Add bold math symbols
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@ -3,7 +3,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2019-12-11 mer. 09:43 -->
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<!-- 2019-12-11 mer. 09:46 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Metrology</title>
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@ -283,25 +283,25 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org2effdbf">1. How do we measure the position of the sample with respect to the granite</a></li>
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<li><a href="#org57e56b3">2. Verify that the function to compute the reference pose is correct</a>
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<li><a href="#orga5a431b">1. How do we measure the position of the sample with respect to the granite</a></li>
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<li><a href="#orgad7f8a4">2. Verify that the function to compute the reference pose is correct</a>
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<ul>
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<li><a href="#org57ba635">2.1. Prepare the Simulation</a></li>
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<li><a href="#org1da3199">2.2. Verify that the pose of the sample is the same as the computed one</a></li>
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<li><a href="#org62ac4b9">2.3. Conclusion</a></li>
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<li><a href="#org7de1029">2.1. Prepare the Simulation</a></li>
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<li><a href="#org20aaa6f">2.2. Verify that the pose of the sample is the same as the computed one</a></li>
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<li><a href="#orgfb913c2">2.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orga7fd6e3">3. Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</a>
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<li><a href="#org0d81df7">3. Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</a>
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<ul>
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<li><a href="#org7abb67d">3.1. Prepare the Simulation</a></li>
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<li><a href="#org41022d3">3.2. Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</a></li>
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<li><a href="#org32b9b63">3.3. Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</a></li>
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<li><a href="#orgfd4276a">3.4. Conclusion</a></li>
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<li><a href="#orgda08fb5">3.1. Prepare the Simulation</a></li>
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<li><a href="#org5f6b52f">3.2. Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</a></li>
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<li><a href="#org2932e43">3.3. Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</a></li>
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<li><a href="#org2512205">3.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org4a7e02c">4. Functions</a>
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<li><a href="#orgefbcdda">4. Functions</a>
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<ul>
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<li><a href="#orgca5b1d1">4.1. computeReferencePose</a></li>
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<li><a href="#org5f1a2be">4.1. computeReferencePose</a></li>
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</ul>
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</li>
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</ul>
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@ -314,25 +314,25 @@ Also, all the stages can be perfectly positioned.
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</p>
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<p>
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First, in section <a href="#orga4b40c4">1</a>, is explained how the measurement of the position of the sample with respect to the granite is performed.
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First, in section <a href="#orgbaf40ae">1</a>, is explained how the measurement of the position of the sample with respect to the granite is performed.
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</p>
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<p>
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In section <a href="#org8d1551f">2</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).
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In section <a href="#orgcb43c20">2</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).
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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.
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</p>
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<p>
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Then, in section <a href="#org869831b">3</a>, we introduce some positioning error in the position stages.
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Then, in section <a href="#org0414be5">3</a>, we introduce some positioning error in the position stages.
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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.
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Finally, 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.
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</p>
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<div id="outline-container-org2effdbf" class="outline-2">
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<h2 id="org2effdbf"><span class="section-number-2">1</span> How do we measure the position of the sample with respect to the granite</h2>
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<div id="outline-container-orga5a431b" class="outline-2">
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<h2 id="orga5a431b"><span class="section-number-2">1</span> How do we measure the position of the sample with respect to the granite</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="orga4b40c4"></a>
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<a id="orgbaf40ae"></a>
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A transform sensor block gives the translation and orientation of the follower frame with respect to the base frame.
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</p>
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@ -358,18 +358,18 @@ We can then determine extract other orientation conventions such that Euler angl
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</div>
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</div>
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<div id="outline-container-org57e56b3" class="outline-2">
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<h2 id="org57e56b3"><span class="section-number-2">2</span> Verify that the function to compute the reference pose is correct</h2>
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<div id="outline-container-orgad7f8a4" class="outline-2">
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<h2 id="orgad7f8a4"><span class="section-number-2">2</span> Verify that the function to compute the reference pose is correct</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org8d1551f"></a>
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<a id="orgcb43c20"></a>
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</p>
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<p>
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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.
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</p>
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</div>
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<div id="outline-container-org57ba635" class="outline-3">
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<h3 id="org57ba635"><span class="section-number-3">2.1</span> Prepare the Simulation</h3>
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<div id="outline-container-org7de1029" class="outline-3">
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<h3 id="org7de1029"><span class="section-number-3">2.1</span> Prepare the Simulation</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We load the configuration.
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@ -454,8 +454,8 @@ And we run the simulation.
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</div>
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</div>
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<div id="outline-container-org1da3199" class="outline-3">
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<h3 id="org1da3199"><span class="section-number-3">2.2</span> Verify that the pose of the sample is the same as the computed one</h3>
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<div id="outline-container-org20aaa6f" class="outline-3">
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<h3 id="org20aaa6f"><span class="section-number-3">2.2</span> Verify that the pose of the sample is the same as the computed one</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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Let's denote:
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@ -475,7 +475,7 @@ We have then computed:
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</ul>
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<p>
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We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix \({}^WT_R\).
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We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix \({}^W\bm{T}_R\).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">n = length<span class="org-rainbow-delimiters-depth-1">(</span>Dref.Dy.Time<span class="org-rainbow-delimiters-depth-1">)</span>;
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@ -488,7 +488,7 @@ WTr = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org
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<p>
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As the displacement is perfect, we also measure in simulation the pose of the sample with respect to the granite.
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From that we can compute the homogeneous transformation matrix \({}^WT_M\).
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From that we can compute the homogeneous transformation matrix \({}^W\bm{T}_M\).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">n = length<span class="org-rainbow-delimiters-depth-1">(</span>Dsm.R.Time<span class="org-rainbow-delimiters-depth-1">)</span>;
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@ -531,8 +531,8 @@ ans =
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</div>
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</div>
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<div id="outline-container-org62ac4b9" class="outline-3">
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<h3 id="org62ac4b9"><span class="section-number-3">2.3</span> Conclusion</h3>
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<div id="outline-container-orgfb913c2" class="outline-3">
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<h3 id="orgfb913c2"><span class="section-number-3">2.3</span> Conclusion</h3>
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<div class="outline-text-3" id="text-2-3">
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<div class="important">
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<p>
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@ -545,11 +545,11 @@ Both the measurement and the theory gives the same result.
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</div>
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</div>
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<div id="outline-container-orga7fd6e3" class="outline-2">
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<h2 id="orga7fd6e3"><span class="section-number-2">3</span> Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</h2>
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<div id="outline-container-org0d81df7" class="outline-2">
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<h2 id="org0d81df7"><span class="section-number-2">3</span> Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="org869831b"></a>
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<a id="org0414be5"></a>
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</p>
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<p>
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We now introduce some positioning error in the stage.
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@ -560,8 +560,8 @@ This will induce a global positioning error of the sample with respect to the de
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We want to verify that we are able to measure this positioning error and convert it in the frame attached to the Nano-hexapod.
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</p>
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</div>
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<div id="outline-container-org7abb67d" class="outline-3">
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<h3 id="org7abb67d"><span class="section-number-3">3.1</span> Prepare the Simulation</h3>
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<div id="outline-container-orgda08fb5" class="outline-3">
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<h3 id="orgda08fb5"><span class="section-number-3">3.1</span> Prepare the Simulation</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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We load the configuration.
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@ -644,8 +644,8 @@ And we run the simulation.
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</div>
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</div>
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<div id="outline-container-org41022d3" class="outline-3">
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<h3 id="org41022d3"><span class="section-number-3">3.2</span> Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</h3>
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<div id="outline-container-org5f6b52f" class="outline-3">
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<h3 id="org5f6b52f"><span class="section-number-3">3.2</span> Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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Now that we have introduced some positioning error, the computed wanted pose and the measured pose will not be the same.
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@ -780,8 +780,8 @@ Rz = <span class="org-rainbow-delimiters-depth-1">[</span>cos<span class="org-ra
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</div>
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</div>
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<div id="outline-container-org32b9b63" class="outline-3">
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<h3 id="org32b9b63"><span class="section-number-3">3.3</span> Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</h3>
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<div id="outline-container-org2932e43" class="outline-3">
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<h3 id="org2932e43"><span class="section-number-3">3.3</span> Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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We now keep the wanted pose but we impose a displacement of the nano hexapod corresponding to the measured position error.
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@ -875,8 +875,8 @@ Verify that the pose error is small.
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</div>
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</div>
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<div id="outline-container-orgfd4276a" class="outline-3">
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<h3 id="orgfd4276a"><span class="section-number-3">3.4</span> Conclusion</h3>
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<div id="outline-container-org2512205" class="outline-3">
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<h3 id="org2512205"><span class="section-number-3">3.4</span> Conclusion</h3>
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<div class="outline-text-3" id="text-3-4">
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<div class="important">
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<p>
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@ -888,15 +888,15 @@ Indeed, we are able to convert the position error in the frame of the NASS and t
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</div>
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</div>
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<div id="outline-container-org4a7e02c" class="outline-2">
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<h2 id="org4a7e02c"><span class="section-number-2">4</span> Functions</h2>
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<div id="outline-container-orgefbcdda" class="outline-2">
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<h2 id="orgefbcdda"><span class="section-number-2">4</span> Functions</h2>
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<div class="outline-text-2" id="text-4">
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</div>
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<div id="outline-container-orgca5b1d1" class="outline-3">
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<h3 id="orgca5b1d1"><span class="section-number-3">4.1</span> computeReferencePose</h3>
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<div id="outline-container-org5f1a2be" class="outline-3">
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<h3 id="org5f1a2be"><span class="section-number-3">4.1</span> computeReferencePose</h3>
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<div class="outline-text-3" id="text-4-1">
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<p>
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<a id="org9e21b70"></a>
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<a id="orgc7ee76b"></a>
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</p>
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<p>
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@ -989,7 +989,7 @@ This Matlab function is accessible <a href="src/computeReferencePose.m">here</a>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2019-12-11 mer. 09:43</p>
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<p class="date">Created: 2019-12-11 mer. 09:46</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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@ -164,7 +164,7 @@ We have then computed:
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- ${}^W\bm{T}_R$ which corresponds to the wanted pose of the sample with respect to the granite
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- ${}^W\bm{T}_M$ which corresponds to the measured pose of the sample with respect to the granite
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We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix ${}^WT_R$.
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We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix ${}^W\bm{T}_R$.
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#+begin_src matlab
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n = length(Dref.Dy.Time);
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WTr = zeros(4, 4, n);
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@ -174,7 +174,7 @@ We load the reference and we compute the desired trajectory of the sample in the
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#+end_src
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As the displacement is perfect, we also measure in simulation the pose of the sample with respect to the granite.
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From that we can compute the homogeneous transformation matrix ${}^WT_M$.
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From that we can compute the homogeneous transformation matrix ${}^W\bm{T}_M$.
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#+begin_src matlab
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n = length(Dsm.R.Time);
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WTm = zeros(4, 4, n);
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