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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>