tdehaeze/nass-simscape
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Finalize flexible joint study and add index link

Browse Source
This commit is contained in:
Thomas Dehaeze 2020-05-05 10:45:02 +02:00
parent 9bc27e1f60
commit 5038151490
6 changed files with 115 additions and 76 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
docs/figs/nass_simple_model.pdf
View File

Binary file not shown.

158
docs/flexible_joints_study.html
View File

@ -1,11 +1,10 @@
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"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. 10:34 -->
<!-- 2020-05-05 mar. 10:44 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Study of the Flexible Joints</title>
<meta name="generator" content="Org mode" />
@ -36,21 +35,21 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org83db6d9">1. Rotational Stiffness</a>
<li><a href="#orge032d30">1. Bending and Torsional Stiffness</a>
<ul>
<li><a href="#orgd487aa8">1.1. Initialization</a></li>
<li><a href="#orgbc5ab48">1.2. Realistic Rotational Stiffness Values</a>
<li><a href="#org4af6fbb">1.1. Initialization</a></li>
<li><a href="#orgde60939">1.2. Realistic Bending Stiffness Values</a>
<ul>
<li><a href="#orgfa496e1">1.2.1. Direct Velocity Feedback</a></li>
<li><a href="#org2cf681e">1.2.2. Primary Plant</a></li>
<li><a href="#org17b7568">1.2.3. Conclusion</a></li>
<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>
</ul>
</li>
<li><a href="#org8ad3f34">1.3. Parametric Study</a>
<ul>
<li><a href="#org067911e">1.3.1. Direct Velocity Feedback</a></li>
<li><a href="#org3d67d1c">1.3.2. Primary Control</a></li>
<li><a href="#org700e2da">1.3.3. Conclusion</a></li>
<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>
</ul>
</li>
</ul>
@ -59,20 +58,22 @@
<ul>
<li><a href="#org969d9e7">2.1. Realistic Translation Stiffness Values</a>
<ul>
<li><a href="#org4af6fbb">2.1.1. Initialization</a></li>
<li><a href="#org5135788">2.1.2. Direct Velocity Feedback</a></li>
<li><a href="#org7eb4054">2.1.3. Primary Plant</a></li>
<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>
</ul>
</li>
<li><a href="#org0275632">2.2. Parametric study</a>
<ul>
<li><a href="#org1f64e69">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orgb35fa00">2.2.2. Primary Control</a></li>
<li><a href="#org5ed48b8">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#org5d9965b">2.2.2. Primary Control</a></li>
</ul>
</li>
<li><a href="#org81a1a77">2.3. Conclusion</a></li>
<li><a href="#org8ee81cd">2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgb8a9692">3. Conclusion</a></li>
</ul>
</div>
</div>
@ -86,16 +87,16 @@ Ideally, we want the x and y rotations to be free and all the translations to be
However, this is never the case and be have to consider:
</p>
<ul class="org-ul">
<li>Finite x and y rotational stiffnesses (Section <a href="#org3eb4121">1</a>)</li>
<li>Translation stiffness in the direction of the legs (Section <a href="#org8f4d83b">2</a>)</li>
<li>Finite bending stiffnesses (Section <a href="#org3eb4121">1</a>)</li>
<li>Axial stiffness in the direction of the legs (Section <a href="#org8f4d83b">2</a>)</li>
</ul>
<p>
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 <a href="#org6614f42">3</a>).
</p>
<div id="outline-container-org83db6d9" class="outline-2">
<h2 id="org83db6d9"><span class="section-number-2">1</span> Rotational Stiffness</h2>
<div id="outline-container-orge032d30" class="outline-2">
<h2 id="orge032d30"><span class="section-number-2">1</span> Bending and Torsional Stiffness</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org3eb4121"></a>
@ -105,8 +106,8 @@ In this section, we wish to study the effect of the rotation flexibility of the
</p>
</div>
<div id="outline-container-orgd487aa8" class="outline-3">
<h3 id="orgd487aa8"><span class="section-number-3">1.1</span> Initialization</h3>
<div id="outline-container-org4af6fbb" class="outline-3">
<h3 id="org4af6fbb"><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.
@ -134,8 +135,8 @@ initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
</div>
</div>
<div id="outline-container-orgbc5ab48" class="outline-3">
<h3 id="orgbc5ab48"><span class="section-number-3">1.2</span> Realistic Rotational Stiffness Values</h3>
<div id="outline-container-orgde60939" class="outline-3">
<h3 id="orgde60939"><span class="section-number-3">1.2</span> Realistic Bending Stiffness Values</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Let&rsquo;s compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:
@ -167,8 +168,8 @@ This corresponds to the optimal identified stiffness.
</p>
</div>
<div id="outline-container-orgfa496e1" class="outline-4">
<h4 id="orgfa496e1"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
<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 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.
@ -188,8 +189,8 @@ It is shown that the adding of stiffness for the flexible joints does increase a
</div>
</div>
<div id="outline-container-org2cf681e" class="outline-4">
<h4 id="org2cf681e"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
<div id="outline-container-org7eb4054" class="outline-4">
<h4 id="org7eb4054"><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).
@ -209,12 +210,12 @@ The plant dynamics is not found to be changing significantly.
</div>
</div>
<div id="outline-container-org17b7568" class="outline-4">
<h4 id="org17b7568"><span class="section-number-4">1.2.3</span> Conclusion</h4>
<div id="outline-container-org81a1a77" class="outline-4">
<h4 id="org81a1a77"><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>
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.
</p>
<p>
@ -235,7 +236,7 @@ This will help to determine the requirements on the joint&rsquo;s stiffness.
</p>
<p>
Let&rsquo;s consider the following rotational stiffness of the flexible joints:
Let&rsquo;s consider the following bending stiffness of the flexible joints:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ks = [1, 5, 10, 50, 100]; % [Nm/rad]
@ -247,8 +248,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p>
</div>
<div id="outline-container-org067911e" class="outline-4">
<h4 id="org067911e"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
<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 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>.
@ -259,7 +260,7 @@ The corresponding Root Locus plot is shown in Figure <a href="#orgb9f3389">4</a>
</p>
<p>
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.
</p>
@ -278,32 +279,32 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha
</div>
</div>
<div id="outline-container-org3d67d1c" class="outline-4">
<h4 id="org3d67d1c"><span class="section-number-4">1.3.2</span> Primary Control</h4>
<div id="outline-container-orgb35fa00" class="outline-4">
<h4 id="orgb35fa00"><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>.
</p>
<p>
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.
</p>
<div id="orgb739560" class="figure">
<p><img src="figs/flex_joints_rot_study_primary_plant.png" alt="flex_joints_rot_study_primary_plant.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered rotational stiffnesses</p>
<p><span class="figure-number">Figure 5: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered bending stiffnesses</p>
</div>
</div>
</div>
<div id="outline-container-org700e2da" class="outline-4">
<h4 id="org700e2da"><span class="section-number-4">1.3.3</span> Conclusion</h4>
<div id="outline-container-org4a1264f" class="outline-4">
<h4 id="org4a1264f"><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>
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.
</p>
</div>
@ -340,8 +341,8 @@ Cz_M = 1*ones(6,1); % [N/(m/s)]
</div>
</div>
<div id="outline-container-org4af6fbb" class="outline-4">
<h4 id="org4af6fbb"><span class="section-number-4">2.1.1</span> Initialization</h4>
<div id="outline-container-org14d57c4" class="outline-4">
<h4 id="org14d57c4"><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.
@ -369,8 +370,8 @@ initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
</div>
</div>
<div id="outline-container-org5135788" class="outline-4">
<h4 id="org5135788"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
<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 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.
@ -389,8 +390,8 @@ The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
</div>
</div>
<div id="outline-container-org7eb4054" class="outline-4">
<h4 id="org7eb4054"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
<div id="outline-container-orgddae25e" class="outline-4">
<h4 id="orgddae25e"><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);
@ -410,6 +411,18 @@ The dynamics is compare with and without the joint flexibility in Figure <a href
<p><img src="figs/flex_joints_trans_primary_plant_L.png" alt="flex_joints_trans_primary_plant_L.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed)</p>
</div>
</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 class="outline-text-4" id="text-2-1-4">
<div class="important">
<p>
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.
</p>
</div>
</div>
</div>
@ -435,8 +448,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p>
</div>
<div id="outline-container-org1f64e69" class="outline-4">
<h4 id="org1f64e69"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
<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 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>.
@ -452,7 +465,7 @@ If the axial stiffness of the flexible joints is \(K_a > 10^7\,[N/m]\) (here \(1
<p>
This is more clear by looking at the root locus (Figures <a href="#org9d43966">9</a> and <a href="#org987d98e">10</a>).
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 \(10^7\,[N/m]\).
</p>
@ -469,12 +482,6 @@ It can be seen that very little active damping can be achieve for rotational joi
<p><span class="figure-number">Figure 9: </span>Root Locus for all the considered axial Stiffnesses</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">xlim([-1e3, 0]);
ylim([0, 1e3]);
</pre>
</div>
<div id="org987d98e" class="figure">
<p><img src="figs/flex_joints_trans_study_root_locus_unzoom.png" alt="flex_joints_trans_study_root_locus_unzoom.png" />
@ -484,8 +491,8 @@ ylim([0, 1e3]);
</div>
</div>
<div id="outline-container-orgb35fa00" class="outline-4">
<h4 id="orgb35fa00"><span class="section-number-4">2.2.2</span> Primary Control</h4>
<div id="outline-container-org5d9965b" class="outline-4">
<h4 id="org5d9965b"><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>.
@ -501,8 +508,8 @@ The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for
</div>
</div>
<div id="outline-container-org81a1a77" class="outline-3">
<h3 id="org81a1a77"><span class="section-number-3">2.3</span> Conclusion</h3>
<div id="outline-container-org8ee81cd" class="outline-3">
<h3 id="org8ee81cd"><span class="section-number-3">2.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-3">
<div class="important">
<p>
@ -510,7 +517,7 @@ The axial stiffness of the flexible joints should be maximized.
</p>
<p>
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 \(10^7\,[N/m]\).
</p>
<p>
@ -522,13 +529,34 @@ We may interpolate the results and say that the axial joint stiffness should be
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb8a9692" class="outline-2">
<h2 id="orgb8a9692"><span class="section-number-2">3</span> Conclusion</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org6614f42"></a>
</p>
<div class="important">
<p>
For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:
</p>
<ul class="org-ul">
<li>Bending Stiffness: \(K_b < 50\,[Nm/rad]\)</li>
<li>Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)</li>
<li>Axial Stiffness: \(K_a > 10^7\,[N/m]\)</li>
</ul>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-05 mar. 10:34</p>
<p class="date">Created: 2020-05-05 mar. 10:44</p>
</div>
</body>
</html>

33
docs/index.html
View File

@ -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>
<!-- 2020-04-17 ven. 09:35 -->
<!-- 2020-05-05 mar. 10:44 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Simscape Model of the Nano-Active-Stabilization-System</title>
<meta name="generator" content="Org mode" />
@ -40,9 +40,10 @@
<li><a href="#orge777d0f">11. Effect of the payload&rsquo;s &ldquo;impedance&rdquo; on the plant dynamics (link)</a></li>
<li><a href="#orga323881">12. Effect of Experimental conditions on the plant dynamics (link)</a></li>
<li><a href="#orge7b9b41">13. Optimal Stiffness of the nano-hexapod to reduce plant uncertainty (link)</a></li>
<li><a href="#org14a10e8">14. Active Damping Techniques on the full Simscape Model (link)</a></li>
<li><a href="#orgd818a00">15. Control of the Nano-Active-Stabilization-System (link)</a></li>
<li><a href="#org361f405">16. Useful Matlab Functions (link)</a></li>
<li><a href="#org5f73af9">14. Effect of flexible joints on the plant dynamics (link)</a></li>
<li><a href="#org14a10e8">15. Active Damping Techniques on the full Simscape Model (link)</a></li>
<li><a href="#orgd818a00">16. Control of the Nano-Active-Stabilization-System (link)</a></li>
<li><a href="#org361f405">17. Useful Matlab Functions (link)</a></li>
</ul>
</div>
</div>
@ -200,18 +201,28 @@ Conclusion are drawn about what experimental conditions are critical on the vari
<h2 id="orge7b9b41"><span class="section-number-2">13</span> Optimal Stiffness of the nano-hexapod to reduce plant uncertainty (<a href="uncertainty_optimal_stiffness.html">link</a>)</h2>
</div>
<div id="outline-container-org14a10e8" class="outline-2">
<h2 id="org14a10e8"><span class="section-number-2">14</span> Active Damping Techniques on the full Simscape Model (<a href="control_active_damping.html">link</a>)</h2>
<div id="outline-container-org5f73af9" class="outline-2">
<h2 id="org5f73af9"><span class="section-number-2">14</span> Effect of flexible joints on the plant dynamics (<a href="flexible_joints_study.html">link</a>)</h2>
<div class="outline-text-2" id="text-14">
<p>
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.
</p>
</div>
</div>
<div id="outline-container-org14a10e8" class="outline-2">
<h2 id="org14a10e8"><span class="section-number-2">15</span> Active Damping Techniques on the full Simscape Model (<a href="control_active_damping.html">link</a>)</h2>
<div class="outline-text-2" id="text-15">
<p>
Active damping techniques are applied to the full Simscape model.
</p>
</div>
</div>
<div id="outline-container-orgd818a00" class="outline-2">
<h2 id="orgd818a00"><span class="section-number-2">15</span> Control of the Nano-Active-Stabilization-System (<a href="control.html">link</a>)</h2>
<div class="outline-text-2" id="text-15">
<h2 id="orgd818a00"><span class="section-number-2">16</span> Control of the Nano-Active-Stabilization-System (<a href="control.html">link</a>)</h2>
<div class="outline-text-2" id="text-16">
<p>
In this file are gathered all studies about the control the Nano-Active-Stabilization-System.
</p>
@ -219,8 +230,8 @@ In this file are gathered all studies about the control the Nano-Active-Stabiliz
</div>
<div id="outline-container-org361f405" class="outline-2">
<h2 id="org361f405"><span class="section-number-2">16</span> Useful Matlab Functions (<a href="./functions.html">link</a>)</h2>
<div class="outline-text-2" id="text-16">
<h2 id="org361f405"><span class="section-number-2">17</span> Useful Matlab Functions (<a href="./functions.html">link</a>)</h2>
<div class="outline-text-2" id="text-17">
<p>
Many matlab functions are shared among all the files of the projects.
</p>
@ -233,7 +244,7 @@ These functions are all defined <a href="./functions.html">here</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-17 ven. 09:35</p>
<p class="date">Created: 2020-05-05 mar. 10:44</p>
</div>
</body>
</html>

BIN
matlab/micro_hexapod_strut.slx
View File

Binary file not shown.

BIN
matlab/nano_hexapod_strut.slx
View File

Binary file not shown.

BIN
matlab/nass_model.slx
View File

Binary file not shown.