Change biber configuration

preamble.tex

@@ -7,6 +7,7 @@
 \usepackage[        % use biblatex for bibliography
     backend=biber,  % use biber backend (bibtex replacement) or bibtex
     style=ieee,     % bib style
+    citestyle=numeric-comp, % bib style
     hyperref=true,  % activate hyperref support
     backref=true,   % activate backrefs
     isbn=false,     % don't show isbn tags
@@ -28,10 +29,10 @@
 \usepackage{caption}
 \usepackage{subcaption}
 
-\captionsetup[figure]{labelfont=bf}
+\captionsetup[figure]{labelfont=bf, justification=raggedright}
 \captionsetup[subfigure]{labelfont=bf}
 \captionsetup[listing]{labelfont=bf}
-\captionsetup[table]{labelfont=bf}
+\captionsetup[table]{labelfont=bf, justification=raggedright}
 
 \usepackage{xcolor}
 

test-bench-apa.bib

@@ -1,3 +1,28 @@
+@article{wehrsdorfer95_large_signal_measur_piezoel_stack,
+  author          = {Wehrsdorfer, E and Borchhardt, G and Karthe, W and Helke,
+                  G},
+  title           = {Large Signal Measurements on Piezoelectric Stacks},
+  journal         = {Ferroelectrics},
+  volume          = 174,
+  number          = 1,
+  pages           = {259--275},
+  year            = 1995,
+  publisher       = {Taylor \& Francis},
+}
+
+
+@book{fleming14_desig_model_contr_nanop_system,
+  author          = {Andrew J. Fleming and Kam K. Leang},
+  title           = {Design, Modeling and Control of Nanopositioning Systems},
+  year            = 2014,
+  publisher       = {Springer International Publishing},
+  url             = {https://doi.org/10.1007/978-3-319-06617-2},
+  doi             = {10.1007/978-3-319-06617-2},
+  series          = {Advances in Industrial Control},
+}
+
+
+
 @book{reza06_piezoel_trans_vibrat_contr_dampin,
   author          = {Reza, Moheimani and Andrew, Fleming},
   title           = {Piezoelectric Transducers for Vibration Control and
@@ -10,6 +35,64 @@
 
 
 
+@book{preumont18_vibrat_contr_activ_struc_fourt_edition,
+  author          = {Andre Preumont},
+  title           = {Vibration Control of Active Structures - Fourth Edition},
+  year            = 2018,
+  publisher       = {Springer International Publishing},
+  url             = {https://doi.org/10.1007/978-3-319-72296-2},
+  doi             = {10.1007/978-3-319-72296-2},
+  keywords        = {favorite, parallel robot},
+  series          = {Solid Mechanics and Its Applications},
+}
+
+
+
+@inproceedings{spanos95_soft_activ_vibrat_isolat,
+  author          = {J. Spanos and Z. Rahman and G. Blackwood},
+  title           = {A Soft 6-axis Active Vibration Isolator},
+  booktitle       = {Proceedings of 1995 American Control Conference - ACC'95},
+  year            = 1995,
+  doi             = {10.1109/acc.1995.529280},
+  url             = {https://doi.org/10.1109/acc.1995.529280},
+  keywords        = {parallel robot},
+}
+
+
+
+@article{thayer02_six_axis_vibrat_isolat_system,
+  author          = {Doug Thayer and Mark Campbell and Juris Vagners and Andrew
+                  von Flotow},
+  title           = {Six-Axis Vibration Isolation System Using Soft Actuators
+                  and Multiple Sensors},
+  journal         = {Journal of Spacecraft and Rockets},
+  volume          = 39,
+  number          = 2,
+  pages           = {206-212},
+  year            = 2002,
+  doi             = {10.2514/2.3821},
+  url             = {https://doi.org/10.2514/2.3821},
+  keywords        = {parallel robot},
+}
+
+
+
+@article{hauge04_sensor_contr_space_based_six,
+  author          = {G.S. Hauge and M.E. Campbell},
+  title           = {Sensors and Control of a Space-Based Six-Axis Vibration
+                  Isolation System},
+  journal         = {Journal of Sound and Vibration},
+  volume          = 269,
+  number          = {3-5},
+  pages           = {913-931},
+  year            = 2004,
+  doi             = {10.1016/s0022-460x(03)00206-2},
+  url             = {https://doi.org/10.1016/s0022-460x(03)00206-2},
+  keywords        = {parallel robot, favorite},
+}
+
+
+
 @article{souleille18_concep_activ_mount_space_applic,
   author          = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and
                   Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues,
@@ -26,18 +109,6 @@
 
 
 
-@book{fleming14_desig_model_contr_nanop_system,
-  author          = {Andrew J. Fleming and Kam K. Leang},
-  title           = {Design, Modeling and Control of Nanopositioning Systems},
-  year            = 2014,
-  publisher       = {Springer International Publishing},
-  url             = {https://doi.org/10.1007/978-3-319-06617-2},
-  doi             = {10.1007/978-3-319-06617-2},
-  series          = {Advances in Industrial Control},
-}
-
-
-
 @article{fleming10_integ_strain_force_feedb_high,
   author          = {Fleming, Andrew J and Leang, Kam K},
   title           = {Integrated Strain and Force Feedback for High-Performance
@@ -53,16 +124,21 @@
 
 
 
-@article{fleming10_integ_strain_force_feedb_high,
+@article{gustavsen99_ration_approx_frequen_domain_respon,
-  author          = {Fleming, Andrew J and Leang, Kam K},
+  author          = {Gustavsen, B.; Semlyen, A.},
-  title           = {Integrated Strain and Force Feedback for High-Performance
+  title           = {Rational Approximation of Frequency Domain Responses By
-                  Control of Piezoelectric Actuators},
+                  Vector Fitting},
-  journal         = {Sensors and Actuators A: Physical},
+  journal         = {IEEE Transactions on Power Delivery},
-  volume          = 161,
+  volume          = 14,
-  number          = {1-2},
+  year            = 1999,
-  pages           = {256--265},
+  doi             = {10.1109/61.772353},
-  year            = 2010,
+  url             = {https://doi.org/10.1109/61.772353},
-  publisher       = {Elsevier},
+  issne           = {1937-4208},
-  keywords        = {flexure,nanostage},
+  issnp           = {0885-8977},
+  issue           = 3,
+  month           = 7,
+  page            = {1052--1061},
+  publisher       = {IEEE},
+  keywords        = {Motors},
 }
 

test-bench-apa.org

@@ -83,6 +83,9 @@
 
 (setq org-export-before-parsing-hook '(org-ref-glossary-before-parsing
                                        org-ref-acronyms-before-parsing))
+
+;; Put the caption below the tables
+(setq org-latex-caption-above nil)
 #+END_SRC
 
 * Notes                                                             :noexport:
@@ -100,7 +103,11 @@ Prefix for figures/section/tables =test_apa=
   - Dynamical measurements (Section 3)
   - Simscape Model (Section 4)
 
-** TODO [#C] Add sitffness of APA shell from FEM                    :@philipp:
+** DONE [#B] Add FEM analysis (APA modes)
+CLOSED: [2024-04-02 Tue 10:17] SCHEDULED: <2024-04-02 Tue>
+
+** DONE [#C] Add sitffness of APA shell from FEM                      :@philipp:
+CLOSED: [2024-04-02 Tue 10:01]
 
 ** TODO [#C] Check things about resistor in parallel with the force sensor
 
@@ -110,18 +117,25 @@ Verify that everything interesting to say about that is either done before in th
 
 * Introduction                                                        :ignore:
 
+In this chapter, the goal is to make sure that the received APA300ML (shown in Figure ref:fig:test_apa_received) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics.
+
 #+name: fig:test_apa_received
 #+attr_latex: :width 0.7\linewidth
 #+caption: Picture of 5 out of the 7 received APA300ML
 [[file:figs/test_apa_received.jpg]]
 
-The first goal is to characterize the APA300ML in terms of:
+In section ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. Flexible modes of the APA300ML are computed with a finite element model and compared with measurements.
-- The, geometric features, electrical capacitance, stroke, hysteresis, spurious resonances.
+
-  This is performed in Section ref:sec:test_apa_basic_meas.
+Using a dedicated test bench, dynamical measurements are performed (Section ref:sec:test_apa_dynamics).
-- The dynamics from the generated DAC voltage (going to the voltage amplifiers and then applied on the actuator stacks) to the induced displacement, and to the measured voltage by the force sensor stack.
+The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
-  Also the "actuator constant" and "sensor constant" are identified.
+Integral Force Feedback is experimentally applied and the damped plants are estimated for several feedback gains.
-  This is done in Section ref:sec:test_apa_dynamics.
+
-- Compare the measurements with the two Simscape models: 2DoF (Section ref:sec:test_apa_model_2dof) Super-Element (Section ref:sec:test_apa_model_flexible)
+Two different models of the APA300ML are then presented.
+First, in Section ref:sec:test_apa_model_2dof, a two degrees of freedom model is presented, tuned and compared with the measured dynamics.
+This model is proven to accurately simulate the APA300ML's axial dynamics.
+
+Then, in Section ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported in Simscape.
+This more complex model is also shown to well capture the axial dynamics of the APA300ML.
 
 #+name: tab:test_apa_section_matlab_code
 #+caption: Report sections and corresponding Matlab files
@@ -142,11 +156,11 @@ The first goal is to characterize the APA300ML in terms of:
 
 ** Introduction                                                      :ignore:
 
-Before using the measurement bench to characterize the APA300ML, first simple measurements are performed:
+Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed.
-- Section ref:ssec:test_apa_geometrical_measurements: the geometric tolerances of the interface planes are checked
+First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section ref:ssec:test_apa_geometrical_measurements.
-- Section ref:ssec:test_apa_electrical_measurements: the capacitance of the piezoelectric stacks is measured
+Then, the capacitance of the piezoelectric stacks is measured in Section ref:ssec:test_apa_electrical_measurements.
-- Section ref:ssec:test_apa_stroke_measurements: the stroke of each APA is measured
+The achievable stroke of the APA300ML is measured using a displacement probe in Section ref:ssec:test_apa_stroke_measurements.
-- Section ref:ssec:test_apa_spurious_resonances: the "spurious" resonances of the APA are investigated
+Finally, in Section ref:ssec:test_apa_spurious_resonances, the flexible modes of the APA are measured and compared with a finite element model.
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -172,16 +186,10 @@ Before using the measurement bench to characterize the APA300ML, first simple me
 ** Geometrical Measurements
 <<ssec:test_apa_geometrical_measurements>>
 
-To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with very good flatness.
+To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness.
-
 As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:3] is used to measure the height of 4 points on each of the APA300ML interfaces.
-
 From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting[fn:4] a plane through all the points.
-
+The measured flatness, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications.
-#+name: fig:test_apa_flatness_setup
-#+attr_latex: :width 0.4\linewidth
-#+caption: Measurement setup for flatness estimation of the two mechanical interfaces
-[[file:figs/test_apa_flatness_setup.png]]
 
 #+begin_src matlab
 %% Measured height for all the APA at the 8 locations
@@ -220,16 +228,24 @@ for i = 1:7
 end
 #+end_src
 
-The measured flatness, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications.
-
 #+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
 
+#+attr_latex: :options [b]{0.49\linewidth}
+#+begin_minipage
+#+name: fig:test_apa_flatness_setup
+#+attr_latex: :width 0.7\linewidth :float nil
+#+caption: Measurement setup for flatness estimation
+[[file:figs/test_apa_flatness_setup.png]]
+#+end_minipage
+\hfill
+#+attr_latex: :options [b]{0.49\linewidth}
+#+begin_minipage
 #+name: tab:test_apa_flatness_meas
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xc
+#+attr_latex: :booktabs t :float nil
 #+caption: Estimated flatness of the APA300ML interfaces
-#+attr_latex: :environment tabularx :width 0.3\linewidth :align Xc
-#+attr_latex: :center t :booktabs t
 #+RESULTS:
 |       | *Flatness* $[\mu m]$ |
 |-------+----------------------|
@@ -240,31 +256,36 @@ data2orgtable(1e6*apa_d', {'APA 1', 'APA 2', 'APA 3', 'APA 4', 'APA 5', 'APA 6',
 | APA 5 |                  1.9 |
 | APA 6 |                  7.1 |
 | APA 7 |                 18.7 |
+#+end_minipage
 
 ** Electrical Measurements
 <<ssec:test_apa_electrical_measurements>>
 
 From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu F$ and $26\,\mu F$ with a nominal capacitance of $20\,\mu F$.
 
-The capacitance of the piezoelectric stacks found in the APA300ML have been measured with the LCR meter[fn:1] shown in Figure ref:fig:test_apa_lcr_meter.
+The piezoelectric stacks capacitance of the APA300ML have been measured with the LCR meter[fn:1] shown in Figure ref:fig:test_apa_lcr_meter.
 The two stacks used as an actuator and the stack used as a force sensor are measured separately.
-
-#+name: fig:test_apa_lcr_meter
-#+caption: LCR Meter used for the measurements
-#+attr_latex: :width 0.6\linewidth
-[[file:figs/test_apa_lcr_meter.jpg]]
-
 The measured capacitance are summarized in Table ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$.
 However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance.
 This may indicate a manufacturing defect.
 
 The measured capacitance is found to be lower than the specified one.
-This may be due to the fact that the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$) while it was here measured with small signals.
+This may be due to the fact that the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$) while it was here measured with small signals [[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]].
 
+#+attr_latex: :options [b]{0.49\linewidth}
+#+begin_minipage
+#+name: fig:test_apa_lcr_meter
+#+attr_latex: :width 0.95\linewidth :float nil
+#+caption: Used LCR meter
+[[file:figs/test_apa_lcr_meter.jpg]]
+#+end_minipage
+\hfill
+#+attr_latex: :options [b]{0.49\linewidth}
+#+begin_minipage
 #+name: tab:test_apa_capacitance
-#+caption: Capacitance measured with the LCR meter. The excitation signal is a sinus at 1kHz
+#+caption: Measured capacitance in $\mu F$
-#+attr_latex: :environment tabularx :width 0.5\linewidth :align lcc
+#+attr_latex: :environment tabularx :width 0.95\linewidth :align lcc
-#+attr_latex: :center t :booktabs t
+#+attr_latex: :center t :booktabs t :float nil
 |       | *Sensor Stack* | *Actuator Stacks* |
 |-------+----------------+-------------------|
 | APA 1 |           5.10 |             10.03 |
@@ -274,28 +295,27 @@ This may be due to the fact that the manufacturer measures the capacitance with
 | APA 5 |           4.90 |              9.66 |
 | APA 6 |           4.99 |              9.91 |
 | APA 7 |           4.85 |              9.85 |
+#+end_minipage
 
 ** Stroke and Hysteresis Measurement
 <<ssec:test_apa_stroke_measurements>>
 
-The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
+In order to verify that the stroke of the APA300ML is as specified in the datasheet, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
-To do so, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
 
 Then, the voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC and a voltage amplifier.
-Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_bench, left).
+Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_voltage).
 
 #+name: fig:test_apa_stroke_bench
 #+caption: Bench to measured the APA stroke
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :width 0.7\linewidth
 [[file:figs/test_apa_stroke_bench.jpg]]
 
-The measured APA displacement is shown as a function of the applied voltage in Figure ref:fig:test_apa_stroke_result, right.
+The measured APA displacement is shown as a function of the applied voltage in Figure ref:fig:test_apa_stroke_hysteresis.
-
 Typical hysteresis curves for piezoelectric stack actuators can be observed.
 The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks, which is enough for the current application.
 This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu m$, therefore $\approx 200\,\mu m$ if only two stacks are used).
 
-It is clear from Figure ref:fig:test_apa_stroke_result that "APA 3" has an issue compared to the other units.
+It is clear from Figure ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared to the other units.
 This confirms the abnormal electrical measurements made in Section ref:ssec:test_apa_electrical_measurements.
 This unit was send sent back to Cedrat and a new one was shipped back.
 From now on, only the six APA that behave as expected will be used.
@@ -306,18 +326,20 @@ load('meas_apa_stroke.mat', 'apa300ml_2s')
 #+end_src
 
 #+begin_src matlab :exports none :results none
-%% Results of the measured APA stroke
+%% Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML
 figure;
-tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
-
-% Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML
-ax1 = nexttile();
 plot(apa300ml_2s{1}.t - apa300ml_2s{1}.t(1), 20*apa300ml_2s{1}.V, 'k-')
 xlabel('Time [s]'); ylabel('Voltage [V]')
 ylim([-20, 160])
+#+end_src
 
-% Measured displacement as a function of the applied voltage
+#+begin_src matlab :tangle no :exports results :results file none
-ax2 = nexttile();
+exportFig('figs/test_apa_stroke_voltage.pdf', 'width', 'half', 'height', 'normal');
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Measured displacement as a function of the applied voltage
+figure;
 hold on;
 for i = 1:7
     plot(20*apa300ml_2s{i}.V, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
@@ -328,40 +350,64 @@ legend('location', 'southwest', 'FontSize', 8)
 xlim([-20, 150]); ylim([-250, 0]);
 #+end_src
 
-#+begin_src matlab :tangle no :exports results :results file replace
+#+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/test_apa_stroke_result.pdf', 'width', 'full', 'height', 'normal');
+exportFig('figs/test_apa_stroke_hysteresis.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
-#+name: fig:test_apa_stroke_result
+#+name: fig:test_apa_stroke
-#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (left). Measured displacement as a function of the applied voltage (right)
+#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of the applied voltage (\subref{fig:test_apa_stroke_hysteresis})
-#+RESULTS:
+#+attr_latex: :options [htbp]
-[[file:figs/test_apa_stroke_result.png]]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :width 0.95\linewidth
+[[file:figs/test_apa_stroke_voltage.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :width 0.95\linewidth
+[[file:figs/test_apa_stroke_hysteresis.png]]
+#+end_subfigure
+#+end_figure
 
 ** Flexible Mode Measurement
 <<ssec:test_apa_spurious_resonances>>
 
 In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
-
+To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_modes).
-To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_torsion).
+A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:7] is used to excite the flexible modes.
-A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer[fn:7] is used to excite the flexible modes.
 Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated.
 
 The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure ref:fig:test_apa_mode_shapes.
 
 #+name: fig:test_apa_mode_shapes
-#+caption: Spurious resonances - Change this with the updated FEM analysis of the APA300ML
+#+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :options [htbp]
-[[file:figs/test_apa_mode_shapes.png]]
+#+begin_figure
-
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode (268Hz)}
-#+name: fig:test_apa_meas_setup_torsion
+#+attr_latex: :options {0.36\textwidth}
-#+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the $Z$ torsion is measured.
+#+begin_subfigure
-#+attr_latex: :width 0.6\linewidth
+#+attr_latex: :height 4.3cm
-[[file:figs/test_apa_meas_setup_torsion.jpg]]
+[[file:figs/test_apa_mode_shapes_1.png]]
-
+#+end_subfigure
-Two other similar measurements are performed to measured the bending of the APA around the $X$ direction and around the $Y$ direction (see Figure ref:fig:test_apa_meas_setup_modes).
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode (399Hz)}
+#+attr_latex: :options {0.28\textwidth}
+#+begin_subfigure
+#+attr_latex: :height 4.3cm
+[[file:figs/test_apa_mode_shapes_2.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode (706Hz)}
+#+attr_latex: :options {0.36\textwidth}
+#+begin_subfigure
+#+attr_latex: :height 4.3cm
+[[file:figs/test_apa_mode_shapes_3.png]]
+#+end_subfigure
+#+end_figure
 
 #+name: fig:test_apa_meas_setup_modes
-#+caption: Experimental setup to measured flexible modes of the APA300ML. For the bending in the $X$ direction, the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
+#+caption: Experimental setup to measured flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
@@ -398,39 +444,26 @@ bending_Y = load('apa300ml_bending_Y_top.mat');
 
 % Compute the transfer function
 [G_bending_Y, ~]  = tfestimate(bending_Y.Track1, bending_Y.Track2, win, Noverlap, Nfft, 1/Ts);
-
-%% Z-Torsion identification
-% Load data
-torsion = load('apa300ml_torsion_top.mat');
-
-% Compute transfer function
-[G_torsion_top, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts);
-
-% Load Data
-torsion = load('apa300ml_torsion_left.mat');
-
-% Compute transfer function
-[G_torsion, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts);
 #+end_src
 
-The three measured frequency response functions are shown in Figure ref:fig:test_apa_meas_freq_compare.
+The measured frequency response functions computed from the experimental setups of figures ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure ref:fig:test_apa_meas_freq_compare.
-- a clear $x$ bending mode at $280\,\text{Hz}$
+The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$.
-- a clear $y$ bending mode at $412\,\text{Hz}$
+These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_meas_setup_modes).
-- for the $z$ torsion test, the $y$ bending mode is also excited and observed, and we may see a mode at $800\,\text{Hz}$
+This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
+This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
+Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
 
 #+begin_src matlab :exports none
 figure;
 hold on;
 plot(f, abs(G_bending_X), 'DisplayName', '$X$ bending');
 plot(f, abs(G_bending_Y), 'DisplayName', '$Y$ bending');
-plot(f, abs(G_torsion),   'DisplayName', '$Z$ torsion');
 text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
 text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
-text(800, 6e-4,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center')
 hold off;
 set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Amplitude');
-xlim([50, 2e3]); ylim([5e-5, 2e-1]);
+xlim([100, 1e3]); ylim([5e-5, 2e-1]);
 legend('location', 'northeast', 'FontSize', 8)
 #+end_src
 
@@ -439,33 +472,20 @@ exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'nor
 #+end_src
 
 #+name: fig:test_apa_meas_freq_compare
-#+caption: Obtained frequency response functions for the 3 tests with the instrumented hammer
+#+caption: Obtained frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$
 #+RESULTS:
 [[file:figs/test_apa_meas_freq_compare.png]]
 
-#+name: tab:test_apa_measured_modes_freq
-#+caption: Measured frequency of the modes
-#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcc
-#+attr_latex: :center t :booktabs t
-| *Mode*      | *FEM* | *Measured Frequency* |
-|-------------+-------+----------------------|
-| $X$ bending |       | 280Hz                |
-| $Y$ bending |       | 410Hz                |
-| $Z$ torsion |       | 800Hz                |
-
-** Conclusion                                                        :ignore:
-
 * Dynamical measurements
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/test_apa_2_dynamics.m
 :END:
 <<sec:test_apa_dynamics>>
-
 ** Introduction                                                      :ignore:
-After the basic measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench is used to better characterize the APA.
+After the basic measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench is used to better characterize the dynamics of the APA300ML.
-
+This test bench, depicted in Figure ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing.
-This test bench is shown in Figure ref:fig:test_bench_apa and consists of the APA300ML fixed on one end to the fixed granite, and on the other end to the 5kg granite vertically guided with an air bearing.
+That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
-An encoder is used to measure the relative motion between the two granites (i.e. the displacement of the APA).
+An encoder[fn:8] is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
 
 #+name: fig:test_bench_apa
 #+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML
@@ -485,32 +505,18 @@ An encoder is used to measure the relative motion between the two granites (i.e.
 #+end_subfigure
 #+end_figure
 
-The bench is schematically shown in Figure ref:fig:test_apa_schematic and the signal used are summarized in Table ref:tab:test_apa_variables.
+The bench is schematically shown in Figure ref:fig:test_apa_schematic with all the associated signals.
+It will be first used to estimate the hysteresis from the piezoelectric stack (Section ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section ref:ssec:test_apa_stiffness).
+Then, the frequency response functions from the DAC voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section ref:ssec:test_apa_meas_dynamics.
+The presence of a non minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section ref:ssec:test_apa_non_minimum_phase.
+In order to limit the low frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section ref:ssec:test_apa_resistance_sensor_stack).
+Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section ref:ssec:test_apa_iff_locus.
 
 #+name: fig:test_apa_schematic
-#+caption: Schematic of the Test Bench
+#+caption: Schematic of the Test Bench used to measured the dynamics of the APA300ML. $u$ is the output DAC voltage, $V_a$ the output amplifier voltage (i.e. voltage applied across the actuator stacks), $d_e$ the measured displacement by the encoder and $V_s$ the measured voltage across the sensor stack.
 #+attr_latex: :scale 1
 [[file:figs/test_apa_schematic.png]]
 
-#+name: tab:test_apa_variables
-#+caption: Variables used during the measurements
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align cXc
-#+attr_latex: :center t :booktabs t
-| *Variable* | *Description*                | *Unit* |
-|------------+------------------------------+--------|
-| $u$        | Output DAC Voltage           | $V$    |
-| $V_a$      | Output Amplifier Voltage     | $V$    |
-| $V_s$      | Measured Stack Voltage (ADC) | $V$    |
-| $d_e$      | Encoder Measurement          | $m$    |
-
-This bench will be used to:
-- ref:ssec:test_apa_hysteresis
-- ref:ssec:test_apa_stiffness
-- measure the dynamics of the APA (section ref:ssec:test_apa_meas_dynamics)
-- estimate the added damping using Integral Force Feedback (Section ref:ssec:test_apa_iff_locus)
-
-These measurements will also be used to tune the developed models of the APA (in Section ref:sec:test_apa_model_2dof for the 2DoF model, and in Section ref:sec:test_apa_model_flexible for the flexible model).
-
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
@@ -536,10 +542,9 @@ These measurements will also be used to tune the developed models of the APA (in
 <<ssec:test_apa_hysteresis>>
 
 As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
-
+Do to so, a quasi static[fn:9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC.
-A quasi static sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$), and an amplitude varying from $4\,V$ up to $80\,V$.
+For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure ref:fig:test_apa_meas_hysteresis.
-
+This is the typical behavior expected from a PZT stack actuator where the hysteresis increases as a function of the applied voltage amplitude [[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]].
-For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage..
 
 #+begin_src matlab
 %% Load measured data - hysteresis
@@ -551,9 +556,6 @@ apa_hyst.t = apa_hyst.t - apa_hyst.t(1);
 ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes
 #+end_src
 
-The measured displacements as a function of the output voltages are shown in Figure ref:fig:test_apa_meas_hysteresis.
-It is interesting to see that the hysteresis is increasing with the excitation amplitude.
-
 #+begin_src matlab :exports none
 %% Measured displacement as a function of the output voltage
 figure;
@@ -584,7 +586,6 @@ exportFig('figs/test_apa_meas_hysteresis.pdf', 'width', 'wide', 'height', 'norma
 <<ssec:test_apa_stiffness>>
 
 In order to estimate the stiffness of the APA, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $d_e$ is measured using the encoder.
-
 The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness.
 
 \begin{equation} \label{eq:test_apa_stiffness}
@@ -608,6 +609,9 @@ The measured displacement $d_e$ as a function of time is shown in Figure ref:fig
 It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
 These two effects induce some uncertainties in the measured stiffness.
 
+The stiffnesses are computed for all the APA from the two displacements $d_1$ and $d_2$ (see Figure ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$.
+These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$.
+
 #+begin_src matlab :exports none
 %% Plot the deflection at a function of time
 figure;
@@ -633,26 +637,28 @@ hold off;
 xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]');
 #+end_src
 
-#+begin_src matlab :tangle no :exports results :results file replace
+#+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/test_apa_meas_stiffness_time.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/test_apa_meas_stiffness_time.pdf', 'width', 'half', 'height', 'normal');
 #+end_src
 
-#+name: fig:test_apa_meas_stiffness_time
-#+caption: Measured displacement when adding the mass (at $t \approx 3\,s$) and removing the mass(at $t \approx 13\,s$)
-#+RESULTS:
-[[file:figs/test_apa_meas_stiffness_time.png]]
-
-The stiffnesses are computed for all the APA from the two displacements $d_1$ and $d_2$ (see Figure ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$.
-These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_stiffnesses and are found to be close to the nominal stiffness $k = 1.8\,N/\mu m$ found in the APA300ML manual.
-
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
   data2orgtable(1e-6*apa_k, cellstr(num2str(apa_nums')), {'APA', '$k_1$', '$k_2$'}, ' %.2f ');
 #+end_src
 
+#+attr_latex: :options [b]{0.57\linewidth}
+#+begin_minipage
+#+name: fig:test_apa_meas_stiffness_time
+#+caption: Measured displacement when adding (at $t \approx 3\,s$) and removing (at $t \approx 13\,s$) the mass
+#+attr_latex: :width 0.9\linewidth :float nil
+[[file:figs/test_apa_meas_stiffness_time.png]]
+#+end_minipage
+\hfill
+#+attr_latex: :options [b]{0.37\linewidth}
+#+begin_minipage
 #+name: tab:test_apa_measured_stiffnesses
-#+caption: Measured stiffnesses (in $N/\mu m$)
+#+caption: Measured axial stiffnesses (in $N/\mu m$)
-#+attr_latex: :environment tabularx :width 0.2\linewidth :align ccc
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xcc
-#+attr_latex: :center t :booktabs t :float t
+#+attr_latex: :center t :booktabs t :float nil
 #+RESULTS:
 | APA | $k_1$ | $k_2$ |
 |-----+-------+-------|
@@ -662,6 +668,7 @@ These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_st
 |   5 |   1.7 |  1.93 |
 |   6 |   1.7 |  1.92 |
 |   8 |  1.73 |  1.98 |
+#+end_minipage
 
 The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$.
 
@@ -672,13 +679,13 @@ The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by
 The obtain stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and by the "static deflection" method.
 
 
-However, changes in the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]].
+It is important to note that changes to the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]].
 
-To estimate this effect, the stiffness of the APA if measured using the "static deflection" method in two cases:
+To estimate this effect for the APA300ML, its stiffness is estimated using the "static deflection" method in two cases:
 - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC)
 - $k_{\text{sc}}$: piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance)
 
-The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ and the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
+It is found that the open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
 
 #+begin_src matlab
 %% Load Data
@@ -697,8 +704,6 @@ apa_k_sc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass
 ** Dynamics
 <<ssec:test_apa_meas_dynamics>>
 
-In this section, the dynamics of the system from the excitation voltage $u$ to encoder measured displacement $d_e$  and to the force sensor voltage $V_s$ is identified.
-
 #+begin_src matlab
 %% Identification using sweep sine (low frequency)
 load('frf_data_sweep.mat');
@@ -746,27 +751,30 @@ save('matlab/mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums')
 save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
 #+end_src
 
-The obtained transfer functions for the 6 APA between the excitation voltage $u$ and the encoder displacement $d_e$ are shown in Figure ref:fig:test_apa_frf_encoder.
+In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified.
-The obtained transfer functions are close to a mass-spring-damper system.
+
-The following can be observed:
+First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_encoder.
+The obtained frequency response functions are similar to that of a (second order) mass-spring-damper system with:
 - A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$.
-  The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA
+  The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA
 - A lightly damped resonance at $95\,\text{Hz}$
-- A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
+- A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
+  Flexible modes studied in section ref:ssec:test_apa_spurious_resonances seems not to impact the measured axial motion of the actuator.
 
-The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ is also identified and shown in Figure ref:fig:test_apa_frf_force.
+The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_force.
 
-A lightly damped resonance is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance at $41\,\text{Hz}$.
+A lightly damped resonance (pole) is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance (zero) at $41\,\text{Hz}$.
-No additional resonances is present up to at least $2\,\text{kHz}$ indicating at Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
+No additional resonances is present up to at least $2\,\text{kHz}$ indicating that Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
+The zero at $41\,\text{Hz}$ seems to be non-minimum phase (the phase /decreases/ by 180 degrees whereas it should have /increased/ by 180 degrees for a minimum phase zero).
+This is investigated in Section ref:ssec:test_apa_non_minimum_phase.
 
-As illustrated by the Root Locus, the poles of the closed-loop system converges to the zeros of the open-loop plant.
+As illustrated by the Root Locus, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases.
-Suppose that a controller with a very high gain is implemented such that the voltage $V_s$ across the sensor stack is zero.
+The significance of this behavior varies on the type of sensor used as explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]].
-In that case, because of the very high controller gain, no stress and strain is present on the sensor stack (and on the actuator stacks are well, as they are both in series).
+Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
-Such closed-loop system would therefore virtually corresponds to a system for which the piezoelectric stacks have been removed and just the mechanical shell is kept.
+Consequently, the closed-loop system would virtually corresponds to one where the piezoelectric stacks are absent, leaving only the mechanical shell.
-From this analysis, the axial stiffness of the shell can be estimated to be $k_{\text{shell}} = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$.
+From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$ (which is close to what is found using a finite element model).
-# TODO - Compare with FEM result
 
-Such reasoning can lead to very interesting insight into the system just from an open-loop identification.
+All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure ref:fig:test_apa_frf_encoder and at the force sensor in Figure ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
 
 #+begin_src matlab :exports none
 %% Plot the FRF from u to de
@@ -860,19 +868,113 @@ exportFig('figs/test_apa_frf_force.pdf', 'width', 'half', 'height', 'tall');
 #+end_subfigure
 #+end_figure
 
-All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure ref:fig:test_apa_frf_encoder and at the force sensor in Figure ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical lever.
+** Non Minimum Phase Zero?
+<<ssec:test_apa_non_minimum_phase>>
+
+It was surprising to observe a non-minimum phase behavior for the zero on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force).
+It was initially thought that this non-minimum phase behavior is an artifact coming from the measurement.
+A longer measurement was performed with different excitation signals (noise, slow sine sweep, etc.) to see it the phase behavior of the zero changes.
+Results of one long measurement is shown in Figure ref:fig:test_apa_non_minimum_phase.
+The coherence (Figure ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior.
+
+Such non-minimum phase zero when using load cells has also been observed on other mechanical systems [[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]].
+It could be induced to small non-linearity in the system, but the reason of this non-minimum phase for the APA300ML is not yet clear.
+
+However, this is not so important here as the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure ref:fig:test_apa_iff_root_locus) should not be unstable except for very large controller gains that will never be applied in practice.
+
+#+begin_src matlab
+%% Long measurement
+long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'u', 'Vs');
+
+% Long window for fine frequency axis
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(10/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+% Transfer function estimation
+[frf_noise, f] = tfestimate(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+[coh_noise, ~] = mscohere(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+#+end_src
+
+#+begin_src matlab :exports none
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+nexttile();
+hold on;
+plot(f, coh_noise, '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+hold off;
+xlim([38, 45]);
+ylim([0, 1]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file none
+exportFig('figs/test_apa_non_minimum_phase_coherence.pdf', 'width', 'half', 'height', 'normal');
+#+end_src
+
+#+begin_src matlab :exports none
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 0]);
+
+linkaxes([ax1,ax2],'x');
+xlim([38, 45]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file none
+exportFig('figs/test_apa_non_minimum_phase_zoom.pdf', 'width', 'half', 'height', 'normal');
+#+end_src
+
+#+name: fig:test_apa_non_minimum_phase
+#+caption: Measurement of the anti-resonance found on the transfer function from $u$ to $V_s$. The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.
+#+attr_latex: :options [htbp]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :width 0.95\linewidth
+[[file:figs/test_apa_non_minimum_phase_coherence.png]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :width 0.95\linewidth
+[[file:figs/test_apa_non_minimum_phase_zoom.png]]
+#+end_subfigure
+#+end_figure
+
 
 ** Effect of the resistor on the IFF Plant
 <<ssec:test_apa_resistance_sensor_stack>>
 
-A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the stack.
+A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu F$).
 
-As explain before, this is done for two reasons:
+As explain before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain.
-1. Limit the voltage offset due to the input bias current of the ADC
-2. Limit the low frequency gain
 
 The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor have been measured and are compared in Figure ref:fig:test_apa_effect_resistance.
-It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of $\approx 0.35\,\text{Hz}$.
+It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$.
 
 #+begin_src matlab
 %% Load the data
@@ -894,7 +996,7 @@ R = 80.6e3; % Parallel Resistor [Ohm]
 
 f0 = 1/(2*pi*R*C); % Crossover frequency of RC HPF [Hz]
 
-G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0);
+G_hpf = 0.6*(s/(2*pi*f0))/(1 + s/(2*pi*f0));
 #+end_src
 
 #+begin_src matlab :exports none
@@ -906,22 +1008,26 @@ ax1 = nexttile();
 hold on;
 plot(f, abs(frf_wo_k), 'DisplayName', 'Without $R$');
 plot(f, abs(frf_wi_k), 'DisplayName', 'With $R$');
+plot(f, abs(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', 'RC model');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-ylim([1e-1, 1e0]);
+ylim([2e-1, 1e0]);
+yticks([0.2, 0.5, 1]);
 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--', 'DisplayName', 'RC');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
-yticks(-360:45:360); ylim([-5, 90]);
+yticks(-360:45:360); ylim([-5, 60]);
+yticks([0, 15, 30, 45, 60]);
 
 linkaxes([ax1,ax2],'x');
 xlim([0.2, 8]);
@@ -933,15 +1039,13 @@ exportFig('figs/test_apa_effect_resistance.pdf', 'width', 'wide', 'height', 600)
 #+end_src
 
 #+name: fig:test_apa_effect_resistance
-#+caption: Transfer function from u to $V_s$ with and without the resistor $R$ in parallel with the piezoelectric stack used as the force sensor
+#+caption: Transfer function from $u$ to $V_s$ with and without the resistor $R$ in parallel with the piezoelectric stack used as the force sensor
 #+RESULTS:
 [[file:figs/test_apa_effect_resistance.png]]
 
 ** Integral Force Feedback
 <<ssec:test_apa_iff_locus>>
 
-This test bench can also be used to estimate the damping added by the implementation of an Integral Force Feedback strategy.
-
 #+begin_src matlab
 %% Load identification Data
 data = load("2023-03-17_11-28_iff_plant.mat");
@@ -956,9 +1060,8 @@ Noverlap = floor(Nfft/2);
 [G_iff, f]  = tfestimate(data.id_plant, data.Vs, win, Noverlap, Nfft, 1/Ts);
 #+end_src
 
-First, the transfer function eqref:eq:test_apa_iff_manual_fit is manually tuned to match the identified dynamics from generated voltage $u$ to the measured sensor stack voltage $V_s$ in Section ref:ssec:test_apa_meas_dynamics.
+In order to implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation eqref:eq:test_apa_iff_manual_fit.
-
+The parameters are manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$.
-The obtained parameter values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$.
 
 \begin{equation} \label{eq:test_apa_iff_manual_fit}
 G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s}
@@ -1061,15 +1164,11 @@ for i = 1:length(i_kept)
 end
 #+end_src
 
-The identified dynamics are then fitted by second order transfer functions.
+The identified dynamics are then fitted by second order transfer functions[fn:10].
 The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure ref:fig:test_apa_identified_damped_plants for different gains $g$.
 It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
 
-
+The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed both using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula and from the fitted transfer functions of the damped plants experimentally identified for several controller gains.
-The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed:
-- using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula
-- from the fitted transfer functions of the damped plants experimentally identified for several controller gains
-
 The two obtained root loci are compared in Figure ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were only fitted using a second order transfer function.
 
 #+begin_src matlab
@@ -1183,12 +1282,6 @@ exportFig('figs/test_apa_iff_root_locus.pdf', 'width', 'half', 'height', 'tall')
 #+end_figure
 
 
-** Conclusion                                                        :ignore:
-
-#+begin_important
-So far, all the measured FRF are showing the dynamical behavior that was expected.
-#+end_important
-
 * APA300ML - 2 Degrees of Freedom Model
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/test_apa_3_model_2dof.m
@@ -1250,23 +1343,22 @@ freqs = 5*logspace(0, 3, 1000);
 ** Two Degrees of Freedom APA Model
 <<ssec:test_apa_2dof_model>>
 
-The APA model shown in Figure ref:fig:test_apa_2dof_model is adapted from cite:souleille18_concep_activ_mount_space_applic.
+The model of the amplified piezoelectric actuator is shown in Figure ref:fig:test_apa_2dof_model.
-
 It can be decomposed into three components:
 - the shell whose axial properties are represented by $k_1$ and $c_1$
 - the actuator stacks whose contribution in the axial stiffness is represented by $k_a$ and $c_a$.
   A force source $\tau$ represents the axial force induced by the force sensor stacks.
   The gain $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $\tau$
-- the actuator stacks whose contribution in the axial stiffness is represented by $k_e$ and $c_e$.
+- the sensor stack whose contribution in the axial stiffness is represented by $k_e$ and $c_e$.
-  A "strain sensor" $d_L$, and a gain $g_s$ (in $V/m$) that converts this strain into a generated voltage
+  A  sensor measures the stack strain $d_L$ which is then converted to a voltage $V_s$ using a gain $g_s$ (in $V/m$)
 
 Such simple model has some limitations:
-- it only represents the axial characteristics of the APA (infinitely rigid in other directions)
+- it only represents the axial characteristics of the APA as it is modelled as infinitely rigid in the other directions
-- some physical insights are lost such as the amplification factor, the real stress and strain on the piezoelectric stacks
+- some physical insights are lost such as the amplification factor, the real stress and strain in the piezoelectric stacks
 - it is fully linear and therefore the creep and hysteresis of the piezoelectric stacks are not modelled
 
 #+name: fig:test_apa_2dof_model
-#+caption: Schematic of the two degrees of freedom model of the APA300ML
+#+caption: Schematic of the two degrees of freedom model of the APA300ML, adapted from cite:souleille18_concep_activ_mount_space_applic
 [[file:figs/test_apa_2dof_model.png]]
 
 ** Tuning of the APA model
@@ -1332,8 +1424,8 @@ fs = 600; % Frequency where the two FRF should match [Hz]
 gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga;
 #+end_src
 
-First, the mass supported by the APA300ML can simply be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
+First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
-Both methods leads to an estimated mass of $5.7\,\text{kg}$.
+Both methods leads to an estimated mass of $m = 5.7\,\text{kg}$.
 
 Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force.
 Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $20\,Ns/m$.
@@ -1355,7 +1447,7 @@ Knowing from eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\te
 Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance.
 $c_a = 100\,Ns/m$ and $c_e = 200\,Ns/m$ are obtained.
 
-Finally, the two gains $g_s$ and $g_a$ can be used to match the gain of the identified transfer functions.
+Finally, the two gains $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions.
 
 The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_simscape are summarized in Table ref:tab:test_apa_2dof_parameters.
 
@@ -1378,9 +1470,8 @@ The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model
 ** Obtained Dynamics
 <<ssec:test_apa_2dof_model_result>>
 
-The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model.
+The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model.
 It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf.
-
 A good match can be observed between the model and the experimental data, both for the encoder (Figure ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure ref:fig:test_apa_2dof_comp_frf_force).
 This indicates that this model represents well the axial dynamics of the APA300ML.
 
@@ -1501,8 +1592,6 @@ exportFig('figs/test_apa_2dof_comp_frf_force.pdf', 'width', 'half', 'height', 't
 #+end_subfigure
 #+end_figure
 
-** Conclusion                                                        :ignore:
-
 * APA300ML - Super Element
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/test_apa_4_model_flexible.m
@@ -1510,14 +1599,20 @@ exportFig('figs/test_apa_2dof_comp_frf_force.pdf', 'width', 'half', 'height', 't
 <<sec:test_apa_model_flexible>>
 
 ** Introduction                                                      :ignore:
-In this section, a /super element/ of the Amplified Piezoelectric Actuator "APA300ML" is extracted using a Finite Element Software.
+In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:11].
-It is then imported in Simscape (using the stiffness and mass matrices) and it is included in the same model that was used in ref:sec:test_apa_model_2dof.
+It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof.
-
 This procedure is illustrated in Figure ref:fig:test_apa_super_element_simscape.
+Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then make accessible in the Simscape model as shown at the right by the "frames" =F1= to =F5=.
+
+For the APA300ML /super element/, 5 /remote points/ are defined.
+Two /remote points/ (=1= and =2=) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used for connecting the APA300ML with other mechanical elements.
+Two /remote points/ (=3= and =4=) are located across two piezoelectric stacks and will be used to apply internal forces representing the actuator stacks.
+Finally, two /remote points/ (=4= and =4=) are located across the third piezoelectric stack.
+It will be used to measure the strain experience by this stack, and model the sensor stack.
 
 #+name: fig:test_apa_super_element_simscape
-#+caption: Finite Element Model of the APA300ML with "remotes points" on the left. Simscape model with included "Reduced Order Flexible Solid" on the right.
 #+attr_latex: :width 1.0\linewidth
+#+caption: Finite Element Model of the APA300ML with "remotes points" on the left. Simscape model with included "Reduced Order Flexible Solid" on the right.
 [[file:figs/test_apa_super_element_simscape.png]]
 
 ** Matlab Init                                              :noexport:ignore:
@@ -1560,19 +1655,11 @@ io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
 freqs = 5*logspace(0, 3, 1000);
 #+end_src
 
-** Extraction of the super-element
-
-- Explain how the "remote points" are chosen
-- Show some parts of the mass and stiffness matrices?
-- Say which materials were used?
-- Maybe this was already explain earlier in the manuscript
-
 ** Identification of the Actuator and Sensor constants
 <<ssec:test_apa_flexible_ga_gs>>
 
-Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be identified.
+Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be extracted.
-The gains $g_a$ and $g_s$ can then be tuned such that the gain of the transfer functions are matching the identified ones.
+The gains $g_a$ and $g_s$ are then be tuned such that the gain of the transfer functions are matching the identified ones.
-
 By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained.
 
 #+begin_src matlab
@@ -1600,7 +1687,6 @@ ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u'));
 gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz'))));
 #+end_src
 
-
 To make sure these "gains" are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
 
 From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force.
@@ -1612,13 +1698,11 @@ From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation bet
 \end{align}
 \end{subequations}
 
-Parameters used in equations eqref:eq:test_apa_piezo_strain_to_voltage and eqref:eq:test_apa_piezo_voltage_to_force are described in Table ref:tab:test_apa_piezo_properties.
-
 Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
 However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties.
 The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties.
 
-From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are very close to the identified constants using the experimentally identified transfer functions.
+From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are close to the identified constants using the experimentally identified transfer functions.
 
 #+name: tab:test_apa_piezo_properties
 #+caption: Piezoelectric properties used for the estimation of the sensor and actuators "gains"
@@ -1659,15 +1743,10 @@ ga_th = d33*n*ka;   % Actuator Constant [N/V]
 <<ssec:test_apa_flexible_comp_frf>>
 
 The obtained dynamics using the /super element/ with the tuned "sensor gain" and "actuator gain" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf.
-
 A good match between the model and the experimental results is observed.
-- the /super element/
+It is however a bit surprising that the model is a bit "softer" than the measured system as finite element models are usually overestimating the stiffness.
 
-This model represents fairly
+Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
-
-The flexible model is a bit "soft" as compared with the experimental results.
-
-This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators.
 
 #+begin_src matlab
 %% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants"
@@ -1779,11 +1858,16 @@ exportFig('figs/test_apa_super_element_comp_frf_force.pdf', 'width', 'half', 'he
 #+end_subfigure
 #+end_figure
 
-** Conclusion                                                        :ignore:
-
 * Conclusion
 <<sec:test_apa_conclusion>>
 
+The main characteristics of the APA300ML such as hysteresis and axial stiffness have been measured and were found to comply with the specifications.
+
+The dynamics of the received APA were measured and found to all be identical (Figure ref:fig:test_apa_frf_dynamics).
+Even tough a non-minimum zero was observed on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_non_minimum_phase), it was not found to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure ref:fig:test_apa_iff).
+
+
+
 - Compare 2DoF and FEM models (usefulness of the two)
 - Good match between all the APA: will simplify the modeling and control of the nano-hexapod
 - No advantage of the FEM model here (as only uniaxial behavior is checked), but may be useful later
@@ -2008,6 +2092,10 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
 * Footnotes
 
+[fn:11]Ansys\textsuperscript{\textregistered} was used
+[fn:10]The transfer function fitting was computed using the =vectfit3= routine, see  [[cite:&gustavsen99_ration_approx_frequen_domain_respon]]
+[fn:9]Frequency of the sinusoidal wave is $1\,\text{Hz}$
+[fn:8]Renishaw Vionic, resolution of $2.5\,nm$
 [fn:7]Kistler 9722A
 [fn:6]Polytec controller 3001 with sensor heads OFV512
 [fn:5]Note that this is not fully correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.

test-bench-apa.tex

@@ -1,4 +1,4 @@
-% Created 2024-03-27 Wed 23:01
+% Created 2024-04-03 Wed 18:15
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -22,24 +22,28 @@
 \tableofcontents
 
 \clearpage
+In this chapter, the goal is to make sure that the received APA300ML (shown in Figure \ref{fig:test_apa_received}) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics.
+
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_received.jpg}
 \caption{\label{fig:test_apa_received}Picture of 5 out of the 7 received APA300ML}
 \end{figure}
 
-The first goal is to characterize the APA300ML in terms of:
+In section \ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. Flexible modes of the APA300ML are computed with a finite element model and compared with measurements.
-\begin{itemize}
+
-\item The, geometric features, electrical capacitance, stroke, hysteresis, spurious resonances.
+Using a dedicated test bench, dynamical measurements are performed (Section \ref{sec:test_apa_dynamics}).
-This is performed in Section \ref{sec:test_apa_basic_meas}.
+The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
-\item The dynamics from the generated DAC voltage (going to the voltage amplifiers and then applied on the actuator stacks) to the induced displacement, and to the measured voltage by the force sensor stack.
+Integral Force Feedback is experimentally applied and the damped plants are estimated for several feedback gains.
-Also the ``actuator constant'' and ``sensor constant'' are identified.
+
-This is done in Section \ref{sec:test_apa_dynamics}.
+Two different models of the APA300ML are then presented.
-\item Compare the measurements with the two Simscape models: 2DoF (Section \ref{sec:test_apa_model_2dof}) Super-Element (Section \ref{sec:test_apa_model_flexible})
+First, in Section \ref{sec:test_apa_model_2dof}, a two degrees of freedom model is presented, tuned and compared with the measured dynamics.
-\end{itemize}
+This model is proven to accurately simulate the APA300ML's axial dynamics.
+
+Then, in Section \ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a finite element model and imported in Simscape.
+This more complex model is also shown to well capture the axial dynamics of the APA300ML.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_apa_section_matlab_code}Report sections and corresponding Matlab files}
 \centering
 \begin{tabularx}{0.6\linewidth}{lX}
 \toprule
@@ -51,38 +55,37 @@ Section \ref{sec:test_apa_model_2dof} & \texttt{test\_apa\_3\_model\_2dof.m}\\
 Section \ref{sec:test_apa_model_flexible} & \texttt{test\_apa\_4\_model\_flexible.m}\\
 \bottomrule
 \end{tabularx}
+\caption{\label{tab:test_apa_section_matlab_code}Report sections and corresponding Matlab files}
+
 \end{table}
 \chapter{First Basic Measurements}
+\label{sec:org399d3a7}
 \label{sec:test_apa_basic_meas}
 
-Before using the measurement bench to characterize the APA300ML, first simple measurements are performed:
+Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed.
-\begin{itemize}
+First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section \ref{ssec:test_apa_geometrical_measurements}.
-\item Section \ref{ssec:test_apa_geometrical_measurements}: the geometric tolerances of the interface planes are checked
+Then, the capacitance of the piezoelectric stacks is measured in Section \ref{ssec:test_apa_electrical_measurements}.
-\item Section \ref{ssec:test_apa_electrical_measurements}: the capacitance of the piezoelectric stacks is measured
+The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}.
-\item Section \ref{ssec:test_apa_stroke_measurements}: the stroke of each APA is measured
+Finally, in Section \ref{ssec:test_apa_spurious_resonances}, the flexible modes of the APA are measured and compared with a finite element model.
-\item Section \ref{ssec:test_apa_spurious_resonances}: the ``spurious'' resonances of the APA are investigated
-\end{itemize}
 \section{Geometrical Measurements}
+\label{sec:org9db80da}
 \label{ssec:test_apa_geometrical_measurements}
 
-To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with very good flatness.
+To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness.
-
 As shown in Figure \ref{fig:test_apa_flatness_setup}, the APA is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)} is used to measure the height of 4 points on each of the APA300ML interfaces.
-
 From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.4\linewidth]{figs/test_apa_flatness_setup.png}
-\caption{\label{fig:test_apa_flatness_setup}Measurement setup for flatness estimation of the two mechanical interfaces}
-\end{figure}
-
 The measured flatness, summarized in Table \ref{tab:test_apa_flatness_meas}, are within the specifications.
 
-\begin{table}[htbp]
+\begin{minipage}[b]{0.49\linewidth}
-\caption{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
+\begin{center}
-\centering
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_flatness_setup.png}
-\begin{tabularx}{0.3\linewidth}{Xc}
+\captionof{figure}{\label{fig:test_apa_flatness_setup}Measurement setup for flatness estimation}
+\end{center}
+\end{minipage}
+\hfill
+\begin{minipage}[b]{0.49\linewidth}
+\begin{center}
+\begin{tabularx}{0.6\linewidth}{Xc}
 \toprule
  & \textbf{Flatness} \([\mu m]\)\\
 \midrule
@@ -95,32 +98,35 @@ APA 6 & 7.1\\
 APA 7 & 18.7\\
 \bottomrule
 \end{tabularx}
-\end{table}
+\captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
+
+\end{center}
+\end{minipage}
 \section{Electrical Measurements}
+\label{sec:orgbec5e3a}
 \label{ssec:test_apa_electrical_measurements}
 
 From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu F\) and \(26\,\mu F\) with a nominal capacitance of \(20\,\mu F\).
 
-The capacitance of the piezoelectric stacks found in the APA300ML have been measured with the LCR meter\footnote{LCR-819 from Gwinstek, specified accuracy of \(0.05\%\), measured frequency is set at \(1\,\text{kHz}\)} shown in Figure \ref{fig:test_apa_lcr_meter}.
+The piezoelectric stacks capacitance of the APA300ML have been measured with the LCR meter\footnote{LCR-819 from Gwinstek, specified accuracy of \(0.05\%\), measured frequency is set at \(1\,\text{kHz}\)} shown in Figure \ref{fig:test_apa_lcr_meter}.
 The two stacks used as an actuator and the stack used as a force sensor are measured separately.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_lcr_meter.jpg}
-\caption{\label{fig:test_apa_lcr_meter}LCR Meter used for the measurements}
-\end{figure}
-
 The measured capacitance are summarized in Table \ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu F\).
 However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance.
 This may indicate a manufacturing defect.
 
 The measured capacitance is found to be lower than the specified one.
-This may be due to the fact that the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)) while it was here measured with small signals.
+This may be due to the fact that the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)) while it was here measured with small signals \cite{wehrsdorfer95_large_signal_measur_piezoel_stack}.
 
-\begin{table}[htbp]
+\begin{minipage}[b]{0.49\linewidth}
-\caption{\label{tab:test_apa_capacitance}Capacitance measured with the LCR meter. The excitation signal is a sinus at 1kHz}
+\begin{center}
-\centering
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_lcr_meter.jpg}
-\begin{tabularx}{0.5\linewidth}{lcc}
+\captionof{figure}{\label{fig:test_apa_lcr_meter}Used LCR meter}
+\end{center}
+\end{minipage}
+\hfill
+\begin{minipage}[b]{0.49\linewidth}
+\begin{center}
+\begin{tabularx}{0.95\linewidth}{lcc}
 \toprule
  & \textbf{Sensor Stack} & \textbf{Actuator Stacks}\\
 \midrule
@@ -133,63 +139,83 @@ APA 6 & 4.99 & 9.91\\
 APA 7 & 4.85 & 9.85\\
 \bottomrule
 \end{tabularx}
-\end{table}
+\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in \(\mu F\)}
+
+\end{center}
+\end{minipage}
 \section{Stroke and Hysteresis Measurement}
+\label{sec:org7997ce8}
 \label{ssec:test_apa_stroke_measurements}
 
-The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
+In order to verify that the stroke of the APA300ML is as specified in the datasheet, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}.
-To do so, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}.
 
 Then, the voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC and a voltage amplifier.
-Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure \ref{fig:test_apa_stroke_bench}, left).
+Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure \ref{fig:test_apa_stroke_voltage}).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_apa_stroke_bench.jpg}
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_stroke_bench.jpg}
 \caption{\label{fig:test_apa_stroke_bench}Bench to measured the APA stroke}
 \end{figure}
 
-The measured APA displacement is shown as a function of the applied voltage in Figure \ref{fig:test_apa_stroke_result}, right.
+The measured APA displacement is shown as a function of the applied voltage in Figure \ref{fig:test_apa_stroke_hysteresis}.
-
 Typical hysteresis curves for piezoelectric stack actuators can be observed.
 The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks, which is enough for the current application.
 This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu m\), therefore \(\approx 200\,\mu m\) if only two stacks are used).
 
-It is clear from Figure \ref{fig:test_apa_stroke_result} that ``APA 3'' has an issue compared to the other units.
+It is clear from Figure \ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared to the other units.
 This confirms the abnormal electrical measurements made in Section \ref{ssec:test_apa_electrical_measurements}.
 This unit was send sent back to Cedrat and a new one was shipped back.
 From now on, only the six APA that behave as expected will be used.
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{0.49\textwidth}
-\includegraphics[scale=1]{figs/test_apa_stroke_result.png}
+\begin{center}
-\caption{\label{fig:test_apa_stroke_result}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (left). Measured displacement as a function of the applied voltage (right)}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_stroke_voltage.png}
+\end{center}
+\subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_stroke_hysteresis.png}
+\end{center}
+\subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
+\end{subfigure}
+\caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of the applied voltage (\subref{fig:test_apa_stroke_hysteresis})}
 \end{figure}
 \section{Flexible Mode Measurement}
+\label{sec:orga2217ed}
 \label{ssec:test_apa_spurious_resonances}
 
 In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
-
+To experimentally estimate these modes, the APA is fixed on one end (see Figure \ref{fig:test_apa_meas_setup_modes}).
-To experimentally estimate these modes, the APA is fixed on one end (see Figure \ref{fig:test_apa_meas_setup_torsion}).
+A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes.
-A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes.
 Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated.
 
 The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure \ref{fig:test_apa_mode_shapes}.
 
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{0.36\textwidth}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_apa_mode_shapes.png}
+\begin{center}
-\caption{\label{fig:test_apa_mode_shapes}Spurious resonances - Change this with the updated FEM analysis of the APA300ML}
+\includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_1.png}
+\end{center}
+\subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode (268Hz)}
+\end{subfigure}
+\begin{subfigure}{0.28\textwidth}
+\begin{center}
+\includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_2.png}
+\end{center}
+\subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode (399Hz)}
+\end{subfigure}
+\begin{subfigure}{0.36\textwidth}
+\begin{center}
+\includegraphics[scale=1,height=4.3cm]{figs/test_apa_mode_shapes_3.png}
+\end{center}
+\subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode (706Hz)}
+\end{subfigure}
+\caption{\label{fig:test_apa_mode_shapes}First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model}
 \end{figure}
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_meas_setup_torsion.jpg}
-\caption{\label{fig:test_apa_meas_setup_torsion}Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the \(Z\) torsion is measured.}
-\end{figure}
-
-Two other similar measurements are performed to measured the bending of the APA around the \(X\) direction and around the \(Y\) direction (see Figure \ref{fig:test_apa_meas_setup_modes}).
-
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
@@ -203,42 +229,28 @@ Two other similar measurements are performed to measured the bending of the APA
 \end{center}
 \subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
 \end{subfigure}
-\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measured flexible modes of the APA300ML. For the bending in the \(X\) direction, the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
+\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measured flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
 \end{figure}
 
-The three measured frequency response functions are shown in Figure \ref{fig:test_apa_meas_freq_compare}.
+The measured frequency response functions computed from the experimental setups of figures \ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure \ref{fig:test_apa_meas_freq_compare}.
-\begin{itemize}
+The \(y\) bending mode is observed at \(280\,\text{Hz}\) and the \(x\) bending mode is at \(412\,\text{Hz}\).
-\item a clear \(x\) bending mode at \(280\,\text{Hz}\)
+These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure \ref{fig:test_apa_meas_setup_modes}).
-\item a clear \(y\) bending mode at \(412\,\text{Hz}\)
+This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
-\item for the \(z\) torsion test, the \(y\) bending mode is also excited and observed, and we may see a mode at \(800\,\text{Hz}\)
+This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
-\end{itemize}
+Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/test_apa_meas_freq_compare.png}
-\caption{\label{fig:test_apa_meas_freq_compare}Obtained frequency response functions for the 3 tests with the instrumented hammer}
+\caption{\label{fig:test_apa_meas_freq_compare}Obtained frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
 \end{figure}
-
-\begin{table}[htbp]
-\caption{\label{tab:test_apa_measured_modes_freq}Measured frequency of the modes}
-\centering
-\begin{tabularx}{0.5\linewidth}{Xcc}
-\toprule
-\textbf{Mode} & \textbf{FEM} & \textbf{Measured Frequency}\\
-\midrule
-\(X\) bending &  & 280Hz\\
-\(Y\) bending &  & 410Hz\\
-\(Z\) torsion &  & 800Hz\\
-\bottomrule
-\end{tabularx}
-\end{table}
 \chapter{Dynamical measurements}
+\label{sec:orgdf7ee61}
 \label{sec:test_apa_dynamics}
-
+After the basic measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the dynamics of the APA300ML.
-After the basic measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the APA.
+This test bench, depicted in Figure \ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing.
-
+That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
-This test bench is shown in Figure \ref{fig:test_bench_apa} and consists of the APA300ML fixed on one end to the fixed granite, and on the other end to the 5kg granite vertically guided with an air bearing.
+An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
-An encoder is used to measure the relative motion between the two granites (i.e. the displacement of the APA).
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.3\textwidth}
@@ -256,49 +268,26 @@ An encoder is used to measure the relative motion between the two granites (i.e.
 \caption{\label{fig:test_bench_apa}Schematic of the test bench used to estimate the dynamics of the APA300ML}
 \end{figure}
 
-The bench is schematically shown in Figure \ref{fig:test_apa_schematic} and the signal used are summarized in Table \ref{tab:test_apa_variables}.
+The bench is schematically shown in Figure \ref{fig:test_apa_schematic} with all the associated signals.
+It will be first used to estimate the hysteresis from the piezoelectric stack (Section \ref{ssec:test_apa_hysteresis}) as well as the axial stiffness of the APA300ML (Section \ref{ssec:test_apa_stiffness}).
+Then, the frequency response functions from the DAC voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section \ref{ssec:test_apa_meas_dynamics}.
+The presence of a non minimum phase zero found on the transfer function from \(u\) to \(V_s\) is investigated in Section \ref{ssec:test_apa_non_minimum_phase}.
+In order to limit the low frequency gain of the transfer function from \(u\) to \(V_s\), a resistor is added across the force sensor stack (Section \ref{ssec:test_apa_resistance_sensor_stack}).
+Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section \ref{ssec:test_apa_iff_locus}.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,scale=1]{figs/test_apa_schematic.png}
-\caption{\label{fig:test_apa_schematic}Schematic of the Test Bench}
+\caption{\label{fig:test_apa_schematic}Schematic of the Test Bench used to measured the dynamics of the APA300ML. \(u\) is the output DAC voltage, \(V_a\) the output amplifier voltage (i.e. voltage applied across the actuator stacks), \(d_e\) the measured displacement by the encoder and \(V_s\) the measured voltage across the sensor stack.}
 \end{figure}
-
-\begin{table}[htbp]
-\caption{\label{tab:test_apa_variables}Variables used during the measurements}
-\centering
-\begin{tabularx}{0.6\linewidth}{cXc}
-\toprule
-\textbf{Variable} & \textbf{Description} & \textbf{Unit}\\
-\midrule
-\(u\) & Output DAC Voltage & \(V\)\\
-\(V_a\) & Output Amplifier Voltage & \(V\)\\
-\(V_s\) & Measured Stack Voltage (ADC) & \(V\)\\
-\(d_e\) & Encoder Measurement & \(m\)\\
-\bottomrule
-\end{tabularx}
-\end{table}
-
-This bench will be used to:
-\begin{itemize}
-\item \ref{ssec:test_apa_hysteresis}
-\item \ref{ssec:test_apa_stiffness}
-\item measure the dynamics of the APA (section \ref{ssec:test_apa_meas_dynamics})
-\item estimate the added damping using Integral Force Feedback (Section \ref{ssec:test_apa_iff_locus})
-\end{itemize}
-
-These measurements will also be used to tune the developed models of the APA (in Section \ref{sec:test_apa_model_2dof} for the 2DoF model, and in Section \ref{sec:test_apa_model_flexible} for the flexible model).
 \section{Hysteresis}
+\label{sec:org7aacb8e}
 \label{ssec:test_apa_hysteresis}
 
 As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
-
+Do to so, a quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)) and with an amplitude varying from \(4\,V\) up to \(80\,V\) is generated using the DAC.
-A quasi static sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)), and an amplitude varying from \(4\,V\) up to \(80\,V\).
+For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage in Figure \ref{fig:test_apa_meas_hysteresis}.
-
+This is the typical behavior expected from a PZT stack actuator where the hysteresis increases as a function of the applied voltage amplitude \cite[chap. 1.4]{fleming14_desig_model_contr_nanop_system}.
-For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage..
-
-The measured displacements as a function of the output voltages are shown in Figure \ref{fig:test_apa_meas_hysteresis}.
-It is interesting to see that the hysteresis is increasing with the excitation amplitude.
 
 \begin{figure}[htbp]
 \centering
@@ -306,10 +295,10 @@ It is interesting to see that the hysteresis is increasing with the excitation a
 \caption{\label{fig:test_apa_meas_hysteresis}Obtained hysteresis curves (displacement as a function of applied voltage) for multiple excitation amplitudes}
 \end{figure}
 \section{Axial stiffness}
+\label{sec:org5cd81d2}
 \label{ssec:test_apa_stiffness}
 
 In order to estimate the stiffness of the APA, a weight with known mass \(m_a = 6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(d_e\) is measured using the encoder.
-
 The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}.
 
 \begin{equation} \label{eq:test_apa_stiffness}
@@ -320,19 +309,19 @@ The measured displacement \(d_e\) as a function of time is shown in Figure \ref{
 It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
 These two effects induce some uncertainties in the measured stiffness.
 
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/test_apa_meas_stiffness_time.png}
-\caption{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding the mass (at \(t \approx 3\,s\)) and removing the mass(at \(t \approx 13\,s\))}
-\end{figure}
-
 The stiffnesses are computed for all the APA from the two displacements \(d_1\) and \(d_2\) (see Figure \ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\).
-These estimated stiffnesses are summarized in Table \ref{tab:test_apa_measured_stiffnesses} and are found to be close to the nominal stiffness \(k = 1.8\,N/\mu m\) found in the APA300ML manual.
+These estimated stiffnesses are summarized in Table \ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,N/\mu m\).
 
-\begin{table}[htbp]
+\begin{minipage}[b]{0.57\linewidth}
-\caption{\label{tab:test_apa_measured_stiffnesses}Measured stiffnesses (in \(N/\mu m\))}
+\begin{center}
-\centering
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_apa_meas_stiffness_time.png}
-\begin{tabularx}{0.2\linewidth}{ccc}
+\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding (at \(t \approx 3\,s\)) and removing (at \(t \approx 13\,s\)) the mass}
+\end{center}
+\end{minipage}
+\hfill
+\begin{minipage}[b]{0.37\linewidth}
+\begin{center}
+\begin{tabularx}{0.6\linewidth}{Xcc}
 \toprule
 APA & \(k_1\) & \(k_2\)\\
 \midrule
@@ -344,7 +333,10 @@ APA & \(k_1\) & \(k_2\)\\
 8 & 1.73 & 1.98\\
 \bottomrule
 \end{tabularx}
-\end{table}
+\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses (in \(N/\mu m\))}
+
+\end{center}
+\end{minipage}
 
 The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 95\,\text{Hz}\) (estimated by the dynamical measurements shown in section \ref{ssec:test_apa_meas_dynamics}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).
 
@@ -355,42 +347,45 @@ The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} b
 The obtain stiffness is \(k \approx 2\,N/\mu m\) which is close to the values found in the documentation and by the ``static deflection'' method.
 
 
-However, changes in the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack \cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}.
+It is important to note that changes to the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack \cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}.
 
-To estimate this effect, the stiffness of the APA if measured using the ``static deflection'' method in two cases:
+To estimate this effect for the APA300ML, its stiffness is estimated using the ``static deflection'' method in two cases:
 \begin{itemize}
 \item \(k_{\text{os}}\): piezoelectric stacks left unconnected (or connect to the high impedance ADC)
 \item \(k_{\text{sc}}\): piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance)
 \end{itemize}
 
-The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) and the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
+It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
 \section{Dynamics}
+\label{sec:org9be9924}
 \label{ssec:test_apa_meas_dynamics}
 
-In this section, the dynamics of the system from the excitation voltage \(u\) to encoder measured displacement \(d_e\)  and to the force sensor voltage \(V_s\) is identified.
+In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified.
 
-The obtained transfer functions for the 6 APA between the excitation voltage \(u\) and the encoder displacement \(d_e\) are shown in Figure \ref{fig:test_apa_frf_encoder}.
+First, the dynamics from \(u\) to \(d_e\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_encoder}.
-The obtained transfer functions are close to a mass-spring-damper system.
+The obtained frequency response functions are similar to that of a (second order) mass-spring-damper system with:
-The following can be observed:
 \begin{itemize}
 \item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu m/V\).
-The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA
+The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA
 \item A lightly damped resonance at \(95\,\text{Hz}\)
-\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
+\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which additional resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
+Flexible modes studied in section \ref{ssec:test_apa_spurious_resonances} seems not to impact the measured axial motion of the actuator.
 \end{itemize}
 
-The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) is also identified and shown in Figure \ref{fig:test_apa_frf_force}.
+The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_force}.
 
-A lightly damped resonance is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance at \(41\,\text{Hz}\).
+A lightly damped resonance (pole) is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance (zero) at \(41\,\text{Hz}\).
-No additional resonances is present up to at least \(2\,\text{kHz}\) indicating at Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
+No additional resonances is present up to at least \(2\,\text{kHz}\) indicating that Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
+The zero at \(41\,\text{Hz}\) seems to be non-minimum phase (the phase \emph{decreases} by 180 degrees whereas it should have \emph{increased} by 180 degrees for a minimum phase zero).
+This is investigated in Section \ref{ssec:test_apa_non_minimum_phase}.
 
-As illustrated by the Root Locus, the poles of the closed-loop system converges to the zeros of the open-loop plant.
+As illustrated by the Root Locus, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases.
-Suppose that a controller with a very high gain is implemented such that the voltage \(V_s\) across the sensor stack is zero.
+The significance of this behavior varies on the type of sensor used as explained in \cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
-In that case, because of the very high controller gain, no stress and strain is present on the sensor stack (and on the actuator stacks are well, as they are both in series).
+Considering the transfer function from \(u\) to \(V_s\), if a controller with a very high gain is applied such that the sensor stack voltage \(V_s\) is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
-Such closed-loop system would therefore virtually corresponds to a system for which the piezoelectric stacks have been removed and just the mechanical shell is kept.
+Consequently, the closed-loop system would virtually corresponds to one where the piezoelectric stacks are absent, leaving only the mechanical shell.
-From this analysis, the axial stiffness of the shell can be estimated to be \(k_{\text{shell}} = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m\).
+From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m\) (which is close to what is found using a finite element model).
 
-Such reasoning can lead to very interesting insight into the system just from an open-loop identification.
+All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure \ref{fig:test_apa_frf_encoder} and at the force sensor in Figure \ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -407,35 +402,58 @@ Such reasoning can lead to very interesting insight into the system just from an
 \end{subfigure}
 \caption{\label{fig:test_apa_frf_dynamics}Measured frequency response function from generated voltage \(u\) to the encoder displacement \(d_e\) (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage \(V_s\) (\subref{fig:test_apa_frf_force}) for the six APA300ML}
 \end{figure}
+\section{Non Minimum Phase Zero?}
+\label{sec:org5d96397}
+\label{ssec:test_apa_non_minimum_phase}
 
-All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure \ref{fig:test_apa_frf_encoder} and at the force sensor in Figure \ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical lever.
+It was surprising to observe a non-minimum phase behavior for the zero on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}).
+It was initially thought that this non-minimum phase behavior is an artifact coming from the measurement.
+A longer measurement was performed with different excitation signals (noise, slow sine sweep, etc.) to see it the phase behavior of the zero changes.
+Results of one long measurement is shown in Figure \ref{fig:test_apa_non_minimum_phase}.
+The coherence (Figure \ref{fig:test_apa_non_minimum_phase_coherence}) is good even in the vicinity of the lightly damped zero, and the phase (Figure \ref{fig:test_apa_non_minimum_phase_zoom}) clearly indicates non-minimum phase behavior.
+
+Such non-minimum phase zero when using load cells has also been observed on other mechanical systems \cite{spanos95_soft_activ_vibrat_isolat,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}.
+It could be induced to small non-linearity in the system, but the reason of this non-minimum phase for the APA300ML is not yet clear.
+
+However, this is not so important here as the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure \ref{fig:test_apa_iff_root_locus}) should not be unstable except for very large controller gains that will never be applied in practice.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_non_minimum_phase_coherence.png}
+\end{center}
+\subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_non_minimum_phase_zoom.png}
+\end{center}
+\subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
+\end{subfigure}
+\caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found on the transfer function from \(u\) to \(V_s\). The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.}
+\end{figure}
 \section{Effect of the resistor on the IFF Plant}
+\label{sec:orge8ed591}
 \label{ssec:test_apa_resistance_sensor_stack}
 
-A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the stack.
+A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the piezoelectric stack (capacitance estimated at \(\approx 5\,\mu F\)).
 
-As explain before, this is done for two reasons:
+As explain before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain.
-\begin{enumerate}
-\item Limit the voltage offset due to the input bias current of the ADC
-\item Limit the low frequency gain
-\end{enumerate}
 
 The (low frequency) transfer function from \(u\) to \(V_s\) with and without this resistor have been measured and are compared in Figure \ref{fig:test_apa_effect_resistance}.
-It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of \(\approx 0.35\,\text{Hz}\).
+It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of \(\approx 0.39\,\text{Hz}\).
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/test_apa_effect_resistance.png}
-\caption{\label{fig:test_apa_effect_resistance}Transfer function from u to \(V_s\) with and without the resistor \(R\) in parallel with the piezoelectric stack used as the force sensor}
+\caption{\label{fig:test_apa_effect_resistance}Transfer function from \(u\) to \(V_s\) with and without the resistor \(R\) in parallel with the piezoelectric stack used as the force sensor}
 \end{figure}
 \section{Integral Force Feedback}
+\label{sec:orge2d1f26}
 \label{ssec:test_apa_iff_locus}
 
-This test bench can also be used to estimate the damping added by the implementation of an Integral Force Feedback strategy.
+In order to implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation \eqref{eq:test_apa_iff_manual_fit}.
-
+The parameters are manually tuned, and the obtained values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\).
-First, the transfer function \eqref{eq:test_apa_iff_manual_fit} is manually tuned to match the identified dynamics from generated voltage \(u\) to the measured sensor stack voltage \(V_s\) in Section \ref{ssec:test_apa_meas_dynamics}.
-
-The obtained parameter values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\).
 
 \begin{equation} \label{eq:test_apa_iff_manual_fit}
 G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s}
@@ -465,17 +483,11 @@ The transfer function from the ``damped'' plant input \(u\prime\) to the encoder
 \caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\)}
 \end{figure}
 
-The identified dynamics are then fitted by second order transfer functions.
+The identified dynamics are then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see  \cite{gustavsen99_ration_approx_frequen_domain_respon}}.
 The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure \ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
 It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
 
-
+The evolution of the pole in the complex plane as a function of the controller gain \(g\) (i.e. the ``root locus'') is computed both using the IFF plant model \eqref{eq:test_apa_iff_manual_fit} and the implemented controller \eqref{eq:test_apa_Kiff_formula} and from the fitted transfer functions of the damped plants experimentally identified for several controller gains.
-The evolution of the pole in the complex plane as a function of the controller gain \(g\) (i.e. the ``root locus'') is computed:
-\begin{itemize}
-\item using the IFF plant model \eqref{eq:test_apa_iff_manual_fit} and the implemented controller \eqref{eq:test_apa_Kiff_formula}
-\item from the fitted transfer functions of the damped plants experimentally identified for several controller gains
-\end{itemize}
-
 The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_locus} and are in good agreement considering that the damped plants were only fitted using a second order transfer function.
 
 \begin{figure}[htbp]
@@ -493,10 +505,8 @@ The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_loc
 \end{subfigure}
 \caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus})}
 \end{figure}
-\begin{important}
-So far, all the measured FRF are showing the dynamical behavior that was expected.
-\end{important}
 \chapter{APA300ML - 2 Degrees of Freedom Model}
+\label{sec:org42297a7}
 \label{sec:test_apa_model_2dof}
 
 In this section, a simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to compare the model of the APA with the measured frequency response functions.
@@ -511,33 +521,34 @@ The obtained model dynamics is compared with the measurements in Section \ref{ss
 \caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model}
 \end{figure}
 \section{Two Degrees of Freedom APA Model}
+\label{sec:org97d98f4}
 \label{ssec:test_apa_2dof_model}
 
-The APA model shown in Figure \ref{fig:test_apa_2dof_model} is adapted from \cite{souleille18_concep_activ_mount_space_applic}.
+The model of the amplified piezoelectric actuator is shown in Figure \ref{fig:test_apa_2dof_model}.
-
 It can be decomposed into three components:
 \begin{itemize}
 \item the shell whose axial properties are represented by \(k_1\) and \(c_1\)
 \item the actuator stacks whose contribution in the axial stiffness is represented by \(k_a\) and \(c_a\).
 A force source \(\tau\) represents the axial force induced by the force sensor stacks.
 The gain \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(\tau\)
-\item the actuator stacks whose contribution in the axial stiffness is represented by \(k_e\) and \(c_e\).
+\item the sensor stack whose contribution in the axial stiffness is represented by \(k_e\) and \(c_e\).
-A ``strain sensor'' \(d_L\), and a gain \(g_s\) (in \(V/m\)) that converts this strain into a generated voltage
+A  sensor measures the stack strain \(d_L\) which is then converted to a voltage \(V_s\) using a gain \(g_s\) (in \(V/m\))
 \end{itemize}
 
 Such simple model has some limitations:
 \begin{itemize}
-\item it only represents the axial characteristics of the APA (infinitely rigid in other directions)
+\item it only represents the axial characteristics of the APA as it is modelled as infinitely rigid in the other directions
-\item some physical insights are lost such as the amplification factor, the real stress and strain on the piezoelectric stacks
+\item some physical insights are lost such as the amplification factor, the real stress and strain in the piezoelectric stacks
 \item it is fully linear and therefore the creep and hysteresis of the piezoelectric stacks are not modelled
 \end{itemize}
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/test_apa_2dof_model.png}
-\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees of freedom model of the APA300ML}
+\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees of freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}}
 \end{figure}
 \section{Tuning of the APA model}
+\label{sec:org29ff205}
 \label{ssec:test_apa_2dof_model_tuning}
 
 9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
@@ -548,8 +559,8 @@ Such simple model has some limitations:
 \caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees of freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
 \end{figure}
 
-First, the mass supported by the APA300ML can simply be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
+First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
-Both methods leads to an estimated mass of \(5.7\,\text{kg}\).
+Both methods leads to an estimated mass of \(m = 5.7\,\text{kg}\).
 
 Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,N/\mu m\) in Section \ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure \ref{fig:test_apa_frf_force}.
 Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(20\,Ns/m\).
@@ -571,12 +582,11 @@ Knowing from \eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{
 Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance.
 \(c_a = 100\,Ns/m\) and \(c_e = 200\,Ns/m\) are obtained.
 
-Finally, the two gains \(g_s\) and \(g_a\) can be used to match the gain of the identified transfer functions.
+Finally, the two gains \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions.
 
 The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
 \centering
 \begin{tabularx}{0.3\linewidth}{cc}
 \toprule
@@ -593,13 +603,15 @@ The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_mode
 \(g_s\) & \(0.46\,V/\mu m\)\\
 \bottomrule
 \end{tabularx}
+\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
+
 \end{table}
 \section{Obtained Dynamics}
+\label{sec:orgbff057f}
 \label{ssec:test_apa_2dof_model_result}
 
-The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
+The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
 It is compared with the experimental data in Figure \ref{fig:test_apa_2dof_comp_frf}.
-
 A good match can be observed between the model and the experimental data, both for the encoder (Figure \ref{fig:test_apa_2dof_comp_frf_enc}) and for the force sensor (Figure \ref{fig:test_apa_2dof_comp_frf_force}).
 This indicates that this model represents well the axial dynamics of the APA300ML.
 
@@ -619,32 +631,31 @@ This indicates that this model represents well the axial dynamics of the APA300M
 \caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
 \end{figure}
 \chapter{APA300ML - Super Element}
+\label{sec:org703026b}
 \label{sec:test_apa_model_flexible}
 
-In this section, a \emph{super element} of the Amplified Piezoelectric Actuator ``APA300ML'' is extracted using a Finite Element Software.
+In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
-It is then imported in Simscape (using the stiffness and mass matrices) and it is included in the same model that was used in \ref{sec:test_apa_model_2dof}.
+It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in \ref{sec:test_apa_model_2dof}.
-
 This procedure is illustrated in Figure \ref{fig:test_apa_super_element_simscape}.
+Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then make accessible in the Simscape model as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}.
+
+For the APA300ML \emph{super element}, 5 \emph{remote points} are defined.
+Two \emph{remote points} (\texttt{1} and \texttt{2}) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used for connecting the APA300ML with other mechanical elements.
+Two \emph{remote points} (\texttt{3} and \texttt{4}) are located across two piezoelectric stacks and will be used to apply internal forces representing the actuator stacks.
+Finally, two \emph{remote points} (\texttt{4} and \texttt{4}) are located across the third piezoelectric stack.
+It will be used to measure the strain experience by this stack, and model the sensor stack.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_simscape.png}
 \caption{\label{fig:test_apa_super_element_simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
 \end{figure}
-\section{Extraction of the super-element}
-
-\begin{itemize}
-\item Explain how the ``remote points'' are chosen
-\item Show some parts of the mass and stiffness matrices?
-\item Say which materials were used?
-\item Maybe this was already explain earlier in the manuscript
-\end{itemize}
 \section{Identification of the Actuator and Sensor constants}
+\label{sec:org1fc96a6}
 \label{ssec:test_apa_flexible_ga_gs}
 
-Once the APA300ML \emph{super element} is included in the Simscape model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be identified.
+Once the APA300ML \emph{super element} is included in the Simscape model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be extracted.
-The gains \(g_a\) and \(g_s\) can then be tuned such that the gain of the transfer functions are matching the identified ones.
+The gains \(g_a\) and \(g_s\) are then be tuned such that the gain of the transfer functions are matching the identified ones.
-
 By doing so, \(g_s = 4.9\,V/\mu m\) and \(g_a = 23.2\,N/V\) are obtained.
 
 To make sure these ``gains'' are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
@@ -658,16 +669,13 @@ From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation betwe
 \end{align}
 \end{subequations}
 
-Parameters used in equations \eqref{eq:test_apa_piezo_strain_to_voltage} and \eqref{eq:test_apa_piezo_voltage_to_force} are described in Table \ref{tab:test_apa_piezo_properties}.
-
 Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
 However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
 The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}.
 
-From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained which are very close to the identified constants using the experimentally identified transfer functions.
+From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained which are close to the identified constants using the experimentally identified transfer functions.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators ``gains''}
 \centering
 \begin{tabularx}{1\linewidth}{ccX}
 \toprule
@@ -682,22 +690,18 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
 \(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
 \bottomrule
 \end{tabularx}
+\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators ``gains''}
+
 \end{table}
 \section{Comparison of the obtained dynamics}
+\label{sec:orgd657b0f}
 \label{ssec:test_apa_flexible_comp_frf}
 
 The obtained dynamics using the \emph{super element} with the tuned ``sensor gain'' and ``actuator gain'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}.
-
 A good match between the model and the experimental results is observed.
-\begin{itemize}
+It is however a bit surprising that the model is a bit ``softer'' than the measured system as finite element models are usually overestimating the stiffness.
-\item the \emph{super element}
-\end{itemize}
 
-This model represents fairly
+Using this simple test bench, it can be concluded that the \emph{super element} model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
-
-The flexible model is a bit ``soft'' as compared with the experimental results.
-
-This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -715,8 +719,16 @@ This method can be used to model piezoelectric stack actuators as well as amplif
 \caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the ``flexible'' model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
 \end{figure}
 \chapter{Conclusion}
+\label{sec:org0ba1df7}
 \label{sec:test_apa_conclusion}
 
+The main characteristics of the APA300ML such as hysteresis and axial stiffness have been measured and were found to comply with the specifications.
+
+The dynamics of the received APA were measured and found to all be identical (Figure \ref{fig:test_apa_frf_dynamics}).
+Even tough a non-minimum zero was observed on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_non_minimum_phase}), it was not found to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure \ref{fig:test_apa_iff}).
+
+
+
 \begin{itemize}
 \item Compare 2DoF and FEM models (usefulness of the two)
 \item Good match between all the APA: will simplify the modeling and control of the nano-hexapod