First version (before grammar review)

matlab/mat/APA300ML_b_mat_K.CSV

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matlab/mat/APA300ML_b_out_nodes_3D.txt

@@ -0,0 +1,54 @@
+
+ LIST ALL SELECTED NODES.   DSYS=      0
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   11:07:02  MAR 25, 2021 CP=      2.062
+
+ Unknown                                                                       
+
+
+
+    NODE        X             Y             Z           THXY     THYZ     THZX
+   840914   0.0000        0.0000       0.28000E-001     0.00     0.00     0.00
+   840915   0.0000        0.0000      -0.28000E-001     0.00     0.00     0.00
+   840916 -0.34000E-001   0.0000        0.0000          0.00     0.00     0.00
+   840917  0.34000E-001   0.0000        0.0000          0.00     0.00     0.00
+
+ LIST MASTERS ON ALL SELECTED NODES.
+ CURRENT DOF SET= UX   UY   UZ   ROTX ROTY ROTZ
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   11:07:02  MAR 25, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
+     NODE  LABEL     SUPPORT
+    840914  UX  
+    840914  UY  
+    840914  UZ  
+    840914  ROTX
+    840914  ROTY
+    840914  ROTZ
+    840915  UX  
+    840915  UY  
+    840915  UZ  
+    840915  ROTX
+    840915  ROTY
+    840915  ROTZ
+    840916  UX  
+    840916  UY  
+    840916  UZ  
+    840916  ROTX
+    840916  ROTY
+    840916  ROTZ
+    840917  UX  
+    840917  UY  
+    840917  UZ  
+    840917  ROTX
+    840917  ROTY
+    840917  ROTZ

matlab/mat/full_APA300ML_K.CSV

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matlab/mat/full_APA300ML_out_nodes_3D.txt

@@ -0,0 +1,61 @@
+
+ LIST ALL SELECTED NODES.   DSYS=      0
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
+
+    NODE        X             Y             Z           THXY     THYZ     THZX
+        1   0.0000        0.0000       0.28000E-001     0.00     0.00     0.00
+  1228810   0.0000        0.0000      -0.28000E-001     0.00     0.00     0.00
+  1228811 -0.30000E-001   0.0000        0.0000          0.00     0.00     0.00
+  1228812  0.10000E-001   0.0000        0.0000          0.00     0.00     0.00
+  1228813  0.30000E-001   0.0000        0.0000          0.00     0.00     0.00
+
+ LIST MASTERS ON ALL SELECTED NODES.
+ CURRENT DOF SET= UX   UY   UZ   ROTX ROTY ROTZ
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
+     NODE  LABEL     SUPPORT
+         1  UX  
+         1  UY  
+         1  UZ  
+         1  ROTX
+         1  ROTY
+         1  ROTZ
+   1228810  UX  
+   1228810  UY  
+   1228810  UZ  
+   1228810  ROTX
+   1228810  ROTY
+   1228810  ROTZ
+   1228811  UX  
+   1228811  UY  
+   1228811  UZ  
+   1228811  ROTX
+   1228811  ROTY
+   1228811  ROTZ
+   1228812  UX  
+   1228812  UY  
+   1228812  UZ  
+   1228812  ROTX
+   1228812  ROTY
+   1228812  ROTZ
+   1228813  UX  
+   1228813  UY  
+   1228813  UZ  
+   1228813  ROTX
+   1228813  ROTY
+   1228813  ROTZ

matlab/src/initializeNanoHexapodFinal.m

@@ -41,9 +41,9 @@ arguments
     args.actuator_k   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*380000
     args.actuator_ke  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*4952605
     args.actuator_ka  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2476302
-    args.actuator_c   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*20
+    args.actuator_c   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
-    args.actuator_ce  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*200
+    args.actuator_ce  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100
-    args.actuator_ca  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100
+    args.actuator_ca  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*50
     args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m]
     % For Flexible Frame
     args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m]

matlab/test_nhexa_1_suspended_table.m

@@ -0,0 +1,67 @@
+% Matlab Init                                              :noexport:ignore:
+
+%% test_nhexa_table.m
+
+%% 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
+
+%% Initialize Parameters for Simscape model
+table_type = 'Rigid'; % On top of vibration table
+device_type = 'None'; % On top of vibration table
+payload_num = 0; % No Payload
+
+% Simulink Model name
+mdl = 'test_bench_nano_hexapod';
+
+%% Colors for the figures
+colors = colororder;
+
+%% Frequency Vector
+freqs = logspace(log10(10), log10(2e3), 1000);
+
+% Simscape Model of the suspended table
+% <<ssec:test_nhexa_table_model>>
+
+% The Simscape model of the suspended table simply consists of two solid bodies connected by 4 springs.
+% The 4 springs are here modelled with "bushing joints" that have stiffness and damping properties in x, y and z directions.
+% The 3D representation of the model is displayed in Figure ref:fig:test_nhexa_suspended_table_simscape where the 4 "bushing joints" are represented by the blue cylinders.
+
+% #+name: fig:test_nhexa_suspended_table_simscape
+% #+caption: 3D representation of the simscape model
+% #+attr_latex: :width 0.8\linewidth
+% [[file:figs/test_nhexa_suspended_table_simscape.png]]
+
+% The model order is 12, and it represents the 6 suspension modes.
+% The inertia properties of the parts are set from the geometry and material densities.
+% The stiffness of the springs was initially set from the datasheet nominal value of $17.8\,N/mm$ and then reduced down to $14\,N/mm$ to better match the measured suspension modes.
+% The stiffness of the springs in the horizontal plane is set at $0.5\,N/mm$.
+% The obtained suspension modes of the simscape model are compared with the measured ones in Table ref:tab:test_nhexa_suspended_table_simscape_modes.
+
+
+%% Configure Simscape Model
+table_type = 'Suspended'; % On top of vibration table
+device_type = 'None'; % No device on the vibration table
+payload_num = 0; % No Payload
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F'],      1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F_v'],  1, 'openoutput'); io_i = io_i + 1;
+
+%% Run the linearization
+G = linearize(mdl, io);
+G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
+G.OutputName  = {'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
+
+%% Compute the resonance frequencies
+ws = eig(G.A);
+ws = ws(imag(ws) > 0);

matlab/test_nhexa_2_dynamics.m

@@ -0,0 +1,332 @@
+% Matlab Init                                              :noexport:ignore:
+
+%% test_nhexa_dynamics.m
+% Identification of the nano-hexapod dynamics from u to dL and to Vs
+% Encoders are fixed to the plates
+
+%% 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
+
+%% Colors for the figures
+colors = colororder;
+
+%% Frequency Vector
+freqs = logspace(log10(10), log10(2e3), 1000);
+
+% Identification of the dynamics
+% <<ssec:test_nhexa_identification>>
+
+% The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) are identified by generating a low pass filtered white noise for each of the command signals, one by one.
+
+% The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure ref:fig:test_nhexa_identified_frf_de.
+% The diagonal terms are displayed using colorful lines, and all the 30 off-diagonal terms are displayed by grey lines.
+
+% All the six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity.
+% Below the first suspension mode, good decoupling can be observed (the amplitude of the all of off-diagonal terms are $\approx 20$ times smaller than the diagonal terms).
+
+% From 10Hz up to 1kHz, around 10 resonance frequencies can be observed.
+% The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section ref:ssec:test_nhexa_enc_struts_modal_analysis.
+% Then, three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section ref:ssec:test_nhexa_comp_model_coupling).
+% Except the mode at 237Hz, their amplitude is rather low.
+% Two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform.
+% Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform.
+
+% Up to at least 1kHz, an alternating pole/zero pattern is observed, which renders the control easier to tune.
+% This would not have been the case if the encoders were fixed to the struts.
+
+
+%% Load identification data
+load('test_nhexa_identification_data_mass_0.mat', 'data');
+
+%% Setup useful variables
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(1/Ts); % Number of points for the FFT computation
+win = hanning(Nfft); % Hanning window
+Noverlap = floor(Nfft/2); % Overlap between frequency analysis
+
+% And we get the frequency vector
+[~, f] = tfestimate(data{1}.u, data{1}.de, win, Noverlap, Nfft, 1/Ts);
+
+%% Transfer function from u to dLm
+G_de = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_de(:,:,i) = tfestimate(data{i}.u, data{i}.de, win, Noverlap, Nfft, 1/Ts);
+end
+
+%% Transfer function from u to Vs
+G_Vs = zeros(length(f), 6, 6);
+
+for i = 1:6
+    G_Vs(:,:,i) = tfestimate(data{i}.u, data{i}.Vs, win, Noverlap, Nfft, 1/Ts);
+end
+
+%% Bode plot for the transfer function from u to dLm
+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(G_de(:, i, j)), 'color', [0, 0, 0, 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i =1:6
+    set(gca,'ColorOrderIndex',i)
+    plot(f, abs(G_de(:,i, i)), ...
+         'DisplayName', sprintf('$d_{e,%i}/u_%i$', i, i));
+end
+plot(f, abs(G_de(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
+     'DisplayName', '$d_{e,i}/u_j$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
+ylim([1e-8, 5e-4]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i =1:6
+    set(gca,'ColorOrderIndex',i)
+    plot(f, 180/pi*angle(G_de(:,i, i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+
+
+% #+name: fig:test_nhexa_identified_frf_de
+% #+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{d}_e$. The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.
+% #+RESULTS:
+% [[file:figs/test_nhexa_identified_frf_de.png]]
+
+
+% Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure ref:fig:test_nhexa_identified_frf_Vs.
+% Alternating poles and zeros is observed up to at least 2kHz, which is a necessary characteristics in order to apply decentralized IFF.
+% Similar to what was observed for the encoder outputs, all the "diagonal" terms are well superimposed, indicating that the same controller can be applied for all the struts.
+% The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF.
+
+
+%% Bode plot of the IFF Plant (transfer function from u to Vs)
+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(G_Vs(:, i, j)), 'color', [0, 0, 0, 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+for i =1:6
+    set(gca,'ColorOrderIndex',i)
+    plot(f, abs(G_Vs(:,i , i)), ...
+         'DisplayName', sprintf('$V_{s%i}/u_%i$', i, i));
+end
+plot(f, abs(G_Vs(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
+     'DisplayName', '$V_{si}/u_j$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 4);
+leg.ItemTokenSize(1) = 15;
+ylim([1e-3, 6e1]);
+
+ax2 = nexttile;
+hold on;
+for i =1:6
+    set(gca,'ColorOrderIndex',i)
+    plot(f, 180/pi*angle(G_Vs(:,i, i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+% Effect of payload mass on the dynamics
+% <<ssec:test_nhexa_added_mass>>
+
+% As one major challenge in the control of the NASS is the wanted robustness to change of payload mass, it is necessary to understand how the dynamics of the nano-hexapod changes with a change of payload mass.
+
+% In order to study this change of dynamics with the payload mass, up to three "cylindrical masses" of $13\,kg$ each can be added for a total of $\approx 40\,kg$.
+% These three cylindrical masses on top of the nano-hexapod are shown in Figure ref:fig:test_nhexa_table_mass_3.
+
+% #+name: fig:test_nhexa_table_mass_3
+% #+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,kg$
+% #+attr_org: :width 800px
+% #+attr_latex: :width 0.8\linewidth
+% [[file:figs/test_nhexa_table_mass_3.jpg]]
+
+
+%% Load identification Data
+meas_added_mass = {...
+    load('test_nhexa_identification_data_mass_0.mat', 'data'), ....
+    load('test_nhexa_identification_data_mass_1.mat', 'data'), ....
+    load('test_nhexa_identification_data_mass_2.mat', 'data'), ....
+    load('test_nhexa_identification_data_mass_3.mat', 'data')};
+
+%% Setup useful variables
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(1/Ts); % Number of points for the FFT computation
+win = hanning(Nfft); % Hanning window
+Noverlap = floor(Nfft/2); % Overlap between frequency analysis
+
+% And we get the frequency vector
+[~, f] = tfestimate(meas_added_mass{1}.data{1}.u, meas_added_mass{1}.data{1}.de, win, Noverlap, Nfft, 1/Ts);
+
+G_de = {};
+
+for i_mass = [0:3]
+    G_de(i_mass+1) = {zeros(length(f), 6, 6)};
+    for i_strut = 1:6
+        G_de{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_mass+1}.data{i_strut}.u, meas_added_mass{i_mass+1}.data{i_strut}.de, win, Noverlap, Nfft, 1/Ts);
+    end
+end
+
+%% IFF Plant (transfer function from u to Vs)
+G_Vs = {};
+
+for i_mass = [0:3]
+    G_Vs(i_mass+1) = {zeros(length(f), 6, 6)};
+    for i_strut = 1:6
+        G_Vs{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_mass+1}.data{i_strut}.u, meas_added_mass{i_mass+1}.data{i_strut}.Vs, win, Noverlap, Nfft, 1/Ts);
+    end
+end
+
+save('./mat/test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de')
+
+%% Load the identified transfer functions
+frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de');
+
+
+
+% The obtained frequency response functions from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure ref:fig:test_nhexa_identified_frf_de_masses.
+% As expected, the frequency of the suspension modes are decreasing with an increase of the payload mass.
+% The low frequency gain does not change as it is linked to the stiffness property of the nano-hexapod, and not to its mass property.
+
+% The frequencies of the two flexible modes of the top plate are first decreased a lot when the first mass is added (from $\approx 700\,Hz$ to $\approx 400\,Hz$).
+% This is due to the fact that the added mass is composed of two half cylinders which are not fixed together.
+% It therefore adds a lot of mass to the top plate without adding stiffness in one direction.
+% When more than one "mass layer" is added, the half cylinders are added with some angles such that rigidity are added in all directions (see how the three mass "layers" are positioned in Figure ref:fig:test_nhexa_table_mass_3).
+% In that case, the frequency of these flexible modes are increased.
+% In practice, the payload should be one solid body, and no decrease of the frequency of this flexible mode should be observed.
+% The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass is increased.
+
+% The measured FRF from $u_i$ to $V_{si}$ are shown in Figure ref:fig:test_nhexa_identified_frf_Vs_masses.
+% For all the tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust way.
+
+
+%% Bode plot for the transfer function from u to dLm - Several payloads
+masses = [0, 13, 26, 39];
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i_mass = [0:3]
+    % Diagonal terms
+    plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.5], ...
+         'DisplayName', sprintf('$d_{ei}/u_i$ - %i kg', masses(i_mass+1)));
+    for i = 2:6
+        plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5], ...
+             'HandleVisibility', 'off');
+    end
+    % % Off-Diagonal terms
+    % for i = 1:5
+    %     for j = i+1:6
+    %         plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ...
+    %              'HandleVisibility', 'off');
+    %     end
+    % end
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
+ylim([1e-8, 5e-4]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i_mass = [0:3]
+    for i =1:6
+        plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5]);
+    end
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+ylim([-90, 180])
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Bode plot for the transfer function from u to dLm
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i_mass = [0:3]
+    % Diagonal terms
+    plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.5], ...
+         'DisplayName', sprintf('$V_{si}/u_i$ - %i kg', masses(i_mass+1)));
+    for i = 2:6
+        plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5], ...
+             'HandleVisibility', 'off');
+    end
+    % % Off-Diagonal terms
+    % for i = 1:5
+    %     for j = i+1:6
+    %         plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ...
+    %              'HandleVisibility', 'off');
+    %     end
+    % end
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
+ylim([1e-2, 1e2]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i_mass = [0:3]
+    for i =1:6
+        plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5]);
+    end
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);

matlab/test_nhexa_3_model.m

@@ -0,0 +1,549 @@
+% Matlab Init                                              :noexport:ignore:
+
+%% test_nhexa_3_model.m
+% Compare the measured dynamics from u to de and to Vs with the Simscape model
+
+%% 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
+
+%% Initialize Parameters for Simscape model
+table_type = 'Rigid'; % On top of vibration table
+device_type = 'None'; % On top of vibration table
+payload_num = 0; % No Payload
+
+% Simulink Model name
+mdl = 'test_bench_nano_hexapod';
+
+%% Colors for the figures
+colors = colororder;
+
+%% Frequency Vector
+freqs = logspace(log10(10), log10(2e3), 1000);
+
+% Extract transfer function matrices from the Simscape Model      :noexport:
+
+%% Extract the transfer function matrix from the Simscape model
+% Initialization of the Simscape model
+table_type = 'Suspended'; % On top of vibration table
+device_type = 'Hexapod'; % Nano-Hexapod
+payload_num = 0; % No Payload
+
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', '2dof');
+
+% Identify the FRF matrix from u to [de,Vs]
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/u'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
+io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoders
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Encoders
+
+G_de = {};
+G_Vs = {};
+
+for i = [0:3]
+    payload_num = i; % Change the payload on the nano-hexapod
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0);
+    G.InputName  = {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'};
+    G.OutputName = {'de1', 'de2', 'de3', 'de4', 'de5', 'de6', ...
+                    'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'};
+    G_de(i+1) = {G({'de1', 'de2', 'de3', 'de4', 'de5', 'de6'},:)};
+    G_Vs(i+1) = {G({'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'},:)};
+end
+
+% Save the identified plants
+save('./mat/test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de')
+
+%% The same identification is performed, but this time with
+% "flexible" model of the APA
+table_type = 'Suspended'; % On top of vibration table
+device_type = 'Hexapod'; % Nano-Hexapod
+payload_num = 0; % No Payload
+
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'plates', ...
+                                       'actuator_type', 'flexible');
+
+G_de = {};
+G_Vs = {};
+
+for i = [0:3]
+    payload_num = i; % Change the payload on the nano-hexapod
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0);
+    G.InputName  = {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'};
+    G.OutputName = {'de1', 'de2', 'de3', 'de4', 'de5', 'de6', ...
+                    'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'};
+    G_de(i+1) = {G({'de1', 'de2', 'de3', 'de4', 'de5', 'de6'},:)};
+    G_Vs(i+1) = {G({'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'},:)};
+end
+
+% Save the identified plants
+save('./mat/test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de')
+
+% Nano-Hexapod model dynamics
+% <<ssec:test_nhexa_comp_model>>
+
+
+%% Load Simscape Model and measured FRF
+sim_ol  = load('test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de');
+frf_ol  = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de');
+
+
+
+% The Simscape model of the nano-hexapod is first configured with 4-DoF flexible joints, 2-DoF APA and rigid top and bottom platforms.
+% The stiffness of the flexible joints are chosen based on the values estimated using the test bench and based on FEM.
+% The parameters of the APA model are the ones determined from the test bench of the APA.
+% The $6 \times 6$ transfer function matrices from $\mathbf{u}$ to $\mathbf{d}_e$ and from $\mathbf{u}$ to $\mathbf{V}_s$ are extracted then from the Simscape model.
+
+% A first feature that should be checked is that the model well represents the "direct" terms of the measured FRF matrix.
+% To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_diag.
+% It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modelled.
+% The three resonances that were attributed to "internal" flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable as the APA are here modelled as a simple uniaxial 2-DoF system.
+% At higher frequencies, no resonances can be seen in the model, as the as the top plate and the encoder supports are modelled as rigid bodies.
+
+
+%% Diagonal elements of the FRF matrix from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:,1, 1)), 'color', [colors(1,:),0.5], ...
+     'DisplayName', '$d_{ei}/u_i$ - FRF')
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.5], ...
+     'DisplayName', '$d_{ei}/u_i$ - Model')
+for i = 2:6
+    plot(frf_ol.f, abs(frf_ol.G_de{1}(:,i, i)), 'color', [colors(1,:),0.5], ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(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/V]'); set(gca, 'XTickLabel',[]);
+ylim([1e-8, 5e-4]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{1}(:,i, i)), 'color', [colors(1,:),0.5]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_de{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Diagonal elements of the FRF matrix from u to Vs
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_Vs{1}(:,1, 1)), 'color', [colors(1,:),0.5], ...
+     'DisplayName', '$V_{si}/u_i$ - FRF')
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{1}(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
+     'DisplayName', '$V_{si}/u_i$ - Model')
+for i = 2:6
+    plot(frf_ol.f, abs(frf_ol.G_Vs{1}(:,i, i)), 'color', [colors(1,:),0.5], ...
+         'HandleVisibility', 'off');
+    plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
+         'HandleVisibility', 'off');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
+legend('location', 'southeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:6
+    plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{1}(:,i, i)), 'color', [colors(1,:),0.5]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_Vs{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+% Modelling dynamical coupling
+% <<ssec:test_nhexa_comp_model_coupling>>
+
+% Another wanted feature of the model is that it well represents the coupling in the system as this is often the limiting factor for the control of MIMO systems.
+% Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\mathbf{u}$ to $\mathbf{d}_e$, only the first "column" is compared (Figure ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$.
+% It can be seen that the coupling in the model is well matching the measurements up to the first un-modelled flexible mode at 237Hz.
+% Similar results are observed for all the other coupling terms, as well as for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$.
+
+
+%% Comparison of the plants (encoder output) when tuning the misalignment
+i_input = 1;
+
+figure;
+tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight');
+
+ax1 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 1, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(1, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
+
+ax2 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 2, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(2, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 3, i_input)), ...
+     'DisplayName', 'Measurements');
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(3, i_input), freqs, 'Hz'))), ...
+     'DisplayName', 'Model (2-DoF APA)');
+text(54, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax4 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 4, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(4, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
+xticks([50, 100, 200, 400])
+
+ax5 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 5, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(5, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([50, 100, 200, 400])
+
+ax6 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 6, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(6, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([50, 100, 200, 400])
+
+linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
+xlim([50, 5e2]); ylim([1e-8, 5e-4]);
+
+
+
+% #+name: fig:test_nhexa_comp_simscape_de_all
+% #+caption: Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$
+% #+RESULTS:
+% [[file:figs/test_nhexa_comp_simscape_de_all.png]]
+
+% The APA300ML are then modelled with a /super-element/ extracted from a FE-software.
+% The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_de_all_flex.
+% While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modelled.
+% Even the mode 395Hz can be observed in the model.
+% Therefore, if the modes of the struts are to be modelled, the /super-element/ of the APA300ML may be used, at the cost of obtaining a much higher order model.
+
+
+%% Load the plant model with Flexible APA
+flex_ol = load('test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de');
+
+%% Comparison of the plants (encoder output) when tuning the misalignment
+i_input = 1;
+
+figure;
+tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight');
+
+ax1 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 1, i_input)));
+plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(1, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
+
+ax2 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 2, i_input)));
+plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(2, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 3, i_input)), ...
+     'DisplayName', 'Measurements');
+plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(3, i_input), freqs, 'Hz'))), ...
+     'DisplayName', 'Model (Flexible APA)');
+text(54, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax4 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 4, i_input)));
+plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(4, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
+xticks([50, 100, 200, 400])
+
+ax5 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 5, i_input)));
+plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(5, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([50, 100, 200, 400])
+
+ax6 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 6, i_input)));
+plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(6, i_input), freqs, 'Hz'))));
+text(54, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([50, 100, 200, 400])
+
+linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
+xlim([50, 5e2]); ylim([1e-8, 5e-4]);
+
+% Modelling the effect of payload mass
+% <<ssec:test_nhexa_comp_model_masses>>
+
+% Another important characteristics of the model is that it should well represents the dynamics of the system for all considered payloads.
+% The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure ref:fig:test_nhexa_comp_simscape_diag_masses.
+% The observed shift to lower frequency of the suspension modes with an increased payload mass is well represented by the Simscape model.
+% The complex conjugate zeros are also well matching with the experiments both for the encoder outputs (Figure ref:fig:test_nhexa_comp_simscape_de_diag_masses) and the force sensor outputs (Figure ref:fig:test_nhexa_comp_simscape_Vs_diag_masses).
+
+% Note that the model displays smaller damping that what is observed experimentally for high values of the payload mass.
+% One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping).
+% However, as decentralized IFF will be applied, the damping will be brought actively, and the open-loop damping value should have very little impact on the obtained plant.
+
+
+%% Bode plot for the transfer function from u to de
+masses = [0, 13, 26, 39];
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i_mass = [0:3]
+    plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ...
+         'DisplayName', sprintf('Meas (%i kg)', masses(i_mass+1)));
+    for i = 2:6
+        plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+    set(gca, 'ColorOrderIndex', i_mass+1)
+    plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--', ...
+         'DisplayName', 'Simscape');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+ylim([5e-8, 1e-3]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i_mass = [0:3]
+    for i =1:6
+        plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]);
+    end
+    set(gca, 'ColorOrderIndex', i_mass+1)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360);
+ylim([-45, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([20, 2e2]);
+xticks([20, 50, 100, 200])
+
+%% Bode plot for the transfer function from u to Vs
+masses = [0, 13, 26, 39];
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i_mass = 0:3
+    plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ...
+         'DisplayName', sprintf('Meas (%i kg)', masses(i_mass+1)));
+    for i = 2:6
+        plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ...
+             'HandleVisibility', 'off');
+    end
+    plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(1,1), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:), ...
+         'DisplayName', 'Simscape');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+ylim([1e-3, 1e2]);
+leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+leg.ItemTokenSize(1) = 15;
+
+ax2 = nexttile;
+hold on;
+for i_mass = 0:3
+    for i =1:6
+        plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]);
+    end
+    plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(i,i), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+
+linkaxes([ax1,ax2],'x');
+xlim([20, 2e2]);
+xticks([20, 50, 100, 200])
+
+
+
+% #+name: fig:test_nhexa_comp_simscape_diag_masses
+% #+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag})
+% #+attr_latex: :options [htbp]
+% #+begin_figure
+% #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$}
+% #+attr_latex: :options {0.49\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :width 0.95\linewidth
+% [[file:figs/test_nhexa_comp_simscape_de_diag_masses.png]]
+% #+end_subfigure
+% #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$}
+% #+attr_latex: :options {0.49\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :width 0.95\linewidth
+% [[file:figs/test_nhexa_comp_simscape_Vs_diag_masses.png]]
+% #+end_subfigure
+% #+end_figure
+
+% In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the 39kg payload in Figure ref:fig:test_nhexa_comp_simscape_de_all_high_mass.
+% Excellent match between the experimental coupling and the model coupling is observed.
+% The model therefore well represents the system dynamical coupling for different considered payloads.
+
+
+%% Comparison of the plants (encoder output) when tuning the misalignment
+i_input = 1;
+
+figure;
+tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight');
+
+ax1 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 1, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(1, i_input), freqs, 'Hz'))));
+text(12, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
+
+ax2 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 2, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(2, i_input), freqs, 'Hz'))));
+text(12, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 3, i_input)), ...
+     'DisplayName', 'Measurements');
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(3, i_input), freqs, 'Hz'))), ...
+     'DisplayName', 'Model (2-DoF APA)');
+text(12, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+leg.ItemTokenSize(1) = 15;
+
+ax4 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 4, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(4, i_input), freqs, 'Hz'))));
+text(12, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
+xticks([10, 50, 100, 200, 400])
+
+ax5 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 5, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(5, i_input), freqs, 'Hz'))));
+text(12, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xticks([10, 50, 100, 200, 400])
+
+ax6 = nexttile();
+hold on;
+plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 6, i_input)));
+plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(6, i_input), freqs, 'Hz'))));
+text(12, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+xticks([10, 50, 100, 200, 400])
+
+linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
+xlim([10, 5e2]); ylim([1e-8, 5e-4]);

test-bench-nano-hexapod.tex

@@ -1,4 +1,4 @@
-% Created 2024-10-27 Sun 14:49
+% Created 2024-10-29 Tue 15:32
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -254,329 +254,322 @@ Simscape & 1.3 Hz & 1.8 Hz & 6.8 Hz & 9.5 Hz\\
 
 \end{table}
 
-\chapter{Nano-Hexapod Dynamics}
+\section*{Conclusion}
+In this section, a suspended table with low frequency suspension modes and high frequency flexible modes was presented.
+This suspended table will be used in Section \ref{sec:test_nhexa_dynamics} for dynamical identification of the Nano-Hexapod.
+The objective is to be able to accurately identify the dynamics of the nano-hexapod, isolated from complex support dynamics.
+The key point of this strategy is to be able to accurately model the suspended table.
+
+To do so, a modal analysis of the suspended table was performed in Section \ref{ssec:test_nhexa_table_identification}, validating the low frequency suspension modes and high frequency flexible modes.
+Then, a multi-body model of this suspended table was tuned to match with the measurements (Section \ref{ssec:test_nhexa_table_model}).
+
+\chapter{Nano-Hexapod Measured Dynamics}
 \label{sec:test_nhexa_dynamics}
-In Figure \ref{fig:test_nhexa_nano_hexapod_signals} is shown a block diagram of the experimental setup.
+The Nano-Hexapod is then mounted on top of the suspended table as shown in Figure \ref{fig:test_nhexa_hexa_suspended_table}.
-When possible, the notations are consistent with this diagram and summarized in Table \ref{tab:list_signals}.
+All the instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) are setup and connected to the nano-hexapod using many cables.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_hexa_suspended_table.jpg}
+\caption{\label{fig:test_nhexa_hexa_suspended_table}Mounted Nano-Hexapod on top of the suspended table}
+\end{figure}
+
+A modal analysis of the nano-hexapod is first performed in Section \ref{ssec:test_nhexa_enc_struts_modal_analysis}.
+It will be used to better understand the measured dynamics from actuators to sensors.
+
+A block diagram schematic of the (open-loop) system is shown in Figure \ref{fig:test_nhexa_nano_hexapod_signals}.
+The transfer function from controlled signals \(\mathbf{u}\) to the force sensors voltages \(\mathbf{V}_s\) and to the encoders measured displacements \(\mathbf{d}_e\) are identified in Section \ref{ssec:test_nhexa_identification}.
+The effect of the payload mass on the dynamics is studied in Section \ref{ssec:test_nhexa_added_mass}.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,scale=1]{figs/test_nhexa_nano_hexapod_signals.png}
-\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the system with named signals}
+\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the system. Command signal generated by the speedgoat is \(\mathbf{u}\), the measured dignals are \(\mathbf{d}_{e}\) and \(\mathbf{V}_s\). Units are indicated in square brackets.}
 \end{figure}
 
+\section{Modal analysis}
+\label{ssec:test_nhexa_enc_struts_modal_analysis}
+
+In order to ease the future analysis of the measured plant dynamics, a basic modal analysis of the nano-hexapod is performed.
+Five 3-axis accelerometers are fixed on the top platform of the nano-hexapod (Figure \ref{fig:test_nhexa_modal_analysis}) and the top platform is excited using an instrumented hammer.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_modal_analysis.jpg}
+\caption{\label{fig:test_nhexa_modal_analysis}Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis}
+\end{figure}
+
+Between 100Hz and 200Hz, 6 suspension modes (i.e. rigid body modes of the top platform) are identified.
+At around 700Hz, two flexible modes of the top plate are observed (see Figure \ref{fig:test_nhexa_hexa_flexible_modes}).
+These modes are summarized in Table \ref{tab:test_nhexa_hexa_modal_modes_list}.
+
 \begin{table}[htbp]
 \centering
-\begin{tabularx}{\linewidth}{Xllll}
+\begin{tabularx}{0.7\linewidth}{ccX}
 \toprule
- & \textbf{Unit} & \textbf{Matlab} & \textbf{Vector} & \textbf{Elements}\\
+\textbf{Mode} & \textbf{Frequency} & \textbf{Description}\\
 \midrule
-Control Input (wanted DAC voltage) & \texttt{[V]} & \texttt{u} & \(\bm{u}\) & \(u_i\)\\
+1 & 120 Hz & Suspension Mode: Y-translation\\
-DAC Output Voltage & \texttt{[V]} & \texttt{u} & \(\tilde{\bm{u}}\) & \(\tilde{u}_i\)\\
+2 & 120 Hz & Suspension Mode: X-translation\\
-PD200 Output Voltage & \texttt{[V]} & \texttt{ua} & \(\bm{u}_a\) & \(u_{a,i}\)\\
+3 & 145 Hz & Suspension Mode: Z-translation\\
-Actuator applied force & \texttt{[N]} & \texttt{tau} & \(\bm{\tau}\) & \(\tau_i\)\\
+4 & 165 Hz & Suspension Mode: Y-rotation\\
-\midrule
+5 & 165 Hz & Suspension Mode: X-rotation\\
-Strut motion & \texttt{[m]} & \texttt{dL} & \(d\bm{\mathcal{L}}\) & \(d\mathcal{L}_i\)\\
+6 & 190 Hz & Suspension Mode: Z-rotation\\
-Encoder measured displacement & \texttt{[m]} & \texttt{dLm} & \(d\bm{\mathcal{L}}_m\) & \(d\mathcal{L}_{m,i}\)\\
+7 & 692 Hz & (flexible) Membrane mode of the top platform\\
-\midrule
+8 & 709 Hz & Second flexible mode of the top platform\\
-Force Sensor strain & \texttt{[m]} & \texttt{epsilon} & \(\bm{\epsilon}\) & \(\epsilon_i\)\\
-Force Sensor Generated Voltage & \texttt{[V]} & \texttt{taum} & \(\tilde{\bm{\tau}}_m\) & \(\tilde{\tau}_{m,i}\)\\
-Measured Generated Voltage & \texttt{[V]} & \texttt{taum} & \(\bm{\tau}_m\) & \(\tau_{m,i}\)\\
-\midrule
-Motion of the top platform & \texttt{[m,rad]} & \texttt{dX} & \(d\bm{\mathcal{X}}\) & \(d\mathcal{X}_i\)\\
-Metrology measured displacement & \texttt{[m,rad]} & \texttt{dXm} & \(d\bm{\mathcal{X}}_m\) & \(d\mathcal{X}_{m,i}\)\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:list_signals}List of signals}
+\caption{\label{tab:test_nhexa_hexa_modal_modes_list}Description of the identified modes of the Nano-Hexapod}
 
 \end{table}
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{\textwidth}
-\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_IMG_20210625_083801.jpg}
+\begin{center}
-\caption{\label{fig:test_nhexa_enc_fixed_to_struts}Nano-Hexapod with encoders fixed to the struts}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_hexa_flexible_mode_1.jpg}
+\end{center}
+\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at 692Hz}
+\end{subfigure}
+\begin{subfigure}{\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_hexa_flexible_mode_2.jpg}
+\end{center}
+\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at 709Hz}
+\end{subfigure}
+\caption{\label{fig:test_nhexa_hexa_flexible_modes}Two identified flexible modes of the top plate of the Nano-Hexapod}
 \end{figure}
 
-It is structured as follow:
-\begin{itemize}
-\item Section \ref{sec:test_nhexa_enc_plates_plant_id}: The dynamics of the nano-hexapod is identified.
-\item Section \ref{sec:test_nhexa_enc_plates_comp_simscape}: The identified dynamics is compared with the Simscape model.
-\end{itemize}
-
 \section{Identification of the dynamics}
-\label{sec:test_nhexa_enc_plates_plant_id}
+\label{ssec:test_nhexa_identification}
-In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is identified.
 
-First, the measurement data are loaded in Section \ref{sec:test_nhexa_enc_plates_plant_id_setup}, then the transfer function matrix from the actuators to the encoders are estimated in Section \ref{sec:test_nhexa_enc_plates_plant_id_dvf}.
+The dynamics of the nano-hexapod from the six command signals (\(u_1\) to \(u_6\)) the six measured displacement by the encoders (\(d_{e1}\) to \(d_{e6}\)) and to the six force sensors (\(V_{s1}\) to \(V_{s6}\)) are identified by generating a low pass filtered white noise for each of the command signals, one by one.
-Finally, the transfer function matrix from the actuators to the force sensors is estimated in Section \ref{sec:test_nhexa_enc_plates_plant_id_iff}.
-\subsection{Data Loading and Spectral Analysis Setup}
-\label{sec:test_nhexa_enc_plates_plant_id_setup}
 
-The actuators are excited one by one using a low pass filtered white noise.
+The \(6 \times 6\) FRF matrix from \(\mathbf{u}\) ot \(\mathbf{d}_e\) is shown in Figure \ref{fig:test_nhexa_identified_frf_de}.
-For each excitation, the 6 force sensors and 6 encoders are measured and saved.
+The diagonal terms are displayed using colorful lines, and all the 30 off-diagonal terms are displayed by grey lines.
-\subsection{Transfer function from Actuator to Encoder}
-\label{sec:test_nhexa_enc_plates_plant_id_dvf}
 
-The 6x6 transfer function matrix from the excitation voltage \(\bm{u}\) and the displacement \(d\bm{\mathcal{L}}_m\) as measured by the encoders is estimated.
+All the six diagonal terms are well superimposed up to at least \(1\,kHz\), indicating good manufacturing and mounting uniformity.
+Below the first suspension mode, good decoupling can be observed (the amplitude of the all of off-diagonal terms are \(\approx 20\) times smaller than the diagonal terms).
 
-The diagonal and off-diagonal terms of this transfer function matrix are shown in Figure \ref{fig:test_nhexa_enc_plates_dvf_frf}.
+From 10Hz up to 1kHz, around 10 resonance frequencies can be observed.
-\begin{figure}[htbp]
+The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section \ref{ssec:test_nhexa_enc_struts_modal_analysis}.
-\centering
+Then, three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section \ref{ssec:test_nhexa_comp_model_coupling}).
-\includegraphics[scale=1]{figs/enc_plates_dvf_frf.png}
+Except the mode at 237Hz, their amplitude is rather low.
-\caption{\label{fig:test_nhexa_enc_plates_dvf_frf}Measured FRF for the transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\)}
+Two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform.
-\end{figure}
+Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform.
 
-\begin{important}
+Up to at least 1kHz, an alternating pole/zero pattern is observed, which renders the control easier to tune.
-From Figure \ref{fig:test_nhexa_enc_plates_dvf_frf}, we can draw few conclusions on the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) when the encoders are fixed to the plates:
+This would not have been the case if the encoders were fixed to the struts.
-\begin{itemize}
-\item the decoupling is rather good at low frequency (below the first suspension mode).
-The low frequency gain is constant for the off diagonal terms, whereas when the encoders where fixed to the struts, the low frequency gain of the off-diagonal terms were going to zero (Figure \ref{fig:test_nhexa_enc_struts_dvf_frf}).
-\item the flexible modes of the struts at 226Hz and 337Hz are indeed shown in the transfer functions, but their amplitudes are rather low.
-\item the diagonal terms have alternating poles and zeros up to at least 600Hz: the flexible modes of the struts are not affecting the alternating pole/zero pattern. This what not the case when the encoders were fixed to the struts (Figure \ref{fig:test_nhexa_enc_struts_dvf_frf}).
-\end{itemize}
-\end{important}
-
-\subsection{Transfer function from Actuator to Force Sensor}
-\label{sec:test_nhexa_enc_plates_plant_id_iff}
-Then the 6x6 transfer function matrix from the excitation voltage \(\bm{u}\) and the voltage \(\bm{\tau}_m\) generated by the Force senors is estimated.
-The bode plot of the diagonal and off-diagonal terms are shown in Figure \ref{fig:test_nhexa_enc_plates_iff_frf}.
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/enc_plates_iff_frf.png}
-\caption{\label{fig:test_nhexa_enc_plates_iff_frf}Measured FRF for the IFF plant}
-\end{figure}
-
-\begin{important}
-It is shown in Figure \ref{fig:test_nhexa_enc_plates_iff_comp_simscape_all} that:
-\begin{itemize}
-\item The IFF plant has alternating poles and zeros
-\item The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF
-\item The decoupling is quite good at low frequency (below the first model) as well as high frequency (above the last suspension mode, except near the flexible modes of the top plate)
-\end{itemize}
-\end{important}
-
-\subsection{Save Identified Plants}
-The identified dynamics is saved for further use.
-\section{Effect of Payload mass on the Dynamics}
-\label{sec:test_nhexa_added_mass}
-In this section, the encoders are fixed to the plates, and we identify the dynamics for several payloads.
-The added payload are half cylinders, and three layers can be added for a total of around 40kg (Figure \ref{fig:test_nhexa_picture_added_3_masses}).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_picture_added_3_masses.jpg}
+\includegraphics[scale=1]{figs/test_nhexa_identified_frf_de.png}
-\caption{\label{fig:test_nhexa_picture_added_3_masses}Picture of the nano-hexapod with added mass}
+\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{d}_e\). The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.}
 \end{figure}
 
-First the dynamics from \(\bm{u}\) to \(d\mathcal{L}_m\) and \(\bm{\tau}_m\) is identified.
+
-Then, the Integral Force Feedback controller is developed and applied as shown in Figure \ref{fig:test_nhexa_nano_hexapod_signals_iff}.
+Similarly, the \(6 \times 6\) FRF matrix from \(\mathbf{u}\) to \(\mathbf{V}_s\) is shown in Figure \ref{fig:test_nhexa_identified_frf_Vs}.
-Finally, the dynamics from \(\bm{u}^\prime\) to \(d\mathcal{L}_m\) is identified and the added damping can be estimated.
+Alternating poles and zeros is observed up to at least 2kHz, which is a necessary characteristics in order to apply decentralized IFF.
+Similar to what was observed for the encoder outputs, all the ``diagonal'' terms are well superimposed, indicating that the same controller can be applied for all the struts.
+The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/test_nhexa_nano_hexapod_signals_iff.png}
+\includegraphics[scale=1]{figs/test_nhexa_identified_frf_Vs.png}
-\caption{\label{fig:test_nhexa_nano_hexapod_signals_iff}Block Diagram of the experimental setup and model}
+\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\). The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.}
 \end{figure}
-\subsection{Measured Frequency Response Functions}
+
-The following data are loaded:
+\section{Effect of payload mass on the dynamics}
-\begin{itemize}
+\label{ssec:test_nhexa_added_mass}
-\item \texttt{Va}: the excitation voltage (corresponding to \(u_i\))
+
-\item \texttt{Vs}: the generated voltage by the 6 force sensors (corresponding to \(\bm{\tau}_m\))
+As one major challenge in the control of the NASS is the wanted robustness to change of payload mass, it is necessary to understand how the dynamics of the nano-hexapod changes with a change of payload mass.
-\item \texttt{de}: the measured motion by the 6 encoders (corresponding to \(d\bm{\mathcal{L}}_m\))
+
-\end{itemize}
+In order to study this change of dynamics with the payload mass, up to three ``cylindrical masses'' of \(13\,kg\) each can be added for a total of \(\approx 40\,kg\).
-The window \texttt{win} and the frequency vector \texttt{f} are defined.
+These three cylindrical masses on top of the nano-hexapod are shown in Figure \ref{fig:test_nhexa_table_mass_3}.
-Finally the \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) and from \(\bm{u}\) to \(\bm{\tau}_m\) are identified:
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_table_mass_3.jpg}
+\caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,kg\)}
+\end{figure}
+
 The identified dynamics are then saved for further use.
-\subsection{Transfer function from Actuators to Encoders}
+The obtained frequency response functions from actuator signal \(u_i\) to the associated encoder \(d_{ei}\) for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure \ref{fig:test_nhexa_identified_frf_de_masses}.
-The transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_{m}\) are shown in Figure \ref{fig:test_nhexa_comp_plant_payloads_dvf}.
+As expected, the frequency of the suspension modes are decreasing with an increase of the payload mass.
+The low frequency gain does not change as it is linked to the stiffness property of the nano-hexapod, and not to its mass property.
 
-\begin{figure}[htbp]
+The frequencies of the two flexible modes of the top plate are first decreased a lot when the first mass is added (from \(\approx 700\,Hz\) to \(\approx 400\,Hz\)).
-\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_plant_payloads_dvf.png}
-\caption{\label{fig:test_nhexa_comp_plant_payloads_dvf}Measured Frequency Response Functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms.}
-\end{figure}
-
-
-\begin{important}
-From Figure \ref{fig:test_nhexa_comp_plant_payloads_dvf}, we can observe few things:
-\begin{itemize}
-\item The obtained dynamics is changing a lot between the case without mass and when there is at least one added mass.
-\item Between 1, 2 and 3 added masses, the dynamics is not much different, and it would be easier to design a controller only for these cases.
-\item The flexible modes of the top plate is first decreased a lot when the first mass is added (from 700Hz to 400Hz).
 This is due to the fact that the added mass is composed of two half cylinders which are not fixed together.
-Therefore is adds a lot of mass to the top plate without adding a lot of rigidity in one direction.
+It therefore adds a lot of mass to the top plate without adding stiffness in one direction.
-When more than 1 mass layer is added, the half cylinders are added with some angles such that rigidity are added in all directions (see Figure \ref{fig:test_nhexa_picture_added_3_masses}).
+When more than one ``mass layer'' is added, the half cylinders are added with some angles such that rigidity are added in all directions (see how the three mass ``layers'' are positioned in Figure \ref{fig:test_nhexa_table_mass_3}).
 In that case, the frequency of these flexible modes are increased.
-In practice, the payload should be one solid body, and we should not see a massive decrease of the frequency of this flexible mode.
+In practice, the payload should be one solid body, and no decrease of the frequency of this flexible mode should be observed.
-\item Flexible modes of the top plate are becoming less problematic as masses are added.
+The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass is increased.
-\item First flexible mode of the strut at 230Hz is not much decreased when mass is added.
-However, its apparent amplitude is much decreased.
-\end{itemize}
-\end{important}
 
-\subsection{Transfer function from Actuators to Force Sensors}
+The measured FRF from \(u_i\) to \(V_{si}\) are shown in Figure \ref{fig:test_nhexa_identified_frf_Vs_masses}.
-The transfer functions from \(\bm{u}\) to \(\bm{\tau}_{m}\) are shown in Figure \ref{fig:test_nhexa_comp_plant_payloads_iff}.
+For all the tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust way.
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{0.49\textwidth}
-\includegraphics[scale=1]{figs/test_nhexa_comp_plant_payloads_iff.png}
+\begin{center}
-\caption{\label{fig:test_nhexa_comp_plant_payloads_iff}Measured Frequency Response Functions from \(u_i\) to \(\tau_{m,i}\) for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms.}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_identified_frf_de_masses.png}
+\end{center}
+\subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_identified_frf_Vs_masses.png}
+\end{center}
+\subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$}
+\end{subfigure}
+\caption{\label{fig:test_struts_mounting}Measured Frequency Response Functions from \(u_i\) to \(d_{ei}\) (\subref{fig:test_nhexa_identified_frf_de_masses}) and from \(u_i\) to \(V_{si}\) (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown.}
 \end{figure}
 
-\begin{important}
+\section*{Conclusion}
-From Figure \ref{fig:test_nhexa_comp_plant_payloads_iff}, we can see that for all added payloads, the transfer function from \(\bm{u}\) to \(\bm{\tau}_{m}\) always has alternating poles and zeros.
+After the Nano-Hexapod was fixed on top of the suspended table, its dynamics was identified.
-\end{important}
 
-\subsection{Coupling of the transfer function from Actuator to Encoders}
+The frequency response functions from the six DAC voltages \(\mathbf{u}\) to the six encoders measured displacements \(\mathbf{d}_e\) displays alternating complex conjugate poles and complex conjugate zeros up to at least 1kHz.
-The RGA-number, which is a measure of the interaction in the system, is computed for the transfer function matrix from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) for all the payloads.
+At low frequency, the coupling is small, indicating correct assembly of all parts.
-The obtained numbers are compared in Figure \ref{fig:test_nhexa_rga_num_ol_masses}.
+This should enables the design of a decentralized positioning controller based on the encoder for relative positioning purposes.
+The suspension modes and flexible modes measured during the modal analysis (Section \ref{ssec:test_nhexa_enc_struts_modal_analysis}) are also observed in the dynamics.
+Lot's of other modes are present above 700Hz, which will inevitably limit the achievable bandwidth.
+The observed effect of the payload's mass on the dynamics is quite large, which also represent a complex control challenge.
 
-\begin{figure}[htbp]
+The frequency response functions from the six DAC voltages \(\mathbf{u}\) to the six force sensors voltages \(\mathbf{V}_s\) all have alternating complex conjugate poles and complex conjugate zeros.
-\centering
+This indicates that it should be possible to implement decentralized Integral Force Feedback in a robust way.
-\includegraphics[scale=1]{figs/test_nhexa_rga_num_ol_masses.png}
+This alternating property holds for all the tested payloads.
-\caption{\label{fig:test_nhexa_rga_num_ol_masses}RGA-number for the open-loop transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\)}
-\end{figure}
 
-\begin{important}
+\chapter{Nano-Hexapod Model Dynamics}
-From Figure \ref{fig:test_nhexa_rga_num_ol_masses}, it is clear that the coupling is quite large starting from the first suspension mode of the nano-hexapod.
-Therefore, is the payload's mass is increase, the coupling in the system start to become unacceptably large at lower frequencies.
-\end{important}
-
-\section{Conclusion}
-\begin{important}
-In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied.
-
-It has been found that:
-\begin{itemize}
-\item The measured dynamics is in agreement with the dynamics of the simscape model, up to the flexible modes of the top plate.
-See figures \ref{fig:test_nhexa_enc_plates_iff_comp_simscape} and \ref{fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape} for the transfer function to the force sensors and Figures \ref{fig:test_nhexa_enc_plates_dvf_comp_simscape} and \ref{fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape} for the transfer functions to the encoders
-\item The Integral Force Feedback strategy is very effective in damping the suspension modes of the nano-hexapod (Figure \ref{fig:test_nhexa_enc_plant_plates_effect_iff}).
-\item The transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) show nice dynamical properties and is a much better candidate for the high-authority-control than when the encoders were fixed to the struts.
-At least up to the flexible modes of the top plate, the diagonal elements of the transfer function matrix have alternating poles and zeros, and the phase is moving smoothly.
-Only the flexible modes of the top plates seems to be problematic for control.
-\end{itemize}
-\end{important}
-
-\chapter{Comparison with the Nano-Hexapod model?}
 \label{sec:test_nhexa_model}
-\section{Comparison with the Simscape Model}
+In this section, the measured dynamics done in Section \ref{sec:test_nhexa_dynamics} is compared with the dynamics estimated from the Simscape model.
-\label{sec:test_nhexa_enc_plates_comp_simscape}
+The nano-hexapod simscape model is therefore added on top of the vibration table Simscape model as shown in Figure \ref{fig:test_nhexa_hexa_simscape}.
-In this section, the measured dynamics done in Section \ref{sec:test_nhexa_enc_plates_plant_id} is compared with the dynamics estimated from the Simscape model.
-
-A configuration is added to be able to put the nano-hexapod on top of the vibration table as shown in Figure \ref{fig:simscape_vibration_table}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/vibration_table_nano_hexapod_simscape.png}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_hexa_simscape.png}
-\caption{\label{fig:simscape_vibration_table}3D representation of the simscape model with the nano-hexapod}
+\caption{\label{fig:test_nhexa_hexa_simscape}3D representation of the simscape model with the nano-hexapod on top of the suspended table. Three mass ``layers'' are here added}
 \end{figure}
-\subsection{Identification with the Simscape Model}
+
-The nano-hexapod is initialized with the APA taken as 2dof models.
+The model should exhibit certain characteristics that are verified in this section.
-Now, the dynamics from the DAC voltage \(\bm{u}\) to the encoders \(d\bm{\mathcal{L}}_m\) is estimated using the Simscape model.
+First, it should match the measured system dynamics from actuators to sensors that were presented in Section \ref{sec:test_nhexa_dynamics}.
-Then the transfer function from \(\bm{u}\) to \(\bm{\tau}_m\) is identified using the Simscape model.
+Both the ``direct'' terms (Section \ref{ssec:test_nhexa_comp_model}) and ``coupling'' terms (Section \ref{ssec:test_nhexa_comp_model_coupling}) of the Simscape model are compared with the measured dynamics.
-The identified dynamics is saved for further use.
+Second, it should also represents how the system dynamics changes when a payload is fixed to the top platform.
-\subsection{Dynamics from Actuator to Force Sensors}
+This is checked in Section \ref{ssec:test_nhexa_comp_model_masses}.
-The identified dynamics is compared with the measured FRF:
+
-\begin{itemize}
+\section{Nano-Hexapod model dynamics}
-\item Figure \ref{fig:test_nhexa_enc_plates_iff_comp_simscape_all}: the individual transfer function from \(u_1\) (the DAC voltage for the first actuator) to the force sensors of all 6 struts are compared
+\label{ssec:test_nhexa_comp_model}
-\item Figure \ref{fig:test_nhexa_enc_plates_iff_comp_simscape}: all the diagonal elements are compared
+
-\item Figure \ref{fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape}: all the off-diagonal elements are compared
+The Simscape model of the nano-hexapod is first configured with 4-DoF flexible joints, 2-DoF APA and rigid top and bottom platforms.
-\end{itemize}
+The stiffness of the flexible joints are chosen based on the values estimated using the test bench and based on FEM.
+The parameters of the APA model are the ones determined from the test bench of the APA.
+The \(6 \times 6\) transfer function matrices from \(\mathbf{u}\) to \(\mathbf{d}_e\) and from \(\mathbf{u}\) to \(\mathbf{V}_s\) are extracted then from the Simscape model.
+
+A first feature that should be checked is that the model well represents the ``direct'' terms of the measured FRF matrix.
+To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure \ref{fig:test_nhexa_comp_simscape_diag}.
+It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modelled.
+The three resonances that were attributed to ``internal'' flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable as the APA are here modelled as a simple uniaxial 2-DoF system.
+At higher frequencies, no resonances can be seen in the model, as the as the top plate and the encoder supports are modelled as rigid bodies.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_de_diag.png}
+\end{center}
+\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_Vs_diag.png}
+\end{center}
+\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$}
+\end{subfigure}
+\caption{\label{fig:test_nhexa_comp_simscape_diag}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})}
+\end{figure}
+
+\section{Modelling dynamical coupling}
+\label{ssec:test_nhexa_comp_model_coupling}
+
+Another wanted feature of the model is that it well represents the coupling in the system as this is often the limiting factor for the control of MIMO systems.
+Instead of comparing the full 36 elements of the \(6 \times 6\) FFR matrix from \(\mathbf{u}\) to \(\mathbf{d}_e\), only the first ``column'' is compared (Figure \ref{fig:test_nhexa_comp_simscape_de_all}), which corresponds to the transfer function from the command \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\).
+It can be seen that the coupling in the model is well matching the measurements up to the first un-modelled flexible mode at 237Hz.
+Similar results are observed for all the other coupling terms, as well as for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/test_nhexa_enc_plates_iff_comp_simscape_all.png}
+\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all.png}
-\caption{\label{fig:test_nhexa_enc_plates_iff_comp_simscape_all}IFF Plant for the first actuator input and all the force senosrs}
+\caption{\label{fig:test_nhexa_comp_simscape_de_all}Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)}
 \end{figure}
 
+The APA300ML are then modelled with a \emph{super-element} extracted from a FE-software.
+The obtained transfer functions from \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\) are compared with the measured FRF in Figure \ref{fig:test_nhexa_comp_simscape_de_all_flex}.
+While the damping of the suspension modes for the \emph{super-element} is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modelled.
+Even the mode 395Hz can be observed in the model.
+Therefore, if the modes of the struts are to be modelled, the \emph{super-element} of the APA300ML may be used, at the cost of obtaining a much higher order model.
+
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/test_nhexa_enc_plates_iff_comp_simscape.png}
+\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all_flex.png}
-\caption{\label{fig:test_nhexa_enc_plates_iff_comp_simscape}Diagonal elements of the IFF Plant}
+\caption{\label{fig:test_nhexa_comp_simscape_de_all_flex}Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)}
 \end{figure}
 
+\section{Modelling the effect of payload mass}
+\label{ssec:test_nhexa_comp_model_masses}
+
+Another important characteristics of the model is that it should well represents the dynamics of the system for all considered payloads.
+The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure \ref{fig:test_nhexa_comp_simscape_diag_masses}.
+The observed shift to lower frequency of the suspension modes with an increased payload mass is well represented by the Simscape model.
+The complex conjugate zeros are also well matching with the experiments both for the encoder outputs (Figure \ref{fig:test_nhexa_comp_simscape_de_diag_masses}) and the force sensor outputs (Figure \ref{fig:test_nhexa_comp_simscape_Vs_diag_masses}).
+
+Note that the model displays smaller damping that what is observed experimentally for high values of the payload mass.
+One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping).
+However, as decentralized IFF will be applied, the damping will be brought actively, and the open-loop damping value should have very little impact on the obtained plant.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_de_diag_masses.png}
+\end{center}
+\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_Vs_diag_masses.png}
+\end{center}
+\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$}
+\end{subfigure}
+\caption{\label{fig:test_nhexa_comp_simscape_diag_masses}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})}
+\end{figure}
+
+In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from \(u_1\) to \(d_{e1}\) to \(d_{e6}\) are compared in the case of the 39kg payload in Figure \ref{fig:test_nhexa_comp_simscape_de_all_high_mass}.
+Excellent match between the experimental coupling and the model coupling is observed.
+The model therefore well represents the system dynamical coupling for different considered payloads.
+
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/test_nhexa_enc_plates_iff_comp_offdiag_simscape.png}
+\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all_high_mass.png}
-\caption{\label{fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape}Off diagonal elements of the IFF Plant}
+\caption{\label{fig:test_nhexa_comp_simscape_de_all_high_mass}Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)}
 \end{figure}
 
-\subsection{Dynamics from Actuator to Encoder}
+\section*{Conclusion}
-The identified dynamics is compared with the measured FRF:
+As illustrated in this section, the developed Simscape model accurately represents the suspension modes of the Nano-Hexapod.
-\begin{itemize}
+Both FRF matrices from \(\mathbf{u}\) to \(\mathbf{V}_s\) and from \(\mathbf{u}\) to \(\mathbf{d}_e\) are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective.
-\item Figure \ref{fig:test_nhexa_enc_plates_dvf_comp_simscape_all}: the individual transfer function from \(u_3\) (the DAC voltage for the actuator number 3) to the six encoders
-\item Figure \ref{fig:test_nhexa_enc_plates_dvf_comp_simscape}: all the diagonal elements are compared
-\item Figure \ref{fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape}: all the off-diagonal elements are compared
-\end{itemize}
 
-\begin{figure}[htbp]
+At frequency above the suspension modes, the Nano-Hexapod model becomes inaccurate as the flexible modes are not modelled.
-\centering
+It was shown that modelling the APA300ML using a ``super-element'' allows to model the internal resonances of the struts.
-\includegraphics[scale=1]{figs/test_nhexa_enc_plates_dvf_comp_simscape_all.png}
+The same could be done with the top platform and the encoder supports, but the model order would be higher and may become unpractical for simulation purposes.
-\caption{\label{fig:test_nhexa_enc_plates_dvf_comp_simscape_all}DVF Plant for the first actuator input and all the encoders}
-\end{figure}
 
-\begin{figure}[htbp]
+\chapter{Conclusion}
-\centering
-\includegraphics[scale=1]{figs/test_nhexa_enc_plates_dvf_comp_simscape.png}
-\caption{\label{fig:test_nhexa_enc_plates_dvf_comp_simscape}Diagonal elements of the DVF Plant}
-\end{figure}
 
-\begin{figure}[htbp]
+The goal of this test bench was to obtain an accurate model of the nano-hexapod that can then be included on top of the micro-station model.
-\centering
-\includegraphics[scale=1]{figs/test_nhexa_enc_plates_dvf_comp_offdiag_simscape.png}
-\caption{\label{fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape}Off diagonal elements of the DVF Plant}
-\end{figure}
 
-\subsection{Conclusion}
+This strategy was to measure the nano-hexapod in conditions where all factors that could have impacted the nano-hexapod dynamics were taken into account.
-\begin{important}
+This was done by developing a suspended table with low frequency suspension modes which can be accurately modelled.
-The Simscape model is quite accurate for the transfer function matrices from \(\bm{u}\) to \(\bm{\tau}_m\) and from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) except at frequencies of the flexible modes of the top-plate.
-The Simscape model can therefore be used to develop the control strategies.
-\end{important}
 
-\section{Comparison with the Simscape model}
+While the dynamics of the nano-hexapod was indeed impacted by the dynamics of the suspended platform, this impact was also taken into account in the Simscape model, and a good match was obtained.
-\label{sec:test_nhexa_added_mass_simscape}
-Let's now compare the identified dynamics with the Simscape model.
-We wish to verify if the Simscape model is still accurate for all the tested payloads.
-\subsection{System Identification}
-Let's initialize the simscape model with the nano-hexapod fixed on top of the vibration table.
-First perform the identification for the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\):
-The identified dynamics are then saved for further use.
-\subsection{Transfer function from Actuators to Encoders}
-The measured FRF and the identified dynamics from \(u_i\) to \(d\mathcal{L}_{m,i}\) are compared in Figure \ref{fig:test_nhexa_comp_masses_model_exp_dvf}.
-A zoom near the ``suspension'' modes is shown in Figure \ref{fig:test_nhexa_comp_masses_model_exp_dvf_zoom}.
 
-\begin{figure}[htbp]
+Obtaining a model accurately representing the complex dynamics of the Nano-Hexapod was made possible by the modelling approach used during this work.
-\centering
+This approach consisted of tuning and validating models of individual components (such as the APA and flexible joints) using dedicated test benches.
-\includegraphics[scale=1]{figs/comp_masses_model_exp_dvf.png}
+Only then, the different models could be combined to form the Nano-Hexapod dynamical model.
-\caption{\label{fig:test_nhexa_comp_masses_model_exp_dvf}Comparison of the transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) - measured FRF and identification from the Simscape model}
+If a model of the nano-hexapod was developed in one time, it would be difficult to tune all model parameters to match the measured dynamics, or even to know if the model ``structure'' would be adequate to represents the system dynamics.
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_masses_model_exp_dvf_zoom.png}
-\caption{\label{fig:test_nhexa_comp_masses_model_exp_dvf_zoom}Comparison of the transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) - measured FRF and identification from the Simscape model (Zoom)}
-\end{figure}
-
-\begin{important}
-The Simscape model is very accurately representing the measured dynamics up.
-Only the flexible modes of the struts and of the top plate are not represented here as these elements are modelled as rigid bodies.
-\end{important}
-
-\subsection{Transfer function from Actuators to Force Sensors}
-The measured FRF and the identified dynamics from \(u_i\) to \(\tau_{m,i}\) are compared in Figure \ref{fig:test_nhexa_comp_masses_model_exp_iff}.
-A zoom near the ``suspension'' modes is shown in Figure \ref{fig:test_nhexa_comp_masses_model_exp_iff_zoom}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_masses_model_exp_iff.png}
-\caption{\label{fig:test_nhexa_comp_masses_model_exp_iff}Comparison of the transfer functions from \(u_i\) to \(\tau_{m,i}\) - measured FRF and identification from the Simscape model}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_masses_model_exp_iff_zoom.png}
-\caption{\label{fig:test_nhexa_comp_masses_model_exp_iff_zoom}Comparison of the transfer functions from \(u_i\) to \(\tau_{m,i}\) - measured FRF and identification from the Simscape model (Zoom)}
-\end{figure}
 
 \printbibliography[heading=bibintoc,title={Bibliography}]