Working Matlab code

matlab/STEPS/12121146_defeature.STEP

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+ Unknown                                                                       
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+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
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+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
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+ Unknown                                                                       
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matlab/src/initializeAPA.m

@@ -0,0 +1,103 @@
+function [actuator] = initializeAPA(args)
+% initializeAPA -
+%
+% Syntax: [actuator] = initializeAPA(args)
+%
+% Inputs:
+%    - args -
+%
+% Outputs:
+%    - actuator -
+
+arguments
+    args.type      char   {mustBeMember(args.type,{'2dof', 'flexible frame', 'flexible'})} = '2dof'
+
+    % Actuator and Sensor constants
+    args.Ga  (1,1) double {mustBeNumeric} = 0
+    args.Gs  (1,1) double {mustBeNumeric} = 0
+
+    % For 2DoF
+    args.k   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*380000
+    args.ke  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*4952605
+    args.ka  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2476302
+
+    args.c   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*20
+    args.ce  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*200
+    args.ca  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100
+
+    args.Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056
+
+    % Force Flexible APA
+    args.xi (1,1) double {mustBeNumeric, mustBePositive} = 0.01
+    args.d_align_bot (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
+    args.d_align_top (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
+
+    % For Flexible Frame
+    args.ks (1,1) double {mustBeNumeric, mustBePositive} = 235e6
+    args.cs (1,1) double {mustBeNumeric, mustBePositive} = 1e1
+end
+
+actuator = struct();
+
+switch args.type
+  case '2dof'
+    actuator.type = 1;
+  case 'flexible frame'
+    actuator.type = 2;
+  case 'flexible'
+    actuator.type = 3;
+end
+
+if args.Ga == 0
+    switch args.type
+      case '2dof'
+        actuator.Ga = -2.5796;
+      case 'flexible frame'
+        actuator.Ga = 1; % TODO
+      case 'flexible'
+        actuator.Ga = 23.2;
+    end
+else
+    actuator.Ga = args.Ga; % Actuator gain [N/V]
+end
+
+if args.Gs == 0
+    switch args.type
+      case '2dof'
+        actuator.Gs = 466664;
+      case 'flexible frame'
+        actuator.Gs = 1; % TODO
+      case 'flexible'
+        actuator.Gs = -4898341;
+    end
+else
+    actuator.Gs = args.Gs; % Sensor gain [V/m]
+end
+
+actuator.k  = args.k;  % [N/m]
+actuator.ke = args.ke; % [N/m]
+actuator.ka = args.ka; % [N/m]
+
+actuator.c  = args.c;  % [N/(m/s)]
+actuator.ce = args.ce; % [N/(m/s)]
+actuator.ca = args.ca; % [N/(m/s)]
+
+actuator.Leq = args.Leq; % [m]
+
+switch args.type
+  case 'flexible frame'
+    actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
+    actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
+    actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
+  case 'flexible'
+    actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix
+    actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix
+    actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m]
+    actuator.d_align_bot = args.d_align_bot;
+    actuator.d_align_top = args.d_align_top;
+end
+
+actuator.xi = args.xi; % Damping ratio
+
+actuator.ks = args.ks; % Stiffness of one stack [N/m]
+actuator.cs = args.cs; % Damping of one stack [N/m]

matlab/src/initializeBotFlexibleJoint.m

@@ -0,0 +1,45 @@
+function [flex_bot] = initializeBotFlexibleJoint(args)
+% initializeBotFlexibleJoint -
+%
+% Syntax: [flex_bot] = initializeBotFlexibleJoint(args)
+%
+% Inputs:
+%    - args -
+%
+% Outputs:
+%    - flex_bot -
+
+arguments
+    args.type      char   {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'
+
+    args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
+    args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
+    args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260
+    args.kz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*7e7
+
+    args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+    args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+    args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+    args.cz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+end
+
+flex_bot = struct();
+
+switch args.type
+  case '2dof'
+    flex_bot.type = 1;
+  case '3dof'
+    flex_bot.type = 2;
+  case '4dof'
+    flex_bot.type = 3;
+end
+
+flex_bot.kRx = args.kRx;
+flex_bot.kRy = args.kRy;
+flex_bot.kRz = args.kRz;
+flex_bot.kz  = args.kz;
+
+flex_bot.cRx = args.cRx;
+flex_bot.cRy = args.cRy;
+flex_bot.cRz = args.cRz;
+flex_bot.cz  = args.cz;

matlab/src/initializeTopFlexibleJoint.m

@@ -0,0 +1,45 @@
+function [flex_top] = initializeTopFlexibleJoint(args)
+% initializeTopFlexibleJoint -
+%
+% Syntax: [flex_top] = initializeTopFlexibleJoint(args)
+%
+% Inputs:
+%    - args -
+%
+% Outputs:
+%    - flex_top -
+
+arguments
+    args.type      char   {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'
+
+    args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
+    args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
+    args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260
+    args.kz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*7e7
+
+    args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+    args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+    args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+    args.cz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
+end
+
+flex_top = struct();
+
+switch args.type
+  case '2dof'
+    flex_top.type = 1;
+  case '3dof'
+    flex_top.type = 2;
+  case '4dof'
+    flex_top.type = 3;
+end
+
+flex_top.kRx = args.kRx;
+flex_top.kRy = args.kRy;
+flex_top.kRz = args.kRz;
+flex_top.kz  = args.kz;
+
+flex_top.cRx = args.cRx;
+flex_top.cRy = args.cRy;
+flex_top.cRz = args.cRz;
+flex_top.cz  = args.cz;

matlab/test_struts_1_flexible_modes.m

@@ -0,0 +1,65 @@
+%% 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
+
+%% Colors for the figures
+colors = colororder;
+
+% Without Encoder
+% When the encoder is not fixed to the strut, the obtained FRF are shown in Figure ref:fig:test_struts_spur_res_frf.
+
+
+%% Load Data (without the encoder)
+bending_X = load('strut_spur_res_x_bending.mat');
+bending_Y = load('strut_spur_res_y_bending.mat');
+torsion_Z = load('strut_spur_res_z_torsion.mat');
+
+%% Plot the responses (without the encoder)
+figure;
+hold on;
+plot(bending_X.FFT1_AvSpc_1_RMS_X_Val, bending_X.FFT1_AvSpc_1_RMS_Y_Val, ...
+     'DisplayName', 'X-Bending')
+plot(bending_Y.FFT1_AvSpc_1_RMS_X_Val, bending_Y.FFT1_AvSpc_1_RMS_Y_Val, ...
+     'DisplayName', 'Y-Bending')
+plot(torsion_Z.FFT1_AvSpc_1_RMS_X_Val, torsion_Z.FFT1_AvSpc_1_RMS_Y_Val, ...
+     'DisplayName', 'Z-torsion');
+text(226, 1.5e-4,{'226Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+text(337, 6e-5,{'337Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+text(398, 1.5e-4,{'398Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 8e2]); ylim([5e-7, 3e-4])
+legend('location', 'northwest');
+
+% With Encoder
+% Then, one encoder is fixed to the strut and the FRF are measured again and shown in Figure ref:fig:test_struts_spur_res_frf_enc.
+
+%% Load Data (with the encoder)
+bending_X_enc = load('strut_spur_res_x_bending_enc.mat');
+bending_Y_enc = load('strut_spur_res_y_bending_enc.mat');
+torsion_Z_enc = load('strut_spur_res_z_torsion_enc.mat');
+
+%% Plot the responses  (with the encoder)
+figure;
+hold on;
+plot(bending_X_enc.FFT1_AvSpc_1_RMS_X_Val, bending_X_enc.FFT1_AvSpc_1_RMS_Y_Val, ...
+     'DisplayName', 'X-Bending')
+plot(bending_Y_enc.FFT1_AvSpc_1_RMS_X_Val, bending_Y_enc.FFT1_AvSpc_1_RMS_Y_Val, ...
+     'DisplayName', 'Y-Bending')
+plot(torsion_Z_enc.FFT1_AvSpc_1_RMS_X_Val, torsion_Z_enc.FFT1_AvSpc_1_RMS_Y_Val, ...
+     'DisplayName', 'Z-torsion');
+text(198, 4e-5,{'198Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+text(293, 6e-5,{'293Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+text(381, 1e-4,{'381Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 8e2]); ylim([5e-7, 2e-4])
+legend('location', 'northwest');

matlab/test_struts_2_dynamical_meas.m

@@ -0,0 +1,687 @@
+%% 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
+
+%% Colors for the figures
+colors = colororder;
+
+% FRF Identification - Setup
+% Similarly to what was done for the identification of the APA, the identification is performed in three steps:
+% 1. White noise excitation with small amplitude.
+%    This is used to determine the main resonance of the system.
+% 2. Sweep sine excitation with the amplitude lowered around the resonance.
+%    The sweep sine is from 10Hz to 400Hz.
+% 3. High frequency noise.
+%    The noise is band-passed between 300Hz and 2kHz.
+
+% Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.
+
+
+%% Sampling frequency/time
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(1/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+%% Load Data
+leg_sweep    = load('frf_data_leg_1_sweep.mat', 'u', 'Vs', 'de', 'da');
+leg_noise_hf = load('frf_data_leg_1_noise_hf.mat', 'u', 'Vs', 'de', 'da');
+
+%% We get the frequency vector that will be the same for all the frequency domain analysis.
+[~, f] = tfestimate(leg_sweep.u, leg_sweep.de, win, Noverlap, Nfft, 1/Ts);
+i_lf = f <= 350; % Indices used for the low frequency
+i_hf = f >  350; % Indices used for the low frequency
+
+% FRF Identification - Interferometer
+% In this section, the dynamics from the excitation voltage $u$ to the interferometer $d_a$ is identified.
+% The transfer function from $u$ to the interferometer measured displacement $d_a$ is estimated and shown in Figure ref:fig:strut_1_frf_dvf_plant_tf.
+
+%% Compute FRF function from u to da
+[frf_sweep, ~]    = tfestimate(leg_sweep.u,    leg_sweep.da,    win, Noverlap, Nfft, 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_noise_hf.u, leg_noise_hf.da, win, Noverlap, Nfft, 1/Ts);
+
+%% Combine the FRF
+int_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+%% Plot the measured FRF
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(int_frf), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-9, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(int_frf), 'k-');
+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([10, 2e3]);
+
+% FRF Identification - IFF
+% In this section, the dynamics from $u$ to $V_s$ is identified.
+% Then the FRF are estimated and shown in Figure ref:fig:strut_1_frf_iff_plant_tf
+
+%% Compute the FRF
+[frf_sweep,    ~] = tfestimate(leg_sweep.u,    leg_sweep.Vs,    win, Noverlap, Nfft, 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_noise_hf.u, leg_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Combine the FRF
+iff_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+%% Plot the measured FRF
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(iff_frf), 'k-');
+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([10, 2e3]);
+
+% Measurement Data
+% The measurements are loaded.
+
+%% Load data
+leg_enc_sweep    = load('frf_data_leg_coder_1_noise.mat',    'u', 'Vs', 'de', 'da');
+leg_enc_noise_hf = load('frf_data_leg_coder_1_noise_hf.mat', 'u', 'Vs', 'de', 'da');
+
+% FRF Identification - Interferometer
+% In this section, the dynamics from $u$ to $d_a$ is identified.
+
+%% Compute FRF function from u to da
+[frf_sweep, ~]    = tfestimate(leg_enc_sweep.u,    leg_enc_sweep.da,    win, Noverlap, Nfft, 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.da, win, Noverlap, Nfft, 1/Ts);
+
+%% Combine the FRF
+int_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+
+
+% The obtained FRF is very close to the one that was obtained when no encoder was fixed to the struts as shown in Figure ref:fig:strut_leg_compare_int_frf.
+
+%% Plot the FRF from u to da with and without the encoder
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(int_with_enc_frf), '-', 'DisplayName', 'With encoder');
+plot(f, abs(int_frf),          '-', 'DisplayName', 'Without encoder');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-7, 1e-3]);
+legend('location', 'northeast')
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(int_with_enc_frf), '-');
+plot(f, 180/pi*angle(int_frf), '-');
+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([10, 2e3]);
+
+% FRF Identification - Encoder
+% In this section, the dynamics from $u$ to $d_e$ (encoder) is identified.
+
+% The FRF from $u$ to the encoder measured displacement $d_e$ is computed and shown in Figure ref:fig:strut_1_enc_frf_dvf_plant_tf.
+
+%% Compute FRF function from u to da
+[frf_sweep, ~]    = tfestimate(leg_enc_sweep.u,    leg_enc_sweep.de,    win, Noverlap, Nfft, 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.de, win, Noverlap, Nfft, 1/Ts);
+
+%% Combine the FRF
+enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+%% Plot the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-7, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(enc_frf), 'k-');
+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([10, 2e3]);
+
+
+
+% #+name: fig:strut_1_enc_frf_dvf_plant_tf
+% #+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the encoder $d_e$)
+% #+RESULTS:
+% [[file:figs/strut_1_enc_frf_dvf_plant_tf.png]]
+
+% The transfer functions from $u$ to $d_e$ (encoder) and to $d_a$ (interferometer) are compared in Figure ref:fig:strut_1_comp_enc_int.
+
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf),          'DisplayName', 'Encoder');
+plot(f, abs(int_with_enc_frf), 'DisplayName', 'Interferometer');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-8, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(enc_frf));
+plot(f, 180/pi*angle(int_with_enc_frf));
+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([10, 2e3]);
+
+% APA Resonances Frequency
+% As shown in Figure ref:fig:strut_1_spurious_resonances, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
+
+
+%% Transfer function from Vs to de with indicated resonances
+figure;
+hold on;
+plot(f, abs(enc_frf), 'k-');
+text(93,  4e-4,  {'93Hz'}, 'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(200, 1.3e-4,{'197Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(300, 4e-6,  {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(400, 1.4e-6,{'376Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); xlabel('Frequency [Hz]');
+hold off;
+ylim([1e-7, 1e-3]); xlim([10, 2e3]);
+
+% FRF Identification - Force Sensor
+% In this section, the dynamics from $u$ to $V_s$ is identified.
+
+
+%% Compute FRF function from u to da
+[frf_sweep, ~]    = tfestimate(leg_enc_sweep.u,    leg_enc_sweep.Vs,    win, Noverlap, Nfft, 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Combine the FRF
+iff_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+
+
+% Let's now compare the IFF plants whether the encoders are fixed to the APA or not (Figure ref:fig:strut_1_frf_iff_comp_enc).
+
+%% Compare the IFF plant with and without the encoders
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_with_enc_frf), 'DisplayName', 'With Encoder');
+plot(f, abs(iff_frf), 'DisplayName', 'Without Encoder');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast', 'FontSize', 8);
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(iff_with_enc_frf));
+plot(f, 180/pi*angle(iff_frf));
+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([10, 2e3]);
+
+% Non-Minimum phase zero?
+
+% In order to determine if the complex conjugate zero of Figure ref:fig:strut_1_enc_frf_iff_plant_tf is minimum phase or non-minimum phase, longer measurements are performed.
+
+
+long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'u', 'Vs');
+
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(10/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+%% Transfer function estimation
+[frf_noise, f] = tfestimate(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 0]);
+
+linkaxes([ax1,ax2],'x');
+xlim([38, 45]);
+
+% FRF Identification - Setup
+% The identification of the struts dynamics is performed in two steps:
+% 1. The excitation signal is a white noise with small amplitude.
+%    This is used to estimate the low frequency dynamics.
+% 2. Then a high frequency noise band-passed between 300Hz and 2kHz is used to estimate the high frequency dynamics.
+
+% Then, the result of the first identification is used between 10Hz and 350Hz and the result of the second identification if used between 350Hz and 2kHz.
+
+% Here are the leg numbers that have been measured.
+
+%% Numnbers of the measured legs
+strut_nums = [1 2 3 4 5];
+
+
+
+% The data are loaded for both the first and second identification:
+
+%% First identification (low frequency noise)
+leg_noise = {};
+for i = 1:length(strut_nums)
+    leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', strut_nums(i)), 'u', 'Vs', 'de', 'da')};
+end
+
+%% Second identification (high frequency noise)
+leg_noise_hf = {};
+for i = 1:length(strut_nums)
+    leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', strut_nums(i)), 'u', 'Vs', 'de', 'da')};
+end
+
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(1/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+
+
+% We get the frequency vector that will be the same for all the frequency domain analysis.
+
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(leg_noise{1}.u, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts);
+i_lf = f <= 350;
+i_hf = f >  350;
+
+% FRF Identification - Encoder
+% In this section, the dynamics from $u$ to $d_e$ (encoder) is identified.
+
+% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the encoder $d_e$ is computed:
+
+%% Transfer function estimation
+enc_frf = zeros(length(f), length(strut_nums));
+
+for i = 1:length(strut_nums)
+    [frf_lf, ~] = tfestimate(leg_noise{i}.u,    detrend(leg_noise{i}.de, 0),    win, Noverlap, Nfft, 1/Ts);
+    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, detrend(leg_noise_hf{i}.de, 0), win, Noverlap, Nfft, 1/Ts);
+    enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+
+
+% The obtained transfer functions are shown in Figure ref:fig:struts_frf_dvf_plant_tf.
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, abs(enc_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', strut_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-8, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)));
+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([10, 2e3]);
+
+% FRF Identification - Interferometer
+% In this section, the dynamics from $u$ to $d_a$ (interferometer) is identified.
+
+% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the Attocube is computed for all the struts and shown in Figure ref:fig:struts_frf_int_plant_tf.
+% All the struts are giving very similar FRF.
+
+%% Transfer function estimation
+int_frf = zeros(length(f), length(strut_nums));
+for i = 1:length(strut_nums)
+    [frf_lf, ~] = tfestimate(leg_noise{i}.u,    leg_noise{i}.da,    win, Noverlap, Nfft, 1/Ts);
+    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, leg_noise_hf{i}.da, win, Noverlap, Nfft, 1/Ts);
+    int_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Plot the FRF from u to de (interferometer)
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, abs(int_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', strut_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-9, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(int_frf(:, i)));
+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([10, 2e3]);
+
+% FRF Identification - Force Sensor
+% In this section, the dynamics from $u$ to $V_s$ is identified.
+% Then the FRF are estimated and shown in Figure ref:fig:struts_frf_iff_plant_tf.
+% They are also shown all to be very similar.
+
+%% FRF estimation of the transfer function from u to Vs
+iff_frf = zeros(length(f), length(strut_nums));
+for i = 1:length(strut_nums)
+    [frf_lf, ~] = tfestimate(leg_noise{i}.u,    leg_noise{i}.Vs,    win, Noverlap, Nfft, 1/Ts);
+    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, leg_noise_hf{i}.Vs, win, Noverlap, Nfft, 1/Ts);
+    iff_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Plot the FRF from u to Vs
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, abs(iff_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', strut_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(iff_frf(:, i)));
+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([10, 2e3]);
+
+% Misalignment of the APA and flexible joints
+
+% The misalignment between the two flexible joints and the APA has been measured for all the struts:
+% - the strut is fixed to the mounting bench
+% - using an indicator, the height difference from the flexible joints and the APA is measured both for the top and bottom joints and on both sides
+% - then it is possible to obtain the misalignment for both flexible joints
+
+% The raw measurements are shown in Table ref:tab:meas_misalignment_struts_raw.
+
+% As the flexible joint's "thickness" is 1mm larger than the APA "thickness", ideally (i.e. if it were perfectly centered) we would measure =-0.50mm= each time.
+
+
+strut_nums = [1, 2, 3, 4, 5];
+
+%             R Top  B Top  R Bot  B Bot
+strut_align = [[-0.40, -0.60, -0.16, -0.82]   % Strut 1
+               [-0.67, -0.30, -0.34, -0.63]   % Strut 2
+               [-0.07, -0.88, -0.16, -0.79]   % Strut 3
+               [-0.48, -0.46,  0.07, -1.00]   % Strut 4
+               [-0.33, -0.64, -0.48, -0.52]]; % Strut 5
+
+%% Save the estimated FRF for further analysis
+save('./mat/meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align');
+
+% Measured misalignment of the APA and flexible joints
+% The misalignment between the APA and the flexible joints are measured.
+
+% The results are defined below and summarized in Table ref:tab:meas_misalignment_struts_new_raw.
+
+
+%               R Top  B Top  R Bot  B Bot
+strut_align = [[-0.54, -0.50, -0.50, -0.52]   % strut 1
+               [-0.44, -0.55, -0.49, -0.49]   % strut 2
+               [-0.48, -0.50, -0.50, -0.46]   % strut 3
+               [-0.45, -0.51, -0.51, -0.45]   % strut 4
+               [-0.50, -0.50, -0.50, -0.50]   % strut 5
+               [-0.50, -0.49, -0.43, -0.54]]; % strut 6
+
+% FRF Identification - Setup
+% The excitation signal is a low pass filtered white noise.
+% Both the encoder and the force sensor voltage are measured.
+
+% Here are the leg numbers that have been measured.
+
+%% Numnbers of the measured legs
+strut_nums = [1 2 3 4 5 6];
+
+%% First identification (low frequency noise)
+leg_noise = {};
+for i = 1:length(strut_nums)
+    leg_noise(i) = {load(sprintf('frf_struts_align_%i_noise.mat', strut_nums(i)), 'u', 'Vs', 'de')};
+end
+
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(1/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+
+
+% We get the frequency vector that will be the same for all the frequency domain analysis.
+
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(leg_noise{1}.u, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts);
+
+% FRF Identification - Encoder
+% In this section, the dynamics from $u$ to $d_e$ (encoder) is identified.
+
+% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the encoder $d_e$ is computed:
+
+%% Transfer function estimation
+enc_frf = zeros(length(f), length(strut_nums));
+
+for i = 1:length(strut_nums)
+    enc_frf(:, i) = tfestimate(leg_noise{i}.u,    leg_noise{i}.de,    win, Noverlap, Nfft, 1/Ts);
+end
+
+%% Transfer function estimation
+iff_frf = zeros(length(f), length(strut_nums));
+
+for i = 1:length(strut_nums)
+    iff_frf(:, i) = tfestimate(leg_noise{i}.u,    leg_noise{i}.Vs,    win, Noverlap, Nfft, 1/Ts);
+end
+
+
+
+% The obtained transfer functions are shown in Figure ref:fig:struts_align_frf_dvf_plant_tf.
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, abs(enc_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', strut_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-8, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)));
+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([10, 2e3]);
+
+% TODO Noise measurement                                          :noexport:
+
+%% Nothing connected to the actuator stacks
+open_circuit  = load('frf_struts_align_3_huddle_open_circuit.mat',      't', 'Vs', 'de');
+
+%% PD200 connected but its input short-circuited
+mid_voltage   = load('frf_struts_align_3_huddle_mid_voltage_dac.mat',   't', 'Vs', 'de');
+
+%% PD200 connected to the DAC that outputs 0V
+zero_voltage  = load('frf_struts_align_3_huddle_dac_zero.mat',          't', 'Vs', 'de');
+
+%% PD200 connected to the DAC that outputs 3.25V
+short_circuit = load('frf_struts_align_3_huddle_amp_short_circuit.mat', 't', 'Vs', 'de');
+
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(2/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+[pxx_oc, f] = pwelch(detrend(open_circuit.Vs, 0),  win, Noverlap, Nfft, 1/Ts);
+[pxx_mv, ~] = pwelch(detrend(mid_voltage.Vs, 0),   win, Noverlap, Nfft, 1/Ts);
+[pxx_zv, ~] = pwelch(detrend(zero_voltage.Vs, 0),  win, Noverlap, Nfft, 1/Ts);
+[pxx_sc, ~] = pwelch(detrend(short_circuit.Vs, 0), win, Noverlap, Nfft, 1/Ts);
+
+figure;
+hold on;
+plot(f, sqrt(pxx_oc), 'DisplayName', 'Open Circuit')
+plot(f, sqrt(pxx_sc), 'DisplayName', 'Amp Short-Circuited')
+plot(f, sqrt(pxx_zv), 'DisplayName', 'Zero Voltage (DAC)')
+plot(f, sqrt(pxx_mv), 'DisplayName', 'Mid Voltage (DAC)')
+hold off;
+xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
+set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+legend('location', 'northeast');
+xlim([1, 5e3]);
+
+[pxx_oc, f] = pwelch(detrend(open_circuit.de, 0),  win, Noverlap, Nfft, 1/Ts);
+[pxx_mv, ~] = pwelch(detrend(mid_voltage.de, 0),   win, Noverlap, Nfft, 1/Ts);
+[pxx_zv, ~] = pwelch(detrend(zero_voltage.de, 0),  win, Noverlap, Nfft, 1/Ts);
+[pxx_sc, ~] = pwelch(detrend(short_circuit.de, 0), win, Noverlap, Nfft, 1/Ts);
+
+figure;
+hold on;
+plot(f, sqrt(pxx_oc), 'DisplayName', 'Open Circuit')
+plot(f, sqrt(pxx_sc), 'DisplayName', 'Amp Short-Circuited')
+plot(f, sqrt(pxx_zv), 'DisplayName', 'Zero Voltage (DAC)')
+plot(f, sqrt(pxx_mv), 'DisplayName', 'Mid Voltage (DAC)')
+hold off;
+xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
+set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+legend('location', 'northeast');
+xlim([1, 5e3])

matlab/test_struts_3_simscape_model.m

@@ -0,0 +1,934 @@
+%% 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 Simscape Model
+
+%% Linearization options
+opts = linearizeOptions;
+opts.SampleTime = 0;
+
+%% Open Simscape Model
+mdl = 'test_struts_simscape'; % Name of the Simulink File
+open(mdl); % Open Simscape Model
+
+%% Colors for the figures
+colors = colororder;
+
+%% Input/Output definition of the Model
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/u'],  1, 'openinput');  io_i = io_i + 1; % DAC Voltage
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
+io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
+io(io_i) = linio([mdl, '/da'], 1, 'openoutput'); io_i = io_i + 1; % Interferometer
+
+freqs = logspace(1, 3, 1000);
+
+% 2Dof model
+% The strut is initialized with default parameters (optimized parameters identified from previous experiments).
+
+%% Initialize structure containing data for the Simscape model
+n_hexapod = struct();
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
+n_hexapod.actuator = initializeAPA('type', '2dof');
+
+c_granite = 0; % Do not take into account damping added by the air bearing
+
+
+
+% The dynamics is identified and shown in Figure ref:fig:strut_bench_model_bode.
+
+%% Run the linearization
+Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+Gs.InputName  = {'u'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+%% Bode plot of the transfer functions
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
+plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'southwest');
+
+ax1b = nexttile([2,1]);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), 'k-')
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))))
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))))
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360);
+ylim([-180, 180])
+
+ax2b = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), 'k-')
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360);
+ylim([0, 180])
+
+linkaxes([ax1,ax2,ax1b,ax2b],'x');
+xlim([10, 2e3]);
+
+
+
+% #+name: fig:strut_bench_model_bode
+% #+caption: Identified transfer function from $u$ to $V_s$ and from $u$ to $d_e,d_a$ using the simple 2DoF model for the APA
+% #+RESULTS:
+% [[file:figs/strut_bench_model_bode.png]]
+
+% The experimentally measured FRF are loaded.
+
+%% Load measured FRF
+load('meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align');
+
+
+
+% The FRF from $u$ to $d_a$ as well as from $u$ to $V_s$ are shown in Figure ref:fig:comp_strut_plant_after_opt and compared with the model.
+% They are both found to match quite well with the model.
+
+%% Compare the FRF and identified dynamics from u to Vs and da
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(int_frf(:, 1)), 'color', [0,0,0,0.2], ...
+     'DisplayName', 'Meas. FRF');
+for i = 2:length(strut_nums)
+    plot(f, abs(int_frf(:, i)), 'color', [0,0,0,0.2], ...
+         'HandleVisibility', 'off');
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax1b = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], ...
+     'DisplayName', 'Meas. FRF');
+for i = 1:length(strut_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], ...
+         'HandleVisibility', 'off');
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(int_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '-')
+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]);
+
+ax2b = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '-')
+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,ax1b,ax2b],'x');
+xlim([10, 2e3]);
+
+
+
+% #+name: fig:comp_strut_plant_after_opt
+% #+caption: Comparison of the measured FRF and the optimized model
+% #+RESULTS:
+% [[file:figs/comp_strut_plant_after_opt.png]]
+
+% The measured FRF from $u$ to $d_e$ (encoder) is compared with the model in Figure ref:fig:comp_strut_plant_iff_after_opt.
+
+%% Compare the FRF and identified dynamics from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], ...
+     'DisplayName', 'Meas. FRF');
+for i = 2:length(strut_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], ...
+         'HandleVisibility', 'off');
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-')
+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([20, 2e3]);
+
+% Comparison with the Flexible Model
+% The strut is initialized with default parameters (optimized parameters identified from previous experiments).
+
+
+%% Initialize structure containing data for the Simscape model
+n_hexapod = struct();
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
+n_hexapod.actuator = initializeAPA('type', 'flexible');
+
+c_granite = 100; % Do not take into account damping added by the air bearing
+
+
+
+% The dynamics is identified and shown in Figure ref:fig:strut_bench_model_bode.
+
+%% Run the linearization
+Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+Gs.InputName  = {'u'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+
+
+% - [ ] Add encoder plot
+
+% The FRF from $u$ to $d_a$ as well as from $u$ to $V_s$ are shown in Figure ref:fig:comp_strut_plant_after_opt and compared with the model.
+% They are both found to match quite well with the model.
+
+%% Compare the FRF and identified dynamics from u to Vs and da
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, 1)), 'color', [colors(2,:), 0.5], ...
+     'DisplayName', 'FRF - Encoder');
+plot(f, abs(int_frf(:, 1)), 'color', [0,0,0, 0.2], ...
+     'DisplayName', 'FRF - Interferometer');
+for i = 2:length(strut_nums)
+    plot(f, abs(int_frf(:, i)), 'color', [0,0,0, 0.2], ...
+         'HandleVisibility', 'off');
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--', ...
+     'DisplayName', 'Model - Interferometer')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'southwest');
+
+ax1b = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf(:, i)), 'color', [colors(2,:), 0.2], ...
+     'DisplayName', 'Meas. FRF');
+for i = 1:length(strut_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0, 0.2], ...
+         'HandleVisibility', 'off');
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '--', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast');
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(enc_frf(:, 1)), 'color', [colors(2,:), 0.5], ...
+     'HandleVisibility', 'off');
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*(angle(int_frf(:, i))), 'color', [0,0,0, 0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--')
+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]);
+
+ax2b = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0, 0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '--')
+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,ax1b,ax2b],'x');
+xlim([10, 2e3]);
+
+% Perfectly aligned APA
+% Let's first consider that the strut is perfectly mounted such that the two flexible joints and the APA are aligned.
+
+%% Initialize Simscape data
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
+n_hexapod.actuator = initializeAPA('type', 'flexible');
+
+
+
+% And define the inputs and outputs of the models:
+% - Input: voltage generated by the DAC
+% - Output: measured displacement by the encoder
+
+% The transfer function is identified and shown in Figure ref:fig:comp_enc_frf_align_perfect.
+
+%% Identification
+Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+Gs.InputName  = {'u'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+
+
+% From Figure ref:fig:comp_enc_frf_align_perfect, it is clear that:
+% 1. The model with perfect alignment is not matching the measured FRF
+% 2. The mode at 200Hz is not present in the identified dynamics of the Simscape model
+% 3. The measured FRF have different shapes
+
+
+%% Measured FRF from Vs to de and identified dynamics using the flexible APA
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], ...
+     'DisplayName', 'Meas. FRF');
+for i = 2:length(strut_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], ...
+         'HandleVisibility', 'off');
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-')
+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([10, 2e3]);
+
+% Effect of a misalignment in y
+% Let's compute the transfer function from output DAC voltage $V_s$ to the measured displacement by the encoder $d_e$ for several misalignment in the $y$ direction:
+
+%% Considered misalignments
+dy_aligns = [-0.5, -0.1, 0, 0.1, 0.5]*1e-3; % [m]
+
+%% Transfer functions from u to de for all the misalignment in y direction
+Gs_align = {zeros(length(dy_aligns), 1)};
+
+for i = 1:length(dy_aligns)
+    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [0; dy_aligns(i); 0], 'd_align_top', [0; dy_aligns(i); 0]);
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs_align(i) = {G};
+end
+
+
+
+% The obtained dynamics are shown in Figure ref:fig:effect_misalignment_y.
+
+
+%% Transfer function from Vs to de - effect of x-misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(dy_aligns)
+    plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$d_y = %.1f$ [mm]', 1e3*dy_aligns(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(dy_aligns)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))));
+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([10, 2e3]);
+
+% Effect of a misalignment in x
+% Let's compute the transfer function from output DAC voltage to the measured displacement by the encoder for several misalignment in the $x$ direction:
+
+%% Considered misalignments
+dx_aligns = [-0.1, -0.05, 0, 0.05, 0.1]*1e-3; % [m]
+
+%% Transfer functions from u to de for all the misalignment in x direction
+Gs_align = {zeros(length(dx_aligns), 1)};
+
+for i = 1:length(dx_aligns)
+    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [dx_aligns(i); 0; 0], 'd_align_top', [dx_aligns(i); 0; 0]);
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs_align(i) = {G};
+end
+
+
+
+% The obtained dynamics are shown in Figure ref:fig:effect_misalignment_x.
+
+%% Transfer function from Vs to de - effect of x-misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(dx_aligns)
+    plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$d_x = %.2f$ [mm]', 1e3*dx_aligns(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(dx_aligns)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))));
+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([10, 2e3]);
+
+% Comparison with identified misalignment
+
+strut_align = 1e-3*[[-0.60, -0.82, -0.40, -0.16]
+                    [-0.30, -0.63, -0.67, -0.34]
+                    [-0.88, -0.79, -0.07, -0.16]
+                    [-0.48,  0.07, -0.46, -1.00]
+                    [-0.33, -0.48, -0.64, -0.52]
+                    [-0.34, -0.42, -0.63, -0.57]];
+
+%% Idenfity the transfer function from actuator to encoder for all cases
+Gs_align = {zeros(size(strut_align,1), 1)};
+
+for i = 1:size(strut_align,1)
+    n_hexapod.actuator = initializeAPA('type', 'flexible', ...
+                                       'd_align_bot', [0; strut_align(i, 2) - strut_align(i, 4); 0], ...
+                                       'd_align_top', [0; strut_align(i, 1) - strut_align(i, 3); 0]);
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs_align(i) = {G};
+end
+
+%% Comparison of the plants (encoder output) when tuning the misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+ax1 = nexttile();
+hold on;
+plot(f, abs(enc_frf(:, 1)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{1}('de', 'u'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
+
+ax2 = nexttile();
+hold on;
+plot(f, abs(enc_frf(:, 2)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{2}('de', 'u'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile(4);
+hold on;
+plot(f, abs(enc_frf(:, 3)), 'DisplayName', 'Meas.');
+plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'u'), freqs, 'Hz'))), ...
+     'DisplayName', 'Model');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
+legend('location', 'southwest', 'FontSize', 8);
+
+ax4 = nexttile(5);
+hold on;
+plot(f, abs(enc_frf(:, 4)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{4}('de', 'u'), freqs, 'Hz'))));
+hold off;
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+
+ax5 = nexttile(6);
+hold on;
+plot(f, abs(enc_frf(:, 5)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{5}('de', 'u'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+
+linkaxes([ax1,ax2,ax3,ax4,ax5],'xy');
+xlim([20, 2e3]); ylim([1e-8, 1e-3]);
+
+% Find the misalignment of each strut
+% From the previous analysis on the effect of a $x$ and $y$ misalignment, it is possible to estimate the $x,y$ misalignment of the measured struts.
+
+% The misalignment that gives the best match for the FRF are defined below.
+
+%% Tuned misalignment [m]
+d_aligns = [[-0.05,  -0.3,  0];
+            [ 0,      0.5,  0];
+            [-0.1,   -0.3,  0];
+            [ 0,      0.3,  0];
+            [-0.05,   0.05, 0]]'*1e-3;
+
+
+
+% For each misalignment, the dynamics from the DAC voltage to the encoder measurement is identified.
+
+%% Idenfity the transfer function from actuator to encoder for all cases
+Gs_align = {zeros(size(d_aligns,2), 1)};
+
+for i = 1:5
+    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_top', d_aligns(:,i), 'd_align_bot', d_aligns(:,i));
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs_align(i) = {G};
+end
+
+
+
+% The results are shown in Figure ref:fig:comp_all_struts_corrected_misalign.
+
+%% Comparison of the plants (encoder output) when tuning the misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
+ax1 = nexttile();
+hold on;
+plot(f, abs(enc_frf(:, 1)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{1}('de', 'u'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
+
+ax2 = nexttile();
+hold on;
+plot(f, abs(enc_frf(:, 2)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{2}('de', 'u'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
+
+ax3 = nexttile();
+hold on;
+plot(f, abs(enc_frf(:, 3)), 'DisplayName', 'Meas.');
+plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'u'), freqs, 'Hz'))), ...
+     'DisplayName', 'Model');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
+legend('location', 'southwest', 'FontSize', 8);
+
+ax4 = nexttile(5);
+hold on;
+plot(f, abs(enc_frf(:, 4)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{4}('de', 'u'), freqs, 'Hz'))));
+hold off;
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+
+ax5 = nexttile(6);
+hold on;
+plot(f, abs(enc_frf(:, 5)));
+plot(freqs, abs(squeeze(freqresp(Gs_align{5}('de', 'u'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+
+linkaxes([ax1,ax2,ax3,ax4,ax5],'xy');
+xlim([20, 2e3]); ylim([1e-8, 1e-3]);
+
+% Paper                                                         :noexport:
+
+%% Comparison of the plants (encoder output) when tuning the misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+colors = colororder;
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:,1)), 'color', [colors(1,:),0.2], ...
+     'DisplayName', 'FRF - $d_{e,i}/V_{a,i}$')
+for i = 2:5
+    plot(f, abs(enc_frf(:,i)), 'color', [colors(1,:),0.2], ...
+         'HandleVisibility', 'off');
+end
+
+plot(f, abs(int_frf(:,1)), 'color', [colors(2,:),0.2], ...
+     'DisplayName', 'FRF - $d_{a,i}/V_{a,i}$')
+for i = 2:5
+    plot(f, abs(int_frf(:,i)), 'color', [colors(2,:),0.2], ...
+         'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(Gs_align{1}('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(1,:), ...
+     'DisplayName', 'Model - $d_{e,i}/V_{a,i}$')
+for i = 2:5
+    plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(1,:), ...
+         'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), ...
+     'DisplayName', 'Model - $d_{a,i}/V_{a,i}$')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
+ylim([1e-8, 1e-3]);
+legend('location', 'southwest')
+
+ax2 = nexttile;
+hold on;
+for i = 1:5
+    plot(f, 180/pi*angle(enc_frf(:,i)), 'color', [colors(1,:),0.2]);
+    plot(f, 180/pi*(angle(int_frf(:, i)) - angle(squeeze(freqresp(exp(-s*2*1e-4), f, 'Hz')))), 'color', [colors(2,:),0.2]);
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(1,:));
+end
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:));
+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([20, 2e3]);
+
+% Effect of bending stiffness of the flexible joints
+% <<sec:struts_effect_bending_stiff_joints>>
+
+% Let's initialize an APA which is a little bit misaligned.
+
+%% APA Initialization
+n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [0.1e-3; 0.5e-3; 0], 'd_align_top', [0.1e-3; 0.5e-3; 0]);
+
+
+
+% The bending stiffnesses for which the dynamics is identified are defined below.
+
+%% Tested bending stiffnesses [Nm/rad]
+kRs = [3, 4, 5, 6, 7];
+
+
+
+% Then the identification is performed for all the values of the bending stiffnesses.
+
+%% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
+Gs = {zeros(length(kRs), 1)};
+
+for i = 1:length(kRs)
+    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
+        'type', '4dof', ...
+        'kRx', kRs(i), ...
+        'kRy', kRs(i));
+    n_hexapod.flex_top = initializeTopFlexibleJoint(...
+        'type', '4dof', ...
+        'kRx', kRs(i), ...
+        'kRy', kRs(i));
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs(i) = {G};
+end
+
+
+
+% The obtained dynamics from DAC voltage to encoder measurements are compared in Figure ref:fig:effect_enc_bending_stiff.
+
+%% Plot the obtained transfer functions for all the bending stiffnesses
+freqs = 2*logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$k_R = %.0f$ [Nm/rad]', kRs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))));
+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([20, 2e3]);
+
+% Effect of axial stiffness of the flexible joints
+% <<sec:struts_effect_axial_stiff_joints>>
+
+% The axial stiffnesses for which the dynamics is identified are defined below.
+
+%% Tested axial stiffnesses [N/m]
+kzs = [5e7 7.5e7 1e8 2.5e8];
+
+
+
+% Then the identification is performed for all the values of the bending stiffnesses.
+
+%% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
+Gs = {zeros(length(kzs), 1)};
+
+for i = 1:length(kzs)
+    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
+        'type', '4dof', ...
+        'kz', kzs(i));
+    n_hexapod.flex_top = initializeTopFlexibleJoint(...
+        'type', '4dof', ...
+        'kz', kzs(i));
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs(i) = {G};
+end
+
+
+
+% The obtained dynamics from DAC voltage to encoder measurements are compared in Figure ref:fig:effect_enc_axial_stiff.
+
+%% Plot the obtained transfer functions for all the axial stiffnesses
+freqs = 2*logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kzs)
+    plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$k_z = %.1e$ [N/m]', kzs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(kzs)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))));
+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([20, 2e3]);
+
+% Effect of bending damping
+% <<sec:struts_effect_bending_damping_joints>>
+% Now let's study the effect of the bending damping of the flexible joints.
+
+% The tested bending damping are defined below:
+
+%% Tested bending dampings [Nm/(rad/s)]
+cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
+
+
+
+% Then the identification is performed for all the values of the bending damping.
+
+%% Idenfity the transfer function from actuator to encoder for all bending dampins
+Gs = {zeros(length(cRs), 1)};
+
+for i = 1:length(cRs)
+    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
+        'type', '4dof', ...
+        'cRx', cRs(i), ...
+        'cRy', cRs(i));
+    n_hexapod.flex_top = initializeTopFlexibleJoint(...
+        'type', '4dof', ...
+        'cRx', cRs(i), ...
+        'cRy', cRs(i));
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs(i) = {G};
+end
+
+
+
+% The results are shown in Figure ref:fig:effect_enc_bending_damp.
+
+%% Plot the obtained transfer functions for all the bending stiffnesses
+freqs = 2*logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(cRs)
+    plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', cRs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'southwest');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(cRs)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))));
+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([20, 2e3]);

test-bench-struts.tex

@@ -1,4 +1,4 @@
-% Created 2024-03-19 Tue 17:29
+% Created 2024-03-25 Mon 10:54
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -24,7 +24,7 @@
 \clearpage
 In this document, a test-bench is used to characterize the struts of the nano-hexapod.
 
-Each strut includes (Figure \ref{fig:picture_strut_top_view}):
+Each strut includes (Figure \ref{fig:test_struts_picture_strut}):
 \begin{itemize}
 \item 2 flexible joints at each ends.
 These flexible joints have been characterized in a separate test bench (see \ldots{}).
@@ -35,67 +35,69 @@ Two stacks are used as an actuator and one stack as a (force) sensor.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/picture_strut_top_view.jpg}
-\caption{\label{fig:picture_strut_top_view}One strut including two flexible joints, an amplified piezoelectric actuator and an encoder}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_struts_picture_strut.jpg}
+\caption{\label{fig:test_struts_picture_strut}One strut including two flexible joints, an amplified piezoelectric actuator and an encoder}
 \end{figure}
 
-Then the struts are mounted (procedure described in Section \ref{sec:test_bench_struts_mounting}), and are fixed to the same measurement bench.
+Then the struts are mounted (procedure described in Section \ref{sec:test_struts_mounting}), and are fixed to the same measurement bench.
 The goals are to:
 \begin{itemize}
-\item Section \ref{sec:test_bench_struts_dynamical_meas}: Identify the dynamics from the generated DAC voltage to:
+\item Section \ref{sec:test_struts_dynamical_meas}: Identify the dynamics from the generated DAC voltage to:
 \begin{itemize}
 \item the sensors stack generated voltage
 \item the measured displacement by the encoder
 \item the measured displacement by the interferometer (representing encoders that would be fixed to the nano-hexapod's plates instead of the struts)
 \end{itemize}
-\item Section \ref{sec:test_bench_struts_simscape}: Compare the measurements with the Simscape model of the struts and tune the models
+\item Section \ref{sec:test_struts_simscape}: Compare the measurements with the Simscape model of the struts and tune the models
 \end{itemize}
 
 The final goal of the work presented in this document is to have an accurate Simscape model of the struts that can then be included in the Simscape model of the nano-hexapod.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_bench_struts_section_matlab_code}Report sections and corresponding Matlab files}
+\caption{\label{tab:test_struts_section_matlab_code}Report sections and corresponding Matlab files}
 \centering
 \begin{tabularx}{0.6\linewidth}{lX}
 \toprule
 \textbf{Sections} & \textbf{Matlab File}\\
 \midrule
-Section \ref{sec:test_bench_struts}\_ & \texttt{test\_bench\_struts\_1\_.m}\\
+Section \ref{sec:test_struts_flexible_modes} & \texttt{test\_struts\_1\_flexible\_modes.m}\\
+Section \ref{sec:test_struts_dynamical_meas} & \texttt{test\_struts\_2\_dynamical\_meas.m}\\
+Section \ref{sec:test_struts_mounting} & \texttt{test\_struts\_3\_simscape\_model.m}\\
 \bottomrule
 \end{tabularx}
 \end{table}
 \chapter{Mounting Procedure}
-\label{sec:test_bench_struts_mounting}
+\label{sec:test_struts_mounting}
 \section{Mounting Bench}
 
 A mounting bench is used to greatly simply the mounting of the struts as well as ensuring the correct strut length and coaxiality of the flexible joint's interfaces.
 This is very important in order to not loose any stroke when the struts will be mounted on the nano-hexapod.
 
-A CAD view of the mounting bench is shown in Figure \ref{fig:strut_mounting_bench_first_concept}.
+A CAD view of the mounting bench is shown in Figure \ref{fig:test_struts_mounting_bench_first_concept}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/strut_mounting_bench_first_concept.png}
-\caption{\label{fig:strut_mounting_bench_first_concept}CAD view of the mounting bench}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_bench_first_concept.png}
+\caption{\label{fig:test_struts_mounting_bench_first_concept}CAD view of the mounting bench}
 \end{figure}
 
-The main part of the bench is here to ensure both the correct strut length and strut coaxiality as shown in Figure \ref{fig:strut_mounting_step_0}.
+The main part of the bench is here to ensure both the correct strut length and strut coaxiality as shown in Figure \ref{fig:test_struts_mounting_step_0}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/strut_mounting_step_0.jpg}
-\caption{\label{fig:strut_mounting_step_0}Useful features of the main mounting element}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_step_0.jpg}
+\caption{\label{fig:test_struts_mounting_step_0}Useful features of the main mounting element}
 \end{figure}
 
-The tight tolerances of this element has been verified as shown in Figure \ref{fig:strut_mounting_bench_first_concept} and were found to comply with the requirements.
+The tight tolerances of this element has been verified as shown in Figure \ref{fig:test_struts_mounting_bench_first_concept} and were found to comply with the requirements.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=\linewidth]{figs/check_dimensions_bench.jpg}
-\caption{\label{fig:strut_mounting_bench_first_concept}Dimensional verifications of the mounting bench tolerances}
+\caption{\label{fig:test_struts_mounting_bench_first_concept}Dimensional verifications of the mounting bench tolerances}
 \end{figure}
 
-The flexible joints are rigidly fixed to cylindrical tools shown in Figure \ref{fig:cylindrical_mounting_part} which are then mounted on the mounting tool shown in Figure \ref{fig:strut_mounting_step_0}.
+The flexible joints are rigidly fixed to cylindrical tools shown in Figure \ref{fig:cylindrical_mounting_part} which are then mounted on the mounting tool shown in Figure \ref{fig:test_struts_mounting_step_0}.
 This cylindrical tool is here to protect the flexible joints when tightening the screws and therefore applying large torque.
 
 \begin{figure}[htbp]
@@ -107,61 +109,72 @@ This cylindrical tool is here to protect the flexible joints when tightening the
 
 The mounting procedure is as follows:
 \begin{enumerate}
-\item Screw flexible joints inside the cylindrical interface element shown in Figure \ref{fig:cylindrical_mounting_part} (Figure \ref{fig:strut_mounting_step_1})
+\item Screw flexible joints inside the cylindrical interface element shown in Figure \ref{fig:cylindrical_mounting_part} (Figure \ref{fig:test_struts_mounting_step_1})
 \item Fix the two interface elements. One of the two should be clamped, the other one should have its axial rotation free.
-Visually align the clamped one horizontally. (Figure \ref{fig:strut_mounting_step_2})
-\item Put cylindrical washers, APA and interface pieces on top of the flexible joints (Figure \ref{fig:strut_mounting_step_3})
+Visually align the clamped one horizontally. (Figure \ref{fig:test_struts_mounting_step_2})
+\item Put cylindrical washers, APA and interface pieces on top of the flexible joints (Figure \ref{fig:test_struts_mounting_step_3})
 \item Put the 4 screws just in contact such that everything is correctly positioned and such that the ``free'' flexible joint is correctly oriented
 \item Put the 8 lateral screws in contact
 \item Tighten the 4 screws to fix the APA on the two flexible joints (using a torque screwdriver)
 \item Remove the 4 laterals screws
-\item (optional) Put the APA horizontally and fix the encoder and align it to maximize the contrast (Figure \ref{fig:strut_mounting_step_4})
+\item (optional) Put the APA horizontally and fix the encoder and align it to maximize the contrast (Figure \ref{fig:test_struts_mounting_step_4})
 \end{enumerate}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.5\linewidth]{figs/strut_mounting_step_1.jpg}
-\caption{\label{fig:strut_mounting_step_1}Step 1 - Flexible joints fixed on the cylindrical interface elements}
+\includegraphics[scale=1,width=0.5\linewidth]{figs/test_struts_mounting_step_1.jpg}
+\caption{\label{fig:test_struts_mounting_step_1}Step 1 - Flexible joints fixed on the cylindrical interface elements}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/strut_mounting_step_2.jpg}
-\caption{\label{fig:strut_mounting_step_2}Step 2 - Cylindrical elements fixed on the bench}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_step_2.jpg}
+\caption{\label{fig:test_struts_mounting_step_2}Step 2 - Cylindrical elements fixed on the bench}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/strut_mounting_step_3.jpg}
-\caption{\label{fig:strut_mounting_step_3}Step 3 - Mount the nuts, washers and APA}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_step_3.jpg}
+\caption{\label{fig:test_struts_mounting_step_3}Step 3 - Mount the nuts, washers and APA}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/strut_mounting_step_4.jpg}
-\caption{\label{fig:strut_mounting_step_4}Last step - Align the encoder on the strut}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_step_4.jpg}
+\caption{\label{fig:test_struts_mounting_step_4}Last step - Align the encoder on the strut}
 \end{figure}
 \section{Mounted Struts}
 
-After removing the strut from the mounting bench, we obtain a strut with ensured coaxiality between the two flexible joint's interfaces (Figure \ref{fig:mounted_strut}).
+After removing the strut from the mounting bench, we obtain a strut with ensured coaxiality between the two flexible joint's interfaces (Figure \ref{fig:test_struts_mounted_strut}).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/mounted_strut.jpg}
-\caption{\label{fig:mounted_strut}Mounted Strut with ensured coaxiality}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounted_strut.jpg}
+\caption{\label{fig:test_struts_mounted_strut}Mounted Strut with ensured coaxiality}
 \end{figure}
 \chapter{Spurious resonances}
-\label{sec:spurious_resonances_struts}
+\label{sec:test_struts_flexible_modes}
 \section{Introduction}
 
-Similarly as in Section \ref{sec:spurious_resonances}, the spurious modes of the struts (Figure \ref{fig:apa_mode_shapes_ter}) are measured.
-
 These modes are present when flexible joints are fixed to the ends of the APA300ML.
 To experimentally measure the frequency of these modes, the struts are mounted (both with and without the encoder).
-Then, each end of the strut is fixed to a vertically guided stage as shown in Figure \ref{fig:meas_spur_res_struts_1_enc}.
+Then, each end of the strut is fixed to a vertically guided stage as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_1_enc}.
+
+From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure \ref{fig:test_struts_mode_shapes}):
+\begin{itemize}
+\item Mode in X-bending at 189Hz
+\item Mode in Y-bending at 285Hz
+\item Mode in Z-torsion at 400Hz
+\end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mode_shapes.png}
+\caption{\label{fig:test_struts_mode_shapes}Spurious resonances of the struts estimated from a Finite Element Model. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz}
+\end{figure}
 \section{Measurement Setup}
 
-A Laser vibrometer is measuring the difference of motion between two points (Figure \ref{fig:meas_spur_res_struts_1_enc}).
+A Laser vibrometer is measuring the difference of motion between two points (Figure \ref{fig:test_struts_meas_spur_res_struts_1_enc}).
 The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.
 
 \begin{note}
@@ -172,54 +185,54 @@ The instrumentation used are:
 \end{itemize}
 \end{note}
 
-The ``X-bending'' mode is measured as shown in Figure \ref{fig:meas_spur_res_struts_1_enc}.
-The ``Y-bending'' mode is measured as shown in Figure \ref{fig:meas_spur_res_struts_2} with the encoder and in Figure \ref{fig:meas_spur_res_struts_2_encoder} with the encoder.
-Finally, the ``Z-torsion'' is measured as shown in Figure \ref{fig:meas_spur_res_struts_3}.
+The ``X-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_1_enc}.
+The ``Y-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_2} with the encoder and in Figure \ref{fig:test_struts_meas_spur_res_struts_2_encoder} with the encoder.
+Finally, the ``Z-torsion'' is measured as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_3}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/meas_spur_res_struts_1_enc.jpg}
-\caption{\label{fig:meas_spur_res_struts_1_enc}Measurement setup for the X-Bending measurement (with the encoder)}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_meas_spur_res_struts_1_enc.jpg}
+\caption{\label{fig:test_struts_meas_spur_res_struts_1_enc}Measurement setup for the X-Bending measurement (with the encoder)}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/meas_spur_res_struts_2.jpg}
-\caption{\label{fig:meas_spur_res_struts_2}Measurement setup for the Y-Bending measurement}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_meas_spur_res_struts_2.jpg}
+\caption{\label{fig:test_struts_meas_spur_res_struts_2}Measurement setup for the Y-Bending measurement}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/meas_spur_res_struts_2_encoder.jpg}
-\caption{\label{fig:meas_spur_res_struts_2_encoder}Measurement setup for the Y-Bending measurement (with the encoder)}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_meas_spur_res_struts_2_encoder.jpg}
+\caption{\label{fig:test_struts_meas_spur_res_struts_2_encoder}Measurement setup for the Y-Bending measurement (with the encoder)}
 \end{figure}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/meas_spur_res_struts_3.jpg}
-\caption{\label{fig:meas_spur_res_struts_3}Measurement setup for the Z-Torsion measurement}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_struts_meas_spur_res_struts_3.jpg}
+\caption{\label{fig:test_struts_meas_spur_res_struts_3}Measurement setup for the Z-Torsion measurement}
 \end{figure}
 \section{Without Encoder}
-When the encoder is not fixed to the strut, the obtained FRF are shown in Figure \ref{fig:struts_spur_res_without_enc}.
+When the encoder is not fixed to the strut, the obtained FRF are shown in Figure \ref{fig:test_struts_spur_res_frf}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/struts_spur_res_without_enc.png}
-\caption{\label{fig:struts_spur_res_without_enc}Obtained FRF for the struts without the encoder}
+\includegraphics[scale=1]{figs/test_struts_spur_res_frf.png}
+\caption{\label{fig:test_struts_spur_res_frf}Obtained FRF for the struts without the encoder}
 \end{figure}
 \section{With Encoder}
-Then, one encoder is fixed to the strut and the FRF are measured again and shown in Figure \ref{fig:struts_spur_res_with_enc}.
+Then, one encoder is fixed to the strut and the FRF are measured again and shown in Figure \ref{fig:test_struts_spur_res_frf_enc}.
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/struts_spur_res_with_enc.png}
-\caption{\label{fig:struts_spur_res_with_enc}Obtained FRF for the struts with encoder}
+\includegraphics[scale=1]{figs/test_struts_spur_res_frf_enc.png}
+\caption{\label{fig:test_struts_spur_res_frf_enc}Obtained FRF for the struts with encoder}
 \end{figure}
 \section{Conclusion}
 
-Table \ref{tab:strut_measured_modes_freq} summarizes the measured resonance frequencies as well as the computed ones using the Finite Element Model.
+Table \ref{tab:test_struts_spur_mode_freqs} summarizes the measured resonance frequencies as well as the computed ones using the Finite Element Model.
 
 \begin{important}
-From the values in Table \ref{tab:strut_measured_modes_freq}, it is shown that:
+From the values in Table \ref{tab:test_struts_spur_mode_freqs}, it is shown that:
 \begin{itemize}
 \item the resonance frequencies of the 3 modes are only slightly increasing when the encoder is removed
 \item the computed resonance frequencies from the FEM are very close to the measured one when the encoder is fixed to the strut
@@ -227,7 +240,7 @@ From the values in Table \ref{tab:strut_measured_modes_freq}, it is shown that:
 \end{important}
 
 \begin{table}[htbp]
-\caption{\label{tab:strut_measured_modes_freq}Measured frequency of the strut spurious modes}
+\caption{\label{tab:test_struts_spur_mode_freqs}Measured frequency of the strut spurious modes}
 \centering
 \begin{tabularx}{0.45\linewidth}{cccc}
 \toprule
@@ -240,35 +253,35 @@ Z-Torsion & 400Hz & 381Hz & 398Hz\\
 \end{tabularx}
 \end{table}
 \chapter{Dynamical measurements}
-\label{sec:test_bench_struts_dynamical_meas}
-The same bench used in Section \ref{sec:dynamical_meas_apa} is here used with the strut instead of only the APA.
+\label{sec:test_struts_dynamical_meas}
+The bench is shown in Figure \ref{fig:test_struts_bench_leg_overview}.
+Measurements are performed either when no encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) or when one encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}).
 
-The bench is shown in Figure \ref{fig:test_bench_leg_overview}.
-Measurements are performed either when no encoder is fixed to the strut (Figure \ref{fig:test_bench_leg_front}) or when one encoder is fixed to the strut (Figure \ref{fig:test_bench_leg_coder}).
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.5\linewidth]{figs/test_bench_leg_overview.jpg}
-\caption{\label{fig:test_bench_leg_overview}Test Bench with Strut - Overview}
+\begin{figure}
+\begin{subfigure}{0.35\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/test_struts_bench_leg_overview.jpg}
+\end{center}
+\subcaption{\label{fig:test_struts_bench_leg_overview}Overview}
+\end{subfigure}
+\begin{subfigure}{0.31\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/test_struts_bench_leg_front.jpg}
+\end{center}
+\subcaption{\label{fig:test_struts_bench_leg_front}Strut without encoder}
+\end{subfigure}
+\begin{subfigure}{0.31\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.99\linewidth]{figs/test_struts_bench_leg_coder.jpg}
+\end{center}
+\subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder}
+\end{subfigure}
+\caption{\label{fig:test_struts_bench_leg}Experimental setup to measured the dynamics of the struts.}
 \end{figure}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.5\linewidth]{figs/test_bench_leg_front.jpg}
-\caption{\label{fig:test_bench_leg_front}Test Bench with Strut - Zoom on the strut}
-\end{figure}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.5\linewidth]{figs/test_bench_leg_coder.jpg}
-\caption{\label{fig:test_bench_leg_coder}Test Bench with Strut - Zoom on the strut with the encoder}
-\end{figure}
-
-Variables are named the same as in Section \ref{sec:dynamical_meas_apa}.
-
-First, only one strut is measured in details (Section \ref{sec:meas_strut_1}), and then all the struts are measured and compared (Section \ref{sec:meas_all_struts}).
+First, only one strut is measured in details (Section \ref{ssec:test_struts_meas_strut_1}), and then all the struts are measured and compared (Section \ref{ssec:test_struts_meas_all_struts}).
 \section{Measurement on Strut 1}
-\label{sec:meas_strut_1}
+\label{ssec:test_struts_meas_strut_1}
 Measurements are first performed on one of the strut that contains:
 \begin{itemize}
 \item the Amplified Piezoelectric Actuator (APA) number 1
@@ -293,7 +306,6 @@ The noise is band-passed between 300Hz and 2kHz.
 Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.
 
 The time is the same for all measurements.
-Then we defined a ``Hanning'' windows that will be used for the spectral analysis:
 We get the frequency vector that will be the same for all the frequency domain analysis.
 \paragraph{FRF Identification - Interferometer}
 In this section, the dynamics from the excitation voltage \(V_a\) to the interferometer \(d_a\) is identified.
@@ -402,21 +414,15 @@ As shown in Figure \ref{fig:strut_1_spurious_resonances}, we can clearly see thr
 
 These resonances correspond to parasitic resonances of the strut itself.
 
-They are very close to what was estimated using a finite element model of the strut (Figure \ref{fig:apa_mode_shapes_bis}):
+They are very close to what was estimated using a finite element model of the strut (Figure \ref{fig:test_struts_mode_shapes}):
 \begin{itemize}
 \item Mode in X-bending at 189Hz
 \item Mode in Y-bending at 285Hz
 \item Mode in Z-torsion at 400Hz
 \end{itemize}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=\linewidth]{figs/apa_mode_shapes.gif}
-\caption{\label{fig:apa_mode_shapes_bis}Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz}
-\end{figure}
-
 \begin{important}
-The resonances seen by the encoder in Figure \ref{fig:strut_1_spurious_resonances} are indeed corresponding to the modes of the strut as shown in Figure \ref{fig:apa_mode_shapes_bis}.
+The resonances seen by the encoder in Figure \ref{fig:strut_1_spurious_resonances} are indeed corresponding to the modes of the strut as shown in Figure \ref{fig:test_struts_mode_shapes}.
 \end{important}
 \paragraph{FRF Identification - Force Sensor}
 In this section, the dynamics from \(V_a\) to \(V_s\) is identified.
@@ -455,7 +461,7 @@ In order to determine if the complex conjugate zero of Figure \ref{fig:strut_1_e
 
 Remove time delay
 \section{Comparison of all the Struts}
-\label{sec:meas_all_struts}
+\label{ssec:test_struts_meas_all_struts}
 Now all struts are measured using the same procedure and test bench as in Section \ref{sec:meas_strut_1}.
 \subsection{FRF Identification - Setup}
 The identification of the struts dynamics is performed in two steps:
@@ -610,7 +616,7 @@ However, the dynamics from \(V_a\) to the encoder measurement \(d_e\) is much mo
 
 The measured FRF are now saved for further use.
 \section{Comparison of all the (re-aligned) Struts}
-\label{sec:meas_all_aligned_struts}
+\label{sec:test_struts_meas_all_aligned_struts}
 The struts are re-aligned and measured using the same test bench.
 \subsection{Measured misalignment of the APA and flexible joints}
 The misalignment between the APA and the flexible joints are measured.
@@ -731,14 +737,13 @@ Having the struts well aligned does not change significantly the obtained dynami
 
 The measured FRF are now saved for further use.
 \chapter{Simscape Model}
-\label{sec:test_bench_struts_simscape}
-The same simscape model that was presented in Section \ref{sec:simscape_bench_apa} is here used.
-However, now the full strut is put instead of only the APA (see Figure \ref{fig:simscape_model_bench_struts}).
+\label{sec:test_struts_simscape}
+However, now the full strut is put instead of only the APA (see Figure \ref{fig:test_struts_simscape_model}).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/simscape_model_bench_struts.png}
-\caption{\label{fig:simscape_model_bench_struts}Screenshot of the Simscape model of the strut fixed to the bench}
+\includegraphics[scale=1,width=\linewidth]{figs/test_struts_simscape_model.png}
+\caption{\label{fig:test_struts_simscape_model}Screenshot of the Simscape model of the strut fixed to the bench}
 \end{figure}
 
 This Simscape model is used to:
@@ -813,13 +818,7 @@ As shown in Figure \ref{fig:struts_frf_dvf_plant_tf}, the dynamics from actuator
 
 This could be explained by a large variability in the alignment of the flexible joints and the APA (at the time, the alignment pins were not used).
 
-Depending on the alignment, the spurious resonances of the struts (Figure \ref{fig:apa_mode_shapes}) can be excited differently.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=\linewidth]{figs/apa_mode_shapes.gif}
-\caption{\label{fig:apa_mode_shapes}Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz}
-\end{figure}
+Depending on the alignment, the spurious resonances of the struts (Figure \ref{fig:test_struts_mode_shapes}) can be excited differently.
 
 For instance, consider Figure \ref{fig:strut_misalign_schematic} where there is a misalignment in the \(y\) direction.
 In such case, the mode at 200Hz is foreseen to be more excited as the misalignment \(d_y\) increases and therefore the dynamics from the actuator to the encoder should also change around 200Hz.
@@ -991,6 +990,6 @@ Not sure is would be effect though.
 \end{question}
 \section{Comparison with identified misalignment}
 \chapter{Conclusion}
-\label{sec:test_bench_struts_conclusion}
+\label{sec:test_struts_conclusion}
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}