#+end_export
#+latex: \clearpage
* Introduction :ignore:
The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML.
This include:
- Stroke
- Stiffness
- Hysteresis
- Gain from the applied voltage $V_a$ to the generated Force $F_a$
- Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$
- Dynamical behavior
#+name: fig:apa300ML
#+caption: Picture of the APA300ML
#+attr_latex: :width 0.8\linewidth
[[file:figs/apa300ML.png]]
* Model of an Amplified Piezoelectric Actuator and Sensor
Consider a schematic of the Amplified Piezoelectric Actuator in Figure [[fig:apa_model_schematic]].
#+name: fig:apa_model_schematic
#+caption: Amplified Piezoelectric Actuator Schematic
[[file:figs/apa_model_schematic.png]]
A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$:
\begin{equation}
F_a = g_a \cdot V_a
\end{equation}
A change of length $dl$ of the sensor stack will induce a voltage $V_s$:
\begin{equation}
V_s = g_s \cdot dl
\end{equation}
We wish here to experimental measure $g_a$ and $g_s$.
The block-diagram model of the piezoelectric actuator is then as shown in Figure [[fig:apa-model-simscape-schematic]].
#+begin_src latex :file apa-model-simscape-schematic.pdf
\begin{tikzpicture}
\node[block={2.0cm}{2.0cm}, align=center] (model) at (0,0){Simscape\\Model};
\node[block, left=1.0 of model] (ga){$g_a(s)$};
\node[block, right=1.0 of model] (gs){$g_s(s)$};
\draw[<-] (ga.west) -- node[midway, above]{$V_a$} node[midway, below]{$[V]$} ++(-1.0, 0);
\draw[->] (ga.east) --node[midway, above]{$F_a$} node[midway, below]{$[N]$} (model.west);
\draw[->] (model.east) --node[midway, above]{$dl$} node[midway, below]{$[m]$} (gs.west);
\draw[->] (gs.east) -- node[midway, above]{$V_s$} node[midway, below]{$[V]$} ++(1.0, 0);
\end{tikzpicture}
#+end_src
#+name: fig:apa-model-simscape-schematic
#+caption: Model of the APA with Simscape/Simulink
#+RESULTS:
[[file:figs/apa-model-simscape-schematic.png]]
* First Basic Measurements
<>
** Introduction :ignore:
- Section [[sec:geometrical_measurements]]:
- Section [[sec:electrical_measurements]]:
- Section [[sec:stroke_measurements]]:
- Section [[sec:spurious_resonances]]:
** Geometrical Measurements
<>
*** Introduction :ignore:
The received APA are shown in Figure [[fig:received_apa]].
#+name: fig:received_apa
#+caption: Received APA
#+attr_latex: :width 0.6\linewidth
[[file:figs/IMG_20210224_143500.jpg]]
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
#+end_src
*** Measurement Setup
The flatness corresponding to the two interface planes are measured as shown in Figure [[fig:flatness_meas_setup]].
#+name: fig:flatness_meas_setup
#+caption: Measurement Setup
#+attr_latex: :width 0.6\linewidth
[[file:figs/IMG_20210224_143809.jpg]]
*** Measurement Results
The height (Z) measurements at the 8 locations (4 points by plane) are defined below.
#+begin_src matlab
apa1 = 1e-6*[0, -0.5 , 3.5 , 3.5 , 42 , 45.5, 52.5 , 46];
apa2 = 1e-6*[0, -2.5 , -3 , 0 , -1.5 , 1 , -2 , -4];
apa3 = 1e-6*[0, -1.5 , 15 , 17.5 , 6.5 , 6.5 , 21 , 23];
apa4 = 1e-6*[0, 6.5 , 14.5 , 9 , 16 , 22 , 29.5 , 21];
apa5 = 1e-6*[0, -12.5, 16.5 , 28.5 , -43 , -52 , -22.5, -13.5];
apa6 = 1e-6*[0, -8 , -2 , 5 , -57.5, -62 , -55.5, -52.5];
apa7 = 1e-6*[0, 19.5 , -8 , -29.5, 75 , 97.5, 70 , 48];
apa7b = 1e-6*[0, 9 , -18.5, -30 , 31 , 46.5, 16.5 , 7.5];
apa = {apa1, apa2, apa3, apa4, apa5, apa6, apa7b};
#+end_src
The X/Y Positions of the 8 measurement points are defined below.
#+begin_src matlab
W = 20e-3; % Width [m]
L = 61e-3; % Length [m]
d = 1e-3; % Distance from border [m]
l = 15.5e-3; % [m]
pos = [[-L/2 + d; W/2 - d], [-L/2 + l - d; W/2 - d], [-L/2 + l - d; -W/2 + d], [-L/2 + d; -W/2 + d], [L/2 - l + d; W/2 - d], [L/2 - d; W/2 - d], [L/2 - d; -W/2 + d], [L/2 - l + d; -W/2 + d]];
#+end_src
Finally, the flatness is estimated by fitting a plane through the 8 points using the =fminsearch= command.
#+begin_src matlab
apa_d = zeros(1, 7);
for i = 1:7
fun = @(x)max(abs(([pos; apa{i}]-[0;0;x(1)])'*([x(2:3);1]/norm([x(2:3);1]))));
x0 = [0;0;0];
[x, min_d] = fminsearch(fun,x0);
apa_d(i) = min_d;
end
#+end_src
The obtained flatness are shown in Table [[tab:flatness_meas]].
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(1e6*apa_d', {'APA 1', 'APA 2', 'APA 3', 'APA 4', 'APA 5', 'APA 6', 'APA 7'}, {'*Flatness* $[\mu m]$'}, ' %.1f ');
#+end_src
#+name: tab:flatness_meas
#+caption: Estimated flatness
#+attr_latex: :environment tabularx :width 0.25\linewidth :align lc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
| | *Flatness* $[\mu m]$ |
|-------+----------------------|
| APA 1 | 8.9 |
| APA 2 | 3.1 |
| APA 3 | 9.1 |
| APA 4 | 3.0 |
| APA 5 | 1.9 |
| APA 6 | 7.1 |
| APA 7 | 18.7 |
** Electrical Measurements
<>
#+begin_note
The capacitance of the stacks is measure with the [[https://www.gwinstek.com/en-global/products/detail/LCR-800][LCR-800 Meter]] ([[file:doc/DS_LCR-800_Series_V2_E.pdf][doc]])
#+end_note
#+name: fig:LCR_meter
#+caption: LCR Meter used for the measurements
#+attr_latex: :width 0.9\linewidth
[[file:figs/IMG_20210312_120337.jpg]]
The excitation frequency is set to be 1kHz.
#+name: tab:apa300ml_capacitance
#+caption: Capacitance measured with the LCR meter. The excitation signal is a sinus at 1kHz
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lcc
#+attr_latex: :center t :booktabs t :float t
| | *Sensor Stack* | *Actuator Stacks* |
|-------+----------------+-------------------|
| APA 1 | 5.10 | 10.03 |
| APA 2 | 4.99 | 9.85 |
| APA 3 | 1.72 | 5.18 |
| APA 4 | 4.94 | 9.82 |
| APA 5 | 4.90 | 9.66 |
| APA 6 | 4.99 | 9.91 |
| APA 7 | 4.85 | 9.85 |
#+begin_warning
There is clearly a problem with APA300ML number 3
#+end_warning
The APA number 3 has ben sent back to Cedrat, and a new APA300ML has been shipped back.
** Stroke measurement
<>
*** Introduction :ignore:
We here wish to estimate the stroke of the APA.
To do so, one side of the APA is fixed, and a displacement probe is located on the other side as shown in Figure [[fig:stroke_test_bench]].
Then, a voltage is applied on either one or two stacks using a DAC and a voltage amplifier.
#+begin_note
Here are the documentation of the equipment used for this test bench:
- *Voltage Amplifier*: [[file:doc/PD200-V7-R1.pdf][PD200]] with a gain of 20
- *16bits DAC*: [[file:doc/IO131-OEM-Datasheet.pdf][IO313 Speedgoat card]]
- *Displacement Probe*: [[file:doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf][Millimar C1216 electronics]] and [[file:doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf][Millimar 1318 probe]]
#+end_note
#+name: fig:stroke_test_bench
#+caption: Bench to measured the APA stroke
#+attr_latex: :width 0.9\linewidth
[[file:figs/CE0EF55E-07B7-461B-8CDB-98590F68D15B.jpeg]]
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
#+end_src
*** Voltage applied on one stack
Let's first look at the relation between the voltage applied to *one* stack to the displacement of the APA as measured by the displacement probe.
#+begin_src matlab :exports none
apa300ml_1s = {};
for i = 1:7
apa300ml_1s(i) = {load(['mat/stroke_apa_1stacks_' num2str(i) '.mat'], 't', 'V', 'd')};
end
#+end_src
#+begin_src matlab :exports none
for i = 1:7
t = apa300ml_1s{i}.t;
apa300ml_1s{i}.d = apa300ml_1s{i}.d - mean(apa300ml_1s{i}.d(t > 1.9 & t < 2.0));
apa300ml_1s{i}.d = apa300ml_1s{i}.d(t > 2.0 & t < 10.0);
apa300ml_1s{i}.V = apa300ml_1s{i}.V(t > 2.0 & t < 10.0);
apa300ml_1s{i}.t = apa300ml_1s{i}.t(t > 2.0 & t < 10.0);
end
#+end_src
The applied voltage is shown in Figure [[fig:apa_stroke_voltage_time]].
#+begin_src matlab :exports none
figure;
plot(apa300ml_1s{1}.t, 20*apa300ml_1s{1}.V)
xlabel('Time [s]'); ylabel('Voltage [V]');
ylim([-20,160]); yticks([-20 0 20 40 60 80 100 120 140 160]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_stroke_voltage_time.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_stroke_voltage_time
#+caption: Applied voltage as a function of time
#+RESULTS:
[[file:figs/apa_stroke_voltage_time.png]]
The obtained displacement is shown in Figure [[fig:apa_stroke_time_1s]].
The displacement is set to zero at initial time when the voltage applied is -20V.
#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
plot(apa300ml_1s{i}.t, 1e6*apa300ml_1s{i}.d, 'DisplayName', sprintf('APA %i', i))
end
hold off;
xlabel('Time [s]'); ylabel('Displacement [$\mu m$]')
legend('location', 'southeast', 'FontSize', 8)
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_stroke_time_1s.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_stroke_time_1s
#+caption: Displacement as a function of time for all the APA300ML
#+RESULTS:
[[file:figs/apa_stroke_time_1s.png]]
Finally, the displacement is shown as a function of the applied voltage in Figure [[fig:apa_d_vs_V_1s]].
We can clearly see that there is a problem with the APA 3.
Also, there is a large hysteresis.
#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
plot(20*apa300ml_1s{i}.V, 1e6*apa300ml_1s{i}.d, 'DisplayName', sprintf('APA %i', i))
end
hold off;
xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
legend('location', 'southwest', 'FontSize', 8)
xlim([-20, 160]); ylim([-140, 0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_d_vs_V_1s.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_d_vs_V_1s
#+caption: Displacement as a function of the applied voltage
#+RESULTS:
[[file:figs/apa_d_vs_V_1s.png]]
#+begin_important
We can clearly see from Figure [[fig:apa_d_vs_V_1s]] that there is a problem with the APA number 3.
#+end_important
*** Voltage applied on two stacks
Now look at the relation between the voltage applied to the *two* other stacks to the displacement of the APA as measured by the displacement probe.
#+begin_src matlab :exports none
apa300ml_2s = {};
for i = 1:7
apa300ml_2s(i) = {load(['mat/stroke_apa_2stacks_' num2str(i) '.mat'], 't', 'V', 'd')};
end
#+end_src
#+begin_src matlab :exports none
for i = 1:7
t = apa300ml_2s{i}.t;
apa300ml_2s{i}.d = apa300ml_2s{i}.d - mean(apa300ml_2s{i}.d(t > 1.9 & t < 2.0));
apa300ml_2s{i}.d = apa300ml_2s{i}.d(t > 2.0 & t < 10.0);
apa300ml_2s{i}.V = apa300ml_2s{i}.V(t > 2.0 & t < 10.0);
apa300ml_2s{i}.t = apa300ml_2s{i}.t(t > 2.0 & t < 10.0);
end
#+end_src
The obtained displacement is shown in Figure [[fig:apa_stroke_time_2s]].
The displacement is set to zero at initial time when the voltage applied is -20V.
#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
plot(apa300ml_2s{i}.t, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
end
hold off;
xlabel('Time [s]'); ylabel('Displacement [$\mu m$]')
legend('location', 'southeast', 'FontSize', 8)
ylim([-250, 0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_stroke_time_2s.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_stroke_time_2s
#+caption: Displacement as a function of time for all the APA300ML
#+RESULTS:
[[file:figs/apa_stroke_time_2s.png]]
Finally, the displacement is shown as a function of the applied voltage in Figure [[fig:apa_d_vs_V_2s]].
We can clearly see that there is a problem with the APA 3.
Also, there is a large hysteresis.
#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
plot(20*apa300ml_2s{i}.V, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
end
hold off;
xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
legend('location', 'southwest', 'FontSize', 8)
xlim([-20, 160]); ylim([-250, 0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_d_vs_V_2s.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_d_vs_V_2s
#+caption: Displacement as a function of the applied voltage
#+RESULTS:
[[file:figs/apa_d_vs_V_2s.png]]
*** Voltage applied on all three stacks
Finally, we can combine the two measurements to estimate the relation between the displacement and the voltage applied to the *three* stacks (Figure [[fig:apa_d_vs_V_3s]]).
#+begin_src matlab :exports none
apa300ml_3s = {};
for i = 1:7
apa300ml_3s(i) = apa300ml_1s(i);
apa300ml_3s{i}.d = apa300ml_1s{i}.d + apa300ml_2s{i}.d;
end
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
plot(20*apa300ml_3s{i}.V, 1e6*apa300ml_3s{i}.d, 'DisplayName', sprintf('APA %i', i))
end
hold off;
xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
legend('location', 'southwest', 'FontSize', 8)
xlim([-20, 160]); ylim([-400, 0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_d_vs_V_3s.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_d_vs_V_3s
#+caption: Displacement as a function of the applied voltage
#+RESULTS:
[[file:figs/apa_d_vs_V_3s.png]]
The obtained maximum stroke for all the APA are summarized in Table [[tab:apa_measured_stroke]].
#+begin_src matlab :exports none
apa300ml_stroke = zeros(1, 7);
for i = 1:7
apa300ml_stroke(i) = max(apa300ml_3s{i}.d) - min(apa300ml_3s{i}.d);
end
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(1e6*apa300ml_stroke', {'APA 1', 'APA 2', 'APA 3', 'APA 4', 'APA 5', 'APA 6', 'APA 7'}, {'*Stroke* $[\mu m]$'}, ' %.1f ');
#+end_src
#+name: tab:apa_measured_stroke
#+caption: Measured maximum stroke
#+attr_latex: :environment tabularx :width 0.25\linewidth :align lc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
| | *Stroke* $[\mu m]$ |
|-------+--------------------|
| APA 1 | 373.2 |
| APA 2 | 365.5 |
| APA 3 | 181.7 |
| APA 4 | 359.7 |
| APA 5 | 361.5 |
| APA 6 | 363.9 |
| APA 7 | 358.4 |
** Spurious resonances
<>
*** Introduction
Three main resonances are foreseen to be problematic for the control of the APA300ML (Figure [[fig:apa_mode_shapes]]):
- Mode in X-bending at 189Hz
- Mode in Y-bending at 285Hz
- Mode in Z-torsion at 400Hz
#+name: fig:apa_mode_shapes
#+caption: Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz
#+attr_latex: :width \linewidth
[[file:figs/apa_mode_shapes.gif]]
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.
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('matlab/');
addpath('matlab/mat/');
#+end_src
#+begin_src matlab :eval no
addpath('mat/');
#+end_src
*** Setup
The measurement setup is shown in Figure [[fig:measurement_setup_torsion]].
A Laser vibrometer is measuring the difference of motion of 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
- Laser Doppler Vibrometer Polytec OFV512
- Instrumented hammer
#+end_note
#+name: fig:measurement_setup_torsion
#+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_torsion.jpg]]
*** Bending - X
The setup to measure the X-bending motion is shown in Figure [[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.
#+name: fig:measurement_setup_X_bending
#+caption: X-Bending measurement setup
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_X_bending.jpg]]
The data is loaded.
#+begin_src matlab
bending_X = load('apa300ml_bending_X_top.mat');
#+end_src
The config for =tfestimate= is performed:
#+begin_src matlab
Ts = bending_X.Track1_X_Resolution; % Sampling frequency [Hz]
win = hann(ceil(1/Ts));
#+end_src
The transfer function from the input force to the output "rotation" (difference between the two measured distances).
#+begin_src matlab
[G_bending_X, f] = tfestimate(bending_X.Track1, bending_X.Track2, win, [], [], 1/Ts);
#+end_src
The result is shown in Figure [[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_src matlab :exports none
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')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_bending_x.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_bending_x
#+caption: Obtained FRF for the X-bending
#+RESULTS:
[[file:figs/apa300ml_meas_freq_bending_x.png]]
*** Bending - Y
The setup to measure the Y-bending is shown in Figure [[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).
#+name: fig:measurement_setup_Y_bending
#+caption: Y-Bending measurement setup
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_Y_bending.jpg]]
The data is loaded, and the transfer function from the force to the measured rotation is computed.
#+begin_src matlab
bending_Y = load('apa300ml_bending_Y_top.mat');
[G_bending_Y, ~] = tfestimate(bending_Y.Track1, bending_Y.Track2, win, [], [], 1/Ts);
#+end_src
The results are shown in Figure [[fig:apa300ml_meas_freq_bending_y]].
The main resonance is at 412Hz, and we also see the third "harmonic" at 1220Hz.
#+begin_src matlab :exports none
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')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_bending_y.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_bending_y
#+caption: Obtained FRF for the Y-bending
#+RESULTS:
[[file:figs/apa300ml_meas_freq_bending_y.png]]
*** Torsion - Z
Finally, we measure the Z-torsion resonance as shown in Figure [[fig:measurement_setup_torsion_bis]].
The excitation is shown on the other side of the APA, on the side to excite the torsion motion.
#+name: fig:measurement_setup_torsion_bis
#+caption: Z-Torsion measurement setup
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_torsion_bis.jpg]]
The data is loaded, and the transfer function computed.
#+begin_src matlab
torsion = load('apa300ml_torsion_left.mat');
[G_torsion, ~] = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);
#+end_src
The results are shown in Figure [[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.
The mode in pure torsion is probably at higher frequency (peak around 1kHz?).
#+begin_src matlab :exports none
figure;
hold on;
plot(f, abs(G_torsion), 'k-');
hold off;
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude');
xlim([50, 2e3]); ylim([1e-5, 2e-2])
text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
text(800, 6e-4,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_torsion_z.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_torsion_z
#+caption: Obtained FRF for the Z-torsion
#+RESULTS:
[[file:figs/apa300ml_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.
#+begin_src matlab
torsion = load('apa300ml_torsion_top.mat');
[G_torsion_top, ~] = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);
#+end_src
The two FRF are compared in Figure [[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_src matlab :exports none
figure;
hold on;
plot(f, abs(G_torsion), 'k-', 'DisplayName', 'Left excitation');
plot(f, abs(G_torsion_top), '-', 'DisplayName', 'Top excitation');
hold off;
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude');
xlim([50, 2e3]); ylim([1e-5, 2e-2])
text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
text(800, 2e-3,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_torsion_z_comp.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_torsion_z_comp
#+caption: Obtained FRF for the Z-torsion
#+RESULTS:
[[file:figs/apa300ml_meas_freq_torsion_z_comp.png]]
*** Compare
The three measurements are shown in Figure [[fig:apa300ml_meas_freq_compare]].
#+begin_src matlab :exports none
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');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_compare.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:apa300ml_meas_freq_compare
#+caption: Obtained FRF - Comparison
#+RESULTS:
[[file:figs/apa300ml_meas_freq_compare.png]]
*** 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?).
#+name: tab:apa300ml_measured_modes_freq
#+caption: Measured frequency of the modes
#+attr_latex: :environment tabularx :width 0.6\linewidth :align ccc
#+attr_latex: :center t :booktabs t :float t
| Mode | Strut Mode | Measured Frequency |
|-----------+------------+--------------------|
| X-Bending | 189Hz | 280Hz |
| Y-Bending | 285Hz | 410Hz |
| Z-Torsion | 400Hz | ? |
* Dynamical measurements - APA
<>
** Introduction :ignore:
In this section, a measurement test bench is used to identify the dynamics of the APA.
The bench is shown in Figure [[fig:picture_apa_bench]], and a zoom picture on the APA and encoder is shown in Figure [[fig:picture_apa_bench_encoder]].
#+name: fig:picture_apa_bench
#+caption: Picture of the test bench
#+attr_latex: :width 0.5\linewidth
[[file:figs/picture_apa_bench.png]]
#+name: fig:picture_apa_bench_encoder
#+caption: Zoom on the APA with the encoder
#+attr_latex: :width 0.5\linewidth
[[file:figs/picture_apa_bench_encoder.png]]
#+begin_note
Here are the documentation of the equipment used for this test bench:
- Voltage Amplifier: [[file:doc/PD200-V7-R1.pdf][PD200]]
- Amplified Piezoelectric Actuator: [[file:doc/APA300ML.pdf][APA300ML]]
- DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]]
- Encoder: [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Renishaw Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]]
- Interferometer: [[https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer][Attocube IDS3010]]
#+end_note
The bench is schematically shown in Figure [[fig:test_bench_apa_alone]] and the signal used are summarized in Table [[tab:test_bench_apa_variables]].
#+name: fig:test_bench_apa_alone
#+caption: Schematic of the Test Bench
#+attr_latex: :width 0.8\linewidth
[[file:figs/test_bench_apa_alone.png]]
#+name: tab:test_bench_apa_variables
#+caption: Variables used during the measurements
#+attr_latex: :environment tabularx :width 0.8\linewidth :align lXXX
#+attr_latex: :center t :booktabs t :float t
| Variable | Description | Unit | Hardware |
|----------+------------------------------+------+-------------------------------|
| =Va= | Output DAC voltage | [V] | DAC - Ch. 1 => PD200 => APA |
| =Vs= | Measured stack voltage (ADC) | [V] | APA => ADC - Ch. 1 |
| =de= | Encoder Measurement | [m] | PEPU Ch. 1 - IO318(1) - Ch. 1 |
| =da= | Attocube Measurement | [m] | PEPU Ch. 2 - IO318(1) - Ch. 2 |
| =t= | Time | [s] | |
This section is structured as follows:
- Section [[sec:meas_apa_speedgoat_setup]]: the Speedgoat setup is described (excitation signals, saved signals, etc.)
- Section [[sec:meas_one_apa]]: the measurements are first performed on one APA.
- Section [[sec:meas_all_apa]]: the same measurements are performed on all the APA and are compared.
** Speedgoat Setup
<>
*** Introduction :ignore:
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./src/');
#+end_src
*** =frf_setup.m= - Measurement Setup
:PROPERTIES:
:header-args:matlab: :tangle matlab/frf_setup.m
:header-args:matlab+: :comments no
:END:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :exports none
addpath('./src/');
#+end_src
First is defined the sampling frequency:
#+begin_src matlab
%% Simulation configuration
Fs = 10e3; % Sampling Frequency [Hz]
Ts = 1/Fs; % Sampling Time [s]
#+end_src
#+begin_src matlab
%% Data record configuration
Trec_start = 5; % Start time for Recording [s]
Trec_dur = 100; % Recording Duration [s]
#+end_src
#+begin_src matlab
Tsim = 2*Trec_start + Trec_dur; % Simulation Time [s]
#+end_src
A white noise excitation signal can be very useful in order to obtain a first idea of the plant FRF.
The gain can be gradually increased until satisfactory output is obtained.
#+begin_src matlab
%% Shaped Noise
V_noise = generateShapedNoise('Ts', 1/Fs, ...
'V_mean', 3.25, ...
't_start', Trec_start, ...
'exc_duration', Trec_dur, ...
'smooth_ends', true, ...
'V_exc', 0.05/(1 + s/2/pi/10));
#+end_src
#+begin_src matlab :exports none :tangle no
figure;
tiledlayout(1, 2, 'TileSpacing', 'Normal', 'Padding', 'None');
ax1 = nexttile;
plot(V_noise(1,:), V_noise(2,:));
xlabel('Time [s]'); ylabel('Amplitude [V]');
ax2 = nexttile;
win = hanning(floor(length(V_noise)/8));
[pxx, f] = pwelch(V_noise(2,:), win, 0, [], Fs);
plot(f, pxx)
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$V^2/Hz$]');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([1, Fs/2]); ylim([1e-10, 1e0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/frf_meas_noise_excitation.pdf', 'width', 'full', 'height', 'normal');
#+end_src
#+name: fig:frf_meas_noise_excitation
#+caption: Example of Shaped noise excitation signal
#+RESULTS:
[[file:figs/frf_meas_noise_excitation.png]]
The maximum excitation voltage at resonance is 9Vrms, therefore corresponding to 0.6V of output DAC voltage.
#+begin_src matlab
%% Sweep Sine
gc = 0.1;
xi = 0.5;
wn = 2*pi*94.3;
% Notch filter at the resonance of the APA
G_sweep = 0.2*(s^2 + 2*gc*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);
V_sweep = generateSweepExc('Ts', Ts, ...
'f_start', 10, ...
'f_end', 400, ...
'V_mean', 3.25, ...
't_start', Trec_start, ...
'exc_duration', Trec_dur, ...
'sweep_type', 'log', ...
'V_exc', G_sweep*1/(1 + s/2/pi/500));
#+end_src
#+begin_src matlab :exports none :tangle no
figure;
tiledlayout(1, 2, 'TileSpacing', 'Normal', 'Padding', 'None');
ax1 = nexttile;
plot(V_sweep(1,:), V_sweep(2,:));
xlabel('Time [s]'); ylabel('Amplitude [V]');
ax2 = nexttile;
win = hanning(floor(length(V_sweep(2,:))/80));
[pxx, f] = pwelch(V_sweep(2,:), win, 0, [], Fs);
plot(f, pxx)
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$V^2/Hz$]');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([1, Fs/2]); ylim([1e-10, 1e0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/frf_meas_sweep_excitation.pdf', 'width', 'full', 'height', 'normal');
#+end_src
#+name: fig:frf_meas_sweep_excitation
#+caption: Example of Sweep Sin excitation signal
#+RESULTS:
[[file:figs/frf_meas_sweep_excitation.png]]
In order to better estimate the high frequency dynamics, a band-limited noise can be used (Figure [[fig:frf_meas_noise_hf_exc]]).
The frequency content of the noise can be precisely controlled.
#+begin_src matlab
%% High Frequency Shaped Noise
[b,a] = cheby1(10, 2, 2*pi*[300 2e3], 'bandpass', 's');
wL = 0.005*tf(b, a);
V_noise_hf = generateShapedNoise('Ts', 1/Fs, ...
'V_mean', 3.25, ...
't_start', Trec_start, ...
'exc_duration', Trec_dur, ...
'smooth_ends', true, ...
'V_exc', wL);
#+end_src
#+begin_src matlab :exports none :tangle no
figure;
tiledlayout(1, 2, 'TileSpacing', 'Normal', 'Padding', 'None');
ax1 = nexttile;
plot(V_noise_hf(1,:), V_noise_hf(2,:));
xlabel('Time [s]'); ylabel('Amplitude [V]');
ax2 = nexttile;
win = hanning(floor(length(V_noise_hf(2,:))/80));
[pxx, f] = pwelch(V_noise_hf(2,:), win, 0, [], Fs);
plot(f, pxx)
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$V^2/Hz$]');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([1, Fs/2]); ylim([1e-10, 1e0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/frf_meas_noise_hf_exc.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:frf_meas_noise_hf_exc
#+caption: Example of band-limited noise excitation signal
#+RESULTS:
[[file:figs/frf_meas_noise_hf_exc.png]]
Then a sinus excitation can be used to estimate the hysteresis.
#+begin_src matlab
%% Sinus excitation with increasing amplitude
V_sin = generateSinIncreasingAmpl('Ts', 1/Fs, ...
'V_mean', 3.25, ...
'sin_ampls', [0.1, 0.2, 0.4, 1, 2, 4], ...
'sin_period', 1, ...
'sin_num', 5, ...
't_start', Trec_start, ...
'smooth_ends', true);
#+end_src
#+begin_src matlab :exports none :tangle no
figure;
plot(V_sin(1,:), V_sin(2,:));
xlabel('Time [s]'); ylabel('Amplitude [V]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/frf_meas_sin_excitation.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:frf_meas_sin_excitation
#+caption: Example of Shaped noise excitation signal
#+RESULTS:
[[file:figs/frf_meas_sin_excitation.png]]
Then, we select the wanted excitation signal.
#+begin_src matlab
%% Select the excitation signal
V_exc = timeseries(V_noise(2,:), V_noise(1,:));
#+end_src
#+begin_src matlab :exports none :eval no
%% Plot
figure;
tiledlayout(1, 2, 'TileSpacing', 'Normal', 'Padding', 'None');
ax1 = nexttile;
plot(V_exc(1,:), V_exc(2,:));
xlabel('Time [s]'); ylabel('Amplitude [V]');
ax2 = nexttile;
win = hanning(floor(length(V_exc)/8));
[pxx, f] = pwelch(V_exc(2,:), win, 0, [], Fs);
plot(f, pxx)
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$V^2/Hz$]');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([1, Fs/2]); ylim([1e-10, 1e0]);
#+end_src
#+begin_src matlab
%% Save data that will be loaded in the Simulink file
save('./frf_data.mat', 'Fs', 'Ts', 'Tsim', 'Trec_start', 'Trec_dur', 'V_exc');
#+end_src
*** =frf_save.m= - Save Data
:PROPERTIES:
:header-args: :tangle matlab/frf_save.m
:header-args:matlab+: :comments no
:END:
First, we get data from the Speedgoat:
#+begin_src matlab
tg = slrt;
f = SimulinkRealTime.openFTP(tg);
mget(f, 'data/data.dat');
close(f);
#+end_src
And we load the data on the Workspace:
#+begin_src matlab
data = SimulinkRealTime.utils.getFileScopeData('data/data.dat').data;
da = data(:, 1); % Excitation Voltage (input of PD200) [V]
de = data(:, 2); % Measured voltage (force sensor) [V]
Vs = data(:, 3); % Measurment displacement (encoder) [m]
Va = data(:, 4); % Measurement displacement (attocube) [m]
t = data(:, end); % Time [s]
#+end_src
And we save this to a =mat= file:
#+begin_src matlab
apa_number = 1;
save(sprintf('mat/frf_data_%i_huddle.mat', apa_number), 't', 'Va', 'Vs', 'de', 'da');
#+end_src
** Measurements on APA 1
<>
*** Introduction :ignore:
Measurements are first performed on only *one* APA.
Once the measurement procedure is validated, it is performed on all the other APA.
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src
*** Excitation Signal
For this first measurement, a basic logarithmic sweep is used between 10Hz and 2kHz.
The data are loaded.
#+begin_src matlab
apa_sweep = load(sprintf('mat/frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'da', 'de');
#+end_src
The initial time is set to zero.
#+begin_src matlab
%% Time vector
t = apa_sweep.t - apa_sweep.t(1) ; % Time vector [s]
#+end_src
The excitation signal is shown in Figure [[fig:apa_bench_exc_sweep]].
It is a sweep sine from 10Hz up to 2kHz filtered with a notch centered with the main resonance of the system and a low pass filter.
#+begin_src matlab :exports none
figure;
plot(t, apa_sweep.Va)
xlabel('Time [s]'); ylabel('Excitation Voltage $V_a$ [V]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_bench_exc_sweep.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_bench_exc_sweep
#+caption: Excitation voltage
#+RESULTS:
[[file:figs/apa_bench_exc_sweep.png]]
*** FRF Identification - Setup
Let's define the sampling time/frequency.
#+begin_src matlab
%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
#+end_src
Then we defined a "Hanning" windows that will be used for the spectral analysis:
#+begin_src matlab
win = hanning(ceil(1*Fs)); % Hannning Windows
#+end_src
We get the frequency vector that will be the same for all the frequency domain analysis.
#+begin_src matlab
% Only used to have the frequency vector "f"
[~, f] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
#+end_src
*** FRF Identification - Displacement
In this section, the transfer function from the excitation voltage $V_a$ to the encoder measured displacement $d_e$ and interferometer measurement $d_a$.
The coherence from $V_a$ to $d_e$ is computed and shown in Figure [[fig:apa_1_coh_dvf]].
It is quite good from 10Hz up to 500Hz.
#+begin_src matlab
%% TF - Encoder
[coh_sweep, ~] = mscohere(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_sweep, 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([5, 5e3]); ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_1_coh_dvf.pdf', 'width', 'normal', 'height', 'normal');
#+end_src
#+name: fig:apa_1_coh_dvf
#+caption: Coherence for the identification from $V_a$ to $d_e$
#+RESULTS:
[[file:figs/apa_1_coh_dvf.png]]
The transfer functions are then estimated and shown in Figure [[fig:apa_1_frf_dvf]].
#+begin_src matlab
%% TF - Encoder
[dvf_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
%% TF - Interferometer
[int_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.da, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(dvf_sweep), 'color', colors(1, :), ...
'DisplayName', 'Encoder');
plot(f, abs(int_sweep), 'color', colors(2, :), ...
'DisplayName', 'Interferometer');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ylim([1e-9, 1e-3]);
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(dvf_sweep), 'color', colors(1, :));
plot(f, 180/pi*angle(int_sweep), 'color', colors(2, :));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_1_frf_dvf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_1_frf_dvf
#+caption: Obtained transfer functions from $V_a$ to both $d_e$ and $d_a$
#+RESULTS:
[[file:figs/apa_1_frf_dvf.png]]
#+begin_important
It seems that using the interferometer, we have a lot more time delay than when using the encoder.
#+end_important
*** FRF Identification - Force Sensor
Now the dynamics from excitation voltage $V_a$ to the force sensor stack voltage $V_s$ is identified.
The coherence is computed and shown in Figure [[fig:apa_1_coh_iff]] and found very good from 10Hz up to 2kHz.
#+begin_src matlab
%% TF - Encoder
[coh_sweep, ~] = mscohere(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_sweep, 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([5, 5e3]); ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_1_coh_iff.pdf', 'width', 'normal', 'height', 'normal');
#+end_src
#+name: fig:apa_1_coh_iff
#+caption: Coherence for the identification from $V_a$ to $V_s$
#+RESULTS:
[[file:figs/apa_1_coh_iff.png]]
The transfer function is estimated and shown in Figure [[fig:apa_1_frf_iff]].
#+begin_src matlab
%% Transfer function estimation
[iff_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(iff_sweep), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(iff_sweep), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_1_frf_iff.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_1_frf_iff
#+caption: Obtained transfer functions from $V_a$ to $V_s$
#+RESULTS:
[[file:figs/apa_1_frf_iff.png]]
*** Hysteresis
We here wish to visually see the amount of hysteresis present in the APA.
To do so, a quasi static sinusoidal excitation $V_a$ at different voltages is used.
The offset is 65V, and the sin amplitude is ranging from 1V up to 80V.
For each excitation amplitude, the vertical displacement $d$ of the mass is measured.
Then, $d$ is plotted as a function of $V_a$ for all the amplitudes.
We expect to obtained something like the hysteresis shown in Figure [[fig:expected_hysteresis]].
#+name: fig:expected_hysteresis
#+caption: Expected Hysteresis cite:poel10_explor_activ_hard_mount_vibrat
#+attr_latex: :width 0.8\linewidth
[[file:figs/expected_hysteresis.png]]
The data is loaded.
#+begin_src matlab
apa_hyst = load('frf_data_1_hysteresis.mat', 't', 'Va', 'de');
% Initial time set to zero
apa_hyst.t = apa_hyst.t - apa_hyst.t(1);
#+end_src
The excitation voltage amplitudes are:
#+begin_src matlab
ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes
#+end_src
The excitation voltage and the measured displacement are shown in Figure [[fig:hyst_exc_signal_time]].
#+begin_src matlab :exports none
figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
plot(apa_hyst.t, apa_hyst.Va)
xlabel('Time [s]'); ylabel('Output Voltage [V]');
ax2 = nexttile;
plot(apa_hyst.t, apa_hyst.de)
xlabel('Time [s]'); ylabel('Measured Displacement [m]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/hyst_exc_signal_time.pdf', 'width', 'full', 'height', 'normal');
#+end_src
#+name: fig:hyst_exc_signal_time
#+caption: Excitation voltage and measured displacement
#+RESULTS:
[[file:figs/hyst_exc_signal_time.png]]
For each amplitude, we only take the last sinus in order to reduce possible transients.
Also, it is centered on zero.
The measured displacement at a function of the output voltage are shown in Figure [[fig:hyst_results_multi_ampl]].
#+begin_src matlab :exports none
figure;
tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([1,2]);
hold on;
for i = flip(1:6)
i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5;
plot(apa_hyst.Va(i_lim) - mean(apa_hyst.Va(i_lim)), apa_hyst.de(i_lim) - mean(apa_hyst.de(i_lim)), ...
'DisplayName', sprintf('$V_a = %.1f [V]$', ampls(i)))
end
hold off;
xlabel('Output Voltage [V]'); ylabel('Measured Displacement [m]');
legend('location', 'northeast');
xlim([-4, 4]); ylim([-1.2e-4, 1.2e-4]);
ax2 = nexttile;
hold on;
for i = flip(1:6)
i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5;
plot(apa_hyst.Va(i_lim) - mean(apa_hyst.Va(i_lim)), apa_hyst.de(i_lim) - mean(apa_hyst.de(i_lim)))
end
hold off;
xlim([-0.4, 0.4]); ylim([-0.8e-5, 0.8e-5]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/hyst_results_multi_ampl.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:hyst_results_multi_ampl
#+caption: Obtained hysteresis for multiple excitation amplitudes
#+RESULTS:
[[file:figs/hyst_results_multi_ampl.png]]
#+begin_important
It is quite clear that hysteresis is increasing with the excitation amplitude.
Also, no hysteresis is found on the sensor stack voltage.
#+end_important
*** Estimation of the APA axial stiffness
In order to estimate the stiffness of the APA, a weight with known mass $m_a$ is added on top of the suspended granite and the deflection $d_e$ is measured using the encoder.
The APA stiffness is then:
\begin{equation}
k_{\text{apa}} = \frac{m_a g}{d}
\end{equation}
Here, a mass of 6.4 kg is used:
#+begin_src matlab
added_mass = 6.4; % Added mass [kg]
#+end_src
The data is loaded, and the measured displacement is shown in Figure [[fig:apa_1_meas_stiffness]].
#+begin_src matlab
apa_mass = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');
apa_mass.de = apa_mass.de - mean(apa_mass.de(apa_mass.t<11));
#+end_src
#+begin_src matlab :exports none
figure;
plot(apa_mass.t, apa_mass.de, 'k-');
xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_1_meas_stiffness.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_1_meas_stiffness
#+caption: Measured displacement when adding the mass and removing the mass
#+RESULTS:
[[file:figs/apa_1_meas_stiffness.png]]
There is some imprecision in the measurement as there are some drifts that are probably due to some creep.
The stiffness is then computed as follows:
#+begin_src matlab
k = 9.8 * added_mass / (mean(apa_mass.de(apa_mass.t > 12 & apa_mass.t < 12.5)) - mean(apa_mass.de(apa_mass.t > 20 & apa_mass.t < 20.5)));
#+end_src
And the stiffness obtained is very close to the one specified in the documentation ($k = 1.794\,[N/\mu m]$).
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('k = %.2f [N/um]', 1e-6*k);
#+end_src
#+RESULTS:
: k = 1.68 [N/um]
*** Stiffness change due to electrical connections
We wish here to see if the stiffness changes when the actuator stacks are not connected to the amplifier and the sensor stacks are not connected to the ADC.
Note here that the resistor in parallel to the sensor stack is present in both cases.
First, the data are loaded.
#+begin_src matlab
add_mass_oc = load(sprintf('frf_data_%i_add_mass_open_circuit.mat', 1), 't', 'de');
add_mass_cc = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');
#+end_src
And the initial displacement is set to zero.
#+begin_src matlab
add_mass_oc.de = add_mass_oc.de - mean(add_mass_oc.de(add_mass_oc.t<11));
add_mass_cc.de = add_mass_cc.de - mean(add_mass_cc.de(add_mass_cc.t<11));
#+end_src
The measured displacements are shown in Figure [[fig:apa_meas_k_time_oc_cc]].
#+begin_src matlab :exports none
figure;
hold on;
plot(add_mass_oc.t, add_mass_oc.de, 'DisplayName', 'Not connected');
plot(add_mass_cc.t, add_mass_cc.de, 'DisplayName', 'Connected');
hold off;
xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]');
legend('location', 'northeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_meas_k_time_oc_cc.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_meas_k_time_oc_cc
#+caption: Measured displacement
#+RESULTS:
[[file:figs/apa_meas_k_time_oc_cc.png]]
And the stiffness is estimated in both case.
The results are shown in Table [[tab:APA_measured_k_oc_cc]].
#+begin_src matlab
apa_k_oc = 9.8 * added_mass / (mean(add_mass_oc.de(add_mass_oc.t > 12 & add_mass_oc.t < 12.5)) - mean(add_mass_oc.de(add_mass_oc.t > 20 & add_mass_oc.t < 20.5)));
apa_k_cc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass_cc.t < 12.5)) - mean(add_mass_cc.de(add_mass_cc.t > 20 & add_mass_cc.t < 20.5)));
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(1e-6*[apa_k_oc; apa_k_cc], {'Not connected', 'Connected'}, {'$k [N/\mu m]$'}, ' %.1f ');
#+end_src
#+name: tab:APA_measured_k_oc_cc
#+caption: Measured stiffnesses on "open" and "closed" circuits
#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
| | $k [N/\mu m]$ |
|---------------+---------------|
| Not connected | 2.3 |
| Connected | 1.7 |
#+begin_important
Clearly, connecting the actuator stacks to the amplified (basically equivalent as to short circuiting them) lowers the stiffness.
#+end_important
*** Effect of the resistor on the IFF Plant
A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack.
This has the effect to form a high pass filter with the capacitance of the stack.
We here measured the low frequency transfer function from $V_a$ to $V_s$ with and without this resistor.
#+begin_src matlab
% With the resistor
wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va');
% Without the resistor
wo_k = load('frf_data_1_sweep_lf.mat', 't', 'Vs', 'Va');
#+end_src
We use a very long "Hanning" window for the spectral analysis in order to estimate the low frequency behavior.
#+begin_src matlab
win = hanning(ceil(50*Fs)); % Hannning Windows
#+end_src
And we estimate the transfer function from $V_a$ to $V_s$ in both cases:
#+begin_src matlab
[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, [], [], 1/Ts);
[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, [], [], 1/Ts);
#+end_src
With the following values of the resistor and capacitance, we obtain a first order high pass filter with a crossover frequency equal to:
#+begin_src matlab
C = 5.1e-6; % Sensor Stack capacitance [F]
R = 80.6e3; % Parallel Resistor [Ohm]
f0 = 1/(2*pi*R*C); % Crossover frequency of RC HPF [Hz]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('f0 = %.2f [Hz]', f0)
#+end_src
#+RESULTS:
: f0 = 0.39 [Hz]
The transfer function of the corresponding high pass filter is:
#+begin_src matlab
G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0);
#+end_src
Let's compare the transfer function from actuator stack to sensor stack with and without the added resistor in Figure [[fig:frf_iff_effect_R]].
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(frf_wo_k), 'DisplayName', 'Without $k$');
plot(f, abs(frf_wi_k), 'DisplayName', 'With $k$');
plot(f, abs(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', sprintf('HPF $f_o = %.2f [Hz]$', f0));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-1, 1e0]);
legend('location', 'southeast')
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(frf_wo_k));
plot(f, 180/pi*angle(frf_wi_k));
plot(f, 180/pi*angle(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360); ylim([-45, 90]);
linkaxes([ax1,ax2],'x');
xlim([0.2, 8]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/frf_iff_effect_R.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:frf_iff_effect_R
#+caption: Transfer function from $V_a$ to $V_s$ with and without the resistor $k$
#+RESULTS:
[[file:figs/frf_iff_effect_R.png]]
#+begin_important
The added resistor has indeed the expected effect.
#+end_important
** Comparison of all the APA
<>
*** Introduction :ignore:
The same measurements that was performed in Section [[sec:meas_one_apa]] are now performed on all the APA and then compared.
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src
*** Axial Stiffnesses - Comparison
Let's first compare the APA axial stiffnesses.
The added mass is:
#+begin_src matlab
added_mass = 6.4; % Added mass [kg]
#+end_src
Here are the number of the APA that have been measured:
#+begin_src matlab
apa_nums = [1 2 4 5 6 7 8];
#+end_src
The data are loaded.
#+begin_src matlab
apa_mass = {};
for i = 1:length(apa_nums)
apa_mass(i) = {load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', apa_nums(i)), 't', 'de')};
% The initial displacement is set to zero
apa_mass{i}.de = apa_mass{i}.de - mean(apa_mass{i}.de(apa_mass{i}.t<11));
end
#+end_src
The raw measurements are shown in Figure [[fig:apa_meas_k_time]].
All the APA seems to have similar stiffness except the APA 7 which should have an higher stiffness.
#+begin_question
It is however strange that the displacement $d_e$ when the mass is removed is higher for the APA 7 than for the other APA.
What could cause that?
#+end_question
#+begin_src matlab :exports none
figure;
hold on;
for i = 1:length(apa_nums)
plot(apa_mass{i}.t, apa_mass{i}.de, 'DisplayName', sprintf('APA %i', apa_nums(i)));
end
hold off;
xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]');
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_meas_k_time.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_meas_k_time
#+caption: Raw measurements for all the APA. A mass of 6.4kg is added at arround 15s and removed at arround 22s
#+RESULTS:
[[file:figs/apa_meas_k_time.png]]
The stiffnesses are computed for all the APA and are summarized in Table [[tab:APA_measured_k]].
#+begin_src matlab :exports none
apa_k = zeros(length(apa_nums), 1);
for i = 1:length(apa_nums)
apa_k(i) = 9.8 * added_mass / (mean(apa_mass{i}.de(apa_mass{i}.t > 12 & apa_mass{i}.t < 12.5)) - mean(apa_mass{i}.de(apa_mass{i}.t > 20 & apa_mass{i}.t < 20.5)));
end
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(1e-6*apa_k, cellstr(num2str(apa_nums')), {'APA Num', '$k [N/\mu m]$'}, ' %.2f ');
#+end_src
#+name: tab:APA_measured_k
#+caption: Measured stiffnesses
#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
| APA Num | $k [N/\mu m]$ |
|---------+---------------|
| 1 | 1.68 |
| 2 | 1.69 |
| 4 | 1.7 |
| 5 | 1.7 |
| 6 | 1.7 |
| 7 | 1.93 |
| 8 | 1.73 |
#+begin_important
The APA300ML manual specifies the nominal stiffness to be $1.8\,[N/\mu m]$ which is very close to what have been measured.
Only the APA number 7 is a little bit off.
#+end_important
*** FRF Identification - Setup
The identification is performed in three steps:
1. White noise excitation with small amplitude.
This is used to determine the main resonance of the system.
2. Sweep sine excitation with the amplitude lowered around the resonance.
The sweep sine is from 10Hz to 400Hz.
3. High frequency noise.
The noise is band-passed between 300Hz and 2kHz.
Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.
Here are the APA numbers that have been measured.
#+begin_src matlab
apa_nums = [1 2 4 5 6 7 8];
#+end_src
The data are loaded for both the second and third identification:
#+begin_src matlab
%% Second identification
apa_sweep = {};
for i = 1:length(apa_nums)
apa_sweep(i) = {load(sprintf('frf_data_%i_sweep.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
%% Third identification
apa_noise_hf = {};
for i = 1:length(apa_nums)
apa_noise_hf(i) = {load(sprintf('frf_data_%i_noise_hf.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
#+end_src
The time is the same for all measurements.
#+begin_src matlab
%% Time vector
t = apa_sweep{1}.t - apa_sweep{1}.t(1) ; % Time vector [s]
%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
#+end_src
Then we defined a "Hanning" windows that will be used for the spectral analysis:
#+begin_src matlab
win = hanning(ceil(0.5*Fs)); % Hannning Windows
#+end_src
We get the frequency vector that will be the same for all the frequency domain analysis.
#+begin_src matlab
% Only used to have the frequency vector "f"
[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts);
#+end_src
*** FRF Identification - DVF
In this section, the dynamics from excitation voltage $V_a$ to encoder measured displacement $d_e$ is identified.
We compute the coherence for 2nd and 3rd identification:
#+begin_src matlab
%% Coherence computation
coh_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
coh_sweep(:, i) = coh;
end
coh_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
coh_noise_hf(:, i) = coh;
end
#+end_src
The coherence is shown in Figure [[fig:apa_frf_dvf_plant_coh]].
It is clear that the Sweep sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency.
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], ...
'DisplayName', 'HF Noise');
plot(f, coh_sweep(:, 1), 'color', [colors(2, :), 0.5], ...
'DisplayName', 'Sweep');
for i = 2:length(apa_nums)
plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ...
'HandleVisibility', 'off');
plot(f, coh_sweep(:, i), 'color', [colors(2, :), 0.5], ...
'HandleVisibility', 'off');
end;
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_frf_dvf_plant_coh
#+caption: Obtained coherence for the plant from $V_a$ to $d_e$
#+RESULTS:
[[file:figs/apa_frf_dvf_plant_coh.png]]
Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoders is computed:
#+begin_src matlab
%% Transfer function estimation
dvf_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
dvf_sweep(:, i) = frf;
end
dvf_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
dvf_noise_hf(:, i) = frf;
end
#+end_src
The obtained transfer functions are shown in Figure [[fig:apa_frf_dvf_plant_tf]].
They are all superimposed except for the APA7.
#+begin_question
Why is the APA7 off?
We could think that the APA7 is stiffer, but also the mass line is off.
It seems that there is a "gain" problem.
The encoder seems fine (it measured the same as the Interferometer).
Maybe it could be due to the amplifier?
#+end_question
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), abs(dvf_noise_hf(f> 350, i)), 'color', colors(i, :), ...
'DisplayName', sprintf('APA %i', apa_nums(i)));
plot(f(f<=350), abs(dvf_sweep( f<=350, i)), 'color', colors(i, :), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-9, 1e-3]);
ax2 = nexttile;
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350, i)), 'color', colors(i, :));
plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350, i)), 'color', colors(i, :));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_frf_dvf_plant_tf
#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)
#+RESULTS:
[[file:figs/apa_frf_dvf_plant_tf.png]]
A zoom on the main resonance is shown in Figure [[fig:apa_frf_dvf_zoom_res_plant_tf]].
It is clear that expect for the APA 7, the response around the resonances are well matching for all the APA.
It is also clear that there is not a single resonance but two resonances, a first one at 95Hz and a second one at 105Hz.
#+begin_question
Why is there a double resonance at around 94Hz?
#+end_question
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(apa_nums)
plot(f(f<=350), abs(dvf_sweep( f<=350, i)), 'color', colors(i, :), ...
'DisplayName', sprintf('APA %i', apa_nums(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([2e-5, 4e-4]);
ax2 = nexttile;
hold on;
for i = 1:length(apa_nums)
plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350, i)), 'color', colors(i, :));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-10, 180]);
linkaxes([ax1,ax2],'x');
xlim([80, 120]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_frf_dvf_zoom_res_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_frf_dvf_zoom_res_plant_tf
#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) - Zoom on the main resonance
#+RESULTS:
[[file:figs/apa_frf_dvf_zoom_res_plant_tf.png]]
*** FRF Identification - IFF
In this section, the dynamics from $V_a$ to $V_s$ is identified.
First the coherence is computed and shown in Figure [[fig:apa_frf_iff_plant_coh]].
The coherence is very nice from 10Hz to 2kHz.
It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).
#+begin_src matlab
%% Coherence
coh_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
coh_sweep(:, i) = coh;
end
coh_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
coh_noise_hf(:, i) = coh;
end
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], 'DisplayName', 'HF Noise');
plot(f, coh_sweep( :, 1), 'color', [colors(2, :), 0.5], 'DisplayName', 'Sweep');
for i = 2:length(apa_nums)
plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ...
'HandleVisibility', 'off');
plot(f, coh_sweep( :, i), 'color', [colors(2, :), 0.5], ...
'HandleVisibility', 'off');
end;
hold off;
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_frf_iff_plant_coh
#+caption: Obtained coherence for the IFF plant
#+RESULTS:
[[file:figs/apa_frf_iff_plant_coh.png]]
Then the FRF are estimated and shown in Figure [[fig:apa_frf_iff_plant_tf]]
#+begin_src matlab
%% FRF estimation of the transfer function from Va to Vs
iff_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
iff_sweep(:, i) = frf;
end
iff_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
iff_noise_hf(:, i) = frf;
end
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', colors(i, :), ...
'DisplayName', sprintf('APA %i', apa_nums(i)));
plot(f(f<=350), abs(iff_sweep( f<=350, i)), 'color', colors(i, :), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ax2 = nexttile;
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', colors(i, :));
plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350, i)), 'color', colors(i, :));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_frf_iff_plant_tf
#+caption:Identified IFF Plant
#+RESULTS:
[[file:figs/apa_frf_iff_plant_tf.png]]
* Dynamical measurements - Struts
<>
** Introduction :ignore:
The same bench used in Section [[sec:dynamical_meas_apa]] is here used with the strut instead of only the APA.
The bench is shown in Figure [[fig:test_bench_leg_overview]].
Measurements are performed either when no encoder is fixed to the strut (Figure [[fig:test_bench_leg_front]]) or when one encoder is fixed to the strut (Figure [[fig:test_bench_leg_overview]]).
#+name: fig:test_bench_leg_overview
#+caption: Test Bench with Strut - Overview
#+attr_latex: :width 0.5\linewidth
[[file:figs/test_bench_leg_overview.png]]
#+name: fig:test_bench_leg_front
#+caption: Test Bench with Strut - Zoom on the strut
#+attr_latex: :width 0.5\linewidth
[[file:figs/test_bench_leg_front.png]]
#+name: fig:test_bench_leg_overview
#+caption: Test Bench with Strut - Zoom on the strut with the encoder
#+attr_latex: :width 0.5\linewidth
[[file:figs/test_bench_leg_coder.png]]
** Measurement on Strut 1
<>
*** Introduction :ignore:
Measurements are first performed on the strut 1 that contains:
- APA 1
- flex 1 and flex 2
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src
*** Without Encoder
<>
**** FRF Identification - Setup
The identification is performed in three steps:
1. White noise excitation with small amplitude.
This is used to determine the main resonance of the system.
2. Sweep sine excitation with the amplitude lowered around the resonance.
The sweep sine is from 10Hz to 400Hz.
3. High frequency noise.
The noise is band-passed between 300Hz and 2kHz.
Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.
#+begin_src matlab
leg_sweep = load(sprintf('frf_data_leg_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
leg_noise_hf = load(sprintf('frf_data_leg_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
#+end_src
The time is the same for all measurements.
#+begin_src matlab
%% Time vector
t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s]
%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
#+end_src
Then we defined a "Hanning" windows that will be used for the spectral analysis:
#+begin_src matlab
win = hanning(ceil(0.5*Fs)); % Hannning Windows
#+end_src
We get the frequency vector that will be the same for all the frequency domain analysis.
#+begin_src matlab
% Only used to have the frequency vector "f"
[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts);
#+end_src
**** FRF Identification - Displacement
In this section, the dynamics from the excitation voltage $V_a$ to the interferometer $d_a$ is identified.
We compute the coherence for 2nd and 3rd identification:
#+begin_src matlab
[coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts);
[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf(:, 1), 'color', colors(1, :), ...
'DisplayName', 'HF Noise');
plot(f, coh_sweep(:, 1), 'color', colors(2, :), ...
'DisplayName', 'Sweep');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:strut_1_frf_dvf_plant_coh
#+caption: Obtained coherence for the plant from $V_a$ to $d_a$
#+RESULTS:
[[file:figs/strut_1_frf_dvf_plant_coh.png]]
The transfer function from $V_a$ to the interferometer measured displacement $d_a$ is estimated and shown in Figure [[fig:strut_1_frf_dvf_plant_tf]].
#+begin_src matlab
[dvf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts);
[dvf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(dvf_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(dvf_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-9, 1e-3]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_frf_dvf_plant_tf
#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the interferometer $d_a$)
#+RESULTS:
[[file:figs/strut_1_frf_dvf_plant_tf.png]]
**** FRF Identification - IFF
In this section, the dynamics from $V_a$ to $V_s$ is identified.
First the coherence is computed and shown in Figure [[fig:strut_1_frf_iff_plant_coh]].
The coherence is very nice from 10Hz to 2kHz.
It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).
#+begin_src matlab
[coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf, 'DisplayName', 'HF Noise');
plot(f, coh_sweep, 'DisplayName', 'Sweep');
hold off;
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:strut_1_frf_iff_plant_coh
#+caption: Obtained coherence for the IFF plant
#+RESULTS:
[[file:figs/strut_1_frf_iff_plant_coh.png]]
Then the FRF are estimated and shown in Figure [[fig:strut_1_frf_iff_plant_tf]]
#+begin_src matlab
[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(iff_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(iff_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_frf_iff_plant_tf
#+caption:Identified IFF Plant for the Strut 1
#+RESULTS:
[[file:figs/strut_1_frf_iff_plant_tf.png]]
*** With Encoder
<>
**** Measurement Data
#+begin_src matlab
leg_enc_sweep = load(sprintf('frf_data_leg_coder_badly_align_%i_noise.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_badly_align_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
#+end_src
**** FRF Identification - DVF
In this section, the dynamics from $V_a$ to $d_e$ is identified.
We compute the coherence for 2nd and 3rd identification:
#+begin_src matlab
[coh_enc_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.de, win, [], [], 1/Ts);
[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_enc_noise_hf(:, 1), 'color', colors(1, :), ...
'DisplayName', 'HF Noise');
plot(f, coh_enc_sweep(:, 1), 'color', colors(2, :), ...
'DisplayName', 'Sweep');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
% xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_enc_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:strut_1_enc_frf_dvf_plant_coh
#+caption: Obtained coherence for the plant from $V_a$ to $d_e$
#+RESULTS:
[[file:figs/strut_1_enc_frf_dvf_plant_coh.png]]
#+begin_src matlab
[dvf_enc_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.de, win, [], [], 1/Ts);
[dvf_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab
[dvf_int_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.da, win, [], [], 1/Ts);
[dvf_int_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);
#+end_src
The obtained transfer functions are shown in Figure [[fig:strut_1_enc_frf_dvf_plant_tf]].
They are all superimposed except for the APA7.
#+begin_question
Why is the APA7 off?
We could think that the APA7 is stiffer, but also the mass line is off.
It seems that there is a "gain" problem.
The encoder seems fine (it measured the same as the Interferometer).
Maybe it could be due to the amplifier?
#+end_question
#+begin_question
Why is there a double resonance at around 94Hz?
#+end_question
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(dvf_enc_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(dvf_enc_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-7, 1e-3]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(dvf_enc_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(dvf_enc_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_enc_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_enc_frf_dvf_plant_tf
#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)
#+RESULTS:
[[file:figs/strut_1_enc_frf_dvf_plant_tf.png]]
**** Comparison of the Encoder and Interferometer
The interferometer could here represent the case where the encoders are fixed to the plates and not the APA.
The dynamics from $V_a$ to $d_e$ and from $V_a$ to $d_a$ are compared in Figure [[fig:strut_1_comp_enc_int]].
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(dvf_enc_noise_hf(f> 350)), 'color', colors(1, :), ...
'DisplayName', 'Encoder');
plot(f(f<=350), abs(dvf_enc_sweep( f<=350)), 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(f(f> 350), abs(dvf_int_noise_hf(f> 350)), 'color', colors(2, :), ...
'DisplayName', 'Interferometer');
plot(f(f<=350), abs(dvf_int_sweep( f<=350)), 'color', colors(2, :), ...
'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-8, 1e-3]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(dvf_enc_noise_hf(f> 350)), 'color', colors(1, :));
plot(f(f<=350), 180/pi*angle(dvf_enc_sweep( f<=350)), 'color', colors(1, :));
plot(f(f> 350), 180/pi*angle(dvf_int_noise_hf(f> 350)), 'color', colors(2, :));
plot(f(f<=350), 180/pi*angle(dvf_int_sweep( f<=350)), '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]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_comp_enc_int.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_comp_enc_int
#+caption: Comparison of the transfer functions from excitation voltage $V_a$ to either the encoder $d_e$ or the interferometer $d_a$
#+RESULTS:
[[file:figs/strut_1_comp_enc_int.png]]
#+begin_important
It will clearly be difficult to do something (except some low frequency positioning) with the encoders fixed to the APA.
#+end_important
**** APA Resonances Frequency
As shown in Figure [[fig:strut_1_spurious_resonances]], we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
#+begin_src matlab :exports none
figure;
hold on;
plot(f(f> 350), abs(dvf_enc_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(dvf_enc_sweep( f<=350)), 'k-');
text(93, 4e-4, {'93Hz'}, 'VerticalAlignment','bottom','HorizontalAlignment','center')
text(197, 1.3e-4,{'197Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
text(290, 4e-6, {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
text(376, 1.4e-6,{'376Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); xlabel('Frequency [Hz]');
hold off;
ylim([1e-7, 1e-3]); xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_spurious_resonances.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_spurious_resonances
#+caption: Magnitude of the transfer function from excitation voltage $V_a$ to encoder measurement $d_e$. The frequency of the resonances are noted.
#+RESULTS:
[[file:figs/strut_1_spurious_resonances.png]]
These resonances correspond to parasitic resonances of the APA itself.
They are very close to what was estimated using the FEM (Figure [[fig:apa_mode_shapes_bis]]):
- Mode in X-bending at 189Hz
- Mode in Y-bending at 285Hz
- Mode in Z-torsion at 400Hz
#+name: fig:apa_mode_shapes_bis
#+caption: Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz
#+attr_latex: :width \linewidth
[[file:figs/apa_mode_shapes.gif]]
#+begin_important
The resonances are indeed due to limited stiffness of the APA.
#+end_important
**** TODO Estimated Flexible Joint axial stiffness
#+begin_src matlab
load(sprintf('frf_data_leg_coder_%i_add_mass_closed_circuit.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
de = de - de(1);
da = da - da(1);
#+end_src
#+begin_src matlab
figure;
hold on;
plot(t, de, 'DisplayName', 'Encoder')
plot(t, da, 'DisplayName', 'Interferometer')
hold off;
xlabel('Time [s]'); ylabel('Displacement [m]');
legend('location', 'northeast');
#+end_src
#+begin_src matlab
de_steps = [mean(de(t > 13 & t < 14));
mean(de(t > 19 & t < 20));
mean(de(t > 24 & t < 25));
mean(de(t > 28.5 & t < 29.5));
mean(de(t > 49 & t < 50))] - mean(de(t > 13 & t < 14));
da_steps = [mean(da(t > 13 & t < 14));
mean(da(t > 19 & t < 20));
mean(da(t > 24 & t < 25));
mean(da(t > 28.5 & t < 29.5));
mean(da(t > 49 & t < 50))] - mean(da(t > 13 & t < 14));
#+end_src
#+begin_src matlab
added_mass = [0; 1; 2; 3; 4];
#+end_src
#+begin_src matlab
lin_fit = (added_mass\abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)));
#+end_src
#+begin_src matlab
figure;
hold on;
plot(added_mass, abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)), 'ok')
plot(added_mass, added_mass*lin_fit, '-r')
hold off;
#+end_src
#+begin_question
What is strange is that the encoder is measuring a larger displacement than the interferometer.
That should be the opposite.
Maybe is is caused by the fact that the APA is experiencing some bending and therefore a larger motion is measured on the encoder.
#+end_question
**** FRF Identification - IFF
In this section, the dynamics from $V_a$ to $V_s$ is identified.
First the coherence is computed and shown in Figure [[fig:strut_1_frf_iff_plant_coh]].
The coherence is very nice from 10Hz to 2kHz.
It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).
#+begin_src matlab
[coh_enc_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, [], [], 1/Ts);
[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_enc_noise_hf, 'color', colors(1, :), 'DisplayName', 'HF Noise');
plot(f, coh_enc_sweep, 'color', colors(2, :), 'DisplayName', 'Sweep');
hold off;
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:strut_1_frf_iff_plant_coh
#+caption: Obtained coherence for the IFF plant
#+RESULTS:
[[file:figs/strut_1_frf_iff_plant_coh.png]]
Then the FRF are estimated and shown in Figure [[fig:strut_1_enc_frf_iff_plant_tf]]
#+begin_src matlab
[iff_enc_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, [], [], 1/Ts);
[iff_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(iff_enc_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(iff_enc_sweep( f<=350)), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(iff_enc_noise_hf(f> 350)), 'k');
plot(f(f<=350), 180/pi*angle(iff_enc_sweep( f<=350)), 'k');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_enc_frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_enc_frf_iff_plant_tf
#+caption:Identified IFF Plant
#+RESULTS:
[[file:figs/strut_1_enc_frf_iff_plant_tf.png]]
Let's now compare the IFF plants whether the encoders are fixed to the APA or not (Figure [[fig:strut_1_frf_iff_comp_enc]]).
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(iff_enc_noise_hf(f> 350)), 'color', colors(1, :), ...
'DisplayName', 'Encoder');
plot(f(f<=350), abs(iff_enc_sweep( f<=350)), 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(f(f> 350), abs(iff_noise_hf(f> 350)), 'color', colors(2, :), ...
'DisplayName', 'Without Encoder');
plot(f(f<=350), abs(iff_sweep( f<=350)), 'color', colors(2, :), ...
'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(iff_enc_noise_hf(f> 350)), 'color', colors(1, :));
plot(f(f<=350), 180/pi*angle(iff_enc_sweep( f<=350)), 'color', colors(1, :));
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350)), 'color', colors(2, :));
plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350)), '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]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_1_frf_iff_effect_enc.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_1_frf_iff_comp_enc
#+caption: Effect of the encoder on the IFF plant
#+RESULTS:
[[file:figs/strut_1_frf_iff_effect_enc.png]]
#+begin_important
We can see that the IFF does not change whether of not the encoder are fixed to the struts.
#+end_important
** Comparison of all the Struts
<>
*** Introduction :ignore:
Now all struts are measured using the same procedure and test bench.
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab
colors = colororder;
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src
*** FRF Identification - Setup
The identification is performed in two steps:
1. White noise excitation with small amplitude.
This is used to estimate the low frequency dynamics.
2. High frequency noise.
The noise is band-passed between 300Hz and 2kHz.
Then, the result of the first identification is used between 10Hz and 350Hz and the result of the second identification if used between 350Hz and 2kHz.
Here are the LEG numbers that have been measured.
#+begin_src matlab
leg_nums = [1 2 3 4 5];
#+end_src
The data are loaded for both the first and second identification:
#+begin_src matlab
%% Second identification
leg_noise = {};
for i = 1:length(leg_nums)
leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
%% Third identification
leg_noise_hf = {};
for i = 1:length(leg_nums)
leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
#+end_src
The time is the same for all measurements.
#+begin_src matlab
%% Time vector
t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s]
%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
#+end_src
Then we defined a "Hanning" windows that will be used for the spectral analysis:
#+begin_src matlab
win = hanning(ceil(0.5*Fs)); % Hannning Windows
#+end_src
We get the frequency vector that will be the same for all the frequency domain analysis.
#+begin_src matlab
% Only used to have the frequency vector "f"
[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, [], [], 1/Ts);
#+end_src
*** FRF Identification - DVF
In this section, the dynamics from $V_a$ to $d_e$ is identified.
We compute the coherence for 2nd and 3rd identification:
#+begin_src matlab
%% Coherence computation
coh_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[coh, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.de, win, [], [], 1/Ts);
coh_noise(:, i) = coh;
end
coh_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[coh, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
coh_noise_hf(:, i) = coh;
end
#+end_src
The coherence is shown in Figure [[fig:struts_frf_dvf_plant_coh]].
It is clear that the Noise sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency.
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], ...
'DisplayName', 'HF Noise');
plot(f, coh_noise(:, 1), 'color', [colors(2, :), 0.5], ...
'DisplayName', 'Noise');
for i = 2:length(leg_nums)
plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ...
'HandleVisibility', 'off');
plot(f, coh_noise(:, i), 'color', [colors(2, :), 0.5], ...
'HandleVisibility', 'off');
end;
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/struts_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:struts_frf_dvf_plant_coh
#+caption: Obtained coherence for the plant from $V_a$ to $d_e$
#+RESULTS:
[[file:figs/struts_frf_dvf_plant_coh.png]]
Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoders is computed:
#+begin_src matlab
%% Transfer function estimation
dvf_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.de, win, [], [], 1/Ts);
dvf_noise(:, i) = frf;
end
dvf_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
dvf_noise_hf(:, i) = frf;
end
#+end_src
The obtained transfer functions are shown in Figure [[fig:struts_frf_dvf_plant_tf]].
They are all superimposed except for the LEG7.
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), abs(dvf_noise_hf(f> 350, i)), 'color', colors(i, :), ...
'DisplayName', sprintf('Leg %i', leg_nums(i)));
plot(f(f<=350), abs(dvf_noise( f<=350, i)), 'color', colors(i, :), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-9, 1e-3]);
ax2 = nexttile;
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350, i)), 'color', colors(i, :));
plot(f(f<=350), 180/pi*angle(dvf_noise( f<=350, i)), 'color', colors(i, :));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/struts_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:struts_frf_dvf_plant_tf
#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)
#+RESULTS:
[[file:figs/struts_frf_dvf_plant_tf.png]]
#+begin_important
Depending on how the APA are mounted with the flexible joints, the dynamics can change a lot as shown in Figure [[fig:struts_frf_dvf_plant_tf]].
In the future, a "pin" will be used to better align the APA with the flexible joints.
We can expect the amplitude of the spurious resonances to decrease.
#+end_important
*** FRF Identification - DVF with interferometer
In this section, the dynamics from $V_a$ to $d_a$ is identified.
We compute the coherence.
#+begin_src matlab
%% Coherence computation
coh_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[coh, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.da, win, [], [], 1/Ts);
coh_noise(:, i) = coh;
end
coh_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[coh, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
coh_noise_hf(:, i) = coh;
end
#+end_src
The coherence is shown in Figure [[fig:struts_frf_int_plant_coh]].
It is clear that the Noise sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency.
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], ...
'DisplayName', 'HF Noise');
plot(f, coh_noise(:, 1), 'color', [colors(2, :), 0.5], ...
'DisplayName', 'Noise');
for i = 2:length(leg_nums)
plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ...
'HandleVisibility', 'off');
plot(f, coh_noise(:, i), 'color', [colors(2, :), 0.5], ...
'HandleVisibility', 'off');
end;
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/struts_frf_int_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:struts_frf_int_plant_coh
#+caption: Obtained coherence for the plant from $V_a$ to $d_e$
#+RESULTS:
[[file:figs/struts_frf_int_plant_coh.png]]
Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the Attocube is computed:
#+begin_src matlab
%% Transfer function estimation
dvf_a_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.da, win, [], [], 1/Ts);
dvf_a_noise(:, i) = frf;
end
dvf_a_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
dvf_a_noise_hf(:, i) = frf;
end
#+end_src
The obtained transfer functions are shown in Figure [[fig:struts_frf_int_plant_tf]].
They are all superimposed except for the LEG7.
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), abs(dvf_a_noise_hf(f> 350, i)), 'color', colors(i, :), ...
'DisplayName', sprintf('Leg %i', leg_nums(i)));
plot(f(f<=350), abs(dvf_a_noise( f<=350, i)), 'color', colors(i, :), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-9, 1e-3]);
ax2 = nexttile;
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), 180/pi*angle(dvf_a_noise_hf(f> 350, i)), 'color', colors(i, :));
plot(f(f<=350), 180/pi*angle(dvf_a_noise( f<=350, i)), 'color', colors(i, :));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/struts_frf_int_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:struts_frf_int_plant_tf
#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)
#+RESULTS:
[[file:figs/struts_frf_int_plant_tf.png]]
*** FRF Identification - IFF
In this section, the dynamics from $V_a$ to $V_s$ is identified.
First the coherence is computed and shown in Figure [[fig:struts_frf_iff_plant_coh]].
The coherence is very nice from 10Hz to 2kHz.
It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).
#+begin_src matlab
%% Coherence
coh_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[coh, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.Vs, win, [], [], 1/Ts);
coh_noise(:, i) = coh;
end
coh_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[coh, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
coh_noise_hf(:, i) = coh;
end
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], 'DisplayName', 'HF Noise');
plot(f, coh_noise( :, 1), 'color', [colors(2, :), 0.5], 'DisplayName', 'Noise');
for i = 2:length(leg_nums)
plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ...
'HandleVisibility', 'off');
plot(f, coh_noise( :, i), 'color', [colors(2, :), 0.5], ...
'HandleVisibility', 'off');
end;
hold off;
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlim([5, 5e3]); ylim([0, 1]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/struts_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:struts_frf_iff_plant_coh
#+caption: Obtained coherence for the IFF plant
#+RESULTS:
[[file:figs/struts_frf_iff_plant_coh.png]]
Then the FRF are estimated and shown in Figure [[fig:struts_frf_iff_plant_tf]]
#+begin_src matlab
%% FRF estimation of the transfer function from Va to Vs
iff_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.Vs, win, [], [], 1/Ts);
iff_noise(:, i) = frf;
end
iff_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
iff_noise_hf(:, i) = frf;
end
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', colors(i, :), ...
'DisplayName', sprintf('Leg %i', leg_nums(i)));
plot(f(f<=350), abs(iff_noise( f<=350, i)), 'color', colors(i, :), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ax2 = nexttile;
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', colors(i, :));
plot(f(f<=350), 180/pi*angle(iff_noise( f<=350, i)), 'color', colors(i, :));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/struts_frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:struts_frf_iff_plant_tf
#+caption:Identified IFF Plant
#+RESULTS:
[[file:figs/struts_frf_iff_plant_tf.png]]
* TODO Compare with the FEM/Simscape Model :noexport:
:PROPERTIES:
:header-args:matlab+: :tangle matlab/APA300ML.m
:END:
** Introduction :ignore:
In this section, the Amplified Piezoelectric Actuator APA300ML ([[file:doc/APA300ML.pdf][doc]]) is modeled using a Finite Element Software.
Then a /super element/ is exported and imported in Simscape where its dynamic is studied.
A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure [[fig:apa300ml_ansys]].
The remote point used are also shown in this figure.
#+name: fig:apa300ml_ansys
#+caption: Ansys FEM of the APA300ML
[[file:figs/apa300ml_ansys.jpg]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no
addpath('matlab/');
addpath('matlab/APA300ML/');
#+end_src
#+begin_src matlab :eval no
addpath('APA300ML/');
#+end_src
#+begin_src matlab
open('APA300ML.slx');
#+end_src
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
We first extract the stiffness and mass matrices.
#+begin_src matlab
K = readmatrix('APA300ML_mat_K.CSV');
M = readmatrix('APA300ML_mat_M.CSV');
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src
#+caption: First 10x10 elements of the Stiffness matrix
#+RESULTS:
| 200000000.0 | 30000.0 | -20000.0 | -70.0 | 300000.0 | 40.0 | 10000000.0 | 10000.0 | -6000.0 | 30.0 |
| 30000.0 | 30000000.0 | 2000.0 | -200000.0 | 60.0 | -10.0 | 4000.0 | 2000000.0 | -500.0 | 9000.0 |
| -20000.0 | 2000.0 | 7000000.0 | -10.0 | -30.0 | 10.0 | 6000.0 | 900.0 | -500000.0 | 3 |
| -70.0 | -200000.0 | -10.0 | 1000.0 | -0.1 | 0.08 | -20.0 | -9000.0 | 3 | -30.0 |
| 300000.0 | 60.0 | -30.0 | -0.1 | 900.0 | 0.1 | 30000.0 | 20.0 | -10.0 | 0.06 |
| 40.0 | -10.0 | 10.0 | 0.08 | 0.1 | 10000.0 | 20.0 | 9 | -5 | 0.03 |
| 10000000.0 | 4000.0 | 6000.0 | -20.0 | 30000.0 | 20.0 | 200000000.0 | 10000.0 | 9000.0 | 50.0 |
| 10000.0 | 2000000.0 | 900.0 | -9000.0 | 20.0 | 9 | 10000.0 | 30000000.0 | -500.0 | 200000.0 |
| -6000.0 | -500.0 | -500000.0 | 3 | -10.0 | -5 | 9000.0 | -500.0 | 7000000.0 | -2 |
| 30.0 | 9000.0 | 3 | -30.0 | 0.06 | 0.03 | 50.0 | 200000.0 | -2 | 1000.0 |
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
#+end_src
#+caption: First 10x10 elements of the Mass matrix
#+RESULTS:
| 0.01 | -2e-06 | 1e-06 | 6e-09 | 5e-05 | -5e-09 | -0.0005 | -7e-07 | 6e-07 | -3e-09 |
| -2e-06 | 0.01 | 8e-07 | -2e-05 | -8e-09 | 2e-09 | -9e-07 | -0.0002 | 1e-08 | -9e-07 |
| 1e-06 | 8e-07 | 0.009 | 5e-10 | 1e-09 | -1e-09 | -5e-07 | 3e-08 | 6e-05 | 1e-10 |
| 6e-09 | -2e-05 | 5e-10 | 3e-07 | 2e-11 | -3e-12 | 3e-09 | 9e-07 | -4e-10 | 3e-09 |
| 5e-05 | -8e-09 | 1e-09 | 2e-11 | 6e-07 | -4e-11 | -1e-06 | -2e-09 | 1e-09 | -8e-12 |
| -5e-09 | 2e-09 | -1e-09 | -3e-12 | -4e-11 | 1e-07 | -2e-09 | -1e-09 | -4e-10 | -5e-12 |
| -0.0005 | -9e-07 | -5e-07 | 3e-09 | -1e-06 | -2e-09 | 0.01 | 1e-07 | -3e-07 | -2e-08 |
| -7e-07 | -0.0002 | 3e-08 | 9e-07 | -2e-09 | -1e-09 | 1e-07 | 0.01 | -4e-07 | 2e-05 |
| 6e-07 | 1e-08 | 6e-05 | -4e-10 | 1e-09 | -4e-10 | -3e-07 | -4e-07 | 0.009 | -2e-10 |
| -3e-09 | -9e-07 | 1e-10 | 3e-09 | -8e-12 | -5e-12 | -2e-08 | 2e-05 | -2e-10 | 3e-07 |
Then, we extract the coordinates of the interface nodes.
#+begin_src matlab
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA300ML_out_nodes_3D.txt');
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src
#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+---------+-------+--------|
| 1.0 | 697783.0 | 0.0 | 0.0 | -0.015 |
| 2.0 | 697784.0 | 0.0 | 0.0 | 0.015 |
| 3.0 | 697785.0 | -0.0325 | 0.0 | 0.0 |
| 4.0 | 697786.0 | -0.0125 | 0.0 | 0.0 |
| 5.0 | 697787.0 | -0.0075 | 0.0 | 0.0 |
| 6.0 | 697788.0 | 0.0125 | 0.0 | 0.0 |
| 7.0 | 697789.0 | 0.0325 | 0.0 | 0.0 |
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src
#+caption: Some extracted parameters of the FEM
#+RESULTS:
| Total number of Nodes | 7 |
| Number of interface Nodes | 7 |
| Number of Modes | 120 |
| Size of M and K matrices | 162 |
Using =K=, =M= and =int_xyz=, we can now use the =Reduced Order Flexible Solid= simscape block.
** Piezoelectric parameters
#+begin_src matlab
Ga = 1; % [N/V]
Gs = 1; % [V/m]
#+end_src
#+begin_src matlab
m = 0.1; % [kg]
#+end_src
** Simscape Model
The flexible element is imported using the =Reduced Order Flexible Solid= simscape block.
Let's say we use two stacks as a force sensor and one stack as an actuator:
- A =Relative Motion Sensor= block is added between the nodes A and C
- An =Internal Force= block is added between the remote points E and B
The interface nodes are shown in Figure [[fig:apa300ml_ansys]].
One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).
** Identification of the APA Characteristics
*** Stiffness
#+begin_src matlab :exports none
m = 0.0001;
#+end_src
The transfer function from vertical external force to the relative vertical displacement is identified.
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
#+end_src
The inverse of its DC gain is the axial stiffness of the APA:
#+begin_src matlab :results replace value
1e-6/dcgain(G) % [N/um]
#+end_src
#+RESULTS:
: 1.753
The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$.
*** Resonance Frequency
The resonance frequency is specified to be between 650Hz and 840Hz.
This is also the case for the FEM model (Figure [[fig:apa300ml_resonance]]).
#+begin_src matlab :exports none
freqs = logspace(2, 4, 5000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude');
hold off;
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_resonance.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_resonance
#+caption: First resonance is around 800Hz
#+RESULTS:
[[file:figs/apa300ml_resonance.png]]
*** Amplification factor
The amplification factor is the ratio of the vertical displacement to the stack displacement.
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
#+end_src
The ratio of the two displacement is computed from the FEM model.
#+begin_src matlab :results replace value
abs(dcgain(G(1,1))./dcgain(G(2,1)))
#+end_src
#+RESULTS:
: 5.0749
This is actually correct and approximately corresponds to the ratio of the piezo height and length:
#+begin_src matlab :results replace value
75/15
#+end_src
#+RESULTS:
: 5
*** Stroke
Estimation of the actuator stroke:
\[ \Delta H = A n \Delta L \]
with:
- $\Delta H$ Axial Stroke of the APA
- $A$ Amplification factor (5 for the APA300ML)
- $n$ Number of stack used
- $\Delta L$ Stroke of the stack (0.1% of its length)
#+begin_src matlab :results replace value
1e6 * 5 * 3 * 20e-3 * 0.1e-2
#+end_src
#+RESULTS:
: 300
This is exactly the specified stroke in the data-sheet.
*** TODO Stroke BIS
- [ ] Identified the stroke form the transfer function from V to z
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/V'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
1e6*170*abs(dcgain(G))
#+end_src
** Identification of the Dynamics from actuator to replace displacement
We first set the mass to be approximately zero.
#+begin_src matlab :exports none
m = 0.01;
#+end_src
The dynamics is identified from the applied force to the measured relative displacement.
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
Gh = -linearize(mdl, io);
#+end_src
The same dynamics is identified for a payload mass of 10Kg.
#+begin_src matlab
m = 10;
#+end_src
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
Ghm = -linearize(mdl, io);
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 4, 5000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gh, freqs, 'Hz')))), '-', ...
'DisplayName', '$m = 0kg$');
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))), '-', ...
'DisplayName', '$m = 10kg$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
ylim([-360 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
legend('location', 'southwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_plant_dynamics.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa300ml_plant_dynamics
#+caption: Transfer function from forces applied by the stack to the axial displacement of the APA
#+RESULTS:
[[file:figs/apa300ml_plant_dynamics.png]]
The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure [[fig:apa300ml_dvf_root_locus]].
#+begin_src matlab :exports none
figure;
gains = logspace(0, 5, 500);
hold on;
plot(real(pole(Ghm)), imag(pole(G)), 'kx');
plot(real(tzero(Ghm)), imag(tzero(G)), 'ko');
for k = 1:length(gains)
cl_poles = pole(feedback(Ghm, gains(k)*s));
plot(real(cl_poles), imag(cl_poles), 'k.');
end
hold off;
axis square;
xlim([-500, 10]); ylim([0, 510]);
xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa300ml_dvf_root_locus
#+caption: Root Locus for Direct Velocity Feedback
#+RESULTS:
[[file:figs/apa300ml_dvf_root_locus.png]]
** Identification of the Dynamics from actuator to force sensor
Let's use 2 stacks as a force sensor and 1 stack as force actuator.
The transfer function from actuator voltage to sensor voltage is identified and shown in Figure [[fig:apa300ml_iff_plant]].
#+begin_src matlab :exports none
m = 10;
#+end_src
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
Giff = -linearize(mdl, io);
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 4, 5000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
ylim([-180 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_iff_plant.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa300ml_iff_plant
#+caption: Transfer function from actuator to force sensor
#+RESULTS:
[[file:figs/apa300ml_iff_plant.png]]
For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]].
#+begin_src matlab :exports none
figure;
gains = logspace(0, 5, 500);
hold on;
plot(real(pole(Giff)), imag(pole(Giff)), 'kx');
plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko');
for k = 1:length(gains)
cl_poles = pole(feedback(Giff, gains(k)/s));
plot(real(cl_poles), imag(cl_poles), 'k.');
end
hold off;
axis square;
xlim([-500, 10]); ylim([0, 510]);
xlabel('Real Part'); ylabel('Imaginary Part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa300ml_iff_root_locus
#+caption: Root Locus for IFF
#+RESULTS:
[[file:figs/apa300ml_iff_root_locus.png]]
** Identification for a simpler model
The goal in this section is to identify the parameters of a simple APA model from the FEM.
This can be useful is a lower order model is to be used for simulations.
The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
The parameters are shown in the table below.
#+name: fig:souleille18_model_piezo
#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
[[file:./figs/souleille18_model_piezo.png]]
#+caption:Parameters used for the model of the APA 100M
| | Meaning |
|-------+----------------------------------------------------------------|
| $k_e$ | Stiffness used to adjust the pole of the isolator |
| $k_1$ | Stiffness of the metallic suspension when the stack is removed |
| $k_a$ | Stiffness of the actuator |
| $c_1$ | Added viscous damping |
The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model.
\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
If we can fix $k_a$, we can determine $k_e$ and $k_1$ with:
\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
#+begin_src matlab :exports none
m = 10;
#+end_src
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m]
G = linearize(mdl, io);
G.InputName = {'Fd', 'w', 'Fa'};
G.OutputName = {'y', 'Fs', 'd'};
#+end_src
From the identified dynamics, compute $\alpha$ and $\beta$
#+begin_src matlab
alpha = abs(dcgain(G('y', 'Fa')));
beta = abs(dcgain(G('y', 'Fd')));
#+end_src
$k_a$ is estimated using the following formula:
#+begin_src matlab
ka = 0.8/abs(dcgain(G('y', 'Fa')));
#+end_src
The factor can be adjusted to better match the curves.
Then $k_e$ and $k_1$ are computed.
#+begin_src matlab
ke = ka/(beta/alpha - 1);
k1 = 1/beta - ke*ka/(ke + ka);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f ');
#+end_src
#+RESULTS:
| | Value [N/um] |
|----+--------------|
| ka | 40.5 |
| ke | 1.5 |
| k1 | 0.4 |
The damping in the system is adjusted to match the FEM model if necessary.
#+begin_src matlab
c1 = 1e2;
#+end_src
The analytical model of the simpler system is defined below:
#+begin_src matlab
Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
[ 1 , k1 + c1*s + ke*ka/(ke + ka) , ke/(ke + ka) ;
-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
Ga.InputName = {'Fd', 'w', 'Fa'};
Ga.OutputName = {'y', 'Fs'};
#+end_src
And the DC gain is adjusted for the force sensor:
#+begin_src matlab
F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
#+end_src
The dynamics of the FEM model and the simpler model are compared in Figure [[fig:apa300ml_comp_simpler_model]].
#+begin_src matlab :exports none
freqs = logspace(0, 5, 1000);
figure;
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/w$ [m/m]');
ylim([1e-6, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/f$ [m/N]');
ylim([1e-14, 1e-6]);
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/F$ [m/N]');
ylim([1e-14, 1e-4]);
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/w$ [m/m]');
ylim([1e2, 1e8]);
ax5 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/f$ [m/N]');
ylim([1e-4, 1e1]);
ax6 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/F$ [m/N]');
ylim([1e-7, 1e2]);
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:apa300ml_comp_simpler_model
#+caption: Comparison of the Dynamics between the FEM model and the simplified one
#+RESULTS:
[[file:figs/apa300ml_comp_simpler_model.png]]
The simplified model has also been implemented in Simscape.
The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure [[fig:apa300ml_comp_simpler_simscape]].
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML_simplified';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
Gs = linearize(mdl, io);
Gs.InputName = {'Fd', 'w', 'Fa'};
Gs.OutputName = {'y', 'Fs'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 5, 1000);
figure;
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/w$ [m/m]');
ylim([1e-6, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/f$ [m/N]');
ylim([1e-14, 1e-6]);
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('$x_1/F$ [m/N]');
ylim([1e-14, 1e-4]);
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/w$ [m/m]');
ylim([1e2, 1e8]);
ax5 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/f$ [m/N]');
ylim([1e-4, 1e1]);
ax6 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fd'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('$F_s/F$ [m/N]');
ylim([1e-7, 1e2]);
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_comp_simpler_simscape.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:apa300ml_comp_simpler_simscape
#+caption: Comparison of the Dynamics between the FEM model and the simplified simscape model
#+RESULTS:
[[file:figs/apa300ml_comp_simpler_simscape.png]]
** Integral Force Feedback
In this section, Integral Force Feedback control architecture is applied on the APA300ML.
First, the plant (dynamics from voltage actuator to voltage sensor is identified).
#+begin_src matlab :exports none
Kiff = tf(0);
#+end_src
The payload mass is set to 10kg.
#+begin_src matlab
m = 10;
#+end_src
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'APA300ML_IFF';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/APA300ML'], 1, 'openoutput'); io_i = io_i + 1;
G_ol = linearize(mdl, io);
G_ol.InputName = {'w', 'f', 'F'};
G_ol.OutputName = {'x1', 'Fs'};
G = G_ol({'Fs'}, {'f'});
#+end_src
The obtained dynamics is shown in Figure [[fig:piezo_amplified_iff_plant]].
#+begin_src matlab :exports none
freqs = logspace(1, 5, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
ylim([-390 30]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/piezo_amplified_iff_plant.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:piezo_amplified_iff_plant
#+caption: IFF Plant
#+RESULTS:
[[file:figs/piezo_amplified_iff_plant.png]]
The controller is defined below and the loop gain is shown in Figure [[fig:piezo_amplified_iff_loop_gain]].
#+begin_src matlab
Kiff = -1e3/s;
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 5, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*Kiff, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*Kiff, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360);
ylim([-180 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/piezo_amplified_iff_loop_gain.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:piezo_amplified_iff_loop_gain
#+caption: IFF Loop Gain
#+RESULTS:
[[file:figs/piezo_amplified_iff_loop_gain.png]]
Now the closed-loop system is identified again and compare with the open loop system in Figure [[fig:piezo_amplified_iff_comp]].
It is the expected behavior as shown in the Figure [[fig:souleille18_results]] (from cite:souleille18_concep_activ_mount_space_applic).
#+begin_src matlab :exports none
clear io; io_i = 1;
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/APA300ML'], 1, 'output'); io_i = io_i + 1;
Giff = linearize(mdl, io);
Giff.InputName = {'w', 'f', 'F'};
Giff.OutputName = {'x1', 'Fs'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('$x_1/w$ [m/m]')
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'f'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('$x_1/f$ [m/N]');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('$x_1/F$ [m/N]');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'w'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('$F_s/w$ [N/m]');
ax5 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'f'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('$F_s/f$ [N/N]');
ax6 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'F'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('$F_s/F$ [N/N]');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/piezo_amplified_iff_comp.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:piezo_amplified_iff_comp
#+caption: OL and CL transfer functions
#+RESULTS:
[[file:figs/piezo_amplified_iff_comp.png]]
#+name: fig:souleille18_results
#+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic
[[file:figs/souleille18_results.png]]
* Test Bench APA300ML - Simscape Model
** Introduction
A simscape model (Figure [[fig:model_bench_apa]]) of the measurement bench is used.
#+name: fig:model_bench_apa
#+caption: Screenshot of the Simscape model
#+attr_latex: :width 0.5\linewidth
[[file:figs/model_bench_apa.png]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no
%% Add useful folders to the path
addpath('matlab/');
addpath('matlab/test_bench_apa300ml/');
addpath('matlab/mat/');
addpath('matlab/src/');
addpath('matlab/png/');
#+end_src
#+begin_src matlab :eval no
%% Add useful folders to the path
addpath('test_bench_apa300ml/');
addpath('png/');
addpath('mat/');
addpath('src/');
#+end_src
#+begin_src matlab
%% Frequency vector used for many plots
freqs = 2*logspace(0, 3, 1000);
#+end_src
#+begin_src matlab
%% Open Simscape Model
options = linearizeOptions;
options.SampleTime = 0;
% Name of the Simulink File
mdl = 'test_bench_apa300ml';
open(mdl)
#+end_src
** First Identification
The APA is first initialized with default parameters and the transfer function from excitation voltage $V_a$ (before the amplification of 20 due to the PD200 amplifier) to the sensor stack voltage $V_s$, encoder $d_L$ and interferometer $z$ is identified.
#+begin_src matlab
%% Initialize the structure containing the data used in the model
n_hexapod = struct();
n_hexapod.actuator = initializeAPA('type', '2dof');
#+end_src
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Motion
%% Run the linearization
Ga = linearize(mdl, io, 0.0, options);
Ga.InputName = {'Va'};
Ga.OutputName = {'Vs', 'dL', 'z'};
#+end_src
The obtain dynamics are shown in Figure [[fig:apa_model_bench_bode_vs]] and [[fig:apa_model_bench_bode_dl_z]].
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to taum
freqs = logspace(1, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'k-')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'k-')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([0, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_model_bench_bode_vs.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa_model_bench_bode_vs
#+caption: Bode plot of the transfer function from $V_a$ to $V_s$
#+RESULTS:
[[file:figs/apa_model_bench_bode_vs.png]]
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm and dZm
freqs = logspace(1, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Ga('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
plot(freqs, abs(squeeze(freqresp(Ga('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('z', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_model_bench_bode_dl_z.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_model_bench_bode_dl_z
#+caption: Bode plot of the transfer function from $V_a$ to $d_L$ and to $z$
#+RESULTS:
[[file:figs/apa_model_bench_bode_dl_z.png]]
** Identify Sensor/Actuator constants and compare with measured FRF
*** How to identify these constants?
**** 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 $d$.
\begin{equation}
d = g_{d/V_a} \cdot V_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 $d$:
\begin{equation}
d = g_{d/F_a} \cdot F_a
\end{equation}
From the two gains, it is then easy to determine $g_a$:
\begin{equation}
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}
**** 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 $V_s$:
\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 piezo capacitor and parallel resistor.
The gain can be computed from the dynamical identification and taking the gain at the wanted frequency (above the first resonance).
Using the simscape model, compute the gain at the same frequency from the actuator force $F_a$ to the strain of the sensor stack $dl$:
\begin{equation}
dl = g_{dl/F_a} F_a
\end{equation}
Then, the "sensor" constant is:
\begin{equation}
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}
*** Identification Data
#+begin_src matlab
%% Load the identification Data
leg_sweep = load(sprintf('frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
leg_noise_hf = load(sprintf('frf_data_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
#+end_src
The time is the same for all measurements.
#+begin_src matlab
%% Time vector
t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s]
%% Sampling Time / Frequency
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
#+end_src
Then we defined a "Hanning" windows that will be used for the spectral analysis:
#+begin_src matlab
%% Window used for spectral analysis
win = hanning(ceil(0.5*Fs)); % Hannning Windows
#+end_src
We get the frequency vector that will be the same for all the frequency domain analysis.
#+begin_src matlab
%% Get the frequency vector "f" common to all further spectral data
[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab
%% Estimate the transfer function from u to dLm
[dvf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts);
[dvf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
#+end_src
#+begin_src matlab
%% Estimate the transfer function from u to taum
[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
#+end_src
*** 2DoF APA
**** 2DoF APA
#+begin_src matlab
%% Initialize a 2DoF APA with Ga=Gs=1
n_hexapod.actuator = initializeAPA('type', '2dof', 'ga', 1, 'gs', 1);
#+end_src
**** Identification without actuator or sensor constants
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Motion
%% Identification
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};
#+end_src
**** Actuator Constant
#+begin_src matlab
%% Estimated Actuator Constant
ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('ga = %.1f [N/V]', ga);
#+end_src
#+RESULTS:
: ga = -32.1 [N/V]
#+begin_src matlab
n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V]
#+end_src
**** Sensor Constant
#+begin_src matlab
%% Estimated Sensor Constant
gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('gs = %.3f [V/m]', gs);
#+end_src
#+RESULTS:
: gs = 0.085 [V/m]
#+begin_src matlab
n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m]
#+end_src
**** Comparison
Identify the dynamics with included constants.
#+begin_src matlab
%% Identify again the dynamics with correct Ga,Gs
Gs = linearize(mdl, io, 0.0, options);
Gs = Gs*exp(-Ts*s);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};
#+end_src
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm (both Simscape and measured FRF)
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(dvf_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(dvf_sweep( f<=350)), 'k-');
plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350)), 'k-');
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_act_constant_comp.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_act_constant_comp
#+caption: Comparison of the experimental data and Simscape model ($u$ to $d\mathcal{L}_m$)
#+RESULTS:
[[file:figs/apa_act_constant_comp.png]]
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to taum (both Simscape and measured FRF)
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(iff_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(iff_sweep( f<=350)), 'k-');
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350)), 'k-');
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_sens_constant_comp.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_sens_constant_comp
#+caption: Comparison of the experimental data and Simscape model ($V_a$ to $V_s$)
#+RESULTS:
[[file:figs/apa_sens_constant_comp.png]]
*** Flexible APA
**** Flexible APA
#+begin_src matlab
%% Initialize the APA as a flexible body
n_hexapod.actuator = initializeAPA('type', 'flexible', 'ga', 1, 'gs', 1);
#+end_src
**** Identification without actuator or sensor constants
#+begin_src matlab
%% Identify the dynamics
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};
#+end_src
**** Actuator Constant
#+begin_src matlab
%% Actuator Constant
ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('ga = %.1f [N/V]', ga);
#+end_src
#+RESULTS:
: ga = 23.4 [N/V]
#+begin_src matlab
n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V]
#+end_src
**** Sensor Constant
#+begin_src matlab
%% Sensor Constant
gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('gs = %.3f [V/m]', gs);
#+end_src
#+RESULTS:
: gs = -4674826.805 [V/m]
#+begin_src matlab
n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m]
#+end_src
**** Comparison
#+begin_src matlab
%% Identify with updated constants
Gs = linearize(mdl, io, 0.0, options);
Gs = Gs*exp(-Ts*s);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};
#+end_src
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm (both Simscape and measured FRF)
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(dvf_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(dvf_sweep( f<=350)), 'k-');
plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-9, 1e-3]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350)), 'k-');
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_act_constant_comp_flex.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_act_constant_comp_flex
#+caption: Comparison of the experimental data and Simscape model ($u$ to $d\mathcal{L}_m$)
#+RESULTS:
[[file:figs/apa_act_constant_comp_flex.png]]
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to taum (both Simscape and measured FRF)
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f(f> 350), abs(iff_noise_hf(f> 350)), 'k-');
plot(f(f<=350), abs(iff_sweep( f<=350)), 'k-');
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350)), 'k-');
plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350)), 'k-');
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_sens_constant_comp_flex.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_sens_constant_comp_flex
#+caption: Comparison of the experimental data and Simscape model ($u$ to $\tau_m$)
#+RESULTS:
[[file:figs/apa_sens_constant_comp_flex.png]]
** Optimize 2-DoF model to fit the experimental Data
#+begin_src matlab
%% Optimized parameters
n_hexapod.actuator = initializeAPA('type', '2dof', ...
'Ga', -30.0, ...
'Gs', 0.098, ...
'k', ones(6,1)*0.38e6, ...
'ke', ones(6,1)*1.75e6, ...
'ka', ones(6,1)*3e7, ...
'c', ones(6,1)*1.3e2, ...
'ce', ones(6,1)*1e1, ...
'ca', ones(6,1)*1e1 ...
);
#+end_src
#+begin_src matlab :exports none
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
%% Identification with optimized parameters
Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL'};
#+end_src
#+begin_src matlab :exports none
apa_nums = [1 2 4 5 6 8];
%% Second identification
apa_sweep = {};
for i = 1:length(apa_nums)
apa_sweep(i) = {load(sprintf('frf_data_%i_sweep.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
%% Third identification
apa_noise_hf = {};
for i = 1:length(apa_nums)
apa_noise_hf(i) = {load(sprintf('frf_data_%i_noise_hf.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
%% Time vector
t = apa_sweep{1}.t - apa_sweep{1}.t(1) ; % Time vector [s]
%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
win = hanning(ceil(0.5*Fs)); % Hannning Windows
% Only used to have the frequency vector "f"
[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts);
%% Transfer function estimation
dvf_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
dvf_sweep(:, i) = frf;
end
dvf_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
dvf_noise_hf(:, i) = frf;
end
%% FRF estimation of the transfer function from Va to Vs
iff_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
iff_sweep(:, i) = frf;
end
iff_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
[frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
iff_noise_hf(:, i) = frf;
end
#+end_src
#+begin_src matlab :exports none
%% Comparison of the experimental data and Simscape Model
freqs = 5*logspace(0, 3, 1000);
figure;
tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), abs(dvf_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), abs(dvf_sweep( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
ax1b = nexttile([2,1]);
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), abs(iff_sweep( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
ax2b = nexttile;
hold on;
for i = 1:length(apa_nums)
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2,ax1b,ax2b],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_apa_plant_after_opt.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:comp_apa_plant_after_opt
#+caption: Comparison of the measured FRF and the optimized model
#+RESULTS:
[[file:figs/comp_apa_plant_after_opt.png]]
** Compare 2-DoF with flexible :noexport:
APA - 2 DoF
#+begin_src matlab
n_hexapod.actuator = initializeAPA('type', '2dof');
G_2dof = linearize(mdl, io, 0.0, options);
G_2dof.InputName = {'Va'};
G_2dof.OutputName = {'Vs', 'dL', 'z'};
#+end_src
APA - Fully Flexible
#+begin_src matlab
n_hexapod.actuator = initializeAPA('type', 'flexible');
G_flex = linearize(mdl, io, 0.0, options);
G_flex.InputName = {'Va'};
G_flex.OutputName = {'Vs', 'dL', 'z'};
#+end_src
Comparison
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', '2DoF')
plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flexible')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southeast');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('Vs', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_effect_joint_comp_vs.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_effect_joint_comp_vs
#+caption: Effect of the joint's flexibility on the transfer function from $V_a$ to $V_s$
#+RESULTS:
[[file:figs/apa_effect_joint_comp_vs.png]]
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_2dof('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', '2DoF')
plot(freqs, abs(squeeze(freqresp(G_flex('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flexible')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('dL', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_effect_joint_comp_dl.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_effect_joint_comp_dl
#+caption: Effect of the joint's flexibility on the transfer function from $V_a$ to $d_L$ (encoder)
#+RESULTS:
[[file:figs/apa_effect_joint_comp_dl.png]]
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_2dof('z', 'Va'), freqs, 'Hz'))), 'DisplayName', '2DoF')
plot(freqs, abs(squeeze(freqresp(G_flex('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flexible')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $z/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('z', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa_effect_joint_comp_z.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:apa_effect_joint_comp_z
#+caption: Effect of the joint's flexibility on the transfer function from $V_a$ to $z$ (interferometer)
#+RESULTS:
[[file:figs/apa_effect_joint_comp_z.png]]
* Test Bench Struts - Simscape Model
** Introduction
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no
%% Add useful folders to the path
addpath('matlab/');
addpath('matlab/test_bench_struts/');
addpath('matlab/mat/');
addpath('matlab/src/');
addpath('matlab/png/');
#+end_src
#+begin_src matlab :eval no
%% Add useful folders to the path
addpath('test_bench_struts/');
addpath('png/');
addpath('mat/');
addpath('src/');
#+end_src
#+begin_src matlab
%% Frequency vector used for many plots
freqs = 2*logspace(0, 3, 1000);
#+end_src
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'test_bench_struts';
%% Open the Simulink File
open(mdl)
#+end_src
** First Identification
The object containing all the data is created.
#+begin_src matlab
%% Initialize structure containing data for the Simscape model
n_hexapod = struct();
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '2dof');
n_hexapod.actuator = initializeAPA('type', '2dof');
#+end_src
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Motion
#+end_src
#+begin_src matlab
%% Run the linearization
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};
#+end_src
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to taum
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), 'k-')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), 'k-')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([0, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_bench_model_iff_bode.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:strut_bench_model_iff_bode
#+caption: Identified transfer function from $V_a$ to $V_s$
#+RESULTS:
[[file:figs/strut_bench_model_iff_bode.png]]
#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm and dZm
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
plot(freqs, abs(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_bench_model_dvf_bode.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_bench_model_dvf_bode
#+caption: Identified transfer function from $V_a$ to $d_L$
#+RESULTS:
[[file:figs/strut_bench_model_dvf_bode.png]]
** Effect of flexible joints
Perfect
#+begin_src matlab
%% Perfect Joints
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');
Gp = linearize(mdl, io, 0.0, options);
Gp.InputName = {'Va'};
Gp.OutputName = {'Vs', 'dL', 'z'};
#+end_src
Top Flexible
#+begin_src matlab
%% Top joint with Axial stiffness
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
Gt = linearize(mdl, io, 0.0, options);
Gt.InputName = {'Va'};
Gt.OutputName = {'Vs', 'dL', 'z'};
#+end_src
Bottom Flexible
#+begin_src matlab
%% Bottom joint with Axial stiffness
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');
Gb = linearize(mdl, io, 0.0, options);
Gb.InputName = {'Va'};
Gb.OutputName = {'Vs', 'dL', 'z'};
#+end_src
Both Flexible
#+begin_src matlab
%% Both joints with Axial stiffness
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
Gf = linearize(mdl, io, 0.0, options);
Gf.InputName = {'Va'};
Gf.OutputName = {'Vs', 'dL', 'z'};
#+end_src
Comparison in Figures [[fig:strut_effect_joint_comp_vs]], [[fig:strut_effect_joint_comp_dl]] and [[fig:strut_effect_joint_comp_z]].
#+begin_src matlab :exports none
%% Transfer function from u to taum - Effect of flexible joints
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gp('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Perfect')
plot(freqs, abs(squeeze(freqresp(Gt('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Top')
plot(freqs, abs(squeeze(freqresp(Gb('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Bot')
plot(freqs, abs(squeeze(freqresp(Gf('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flex')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southeast');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vs', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_effect_joint_comp_vs.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_effect_joint_comp_vs
#+caption: Effect of the joint's flexibility on the transfer function from $V_a$ to $V_s$
#+RESULTS:
[[file:figs/strut_effect_joint_comp_vs.png]]
#+begin_src matlab :exports none
%% Transfer function from u to dLm - Effect of flexible joints
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gp('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Perfect')
plot(freqs, abs(squeeze(freqresp(Gt('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Top')
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Bot')
plot(freqs, abs(squeeze(freqresp(Gf('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flex')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('dL', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_effect_joint_comp_dl.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_effect_joint_comp_dl
#+caption: Effect of the joint's flexibility on the transfer function from $V_a$ to $d_L$ (encoder)
#+RESULTS:
[[file:figs/strut_effect_joint_comp_dl.png]]
#+begin_src matlab :exports none
%% Transfer function from u to dZm - Effect of flexible joints
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gp('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Perfect')
plot(freqs, abs(squeeze(freqresp(Gt('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Top')
plot(freqs, abs(squeeze(freqresp(Gb('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Bot')
plot(freqs, abs(squeeze(freqresp(Gf('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flex')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $z/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('z', 'Va'), freqs, 'Hz'))))
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strut_effect_joint_comp_z.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:strut_effect_joint_comp_z
#+caption: Effect of the joint's flexibility on the transfer function from $V_a$ to $z$ (interferometer)
#+RESULTS:
[[file:figs/strut_effect_joint_comp_z.png]]
#+begin_important
Imperfection of the flexible joints has the largest impact on the transfer function from $V_a$ to $d_L$.
However, it has relatively small impact on the transfer functions from $V_a$ to $V_s$ and to $z$.
#+end_important
** Integral Force Feedback
*** Initialize the system
#+begin_src matlab
%% Initialize typical system
n_hexapod = struct();
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');
n_hexapod.actuator = initializeAPA('type', '2dof');
#+end_src
*** Plant Identification
#+begin_src matlab
%% Identify th edynamics
G = linearize(mdl, io, 0.0, options);
G.InputName = {'Va'};
G.OutputName = {'Vs', 'dL', 'z'};
#+end_src
#+begin_src matlab
%% IFF Plant (transfer function from u to taum)
Giff = G('Vs', 'Va');
#+end_src
*** Root Locus
\begin{equation}
K_{\text{IFF}} = \frac{g}{s + \omega_c}
\end{equation}
#+begin_src matlab
%% Cut-off frequency of the LPF
wc = 2*pi*20;
#+end_src
#+begin_src matlab :exports none
% Gains used for the Root Locus plot
gains = logspace(1, 5, 300);
%% Root Locus for the IFF
figure;
hold on;
% Pure Integrator
set(gca,'ColorOrderIndex',1);
plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', '$g = 0$');
set(gca,'ColorOrderIndex',1);
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(Giff, (g/(s + wc))));
set(gca,'ColorOrderIndex',1);
plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
end
g = 2e2;
clpoles = pole(feedback(Giff, (g/(s + wc))));
set(gca,'ColorOrderIndex',2);
plot(real(clpoles), imag(clpoles), 'x', 'DisplayName', sprintf('$g=%.0f$', g));
axis square;
xlim([-700, 0]); ylim([-0, 700]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
** Comparison with the experimental Data
#+begin_src matlab :exports none
%% Measured dynamics
leg_nums = [1 2 3 4 5];
% Second identification
leg_noise = {};
for i = 1:length(leg_nums)
leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
% Third identification
leg_noise_hf = {};
for i = 1:length(leg_nums)
leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end
t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s]
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]
win = hanning(ceil(0.5*Fs)); % Hannning Windows
% Only used to have the frequency vector "f"
[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, [], [], 1/Ts);
% Transfer function estimation
dvf_a_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.da, win, [], [], 1/Ts);
dvf_a_noise(:, i) = frf;
end
dvf_a_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
dvf_a_noise_hf(:, i) = frf;
end
% FRF estimation of the transfer function from Va to Vs
iff_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.Vs, win, [], [], 1/Ts);
iff_noise(:, i) = frf;
end
iff_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
[frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
iff_noise_hf(:, i) = frf;
end
#+end_src
#+begin_src matlab
%% Initialize Simscape data
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
n_hexapod.actuator = initializeAPA('type', '2dof');
#+end_src
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
%% Identification
Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'z'};
#+end_src
#+begin_src matlab :exports none
freqs = 5*logspace(0, 3, 1000);
figure;
tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), abs(dvf_a_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), abs(dvf_a_noise( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))), '-')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
ax1b = nexttile([2,1]);
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), abs(iff_noise( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), '-')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), 180/pi*angle(dvf_a_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), 180/pi*angle(dvf_a_noise( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))), '-')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
ax2b = nexttile;
hold on;
for i = 1:length(leg_nums)
plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]);
plot(f(f<=350), 180/pi*angle(iff_noise( f<=350, i)), 'color', [0,0,0,0.2]);
end
set(gca,'ColorOrderIndex',2);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), '-')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);
linkaxes([ax1,ax2,ax1b,ax2b],'x');
xlim([10, 2e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_strut_plant_after_opt.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:comp_strut_plant_after_opt
#+caption: Comparison of the measured FRF and the optimized model
#+RESULTS:
[[file:figs/comp_strut_plant_after_opt.png]]
* Function
** =initializeBotFlexibleJoint= - Initialize Flexible Joint
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeBotFlexibleJoint.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [flex_bot] = initializeBotFlexibleJoint(args)
% initializeBotFlexibleJoint -
%
% Syntax: [flex_bot] = initializeBotFlexibleJoint(args)
%
% Inputs:
% - args -
%
% Outputs:
% - flex_bot -
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.type char {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'
args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260
args.kz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1e8
args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.1
args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.1
args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.1
args.cz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1e2
end
#+end_src
*** Initialize the structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
flex_bot = struct();
#+end_src
*** Set the Joint's type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
switch args.type
case '2dof'
flex_bot.type = 1;
case '3dof'
flex_bot.type = 2;
case '4dof'
flex_bot.type = 3;
end
#+end_src
*** Set parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
flex_bot.kRx = args.kRx;
flex_bot.kRy = args.kRy;
flex_bot.kRz = args.kRz;
flex_bot.kz = args.kz;
#+end_src
#+begin_src matlab
flex_bot.cRx = args.cRx;
flex_bot.cRy = args.cRy;
flex_bot.cRz = args.cRz;
flex_bot.cz = args.cz;
#+end_src
** =initializeTopFlexibleJoint= - Initialize Flexible Joint
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeTopFlexibleJoint.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [flex_top] = initializeTopFlexibleJoint(args)
% initializeTopFlexibleJoint -
%
% Syntax: [flex_top] = initializeTopFlexibleJoint(args)
%
% Inputs:
% - args -
%
% Outputs:
% - flex_top -
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.type char {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'
args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260
args.kz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1e8
args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.1
args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.1
args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.1
args.cz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1e2
end
#+end_src
*** Initialize the structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
flex_top = struct();
#+end_src
*** Set the Joint's type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
switch args.type
case '2dof'
flex_top.type = 1;
case '3dof'
flex_top.type = 2;
case '4dof'
flex_top.type = 3;
end
#+end_src
*** Set parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
flex_top.kRx = args.kRx;
flex_top.kRy = args.kRy;
flex_top.kRz = args.kRz;
flex_top.kz = args.kz;
#+end_src
#+begin_src matlab
flex_top.cRx = args.cRx;
flex_top.cRy = args.cRy;
flex_top.cRz = args.cRz;
flex_top.cz = args.cz;
#+end_src
** =initializeAPA= - Initialize APA
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeAPA.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [actuator] = initializeAPA(args)
% initializeAPA -
%
% Syntax: [actuator] = initializeAPA(args)
%
% Inputs:
% - args -
%
% Outputs:
% - actuator -
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.type char {mustBeMember(args.type,{'2dof', 'flexible frame', 'flexible'})} = '2dof'
args.Ga (1,1) double {mustBeNumeric} = 0
args.Gs (1,1) double {mustBeNumeric} = 0
% For 2DoF
args.k (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.38e6
args.ke (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.75e6
args.ka (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e7
args.c (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e1
args.ce (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1
args.ca (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1
args.Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056
% Force Flexible APA
args.xi (1,1) double {mustBeNumeric, mustBePositive} = 0.5
% For Flexible Frame
args.ks (1,1) double {mustBeNumeric, mustBePositive} = 235e6
args.cs (1,1) double {mustBeNumeric, mustBePositive} = 1e1
end
#+end_src
*** Initialize Structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
actuator = struct();
#+end_src
*** Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
switch args.type
case '2dof'
actuator.type = 1;
case 'flexible frame'
actuator.type = 2;
case 'flexible'
actuator.type = 3;
end
#+end_src
*** Actuator/Sensor Constants
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
if args.Ga == 0
switch args.type
case '2dof'
actuator.Ga = -30.0;
case 'flexible frame'
actuator.Ga = 1; % TODO
case 'flexible'
actuator.Ga = 23.4;
end
else
actuator.Ga = args.Ga; % Actuator gain [N/V]
end
#+end_src
#+begin_src matlab
if args.Gs == 0
switch args.type
case '2dof'
actuator.Gs = 0.098;
case 'flexible frame'
actuator.Gs = 1; % TODO
case 'flexible'
actuator.Gs = -4674824;
end
else
actuator.Gs = args.Gs; % Sensor gain [V/m]
end
#+end_src
*** 2DoF parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
actuator.k = args.k; % [N/m]
actuator.ke = args.ke; % [N/m]
actuator.ka = args.ka; % [N/m]
actuator.c = args.c; % [N/(m/s)]
actuator.ce = args.ce; % [N/(m/s)]
actuator.ca = args.ca; % [N/(m/s)]
actuator.Leq = args.Leq; % [m]
#+end_src
*** Flexible frame and fully flexible
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
switch args.type
case 'flexible frame'
actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
case 'flexible'
actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix
actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix
actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m]
end
actuator.xi = args.xi; % Damping ratio
actuator.ks = args.ks; % Stiffness of one stack [N/m]
actuator.cs = args.cs; % Damping of one stack [N/m]
#+end_src
** =generateSweepExc=: Generate sweep sinus excitation
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/src/generateSweepExc.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [U_exc] = generateSweepExc(args)
% generateSweepExc - Generate a Sweep Sine excitation signal
%
% Syntax: [U_exc] = generateSweepExc(args)
%
% Inputs:
% - args - Optinal arguments:
% - Ts - Sampling Time - [s]
% - f_start - Start frequency of the sweep - [Hz]
% - f_end - End frequency of the sweep - [Hz]
% - V_mean - Mean value of the excitation voltage - [V]
% - V_exc - Excitation Amplitude for the Sweep, could be numeric or TF - [V]
% - t_start - Time at which the sweep begins - [s]
% - exc_duration - Duration of the sweep - [s]
% - sweep_type - 'logarithmic' or 'linear' - [-]
% - smooth_ends - 'true' or 'false': smooth transition between 0 and V_mean - [-]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-4
args.f_start (1,1) double {mustBeNumeric, mustBePositive} = 1
args.f_end (1,1) double {mustBeNumeric, mustBePositive} = 1e3
args.V_mean (1,1) double {mustBeNumeric} = 0
args.V_exc = 1
args.t_start (1,1) double {mustBeNumeric, mustBeNonnegative} = 5
args.exc_duration (1,1) double {mustBeNumeric, mustBePositive} = 10
args.sweep_type char {mustBeMember(args.sweep_type,{'log', 'lin'})} = 'lin'
args.smooth_ends logical {mustBeNumericOrLogical} = true
end
#+end_src
*** Sweep Sine part
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
t_sweep = 0:args.Ts:args.exc_duration;
if strcmp(args.sweep_type, 'log')
V_exc = sin(2*pi*args.f_start * args.exc_duration/log(args.f_end/args.f_start) * (exp(log(args.f_end/args.f_start)*t_sweep/args.exc_duration) - 1));
elseif strcmp(args.sweep_type, 'lin')
V_exc = sin(2*pi*(args.f_start + (args.f_end - args.f_start)/2/args.exc_duration*t_sweep).*t_sweep);
else
error('sweep_type should either be equal to "log" or to "lin"');
end
#+end_src
#+begin_src matlab
if isnumeric(args.V_exc)
V_sweep = args.V_mean + args.V_exc*V_exc;
elseif isct(args.V_exc)
if strcmp(args.sweep_type, 'log')
V_sweep = args.V_mean + abs(squeeze(freqresp(args.V_exc, args.f_start*(args.f_end/args.f_start).^(t_sweep/args.exc_duration), 'Hz')))'.*V_exc;
elseif strcmp(args.sweep_type, 'lin')
V_sweep = args.V_mean + abs(squeeze(freqresp(args.V_exc, args.f_start+(args.f_end-args.f_start)/args.exc_duration*t_sweep, 'Hz')))'.*V_exc;
end
end
#+end_src
*** Smooth Ends
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
if args.t_start > 0
t_smooth_start = args.Ts:args.Ts:args.t_start;
V_smooth_start = zeros(size(t_smooth_start));
V_smooth_end = zeros(size(t_smooth_start));
if args.smooth_ends
Vd_max = args.V_mean/(0.7*args.t_start);
V_d = zeros(size(t_smooth_start));
V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);
V_smooth_start = cumtrapz(V_d)*args.Ts;
V_smooth_end = args.V_mean - V_smooth_start;
end
else
V_smooth_start = [];
V_smooth_end = [];
end
#+end_src
*** Combine Excitation signals
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
V_exc = [V_smooth_start, V_sweep, V_smooth_end];
t_exc = args.Ts*[0:1:length(V_exc)-1];
#+end_src
#+begin_src matlab
U_exc = [t_exc; V_exc];
#+end_src
** =generateShapedNoise=: Generate Shaped Noise excitation
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/src/generateShapedNoise.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [U_exc] = generateShapedNoise(args)
% generateShapedNoise - Generate a Shaped Noise excitation signal
%
% Syntax: [U_exc] = generateShapedNoise(args)
%
% Inputs:
% - args - Optinal arguments:
% - Ts - Sampling Time - [s]
% - V_mean - Mean value of the excitation voltage - [V]
% - V_exc - Excitation Amplitude, could be numeric or TF - [V rms]
% - t_start - Time at which the noise begins - [s]
% - exc_duration - Duration of the noise - [s]
% - smooth_ends - 'true' or 'false': smooth transition between 0 and V_mean - [-]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-4
args.V_mean (1,1) double {mustBeNumeric} = 0
args.V_exc = 1
args.t_start (1,1) double {mustBeNumeric, mustBePositive} = 5
args.exc_duration (1,1) double {mustBeNumeric, mustBePositive} = 10
args.smooth_ends logical {mustBeNumericOrLogical} = true
end
#+end_src
*** Shaped Noise
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
t_noise = 0:args.Ts:args.exc_duration;
#+end_src
#+begin_src matlab
if isnumeric(args.V_exc)
V_noise = args.V_mean + args.V_exc*sqrt(1/args.Ts/2)*randn(length(t_noise), 1)';
elseif isct(args.V_exc)
V_noise = args.V_mean + lsim(args.V_exc, sqrt(1/args.Ts/2)*randn(length(t_noise), 1), t_noise)';
end
#+end_src
*** Smooth Ends
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
t_smooth_start = args.Ts:args.Ts:args.t_start;
V_smooth_start = zeros(size(t_smooth_start));
V_smooth_end = zeros(size(t_smooth_start));
if args.smooth_ends
Vd_max = args.V_mean/(0.7*args.t_start);
V_d = zeros(size(t_smooth_start));
V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);
V_smooth_start = cumtrapz(V_d)*args.Ts;
V_smooth_end = args.V_mean - V_smooth_start;
end
#+end_src
*** Combine Excitation signals
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
V_exc = [V_smooth_start, V_noise, V_smooth_end];
t_exc = args.Ts*[0:1:length(V_exc)-1];
#+end_src
#+begin_src matlab
U_exc = [t_exc; V_exc];
#+end_src
** =generateSinIncreasingAmpl=: Generate Sinus with increasing amplitude
:PROPERTIES:
:header-args:matlab+: :tangle ./matlab/src/generateSinIncreasingAmpl.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [U_exc] = generateSinIncreasingAmpl(args)
% generateSinIncreasingAmpl - Generate Sinus with increasing amplitude
%
% Syntax: [U_exc] = generateSinIncreasingAmpl(args)
%
% Inputs:
% - args - Optinal arguments:
% - Ts - Sampling Time - [s]
% - V_mean - Mean value of the excitation voltage - [V]
% - sin_ampls - Excitation Amplitudes - [V]
% - sin_freq - Excitation Frequency - [Hz]
% - sin_num - Number of period for each amplitude - [-]
% - t_start - Time at which the excitation begins - [s]
% - smooth_ends - 'true' or 'false': smooth transition between 0 and V_mean - [-]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-4
args.V_mean (1,1) double {mustBeNumeric} = 0
args.sin_ampls double {mustBeNumeric, mustBePositive} = [0.1, 0.2, 0.3]
args.sin_period (1,1) double {mustBeNumeric, mustBePositive} = 1
args.sin_num (1,1) double {mustBeNumeric, mustBePositive, mustBeInteger} = 3
args.t_start (1,1) double {mustBeNumeric, mustBePositive} = 5
args.smooth_ends logical {mustBeNumericOrLogical} = true
end
#+end_src
*** Sinus excitation
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
t_noise = 0:args.Ts:args.sin_period*args.sin_num;
sin_exc = [];
#+end_src
#+begin_src matlab
for sin_ampl = args.sin_ampls
sin_exc = [sin_exc, args.V_mean + sin_ampl*sin(2*pi/args.sin_period*t_noise)];
end
#+end_src
*** Smooth Ends
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
t_smooth_start = args.Ts:args.Ts:args.t_start;
V_smooth_start = zeros(size(t_smooth_start));
V_smooth_end = zeros(size(t_smooth_start));
if args.smooth_ends
Vd_max = args.V_mean/(0.7*args.t_start);
V_d = zeros(size(t_smooth_start));
V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);
V_smooth_start = cumtrapz(V_d)*args.Ts;
V_smooth_end = args.V_mean - V_smooth_start;
end
#+end_src
*** Combine Excitation signals
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
V_exc = [V_smooth_start, sin_exc, V_smooth_end];
t_exc = args.Ts*[0:1:length(V_exc)-1];
#+end_src
#+begin_src matlab
U_exc = [t_exc; V_exc];
#+end_src
* Bibliography :ignore:
#+latex: \printbibliography