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Thomas Dehaeze 2019-12-13 11:06:40 +01:00
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"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>
<!-- 2019-12-12 jeu. 13:46 --> <!-- 2019-12-13 ven. 11:05 -->
<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>Computation of the Positioning Error with respect to the nano-hexapod</title> <title>Computation of the Positioning Error with respect to the nano-hexapod</title>
@ -283,19 +283,19 @@ 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="#org05554ac">1. Verify that the function to compute the reference pose is correct</a> <li><a href="#orgfec69a9">1. Verify that the function to compute the reference pose is correct</a>
<ul> <ul>
<li><a href="#org615b133">1.1. Prepare the Simulation</a></li> <li><a href="#orgf92ae6f">1.1. Prepare the Simulation</a></li>
<li><a href="#org218866c">1.2. Verify that the pose of the sample is the same as the computed one</a></li> <li><a href="#orgf02914f">1.2. Verify that the pose of the sample is the same as the computed one</a></li>
<li><a href="#org9fdab3a">1.3. Conclusion</a></li> <li><a href="#org694b331">1.3. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org6581a39">2. Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</a> <li><a href="#orgde75002">2. Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</a>
<ul> <ul>
<li><a href="#org7b48080">2.1. Prepare the Simulation</a></li> <li><a href="#orgdd3b23d">2.1. Prepare the Simulation</a></li>
<li><a href="#orgab016c7">2.2. Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</a></li> <li><a href="#org919e4c0">2.2. Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</a></li>
<li><a href="#org8defab7">2.3. Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</a></li> <li><a href="#org8c406f5">2.3. Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</a></li>
<li><a href="#orgc0eac97">2.4. Conclusion</a></li> <li><a href="#org1025ff4">2.4. Conclusion</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -303,11 +303,11 @@ for the JavaScript code in this tag.
</div> </div>
<p> <p>
The global measurement and control schematic is shown in figure <a href="#org98c141e">1</a>. The global measurement and control schematic is shown in figure <a href="#org4153b3d">1</a>.
</p> </p>
<div id="org98c141e" class="figure"> <div id="org4153b3d" class="figure">
<p><img src="figs/control-schematic-nass.png" alt="control-schematic-nass.png" /> <p><img src="figs/control-schematic-nass.png" alt="control-schematic-nass.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Global Control Schematic for the Station</p> <p><span class="figure-number">Figure 1: </span>Global Control Schematic for the Station</p>
@ -329,37 +329,29 @@ Also, all the stages can be perfectly positioned.
</p> </p>
<p> <p>
In section <a href="#org9f71d9e">1</a>, we verify that the function developed to compute the wanted pose (translation and orientation) of the sample with respect to the granite can be determined from the wanted position of each stage (translation stage, tilt stage, spindle and micro-hexapod). This corresponds to the bloc "Compute Wanted Sample Position w.r.t. Granite" in figure <a href="#org98c141e">1</a>. In section <a href="#org9d27621">1</a>, we verify that the function developed to compute the wanted pose (translation and orientation) of the sample with respect to the granite can be determined from the wanted position of each stage (translation stage, tilt stage, spindle and micro-hexapod). This corresponds to the bloc "Compute Wanted Sample Position w.r.t. Granite" in figure <a href="#org4153b3d">1</a>.
To do so, we impose a perfect displacement and all the stage, we perfectly measure the position of the sample with respect to the granite, and we verify that this measured position corresponds to the computed wanted pose of the sample. To do so, we impose a perfect displacement and all the stage, we perfectly measure the position of the sample with respect to the granite, and we verify that this measured position corresponds to the computed wanted pose of the sample.
</p> </p>
<p> <p>
Then, in section <a href="#orgff651b1">2</a>, we introduce some positioning error in the micro-station's stages. Then, in section <a href="#orgaef8c87">2</a>, we introduce some positioning error in the micro-station's stages.
The positioning error of the sample expressed with respect to the granite frame (the one measured) is expressed in a frame connected to the NASS top platform (corresponding to the green bloc "Compute Sample Position Error w.r.t. NASS" in figure <a href="#org98c141e">1</a>). The positioning error of the sample expressed with respect to the granite frame (the one measured) is expressed in a frame connected to the NASS top platform (corresponding to the green bloc "Compute Sample Position Error w.r.t. NASS" in figure <a href="#org4153b3d">1</a>).
Then, we move the NASS such that it compensate for the positioning error that are expressed in the frame of the NASS, and we verify that the positioning error of the sample is well compensated. Then, we move the NASS such that it compensate for the positioning error that are expressed in the frame of the NASS, and we verify that the positioning error of the sample is well compensated.
</p> </p>
<div id="outline-container-org05554ac" class="outline-2"> <div id="outline-container-orgfec69a9" class="outline-2">
<h2 id="org05554ac"><span class="section-number-2">1</span> Verify that the function to compute the reference pose is correct</h2> <h2 id="orgfec69a9"><span class="section-number-2">1</span> Verify that the function to compute the reference pose is correct</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="org9f71d9e"></a> <a id="org9d27621"></a>
</p> </p>
<p> <p>
The goal here is to perfectly move the station and verify that there is no mismatch between the metrology measurement and the computation of the reference pose. The goal here is to perfectly move the station and verify that there is no mismatch between the metrology measurement and the computation of the reference pose.
</p> </p>
</div> </div>
<div id="outline-container-org615b133" class="outline-3"> <div id="outline-container-orgf92ae6f" class="outline-3">
<h3 id="org615b133"><span class="section-number-3">1.1</span> Prepare the Simulation</h3> <h3 id="orgf92ae6f"><span class="section-number-3">1.1</span> Prepare the Simulation</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p>
We load the configuration.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/conf_simscape.mat'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p> <p>
We set a small <code>StopTime</code>. We set a small <code>StopTime</code>.
</p> </p>
@ -409,7 +401,7 @@ We setup the reference path to be constant.
<span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-rainbow-delimiters-depth-1">)</span>;
initializeReferences<span class="org-rainbow-delimiters-depth-1">()</span>; initializeReferences<span class="org-rainbow-delimiters-depth-1">(</span>opts<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre> </pre>
</div> </div>
@ -436,8 +428,8 @@ And we run the simulation.
</div> </div>
</div> </div>
<div id="outline-container-org218866c" class="outline-3"> <div id="outline-container-orgf02914f" class="outline-3">
<h3 id="org218866c"><span class="section-number-3">1.2</span> Verify that the pose of the sample is the same as the computed one</h3> <h3 id="orgf02914f"><span class="section-number-3">1.2</span> Verify that the pose of the sample is the same as the computed one</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
Let's denote: Let's denote:
@ -501,20 +493,20 @@ WTr<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlig
<pre class="example"> <pre class="example">
WTr(1:3, 4, end)-WTm(1:3, 4, end) WTr(1:3, 4, end)-WTm(1:3, 4, end)
ans = ans =
-8.22065745307538e-15 8.53830894875784e-15
-1.74128279577812e-15 -5.58580959264532e-15
-8.3754490393689e-16 -5.89805981832114e-17
WTr(1:3, 1:3, end)'*WTm(1:3, 1:3, end)-eye(3) WTr(1:3, 1:3, end)'*WTm(1:3, 1:3, end)-eye(3)
ans = ans =
2.66453525910038e-15 6.12072360433062e-16 2.08519182823275e-16 1.62092561595273e-14 -1.59832000065641e-14 1.11022302462516e-16
-6.12072360433062e-16 2.66453525910038e-15 3.83905507244395e-16 1.61295672998496e-14 1.62092561595273e-14 -1.72431513512095e-15
-2.08519182823275e-16 -3.83905507244395e-16 2.66453525910038e-15 -1.65492619608187e-15 -9.8879238130678e-16 9.41469124882133e-14
</pre> </pre>
</div> </div>
</div> </div>
<div id="outline-container-org9fdab3a" class="outline-3"> <div id="outline-container-org694b331" class="outline-3">
<h3 id="org9fdab3a"><span class="section-number-3">1.3</span> Conclusion</h3> <h3 id="org694b331"><span class="section-number-3">1.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<div class="important"> <div class="important">
<p> <p>
@ -527,11 +519,11 @@ Both the measurement and the theory gives the same result.
</div> </div>
</div> </div>
<div id="outline-container-org6581a39" class="outline-2"> <div id="outline-container-orgde75002" class="outline-2">
<h2 id="org6581a39"><span class="section-number-2">2</span> Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</h2> <h2 id="orgde75002"><span class="section-number-2">2</span> Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="orgff651b1"></a> <a id="orgaef8c87"></a>
</p> </p>
<p> <p>
We now introduce some positioning error in the stage. We now introduce some positioning error in the stage.
@ -542,8 +534,8 @@ This will induce a global positioning error of the sample with respect to the de
We want to verify that we are able to measure this positioning error and convert it in the frame attached to the Nano-hexapod. We want to verify that we are able to measure this positioning error and convert it in the frame attached to the Nano-hexapod.
</p> </p>
</div> </div>
<div id="outline-container-org7b48080" class="outline-3"> <div id="outline-container-orgdd3b23d" class="outline-3">
<h3 id="org7b48080"><span class="section-number-3">2.1</span> Prepare the Simulation</h3> <h3 id="orgdd3b23d"><span class="section-number-3">2.1</span> Prepare the Simulation</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
We load the configuration. We load the configuration.
@ -625,8 +617,8 @@ And we run the simulation.
</div> </div>
</div> </div>
<div id="outline-container-orgab016c7" class="outline-3"> <div id="outline-container-org919e4c0" class="outline-3">
<h3 id="orgab016c7"><span class="section-number-3">2.2</span> Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</h3> <h3 id="org919e4c0"><span class="section-number-3">2.2</span> Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
Now that we have introduced some positioning error, the computed wanted pose and the measured pose will not be the same. Now that we have introduced some positioning error, the computed wanted pose and the measured pose will not be the same.
@ -761,8 +753,8 @@ Rz = <span class="org-rainbow-delimiters-depth-1">[</span>cos<span class="org-ra
</div> </div>
</div> </div>
<div id="outline-container-org8defab7" class="outline-3"> <div id="outline-container-org8c406f5" class="outline-3">
<h3 id="org8defab7"><span class="section-number-3">2.3</span> Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</h3> <h3 id="org8c406f5"><span class="section-number-3">2.3</span> Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
We now keep the wanted pose but we impose a displacement of the nano hexapod corresponding to the measured position error. We now keep the wanted pose but we impose a displacement of the nano hexapod corresponding to the measured position error.
@ -844,20 +836,20 @@ Verify that the pose error is small.
<tbody> <tbody>
<tr> <tr>
<td class="org-left">Error</td> <td class="org-left">Error</td>
<td class="org-right">1.0e-16</td> <td class="org-right">2.0e-16</td>
<td class="org-right">-1.1e-18</td> <td class="org-right">1.1e-16</td>
<td class="org-right">3.3e-20</td> <td class="org-right">3.2e-18</td>
<td class="org-right">1.5e-16</td> <td class="org-right">-1.1e-17</td>
<td class="org-right">5.8e-17</td> <td class="org-right">1.0e-17</td>
<td class="org-right">-6.1e-16</td> <td class="org-right">-9.5e-16</td>
</tr> </tr>
</tbody> </tbody>
</table> </table>
</div> </div>
</div> </div>
<div id="outline-container-orgc0eac97" class="outline-3"> <div id="outline-container-org1025ff4" class="outline-3">
<h3 id="orgc0eac97"><span class="section-number-3">2.4</span> Conclusion</h3> <h3 id="org1025ff4"><span class="section-number-3">2.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<div class="important"> <div class="important">
<p> <p>
@ -871,7 +863,7 @@ Indeed, we are able to convert the position error in the frame of the NASS and t
</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: 2019-12-12 jeu. 13:46</p> <p class="date">Created: 2019-12-13 ven. 11:05</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>