Add study about APA300ML

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@@ -664,6 +664,12 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 #+end_src
 
 * APA300ML
+** Introduction                                                      :ignore:
+
+#+name: fig:apa300ml_ansys
+#+caption: Ansys FEM of the APA300ML
+[[file:figs/apa300ml_ansys.jpg]]
+
 ** Matlab Init                                             :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
   <<matlab-dir>>
@@ -715,8 +721,8 @@ Then, we extract the coordinates of the interface nodes.
 #+RESULTS:
 | Total number of Nodes     |  7 |
 | Number of interface Nodes |  7 |
-| Number of Modes           | 38 |
+| Number of Modes           |  6 |
-| Size of M and K matrices  | 80 |
+| Size of M and K matrices  | 48 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
@@ -814,6 +820,112 @@ where:
 #+RESULTS:
 : 5.8594
 
+** Identification of the APA Characteristics
+*** Stiffness
+#+begin_src matlab :exports none
+  m = 0.001;
+#+end_src
+
+The transfer function from vertical external force to the relative vertical displacement is identified.
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML_test_bench';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+
+  G = linearize(mdl, io);
+#+end_src
+
+The inverse of its DC gain is the axial stiffness of the APA:
+#+begin_src matlab :results replace value
+  1e-6/dcgain(G) % [N/um]
+#+end_src
+
+#+RESULTS:
+: 1.8634
+
+The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$.
+
+*** Resonance Frequency
+The resonance frequency is specified to be between 650Hz and 840Hz.
+This is also the case for the FEM model (Figure [[fig:apa300ml_resonance]]).
+
+#+begin_src matlab :exports none
+  freqs = logspace(2, 4, 5000);
+
+  figure;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Amplitude');
+  hold off;
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_resonance.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:apa300ml_resonance
+#+caption: First resonance is around 800Hz
+#+RESULTS:
+[[file:figs/apa300ml_resonance.png]]
+
+*** Amplification factor
+The amplification factor is the ratio of the axial displacement to the stack displacement.
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML_test_bench';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
+
+  G = linearize(mdl, io);
+#+end_src
+
+The ratio of the two displacement is computed from the FEM model.
+#+begin_src matlab :results replace value
+  -dcgain(G(1,1))./dcgain(G(2,1))
+#+end_src
+
+#+RESULTS:
+: 4.936
+
+If we take the ratio of the piezo height and length (approximation of the amplification factor):
+#+begin_src matlab :results replace value
+  75/15
+#+end_src
+
+#+RESULTS:
+: 5
+
+*** Stroke
+
+Estimation of the actuator stroke:
+\[ \Delta H = A n \Delta L \]
+with:
+- $\Delta H$ Axial Stroke of the APA
+- $A$ Amplification factor (5 for the APA300ML)
+- $n$ Number of stack used
+- $\Delta L$ Stroke of the stack (0.1% of its length)
+
+#+begin_src matlab :results replace value
+  1e6 * 5 * 3 * 20e-3 * 0.1e-2
+#+end_src
+
+#+RESULTS:
+: 300
+
+This is exactly the specified stroke in the data-sheet.
+
 ** Identification of the Dynamics
 The flexible element is imported using the =Reduced Order Flexible Solid= simscape block.
 
@@ -823,12 +935,12 @@ A =Relative Motion Sensor= block is added between the nodes 1 and 2 to measure t
 One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
 
 We first set the mass to be zero.
-#+begin_src matlab
+#+begin_src matlab :exports none
   m = 0.01;
 #+end_src
 
 The dynamics is identified from the applied force to the measured relative displacement.
-#+begin_src matlab
+#+begin_src matlab :exports none
   %% Name of the Simulink File
   mdl = 'APA300ML_test_bench';
 
@@ -840,15 +952,12 @@ The dynamics is identified from the applied force to the measured relative displ
   Gh = -linearize(mdl, io);
 #+end_src
 
-Then, we add 10Kg of mass:
+The same dynamics is identified for a payload mass of 10Kg.
 #+begin_src matlab
   m = 10;
 #+end_src
 
-And the dynamics is identified.
+#+begin_src matlab :exports none
-
-The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_mass]].
-#+begin_src matlab
   %% Name of the Simulink File
   mdl = 'APA300ML_test_bench';
 
@@ -890,16 +999,24 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
   legend('location', 'southwest');
 #+end_src
 
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_plant_dynamics.pdf', 'width', 'full', 'height', 'full');
+#+end_src
+
+#+name: fig:apa300ml_plant_dynamics
+#+caption: Transfer function from forces applied by the stack to the axial displacement of the APA
+#+RESULTS:
+[[file:figs/apa300ml_plant_dynamics.png]]
+
 ** IFF
-Then, we add 10Kg of mass:
+Let's use 2 stacks as actuators and 1 stack as force sensor.
-#+begin_src matlab
+
+The transfer function from actuator to sensors is identified and shown in Figure [[fig:apa300ml_iff_plant]].
+#+begin_src matlab :exports none
   m = 10;
 #+end_src
 
-And the dynamics is identified.
+#+begin_src matlab :exports none
-
-The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_mass]].
-#+begin_src matlab
   %% Name of the Simulink File
   mdl = 'APA300ML_test_bench';
 
@@ -908,7 +1025,7 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
   io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1;
   io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
 
-  G = -linearize(mdl, io);
+  Giff = -linearize(mdl, io);
 #+end_src
 
 #+begin_src matlab :exports none
@@ -918,7 +1035,7 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 
   ax1 = subplot(2,1,1);
   hold on;
-  plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), '-');
+  plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-');
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
@@ -926,26 +1043,37 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 
   ax2 = subplot(2,1,2);
   hold on;
-  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))), '-');
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
   yticks(-360:90:360);
-  ylim([-360 0]);
+  ylim([-180 180]);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
   linkaxes([ax1,ax2],'x');
   xlim([freqs(1), freqs(end)]);
 #+end_src
 
-#+begin_src matlab
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_iff_plant.pdf', 'width', 'full', 'height', 'full');
+#+end_src
+
+#+name: fig:apa300ml_iff_plant
+#+caption: Transfer function from actuator to force sensor
+#+RESULTS:
+[[file:figs/apa300ml_iff_plant.png]]
+
+For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]].
+
+#+begin_src matlab :exports none
   figure;
 
   gains = logspace(0, 5, 500);
 
   hold on;
-  plot(real(pole(G)),  imag(pole(G)), 'kx');
+  plot(real(pole(Giff)),  imag(pole(Giff)), 'kx');
-  plot(real(tzero(G)),  imag(tzero(G)), 'ko');
+  plot(real(tzero(Giff)),  imag(tzero(Giff)), 'ko');
   for k = 1:length(gains)
-      cl_poles = pole(feedback(G, gains(k)/s));
+      cl_poles = pole(feedback(Giff, gains(k)/s));
       plot(real(cl_poles), imag(cl_poles), 'k.');
   end
   hold off;
@@ -955,15 +1083,22 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
   xlabel('Real Part'); ylabel('Imaginary Part');
 #+end_src
 
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:apa300ml_iff_root_locus
+#+caption: Root Locus for IFF
+#+RESULTS:
+[[file:figs/apa300ml_iff_root_locus.png]]
+
 ** DVF
-#+begin_src matlab
+Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure [[fig:apa300ml_dvf_plant]].
+#+begin_src matlab :exports none
   m = 10;
 #+end_src
 
-And the dynamics is identified.
+#+begin_src matlab :exports none
-
-The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_mass]].
-#+begin_src matlab
   %% Name of the Simulink File
   mdl = 'APA300ML_test_bench';
 
@@ -1000,7 +1135,18 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
   xlim([freqs(1), freqs(end)]);
 #+end_src
 
-#+begin_src matlab
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_dvf_plant.pdf', 'width', 'full', 'height', 'full');
+#+end_src
+
+#+name: fig:apa300ml_dvf_plant
+#+caption: Transfer function from stack actuator to relative motion sensor
+#+RESULTS:
+[[file:figs/apa300ml_dvf_plant.png]]
+
+The root locus for DVF is shown in Figure [[fig:apa300ml_dvf_root_locus]].
+
+#+begin_src matlab :exports none
   figure;
 
   gains = logspace(0, 5, 500);
@@ -1019,7 +1165,16 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
   xlabel('Real Part'); ylabel('Imaginary Part');
 #+end_src
 
-** Sensor Fusion
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:apa300ml_dvf_root_locus
+#+caption: Root Locus for Direct Velocity Feedback
+#+RESULTS:
+[[file:figs/apa300ml_dvf_root_locus.png]]
+
+** TODO Sensor Fusion                                              :noexport:
 - [ ] What is the goal of that? Special control properties, lower the sensor noise?
 
 Use the relative motion sensor at low frequency and the force sensor at high frequency.
@@ -1203,6 +1358,12 @@ Root locus
 #+end_src
 
 * Flexible Joint
+** Introduction                                                      :ignore:
+
+#+name: fig:flexor_id16_screenshot
+#+caption: Flexor studied
+[[file:figs/flexor_id16_screenshot.png]]
+
 ** Matlab Init                                             :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
   <<matlab-dir>>
@@ -1302,16 +1463,16 @@ Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= sims
 
 ** Flexible Joint Characteristics
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6)]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %0.f ');
+  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6); 60, 15, 15, 20]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Estimation by Francois*'}, ' %0.f ');
 #+end_src
 
 #+RESULTS:
-| *Caracteristic*            | *Value* |
+| *Caracteristic*            | *Value* | *Estimation by Francois* |
-|----------------------------+---------|
+|----------------------------+---------+--------------------------|
-| Axial Stiffness [N/um]     |     119 |
+| Axial Stiffness [N/um]     |     119 |                       60 |
-| Bending Stiffness [Nm/rad] |      33 |
+| Bending Stiffness [Nm/rad] |      33 |                       15 |
-| Bending Stiffness [Nm/rad] |      33 |
+| Bending Stiffness [Nm/rad] |      33 |                       15 |
-| Torsion Stiffness [Nm/rad] |     236 |
+| Torsion Stiffness [Nm/rad] |     236 |                       20 |
 
 ** Identification
 #+begin_src matlab