Update analysis

.gitignore

@@ -1,6 +1,7 @@
 mat/
 figures/
 ltximg/
+*.autosave
 slprj/
 matlab/slprj/
 *.slxc

matlab/STEPS/full_APA300ML_K.CSV

@@ -0,0 +1,36 @@
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matlab/STEPS/full_APA300ML_out_nodes_3D.txt

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

matlab/test_apa_1_basic_meas.m

@@ -61,7 +61,7 @@ for i = 1:7
     apa_d(i) = min_d;
 end
 
-% Stroke Measurement
+% Stroke and Hysteresis Measurement
 % <<sec:test_apa_stroke_measurements>>
 
 % The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
@@ -111,177 +111,99 @@ xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
 legend('location', 'southwest', 'FontSize', 8)
 xlim([-20, 150]); ylim([-250, 0]);
 
-% X-Bending Mode
+% Flexible Mode Measurement
+% SCHEDULED: <2024-03-27 Wed>
+% <<sec:test_apa_spurious_resonances>>
 
-% The vibrometer is setup to measure the X-bending motion is shown in Figure ref:fig:test_apa_meas_setup_X_bending.
+% In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
-% The APA is excited with an instrumented hammer having a solid metallic tip.
-% The impact point is on the back-side of the APA aligned with the top measurement point.
 
-% #+name: fig:test_apa_meas_setup_X_bending
+% To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_torsion).
-% #+caption: X-Bending measurement setup
+% A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer[fn:7] is used to excite the flexible modes.
-% #+attr_latex: :width 0.7\linewidth
+% Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated.
+
+% The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure ref:fig:test_apa_mode_shapes.
+
+% #+name: fig:test_apa_mode_shapes
+% #+caption: Spurious resonances - Change this with the updated FEM analysis of the APA300ML
+% #+attr_latex: :width 0.9\linewidth
+% [[file:figs/test_apa_mode_shapes.png]]
+
+% #+name: fig:test_apa_meas_setup_torsion
+% #+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the $Z$ torsion is measured.
+% #+attr_latex: :width 0.6\linewidth
+% [[file:figs/test_apa_meas_setup_torsion.jpg]]
+
+% Two other similar measurements are performed to measured the bending of the APA around the $X$ direction and around the $Y$ direction (see Figure ref:fig:test_apa_meas_setup_modes).
+
+% #+name: fig:test_apa_meas_setup_modes
+% #+caption: Experimental setup to measured flexible modes of the APA300ML. For the bending in the $X$ direction, the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
+% #+begin_figure
+% #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
+% #+attr_latex: :options {0.49\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :width 0.95\linewidth
 % [[file:figs/test_apa_meas_setup_X_bending.jpg]]
+% #+end_subfigure
+% #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
+% #+attr_latex: :options {0.49\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :width 0.95\linewidth
+% [[file:figs/test_apa_meas_setup_Y_bending.jpg]]
+% #+end_subfigure
+% #+end_figure
 
-% The data is loaded.
 
-%% Load Data
+%% X-Bending Identification
+% Load Data
 bending_X = load('apa300ml_bending_X_top.mat');
 
-
+% Spectral Analysis setup
-
-% The configuration (Sampling time and windows) for =tfestimate= is done:
-
-%% Spectral Analysis setup
 Ts = bending_X.Track1_X_Resolution; % Sampling Time [s]
 Nfft = floor(1/Ts);
 win = hanning(Nfft);
 Noverlap = floor(Nfft/2);
 
-
+% Compute the transfer function from applied force to measured rotation
-
-% The transfer function from the input force to the output "rotation" (difference between the two measured distances).
-
-%% Compute the transfer function from applied force to measured rotation
 [G_bending_X, f]  = tfestimate(bending_X.Track1, bending_X.Track2, win, Noverlap, Nfft, 1/Ts);
 
-
+%% Y-Bending identification
-
+% Load Data
-% The result is shown in Figure ref:fig:test_apa_meas_freq_bending_x.
-
-% The can clearly observe a nice peak at 280Hz, and then peaks at the odd "harmonics" (third "harmonic" at 840Hz, and fifth "harmonic" at 1400Hz).
-
-%% 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')
-
-% Y-Bending Mode
-
-% The setup to measure the Y-bending is shown in Figure ref:fig:test_apa_meas_setup_Y_bending.
-
-% The impact point of the instrumented hammer is located on the back surface of the top interface (on the back of the 2 measurements points).
-
-% #+name: fig:test_apa_meas_setup_Y_bending
-% #+caption: Y-Bending measurement setup
-% #+attr_latex: :width 0.7\linewidth
-% [[file:figs/test_apa_meas_setup_Y_bending.jpg]]
-
-% The data is loaded, and the transfer function from the force to the measured rotation is computed.
-
-%% Load Data
 bending_Y = load('apa300ml_bending_Y_top.mat');
 
-%% Compute the transfer function
+% Compute the transfer function
 [G_bending_Y, ~]  = tfestimate(bending_Y.Track1, bending_Y.Track2, win, Noverlap, Nfft, 1/Ts);
 
+%% Z-Torsion identification
+% Load data
+torsion = load('apa300ml_torsion_top.mat');
 
+% Compute transfer function
+[G_torsion_top, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts);
 
-% The results are shown in Figure ref:fig:test_apa_meas_freq_bending_y.
+% Load Data
-% The main resonance is at 412Hz, and we also see the third "harmonic" at 1220Hz.
-
-
-%% 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')
-
-% Z-Torsion Mode
-
-% Finally, we measure the Z-torsion resonance as shown in Figure ref:fig:test_apa_meas_setup_torsion_bis.
-
-% The excitation is shown on the other side of the APA, on the side to excite the torsion motion.
-
-% #+name: fig:test_apa_meas_setup_torsion_bis
-% #+caption: Z-Torsion measurement setup
-% #+attr_latex: :width 0.7\linewidth
-% [[file:figs/test_apa_meas_setup_torsion_bis.jpg]]
-
-% The data is loaded, and the transfer function computed.
-
-%% Load Data
 torsion = load('apa300ml_torsion_left.mat');
 
-%% Compute transfer function
+% Compute transfer function
 [G_torsion, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts);
 
 
 
-% The results are shown in Figure ref:fig:test_apa_meas_freq_torsion_z.
+% The three measured frequency response functions are shown in Figure ref:fig:test_apa_meas_freq_compare.
-% We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz.
+% - a clear $x$ bending mode at $280\,\text{Hz}$
-% And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz.
+% - a clear $y$ bending mode at $412\,\text{Hz}$
-% A third mode at 800Hz could correspond to this torsion mode.
+% - for the $z$ torsion test, the $y$ bending mode is also excited and observed, and we may see a mode at $800\,\text{Hz}$
 
 
-%% Plot the transfer function
 figure;
 hold on;
-plot(f, abs(G_torsion), 'k-');
+plot(f, abs(G_bending_X), 'DisplayName', '$X$ bending');
-hold off;
+plot(f, abs(G_bending_Y), 'DisplayName', '$Y$ bending');
-set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+plot(f, abs(G_torsion),   'DisplayName', '$Z$ torsion');
-xlabel('Frequency [Hz]'); ylabel('Amplitude');
+text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
-xlim([50, 2e3]); ylim([1e-5, 2e-2])
+text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
-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')
-
-
-
-% #+name: fig:test_apa_meas_freq_torsion_z
-% #+caption: Obtained FRF for the Z-torsion
-% #+RESULTS:
-% [[file:figs/test_apa_meas_freq_torsion_z.png]]
-
-% In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited.
-
-%% Load data
-torsion = load('apa300ml_torsion_top.mat');
-
-%% Compute transfer function
-[G_torsion_top, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts);
-
-
-
-% The two FRF are compared in Figure ref:fig:test_apa_meas_freq_torsion_z_comp.
-% It is clear that the first two modes does not correspond to the torsional mode.
-% Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here.
-
-%% 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])
+xlim([50, 2e3]); ylim([5e-5, 2e-1]);
-text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+legend('location', 'northeast', 'FontSize', 8)
-text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
-text(800, 2e-3,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
-legend('location', 'northwest');
-
-% Compare
-% The three measurements are shown in Figure ref:fig:test_apa_meas_freq_compare.
-
-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/test_apa_2_dynamics.m

@@ -119,7 +119,7 @@ xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]');
 % |   6 |   1.7 |  1.92 |
 % |   8 |  1.73 |  1.98 |
 
-% The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 94\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_frf_disp) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$.
+% The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$.
 
 % \begin{equation} \label{eq:test_apa_res_freq}
 % \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
@@ -204,7 +204,7 @@ save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
 % - A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$.
 %   The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA
 % - A lightly damped resonance at $95\,\text{Hz}$
-% - A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear
+% - A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
 
 
 %% Plot the FRF from u to de
@@ -305,17 +305,17 @@ xlim([10, 2e3]);
 
 
 %% Load the data
-wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va'); % With the resistor
+wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'u'); % With the resistor
-wo_k = load('frf_data_1_sweep_lf.mat',        't', 'Vs', 'Va'); % Without the resistor
+wo_k = load('frf_data_1_sweep_lf.mat',        't', 'Vs', 'u'); % Without the resistor
 
 %% Large Hanning window for good low frequency estimate
 Nfft = floor(50/Ts);
 win = hanning(Nfft);
 Noverlap = floor(Nfft/2);
 
-%% Compute the transfer functions from Va to Vs
+%% Compute the transfer functions from u to Vs
-[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, Noverlap, Nfft, 1/Ts);
+[frf_wo_k, f] = tfestimate(wo_k.u, wo_k.Vs, win, Noverlap, Nfft, 1/Ts);
-[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, Noverlap, Nfft, 1/Ts);
+[frf_wi_k, ~] = tfestimate(wi_k.u, wi_k.Vs, win, Noverlap, Nfft, 1/Ts);
 
 %% Model for the high pass filter
 C = 5.1e-6; % Sensor Stack capacitance [F]

matlab/test_apa_3_model_2dof.m

@@ -0,0 +1,177 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./src/'); % Path for scripts
+addpath('./mat/'); % Path for data
+
+addpath('./STEPS/'); % Path for Simscape Model
+
+%% Linearization options
+opts = linearizeOptions;
+opts.SampleTime = 0;
+
+%% Open Simscape Model
+mdl = 'test_apa_simscape'; % Name of the Simulink File
+open(mdl); % Open Simscape Model
+
+%% Colors for the figures
+colors = colororder;
+
+%% Input/Output definition of the Model
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/u'],  1, 'openinput');  io_i = io_i + 1; % DAC Voltage
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
+io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
+
+% Tuning of the APA model
+% <<ssec:test_apa_2dof_model_tuning>>
+
+% 9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
+
+% #+name: fig:test_apa_2dof_model_simscape
+% #+caption: Schematic of the two degrees of freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$
+% [[file:figs/test_apa_2dof_model_simscape.png]]
+
+
+%% Stiffness values for the 2DoF APA model
+k1 = 0.38e6; % Estimated Shell Stiffness [N/m]
+
+w0 = 2*pi*95; % Resonance frequency [rad/s]
+m = 5.7; % Suspended mass [kg]
+ktot = m*(w0)^2; % Total Axial Stiffness to have to wanted resonance frequency [N/m]
+
+ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m]
+ke = 2*ka; % Stiffness of the Sensor stack [N/m]
+
+%% Damping values for the 2DoF APA model
+c1 = 20;   % Damping for the Shell [N/(m/s)]
+ca = 100;  % Damping of the actuators stacks [N/(m/s)]
+ce = 2*ca; % Damping of the sensor stack [N/(m/s)]
+
+%% Estimation ot the sensor and actuator gains
+% Initialize the structure with unitary sensor and actuator "gains"
+n_hexapod = struct();
+n_hexapod.actuator = initializeAPA(...
+    'type', '2dof', ...
+    'k',  k1,  ...
+    'ka', ka,  ...
+    'ke', ke,  ...
+    'c',  c1, ...
+    'ca', ca, ...
+    'ce', ce, ...
+    'Ga', 1,   ... % Actuator constant [N/V]
+    'Gs', 1    ... % Sensor constant [V/m]
+);
+
+c_granite = 0; % Do not take into account damping added by the air bearing
+
+% Run the linearization
+G_norm = linearize(mdl, io, 0.0, opts);
+G_norm.InputName  = {'u'};
+G_norm.OutputName = {'Vs', 'de'};
+
+% Load Identification Data to estimate the two gains
+load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
+
+% Estimation ot the Actuator Gain
+fa = 10; % Frequency where the two FRF should match [Hz]
+[~, i_f] = min(abs(f - fa));
+ga = -abs(enc_frf(i_f,1))./abs(evalfr(G_norm('de', 'u'), 1i*2*pi*fa));
+
+% Estimation ot the Sensor Gain
+fs = 600; % Frequency where the two FRF should match [Hz]
+[~, i_f] = min(abs(f - fs))
+gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga;
+
+% Obtained Dynamics
+% <<ssec:test_apa_2dof_model_result>>
+
+% The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model.
+% It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf.
+
+% A good match can be observed between the model and the experimental data, both for the encoder and for the force sensor.
+% This indicates that this model represents well the axial dynamics of the APA300ML.
+
+
+%% 2DoF APA300ML with optimized parameters
+n_hexapod = struct();
+n_hexapod.actuator = initializeAPA(...
+    'type', '2dof', ...
+    'k',  k1,  ...
+    'ka', ka,  ...
+    'ke', ke,  ...
+    'c',  c1, ...
+    'ca', ca, ...
+    'ce', ce, ...
+    'Ga', ga, ...
+    'Gs', gs  ...
+);
+
+%% Identification of the APA300ML with optimized parameters
+G_2dof = exp(-s*Ts)*linearize(mdl, io, 0.0, opts);
+G_2dof.InputName  = {'u'};
+G_2dof.OutputName = {'Vs', 'de'};
+
+%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML
+freqs = 5*logspace(0, 3, 1000);
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax1b = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 2:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+
+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
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', '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); 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
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', '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); ylim([-180, 180]);
+
+linkaxes([ax1,ax2,ax1b,ax2b],'x');
+xlim([10, 2e3]);

matlab/test_apa_4_model_flexible.m

@@ -0,0 +1,202 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./src/'); % Path for scripts
+addpath('./mat/'); % Path for data
+
+addpath('./STEPS/'); % Path for Simscape Model
+
+%% Linearization options
+opts = linearizeOptions;
+opts.SampleTime = 0;
+
+%% Open Simscape Model
+mdl = 'test_apa_simscape'; % Name of the Simulink File
+open(mdl); % Open Simscape Model
+
+%% Colors for the figures
+colors = colororder;
+
+%% Input/Output definition of the Model
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/u'],  1, 'openinput');  io_i = io_i + 1; % DAC Voltage
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
+io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
+
+% Identification of the Actuator and Sensor constants
+% <<ssec:test_apa_flexible_ga_gs>>
+
+% Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be identified.
+% The gains $g_a$ and $g_s$ can then be tuned such that the gain of the transfer functions are matching the identified ones.
+
+% By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained.
+
+
+%% Identification of the actuator and sensor "constants"
+% Initialize the APA as a flexible body with unity "constants"
+n_hexapod.actuator = initializeAPA(...
+    'type', 'flexible', ...
+    'ga', 1, ...
+    'gs', 1);
+
+c_granite = 100; % Rought estimation of the damping added by the air bearing
+
+% Identify the dynamics
+G_norm = linearize(mdl, io, 0.0, opts);
+G_norm.InputName  = {'u'};
+G_norm.OutputName = {'Vs', 'de'};
+
+% Load Identification Data to estimate the two gains
+load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
+
+% Actuator Constant in [N/V]
+ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u'));
+
+% Sensor Constant in [V/m]
+gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz'))));
+
+
+
+
+% To make sure these "gains" are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
+
+% From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force.
+
+% \begin{subequations}
+% \begin{align}
+%   V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\
+%   F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force}
+% \end{align}
+% \end{subequations}
+
+% Parameters used in equations eqref:eq:test_apa_piezo_strain_to_voltage and eqref:eq:test_apa_piezo_voltage_to_force are described in Table ref:tab:test_apa_piezo_properties.
+
+% Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
+% However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties.
+% The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties.
+
+% From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are very close to the identified constants using the experimentally identified transfer functions.
+
+% #+name: tab:test_apa_piezo_properties
+% #+caption: Piezoelectric properties used for the estimation of the sensor and actuators "gains"
+% #+attr_latex: :environment tabularx :width 1\linewidth :align ccX
+% #+attr_latex: :center t :booktabs t
+% | *Parameter*    | *Value*                    | *Description*                                                |
+% |----------------+----------------------------+--------------------------------------------------------------|
+% | $d_{33}$       | $680 \cdot 10^{-12}\,m/V$  | Piezoelectric constant                                       |
+% | $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,F/m$   | Permittivity under constant stress                           |
+% | $s^{D}$        | $21 \cdot 10^{-12}\,m^2/N$ | Elastic compliance understand constant electric displacement |
+% | $c^{E}$        | $48 \cdot 10^{9}\,N/m^2$   | Young's modulus of elasticity                                |
+% | $L$            | $20\,mm$ per stack         | Length of the stack                                          |
+% | $A$            | $10^{-4}\,m^2$             | Area of the piezoelectric stack                              |
+% | $n$            | $160$ per stack            | Number of layers in the piezoelectric stack                  |
+
+
+%% Estimate "Sensor Constant" - (THP5H)
+d33 = 680e-12;        % Strain constant [m/V]
+n   = 160;            % Number of layers per stack
+eT  = 4500*8.854e-12; % Permittivity under constant stress [F/m]
+sD  = 21e-12;         % Compliance under constant electric displacement [m2/N]
+
+gs = d33/(eT*sD*n);   % Sensor Constant [V/m]
+
+%% Estimate "Actuator Constant" - (THP5H)
+d33 = 680e-12;   % Strain constant [m/V]
+n   = 320;       % Number of layers
+
+cE  = 1/sD;      % Youngs modulus [N/m^2]
+A   = (10e-3)^2; % Area of the stacks [m^2]
+L   = 40e-3;     % Length of the two stacks [m]
+ka  = cE*A/L;    % Stiffness of the two stacks [N/m]
+
+ga = d33*n*ka;   % Actuator Constant [N/V]
+
+% Comparison of the obtained dynamics
+
+% The obtained dynamics using the /super element/ with the tuned "sensor gain" and "actuator gain" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf.
+
+% A good match between the model and the experimental results is observed.
+% - the /super element/
+
+% This model represents fairly
+
+% The flexible model is a bit "soft" as compared with the experimental results.
+
+% This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators.
+
+
+%% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants"
+% Initialize the APA
+n_hexapod.actuator = initializeAPA(...
+    'type', 'flexible', ...
+    'ga',   23.2, ... % Actuator gain [N/V]
+    'gs',   -4.9e6);    % Sensor gain [V/m]
+
+% Identify with updated constants
+G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts);
+G_flex.InputName  = {'u'};
+G_flex.OutputName = {'Vs', 'de'};
+
+%% Comparison of the measured FRF and the "Flexible" model of the APA300ML
+freqs = 5*logspace(0, 3, 1000);
+figure;
+tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax1b = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 2:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+
+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
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', '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); 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
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', '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); ylim([-180, 180]);
+
+linkaxes([ax1,ax2,ax1b,ax2b],'x');
+xlim([10, 2e3]);

test-bench-apa.bib

@@ -10,26 +10,6 @@
 
 
 
-@article{gustavsen99_ration_approx_frequen_domain_respon,
-  author          = {Gustavsen, B.; Semlyen, A.},
-  title           = {Rational Approximation of Frequency Domain Responses By
-                  Vector Fitting},
-  journal         = {IEEE Transactions on Power Delivery},
-  volume          = 14,
-  year            = 1999,
-  doi             = {10.1109/61.772353},
-  url             = {https://doi.org/10.1109/61.772353},
-  issne           = {1937-4208},
-  issnp           = {0885-8977},
-  issue           = 3,
-  month           = 7,
-  page            = {1052--1061},
-  publisher       = {IEEE},
-  keywords        = {Motors},
-}
-
-
-
 @article{souleille18_concep_activ_mount_space_applic,
   author          = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and
                   Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues,
@@ -44,3 +24,45 @@
   keywords        = {parallel robot, iff},
 }
 
+
+
+@book{fleming14_desig_model_contr_nanop_system,
+  author          = {Andrew J. Fleming and Kam K. Leang},
+  title           = {Design, Modeling and Control of Nanopositioning Systems},
+  year            = 2014,
+  publisher       = {Springer International Publishing},
+  url             = {https://doi.org/10.1007/978-3-319-06617-2},
+  doi             = {10.1007/978-3-319-06617-2},
+  series          = {Advances in Industrial Control},
+}
+
+
+
+@article{fleming10_integ_strain_force_feedb_high,
+  author          = {Fleming, Andrew J and Leang, Kam K},
+  title           = {Integrated Strain and Force Feedback for High-Performance
+                  Control of Piezoelectric Actuators},
+  journal         = {Sensors and Actuators A: Physical},
+  volume          = 161,
+  number          = {1-2},
+  pages           = {256--265},
+  year            = 2010,
+  publisher       = {Elsevier},
+  keywords        = {flexure,nanostage},
+}
+
+
+
+@article{fleming10_integ_strain_force_feedb_high,
+  author          = {Fleming, Andrew J and Leang, Kam K},
+  title           = {Integrated Strain and Force Feedback for High-Performance
+                  Control of Piezoelectric Actuators},
+  journal         = {Sensors and Actuators A: Physical},
+  volume          = 161,
+  number          = {1-2},
+  pages           = {256--265},
+  year            = 2010,
+  publisher       = {Elsevier},
+  keywords        = {flexure,nanostage},
+}
+

test-bench-apa.tex

@@ -1,4 +1,4 @@
-% Created 2024-03-21 Thu 18:12
+% Created 2024-03-22 Fri 19:12
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -47,13 +47,15 @@ This is explained in Section \ref{sec:test_apa_simscape}.
 \textbf{Sections} & \textbf{Matlab File}\\
 \midrule
 Section \ref{sec:test_apa_basic_meas} & \texttt{test\_apa\_1\_basic\_meas.m}\\
-Section \ref{sec:test_apa_dynamics} & \texttt{test\_apa\_2\_.m}\\
+Section \ref{sec:test_apa_dynamics} & \texttt{test\_apa\_2\_dynamics.m}\\
-Section \ref{sec:test_apa_simscape} & \texttt{test\_apa\_3\_.m}\\
+Section \ref{sec:test_apa_model_2dof} & \texttt{test\_apa\_3\_model\_2dof.m}\\
+Section \ref{sec:test_apa_model_flexible} & \texttt{test\_apa\_4\_model\_flexible.m}\\
 \bottomrule
 \end{tabularx}
 \end{table}
 \chapter{First Basic Measurements}
 \label{sec:test_apa_basic_meas}
+
 Before using the measurement bench to characterize the APA300ML, first simple measurements are performed:
 \begin{itemize}
 \item Section \ref{sec:test_apa_geometrical_measurements}: the geometric tolerances of the interface planes are checked
@@ -133,7 +135,7 @@ APA 7 & 4.85 & 9.85\\
 \bottomrule
 \end{tabularx}
 \end{table}
-\section{Stroke Measurement}
+\section{Stroke and Hysteresis Measurement}
 \label{sec:test_apa_stroke_measurements}
 
 The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
@@ -164,161 +166,74 @@ From now on, only the six APA that behave as expected will be used.
 \includegraphics[scale=1]{figs/test_apa_stroke_result.png}
 \caption{\label{fig:test_apa_stroke_result}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (left). Measured displacement as a function of the applied voltage (right)}
 \end{figure}
-\section{Spurious resonances - APA\hfill{}\textsc{@philipp}}
+\section{Flexible Mode Measurement}
 \label{sec:test_apa_spurious_resonances}
-\subsection{Introduction}
 
-From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure \ref{fig:apa_mode_shapes_ter}):
+In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
+
+To experimentally estimate these modes, the APA is fixed on one end (see Figure \ref{fig:test_apa_meas_setup_torsion}).
+A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes.
+Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated.
+
+The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure \ref{fig:test_apa_mode_shapes}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_apa_mode_shapes.png}
+\caption{\label{fig:test_apa_mode_shapes}Spurious resonances - Change this with the updated FEM analysis of the APA300ML}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_meas_setup_torsion.jpg}
+\caption{\label{fig:test_apa_meas_setup_torsion}Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the \(Z\) torsion is measured.}
+\end{figure}
+
+Two other similar measurements are performed to measured the bending of the APA around the \(X\) direction and around the \(Y\) direction (see Figure \ref{fig:test_apa_meas_setup_modes}).
+
+\begin{figure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_meas_setup_X_bending.jpg}
+\end{center}
+\subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_meas_setup_Y_bending.jpg}
+\end{center}
+\subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
+\end{subfigure}
+\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measured flexible modes of the APA300ML. For the bending in the \(X\) direction, the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
+\end{figure}
+
+The three measured frequency response functions are shown in Figure \ref{fig:test_apa_meas_freq_compare}.
 \begin{itemize}
-\item Mode in X-bending at 189Hz
+\item a clear \(x\) bending mode at \(280\,\text{Hz}\)
-\item Mode in Y-bending at 285Hz
+\item a clear \(y\) bending mode at \(412\,\text{Hz}\)
-\item Mode in Z-torsion at 400Hz
+\item for the \(z\) torsion test, the \(y\) bending mode is also excited and observed, and we may see a mode at \(800\,\text{Hz}\)
 \end{itemize}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/apa_mode_shapes.gif}
+\includegraphics[scale=1]{figs/test_apa_meas_freq_compare.png}
-\caption{\label{fig:apa_mode_shapes_ter}Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz}
+\caption{\label{fig:test_apa_meas_freq_compare}Obtained frequency response functions for the 3 tests with the instrumented hammer}
 \end{figure}
 
-These modes are present when flexible joints are fixed to the ends of the APA300ML.
-
-In this section, we try to find the resonance frequency of these modes when one end of the APA is fixed and the other is free.
-
-In the section \ref{sec:spurious_resonances_struts}, a similar measurement will be performed directly on the struts.
-\subsection{Measurement Setup}
-
-The measurement setup is shown in Figure \ref{fig:measurement_setup_torsion}.
-A Laser vibrometer is measuring the difference of motion between two points.
-The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.
-
-\begin{note}
-The instrumentation used are:
-\begin{itemize}
-\item Laser Doppler Vibrometer Polytec OFV512
-\item Instrumented hammer
-\end{itemize}
-\end{note}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_torsion.jpg}
-\caption{\label{fig:measurement_setup_torsion}Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer}
-\end{figure}
-\subsection{X-Bending Mode}
-
-The vibrometer is setup to measure the X-bending motion is shown in Figure \ref{fig:measurement_setup_X_bending}.
-The APA is excited with an instrumented hammer having a solid metallic tip.
-The impact point is on the back-side of the APA aligned with the top measurement point.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_X_bending.jpg}
-\caption{\label{fig:measurement_setup_X_bending}X-Bending measurement setup}
-\end{figure}
-
-The data is loaded.
-The configuration (Sampling time and windows) for \texttt{tfestimate} is done:
-The transfer function from the input force to the output ``rotation'' (difference between the two measured distances).
-The result is shown in Figure \ref{fig:apa300ml_meas_freq_bending_x}.
-
-The can clearly observe a nice peak at 280Hz, and then peaks at the odd ``harmonics'' (third ``harmonic'' at 840Hz, and fifth ``harmonic'' at 1400Hz).
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/apa300ml_meas_freq_bending_x.png}
-\caption{\label{fig:apa300ml_meas_freq_bending_x}Obtained FRF for the X-bending}
-\end{figure}
-
-Then the APA is in the ``free-free'' condition, this bending mode is foreseen to be at 200Hz (Figure \ref{fig:apa_mode_shapes_ter}).
-We are here in the ``fixed-free'' condition.
-If we consider that we therefore double the stiffness associated with this mode, we should obtain a resonance a factor \(\sqrt{2}\) higher than 200Hz which is indeed 280Hz.
-Not sure this reasoning is correct though.
-\subsection{Y-Bending Mode}
-
-The setup to measure the Y-bending is shown in Figure \ref{fig:measurement_setup_Y_bending}.
-
-The impact point of the instrumented hammer is located on the back surface of the top interface (on the back of the 2 measurements points).
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_Y_bending.jpg}
-\caption{\label{fig:measurement_setup_Y_bending}Y-Bending measurement setup}
-\end{figure}
-
-The data is loaded, and the transfer function from the force to the measured rotation is computed.
-The results are shown in Figure \ref{fig:apa300ml_meas_freq_bending_y}.
-The main resonance is at 412Hz, and we also see the third ``harmonic'' at 1220Hz.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/apa300ml_meas_freq_bending_y.png}
-\caption{\label{fig:apa300ml_meas_freq_bending_y}Obtained FRF for the Y-bending}
-\end{figure}
-
-We can apply the same reasoning as in the previous section and estimate the mode to be a factor \(\sqrt{2}\) higher than the mode estimated in the ``free-free'' condition.
-We would obtain a mode at 403Hz which is very close to the one estimated here.
-\subsection{Z-Torsion Mode}
-
-Finally, we measure the Z-torsion resonance as shown in Figure \ref{fig:measurement_setup_torsion_bis}.
-
-The excitation is shown on the other side of the APA, on the side to excite the torsion motion.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_torsion_bis.jpg}
-\caption{\label{fig:measurement_setup_torsion_bis}Z-Torsion measurement setup}
-\end{figure}
-
-The data is loaded, and the transfer function computed.
-The results are shown in Figure \ref{fig:apa300ml_meas_freq_torsion_z}.
-We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz.
-And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz.
-A third mode at 800Hz could correspond to this torsion mode.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/apa300ml_meas_freq_torsion_z.png}
-\caption{\label{fig:apa300ml_meas_freq_torsion_z}Obtained FRF for the Z-torsion}
-\end{figure}
-
-In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited.
-The two FRF are compared in Figure \ref{fig:apa300ml_meas_freq_torsion_z_comp}.
-It is clear that the first two modes does not correspond to the torsional mode.
-Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here.
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/apa300ml_meas_freq_torsion_z_comp.png}
-\caption{\label{fig:apa300ml_meas_freq_torsion_z_comp}Obtained FRF for the Z-torsion}
-\end{figure}
-\subsection{Compare}
-The three measurements are shown in Figure \ref{fig:apa300ml_meas_freq_compare}.
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/apa300ml_meas_freq_compare.png}
-\caption{\label{fig:apa300ml_meas_freq_compare}Obtained FRF - Comparison}
-\end{figure}
-\subsection{Conclusion}
-
-When two flexible joints are fixed at each ends of the APA, the APA is mostly in a free/free condition in terms of bending/torsion (the bending/torsional stiffness of the joints being very small).
-
-In the current tests, the APA are in a fixed/free condition.
-Therefore, it is quite obvious that we measured higher resonance frequencies than what is foreseen for the struts.
-It is however quite interesting that there is a factor \(\approx \sqrt{2}\) between the two (increased of the stiffness by a factor 2?).
-
 \begin{table}[htbp]
-\caption{\label{tab:apa300ml_measured_modes_freq}Measured frequency of the modes}
+\caption{\label{tab:test_apa_measured_modes_freq}Measured frequency of the modes}
 \centering
-\begin{tabularx}{0.7\linewidth}{Xcc}
+\begin{tabularx}{0.5\linewidth}{Xcc}
 \toprule
-\textbf{Mode} & \textbf{FEM - Strut mode} & \textbf{Measured Frequency}\\
+\textbf{Mode} & \textbf{FEM} & \textbf{Measured Frequency}\\
 \midrule
-X-Bending & 189Hz & 280Hz\\
+\(X\) bending &  & 280Hz\\
-Y-Bending & 285Hz & 410Hz\\
+\(Y\) bending &  & 410Hz\\
-Z-Torsion & 400Hz & 800Hz?\\
+\(Z\) torsion &  & 800Hz\\
 \bottomrule
 \end{tabularx}
 \end{table}
-\chapter{Dynamical measurements - APA}
+\chapter{Dynamical measurements}
 \label{sec:test_apa_dynamics}
 
 After the basic measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the APA.
@@ -430,7 +345,7 @@ APA & \(k_1\) & \(k_2\)\\
 \end{tabularx}
 \end{table}
 
-The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 94\,\text{Hz}\) (estimated by the dynamical measurements shown in section \ref{ssec:test_apa_meas_frf_disp}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).
+The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 95\,\text{Hz}\) (estimated by the dynamical measurements shown in section \ref{ssec:test_apa_meas_dynamics}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).
 
 \begin{equation} \label{eq:test_apa_res_freq}
 \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
@@ -460,7 +375,7 @@ The following can be observed:
 \item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu m/V\).
 The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA
 \item A lightly damped resonance at \(95\,\text{Hz}\)
-\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which some resonances appear
+\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
 \end{itemize}
 
 \begin{figure}[htbp]
@@ -544,7 +459,7 @@ The transfer function from the ``damped'' plant input \(u\prime\) to the encoder
 \caption{\label{fig:test_apa_iff_schematic}Figure caption}
 \end{figure}
 
-The identified dynamics are then fitted by second order transfer functions using the ``Vector Fitting'' toolbox \cite{gustavsen99_ration_approx_frequen_domain_respon}.
+The identified dynamics are then fitted by second order transfer functions.
 The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure \ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
 It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
 
@@ -567,14 +482,13 @@ The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_loc
 \includegraphics[scale=1]{figs/test_apa_iff_root_locus.png}
 \caption{\label{fig:test_apa_iff_root_locus}Root Locus of the APA300ML with Integral Force Feedback - Comparison between the computed root locus from the plant model (black line) and the root locus estimated from the damped plant pole identification (colorful crosses)}
 \end{figure}
-\section{Conclusion}
-
 \begin{important}
 So far, all the measured FRF are showing the dynamical behavior that was expected.
 \end{important}
-\chapter{Test Bench APA300ML - Simscape Model}
+\chapter{2 Degrees of Freedom Model}
-\label{sec:test_apa_simscape}
+\label{sec:test_apa_model_2dof}
-In this section, a simscape model (Figure \ref{fig:model_bench_apa}) of the measurement bench is used to compare the model of the APA with the measured FRF.
+
+In this section, a simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to compare the model of the APA with the measured frequency response functions.
 
 After the transfer functions are extracted from the model (Section \ref{sec:simscape_bench_apa_first_id}), the comparison of the obtained dynamics with the measured FRF will permit to:
 \begin{enumerate}
@@ -586,176 +500,199 @@ After the transfer functions are extracted from the model (Section \ref{sec:sims
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.5\linewidth]{figs/model_bench_apa.png}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_apa_bench_model.png}
-\caption{\label{fig:model_bench_apa}Screenshot of the Simscape model}
+\caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model}
 \end{figure}
-\section{First Identification}
+\section{Two Degrees of Freedom APA Model}
-\label{sec:simscape_bench_apa_first_id}
+\label{ssec:test_apa_2dof_model}
 
-The APA is first initialized with default parameters:
+The APA model shown in Figure \ref{fig:test_apa_2dof_model} is adapted from \cite{souleille18_concep_activ_mount_space_applic}.
-The transfer function from excitation voltage \(V_a\) (before the amplification of \(20\) due to the PD200 amplifier) to:
-\begin{enumerate}
-\item the sensor stack voltage \(V_s\)
-\item the measured displacement by the encoder \(d_e\)
-\end{enumerate}
 
-The obtain dynamics are shown in Figure \ref{fig:apa_model_bench_bode_vs} and \ref{fig:apa_model_bench_bode_dl_z}.
+It can be decomposed into three components:
-It can be seen that:
 \begin{itemize}
-\item the shape of these bode plots are very similar to the one measured in Section \ref{sec:dynamical_meas_apa} expect from a change in gain and exact location of poles and zeros
+\item the shell whose axial properties are represented by \(k_1\) and \(c_1\)
-\item there is a sign error for the transfer function from \(V_a\) to \(V_s\).
+\item the actuator stacks whose contribution in the axial stiffness is represented by \(k_a\) and \(c_a\).
-This will be corrected by taking a negative ``sensor gain''.
+A force source \(\tau\) represents the axial force induced by the force sensor stacks.
-\item the low frequency zero of the transfer function from \(V_a\) to \(V_s\) is minimum phase as expected.
+The gain \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(\tau\)
-The measured FRF are showing non-minimum phase zero, but it is most likely due to measurements artifacts.
+\item the actuator stacks whose contribution in the axial stiffness is represented by \(k_e\) and \(c_e\).
+A ``strain sensor'' \(d_L\), and a gain \(g_s\) (in \(V/m\)) that converts this strain into a generated voltage
+\end{itemize}
+
+Such simple model has some limitations:
+\begin{itemize}
+\item it only represents the axial characteristics of the APA (infinitely rigid in other directions)
+\item some physical insights are lost such as the amplification factor, the real stress and strain on the piezoelectric stacks
+\item it is fully linear and therefore the creep and hysteresis of the piezoelectric stacks are not modelled
 \end{itemize}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/apa_model_bench_bode_vs.png}
+\includegraphics[scale=1]{figs/test_apa_2dof_model.png}
-\caption{\label{fig:apa_model_bench_bode_vs}Bode plot of the transfer function from \(V_a\) to \(V_s\)}
+\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees of freedom model of the APA300ML}
 \end{figure}
+\section{Tuning of the APA model}
+\label{ssec:test_apa_2dof_model_tuning}
+
+9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/apa_model_bench_bode_dl_z.png}
+\includegraphics[scale=1]{figs/test_apa_2dof_model_simscape.png}
-\caption{\label{fig:apa_model_bench_bode_dl_z}Bode plot of the transfer function from \(V_a\) to \(d_L\) and to \(z\)}
+\caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees of freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
 \end{figure}
-\section{Identify Sensor/Actuator constants and compare with measured FRF}
-\label{sec:simscape_bench_apa_id_constants}
-\subsection{How to identify these constants?}
-\paragraph{Piezoelectric Actuator Constant}
 
-Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\).
+First, the mass supported by the APA300ML can simply be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
-\begin{equation}
+Both methods leads to an estimated mass of \(5.7\,\text{kg}\).
-  d = g_{d/V_a} \cdot V_a
+
+Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,N/\mu m\) in Section \ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure \ref{fig:test_apa_frf_force}.
+Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(20\,Ns/m\).
+
+Then, it is reasonable to make the assumption that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not fully correct as it was shown in Section \ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}.
+Therefore, we have \(k_e = 2 k_a\) and \(c_e = 2 c_a\) as the actuator stack is composed of two stacks in series.
+In that case, the total stiffness of the APA model is described by \eqref{eq:test_apa_2dof_stiffness}.
+
+\begin{equation}\label{eq:test_apa_2dof_stiffness}
+k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a
 \end{equation}
 
-Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force \(F_a\) to the induced displacement \(d\):
+Knowing from \eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{\text{tot}} = 2\,N/\mu m\), we get from \eqref{eq:test_apa_2dof_stiffness} that \(k_a = 2.5\,N/\mu m\) and \(k_e = 5\,N/\mu m\).
-\begin{equation}
+
-  d = g_{d/F_a} \cdot F_a
+\begin{equation}\label{eq:test_apa_tot_stiffness}
+\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,N/\mu m \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s
 \end{equation}
 
-From the two gains, it is then easy to determine \(g_a\):
+Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance.
-\begin{equation} \label{eq:actuator_constant_formula}
+\(c_a = 100\,Ns/m\) and \(c_e = 200\,Ns/m\) are obtained.
-  \boxed{g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}}}
-\end{equation}
-\paragraph{Piezoelectric Sensor Constant}
 
-Similarly, it is easy to determine the gain from the excitation voltage \(V_a\) to the voltage generated by the sensor stack \(V_s\):
+Finally, the two gains \(g_s\) and \(g_a\) can be used to match the gain of the identified transfer functions.
-\begin{equation}
-  V_s = g_{V_s/V_a} V_a
-\end{equation}
 
-Note here that there is an high pass filter formed by the piezoelectric capacitor and parallel resistor.
+The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
 
-The gain can be computed from the dynamical identification and taking the gain at the wanted frequency (above the first resonance).
+\begin{table}[htbp]
+\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
+\centering
+\begin{tabularx}{0.3\linewidth}{cc}
+\toprule
+\textbf{Parameter} & \textbf{Value}\\
+\midrule
+\(m\) & \(5.7\,\text{kg}\)\\
+\(k_1\) & \(0.38\,N/\mu m\)\\
+\(k_e\) & \(5.0\, N/\mu m\)\\
+\(k_a\) & \(2.5\,N/\mu m\)\\
+\(c_1\) & \(20\,Ns/m\)\\
+\(c_e\) & \(200\,Ns/m\)\\
+\(c_a\) & \(100\,Ns/m\)\\
+\(g_a\) & \(-2.58\,N/V\)\\
+\(g_s\) & \(4.6\,V/\mu m\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Obtained Dynamics}
+\label{ssec:test_apa_2dof_model_result}
 
-Using the simscape model, compute the gain at the same frequency from the actuator force \(F_a\) to the strain of the sensor stack \(dl\):
+The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
-\begin{equation}
+It is compared with the experimental data in Figure \ref{fig:test_apa_2dof_comp_frf}.
-  dl = g_{dl/F_a} F_a
-\end{equation}
 
-Then, the ``sensor'' constant is:
+A good match can be observed between the model and the experimental data, both for the encoder and for the force sensor.
-\begin{equation} \label{eq:sensor_constant_formula}
+This indicates that this model represents well the axial dynamics of the APA300ML.
-  \boxed{g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}}}
-\end{equation}
-\subsection{Identification Data}
-Let's load the measured FRF from the DAC voltage to the measured encoder and to the sensor stack voltage.
-\subsection{2DoF APA}
-\paragraph{2DoF APA}
-Let's initialize the APA as a 2DoF model with unity sensor and actuator gains.
-\paragraph{Identification without actuator or sensor constants}
-The transfer function from \(V_a\) to \(V_s\), \(d_e\) and \(d_a\) is identified.
-\paragraph{Actuator Constant}
-Then, the actuator constant can be computed as shown in Eq. \eqref{eq:actuator_constant_formula} by dividing the measured DC gain of the transfer function from \(V_a\) to \(d_e\) by the estimated DC gain of the transfer function from \(V_a\) (in truth the actuator force called \(F_a\)) to \(d_e\) using the Simscape model.
-\begin{verbatim}
-ga = -32.2 [N/V]
-\end{verbatim}
-\paragraph{Sensor Constant}
-Similarly, the sensor constant can be estimated using Eq. \eqref{eq:sensor_constant_formula}.
-
-\begin{verbatim}
-gs = 0.088 [V/m]
-\end{verbatim}
-\paragraph{Comparison}
-Let's now initialize the APA with identified sensor and actuator constant:
-And identify the dynamics with included constants.
-The transfer functions from \(V_a\) to \(d_e\) are compared in Figure \ref{fig:apa_act_constant_comp} and the one from \(V_a\) to \(V_s\) are compared in Figure \ref{fig:apa_sens_constant_comp}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/apa_act_constant_comp.png}
+\includegraphics[scale=1]{figs/test_apa_2dof_comp_frf.png}
-\caption{\label{fig:apa_act_constant_comp}Comparison of the experimental data and Simscape model (\(V_a\) to \(d_e\))}
+\caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured FRF and the optimized 2DoF model of the APA300ML}
 \end{figure}
+\chapter{APA300ML - Super Element}
+\label{sec:test_apa_model_flexible}
+
+In this section, a \emph{super element} of the Amplified Piezoelectric Actuator ``APA300ML'' is extracted using a Finite Element Software.
+It is then imported in Simscape (using the stiffness and mass matrices) and it is included in the same model that was used in \ref{sec:test_apa_model_2dof}.
+
+This procedure is illustrated in Figure \ref{fig:test_apa_super_element_simscape}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/apa_sens_constant_comp.png}
+\includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_simscape.png}
-\caption{\label{fig:apa_sens_constant_comp}Comparison of the experimental data and Simscape model (\(V_a\) to \(V_s\))}
+\caption{\label{fig:test_apa_super_element_simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
 \end{figure}
+\section{Extraction of the super-element}
 
-\begin{figure}[htbp]
+\begin{itemize}
+\item Explain how the ``remote points'' are chosen
+\item Show some parts of the mass and stiffness matrices?
+\item Say which materials were used?
+\item Maybe this was already explain earlier in the manuscript
+\end{itemize}
+\section{Identification of the Actuator and Sensor constants}
+\label{ssec:test_apa_flexible_ga_gs}
+
+Once the APA300ML \emph{super element} is included in the Simscape model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be identified.
+The gains \(g_a\) and \(g_s\) can then be tuned such that the gain of the transfer functions are matching the identified ones.
+
+By doing so, \(g_s = 4.9\,V/\mu m\) and \(g_a = 23.2\,N/V\) are obtained.
+
+To make sure these ``gains'' are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
+
+From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:test_apa_piezo_strain_to_voltage} and from \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:test_apa_piezo_voltage_to_force}.
+
+\begin{subequations}
+\begin{align}
+  V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\
+  F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force}
+\end{align}
+\end{subequations}
+
+Parameters used in equations \eqref{eq:test_apa_piezo_strain_to_voltage} and \eqref{eq:test_apa_piezo_voltage_to_force} are described in Table \ref{tab:test_apa_piezo_properties}.
+
+Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
+However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
+The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}.
+
+From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained which are very close to the identified constants using the experimentally identified transfer functions.
+
+\begin{table}[htbp]
+\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators ``gains''}
 \centering
-\includegraphics[scale=1]{figs/apa_comp_model_frf.png}
+\begin{tabularx}{1\linewidth}{ccX}
-\label{fig:apa_comp_model_frf}
+\toprule
-\end{figure}
+\textbf{Parameter} & \textbf{Value} & \textbf{Description}\\
+\midrule
+\(d_{33}\) & \(680 \cdot 10^{-12}\,m/V\) & Piezoelectric constant\\
+\(\epsilon^{T}\) & \(4.0 \cdot 10^{-8}\,F/m\) & Permittivity under constant stress\\
+\(s^{D}\) & \(21 \cdot 10^{-12}\,m^2/N\) & Elastic compliance understand constant electric displacement\\
+\(c^{E}\) & \(48 \cdot 10^{9}\,N/m^2\) & Young's modulus of elasticity\\
+\(L\) & \(20\,mm\) per stack & Length of the stack\\
+\(A\) & \(10^{-4}\,m^2\) & Area of the piezoelectric stack\\
+\(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Comparison of the obtained dynamics}
 
+The obtained dynamics using the \emph{super element} with the tuned ``sensor gain'' and ``actuator gain'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}.
 
-\begin{important}
+A good match between the model and the experimental results is observed.
-The ``actuator constant'' and ``sensor constant'' can indeed be identified using this test bench.
+\begin{itemize}
-After identifying these constants, the 2DoF model shows good agreement with the measured dynamics.
+\item the \emph{super element}
-\end{important}
+\end{itemize}
-\subsection{Flexible APA}
-In this section, the sensor and actuator ``constants'' are also estimated for the flexible model of the APA.
-\paragraph{Flexible APA}
-The Simscape APA model is initialized as a flexible one with unity ``constants''.
-\paragraph{Identification without actuator or sensor constants}
-The dynamics from \(V_a\) to \(V_s\), \(d_e\) and \(d_a\) is identified.
-\paragraph{Actuator Constant}
-Then, the actuator constant can be computed as shown in Eq. \eqref{eq:actuator_constant_formula}:
-\begin{verbatim}
-ga = 23.5 [N/V]
-\end{verbatim}
-\paragraph{Sensor Constant}
-\begin{verbatim}
-gs = -4839841.756 [V/m]
-\end{verbatim}
-\paragraph{Comparison}
-Let's now initialize the flexible APA with identified sensor and actuator constant:
-And identify the dynamics with included constants.
-The obtained dynamics is compared with the measured one in Figures \ref{fig:apa_act_constant_comp_flex} and \ref{fig:apa_sens_constant_comp_flex}.
 
-\begin{figure}[htbp]
+This model represents fairly
-\centering
-\includegraphics[scale=1]{figs/apa_act_constant_comp_flex.png}
-\caption{\label{fig:apa_act_constant_comp_flex}Comparison of the experimental data and Simscape model (\(u\) to \(d\mathcal{L}_m\))}
-\end{figure}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/apa_sens_constant_comp_flex.png}
-\caption{\label{fig:apa_sens_constant_comp_flex}Comparison of the experimental data and Simscape model (\(u\) to \(\tau_m\))}
-\end{figure}
-
-\begin{important}
 The flexible model is a bit ``soft'' as compared with the experimental results.
-\end{important}
-\section{Optimize 2-DoF model to fit the experimental Data}
-\label{sec:simscape_bench_apa_tune_2dof_model}
-The parameters of the 2DoF model presented in Section \ref{sec:apa_2dof_model} are now optimize such that the model best matches the measured FRF.
 
-After optimization, the following parameters are used:
+This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators.
-The dynamics is identified using the Simscape model and compared with the measured FRF in Figure \ref{fig:comp_apa_plant_after_opt}.
+
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/comp_apa_plant_after_opt.png}
+\includegraphics[scale=1]{figs/test_apa_super_element_comp_frf.png}
-\caption{\label{fig:comp_apa_plant_after_opt}Comparison of the measured FRF and the optimized model}
+\caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured FRF and the ``Flexible'' model of the APA300ML}
 \end{figure}
-
-\begin{important}
-The tuned 2DoF is very well representing the (axial) dynamics of the APA.
-\end{important}
 \chapter{Conclusion}
 \label{sec:test_apa_conclusion}
+
+\begin{itemize}
+\item Compare 2DoF and FEM models (usefulness of the two)
+\item Good match between all the APA: will simplify the modeling and control of the nano-hexapod
+\item No advantage of the FEM model here (as only uniaxial behavior is checked), but may be useful later
+\end{itemize}
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}