Update few figures

2020-05-05 11:27:37 +02:00
parent 5038151490
commit 135a1f4fe6
14 changed files with 374 additions and 310 deletions
--- a/org/flexible_joints_study.org
+++ b/org/flexible_joints_study.org
@@ -6,12 +6,12 @@ In this document is studied the effect of the mechanical behavior of the flexibl

 Ideally, we want the x and y rotations to be free and all the translations to be blocked.
 However, this is never the case and be have to consider:
- Finite x and y rotational stiffnesses (Section [[sec:rot_stiffness]])
- Translation stiffness in the direction of the legs (Section [[sec:trans_stiffness]])
+- Finite bending stiffnesses (Section [[sec:rot_stiffness]])
+- Axial stiffness in the direction of the legs (Section [[sec:trans_stiffness]])

-This may impose some limitations, also, the goal is to specify the required joints stiffnesses.
+This may impose some limitations, also, the goal is to specify the required joints stiffnesses (summarized in Section [[sec:conclusion]]).

-* Rotational Stiffness
+* Bending and Torsional Stiffness
 <<sec:rot_stiffness>>

 ** Introduction                                                      :ignore:
@@ -67,7 +67,7 @@ Let's consider the heaviest mass which should we the most problematic with it co
  Kdvf = tf(zeros(6));
 #+end_src

-** Realistic Rotational Stiffness Values
+** Realistic Bending Stiffness Values
 *** Introduction                                                    :ignore:
 Let's compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:
 - $K_{\theta, \phi} = 15\,[Nm/rad]$ stiffness in flexion
@@ -231,7 +231,7 @@ The plant dynamics is not found to be changing significantly.
  end
  for j = 1:6
      set(gca,'ColorOrderIndex',2);
-      plot(freqs, abs(squeeze(freqresp(Gl_p(j, j), freqs, 'Hz'))), '--');
+      plot(freqs, abs(squeeze(freqresp(Gl_p(j, j), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -241,17 +241,30 @@ The plant dynamics is not found to be changing significantly.
  hold on;
  for j = 1:6
      set(gca,'ColorOrderIndex',1);
-      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl(j, j), freqs, 'Hz')))));
+      if j == 1
+          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl(j, j), freqs, 'Hz')))), ...
+               'DisplayName', 'Flexible Joints');
+      else
+          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl(j, j), freqs, 'Hz')))), ...
+               'HandleVisibility', 'off');
+      end
  end
  for j = 1:6
      set(gca,'ColorOrderIndex',2);
-      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_p(j, j), freqs, 'Hz')))), '--');
+      if j == 1
+          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_p(j, j), freqs, 'Hz')))), ...
+               'DisplayName', 'Perfect Joints');
+      else
+          plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_p(j, j), freqs, 'Hz')))), ...
+               'HandleVisibility', 'off');
+      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-270, 90]);
  yticks([-360:90:360]);
+  legend('location', 'southwest');

  linkaxes([ax1,ax2],'x');
 #+end_src
@@ -261,13 +274,13 @@ The plant dynamics is not found to be changing significantly.
 #+end_src

 #+name: fig:flex_joints_rot_primary_plant_L
-#+caption: Dynamics from $\bm{\tau}^\prime_i$ to $\bm{\epsilon}_{\mathcal{X}_n,i}$ with perfect joints (dashed) and with flexible joints (solid)
+#+caption: Dynamics from $\bm{\tau}^\prime_i$ to $\bm{\epsilon}_{\mathcal{X}_n,i}$ with perfect joints and with flexible joints
 #+RESULTS:
 [[file:figs/flex_joints_rot_primary_plant_L.png]]

 *** Conclusion
 #+begin_important
-Considering realistic flexible joint rotational stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.
+Considering realistic flexible joint bending stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.

 It only increases a little bit the suspension modes of the sample on top of the nano-hexapod.
 #+end_important
@@ -277,7 +290,7 @@ It only increases a little bit the suspension modes of the sample on top of the
 We wish now to see what is the impact of the rotation stiffness of the flexible joints on the dynamics.
 This will help to determine the requirements on the joint's stiffness.

-Let's consider the following rotational stiffness of the flexible joints:
+Let's consider the following bending stiffness of the flexible joints:
 #+begin_src matlab
  Ks = [1, 5, 10, 50, 100]; % [Nm/rad]
 #+end_src
@@ -294,7 +307,7 @@ The dynamics from the actuators to the relative displacement sensor in each leg

 The corresponding Root Locus plot is shown in Figure [[fig:flex_joints_rot_study_dvf_root_locus]].

-It is shown that the rotational stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback.
+It is shown that the bending stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback.

 #+begin_src matlab :exports none
  %% Name of the Simulink File
@@ -343,7 +356,7 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha
  hold on;
  for i = 1:length(Ks)
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_s{i}(1, 1), freqs, 'Hz')))), ...
-           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+           'DisplayName', sprintf('$k = %.0f$ [Nm/rad]', Ks(i)));
  end
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_p(1, 1), freqs, 'Hz')))), 'k--', ...
       'DisplayName', 'Ideal Joint');
@@ -375,7 +388,7 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha
  for i = 1:length(Ks)
      set(gca,'ColorOrderIndex',i);
      plot(real(pole(G_dvf_s{i})),  imag(pole(G_dvf_s{i})), 'x', ...
-         'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+         'DisplayName', sprintf('$k = %.0f$ [Nm/rad]', Ks(i)));
      set(gca,'ColorOrderIndex',i);
      plot(real(tzero(G_dvf_s{i})),  imag(tzero(G_dvf_s{i})), 'o', ...
           'HandleVisibility', 'off');
@@ -406,7 +419,7 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha
 *** Primary Control
 The dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the primary controller designed in the frame of the legs) is shown in Figure [[fig:flex_joints_rot_study_primary_plant]].

-It is shown that the rotational stiffness of the flexible joints have very little impact on the dynamics.
+It is shown that the bending stiffness of the flexible joints have very little impact on the dynamics.

 #+begin_src matlab :exports none
  Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
@@ -463,7 +476,7 @@ It is shown that the rotational stiffness of the flexible joints have very littl
  hold on;
  for i = 1:length(Ks)
      plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_s{i}(1, 1), freqs, 'Hz')))), ...
-           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+           'DisplayName', sprintf('$k = %.0f$ [Nm/rad]', Ks(i)));
  end
  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_p(1, 1), freqs, 'Hz')))), 'k--', ...
       'DisplayName', 'Ideal Joint');
@@ -482,13 +495,13 @@ It is shown that the rotational stiffness of the flexible joints have very littl
 #+end_src

 #+name: fig:flex_joints_rot_study_primary_plant
-#+caption: Diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the considered rotational stiffnesses
+#+caption: Diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the considered bending stiffnesses
 #+RESULTS:
 [[file:figs/flex_joints_rot_study_primary_plant.png]]

 *** Conclusion
 #+begin_important
-  The rotational stiffness of the flexible joint does not significantly change the dynamics.
+  The bending stiffness of the flexible joint does not significantly change the dynamics.
 #+end_important

 * Translation Stiffness
@@ -735,6 +748,11 @@ The dynamics is compare with and without the joint flexibility in Figure [[fig:f
 #+RESULTS:
 [[file:figs/flex_joints_trans_primary_plant_L.png]]

+*** Conclusion
+#+begin_important
+  For the realistic value of the flexible joint axial stiffness, the dynamics is not much impact, and this should not be a problem for control.
+#+end_important
+
 ** Parametric study
 *** Introduction                                                    :ignore:
 We wish now to see what is the impact of the *axial* stiffness of the flexible joints on the dynamics.
@@ -759,7 +777,7 @@ It is shown that the axial stiffness of the flexible joints does have a huge imp
 If the axial stiffness of the flexible joints is $K_a > 10^7\,[N/m]$ (here $100$ times higher than the actuator stiffness), then the change of dynamics stays reasonably small.

 This is more clear by looking at the root locus (Figures [[fig:flex_joints_trans_study_dvf_root_locus]] and [[fig:flex_joints_trans_study_root_locus_unzoom]]).
-It can be seen that very little active damping can be achieve for rotational joint axial stiffnesses below $10^7\,[N/m]$.
+It can be seen that very little active damping can be achieve for axial stiffnesses below $10^7\,[N/m]$.

 #+begin_src matlab :exports none
  %% Name of the Simulink File
@@ -868,7 +886,7 @@ It can be seen that very little active damping can be achieve for rotational joi
 #+RESULTS:
 [[file:figs/flex_joints_trans_study_dvf_root_locus.png]]

-#+begin_src matlab
+#+begin_src matlab :exports none
  xlim([-1e3, 0]);
  ylim([0, 1e3]);
 #+end_src
@@ -964,9 +982,19 @@ The dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the
 #+begin_important
  The axial stiffness of the flexible joints should be maximized.

-  For the considered actuator stiffness $k = 10^5\,[N/m]$, the axial stiffness of the rotational joints should ideally be above $10^7\,[N/m]$.
+  For the considered actuator stiffness $k = 10^5\,[N/m]$, the axial stiffness of the flexible joints should ideally be above $10^7\,[N/m]$.

  This is a reasonable stiffness value for such joints.

  We may interpolate the results and say that the axial joint stiffness should be 100 times higher than the actuator stiffness, but this should be confirmed with further analysis.
 #+end_important
+
+* Conclusion
+<<sec:conclusion>>
+
+#+begin_important
+For the identified optimal actuator stiffness $k = 10^5\,[N/m]$, the flexible joint should have the following stiffness properties:
+- Bending Stiffness: $K_b < 50\,[Nm/rad]$
+- Torsion Stiffness: $K_t < 50\,[Nm/rad]$
+- Axial Stiffness: $K_a > 10^7\,[N/m]$
+#+end_important
--- a/org/index.org
+++ b/org/index.org
@@ -65,6 +65,10 @@ Conclusion are drawn about what experimental conditions are critical on the vari

 * Optimal Stiffness of the nano-hexapod to reduce plant uncertainty ([[file:uncertainty_optimal_stiffness.org][link]])

+* Effect of flexible joints on the plant dynamics ([[file:flexible_joints_study.org][link]])
+In this document is studied how the flexible joint stiffnesses (in flexion, torsion and compression) is affecting the plant dynamics.
+Conclusion are drawn on the required stiffness properties of the flexible joints.
+
 * Active Damping Techniques on the full Simscape Model ([[file:control_active_damping.org][link]])
 Active damping techniques are applied to the full Simscape model.

--- a/org/setup/org-setup-file.org
+++ b/org/setup/org-setup-file.org
@@ -25,7 +25,7 @@

 #+PROPERTY: header-args:shell  :eval no-export

-#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
+#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
 #+PROPERTY: header-args:latex+ :imagemagick t :fit yes
 #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100