Dist. sensitivity / optimal stiffness analysis

2020-04-07 15:58:26 +02:00
parent ae89d0ea19
commit ca42338fe3
12 changed files with 269 additions and 118 deletions
--- a/org/optimal_stiffness_disturbances.org
+++ b/org/optimal_stiffness_disturbances.org
@@ -131,6 +131,9 @@ In this study, the expected frequency content of the direct forces applied to th
 * Effect of disturbances on the position error
 <<sec:effect_disturbances>>
 ** Introduction                                                      :ignore:
+In this section, we use the Simscape model to identify the transfer function from disturbances to the position error of the sample.
+We do that for a wide range of nano-hexapod stiffnesses and we compare the obtained results.
+
 ** 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>>
@@ -167,16 +170,16 @@ We use a sample mass of 10kg.
  initializeSample('mass', 10);
 #+end_src

+We include gravity, and we use no controller.
 #+begin_src matlab
  initializeSimscapeConfiguration('gravity', true);
+  initializeController();
  initializeDisturbances('enable', false);
  initializeLoggingConfiguration('log', 'none');
-  initializeController();
 #+end_src

-
 ** Identification
-Inputs:
+The considered inputs are:
 - =Dwx=: Ground displacement in the $x$ direction
 - =Dwy=: Ground displacement in the $y$ direction
 - =Dwz=: Ground displacement in the $z$ direction
@@ -185,15 +188,14 @@ Inputs:
 - =Frz_z=: Forces applied by the Spindle in the $z$ direction
 - =Fd=: Direct forces applied at the center of mass of the Payload

+The outputs are =Ex=, =Ey=, =Ez=, =Erx=, =Ery=, =Erz= which are the 3 positions and 3 orientations errors of the sample.
+
+We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.
 #+begin_src matlab
  Ks = logspace(3,9,7); % [N/m]
 #+end_src

 #+begin_src matlab :exports none
-  Gd = {zeros(length(Ks), 1)};
-#+end_src
-
-#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'nass_model';

@@ -210,7 +212,9 @@ Inputs:
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En');   io_i = io_i + 1; % Position Error
 #+end_src

-#+begin_src matlab
+#+begin_src matlab :exports none
+  Gd = {zeros(length(Ks), 1)};
+
  for i = 1:length(Ks)
      initializeNanoHexapod('k', Ks(i));

@@ -222,8 +226,12 @@ Inputs:
  end
 #+end_src

-** Plots
-Effect of Stages vibration (Filtering).
+** Sensitivity to Stages vibration (Filtering)
+The sensitivity the stage vibrations are displayed:
+- Figure [[fig:opt_stiff_sensitivity_Frz]]: sensitivity to vertical spindle vibrations
+- Figure [[fig:opt_stiff_sensitivity_Fty_z]]: sensitivity to vertical translation stage vibrations
+- Figure [[fig:opt_stiff_sensitivity_Fty_x]]: sensitivity to horizontal (x) translation stage vibrations
+
 #+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

@@ -235,26 +243,76 @@ Effect of Stages vibration (Filtering).
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
+  ylabel('Effect of $F_{rz}$ on $E_z$ [m/N]'); xlabel('Frequency [Hz]');
+  legend('location', 'southwest');
 #+end_src

-Effect of Ground motion (Transmissibility).
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_sensitivity_Frz
+#+caption: Sensitivity to Spindle vertical motion error ($F_{rz}$) to the vertical error position of the sample ($E_z$) ([[./figs/opt_stiff_sensitivity_Frz.png][png]], [[./figs/opt_stiff_sensitivity_Frz.pdf][pdf]])
+[[file:figs/opt_stiff_sensitivity_Frz.png]]
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 3, 1000);
+
+  figure;
+  hold on;
+  for i = 1:length(Ks)
+      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '-', ...
+           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Effect of $F_{ty}$ on $E_z$ [m/N]'); xlabel('Frequency [Hz]');
+  legend('location', 'southwest');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Fty_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_sensitivity_Fty_z
+#+caption: Sensitivity to Translation stage vertical motion error ($F_{ty,z}$) to the vertical error position of the sample ($E_z$) ([[./figs/opt_stiff_sensitivity_Fty_z.png][png]], [[./figs/opt_stiff_sensitivity_Fty_z.pdf][pdf]])
+[[file:figs/opt_stiff_sensitivity_Fty_z.png]]
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 3, 1000);
+
+  figure;
+  hold on;
+  for i = 1:length(Ks)
+      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ex', 'Fty_x'), freqs, 'Hz'))), '-', ...
+           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Effect of $F_{ty}$ on $E_x$ [m/N]'); xlabel('Frequency [Hz]');
+  legend('location', 'northeast');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Fty_x.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_sensitivity_Fty_x
+#+caption: Sensitivity to Translation stage $x$ motion error ($F_{ty,x}$) to the error position of the sample in the $x$ direction ($E_x$) ([[./figs/opt_stiff_sensitivity_Fty_x.png][png]], [[./figs/opt_stiff_sensitivity_Fty_x.pdf][pdf]])
+[[file:figs/opt_stiff_sensitivity_Fty_x.png]]
+
+** Effect of Ground motion (Transmissibility).
+The effect of Ground motion on the position error of the sample is shown in Figure [[fig:opt_stiff_sensitivity_Dw]].
+
 #+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

-  ax1 = subplot(3, 1, 1);
-  hold on;
-  for i = 1:length(Ks)
-      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ex', 'Dwx'), freqs, 'Hz'))), '-', ...
-           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
-  end
-  hold off;
-  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Amplitude [m/m]');
-
-  ax1 = subplot(3, 1, 2);
+  ax1 = subplot(1, 2, 1);
  hold on;
  for i = 1:length(Ks)
      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ey', 'Dwy'), freqs, 'Hz'))), '-', ...
@@ -262,9 +320,9 @@ Effect of Ground motion (Transmissibility).
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Amplitude [m/m]');
+  ylabel('$E_y/D_{wy}$ [m/m]'); xlabel('Frequency [Hz]');

-  ax1 = subplot(3, 1, 3);
+  ax2 = subplot(1, 2, 2);
  hold on;
  for i = 1:length(Ks)
      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), '-', ...
@@ -272,14 +330,36 @@ Effect of Ground motion (Transmissibility).
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
+  ylabel('$E_z/D_{wz}$ [m/m]'); xlabel('Frequency [Hz]');
 #+end_src

-Direct Forces (Compliance).
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Dw.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_sensitivity_Dw
+#+caption: Sensitivity to Ground motion ($D_{w}$) to the position error of the sample ($E_y$ and $E_z$) ([[./figs/opt_stiff_sensitivity_Dw.png][png]], [[./figs/opt_stiff_sensitivity_Dw.pdf][pdf]])
+[[file:figs/opt_stiff_sensitivity_Dw.png]]
+
+** Direct Forces (Compliance).
+The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure [[fig:opt_stiff_sensitivity_Fd]].
+
 #+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;
+
+  ax1 = subplot(1, 2, 1);
+  hold on;
+  for i = 1:length(Ks)
+      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ery', 'Mdy'), freqs, 'Hz'))), '-');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('$E_{ry}/M_{d,y}\ \left[\frac{rad}{N m}\right]$'); xlabel('Frequency [Hz]');
+
+  ax2 = subplot(1, 2, 2);
  hold on;
  for i = 1:length(Ks)
      plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))), '-', ...
@@ -287,14 +367,31 @@ Direct Forces (Compliance).
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
+  ylabel('$E_{z}/F_{d,z}$ [m/N]'); xlabel('Frequency [Hz]');
+  legend('location', 'northeast');
+
+  linkaxes([ax1 ax2], 'xy')
 #+end_src

-** Save
-#+begin_src matlab
-  save('./mat/opt_stiffness_disturbances.mat', 'Gd')
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Fd.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
 #+end_src

+#+name: fig:opt_stiff_sensitivity_Fd
+#+caption: Sensitivity to Direct forces and torques applied to the sample ($F_d$, $M_d$) to the position error of the sample ([[./figs/opt_stiff_sensitivity_Fd.png][png]], [[./figs/opt_stiff_sensitivity_Fd.pdf][pdf]])
+[[file:figs/opt_stiff_sensitivity_Fd.png]]
+
+** Save                                                            :noexport:
+#+begin_src matlab
+  save('./mat/opt_stiffness_disturbances.mat', 'Ks', 'Gd')
+#+end_src
+
+** Conclusion
+#+begin_important
+
+#+end_important
+
 * Effect of granite stiffness
 <<sec:granite_stiffness>>
 ** Analytical Analysis