Update figures and tangle matlab code

.gitignore

@@ -0,0 +1,38 @@
+# Windows default autosave extension
+*.asv
+*rtw/
+test_at**.m
+data/
+
+# OSX / *nix default autosave extension
+*.m~
+
+# Compiled MEX binaries (all platforms)
+*.mex*
+
+# Packaged app and toolbox files
+*.mlappinstall
+*.slxc
+*.mldatx
+*.mltbx
+*.xml
+piezoapa_slrt_rtw/
+
+# Generated helpsearch folders
+helpsearch*/
+
+# Simulink code generation folders
+slprj/
+sccprj/
+
+# Matlab code generation folders
+codegen/
+
+# Simulink autosave extension
+*.autosave
+
+# Simulink cache files
+*.slxc
+
+# Octave session info
+octave-workspace

figs/iff_plant_identification_apa95ml.pdf

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matlab/force_sensor_identification.m

@@ -0,0 +1,95 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Load Data                                                         :ignore:
+% The data are loaded:
+
+load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
+
+
+
+% Any offset is removed.
+
+u = detrend(u, 0); % Speedgoat DAC output Voltage [V]
+v = detrend(v, 0); % Speedgoat ADC input Voltage (sensor stack) [V]
+
+
+
+% Here, the amplifier gain is 20.
+
+u = 20*u; % Actuator Stack Voltage [V]
+
+% Compute TF estimate and Coherence                                 :ignore:
+% The transfer function from the actuator voltage $V_a$ to the force sensor stack voltage $V_s$ is computed.
+
+
+Ts = t(end)/(length(t)-1);
+Fs = 1/Ts;
+
+win = hann(ceil(5/Ts));
+
+[tf_est,  f] = tfestimate(u, v, win, [], [], 1/Ts);
+[coh,     ~] = mscohere(  u, v, win, [], [], 1/Ts);
+
+
+
+% The coherence is shown in Figure [[fig:apa95ml_5kg_cedrat_coh]].
+
+
+figure;
+
+hold on;
+plot(f, coh, 'k-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Coherence'); xlabel('Frequency [Hz]');
+hold off;
+xlim([10, 5e3]);
+
+
+
+% #+name: fig:apa95ml_5kg_cedrat_coh
+% #+caption: Coherence
+% #+RESULTS:
+% [[file:figs/apa95ml_5kg_cedrat_coh.png]]
+
+% The Simscape model is loaded and compared with the identified dynamics in Figure [[fig:bode_plot_force_sensor_voltage_comp_fem]].
+% The non-minimum phase zero is just a side effect of the not so great identification.
+% Taking longer measurements would results in a minimum phase zero.
+
+
+load('mat/fem_simscape_models.mat', 'Gfm');
+
+freqs = logspace(1, 4, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(tf_est))
+plot(freqs, abs(squeeze(freqresp(Gfm, freqs, 'Hz'))))
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-3, 1e1]);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(tf_est), ...
+     'DisplayName', 'Identification')
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gfm, freqs, 'Hz'))), ...
+     'DisplayName', 'FEM')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+hold off;
+ylim([-180, 180]);
+yticks(-180:90:180);
+legend('location', 'northeast')
+
+linkaxes([ax1,ax2], 'x');
+xlim([10, 5e3]);

matlab/huddle_test.m

@@ -0,0 +1,34 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Load Data                                                       :noexport:
+
+load('huddle_test.mat', 't', 'y');
+
+y = y - mean(y(1000:end));
+
+% Time Domain Data
+
+figure;
+plot(t(1000:end), y(1000:end))
+ylabel('Output Displacement [m]'); xlabel('Time [s]');
+
+% PSD of Measurement Noise
+
+Ts = t(end)/(length(t)-1);
+Fs = 1/Ts;
+
+win = hanning(ceil(1*Fs));
+
+[pxx, f] = pwelch(y(1000:end), win, [], [], Fs);
+
+figure;
+plot(f, sqrt(pxx));
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
+xlim([1, Fs/2]);

matlab/integral_force_feedback.m

@@ -0,0 +1,307 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% IFF Plant
+% From the identified plant, a model of the transfer function from the actuator stack voltage to the force sensor generated voltage is developed.
+
+
+w_z = 2*pi*111; % Zeros frequency [rad/s]
+w_p = 2*pi*255; % Pole frequency [rad/s]
+xi_z = 0.05;
+xi_p = 0.015;
+G_inf = 0.1;
+
+Gi = G_inf*(s^2 + 2*xi_z*w_z*s + w_z^2)/(s^2 + 2*xi_p*w_p*s + w_p^2);
+
+
+
+% Its bode plot is shown in Figure [[fig:iff_plant_identification_apa95ml]].
+
+
+freqs = logspace(1, 4, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Gi, freqs, 'Hz'))), 'k-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel', []);
+hold off;
+ylim([1e-3, 1e1]);
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gi, freqs, 'Hz'))), 'k-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+hold off;
+ylim([-180, 180]);
+yticks(-180:90:180);
+
+linkaxes([ax1,ax2], 'x');
+xlim([10, 5e3]);
+
+
+
+% #+name: fig:iff_plant_identification_apa95ml
+% #+caption: Bode plot of the IFF plant
+% #+RESULTS:
+% [[file:figs/iff_plant_identification_apa95ml.png]]
+
+% The controller used in the Integral Force Feedback Architecture is:
+% \begin{equation}
+%   K_{\text{IFF}}(s) = \frac{g}{s + 2\cdot 2\pi} \cdot \frac{s}{s + 0.5 \cdot 2\pi}
+% \end{equation}
+% where $g$ is a gain that can be tuned.
+
+% Above 2 Hz the controller is basically an integrator, whereas an high pass filter is added at 0.5Hz to further reduce the low frequency gain.
+
+% In the frequency band of interest, this controller should mostly act as a pure integrator.
+
+% The Root Locus corresponding to this controller is shown in Figure [[fig:root_locus_iff_apa95ml_identification]].
+
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+plot(real(pole(Gi)),  imag(pole(Gi)),  'kx');
+plot(real(tzero(Gi)),  imag(tzero(Gi)),  'ko');
+for i = 1:length(gains)
+  cl_poles = pole(feedback(Gi, (gains(i)/(s + 2*2*pi)*s/(s + 0.5*2*pi))));
+  plot(real(cl_poles), imag(cl_poles), 'k.');
+end
+ylim([0, 1800]);
+xlim([-1600,200]);
+xlabel('Real Part')
+ylabel('Imaginary Part')
+axis square
+
+% First tests with few gains
+% The controller is now implemented in practice, and few controller gains are tested: $g = 0$, $g = 10$ and $g = 100$.
+
+% For each controller gain, the identification shown in Figure [[fig:test_bench_apa_schematic_iff]] is performed.
+
+iff_g10  = load('apa95ml_iff_g10_res.mat',  'u', 't', 'y', 'v');
+iff_g100 = load('apa95ml_iff_g100_res.mat', 'u', 't', 'y', 'v');
+iff_of   = load('apa95ml_iff_off_res.mat',  'u', 't', 'y', 'v');
+
+Ts = 1e-4;
+win = hann(ceil(10/Ts));
+
+[tf_iff_g10, f] = tfestimate(iff_g10.u, iff_g10.y, win, [], [], 1/Ts);
+[co_iff_g10, ~] = mscohere(  iff_g10.u, iff_g10.y, win, [], [], 1/Ts);
+
+[tf_iff_g100, ~] = tfestimate(iff_g100.u, iff_g100.y, win, [], [], 1/Ts);
+[co_iff_g100, ~] = mscohere(  iff_g100.u, iff_g100.y, win, [], [], 1/Ts);
+
+[tf_iff_of, ~] = tfestimate(iff_of.u, iff_of.y, win, [], [], 1/Ts);
+[co_iff_of, ~] = mscohere(  iff_of.u, iff_of.y, win, [], [], 1/Ts);
+
+
+
+% The coherence between the excitation signal and the mass displacement as measured by the interferometer is shown in Figure [[fig:iff_first_test_coherence]].
+
+
+figure;
+
+hold on;
+plot(f, co_iff_of, '-', 'DisplayName', 'g=0')
+plot(f, co_iff_g10, '-', 'DisplayName', 'g=10')
+plot(f, co_iff_g100, '-', 'DisplayName', 'g=100')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Coherence'); xlabel('Frequency [Hz]');
+hold off;
+legend();
+xlim([60, 600])
+
+
+
+% #+name: fig:iff_first_test_coherence
+% #+caption: Coherence
+% #+RESULTS:
+% [[file:figs/iff_first_test_coherence.png]]
+
+% The obtained transfer functions are shown in Figure [[fig:iff_first_test_bode_plot]].
+% It is clear that the IFF architecture can actively damp the main resonance of the system.
+
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2, 1]);
+hold on;
+plot(f, abs(tf_iff_of), '-', 'DisplayName', 'g=0')
+plot(f, abs(tf_iff_g10), '-', 'DisplayName', 'g=10')
+plot(f, abs(tf_iff_g100), '-', 'DisplayName', 'g=100')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+legend();
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(-tf_iff_of), '-')
+plot(f, 180/pi*angle(-tf_iff_g10), '-')
+plot(f, 180/pi*angle(-tf_iff_g100), '-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+hold off;
+yticks(-360:90:360);
+
+linkaxes([ax1,ax2], 'x');
+xlim([60, 600]);
+
+% Second test with many Gains
+% Then, the same identification test is performed for many more gains.
+
+load('apa95ml_iff_test.mat', 'results');
+
+Ts = 1e-4;
+win = hann(ceil(10/Ts));
+
+tf_iff = {zeros(1, length(results))};
+co_iff = {zeros(1, length(results))};
+g_iff = [0, 1, 5, 10, 50, 100];
+
+for i=1:length(results)
+    [tf_est, f] = tfestimate(results{i}.u, results{i}.y, win, [], [], 1/Ts);
+    [co_est, ~] = mscohere(results{i}.u, results{i}.y, win, [], [], 1/Ts);
+
+    tf_iff(i) = {tf_est};
+    co_iff(i) = {co_est};
+end
+
+
+
+% The obtained dynamics are shown in Figure [[fig:iff_results_bode_plots]].
+
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2, 1]);
+hold on;
+for i = 1:length(results)
+  plot(f, abs(tf_iff{i}), '-', 'DisplayName', sprintf('g = %0.f', g_iff(i)))
+end
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+legend();
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(results)
+  plot(f, 180/pi*angle(-tf_iff{i}), '-')
+end
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+yticks(-360:90:360);
+axis padded 'auto x'
+
+linkaxes([ax1,ax2], 'x');
+xlim([60, 600]);
+
+
+
+% #+name: fig:iff_results_bode_plots
+% #+caption: Identified dynamics from excitation voltage to the mass displacement
+% #+RESULTS:
+% [[file:figs/iff_results_bode_plots.png]]
+
+% For each gain, the parameters of a second order resonant system that best fits the data are estimated and are compared with the data in Figure [[fig:iff_results_bode_plots_identification]].
+
+
+G_id = {zeros(1,length(results))};
+
+f_start = 70; % [Hz]
+f_end = 500; % [Hz]
+
+for i = 1:length(results)
+    tf_id = tf_iff{i}(sum(f<f_start):length(f)-sum(f>f_end));
+    f_id = f(sum(f<f_start):length(f)-sum(f>f_end));
+
+    gfr = idfrd(tf_id, 2*pi*f_id, Ts);
+    G_id(i) = {procest(gfr,'P2UDZ')};
+end
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2, 1]);
+hold on;
+for i = 1:length(results)
+  set(gca,'ColorOrderIndex',i)
+  plot(f, abs(tf_iff{i}), '-', 'DisplayName', sprintf('g = %0.f', g_iff(i)))
+  set(gca,'ColorOrderIndex',i)
+  plot(f, abs(squeeze(freqresp(G_id{i}, f, 'Hz'))), '--', 'HandleVisibility', 'off')
+end
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+legend();
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(results)
+  set(gca,'ColorOrderIndex',i)
+  plot(f, 180/pi*angle(tf_iff{i}), '-')
+  set(gca,'ColorOrderIndex',i)
+  plot(f, 180/pi*angle(squeeze(freqresp(G_id{i}, f, 'Hz'))), '--')
+end
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+hold off;
+yticks(-360:90:360);
+axis padded 'auto x'
+
+linkaxes([ax1,ax2], 'x');
+xlim([60, 600]);
+
+
+
+% #+name: fig:iff_results_bode_plots_identification
+% #+caption: Comparison of the measured dynamic and the identified 2nd order resonant systems that best fits the data
+% #+RESULTS:
+% [[file:figs/iff_results_bode_plots_identification.png]]
+
+% Finally, we can represent the position of the poles of the 2nd order systems on the Root Locus plot (Figure [[fig:iff_results_root_locus]]).
+
+
+w_z = 2*pi*111; % Zeros frequency [rad/s]
+w_p = 2*pi*255; % Pole frequency [rad/s]
+xi_z = 0.05;
+xi_p = 0.015;
+G_inf = 2;
+
+Gi = G_inf*(s^2 - 2*xi_z*w_z*s + w_z^2)/(s^2 + 2*xi_p*w_p*s + w_p^2);
+
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+plot(real(pole(Gi)),  imag(pole(Gi)),  'kx', 'HandleVisibility', 'off');
+plot(real(tzero(Gi)),  imag(tzero(Gi)),  'ko', 'HandleVisibility', 'off');
+for i = 1:length(results)
+  set(gca,'ColorOrderIndex',i)
+  plot(real(pole(G_id{i})), imag(pole(G_id{i})), 'o', 'DisplayName', sprintf('g = %0.f', g_iff(i)));
+end
+for i = 1:length(gains)
+  cl_poles = pole(feedback(Gi, (gains(i)/(s + 2*pi*2))));
+  plot(real(cl_poles), imag(cl_poles), 'k.', 'HandleVisibility', 'off');
+end
+ylim([0, 1800]);
+xlim([-1600,200]);
+xlabel('Real Part')
+ylabel('Imaginary Part')
+axis square
+legend('location', 'northwest');

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matlab/mat/APA95ML_out_nodes_3D.txt

@@ -0,0 +1,75 @@
+
+ LIST ALL SELECTED NODES.   DSYS=      0
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Academic Research                           
+
+ 00208316  VERSION=WINDOWS x64   15:24:46  NOV 18, 2020 CP=      2.031
+
+ Piezo-APA95ML-3DElm--Modal (A5)                                               
+
+
+
+    NODE        X             Y             Z           THXY     THYZ     THZX
+    40467   0.0000        0.0000       0.29997E-001     0.00     0.00     0.00
+    40469   0.0000        0.0000      -0.29997E-001     0.00     0.00     0.00
+    40470 -0.35000E-001   0.0000        0.0000          0.00     0.00     0.00
+    40475 -0.15000E-001   0.0000        0.0000          0.00     0.00     0.00
+    40476 -0.50000E-002   0.0000        0.0000          0.00     0.00     0.00
+    40477  0.15000E-001   0.0000        0.0000          0.00     0.00     0.00
+    40478  0.35000E-001   0.0000        0.0000          0.00     0.00     0.00
+
+ LIST MASTERS ON ALL SELECTED NODES.
+ CURRENT DOF SET= UX   UY   UZ   ROTX ROTY ROTZ
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Academic Research                           
+
+ 00208316  VERSION=WINDOWS x64   15:24:46  NOV 18, 2020 CP=      2.156
+
+ Piezo-APA95ML-3DElm--Modal (A5)                                               
+
+
+     NODE  LABEL     SUPPORT
+     40467  UX  
+     40467  UY  
+     40467  UZ  
+     40467  ROTX
+     40467  ROTY
+     40467  ROTZ
+     40469  UX  
+     40469  UY  
+     40469  UZ  
+     40469  ROTX
+     40469  ROTY
+     40469  ROTZ
+     40470  UX  
+     40470  UY  
+     40470  UZ  
+     40470  ROTX
+     40470  ROTY
+     40470  ROTZ
+     40475  UX  
+     40475  UY  
+     40475  UZ  
+     40475  ROTX
+     40475  ROTY
+     40475  ROTZ
+     40476  UX  
+     40476  UY  
+     40476  UZ  
+     40476  ROTX
+     40476  ROTY
+     40476  ROTZ
+     40477  UX  
+     40477  UY  
+     40477  UZ  
+     40477  ROTX
+     40477  ROTY
+     40477  ROTZ
+     40478  UX  
+     40478  UY  
+     40478  UZ  
+     40478  ROTX
+     40478  ROTY
+     40478  ROTZ

matlab/motion_identification.m

@@ -0,0 +1,100 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Load Data
+% The data from the "noise test" and the identification test are loaded.
+
+ht = load('huddle_test.mat', 't', 'y');
+load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
+
+
+
+% Any offset value is removed:
+
+um = detrend(um, 0); % Generated DAC Voltage [V]
+y  = detrend(y , 0); % Mass displacement [m]
+
+ht.y  = detrend(ht.y, 0);
+
+
+
+% Now we add a factor 10 to take into account the gain of the voltage amplifier.
+
+Va = 10*um;
+
+% Comparison of the PSD with Huddle Test
+
+Ts = t(end)/(length(t)-1);
+Fs = 1/Ts;
+
+win = hanning(ceil(1*Fs));
+
+[pxx, f] = pwelch(y, win, [], [], Fs);
+[pht, ~] = pwelch(ht.y, win, [], [], Fs);
+
+figure;
+hold on;
+plot(f, sqrt(pxx), 'DisplayName', '5kg');
+plot(f, sqrt(pht), 'DisplayName', 'Huddle Test');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
+legend('location', 'northeast');
+xlim([1, Fs/2]);
+
+% Compute TF estimate and Coherence
+
+[tf_est, f] = tfestimate(Va, -y, win, [], [], 1/Ts);
+[co_est, ~] = mscohere(  Va, -y, win, [], [], 1/Ts);
+
+figure;
+
+hold on;
+plot(f, co_est, 'k-')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Coherence'); xlabel('Frequency [Hz]');
+hold off;
+xlim([10, 5e3]);
+
+
+
+% #+name: fig:apa95ml_5kg_PI_coh
+% #+caption: Coherence
+% #+RESULTS:
+% [[file:figs/apa95ml_5kg_PI_coh.png]]
+
+% Comparison with the FEM model
+
+load('mat/fem_simscape_models.mat', 'Ghm');
+
+freqs = logspace(0, 4, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(tf_est), 'DisplayName', 'Identification')
+plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), 'DisplayName', 'FEM')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel', []);
+legend('location', 'northeast')
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(f,     180/pi*angle(tf_est))
+plot(freqs, 180/pi*angle(squeeze(freqresp(Ghm, freqs, 'Hz'))))
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
+ylabel('Phase'); xlabel('Frequency [Hz]');
+hold off;
+ylim([-180, 180]);
+yticks(-180:90:180);
+
+linkaxes([ax1,ax2], 'x');
+xlim([10, 5e3]);

matlab/piezo_parameters.m

@@ -0,0 +1,317 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Estimation from Data-sheet
+% <<sec:estimation_datasheet>>
+
+% The stack parameters taken from the data-sheet are shown in Table [[tab:stack_parameters]].
+
+% #+name: tab:stack_parameters
+% #+caption: Stack Parameters
+% | Parameter      | Unit      |    Value |
+% |----------------+-----------+----------|
+% | Nominal Stroke | $\mu m$   |       20 |
+% | Blocked force  | $N$       |     4700 |
+% | Stiffness      | $N/\mu m$ |      235 |
+% | Voltage Range  | $V$       | -20..150 |
+% | Capacitance    | $\mu F$   |      4.4 |
+% | Length         | $mm$      |       20 |
+% | Stack Area     | $mm^2$    |    10x10 |
+
+% Let's compute the generated force
+
+% The stroke is $L_{\max} = 20\mu m$ for a voltage range of $V_{\max} = 170 V$.
+% Furthermore, the stiffness is $k_a = 235 \cdot 10^6 N/m$.
+
+% The relation between the applied voltage and the generated force can be estimated as follows:
+% \begin{equation}
+%   g_a = k_a \frac{L_{\max}}{V_{\max}}
+% \end{equation}
+
+
+ka = 235e6; % [N/m]
+Lmax = 20e-6; % [m]
+Vmax = 170; % [V]
+
+ga = ka*Lmax/Vmax; % [N/V]
+
+% Estimation from Piezoelectric parameters
+% <<sec:estimation_piezo_params>>
+
+% In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
+
+d33 = 300e-12; % Strain constant [m/V]
+n   = 80;      % Number of layers per stack
+ka  = 235e6;   % Stack stiffness [N/m]
+
+
+
+% The ratio of the developed force to applied voltage is:
+% #+name: eq:piezo_voltage_to_force
+% \begin{equation}
+%   F_a = g_a V_a, \quad g_a = d_{33} n k_a
+% \end{equation}
+% where:
+% - $F_a$: developed force in [N]
+% - $n$: number of layers of the actuator stack
+% - $d_{33}$: strain constant in [m/V]
+% - $k_a$: actuator stack stiffness in [N/m]
+% - $V_a$: applied voltage in [V]
+
+% If we take the numerical values, we obtain:
+
+ga = d33*n*ka; % [N/V]
+
+
+
+% #+RESULTS:
+% : ga = 5.6 [N/V]
+
+
+% From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
+% #+name: eq:piezo_strain_to_voltage
+% \begin{equation}
+%   V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
+% \end{equation}
+% where:
+% - $V_s$: measured voltage in [V]
+% - $d_{33}$: strain constant in [m/V]
+% - $\epsilon^T$: permittivity under constant stress in [F/m]
+% - $s^D$: elastic compliance under constant electric displacement in [m^2/N]
+% - $n$: number of layers of the sensor stack
+% - $\Delta h$: relative displacement in [m]
+
+% If we take the numerical values, we obtain:
+
+d33 = 300e-12; % Strain constant [m/V]
+n   = 80;      % Number of layers per stack
+eT  = 5.3e-9;  % Permittivity under constant stress [F/m]
+sD  = 2e-11;   % Compliance under constant electric displacement [m2/N]
+
+gs = d33/(eT*sD*n); % [V/m]
+
+% From actuator voltage $V_a$ to actuator force $F_a$
+% The data from the identification test is loaded.
+
+load('apa95ml_5kg_Amp_E505.mat', 't', 'um', 'y');
+
+% Any offset value is removed:
+um = detrend(um, 0); % Amplifier Input Voltage [V]
+y  = detrend(y , 0); % Mass displacement [m]
+
+
+
+% Now we add a factor 10 to take into account the gain of the voltage amplifier and thus obtain the voltage across the piezoelectric stack.
+
+um = 10*um; % Stack Actuator Input Voltage [V]
+
+
+
+% Then, the transfer function from the stack voltage to the vertical displacement is computed.
+
+Ts = t(end)/(length(t)-1);
+Fs = 1/Ts;
+
+win = hanning(ceil(1*Fs));
+
+[tf_est, f] = tfestimate(um, y, win, [], [], 1/Ts);
+
+
+
+% The gain from input voltage of the stack to the vertical displacement is determined:
+
+g_d_Va = 4e-7; % [m/V]
+
+freqs = logspace(0, 4, 1000);
+
+figure;
+hold on;
+plot(f, abs(tf_est), 'DisplayName', 'Identification')
+plot([f(2) f(end)], [g_d_Va g_d_Va], '--', 'DisplayName', sprintf('$d/V_a \\approx %.0g\\ m/V$', g_d_Va))
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude $d/V_a$ [m/V]');
+xlim([10, 5e3]); ylim([1e-8, 1e-5]);
+legend('location', 'southwest');
+
+
+
+% #+name: fig:gain_Va_to_d
+% #+caption: Transfer function from actuator stack voltage $V_a$ to vertical displacement of the mass $d$
+% #+RESULTS:
+% [[file:figs/gain_Va_to_d.png]]
+
+% Then, the transfer function from forces applied by the stack actuator to the vertical displacement of the mass is identified from the Simscape model.
+
+% Load Parameters
+K = readmatrix('APA95ML_K.CSV');
+M = readmatrix('APA95ML_M.CSV');
+[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
+
+% Define parameters just for the simulation
+ga = 1; % [N/V]
+gs = 1; % [V/m]
+
+m = 5.5;
+
+%% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force [N]
+io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
+
+Gd = linearize(mdl, io);
+
+
+
+% The DC gain the the identified dynamics
+
+g_d_Fa = abs(dcgain(Gd)); % [m/N]
+
+
+
+% #+RESULTS:
+% : G_d_Fa = 1.2e-08 [m/N]
+
+
+% And finally, the gain $g_a$ from the the actuator voltage $V_a$ to the generated force $F_a$ can be computed:
+% \begin{equation}
+%   g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \frac{d}{V_a}
+% \end{equation}
+
+
+ga = g_d_Va/g_d_Fa;
+
+
+
+% #+RESULTS:
+% : ga = 33.7 [N/V]
+
+% The obtained comparison between the Simscape model and the identified dynamics is shown in Figure [[fig:compare_Gd_id_simscape]].
+
+
+freqs = logspace(1, 4, 1000);
+
+figure;
+hold on;
+plot(f, abs(tf_est), 'DisplayName', 'Identification')
+plot(f, ga*abs(squeeze(freqresp(Gd, f, 'Hz'))), 'DisplayName', 'Simscape Model')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude $d/V_a$ [m/V]'); xlabel('Frequency [Hz]');
+hold off;
+xlim([1, 5e3]); ylim([1e-9, 1e-5]);
+legend('location', 'southwest');
+
+% From stack strain $\Delta h$ to generated voltage $V_s$
+% Now, the gain from the stack strain $\Delta h$ to the generated voltage $V_s$ is estimated.
+
+% We can determine the gain from actuator voltage $V_a$ to sensor voltage $V_s$ thanks to the identification.
+% Using the simscape model, we can have the transfer function from the actuator voltage $V_a$ (using the previously estimated gain $g_a$) to the sensor stack strain $\Delta h$.
+
+% Finally, using these two values, we can compute the gain $g_s$ from the stack strain $\Delta h$ to the generated Voltage $V_s$.
+
+% Identification data is loaded.
+
+load('apa95ml_5kg_2a_1s.mat', 't', 'u', 'v');
+
+u = detrend(u, 0); % Input Voltage of the Amplifier [V]
+v = detrend(v, 0); % Voltage accross the stack sensor [V]
+
+
+
+% Here, an amplifier with a gain of 20 is used.
+
+u = 20*u; % Input Voltage of the Amplifier [V]
+
+
+
+% Then, the transfer function from $V_a$ to $V_s$ is identified and its DC gain is estimated (Figure [[fig:gain_Va_to_Vs]]).
+
+Ts = t(end)/(length(t)-1);
+Fs = 1/Ts;
+
+win = hann(ceil(10/Ts));
+
+[tf_est, f] = tfestimate(u, v, win, [], [], 1/Ts);
+
+g_Vs_Va = 0.022; % [V/V]
+
+freqs = logspace(1, 4, 1000);
+
+figure;
+hold on;
+plot(f, abs(tf_est), 'DisplayName', 'Identification')
+plot([f(2); f(end)], [g_Vs_Va; g_Vs_Va], '--', 'DisplayName', sprintf('$V_s/V_a \\approx %.0g\\ m/V$', g_Vs_Va))
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); xlabel('Frequency [Hz]');
+xlim([5e-1 5e3]); ylim([1e-3, 1e1]);
+legend('location', 'northwest');
+
+
+
+% #+name: fig:gain_Va_to_Vs
+% #+caption: Transfer function from actuator stack voltage $V_a$ to sensor stack voltage $V_s$
+% #+RESULTS:
+% [[file:figs/gain_Va_to_Vs.png]]
+
+
+% Now the transfer function from the actuator stack voltage to the sensor stack strain is estimated using the Simscape model.
+
+m = 5.5;
+
+%% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Actuator Voltage [V]
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Stack displacement [m]
+
+Gf = linearize(mdl, io);
+
+
+
+% The gain from the actuator stack voltage to the sensor stack strain is estimated below.
+
+G_dh_Va = abs(dcgain(Gf));
+
+
+
+% #+RESULTS:
+% : G_dh_Va = 6.2e-09 [m/V]
+
+% And finally, the gain $g_s$ from the sensor stack strain to the generated voltage can be estimated:
+% \begin{equation}
+%   g_s = \frac{V_s}{\Delta h} = \frac{V_s}{V_a} \frac{V_a}{\Delta h}
+% \end{equation}
+
+
+gs = g_Vs_Va/G_dh_Va; % [V/m]
+
+
+
+% #+RESULTS:
+% : gs = 3.5 [V/um]
+
+
+freqs = logspace(1, 4, 1000);
+
+figure;
+hold on;
+plot(f, abs(tf_est), 'DisplayName', 'Identification')
+plot(f, gs*abs(squeeze(freqresp(Gf, f, 'Hz'))), 'DisplayName', 'Simscape Model')
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+ylabel('Amplitude $V_s/V_a$ [V/V]'); xlabel('Frequency [Hz]');
+hold off;
+xlim([1, 5e3]); ylim([1e-3, 1e1]);
+legend('location', 'southwest');
+
+save('./mat/apa95ml_params.mat', 'ga', 'gs');

matlab/simscape_model.m

@@ -0,0 +1,120 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+addpath('./mat/');
+
+% Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
+% We first extract the stiffness and mass matrices.
+
+K = readmatrix('APA95ML_K.CSV');
+M = readmatrix('APA95ML_M.CSV');
+
+
+
+% #+caption: First 10x10 elements of the Mass matrix
+% #+RESULTS:
+% |    0.03 |   7e-08 |  2e-06 | -3e-09 | -0.0002 | -6e-08 | -0.001 |   8e-07 |  6e-07 | -8e-09 |
+% |   7e-08 |    0.02 | -1e-06 |  9e-05 |  -3e-09 | -4e-09 | -1e-06 | -0.0006 | -4e-08 |  5e-06 |
+% |   2e-06 |  -1e-06 |   0.02 | -3e-08 |  -4e-08 |  1e-08 |  1e-07 |  -2e-07 | 0.0003 |  1e-09 |
+% |  -3e-09 |   9e-05 | -3e-08 |  1e-06 |  -3e-11 | -3e-13 | -7e-09 |  -5e-06 | -3e-10 |  3e-08 |
+% | -0.0002 |  -3e-09 | -4e-08 | -3e-11 |   2e-06 |  6e-10 |  2e-06 |  -7e-09 | -2e-09 |  7e-11 |
+% |  -6e-08 |  -4e-09 |  1e-08 | -3e-13 |   6e-10 |  1e-06 |  1e-08 |   3e-09 | -2e-09 |  2e-13 |
+% |  -0.001 |  -1e-06 |  1e-07 | -7e-09 |   2e-06 |  1e-08 |   0.03 |   4e-08 | -2e-06 |  8e-09 |
+% |   8e-07 | -0.0006 | -2e-07 | -5e-06 |  -7e-09 |  3e-09 |  4e-08 |    0.02 | -9e-07 | -9e-05 |
+% |   6e-07 |  -4e-08 | 0.0003 | -3e-10 |  -2e-09 | -2e-09 | -2e-06 |  -9e-07 |   0.02 |  2e-08 |
+% |  -8e-09 |   5e-06 |  1e-09 |  3e-08 |   7e-11 |  2e-13 |  8e-09 |  -9e-05 |  2e-08 |  1e-06 |
+
+% Then, we extract the coordinates of the interface nodes.
+
+[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
+
+% Simscape Model
+% The flexible element is imported using the =Reduced Order Flexible Solid= Simscape block.
+
+% To model the actuator, an =Internal Force= block is added between the nodes 3 and 12.
+% A =Relative Motion Sensor= block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
+
+% One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
+
+m = 5.5;
+
+load('apa95ml_params.mat', 'ga', 'gs');
+
+% Dynamics from Actuator Voltage to Vertical Mass Displacement
+% The identified dynamics is shown in Figure [[fig:dynamics_act_disp_comp_mass]].
+
+
+%% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Actuator Voltage [V]
+io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
+
+Ghm = linearize(mdl, io);
+
+freqs = logspace(0, 4, 5000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))));
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+yticks(-360:90:360);
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+% Dynamics from Actuator Voltage to Force Sensor Voltage
+% The obtained dynamics is shown in Figure [[fig:dynamics_force_force_sensor_comp_mass]].
+
+
+%% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1; % Voltage Actuator [V]
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage [V]
+
+Gfm = linearize(mdl, io);
+
+freqs = logspace(1, 5, 1000);
+
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(Gfm, 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;
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gfm, freqs, 'Hz')))));
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+yticks(-360:90:360); ylim([-360, 180]);
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+
+save('mat/fem_simscape_models.mat', 'Ghm', 'Gfm')