Remove introduction from TOC

figs/inkscape/convert_svg.sh

@@ -0,0 +1,18 @@
+#!/bin/bash
+
+# Directory containing SVG files
+INPUT_DIR="."
+
+# Loop through all SVG files in the directory
+for svg_file in "$INPUT_DIR"/*.svg; do
+    # Check if there are SVG files in the directory
+    if [ -f "$svg_file" ]; then
+        # Output PDF file name
+        pdf_file="../${svg_file%.svg}.pdf"
+        png_file="../${svg_file%.svg}"
+        
+        # Convert SVG to PDF using Inkscape
+        inkscape "$svg_file" --export-filename="$pdf_file" && \
+            pdftocairo -png -singlefile -cropbox "$pdf_file" "$png_file"
+    fi
+done

figs/nass_iff_loop_gain.pdf

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figs/nass_iff_plant_effect_payload.pdf

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figs/nass_iff_plant_kp.pdf

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figs/nass_soft_nano_hexapod_effect_Wz.pdf

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matlab/nass_1_active_damping.m

@@ -0,0 +1,451 @@
+%% 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
+addpath('./src/'); % Path for functions
+addpath('./STEPS/'); % Path for STEPS
+addpath('./subsystems/'); % Path for Subsystems Simulink files
+
+%% Data directory
+data_dir = './mat/';
+
+% Simulink Model name
+mdl = 'nass_model';
+
+%% Colors for the figures
+colors = colororder;
+
+%% Frequency Vector [Hz]
+freqs = logspace(0, 3, 1000);
+
+%% Identify the IFF plant dynamics using the Simscape model
+
+% Initialize each Simscape model elements
+initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeSimplifiedNanoHexapod();
+
+% 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, '/Controller'], 1, 'openinput');             io_i = io_i + 1; % Actuator Inputs [N]
+io(io_i) = linio([mdl, '/NASS'],       3, 'openoutput', [], 'fn');  io_i = io_i + 1; % Force Sensors [N]
+
+% Identify for multi payload masses (no rotation)
+initializeReferences(); % No Spindle Rotation
+% 1kg Sample
+initializeSample('type', 'cylindrical', 'm', 1);
+G_iff_m1 = linearize(mdl, io);
+G_iff_m1.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_iff_m1.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
+
+% 25kg Sample
+initializeSample('type', 'cylindrical', 'm', 25);
+G_iff_m25 = linearize(mdl, io);
+G_iff_m25.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_iff_m25.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
+
+% 50kg Sample
+initializeSample('type', 'cylindrical', 'm', 50);
+G_iff_m50 = linearize(mdl, io);
+G_iff_m50.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_iff_m50.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
+
+% Effect of Rotation
+initializeReferences(...
+    'Rz_type', 'rotating', ...
+    'Rz_period', 1); % 360 deg/s
+initializeSample('type', 'cylindrical', 'm', 25);
+G_iff_m25_Rz = linearize(mdl, io, 0.1);
+G_iff_m25_Rz.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_iff_m25_Rz.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
+
+% Effect of Rotation - No added parallel stiffness
+initializeSimplifiedNanoHexapod('actuator_kp', 0);
+initializeReferences(...
+    'Rz_type', 'rotating', ...
+    'Rz_period', 1); % 360 deg/s
+initializeSample('type', 'cylindrical', 'm', 25);
+G_iff_m25_Rz_no_kp = linearize(mdl, io, 0.1);
+G_iff_m25_Rz_no_kp.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_iff_m25_Rz_no_kp.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
+
+%% IFF Plant - Without parallel stiffness
+f = logspace(-1,3,1000);
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:5
+    for j = i+1:6
+        plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(i,j), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(1,1), f, 'Hz'))), 'color', colors(1,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $k_p = 0$')
+for i = 2:6
+    plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(i,i), f, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+end
+plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(1,2), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
+     'DisplayName', '$f_{ni}/f_j$ - $k_p = 0$')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-4, 1e1]);
+leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(f, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_m25_Rz_no_kp(i,i), f, 'Hz')))), 'color', colors(1,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-20, 200]);
+yticks([0:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([f(1), f(end)]);
+
+%% IFF Plant - With added parallel stiffness
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:5
+    for j = i+1:6
+        plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(i,j), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(1,1), f, 'Hz'))), 'color', colors(1,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $k_p = 50N/mm$')
+for i = 2:6
+    plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(i,i), f, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+end
+plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(1,2), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
+     'DisplayName', '$f_{ni}/f_j$ - $k_p = 50N/mm$')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-4, 1e1]);
+leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(f, 180/pi*angle(squeeze(freqresp(G_iff_m25_Rz(i,i), f, 'Hz'))), 'color', colors(1,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-20, 200]);
+yticks([0:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([f(1), f(end)]);
+
+%% Effect of spindle's rotation on the IFF Plant
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:5
+    for j = i+1:6
+        plot(freqs, abs(squeeze(freqresp(G_iff_m25(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(G_iff_m25(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 0$ deg/s')
+plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 360$ deg/s')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+end
+% plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
+%      'DisplayName', '$f_{ni}/f_j$')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-4, 1e2]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', colors(1,:));
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m25_Rz(i,i), freqs, 'Hz'))), 'color', colors(2,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-20, 200]);
+yticks([0:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Effect of the payload's mass on the IFF Plant
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_iff_m1(1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
+     'DisplayName', '$f_{ni}/f_i$ - 1kg')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_iff_m25(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
+     'DisplayName', '$f_{ni}/f_i$ - 25kg')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_iff_m50(1,1), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
+     'DisplayName', '$f_{ni}/f_i$ - 50kg')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-4, 1e2]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+end
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
+end
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-20, 200]);
+yticks([0:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Verify that parallel stiffness permits to have a stable plant
+Kiff_pure_int = -200/s*eye(6);
+
+if isstable(feedback(G_iff_m25_Rz_no_kp, Kiff_pure_int, 1))
+    disp("Decentralized IFF is not stable with rotation")
+end
+
+if not(isstable(feedback(G_iff_m25_Rz, Kiff_pure_int, 1)))
+    disp("Parallel stiffness makes the decentralized IFF stable")
+else
+    warning("Decentralized IFF is not stable even with the parallel stiffness")
+end
+
+%% IFF Controller Design
+% Second order high pass filter
+wz = 2*pi*2;
+xiz = 0.7;
+Ghpf = (s^2/wz^2)/(s^2/wz^2 + 2*xiz*s/wz + 1);
+
+Kiff = -200 * ...              % Gain
+       1/(0.01*2*pi + s) * ... % LPF: provides integral action
+       Ghpf * ...              % 2nd order HPF (limit low frequency gain)
+       eye(6);                 % Diagonal 6x6 controller (i.e. decentralized)
+
+Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
+Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+
+% The designed IFF controller is saved
+save('./mat/nass_K_iff.mat', 'Kiff');
+
+%% Loop gain for the decentralized IFF
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m1(1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
+     'DisplayName', '1kg')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m25(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
+     'DisplayName', '25kg')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m50(1,1), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
+     'DisplayName', '50kg')
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
+ylim([1e-4, 1e2]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(-Kiff(1,1)*G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+end
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(-Kiff(1,1)*G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
+end
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(-Kiff(1,1)*G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-110, 200]);
+yticks([-180, -90, 0, 90, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Root Locus for the Decentralized IFF controller - 1kg Payload
+figure;
+gains = logspace(-2, 1, 200);
+
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+nexttile();
+hold on;
+
+plot(real(pole(G_iff_m1)),  imag(pole(G_iff_m1)),  'x', 'color', colors(1,:), ...
+    'DisplayName', '$g = 0$');
+plot(real(tzero(G_iff_m1)), imag(tzero(G_iff_m1)), 'o', 'color', colors(1,:), ...
+    'HandleVisibility', 'off');
+
+for g = gains
+    clpoles = pole(feedback(G_iff_m1, g*Kiff, +1));
+    plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
+        'HandleVisibility', 'off');
+end
+
+% Optimal gain
+clpoles = pole(feedback(G_iff_m1, Kiff, +1));
+plot(real(clpoles), imag(clpoles), 'kx', ...
+    'DisplayName', '$g_{opt}$');
+
+xline(0);
+yline(0);
+hold off;
+axis equal;
+xlim([-900, 100]); ylim([-100, 900]);
+xticks([-900:100:0]);
+yticks([0:100:900]);
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+xlabel('Real part'); ylabel('Imaginary part');
+
+%% Root Locus for the Decentralized IFF controller - 25kg Payload
+gains = logspace(-2, 1, 200);
+
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+nexttile();
+hold on;
+
+plot(real(pole(G_iff_m25)),  imag(pole(G_iff_m25)),  'x', 'color', colors(2,:), ...
+    'DisplayName', '$g = 0$');
+plot(real(tzero(G_iff_m25)), imag(tzero(G_iff_m25)), 'o', 'color', colors(2,:), ...
+    'HandleVisibility', 'off');
+
+for g = gains
+    clpoles = pole(feedback(G_iff_m25, g*Kiff, +1));
+    plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
+        'HandleVisibility', 'off');
+end
+
+% Optimal gain
+clpoles = pole(feedback(G_iff_m25, Kiff, +1));
+plot(real(clpoles), imag(clpoles), 'kx', ...
+    'DisplayName', '$g_{opt}$');
+
+xline(0);
+yline(0);
+hold off;
+axis equal;
+xlim([-900, 100]); ylim([-100, 900]);
+xticks([-900:100:0]);
+yticks([0:100:900]);
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+xlabel('Real part'); ylabel('Imaginary part');
+
+%% Root Locus for the Decentralized IFF controller - 50kg Payload
+gains = logspace(-2, 1, 200);
+
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+nexttile();
+hold on;
+
+plot(real(pole(G_iff_m50)),  imag(pole(G_iff_m50)),  'x', 'color', colors(3,:), ...
+    'DisplayName', '$g = 0$');
+plot(real(tzero(G_iff_m50)), imag(tzero(G_iff_m50)), 'o', 'color', colors(3,:), ...
+    'HandleVisibility', 'off');
+
+for g = gains
+    clpoles = pole(feedback(G_iff_m50, g*Kiff, +1));
+    plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
+        'HandleVisibility', 'off');
+end
+
+% Optimal gain
+clpoles = pole(feedback(G_iff_m50, Kiff, +1));
+plot(real(clpoles), imag(clpoles), 'kx', ...
+    'DisplayName', '$g_{opt}$');
+
+xline(0);
+yline(0);
+hold off;
+axis equal;
+xlim([-900, 100]); ylim([-100, 900]);
+xticks([-900:100:0]);
+yticks([0:100:900]);
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+xlabel('Real part'); ylabel('Imaginary part');

matlab/nass_2_hac.m

@@ -0,0 +1,846 @@
+%% 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
+addpath('./src/'); % Path for functions
+addpath('./STEPS/'); % Path for STEPS
+addpath('./subsystems/'); % Path for Subsystems Simulink files
+
+%% Data directory
+data_dir = './mat/';
+
+% Simulink Model name
+mdl = 'nass_model';
+
+%% Colors for the figures
+colors = colororder;
+
+%% Frequency Vector [Hz]
+freqs = logspace(0, 3, 1000);
+
+%% Identify the IFF plant dynamics using the Simscape model
+
+% 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, '/Controller'],     1, 'input');                  io_i = io_i + 1; % Actuator Inputs [N]
+io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL');  io_i = io_i + 1; % Strut errors [m]
+
+% Identify HAC Plant without using IFF
+initializeSample('type', 'cylindrical', 'm', 1);
+G_m1 = linearize(mdl, io);
+G_m1.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m1.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+initializeSample('type', 'cylindrical', 'm', 25);
+G_m25 = linearize(mdl, io);
+G_m25.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m25.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+initializeSample('type', 'cylindrical', 'm', 50);
+G_m50 = linearize(mdl, io);
+G_m50.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m50.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% Effect of Rotation
+initializeSample('type', 'cylindrical', 'm', 1);
+initializeReferences(...
+    'Rz_type', 'rotating', ...
+    'Rz_period', 1); % 360 deg/s
+
+G_m1_Rz = linearize(mdl, io, 0.1);
+G_m1_Rz.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_m1_Rz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+%% Effect of rotation on the HAC plant
+figure;
+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', '$\epsilon_{\mathcal{L}i}/f_i$, $\Omega = 0$')
+plot(freqs, abs(squeeze(freqresp(G_m1_Rz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, $\Omega = 360$ deg/s')
+plot(freqs, abs(squeeze(freqresp(G_m1(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$')
+plot(freqs, abs(squeeze(freqresp(G_m1_Rz(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$')
+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], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_m1_Rz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_m1(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_m1_Rz(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-11, 2e-5]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1(i,i), freqs, 'Hz')))), 'color', colors(1,:));
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1_Rz(i,i), freqs, 'Hz')))), 'color', colors(2,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-200, 20]);
+yticks([-180:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Effect of payload's mass on the HAC plant
+figure;
+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', '$\epsilon_{\mathcal{L}i}/f_i$, 1 kg')
+plot(freqs, abs(squeeze(freqresp(G_m25(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, 25 kg')
+plot(freqs, abs(squeeze(freqresp(G_m50(1,1), freqs, 'Hz'))), 'color', colors(3,:), ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, 50 kg')
+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], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_m25(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_m50(i,j), freqs, 'Hz'))), 'color', [colors(3,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_m50(i,i), freqs, 'Hz'))), 'color', colors(3,:), ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-11, 2e-5]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1(i,i), freqs, 'Hz')))), 'color', colors(1,:));
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25(i,i), freqs, 'Hz')))), 'color', colors(2,:));
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m50(i,i), freqs, 'Hz')))), 'color', colors(3,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-200, 20]);
+yticks([-180:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Identify HAC Plant with IFF
+initializeReferences(); % No Spindle Rotation
+initializeController('type', 'iff'); % Implemented IFF controller
+load('nass_K_iff.mat', 'Kiff'); % Load designed IFF controller
+
+% 1kg payload
+initializeSample('type', 'cylindrical', 'm', 1);
+G_hac_m1 = linearize(mdl, io);
+G_hac_m1.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_hac_m1.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% 25kg payload
+initializeSample('type', 'cylindrical', 'm', 25);
+G_hac_m25 = linearize(mdl, io);
+G_hac_m25.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_hac_m25.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% 50kg payload
+initializeSample('type', 'cylindrical', 'm', 50);
+G_hac_m50 = linearize(mdl, io);
+G_hac_m50.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_hac_m50.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% Check stability
+if not(isstable(G_hac_m1) && isstable(G_hac_m25) && isstable(G_hac_m50))
+    warning('One of HAC plant is not stable')
+end
+
+%% Comparison of the OL plant and the plant with IFF - 1kg payload
+figure;
+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')
+plot(freqs, abs(squeeze(freqresp(G_hac_m1(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
+     '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], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_hac_m1(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_hac_m1(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-10, 5e-5]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1(i,i), freqs, 'Hz')))), 'color', colors(1,:));
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_hac_m1(i,i), freqs, 'Hz')))), 'color', colors(2,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-200, 20]);
+yticks([-180:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Comparison of all the undamped FRF and all the damped FRF
+figure;
+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', '$\epsilon\mathcal{L}_i/f_i$ - 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');
+    plot(freqs, abs(squeeze(freqresp(G_m50(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
+end
+for i = 1:6
+    plot(freqs, abs(squeeze(freqresp(G_hac_m1( i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_hac_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_hac_m50(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
+end
+hold off;
+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-10, 5e-5]);
+
+ax2 = nexttile;
+hold on;
+for i =1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_m50(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
+end
+for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_m1( i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_m50(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+ylim([-200, 20]);
+yticks([-180:45:180]);
+
+linkaxes([ax1,ax2],'x');
+% xlim([1, 5e2]);
+
+%% Identify plant with "rigid" micro-station
+initializeGround('type', 'rigid');
+initializeGranite('type', 'rigid');
+initializeTy('type', 'rigid');
+initializeRy('type', 'rigid');
+initializeRz('type', 'rigid');
+initializeMicroHexapod('type', 'rigid');
+initializeSimplifiedNanoHexapod();
+initializeSample('type', 'cylindrical', 'm', 25);
+
+initializeReferences();
+initializeController('type', 'open-loop'); % Implemented IFF controller
+load('nass_K_iff.mat', 'Kiff'); % Load designed IFF controller
+
+% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Controller'],     1, 'input');                  io_i = io_i + 1; % Actuator Inputs [N]
+io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL');  io_i = io_i + 1; % Strut errors [m]
+
+G_m25_rigid = linearize(mdl, io);
+G_m25_rigid.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m25_rigid.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+%% Effect of the micro-station limited compliance on the plant dynamics
+figure;
+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', '$\epsilon_{\mathcal{L}i}/f_i$ - Rigid support')
+plot(freqs, abs(squeeze(freqresp(G_m25(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
+     '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], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_m25(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_m25_rigid( i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-10, 5e-5]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_rigid(i,i), freqs, 'Hz')))), 'color', colors(1,:));
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25(i,i), freqs, 'Hz')))), 'color', colors(2,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-200, 20]);
+yticks([-180:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Identify Dynamics with a Stiff nano-hexapod (100N/um)
+
+% Initialize each Simscape model elements
+initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeSimplifiedNanoHexapod('actuator_k', 1e8, 'actuator_kp', 0, 'actuator_c', 1e3);
+initializeSample('type', 'cylindrical', 'm', 25);
+
+% 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, '/Controller'],     1, 'input');                  io_i = io_i + 1; % Actuator Inputs [N]
+io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL');  io_i = io_i + 1; % Strut errors [m]
+
+% Identify Plant
+G_m25_pz = linearize(mdl, io);
+G_m25_pz.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m25_pz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% Compare with Nano-Hexapod alone (rigid micro-station)
+initializeGround('type', 'rigid');
+initializeGranite('type', 'rigid');
+initializeTy('type', 'rigid');
+initializeRy('type', 'rigid');
+initializeRz('type', 'rigid');
+initializeMicroHexapod('type', 'rigid');
+
+% Identify Plant
+G_m25_pz_rigid = linearize(mdl, io);
+G_m25_pz_rigid.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m25_pz_rigid.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+%% Stiff nano-hexapod - Coupling with the micro-station
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - Rigid')
+plot(freqs, abs(squeeze(freqresp(G_m25_pz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - $\mu$-station')
+plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$ - Rigid')
+plot(freqs, abs(squeeze(freqresp(G_m25_pz(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
+     'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$ - $\mu$-station')
+for i = 1:5
+    for j = i+1:6
+        plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
+             'HandleVisibility', 'off');
+        plot(freqs, abs(squeeze(freqresp(G_m25_pz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i = 2:6
+    plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(G_m25_pz(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-12, 3e-7]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz_rigid(i,i), freqs, 'Hz')))), 'color', colors(1,:));
+    plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz(i,i), freqs, 'Hz')))), 'color', colors(2,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-200, 20]);
+yticks([-180:45:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Identify Dynamics with a Soft nano-hexapod (0.01N/um)
+initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeSimplifiedNanoHexapod('actuator_k', 1e4, 'actuator_kp', 0, 'actuator_c', 1);
+
+% Initialize each Simscape model elements
+initializeSample('type', 'cylindrical', 'm', 25); % 25kg payload
+initializeController('type', 'open-loop');
+
+% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Controller'],     1, 'input');                  io_i = io_i + 1; % Actuator Inputs [N]
+io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL');  io_i = io_i + 1; % Strut errors [m]
+
+% Identify the dynamics without rotation
+initializeReferences();
+G_m1_vc = linearize(mdl, io);
+G_m1_vc.InputName  = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
+G_m1_vc.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% Identify the dynamics with 36 deg/s rotation
+initializeReferences(...
+    'Rz_type', 'rotating', ...
+    'Rz_period', 10); % 36 deg/s
+
+G_m1_vc_Rz_slow = linearize(mdl, io, 0.1);
+G_m1_vc_Rz_slow.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_m1_vc_Rz_slow.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+% Identify the dynamics with 360 deg/s rotation
+initializeReferences(...
+    'Rz_type', 'rotating', ...
+    'Rz_period', 1); % 360 deg/s
+
+G_m1_vc_Rz_fast = linearize(mdl, io, 0.1);
+G_m1_vc_Rz_fast.InputName  = {'f1',  'f2',  'f3',  'f4',  'f5',  'f6'};
+G_m1_vc_Rz_fast.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
+
+%% Soft Nano-Hexapod - effect of rotational velocity on the dynamics
+f = logspace(-1,2,200);
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(squeeze(freqresp(G_m1_vc(1,1), f, 'Hz'))), 'color', colors(1,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 0$')
+plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_slow(1,1), f, 'Hz'))), 'color', colors(2,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 36$ deg/s')
+plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_fast(1,1), f, 'Hz'))), 'color', colors(3,:), ...
+     'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 360$ deg/s')
+for i = 1:5
+    for j = i+1:6
+        plot(f, abs(squeeze(freqresp(G_m1_vc(i,j), f, 'Hz'))), 'color', [colors(1,:), 0.2], ...
+             'HandleVisibility', 'off');
+        plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_slow(i,j), f, 'Hz'))), 'color', [colors(2,:), 0.2], ...
+             'HandleVisibility', 'off');
+        plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_fast(i,j), f, 'Hz'))), 'color', [colors(3,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i = 2:6
+    plot(f, abs(squeeze(freqresp(G_m1_vc(i,i), f, 'Hz'))), 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+end
+for i = 2:6
+    plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_slow(i,i), f, 'Hz'))), 'color', colors(2,:), ...
+         'HandleVisibility', 'off');
+end
+for i = 2:6
+    plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_fast(i,i), f, 'Hz'))), 'color', colors(3,:), ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+ylim([1e-9, 1e-2]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(f, 180/pi*angle(squeeze(freqresp(G_m1_vc(i,i), f, 'Hz'))), 'color', colors(1,:));
+    plot(f, 180/pi*angle(squeeze(freqresp(G_m1_vc_Rz_slow(i,i), f, 'Hz'))), 'color', colors(2,:));
+    plot(f, 180/pi*angle(squeeze(freqresp(G_m1_vc_Rz_fast(i,i), f, 'Hz'))), 'color', colors(3,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180:90:180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([f(1), f(end)]);
+
+%% HAC Design
+% Wanted crossover
+wc = 2*pi*10; % [rad/s]
+
+% Integrator
+H_int = wc/s;
+
+% Lead to increase phase margin
+a  = 2;  % Amount of phase lead / width of the phase lead / high frequency gain
+H_lead = 1/sqrt(a)*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+
+% Low Pass filter to increase robustness
+H_lpf = 1/(1 + s/2/pi/80);
+
+% Gain to have unitary crossover at wc
+H_gain = 1./abs(evalfr(G_hac_m50(1,1), 1j*wc));
+
+% Decentralized HAC
+Khac = -H_gain  * ... % Gain
+        H_int   * ... % Integrator
+        H_lead   * ... % Low Pass filter
+        H_lpf   * ... % Low Pass filter
+        eye(6);       % 6x6 Diagonal
+
+% The designed HAC controller is saved
+save('./mat/nass_K_hac.mat', 'Khac');
+
+%% "Diagonal" loop gain for the High Authority Controller
+f = logspace(-1, 2, 1000);
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m1( i,i), f, 'Hz'))), ...
+     'color', [colors(1,:), 0.5], 'DisplayName', '1kg');
+plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m25(i,i), f, 'Hz'))), ...
+     'color', [colors(2,:), 0.5], 'DisplayName', '25kg');
+plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m50(i,i), f, 'Hz'))), ...
+     'color', [colors(3,:), 0.5], 'DisplayName', '50kg');
+for i = 2:6
+    plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m1( i,i), f, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
+    plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m25(i,i), f, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
+    plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m50(i,i), f, 'Hz'))), 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
+ylim([1e-2, 1e2]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(f, 180/pi*angle(squeeze(freqresp(-Khac(i,i)*G_hac_m1( i,i), f, 'Hz'))), 'color', [colors(1,:), 0.5]);
+    plot(f, 180/pi*angle(squeeze(freqresp(-Khac(i,i)*G_hac_m25(i,i), f, 'Hz'))), 'color', [colors(2,:), 0.5]);
+    plot(f, 180/pi*angle(squeeze(freqresp(-Khac(i,i)*G_hac_m50(i,i), f, 'Hz'))), 'color', [colors(3,:), 0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+ylim([-180, 180])
+
+linkaxes([ax1,ax2],'x');
+xlim([0.1, 100]);
+
+%% Characteristic Loci for the High Authority Controller
+Ldet_m1 = zeros(6, length(freqs));
+Lmimo_m1 = squeeze(freqresp(-G_hac_m1*Khac, freqs, 'Hz'));
+for i_f = 2:length(freqs)
+    Ldet_m1(:, i_f) = eig(squeeze(Lmimo_m1(:,:,i_f)));
+end
+Ldet_m25 = zeros(6, length(freqs));
+Lmimo_m25 = squeeze(freqresp(-G_hac_m25*Khac, freqs, 'Hz'));
+for i_f = 2:length(freqs)
+    Ldet_m25(:, i_f) = eig(squeeze(Lmimo_m25(:,:,i_f)));
+end
+Ldet_m50 = zeros(6, length(freqs));
+Lmimo_m50 = squeeze(freqresp(-G_hac_m50*Khac, freqs, 'Hz'));
+for i_f = 2:length(freqs)
+    Ldet_m50(:, i_f) = eig(squeeze(Lmimo_m50(:,:,i_f)));
+end
+
+figure;
+hold on;
+plot(real(squeeze(Ldet_m1(1,:))), imag(squeeze(Ldet_m1(1,:))), ...
+        '.', 'color', colors(1, :), ...
+        'DisplayName', '1kg');
+plot(real(squeeze(Ldet_m1(1,:))),-imag(squeeze(Ldet_m1(1,:))), ...
+        '.', 'color', colors(1, :), ...
+        'HandleVisibility', 'off');
+plot(real(squeeze(Ldet_m25(1,:))), imag(squeeze(Ldet_m25(1,:))), ...
+        '.', 'color', colors(2, :), ...
+        'DisplayName', '25kg');
+plot(real(squeeze(Ldet_m25(1,:))),-imag(squeeze(Ldet_m25(1,:))), ...
+        '.', 'color', colors(2, :), ...
+        'HandleVisibility', 'off');
+plot(real(squeeze(Ldet_m50(1,:))), imag(squeeze(Ldet_m50(1,:))), ...
+        '.', 'color', colors(3, :), ...
+        'DisplayName', '50kg');
+plot(real(squeeze(Ldet_m50(1,:))),-imag(squeeze(Ldet_m50(1,:))), ...
+        '.', 'color', colors(3, :), ...
+        'HandleVisibility', 'off');
+for i = 2:6
+    plot(real(squeeze(Ldet_m1(i,:))), imag(squeeze(Ldet_m1(i,:))), ...
+            '.', 'color', colors(1, :), ...
+            'HandleVisibility', 'off');
+    plot(real(squeeze(Ldet_m1(i,:))), -imag(squeeze(Ldet_m1(i,:))), ...
+            '.', 'color', colors(1, :), ...
+            'HandleVisibility', 'off');
+
+    plot(real(squeeze(Ldet_m25(i,:))), imag(squeeze(Ldet_m25(i,:))), ...
+            '.', 'color', colors(2, :), ...
+            'HandleVisibility', 'off');
+    plot(real(squeeze(Ldet_m25(i,:))), -imag(squeeze(Ldet_m25(i,:))), ...
+            '.', 'color', colors(2, :), ...
+            'HandleVisibility', 'off');
+
+    plot(real(squeeze(Ldet_m50(i,:))), imag(squeeze(Ldet_m50(i,:))), ...
+            '.', 'color', colors(3, :), ...
+            'HandleVisibility', 'off');
+    plot(real(squeeze(Ldet_m50(i,:))), -imag(squeeze(Ldet_m50(i,:))), ...
+            '.', 'color', colors(3, :), ...
+            'HandleVisibility', 'off');
+end
+plot(-1, 0, 'kx', 'HandleVisibility', 'off');
+hold off;
+set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
+xlabel('Real Part'); ylabel('Imaginary Part');
+axis square
+xlim([-1.8, 0.2]); ylim([-1, 1]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+%% Simulation of tomography experiments
+% Sample is not centered with the rotation axis
+% This is done by offsetfing the micro-hexapod by 0.9um
+P_micro_hexapod = [0.9e-6; 0; 0]; % [m]
+
+open(mdl);
+set_param(mdl, 'StopTime', '2');
+
+initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod('AP', P_micro_hexapod);
+initializeSample('type', 'cylindrical', 'm', 1);
+
+initializeSimscapeConfiguration('gravity', false);
+initializeLoggingConfiguration('log', 'all', 'Ts', 1e-3);
+
+initializeDisturbances(...
+    'Dw_x',  true, ... % Ground Motion - X direction
+    'Dw_y',  true, ... % Ground Motion - Y direction
+    'Dw_z',  true, ... % Ground Motion - Z direction
+    'Fdy_x', false, ... % Translation Stage - X direction
+    'Fdy_z', false, ... % Translation Stage - Z direction
+    'Frz_x', true, ... % Spindle - X direction
+    'Frz_y', true, ... % Spindle - Y direction
+    'Frz_z', true);    % Spindle - Z direction
+
+initializeReferences(...
+    'Rz_type', 'rotating', ...
+    'Rz_period', 1, ...
+    'Dh_pos', [P_micro_hexapod; 0; 0; 0]);
+
+% Open-Loop Simulation without Nano-Hexapod - 1kg payload
+initializeSimplifiedNanoHexapod('type', 'none');
+initializeController('type', 'open-loop');
+sim(mdl);
+exp_tomo_ol_m1 = simout;
+
+% Closed-Loop Simulation with NASS
+initializeSimplifiedNanoHexapod();
+initializeController('type', 'hac-iff');
+load('nass_K_iff.mat', 'Kiff');
+load('nass_K_hac.mat', 'Khac');
+
+% 1kg payload
+initializeSample('type', 'cylindrical', 'm', 1);
+sim(mdl);
+exp_tomo_cl_m1 = simout;
+
+% 25kg payload
+initializeSample('type', 'cylindrical', 'm', 25);
+sim(mdl);
+exp_tomo_cl_m25 = simout;
+
+% 50kg payload
+initializeSample('type', 'cylindrical', 'm', 50);
+sim(mdl);
+exp_tomo_cl_m50 = simout;
+
+%% Simulation of tomography experiment - 1kg payload - 360deg/s - XY errors
+figure;
+hold on;
+plot(1e6*exp_tomo_ol_m1.y.x.Data, 1e6*exp_tomo_ol_m1.y.y.Data, 'DisplayName', 'OL')
+plot(1e6*exp_tomo_cl_m1.y.x.Data(1e3:end), 1e6*exp_tomo_cl_m1.y.y.Data(1e3:end), 'color', colors(2,:), 'DisplayName', 'CL')
+hold off;
+xlabel('$D_x$ [$\mu$m]'); ylabel('$D_y$ [$\mu$m]');
+axis equal
+xlim([-2, 2]); ylim([-2, 2]);
+xticks([-2:1:2]);
+yticks([-2:1:2]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+%% Simulation of tomography experiment - 1kg payload - 360deg/s - YZ errors
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(1e6*exp_tomo_ol_m1.y.y.Data,          1e6*exp_tomo_ol_m1.y.z.Data, 'DisplayName', 'OL')
+plot(1e6*exp_tomo_cl_m1.y.y.Data(1e3:end), 1e6*exp_tomo_cl_m1.y.z.Data(1e3:end), 'color', colors(2,:), 'DisplayName', 'CL')
+hold off;
+xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
+axis equal
+xlim([-2, 2]); ylim([-0.4, 0.4]);
+xticks([-2:1:2]);
+yticks([-2:0.2:2]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile();
+hold on;
+plot(1e9*exp_tomo_cl_m1.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m1.y.z.Data(1e3:end), 'color', colors(2,:), 'DisplayName', 'CL')
+theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
+plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
+hold off;
+xlabel('$D_y$ [nm]'); ylabel('$D_z$ [nm]');
+axis equal
+xlim([-500, 500]); ylim([-100, 100]);
+xticks([-500:100:500]);
+yticks([-100:50:100]);
+leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+%% Simulation of tomography experiment - 1kg payload - 360deg/s - YZ errors
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(1e9*exp_tomo_cl_m1.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m1.y.z.Data(1e3:end), 'color', colors(1,:), 'DisplayName', '$m = 1$ kg')
+theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
+plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
+hold off;
+xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
+axis equal
+xlim([-200, 200]); ylim([-100, 100]);
+xticks([-200:50:200]); yticks([-100:50:100]);
+
+%% Simulation of tomography experiment - 25kg payload - 360deg/s - YZ errors
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(1e9*exp_tomo_cl_m25.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m25.y.z.Data(1e3:end), 'color', colors(2,:), 'DisplayName', '$m = 25$ kg')
+theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
+plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
+hold off;
+xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
+axis equal
+xlim([-200, 200]); ylim([-100, 100]);
+xticks([-200:50:200]); yticks([-100:50:100]);
+
+%% Simulation of tomography experiment - 50kg payload - 360deg/s - YZ errors
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(1e9*exp_tomo_cl_m50.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m50.y.z.Data(1e3:end), 'color', colors(3,:), 'DisplayName', '$m = 50$ kg')
+theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
+plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
+hold off;
+xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
+axis equal
+xlim([-200, 200]); ylim([-100, 100]);
+xticks([-200:50:200]); yticks([-100:50:100]);

matlab/src/initializeSample.m

@@ -2,7 +2,7 @@ function [sample] = initializeSample(args)
 
     arguments
         args.type char {mustBeMember(args.type,{'none', 'cylindrical'})} = 'none'
-        args.H (1,1) double {mustBeNumeric, mustBePositive} = 350e-3 % Height [m]
+        args.H (1,1) double {mustBeNumeric, mustBePositive} = 250e-3 % Height [m]
         args.R (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius [m]
         args.m (1,1) double {mustBeNumeric, mustBePositive} = 1 % Mass [kg]
     end

simscape-nass.bib

@@ -0,0 +1,28 @@
+@article{hauge04_sensor_contr_space_based_six,
+  author          = {G.S. Hauge and M.E. Campbell},
+  title           = {Sensors and Control of a Space-Based Six-Axis Vibration
+                  Isolation System},
+  journal         = {Journal of Sound and Vibration},
+  volume          = 269,
+  number          = {3-5},
+  pages           = {913-931},
+  year            = 2004,
+  doi             = {10.1016/s0022-460x(03)00206-2},
+  url             = {https://doi.org/10.1016/s0022-460x(03)00206-2},
+  keywords        = {parallel robot, favorite},
+}
+
+
+
+@article{preumont08_trans_zeros_struc_contr_with,
+  author          = {Preumont, Andr{\'e} and De Marneffe, Bruno and Krenk,
+                  Steen},
+  title           = {Transmission Zeros in Structural Control With Collocated
+                  Multi-Input/multi-Output Pairs},
+  journal         = {Journal of guidance, control, and dynamics},
+  volume          = 31,
+  number          = 2,
+  pages           = {428--432},
+  year            = 2008,
+}
+

simscape-nass.tex

@@ -1,4 +1,4 @@
-% Created 2025-02-17 Mon 22:37
+% Created 2025-04-16 Wed 12:11
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -13,7 +13,7 @@
  pdftitle={Simscape Model - Nano Active Stabilization System},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.4 (Org mode 9.6)}, 
+ pdfcreator={Emacs 30.1 (Org mode 9.7.26)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -23,53 +23,47 @@
 \tableofcontents
 
 \clearpage
-
+The previous chapters have established crucial foundational elements for the development of the Nano Active Stabilization System (NASS).
-From last sections:
+The uniaxial model study demonstrated that very stiff nano-hexapod configurations should be avoided due to their high coupling with the micro-station dynamics.
-\begin{itemize}
+A rotating three-degree-of-freedom model revealed that soft nano-hexapod designs prove unsuitable due to gyroscopic effect induced by the spindle rotation.
-\item Uniaxial: No stiff nano-hexapod (should also demonstrate that here)
+To further improve the model accuracy, a multi-body model of the micro-station was developed, which was carefully tuned using experimental modal analysis.
-\item Rotating: No soft nano-hexapod, Decentralized IFF can be used robustly by adding parallel stiffness
+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}.
-\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}
-
-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}
 
 \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 used in the control architecture.
+The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section \ref{sec:nass_active_damping}), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section \ref{sec:nass_hac}).
+
+The robustness of the proposed control scheme was evaluated under various operational conditions.
+Particular attention was paid to system performance under changing payload masses and varying spindle rotational velocities.
+
+This chapter concludes the conceptual design phase, with the simulation of tomography experiments providing strong evidence for the viability of the proposed NASS architecture.
 \chapter{Control Kinematics}
 \label{sec:nass_kinematics}
 Figure \ref{fig:nass_concept_schematic} presents a schematic overview of the NASS.
-This section focuses specifically on the components of the ``Instrumentation and Real-Time Control'' block.
+This section focuses on the components of the ``Instrumentation and Real-Time Control'' block.
 
 \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}
 
 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.
 
 For the Nano Active Stabilization System, computing the positioning errors in the frame of the struts involves three key steps.
-First, the system computes the desired sample pose relative to a frame representing the point where the X-ray light is focused using micro-station kinematics, as detailed in Section \ref{ssec:nass_ustation_kinematics}.
+First, desired sample pose with respect to a fixed reference frame is computed using the micro-station kinematics as detailed in Section \ref{ssec:nass_ustation_kinematics}.
+This fixed frame is located at the X-ray beam focal point, as it is where the point of interest needs to be positioned.
 Second, it measures the actual sample pose relative to the same fix frame, described in Section \ref{ssec:nass_sample_pose_error}.
 Finally, it determines the sample pose error and maps these errors to the nano-hexapod struts, as explained in Section \ref{ssec:nass_error_struts}.
 
-The complete control architecture is detailed in Section \ref{ssec:nass_control_architecture}.
+The complete control architecture is described in Section \ref{ssec:nass_control_architecture}.
 \section{Micro Station Kinematics}
 \label{ssec:nass_ustation_kinematics}
 
@@ -113,8 +107,6 @@ Using these reference signals, the desired sample position relative to the fixed
   \end{bmatrix}
   \end{align}
 \end{equation}
-
-
 \section{Computation of the sample's pose error}
 \label{ssec:nass_sample_pose_error}
 
@@ -122,7 +114,7 @@ The external metrology system measures the sample position relative to the fixed
 Due to the system's symmetry, this metrology provides measurements for five degrees of freedom: three translations (\(D_x\), \(D_y\), \(D_z\)) and two rotations (\(R_x\), \(R_y\)).
 
 The sixth degree of freedom (\(R_z\)) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics).
-This \(R_z\) rotation is estimated by combining measurements from the spindle encoder and the nano-hexapod's internal metrology, which consists of relative motion sensors in each strut (note that the micro-hexapod is not used for \(R_z\) rotation, and is therefore ignore for \(R_z\) estimation).
+This \(R_z\) rotation is estimated by combining measurements from the spindle encoder and the nano-hexapod's internal metrology, which consists of relative motion sensors in each strut (note that the micro-hexapod is not used for \(R_z\) rotation, and is therefore ignored for \(R_z\) estimation).
 
 The measured sample pose is represented by the homogeneous transformation matrix \(\bm{T}_{\text{sample}}\), as shown in equation \eqref{eq:nass_sample_pose}.
 
@@ -136,11 +128,9 @@ The measured sample pose is represented by the homogeneous transformation matrix
   0 & 0 & 0 & 1
 \end{array} \right]
 \end{equation}
-
 \section{Position error in the frame of the struts}
 \label{ssec:nass_error_struts}
 
-
 The homogeneous transformation formalism enables straightforward computation of the sample position error.
 This computation involves the previously computed homogeneous \(4 \times 4\) matrices: \(\bm{T}_{\mu\text{-station}}\) representing the desired pose, and \(\bm{T}_{\text{sample}}\) representing the measured pose.
 Their combination yields \(\bm{T}_{\text{error}}\), which expresses the position error of the sample in the frame of the rotating nano-hexapod, as shown in equation \eqref{eq:nass_transformation_error}.
@@ -149,7 +139,7 @@ Their combination yields \(\bm{T}_{\text{error}}\), which expresses the position
   \bm{T}_{\text{error}} = \bm{T}_{\mu\text{-station}}^{-1} \cdot \bm{T}_{\text{sample}}
 \end{equation}
 
-The known structure of the homogeneous transformation matrix facilitates efficient real-time computation of the inverse.
+The known structure of the homogeneous transformation matrix facilitates efficient real-time inverse computation.
 From \(\bm{T}_{\text{error}}\), the position and orientation errors \(\bm{\epsilon}_{\mathcal{X}} = [\epsilon_{D_x},\ \epsilon_{D_y},\ \epsilon_{D_z},\ \epsilon_{R_x},\ \epsilon_{R_y},\ \epsilon_{R_z}]\) of the sample are extracted using equation \eqref{eq:nass_compute_errors}:
 
 \begin{equation}\label{eq:nass_compute_errors}
@@ -163,12 +153,11 @@ From \(\bm{T}_{\text{error}}\), the position and orientation errors \(\bm{\epsil
   \end{align}
 \end{equation}
 
-Finally, these errors are mapped to the strut space through the nano-hexapod Jacobian matrix \eqref{eq:nass_inverse_kinematics}.
+Finally, these errors are mapped to the strut space using the nano-hexapod Jacobian matrix \eqref{eq:nass_inverse_kinematics}.
 
 \begin{equation}\label{eq:nass_inverse_kinematics}
   \bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}}
 \end{equation}
-
 \section{Control Architecture - Summary}
 \label{ssec:nass_control_architecture}
 
@@ -176,35 +165,35 @@ The complete control architecture is summarized in Figure \ref{fig:nass_control_
 The sample pose is measured using external metrology for five degrees of freedom, while the sixth degree of freedom (Rz) is estimated by combining measurements from the nano-hexapod encoders and spindle encoder.
 
 The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and micro-hexapod.
-Position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the nano-hexapod frame, then the Jacobian matrix \(\bm{J}\) maps these errors to individual strut coordinates.
+The position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the nano-hexapod frame.
+Then, the Jacobian matrix \(\bm{J}\) maps these errors to individual strut coordinates.
 
-For control purposes, force sensors mounted on each strut are used in a decentralized way for active damping, as detailed in Section \ref{sec:nass_active_damping}.
+For control purposes, force sensors mounted on each strut are used in a decentralized manner for active damping, as detailed in Section \ref{sec:nass_active_damping}.
 Then, the high authority controller uses the computed errors in the frame of the struts to provides real-time stabilization of the sample position (Section \ref{sec:nass_hac}).
 
 \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.}
+\caption{\label{fig:nass_control_architecture}Control architecture for the NASS. Physical systems are shown in blue, control kinematics elements in red, decentralized Integral Force Feedback controller in yellow, and centralized high authority controller in green.}
 \end{figure}
-
 \chapter{Decentralized Active Damping}
 \label{sec:nass_active_damping}
-Building upon the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the HAC-LAC strategy.
+Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the HAC-LAC strategy.
-Springs in parallel to the force sensors are used to guarantee the control robustness as was found using the 3DoF rotating model.
+The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3DoF rotating model.
 The objective here is to design a decentralized IFF controller that provides good damping of the nano-hexapod modes across payload masses ranging from \(1\) to \(50\,\text{kg}\) and rotational velocity up to \(360\,\text{deg/s}\).
-Used payloads have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg.
+The payloads used for validation have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg.
 \section{IFF Plant}
 \label{ssec:nass_active_damping_plant}
 
 Transfer functions from actuator forces \(f_i\) to force sensor measurements \(f_{mi}\) are computed using the multi-body model.
-Figure \ref{fig:nass_iff_plant_effect_kp} examines how parallel stiffness affects the plant dynamics, with identification performed at maximum spindle velocity \(\Omega_z = 360\,\text{deg/s}\) and with a payload mass of 25 kg.
+Figure \ref{fig:nass_iff_plant_effect_kp} examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity \(\Omega_z = 360\,\text{deg/s}\) and with a payload mass of 25 kg.
 
-Without parallel stiffness (Figure \ref{fig:nass_iff_plant_no_kp}), the dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model.
+Without parallel stiffness (Figure \ref{fig:nass_iff_plant_no_kp}), the plant dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model.
 Adding parallel stiffness (Figure \ref{fig:nass_iff_plant_kp}) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation.
 
-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}.
+Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the 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}
@@ -217,15 +206,16 @@ Though both cases show significant coupling around resonances, stability is guar
 \end{center}
 \subcaption{\label{fig:nass_iff_plant_kp}with parallel stiffness}
 \end{subfigure}
-\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and 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})}
+\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and a payload mass of 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 into complex conjugate zeros (\subref{fig:nass_iff_plant_kp})}
 \end{figure}
 
-The effect of rotation, shown in Figure \ref{fig:nass_iff_plant_effect_rotation}, is negligible as the actuator stiffness (\(k_a = 1\,N/\mu m\)) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model).
+The effect of rotation, as shown in Figure \ref{fig:nass_iff_plant_effect_rotation}, is negligible as the actuator stiffness (\(k_a = 1\,N/\mu m\)) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model).
 
 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.
+The poles and zeros shift in frequency as the payload mass varies.
+However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining robust stability 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}
@@ -240,16 +230,15 @@ While the poles and zeros are shifting with payload mass, the alternating patter
 \end{subfigure}
 \caption{\label{fig:nass_iff_plant_effect_rotation_payload}Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload})}
 \end{figure}
-
 \section{Controller Design}
 \label{ssec:nass_active_damping_control}
 
-Previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to gyroscopic effects caused by spindle rotation.
+The previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation.
-This finding is also confirmed with the multi-body model of the NASS: the system is unstable when using pure integrators and without parallel stiffness.
+This finding was also confirmed with the multi-body model of the NASS: the system was unstable when using pure integrators and without parallel stiffness.
 
 This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors.
 However, as illustrated in Figure \ref{fig:nass_iff_plant_kp}, adding parallel stiffness increases the low frequency gain.
-If using pure integrators, this would results in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable.
+Using pure integrators would result in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable.
 To resolve this issue, a second-order high-pass filter is introduced to limit the low frequency gain, as shown in Equation \eqref{eq:nass_kiff}.
 
 \begin{equation}\label{eq:nass_kiff}
@@ -260,19 +249,19 @@ To resolve this issue, a second-order high-pass filter is introduced to limit th
   \end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s} \cdot \frac{\frac{s^2}{\omega_z^2}}{\frac{s^2}{\omega_z^2} + 2 \xi_z \frac{s}{\omega_z} + 1}
 \end{equation}
 
-The cut-off frequency of the second-order high-pass filter is tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at \(5\,\text{Hz}\).
+The cut-off frequency of the second-order high-pass filter was tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at \(5\,\text{Hz}\).
-The overall gain is then increased to have large loop gain around resonances to be damped, as illustrated in Figure \ref{fig:nass_iff_loop_gain}.
+The overall gain was then increased to obtain a large loop gain around the resonances to be damped, as illustrated in Figure \ref{fig:nass_iff_loop_gain}.
 
 \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}.
+To verify stability, the root loci for the three payload configurations were computed, as 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.
+The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability 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}
@@ -291,40 +280,38 @@ The results demonstrate that the closed-loop poles remain within the left-half p
 \end{center}
 \subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$}
 \end{subfigure}
-\caption{\label{fig:nass_iff_root_locus}Root Loci for Decentralized IFF for three payload masses. Closed-loop poles are shown by the black crosses.}
+\caption{\label{fig:nass_iff_root_locus}Root Loci for Decentralized IFF for three payload masses. The closed-loop poles are shown by the black crosses.}
 \end{figure}
-
 \chapter{Centralized Active Vibration Control}
 \label{sec:nass_hac}
-\begin{itemize}
+The implementation of high-bandwidth position control for the nano-hexapod presents several technical challenges.
-\item[{$\square$}] Effect of micro-station compliance
+The plant dynamics exhibits complex behavior influenced by multiple factors, including payload mass, rotational velocity, and the mechanical coupling between the nano-hexapod and the micro-station.
-Compare plant with ``rigid'' u-station and normal u-station
+This section presents the development and validation of a centralized control strategy designed to achieve precise sample positioning during high-speed tomography experiments.
-\item Effect of IFF
-\item Effect of payload mass
-\item Decoupled plant
-\item Controller design
-\end{itemize}
 
-From control kinematics:
+First, a comprehensive analysis of the plant dynamics is presented in Section \ref{ssec:nass_hac_plant}, examining the effects of spindle rotation, payload mass variation, and the implementation of Integral Force Feedback (IFF).
-\begin{itemize}
+Section \ref{ssec:nass_hac_stiffness} validates previous modeling predictions that both overly stiff and compliant nano-hexapod configurations lead to degraded performance.
-\item Talk about issue of not estimating Rz from external metrology? (maybe could be nice to discuss that during the experiments!)
+Building upon these findings, Section \ref{ssec:nass_hac_controller} presents the design of a robust high-authority controller that maintains stability across varying payload masses while achieving the desired control bandwidth.
-\item Show what happens is Rz is not estimated (for instance supposed equaled to zero => increased coupling)
+
-\end{itemize}
+The performance of the developed control strategy was validated through simulations of tomography experiments in Section \ref{ssec:nass_hac_tomography}.
+These simulations included realistic disturbance sources and were used to evaluate the system performance against the stringent positioning requirements imposed by future beamline specifications.
+Particular attention was 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}
+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 the implementation of the decentralized IFF.
-\item Effect of rotation: \ref{fig:nass_undamped_plant_effect_Wz}
+The influence of spindle rotation on plant dynamics was investigated, and the results are presented in Figure \ref{fig:nass_undamped_plant_effect_Wz}.
-Add some coupling at low frequency, but still small at the considered velocity.
+While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly.
-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}
 
-\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 the payload mass decreased the 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, which can be attributed 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 strategy.
+
+\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 +327,18 @@ 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}
+The effectiveness of the IFF implementation was first evaluated with a \(1\,\text{kg}\) payload, as demonstrated in Figure \ref{fig:nass_comp_undamped_damped_plant_m1}.
-\item Effect of IFF on the plant \ref{fig:nass_comp_undamped_damped_plant_m1}
+The results indicate successful damping of the nano-hexapod resonance modes, although a minor increase in low-frequency coupling was observed.
-Modes are well damped
+This trade-off was considered acceptable, given the overall improvement in system behavior.
-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}
 
-\begin{figure}[htbp]
+The benefits of IFF implementation were further assessed across the full range of payload configurations, and the results are presented in Figure \ref{fig:nass_hac_plants}.
+For all tested payloads (\(1\,\text{kg}\), \(25\,\text{kg}\) and \(50\,\text{kg}\)), the decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics.
+More importantly, in the vicinity of the desired 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 plants (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically impossible.
+
+\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}
@@ -363,84 +351,67 @@ Between 10 and 50Hz, the plant dynamics does not vary a lot with the frequency,
 \end{center}
 \subcaption{\label{fig:nass_hac_plants}Effect of IFF on the set of plants to control}
 \end{subfigure}
-\caption{\label{fig:nass_hac_plant}Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})}
+\caption{\label{fig:nass_hac_plant}Effect of Decentralized Integral Force Feedback on the positioning plant for a \(1\,\text{kg}\) sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}).}
 \end{figure}
 
-\section{Effect of micro-station compliance}
+The coupling between the nano-hexapod and the micro-station was evaluated through a 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}):
+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.
-\begin{itemize}
+The only observable difference manifests as additional alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance.
-\item adds some alternating poles and zeros above 100Hz, which should not be an issue for control
+This result confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure.
-\end{itemize}
 
 \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{Effect of Nano-Hexapod Stiffness on System Dynamics}
+\label{ssec:nass_hac_stiffness}
 
-\section{Higher or lower nano-hexapod stiffness?}
+The influence of nano-hexapod stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models.
+These models suggest that a moderate stiffness of approximately \(1\,N/\mu m\) would provide better performance than either very stiff or very soft configurations.
 
-\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.
+For the stiff nano-hexapod analysis, a system with an 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 is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model.
 
-\begin{itemize}
+The soft nano-hexapod configuration was evaluated using a stiffness of \(0.01\,N/\mu m\) with a \(25\,\text{kg}\) payload.
-\item \textbf{Stiff nano-hexapod}:
+The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s.
-uniaxial model: high nano-hexapod stiffness induce coupling between the nano-hexapod and the micro-station dynamics.
+Figure \ref{fig:nass_soft_nano_hexapod_effect_Wz} demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies.
-considering the complex dynamics of the micro-station as shown by the modal analysis, that would result in a complex system to control
+The current approach of controlling the position in the strut frame is inadequate for soft nano-hexapods; but even shifting control to a frame matching the payload's center of mass would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects.
-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}
 
-\begin{figure}[htbp]
+\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}
 \end{center}
-\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}Stiff nano-hexapod - Coupling with the micro-station}
+\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,N/\mu m$ - Coupling with the micro-station}
 \end{subfigure}
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/nass_soft_nano_hexapod_effect_Wz.png}
 \end{center}
-\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}Soft nano-hexapod - Effect of Spindle rotational velocity}
+\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,N/\mu m$ - Effect of Spindle rotation}
 \end{subfigure}
-\caption{\label{fig:nass_soft_stiff_hexapod}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})}
+\caption{\label{fig:nass_soft_stiff_hexapod}Coupling between a stiff nano-hexapod (\(k_a = 100\,N/\mu m\)) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance (\(k_a = 0.01\,N/\mu m\)) nano-hexapod (\subref{fig:nass_soft_nano_hexapod_effect_Wz})}
 \end{figure}
-
 \section{Controller design}
+\label{ssec:nass_hac_controller}
 
-In this section, a high authority controller is design such that:
+A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure \ref{fig:nass_hac_plants}), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation.
-\begin{itemize}
+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.
-\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}
 
 \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}
+The controller performance was evaluated through two complementary analyses.
-\item ``Decentralized'' Loop Gain:
+First, the decentralized loop gain shown in Figure \ref{fig:nass_hac_loop_gain}, confirms the achievement of the desired 10Hz bandwidth.
-Bandwidth around 10Hz
+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.
-\item Characteristic Loci:
-Stable for all payloads with acceptable stability margins
-\end{itemize}
 
-\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}
@@ -455,19 +426,21 @@ Stable for all payloads with acceptable stability margins
 \end{subfigure}
 \caption{\label{fig:nass_hac_controller}High Authority Controller - ``Diagonal Loop Gain'' (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci})}
 \end{figure}
-
 \section{Tomography experiment}
+\label{ssec:nass_hac_tomography}
 
-\begin{itemize}
+The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with a particular focus on tomography scanning because of its demanding performance requirements.
-\item Validation of concept with tomography scans at the highest rotational velocity of \(\Omega_z = 360\,\text{deg/s}\)
+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.
-\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 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) by 100nm (vertical).
+The primary requirement stipulates that the point of interest must remain within beam dimensions throughout operation.
+The simulation included 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 the sample's position within the specified beam dimensions, thus 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 +456,14 @@ 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}
+The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and 50kg payloads, complementing the initial 1kg test case.
-\item Effect of payload mass (Figure \ref{fig:nass_tomography_hac_iff}):
+As illustrated in Figure \ref{fig:nass_tomography_hac_iff}, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis.
-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.
+While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions.
-\end{itemize}
 
-\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 expected 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}
@@ -507,14 +482,24 @@ Worse performance for high masses, as expected from the control analysis, but st
 \end{center}
 \subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,kg$}
 \end{subfigure}
-\caption{\label{fig:nass_tomography_hac_iff}Simulation of tomography experiments - 360deg/s. Beam size shown by dashed black}
+\caption{\label{fig:nass_tomography_hac_iff}Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse}
 \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 control strategy implements a High Authority Control - Low Authority Control architecture - a proven approach that has been specifically adapted to meet the unique requirements of the rotating NASS.
 
+The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping under various operating conditions.
+The addition of parallel springs to the force sensors has been shown to ensure stability during spindle rotation.
+The centralized High Authority Controller, operating in the frame of the struts for simplicity, has successfully achieved the desired performance objectives of maintaining a bandwidth of \(10\,\text{Hz}\) while maintaining robustness against payload mass variations.
+This investigation has confirmed that the moderate actuator stiffness of \(1\,N/\mu m\) represents an adequate choice for the nano-hexapod, as both very stiff and very compliant configurations introduce significant performance limitations.
+
+Simulations of tomography experiments have been performed, with positioning accuracy requirements defined by the expected minimum beam dimensions of \(200\,\text{nm}\) by \(100\,\text{nm}\).
+The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples.
+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}