Rework all sections

simscape-nass.org

@@ -158,6 +158,8 @@ The goals of this report are:
 - Tomography + lateral scans (same as what was done in open loop [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A4-simscape-micro-station/simscape-micro-station.org::*Simulation of Scientific Experiments][here]])
 - Validation of concept
 
+** Backup
+*** Sensitivity to disturbances
 ** DONE Old Outline
 CLOSED: [2024-11-07 Thu 16:19]
 *** Introduction                                                    :ignore:
@@ -275,6 +277,14 @@ J_L_to_X = inv(nano_hexapod.geometry.J);
 - [ ] matlab/mat/nass_references.mat
 - [ ] matlab/mat/nass_stages.mat
 
+** TODO [#B] Check all figures
+
+- [ ] Caption
+- [ ] Units
+- [ ] Legend
+
+** TODO [#B] Check all matlab files
+
 ** TODO [#B] Check if things are compatible to results of uniaxial model
 
 ** DONE [#C] Check if it would be interesting to show soft/stiff nano-hexapod plants
@@ -543,28 +553,31 @@ This can then be used to compare with obtained performance with the nano-hexapod
 
 This should be done in the ustation report (A4).
 
-* Introduction                                                        :ignore:
+* Introduction
 
-From last sections:
-- Uniaxial: No stiff nano-hexapod (should also demonstrate that here)
-- Rotating: No soft nano-hexapod, Decentralized IFF can be used robustly by adding parallel stiffness
-- Micro-Station multi body model tuned from a modal analysis
-- Multi-body model of a nano-hexapod that can be merged with the multi-body model of the micro-station
-
-In this section:
-- Take the model of the nano-hexapod described in previous section (stiffness 1um/N)
-- Control kinematics: how the external metrology, the nano-hexapod metrology are used to control the sample's position (Section ref:sec:nass_kinematics)
-- Apply decentralized IFF (Section ref:sec:nass_active_damping)
-- Apply HAC-LAC (Section ref:sec:nass_hac)
-  - Check robustness to change of payload and to spindle rotation
-  - Simulation of experiments
-- Conclusion of the conceptual phase, validation with simulations
+The preceding chapters have established crucial foundational elements for the development of the Nano Active Stabilization System (NASS).
+The uniaxial model study demonstrated that very stiff nano-hexapod configurations should be avoided due to their high coupling with the micro-station's dynamics.
+A rotating three-degree-of-freedom model revealed that soft nano-hexapod designs prove unsuitable for rotating applications due to gyroscopic effect.
+To further improve the model accuracy, a multi-body model of the micro-station was developed, which was carefully tuned using experimental modal analysis.
+Furthermore, a multi-body model of the nano-hexapod was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure ref:fig:nass_simscape_model.
 
 #+name: fig:nass_simscape_model
 #+caption: 3D view of the NASS multi-body model
+#+attr_latex: :options [h!tbp]
 #+attr_latex: :width 0.8\linewidth
 [[file:figs/nass_simscape_model.jpg]]
 
+Building upon these foundations, this chapter presents the validation of the NASS concept.
+The investigation begins with the previously established nano-hexapod model, with actuator stiffness $k_a = 1\,N/\mu m$.
+A thorough examination of the control kinematics is presented in Section ref:sec:nass_kinematics, detailing how both external metrology and nano-hexapod internal sensors are utilized in the control architecture.
+The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section ref:sec:nass_active_damping), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section ref:sec:nass_hac).
+
+The robustness of the proposed control scheme is rigorously evaluated across various operational conditions.
+Particular attention is paid to system performance under changing payload masses and varying spindle rotational velocities, as these represent critical operational parameters in practical applications.
+
+This chapter marks the conclusion of the conceptual design phase, with simulation of tomography experiments providing strong evidence for the viability of the proposed NASS architecture.
+The findings presented here establish a solid foundation for subsequent detailed design and experimental validation phases.
+
 * Control Kinematics
 :PROPERTIES:
 :HEADER-ARGS:matlab+: :tangle matlab/nass_1_kinematics.m
@@ -577,6 +590,7 @@ This section focuses specifically on the components of the "Instrumentation and
 
 #+name: fig:nass_concept_schematic
 #+caption: Schematic of the Nano Active Stabilization System
+#+attr_latex: :options [h!tbp]
 [[file:figs/nass_concept_schematic.png]]
 
 As established in the previous section on Stewart platforms, the proposed control strategy combines Decentralized Integral Force Feedback with a High Authority Controller performed in the frame of the struts.
@@ -797,6 +811,7 @@ Then, the high authority controller uses the computed errors in the frame of the
 
 #+name: fig:nass_control_architecture
 #+caption: The physical systems are shown in blue, the control kinematics in red, the decentralized Integral Force Feedback in yellow and the centralized High Authority Controller in green.
+#+attr_latex: :options [h!tbp]
 #+attr_latex: :width \linewidth
 #+RESULTS:
 [[file:figs/nass_control_architecture.png]]
@@ -1010,7 +1025,7 @@ exportFig('figs/nass_iff_plant_kp.pdf', 'width', 'half', 'height', 600);
 
 #+name: fig:nass_iff_plant_effect_kp
 #+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and payload mass of 25kg. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros to complex conjugate zeros (\subref{fig:nass_iff_plant_kp})
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_no_kp}without parallel stiffness}
 #+attr_latex: :options {0.48\textwidth}
@@ -1144,7 +1159,7 @@ exportFig('figs/nass_iff_plant_effect_payload.pdf', 'width', 'half', 'height', 6
 
 #+name: fig:nass_iff_plant_effect_rotation_payload
 #+caption: Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload})
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_effect_rotation}Effect of Spindle rotation}
 #+attr_latex: :options {0.48\textwidth}
@@ -1274,6 +1289,7 @@ exportFig('figs/nass_iff_loop_gain.pdf', 'width', 'wide', 'height', 'normal');
 
 #+name: fig:nass_iff_loop_gain
 #+caption: Loop gain for the decentralized IFF: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$
+#+attr_latex: :options [h!tbp]
 #+RESULTS:
 [[file:figs/nass_iff_loop_gain.png]]
 
@@ -1403,7 +1419,7 @@ exportFig('figs/nass_iff_root_locus_50kg.pdf', 'width', 'third', 'height', 'norm
 
 #+name: fig:nass_iff_root_locus
 #+caption: Root Loci for Decentralized IFF for three payload masses. Closed-loop poles are shown by the black crosses.
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$}
 #+attr_latex: :options {0.33\textwidth}
@@ -1432,18 +1448,17 @@ exportFig('figs/nass_iff_root_locus_50kg.pdf', 'width', 'third', 'height', 'norm
 <<sec:nass_hac>>
 ** Introduction                                                      :ignore:
 
-# - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/uncertainty_experiment.org][uncertainty_experiment]]: Effect of experimental conditions on the plant (payload mass, Ry position, Rz position, Rz velocity, etc...)
+The implementation of high-bandwidth position control for the nano-hexapod presents several technical challenges.
+The plant dynamics exhibit complex behavior influenced by multiple factors including payload mass, rotational velocity, and the mechanical coupling between the nano-hexapod and the micro-station.
+This section presents the development and validation of a centralized control strategy designed to achieve precise sample positioning during high-speed tomography experiments.
 
-- [ ] Effect of micro-station compliance
-  Compare plant with "rigid" u-station and normal u-station
-- Effect of IFF
-- Effect of payload mass
-- Decoupled plant
-- Controller design
+First, a comprehensive analysis of the plant dynamics is conducted in Section ref:ssec:nass_hac_plant, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF).
+Section ref:ssec:nass_hac_stiffness validates previous modeling predictions that both overly stiff and overly compliant nano-hexapod configurations lead to degraded performance, through detailed analysis using the multi-body model.
+Building upon these findings, Section ref:ssec:nass_hac_controller presents the design of a robust high-authority controller capable of maintaining stability across varying payload masses while achieving the desired control bandwidth.
 
-From control kinematics:
-- Talk about issue of not estimating Rz from external metrology? (maybe could be nice to discuss that during the experiments!)
-- Show what happens is Rz is not estimated (for instance supposed equaled to zero => increased coupling)
+The performance of the developed control strategy is validated through simulations of tomography experiments in Section ref:ssec:nass_hac_tomography.
+These simulations incorporate realistic disturbance sources and evaluate system performance against the stringent positioning requirements imposed by future beamline specifications.
+Particular attention is paid to the system's behavior under maximum rotational velocity conditions and its ability to accommodate varying payload masses, demonstrating the practical viability of the proposed control approach.
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1471,6 +1486,20 @@ From control kinematics:
 #+end_src
 
 ** HAC Plant
+<<ssec:nass_hac_plant>>
+
+The plant dynamics from force inputs $\bm{f}$ to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were first extracted from the multi-body model without implementation of the decentralized IFF.
+The influence of spindle rotation on plant dynamics was investigated, with results presented in Figure ref:fig:nass_undamped_plant_effect_Wz.
+While rotational motion introduces coupling effects at low frequencies, these remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly.
+
+Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure ref:fig:nass_undamped_plant_effect_mass.
+As expected, increasing payload mass was found to decrease resonance frequencies while amplifying coupling at low frequency.
+These mass-dependent dynamic changes present considerable challenges for control system design, particularly for configurations with high payload masses.
+
+Additional operational parameters were systematically evaluated, including the $R_y$ tilt angle, $R_z$ spindle position, and micro-hexapod position.
+These factors were found to exert negligible influence on the plant dynamics, attributable to the effective mechanical decoupling achieved between the plant and micro-station dynamics.
+This decoupling characteristic ensures consistent performance across various operational configurations.
+This also validates the developed control kinematics.
 
 #+begin_src matlab
 %% Identify the IFF plant dynamics using the Simscape model
@@ -1524,16 +1553,6 @@ G_m1_Rz.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
 G_m1_Rz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
 #+end_src
 
-- Effect of rotation: ref:fig:nass_undamped_plant_effect_Wz
-  Add some coupling at low frequency, but still small at the considered velocity.
-  This is thanks to the relatively stiff nano-hexapod (CF rotating model)
-- Effect of payload mass:
-  Decrease resonance frequencies
-  Increase coupling: ref:fig:nass_undamped_plant_effect_mass
-  => control challenge for high payload masses
-- Other effects such as: Ry tilt angle, Rz spindle position, micro-hexapod position are found to have negligible effect on the plant dynamics.
-  This is thanks to the fact the the plant dynamics is well decoupled from the micro-station dynamics.
-
 #+begin_src matlab :exports none :results none
 figure;
 tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
@@ -1649,7 +1668,7 @@ exportFig('figs/nass_undamped_plant_effect_mass.pdf', 'width', 'half', 'height',
 
 #+name: fig:nass_undamped_plant_effect
 #+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_Wz}Effect of rotational velocity $\Omega_z$}
 #+attr_latex: :options {0.48\textwidth}
@@ -1665,13 +1684,16 @@ exportFig('figs/nass_undamped_plant_effect_mass.pdf', 'width', 'half', 'height',
 #+end_subfigure
 #+end_figure
 
+The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs $\bm{f}^\prime$ of the damped plant to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were extracted from this model.
 
-- Effect of IFF on the plant ref:fig:nass_comp_undamped_damped_plant_m1
-  Modes are well damped
-  Small coupling increase at low frequency
-- Benefits of using IFF ref:fig:nass_hac_plants
-  with added damping, the set of plants to be controlled (with payloads from 1kg to 50kg) is more easily controlled.
-  Between 10 and 50Hz, the plant dynamics does not vary a lot with the frequency, whereas without active damping, it would be impossible to design a robust controller with bandwidth above 10Hz that is robust to the change of payload
+The effectiveness of IFF implementation was first evaluated with a $1\,\text{kg}$ payload, as demonstrated in Figure ref:fig:nass_comp_undamped_damped_plant_m1.
+The results indicate successful damping of the nano-hexapod resonance modes, though a minor increase in low-frequency coupling was observed.
+This trade-off was considered acceptable given the overall improvement in system behavior.
+
+The benefits of IFF implementation were further assessed across the full range of payload configurations, with results presented in Figure ref:fig:nass_hac_plants.
+For all tested payloads ($1\,\text{kg}$, $25\,\text{kg}$ and $50\,\text{kg}$), decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics.
+More importantly, is the fact that in the vicinity of the wanted high authority control bandwidth (i.e. between $10\,\text{Hz}$ and $50\,\text{Hz}$), the damped dynamics (shown in red) exhibited minimal gain and phase variations with frequency.
+For the undamped system (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically unfeasible.
 
 #+begin_src matlab
 %% Identify HAC Plant without using IFF
@@ -1710,9 +1732,9 @@ tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 ax1 = nexttile([2,1]);
 hold on;
 plot(freqs, abs(squeeze(freqresp(G_m1( 1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
-     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, OL')
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - OL')
 plot(freqs, abs(squeeze(freqresp(G_hac_m1(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
-     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, with IFF')
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - IFF')
 for i = 1:5
     for j = i+1:6
         plot(freqs, abs(squeeze(freqresp(G_m1(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
@@ -1730,7 +1752,7 @@ end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-10, 2e-5]);
+ylim([1e-10, 5e-5]);
 leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
 leg.ItemTokenSize(1) = 15;
 
@@ -1761,8 +1783,8 @@ tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
 
 ax1 = nexttile([2,1]);
 hold on;
-plot(freqs, abs(squeeze(freqresp(G_m1(    1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'DisplayName', 'Undamped - $\epsilon\mathcal{L}_i/f_i$');
-plot(freqs, abs(squeeze(freqresp(G_hac_m1(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'DisplayName', 'Damped - $\epsilon\mathcal{L}_i/f_i^\prime$');
+plot(freqs, abs(squeeze(freqresp(G_m1(    1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'DisplayName', '$\epsilon\mathcal{L}_i/f_i$ - OL');
+plot(freqs, abs(squeeze(freqresp(G_hac_m1(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'DisplayName', '$\epsilon\mathcal{L}_i/f_i^\prime$ - IFF');
 for i = 1:6
     plot(freqs, abs(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
     plot(freqs, abs(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
@@ -1778,7 +1800,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
 leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
 leg.ItemTokenSize(1) = 15;
-% ylim([1e-8, 1e-4]);
+ylim([1e-10, 5e-5]);
 
 ax2 = nexttile;
 hold on;
@@ -1809,7 +1831,7 @@ exportFig('figs/nass_hac_plants.pdf', 'width', 'half', 'height', 600);
 
 #+name: fig:nass_hac_plant
 #+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_comp_undamped_damped_plant_m1}Effect of IFF - $m = 1\,\text{kg}$}
 #+attr_latex: :options {0.48\textwidth}
@@ -1825,10 +1847,12 @@ exportFig('figs/nass_hac_plants.pdf', 'width', 'half', 'height', 600);
 #+end_subfigure
 #+end_figure
 
-** Effect of micro-station compliance
+The coupling between the nano-hexapod and micro-station was evaluated through comparative analysis of plant dynamics under two mounting conditions.
+In the first configuration, the nano-hexapod was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance.
 
-Micro-Station complex dynamics has almost no effect on the plant dynamics (Figure ref:fig:nass_effect_ustation_compliance):
-- adds some alternating poles and zeros above 100Hz, which should not be an issue for control
+As illustrated in Figure ref:fig:nass_effect_ustation_compliance, the complex dynamics of the micro-station were found to have little impact on the plant dynamics.
+The only observable difference manifests as alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with system performance.
+This finding confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure.
 
 #+begin_src matlab
 %% Identify plant with "rigid" micro-station
@@ -1863,9 +1887,9 @@ tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 ax1 = nexttile([2,1]);
 hold on;
 plot(freqs, abs(squeeze(freqresp(G_m25_rigid( 1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
-     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, OL')
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - Rigid support')
 plot(freqs, abs(squeeze(freqresp(G_m25(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
-     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, with IFF')
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - $\mu$-station support')
 for i = 1:5
     for j = i+1:6
         plot(freqs, abs(squeeze(freqresp(G_m25_rigid(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
@@ -1883,7 +1907,7 @@ end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-10, 2e-5]);
+ylim([1e-10, 5e-5]);
 leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
 leg.ItemTokenSize(1) = 15;
 
@@ -1909,26 +1933,25 @@ exportFig('figs/nass_effect_ustation_compliance.pdf', 'width', 'wide', 'height',
 
 #+name: fig:nass_effect_ustation_compliance
 #+caption: Effect of the micro-station limited compliance on the plant dynamics
+#+attr_latex: :options [h!tbp]
 #+RESULTS:
 [[file:figs/nass_effect_ustation_compliance.png]]
 
-** Higher or lower nano-hexapod stiffness?
+** Effect of Nano-Hexapod Stiffness on System Dynamics
+<<ssec:nass_hac_stiffness>>
 
-*Goal*: confirm the analysis with simpler models (uniaxial and 3DoF) that a nano-hexapod stiffness of $\approx 1\,N/\mu m$ should give better performances than a very stiff or very soft nano-hexapod.
+The influence of nano-hexapod stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models.
+These models suggested that a moderate stiffness of approximately $1\,N/\mu m$ would provide better performance compared to either very stiff or very soft configurations.
 
-- *Stiff nano-hexapod*:
-  uniaxial model: high nano-hexapod stiffness induce coupling between the nano-hexapod and the micro-station dynamics.
-  considering the complex dynamics of the micro-station as shown by the modal analysis, that would result in a complex system to control
-  To show that, a nano-hexapod with actuator stiffness equal to 100N/um is initialized, payload of 25kg.
-  The dynamics from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ is identified and compared to the case where the micro-station is infinitely rigid (figure ref:fig:nass_stiff_nano_hexapod_coupling_ustation):
-  - Coupling induced by the micro-station: much more complex and difficult to model / predict
-  - Similar to what was predicted using the uniaxial model
-- *Soft nano-hexapod*:
-  Nano-hexapod with stiffness of 0.01N/um is initialized, payload of 25kg.
-  Dynamics is identified with no spindle rotation, and with spindle rotation of 36deg/s and 360deg/s (Figure ref:fig:nass_soft_nano_hexapod_effect_Wz)
-  - Rotation as huge effect on the dynamics: unstable for high rotational velocities, added coupling due to gyroscopic effects, and change of resonance frequencies as a function of the rotational velocity
-  - Simple 3DoF rotating model is helpful to understand the complex effect of the rotation => similar conclusion
-  - Say that controlling the frame of the struts is not adapted with a soft nano-hexapod, but we should rather control in the frame matching the center of mass of the payload, but we would still obtain large coupling and change of dynamics due to gyroscopic effects.
+For the stiff nano-hexapod analysis, a system with actuator stiffness of $100\,N/\mu m$ was simulated with a $25\,\text{kg}$ payload.
+The transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station.
+As shown in Figure ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the nano-hexapod and micro-station dynamics.
+This coupling introduces complex behavior that proves difficult to model and predict accurately, corroborating the predictions of the simplified uniaxial model.
+
+The soft nano-hexapod configuration was evaluated using a stiffness of $0.01\,N/\mu m$ with a $25\,\text{kg}$ payload.
+Dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s.
+Figure ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation substantially impacts system dynamics, manifesting as instability at high rotational velocities, increased coupling from gyroscopic effects, and rotation-dependent resonance frequencies.
+The current approach of controlling the motion in the strut frame proves inadequate for soft nano-hexapods; but even shifting control to the payload's center of mass frame would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects.
 
 #+begin_src matlab
 %% Identify Dynamics with a Stiff nano-hexapod (100N/um)
@@ -2140,7 +2163,7 @@ exportFig('figs/nass_soft_nano_hexapod_effect_Wz.pdf', 'width', 'half', 'height'
 
 #+name: fig:nass_soft_stiff_hexapod
 #+caption: Plant dynamics of a stiff ($k_a = 100\,N/\mu m$) nano-hexapod (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}) and of a soft ($k_a = 0.01\,N/\mu m$) nano-hexapod (\subref{fig:nass_soft_nano_hexapod_effect_Wz})
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}Stiff nano-hexapod - Coupling with the micro-station}
 #+attr_latex: :options {0.48\textwidth}
@@ -2157,12 +2180,10 @@ exportFig('figs/nass_soft_nano_hexapod_effect_Wz.pdf', 'width', 'half', 'height'
 #+end_figure
 
 ** Controller design
+<<ssec:nass_hac_controller>>
 
-In this section, a high authority controller is design such that:
-- it is robust to the change of payload mass (i.e. is should be stable for all the damped plants of Figure ref:fig:nass_hac_plants)
-- it has reasonably high bandwidth to give good performances (here 10Hz)
-
-eqref:eq:nass_robust_hac
+A high authority controller was designed to meet two key requirements: stable for all payload masses (i.e. for all the damped plants of Figure ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation.
+The controller structure is defined in Equation eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes.
 
 \begin{equation}\label{eq:nass_robust_hac}
   K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right)
@@ -2204,10 +2225,9 @@ save('./matlab/mat/nass_K_hac.mat', 'Khac');
 save('./mat/nass_K_hac.mat', 'Khac');
 #+end_src
 
-- "Decentralized" Loop Gain:
-  Bandwidth around 10Hz
-- Characteristic Loci:
-  Stable for all payloads with acceptable stability margins
+The controller's performance was evaluated through two complementary analyses.
+First, the decentralized loop gain, shown in Figure ref:fig:nass_hac_loop_gain, confirms the achievement of the desired 10Hz bandwidth.
+Second, the characteristic loci analysis presented in Figure ref:fig:nass_hac_loci demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope.
 
 #+begin_src matlab :exports none :results none
 %% "Diagonal" loop gain for the High Authority Controller
@@ -2333,7 +2353,7 @@ exportFig('figs/nass_hac_loci.pdf', 'width', 'half', 'height', 600);
 
 #+name: fig:nass_hac_controller
 #+caption: High Authority Controller - "Diagonal Loop Gain" (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci})
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_hac_loop_gain}Loop Gain}
 #+attr_latex: :options {0.48\textwidth}
@@ -2349,140 +2369,19 @@ exportFig('figs/nass_hac_loci.pdf', 'width', 'half', 'height', 600);
 #+end_subfigure
 #+end_figure
 
-** TODO Sensitivity to disturbances                                :noexport:
-
-- Compute transfer functions from spindle vertical error to sample vertical error with HAC-IFF
-  Compare without the NASS, and with just IFF
-- Same for horizontal
-
-#+begin_src matlab
-% Initialize each Simscape model elements
-initializeGround();
-initializeGranite();
-initializeTy();
-initializeRy();
-initializeRz();
-initializeMicroHexapod();
-initializeSimplifiedNanoHexapod();
-initializeSample('type', 'cylindrical', 'm', 1);
-
-% Initial Simscape Configuration
-initializeSimscapeConfiguration('gravity', false);
-initializeDisturbances('enable', false);
-initializeLoggingConfiguration('log', 'none');
-initializeController('type', 'open-loop');
-initializeReferences();
-
-% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_y'); io_i = io_i + 1; % Spindle Lateral Vibration [N]
-io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Spindle Vertical Vibration [N]
-io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fdy_z'); io_i = io_i + 1; % Vertical Ground Motion [m]
-io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwy');   io_i = io_i + 1; % Vertical Ground Motion [m]
-io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz');   io_i = io_i + 1; % Vertical Ground Motion [m]
-io(io_i) = linio([mdl, '/NASS'],         2, 'output',    [], 'y');     io_i = io_i + 1; % Lateral Displacement [m]
-io(io_i) = linio([mdl, '/NASS'],         2, 'output',    [], 'z');     io_i = io_i + 1; % Vertical Displacement [m]
-
-Gd_ol = linearize(mdl, io);
-Gd_ol.InputName  = {'Frz_y', 'Frz_z', 'Fdy_z', 'Dwy', 'Dwz'};
-Gd_ol.OutputName  = {'Dy', 'Dz'};
-
-initializeController('type', 'iff'); % Implemented IFF controller
-load('nass_K_iff.mat', 'Kiff'); % Load designed IFF controller
-
-Gd_iff = linearize(mdl, io);
-Gd_iff.InputName  = {'Frz_y', 'Frz_z', 'Fdy_z', 'Dwy', 'Dwz'};
-Gd_iff.OutputName  = {'Dy', 'Dz'};
-
-initializeController('type', 'hac-iff'); % Implemented IFF controller
-load('nass_K_hac.mat', 'Khac'); % Load designed HAC controller
-
-Gd_hac_iff = linearize(mdl, io);
-Gd_hac_iff.InputName  = {'Frz_y', 'Frz_z', 'Fdy_z', 'Dwy', 'Dwz'};
-Gd_hac_iff.OutputName  = {'Dy', 'Dz'};
-#+end_src
-
-#+begin_src matlab
-dist = load('ustation_disturbance_psd.mat');
-#+end_src
-
-Spindle, lateral:
-#+begin_src matlab
-figure;
-hold on;
-plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dy', 'Frz_y'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_iff('Dy', 'Frz_y'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dy', 'Frz_y'), freqs, 'Hz'))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude $D_z/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
-xticks([1e0, 1e1, 1e2]);
-xlim([1, 500]);
-#+end_src
-
-Spindle, vertical:
-#+begin_src matlab
-freqs = logspace(-1,3,1000);
-figure;
-hold on;
-plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dz', 'Frz_z'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_iff('Dz', 'Frz_z'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dz', 'Frz_z'), freqs, 'Hz'))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude $D_z/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
-#+end_src
-
-Ground motion, vertical:
-#+begin_src matlab
-freqs = logspace(-1,3,1000);
-figure;
-hold on;
-plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dz', 'Dwz'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_iff('Dz', 'Dwz'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dz', 'Dwz'), freqs, 'Hz'))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude $D_z/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
-xticks([1e0, 1e1, 1e2]);
-% xlim([1, 500]);
-#+end_src
-
-Ground motion, lateral:
-#+begin_src matlab
-figure;
-hold on;
-plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dy', 'Dwy'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_iff('Dy', 'Dwy'), freqs, 'Hz'))));
-plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dy', 'Dwy'), freqs, 'Hz'))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude $D_y/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
-xticks([1e0, 1e1, 1e2]);
-xlim([1, 500]);
-#+end_src
-
-Noise Budget:
-#+begin_src matlab
-figure;
-hold on;
-plot(dist.gm_dist.f, sqrt(flip(-cumtrapz(flip(dist.gm_dist.f), flip(dist.gm_dist.pxx_y.*abs(squeeze(freqresp(Gd_ol(  'Dy', 'Dwy'), dist.gm_dist.f, 'Hz'))).^2)))));
-plot(dist.gm_dist.f, sqrt(flip(-cumtrapz(flip(dist.gm_dist.f), flip(dist.gm_dist.pxx_y.*abs(squeeze(freqresp(Gd_iff(  'Dy', 'Dwy'), dist.gm_dist.f, 'Hz'))).^2)))));
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('ASD [m/sqrt(Hz)]'); xlabel('Frequency [Hz]');
-xticks([1e0, 1e1, 1e2]);
-xlim([1, 500]);
-#+end_src
-
 ** Tomography experiment
+<<ssec:nass_hac_tomography>>
 
-- Validation of concept with tomography scans at the highest rotational velocity of $\Omega_z = 360\,\text{deg/s}$
-- Compare obtained results with the smallest beam size that is expected with future beamline upgrade: 200nm (horizontal size) x 100nm (vertical size)
-- Take into account the two main sources of disturbances: ground motion, spindle vibrations
-  Other noise sources are not taken into account here as they will be optimized latter (detail design phase): measurement noise, electrical noise for DAC and voltage amplifiers, ...
+The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with particular focus on tomography scanning due to its demanding performance requirements.
+Simulations were conducted at the maximum operational rotational velocity of $\Omega_z = 360\,\text{deg/s}$ to evaluate system performance under the most challenging conditions.
 
-The open-loop errors and the closed-loop errors for the tomography scan with the light sample $1\,kg$ are shown in Figure ref:fig:nass_tomo_1kg_60rpm.
+Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of 200nm (horizontal) × 100nm (vertical).
+The primary requirement stipulates that the point of interest must remain within these beam dimensions throughout operation.
+The simulation incorporated two principal disturbance sources: ground motion and spindle vibrations.
+Additional noise sources, including measurement noise and electrical noise from DAC and voltage amplifiers, were not included in this analysis as these parameters will be optimized during the detailed design phase.
+
+Figure ref:fig:nass_tomo_1kg_60rpm presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg).
+The results demonstrate the system's capability to maintain position within the specified beam dimensions, validating the fundamental concept of the stabilization system.
 
 #+begin_src matlab
 % Sample is not centered with the rotation axis
@@ -2615,7 +2514,7 @@ exportFig('figs/nass_tomo_1kg_60rpm_yz.pdf', 'width', 'half', 'height', 'normal'
 
 #+name: fig:nass_tomo_1kg_60rpm
 #+caption: Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at $360\,\text{deg/s}$. 1kg payload is placed on top of the nano-hexapod.
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_xy}XY plane}
 #+attr_latex: :options {0.48\textwidth}
@@ -2631,8 +2530,12 @@ exportFig('figs/nass_tomo_1kg_60rpm_yz.pdf', 'width', 'half', 'height', 'normal'
 #+end_subfigure
 #+end_figure
 
-- Effect of payload mass (Figure ref:fig:nass_tomography_hac_iff):
-  Worse performance for high masses, as expected from the control analysis, but still acceptable considering that the rotational velocity of 360deg/s is only used for light payloads.
+The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25kg and 50kg payloads, complementing the initial 1kg test case.
+As illustrated in Figure ref:fig:nass_tomography_hac_iff, system performance exhibits some degradation with increasing payload mass, aligning with predictions from the control analysis.
+While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions.
+
+It should be noted that the maximum rotational velocity of 360deg/s is primarily intended for lightweight payload applications.
+For higher mass configurations, rotational velocities are foreseen to be below 36deg/s.
 
 #+begin_src matlab :exports none :results none
 %% Simulation of tomography experiment - no payload, 30rpm - YZ errors
@@ -2699,7 +2602,7 @@ exportFig('figs/nass_tomography_hac_iff_m50.pdf', 'width', 'third', 'height', 'n
 
 #+name: fig:nass_tomography_hac_iff
 #+caption: Simulation of tomography experiments - 360deg/s. Beam size shown by dashed black
-#+attr_latex: :options [htbp]
+#+attr_latex: :options [h!tbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,kg$}
 #+attr_latex: :options {0.33\textwidth}
@@ -2721,17 +2624,27 @@ exportFig('figs/nass_tomography_hac_iff_m50.pdf', 'width', 'third', 'height', 'n
 #+end_subfigure
 #+end_figure
 
-** Conclusion
-:PROPERTIES:
-:UNNUMBERED: t
-:END:
-
 * Conclusion
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 <<sec:nass_conclusion>>
 
+The development and analysis presented in this chapter have successfully validated the Nano Active Stabilization System concept, marking the completion of the conceptual design phase.
+A comprehensive control strategy has been established, effectively combining external metrology with nano-hexapod sensor measurements to achieve precise position control.
+The implementation follows the proven High Authority Control - Low Authority Control architecture, which has been successfully adapted for this specific application.
+
+The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping across various operating conditions.
+The addition of parallel springs to the force sensors, has been shown to ensure stability during spindle rotation.
+The centralized High Authority Controller, operating in the frame of the struts for simplicity, has achieved the desired performance objectives.
+This investigation has confirmed that the moderate actuator stiffness of $1\,N/\mu m$ represents an adequate choice for the nano-hexapod, as both very stiff and very compliant configurations have been shown to introduce significant performance limitations.
+
+The control system has achieved the targeted bandwidth of 10 Hz while maintaining robustness to payload mass variations.
+These theoretical predictions have been validated through simulations of tomography experiments, where positioning accuracy requirements were defined by the expected minimum beam dimensions of 200 nm × 100 nm.
+The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples.
+While some degradation in positioning accuracy has been observed with heavier payloads, as anticipated by the control analysis, the overall performance remains sufficient to validate the fundamental concept of the NASS.
+
+These results provide a solid foundation for advancing to the subsequent detailed design phase and experimental implementation.
 
 * Bibliography                                                        :ignore:
 #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

simscape-nass.tex

@@ -1,4 +1,4 @@
-% Created 2025-02-17 Mon 22:37
+% Created 2025-02-18 Tue 10:43
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -24,33 +24,31 @@
 
 \clearpage
 
-From last sections:
-\begin{itemize}
-\item Uniaxial: No stiff nano-hexapod (should also demonstrate that here)
-\item Rotating: No soft nano-hexapod, Decentralized IFF can be used robustly by adding parallel stiffness
-\item Micro-Station multi body model tuned from a modal analysis
-\item Multi-body model of a nano-hexapod that can be merged with the multi-body model of the micro-station
-\end{itemize}
+\chapter{Introduction}
 
-In this section:
-\begin{itemize}
-\item Take the model of the nano-hexapod described in previous section (stiffness 1um/N)
-\item Control kinematics: how the external metrology, the nano-hexapod metrology are used to control the sample's position (Section \ref{sec:nass_kinematics})
-\item Apply decentralized IFF (Section \ref{sec:nass_active_damping})
-\item Apply HAC-LAC (Section \ref{sec:nass_hac})
-\begin{itemize}
-\item Check robustness to change of payload and to spindle rotation
-\item Simulation of experiments
-\end{itemize}
-\item Conclusion of the conceptual phase, validation with simulations
-\end{itemize}
+The preceding chapters have established crucial foundational elements for the development of the Nano Active Stabilization System (NASS).
+The uniaxial model study demonstrated that very stiff nano-hexapod configurations should be avoided due to their high coupling with the micro-station's dynamics.
+A rotating three-degree-of-freedom model revealed that soft nano-hexapod designs prove unsuitable for rotating applications due to gyroscopic effect.
+To further improve the model accuracy, a multi-body model of the micro-station was developed, which was carefully tuned using experimental modal analysis.
+Furthermore, a multi-body model of the nano-hexapod was created, that can then be seamlessly integrated with the micro-station model, as illustrated in Figure \ref{fig:nass_simscape_model}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/nass_simscape_model.jpg}
+\includegraphics[h!tbp,width=0.8\linewidth]{figs/nass_simscape_model.jpg}
 \caption{\label{fig:nass_simscape_model}3D view of the NASS multi-body model}
 \end{figure}
 
+Building upon these foundations, this chapter presents the validation of the NASS concept.
+The investigation begins with the previously established nano-hexapod model, with actuator stiffness \(k_a = 1\,N/\mu m\).
+A thorough examination of the control kinematics is presented in Section \ref{sec:nass_kinematics}, detailing how both external metrology and nano-hexapod internal sensors are utilized in the control architecture.
+The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section \ref{sec:nass_active_damping}), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section \ref{sec:nass_hac}).
+
+The robustness of the proposed control scheme is rigorously evaluated across various operational conditions.
+Particular attention is paid to system performance under changing payload masses and varying spindle rotational velocities, as these represent critical operational parameters in practical applications.
+
+This chapter marks the conclusion of the conceptual design phase, with simulation of tomography experiments providing strong evidence for the viability of the proposed NASS architecture.
+The findings presented here establish a solid foundation for subsequent detailed design and experimental validation phases.
+
 \chapter{Control Kinematics}
 \label{sec:nass_kinematics}
 Figure \ref{fig:nass_concept_schematic} presents a schematic overview of the NASS.
@@ -58,7 +56,7 @@ This section focuses specifically on the components of the ``Instrumentation and
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/nass_concept_schematic.png}
+\includegraphics[h!tbp]{figs/nass_concept_schematic.png}
 \caption{\label{fig:nass_concept_schematic}Schematic of the Nano Active Stabilization System}
 \end{figure}
 
@@ -183,7 +181,7 @@ Then, the high authority controller uses the computed errors in the frame of the
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/nass_control_architecture.png}
+\includegraphics[h!tbp,width=\linewidth]{figs/nass_control_architecture.png}
 \caption{\label{fig:nass_control_architecture}The physical systems are shown in blue, the control kinematics in red, the decentralized Integral Force Feedback in yellow and the centralized High Authority Controller in green.}
 \end{figure}
 
@@ -204,7 +202,7 @@ Adding parallel stiffness (Figure \ref{fig:nass_iff_plant_kp}) transforms these
 
 Though both cases show significant coupling around resonances, stability is guaranteed by the collocated arrangement of actuators and sensors \cite{preumont08_trans_zeros_struc_contr_with}.
 
-\begin{figure}[htbp]
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_iff_plant_no_kp.png}
@@ -225,7 +223,7 @@ The effect of rotation, shown in Figure \ref{fig:nass_iff_plant_effect_rotation}
 Figure \ref{fig:nass_iff_plant_effect_payload} illustrate the effect of payload mass on the plant dynamics.
 While the poles and zeros are shifting with payload mass, the alternating pattern of poles and zeros is maintained, ensuring that the phase remains bounded between 0 and 180 degrees, and thus good robustness properties.
 
-\begin{figure}[htbp]
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_iff_plant_effect_rotation.png}
@@ -265,14 +263,14 @@ The overall gain is then increased to have large loop gain around resonances to
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/nass_iff_loop_gain.png}
+\includegraphics[h!tbp]{figs/nass_iff_loop_gain.png}
 \caption{\label{fig:nass_iff_loop_gain}Loop gain for the decentralized IFF: \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)}
 \end{figure}
 
 To verify stability, root loci for the three payload configurations are computed and shown in Figure \ref{fig:nass_iff_root_locus}.
 The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability properties of the applied decentralized IFF.
 
-\begin{figure}[htbp]
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.9\linewidth]{figs/nass_iff_root_locus_1kg.png}
@@ -296,35 +294,34 @@ The results demonstrate that the closed-loop poles remain within the left-half p
 
 \chapter{Centralized Active Vibration Control}
 \label{sec:nass_hac}
-\begin{itemize}
-\item[{$\square$}] Effect of micro-station compliance
-Compare plant with ``rigid'' u-station and normal u-station
-\item Effect of IFF
-\item Effect of payload mass
-\item Decoupled plant
-\item Controller design
-\end{itemize}
+The implementation of high-bandwidth position control for the nano-hexapod presents several technical challenges.
+The plant dynamics exhibit complex behavior influenced by multiple factors including payload mass, rotational velocity, and the mechanical coupling between the nano-hexapod and the micro-station.
+This section presents the development and validation of a centralized control strategy designed to achieve precise sample positioning during high-speed tomography experiments.
 
-From control kinematics:
-\begin{itemize}
-\item Talk about issue of not estimating Rz from external metrology? (maybe could be nice to discuss that during the experiments!)
-\item Show what happens is Rz is not estimated (for instance supposed equaled to zero => increased coupling)
-\end{itemize}
+First, a comprehensive analysis of the plant dynamics is conducted in Section \ref{ssec:nass_hac_plant}, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF).
+Section \ref{ssec:nass_hac_stiffness} validates previous modeling predictions that both overly stiff and overly compliant nano-hexapod configurations lead to degraded performance, through detailed analysis using the multi-body model.
+Building upon these findings, Section \ref{ssec:nass_hac_controller} presents the design of a robust high-authority controller capable of maintaining stability across varying payload masses while achieving the desired control bandwidth.
+
+The performance of the developed control strategy is validated through simulations of tomography experiments in Section \ref{ssec:nass_hac_tomography}.
+These simulations incorporate realistic disturbance sources and evaluate system performance against the stringent positioning requirements imposed by future beamline specifications.
+Particular attention is paid to the system's behavior under maximum rotational velocity conditions and its ability to accommodate varying payload masses, demonstrating the practical viability of the proposed control approach.
 \section{HAC Plant}
+\label{ssec:nass_hac_plant}
 
-\begin{itemize}
-\item Effect of rotation: \ref{fig:nass_undamped_plant_effect_Wz}
-Add some coupling at low frequency, but still small at the considered velocity.
-This is thanks to the relatively stiff nano-hexapod (CF rotating model)
-\item Effect of payload mass:
-Decrease resonance frequencies
-Increase coupling: \ref{fig:nass_undamped_plant_effect_mass}
-=> control challenge for high payload masses
-\item Other effects such as: Ry tilt angle, Rz spindle position, micro-hexapod position are found to have negligible effect on the plant dynamics.
-This is thanks to the fact the the plant dynamics is well decoupled from the micro-station dynamics.
-\end{itemize}
+The plant dynamics from force inputs \(\bm{f}\) to the strut errors \(\bm{\epsilon}_{\mathcal{L}}\) were first extracted from the multi-body model without implementation of the decentralized IFF.
+The influence of spindle rotation on plant dynamics was investigated, with results presented in Figure \ref{fig:nass_undamped_plant_effect_Wz}.
+While rotational motion introduces coupling effects at low frequencies, these remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly.
 
-\begin{figure}[htbp]
+Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure \ref{fig:nass_undamped_plant_effect_mass}.
+As expected, increasing payload mass was found to decrease resonance frequencies while amplifying coupling at low frequency.
+These mass-dependent dynamic changes present considerable challenges for control system design, particularly for configurations with high payload masses.
+
+Additional operational parameters were systematically evaluated, including the \(R_y\) tilt angle, \(R_z\) spindle position, and micro-hexapod position.
+These factors were found to exert negligible influence on the plant dynamics, attributable to the effective mechanical decoupling achieved between the plant and micro-station dynamics.
+This decoupling characteristic ensures consistent performance across various operational configurations.
+This also validates the developed control kinematics.
+
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_undamped_plant_effect_Wz.png}
@@ -340,17 +337,18 @@ This is thanks to the fact the the plant dynamics is well decoupled from the mic
 \caption{\label{fig:nass_undamped_plant_effect}Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})}
 \end{figure}
 
+The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs \(\bm{f}^\prime\) of the damped plant to the strut errors \(\bm{\epsilon}_{\mathcal{L}}\) were extracted from this model.
 
-\begin{itemize}
-\item Effect of IFF on the plant \ref{fig:nass_comp_undamped_damped_plant_m1}
-Modes are well damped
-Small coupling increase at low frequency
-\item Benefits of using IFF \ref{fig:nass_hac_plants}
-with added damping, the set of plants to be controlled (with payloads from 1kg to 50kg) is more easily controlled.
-Between 10 and 50Hz, the plant dynamics does not vary a lot with the frequency, whereas without active damping, it would be impossible to design a robust controller with bandwidth above 10Hz that is robust to the change of payload
-\end{itemize}
+The effectiveness of IFF implementation was first evaluated with a \(1\,\text{kg}\) payload, as demonstrated in Figure \ref{fig:nass_comp_undamped_damped_plant_m1}.
+The results indicate successful damping of the nano-hexapod resonance modes, though a minor increase in low-frequency coupling was observed.
+This trade-off was considered acceptable given the overall improvement in system behavior.
 
-\begin{figure}[htbp]
+The benefits of IFF implementation were further assessed across the full range of payload configurations, with results presented in Figure \ref{fig:nass_hac_plants}.
+For all tested payloads (\(1\,\text{kg}\), \(25\,\text{kg}\) and \(50\,\text{kg}\)), decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics.
+More importantly, is the fact that in the vicinity of the wanted high authority control bandwidth (i.e. between \(10\,\text{Hz}\) and \(50\,\text{Hz}\)), the damped dynamics (shown in red) exhibited minimal gain and phase variations with frequency.
+For the undamped system (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically unfeasible.
+
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_comp_undamped_damped_plant_m1.png}
@@ -366,44 +364,36 @@ Between 10 and 50Hz, the plant dynamics does not vary a lot with the frequency,
 \caption{\label{fig:nass_hac_plant}Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})}
 \end{figure}
 
-\section{Effect of micro-station compliance}
+The coupling between the nano-hexapod and micro-station was evaluated through comparative analysis of plant dynamics under two mounting conditions.
+In the first configuration, the nano-hexapod was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance.
 
-Micro-Station complex dynamics has almost no effect on the plant dynamics (Figure \ref{fig:nass_effect_ustation_compliance}):
-\begin{itemize}
-\item adds some alternating poles and zeros above 100Hz, which should not be an issue for control
-\end{itemize}
+As illustrated in Figure \ref{fig:nass_effect_ustation_compliance}, the complex dynamics of the micro-station were found to have little impact on the plant dynamics.
+The only observable difference manifests as alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with system performance.
+This finding confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/nass_effect_ustation_compliance.png}
+\includegraphics[h!tbp]{figs/nass_effect_ustation_compliance.png}
 \caption{\label{fig:nass_effect_ustation_compliance}Effect of the micro-station limited compliance on the plant dynamics}
 \end{figure}
 
-\section{Higher or lower nano-hexapod stiffness?}
+\section{Effect of Nano-Hexapod Stiffness on System Dynamics}
+\label{ssec:nass_hac_stiffness}
 
-\textbf{Goal}: confirm the analysis with simpler models (uniaxial and 3DoF) that a nano-hexapod stiffness of \(\approx 1\,N/\mu m\) should give better performances than a very stiff or very soft nano-hexapod.
+The influence of nano-hexapod stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models.
+These models suggested that a moderate stiffness of approximately \(1\,N/\mu m\) would provide better performance compared to either very stiff or very soft configurations.
 
-\begin{itemize}
-\item \textbf{Stiff nano-hexapod}:
-uniaxial model: high nano-hexapod stiffness induce coupling between the nano-hexapod and the micro-station dynamics.
-considering the complex dynamics of the micro-station as shown by the modal analysis, that would result in a complex system to control
-To show that, a nano-hexapod with actuator stiffness equal to 100N/um is initialized, payload of 25kg.
-The dynamics from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) is identified and compared to the case where the micro-station is infinitely rigid (figure \ref{fig:nass_stiff_nano_hexapod_coupling_ustation}):
-\begin{itemize}
-\item Coupling induced by the micro-station: much more complex and difficult to model / predict
-\item Similar to what was predicted using the uniaxial model
-\end{itemize}
-\item \textbf{Soft nano-hexapod}:
-Nano-hexapod with stiffness of 0.01N/um is initialized, payload of 25kg.
-Dynamics is identified with no spindle rotation, and with spindle rotation of 36deg/s and 360deg/s (Figure \ref{fig:nass_soft_nano_hexapod_effect_Wz})
-\begin{itemize}
-\item Rotation as huge effect on the dynamics: unstable for high rotational velocities, added coupling due to gyroscopic effects, and change of resonance frequencies as a function of the rotational velocity
-\item Simple 3DoF rotating model is helpful to understand the complex effect of the rotation => similar conclusion
-\item Say that controlling the frame of the struts is not adapted with a soft nano-hexapod, but we should rather control in the frame matching the center of mass of the payload, but we would still obtain large coupling and change of dynamics due to gyroscopic effects.
-\end{itemize}
-\end{itemize}
+For the stiff nano-hexapod analysis, a system with actuator stiffness of \(100\,N/\mu m\) was simulated with a \(25\,\text{kg}\) payload.
+The transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station.
+As shown in Figure \ref{fig:nass_stiff_nano_hexapod_coupling_ustation}, significant coupling was observed between the nano-hexapod and micro-station dynamics.
+This coupling introduces complex behavior that proves difficult to model and predict accurately, corroborating the predictions of the simplified uniaxial model.
 
-\begin{figure}[htbp]
+The soft nano-hexapod configuration was evaluated using a stiffness of \(0.01\,N/\mu m\) with a \(25\,\text{kg}\) payload.
+Dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s.
+Figure \ref{fig:nass_soft_nano_hexapod_effect_Wz} demonstrates that rotation substantially impacts system dynamics, manifesting as instability at high rotational velocities, increased coupling from gyroscopic effects, and rotation-dependent resonance frequencies.
+The current approach of controlling the motion in the strut frame proves inadequate for soft nano-hexapods; but even shifting control to the payload's center of mass frame would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects.
+
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_stiff_nano_hexapod_coupling_ustation.png}
@@ -420,27 +410,20 @@ Dynamics is identified with no spindle rotation, and with spindle rotation of 36
 \end{figure}
 
 \section{Controller design}
+\label{ssec:nass_hac_controller}
 
-In this section, a high authority controller is design such that:
-\begin{itemize}
-\item it is robust to the change of payload mass (i.e. is should be stable for all the damped plants of Figure \ref{fig:nass_hac_plants})
-\item it has reasonably high bandwidth to give good performances (here 10Hz)
-\end{itemize}
-
-\eqref{eq:nass_robust_hac}
+A high authority controller was designed to meet two key requirements: stable for all payload masses (i.e. for all the damped plants of Figure \ref{fig:nass_hac_plants}), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation.
+The controller structure is defined in Equation \eqref{eq:nass_robust_hac}, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes.
 
 \begin{equation}\label{eq:nass_robust_hac}
   K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right)
 \end{equation}
 
-\begin{itemize}
-\item ``Decentralized'' Loop Gain:
-Bandwidth around 10Hz
-\item Characteristic Loci:
-Stable for all payloads with acceptable stability margins
-\end{itemize}
+The controller's performance was evaluated through two complementary analyses.
+First, the decentralized loop gain, shown in Figure \ref{fig:nass_hac_loop_gain}, confirms the achievement of the desired 10Hz bandwidth.
+Second, the characteristic loci analysis presented in Figure \ref{fig:nass_hac_loci} demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope.
 
-\begin{figure}[htbp]
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_hac_loop_gain.png}
@@ -457,17 +440,20 @@ Stable for all payloads with acceptable stability margins
 \end{figure}
 
 \section{Tomography experiment}
+\label{ssec:nass_hac_tomography}
 
-\begin{itemize}
-\item Validation of concept with tomography scans at the highest rotational velocity of \(\Omega_z = 360\,\text{deg/s}\)
-\item Compare obtained results with the smallest beam size that is expected with future beamline upgrade: 200nm (horizontal size) x 100nm (vertical size)
-\item Take into account the two main sources of disturbances: ground motion, spindle vibrations
-Other noise sources are not taken into account here as they will be optimized latter (detail design phase): measurement noise, electrical noise for DAC and voltage amplifiers, \ldots{}
-\end{itemize}
+The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with particular focus on tomography scanning due to its demanding performance requirements.
+Simulations were conducted at the maximum operational rotational velocity of \(\Omega_z = 360\,\text{deg/s}\) to evaluate system performance under the most challenging conditions.
 
-The open-loop errors and the closed-loop errors for the tomography scan with the light sample \(1\,kg\) are shown in Figure \ref{fig:nass_tomo_1kg_60rpm}.
+Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of 200nm (horizontal) × 100nm (vertical).
+The primary requirement stipulates that the point of interest must remain within these beam dimensions throughout operation.
+The simulation incorporated two principal disturbance sources: ground motion and spindle vibrations.
+Additional noise sources, including measurement noise and electrical noise from DAC and voltage amplifiers, were not included in this analysis as these parameters will be optimized during the detailed design phase.
 
-\begin{figure}[htbp]
+Figure \ref{fig:nass_tomo_1kg_60rpm} presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg).
+The results demonstrate the system's capability to maintain position within the specified beam dimensions, validating the fundamental concept of the stabilization system.
+
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=0.9]{figs/nass_tomo_1kg_60rpm_xy.png}
@@ -483,12 +469,14 @@ The open-loop errors and the closed-loop errors for the tomography scan with the
 \caption{\label{fig:nass_tomo_1kg_60rpm}Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at \(360\,\text{deg/s}\). 1kg payload is placed on top of the nano-hexapod.}
 \end{figure}
 
-\begin{itemize}
-\item Effect of payload mass (Figure \ref{fig:nass_tomography_hac_iff}):
-Worse performance for high masses, as expected from the control analysis, but still acceptable considering that the rotational velocity of 360deg/s is only used for light payloads.
-\end{itemize}
+The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25kg and 50kg payloads, complementing the initial 1kg test case.
+As illustrated in Figure \ref{fig:nass_tomography_hac_iff}, system performance exhibits some degradation with increasing payload mass, aligning with predictions from the control analysis.
+While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions.
 
-\begin{figure}[htbp]
+It should be noted that the maximum rotational velocity of 360deg/s is primarily intended for lightweight payload applications.
+For higher mass configurations, rotational velocities are foreseen to be below 36deg/s.
+
+\begin{figure}[h!tbp]
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/nass_tomography_hac_iff_m1.png}
@@ -510,11 +498,24 @@ Worse performance for high masses, as expected from the control analysis, but st
 \caption{\label{fig:nass_tomography_hac_iff}Simulation of tomography experiments - 360deg/s. Beam size shown by dashed black}
 \end{figure}
 
-\section*{Conclusion}
-
 \chapter*{Conclusion}
 \label{sec:nass_conclusion}
 
+The development and analysis presented in this chapter have successfully validated the Nano Active Stabilization System concept, marking the completion of the conceptual design phase.
+A comprehensive control strategy has been established, effectively combining external metrology with nano-hexapod sensor measurements to achieve precise position control.
+The implementation follows the proven High Authority Control - Low Authority Control architecture, which has been successfully adapted for this specific application.
+
+The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping across various operating conditions.
+The addition of parallel springs to the force sensors, has been shown to ensure stability during spindle rotation.
+The centralized High Authority Controller, operating in the frame of the struts for simplicity, has achieved the desired performance objectives.
+This investigation has confirmed that the moderate actuator stiffness of \(1\,N/\mu m\) represents an adequate choice for the nano-hexapod, as both very stiff and very compliant configurations have been shown to introduce significant performance limitations.
+
+The control system has achieved the targeted bandwidth of 10 Hz while maintaining robustness to payload mass variations.
+These theoretical predictions have been validated through simulations of tomography experiments, where positioning accuracy requirements were defined by the expected minimum beam dimensions of 200 nm × 100 nm.
+The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples.
+While some degradation in positioning accuracy has been observed with heavier payloads, as anticipated by the control analysis, the overall performance remains sufficient to validate the fundamental concept of the NASS.
+
+These results provide a solid foundation for advancing to the subsequent detailed design phase and experimental implementation.
 
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}