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Update few figures

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Thomas Dehaeze
2020-05-05 11:27:37 +02:00
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@@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-05-05 mar. 10:44 -->
<!-- 2020-05-05 mar. 11:26 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Study of the Flexible Joints</title>
<meta name="generator" content="Org mode" />
@@ -37,19 +37,19 @@
<ul>
<li><a href="#orge032d30">1. Bending and Torsional Stiffness</a>
<ul>
<li><a href="#org4af6fbb">1.1. Initialization</a></li>
<li><a href="#org14d57c4">1.1. Initialization</a></li>
<li><a href="#orgde60939">1.2. Realistic Bending Stiffness Values</a>
<ul>
<li><a href="#org1f64e69">1.2.1. Direct Velocity Feedback</a></li>
<li><a href="#org7eb4054">1.2.2. Primary Plant</a></li>
<li><a href="#org81a1a77">1.2.3. Conclusion</a></li>
<li><a href="#org5ed48b8">1.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orgddae25e">1.2.2. Primary Plant</a></li>
<li><a href="#orgb8a9692">1.2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org8ad3f34">1.3. Parametric Study</a>
<ul>
<li><a href="#org1575b3d">1.3.1. Direct Velocity Feedback</a></li>
<li><a href="#orgb35fa00">1.3.2. Primary Control</a></li>
<li><a href="#org4a1264f">1.3.3. Conclusion</a></li>
<li><a href="#org44ccdbe">1.3.1. Direct Velocity Feedback</a></li>
<li><a href="#org5d9965b">1.3.2. Primary Control</a></li>
<li><a href="#org0f9f990">1.3.3. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -58,22 +58,22 @@
<ul>
<li><a href="#org969d9e7">2.1. Realistic Translation Stiffness Values</a>
<ul>
<li><a href="#org14d57c4">2.1.1. Initialization</a></li>
<li><a href="#org790d5e4">2.1.2. Direct Velocity Feedback</a></li>
<li><a href="#orgddae25e">2.1.3. Primary Plant</a></li>
<li><a href="#org7ebf071">2.1.4. Conclusion</a></li>
<li><a href="#org8fdef7f">2.1.1. Initialization</a></li>
<li><a href="#orgc087bb9">2.1.2. Direct Velocity Feedback</a></li>
<li><a href="#org4069e58">2.1.3. Primary Plant</a></li>
<li><a href="#org3d8a1a7">2.1.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org0275632">2.2. Parametric study</a>
<ul>
<li><a href="#org5ed48b8">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#org5d9965b">2.2.2. Primary Control</a></li>
<li><a href="#orgdb214f9">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#org53e5f08">2.2.2. Primary Control</a></li>
</ul>
</li>
<li><a href="#org8ee81cd">2.3. Conclusion</a></li>
<li><a href="#org1ddd8bf">2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgb8a9692">3. Conclusion</a></li>
<li><a href="#orga32adf0">3. Conclusion</a></li>
</ul>
</div>
</div>
@@ -106,8 +106,8 @@ In this section, we wish to study the effect of the rotation flexibility of the
</p>
</div>
<div id="outline-container-org4af6fbb" class="outline-3">
<h3 id="org4af6fbb"><span class="section-number-3">1.1</span> Initialization</h3>
<div id="outline-container-org14d57c4" class="outline-3">
<h3 id="org14d57c4"><span class="section-number-3">1.1</span> Initialization</h3>
<div class="outline-text-3" id="text-1-1">
<p>
Let&rsquo;s initialize all the stages with default parameters.
@@ -168,8 +168,8 @@ This corresponds to the optimal identified stiffness.
</p>
</div>
<div id="outline-container-org1f64e69" class="outline-4">
<h4 id="org1f64e69"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
<div id="outline-container-org5ed48b8" class="outline-4">
<h4 id="org5ed48b8"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-1-2-1">
<p>
We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness.
@@ -189,8 +189,8 @@ It is shown that the adding of stiffness for the flexible joints does increase a
</div>
</div>
<div id="outline-container-org7eb4054" class="outline-4">
<h4 id="org7eb4054"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
<div id="outline-container-orgddae25e" class="outline-4">
<h4 id="orgddae25e"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
<div class="outline-text-4" id="text-1-2-2">
<p>
Let&rsquo;s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
@@ -205,13 +205,13 @@ The plant dynamics is not found to be changing significantly.
<div id="org4322feb" class="figure">
<p><img src="figs/flex_joints_rot_primary_plant_L.png" alt="flex_joints_rot_primary_plant_L.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints (dashed) and with flexible joints (solid)</p>
<p><span class="figure-number">Figure 2: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints and with flexible joints</p>
</div>
</div>
</div>
<div id="outline-container-org81a1a77" class="outline-4">
<h4 id="org81a1a77"><span class="section-number-4">1.2.3</span> Conclusion</h4>
<div id="outline-container-orgb8a9692" class="outline-4">
<h4 id="orgb8a9692"><span class="section-number-4">1.2.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-1-2-3">
<div class="important">
<p>
@@ -248,8 +248,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p>
</div>
<div id="outline-container-org1575b3d" class="outline-4">
<h4 id="org1575b3d"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
<div id="outline-container-org44ccdbe" class="outline-4">
<h4 id="org44ccdbe"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-1-3-1">
<p>
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#org8fbbf9d">3</a>.
@@ -279,8 +279,8 @@ It is shown that the bending stiffness of the flexible joints does indeed change
</div>
</div>
<div id="outline-container-orgb35fa00" class="outline-4">
<h4 id="orgb35fa00"><span class="section-number-4">1.3.2</span> Primary Control</h4>
<div id="outline-container-org5d9965b" class="outline-4">
<h4 id="org5d9965b"><span class="section-number-4">1.3.2</span> Primary Control</h4>
<div class="outline-text-4" id="text-1-3-2">
<p>
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 <a href="#orgb739560">5</a>.
@@ -299,8 +299,8 @@ It is shown that the bending stiffness of the flexible joints have very little i
</div>
</div>
<div id="outline-container-org4a1264f" class="outline-4">
<h4 id="org4a1264f"><span class="section-number-4">1.3.3</span> Conclusion</h4>
<div id="outline-container-org0f9f990" class="outline-4">
<h4 id="org0f9f990"><span class="section-number-4">1.3.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-1-3-3">
<div class="important">
<p>
@@ -341,8 +341,8 @@ Cz_M = 1*ones(6,1); % [N/(m/s)]
</div>
</div>
<div id="outline-container-org14d57c4" class="outline-4">
<h4 id="org14d57c4"><span class="section-number-4">2.1.1</span> Initialization</h4>
<div id="outline-container-org8fdef7f" class="outline-4">
<h4 id="org8fdef7f"><span class="section-number-4">2.1.1</span> Initialization</h4>
<div class="outline-text-4" id="text-2-1-1">
<p>
Let&rsquo;s initialize all the stages with default parameters.
@@ -370,8 +370,8 @@ initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
</div>
</div>
<div id="outline-container-org790d5e4" class="outline-4">
<h4 id="org790d5e4"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
<div id="outline-container-orgc087bb9" class="outline-4">
<h4 id="orgc087bb9"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-2-1-2">
<p>
The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified.
@@ -390,8 +390,8 @@ The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
</div>
</div>
<div id="outline-container-orgddae25e" class="outline-4">
<h4 id="orgddae25e"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
<div id="outline-container-org4069e58" class="outline-4">
<h4 id="org4069e58"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
<div class="outline-text-4" id="text-2-1-3">
<div class="org-src-container">
<pre class="src src-matlab">Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
@@ -415,8 +415,8 @@ The dynamics is compare with and without the joint flexibility in Figure <a href
</div>
</div>
<div id="outline-container-org7ebf071" class="outline-4">
<h4 id="org7ebf071"><span class="section-number-4">2.1.4</span> Conclusion</h4>
<div id="outline-container-org3d8a1a7" class="outline-4">
<h4 id="org3d8a1a7"><span class="section-number-4">2.1.4</span> Conclusion</h4>
<div class="outline-text-4" id="text-2-1-4">
<div class="important">
<p>
@@ -448,8 +448,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p>
</div>
<div id="outline-container-org5ed48b8" class="outline-4">
<h4 id="org5ed48b8"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
<div id="outline-container-orgdb214f9" class="outline-4">
<h4 id="orgdb214f9"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-2-2-1">
<p>
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#orgab9ab86">8</a>.
@@ -491,8 +491,8 @@ It can be seen that very little active damping can be achieve for axial stiffnes
</div>
</div>
<div id="outline-container-org5d9965b" class="outline-4">
<h4 id="org5d9965b"><span class="section-number-4">2.2.2</span> Primary Control</h4>
<div id="outline-container-org53e5f08" class="outline-4">
<h4 id="org53e5f08"><span class="section-number-4">2.2.2</span> Primary Control</h4>
<div class="outline-text-4" id="text-2-2-2">
<p>
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 <a href="#org6070692">11</a>.
@@ -508,8 +508,8 @@ The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for
</div>
</div>
<div id="outline-container-org8ee81cd" class="outline-3">
<h3 id="org8ee81cd"><span class="section-number-3">2.3</span> Conclusion</h3>
<div id="outline-container-org1ddd8bf" class="outline-3">
<h3 id="org1ddd8bf"><span class="section-number-3">2.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-3">
<div class="important">
<p>
@@ -533,8 +533,8 @@ We may interpolate the results and say that the axial joint stiffness should be
</div>
</div>
<div id="outline-container-orgb8a9692" class="outline-2">
<h2 id="orgb8a9692"><span class="section-number-2">3</span> Conclusion</h2>
<div id="outline-container-orga32adf0" class="outline-2">
<h2 id="orga32adf0"><span class="section-number-2">3</span> Conclusion</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org6614f42"></a>
@@ -556,7 +556,7 @@ For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-05 mar. 10:44</p>
<p class="date">Created: 2020-05-05 mar. 11:26</p>
</div>
</body>
</html>

70
org/flexible_joints_study.org
View File

@@ -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

4
org/index.org
View File

@@ -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.

2
org/setup/org-setup-file.org
View File

@@ -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

24
src/initializeJointDynamics.m
View File

@@ -11,10 +11,14 @@ function [stewart] = initializeJointDynamics(stewart, args)
% - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
% - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
% - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
% - Kz_M [6x1] - Translation (Tz) Stiffness for each top joints [N/m]
% - Cz_M [6x1] - Translation (Tz) Damping of each top joint [N/m]
% - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
% - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
% - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
% - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
% - Kz_F [6x1] - Translation (Tz) Stiffness for each bottom joints [N/m]
% - Cz_F [6x1] - Translation (Tz) Damping of each bottom joint [N/m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
@@ -29,16 +33,20 @@ function [stewart] = initializeJointDynamics(stewart, args)
arguments
stewart
args.type_F char {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'universal'
args.type_M char {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'spherical'
args.type_F char {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof'})} = 'universal'
args.type_M char {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'spherical_3dof'})} = 'spherical'
args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
args.Kz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
args.Cz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
args.Kz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
args.Cz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
end
switch args.type_F
@@ -50,6 +58,8 @@ switch args.type_F
stewart.joints_F.type = 3;
case 'spherical_p'
stewart.joints_F.type = 4;
case 'universal_3dof'
stewart.joints_F.type = 5;
end
switch args.type_M
@@ -61,23 +71,25 @@ switch args.type_M
stewart.joints_M.type = 3;
case 'spherical_p'
stewart.joints_M.type = 4;
case 'spherical_3dof'
stewart.joints_M.type = 6;
end
stewart.joints_M.Kx = zeros(6,1);
stewart.joints_M.Ky = zeros(6,1);
stewart.joints_M.Kz = zeros(6,1);
stewart.joints_M.Kz = args.Kz_M;
stewart.joints_F.Kx = zeros(6,1);
stewart.joints_F.Ky = zeros(6,1);
stewart.joints_F.Kz = zeros(6,1);
stewart.joints_F.Kz = args.Kz_F;
stewart.joints_M.Cx = zeros(6,1);
stewart.joints_M.Cy = zeros(6,1);
stewart.joints_M.Cz = zeros(6,1);
stewart.joints_M.Cz = args.Cz_M;
stewart.joints_F.Cx = zeros(6,1);
stewart.joints_F.Cy = zeros(6,1);
stewart.joints_F.Cz = zeros(6,1);
stewart.joints_F.Cz = args.Cz_F;
stewart.joints_M.Kf = args.Kf_M;
stewart.joints_M.Kt = args.Kf_M;

14
src/initializeNanoHexapod.m
View File

@@ -17,16 +17,20 @@ arguments
args.k (1,1) double {mustBeNumeric} = -1
args.c (1,1) double {mustBeNumeric} = -1
% initializeJointDynamics
args.type_F char {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'universal'
args.type_M char {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p'})} = 'spherical'
args.type_F char {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof'})} = 'universal'
args.type_M char {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'spherical_3dof'})} = 'spherical'
args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
args.Kz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
args.Cz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
args.Kz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
args.Cz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
% initializeCylindricalPlatforms
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
@@ -77,10 +81,14 @@ stewart = initializeJointDynamics(stewart, ...
'Cf_M' , args.Cf_M, ...
'Kt_M' , args.Kt_M, ...
'Ct_M' , args.Ct_M, ...
'Kz_M' , args.Kz_M, ...
'Cz_M' , args.Cz_M, ...
'Kf_F' , args.Kf_F, ...
'Cf_F' , args.Cf_F, ...
'Kt_F' , args.Kt_F, ...
'Ct_F' , args.Ct_F);
'Ct_F' , args.Ct_F, ...
'Kz_F' , args.Kz_F, ...
'Cz_F' , args.Cz_F);
stewart = initializeCylindricalPlatforms(stewart, 'Fpm', args.Fpm, 'Fph', args.Fph, 'Fpr', args.Fpr, 'Mpm', args.Mpm, 'Mph', args.Mph, 'Mpr', args.Mpr);