Correct typo

matlab/STEPS/new_APA300ML_K.CSV

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matlab/STEPS/new_APA300ML_out_nodes_3D.txt

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

matlab/src/initializeAPA.m

@@ -53,26 +53,22 @@ if args.Ga == 0
     switch args.type
       case '2dof'
         actuator.Ga = -2.5796;
-      case 'flexible frame'
-        actuator.Ga = 1; % TODO
       case 'flexible'
         actuator.Ga = 23.2;
     end
 else
-    actuator.Ga = args.Ga; % Actuator gain [N/V]
+    actuator.Ga = args.Ga; % Actuator sensitivity [N/V]
 end
 
 if args.Gs == 0
     switch args.type
       case '2dof'
         actuator.Gs = 466664;
-      case 'flexible frame'
-        actuator.Gs = 1; % TODO
       case 'flexible'
         actuator.Gs = -4898341;
     end
 else
-    actuator.Gs = args.Gs; % Sensor gain [V/m]
+    actuator.Gs = args.Gs; % Sensor sensitivity [V/m]
 end
 
 actuator.k  = args.k;  % [N/m]

matlab/test_apa_1_basic_meas.m

@@ -12,18 +12,12 @@ addpath('./mat/'); % Path for data
 colors = colororder;
 
 % Geometrical Measurements
-% <<sec:test_apa_geometrical_measurements>>
+% <<ssec:test_apa_geometrical_measurements>>
 
-% To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with very good flatness.
+% To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness.
-
+% As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:3] is used to measure the height of four points on each of the APA300ML interfaces.
-% As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:3] is used to measure the height of 4 points on each of the APA300ML interfaces.
+% From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting[fn:4] a plane through all the points.
-
+% The measured flatness values, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications.
-% From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting[fn:4] a plane through all the points.
-
-% #+name: fig:test_apa_flatness_setup
-% #+attr_latex: :width 0.4\linewidth
-% #+caption: Measurement setup for flatness estimation of the two mechanical interfaces
-% [[file:figs/test_apa_flatness_setup.png]]
 
 
 %% Measured height for all the APA at the 8 locations
@@ -62,46 +56,41 @@ for i = 1:7
 end
 
 % Stroke and Hysteresis Measurement
-% <<sec:test_apa_stroke_measurements>>
+% <<ssec:test_apa_stroke_measurements>>
 
-% The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
+% To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
-% To do so, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
 
-% Then, the voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC and a voltage amplifier.
+% The voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:12] and a voltage amplifier[fn:13].
-% Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_bench, left).
+% Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_voltage).
 
 % #+name: fig:test_apa_stroke_bench
-% #+caption: Bench to measured the APA stroke
+% #+caption: Bench to measure the APA stroke
-% #+attr_latex: :width 0.9\linewidth
+% #+attr_latex: :width 0.7\linewidth
 % [[file:figs/test_apa_stroke_bench.jpg]]
 
-% The measured APA displacement is shown as a function of the applied voltage in Figure ref:fig:test_apa_stroke_result, right.
+% The measured APA displacement is shown as a function of the applied voltage in Figure ref:fig:test_apa_stroke_hysteresis.
-
 % Typical hysteresis curves for piezoelectric stack actuators can be observed.
-% The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks, which is enough for the current application.
+% The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks.
 % This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu m$, therefore $\approx 200\,\mu m$ if only two stacks are used).
+% For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu m$.
 
-% It is clear from Figure ref:fig:test_apa_stroke_result that "APA 3" has an issue compared to the other units.
+% It is clear from Figure ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units.
-% This confirms the abnormal electrical measurements made in Section ref:sec:test_apa_electrical_measurements.
+% This confirms the abnormal electrical measurements made in Section ref:ssec:test_apa_electrical_measurements.
-% This unit was send sent back to Cedrat and a new one was shipped back.
+% This unit was sent sent back to Cedrat, and a new one was shipped back.
-% From now on, only the six APA that behave as expected will be used.
+% From now on, only the six remaining amplified piezoelectric actuators that behave as expected will be used.
 
 
 %% Load the measured strokes
 load('meas_apa_stroke.mat', 'apa300ml_2s')
 
-%% Results of the measured APA stroke
+%% Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML
 figure;
-tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
-
-% Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML
-ax1 = nexttile();
 plot(apa300ml_2s{1}.t - apa300ml_2s{1}.t(1), 20*apa300ml_2s{1}.V, 'k-')
 xlabel('Time [s]'); ylabel('Voltage [V]')
 ylim([-20, 160])
 
-% Measured displacement as a function of the applied voltage
+%% Measured displacement as a function of the applied voltage
-ax2 = nexttile();
+figure;
 hold on;
 for i = 1:7
     plot(20*apa300ml_2s{i}.V, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
@@ -112,31 +101,42 @@ legend('location', 'southwest', 'FontSize', 8)
 xlim([-20, 150]); ylim([-250, 0]);
 
 % Flexible Mode Measurement
-% SCHEDULED: <2024-03-27 Wed>
+% <<ssec:test_apa_spurious_resonances>>
-% <<sec:test_apa_spurious_resonances>>
 
 % 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 at one end (see Figure ref:fig:test_apa_meas_setup_modes).
+% A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:7] is used to excite the flexible modes.
+% Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated.
 
-% To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_torsion).
+% 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.
-% 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.
-% 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
+% #+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model
-% #+attr_latex: :width 0.9\linewidth
+% #+attr_latex: :options [htbp]
-% [[file:figs/test_apa_mode_shapes.png]]
+% #+begin_figure
-
+% #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode (268Hz)}
-% #+name: fig:test_apa_meas_setup_torsion
+% #+attr_latex: :options {0.36\textwidth}
-% #+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the $Z$ torsion is measured.
+% #+begin_subfigure
-% #+attr_latex: :width 0.6\linewidth
+% #+attr_latex: :height 4.3cm
-% [[file:figs/test_apa_meas_setup_torsion.jpg]]
+% [[file:figs/test_apa_mode_shapes_1.png]]
-
+% #+end_subfigure
-% 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).
+% #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode (399Hz)}
+% #+attr_latex: :options {0.28\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :height 4.3cm
+% [[file:figs/test_apa_mode_shapes_2.png]]
+% #+end_subfigure
+% #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode (706Hz)}
+% #+attr_latex: :options {0.36\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :height 4.3cm
+% [[file:figs/test_apa_mode_shapes_3.png]]
+% #+end_subfigure
+% #+end_figure
 
 % #+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).
+% #+caption: Experimental setup to measure the flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
+% #+attr_latex: :options [htbp]
 % #+begin_figure
 % #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
 % #+attr_latex: :options {0.49\textwidth}
@@ -173,37 +173,24 @@ bending_Y = load('apa300ml_bending_Y_top.mat');
 % 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);
-
-% Load Data
-torsion = load('apa300ml_torsion_left.mat');
-
-% Compute transfer function
-[G_torsion, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts);
 
 
-
+% The measured frequency response functions computed from the experimental setups of figures ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure ref:fig:test_apa_meas_freq_compare.
-% The three measured frequency response functions are shown in Figure ref:fig:test_apa_meas_freq_compare.
+% The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$.
-% - a clear $x$ bending mode at $280\,\text{Hz}$
+% These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_mode_shapes).
-% - a clear $y$ bending mode at $412\,\text{Hz}$
+% This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
-% - for the $z$ torsion test, the $y$ bending mode is also excited and observed, and we may see a mode at $800\,\text{Hz}$
+% This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
+% Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
 
 
 figure;
 hold on;
 plot(f, abs(G_bending_X), 'DisplayName', '$X$ bending');
 plot(f, abs(G_bending_Y), 'DisplayName', '$Y$ bending');
-plot(f, abs(G_torsion),   'DisplayName', '$Z$ torsion');
 text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
 text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
-text(800, 6e-4,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
 hold off;
 set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Amplitude');
-xlim([50, 2e3]); ylim([5e-5, 2e-1]);
+xlim([100, 1e3]); ylim([5e-5, 2e-1]);
 legend('location', 'northeast', 'FontSize', 8)

matlab/test_apa_2_dynamics.m

@@ -14,11 +14,10 @@ colors = colororder;
 % Hysteresis
 % <<ssec:test_apa_hysteresis>>
 
-% As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
+% Because the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
-
+% A quasi static[fn:9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC.
-% A quasi static sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$), and an amplitude varying from $4\,V$ up to $80\,V$.
+% For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure ref:fig:test_apa_meas_hysteresis.
-
+% This is the typical behavior expected from a PZT stack actuator, where the hysteresis increases as a function of the applied voltage amplitude [[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]].
-% For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage..
 
 
 %% Load measured data - hysteresis
@@ -29,12 +28,6 @@ apa_hyst.t = apa_hyst.t - apa_hyst.t(1);
 
 ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes
 
-
-
-% The measured displacements as a function of the output voltages are shown in Figure ref:fig:test_apa_meas_hysteresis.
-% It is interesting to see that the hysteresis is increasing with the excitation amplitude.
-
-
 %% Measured displacement as a function of the output voltage
 figure;
 
@@ -53,9 +46,8 @@ ylim([-120, 120]);
 % Axial stiffness
 % <<ssec:test_apa_stiffness>>
 
-% In order to estimate the stiffness of the APA, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $d_e$ is measured using the encoder.
+% To estimate the stiffness of the APA, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $\Delta d_e$ is measured using the encoder.
-
+% The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity.
-% The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness.
 
 % \begin{equation} \label{eq:test_apa_stiffness}
 %   k_{\text{apa}} = \frac{m_a g}{\Delta d_e}
@@ -76,9 +68,12 @@ added_mass = 6.4; % Added mass [kg]
 
 
 % The measured displacement $d_e$ as a function of time is shown in Figure ref:fig:test_apa_meas_stiffness_time.
-% It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
+% It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep), and that the displacement does not return to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
 % These two effects induce some uncertainties in the measured stiffness.
 
+% The stiffnesses are computed for all APAs from the two displacements $d_1$ and $d_2$ (see Figure ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$.
+% These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$.
+
 
 %% Plot the deflection at a function of time
 figure;
@@ -105,10 +100,20 @@ xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]');
 
 
 
+% #+attr_latex: :options [b]{0.57\linewidth}
+% #+begin_minipage
+% #+name: fig:test_apa_meas_stiffness_time
+% #+caption: Measured displacement when adding (at $t \approx 3\,s$) and removing (at $t \approx 13\,s$) the mass
+% #+attr_latex: :width 0.9\linewidth :float nil
+% [[file:figs/test_apa_meas_stiffness_time.png]]
+% #+end_minipage
+% \hfill
+% #+attr_latex: :options [b]{0.37\linewidth}
+% #+begin_minipage
 % #+name: tab:test_apa_measured_stiffnesses
-% #+caption: Measured stiffnesses (in $N/\mu m$)
+% #+caption: Measured axial stiffnesses (in $N/\mu m$)
-% #+attr_latex: :environment tabularx :width 0.2\linewidth :align ccc
+% #+attr_latex: :environment tabularx :width 0.6\linewidth :align Xcc
-% #+attr_latex: :center t :booktabs t :float t
+% #+attr_latex: :center t :booktabs t :float nil
 % #+RESULTS:
 % | APA | $k_1$ | $k_2$ |
 % |-----+-------+-------|
@@ -118,6 +123,7 @@ xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]');
 % |   5 |   1.7 |  1.93 |
 % |   6 |   1.7 |  1.92 |
 % |   8 |  1.73 |  1.98 |
+% #+end_minipage
 
 % 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 $m_{\text{sus}} = 5.7\,\text{kg}$.
 
@@ -125,16 +131,16 @@ xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]');
 % \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
 % \end{equation}
 
-% The obtain stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and by the "static deflection" method.
+% The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and using the "static deflection" method.
 
 
-% However, changes in the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]].
+% It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]].
 
-% To estimate this effect, the stiffness of the APA if measured using the "static deflection" method in two cases:
+% To estimate this effect for the APA300ML, its stiffness is estimated using the "static deflection" method in two cases:
 % - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC)
-% - $k_{\text{sc}}$: piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance)
+% - $k_{\text{sc}}$: piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance)
 
-% The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ and the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
+% The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
 
 
 %% Load Data
@@ -152,8 +158,6 @@ apa_k_sc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass
 % Dynamics
 % <<ssec:test_apa_meas_dynamics>>
 
-% In this section, the dynamics of the system from the excitation voltage $u$ to encoder measured displacement $d_e$  and to the force sensor voltage $V_s$ is identified.
-
 
 %% Identification using sweep sine (low frequency)
 load('frf_data_sweep.mat');
@@ -195,13 +199,30 @@ save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
 
 
 
-% The obtained transfer functions for the 6 APA between the excitation voltage $u$ and the encoder displacement $d_e$ are shown in Figure ref:fig:test_apa_frf_encoder.
+% In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified.
-% The obtained transfer functions are close to a mass-spring-damper system.
+
-% The following can be observed:
+% First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_encoder.
+% The obtained frequency response functions are similar to those of a (second order) mass-spring-damper system with:
 % - 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
+%   The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which 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. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
+% - A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the APA support.
+%   The flexible modes studied in section ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator.
+
+% The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_force.
+
+% A lightly damped resonance (pole) is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance (zero) at $41\,\text{Hz}$.
+% No additional resonances are present up to at least $2\,\text{kHz}$ indicating that Integral Force Feedback can be applied without stability issues from high-frequency flexible modes.
+% The zero at $41\,\text{Hz}$ seems to be non-minimum phase (the phase /decreases/ by 180 degrees whereas it should have /increased/ by 180 degrees for a minimum phase zero).
+% This is investigated in Section ref:ssec:test_apa_non_minimum_phase.
+
+% As illustrated by the Root Locus plot, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases.
+% The significance of this behavior varies with the type of sensor used, as explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]].
+% Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
+% Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell.
+% From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$ (which is close to what is found using a finite element model).
+
+% All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure ref:fig:test_apa_frf_encoder and at the force sensor in Figure ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
 
 
 %% Plot the FRF from u to de
@@ -218,7 +239,7 @@ hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
 ylim([1e-8, 1e-3]);
 
 ax2 = nexttile;
@@ -235,28 +256,6 @@ yticks(-360:90:360);
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
 
-
-
-% #+name: fig:test_apa_frf_encoder
-% #+caption: Estimated Frequency Response Function from generated voltage $u$ to the encoder displacement $d_e$ for the 6 APA300ML
-% #+RESULTS:
-% [[file:figs/test_apa_frf_encoder.png]]
-
-% The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ is also identified and shown in Figure ref:fig:test_apa_frf_force.
-
-% A lightly damped resonance is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance at $41\,\text{Hz}$.
-% No additional resonances is present up to at least $2\,\text{kHz}$ indicating at Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
-
-% As illustrated by the Root Locus, the poles of the closed-loop system converges to the zeros of the open-loop plant.
-% Suppose that a controller with a very high gain is implemented such that the voltage $V_s$ across the sensor stack is zero.
-% In that case, because of the very high controller gain, no stress and strain is present on the sensor stack (and on the actuator stacks are well, as they are both in series).
-% Such closed-loop system would therefore virtually corresponds to a system for which the piezoelectric stacks have been removed and just the mechanical shell is kept.
-% From this analysis, the axial stiffness of the shell can be estimated to be $k_{\text{shell}} = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$.
-% # TODO - Compare with FEM result
-
-% Such reasoning can lead to very interesting insight into the system just from an open-loop identification.
-
-
 %% Plot the FRF from u to Vs
 figure;
 tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
@@ -288,17 +287,80 @@ yticks(-360:90:360); ylim([-180, 180]);
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
 
+% Non Minimum Phase Zero?
+% <<ssec:test_apa_non_minimum_phase>>
+
+% It was surprising to observe a non-minimum phase zero on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force).
+% It was initially thought that this non-minimum phase behavior was an artifact arising from the measurement.
+% A longer measurement was performed using different excitation signals (noise, slow sine sweep, etc.) to determine if the phase behavior of the zero changes (Figure ref:fig:test_apa_non_minimum_phase).
+% The coherence (Figure ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior.
+
+% Such non-minimum phase zero when using load cells has also been observed on other mechanical systems [[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]].
+% It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear.
+
+% However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice.
+
+
+%% Long measurement
+long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'u', 'Vs');
+
+% Long window for fine frequency axis
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(10/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+% Transfer function estimation
+[frf_noise, f] = tfestimate(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+[coh_noise, ~] = mscohere(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+nexttile();
+hold on;
+plot(f, coh_noise, '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+hold off;
+xlim([38, 45]);
+ylim([0, 1]);
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 0]);
+
+linkaxes([ax1,ax2],'x');
+xlim([38, 45]);
+
 % Effect of the resistor on the IFF Plant
 % <<ssec:test_apa_resistance_sensor_stack>>
 
-% A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the stack.
+% A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu F$).
 
-% As explain before, this is done for two reasons:
+% As explained before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain.
-% 1. Limit the voltage offset due to the input bias current of the ADC
-% 2. Limit the low frequency gain
 
-% The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor have been measured and are compared in Figure ref:fig:test_apa_effect_resistance.
+% The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor were measured and compared in Figure ref:fig:test_apa_effect_resistance.
-% It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of $\approx 0.35\,\text{Hz}$.
+% It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$.
 
 
 %% Load the data
@@ -320,41 +382,44 @@ R = 80.6e3; % Parallel Resistor [Ohm]
 
 f0 = 1/(2*pi*R*C); % Crossover frequency of RC HPF [Hz]
 
-G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0);
+G_hpf = 0.6*(s/(2*pi*f0))/(1 + s/(2*pi*f0));
 
 %% Compare the HPF model and the measured FRF
 figure;
-tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 
-ax1 = nexttile([2,1]);
+ax1 = nexttile();
 hold on;
 plot(f, abs(frf_wo_k), 'DisplayName', 'Without $R$');
 plot(f, abs(frf_wi_k), 'DisplayName', 'With $R$');
+plot(f, abs(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', 'RC model');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-ylim([1e-1, 1e0]);
+ylim([2e-1, 1e0]);
-legend('location', 'southeast')
+yticks([0.2, 0.5, 1]);
+legend('location', 'southeast', 'FontSize', 8);
 
 ax2 = nexttile;
 hold on;
 plot(f, 180/pi*angle(frf_wo_k));
 plot(f, 180/pi*angle(frf_wi_k));
+plot(f, 180/pi*angle(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', 'RC');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
-yticks(-360:45:360); ylim([-45, 90]);
+yticks(-360:45:360); ylim([-5, 60]);
+yticks([0, 15, 30, 45, 60]);
 
 linkaxes([ax1,ax2],'x');
 xlim([0.2, 8]);
+xticks([0.2, 0.5, 1, 2, 5]);
 
 % Integral Force Feedback
 % <<ssec:test_apa_iff_locus>>
 
-% This test bench can also be used to estimate the damping added by the implementation of an Integral Force Feedback strategy.
-
 
 %% Load identification Data
 data = load("2023-03-17_11-28_iff_plant.mat");
@@ -370,15 +435,14 @@ Noverlap = floor(Nfft/2);
 
 
 
-% First, the transfer function eqref:eq:test_apa_iff_manual_fit is manually tuned to match the identified dynamics from generated voltage $u$ to the measured sensor stack voltage $V_s$ in Section ref:ssec:test_apa_meas_dynamics.
+% To implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation eqref:eq:test_apa_iff_manual_fit.
-
+% The parameters were manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$.
-% The obtained parameter values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$.
 
 % \begin{equation} \label{eq:test_apa_iff_manual_fit}
 % G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s}
 % \end{equation}
 
-% The comparison between the identified plant and the manually tuned transfer function is done in Figure ref:fig:test_apa_iff_plant_comp_manual_fit.
+% A comparison between the identified plant and the manually tuned transfer function is shown in Figure ref:fig:test_apa_iff_plant_comp_manual_fit.
 
 
 %% Basic manually tuned model
@@ -420,15 +484,15 @@ xlim([0.2, 1e3]);
 
 
 % #+name: fig:test_apa_iff_plant_comp_manual_fit
-% #+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase)
+% #+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.
 % #+RESULTS:
 % [[file:figs/test_apa_iff_plant_comp_manual_fit.png]]
 
 % The implemented Integral Force Feedback Controller transfer function is shown in equation eqref:eq:test_apa_Kiff_formula.
-% It contains an high pass filter (cut-off frequency of $2\,\text{Hz}$) to limit the low frequency gain, a low pass filter to add integral action above $20\,\text{Hz}$, a second low pass filter to add robustness to high frequency resonances and a tunable gain $g$.
+% It contains a high-pass filter (cut-off frequency of $2\,\text{Hz}$) to limit the low-frequency gain, a low-pass filter to add integral action above $20\,\text{Hz}$, a second low-pass filter to add robustness to high-frequency resonances, and a tunable gain $g$.
 
 % \begin{equation} \label{eq:test_apa_Kiff_formula}
-% K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{1 + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000}
+% K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000}
 % \end{equation}
 
 
@@ -443,7 +507,7 @@ K_iff = -10*(1/(s + 2*pi*20)) * ... % LPF: provides integral action above 20Hz
 % The transfer function from the "damped" plant input $u\prime$ to the encoder displacement $d_e$ is identified for several IFF controller gains $g$.
 
 % #+name: fig:test_apa_iff_schematic
-% #+caption: Figure caption
+% #+caption: Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input $u\prime$
 % [[file:figs/test_apa_iff_schematic.png]]
 
 
@@ -460,7 +524,7 @@ Noverlap = floor(Nfft/2);
 [~, f]  = tfestimate(data.data(1).id_plant(1:end), data.data(1).dL(1:end), win, Noverlap, Nfft, 1/Ts);
 
 %% Gains used for analysis are between 1 and 50
-i_kept = [5:10]
+i_kept = [5:10];
 
 %% Identify the damped plant from u' to de for different IFF gains
 G_dL_frf = {zeros(1,length(i_kept))};
@@ -472,9 +536,14 @@ end
 
 
 
-% The identified dynamics are then fitted by second order transfer functions.
+% The identified dynamics were then fitted by second order transfer functions[fn:10].
-% 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 $g$.
+% A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown 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.
+% It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
+
+% The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed in two cases.
+% First using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula.
+% Second using the fitted transfer functions of the damped plants experimentally identified for several controller gains.
+% The two obtained root loci are compared in Figure ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were fitted using only a second-order transfer function.
 
 
 %% Fit the data with 2nd order transfer function using vectfit3
@@ -519,24 +588,10 @@ for i = 1:length(i_kept)
 end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Amplitude $d_L/V_a$ [m/V]');
+xlabel('Frequency [Hz]'); ylabel('Amplitude $d_e/u^\prime$ [m/V]');
 xlim([10, 1e3]);
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
-
-
-% #+name: fig:test_apa_identified_damped_plants
-% #+caption: Identified dynamics (solid lines) and fitted transfer functions (dashed lines) from $u\prime$ to $d_e$ for different IFF gains
-% #+RESULTS:
-% [[file:figs/test_apa_identified_damped_plants.png]]
-
-% The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed:
-% - using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula
-% - from the fitted transfer functions of the damped plants experimentally identified for several controller gains
-
-% The two obtained root loci are compared in Figure ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were only fitted using a second order transfer function.
-
-
 %% Root Locus of the APA300ML with Integral Force Feedback
 % Comparison between the computed root locus from the plant model and the root locus estimated from the damped plant pole identification
 gains = logspace(-1, 3, 1000);
@@ -562,9 +617,9 @@ end
 for i = 1:length(i_kept)
     plot(real(pole(G_dL_id{i})), imag(pole(G_dL_id{i})), 'x', 'color', [colors(i,:), 1], 'DisplayName', sprintf('g = %1.f', data.gains(i_kept(i))));
 end
-ylim([0, 700]);
-xlim([-600,100]);
 xlabel('Real Part')
 ylabel('Imaginary Part')
-axis square
+axis equal
-legend('location', 'northwest');
+ylim([0, 610]);
+xlim([-300,0]);
+legend('location', 'southwest', 'FontSize', 8);

matlab/test_apa_3_model_2dof.m

@@ -27,14 +27,16 @@ 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
+%% Frequency vector for analysis
-% <<ssec:test_apa_2dof_model_tuning>>
+freqs = 5*logspace(0, 3, 1000);
 
-% 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.
+% Tuning of the APA model                                       :ignore:
 
-% #+name: fig:test_apa_2dof_model_simscape
+% 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.
-% #+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]]
+% #+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
@@ -52,8 +54,8 @@ 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
+%% Estimation ot the sensor and actuator sensitivities
-% Initialize the structure with unitary sensor and actuator "gains"
+% Initialize the structure with unitary sensor and actuator "sensitivities"
 n_hexapod = struct();
 n_hexapod.actuator = initializeAPA(...
     'type', '2dof', ...
@@ -74,32 +76,30 @@ 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 Identification Data to estimate the two sensitivities
 load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
 
-% Estimation ot the Actuator Gain
+% Estimation ot the Actuator sensitivity
 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
+% Estimation ot the Sensor sensitivity
 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
+% Obtained Dynamics                                             :ignore:
-% <<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.
+% The dynamics of the two degrees-of-freedom model of the APA300ML are 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.
+% This 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 (Figure ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure ref:fig:test_apa_2dof_comp_frf_force).
-% 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(...
+n_hexapod.actuator = initializeAPA( ...
     'type', '2dof', ...
     'k',  k1,  ...
     'ka', ka,  ...
@@ -117,9 +117,8 @@ 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');
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 
 ax1 = nexttile([2,1]);
 hold on;
@@ -135,7 +134,26 @@ hold off;
 ylim([1e-8, 1e-3]);
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
-ax1b = nexttile([2,1]);
+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]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = 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)
@@ -151,18 +169,6 @@ 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
@@ -173,5 +179,5 @@ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
 yticks(-360:90:360); ylim([-180, 180]);
 
-linkaxes([ax1,ax2,ax1b,ax2b],'x');
+linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);

matlab/test_apa_4_model_flexible.m

@@ -27,12 +27,13 @@ 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
 
+%% Frequency vector for analysis
+freqs = 5*logspace(0, 3, 1000);
+
 % 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.
 
+% 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 extracted.
+% The gains $g_a$ and $g_s$ are then tuned such that the gains of the transfer functions match the identified ones.
 % By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained.
 
 
@@ -61,8 +62,7 @@ gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', '
 
 
 
-
+% To ensure that the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material.
-% 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.
 
@@ -73,16 +73,14 @@ gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', '
 % \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.
+% However, based on the 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.
+% From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, which are close to the constants identified 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"
+% #+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities
 % #+attr_latex: :environment tabularx :width 1\linewidth :align ccX
 % #+attr_latex: :center t :booktabs t
 % | *Parameter*    | *Value*                    | *Description*                                                |
@@ -102,7 +100,7 @@ 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]
+gs_th = d33/(eT*sD*n);   % Sensor Constant [V/m]
 
 %% Estimate "Actuator Constant" - (THP5H)
 d33 = 680e-12;   % Strain constant [m/V]
@@ -113,29 +111,23 @@ 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]
+ga_th = d33*n*ka;   % Actuator Constant [N/V]
 
 % Comparison of the obtained dynamics
-% <<ssec:test_apa_flexible_comp_frf>>
 
-% 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.
+% The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" 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 was observed.
+% It is however surprising that the model is "softer" than the measured system, as finite element models usually overestimate the stiffness (see Section ref:ssec:test_apa_spurious_resonances for possible explanations).
 
-% A good match between the model and the experimental results is observed.
+% Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
-% - 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]
+    'ga',   23.2, ... % Actuator sensitivity [N/V]
-    'gs',   -4.9e6);    % Sensor gain [V/m]
+    'gs',   -4.9e6);    % Sensor sensitivity [V/m]
 
 % Identify with updated constants
 G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts);
@@ -143,9 +135,8 @@ 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');
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 
 ax1 = nexttile([2,1]);
 hold on;
@@ -161,7 +152,26 @@ hold off;
 ylim([1e-8, 1e-3]);
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 
-ax1b = nexttile([2,1]);
+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]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Comparison of the measured FRF and the "Flexible" model of the APA300ML
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = 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)
@@ -177,18 +187,6 @@ 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
@@ -199,5 +197,5 @@ xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
 yticks(-360:90:360); ylim([-180, 180]);
 
-linkaxes([ax1,ax2,ax1b,ax2b],'x');
+linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);

test-bench-apa.org

@@ -1939,7 +1939,7 @@ opts = linearizeOptions;
 opts.SampleTime = 0;
 
 %% Open Simscape Model
-mdl = 'test_apa_Simscape'; % Name of the Simulink File
+mdl = 'test_apa_simscape'; % Name of the Simulink File
 open(mdl); % Open Simscape Model
 #+END_SRC
 
@@ -1952,7 +1952,7 @@ opts = linearizeOptions;
 opts.SampleTime = 0;
 
 %% Open Simscape Model
-mdl = 'test_apa_Simscape'; % Name of the Simulink File
+mdl = 'test_apa_simscape'; % Name of the Simulink File
 open(mdl); % Open Simscape Model
 #+END_SRC