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Add display for each cubic configuration

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Thomas Dehaeze 2020-02-07 17:31:52 +01:00
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cubic-configuration.html
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@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-02-06 jeu. 18:22 --> <!-- 2020-02-07 ven. 17:31 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Cubic configuration for the Stewart Platform</title> <title>Cubic configuration for the Stewart Platform</title>
@ -268,25 +268,25 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org8350a45">1. Configuration Analysis - Stiffness Matrix</a> <li><a href="#org16a66cd">1. Configuration Analysis - Stiffness Matrix</a>
<ul> <ul>
<li><a href="#org6b34363">1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li> <li><a href="#orgd83749a">1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
<li><a href="#orgd59e9f0">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li> <li><a href="#org84204db">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
<li><a href="#org27bd91a">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li> <li><a href="#orgada859c">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
<li><a href="#org4adea52">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li> <li><a href="#org8b2c367">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
<li><a href="#org5defe19">1.5. Conclusion</a></li> <li><a href="#org34d7a0d">1.5. Conclusion</a></li>
<li><a href="#org905f726">1.6. Having Cube&rsquo;s center above the top platform</a></li> <li><a href="#org265d677">1.6. Having Cube&rsquo;s center above the top platform</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org6746f61">2. Functions</a> <li><a href="#org93a8538">2. Functions</a>
<ul> <ul>
<li><a href="#org80fcb20">2.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a> <li><a href="#org8786798">2.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<ul> <ul>
<li><a href="#orgd3654f4">Function description</a></li> <li><a href="#org0b63bb1">Function description</a></li>
<li><a href="#orgf1c4374">Documentation</a></li> <li><a href="#orgaac7da3">Documentation</a></li>
<li><a href="#org7703a4a">Optional Parameters</a></li> <li><a href="#org747c61c">Optional Parameters</a></li>
<li><a href="#org43039c1">Position of the Cube</a></li> <li><a href="#orgec4e738">Position of the Cube</a></li>
<li><a href="#orga18e804">Compute the pose</a></li> <li><a href="#org97e1d45">Compute the pose</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -313,7 +313,7 @@ According to <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage
</p> </p>
<p> <p>
To generate and study the Cubic configuration, <code>generateCubicConfiguration</code> is used (description in section <a href="#orgfe8cdfe">2.1</a>). To generate and study the Cubic configuration, <code>generateCubicConfiguration</code> is used (description in section <a href="#orgf3011db">2.1</a>).
The goal is to study the benefits of using a cubic configuration: The goal is to study the benefits of using a cubic configuration:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
@ -322,15 +322,15 @@ The goal is to study the benefits of using a cubic configuration:
<li>Is the center of the cube an important point?</li> <li>Is the center of the cube an important point?</li>
</ul> </ul>
<div id="outline-container-org8350a45" class="outline-2"> <div id="outline-container-org16a66cd" class="outline-2">
<h2 id="org8350a45"><span class="section-number-2">1</span> Configuration Analysis - Stiffness Matrix</h2> <h2 id="org16a66cd"><span class="section-number-2">1</span> Configuration Analysis - Stiffness Matrix</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org6b34363" class="outline-3"> <div id="outline-container-orgd83749a" class="outline-3">
<h3 id="org6b34363"><span class="section-number-3">1.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3> <h3 id="orgd83749a"><span class="section-number-3">1.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
We create a cubic Stewart platform (figure <a href="#org964919a">1</a>) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot). We create a cubic Stewart platform (figure <a href="#org0e6176c">1</a>) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
The Jacobian matrix is estimated at the location of the center of the cube. The Jacobian matrix is estimated at the location of the center of the cube.
</p> </p>
@ -340,16 +340,24 @@ stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</spa
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1)); stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
</pre> </pre>
</div> </div>
<div id="org964919a" class="figure"> <div id="org0e6176c" class="figure">
<p><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" /> <p><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p> <p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p>
</div> </div>
<div id="orgf0479ad" class="figure">
<p><img src="figs/cubic_conf_centered_J_center.png" alt="cubic_conf_centered_J_center.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_center.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -425,11 +433,11 @@ stewart = computeJacobian(stewart);
</div> </div>
</div> </div>
<div id="outline-container-orgd59e9f0" class="outline-3"> <div id="outline-container-org84204db" class="outline-3">
<h3 id="orgd59e9f0"><span class="section-number-3">1.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3> <h3 id="org84204db"><span class="section-number-3">1.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org964919a">1</a>). We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org0e6176c">1</a>).
The Jacobian matrix is not estimated at the location of the center of the cube. The Jacobian matrix is not estimated at the location of the center of the cube.
</p> </p>
@ -439,9 +447,17 @@ stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</spa
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1)); stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
</pre> </pre>
</div> </div>
<div id="org7a8523c" class="figure">
<p><img src="figs/cubic_conf_centered_J_not_center.png" alt="cubic_conf_centered_J_not_center.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (<a href="./figs/cubic_conf_centered_J_not_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_not_center.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -517,19 +533,19 @@ stewart = computeJacobian(stewart);
</div> </div>
</div> </div>
<div id="outline-container-org27bd91a" class="outline-3"> <div id="outline-container-orgada859c" class="outline-3">
<h3 id="org27bd91a"><span class="section-number-3">1.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3> <h3 id="orgada859c"><span class="section-number-3">1.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
Here, the &ldquo;center&rdquo; of the Stewart platform is not at the cube center (figure <a href="#orgeceb55c">2</a>). Here, the &ldquo;center&rdquo; of the Stewart platform is not at the cube center (figure <a href="#org9e92807">4</a>).
The Jacobian is estimated at the cube center. The Jacobian is estimated at the cube center.
</p> </p>
<div id="orgeceb55c" class="figure"> <div id="org9e92807" class="figure">
<p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" /> <p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p> <p><span class="figure-number">Figure 4: </span>Not centered cubic configuration</p>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
@ -538,9 +554,17 @@ stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</spa
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1)); stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
</pre> </pre>
</div> </div>
<div id="orgcc43044" class="figure">
<p><img src="figs/cubic_conf_not_centered_J_center.png" alt="cubic_conf_not_centered_J_center.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_not_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_center.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -561,9 +585,9 @@ stewart = computeJacobian(stewart);
<tr> <tr>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-1.5e-16</td> <td class="org-right">-1.7e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">0.04</td> <td class="org-right">0.02</td>
<td class="org-right">0</td> <td class="org-right">0</td>
</tr> </tr>
@ -571,44 +595,44 @@ stewart = computeJacobian(stewart);
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-0.04</td> <td class="org-right">-0.02</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-2.8e-17</td> <td class="org-right">2.8e-17</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">-1.5e-16</td> <td class="org-right">-1.7e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">1.2e-18</td> <td class="org-right">1.2e-19</td>
<td class="org-right">-1e-17</td> <td class="org-right">-1.4e-17</td>
<td class="org-right">0</td> <td class="org-right">1.4e-17</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-0.04</td> <td class="org-right">-0.02</td>
<td class="org-right">1.2e-18</td> <td class="org-right">1.2e-19</td>
<td class="org-right">0.016</td> <td class="org-right">0.015</td>
<td class="org-right">0</td> <td class="org-right">-4.3e-19</td>
<td class="org-right">8.7e-19</td> <td class="org-right">1.7e-18</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">0.04</td> <td class="org-right">0.02</td>
<td class="org-right">0</td>
<td class="org-right">-7.3e-18</td>
<td class="org-right">-3.3e-21</td>
<td class="org-right">0.015</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-6.2e-18</td>
<td class="org-right">-1.1e-19</td>
<td class="org-right">0.016</td>
<td class="org-right">8.7e-19</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">-3.7e-19</td> <td class="org-right">6.6e-18</td>
<td class="org-right">-2.5e-17</td> <td class="org-right">2.5e-17</td>
<td class="org-right">0</td>
<td class="org-right">2e-18</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">1.2e-18</td>
<td class="org-right">8.7e-19</td>
<td class="org-right">0.06</td> <td class="org-right">0.06</td>
</tr> </tr>
</tbody> </tbody>
@ -620,8 +644,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
</div> </div>
</div> </div>
<div id="outline-container-org4adea52" class="outline-3"> <div id="outline-container-org8b2c367" class="outline-3">
<h3 id="org4adea52"><span class="section-number-3">1.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3> <h3 id="org8b2c367"><span class="section-number-3">1.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
<p> <p>
Here, the &ldquo;center&rdquo; of the Stewart platform is not at the cube center. Here, the &ldquo;center&rdquo; of the Stewart platform is not at the cube center.
@ -636,14 +660,22 @@ The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 80e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, <span class="org-type">-</span>30e<span class="org-type">-</span>3); <pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 80e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, <span class="org-type">-</span>40e<span class="org-type">-</span>3);
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0); stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 100e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1)); stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
</pre> </pre>
</div> </div>
<div id="org57db017" class="figure">
<p><img src="figs/cubic_conf_not_centered_J_stewart_center.png" alt="cubic_conf_not_centered_J_stewart_center.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (<a href="./figs/cubic_conf_not_centered_J_stewart_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_stewart_center.pdf">pdf</a>)</p>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -666,7 +698,7 @@ stewart = computeJacobian(stewart);
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-1.7e-16</td> <td class="org-right">-1.7e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">4.9e-17</td> <td class="org-right">0.02</td>
<td class="org-right">0</td> <td class="org-right">0</td>
</tr> </tr>
@ -674,7 +706,7 @@ stewart = computeJacobian(stewart);
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-2.2e-17</td> <td class="org-right">-0.02</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2.8e-17</td> <td class="org-right">2.8e-17</td>
</tr> </tr>
@ -683,48 +715,44 @@ stewart = computeJacobian(stewart);
<td class="org-right">-1.7e-16</td> <td class="org-right">-1.7e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">1.1e-18</td> <td class="org-right">1.2e-19</td>
<td class="org-right">-1.4e-17</td> <td class="org-right">-1.4e-17</td>
<td class="org-right">1.4e-17</td> <td class="org-right">1.4e-17</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-2.2e-17</td> <td class="org-right">-0.02</td>
<td class="org-right">1.1e-18</td> <td class="org-right">1.2e-19</td>
<td class="org-right">0.015</td> <td class="org-right">0.015</td>
<td class="org-right">0</td> <td class="org-right">-4.3e-19</td>
<td class="org-right">3.5e-18</td> <td class="org-right">1.7e-18</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">4.4e-17</td> <td class="org-right">0.02</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-1.4e-17</td> <td class="org-right">-7.3e-18</td>
<td class="org-right">-5.7e-20</td> <td class="org-right">-3.3e-21</td>
<td class="org-right">0.015</td> <td class="org-right">0.015</td>
<td class="org-right">-8.7e-19</td> <td class="org-right">0</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">6.6e-18</td> <td class="org-right">6.6e-18</td>
<td class="org-right">2.5e-17</td> <td class="org-right">2.5e-17</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">3.5e-18</td> <td class="org-right">2e-18</td>
<td class="org-right">-8.7e-19</td> <td class="org-right">0</td>
<td class="org-right">0.06</td> <td class="org-right">0.06</td>
</tr> </tr>
</tbody> </tbody>
</table> </table>
<p>
We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), and the Stiffness matrix is diagonal.
</p>
</div> </div>
</div> </div>
<div id="outline-container-org5defe19" class="outline-3"> <div id="outline-container-org34d7a0d" class="outline-3">
<h3 id="org5defe19"><span class="section-number-3">1.5</span> Conclusion</h3> <h3 id="org34d7a0d"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
<div class="important"> <div class="important">
<ul class="org-ul"> <ul class="org-ul">
@ -736,8 +764,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), and the Stiff
</div> </div>
</div> </div>
<div id="outline-container-org905f726" class="outline-3"> <div id="outline-container-org265d677" class="outline-3">
<h3 id="org905f726"><span class="section-number-3">1.6</span> Having Cube&rsquo;s center above the top platform</h3> <h3 id="org265d677"><span class="section-number-3">1.6</span> Having Cube&rsquo;s center above the top platform</h3>
<div class="outline-text-3" id="text-1-6"> <div class="outline-text-3" id="text-1-6">
<p> <p>
Let&rsquo;s say we want to have a decouple dynamics above the top platform. Let&rsquo;s say we want to have a decouple dynamics above the top platform.
@ -785,9 +813,9 @@ stewart = initializeStewartPose(stewart);
<tr> <tr>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-4.6e-16</td> <td class="org-right">-3.2e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">4e-17</td> <td class="org-right">3.1e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
</tr> </tr>
@ -795,45 +823,45 @@ stewart = initializeStewartPose(stewart);
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-4.8e-17</td> <td class="org-right">-1.2e-16</td>
<td class="org-right">0</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-3.5e-17</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">-4.6e-16</td> <td class="org-right">-3.2e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">2</td> <td class="org-right">2</td>
<td class="org-right">1.5e-20</td> <td class="org-right">5e-18</td>
<td class="org-right">4e-17</td> <td class="org-right">-5.6e-17</td>
<td class="org-right">0</td> <td class="org-right">0</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">-4.8e-17</td> <td class="org-right">-1.2e-16</td>
<td class="org-right">1.5e-20</td> <td class="org-right">5e-18</td>
<td class="org-right">0.00034</td> <td class="org-right">0.14</td>
<td class="org-right">6.8e-21</td> <td class="org-right">3.5e-18</td>
<td class="org-right">4.2e-19</td> <td class="org-right">1.4e-17</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">4e-17</td> <td class="org-right">3e-16</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">4e-17</td> <td class="org-right">-5.4e-17</td>
<td class="org-right">-3e-21</td> <td class="org-right">2.1e-19</td>
<td class="org-right">0.00034</td> <td class="org-right">0.14</td>
<td class="org-right">-2.7e-20</td> <td class="org-right">-6.9e-18</td>
</tr> </tr>
<tr> <tr>
<td class="org-right">-1.7e-19</td> <td class="org-right">7.4e-19</td>
<td class="org-right">-3.6e-17</td> <td class="org-right">-2.6e-17</td>
<td class="org-right">0</td> <td class="org-right">0</td>
<td class="org-right">4.2e-19</td> <td class="org-right">1.3e-17</td>
<td class="org-right">-2.7e-20</td> <td class="org-right">-6.9e-18</td>
<td class="org-right">0.0014</td> <td class="org-right">0.54</td>
</tr> </tr>
</tbody> </tbody>
</table> </table>
@ -845,19 +873,19 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
</div> </div>
</div> </div>
<div id="outline-container-org6746f61" class="outline-2"> <div id="outline-container-org93a8538" class="outline-2">
<h2 id="org6746f61"><span class="section-number-2">2</span> Functions</h2> <h2 id="org93a8538"><span class="section-number-2">2</span> Functions</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="org2a4885e"></a> <a id="org44c44dc"></a>
</p> </p>
</div> </div>
<div id="outline-container-org80fcb20" class="outline-3"> <div id="outline-container-org8786798" class="outline-3">
<h3 id="org80fcb20"><span class="section-number-3">2.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3> <h3 id="org8786798"><span class="section-number-3">2.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
<a id="orgfe8cdfe"></a> <a id="orgf3011db"></a>
</p> </p>
<p> <p>
@ -865,9 +893,9 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
</p> </p>
</div> </div>
<div id="outline-container-orgd3654f4" class="outline-4"> <div id="outline-container-org0b63bb1" class="outline-4">
<h4 id="orgd3654f4">Function description</h4> <h4 id="org0b63bb1">Function description</h4>
<div class="outline-text-4" id="text-orgd3654f4"> <div class="outline-text-4" id="text-org0b63bb1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>) <pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span> <span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
@ -892,21 +920,21 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
</div> </div>
</div> </div>
<div id="outline-container-orgf1c4374" class="outline-4"> <div id="outline-container-orgaac7da3" class="outline-4">
<h4 id="orgf1c4374">Documentation</h4> <h4 id="orgaac7da3">Documentation</h4>
<div class="outline-text-4" id="text-orgf1c4374"> <div class="outline-text-4" id="text-orgaac7da3">
<div id="org8d1a2f5" class="figure"> <div id="orgc2715ee" class="figure">
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" /> <p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Cubic Configuration</p> <p><span class="figure-number">Figure 7: </span>Cubic Configuration</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org7703a4a" class="outline-4"> <div id="outline-container-org747c61c" class="outline-4">
<h4 id="org7703a4a">Optional Parameters</h4> <h4 id="org747c61c">Optional Parameters</h4>
<div class="outline-text-4" id="text-org7703a4a"> <div class="outline-text-4" id="text-org747c61c">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">arguments <pre class="src src-matlab">arguments
stewart stewart
@ -920,9 +948,9 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
</div> </div>
</div> </div>
<div id="outline-container-org43039c1" class="outline-4"> <div id="outline-container-orgec4e738" class="outline-4">
<h4 id="org43039c1">Position of the Cube</h4> <h4 id="orgec4e738">Position of the Cube</h4>
<div class="outline-text-4" id="text-org43039c1"> <div class="outline-text-4" id="text-orgec4e738">
<p> <p>
We define the useful points of the cube with respect to the Cube&rsquo;s center. We define the useful points of the cube with respect to the Cube&rsquo;s center.
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is \({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
@ -947,9 +975,9 @@ CCm = [Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>
</div> </div>
</div> </div>
<div id="outline-container-orga18e804" class="outline-4"> <div id="outline-container-org97e1d45" class="outline-4">
<h4 id="orga18e804">Compute the pose</h4> <h4 id="org97e1d45">Compute the pose</h4>
<div class="outline-text-4" id="text-orga18e804"> <div class="outline-text-4" id="text-org97e1d45">
<p> <p>
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)). We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
</p> </p>
@ -982,7 +1010,7 @@ stewart.Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-06 jeu. 18:22</p> <p class="date">Created: 2020-02-07 ven. 17:31</p>
</div> </div>
</body> </body>
</html> </html>

105
cubic-configuration.org
View File

@ -61,12 +61,27 @@ The Jacobian matrix is estimated at the location of the center of the cube.
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
#+end_src #+end_src
#+name: fig:3d-cubic-stewart-aligned #+name: fig:3d-cubic-stewart-aligned
#+caption: Centered cubic configuration #+caption: Centered cubic configuration
[[file:./figs/3d-cubic-stewart-aligned.png]] [[file:./figs/3d-cubic-stewart-aligned.png]]
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false);
scatter3(0, 0, 50e-3, 200, 'kh');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/cubic_conf_centered_J_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:cubic_conf_centered_J_center
#+caption: Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center ([[./figs/cubic_conf_centered_J_center.png][png]], [[./figs/cubic_conf_centered_J_center.pdf][pdf]])
[[file:figs/cubic_conf_centered_J_center.png]]
#+begin_src matlab :exports results :results value table replace :tangle no #+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(stewart.K, {}, {}, ' %.2g '); data2orgtable(stewart.K, {}, {}, ' %.2g ');
#+end_src #+end_src
@ -89,8 +104,23 @@ The Jacobian matrix is not estimated at the location of the center of the cube.
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
#+end_src #+end_src
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false);
scatter3(0, 0, 50e-3, 200, 'kh');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/cubic_conf_centered_J_not_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:cubic_conf_centered_J_not_center
#+caption: Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center ([[./figs/cubic_conf_centered_J_not_center.png][png]], [[./figs/cubic_conf_centered_J_not_center.pdf][pdf]])
[[file:figs/cubic_conf_centered_J_not_center.png]]
#+begin_src matlab :exports results :results value table replace :tangle no #+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(stewart.K, {}, {}, ' %.2g '); data2orgtable(stewart.K, {}, {}, ' %.2g ');
#+end_src #+end_src
@ -117,19 +147,34 @@ The Jacobian is estimated at the cube center.
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
#+end_src #+end_src
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false);
scatter3(0, 0, 50e-3, 200, 'kh');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/cubic_conf_not_centered_J_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:cubic_conf_not_centered_J_center
#+caption: Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center ([[./figs/cubic_conf_not_centered_J_center.png][png]], [[./figs/cubic_conf_not_centered_J_center.pdf][pdf]])
[[file:figs/cubic_conf_not_centered_J_center.png]]
#+begin_src matlab :exports results :results value table replace :tangle no #+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(stewart.K, {}, {}, ' %.2g '); data2orgtable(stewart.K, {}, {}, ' %.2g ');
#+end_src #+end_src
#+RESULTS: #+RESULTS:
| 2 | 0 | -1.5e-16 | 0 | 0.04 | 0 | | 2 | 0 | -1.7e-16 | 0 | 0.02 | 0 |
| 0 | 2 | 0 | -0.04 | 0 | -2.8e-17 | | 0 | 2 | 0 | -0.02 | 0 | 2.8e-17 |
| -1.5e-16 | 0 | 2 | 1.2e-18 | -1e-17 | 0 | | -1.7e-16 | 0 | 2 | 1.2e-19 | -1.4e-17 | 1.4e-17 |
| 0 | -0.04 | 1.2e-18 | 0.016 | 0 | 8.7e-19 | | 0 | -0.02 | 1.2e-19 | 0.015 | -4.3e-19 | 1.7e-18 |
| 0.04 | 0 | -6.2e-18 | -1.1e-19 | 0.016 | 8.7e-19 | | 0.02 | 0 | -7.3e-18 | -3.3e-21 | 0.015 | 0 |
| -3.7e-19 | -2.5e-17 | 0 | 1.2e-18 | 8.7e-19 | 0.06 | | 6.6e-18 | 2.5e-17 | 0 | 2e-18 | 0 | 0.06 |
We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal. We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.
@ -143,26 +188,39 @@ The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$
The center of the cube from the top platform is at $z = 110 - 175 = -65$. The center of the cube from the top platform is at $z = 110 - 175 = -65$.
#+begin_src matlab #+begin_src matlab
stewart = initializeFramesPositions('H', 80e-3, 'MO_B', -30e-3); stewart = initializeFramesPositions('H', 80e-3, 'MO_B', -40e-3);
stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0); stewart = generateCubicConfiguration(stewart, 'Hc', 100e-3, 'FOc', 50e-3, 'FHa', 0, 'MHb', 0);
stewart = computeJointsPose(stewart); stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1)); stewart = initializeStrutDynamics(stewart, 'Ki', ones(6,1));
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
#+end_src #+end_src
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false);
scatter3(0, 0, 50e-3, 200, 'kh');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/cubic_conf_not_centered_J_stewart_center.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:cubic_conf_not_centered_J_stewart_center
#+caption: Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center ([[./figs/cubic_conf_not_centered_J_stewart_center.png][png]], [[./figs/cubic_conf_not_centered_J_stewart_center.pdf][pdf]])
[[file:figs/cubic_conf_not_centered_J_stewart_center.png]]
#+begin_src matlab :exports results :results value table replace :tangle no #+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(stewart.K, {}, {}, ' %.2g '); data2orgtable(stewart.K, {}, {}, ' %.2g ');
#+end_src #+end_src
#+RESULTS: #+RESULTS:
| 2 | 0 | -1.7e-16 | 0 | 4.9e-17 | 0 | | 2 | 0 | -1.7e-16 | 0 | 0.02 | 0 |
| 0 | 2 | 0 | -2.2e-17 | 0 | 2.8e-17 | | 0 | 2 | 0 | -0.02 | 0 | 2.8e-17 |
| -1.7e-16 | 0 | 2 | 1.1e-18 | -1.4e-17 | 1.4e-17 | | -1.7e-16 | 0 | 2 | 1.2e-19 | -1.4e-17 | 1.4e-17 |
| 0 | -2.2e-17 | 1.1e-18 | 0.015 | 0 | 3.5e-18 | | 0 | -0.02 | 1.2e-19 | 0.015 | -4.3e-19 | 1.7e-18 |
| 4.4e-17 | 0 | -1.4e-17 | -5.7e-20 | 0.015 | -8.7e-19 | | 0.02 | 0 | -7.3e-18 | -3.3e-21 | 0.015 | 0 |
| 6.6e-18 | 2.5e-17 | 0 | 3.5e-18 | -8.7e-19 | 0.06 | | 6.6e-18 | 2.5e-17 | 0 | 2e-18 | 0 | 0.06 |
We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, and the Stiffness matrix is diagonal.
** Conclusion ** Conclusion
#+begin_important #+begin_important
@ -192,17 +250,22 @@ This is possible, to do so:
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
#+end_src #+end_src
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false);
scatter3(0, 0, 50e-3, 200, 'kh');
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no #+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(stewart.K, {}, {}, ' %.2g '); data2orgtable(stewart.K, {}, {}, ' %.2g ');
#+end_src #+end_src
#+RESULTS: #+RESULTS:
| 2 | 0 | -4.6e-16 | 0 | 4e-17 | 0 | | 2 | 0 | -3.2e-16 | 0 | 3.1e-16 | 0 |
| 0 | 2 | 0 | -4.8e-17 | 0 | -3.5e-17 | | 0 | 2 | 0 | -1.2e-16 | 0 | 0 |
| -4.6e-16 | 0 | 2 | 1.5e-20 | 4e-17 | 0 | | -3.2e-16 | 0 | 2 | 5e-18 | -5.6e-17 | 0 |
| 0 | -4.8e-17 | 1.5e-20 | 0.00034 | 6.8e-21 | 4.2e-19 | | 0 | -1.2e-16 | 5e-18 | 0.14 | 3.5e-18 | 1.4e-17 |
| 4e-17 | 0 | 4e-17 | -3e-21 | 0.00034 | -2.7e-20 | | 3e-16 | 0 | -5.4e-17 | 2.1e-19 | 0.14 | -6.9e-18 |
| -1.7e-19 | -3.6e-17 | 0 | 4.2e-19 | -2.7e-20 | 0.0014 | | 7.4e-19 | -2.6e-17 | 0 | 1.3e-17 | -6.9e-18 | 0.54 |
We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal. We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.

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