Add tangled matlab scripts

.gitignore

@@ -4,6 +4,7 @@ auto/
 *.synctex.gz
 .auctex-auto/
 _minted*
+flycheck_*
 
 # Emacs
 auto/

figs/apa_act_constant_comp.pdf

@@ -3,7 +3,7 @@
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matlab/apa_meas_analysis_1.m

@@ -0,0 +1,247 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+Fs = 1e4; % Sampling Frequency [Hz]
+Ts = 1/Fs; % Sampling Time [s]
+
+addpath('./mat/');
+addpath('./src/');
+
+%% Load data
+apa_sweep = load(sprintf('mat/frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'da', 'de');
+
+%% Time vector
+t = apa_sweep.t - apa_sweep.t(1) ; % Time vector [s]
+
+%% Plot the excitation signal
+figure;
+plot(t, apa_sweep.Va)
+xlabel('Time [s]'); ylabel('Excitation Voltage $V_a$ [V]');
+
+%% Sampling Frequency / Time
+Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
+Fs = 1/Ts; % Sampling Frequency [Hz]
+
+win = hanning(ceil(1*Fs)); % Hannning Windows
+
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
+
+%% Compute the coherence
+[enc_coh, ~] = mscohere(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
+[int_coh, ~] = mscohere(apa_sweep.Va, apa_sweep.da, win, [], [], 1/Ts);
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, enc_coh, 'DisplayName', '$d_e/V_a$');
+plot(f, int_coh, 'DisplayName', '$d_a/V_a$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([5, 5e3]); ylim([0, 1]);
+
+%% TF - Encoder and interferometer
+[frf_enc, ~] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
+[frf_int, ~] = tfestimate(apa_sweep.Va, apa_sweep.da, win, [], [], 1/Ts);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(frf_enc), 'color', colors(1, :), ...
+     'DisplayName', 'Encoder');
+plot(f, abs(frf_int), 'color', colors(2, :), ...
+     'DisplayName', 'Interferometer');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast');
+ylim([1e-9, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_enc), 'color', colors(1, :));
+plot(f, 180/pi*angle(frf_int), 'color', colors(2, :));
+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]);
+
+%% Compute the coherence from Va to Vs
+[iff_coh, ~] = mscohere(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, iff_coh, 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([5, 5e3]); ylim([0, 1]);
+
+%% Compute the TF from Va to Vs
+[iff_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);
+
+%% Plot the TF
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_sweep), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(iff_sweep), 'k-');
+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]);
+
+%% Load measured data - hysteresis
+apa_hyst = load('frf_data_1_hysteresis.mat', 't', 'Va', 'de');
+% Initial time set to zero
+apa_hyst.t = apa_hyst.t - apa_hyst.t(1);
+
+ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes
+
+%% Plot the excitation voltages and measured displacements
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile;
+plot(apa_hyst.t, apa_hyst.Va)
+xlabel('Time [s]'); ylabel('Output Voltage [V]');
+
+ax2 = nexttile;
+plot(apa_hyst.t, apa_hyst.de)
+xlabel('Time [s]'); ylabel('Measured Displacement [m]');
+
+%% Measured displacement as a function of the output voltage
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([1,2]);
+hold on;
+for i = flip(1:6)
+    i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5;
+    plot(apa_hyst.Va(i_lim) - mean(apa_hyst.Va(i_lim)), apa_hyst.de(i_lim) - mean(apa_hyst.de(i_lim)), ...
+         'DisplayName', sprintf('$V_a = %.1f [V]$', ampls(i)))
+end
+hold off;
+xlabel('Output Voltage [V]'); ylabel('Measured Displacement [m]');
+legend('location', 'northeast');
+xlim([-4, 4]); ylim([-1.2e-4, 1.2e-4]);
+
+ax2 = nexttile;
+hold on;
+for i = flip(1:6)
+    i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5;
+    plot(apa_hyst.Va(i_lim) - mean(apa_hyst.Va(i_lim)), apa_hyst.de(i_lim) - mean(apa_hyst.de(i_lim)))
+end
+hold off;
+xlim([-0.4, 0.4]); ylim([-0.8e-5, 0.8e-5]);
+
+%% Load data for stiffness measurement
+apa_mass = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');
+apa_mass.de = apa_mass.de - mean(apa_mass.de(apa_mass.t<11));
+
+%% Plot the deflection at a function of time
+figure;
+plot(apa_mass.t, apa_mass.de, 'k-');
+xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]');
+
+added_mass = 6.4; % Added mass [kg]
+
+k = 9.8 * added_mass / (mean(apa_mass.de(apa_mass.t > 12 & apa_mass.t < 12.5)) - mean(apa_mass.de(apa_mass.t > 20 & apa_mass.t < 20.5)));
+
+wz = 2*pi*94; % [rad/s]
+msus = 5.7; % [kg]
+k = msus * wz^2;
+
+%% Load Data
+add_mass_oc = load(sprintf('frf_data_%i_add_mass_open_circuit.mat',   1), 't', 'de');
+add_mass_cc = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');
+
+%% Zero displacement at initial time
+add_mass_oc.de = add_mass_oc.de - mean(add_mass_oc.de(add_mass_oc.t<11));
+add_mass_cc.de = add_mass_cc.de - mean(add_mass_cc.de(add_mass_cc.t<11));
+
+figure;
+hold on;
+plot(add_mass_oc.t, add_mass_oc.de, 'DisplayName', 'Not connected');
+plot(add_mass_cc.t, add_mass_cc.de, 'DisplayName', 'Connected');
+hold off;
+xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]');
+legend('location', 'northeast');
+
+apa_k_oc = 9.8 * added_mass / (mean(add_mass_oc.de(add_mass_oc.t > 12 & add_mass_oc.t < 12.5)) - mean(add_mass_oc.de(add_mass_oc.t > 20 & add_mass_oc.t < 20.5)));
+apa_k_cc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass_cc.t < 12.5)) - mean(add_mass_cc.de(add_mass_cc.t > 20 & add_mass_cc.t < 20.5)));
+
+%% Load the data
+wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va'); % With the resistor
+wo_k = load('frf_data_1_sweep_lf.mat',        't', 'Vs', 'Va'); % Without the resistor
+
+win = hanning(ceil(50*Fs)); % Hannning Windows
+
+%% Compute the transfer functions from Va to Vs
+[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, [], [], 1/Ts);
+[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, [], [], 1/Ts);
+
+%% Model for the high pass filter
+C = 5.1e-6; % Sensor Stack capacitance [F]
+R = 80.6e3; % Parallel Resistor [Ohm]
+
+f0 = 1/(2*pi*R*C); % Crossover frequency of RC HPF [Hz]
+
+G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0);
+
+%% Compare the HPF model and the measured FRF
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(frf_wo_k), 'DisplayName', 'Without $k$');
+plot(f, abs(frf_wi_k), 'DisplayName', 'With $k$');
+plot(f, abs(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', sprintf('HPF $f_o = %.2f [Hz]$', f0));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-1, 1e0]);
+legend('location', 'southeast')
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_wo_k));
+plot(f, 180/pi*angle(frf_wi_k));
+plot(f, 180/pi*angle(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360); ylim([-45, 90]);
+
+linkaxes([ax1,ax2],'x');
+xlim([0.2, 8]);

matlab/apa_meas_analysis_all.m

@@ -0,0 +1,217 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+
+addpath('./mat/');
+addpath('./src/');
+
+added_mass = 6.4; % Added mass [kg]
+
+apa_nums = [1 2 4 5 6 7 8];
+
+%% Load Data
+apa_mass = {};
+for i = 1:length(apa_nums)
+    apa_mass(i) = {load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', apa_nums(i)), 't', 'de')};
+    % The initial displacement is set to zero
+    apa_mass{i}.de = apa_mass{i}.de - mean(apa_mass{i}.de(apa_mass{i}.t<11));
+end
+
+%% Plot the time domain measured deflection
+figure;
+hold on;
+for i = 1:length(apa_nums)
+  plot(apa_mass{i}.t, apa_mass{i}.de, 'DisplayName', sprintf('APA %i', apa_nums(i)));
+end
+hold off;
+xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]');
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+
+%% Compute the stiffness
+apa_k = zeros(length(apa_nums), 1);
+for i = 1:length(apa_nums)
+    apa_k(i) = 9.8 * added_mass / (mean(apa_mass{i}.de(apa_mass{i}.t > 12 & apa_mass{i}.t < 12.5)) - mean(apa_mass{i}.de(apa_mass{i}.t > 20 & apa_mass{i}.t < 20.5)));
+end
+
+%% Second identification
+apa_sweep = {};
+for i = 1:length(apa_nums)
+    apa_sweep(i) = {load(sprintf('frf_data_%i_sweep.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
+end
+
+%% Third identification
+apa_noise_hf = {};
+for i = 1:length(apa_nums)
+    apa_noise_hf(i) = {load(sprintf('frf_data_%i_noise_hf.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
+end
+
+%% Time vector
+t = apa_sweep{1}.t - apa_sweep{1}.t(1) ; % Time vector [s]
+
+%% Sampling
+Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
+Fs = 1/Ts; % Sampling Frequency [Hz]
+
+win = hanning(ceil(0.5*Fs)); % Hannning Windows
+
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts);
+i_lf = f <= 350;
+i_hf = f >  350;
+
+%% Coherence computation
+coh_enc = zeros(length(f), length(apa_nums));
+for i = 1:length(apa_nums)
+    [coh_lf, ~] = mscohere(apa_sweep{i}.Va,    apa_sweep{i}.de,    win, [], [], 1/Ts);
+    [coh_hf, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
+    coh_enc(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
+end
+
+figure;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, coh_enc(:, i));
+end;
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([5, 5e3]); ylim([0, 1]);
+
+%% Transfer function estimation
+enc_frf = zeros(length(f), length(apa_nums));
+for i = 1:length(apa_nums)
+    [frf_lf, ~] = tfestimate(apa_sweep{i}.Va,    apa_sweep{i}.de,    win, [], [], 1/Ts);
+    [frf_hf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
+    enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), ...
+         'DisplayName', sprintf('APA %i', apa_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [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(apa_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);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), ...
+         'DisplayName', sprintf('APA %i', apa_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([2e-5, 4e-4]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_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([-10, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([80, 120]);
+
+%% Compute the Coherence
+coh_iff = zeros(length(f), length(apa_nums));
+for i = 1:length(apa_nums)
+    [coh_lf, ~] = mscohere(apa_sweep{i}.Va,    apa_sweep{i}.Vs,    win, [], [], 1/Ts);
+    [coh_hf, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
+    coh_iff(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
+end
+
+%% Plot the coherence
+figure;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, coh_iff(:, i));
+end;
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlim([5, 5e3]); ylim([0, 1]);
+
+%% FRF estimation of the transfer function from Va to Vs
+iff_frf = zeros(length(f), length(apa_nums));
+for i = 1:length(apa_nums)
+    [frf_lf, ~] = tfestimate(apa_sweep{i}.Va,    apa_sweep{i}.Vs,    win, [], [], 1/Ts);
+    [frf_hf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
+    iff_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Plot the FRF from Va to Vs
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), ...
+         'DisplayName', sprintf('APA %i', apa_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [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(apa_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]);
+
+%% Remove the APA 7 (6 in the list) from measurements
+apa_nums(6) = [];
+enc_frf(:,6) = [];
+iff_frf(:,6) = [];
+
+%% Save the measured FRF
+save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');

matlab/apa_simscape_model_comp.m

@@ -0,0 +1,383 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Add useful folders to the path
+addpath('test_bench_apa300ml/');
+addpath('png/');
+addpath('mat/');
+addpath('src/');
+
+%% Frequency vector used for many plots
+freqs = 2*logspace(0, 3, 1000);
+
+%% Open Simscape Model
+options = linearizeOptions;
+options.SampleTime = 0;
+
+% Name of the Simulink File
+mdl = 'test_bench_apa300ml';
+
+open(mdl)
+
+%% Initialize the structure with default values
+n_hexapod = struct();
+n_hexapod.actuator = initializeAPA(...
+    'type', '2dof', ...
+    'Ga', 1, ... % Actuator constant [N/V]
+    'Gs', 1);    % Sensor constant [V/m]
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'],  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
+
+%% Run the linearization
+Ga = linearize(mdl, io, 0.0, options);
+Ga.InputName  = {'Va'};
+Ga.OutputName = {'Vs', 'de', 'da'};
+
+%% Bode plot of the transfer function from u to taum
+freqs = logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'k-')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'k-')
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360);
+ylim([-180, 0])
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+%% Bode plot of the transfer function from Va to de and da
+freqs = logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Ga('de', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
+plot(freqs, abs(squeeze(freqresp(Ga('da', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'southwest');
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('de', 'Va'), freqs, 'Hz'))))
+plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('da', 'Va'), freqs, 'Hz'))))
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360);
+ylim([-180, 0])
+
+linkaxes([ax1,ax2],'x');
+
+%% Load Data
+load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
+
+%% Initialize a 2DoF APA with Ga=Gs=1
+n_hexapod.actuator = initializeAPA(...
+    'type', '2dof', ...
+    'ga', 1, ...
+    'gs', 1);
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Actuator 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; % Attocube
+
+%% Identification
+Gs = linearize(mdl, io, 0.0, options);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+%% Estimated Actuator Constant
+ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('de', 'Va')); % [N/V]
+
+%% Estimated Sensor Constant
+gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
+
+%% Set the identified constants
+n_hexapod.actuator = initializeAPA(...
+    'type', '2dof', ...
+    'ga', ga, ... % Actuator gain [N/V]
+    'gs', gs);    % Sensor gain [V/m]
+
+%% Identify again the dynamics with correct Ga,Gs
+Gs = linearize(mdl, io, 0.0, options);
+Gs = Gs*exp(-Ts*s);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+%% Bode plot of the transfer function from u to de
+freqs = logspace(1,4,1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(enc_frf(:,1)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'Va'), 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]);
+
+%% Bode plot of the transfer function from Va to Vs (both Simscape and measured FRF)
+freqs = logspace(1,4,1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(iff_frf(:,1)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), 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]);
+
+%% Initialize the APA as a flexible body
+n_hexapod.actuator = initializeAPA(...
+    'type', 'flexible', ...
+    'ga', 1, ...
+    'gs', 1);
+
+%% Identify the dynamics
+Gs = linearize(mdl, io, 0.0, options);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+%% Actuator Constant
+ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('de', 'Va')); % [N/V]
+
+%% Sensor Constant
+gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
+
+%% Set the identified constants
+n_hexapod.actuator = initializeAPA(...
+    'type', 'flexible', ...
+    'ga', ga, ... % Actuator gain [N/V]
+    'gs', gs);    % Sensor gain [V/m]
+
+%% Identify with updated constants
+Gs = linearize(mdl, io, 0.0, options);
+Gs = Gs*exp(-Ts*s);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'Vs', 'de', 'da'};
+
+%% Bode plot of the transfer function from V_a to d_e (both Simscape and measured FRF)
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-9, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(enc_frf(:,1)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'Va'), 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]);
+
+%% Bode plot of the transfer function from Va to Vs (both Simscape and measured FRF)
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(iff_frf(:,1)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), 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]);
+
+%% Optimized parameters
+n_hexapod.actuator = initializeAPA('type', '2dof', ...
+                                   'Ga', -32.2, ...
+                                   'Gs', 0.088, ...
+                                   'k',  ones(6,1)*0.38e6, ...
+                                   'ke', ones(6,1)*1.75e6, ...
+                                   'ka', ones(6,1)*3e7, ...
+                                   'c',  ones(6,1)*1.3e2, ...
+                                   'ce', ones(6,1)*1e1, ...
+                                   'ca', ones(6,1)*1e1 ...
+                                   );
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator 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
+
+%% Identification with optimized parameters
+Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'Vs', 'de'};
+
+%% Comparison of the experimental data and Simscape Model
+freqs = 5*logspace(0, 3, 1000);
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+
+ax1b = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_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', 'Va'), 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(apa_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', 'Va'), 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]);

matlab/basic_meas_geometrical.m

@@ -0,0 +1,45 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+
+addpath('./mat/');
+
+%% Measured height for all the APA at the 8 locations
+apa1  = 1e-6*[0, -0.5 , 3.5  , 3.5  , 42   , 45.5, 52.5 , 46];
+apa2  = 1e-6*[0, -2.5 , -3   , 0    , -1.5 , 1   , -2   , -4];
+apa3  = 1e-6*[0, -1.5 , 15   , 17.5 , 6.5  , 6.5 , 21   , 23];
+apa4  = 1e-6*[0, 6.5  , 14.5 , 9    , 16   , 22  , 29.5 , 21];
+apa5  = 1e-6*[0, -12.5, 16.5 , 28.5 , -43  , -52 , -22.5, -13.5];
+apa6  = 1e-6*[0, -8   , -2   , 5    , -57.5, -62 , -55.5, -52.5];
+apa7  = 1e-6*[0, 19.5 , -8   , -29.5, 75   , 97.5, 70   , 48];
+apa7b = 1e-6*[0, 9    , -18.5, -30  , 31   , 46.5, 16.5 , 7.5];
+
+apa = {apa1, apa2, apa3, apa4, apa5, apa6, apa7b};
+
+%% X-Y positions of the measurements points
+W = 20e-3;   % Width [m]
+L = 61e-3;   % Length [m]
+d = 1e-3;    % Distance from border [m]
+l = 15.5e-3; % [m]
+
+pos = [[-L/2 + d;      W/2 - d],
+       [-L/2 + l - d;  W/2 - d],
+       [-L/2 + l - d; -W/2 + d],
+       [-L/2 + d;     -W/2 + d],
+       [L/2 - l + d;   W/2 - d],
+       [L/2 - d;       W/2 - d],
+       [L/2 - d;      -W/2 + d],
+       [L/2 - l + d;  -W/2 + d]];
+
+%% Using fminsearch to find the best fitting plane
+apa_d = zeros(1, 7);
+for i = 1:7
+    fun = @(x)max(abs(([pos; apa{i}]-[0;0;x(1)])'*([x(2:3);1]/norm([x(2:3);1]))));
+    x0 = [0;0;0];
+    [x, min_d] = fminsearch(fun,x0);
+    apa_d(i) = min_d;
+end

matlab/basic_meas_spurious_res.m

@@ -0,0 +1,97 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+
+addpath('mat/');
+
+%% Load Data
+bending_X = load('apa300ml_bending_X_top.mat');
+
+%% Spectral Analysis setup
+Ts = bending_X.Track1_X_Resolution; % Sampling Time [s]
+win = hann(ceil(1/Ts));
+
+%% Compute the transfer function from applied force to measured rotation
+[G_bending_X, f]  = tfestimate(bending_X.Track1, bending_X.Track2, win, [], [], 1/Ts);
+
+%% Plot the transfer function
+figure;
+hold on;
+plot(f, abs(G_bending_X), 'k-');
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 2e3]); ylim([1e-5, 2e-1]);
+text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(840, 2.0e-3,{'840Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(1400, 7.0e-3,{'1400Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+
+%% Load Data
+bending_Y = load('apa300ml_bending_Y_top.mat');
+
+%% Compute the transfer function
+[G_bending_Y, ~]  = tfestimate(bending_Y.Track1, bending_Y.Track2, win, [], [], 1/Ts);
+
+%% Plot the transfer function
+figure;
+hold on;
+plot(f, abs(G_bending_Y), 'k-');
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 2e3]); ylim([1e-5, 3e-2])
+text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(1218, 1.5e-2,{'1220Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+
+%% Load Data
+torsion = load('apa300ml_torsion_left.mat');
+
+%% Compute transfer function
+[G_torsion, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);
+
+%% Plot the transfer function
+figure;
+hold on;
+plot(f, abs(G_torsion), 'k-');
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 2e3]); ylim([1e-5, 2e-2])
+text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+text(800, 6e-4,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+
+%% Load data
+torsion = load('apa300ml_torsion_top.mat');
+
+%% Compute transfer function
+[G_torsion_top, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);
+
+%% Plot the two transfer functions
+figure;
+hold on;
+plot(f, abs(G_torsion), 'k-', 'DisplayName', 'Left excitation');
+plot(f, abs(G_torsion_top), '-', 'DisplayName', 'Top excitation');
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 2e3]); ylim([1e-5, 2e-2])
+text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+text(800, 2e-3,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
+legend('location', 'northwest');
+
+figure;
+hold on;
+plot(f, abs(G_torsion),    'DisplayName', 'Torsion');
+plot(f, abs(G_bending_X),  'DisplayName', 'Bending - X');
+plot(f, abs(G_bending_Y),  'DisplayName', 'Bending - Y');
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([50, 2e3]); ylim([1e-5, 1e-1]);
+legend('location', 'southeast');

matlab/basic_meas_stroke.m

@@ -0,0 +1,112 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+
+addpath('./mat/');
+
+%% Load the measurements
+apa300ml_1s = {};
+for i = 1:7
+    apa300ml_1s(i) = {load(['mat/stroke_apa_1stacks_' num2str(i) '.mat'], 't', 'V', 'd')};
+end
+
+%% Only take the data between t=2 and t=10 and reset the measured displacement at t=2
+for i = 1:7
+    t = apa300ml_1s{i}.t;
+    apa300ml_1s{i}.d = apa300ml_1s{i}.d - mean(apa300ml_1s{i}.d(t > 1.9 & t < 2.0));
+    apa300ml_1s{i}.d = apa300ml_1s{i}.d(t > 2.0 & t < 10.0);
+    apa300ml_1s{i}.V = apa300ml_1s{i}.V(t > 2.0 & t < 10.0);
+    apa300ml_1s{i}.t = apa300ml_1s{i}.t(t > 2.0 & t < 10.0);
+end
+
+%% Applied voltage as a function of time
+figure;
+plot(apa300ml_1s{1}.t, 20*apa300ml_1s{1}.V)
+xlabel('Time [s]'); ylabel('Voltage [V]');
+ylim([-20,160]); yticks([-20     0    20    40    60    80   100   120   140   160]);
+
+%% Measured motion for all the APA300ML
+figure;
+hold on;
+for i = 1:7
+    plot(apa300ml_1s{i}.t, 1e6*apa300ml_1s{i}.d, 'DisplayName', sprintf('APA %i', i))
+end
+hold off;
+xlabel('Time [s]'); ylabel('Displacement [$\mu m$]')
+legend('location', 'southeast', 'FontSize', 8)
+
+%% Displacement as a function of the applied voltage
+figure;
+hold on;
+for i = 1:7
+    plot(20*apa300ml_1s{i}.V, 1e6*apa300ml_1s{i}.d, 'DisplayName', sprintf('APA %i', i))
+end
+hold off;
+xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
+legend('location', 'southwest', 'FontSize', 8)
+xlim([-20, 160]); ylim([-140, 0]);
+
+%% Load the measurements
+apa300ml_2s = {};
+for i = 1:7
+    apa300ml_2s(i) = {load(['mat/stroke_apa_2stacks_' num2str(i) '.mat'], 't', 'V', 'd')};
+end
+
+%% Only take the data between t=2 and t=10 and reset the measured displacement at t=2
+for i = 1:7
+    t = apa300ml_2s{i}.t;
+    apa300ml_2s{i}.d = apa300ml_2s{i}.d - mean(apa300ml_2s{i}.d(t > 1.9 & t < 2.0));
+    apa300ml_2s{i}.d = apa300ml_2s{i}.d(t > 2.0 & t < 10.0);
+    apa300ml_2s{i}.V = apa300ml_2s{i}.V(t > 2.0 & t < 10.0);
+    apa300ml_2s{i}.t = apa300ml_2s{i}.t(t > 2.0 & t < 10.0);
+end
+
+%% Measured motion for all the APA300ML
+figure;
+hold on;
+for i = 1:7
+    plot(apa300ml_2s{i}.t, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
+end
+hold off;
+xlabel('Time [s]'); ylabel('Displacement [$\mu m$]')
+legend('location', 'southeast', 'FontSize', 8)
+ylim([-250, 0]);
+
+%% Displacement as a function of the applied voltage
+figure;
+hold on;
+for i = 1:7
+    plot(20*apa300ml_2s{i}.V, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
+end
+hold off;
+xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
+legend('location', 'southwest', 'FontSize', 8)
+xlim([-20, 160]); ylim([-250, 0]);
+
+%% Motion induced by applying a voltage to the three stack is the sum to the previous two measured displacements
+apa300ml_3s = {};
+for i = 1:7
+    apa300ml_3s(i) = apa300ml_1s(i);
+    apa300ml_3s{i}.d = apa300ml_1s{i}.d + apa300ml_2s{i}.d;
+end
+
+%% Displacement as a function of the applied voltage
+figure;
+hold on;
+for i = 1:7
+    plot(20*apa300ml_3s{i}.V, 1e6*apa300ml_3s{i}.d, 'DisplayName', sprintf('APA %i', i))
+end
+hold off;
+xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
+legend('location', 'southwest', 'FontSize', 8)
+xlim([-20, 160]); ylim([-400, 0]);
+
+%% Estimate the maximum stroke
+apa300ml_stroke = zeros(1, 7);
+for i = 1:7
+    apa300ml_stroke(i) = max(apa300ml_3s{i}.d) - min(apa300ml_3s{i}.d);
+end

matlab/strut_meas_analysis_1.m

@@ -0,0 +1,369 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+
+addpath('./mat/');
+addpath('./src/');
+
+%% Load Data
+leg_sweep    = load(sprintf('frf_data_leg_%i_sweep.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
+leg_noise_hf = load(sprintf('frf_data_leg_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
+
+%% Time vector
+t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s]
+
+%% Sampling frequency/time
+Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
+Fs = 1/Ts; % Sampling Frequency [Hz]
+
+win = hanning(ceil(0.5*Fs)); % Hannning Windows
+
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts);
+i_lf = f <= 350; % Indices used for the low frequency
+i_hf = f >  350; % Indices used for the low frequency
+
+%% Compute the coherence for both excitation signals
+[int_coh_sweep, ~]    = mscohere(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
+[int_coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
+
+%% Combine the coherence
+int_coh = [int_coh_sweep(i_lf); int_coh_noise_hf(i_hf)];
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, int_coh(:, 1), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Compute FRF function from Va to da
+[frf_sweep, ~]    = tfestimate(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 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', 'None', '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 $d_e/V_a$ [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]);
+
+%% Compute the coherence for both excitation signals
+[iff_coh_sweep, ~]    = mscohere(leg_sweep.Va,    leg_sweep.Vs,    win, [], [], 1/Ts);
+[iff_coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
+
+%% Combine the coherence
+iff_coh = [iff_coh_sweep(i_lf); iff_coh_noise_hf(i_hf)];
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, iff_coh, 'k-');
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Compute the FRF
+[frf_sweep,    ~] = tfestimate(leg_sweep.Va,    leg_sweep.Vs,    win, [], [], 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 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', 'None', '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 $V_s/V_a$ [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]);
+
+%% Load data
+leg_enc_sweep    = load(sprintf('frf_data_leg_coder_badly_align_%i_noise.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
+leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_badly_align_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
+
+%% Compute the coherence for both excitation signals
+[int_coh_sweep, ~]    = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.da,    win, [], [], 1/Ts);
+[int_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);
+
+%% Combine the coherinte
+int_coh = [int_coh_sweep(i_lf); int_coh_noise_hf(i_hf)];
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, int_coh(:, 1), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Compute FRF function from Va to da
+[frf_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.da,    win, [], [], 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);
+
+%% Combine the FRF
+int_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+%% Plot the FRF from Va to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(int_with_enc_frf), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-7, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(int_with_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]);
+
+%% Plot the FRF from Va to da with and without the encoder
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', '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 $d_a/V_a$ [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]);
+
+%% Compute the coherence for both excitation signals
+[enc_coh_sweep, ~]    = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.de,    win, [], [], 1/Ts);
+[enc_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
+
+%% Combine the coherence
+enc_coh = [enc_coh_sweep(i_lf); enc_coh_noise_hf(i_hf)];
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, enc_coh(:, 1), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Compute FRF function from Va to da
+[frf_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.de,    win, [], [], 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
+
+%% Combine the FRF
+enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+%% Plot the FRF from Va to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', '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 $d_e/V_a$ [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]);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', '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 $d/V_a$ [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]);
+
+%% 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 $d_e/V_a$ [m/V]'); xlabel('Frequency [Hz]');
+hold off;
+ylim([1e-7, 1e-3]); xlim([10, 2e3]);
+
+%% Compute the coherence for both excitation signals
+[iff_coh_sweep, ~]    = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
+[iff_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
+
+%% Combine the coherence
+iff_coh = [iff_coh_sweep(i_lf); iff_coh_noise_hf(i_hf)];
+
+%% Plot the coherence
+figure;
+hold on;
+plot(f, iff_coh, 'k-');
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Compute FRF function from Va to da
+[frf_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
+[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
+
+%% Combine the FRF
+iff_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
+
+%% Plot FRF of the transfer function from Va to Vs
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_with_enc_frf), 'k-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(iff_with_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]);
+
+%% Compare the IFF plant with and without the encoders
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', '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 $V_s/V_a$ [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]);

matlab/strut_meas_analysis_all.m

@@ -0,0 +1,217 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+colors = colororder;
+
+addpath('./mat/');
+addpath('./src/');
+
+%% Numnbers of the measured legs
+leg_nums = [1 2 3 4 5];
+
+%% First identification (low frequency noise)
+leg_noise = {};
+for i = 1:length(leg_nums)
+    leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
+end
+
+%% Second identification (high frequency noise)
+leg_noise_hf = {};
+for i = 1:length(leg_nums)
+    leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
+end
+
+%% Time vector
+t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s]
+
+%% Sampling
+Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
+Fs = 1/Ts; % Sampling Frequency [Hz]
+
+win = hanning(ceil(0.5*Fs)); % Hannning Windows
+
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, [], [], 1/Ts);
+i_lf = f <= 350;
+i_hf = f >  350;
+
+%% Coherence computation
+coh_enc = zeros(length(f), length(leg_nums));
+for i = 1:length(leg_nums)
+    [coh_lf, ~] = mscohere(leg_noise{i}.Va,    leg_noise{i}.de,    win, [], [], 1/Ts);
+    [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
+    coh_enc(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
+end
+
+%% Plot the coherence
+figure;
+hold on;
+for i = 1:length(leg_nums)
+    plot(f, coh_enc(:, i));
+end;
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Transfer function estimation
+enc_frf = zeros(length(f), length(leg_nums));
+
+for i = 1:length(leg_nums)
+    [frf_lf, ~] = tfestimate(leg_noise{i}.Va,    leg_noise{i}.de,    win, [], [], 1/Ts);
+    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
+    enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Bode plot of the FRF from Va to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(leg_nums)
+    plot(f, abs(enc_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', leg_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [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(leg_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]);
+
+%% Coherence computation
+coh_int = zeros(length(f), length(leg_nums));
+for i = 1:length(leg_nums)
+    [coh_lf, ~] = mscohere(leg_noise{i}.Va,    leg_noise{i}.da,    win, [], [], 1/Ts);
+    [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
+    coh_int(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
+end
+
+%% Plot coherence
+figure;
+hold on;
+for i = 1:length(leg_nums)
+    plot(f, coh_int(:, i));
+end;
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% Transfer function estimation
+int_frf = zeros(length(f), length(leg_nums));
+for i = 1:length(leg_nums)
+    [frf_lf, ~] = tfestimate(leg_noise{i}.Va,    leg_noise{i}.da,    win, [], [], 1/Ts);
+    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
+    int_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Plot the FRF from Va to de (interferometer)
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(leg_nums)
+    plot(f, abs(int_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', leg_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/V_a$ [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(leg_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]);
+
+%% Coherence
+coh_iff = zeros(length(f), length(leg_nums));
+for i = 1:length(leg_nums)
+    [coh_lf, ~] = mscohere(leg_noise{i}.Va,    leg_noise{i}.Vs,    win, [], [], 1/Ts);
+    [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
+    coh_iff(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
+end
+
+%% Plot the coherence
+figure;
+hold on;
+for i = 1:length(leg_nums)
+    plot(f, coh_iff(:, i));
+end;
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlim([10, 2e3]); ylim([0, 1]);
+
+%% FRF estimation of the transfer function from Va to Vs
+iff_frf = zeros(length(f), length(leg_nums));
+for i = 1:length(leg_nums)
+    [frf_lf, ~] = tfestimate(leg_noise{i}.Va,    leg_noise{i}.Vs,    win, [], [], 1/Ts);
+    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
+    iff_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Plot the FRF from Va to Vs
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(leg_nums)
+    plot(f, abs(iff_frf(:, i)), ...
+         'DisplayName', sprintf('Leg %i', leg_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [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(leg_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]);
+
+%% Save the estimated FRF for further analysis
+save('mat/meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');

matlab/strut_simscape_model_comp.m

@@ -0,0 +1,597 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Add useful folders to the path
+addpath('test_bench_struts/');
+addpath('png/');
+addpath('mat/');
+addpath('src/');
+
+%% Frequency vector used for many plots
+freqs = 2*logspace(0, 3, 1000);
+
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'test_bench_struts';
+
+%% Open the Simulink File
+open(mdl)
+
+%% 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');
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Actuator 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
+
+%% Run the linearization
+Gs = linearize(mdl, io, 0.0, options);
+Gs.InputName  = {'Va'};
+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', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
+plot(freqs, abs(squeeze(freqresp(Gs('da', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'southwest');
+
+ax1b = nexttile([2,1]);
+plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), 'k-')
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))))
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'Va'), 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', 'Va'), 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]);
+
+%% Load measured FRF
+load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');
+
+%% Add time delay to the Simscape model
+Gs = exp(-s*Ts)*Gs;
+
+%% Compare the FRF and identified dynamics from Va 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(leg_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', 'Va'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_a/V_a$ [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(leg_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', 'Va'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(leg_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', 'Va'), 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(leg_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', 'Va'), 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]);
+
+%% Compare the FRF and identified dynamics from Va to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', '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(leg_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', 'Va'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(leg_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', 'Va'), 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]);
+
+%% Load measured FRF of the struts
+load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');
+
+%% Initialize Simscape data
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
+n_hexapod.actuator = initializeAPA('type', 'flexible');
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator Voltage
+io(io_i) = linio([mdl, '/de'],  1, 'openoutput'); io_i = io_i + 1; % Encoder
+
+%% Identification
+Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'de'};
+
+%% Measured FRF from Vs to de and identified dynamics using the flexible APA
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', '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(leg_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', 'Va'), freqs, 'Hz'))), '-', ...
+     'DisplayName', 'Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(leg_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', 'Va'), 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]);
+
+%% 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', [0; dy_aligns(i); 0]);
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va'};
+    G.OutputName = {'de'};
+
+    Gs_align(i) = {G};
+end
+
+%% Transfer function from Vs to de - effect of x-misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(dy_aligns)
+    plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'Va'), 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 $d_e/V_a$ [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', 'Va'), 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]);
+
+%% 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', [dx_aligns(i); 0; 0]);
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va'};
+    G.OutputName = {'de'};
+
+    Gs_align(i) = {G};
+end
+
+%% Transfer function from Vs to de - effect of x-misalignment
+freqs = 2*logspace(0, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(dx_aligns)
+    plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'Va'), 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 $d_e/V_a$ [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', 'Va'), 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]);
+
+%% 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;
+
+%% Idenfity the transfer function from actuator to encoder for all cases
+Gs_align = {zeros(size(d_aligns,2), 1)};
+
+for i = 1:size(d_aligns,2)
+    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', d_aligns(:,i));
+
+    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va'};
+    G.OutputName = {'de'};
+
+    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', 'Va'), 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', 'Va'), 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', 'Va'), 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', 'Va'), 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', 'Va'), 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]);
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator Voltage
+io(io_i) = linio([mdl, '/de'],  1, 'openoutput'); io_i = io_i + 1; % Encoder
+
+%% APA Initialization
+n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [0.1e-3; 0.5e-3; 0]);
+
+%% Tested bending stiffnesses [Nm/rad]
+kRs = [3, 4, 5, 6, 7];
+
+%% 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*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va'};
+    G.OutputName = {'de'};
+
+    Gs(i) = {G};
+end
+
+%% Plot the obtained transfer functions for all the bending stiffnesses
+freqs = 2*logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'Va'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$k_R = %.0f$ [Nm/rad]', kRs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [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', 'Va'), 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]);
+
+%% Tested axial stiffnesses [N/m]
+kzs = [5e7 7.5e7 1e8 2.5e8];
+
+%% 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*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va'};
+    G.OutputName = {'de'};
+
+    Gs(i) = {G};
+end
+
+%% Plot the obtained transfer functions for all the axial stiffnesses
+freqs = 2*logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kzs)
+    plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'Va'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$k_z = %.1e$ [N/m]', kzs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/V_a$ [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', 'Va'), 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]);
+
+%% Tested bending dampings [Nm/(rad/s)]
+cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
+
+%% Idenfity the transfer function from actuator to encoder for all bending dampins
+Gs = {zeros(length(kRs), 1)};
+
+for i = 1:length(kRs)
+    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*Ts)*linearize(mdl, io, 0.0, options);
+    G.InputName  = {'Va'};
+    G.OutputName = {'de'};
+
+    Gs(i) = {G};
+end
+
+%% Plot the obtained transfer functions for all the bending stiffnesses
+freqs = 2*logspace(1, 3, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'Va'), 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 $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'southwest');
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'Va'), 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]);

ref.bib

@@ -8,6 +8,8 @@
   number          = 2,
   pages           = {157--165},
   year            = 2018,
+  doi             = {10.1007/s12567-017-0180-6},
+  url             = {https://doi.org/10.1007/s12567-017-0180-6},
   publisher       = {Springer},
 }