_compliance.m
 create mode 100644 matlab/uniaxial_8_payload_dynamics.m
 create mode 100644 nass-uniaxial-model.bib

diff --git a/.latexmkrc b/.latexmkrc
new file mode 100644
index 0000000..ba5b929
--- /dev/null
+++ b/.latexmkrc
@@ -0,0 +1,111 @@
+#!/bin/env perl
+
+# Shebang is only to get syntax highlighting right across GitLab, GitHub and IDEs.
+# This file is not meant to be run, but read by `latexmk`.
+
+# ======================================================================================
+# Perl `latexmk` configuration file
+# ======================================================================================
+
+# ======================================================================================
+# PDF Generation/Building/Compilation
+# ======================================================================================
+
+@default_files=('nass-uniaxial-model.tex');
+
+# PDF-generating modes are:
+# 1: pdflatex, as specified by $pdflatex variable (still largely in use)
+# 2: postscript conversion, as specified by the $ps2pdf variable (useless)
+# 3: dvi conversion, as specified by the $dvipdf variable (useless)
+# 4: lualatex, as specified by the $lualatex variable (best)
+# 5: xelatex, as specified by the $xelatex variable (second best)
+$pdf_mode = 1;
+
+# Treat undefined references and citations as well as multiply defined references as
+# ERRORS instead of WARNINGS.
+# This is only checked in the *last* run, since naturally, there are undefined references
+# in initial runs.
+# This setting is potentially annoying when debugging/editing, but highly desirable
+# in the CI pipeline, where such a warning should result in a failed pipeline, since the
+# final document is incomplete/corrupted.
+#
+# However, I could not eradicate all warnings, so that `latexmk` currently fails with
+# this option enabled.
+# Specifically, `microtype` fails together with `fontawesome`/`fontawesome5`, see:
+# https://tex.stackexchange.com/a/547514/120853
+# The fix in that answer did not help.
+# Setting `verbose=silent` to mute `microtype` warnings did not work.
+# Switching between `fontawesome` and `fontawesome5` did not help.
+$warnings_as_errors = 0;
+
+# Show used CPU time. Looks like: https://tex.stackexchange.com/a/312224/120853
+$show_time = 1;
+
+# Default is 5; we seem to need more owed to the complexity of the document.
+# Actual documents probably don't need this many since they won't use all features,
+# plus won't be compiling from cold each time.
+$max_repeat=7;
+
+# --shell-escape option (execution of code outside of latex) is required for the
+#'svg' package.
+# It converts raw SVG files to the PDF+PDF_TEX combo using InkScape.
+#
+# SyncTeX allows to jump between source (code) and output (PDF) in IDEs with support
+# (many have it). A value of `1` is enabled (gzipped), `-1` is enabled but uncompressed,
+# `0` is off.
+# Testing in VSCode w/ LaTeX Workshop only worked for the compressed version.
+# Adjust this as needed. Of course, only relevant for local use, no effect on a remote
+# CI pipeline (except for slower compilation, probably).
+#
+# %O and %S will forward Options and the Source file, respectively, given to latexmk.
+#
+# `set_tex_cmds` applies to all *latex commands (latex, xelatex, lualatex, ...), so
+# no need to specify these each. This allows to simply change `$pdf_mode` to get a
+# different engine. Check if this works with `latexmk --commands`.
+set_tex_cmds("--shell-escape -interaction=nonstopmode --synctex=1 %O %S");
+
+# Use default pdf viewer
+$pdf_previewer = 'zathura';
+
+# option 2 is same as 1 (run biber when necessary), but also deletes the
+# regeneratable bbl-file in a clenaup (`latexmk -c`). Do not use if original
+# bib file is not available!
+$bibtex_use = 2;  # default: 1
+
+# Change default `biber` call, help catch errors faster/clearer. See
+# https://web.archive.org/web/20200526101657/https://www.semipol.de/2018/06/12/latex-best-practices.html#database-entries
+$biber = "biber --validate-datamodel %O %S";
+
+# Glossaries
+add_cus_dep('glo', 'gls', 0, 'run_makeglossaries');
+add_cus_dep('acn', 'acr', 0, 'run_makeglossaries');
+
+sub run_makeglossaries {
+  if ( $silent ) {
+    system "makeglossaries -q -s '$_[0].ist' '$_[0]'";
+  }
+  else {
+    system "makeglossaries -s '$_[0].ist' '$_[0]'";
+  };
+}
+
+# ======================================================================================
+# Auxiliary Files
+# ======================================================================================
+
+# Let latexmk know about generated files, so they can be used to detect if a
+# rerun is required, or be deleted in a cleanup.
+# loe: List of Examples (KOMAScript)
+# lol: List of Listings (`listings` and `minted` packages)
+# run.xml: biber runs
+# glg: glossaries log
+# glstex: generated from glossaries-extra
+push @generated_exts, 'loe', 'lol', 'run.xml', 'glstex', 'glo', 'gls', 'glg', 'acn', 'acr', 'alg';
+
+# Also delete the *.glstex files from package glossaries-extra. Problem is,
+# that that package generates files of the form "basename-digit.glstex" if
+# multiple glossaries are present. Latexmk looks for "basename.glstex" and so
+# does not find those. For that purpose, use wildcard.
+# Also delete files generated by gnuplot/pgfplots contour plots
+# (.dat, .script, .table).
+$clean_ext = "%R-*.glstex %R_contourtmp*.*";
diff --git a/figs/micro_station_meas_dynamics_schematic.pdf b/figs/micro_station_meas_dynamics_schematic.pdf
index a1188eef93832ec559fa75c9bca076f1870b4619..de87770eb761ee2a6857bdfa9cde176894031c14 100644
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diff --git a/figs/uniaxial_overview_model_sections.pdf b/figs/uniaxial_overview_model_sections.pdf
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diff --git a/figs/uniaxial_paylaod_dynamics_schematic_beam.pdf b/figs/uniaxial_paylaod_dynamics_schematic_beam.pdf
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diff --git a/figs/uniaxial_support_compliance_test_system.pdf b/figs/uniaxial_support_compliance_test_system.pdf
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diff --git a/matlab/nass_uniaxial_model.slx b/matlab/nass_uniaxial_model.slx
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diff --git a/matlab/uniaxial_1_micro_station_model.m b/matlab/uniaxial_1_micro_station_model.m
index 2aa2c5f..4e7c6e1 100644
--- a/matlab/uniaxial_1_micro_station_model.m
+++ b/matlab/uniaxial_1_micro_station_model.m
@@ -18,56 +18,30 @@ freqs = logspace(0, 3, 1000);
 
 
 
-% #+name: fig:micro_station_meas_dynamics_schematic
-% #+caption: Measurement setup - Schematic
-% #+RESULTS:
+% #+name: fig:micro_station_uniaxial_model
+% #+caption: Schematic of the Micro-Station measurement setup and uniaxial model.
+% #+begin_figure
+% #+attr_latex: :caption \subcaption{\label{fig:micro_station_meas_dynamics_schematic}Measurement setup - Schematic}
+% #+attr_latex: :options {0.69\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :scale 1
 % [[file:figs/micro_station_meas_dynamics_schematic.png]]
+% #+end_subfigure
+% #+attr_latex: :caption \subcaption{\label{fig:uniaxial_model_micro_station}Uniaxial model of the micro-station}
+% #+attr_latex: :options {0.29\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :scale 1
+% [[file:figs/uniaxial_model_micro_station.png]]
+% #+end_subfigure
+% #+end_figure
 
-% Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure ref:fig:uniaxial_measured_frf_vertical).
+% Due to the bad coherence at low frequency, the frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure ref:fig:uniaxial_comp_frf_meas_model).
 
 
 %% Load measured FRF
 load('meas_microstation_frf.mat');
 
-%% Measured Frequency Response Functions in the vertical direction
-figure;
-tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 
-ax1 = nexttile([2,1]);
-hold on;
-plot(f(f>20), abs(frf_Fhx_to_Dhx(f>20)), ...
-        'DisplayName', '$D_{h}/F_{h}$');
-plot(f(f>20), abs(frf_Fgx_to_Dhx(f>20)), ...
-        'DisplayName', '$D_{h}/F_{g}$');
-plot(f(f>20), abs(frf_Fgx_to_Dgx(f>20)), ...
-        'DisplayName', '$D_{g}/F_{g}$');
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-10, 2e-6]);
-legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
-
-ax2 = nexttile;
-hold on;
-plot(f(f>20), 180/pi*unwrap(angle(frf_Fhx_to_Dhx(f>20))));
-plot(f(f>30), 180/pi*unwrap(angle(frf_Fgx_to_Dhx(f>30))));
-plot(f(f>20), 180/pi*unwrap(angle(frf_Fgx_to_Dgx(f>20))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
-xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
-hold off;
-yticks(-360:90:360);
-ylim([-360, 10]);
-
-linkaxes([ax1,ax2],'x');
-xlim([1, 500]);
-
-
-
-% #+name: fig:uniaxial_model_micro_station
-% #+caption: Uniaxial model of the micro-station
-% #+RESULTS:
-% [[file:figs/uniaxial_model_micro_station.png]]
 
 % Masses are estimated from the CAD.
 
@@ -115,7 +89,9 @@ G_id.InputName = {'Fg', 'Fh'};
 G_id.OutputName = {'Dg', 'Dh'};
 
 % Comparison of the model and measurements
-% The comparison between the measurements and the model is done in Figure ref:fig:uniaxial_comp_frf_meas_model.
+% The comparison between the measurements and the model is shown in Figure ref:fig:uniaxial_comp_frf_meas_model.
+
+% Only three modes are modelled with frequencies at 70Hz, 140Hz and 320Hz.
 
 % As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation.
 % More accurate models will be used later on.
diff --git a/matlab/uniaxial_2_nano_hexapod_model.m b/matlab/uniaxial_2_nano_hexapod_model.m
index a4faf61..36ab6c4 100644
--- a/matlab/uniaxial_2_nano_hexapod_model.m
+++ b/matlab/uniaxial_2_nano_hexapod_model.m
@@ -36,11 +36,13 @@ cn = 2*0.01*sqrt(mn*kn); % [N/(m/s)]
 %% Sample Mass
 ms = 10; % [kg]
 
-% Obtained Dynamics
+% Obtained Dynamic Response
 % The sensitivity to disturbances (i.e. $x_f$, $f_t$ and $f_s$) are shown in Figure ref:fig:uniaxial_sensitivity_dist_first_params.
 % The /plant/ (i.e. the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure ref:fig:uniaxial_plant_first_params.
 
+% #+begin_important
 % For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) combined with three sample's masses ($m_s = 1\,kg$, $m_s = 25\,kg$ and $m_s = 50\,kg$).
+% #+end_important
 
 
 %% Use 1DoF Nano-Hexpod model
@@ -95,7 +97,7 @@ xlim([1, 500]);
 
 
 % #+name: fig:uniaxial_sensitivity_dist_first_params
-% #+caption: Sensitivity to disturbances
+% #+caption: Sensitivity of the relative motion $d$ to disturbances: $f_s$ the direct forces applied on the sample, $f_t$ disturbances from the micro-station stages and $x_f$ the floor motion (from left to right)
 % #+RESULTS:
 % [[file:figs/uniaxial_sensitivity_dist_first_params.png]]
 
diff --git a/matlab/uniaxial_3_disturbances.m b/matlab/uniaxial_3_disturbances.m
index 511a601..ea16b9c 100644
--- a/matlab/uniaxial_3_disturbances.m
+++ b/matlab/uniaxial_3_disturbances.m
@@ -76,7 +76,7 @@ psd_xf = psd_V.*abs(squeeze(freqresp(G_geo, f, 'Hz'))).^2; % [m^2/Hz]
 % The amplitude spectral density $\Gamma_{x_f}$ of the measured displacement $x_f$ is shown in Figure ref:fig:asd_floor_motion_id31.
 
 
-%% Measured Amplitude Spectral Density of the Floor motion on ID31
+%% Amplitude Spectral Density of the measured Floor motion on ID31
 figure;
 hold on;
 plot(f, sqrt(psd_xf), 'DisplayName', '$\Gamma_{x_{f}}$');
@@ -114,7 +114,7 @@ win = hanning(ceil(2*Fs)); % Hanning window
 % It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.
 
 
-%% Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating
+%% Amplitude Spectral Density of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating
 figure;
 hold on;
 plot(f, sqrt(psd_vft)./(2*pi*f), 'DisplayName', '6rpm');
@@ -128,7 +128,7 @@ xlim([1, 500]); ylim([1e-12, 1e-7])
 
 
 % #+name: fig:asd_vibration_spindle_rotation
-% #+caption: Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating
+% #+caption: Amplitude Spectral Density of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating
 % #+RESULTS:
 % [[file:figs/asd_vibration_spindle_rotation.png]]
 
diff --git a/matlab/uniaxial_4_dynamic_noise_budget.m b/matlab/uniaxial_4_dynamic_noise_budget.m
index e94fa60..65922f7 100644
--- a/matlab/uniaxial_4_dynamic_noise_budget.m
+++ b/matlab/uniaxial_4_dynamic_noise_budget.m
@@ -32,10 +32,10 @@ load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
 % The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the light sample (same conclusions can be drawn with the heavy one).
 
 % #+begin_important
-% From Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses, following can be concluded for the *soft nano-hexapod*:
+% From Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses, the following can be concluded for the *soft nano-hexapod*:
 % - It is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to the lower stiffness
 % - Between the suspension mode of the nano-hexapod (here at 5Hz) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out.
-% - Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is almost $1$.
+% - Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is close to $1$.
 % #+end_important
 
 
diff --git a/matlab/uniaxial_5_active_damping.m b/matlab/uniaxial_5_active_damping.m
index dffff45..c99388a 100644
--- a/matlab/uniaxial_5_active_damping.m
+++ b/matlab/uniaxial_5_active_damping.m
@@ -23,11 +23,12 @@ load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
 
 % 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 *alternating poles and zeros* meaning that the phase is bounded to $\pm 90\,\text{deg}$ which makes the controller very robust.
+% All have *alternating poles and zeros* meaning that the phase do not vary by more than $\pm 90\,\text{deg}$ which makes the design of a robust controller very easy.
+% The reason all three plants in Figure ref:fig:uniaxial_plant_active_damping_techniques have alternating poles and zeros is because the three sensors are all collocated with the actuator [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7]].
 
-% When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
+% When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and DVF transfer functions.
 
-% For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques.
+% For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques.
 % Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
 
 
@@ -145,13 +146,21 @@ linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
 xlim([1, 1000]);
 
 % Achievable Damping - Root Locus
+% <<ssec:uniaxial_active_damping_achievable_damping>>
 % The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
 % They are shown in Figure ref:fig:uniaxial_root_locus_damping_techniques.
 
 % All three active damping approach can lead to *critical damping* of the nano-hexapod suspension mode.
 
-% There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) and with RDC for a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode).
-% This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
+% There is even some damping authority on micro-station modes in the following cases:
+% - IFF with a stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) ::
+%   This can be understood from the mechanical equivalent of IFF shown in Figure ref:fig:uniaxial_active_damping_iff_equiv (right) considering an high stiffness $k$.
+%   The micro-station top platform is connected to an inertial mass (the nano-hexapod) through a damper, which damps the micro-station suspension suspension mode.
+% - DVF with a stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) ::
+%   In that case, the "sky hook damper" (see mechanical equivalent of IFF in Figure ref:fig:uniaxial_active_damping_dvf_equiv, right) is connected to the micro-station top platform through the stiff nano-hexapod.
+% - RDC with a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) ::
+%   At the frequency of the micro-station mode, the nano-hexapod top mass is behaving as an inertial reference as the suspension mode of the soft nano-hexapod is at much lower frequency.
+%   The micro-station and the nano-hexapod masses are connected through a large damper induced by RDC (see mechanical equivalent in Figure ref:fig:uniaxial_active_damping_rdc_equiv, right) which allows some damping of the micro-station.
 
 
 %% Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses
@@ -286,7 +295,7 @@ xlim([-4000, 0]); ylim([0, 4000]);
 
 
 % #+name: fig:uniaxial_root_locus_damping_techniques
-% #+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.
+% #+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses. The Root Loci are zoomed on the suspension mode of the nano-hexapod.
 % #+RESULTS:
 % [[file:figs/uniaxial_root_locus_damping_techniques.png]]
 
@@ -425,7 +434,7 @@ isstable(G_dvf_vc_light) && isstable(G_dvf_vc_mid) && isstable(G_dvf_vc_heavy) &
 isstable(G_dvf_md_light) && isstable(G_dvf_md_mid) && isstable(G_dvf_md_heavy) && ...
 isstable(G_dvf_pz_light) && isstable(G_dvf_pz_mid) && isstable(G_dvf_pz_heavy)
 
-%% Save Active Damping Controller
+%% Save Damped Plants
 save('./mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz_light', ...
                                          'G_rdc_vc_light', 'G_rdc_md_light', 'G_rdc_pz_light', ...
                                          'G_dvf_vc_light', 'G_dvf_md_light', 'G_dvf_pz_light', ...
@@ -437,6 +446,8 @@ save('./mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_
                                          'G_dvf_vc_heavy', 'G_dvf_md_heavy', 'G_dvf_pz_heavy');
 
 % Change of sensitivity to disturbances
+% <<ssec:uniaxial_active_damping_sensitivity_disturbances>>
+
 % The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure ref:fig:uniaxial_sensitivity_dist_active_damping.
 % The comparison is done with the nano-hexapod having a stiffness $k_n = 1\,N/\mu m$.
 
@@ -490,10 +501,11 @@ linkaxes([ax1,ax2,ax3],'x');
 xlim([1, 500]);
 
 % Noise Budgeting after Active Damping
+% <<ssec:uniaxial_active_damping_noise_budget>>
 % Cumulative Amplitude Spectrum of the distance $d$ with all three active damping techniques are compared in Figure ref:fig:uniaxial_cas_active_damping.
 % All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
 
-% Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
+% Compared to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
 
 
 %% Cumulative Amplitude Spectrum of the distance d with all three active damping techniques
@@ -565,7 +577,7 @@ linkaxes([ax1,ax2,ax3], 'xy')
 xlim([1, 500]);
 ylim([2e-10, 3e-6])
 
-% Obtained Damped Plant
+% Obtained Closed Loop Response
 % The transfer functions from the plant input $f$ to the relative displacement $d$ while the active damping is implemented are shown in Figure ref:fig:uniaxial_damped_plant_three_active_damping_techniques.
 % All three active damping techniques yield similar damped plants.
 
@@ -744,6 +756,8 @@ linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
 xlim([1, 1e3]);
 
 % Robustness to change of payload's mass
+% <<ssec:uniaxial_active_damping_robustness>>
+
 % The Root Locus for the three damping techniques are shown in Figure ref:fig:uniaxial_active_damping_robustness_mass_root_locus for three sample's mass (1kg, 25kg and 50kg).
 % The closed-loop poles are shown by the squares for a specific gain.
 
diff --git a/matlab/uniaxial_6_hac_lac.m b/matlab/uniaxial_6_hac_lac.m
index 2c92676..c0abe53 100644
--- a/matlab/uniaxial_6_hac_lac.m
+++ b/matlab/uniaxial_6_hac_lac.m
@@ -33,10 +33,17 @@ load('uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz
                                    'G_dvf_vc_heavy', 'G_dvf_md_heavy', 'G_dvf_pz_heavy');
 
 % Damped Plant Dynamics
+% <<ssec:uniaxial_position_control_damped_dynamics>>
+
 % As was shown in Section ref:sec:uniaxial_active_damping, all three proposed active damping techniques yield similar damping plants.
-% Therefore, /Integral Force Feedback/ will be used in this section to study the HAC-LAC performances.
+% Therefore, /Integral Force Feedback/ will be used in this section to study the HAC-LAC performance.
 
 % The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_hac_iff_damped_plants_masses.
+% For $k_n = 0.01\,N/\mu m$ and $k_n = 1\,N/\mu m$, except from the nano-hexapod mode, the dynamics is quite simple and can be well approximated by a second order plant.
+% However, this is not the case for the stiff nano-hexapod ($k_n = 100\,N/\mu m$) where two modes can be seen (Figure ref:fig:uniaxial_hac_iff_damped_plants_masses, right).
+
+% This is due to the interaction between the micro-station (modelled modes at 70Hz, 140Hz and 320Hz) and the nano-hexapod.
+% Such effect will be explained in Section ref:sec:uniaxial_support_compliance.
 
 
 %% Damped plant - Robustness to change of sample's mass
@@ -45,22 +52,28 @@ tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
 
 ax1 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 1\,kg$');
-plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 25\,kg$');
-plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 50\,kg$');
+plot(freqs, abs(squeeze(freqresp(G_vc_light('d', 'f'), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz'))), 'color', colors(2,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:));
+loglog(10.^(0.4*cos([0:0.01:2*pi])+log10(100)), ...
+       10.^(0.8*sin([0:0.01:2*pi]-pi/4)+log10(8e-8)), 'k--');
+text(20, 4e-8, sprintf('Small\nInteraction'), 'horizontalalignment', 'center');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
 title('$k_n = 0.01\,N/\mu m$');
 ylim([5e-10, 1e-3]);
-ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-ldg.ItemTokenSize = [20, 1];
 
 ax2 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_md_light('d', 'f'), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz'))), 'color', colors(2,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:));
+loglog(10.^(0.4*cos([0:0.01:2*pi])+log10(200)), ...
+       10.^(0.8*sin([0:0.01:2*pi]-pi/4)+log10(2e-8)), 'k--');
+text(40, 1e-8, sprintf('Small\nInteraction'), 'horizontalalignment', 'center');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
@@ -69,20 +82,27 @@ ylim([5e-10, 1e-3]);
 
 ax3 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_pz_light('d', 'f'), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'DisplayName', '$m_s = 1\,kg$, OL');
+plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$, IFF');
+plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$m_s = 25\,kg$, IFF');
+plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$m_s = 50\,kg$, IFF');
+loglog(10.^(0.8*cos([0:0.01:2*pi])+log10(350)), ...
+       10.^(1.2*sin([0:0.01:2*pi])+log10(8e-9)), 'k--', 'HandleVisibility', 'off');
+text(200, 5e-7, sprintf('$\\mu$ Station\nCoupling'), 'horizontalalignment', 'center');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
 title('$k_n = 100\,N/\mu m$');
 ylim([5e-10, 1e-3]);
+ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [20, 1];
 
 ax1b = nexttile();
 hold on;
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('d', 'f'), freqs, 'Hz')))), 'color', [colors(1,:), 0.5]);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
@@ -92,9 +112,10 @@ ylim([-200, 20]);
 
 ax2b = nexttile();
 hold on;
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('d', 'f'), freqs, 'Hz')))), 'color', [colors(1,:), 0.5]);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
@@ -104,9 +125,10 @@ ylim([-200, 20]);
 
 ax3b = nexttile();
 hold on;
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('d', 'f'), freqs, 'Hz')))), 'color', [colors(1,:), 0.5]);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
@@ -118,13 +140,15 @@ linkaxes([ax1,ax2,ax3,ax1b,ax2b,ax3b],'x');
 xlim([1, 1e3]);
 
 % Position Feedback Controller
+% <<ssec:uniaxial_position_control_design>>
+
 % The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample's mass.
 
-% The required feedback bandwidth was approximately determined un Section ref:sec:uniaxial_noise_budgeting:
+% The required feedback bandwidth was approximately determined in Section ref:sec:uniaxial_noise_budgeting:
 % - $\approx 10\,\text{Hz}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$).
 %   Near this frequency, the plants are equivalent to a mass line.
 %   The gain of the mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,kg$ up to $65\,kg$).
-%   This mean that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
+%   This means that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
 % - $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$).
 %   Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line.
 %   It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.
@@ -136,7 +160,37 @@ xlim([1, 1e3]);
 
 
 % Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure ref:fig:uniaxial_nyquist_hac).
-% These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
+% These high authority controllers are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
+% The controllers used for the three nano-hexapod are shown in Equation eqref:eq:uniaxial_hac_formulas, and the parameters are summarized in Table ref:tab:uniaxial_feedback_controller_parameters.
+
+% \begin{subequations} \label{eq:uniaxial_hac_formulas}
+% \begin{align}
+% K_{\text{soft}}(s) &= g \cdot
+%        \underbrace{\frac{s + \omega_0}{s + \omega_i}}_{\text{lag}} \cdot
+%        \underbrace{\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}}_{\text{lead}} \cdot
+%        \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}} \\
+% K_{\text{mid}}(s) &= g \cdot
+%        \underbrace{\left(\frac{s + \omega_0}{s + \omega_i}\right)^2}_{\text{2 lags}} \cdot
+%        \underbrace{\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}}_{\text{lead}} \cdot
+%        \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}} \\
+% K_{\text{stiff}}(s) &= g \cdot
+%        \underbrace{\left(\frac{1}{s + \omega_i}\right)^2}_{\text{2 lags}} \cdot
+%        \underbrace{\left(\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}\right)^2}_{\text{2 leads}} \cdot
+%        \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}}
+% \end{align}
+% \end{subequations}
+
+% #+name: tab:uniaxial_feedback_controller_parameters
+% #+caption: Parameters used for the position feedback controllers
+% #+attr_latex: :environment tabularx :width \linewidth :align lXXX
+% #+attr_latex: :center t :booktabs t
+% |        | *Soft*                                    | *Moderately stiff*                         | *Stiff*                                  |
+% |--------+-------------------------------------------+--------------------------------------------+------------------------------------------|
+% | *Gain* | $g = 4 \cdot 10^5$                        | $g = 3 \cdot 10^6$                         | $g = 6 \cdot 10^12$                      |
+% | *Lead* | $a = 5$, $\omega_c = 20\,Hz$              | $a = 4$, $\omega_c = 70\,Hz$               | $a = 5$, $\omega_c = 100\,Hz$            |
+% | *Lag*  | $\omega_0 = 5\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_0 = 20\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_i = 0.01\,Hz$   |
+% | *LPF*  | $\omega_l = 200\,Hz$                      | $\omega_l = 300\,Hz$                       | $\omega_l = 500\,Hz$                     |
+
 % The loop gains for the three nano-hexapod are shown in Figure ref:fig:uniaxial_loop_gain_hac.
 % We can see that:
 % - for the soft and moderately stiff nano-hexapod, the crossover frequency varies a lot with the sample mass.
@@ -144,8 +198,8 @@ xlim([1, 1e3]);
 % - for the stiff nano-hexapod, the obtained crossover frequency is not at high as what was estimated necessary.
 %   The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.
 
-% Note that these controller were quickly tuned by hand and not designed using any optimization methods.
-% The goal is just to have a first estimation of the attainable performances.
+% Note that these controllers were quickly tuned by hand and not designed using any optimization methods.
+% The goal is just to have a first estimation of the attainable performance.
 
 
 %% High Authority Controller - Soft Nano-Hexapod
@@ -154,61 +208,65 @@ a  = 5;  % Amount of phase lead / width of the phase lead / high frequency gain
 wc = 2*pi*20; % Frequency with the maximum phase lead [rad/s]
 H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
 
+% Lag at low frequency
+H_lag = (s + 2*pi*5)/(s + 2*pi*0.01);
+
 % Low Pass filter to increase robustness
 H_lpf = 1/(1 + s/2/pi/200);
 
-% Added integrator at low frequency
-H_int = (s + 2*pi*5)/(s + 2*pi*0.01);
-
 % High Authority Controller
 K_hac_vc = 4e5    * ... % Gain
            H_lead * ... % Lead
-           H_int * ...  % Integrator
+           H_lag  * ... % Lag
            H_lpf;       % LPF
 K_hac_vc.InputName = {'d'};
 K_hac_vc.OutputName = {'f'};
 
 %% High Authority Controller - Mid Stiffness Nano-Hexapod
-% Integrator as low frequency
-H_int = (s + 2*pi*10)/(s + 2*pi*0.01) * (s + 2*pi*20)/(s + 2*pi*0.01);
-
 % Lead to increase phase margin
 a  = 4;  % Amount of phase lead / width of the phase lead / high frequency gain
 wc = 2*pi*70; % Frequency with the maximum phase lead [rad/s]
 
 H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
 
+% Lag at low frequency
+H_lag = ((s + 2*pi*15)/(s + 2*pi*0.01))^2;
+
 % Low Pass filter to increase robustness
 H_lpf = 1/(1 + s/2/pi/300);
 
 % High Authority Controller
 K_hac_md = 3e6    * ... % Gain
            H_lead * ... % Lead
-           H_lpf  * ... % Low Pass Filter
-           H_int;         % Integrator
+           H_lag  * ... % Lag
+           H_lpf;       % LPF
 K_hac_md.InputName = {'d'};
 K_hac_md.OutputName = {'f'};
 
 %% High Authority Controller - Stiff Nano-Hexapod
-% Integrator as low frequency
-H_int = 1/(s + 2*pi*0.01) * 1/(s + 2*pi*0.01);
-
 % Lead to increase phase margin
 a  = 5;  % Amount of phase lead / width of the phase lead / high frequency gain
 wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s]
-H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+H_lead = ((1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))))^2;
+
+% Integrator
+H_int = 1/(s + 2*pi*0.01)^2;
 
 % Low Pass filter to increase robustness
 H_lpf = 1/(1 + s/2/pi/500);
 
 % High Authority Controller
-K_hac_pz = 6e12      * ... % Gain
-           H_lead^2 * ... % Lead
-           H_lpf    * ... % Low Pass Filter
-           H_int;         % Integrator
+K_hac_pz = 6e12   * ... % Gain
+           H_lead * ... % Lead
+           H_int  * ... % Lag
+           H_lpf;       % LPF
 K_hac_pz.InputName = {'d'};
 K_hac_pz.OutputName = {'f'};
 
+%% Save High Authority Controllers
+save('/mat/uniaxial_high_authority_controllers.mat', ...
+        'K_hac_vc', 'K_hac_md', 'K_hac_pz');
+
 %% Compute Loop gain for Nyquist Plot
 L_vc_light = squeeze(freqresp(K_hac_vc*G_iff_vc_light('d', 'f'), freqs, 'Hz'));
 L_vc_mid   = squeeze(freqresp(K_hac_vc*G_iff_vc_mid(  'd', 'f'), freqs, 'Hz'));
@@ -304,6 +362,8 @@ linkaxes([ax1,ax2],'x');
 xlim([1, 500]);
 
 % Closed-Loop Noise Budgeting
+% <<ssec:uniaxial_position_control_cl_noise_budget>>
+
 % The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
 % These are compared with the open-loop and damped plants cases in Figure ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass).
 % As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
@@ -370,7 +430,7 @@ xlim([1, 500]);
 
 
 % #+name: fig:uniaxial_sensitivity_dist_hac_lac
-% #+caption: Change of sensitivity to disturbances with LAC and with HAC-LAC
+% #+caption: Change of sensitivity to disturbances with LAC and with HAC-LAC. Nano-Hexapod with $k_n = 1\,N/\mu m$ and sample mass of $25\,kg$ are used.
 % #+RESULTS:
 % [[file:figs/uniaxial_sensitivity_dist_hac_lac.png]]
 
diff --git a/matlab/uniaxial_7_support_compliance.m b/matlab/uniaxial_7_support_compliance.m
new file mode 100644
index 0000000..2c550b7
--- /dev/null
+++ b/matlab/uniaxial_7_support_compliance.m
@@ -0,0 +1,245 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Load the micro-station parameters
+load('uniaxial_micro_station_parameters.mat')
+
+%% Frequency Vector [Hz]
+freqs = logspace(0, 3, 1000);
+
+% Neglected support compliance
+
+% Let's first neglect the limited compliance of the micro-station and use the uniaxial model show in Figure ref:fig:uniaxial_support_compliance_nano_hexapod_only.
+% Let's choose a nano-hexapod mass (including the payload) of $20\,\text{kg}$ and three hexapod stiffnesses such that their resonance frequencies are at $\omega_{\nu} = 10\,\text{Hz}$, $\omega_{\nu} = 70\,\text{Hz}$ and $\omega_{\nu} = 400\,\text{Hz}$.
+
+% The obtained transfer functions from $F$ to $L^\prime$ are shown in Figure ref:fig:uniaxial_effect_support_compliance_neglected.
+% When neglecting the support compliance, large feedback bandwidth can be achieve for all three Nano-Hexapod.
+
+
+%% Nano-Hexapod Parameters
+m = 20; % Mass [kg]
+
+% "Soft" Nano-Hexapod
+k_soft = m*(2*pi*10)^2; % Stiffness [N/m]
+c_soft = 0.1*2*sqrt(m*k_soft); % Damping [N/(m/s)]
+
+% "Mid" Nano-Hexapod
+k_mid = m*(2*pi*70)^2; % Stiffness [N/m]
+c_mid = 0.1*2*sqrt(m*k_mid); % Damping [N/(m/s)]
+
+% "Stiff" Nano-Hexapod
+k_stiff = m*(2*pi*400)^2; % Stiffness [N/m]
+c_stiff = 0.1*2*sqrt(m*k_stiff); % Damping [N/(m/s)]
+
+%% Compute the transfer functions for considered nano-hexapods - From F to L'
+% "Soft" Nano-Hexapod
+G_soft_a = 1/(m*s^2 + c_soft*s + k_soft); % Transfer function from F to L'
+
+% "Mid" Nano-Hexapod
+G_mid_a = 1/(m*s^2 + c_mid*s + k_mid); % Transfer function from F to L'
+
+% "Stiff" Nano-Hexapod
+G_stiff_a = 1/(m*s^2 + c_stiff*s + k_stiff); % Transfer function from F to L'
+
+%% Obtained transfer functions from F to L when neglecing support compliance
+freqs = logspace(0, 3, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_soft_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+text(50, 5e-5, '$\omega_\nu =$ 10Hz', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude [m/N]');
+xticks([1e0, 1e1, 1e2]);
+title('"Soft" $\nu$-hexapod');
+
+ax2 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_mid_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+text(70, 3e-6, '$\omega_\nu =$ 70Hz', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]');
+set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+title('"Mid" $\nu$-hexapod');
+
+ax3 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_stiff_a, freqs, 'Hz'))), '-', 'color', colors(1,:), ...
+     'DisplayName', '$L^\prime/F$');
+text(200, 8e-8, '$\omega_\nu =$ 400Hz', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]);
+legend('location', 'northeast');
+xticks([1e0, 1e1, 1e2]);
+title('"Stiff" $\nu$-hexapod');
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-9, 1e-4]);
+
+% Effect of support compliance on $L/F$
+
+% Let's now add some support compliance and use the model shown in Figure ref:fig:uniaxial_support_compliance_test_system.
+% The parameters of the support (i.e. $m^{\prime}$, $c^{\prime}$ and $k^{\prime}$) are chosen in such a way that the main vertical mode at $\omega_\mu = 70\,\text{Hz}$ seen on the micro-station is matched (Figure ref:fig:uniaxial_comp_frf_meas_model, yellow curve).
+
+
+%% Parameters of the support compliance
+w0h = 2*pi*70; % [rad/s]
+xih = 0.1; % [-]
+
+mh = 20; % [kg]
+kh = mh*w0h^2;
+ch = xih*2*sqrt(kh*mh);
+
+%% Compute the transfer functions from F to L and from F to d for considered Nano-Hexapods
+% "Soft" Nano-Hexapod
+G_soft = (mh*s^2 + ch*s + kh)/(m*s^2*(c_soft*s + k_soft) + (m*s^2 + c_soft*s + k_soft)*(mh*s^2 + ch*s + kh)); % d/F
+G_soft_r = (1 - m*s^2*G_soft)/(c_soft*s + k_soft); % L/F
+
+% "Mid" Nano-Hexapod
+G_mid = (mh*s^2 + ch*s + kh)/(m*s^2*(c_mid*s + k_mid) + (m*s^2 + c_mid*s + k_mid)*(mh*s^2 + ch*s + kh)); % d/F
+G_mid_r = (1 - m*s^2*G_mid)/(c_mid*s + k_mid); % L/F
+
+% "Stiff" Nano-Hexapod
+G_stiff = (mh*s^2 + ch*s + kh)/(m*s^2*(c_stiff*s + k_stiff) + (m*s^2 + c_stiff*s + k_stiff)*(mh*s^2 + ch*s + kh)); % d/F
+G_stiff_r = (1 - m*s^2*G_stiff)/(c_stiff*s + k_stiff); % L/F
+
+
+
+% The transfer functions from $F$ to $L$ (i.e. control of the relative motion of the nano-hexapod) and from $L$ to $d$ (i.e. control of the position between the nano-hexapod and the fixed granite) can then be computed.
+
+% When the relative displacement of the nano-hexapod $L$ is to be controlled (dynamics shown in Figure ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff nano-hexapod (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, right).
+
+% This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages.
+% In such 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 is it fixed.
+% # Add references of such stations with piezo stages on top
+
+% If a soft nano-hexapod is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure ref:fig:uniaxial_effect_support_compliance_dynamics, left) which will impact the control robustness and performance.
+
+
+%% Effect of the support compliance on the transfer functions from F to L and from F to d
+figure;
+freqs = logspace(0, 3, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_soft_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_soft_r, freqs, 'Hz'))), '-', 'color', colors(2,:));
+loglog(10.^(0.3*cos(0:0.01:2*pi)+log10(60)), ...
+       10.^(0.6*sin(0:0.01:2*pi)+log10(4e-7)), 'k--');
+text(8, 3e-7, sprintf('Support\nDynamics'), 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+title('"Soft" $\nu$-hexapod');
+ylabel('Magnitude [m/N]');
+
+
+ax2 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_mid_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_mid_r, freqs, 'Hz'))), '-', 'color', colors(2,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]');
+xticks([1e0, 1e1, 1e2]);
+title('"Mid" $\nu$-hexapod');
+set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_stiff_a, freqs, 'Hz'))), '-', 'color', colors(1,:), ...
+     'DisplayName', '$L^\prime/F$');
+plot(freqs, abs(squeeze(freqresp(G_stiff_r, freqs, 'Hz'))), '-', 'color', colors(2,:), ...
+     'DisplayName', '$L/F$');
+loglog(10.^(0.3*cos(0:0.01:2*pi)+log10(50)), ...
+       10.^(0.3*sin(0:0.01:2*pi)+log10(8e-9)), 'k--', 'HandleVisibility', 'off');
+text(50, 3e-8, 'No effect', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+title('"Stiff" $\nu$-hexapod');
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-9, 1e-4]);
+
+% Effect of support compliance on $d/F$
+
+% When the motion to be controlled is the relative displacement $d$ between the granite and the nano-hexapod's top platform, the effect of the support compliance on the plant dynamics is opposite than what was previously observed.
+% Indeed, using a "soft" nano-hexapod (i.e. with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_d, left).
+% On the contrary, if a "stiff" nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_d, right).
+
+
+%% Effect of the support compliance on the transfer functions from F to L and from F to d
+figure;
+freqs = logspace(0, 3, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_soft_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_soft,   freqs, 'Hz'))), '-', 'color', colors(3,:));
+loglog(10.^(0.3*cos(0:0.01:2*pi)+log10(60)), ...
+       10.^(0.6*sin(0:0.01:2*pi)+log10(4e-7)), 'k--');
+text(8, 3e-7, 'No effect', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+title('"Soft" $\nu$-hexapod');
+ylabel('Magnitude [m/N]');
+
+
+ax2 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_mid_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_mid,   freqs, 'Hz'))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]');
+xticks([1e0, 1e1, 1e2]);
+title('"Mid" $\nu$-hexapod');
+set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_stiff_a, freqs, 'Hz'))), '-', 'color', colors(1,:), ...
+     'DisplayName', '$L^\prime/F$');
+plot(freqs, abs(squeeze(freqresp(G_stiff,   freqs, 'Hz'))), '-', 'color', colors(3,:), ...
+     'DisplayName', '$d/F$');
+loglog(10.^(0.4*cos(0:0.01:2*pi)+log10(50)), ...
+       10.^(0.8*sin(0:0.01:2*pi)+log10(8e-9)), 'k--', 'HandleVisibility', 'off');
+text(50, 2e-7, sprintf('Support\nDynamics'), 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+title('"Stiff" $\nu$-hexapod');
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-9, 1e-4]);
diff --git a/matlab/uniaxial_8_payload_dynamics.m b/matlab/uniaxial_8_payload_dynamics.m
new file mode 100644
index 0000000..acbffc0
--- /dev/null
+++ b/matlab/uniaxial_8_payload_dynamics.m
@@ -0,0 +1,479 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Uniaxial Simscape model name
+mdl = 'nass_uniaxial_model';
+
+%% Load the micro-station parameters
+load('uniaxial_micro_station_parameters.mat')
+
+%% Load the PSD of disturbances
+load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
+
+%% Load Active Damping Controller
+load('uniaxial_active_damping_controllers.mat', 'K_iff_vc', 'K_iff_md', 'K_iff_pz', ...
+                                                'K_rdc_vc', 'K_rdc_md', 'K_rdc_pz', ...
+                                                'K_dvf_vc', 'K_dvf_md', 'K_dvf_pz');
+
+%% Load High Authority Controllers
+load('uniaxial_high_authority_controllers.mat', 'K_hac_vc', 'K_hac_md', 'K_hac_pz');
+
+%% Frequency Vector [Hz]
+freqs = logspace(0, 3, 1000);
+
+% Impact on the plant dynamics
+% <<ssec:uniaxial_payload_dynamics_effect_dynamics>>
+
+% To study the impact of sample dynamics, the following sample configurations are considered:
+% - rigid sample, corresponding to Figure ref:fig:uniaxial_paylaod_dynamics_rigid_schematic
+% - two flexible samples with a resonance at $\omega_s = 200\,\text{Hz}$ and at $\omega_s = 20\,\text{Hz}$ respectively, corresponding to Figure ref:fig:uniaxial_paylaod_dynamics_schematic
+% - for all cases, two sample masses are considered: $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$
+
+% The transfer functions from the nano-hexapod force to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod $k_n = 0.01\,N/\mu m$ 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 $\frac{1}{m_ns^2}$).
+
+
+%% Soft Nano-Hexapod
+% Light payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 1; % Sample Mass [kg]
+kn = 1e4; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_vc_rigid_light = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_soft_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_stiff_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Heavy payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 50; % Sample Mass [kg]
+kn = 1e4; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_vc_rigid_heavy = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_soft_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_stiff_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+%% Effect of the payload dynamics on the soft Nano-Hexapod. Light sample on the right, and heavy sample on the left
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_vc_rigid_light, freqs, 'Hz'))), '-',  'color', colors(1,:), 'DisplayName', 'Rigid sample');
+plot(freqs, abs(squeeze(freqresp(G_vc_stiff_light, freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$\omega_s = 200\,Hz$');
+plot(freqs, abs(squeeze(freqresp(G_vc_soft_light, freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$\omega_s = 20\,Hz$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-10, 1e-2])
+title('$k_n = 0.01\,N/\mu m$, $m_s = 1\,kg$');
+
+ax2 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_vc_rigid_heavy, freqs, 'Hz'))), '-',  'color', colors(1,:), 'DisplayName', 'Rigid sample');
+plot(freqs, abs(squeeze(freqresp(G_vc_stiff_heavy, freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$\omega_s = 200\,Hz$');
+plot(freqs, abs(squeeze(freqresp(G_vc_soft_heavy, freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$\omega_s = 20\,Hz$');
+plot(freqs, abs(squeeze(freqresp(1/(mn*s^2), freqs, 'Hz'))), '-', 'color', [0,0,0,0.5], 'DisplayName', '$\frac{1}{m_n s^2}$');
+plot(freqs, abs(squeeze(freqresp(1/((mn + ms)*s^2), freqs, 'Hz'))), '--', 'color', [0,0,0,0.5], 'DisplayName', '$\frac{1}{(m_n + m_s) s^2}$');
+text(2.2, 3e-3, '$\omega_n = \sqrt{\frac{k_n}{m_n + m_s}}$', 'horizontalalignment', 'left');
+text(20, 1e-8, '$\omega_s = \sqrt{\frac{k_s}{m_s}}$', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+title('$k_n = 0.01\,N/\mu m$, $m_s = 50\,kg$');
+ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [20, 1];
+ylim([1e-10, 1e-2])
+
+ax1b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_rigid_light, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_stiff_light, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_soft_light, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+ax2b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_rigid_heavy, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_stiff_heavy, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_soft_heavy, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+linkaxes([ax1,ax1b],'x');
+xlim([1, 1000]);
+
+
+
+% #+name: fig:uniaxial_payload_dynamics_soft_nano_hexapod
+% #+caption: Effect of the payload dynamics on the soft Nano-Hexapod. Light sample on the right, and heavy sample on the left
+% #+RESULTS:
+% [[file:figs/uniaxial_payload_dynamics_soft_nano_hexapod.png]]
+
+% The same transfer functions are now compared when using a stiff nano-hexapod ($k_n = 100\,N/\mu m$) in Figure ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod.
+% In that case, the sample's resonance $\omega_n$ is smaller than the nano-hexapod 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 tough the added sample's flexibility still changes the high frequency mass line from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$, the overall dynamics is much less impacted, even if the sample mass is high (see yellow curve in Figure ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod, right).
+
+
+%% Stiff Nano-Hexapod
+% Light payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 1; % Sample Mass [kg]
+kn = 1e8; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_pz_rigid_light = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_soft_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_stiff_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Heavy payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 50; % Sample Mass [kg]
+kn = 1e8; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_pz_rigid_heavy = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_soft_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_stiff_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+%% Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample on the right, and heavy sample on the left
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_pz_rigid_light, freqs, 'Hz'))), '-',  'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_pz_stiff_light, freqs, 'Hz'))), '-', 'color', colors(2,:));
+plot(freqs, abs(squeeze(freqresp(G_pz_soft_light, freqs, 'Hz'))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-10, 1e-2])
+title('$k_n = 100\,N/\mu m$, $m_s = 1\,kg$');
+
+ax2 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_pz_rigid_heavy, freqs, 'Hz'))), '-',  'color', colors(1,:), 'DisplayName', 'Rigid sample');
+plot(freqs, abs(squeeze(freqresp(G_pz_stiff_heavy, freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', 'Stiff sample: $\omega_s = 200\,Hz$');
+plot(freqs, abs(squeeze(freqresp(G_pz_soft_heavy, freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', 'Soft sample: $\omega_s = 20\,Hz$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+title('$k_n = 100\,N/\mu m$, $m_s = 50\,kg$');
+ylim([1e-10, 1e-2])
+ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [20, 1];
+
+ax1b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_rigid_light, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_stiff_light, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_soft_light, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+ax2b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_rigid_heavy, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_stiff_heavy, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_soft_heavy, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+linkaxes([ax1,ax1b],'x');
+xlim([1, 1000]);
+
+
+
+% #+name: fig:uniaxial_sample_flexibility_control
+% #+caption: Uniaxial model considering a flexibility between the nano-hexapod top platform and the sample. In that case the measured and controlled distance $d$ is different from the distance $y$ that is wish to be controlled
+% #+RESULTS:
+% [[file:figs/uniaxial_sample_flexibility_control.png]]
+
+% After the system dynamics extracted from the model, IFF is applied using the same gains as the ones used in Section ref:sec:uniaxial_active_damping.
+% Thanks to the collocation between the nano-hexapod and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample.
+
+% The High Authority Controllers used in Section ref:sec:uniaxial_position_control are then implemented on the damped plants.
+% The obtained closed-loop systems are stable, indicating good robustness.
+
+% Finally, closed-loop noise budgeting is computed for the obtained the closed-loop system and the cumulative amplitude spectrum of $d$ and $y$ are shown in Figure ref:fig:uniaxial_sample_flexibility_noise_budget_y.
+% The cumulative amplitude spectrum of the measured distance $d$ (Figure ref:fig:uniaxial_sample_flexibility_noise_budget_d) shows that the added flexibility at the sample location have very little effect on the control performance.
+% However, the cumulative amplitude spectrum of the distance $y$ (Figure ref:fig:uniaxial_sample_flexibility_noise_budget_y) shows that the stability of $y$ is degraded when the sample flexibility is considered and is degraded as $\omega_s$ is lowered.
+
+% What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the nano-hexapod and the control on $y$ is not effective.
+
+
+%% Nano-Hexpod model
+model_config = struct();
+model_config.controller = "open_loop";
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 1; % Sample Mass [kg]
+
+%% Identification
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/controller'],       1, 'openinput');  io_i = io_i + 1; % Actuator Force
+io(io_i) = linio([mdl, '/micro_station/xf'], 1, 'openinput');  io_i = io_i + 1; % Floor Motion
+io(io_i) = linio([mdl, '/micro_station/ft'], 1, 'openinput');  io_i = io_i + 1; % Stage vibrations
+io(io_i) = linio([mdl, '/fs'],               1, 'openinput');  io_i = io_i + 1; % Direct sample forces
+io(io_i) = linio([mdl, '/dL'],               1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor
+io(io_i) = linio([mdl, '/fm'],               1, 'openoutput'); io_i = io_i + 1; % Force Sensor
+io(io_i) = linio([mdl, '/vn'] ,              1, 'openoutput'); io_i = io_i + 1; % Geophone
+io(io_i) = linio([mdl, '/d'] ,               1, 'openoutput'); io_i = io_i + 1; % Metrology Output
+io(io_i) = linio([mdl, '/y'] ,               1, 'openoutput'); io_i = io_i + 1; % Sample's position
+
+%% Soft Nano-Hexapod
+% Light payload mass
+kn = 1e4; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid Sample
+model_config.nhexa = "1dof";
+G_vc_light_rigid = linearize(mdl, io, 0.0);
+G_vc_light_rigid.InputName = {'f', 'xf', 'ft', 'fs'};
+G_vc_light_rigid.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Soft Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_vc_light_soft = linearize(mdl, io, 0.0);
+G_vc_light_soft.InputName = {'f', 'xf', 'ft', 'fs'};
+G_vc_light_soft.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Rigid Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_vc_light_stiff = linearize(mdl, io, 0.0);
+G_vc_light_stiff.InputName = {'f', 'xf', 'ft', 'fs'};
+G_vc_light_stiff.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+%% Stiff Nano-Hexapod
+% Light payload mass
+kn = 1e8; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid Sample
+model_config.nhexa = "1dof";
+G_pz_light_rigid = linearize(mdl, io, 0.0);
+G_pz_light_rigid.InputName = {'f', 'xf', 'ft', 'fs'};
+G_pz_light_rigid.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Soft Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_pz_light_soft = linearize(mdl, io, 0.0);
+G_pz_light_soft.InputName = {'f', 'xf', 'ft', 'fs'};
+G_pz_light_soft.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Rigid Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_pz_light_stiff = linearize(mdl, io, 0.0);
+G_pz_light_stiff.InputName = {'f', 'xf', 'ft', 'fs'};
+G_pz_light_stiff.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+%% Apply IFF and verify stability
+% Soft Nano-Hexapod
+G_iff_vc_light_rigid = feedback(G_vc_light_rigid, K_iff_vc, 'name', +1);
+G_iff_vc_light_soft  = feedback(G_vc_light_soft , K_iff_vc, 'name', +1);
+G_iff_vc_light_stiff = feedback(G_vc_light_stiff, K_iff_vc, 'name', +1);
+
+isstable(G_iff_vc_light_rigid)
+isstable(G_iff_vc_light_soft)
+isstable(G_iff_vc_light_stiff)
+
+% Stiff Nano-Hexapod
+G_iff_pz_light_rigid = feedback(G_pz_light_rigid, K_iff_pz, 'name', +1);
+G_iff_pz_light_soft  = feedback(G_pz_light_soft , K_iff_pz, 'name', +1);
+G_iff_pz_light_stiff = feedback(G_pz_light_stiff, K_iff_pz, 'name', +1);
+
+isstable(G_iff_pz_light_rigid)
+isstable(G_iff_pz_light_soft)
+isstable(G_iff_pz_light_stiff)
+
+%% Compute closed-loop plants and verify stability
+% Soft Nano-Hexapod
+G_hac_iff_vc_light_rigid = feedback(G_iff_vc_light_rigid, K_hac_vc, 'name', -1);
+G_hac_iff_vc_light_soft  = feedback(G_iff_vc_light_soft , K_hac_vc, 'name', -1);
+G_hac_iff_vc_light_stiff = feedback(G_iff_vc_light_stiff, K_hac_vc, 'name', -1);
+
+isstable(G_hac_iff_vc_light_rigid)
+isstable(G_hac_iff_vc_light_soft)
+isstable(G_hac_iff_vc_light_stiff)
+
+% Stiff Nano-Hexapod
+G_hac_iff_pz_light_rigid = feedback(G_iff_pz_light_rigid, K_hac_pz, 'name', -1);
+G_hac_iff_pz_light_soft  = feedback(G_iff_pz_light_soft , K_hac_pz, 'name', -1);
+G_hac_iff_pz_light_stiff = feedback(G_iff_pz_light_stiff, K_hac_pz, 'name', -1);
+
+isstable(G_hac_iff_pz_light_rigid)
+isstable(G_hac_iff_pz_light_soft)
+isstable(G_hac_iff_pz_light_stiff)
+
+%% Cumulative Amplitude Spectrum of d - Effect of Sample's flexibility
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Rigid sample');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Stiff $\omega_s = 200\,$Hz');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Soft $\omega_s = 20\,$Hz');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+ylabel('CAS of $d$ [m]'); xlabel('Frequency [Hz]');
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+title('$k_n = 0.01\,N/\mu m$');
+
+ax2 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('d', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('d', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('d', 'xf'), f, 'Hz'))).^2)))), '-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+title('$k_n = 100\,N/\mu m$');
+xlim([1, 500]);
+
+linkaxes([ax1,ax2],'xy');
+xlim([1, 500]);
+ylim([2e-10, 2e-7])
+
+%% Cumulative Amplitude Spectrum - Effect of Sample's flexibility
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('y', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Rigid sample');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('y', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Stiff $\omega_s = 200\,$Hz');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('y', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Soft $\omega_s = 20\,$Hz');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+ylabel('CAS of $y$ [m]'); xlabel('Frequency [Hz]');
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+title('$k_n = 0.01\,N/\mu m$');
+
+ax2 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('y', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('y', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('y', 'xf'), f, 'Hz'))).^2)))), '-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+title('$k_n = 100\,N/\mu m$');
+xlim([1, 500]);
+
+linkaxes([ax1,ax2],'xy');
+xlim([1, 500]);
+ylim([2e-10, 2e-7])
diff --git a/nass-uniaxial-model.bib b/nass-uniaxial-model.bib
new file mode 100644
index 0000000..14db4af
--- /dev/null
+++ b/nass-uniaxial-model.bib
@@ -0,0 +1,81 @@
+@inproceedings{preumont91_activ,
+  author          = {Andre Preumont and Jean-Paul Dufour and Christian Malekian},
+  title           = {Active damping by a local force feedback with piezoelectric
+                  actuators},
+  booktitle       = {32nd Structures, Structural Dynamics, and Materials
+                  Conference},
+  year            = 1991,
+  doi             = {10.2514/6.1991-989},
+  url             = {https://doi.org/10.2514/6.1991-989},
+  month           = {apr},
+  publisher       = {American Institute of Aeronautics and Astronautics},
+  keywords        = {active damping},
+}
+
+
+
+@book{preumont18_vibrat_contr_activ_struc_fourt_edition,
+  author          = {Andre Preumont},
+  title           = {Vibration Control of Active Structures - Fourth Edition},
+  year            = 2018,
+  publisher       = {Springer International Publishing},
+  url             = {https://doi.org/10.1007/978-3-319-72296-2},
+  doi             = {10.1007/978-3-319-72296-2},
+  keywords        = {favorite, parallel robot},
+  series          = {Solid Mechanics and Its Applications},
+}
+
+
+
+@article{karnopp74_vibrat_contr_using_semi_activ_force_gener,
+  author          = {Karnopp, Dean and Crosby, Michael J and Harwood, RA},
+  title           = {Vibration Control Using Semi-Active Force Generators},
+  journal         = {Journal of Engineering for Industry},
+  volume          = 96,
+  pages           = {619-626},
+  year            = 1974,
+  doi             = {10.1115/1.3438373},
+  url             = {https://doi.org/10.1115/1.3438373},
+}
+
+
+
+@article{serrand00_multic_feedb_contr_isolat_base_excit_vibrat,
+  author          = {Serrand, M and Elliott, SJ},
+  title           = {Multichannel Feedback Control for the Isolation of
+                  Base-Excited Vibration},
+  journal         = {Journal of Sound and Vibration},
+  volume          = 234,
+  number          = 4,
+  pages           = {681--704},
+  year            = 2000,
+  publisher       = {Elsevier},
+}
+
+
+
+@article{preumont02_force_feedb_versus_accel_feedb,
+  author          = {A. Preumont and A. Fran{\c{c}}ois and F. Bossens and A.
+                  Abu-Hanieh},
+  title           = {Force Feedback Versus Acceleration Feedback in Active
+                  Vibration Isolation},
+  journal         = {Journal of Sound and Vibration},
+  volume          = 257,
+  number          = 4,
+  pages           = {605-613},
+  year            = 2002,
+  doi             = {10.1006/jsvi.2002.5047},
+  url             = {https://doi.org/10.1006/jsvi.2002.5047},
+}
+
+
+
+@phdthesis{rankers98_machin,
+  author          = {Rankers, Adrian Mathias},
+  keywords        = {favorite},
+  school          = {University of Twente},
+  title           = {Machine dynamics in mechatronic systems: An engineering
+                  approach.},
+  year            = 1998,
+}
+
diff --git a/nass-uniaxial-model.org b/nass-uniaxial-model.org
index 79029d1..6a193b9 100644
--- a/nass-uniaxial-model.org
+++ b/nass-uniaxial-model.org
@@ -1,4 +1,4 @@
-#+TITLE: NASS - Uniaxial Model
+#+TITLE: Nano Active Stabilization System - Uniaxial Model
 :DRAWER:
 #+LANGUAGE: en
 #+EMAIL: dehaeze.thomas@gmail.com
@@ -14,12 +14,15 @@
 #+BIND: org-latex-image-default-width ""
 
 #+LaTeX_CLASS: scrreprt
-#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
+#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]
 #+LaTeX_HEADER_EXTRA: \input{preamble.tex}
+#+LATEX_HEADER_EXTRA: \bibliography{nass-uniaxial-model.bib}
+
+#+BIND: org-latex-bib-compiler "biber"
 
 #+PROPERTY: header-args:matlab  :session *MATLAB*
 #+PROPERTY: header-args:matlab+ :comments org
-#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :exports none
 #+PROPERTY: header-args:matlab+ :results none
 #+PROPERTY: header-args:matlab+ :eval no-export
 #+PROPERTY: header-args:matlab+ :noweb yes
@@ -49,6 +52,50 @@
 
 #+latex: \clearpage
 
+* Build                                                             :noexport:
+#+NAME: startblock
+#+BEGIN_SRC emacs-lisp :results none :tangle no
+(add-to-list 'org-latex-classes
+             '("scrreprt"
+               "\\documentclass{scrreprt}"
+               ("\\chapter{%s}" . "\\chapter*{%s}")
+               ("\\section{%s}" . "\\section*{%s}")
+               ("\\subsection{%s}" . "\\subsection*{%s}")
+               ("\\paragraph{%s}" . "\\paragraph*{%s}")
+               ))
+
+
+;; Remove automatic org heading labels
+(defun my-latex-filter-removeOrgAutoLabels (text backend info)
+  "Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
+  (when (org-export-derived-backend-p backend 'latex)
+    (replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
+(add-to-list 'org-export-filter-headline-functions
+             'my-latex-filter-removeOrgAutoLabels)
+
+;; Remove all org comments in the output LaTeX file
+(defun delete-org-comments (backend)
+  (loop for comment in (reverse (org-element-map (org-element-parse-buffer)
+                                    'comment 'identity))
+        do
+        (setf (buffer-substring (org-element-property :begin comment)
+                                (org-element-property :end comment))
+              "")))
+(add-hook 'org-export-before-processing-hook 'delete-org-comments)
+
+;; Use no package by default
+(setq org-latex-packages-alist nil)
+(setq org-latex-default-packages-alist nil)
+
+;; Do not include the subtitle inside the title
+(setq org-latex-subtitle-separate t)
+(setq org-latex-subtitle-format "\\subtitle{%s}")
+
+(setq org-export-before-parsing-hook '(org-ref-glossary-before-parsing
+                                       org-ref-acronyms-before-parsing
+                                       tdh-org-ref-extract-bibtex-to-file))
+#+END_SRC
+
 * Introduction                                                        :ignore:
 
 In this report, a uniaxial model of the Nano Active Stabilization System (NASS) is developed and used to have a first idea of the challenges involved in this complex system.
@@ -59,14 +106,16 @@ In order to have a relevant model, the micro-station dynamics is first identifie
 Then, a model of the nano-hexapod is added on top of the micro-station.
 With added sample and sensors, this gives a uniaxial dynamical model of the NASS that will be used for further analysis (Section ref:sec:nano_station_model).
 
-The disturbances affecting the position accuracy are identified experimentally (Section ref:sec:uniaxial_disturbances) and included in the model for dynamical noise budgeting (Section ref:sec:uniaxial_noise_budgeting).
-In all the following analysis, there nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate stiffness.
+The disturbances affecting the position stability are identified experimentally (Section ref:sec:uniaxial_disturbances) and included in the model for dynamical noise budgeting (Section ref:sec:uniaxial_noise_budgeting).
+In all the following analysis, there nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate nano-hexapod design.
 Three sample masses are also considered to verify the robustness of the applied control strategies to a change of sample.
 
 Three active damping techniques are then applied on the nano-hexapod.
 This helps to reduce the effect of disturbances as well as render the system easier to control afterwards (Section ref:sec:uniaxial_active_damping).
 
-Once the system is well damped, a feedback position controller is applied, and the obtained performances are compared (Section ref:sec:uniaxial_position_control).
+Once the system is well damped, a feedback position controller is applied, and the obtained performance are compared (Section ref:sec:uniaxial_position_control).
+
+Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section ref:sec:uniaxial_support_compliance) and the presence of dynamics between the nano-hexapod and the sample's point of interest (Section ref:sec:uniaxial_payload_dynamics).
 
 Conclusion remarks are given in Section ref:sec:conclusion.
 
@@ -74,14 +123,16 @@ Conclusion remarks are given in Section ref:sec:conclusion.
 #+caption: Report sections and corresponding Matlab files
 #+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
 #+attr_latex: :center t :booktabs t
-| *Sections*                                | *Matlab File*                       |
-|-------------------------------------------+-------------------------------------|
-| Section ref:sec:micro_station_model       | =uniaxial_1_micro_station_model.m=  |
-| Section ref:sec:nano_station_model        | =uniaxial_2_nano_hexapod_model.m=   |
-| Section ref:sec:uniaxial_disturbances     | =uniaxial_3_disturbances.m=         |
-| Section ref:sec:uniaxial_noise_budgeting  | =uniaxial_4_dynamic_noise_budget.m= |
-| Section ref:sec:uniaxial_active_damping   | =uniaxial_5_active_damping.m=       |
-| Section ref:sec:uniaxial_position_control | =uniaxial_6_hac_lac.m=              |
+| *Sections*                                  | *Matlab File*                       |
+|---------------------------------------------+-------------------------------------|
+| Section ref:sec:micro_station_model         | =uniaxial_1_micro_station_model.m=  |
+| Section ref:sec:nano_station_model          | =uniaxial_2_nano_hexapod_model.m=   |
+| Section ref:sec:uniaxial_disturbances       | =uniaxial_3_disturbances.m=         |
+| Section ref:sec:uniaxial_noise_budgeting    | =uniaxial_4_dynamic_noise_budget.m= |
+| Section ref:sec:uniaxial_active_damping     | =uniaxial_5_active_damping.m=       |
+| Section ref:sec:uniaxial_position_control   | =uniaxial_6_hac_lac.m=              |
+| Section ref:sec:uniaxial_support_compliance | =uniaxial_7_support_compliance=     |
+| Section ref:sec:uniaxial_payload_dynamics   | =uniaxial_8_payload_dynamics=       |
 
 #+begin_src latex :file uniaxial_overview_model_sections.pdf
 \begin{tikzpicture}
@@ -193,10 +244,19 @@ Conclusion remarks are given in Section ref:sec:conclusion.
   % sample
   \begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
     % Mass
-    \draw[draw=colorred,fill=colorred!10!white] (-0.3*\massw, 0) rectangle (0.3*\massw, \massh) node[pos=0.5,color=colorred]{$m_{s}$};
+    \draw[draw=colorcyan,fill=colorcyan!10!white] (-0.3*\massw, \spaceh) rectangle (0.3*\massw, \spaceh+\massh) node[pos=0.5, color=colorcyan]{$m_{s}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorcyan] (-0.2*\massw, 0) -- (-0.2*\massw, \spaceh) node[midway, left=0.1, color=colorcyan]{$k_{s}$};
+    \draw[damper={colorcyan}{}{}] ( 0.2*\massw, 0) -- ( 0.2*\massw, \spaceh) node[midway, left=0.2, color=colorcyan]{$c_{s}$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Sample};
 
     % External Force
-    \draw[->, color=coloryellow] (0, \massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right, color=coloryellow]{$f_s$};
+    \draw[->, color=coloryellow] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right, color=coloryellow]{$f_s$};
   \end{scope}
   % ====================
 
@@ -220,7 +280,7 @@ Conclusion remarks are given in Section ref:sec:conclusion.
 #+end_src
 
 #+name: fig:uniaxial_overview_model_sections
-#+caption: Uniaxial Micro-Station model in blue (Section ref:sec:micro_station_model), Nano-Hexapod and sample models in red (Section ref:sec:nano_station_model), Disturbances in yellow (Section ref:sec:uniaxial_disturbances), Active Damping in green (Section ref:sec:uniaxial_active_damping) and Position control in purple (Section ref:sec:uniaxial_position_control)
+#+caption: Uniaxial Micro-Station model in blue (Section ref:sec:micro_station_model), Nano-Hexapod models in red (Section ref:sec:nano_station_model), Disturbances in yellow (Section ref:sec:uniaxial_disturbances), Active Damping in green (Section ref:sec:uniaxial_active_damping), Position control in purple (Section ref:sec:uniaxial_position_control) and Sample dynamics in cyan (Section ref:sec:uniaxial_payload_dynamics)
 #+RESULTS:
 [[file:figs/uniaxial_overview_model_sections.png]]
 
@@ -283,7 +343,7 @@ From the forces applied by the instrumented hammer and the responses of the geop
 - from $F_g$ to $d_h$ (or from $F_h$ to $d_g$)
 - from $F_g$ to $d_g$
 
-#+begin_src latex :file micro_station_meas_dynamics_schematic.pdf
+#+begin_src latex :file micro_station_meas_dynamics_schematic.pdf :results file raw silent
 \begin{tikzpicture}
   % Parameters
   \def\blockw{6.0cm}
@@ -424,71 +484,39 @@ From the forces applied by the instrumented hammer and the responses of the geop
 \end{tikzpicture}
 #+end_src
 
-#+name: fig:micro_station_meas_dynamics_schematic
-#+caption: Measurement setup - Schematic
-#+RESULTS:
+#+name: fig:micro_station_uniaxial_model
+#+caption: Schematic of the Micro-Station measurement setup and uniaxial model.
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:micro_station_meas_dynamics_schematic}Measurement setup - Schematic}
+#+attr_latex: :options {0.69\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
 [[file:figs/micro_station_meas_dynamics_schematic.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_model_micro_station}Uniaxial model of the micro-station}
+#+attr_latex: :options {0.29\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_model_micro_station.png]]
+#+end_subfigure
+#+end_figure
 
-Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure ref:fig:uniaxial_measured_frf_vertical).
+Due to the bad coherence at low frequency, the frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure ref:fig:uniaxial_comp_frf_meas_model).
 
 #+begin_src matlab
 %% Load measured FRF
 load('meas_microstation_frf.mat');
 #+end_src
 
-#+begin_src matlab :exports none :results none
-%% Measured Frequency Response Functions in the vertical direction
-figure;
-tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
-
-ax1 = nexttile([2,1]);
-hold on;
-plot(f(f>20), abs(frf_Fhx_to_Dhx(f>20)), ...
-        'DisplayName', '$D_{h}/F_{h}$');
-plot(f(f>20), abs(frf_Fgx_to_Dhx(f>20)), ...
-        'DisplayName', '$D_{h}/F_{g}$');
-plot(f(f>20), abs(frf_Fgx_to_Dgx(f>20)), ...
-        'DisplayName', '$D_{g}/F_{g}$');
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-10, 2e-6]);
-legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
-
-ax2 = nexttile;
-hold on;
-plot(f(f>20), 180/pi*unwrap(angle(frf_Fhx_to_Dhx(f>20))));
-plot(f(f>30), 180/pi*unwrap(angle(frf_Fgx_to_Dhx(f>30))));
-plot(f(f>20), 180/pi*unwrap(angle(frf_Fgx_to_Dgx(f>20))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
-xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
-hold off;
-yticks(-360:90:360);
-ylim([-360, 10]);
-
-linkaxes([ax1,ax2],'x');
-xlim([1, 500]);
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/uniaxial_measured_frf_vertical.pdf', 'width', 'wide', 'height', 'tall');
-#+end_src
-
-#+name: fig:uniaxial_measured_frf_vertical
-#+caption: Measured Frequency Response Functions in the vertical direction
-#+RESULTS:
-[[file:figs/uniaxial_measured_frf_vertical.png]]
-
 ** Uniaxial Model
-The uni-axial model of the micro-station is shown in Figure ref:fig:uniaxial_comp_frf_meas_model, with:
+The uni-axial model of the micro-station is shown in Figure ref:fig:uniaxial_model_micro_station, with:
 - Disturbances:
   - $x_f$: Floor motion
   - $f_t$: Stage vibrations
 - Hammer impacts: $F_h$ and $F_g$.
 - Geophones: $x_h$ and $x_g$
 
-#+begin_src latex :file uniaxial_model_micro_station.pdf
+#+begin_src latex :file uniaxial_model_micro_station.pdf :results file raw silent
 \begin{tikzpicture}
   % ====================
   % Parameters
@@ -586,11 +614,6 @@ The uni-axial model of the micro-station is shown in Figure ref:fig:uniaxial_com
 \end{tikzpicture}
 #+end_src
 
-#+name: fig:uniaxial_model_micro_station
-#+caption: Uniaxial model of the micro-station
-#+RESULTS:
-[[file:figs/uniaxial_model_micro_station.png]]
-
 Masses are estimated from the CAD.
 #+BEGIN_SRC matlab
 %% Parameters - Mass
@@ -645,7 +668,9 @@ G_id.OutputName = {'Dg', 'Dh'};
 #+end_src
 
 ** Comparison of the model and measurements
-The comparison between the measurements and the model is done in Figure ref:fig:uniaxial_comp_frf_meas_model.
+The comparison between the measurements and the model is shown in Figure ref:fig:uniaxial_comp_frf_meas_model.
+
+Only three modes are modelled with frequencies at 70Hz, 140Hz and 320Hz.
 
 As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation.
 More accurate models will be used later on.
@@ -708,7 +733,7 @@ A model of the nano-hexapod and sample is now added on top of the uni-axial mode
 
 Disturbances (shown in red) are:
 - $f_s$: direct forces applied to the sample (for instance cable forces)
-- $f_t$: disturbances coming from the imperfect stage scanning performances
+- $f_t$: disturbances coming from the imperfect stage scanning performance
 - $x_f$: floor motion
 
 The control signal is the force applied by the nano-hexapod $f$ and the measurement is the relative motion between the sample and the granite $d$.
@@ -893,11 +918,13 @@ cn = 2*0.01*sqrt(mn*kn); % [N/(m/s)]
 ms = 10; % [kg]
 #+end_src
 
-** Obtained Dynamics
+** Obtained Dynamic Response
 The sensitivity to disturbances (i.e. $x_f$, $f_t$ and $f_s$) are shown in Figure ref:fig:uniaxial_sensitivity_dist_first_params.
 The /plant/ (i.e. the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure ref:fig:uniaxial_plant_first_params.
 
+#+begin_important
 For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) combined with three sample's masses ($m_s = 1\,kg$, $m_s = 25\,kg$ and $m_s = 50\,kg$).
+#+end_important
 
 #+begin_src matlab :exports none
 %% Use 1DoF Nano-Hexpod model
@@ -957,7 +984,7 @@ exportFig('figs/uniaxial_sensitivity_dist_first_params.pdf', 'width', 'full', 'h
 #+end_src
 
 #+name: fig:uniaxial_sensitivity_dist_first_params
-#+caption: Sensitivity to disturbances
+#+caption: Sensitivity of the relative motion $d$ to disturbances: $f_s$ the direct forces applied on the sample, $f_t$ disturbances from the micro-station stages and $x_f$ the floor motion (from left to right)
 #+RESULTS:
 [[file:figs/uniaxial_sensitivity_dist_first_params.png]]
 
@@ -1352,7 +1379,7 @@ psd_xf = psd_V.*abs(squeeze(freqresp(G_geo, f, 'Hz'))).^2; % [m^2/Hz]
 The amplitude spectral density $\Gamma_{x_f}$ of the measured displacement $x_f$ is shown in Figure ref:fig:asd_floor_motion_id31.
 
 #+begin_src matlab :exports none :results none
-%% Measured Amplitude Spectral Density of the Floor motion on ID31
+%% Amplitude Spectral Density of the measured Floor motion on ID31
 figure;
 hold on;
 plot(f, sqrt(psd_xf), 'DisplayName', '$\Gamma_{x_{f}}$');
@@ -1368,7 +1395,7 @@ exportFig('figs/asd_floor_motion_id31.pdf', 'width', 'wide', 'height', 'normal')
 #+end_src
 
 #+name: fig:asd_floor_motion_id31
-#+caption: Measured Amplitude Spectral Density of the Floor motion on ID31
+#+caption: Amplitude Spectral Density of the measured Floor motion on ID31
 #+RESULTS:
 [[file:figs/asd_floor_motion_id31.png]]
 
@@ -1398,7 +1425,7 @@ It is then integrated to obtain the Amplitude Spectral Density of the relative m
 It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.
 
 #+begin_src matlab :exports none :results none
-%% Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating
+%% Amplitude Spectral Density of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating
 figure;
 hold on;
 plot(f, sqrt(psd_vft)./(2*pi*f), 'DisplayName', '6rpm');
@@ -1415,7 +1442,7 @@ exportFig('figs/asd_vibration_spindle_rotation.pdf', 'width', 'wide', 'height',
 #+end_src
 
 #+name: fig:asd_vibration_spindle_rotation
-#+caption: Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating
+#+caption: Amplitude Spectral Density of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating
 #+RESULTS:
 [[file:figs/asd_vibration_spindle_rotation.png]]
 
@@ -1473,7 +1500,7 @@ exportFig('figs/asd_disturbance_force.pdf', 'width', 'wide', 'height', 'normal')
 #+RESULTS:
 [[file:figs/asd_disturbance_force.png]]
 
-The vibrations induced by the $T_y$ stage are not considered here as:
+The vibrations induced by the $T_y$ stage are not considered here because:
 - the induced vibrations have less amplitude than the vibrations induced by the $R_z$ stage
 - it can be scanned at lower velocities if the induced vibrations are an issue
 
@@ -1542,10 +1569,10 @@ This is done for *two extreme sample masses* $m_s = 1\,\text{kg}$ and $m_s = 50\
 The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the light sample (same conclusions can be drawn with the heavy one).
 
 #+begin_important
-From Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses, following can be concluded for the *soft nano-hexapod*:
+From Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses, the following can be concluded for the *soft nano-hexapod*:
 - It is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to the lower stiffness
 - Between the suspension mode of the nano-hexapod (here at 5Hz) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out.
-- Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is almost $1$.
+- Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is close to $1$.
 #+end_important
 
 #+begin_src matlab :exports none :results none
@@ -1682,13 +1709,14 @@ exportFig('figs/uniaxial_cas_d_disturbances_payload_masses.pdf', 'width', 'wide'
 [[file:figs/uniaxial_cas_d_disturbances_payload_masses.png]]
 
 ** Conclusion
+
 #+begin_important
-The conclusion is that in order to have a closed-loop residual vibration $d \approx 20\,nm\text{ rms}$, if a simple feedback controller is used, the required closed-loop bandwidth would be:
+In order to have a closed-loop residual vibration $d \approx 20\,nm\text{ rms}$, if a simple feedback controller is used, the required closed-loop bandwidth would be:
 - $\approx 10\,\text{Hz}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$)
 - $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$)
 - $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$)
 
-This can be explain by the fact that above the suspension mode of the nano-hexapod, the stage vibrations are filtered out (see Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses).
+This can be explained by the fact that for frequencies larger than the suspension mode of the nano-hexapod, the stage vibrations are filtered out (see Figure ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses).
 
 This gives a first advantage to having a soft nano-hexapod.
 #+end_important
@@ -1699,13 +1727,14 @@ This gives a first advantage to having a soft nano-hexapod.
 :END:
 <<sec:uniaxial_active_damping>>
 ** Introduction                                                      :ignore:
+In this section, three active damping are applied on the nano-hexapod (see Figure ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF) cite:preumont91_activ, Relative Damping Control (RDC) [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7.2]] 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 this section, three active damping are applied on the nano-hexapod (see Figure ref:fig:uniaxial_active_damping_strategies): Integral Force Feedback (IFF), Relative Damping Control (RDC) and Direct Velocity Feedback (DVF).
+These damping strategies will be described in Section ref:ssec:uniaxial_active_damping_strategies.
 
 These active damping techniques are compared in terms of:
-- Reduction of the effect of disturbances (i.e. $x_f$, $f_t$ and $f_s$) on the displacement $d$
-- Achievable damping
-- Robustness to a change of sample's mass
+- Achievable damping of the nano-hexapod 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 and ref:ssec:uniaxial_active_damping_noise_budget)
+- Robustness to a change of sample's mass (Section ref:ssec:uniaxial_active_damping_robustness)
 
 #+begin_src latex :file uniaxial_active_damping_strategies.pdf
 \begin{tikzpicture}
@@ -2044,12 +2073,13 @@ load('uniaxial_plants.mat', 'G_vc_light', 'G_md_light', 'G_pz_light', ...
 #+end_src
 
 ** Active Damping Strategies
-
+<<ssec:uniaxial_active_damping_strategies>>
+**** 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_equiv, left).
 
 The control strategy consists of integrating the measured force and feeding it back to the actuator:
 \begin{equation}
-K_{\text{IFF}}(s) = \frac{g}{s}
+\boxed{K_{\text{IFF}}(s) = \frac{g}{s}}
 \end{equation}
 
 The mechanical equivalent of this strategy is to add a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain $k/g$ (see Figure ref:fig:uniaxial_active_damping_iff_equiv, right).
@@ -2160,11 +2190,12 @@ The mechanical equivalent of this strategy is to add a dashpot in series with th
 [[file:figs/uniaxial_active_damping_iff_equiv.png]]
 
 
+**** 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_equiv, left).
 
 The derivative of this relative motion is used for the feedback signal:
 \begin{equation}
-K_{\text{RDC}}(s) = - g \cdot s
+\boxed{K_{\text{RDC}}(s) = - g \cdot s}
 \end{equation}
 
 The mechanical equivalent is to add a dashpot in parallel with the actuator with a damping coefficient equal to the controller gain $g$ (see Figure ref:fig:uniaxial_active_damping_rdc_equiv, right).
@@ -2280,11 +2311,12 @@ The mechanical equivalent is to add a dashpot in parallel with the actuator with
 #+RESULTS:
 [[file:figs/uniaxial_active_damping_rdc_equiv.png]]
 
+**** Direct Velocity Feedback (DVF)
 Finally, the Direct Velocity Feedback strategy consists of using an inertial sensor (usually a geophone), that measured the "absolute" velocity of the body fixed on top of the actuator (se Figure ref:fig:uniaxial_active_damping_dvf_equiv, left).
 
 The measured velocity is then fed back to the actuator:
 \begin{equation}
-K_{\text{DVF}}(s) = - g
+\boxed{K_{\text{DVF}}(s) = - g}
 \end{equation}
 
 This is equivalent as to fix a dashpot (with a damping coefficient equal to the controller gain $g$) between the body (one which the inertial sensor is fixed) and an inertial reference frame (see Figure ref:fig:uniaxial_active_damping_dvf_equiv, right).
@@ -2406,11 +2438,12 @@ This is usually refers to as "/sky hook damper/".
 
 ** 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 *alternating poles and zeros* meaning that the phase is bounded to $\pm 90\,\text{deg}$ which makes the controller very robust.
+All have *alternating poles and zeros* meaning that the phase do not vary by more than $\pm 90\,\text{deg}$ which makes the design of a robust controller very easy.
+The reason all three plants in Figure ref:fig:uniaxial_plant_active_damping_techniques have alternating poles and zeros is because the three sensors are all collocated with the actuator [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition Chapter 7]].
 
-When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
+When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure ref:fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the IFF and DVF transfer functions.
 
-For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques.
+For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure ref:fig:uniaxial_plant_active_damping_techniques.
 Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
 
 #+begin_src matlab :exports none :results none
@@ -2538,13 +2571,21 @@ exportFig('figs/uniaxial_plant_active_damping_techniques.pdf', 'width', 'full',
 [[file:figs/uniaxial_plant_active_damping_techniques.png]]
 
 ** Achievable Damping - Root Locus
+<<ssec:uniaxial_active_damping_achievable_damping>>
 The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
 They are shown in Figure ref:fig:uniaxial_root_locus_damping_techniques.
 
 All three active damping approach can lead to *critical damping* of the nano-hexapod suspension mode.
 
-There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) and with RDC for a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode).
-This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
+There is even some damping authority on micro-station modes in the following cases:
+- IFF with a stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) ::
+  This can be understood from the mechanical equivalent of IFF shown in Figure ref:fig:uniaxial_active_damping_iff_equiv (right) considering an high stiffness $k$.
+  The micro-station top platform is connected to an inertial mass (the nano-hexapod) through a damper, which damps the micro-station suspension suspension mode.
+- DVF with a stiff nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques, right) ::
+  In that case, the "sky hook damper" (see mechanical equivalent of IFF in Figure ref:fig:uniaxial_active_damping_dvf_equiv, right) is connected to the micro-station top platform through the stiff nano-hexapod.
+- RDC with a soft nano-hexapod (Figure ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) ::
+  At the frequency of the micro-station mode, the nano-hexapod top mass is behaving as an inertial reference as the suspension mode of the soft nano-hexapod is at much lower frequency.
+  The micro-station and the nano-hexapod masses are connected through a large damper induced by RDC (see mechanical equivalent in Figure ref:fig:uniaxial_active_damping_rdc_equiv, right) which allows some damping of the micro-station.
 
 #+begin_src matlab :exports none
 %% Active Damping Robustness to change of sample's mass - Root Locus for all three damping techniques with 3 different sample's masses
@@ -2682,7 +2723,7 @@ exportFig('figs/uniaxial_root_locus_damping_techniques.pdf', 'width', 'full', 'h
 #+end_src
 
 #+name: fig:uniaxial_root_locus_damping_techniques
-#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.
+#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses. The Root Loci are zoomed on the suspension mode of the nano-hexapod.
 #+RESULTS:
 [[file:figs/uniaxial_root_locus_damping_techniques.png]]
 
@@ -2736,7 +2777,7 @@ exportFig('figs/uniaxial_root_locus_damping_techniques_micro_station_mode.pdf',
 #+end_src
 
 #+name: fig:uniaxial_root_locus_damping_techniques_micro_station_mode
-#+caption: Root Locus for the three damping techniques. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.
+#+caption: Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.
 #+RESULTS:
 [[file:figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png]]
 
@@ -2846,7 +2887,7 @@ isstable(G_dvf_pz_light) && isstable(G_dvf_pz_mid) && isstable(G_dvf_pz_heavy)
 #+end_src
 
 #+begin_src matlab :exports none :tangle no
-%% Save Active Damping Controllers
+%% Save Damped Plants
 save('./matlab/mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz_light', ...
                                                 'G_rdc_vc_light', 'G_rdc_md_light', 'G_rdc_pz_light', ...
                                                 'G_dvf_vc_light', 'G_dvf_md_light', 'G_dvf_pz_light', ...
@@ -2859,7 +2900,7 @@ save('./matlab/mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_ligh
 #+end_src
 
 #+begin_src matlab :exports none :eval no
-%% Save Active Damping Controller
+%% Save Damped Plants
 save('./mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz_light', ...
                                          'G_rdc_vc_light', 'G_rdc_md_light', 'G_rdc_pz_light', ...
                                          'G_dvf_vc_light', 'G_dvf_md_light', 'G_dvf_pz_light', ...
@@ -2872,6 +2913,8 @@ save('./mat/uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_
 #+end_src
 
 ** Change of sensitivity to disturbances
+<<ssec:uniaxial_active_damping_sensitivity_disturbances>>
+
 The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure ref:fig:uniaxial_sensitivity_dist_active_damping.
 The comparison is done with the nano-hexapod having a stiffness $k_n = 1\,N/\mu m$.
 
@@ -2935,10 +2978,11 @@ exportFig('figs/uniaxial_sensitivity_dist_active_damping.pdf', 'width', 'full',
 [[file:figs/uniaxial_sensitivity_dist_active_damping.png]]
 
 ** Noise Budgeting after Active Damping
+<<ssec:uniaxial_active_damping_noise_budget>>
 Cumulative Amplitude Spectrum of the distance $d$ with all three active damping techniques are compared in Figure ref:fig:uniaxial_cas_active_damping.
 All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
 
-Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
+Compared to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
 
 #+begin_src matlab :exports none :results none
 %% Cumulative Amplitude Spectrum of the distance d with all three active damping techniques
@@ -3020,7 +3064,7 @@ exportFig('figs/uniaxial_cas_active_damping.pdf', 'width', 'full', 'height', 'no
 #+RESULTS:
 [[file:figs/uniaxial_cas_active_damping.png]]
 
-** Obtained Damped Plant
+** Obtained Closed Loop Response
 The transfer functions from the plant input $f$ to the relative displacement $d$ while the active damping is implemented are shown in Figure ref:fig:uniaxial_damped_plant_three_active_damping_techniques.
 All three active damping techniques yield similar damped plants.
 
@@ -3212,6 +3256,8 @@ exportFig('figs/uniaxial_damped_plant_change_sample_mass.pdf', 'width', 'full',
 [[file:figs/uniaxial_damped_plant_change_sample_mass.png]]
 
 ** Robustness to change of payload's mass
+<<ssec:uniaxial_active_damping_robustness>>
+
 The Root Locus for the three damping techniques are shown in Figure ref:fig:uniaxial_active_damping_robustness_mass_root_locus for three sample's mass (1kg, 25kg and 50kg).
 The closed-loop poles are shown by the squares for a specific gain.
 
@@ -3395,7 +3441,7 @@ Conclusions for Active Damping:
 
 #+name: tab:comp_active_damping
 #+caption: Comparison of active damping strategies
-#+attr_latex: :environment tabularx :width \linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc
 #+attr_latex: :center t :booktabs t :font \scriptsize
 |                     | *IFF*                       | *RDC*                     | *DVF*                           |
 |---------------------+-----------------------------+---------------------------+---------------------------------|
@@ -3419,7 +3465,7 @@ The High Authority Control - Low Authority Control (HAC-LAC) architecture is sho
 It corresponds to a /two step/ control strategy:
 - First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section ref:sec:uniaxial_active_damping).
   It allows to reduce the vibration level, and it also makes the damped plant (transfer function from $u^{\prime}$ to $y$) easier to control than the undamped plant (transfer function from $u$ to $y$).
-- Then, a position controller $\bm{K}_{\textsc{HAC}}$ is implemented.
+- Then, a position controller $\bm{K}_{\textsc{HAC}}$ is implemented which is used to control the position $d$.
 
 #+begin_src latex :file uniaxial_hac_lac_architecture.pdf :results none
 \begin{tikzpicture}
@@ -3453,7 +3499,7 @@ It corresponds to a /two step/ control strategy:
 \end{tikzpicture}
 #+end_src
 
-Combined with the uniaxial model, it is shown in Figure ref:fig:uniaxial_hac_lac_model.
+When applied on the uniaxial model with IFF used as the Low Authority Control, the schematic shown in Figure ref:fig:uniaxial_hac_lac_model is obtained.
 
 #+begin_src latex :file uniaxial_hac_lac_model.pdf :results silent
 \begin{tikzpicture}
@@ -3598,8 +3644,8 @@ Combined with the uniaxial model, it is shown in Figure ref:fig:uniaxial_hac_lac
 #+attr_latex: :width 1.0\linewidth
 [[file:figs/uniaxial_hac_lac_architecture.png]]
 #+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-IFF strategy}
 #+attr_latex: :options {0.45\textwidth}
-#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-LAC strategy}
 #+begin_subfigure
 #+attr_latex: :scale 1
 [[file:figs/uniaxial_hac_lac_model.png]]
@@ -3659,10 +3705,17 @@ load('uniaxial_damped_plants.mat', 'G_iff_vc_light', 'G_iff_md_light', 'G_iff_pz
 #+end_src
 
 ** Damped Plant Dynamics
+<<ssec:uniaxial_position_control_damped_dynamics>>
+
 As was shown in Section ref:sec:uniaxial_active_damping, all three proposed active damping techniques yield similar damping plants.
-Therefore, /Integral Force Feedback/ will be used in this section to study the HAC-LAC performances.
+Therefore, /Integral Force Feedback/ will be used in this section to study the HAC-LAC performance.
 
 The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure ref:fig:uniaxial_hac_iff_damped_plants_masses.
+For $k_n = 0.01\,N/\mu m$ and $k_n = 1\,N/\mu m$, except from the nano-hexapod mode, the dynamics is quite simple and can be well approximated by a second order plant.
+However, this is not the case for the stiff nano-hexapod ($k_n = 100\,N/\mu m$) where two modes can be seen (Figure ref:fig:uniaxial_hac_iff_damped_plants_masses, right).
+
+This is due to the interaction between the micro-station (modelled modes at 70Hz, 140Hz and 320Hz) and the nano-hexapod.
+Such effect will be explained in Section ref:sec:uniaxial_support_compliance.
 
 #+begin_src matlab :exports none :results none
 %% Damped plant - Robustness to change of sample's mass
@@ -3671,22 +3724,28 @@ tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
 
 ax1 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 1\,kg$');
-plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 25\,kg$');
-plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), '-', 'DisplayName', '$m_s = 50\,kg$');
+plot(freqs, abs(squeeze(freqresp(G_vc_light('d', 'f'), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+plot(freqs, abs(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz'))), 'color', colors(2,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:));
+loglog(10.^(0.4*cos([0:0.01:2*pi])+log10(100)), ...
+       10.^(0.8*sin([0:0.01:2*pi]-pi/4)+log10(8e-8)), 'k--');
+text(20, 4e-8, sprintf('Small\nInteraction'), 'horizontalalignment', 'center');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
 title('$k_n = 0.01\,N/\mu m$');
 ylim([5e-10, 1e-3]);
-ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-ldg.ItemTokenSize = [20, 1];
 
 ax2 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_md_light('d', 'f'), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+plot(freqs, abs(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz'))), 'color', colors(2,:));
+plot(freqs, abs(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:));
+loglog(10.^(0.4*cos([0:0.01:2*pi])+log10(200)), ...
+       10.^(0.8*sin([0:0.01:2*pi]-pi/4)+log10(2e-8)), 'k--');
+text(40, 1e-8, sprintf('Small\nInteraction'), 'horizontalalignment', 'center');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
@@ -3695,20 +3754,27 @@ ylim([5e-10, 1e-3]);
 
 ax3 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_pz_light('d', 'f'), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'DisplayName', '$m_s = 1\,kg$, OL');
+plot(freqs, abs(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$m_s = 1\,kg$, IFF');
+plot(freqs, abs(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$m_s = 25\,kg$, IFF');
+plot(freqs, abs(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$m_s = 50\,kg$, IFF');
+loglog(10.^(0.8*cos([0:0.01:2*pi])+log10(350)), ...
+       10.^(1.2*sin([0:0.01:2*pi])+log10(8e-9)), 'k--', 'HandleVisibility', 'off');
+text(200, 5e-7, sprintf('$\\mu$ Station\nCoupling'), 'horizontalalignment', 'center');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
 title('$k_n = 100\,N/\mu m$');
 ylim([5e-10, 1e-3]);
+ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [20, 1];
 
 ax1b = nexttile();
 hold on;
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_light('d', 'f'), freqs, 'Hz')))), 'color', [colors(1,:), 0.5]);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_light('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_mid(  'd', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_vc_heavy('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
@@ -3718,9 +3784,10 @@ ylim([-200, 20]);
 
 ax2b = nexttile();
 hold on;
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_md_light('d', 'f'), freqs, 'Hz')))), 'color', [colors(1,:), 0.5]);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_light('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_mid(  'd', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_md_heavy('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
@@ -3730,9 +3797,10 @@ ylim([-200, 20]);
 
 ax3b = nexttile();
 hold on;
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz')))));
-plot(freqs, unwrap(180/pi*angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_light('d', 'f'), freqs, 'Hz')))), 'color', [colors(1,:), 0.5]);
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_light('d', 'f'), freqs, 'Hz')))), 'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_mid(  'd', 'f'), freqs, 'Hz')))), 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_pz_heavy('d', 'f'), freqs, 'Hz')))), 'color', colors(3,:));
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
@@ -3754,13 +3822,15 @@ exportFig('figs/uniaxial_hac_iff_damped_plants_masses.pdf', 'width', 'full', 'he
 [[file:figs/uniaxial_hac_iff_damped_plants_masses.png]]
 
 ** Position Feedback Controller
+<<ssec:uniaxial_position_control_design>>
+
 The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample's mass.
 
-The required feedback bandwidth was approximately determined un Section ref:sec:uniaxial_noise_budgeting:
+The required feedback bandwidth was approximately determined in Section ref:sec:uniaxial_noise_budgeting:
 - $\approx 10\,\text{Hz}$ for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$).
   Near this frequency, the plants are equivalent to a mass line.
   The gain of the mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,kg$ up to $65\,kg$).
-  This mean that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
+  This means that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
 - $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$).
   Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line.
   It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.
@@ -3772,7 +3842,37 @@ The required feedback bandwidth was approximately determined un Section ref:sec:
 
 
 Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure ref:fig:uniaxial_nyquist_hac).
-These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
+These high authority controllers are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
+The controllers used for the three nano-hexapod are shown in Equation eqref:eq:uniaxial_hac_formulas, and the parameters are summarized in Table ref:tab:uniaxial_feedback_controller_parameters.
+
+\begin{subequations} \label{eq:uniaxial_hac_formulas}
+\begin{align}
+K_{\text{soft}}(s) &= g \cdot
+       \underbrace{\frac{s + \omega_0}{s + \omega_i}}_{\text{lag}} \cdot
+       \underbrace{\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}}_{\text{lead}} \cdot
+       \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}} \\
+K_{\text{mid}}(s) &= g \cdot
+       \underbrace{\left(\frac{s + \omega_0}{s + \omega_i}\right)^2}_{\text{2 lags}} \cdot
+       \underbrace{\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}}_{\text{lead}} \cdot
+       \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}} \\
+K_{\text{stiff}}(s) &= g \cdot
+       \underbrace{\left(\frac{1}{s + \omega_i}\right)^2}_{\text{2 lags}} \cdot
+       \underbrace{\left(\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}\right)^2}_{\text{2 leads}} \cdot
+       \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}}
+\end{align}
+\end{subequations}
+
+#+name: tab:uniaxial_feedback_controller_parameters
+#+caption: Parameters used for the position feedback controllers
+#+attr_latex: :environment tabularx :width \linewidth :align lXXX
+#+attr_latex: :center t :booktabs t
+|        | *Soft*                                    | *Moderately stiff*                         | *Stiff*                                  |
+|--------+-------------------------------------------+--------------------------------------------+------------------------------------------|
+| *Gain* | $g = 4 \cdot 10^5$                        | $g = 3 \cdot 10^6$                         | $g = 6 \cdot 10^12$                      |
+| *Lead* | $a = 5$, $\omega_c = 20\,Hz$              | $a = 4$, $\omega_c = 70\,Hz$               | $a = 5$, $\omega_c = 100\,Hz$            |
+| *Lag*  | $\omega_0 = 5\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_0 = 20\,Hz$, $\omega_i = 0.01\,Hz$ | $\omega_i = 0.01\,Hz$   |
+| *LPF*  | $\omega_l = 200\,Hz$                      | $\omega_l = 300\,Hz$                       | $\omega_l = 500\,Hz$                     |
+
 The loop gains for the three nano-hexapod are shown in Figure ref:fig:uniaxial_loop_gain_hac.
 We can see that:
 - for the soft and moderately stiff nano-hexapod, the crossover frequency varies a lot with the sample mass.
@@ -3780,8 +3880,8 @@ We can see that:
 - for the stiff nano-hexapod, the obtained crossover frequency is not at high as what was estimated necessary.
   The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.
 
-Note that these controller were quickly tuned by hand and not designed using any optimization methods.
-The goal is just to have a first estimation of the attainable performances.
+Note that these controllers were quickly tuned by hand and not designed using any optimization methods.
+The goal is just to have a first estimation of the attainable performance.
 
 #+begin_src matlab :exports none
 %% High Authority Controller - Soft Nano-Hexapod
@@ -3790,16 +3890,16 @@ a  = 5;  % Amount of phase lead / width of the phase lead / high frequency gain
 wc = 2*pi*20; % Frequency with the maximum phase lead [rad/s]
 H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
 
+% Lag at low frequency
+H_lag = (s + 2*pi*5)/(s + 2*pi*0.01);
+
 % Low Pass filter to increase robustness
 H_lpf = 1/(1 + s/2/pi/200);
 
-% Added integrator at low frequency
-H_int = (s + 2*pi*5)/(s + 2*pi*0.01);
-
 % High Authority Controller
 K_hac_vc = 4e5    * ... % Gain
            H_lead * ... % Lead
-           H_int * ...  % Integrator
+           H_lag  * ... % Lag
            H_lpf;       % LPF
 K_hac_vc.InputName = {'d'};
 K_hac_vc.OutputName = {'f'};
@@ -3807,49 +3907,61 @@ K_hac_vc.OutputName = {'f'};
 
 #+begin_src matlab :exports none
 %% High Authority Controller - Mid Stiffness Nano-Hexapod
-% Integrator as low frequency
-H_int = (s + 2*pi*10)/(s + 2*pi*0.01) * (s + 2*pi*20)/(s + 2*pi*0.01);
-
 % Lead to increase phase margin
 a  = 4;  % Amount of phase lead / width of the phase lead / high frequency gain
 wc = 2*pi*70; % Frequency with the maximum phase lead [rad/s]
 
 H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
 
+% Lag at low frequency
+H_lag = ((s + 2*pi*15)/(s + 2*pi*0.01))^2;
+
 % Low Pass filter to increase robustness
 H_lpf = 1/(1 + s/2/pi/300);
 
 % High Authority Controller
 K_hac_md = 3e6    * ... % Gain
            H_lead * ... % Lead
-           H_lpf  * ... % Low Pass Filter
-           H_int;         % Integrator
+           H_lag  * ... % Lag
+           H_lpf;       % LPF
 K_hac_md.InputName = {'d'};
 K_hac_md.OutputName = {'f'};
 #+end_src
 
 #+begin_src matlab :exports none
 %% High Authority Controller - Stiff Nano-Hexapod
-% Integrator as low frequency
-H_int = 1/(s + 2*pi*0.01) * 1/(s + 2*pi*0.01);
-
 % Lead to increase phase margin
 a  = 5;  % Amount of phase lead / width of the phase lead / high frequency gain
 wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s]
-H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+H_lead = ((1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))))^2;
+
+% Integrator
+H_int = 1/(s + 2*pi*0.01)^2;
 
 % Low Pass filter to increase robustness
 H_lpf = 1/(1 + s/2/pi/500);
 
 % High Authority Controller
-K_hac_pz = 6e12      * ... % Gain
-           H_lead^2 * ... % Lead
-           H_lpf    * ... % Low Pass Filter
-           H_int;         % Integrator
+K_hac_pz = 6e12   * ... % Gain
+           H_lead * ... % Lead
+           H_int  * ... % Lag
+           H_lpf;       % LPF
 K_hac_pz.InputName = {'d'};
 K_hac_pz.OutputName = {'f'};
 #+end_src
 
+#+begin_src matlab :exports none :tangle no
+%% Save High Authority Controllers
+save('./matlab/mat/uniaxial_high_authority_controllers.mat', ...
+        'K_hac_vc', 'K_hac_md', 'K_hac_pz');
+#+end_src
+
+#+begin_src matlab :exports none :eval no
+%% Save High Authority Controllers
+save('/mat/uniaxial_high_authority_controllers.mat', ...
+        'K_hac_vc', 'K_hac_md', 'K_hac_pz');
+#+end_src
+
 #+begin_src matlab :exports none
 %% Compute Loop gain for Nyquist Plot
 L_vc_light = squeeze(freqresp(K_hac_vc*G_iff_vc_light('d', 'f'), freqs, 'Hz'));
@@ -3961,6 +4073,8 @@ exportFig('figs/uniaxial_loop_gain_hac.pdf', 'width', 'wide', 'height', 'tall');
 [[file:figs/uniaxial_loop_gain_hac.png]]
 
 ** Closed-Loop Noise Budgeting
+<<ssec:uniaxial_position_control_cl_noise_budget>>
+
 The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
 These are compared with the open-loop and damped plants cases in Figure ref:fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass).
 As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
@@ -4034,7 +4148,7 @@ exportFig('figs/uniaxial_sensitivity_dist_hac_lac.pdf', 'width', 'full', 'height
 #+end_src
 
 #+name: fig:uniaxial_sensitivity_dist_hac_lac
-#+caption: Change of sensitivity to disturbances with LAC and with HAC-LAC
+#+caption: Change of sensitivity to disturbances with LAC and with HAC-LAC. Nano-Hexapod with $k_n = 1\,N/\mu m$ and sample mass of $25\,kg$ are used.
 #+RESULTS:
 [[file:figs/uniaxial_sensitivity_dist_hac_lac.png]]
 
@@ -4167,6 +4281,1318 @@ exportFig('figs/uniaxial_cas_hac_lac.pdf', 'width', 'full', 'height', 'normal');
 #+RESULTS:
 [[file:figs/uniaxial_cas_hac_lac.png]]
 
+** Conclusion
+#+begin_important
+Based on the open-loop noise budgeting made in Section ref:sec:uniaxial_noise_budgeting, the closed-loop bandwidth required to obtain acceptable vibration levels was estimated.
+In order to obtain such bandwidth, the HAC-LAC strategy was followed which consists of applying an active damping controller and then a high authority position feedback controller.
+
+In this section, feedback controllers were design in such a way that the required closed-loop bandwidth was reached while being robust to a change of payload mass.
+The attainable vibration control performances were estimated for the three nano-hexapod stiffnesses and were found to be close to the required values.
+A slight advantage can be given to the soft nano-hexapod as it requires less feedback bandwidth while giving better stability results.
+#+end_important
+
+* Effect of limited micro-station compliance
+:PROPERTIES:
+:HEADER-ARGS:matlab+: :tangle matlab/uniaxial_7_support_compliance.m
+:END:
+<<sec:uniaxial_support_compliance>>
+** Introduction                                                      :ignore:
+
+In this section, the impact of the compliance of the support (i.e. the micro-station) on the dynamics of the plant to control will be studied.
+
+This is a critical point as 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 which is not impacted by the micro-station.
+
+Therefore, the objective in this section is to obtain some guidance for the design of a nano-hexapod that will not by impacted by the complex micro-station dynamics.
+
+In order to study this, two models are used (Figure ref:fig:uniaxial_support_compliance_models) with:
+- the nano-hexapod directly fixed on top of the granite, therefore neglecting any support compliance (Figure ref:fig:uniaxial_support_compliance_nano_hexapod_only)
+- the nano-hexapod fixed on top of the micro-station having some limited compliance (Figure ref:fig:uniaxial_support_compliance_test_system)
+
+#+begin_src latex :file uniaxial_support_compliance_nano_hexapod_only.pdf :results file raw silent
+\begin{tikzpicture}
+  % Parameters
+  \def\massw{2.2}  % Width of the masses
+  \def\massh{0.8}  % Height of the masses
+  \def\spaceh{1.2} % Height of the springs/dampers
+  \def\dispw{0.3}  % Width of the dashed line for the displacement
+  \def\disph{0.5}  % Height of the arrow for the displacements
+  \def\bracs{0.05} % Brace spacing vertically
+  \def\brach{-10pt} % Brace shift horizontaly
+
+  % Granite
+  \draw[fill=white!70!black] (-1*\massw, -\massh) rectangle (\massw, 0) node[pos=0.5]{Granite};
+  \coordinate(origin) at (0,0);
+
+  % Nano Hexapod
+  \begin{scope}[shift={(0, 0)}]
+    % Mass
+    \draw[draw=colorred,fill=colorred!10!white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5, color=colorred]{$m_{n}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorred] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred]{$k_{n}$};
+    \draw[damper={colorred}{}{}] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2, color=colorred]{$c_{n}$};
+    \draw[actuator={0.4}{0.2}{colorred}] ( 0.4*\massw, 0) -- (0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred](F){$f$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh) -- coordinate[near end](dhigh) ++(1.2, 0);
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{$\nu$-hexapod};
+  \end{scope}
+
+  \draw[<->, dashed, colorblue] (dhigh) --node[midway, right]{$L^\prime$} (origin-|dhigh);
+\end{tikzpicture}
+#+end_src
+
+#+begin_src latex :file uniaxial_support_compliance_test_system.pdf :results file raw silent
+\begin{tikzpicture}
+  % Parameters
+  \def\massw{2.2}  % Width of the masses
+  \def\massh{0.8}  % Height of the masses
+  \def\spaceh{1.2} % Height of the springs/dampers
+  \def\dispw{0.3}  % Width of the dashed line for the displacement
+  \def\disph{0.5}  % Height of the arrow for the displacements
+  \def\bracs{0.05} % Brace spacing vertically
+  \def\brach{-10pt} % Brace shift horizontaly
+
+  % Granite
+  \draw[fill=white!70!black] (-1*\massw, -\massh) rectangle (\massw, 0) node[pos=0.5]{Granite};
+  \coordinate(origin) at (0,0);
+
+  % Micro Station
+  \begin{scope}[shift={(0, 0)}]
+    % Mass
+    \draw[draw=colorblue,fill=colorblue!10!white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5, color=colorblue]{$m_{\mu}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorblue] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1, color=colorblue]{$k_{\mu}$};
+    \draw[damper={colorblue}{}{}] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2, color=colorblue]{$c_{\mu}$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh+\massh) -- coordinate[near end](dmid) ++(0.3, 0);
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt]{$\mu$-station};
+  \end{scope}
+
+  % Nano Station
+  \begin{scope}[shift={(0, \spaceh+\massh)}]
+    % Mass
+    \draw[draw=colorred,fill=colorred!10!white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5, color=colorred]{$m_{n}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorred] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred]{$k_{n}$};
+    \draw[damper={colorred}{}{}] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2, color=colorred]{$c_{n}$};
+    \draw[actuator={0.4}{0.2}{colorred}] ( 0.4*\massw, 0) -- (0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred](F){$f$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh) -- coordinate[near end](dhigh) ++(1.2, 0);
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{$\nu$-hexapod};
+  \end{scope}
+
+  \draw[<->, dashed, coloryellow]    (dhigh) --node[midway, right]{$d$} (origin-|dhigh);
+  \draw[<->, dashed, colorred] (dmid)  --node[midway, right]{$L$} (dhigh-|dmid);
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:uniaxial_support_compliance_models
+#+caption: Models used to study the effect of limited support compliance
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Nano-Hexapod fixed directly on the Granite}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_support_compliance_nano_hexapod_only.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_support_compliance_test_system}Nano-Hexapod fixed on top of the Micro-Station}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_support_compliance_test_system.png]]
+#+end_subfigure
+#+end_figure
+
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+<<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+<<matlab-init>>
+#+end_src
+
+#+begin_src matlab :tangle no :noweb yes
+<<m-init-path>>
+#+end_src
+
+#+begin_src matlab :eval no :noweb yes
+<<m-init-path-tangle>>
+#+end_src
+
+#+begin_src matlab :noweb yes
+<<m-init-other>>
+#+end_src
+
+#+begin_src matlab
+%% Load the micro-station parameters
+load('uniaxial_micro_station_parameters.mat')
+#+end_src
+
+#+begin_src matlab
+%% Frequency Vector [Hz]
+freqs = logspace(0, 3, 1000);
+#+end_src
+
+** Neglected support compliance
+
+Let's first neglect the limited compliance of the micro-station and use the uniaxial model show in Figure ref:fig:uniaxial_support_compliance_nano_hexapod_only.
+Let's choose a nano-hexapod mass (including the payload) of $20\,\text{kg}$ and three hexapod stiffnesses such that their resonance frequencies are at $\omega_{\nu} = 10\,\text{Hz}$, $\omega_{\nu} = 70\,\text{Hz}$ and $\omega_{\nu} = 400\,\text{Hz}$.
+
+The obtained transfer functions from $F$ to $L^\prime$ are shown in Figure ref:fig:uniaxial_effect_support_compliance_neglected.
+When neglecting the support compliance, large feedback bandwidth can be achieve for all three Nano-Hexapod.
+
+#+begin_src matlab :exports none
+%% Nano-Hexapod Parameters
+m = 20; % Mass [kg]
+
+% "Soft" Nano-Hexapod
+k_soft = m*(2*pi*10)^2; % Stiffness [N/m]
+c_soft = 0.1*2*sqrt(m*k_soft); % Damping [N/(m/s)]
+
+% "Mid" Nano-Hexapod
+k_mid = m*(2*pi*70)^2; % Stiffness [N/m]
+c_mid = 0.1*2*sqrt(m*k_mid); % Damping [N/(m/s)]
+
+% "Stiff" Nano-Hexapod
+k_stiff = m*(2*pi*400)^2; % Stiffness [N/m]
+c_stiff = 0.1*2*sqrt(m*k_stiff); % Damping [N/(m/s)]
+#+end_src
+
+#+begin_src matlab :exports none
+%% Compute the transfer functions for considered nano-hexapods - From F to L'
+% "Soft" Nano-Hexapod
+G_soft_a = 1/(m*s^2 + c_soft*s + k_soft); % Transfer function from F to L'
+
+% "Mid" Nano-Hexapod
+G_mid_a = 1/(m*s^2 + c_mid*s + k_mid); % Transfer function from F to L'
+
+% "Stiff" Nano-Hexapod
+G_stiff_a = 1/(m*s^2 + c_stiff*s + k_stiff); % Transfer function from F to L'
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Obtained transfer functions from F to L when neglecing support compliance
+freqs = logspace(0, 3, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_soft_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+text(50, 5e-5, '$\omega_\nu =$ 10Hz', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude [m/N]');
+xticks([1e0, 1e1, 1e2]);
+title('"Soft" $\nu$-hexapod');
+
+ax2 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_mid_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+text(70, 3e-6, '$\omega_\nu =$ 70Hz', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]');
+set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+title('"Mid" $\nu$-hexapod');
+
+ax3 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_stiff_a, freqs, 'Hz'))), '-', 'color', colors(1,:), ...
+     'DisplayName', '$L^\prime/F$');
+text(200, 8e-8, '$\omega_\nu =$ 400Hz', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]);
+legend('location', 'northeast');
+xticks([1e0, 1e1, 1e2]);
+title('"Stiff" $\nu$-hexapod');
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-9, 1e-4]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/uniaxial_effect_support_compliance_neglected.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:uniaxial_effect_support_compliance_neglected
+#+caption: Obtained transfer functions from $F$ to $L^{\prime}$ when neglecing support compliance
+#+RESULTS:
+[[file:figs/uniaxial_effect_support_compliance_neglected.png]]
+
+** Effect of support compliance on $L/F$
+
+Let's now add some support compliance and use the model shown in Figure ref:fig:uniaxial_support_compliance_test_system.
+The parameters of the support (i.e. $m^{\prime}$, $c^{\prime}$ and $k^{\prime}$) are chosen in such a way that the main vertical mode at $\omega_\mu = 70\,\text{Hz}$ seen on the micro-station is matched (Figure ref:fig:uniaxial_comp_frf_meas_model, yellow curve).
+
+#+begin_src matlab
+%% Parameters of the support compliance
+w0h = 2*pi*70; % [rad/s]
+xih = 0.1; % [-]
+
+mh = 20; % [kg]
+kh = mh*w0h^2;
+ch = xih*2*sqrt(kh*mh);
+#+end_src
+
+#+begin_src matlab :exports none
+%% Compute the transfer functions from F to L and from F to d for considered Nano-Hexapods
+% "Soft" Nano-Hexapod
+G_soft = (mh*s^2 + ch*s + kh)/(m*s^2*(c_soft*s + k_soft) + (m*s^2 + c_soft*s + k_soft)*(mh*s^2 + ch*s + kh)); % d/F
+G_soft_r = (1 - m*s^2*G_soft)/(c_soft*s + k_soft); % L/F
+
+% "Mid" Nano-Hexapod
+G_mid = (mh*s^2 + ch*s + kh)/(m*s^2*(c_mid*s + k_mid) + (m*s^2 + c_mid*s + k_mid)*(mh*s^2 + ch*s + kh)); % d/F
+G_mid_r = (1 - m*s^2*G_mid)/(c_mid*s + k_mid); % L/F
+
+% "Stiff" Nano-Hexapod
+G_stiff = (mh*s^2 + ch*s + kh)/(m*s^2*(c_stiff*s + k_stiff) + (m*s^2 + c_stiff*s + k_stiff)*(mh*s^2 + ch*s + kh)); % d/F
+G_stiff_r = (1 - m*s^2*G_stiff)/(c_stiff*s + k_stiff); % L/F
+#+end_src
+
+The transfer functions from $F$ to $L$ (i.e. control of the relative motion of the nano-hexapod) and from $L$ to $d$ (i.e. control of the position between the nano-hexapod and the fixed granite) can then be computed.
+
+When the relative displacement of the nano-hexapod $L$ is to be controlled (dynamics shown in Figure ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff nano-hexapod (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, right).
+
+This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages.
+In such 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 is it fixed.
+# Add references of such stations with piezo stages on top
+
+If a soft nano-hexapod is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure ref:fig:uniaxial_effect_support_compliance_dynamics, left) which will impact the control robustness and performance.
+
+#+begin_src matlab :exports none :results none
+%% Effect of the support compliance on the transfer functions from F to L and from F to d
+figure;
+freqs = logspace(0, 3, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_soft_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_soft_r, freqs, 'Hz'))), '-', 'color', colors(2,:));
+loglog(10.^(0.3*cos(0:0.01:2*pi)+log10(60)), ...
+       10.^(0.6*sin(0:0.01:2*pi)+log10(4e-7)), 'k--');
+text(8, 3e-7, sprintf('Support\nDynamics'), 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+title('"Soft" $\nu$-hexapod');
+ylabel('Magnitude [m/N]');
+
+
+ax2 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_mid_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_mid_r, freqs, 'Hz'))), '-', 'color', colors(2,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]');
+xticks([1e0, 1e1, 1e2]);
+title('"Mid" $\nu$-hexapod');
+set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_stiff_a, freqs, 'Hz'))), '-', 'color', colors(1,:), ...
+     'DisplayName', '$L^\prime/F$');
+plot(freqs, abs(squeeze(freqresp(G_stiff_r, freqs, 'Hz'))), '-', 'color', colors(2,:), ...
+     'DisplayName', '$L/F$');
+loglog(10.^(0.3*cos(0:0.01:2*pi)+log10(50)), ...
+       10.^(0.3*sin(0:0.01:2*pi)+log10(8e-9)), 'k--', 'HandleVisibility', 'off');
+text(50, 3e-8, 'No effect', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+title('"Stiff" $\nu$-hexapod');
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-9, 1e-4]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/uniaxial_effect_support_compliance_dynamics.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:uniaxial_effect_support_compliance_dynamics
+#+caption: Effect of the support compliance on the transfer functions from $F$ to $L$
+#+RESULTS:
+[[file:figs/uniaxial_effect_support_compliance_dynamics.png]]
+
+** Effect of support compliance on $d/F$
+
+When the motion to be controlled is the relative displacement $d$ between the granite and the nano-hexapod's top platform, the effect of the support compliance on the plant dynamics is opposite than what was previously observed.
+Indeed, using a "soft" nano-hexapod (i.e. with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_d, left).
+On the contrary, if a "stiff" nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure ref:fig:uniaxial_effect_support_compliance_dynamics_d, right).
+
+#+begin_src matlab :exports none :results none
+%% Effect of the support compliance on the transfer functions from F to L and from F to d
+figure;
+freqs = logspace(0, 3, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_soft_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_soft,   freqs, 'Hz'))), '-', 'color', colors(3,:));
+loglog(10.^(0.3*cos(0:0.01:2*pi)+log10(60)), ...
+       10.^(0.6*sin(0:0.01:2*pi)+log10(4e-7)), 'k--');
+text(8, 3e-7, 'No effect', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+title('"Soft" $\nu$-hexapod');
+ylabel('Magnitude [m/N]');
+
+
+ax2 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_mid_a, freqs, 'Hz'))), '-', 'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_mid,   freqs, 'Hz'))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]');
+xticks([1e0, 1e1, 1e2]);
+title('"Mid" $\nu$-hexapod');
+set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_stiff_a, freqs, 'Hz'))), '-', 'color', colors(1,:), ...
+     'DisplayName', '$L^\prime/F$');
+plot(freqs, abs(squeeze(freqresp(G_stiff,   freqs, 'Hz'))), '-', 'color', colors(3,:), ...
+     'DisplayName', '$d/F$');
+loglog(10.^(0.4*cos(0:0.01:2*pi)+log10(50)), ...
+       10.^(0.8*sin(0:0.01:2*pi)+log10(8e-9)), 'k--', 'HandleVisibility', 'off');
+text(50, 2e-7, sprintf('Support\nDynamics'), 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+title('"Stiff" $\nu$-hexapod');
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-9, 1e-4]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/uniaxial_effect_support_compliance_dynamics_d.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:uniaxial_effect_support_compliance_dynamics_d
+#+caption: Effect of the support compliance on the transfer functions from $F$ to $d$
+#+RESULTS:
+[[file:figs/uniaxial_effect_support_compliance_dynamics_d.png]]
+
+** Conclusion
+#+begin_important
+In order to study the impact of the support compliance on the plant dynamics, simple models shown in Figure ref:fig:uniaxial_support_compliance_models were used.
+
+Depending on the quantity to be controlled ($L$ or $d$ in Figure ref:fig:uniaxial_support_compliance_test_system) and on the relative location of $\omega_\nu$ (suspension mode of the nano-hexapod) and $\omega_\mu$ (modes of the support), the interaction between the support and the nano-hexapod dynamics can change drastically (observations made are summarized in Table ref:tab:uniaxial_effect_compliance).
+
+For the NASS, having the suspension mode of the nano-hexapod at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control.
+
+Note that observations made in this section are also affected by the ratio between the support mass $m^{\prime}$ and the nano-hexapod mass $m$ (the effect is more pronounced when the ratio $m/m^{\prime}$ increases).
+#+end_important
+
+#+name: tab:uniaxial_effect_compliance
+#+caption: Impact of the support dynamics on the plant dynamics
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc
+#+attr_latex: :center t :booktabs t
+|        | $\omega_{\nu} \ll \omega_{\mu}$ | $\omega_{\nu} \approx \omega_{\mu}$ | $\omega_{\nu} \gg \omega_{\mu}$ |
+|--------+---------------------------------+-------------------------------------+---------------------------------|
+| $d/F$: | small                           | large                               | large                           |
+| $L/F$: | large                           | large                               | small                           |
+
+* Effect of Payload Dynamics
+:PROPERTIES:
+:HEADER-ARGS:matlab+: :tangle matlab/uniaxial_8_payload_dynamics.m
+:END:
+<<sec:uniaxial_payload_dynamics>>
+
+** Introduction                                                     :ignore:
+
+Up to this section, the sample was modelled as a mass rigidly fixed to the nano-hexapod (as shown in Figure ref:fig:uniaxial_paylaod_dynamics_rigid_schematic).
+However, such sample may present internal dynamics and its fixation to the nano-hexapod may have limited stiffness.
+
+To study the effect of the sample dynamics, models shown in Figure ref:fig:uniaxial_paylaod_dynamics_schematic are used is this Section.
+
+#+begin_src latex :file uniaxial_paylaod_dynamics_rigid_schematic.pdf :results file raw silent
+\begin{tikzpicture}
+  % Parameters
+  \def\massw{2.2}  % Width of the masses
+  \def\massh{0.8}  % Height of the masses
+  \def\spaceh{1.2} % Height of the springs/dampers
+  \def\dispw{0.3}  % Width of the dashed line for the displacement
+  \def\disph{0.5}  % Height of the arrow for the displacements
+  \def\bracs{0.05} % Brace spacing vertically
+  \def\brach{-10pt} % Brace shift horizontaly
+
+  % Granite
+  \draw[fill=white!70!black] (-1*\massw, -\massh) rectangle (\massw, 0) node[pos=0.5]{Granite};
+  \coordinate(origin) at (0,0);
+
+  % Nano Station
+  \begin{scope}[shift={(0, 0)}]
+    % Mass
+    \draw[draw=colorred,fill=colorred!10!white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5, color=colorred]{$m_{n}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorred] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred]{$k_{n}$};
+    \draw[damper={colorred}{}{}] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2, color=colorred]{$c_{n}$};
+    \draw[actuator={0.4}{0.2}{colorred}] ( 0.4*\massw, 0) -- (0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred](F){$f$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh) -- coordinate[near end](dhigh) ++(1.2, 0);
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{$\nu$-hexapod};
+  \end{scope}
+
+  % Sample
+  \begin{scope}[shift={(0, \spaceh+\massh)}]
+    % Mass
+    \draw[draw=colorblue,fill=colorblue!10!white] (-0.3*\massw, 0) rectangle (0.3*\massw, \massh) node[pos=0.5, color=colorblue]{$m_{s}$};
+  \end{scope}
+
+
+  \draw[<->, dashed] (dhigh) --node[midway, right]{$L$} (origin-|dhigh);
+\end{tikzpicture}
+#+end_src
+
+#+begin_src latex :file uniaxial_paylaod_dynamics_schematic.pdf :results file raw silent
+\begin{tikzpicture}
+  % Parameters
+  \def\massw{2.2}  % Width of the masses
+  \def\massh{0.8}  % Height of the masses
+  \def\spaceh{1.2} % Height of the springs/dampers
+  \def\dispw{0.3}  % Width of the dashed line for the displacement
+  \def\disph{0.5}  % Height of the arrow for the displacements
+  \def\bracs{0.05} % Brace spacing vertically
+  \def\brach{-10pt} % Brace shift horizontaly
+
+  % Granite
+  \draw[fill=white!70!black] (-1*\massw, -\massh) rectangle (\massw, 0) node[pos=0.5]{Granite};
+  \coordinate(origin) at (0,0);
+
+  % Nano Station
+  \begin{scope}[shift={(0, 0)}]
+    % Mass
+    \draw[draw=colorred,fill=colorred!10!white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5, color=colorred]{$m_{n}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorred] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred]{$k_{n}$};
+    \draw[damper={colorred}{}{}] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2, color=colorred]{$c_{n}$};
+    \draw[actuator={0.4}{0.2}{colorred}] ( 0.4*\massw, 0) -- (0.4*\massw, \spaceh) node[midway, left=0.1, color=colorred](F){$f$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh) -- coordinate[near end](dhigh) ++(1.2, 0);
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{$\nu$-hexapod};
+  \end{scope}
+
+  % Sample
+  \begin{scope}[shift={(0, \spaceh+\massh)}]
+    % Mass
+    \draw[draw=colorblue,fill=colorblue!10!white] (-0.3*\massw, \spaceh) rectangle (0.3*\massw, \spaceh+\massh) node[pos=0.5, color=colorblue]{$m_{s}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring,draw=colorblue] (-0.2*\massw, 0) -- (-0.2*\massw, \spaceh) node[midway, left=0.1, color=colorblue]{$k_{s}$};
+    \draw[damper={colorblue}{}{}] ( 0.2*\massw, 0) -- ( 0.2*\massw, \spaceh) node[midway, left=0.2, color=colorblue]{$c_{s}$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Payload};
+  \end{scope}
+
+
+  \draw[<->, dashed] (dhigh) --node[midway, right]{$L^\prime$} (origin-|dhigh);
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:uniaxial_payload_dynamics_models
+#+caption: Models used to study the effect of payload dynamics
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_rigid_schematic}Rigid payload}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_paylaod_dynamics_rigid_schematic.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Payload with some flexibility}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_paylaod_dynamics_schematic.png]]
+#+end_subfigure
+#+end_figure
+
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+<<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+<<matlab-init>>
+#+end_src
+
+#+begin_src matlab :tangle no :noweb yes
+<<m-init-path>>
+#+end_src
+
+#+begin_src matlab :eval no :noweb yes
+<<m-init-path-tangle>>
+#+end_src
+
+#+begin_src matlab :noweb yes
+<<m-init-other>>
+#+end_src
+
+#+begin_src matlab
+%% Uniaxial Simscape model name
+mdl = 'nass_uniaxial_model';
+#+end_src
+
+#+begin_src matlab
+%% Load the micro-station parameters
+load('uniaxial_micro_station_parameters.mat')
+#+end_src
+
+#+begin_src matlab
+%% Load the PSD of disturbances
+load('uniaxial_disturbance_psd.mat', 'f', 'psd_ft', 'psd_xf');
+#+end_src
+
+#+begin_src matlab :exports none
+%% Load Active Damping Controller
+load('uniaxial_active_damping_controllers.mat', 'K_iff_vc', 'K_iff_md', 'K_iff_pz', ...
+                                                'K_rdc_vc', 'K_rdc_md', 'K_rdc_pz', ...
+                                                'K_dvf_vc', 'K_dvf_md', 'K_dvf_pz');
+#+end_src
+
+#+begin_src matlab
+%% Load High Authority Controllers
+load('uniaxial_high_authority_controllers.mat', 'K_hac_vc', 'K_hac_md', 'K_hac_pz');
+#+end_src
+
+#+begin_src matlab
+%% Frequency Vector [Hz]
+freqs = logspace(0, 3, 1000);
+#+end_src
+
+** Impact on the plant dynamics
+<<ssec:uniaxial_payload_dynamics_effect_dynamics>>
+
+To study the impact of sample dynamics, the following sample configurations are considered:
+- rigid sample, corresponding to Figure ref:fig:uniaxial_paylaod_dynamics_rigid_schematic
+- two flexible samples with a resonance at $\omega_s = 200\,\text{Hz}$ and at $\omega_s = 20\,\text{Hz}$ respectively, corresponding to Figure ref:fig:uniaxial_paylaod_dynamics_schematic
+- for all cases, two sample masses are considered: $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$
+
+The transfer functions from the nano-hexapod force to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod $k_n = 0.01\,N/\mu m$ 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 $\frac{1}{m_ns^2}$).
+
+#+begin_src matlab :exports none
+%% Soft Nano-Hexapod
+% Light payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 1; % Sample Mass [kg]
+kn = 1e4; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_vc_rigid_light = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_soft_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_stiff_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Heavy payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 50; % Sample Mass [kg]
+kn = 1e4; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_vc_rigid_heavy = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_soft_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_vc_stiff_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Effect of the payload dynamics on the soft Nano-Hexapod. Light sample on the right, and heavy sample on the left
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_vc_rigid_light, freqs, 'Hz'))), '-',  'color', colors(1,:), 'DisplayName', 'Rigid sample');
+plot(freqs, abs(squeeze(freqresp(G_vc_stiff_light, freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$\omega_s = 200\,Hz$');
+plot(freqs, abs(squeeze(freqresp(G_vc_soft_light, freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$\omega_s = 20\,Hz$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-10, 1e-2])
+title('$k_n = 0.01\,N/\mu m$, $m_s = 1\,kg$');
+
+ax2 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_vc_rigid_heavy, freqs, 'Hz'))), '-',  'color', colors(1,:), 'DisplayName', 'Rigid sample');
+plot(freqs, abs(squeeze(freqresp(G_vc_stiff_heavy, freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', '$\omega_s = 200\,Hz$');
+plot(freqs, abs(squeeze(freqresp(G_vc_soft_heavy, freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', '$\omega_s = 20\,Hz$');
+plot(freqs, abs(squeeze(freqresp(1/(mn*s^2), freqs, 'Hz'))), '-', 'color', [0,0,0,0.5], 'DisplayName', '$\frac{1}{m_n s^2}$');
+plot(freqs, abs(squeeze(freqresp(1/((mn + ms)*s^2), freqs, 'Hz'))), '--', 'color', [0,0,0,0.5], 'DisplayName', '$\frac{1}{(m_n + m_s) s^2}$');
+text(2.2, 3e-3, '$\omega_n = \sqrt{\frac{k_n}{m_n + m_s}}$', 'horizontalalignment', 'left');
+text(20, 1e-8, '$\omega_s = \sqrt{\frac{k_s}{m_s}}$', 'horizontalalignment', 'center');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+title('$k_n = 0.01\,N/\mu m$, $m_s = 50\,kg$');
+ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [20, 1];
+ylim([1e-10, 1e-2])
+
+ax1b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_rigid_light, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_stiff_light, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_soft_light, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+ax2b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_rigid_heavy, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_stiff_heavy, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_soft_heavy, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+linkaxes([ax1,ax1b],'x');
+xlim([1, 1000]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/uniaxial_payload_dynamics_soft_nano_hexapod.pdf', 'width', 'full', 'height', 'tall');
+#+end_src
+
+#+name: fig:uniaxial_payload_dynamics_soft_nano_hexapod
+#+caption: Effect of the payload dynamics on the soft Nano-Hexapod. Light sample on the right, and heavy sample on the left
+#+RESULTS:
+[[file:figs/uniaxial_payload_dynamics_soft_nano_hexapod.png]]
+
+The same transfer functions are now compared when using a stiff nano-hexapod ($k_n = 100\,N/\mu m$) in Figure ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod.
+In that case, the sample's resonance $\omega_n$ is smaller than the nano-hexapod 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 tough the added sample's flexibility still changes the high frequency mass line from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$, the overall dynamics is much less impacted, even if the sample mass is high (see yellow curve in Figure ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod, right).
+
+#+begin_src matlab :exports none
+%% Stiff Nano-Hexapod
+% Light payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 1; % Sample Mass [kg]
+kn = 1e8; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_pz_rigid_light = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_soft_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_stiff_light = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Heavy payload mass
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 50; % Sample Mass [kg]
+kn = 1e8; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid sample
+G_pz_rigid_heavy = 1/((mn + ms)*s^2 + cn*s + kn);
+
+% Soft Sample
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_soft_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+
+% Stiff Sample
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+G_pz_stiff_heavy = (ms*s^2 + cs*s + ks)/((mn*s^2 + cn*s + kn)*(ms*s^2 + cs*s + ks) + ms*s^2*(cs*s + ks));
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample on the right, and heavy sample on the left
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_pz_rigid_light, freqs, 'Hz'))), '-',  'color', colors(1,:));
+plot(freqs, abs(squeeze(freqresp(G_pz_stiff_light, freqs, 'Hz'))), '-', 'color', colors(2,:));
+plot(freqs, abs(squeeze(freqresp(G_pz_soft_light, freqs, 'Hz'))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-10, 1e-2])
+title('$k_n = 100\,N/\mu m$, $m_s = 1\,kg$');
+
+ax2 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_pz_rigid_heavy, freqs, 'Hz'))), '-',  'color', colors(1,:), 'DisplayName', 'Rigid sample');
+plot(freqs, abs(squeeze(freqresp(G_pz_stiff_heavy, freqs, 'Hz'))), '-', 'color', colors(2,:), 'DisplayName', 'Stiff sample: $\omega_s = 200\,Hz$');
+plot(freqs, abs(squeeze(freqresp(G_pz_soft_heavy, freqs, 'Hz'))), '-', 'color', colors(3,:), 'DisplayName', 'Soft sample: $\omega_s = 20\,Hz$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+title('$k_n = 100\,N/\mu m$, $m_s = 50\,kg$');
+ylim([1e-10, 1e-2])
+ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ldg.ItemTokenSize = [20, 1];
+
+ax1b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_rigid_light, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_stiff_light, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_soft_light, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+ax2b = nexttile();
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_rigid_heavy, freqs, 'Hz')))), '-',  'color', colors(1,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_stiff_heavy, freqs, 'Hz')))), '-', 'color', colors(2,:));
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_soft_heavy, freqs, 'Hz')))), '-', 'color', colors(3,:));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([1e0, 1e1, 1e2]);
+yticks(-360:90:360);
+ylim([-200, 20]);
+
+linkaxes([ax1,ax1b],'x');
+xlim([1, 1000]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/uniaxial_payload_dynamics_stiff_nano_hexapod.pdf', 'width', 'full', 'height', 'tall');
+#+end_src
+
+#+name: fig:uniaxial_payload_dynamics_stiff_nano_hexapod
+#+caption: Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample on the right, and heavy sample on the left
+#+RESULTS:
+[[file:figs/uniaxial_payload_dynamics_stiff_nano_hexapod.png]]
+
+
+#+begin_important
+The impact of the sample dynamics on the plant varies with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the nano-hexapod $\omega_n$.
+The larger the sample mass, the larger the effect (i.e. change of high frequency gain, appearance of additional resonances and anti-resonances).
+A zero/pole pattern is observed if $\omega_s > \omega_n$ and a pole/zero pattern if $\omega_s > \omega_n$.
+Such additional dynamics can induce stability issues depending on their position relative to the wanted feedback bandwidth as explained in [[cite:&rankers98_machin Section 4.2]].
+
+The general conclusion is that the stiffer the nano-hexapod, 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, ...), while stiff positioning systems can more easily accept various payloads.
+#+end_important
+
+** Impact on the positioning stability
+<<ssec:uniaxial_payload_dynamics_effect_stability>>
+
+Having a flexibility between the measured position (i.e. the top platform of the nano-hexapod) and the point-of-interest to be positioned relative to the x-ray can also impact the positioning stability.
+This is due to the fact that the measured (and therefore controlled) distance is no longer equal to the important distance for the experiment.
+
+In order to estimate if the sample flexibility is critical for the position stability of the point of interest, the model shown in Figure ref:fig:uniaxial_sample_flexibility_control is used.
+This is the same model that was used in Section ref:sec:uniaxial_position_control but with an added flexibility between the nano-hexapod and the sample (considered sample modes are at $\omega_s = 20\,\text{Hz}$ and $\omega_n = 200\,\text{Hz}$).
+
+#+begin_src latex :file uniaxial_sample_flexibility_control.pdf
+\begin{tikzpicture}
+  % Parameters
+  \def\massw{2.2}  % Width of the masses
+  \def\massh{0.8}  % Height of the masses
+  \def\spaceh{1.2} % Height of the springs/dampers
+  \def\dispw{0.4}  % Width of the dashed line for the displacement
+  \def\disph{0.3}  % Height of the arrow for the displacements
+  \def\bracs{0.05} % Brace spacing vertically
+  \def\brach{-12pt} % Brace shift horizontaly
+  \def\fsensh{0.2} % Height of the force sensor
+  \def\velsize{0.2} % Size of the velocity sensor
+
+  % Ground
+  \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
+  \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0);
+  \draw[->, color=colorpurple] (0.5*\massw+0.5*\dispw, 0) -- ++(0, 2*\disph) node[right, color=colorpurple]{$x_{f}$};
+
+  % ====================
+  % Granite
+  \begin{scope}[shift={(0, 0)}]
+    % Mass
+    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{g}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{g}$};
+    \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c_{g}$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Granite};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(9.0*\dispw, 0) coordinate(xg);
+  \end{scope}
+  % ====================
+
+
+  % ====================
+  % Stages
+  \begin{scope}[shift={(0, \spaceh+\massh)}]
+    % Mass
+    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$};
+    \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c_{t}$};
+    \draw[actuator={0.45}{0.2}{colorpurple}] ( 0.3*\massw, 0) -- (0.3*\massw, \spaceh) node[midway, right=0.1, color=colorpurple](ft){$f_{t}$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{$T_y$/$R_y$/$R_z$};
+  \end{scope}
+  % ====================
+
+
+  % ====================
+  % Hexapod
+  \begin{scope}[shift={(0, 2*(\spaceh+\massh))}]
+    % Mass
+    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_h$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_h$};
+    \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c_h$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{$\mu\text{-hexa}$};
+  \end{scope}
+  % ====================
+
+
+  % ====================
+  % NASS
+  \begin{scope}[shift={(0, 3*(\spaceh+\massh))}]
+    % Mass
+    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$};
+
+    % Force Sensor
+    \node[forcesensor={\massw}{\fsensh}{colorblue}] (fsensn) at (0, \spaceh-\fsensh){};
+    \node[right, color=colorblue] (fn) at (fsensn.east) {$f_n$};
+
+    % Spring, Damper, and Actuator
+    \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh-\fsensh) node[midway, left=0.1]{$k_{n}$};
+    \draw[damper] (0, 0)           -- ( 0, \spaceh-\fsensh)          node[midway, left=0.2]{$c_{n}$};
+    \draw[actuator={0.4}{0.2}{black}] ( 0.3*\massw, 0) -- (0.3*\massw, \spaceh-\fsensh) node[midway, right=0.1](f){$f$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{$\nu\text{-hexa}$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(7.5*\dispw, 0) coordinate(xnpos);
+  \end{scope}
+  % ====================
+
+  % ====================
+  % sample
+  \begin{scope}[shift={(0, 4*(\spaceh+\massh))}]
+    % Mass
+    \draw[fill=white] (-0.3*\massw, \spaceh) rectangle (0.3*\massw, \spaceh+\massh) coordinate[pos=0.5](poi);
+
+    % Spring, Damper, and Actuator
+    \draw[spring] (-0.2*\massw, 0) -- (-0.2*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$};
+    \draw[damper] ( 0.2*\massw, 0) -- ( 0.2*\massw, \spaceh) node[midway, left=0.2]{$c_{s}$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+    (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+    node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Sample};
+
+    % X-ray
+    \draw[dashed, draw=red, -<-=.5] (poi)node[branch, color=red]{} -- node[near end, above]{x-ray} ++(2.5, 0);
+
+    % Displacements
+    \draw[dashed] (0.2*\massw, \spaceh)coordinate(a) -- (xg|-a) coordinate(xspos);
+  \end{scope}
+  % ====================
+
+  % Measured displacement
+  \draw[<->, dashed, draw=colorred] ($(xnpos)+(-0.1, 0)$)coordinate(a) -- node[left, color=colorred](d){$d$} (xg-|a);
+  \draw[<->, dashed] ($(xg)+(-0.1, 0)$) -- node[left]{$y$} ($(xspos)+(-0.1, 0)$);
+
+  % ====================
+  % IFF Control
+  \node[block={2em}{2em}, right=1.0 of fn, color=colorblue, fill=colorblue!10!white] (iff) {$K_{\textsc{IFF}}$};
+  \node[addb, right=0.5 of f.east] (ctrladd) {};
+  \node[block={2em}{2em}, color=colorred, fill=colorred!10!white] (Khac) at (ctrladd|-d) {$K_{\textsc{HAC}}$};
+
+  \draw[->, draw=colorblue] (fn.east) -- (iff.west);
+  \draw[->, draw=colorblue] (iff.south) |- (ctrladd.east);
+  \draw[->] (ctrladd.west) -- (f.east);
+
+  \draw[->, draw=colorred] (d.west) -- (Khac.east);
+  \draw[->, draw=colorred] (Khac.north) -- (ctrladd.south) node[below right, color=colorred]{$f^{\prime}$};
+  % ====================
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:uniaxial_sample_flexibility_control
+#+caption: Uniaxial model considering a flexibility between the nano-hexapod top platform and the sample. In that case the measured and controlled distance $d$ is different from the distance $y$ that is wish to be controlled
+#+RESULTS:
+[[file:figs/uniaxial_sample_flexibility_control.png]]
+
+After the system dynamics extracted from the model, IFF is applied using the same gains as the ones used in Section ref:sec:uniaxial_active_damping.
+Thanks to the collocation between the nano-hexapod and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample.
+
+The High Authority Controllers used in Section ref:sec:uniaxial_position_control are then implemented on the damped plants.
+The obtained closed-loop systems are stable, indicating good robustness.
+
+Finally, closed-loop noise budgeting is computed for the obtained the closed-loop system and the cumulative amplitude spectrum of $d$ and $y$ are shown in Figure ref:fig:uniaxial_sample_flexibility_noise_budget_y.
+The cumulative amplitude spectrum of the measured distance $d$ (Figure ref:fig:uniaxial_sample_flexibility_noise_budget_d) shows that the added flexibility at the sample location have very little effect on the control performance.
+However, the cumulative amplitude spectrum of the distance $y$ (Figure ref:fig:uniaxial_sample_flexibility_noise_budget_y) shows that the stability of $y$ is degraded when the sample flexibility is considered and is degraded as $\omega_s$ is lowered.
+
+What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the nano-hexapod and the control on $y$ is not effective.
+
+#+begin_src matlab :exports none
+%% Nano-Hexpod model
+model_config = struct();
+model_config.controller = "open_loop";
+mn = 15; % Nano-Hexapod mass [kg]
+ms = 1; % Sample Mass [kg]
+
+%% Identification
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/controller'],       1, 'openinput');  io_i = io_i + 1; % Actuator Force
+io(io_i) = linio([mdl, '/micro_station/xf'], 1, 'openinput');  io_i = io_i + 1; % Floor Motion
+io(io_i) = linio([mdl, '/micro_station/ft'], 1, 'openinput');  io_i = io_i + 1; % Stage vibrations
+io(io_i) = linio([mdl, '/fs'],               1, 'openinput');  io_i = io_i + 1; % Direct sample forces
+io(io_i) = linio([mdl, '/dL'],               1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor
+io(io_i) = linio([mdl, '/fm'],               1, 'openoutput'); io_i = io_i + 1; % Force Sensor
+io(io_i) = linio([mdl, '/vn'] ,              1, 'openoutput'); io_i = io_i + 1; % Geophone
+io(io_i) = linio([mdl, '/d'] ,               1, 'openoutput'); io_i = io_i + 1; % Metrology Output
+io(io_i) = linio([mdl, '/y'] ,               1, 'openoutput'); io_i = io_i + 1; % Sample's position
+#+end_src
+
+#+begin_src matlab :exports none
+%% Soft Nano-Hexapod
+% Light payload mass
+kn = 1e4; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid Sample
+model_config.nhexa = "1dof";
+G_vc_light_rigid = linearize(mdl, io, 0.0);
+G_vc_light_rigid.InputName = {'f', 'xf', 'ft', 'fs'};
+G_vc_light_rigid.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Soft Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_vc_light_soft = linearize(mdl, io, 0.0);
+G_vc_light_soft.InputName = {'f', 'xf', 'ft', 'fs'};
+G_vc_light_soft.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Rigid Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_vc_light_stiff = linearize(mdl, io, 0.0);
+G_vc_light_stiff.InputName = {'f', 'xf', 'ft', 'fs'};
+G_vc_light_stiff.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+#+end_src
+
+#+begin_src matlab :exports none
+%% Stiff Nano-Hexapod
+% Light payload mass
+kn = 1e8; % Nano-Hexapod (soft) Stiffness [N/m]
+cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)]
+
+% Rigid Sample
+model_config.nhexa = "1dof";
+G_pz_light_rigid = linearize(mdl, io, 0.0);
+G_pz_light_rigid.InputName = {'f', 'xf', 'ft', 'fs'};
+G_pz_light_rigid.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Soft Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*20;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_pz_light_soft = linearize(mdl, io, 0.0);
+G_pz_light_soft.InputName = {'f', 'xf', 'ft', 'fs'};
+G_pz_light_soft.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+
+% Rigid Sample
+model_config.nhexa = "2dof";
+ws = 2*pi*200;
+ks = ms * ws^2;
+cs = 2*0.01*sqrt(ms*ks);
+
+G_pz_light_stiff = linearize(mdl, io, 0.0);
+G_pz_light_stiff.InputName = {'f', 'xf', 'ft', 'fs'};
+G_pz_light_stiff.OutputName = {'dL', 'fm', 'vn', 'd', 'y'};
+#+end_src
+
+#+begin_src matlab
+%% Apply IFF and verify stability
+% Soft Nano-Hexapod
+G_iff_vc_light_rigid = feedback(G_vc_light_rigid, K_iff_vc, 'name', +1);
+G_iff_vc_light_soft  = feedback(G_vc_light_soft , K_iff_vc, 'name', +1);
+G_iff_vc_light_stiff = feedback(G_vc_light_stiff, K_iff_vc, 'name', +1);
+
+isstable(G_iff_vc_light_rigid)
+isstable(G_iff_vc_light_soft)
+isstable(G_iff_vc_light_stiff)
+
+% Stiff Nano-Hexapod
+G_iff_pz_light_rigid = feedback(G_pz_light_rigid, K_iff_pz, 'name', +1);
+G_iff_pz_light_soft  = feedback(G_pz_light_soft , K_iff_pz, 'name', +1);
+G_iff_pz_light_stiff = feedback(G_pz_light_stiff, K_iff_pz, 'name', +1);
+
+isstable(G_iff_pz_light_rigid)
+isstable(G_iff_pz_light_soft)
+isstable(G_iff_pz_light_stiff)
+#+end_src
+
+#+begin_src matlab
+%% Compute closed-loop plants and verify stability
+% Soft Nano-Hexapod
+G_hac_iff_vc_light_rigid = feedback(G_iff_vc_light_rigid, K_hac_vc, 'name', -1);
+G_hac_iff_vc_light_soft  = feedback(G_iff_vc_light_soft , K_hac_vc, 'name', -1);
+G_hac_iff_vc_light_stiff = feedback(G_iff_vc_light_stiff, K_hac_vc, 'name', -1);
+
+isstable(G_hac_iff_vc_light_rigid)
+isstable(G_hac_iff_vc_light_soft)
+isstable(G_hac_iff_vc_light_stiff)
+
+% Stiff Nano-Hexapod
+G_hac_iff_pz_light_rigid = feedback(G_iff_pz_light_rigid, K_hac_pz, 'name', -1);
+G_hac_iff_pz_light_soft  = feedback(G_iff_pz_light_soft , K_hac_pz, 'name', -1);
+G_hac_iff_pz_light_stiff = feedback(G_iff_pz_light_stiff, K_hac_pz, 'name', -1);
+
+isstable(G_hac_iff_pz_light_rigid)
+isstable(G_hac_iff_pz_light_soft)
+isstable(G_hac_iff_pz_light_stiff)
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Cumulative Amplitude Spectrum of d - Effect of Sample's flexibility
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Rigid sample');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Stiff $\omega_s = 200\,$Hz');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('d', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Soft $\omega_s = 20\,$Hz');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+ylabel('CAS of $d$ [m]'); xlabel('Frequency [Hz]');
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+title('$k_n = 0.01\,N/\mu m$');
+
+ax2 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('d', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('d', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('d', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('d', 'xf'), f, 'Hz'))).^2)))), '-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+title('$k_n = 100\,N/\mu m$');
+xlim([1, 500]);
+
+linkaxes([ax1,ax2],'xy');
+xlim([1, 500]);
+ylim([2e-10, 2e-7])
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file raw silent
+exportFig('figs/uniaxial_sample_flexibility_noise_budget_d.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Cumulative Amplitude Spectrum - Effect of Sample's flexibility
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_rigid('y', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Rigid sample');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_stiff('y', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Stiff $\omega_s = 200\,$Hz');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_vc_light_soft('y', 'xf'), f, 'Hz'))).^2)))), '-', ...
+     'DisplayName', 'Soft $\omega_s = 20\,$Hz');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+ylabel('CAS of $y$ [m]'); xlabel('Frequency [Hz]');
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+title('$k_n = 0.01\,N/\mu m$');
+
+ax2 = nexttile();
+hold on;
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_rigid('y', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_stiff('y', 'xf'), f, 'Hz'))).^2)))), '-');
+plot(f, sqrt(flip(-cumtrapz(flip(f), flip(psd_ft.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('y', 'ft'), f, 'Hz'))).^2 + ...
+                                     psd_xf.*abs(squeeze(freqresp(G_hac_iff_pz_light_soft('y', 'xf'), f, 'Hz'))).^2)))), '-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([1e0, 1e1, 1e2]);
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+title('$k_n = 100\,N/\mu m$');
+xlim([1, 500]);
+
+linkaxes([ax1,ax2],'xy');
+xlim([1, 500]);
+ylim([2e-10, 2e-7])
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file raw silent
+exportFig('figs/uniaxial_sample_flexibility_noise_budget_y.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:uniaxial_sample_flexibility_noise_budget
+#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affects much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ is used for the simulations.
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$}
+#+attr_latex: :options {0.95\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_sample_flexibility_noise_budget_d.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_y}Cumulative Amplitude Spectrum of $y$}
+#+attr_latex: :options {0.95\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/uniaxial_sample_flexibility_noise_budget_y.png]]
+#+end_subfigure
+#+end_figure
+
+** Conclusion
+#+begin_important
+Payload dynamics is usually a major concern when designing a positioning system.
+For the NASS, the sample may present internal dynamics and limited fixation stiffness to the nano-hexapod platform.
+On top of that, its mass can vary from 1kg up to 50kg which is a first challenge.
+
+This sample dynamics can add pole/zero or zero/pole pattern in the plant as shown in Section ref:ssec:uniaxial_payload_dynamics_effect_dynamics which may impact the stability and makes the design of the controller more complex.
+However, it was shown that this issue is less critical if a stiff nano-hexapod is used.
+
+Having some flexibility between the measurement point and the point of interest (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.
+
+It will be therefore important to take special care when designing sampling environments, especially is a soft nano-hexapod is used.
+#+end_important
+
 * Conclusion
 <<sec:conclusion>>
 
@@ -4179,7 +5605,12 @@ Finally, position feedback controllers have been developed for three considered
 These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances.
 It has been found that having a soft nano-hexapod makes the plant dynamics easier to control (because decoupled from the micro-station dynamics) and requires less position feedback bandwidth to fulfill the requirements.
 The moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$) is requiring a bit more position feedback bandwidth, but it still seems to give acceptable results.
-However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performances.
+However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performance.
+
+# TODO - Add summary table of advantages and disadvantages of nano-hexapod stiffnesses
+
+* Bibliography                                                        :ignore:
+#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
 
 * Helping Functions                                                 :noexport:
 ** Initialize Path
diff --git a/nass-uniaxial-model.pdf b/nass-uniaxial-model.pdf
index 4226c81339ee906b43832e51a87b2055a9da51bd..86b6b4b01f6871e0bd57f2ca5bba063f89818d1a 100644
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diff --git a/nass-uniaxial-model.tex b/nass-uniaxial-model.tex
index 3e5af7f..9d789e9 100644
--- a/nass-uniaxial-model.tex
+++ b/nass-uniaxial-model.tex
@@ -1,32 +1,15 @@
-% Created 2023-02-20 Mon 10:57
+% Created 2023-06-30 Fri 20:00
 % Intended LaTeX compiler: pdflatex
-\documentclass[a4paper, 10pt, DIV=12, parskip=full]{scrreprt}
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{graphicx}
-\usepackage{grffile}
-\usepackage{longtable}
-\usepackage{wrapfig}
-\usepackage{rotating}
-\usepackage[normalem]{ulem}
-\usepackage{amsmath}
-\usepackage{textcomp}
-\usepackage{amssymb}
-\usepackage{capt-of}
-\usepackage{hyperref}
-\usepackage[most]{tcolorbox}
-\usepackage{bm}
-\usepackage{booktabs}
-\usepackage{tabularx}
-\usepackage{array}
-\usepackage{siunitx}
+\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
+
 \input{preamble.tex}
+\bibliography{nass-uniaxial-model.bib}
 \author{Dehaeze Thomas}
 \date{\today}
-\title{NASS - Uniaxial Model}
+\title{Nano Active Stabilization System - Uniaxial Model}
 \hypersetup{
  pdfauthor={Dehaeze Thomas},
- pdftitle={NASS - Uniaxial Model},
+ pdftitle={Nano Active Stabilization System - Uniaxial Model},
  pdfkeywords={},
  pdfsubject={},
  pdfcreator={Emacs 28.2 (Org mode 9.5.2)}, 
@@ -48,21 +31,23 @@ In order to have a relevant model, the micro-station dynamics is first identifie
 Then, a model of the nano-hexapod is added on top of the micro-station.
 With added sample and sensors, this gives a uniaxial dynamical model of the NASS that will be used for further analysis (Section \ref{sec:nano_station_model}).
 
-The disturbances affecting the position accuracy are identified experimentally (Section \ref{sec:uniaxial_disturbances}) and included in the model for dynamical noise budgeting (Section \ref{sec:uniaxial_noise_budgeting}).
-In all the following analysis, there nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate stiffness.
+The disturbances affecting the position stability are identified experimentally (Section \ref{sec:uniaxial_disturbances}) and included in the model for dynamical noise budgeting (Section \ref{sec:uniaxial_noise_budgeting}).
+In all the following analysis, there nano-hexapod stiffnesses are considered to better understand the trade-offs and to find the most adequate nano-hexapod design.
 Three sample masses are also considered to verify the robustness of the applied control strategies to a change of sample.
 
 Three active damping techniques are then applied on the nano-hexapod.
 This helps to reduce the effect of disturbances as well as render the system easier to control afterwards (Section \ref{sec:uniaxial_active_damping}).
 
-Once the system is well damped, a feedback position controller is applied, and the obtained performances are compared (Section \ref{sec:uniaxial_position_control}).
+Once the system is well damped, a feedback position controller is applied, and the obtained performance are compared (Section \ref{sec:uniaxial_position_control}).
+
+Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section \ref{sec:uniaxial_support_compliance}) and the presence of dynamics between the nano-hexapod and the sample's point of interest (Section \ref{sec:uniaxial_payload_dynamics}).
 
 Conclusion remarks are given in Section \ref{sec:conclusion}.
 
 \begin{table}[htbp]
 \caption{\label{tab:section_matlab_code}Report sections and corresponding Matlab files}
 \centering
-\begin{tabularx}{0.5\linewidth}{lX}
+\begin{tabularx}{0.6\linewidth}{lX}
 \toprule
 \textbf{Sections} & \textbf{Matlab File}\\
 \midrule
@@ -72,6 +57,8 @@ Section \ref{sec:uniaxial_disturbances} & \texttt{uniaxial\_3\_disturbances.m}\\
 Section \ref{sec:uniaxial_noise_budgeting} & \texttt{uniaxial\_4\_dynamic\_noise\_budget.m}\\
 Section \ref{sec:uniaxial_active_damping} & \texttt{uniaxial\_5\_active\_damping.m}\\
 Section \ref{sec:uniaxial_position_control} & \texttt{uniaxial\_6\_hac\_lac.m}\\
+Section \ref{sec:uniaxial_support_compliance} & \texttt{uniaxial\_7\_support\_compliance}\\
+Section \ref{sec:uniaxial_payload_dynamics} & \texttt{uniaxial\_8\_payload\_dynamics}\\
 \bottomrule
 \end{tabularx}
 \end{table}
@@ -79,11 +66,10 @@ Section \ref{sec:uniaxial_position_control} & \texttt{uniaxial\_6\_hac\_lac.m}\\
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/uniaxial_overview_model_sections.png}
-\caption{\label{fig:uniaxial_overview_model_sections}Uniaxial Micro-Station model in blue (Section \ref{sec:micro_station_model}), Nano-Hexapod and sample models in red (Section \ref{sec:nano_station_model}), Disturbances in yellow (Section \ref{sec:uniaxial_disturbances}), Active Damping in green (Section \ref{sec:uniaxial_active_damping}) and Position control in purple (Section \ref{sec:uniaxial_position_control})}
+\caption{\label{fig:uniaxial_overview_model_sections}Uniaxial Micro-Station model in blue (Section \ref{sec:micro_station_model}), Nano-Hexapod models in red (Section \ref{sec:nano_station_model}), Disturbances in yellow (Section \ref{sec:uniaxial_disturbances}), Active Damping in green (Section \ref{sec:uniaxial_active_damping}), Position control in purple (Section \ref{sec:uniaxial_position_control}) and Sample dynamics in cyan (Section \ref{sec:uniaxial_payload_dynamics})}
 \end{figure}
 
 \chapter{Micro Station Model}
-\label{sec:org4e07986}
 \label{sec:micro_station_model}
 In this section, a uni-axial model of the micro-station is tuned in order to match measurements made on the micro-station
 The measurement setup is shown in Figure \ref{fig:micro_station_first_meas_dynamics} where several geophones are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station.
@@ -96,7 +82,6 @@ From the measured frequency response functions (FRF), the model can be tuned to
 \caption{\label{fig:micro_station_first_meas_dynamics}Experimental Setup for the first dynamical measurements on the Micro-Station. Geophones are fixed to the micro-station, and the granite as well as the micro-hexapod's top platform are impact with an instrumented hammer}
 \end{figure}
 \section{Measured dynamics}
-\label{sec:org525a6c0}
 
 The measurement setup is schematically shown in Figure \ref{fig:micro_station_meas_dynamics_schematic} where:
 \begin{itemize}
@@ -111,28 +96,26 @@ From the forces applied by the instrumented hammer and the responses of the geop
 \item from \(F_g\) to \(d_g\)
 \end{itemize}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/micro_station_meas_dynamics_schematic.png}
-\caption{\label{fig:micro_station_meas_dynamics_schematic}Measurement setup - Schematic}
+\begin{figure}
+\begin{subfigure}{0.69\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/micro_station_meas_dynamics_schematic.png}
+\end{center}
+\subcaption{\label{fig:micro_station_meas_dynamics_schematic}Measurement setup - Schematic}
+\end{subfigure}
+\begin{subfigure}{0.29\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_model_micro_station.png}
+\end{center}
+\subcaption{\label{fig:uniaxial_model_micro_station}Uniaxial model of the micro-station}
+\end{subfigure}
+\caption{\label{fig:micro_station_uniaxial_model}Schematic of the Micro-Station measurement setup and uniaxial model.}
 \end{figure}
 
-Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure \ref{fig:uniaxial_measured_frf_vertical}).
-
-\begin{minted}[]{matlab}
-%% Load measured FRF
-load('meas_microstation_frf.mat');
-\end{minted}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/uniaxial_measured_frf_vertical.png}
-\caption{\label{fig:uniaxial_measured_frf_vertical}Measured Frequency Response Functions in the vertical direction}
-\end{figure}
+Due to the bad coherence at low frequency, the frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure \ref{fig:uniaxial_comp_frf_meas_model}).
 
 \section{Uniaxial Model}
-\label{sec:orge223677}
-The uni-axial model of the micro-station is shown in Figure \ref{fig:uniaxial_comp_frf_meas_model}, with:
+The uni-axial model of the micro-station is shown in Figure \ref{fig:uniaxial_model_micro_station}, with:
 \begin{itemize}
 \item Disturbances:
 \begin{itemize}
@@ -143,39 +126,13 @@ The uni-axial model of the micro-station is shown in Figure \ref{fig:uniaxial_co
 \item Geophones: \(x_h\) and \(x_g\)
 \end{itemize}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/uniaxial_model_micro_station.png}
-\caption{\label{fig:uniaxial_model_micro_station}Uniaxial model of the micro-station}
-\end{figure}
-
 Masses are estimated from the CAD.
-\begin{minted}[]{matlab}
-%% Parameters - Mass
-mh = 15;   % Micro Hexapod [kg]
-mt = 1200; % Ty + Ry + Rz [kg]
-mg = 2500; % Granite [kg]
-\end{minted}
-
 And stiffnesses from the data-sheet of stage manufacturers.
-\begin{minted}[]{matlab}
-%% Parameters - Stiffnesses
-kh = 6.11e+07; % [N/m]
-kt = 5.19e+08; % [N/m]
-kg = 9.50e+08; % [N/m]
-\end{minted}
-
 The damping coefficients are tuned to match the identified damping from the measurements.
-\begin{minted}[]{matlab}
-%% Parameters - damping
-ch = 2*0.05*sqrt(kh*mh); % [N/(m/s)]
-ct = 2*0.05*sqrt(kt*mt); % [N/(m/s)]
-cg = 2*0.08*sqrt(kg*mg); % [N/(m/s)]
-\end{minted}
-
 \section{Comparison of the model and measurements}
-\label{sec:orgfa10aaa}
-The comparison between the measurements and the model is done in Figure \ref{fig:uniaxial_comp_frf_meas_model}.
+The comparison between the measurements and the model is shown in Figure \ref{fig:uniaxial_comp_frf_meas_model}.
+
+Only three modes are modelled with frequencies at 70Hz, 140Hz and 320Hz.
 
 As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation.
 More accurate models will be used later on.
@@ -187,14 +144,13 @@ More accurate models will be used later on.
 \end{figure}
 
 \chapter{Nano-Hexapod Model}
-\label{sec:org5ef4103}
 \label{sec:nano_station_model}
 A model of the nano-hexapod and sample is now added on top of the uni-axial model of the micro-station (Figure \ref{fig:uniaxial_model_micro_station-nass}).
 
 Disturbances (shown in red) are:
 \begin{itemize}
 \item \(f_s\): direct forces applied to the sample (for instance cable forces)
-\item \(f_t\): disturbances coming from the imperfect stage scanning performances
+\item \(f_t\): disturbances coming from the imperfect stage scanning performance
 \item \(x_f\): floor motion
 \end{itemize}
 
@@ -209,7 +165,6 @@ The effect of having resonances between the sample's point of interest and the n
 \caption{\label{fig:uniaxial_model_micro_station-nass}Uni-axial model of the micro-station with added nano-hexapod (represented in blue) and sample (represented in green)}
 \end{figure}
 \section{Nano-Hexapod Parameters}
-\label{sec:orge14d0af}
 The parameters for the nano-hexapod and sample are:
 \begin{itemize}
 \item \(m_s\) the sample mass that can vary from 1kg up to 50kg
@@ -219,27 +174,18 @@ The parameters for the nano-hexapod and sample are:
 
 As a first example, let's choose a nano-hexapod stiffness of \(10\,N/\mu m\) and a sample mass of 10kg.
 
-\begin{minted}[]{matlab}
-%% Nano-Hexapod Parameters
-mn = 15; % [kg]
-kn = 1e7; % [N/m]
-cn = 2*0.01*sqrt(mn*kn); % [N/(m/s)]
-
-%% Sample Mass
-ms = 10; % [kg]
-\end{minted}
-
-\section{Obtained Dynamics}
-\label{sec:orgddb8cf5}
+\section{Obtained Dynamic Response}
 The sensitivity to disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) 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}.
 
+\begin{important}
 For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) combined with three sample's masses (\(m_s = 1\,kg\), \(m_s = 25\,kg\) and \(m_s = 50\,kg\)).
+\end{important}
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/uniaxial_sensitivity_dist_first_params.png}
-\caption{\label{fig:uniaxial_sensitivity_dist_first_params}Sensitivity to disturbances}
+\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, \(f_t\) disturbances from the micro-station stages and \(x_f\) the floor motion (from left to right)}
 \end{figure}
 
 \begin{figure}[htbp]
@@ -248,7 +194,6 @@ For further analysis, 9 configurations are considered: three nano-hexapod stiffn
 \caption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces to measured displacement by the metrology}
 \end{figure}
 \chapter{Disturbance Identification}
-\label{sec:org2036233}
 \label{sec:uniaxial_disturbances}
 In order to measure disturbances, two geophones are used, on located on the floor and on on the micro-hexapod's top platform (see Figure \ref{fig:micro_station_meas_disturbances}).
 
@@ -266,42 +211,11 @@ The geophone on the floor is used to measured the floor motion \(x_f\) while the
 \caption{\label{fig:micro_station_dynamical_id_setup}Two geophones are used to measure the micro-station vibrations induced by the scanning of the \(T_y\) and \(R_z\) stages}
 \end{figure}
 \section{Ground Motion}
-\label{sec:org931b2ea}
 The geophone fixed to the floor to measure the floor motion.
 
-\begin{minted}[]{matlab}
-%% Load floor motion data
-% t: time in [s]
-% V: measured voltage genrated by the geophone and amplified by a 60dB gain voltage amplifier [V]
-load('ground_motion_measurement.mat', 't', 'V');
-\end{minted}
-
 The voltage generated by each geophone is amplified using a voltage amplifier (gain of 60dB) before going to the ADC.
 The sensitivity of the geophone as well as the gain of the voltage amplifier are then taken into account to reconstruct the floor displacement.
-\begin{minted}[]{matlab}
-%% Sensitivity of the geophone
-S0 = 88; % Sensitivity [V/(m/s)]
-f0 = 2; % Cut-off frequency [Hz]
-
-S = S0*(s/2/pi/f0)/(1+s/2/pi/f0); % Geophone's transfer function [V/(m/s)]
-
-%% Gain of the voltage amplifier
-G0_db = 60; % [dB]
-G0 = 10^(G0_db/20); % [abs]
-
-%% Transfer function from measured voltage to displacement
-G_geo = 1/S/G0/s; % [m/V]
-\end{minted}
-
 The PSD \(S_{V_f}\) of the measured voltage \(V_f\) is computed.
-\begin{minted}[]{matlab}
-%% Compute measured voltage PSD
-Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
-win = hanning(ceil(2*Fs)); % Hanning window
-
-[psd_V, f] = pwelch(V, win, [], [], Fs); % [V^2/Hz]
-\end{minted}
-
 The PSD of the corresponding displacement can be computed as follows:
 \begin{equation}
 S_{x_f}(\omega) = \frac{S_{V_f}(\omega)}{|S_{\text{geo}}(j\omega)| \cdot G_{\text{amp}} \cdot \omega}
@@ -313,49 +227,25 @@ with:
 \item \(\omega\) is here to integrate and have the displacement instead of the velocity
 \end{itemize}
 
-\begin{minted}[]{matlab}
-%% Ground Motion ASD
-psd_xf = psd_V.*abs(squeeze(freqresp(G_geo, f, 'Hz'))).^2; % [m^2/Hz]
-\end{minted}
-
 The amplitude spectral density \(\Gamma_{x_f}\) of the measured displacement \(x_f\) is shown in Figure \ref{fig:asd_floor_motion_id31}.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/asd_floor_motion_id31.png}
-\caption{\label{fig:asd_floor_motion_id31}Measured Amplitude Spectral Density of the Floor motion on ID31}
+\caption{\label{fig:asd_floor_motion_id31}Amplitude Spectral Density of the measured Floor motion on ID31}
 \end{figure}
 
 \section{Stage Vibration}
-\label{sec:orga7b3779}
 During Spindle rotation (here at 6rpm), the granite velocity and micro-hexapod's top platform velocity are measured with the geophones.
 
-\begin{minted}[]{matlab}
-%% Measured velocity of granite and hexapod during spindle rotation
-% t: time in [s]
-% vg: measured granite velocity [m/s]
-% vg: measured micro-hexapod's top platform velocity [m/s]
-load('meas_spindle_on.mat', 't', 'vg', 'vh');
-spindle_off = load('meas_spindle_off.mat', 't', 'vg', 'vh'); % No Rotation
-\end{minted}
-
 The Power Spectral Density of the relative velocity between the hexapod and the granite is computed.
-\begin{minted}[]{matlab}
-%% Compute Power Spectral Density of the relative velocity between granite and hexapod during spindle rotation
-Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
-win = hanning(ceil(2*Fs)); % Hanning window
-
-[psd_vft, f] = pwelch(vh-vg, win, [], [], Fs); % [(m/s)^2/Hz]
-[psd_off, ~] = pwelch(spindle_off.vh-spindle_off.vg, win, [], [], Fs); % [(m/s)^2/Hz]
-\end{minted}
-
 It is then integrated to obtain the Amplitude Spectral Density of the relative motion which is compared with a non-rotating case (Figure \ref{fig:asd_vibration_spindle_rotation}).
 It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/asd_vibration_spindle_rotation.png}
-\caption{\label{fig:asd_vibration_spindle_rotation}Measured Amplitude Spectral Density of the relative motion between the granite and the micro-hexapod's top platform during Spindle rotating}
+\caption{\label{fig:asd_vibration_spindle_rotation}Amplitude Spectral Density of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating}
 \end{figure}
 
 In order to compute the equivalent disturbance force \(f_t\) that induces such motion, the transfer function from \(f_t\) to the relative motion of the hexapod's top platform and the granite is extracted from the model.
@@ -376,18 +266,16 @@ The obtained amplitude spectral density of the disturbance force \(f_t\) is show
 \caption{\label{fig:asd_disturbance_force}Estimated disturbance force ft from measurement and uniaxial model}
 \end{figure}
 
-The vibrations induced by the \(T_y\) stage are not considered here as:
+The vibrations induced by the \(T_y\) stage are not considered here because:
 \begin{itemize}
 \item the induced vibrations have less amplitude than the vibrations induced by the \(R_z\) stage
 \item it can be scanned at lower velocities if the induced vibrations are an issue
 \end{itemize}
 
 \chapter{Open-Loop Dynamic Noise Budgeting}
-\label{sec:org11bfa6e}
 \label{sec:uniaxial_noise_budgeting}
 Now that we have a model of the NASS and an estimation of the power spectral density of the disturbances, it is possible to perform an \emph{open-loop dynamic noise budgeting}.
 \section{Sensitivity to disturbances}
-\label{sec:org469c500}
 From the Uni-axial model, the transfer function from the disturbances (\(f_s\), \(x_f\) and \(f_t\)) to the displacement \(d\) are computed.
 
 This is done for \textbf{two extreme sample masses} \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and \textbf{three nano-hexapod stiffnesses}:
@@ -400,11 +288,11 @@ This is done for \textbf{two extreme sample masses} \(m_s = 1\,\text{kg}\) and \
 The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses} for the light sample (same conclusions can be drawn with the heavy one).
 
 \begin{important}
-From Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}, following can be concluded for the \textbf{soft nano-hexapod}:
+From Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}, the following can be concluded for the \textbf{soft nano-hexapod}:
 \begin{itemize}
 \item It is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to the lower stiffness
 \item Between the suspension mode of the nano-hexapod (here at 5Hz) and the first mode of the micro-station (here at 70Hz), the disturbances induced by the stage vibrations are filtered out.
-\item Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is almost \(1\).
+\item Above the suspension mode of the nano-hexapod, the sample's motion is unaffected by the floor motion, and therefore the sensitivity to floor motion is close to \(1\).
 \end{itemize}
 \end{important}
 
@@ -415,7 +303,6 @@ From Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}
 \end{figure}
 
 \section{Open-Loop Dynamic Noise Budgeting}
-\label{sec:org99b18ec}
 Now, the power spectral density of the disturbances is taken into account to estimate the residual motion \(d\) in each case.
 
 The Cumulative Amplitude Spectrum of the relative motion \(d\) due to both the floor motion \(x_f\) and the stage vibrations \(f_t\) are shown in Figure \ref{fig:uniaxial_cas_d_disturbances_stiffnesses} for the three nano-hexapod stiffnesses.
@@ -438,30 +325,31 @@ The conclusion is that the sample's mass has little effect on the cumulative amp
 \end{figure}
 
 \section{Conclusion}
-\label{sec:org6160aae}
+
 \begin{important}
-The conclusion is that in order to have a closed-loop residual vibration \(d \approx 20\,nm\text{ rms}\), if a simple feedback controller is used, the required closed-loop bandwidth would be:
+In order to have a closed-loop residual vibration \(d \approx 20\,nm\text{ rms}\), if a simple feedback controller is used, the required closed-loop bandwidth would be:
 \begin{itemize}
 \item \(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\))
 \item \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\))
 \item \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\))
 \end{itemize}
 
-This can be explain by the fact that above the suspension mode of the nano-hexapod, the stage vibrations are filtered out (see Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}).
+This can be explained by the fact that for frequencies larger than the suspension mode of the nano-hexapod, the stage vibrations are filtered out (see Figure \ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses}).
 
 This gives a first advantage to having a soft nano-hexapod.
 \end{important}
 
 \chapter{Active Damping}
-\label{sec:orgc25f9e5}
 \label{sec:uniaxial_active_damping}
-In this section, three active damping are applied on the nano-hexapod (see Figure \ref{fig:uniaxial_active_damping_strategies}): Integral Force Feedback (IFF), Relative Damping Control (RDC) and Direct Velocity Feedback (DVF).
+In this section, three active damping are applied on the nano-hexapod (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 will be described in Section \ref{ssec:uniaxial_active_damping_strategies}.
 
 These active damping techniques are compared in terms of:
 \begin{itemize}
-\item Reduction of the effect of disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) on the displacement \(d\)
-\item Achievable damping
-\item Robustness to a change of sample's mass
+\item Achievable damping of the nano-hexapod mode (Section \ref{ssec:uniaxial_active_damping_achievable_damping})
+\item 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} and \ref{ssec:uniaxial_active_damping_noise_budget})
+\item Robustness to a change of sample's mass (Section \ref{ssec:uniaxial_active_damping_robustness})
 \end{itemize}
 
 \begin{figure}[htbp]
@@ -470,13 +358,13 @@ These active damping techniques are compared in terms of:
 \caption{\label{fig:uniaxial_active_damping_strategies}Three active damping strategies: Integral Force Feedback (IFF) using a force sensor, Relative Damping Control (RDC) using a relative displacement sensor, and Direct Velocity Feedback (DVF) using a geophone}
 \end{figure}
 \section{Active Damping Strategies}
-\label{sec:org224d0f3}
-
+\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_equiv}, left).
 
 The control strategy consists of integrating the measured force and feeding it back to the actuator:
 \begin{equation}
-K_{\text{IFF}}(s) = \frac{g}{s}
+\boxed{K_{\text{IFF}}(s) = \frac{g}{s}}
 \end{equation}
 
 The mechanical equivalent of this strategy is to add a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain \(k/g\) (see Figure \ref{fig:uniaxial_active_damping_iff_equiv}, right).
@@ -488,11 +376,12 @@ The mechanical equivalent of this strategy is to add a dashpot in series with th
 \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_equiv}, left).
 
 The derivative of this relative motion is used for the feedback signal:
 \begin{equation}
-K_{\text{RDC}}(s) = - g \cdot s
+\boxed{K_{\text{RDC}}(s) = - g \cdot s}
 \end{equation}
 
 The mechanical equivalent is to add a dashpot in parallel with the actuator with a damping coefficient equal to the controller gain \(g\) (see Figure \ref{fig:uniaxial_active_damping_rdc_equiv}, right).
@@ -503,11 +392,12 @@ The mechanical equivalent is to add a dashpot in parallel with the actuator with
 \caption{\label{fig:uniaxial_active_damping_rdc_equiv}Relative Damping Control is equivalent as adding a damper in parallel with the actuator/relative motion sensor}
 \end{figure}
 
+\paragraph{Direct Velocity Feedback (DVF)}
 Finally, the Direct Velocity Feedback strategy consists of using an inertial sensor (usually a geophone), that measured the ``absolute'' velocity of the body fixed on top of the actuator (se Figure \ref{fig:uniaxial_active_damping_dvf_equiv}, left).
 
 The measured velocity is then fed back to the actuator:
 \begin{equation}
-K_{\text{DVF}}(s) = - g
+\boxed{K_{\text{DVF}}(s) = - g}
 \end{equation}
 
 This is equivalent as to fix a dashpot (with a damping coefficient equal to the controller gain \(g\)) between the body (one which the inertial sensor is fixed) and an inertial reference frame (see Figure \ref{fig:uniaxial_active_damping_dvf_equiv}, right).
@@ -520,13 +410,13 @@ This is usually refers to as ``\emph{sky hook damper}''.
 \end{figure}
 
 \section{Plant Dynamics for Active Damping}
-\label{sec:orga11bdaf}
 The plant dynamics for all three active damping techniques are shown in Figure \ref{fig:uniaxial_plant_active_damping_techniques}.
-All have \textbf{alternating poles and zeros} meaning that the phase is bounded to \(\pm 90\,\text{deg}\) which makes the controller very robust.
+All have \textbf{alternating poles and zeros} meaning that the phase do not vary by more than \(\pm 90\,\text{deg}\) which makes the design of a robust controller very easy.
+The reason all three plants in Figure \ref{fig:uniaxial_plant_active_damping_techniques} have alternating poles and zeros is because the three sensors are all collocated with the actuator \cite[Chapter 7]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
 
-When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure \ref{fig:uniaxial_plant_active_damping_techniques}), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.
+When the nano-hexapod's suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure \ref{fig:uniaxial_plant_active_damping_techniques}), the resonances of the micro-stations have little impact on the IFF and DVF transfer functions.
 
-For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure \ref{fig:uniaxial_plant_active_damping_techniques}.
+For the stiff nano-hexapod (yellow curves), the micro-station dynamics can be seen on the transfer functions in Figure \ref{fig:uniaxial_plant_active_damping_techniques}.
 Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.
 
 \begin{figure}[htbp]
@@ -536,29 +426,36 @@ Therefore, it is expected that the micro-station dynamics might impact the achie
 \end{figure}
 
 \section{Achievable Damping - Root Locus}
-\label{sec:org682230b}
+\label{ssec:uniaxial_active_damping_achievable_damping}
 The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques.
 They are shown in Figure \ref{fig:uniaxial_root_locus_damping_techniques}.
 
 All three active damping approach can lead to \textbf{critical damping} of the nano-hexapod suspension mode.
 
-There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques}, right) and with RDC for a soft nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}).
-This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).
+There is even some damping authority on micro-station modes in the following cases:
+\begin{description}
+\item[{IFF with a stiff nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques}, right)}] This can be understood from the mechanical equivalent of IFF shown in Figure \ref{fig:uniaxial_active_damping_iff_equiv} (right) considering an high stiffness \(k\).
+The micro-station top platform is connected to an inertial mass (the nano-hexapod) through a damper, which damps the micro-station suspension suspension mode.
+\item[{DVF with a stiff nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques}, right)}] In that case, the ``sky hook damper'' (see mechanical equivalent of IFF in Figure \ref{fig:uniaxial_active_damping_dvf_equiv}, right) is connected to the micro-station top platform through the stiff nano-hexapod.
+\item[{RDC with a soft nano-hexapod (Figure \ref{fig:uniaxial_root_locus_damping_techniques_micro_station_mode})}] At the frequency of the micro-station mode, the nano-hexapod top mass is behaving as an inertial reference as the suspension mode of the soft nano-hexapod is at much lower frequency.
+The micro-station and the nano-hexapod masses are connected through a large damper induced by RDC (see mechanical equivalent in Figure \ref{fig:uniaxial_active_damping_rdc_equiv}, right) which allows some damping of the micro-station.
+\end{description}
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/uniaxial_root_locus_damping_techniques.png}
-\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses.}
+\caption{\label{fig:uniaxial_root_locus_damping_techniques}Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for three nano-hexapod stiffnesses. The Root Loci are zoomed on the suspension mode of the nano-hexapod.}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png}
-\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.}
+\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.}
 \end{figure}
 
 \section{Change of sensitivity to disturbances}
-\label{sec:org240dd20}
+\label{ssec:uniaxial_active_damping_sensitivity_disturbances}
+
 The sensitivity to disturbances (direct forces \(f_s\), stage vibrations \(f_t\) and floor motion \(x_f\)) for all three active damping techniques are compared in Figure \ref{fig:uniaxial_sensitivity_dist_active_damping}.
 The comparison is done with the nano-hexapod having a stiffness \(k_n = 1\,N/\mu m\).
 
@@ -578,11 +475,11 @@ Conclusions from Figure \ref{fig:uniaxial_sensitivity_dist_active_damping} are:
 \end{figure}
 
 \section{Noise Budgeting after Active Damping}
-\label{sec:org59aa5fa}
+\label{ssec:uniaxial_active_damping_noise_budget}
 Cumulative Amplitude Spectrum of the distance \(d\) with all three active damping techniques are compared in Figure \ref{fig:uniaxial_cas_active_damping}.
 All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).
 
-Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
+Compared to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.
 
 \begin{figure}[htbp]
 \centering
@@ -590,8 +487,7 @@ Compare to the open-loop case, the active damping helps to lower the vibrations
 \caption{\label{fig:uniaxial_cas_active_damping}Comparison of the cumulative amplitude spectrum (CAS) of the distance \(d\) for all three active damping techniques (OL in black, IFF in blue, RDC in red and DVF in yellow).}
 \end{figure}
 
-\section{Obtained Damped Plant}
-\label{sec:org534205b}
+\section{Obtained Closed Loop Response}
 The transfer functions from the plant input \(f\) to the relative displacement \(d\) while the active damping is implemented are shown in Figure \ref{fig:uniaxial_damped_plant_three_active_damping_techniques}.
 All three active damping techniques yield similar damped plants.
 
@@ -609,7 +505,8 @@ The damped plants are shown in Figure \ref{fig:uniaxial_damped_plant_change_samp
 \end{figure}
 
 \section{Robustness to change of payload's mass}
-\label{sec:org1194b7f}
+\label{ssec:uniaxial_active_damping_robustness}
+
 The Root Locus for the three damping techniques are shown in Figure \ref{fig:uniaxial_active_damping_robustness_mass_root_locus} for three sample's mass (1kg, 25kg and 50kg).
 The closed-loop poles are shown by the squares for a specific gain.
 
@@ -622,7 +519,6 @@ We can see that having heavier samples yields larger damping for IFF and smaller
 \end{figure}
 
 \section{Conclusion}
-\label{sec:orgd9c89ae}
 
 \begin{important}
 Conclusions for Active Damping:
@@ -637,7 +533,7 @@ Conclusions for Active Damping:
 \caption{\label{tab:comp_active_damping}Comparison of active damping strategies}
 \centering
 \scriptsize
-\begin{tabularx}{\linewidth}{Xccc}
+\begin{tabularx}{0.9\linewidth}{Xccc}
 \toprule
  & \textbf{IFF} & \textbf{RDC} & \textbf{DVF}\\
 \midrule
@@ -655,17 +551,16 @@ Conclusions for Active Damping:
 \end{table}
 
 \chapter{Position Feedback Controller}
-\label{sec:orgb8f8fbb}
 \label{sec:uniaxial_position_control}
 The High Authority Control - Low Authority Control (HAC-LAC) architecture is shown in Figure \ref{fig:uniaxial_hac_lac_architecture}.
 It corresponds to a \emph{two step} control strategy:
 \begin{itemize}
 \item First, an active damping controller \(\bm{K}_{\textsc{LAC}}\) is implemented (see Section \ref{sec:uniaxial_active_damping}).
 It allows to reduce the vibration level, and it also makes the damped plant (transfer function from \(u^{\prime}\) to \(y\)) easier to control than the undamped plant (transfer function from \(u\) to \(y\)).
-\item Then, a position controller \(\bm{K}_{\textsc{HAC}}\) is implemented.
+\item Then, a position controller \(\bm{K}_{\textsc{HAC}}\) is implemented which is used to control the position \(d\).
 \end{itemize}
 
-Combined with the uniaxial model, it is shown in Figure \ref{fig:uniaxial_hac_lac_model}.
+When applied on the uniaxial model with IFF used as the Low Authority Control, the schematic shown in Figure \ref{fig:uniaxial_hac_lac_model} is obtained.
 
 \begin{figure}
 \begin{subfigure}{0.54\textwidth}
@@ -678,16 +573,22 @@ Combined with the uniaxial model, it is shown in Figure \ref{fig:uniaxial_hac_la
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/uniaxial_hac_lac_model.png}
 \end{center}
-\subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-LAC strategy}
+\subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-IFF strategy}
 \end{subfigure}
 \caption{\label{fig:uniaxial_hac_lac}High Authority Control - Low Authority Control (HAC-LAC)}
 \end{figure}
 \section{Damped Plant Dynamics}
-\label{sec:orge78415e}
+\label{ssec:uniaxial_position_control_damped_dynamics}
+
 As was shown in Section \ref{sec:uniaxial_active_damping}, all three proposed active damping techniques yield similar damping plants.
-Therefore, \emph{Integral Force Feedback} will be used in this section to study the HAC-LAC performances.
+Therefore, \emph{Integral Force Feedback} will be used in this section to study the HAC-LAC performance.
 
 The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:uniaxial_hac_iff_damped_plants_masses}.
+For \(k_n = 0.01\,N/\mu m\) and \(k_n = 1\,N/\mu m\), except from the nano-hexapod mode, the dynamics is quite simple and can be well approximated by a second order plant.
+However, this is not the case for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)) where two modes can be seen (Figure \ref{fig:uniaxial_hac_iff_damped_plants_masses}, right).
+
+This is due to the interaction between the micro-station (modelled modes at 70Hz, 140Hz and 320Hz) and the nano-hexapod.
+Such effect will be explained in Section \ref{sec:uniaxial_support_compliance}.
 
 \begin{figure}[htbp]
 \centering
@@ -696,15 +597,16 @@ The obtained damped plants for the three nano-hexapod stiffnesses are shown in F
 \end{figure}
 
 \section{Position Feedback Controller}
-\label{sec:orgbfebc9e}
+\label{ssec:uniaxial_position_control_design}
+
 The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample's mass.
 
-The required feedback bandwidth was approximately determined un Section \ref{sec:uniaxial_noise_budgeting}:
+The required feedback bandwidth was approximately determined in Section \ref{sec:uniaxial_noise_budgeting}:
 \begin{itemize}
 \item \(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)).
 Near this frequency, the plants are equivalent to a mass line.
 The gain of the mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,kg\) up to \(65\,kg\)).
-This mean that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
+This means that the designed controller will need to have large gain margins to be robust to the change of sample's mass.
 \item \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)).
 Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line.
 It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.
@@ -717,7 +619,41 @@ This makes the robust design of the controller a little bit more complicated.
 
 
 Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure \ref{fig:uniaxial_nyquist_hac}).
-These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
+These high authority controllers are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics.
+The controllers used for the three nano-hexapod are shown in Equation \eqref{eq:uniaxial_hac_formulas}, and the parameters are summarized in Table \ref{tab:uniaxial_feedback_controller_parameters}.
+
+\begin{subequations} \label{eq:uniaxial_hac_formulas}
+\begin{align}
+K_{\text{soft}}(s) &= g \cdot
+       \underbrace{\frac{s + \omega_0}{s + \omega_i}}_{\text{lag}} \cdot
+       \underbrace{\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}}_{\text{lead}} \cdot
+       \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}} \\
+K_{\text{mid}}(s) &= g \cdot
+       \underbrace{\left(\frac{s + \omega_0}{s + \omega_i}\right)^2}_{\text{2 lags}} \cdot
+       \underbrace{\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}}_{\text{lead}} \cdot
+       \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}} \\
+K_{\text{stiff}}(s) &= g \cdot
+       \underbrace{\left(\frac{1}{s + \omega_i}\right)^2}_{\text{2 lags}} \cdot
+       \underbrace{\left(\frac{1 + \frac{s}{\omega_c/\sqrt{a}}}{1 + \frac{s}{\omega_c \sqrt{a}}}\right)^2}_{\text{2 leads}} \cdot
+       \underbrace{\frac{1}{1 + \frac{s}{\omega_l}}}_{\text{LPF}}
+\end{align}
+\end{subequations}
+
+\begin{table}[htbp]
+\caption{\label{tab:uniaxial_feedback_controller_parameters}Parameters used for the position feedback controllers}
+\centering
+\begin{tabularx}{\linewidth}{lXXX}
+\toprule
+ & \textbf{Soft} & \textbf{Moderately stiff} & \textbf{Stiff}\\
+\midrule
+\textbf{Gain} & \(g = 4 \cdot 10^5\) & \(g = 3 \cdot 10^6\) & \(g = 6 \cdot 10^12\)\\
+\textbf{Lead} & \(a = 5\), \(\omega_c = 20\,Hz\) & \(a = 4\), \(\omega_c = 70\,Hz\) & \(a = 5\), \(\omega_c = 100\,Hz\)\\
+\textbf{Lag} & \(\omega_0 = 5\,Hz\), \(\omega_i = 0.01\,Hz\) & \(\omega_0 = 20\,Hz\), \(\omega_i = 0.01\,Hz\) & \(\omega_i = 0.01\,Hz\)\\
+\textbf{LPF} & \(\omega_l = 200\,Hz\) & \(\omega_l = 300\,Hz\) & \(\omega_l = 500\,Hz\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
 The loop gains for the three nano-hexapod are shown in Figure \ref{fig:uniaxial_loop_gain_hac}.
 We can see that:
 \begin{itemize}
@@ -727,8 +663,8 @@ This is due to the fact that the crossover frequency corresponds to the mass lin
 The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.
 \end{itemize}
 
-Note that these controller were quickly tuned by hand and not designed using any optimization methods.
-The goal is just to have a first estimation of the attainable performances.
+Note that these controllers were quickly tuned by hand and not designed using any optimization methods.
+The goal is just to have a first estimation of the attainable performance.
 
 \begin{figure}[htbp]
 \centering
@@ -743,7 +679,8 @@ The goal is just to have a first estimation of the attainable performances.
 \end{figure}
 
 \section{Closed-Loop Noise Budgeting}
-\label{sec:org6f05561}
+\label{ssec:uniaxial_position_control_cl_noise_budget}
+
 The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed.
 These are compared with the open-loop and damped plants cases in Figure \ref{fig:uniaxial_sensitivity_dist_hac_lac} for just one configuration (moderately stiff nano-hexapod with 25kg sample's mass).
 As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.
@@ -751,7 +688,7 @@ As expected, the sensitivity to disturbances is decreased in the controller band
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/uniaxial_sensitivity_dist_hac_lac.png}
-\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with LAC and with HAC-LAC}
+\caption{\label{fig:uniaxial_sensitivity_dist_hac_lac}Change of sensitivity to disturbances with LAC and with HAC-LAC. Nano-Hexapod with \(k_n = 1\,N/\mu m\) and sample mass of \(25\,kg\) are used.}
 \end{figure}
 
 The cumulative amplitude spectrum of the motion \(d\) is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases.
@@ -764,8 +701,241 @@ Obtained root mean square values of the distance \(d\) are better for the soft n
 \caption{\label{fig:uniaxial_cas_hac_lac}Cumulative Amplitude Spectrum for all three nano-hexapod stiffnesses - Comparison of OL, IFF and HAC-LAC cases}
 \end{figure}
 
+\section{Conclusion}
+\begin{important}
+Based on the open-loop noise budgeting made in Section \ref{sec:uniaxial_noise_budgeting}, the closed-loop bandwidth required to obtain acceptable vibration levels was estimated.
+In order to obtain such bandwidth, the HAC-LAC strategy was followed which consists of applying an active damping controller and then a high authority position feedback controller.
+
+In this section, feedback controllers were design in such a way that the required closed-loop bandwidth was reached while being robust to a change of payload mass.
+The attainable vibration control performances were estimated for the three nano-hexapod stiffnesses and were found to be close to the required values.
+A slight advantage can be given to the soft nano-hexapod as it requires less feedback bandwidth while giving better stability results.
+\end{important}
+
+\chapter{Effect of limited micro-station 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 will be studied.
+
+This is a critical point as 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 which is not impacted by the micro-station.
+
+Therefore, the objective in this section is to obtain some guidance for the design of a nano-hexapod that will not by impacted by the complex micro-station dynamics.
+
+In order to study this, two models are used (Figure \ref{fig:uniaxial_support_compliance_models}) with:
+\begin{itemize}
+\item the nano-hexapod directly fixed on top of the granite, therefore neglecting any support compliance (Figure \ref{fig:uniaxial_support_compliance_nano_hexapod_only})
+\item the nano-hexapod fixed on top of the micro-station having some limited compliance (Figure \ref{fig:uniaxial_support_compliance_test_system})
+\end{itemize}
+
+\begin{figure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_support_compliance_nano_hexapod_only.png}
+\end{center}
+\subcaption{\label{fig:uniaxial_support_compliance_nano_hexapod_only}Nano-Hexapod fixed directly on the Granite}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_support_compliance_test_system.png}
+\end{center}
+\subcaption{\label{fig:uniaxial_support_compliance_test_system}Nano-Hexapod 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}
+\end{figure}
+\section{Neglected support compliance}
+
+Let's first neglect the limited compliance of the micro-station and use the uniaxial model show in Figure \ref{fig:uniaxial_support_compliance_nano_hexapod_only}.
+Let's choose a nano-hexapod mass (including the payload) of \(20\,\text{kg}\) and three hexapod stiffnesses such that their resonance frequencies are at \(\omega_{\nu} = 10\,\text{Hz}\), \(\omega_{\nu} = 70\,\text{Hz}\) and \(\omega_{\nu} = 400\,\text{Hz}\).
+
+The obtained transfer functions from \(F\) to \(L^\prime\) are shown in Figure \ref{fig:uniaxial_effect_support_compliance_neglected}.
+When neglecting the support compliance, large feedback bandwidth can be achieve for all three Nano-Hexapod.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/uniaxial_effect_support_compliance_neglected.png}
+\caption{\label{fig:uniaxial_effect_support_compliance_neglected}Obtained transfer functions from \(F\) to \(L^{\prime}\) when neglecing support compliance}
+\end{figure}
+
+\section{Effect of support compliance on \(L/F\)}
+
+Let's now add some support compliance and use the model shown in Figure \ref{fig:uniaxial_support_compliance_test_system}.
+The parameters of the support (i.e. \(m^{\prime}\), \(c^{\prime}\) and \(k^{\prime}\)) are chosen in such a way that the main vertical mode at \(\omega_\mu = 70\,\text{Hz}\) seen on the micro-station is matched (Figure \ref{fig:uniaxial_comp_frf_meas_model}, yellow curve).
+
+The transfer functions from \(F\) to \(L\) (i.e. control of the relative motion of the nano-hexapod) and from \(L\) to \(d\) (i.e. control of the position between the nano-hexapod and the fixed granite) can then be computed.
+
+When the relative displacement of the nano-hexapod \(L\) is to be controlled (dynamics shown in Figure \ref{fig:uniaxial_effect_support_compliance_dynamics}), having a stiff nano-hexapod (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}, right).
+
+This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages.
+In such 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 is it fixed.
+If a soft nano-hexapod is used, the support dynamics appears in the dynamics between \(F\) and \(L\) (see Figure \ref{fig:uniaxial_effect_support_compliance_dynamics}, left) which will impact the control robustness and performance.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/uniaxial_effect_support_compliance_dynamics.png}
+\caption{\label{fig:uniaxial_effect_support_compliance_dynamics}Effect of the support compliance on the transfer functions from \(F\) to \(L\)}
+\end{figure}
+
+\section{Effect of support compliance on \(d/F\)}
+
+When the motion to be controlled is the relative displacement \(d\) between the granite and the nano-hexapod's top platform, the effect of the support compliance on the plant dynamics is opposite than what was previously observed.
+Indeed, using a ``soft'' nano-hexapod (i.e. with a suspension mode at lower frequency than the mode of the support) makes the dynamics less affected by the support dynamics (Figure \ref{fig:uniaxial_effect_support_compliance_dynamics_d}, left).
+On the contrary, if a ``stiff'' nano-hexapod is used, the support dynamics appears in the plant dynamics (Figure \ref{fig:uniaxial_effect_support_compliance_dynamics_d}, right).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/uniaxial_effect_support_compliance_dynamics_d.png}
+\caption{\label{fig:uniaxial_effect_support_compliance_dynamics_d}Effect of the support compliance on the transfer functions from \(F\) to \(d\)}
+\end{figure}
+
+\section{Conclusion}
+\begin{important}
+In order to study the impact of the support compliance on the plant dynamics, simple models shown in Figure \ref{fig:uniaxial_support_compliance_models} were used.
+
+Depending on the quantity to be controlled (\(L\) or \(d\) in Figure \ref{fig:uniaxial_support_compliance_test_system}) and on the relative location of \(\omega_\nu\) (suspension mode of the nano-hexapod) and \(\omega_\mu\) (modes of the support), the interaction between the support and the nano-hexapod dynamics can change drastically (observations made are summarized in Table \ref{tab:uniaxial_effect_compliance}).
+
+For the NASS, having the suspension mode of the nano-hexapod at lower frequencies than the suspension modes of the micro-station would make the plant less dependent on the micro-station dynamics, and therefore easier to control.
+
+Note that observations made in this section are also affected by the ratio between the support mass \(m^{\prime}\) and the nano-hexapod mass \(m\) (the effect is more pronounced when the ratio \(m/m^{\prime}\) increases).
+\end{important}
+
+\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
+ & \(\omega_{\nu} \ll \omega_{\mu}\) & \(\omega_{\nu} \approx \omega_{\mu}\) & \(\omega_{\nu} \gg \omega_{\mu}\)\\
+\midrule
+\(d/F\): & small & large & large\\
+\(L/F\): & large & large & small\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\chapter{Effect of Payload Dynamics}
+\label{sec:uniaxial_payload_dynamics}
+Up to this section, the sample was modelled as a mass rigidly fixed to the nano-hexapod (as shown in Figure \ref{fig:uniaxial_paylaod_dynamics_rigid_schematic}).
+However, such sample may present internal dynamics and its fixation to the nano-hexapod may have limited stiffness.
+
+To study the effect of the sample dynamics, models shown in Figure \ref{fig:uniaxial_paylaod_dynamics_schematic} are used is this Section.
+
+\begin{figure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_paylaod_dynamics_rigid_schematic.png}
+\end{center}
+\subcaption{\label{fig:uniaxial_paylaod_dynamics_rigid_schematic}Rigid payload}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\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}
+\end{subfigure}
+\caption{\label{fig:uniaxial_payload_dynamics_models}Models used to study the effect of payload dynamics}
+\end{figure}
+
+\section{Impact on the plant dynamics}
+\label{ssec:uniaxial_payload_dynamics_effect_dynamics}
+
+To study the impact of sample dynamics, the following sample configurations are considered:
+\begin{itemize}
+\item rigid sample, corresponding to Figure \ref{fig:uniaxial_paylaod_dynamics_rigid_schematic}
+\item two flexible samples with a resonance at \(\omega_s = 200\,\text{Hz}\) and at \(\omega_s = 20\,\text{Hz}\) respectively, corresponding to Figure \ref{fig:uniaxial_paylaod_dynamics_schematic}
+\item for all cases, two sample masses are considered: \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\)
+\end{itemize}
+
+The transfer functions from the nano-hexapod force to the motion of the nano-hexapod top platform are computed for all the above configurations and are compared for a soft Nano-Hexapod \(k_n = 0.01\,N/\mu m\) 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 \(\frac{1}{m_ns^2}\)).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/uniaxial_payload_dynamics_soft_nano_hexapod.png}
+\caption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod}Effect of the payload dynamics on the soft Nano-Hexapod. Light sample on the right, and heavy sample on the left}
+\end{figure}
+
+The same transfer functions are now compared when using a stiff nano-hexapod (\(k_n = 100\,N/\mu m\)) in Figure \ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}.
+In that case, the sample's resonance \(\omega_n\) is smaller than the nano-hexapod 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 tough the added sample's flexibility still changes the high frequency mass line from \(\frac{1}{(m_n + m_s)s^2}\) to \(\frac{1}{m_ns^2}\), the overall dynamics is much less impacted, even if the sample mass is high (see yellow curve in Figure \ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}, right).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod.png}
+\caption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}Effect of the payload dynamics on the stiff Nano-Hexapod. Light sample on the right, and heavy sample on the left}
+\end{figure}
+
+
+\begin{important}
+The impact of the sample dynamics on the plant varies with the sample mass and the relative resonance frequency of the sample \(\omega_s\) and of the nano-hexapod \(\omega_n\).
+The larger the sample mass, the larger the effect (i.e. change of high frequency gain, appearance of additional resonances and anti-resonances).
+A zero/pole pattern is observed if \(\omega_s > \omega_n\) and a pole/zero pattern if \(\omega_s > \omega_n\).
+Such additional dynamics can induce stability issues depending on their position relative to the wanted feedback bandwidth as explained in \cite[Section 4.2]{rankers98_machin}.
+
+The general conclusion is that the stiffer the nano-hexapod, 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{}), while stiff positioning systems can more easily accept various payloads.
+\end{important}
+
+\section{Impact on the positioning stability}
+\label{ssec:uniaxial_payload_dynamics_effect_stability}
+
+Having a flexibility between the measured position (i.e. the top platform of the nano-hexapod) and the point-of-interest to be positioned relative to the x-ray can also impact the positioning stability.
+This is due to the fact that the measured (and therefore controlled) distance is no longer equal to the important distance for the experiment.
+
+In order to estimate if the sample flexibility is critical for the position stability of the point of interest, the model shown in Figure \ref{fig:uniaxial_sample_flexibility_control} is used.
+This is the same model that was used in Section \ref{sec:uniaxial_position_control} but with an added flexibility between the nano-hexapod and the sample (considered sample modes are at \(\omega_s = 20\,\text{Hz}\) and \(\omega_n = 200\,\text{Hz}\)).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/uniaxial_sample_flexibility_control.png}
+\caption{\label{fig:uniaxial_sample_flexibility_control}Uniaxial model considering a flexibility between the nano-hexapod top platform and the sample. In that case the measured and controlled distance \(d\) is different from the distance \(y\) that is wish to be controlled}
+\end{figure}
+
+After the system dynamics extracted from the model, IFF is applied using the same gains as the ones used in Section \ref{sec:uniaxial_active_damping}.
+Thanks to the collocation between the nano-hexapod and the force sensor used for IFF, the damped plants are still stable and similar damping values are obtained than when considering a rigid sample.
+
+The High Authority Controllers used in Section \ref{sec:uniaxial_position_control} are then implemented on the damped plants.
+The obtained closed-loop systems are stable, indicating good robustness.
+
+Finally, closed-loop noise budgeting is computed for the obtained the closed-loop system and the cumulative amplitude spectrum of \(d\) and \(y\) are shown in Figure \ref{fig:uniaxial_sample_flexibility_noise_budget_y}.
+The cumulative amplitude spectrum of the measured distance \(d\) (Figure \ref{fig:uniaxial_sample_flexibility_noise_budget_d}) shows that the added flexibility at the sample location have very little effect on the control performance.
+However, the cumulative amplitude spectrum of the distance \(y\) (Figure \ref{fig:uniaxial_sample_flexibility_noise_budget_y}) shows that the stability of \(y\) is degraded when the sample flexibility is considered and is degraded as \(\omega_s\) is lowered.
+
+What happens is that above \(\omega_s\), even though the motion \(d\) can be controlled perfectly, the sample's mass is ``isolated'' from the motion of the nano-hexapod and the control on \(y\) is not effective.
+
+\begin{figure}
+\begin{subfigure}{0.95\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_sample_flexibility_noise_budget_d.png}
+\end{center}
+\subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$}
+\end{subfigure}
+\begin{subfigure}{0.95\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_sample_flexibility_noise_budget_y.png}
+\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 affects much \(d\) but is detrimental to the stability of \(y\). A sample mass \(m_s = 1\,\text{kg}\) is used for the simulations.}
+\end{figure}
+
+\section{Conclusion}
+\begin{important}
+Payload dynamics is usually a major concern when designing a positioning system.
+For the NASS, the sample may present internal dynamics and limited fixation stiffness to the nano-hexapod platform.
+On top of that, its mass can vary from 1kg up to 50kg which is a first challenge.
+
+This sample dynamics can add pole/zero or zero/pole pattern in the plant as shown in Section \ref{ssec:uniaxial_payload_dynamics_effect_dynamics} which may impact the stability and makes the design of the controller more complex.
+However, it was shown that this issue is less critical if a stiff nano-hexapod is used.
+
+Having some flexibility between the measurement point and the point of interest (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}.
+
+It will be therefore important to take special care when designing sampling environments, especially is a soft nano-hexapod is used.
+\end{important}
+
 \chapter{Conclusion}
-\label{sec:org1cd0a09}
 \label{sec:conclusion}
 
 In this study, a uniaxial model of the nano-active-stabilization-system has been tuned both from dynamical measurements (Section \ref{sec:micro_station_model}) and from disturbances measurements (Section \ref{sec:uniaxial_disturbances}).
@@ -777,5 +947,8 @@ Finally, position feedback controllers have been developed for three considered
 These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances.
 It has been found that having a soft nano-hexapod makes the plant dynamics easier to control (because decoupled from the micro-station dynamics) and requires less position feedback bandwidth to fulfill the requirements.
 The moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)) is requiring a bit more position feedback bandwidth, but it still seems to give acceptable results.
-However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performances.
+However, the stiff nano-hexapod is the most complex to control and gives the worst positioning performance.
+
+
+\printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}
diff --git a/preamble.tex b/preamble.tex
index bf18bb9..d18dbd9 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -3,6 +3,7 @@
 \usepackage{caption,tabularx,booktabs}
 \usepackage{bm}
 
+\usepackage{xpatch} % Recommanded for biblatex
 \usepackage[        % use biblatex for bibliography
     backend=biber,  % use biber backend (bibtex replacement) or bibtex
     style=ieee,     % bib style
@@ -20,6 +21,10 @@
     mincitenames=1  %
     ]{biblatex}
 
+\setlength\bibitemsep{1.1\itemsep}
+
+% \renewcommand*{\bibfont}{\footnotesize}
+
 \usepackage{fontawesome}
 
 \usepackage{caption}