1912 lines
60 KiB
HTML
1912 lines
60 KiB
HTML
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<!-- 2021-03-10 mer. 11:41 -->
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<title>Flexible Joints - Test Bench</title>
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Flexible Joints - Test Bench</h1>
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<div id="table-of-contents">
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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="#orgba228d0">1. Flexible Joints</a></li>
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<li><a href="#orgc12e6d4">2. Dimensional Measurements</a>
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<ul>
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<li><a href="#org481b14d">2.1. Measurement Bench</a></li>
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<li><a href="#org2a4e284">2.2. Measurement Results</a></li>
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</ul>
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</li>
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<li><a href="#orgee2f239">3. Measurement Test Bench - Bending Stiffness</a>
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<ul>
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<li><a href="#orgc946d23">3.1. Flexible joint Geometry</a></li>
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<li><a href="#org96fb02e">3.2. Required external applied force</a></li>
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<li><a href="#orgcd7c170">3.3. Required actuator stroke and sensors range</a></li>
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<li><a href="#orgfd51ca6">3.4. Test Bench</a></li>
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</ul>
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</li>
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<li><a href="#orgdf7ba71">4. Error budget</a>
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<ul>
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<li><a href="#orgd3136d0">4.1. Finite Element Model</a></li>
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<li><a href="#org4ca82ad">4.2. Setup</a></li>
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<li><a href="#org95d9447">4.3. Effect of Bending</a></li>
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<li><a href="#org989ae59">4.4. Computation of the bending stiffness</a></li>
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<li><a href="#org82d111d">4.5. Estimation error due to force and displacement sensors accuracy</a></li>
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<li><a href="#org30e4c30">4.6. Estimation error due to Shear</a></li>
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<li><a href="#org8c74234">4.7. Estimation error due to force sensor compression</a></li>
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<li><a href="#orga4639c7">4.8. Estimation error due to height estimation error</a></li>
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<li><a href="#orgfdab5b0">4.9. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgadd76ee">5. First Measurements</a>
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<ul>
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<li><a href="#org48ad99b">5.1. Agreement between the probe and the encoder</a></li>
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<li><a href="#org9afc147">5.2. Measurement of the Millimar 1318 probe stiffness</a></li>
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<li><a href="#org412d759">5.3. Force Sensor Calibration</a></li>
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<li><a href="#org709b570">5.4. Force Sensor Noise</a></li>
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<li><a href="#org9cde23b">5.5. Force Sensor Stiffness</a></li>
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</ul>
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</li>
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<li><a href="#orgfddad67">6. Bending Stiffness Measurement</a>
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<ul>
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<li><a href="#org366c038">6.1. Introduction</a></li>
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<li><a href="#org2174e74">6.2. Analysis of one measurement</a></li>
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<li><a href="#org134bc98">6.3. Bending stiffness and bending stroke of all the flexible joints</a></li>
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<li><a href="#orgbb78147">6.4. Analysis</a></li>
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<li><a href="#org1c10279">6.5. Conclusion</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<hr>
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<p>This report is also available as a <a href="./test-bench-flexible-joints.pdf">pdf</a>.</p>
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<hr>
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<p>
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In this document, we present a test-bench that has been developed in order to measure the bending stiffness of flexible joints.
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</p>
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<p>
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It is structured as follow:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#orgd5f05d0">1</a>: the geometry of the flexible joints and the expected stiffness and stroke are presented</li>
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<li>Section <a href="#orgab94b61">2</a>: each flexible joint is measured using a profile projector</li>
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<li>Section <a href="#org7938a31">3</a>: the stiffness measurement bench is presented</li>
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<li>Section <a href="#orgbd6228c">4</a>: an error budget is performed in order to estimate the accuracy of the measured stiffness</li>
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<li>Section <a href="#orge162618">5</a>: first measurements are performed</li>
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<li>Section <a href="#orgcf7ec01">6</a>: the bending stiffness of the flexible joints are measured</li>
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</ul>
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<div id="outline-container-orgba228d0" class="outline-2">
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<h2 id="orgba228d0"><span class="section-number-2">1</span> Flexible Joints</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="orgd5f05d0"></a>
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</p>
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<p>
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The flexible joints that are going to be measured in this document have been design to be used with a Nano-Hexapod (Figure <a href="#org00ed786">1</a>).
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</p>
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<div id="org00ed786" class="figure">
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<p><img src="figs/nano_hexapod.png" alt="nano_hexapod.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>CAD view of the Nano-Hexapod containing the flexible joints</p>
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</div>
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<p>
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Ideally, these flexible joints would behave as perfect ball joints, that is to say:
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</p>
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<ul class="org-ul">
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<li>no bending and torsional stiffnesses</li>
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<li>infinite shear and axial stiffnesses</li>
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<li>un-limited bending and torsional stroke</li>
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<li>no friction, no backlash</li>
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</ul>
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<p>
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The real characteristics of the flexible joints will influence the dynamics of the Nano-Hexapod.
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Using a multi-body dynamical model of the nano-hexapod, the specifications in term of stiffness and stroke of the flexible joints have been determined and summarized in Table <a href="#orgd7b968a">1</a>.
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</p>
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<table id="orgd7b968a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Specifications for the flexible joints and estimated characteristics from the Finite Element Model</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-left"><b>Specification</b></th>
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<th scope="col" class="org-right"><b>FEM</b></th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">Axial Stiffness</td>
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<td class="org-left">> 100 [N/um]</td>
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<td class="org-right">94</td>
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</tr>
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<tr>
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<td class="org-left">Shear Stiffness</td>
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<td class="org-left">> 1 [N/um]</td>
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<td class="org-right">13</td>
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</tr>
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<tr>
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<td class="org-left">Bending Stiffness</td>
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<td class="org-left">< 100 [Nm/rad]</td>
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<td class="org-right">5</td>
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</tr>
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<tr>
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<td class="org-left">Torsion Stiffness</td>
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<td class="org-left">< 500 [Nm/rad]</td>
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<td class="org-right">260</td>
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</tr>
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<tr>
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<td class="org-left">Bending Stroke</td>
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<td class="org-left">> 1 [mrad]</td>
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<td class="org-right">24.5</td>
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</tr>
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<tr>
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<td class="org-left">Torsion Stroke</td>
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<td class="org-left">> 5 [urad]</td>
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<td class="org-right"> </td>
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</tr>
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</tbody>
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</table>
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<p>
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Then, the classical geometry of a flexible ball joint shown in Figure <a href="#org1ea4826">2</a> has been optimized in order to meet the requirements.
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This has been done using a Finite Element Software and the obtained joint’s characteristics are summarized in Table <a href="#orgd7b968a">1</a>.
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</p>
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<div id="org1ea4826" class="figure">
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<p><img src="figs/flexible_joint_fem_geometry.png" alt="flexible_joint_fem_geometry.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Flexible part of the Joint used for FEM - CAD view</p>
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</div>
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<p>
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The obtained geometry are defined in the <a href="doc/flex_joints.pdf">drawings of the flexible joints</a>.
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The material is a special kind of stainless steel called “F16PH”.
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</p>
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<p>
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The flexible joints can be seen on Figure <a href="#org43b29c2">3</a>.
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</p>
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<div id="org43b29c2" class="figure">
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<p><img src="figs/IMG_20210302_173619.jpg" alt="IMG_20210302_173619.jpg" />
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</p>
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<p><span class="figure-number">Figure 3: </span>15 of the 16 flexible joints</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgc12e6d4" class="outline-2">
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<h2 id="orgc12e6d4"><span class="section-number-2">2</span> Dimensional Measurements</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orgab94b61"></a>
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</p>
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</div>
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<div id="outline-container-org481b14d" class="outline-3">
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<h3 id="org481b14d"><span class="section-number-3">2.1</span> Measurement Bench</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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The axis corresponding to the flexible joints are defined in Figure <a href="#orgdbf1f86">4</a>.
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</p>
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<div id="orgdbf1f86" class="figure">
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<p><img src="figs/flexible_joint_axis.png" alt="flexible_joint_axis.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Define axis for the flexible joints</p>
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</div>
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<p>
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The dimensions of the flexible part in the Y-Z plane will contribute to the X-bending stiffness.
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Similarly, the dimensions of the flexible part in the X-Z plane will contribute to the Y-bending stiffness.
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</p>
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<p>
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The setup to measure the dimension of the “Y” flexible beam is shown in Figure <a href="#orgf2666ad">5</a>.
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</p>
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<div id="orgf2666ad" class="figure">
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<p><img src="figs/flexible_joint_y_flex_meas_setup.png" alt="flexible_joint_y_flex_meas_setup.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness</p>
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</div>
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<p>
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What we typically observe is shown in Figure <a href="#org4e5ee22">6</a>.
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It is then possible to estimate to dimension of the flexible beam with an accuracy of \(\approx 5\,\mu m\),
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</p>
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<div id="org4e5ee22" class="figure">
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<p><img src="figs/soft_measure_flex_size.jpg" alt="soft_measure_flex_size.jpg" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Image used to measure the flexible joint’s dimensions</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org2a4e284" class="outline-3">
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<h3 id="org2a4e284"><span class="section-number-3">2.2</span> Measurement Results</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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The expected flexible beam thickness is \(250\,\mu m\).
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However, it is more important that the thickness of all beams are close to each other.
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</p>
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<p>
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The dimension of the beams are been measured at each end to be able to estimate the mean of the beam thickness.
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</p>
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<p>
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All the measured dimensions are summarized in Table <a href="#org870172e">2</a>.
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</p>
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<table id="org870172e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> Measured Dimensions of the flexible beams in \(\mu m\)</caption>
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<colgroup>
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<col class="org-right" />
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|
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<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
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|
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-right"> </th>
|
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<th scope="col" class="org-right">X1</th>
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<th scope="col" class="org-right">X2</th>
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<th scope="col" class="org-right">X3</th>
|
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<th scope="col" class="org-right">X4</th>
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</tr>
|
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</thead>
|
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<tbody>
|
|
<tr>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">223</td>
|
|
<td class="org-right">226</td>
|
|
<td class="org-right">224</td>
|
|
<td class="org-right">214</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">229</td>
|
|
<td class="org-right">231</td>
|
|
<td class="org-right">237</td>
|
|
<td class="org-right">224</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3</td>
|
|
<td class="org-right">234</td>
|
|
<td class="org-right">230</td>
|
|
<td class="org-right">239</td>
|
|
<td class="org-right">231</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4</td>
|
|
<td class="org-right">233</td>
|
|
<td class="org-right">227</td>
|
|
<td class="org-right">229</td>
|
|
<td class="org-right">232</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5</td>
|
|
<td class="org-right">225</td>
|
|
<td class="org-right">212</td>
|
|
<td class="org-right">228</td>
|
|
<td class="org-right">228</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6</td>
|
|
<td class="org-right">220</td>
|
|
<td class="org-right">221</td>
|
|
<td class="org-right">224</td>
|
|
<td class="org-right">220</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7</td>
|
|
<td class="org-right">206</td>
|
|
<td class="org-right">207</td>
|
|
<td class="org-right">228</td>
|
|
<td class="org-right">226</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">8</td>
|
|
<td class="org-right">230</td>
|
|
<td class="org-right">224</td>
|
|
<td class="org-right">224</td>
|
|
<td class="org-right">223</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">9</td>
|
|
<td class="org-right">223</td>
|
|
<td class="org-right">231</td>
|
|
<td class="org-right">228</td>
|
|
<td class="org-right">233</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">10</td>
|
|
<td class="org-right">228</td>
|
|
<td class="org-right">230</td>
|
|
<td class="org-right">235</td>
|
|
<td class="org-right">231</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">11</td>
|
|
<td class="org-right">197</td>
|
|
<td class="org-right">207</td>
|
|
<td class="org-right">211</td>
|
|
<td class="org-right">204</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">12</td>
|
|
<td class="org-right">227</td>
|
|
<td class="org-right">226</td>
|
|
<td class="org-right">225</td>
|
|
<td class="org-right">226</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">13</td>
|
|
<td class="org-right">215</td>
|
|
<td class="org-right">228</td>
|
|
<td class="org-right">231</td>
|
|
<td class="org-right">220</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">14</td>
|
|
<td class="org-right">216</td>
|
|
<td class="org-right">224</td>
|
|
<td class="org-right">224</td>
|
|
<td class="org-right">221</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">15</td>
|
|
<td class="org-right">209</td>
|
|
<td class="org-right">214</td>
|
|
<td class="org-right">220</td>
|
|
<td class="org-right">221</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">16</td>
|
|
<td class="org-right">213</td>
|
|
<td class="org-right">210</td>
|
|
<td class="org-right">230</td>
|
|
<td class="org-right">229</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
An histogram of these measured dimensions is shown in Figure <a href="#orgdd64c7d">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgdd64c7d" class="figure">
|
|
<p><img src="figs/beam_dim_histogram.png" alt="beam_dim_histogram.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Histogram for the (16x2) measured beams’ thickness</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgee2f239" class="outline-2">
|
|
<h2 id="orgee2f239"><span class="section-number-2">3</span> Measurement Test Bench - Bending Stiffness</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org7938a31"></a>
|
|
</p>
|
|
<p>
|
|
The most important characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
|
|
</p>
|
|
|
|
<p>
|
|
To do so, we have to apply a torque \(T_x\) on the flexible joint and measure its angular deflection \(\theta_x\).
|
|
The stiffness is then
|
|
</p>
|
|
\begin{equation}
|
|
k_{R_x} = \frac{T_x}{\theta_x}
|
|
\end{equation}
|
|
|
|
<p>
|
|
As it is quite difficult to apply a pure torque, a force will be applied instead.
|
|
The application point of the force should far enough from the flexible part such that the obtained bending is much larger than the displacement in shear.
|
|
</p>
|
|
|
|
<p>
|
|
The working principle of the bench is schematically shown in Figure <a href="#org447267e">8</a>.
|
|
One part of the flexible joint is fixed. On the mobile part, a force \(F_x\) is applied which is equivalent to a torque applied on the flexible joint center.
|
|
The induced rotation is measured with a displacement sensor \(d_x\).
|
|
</p>
|
|
|
|
|
|
<div id="org447267e" class="figure">
|
|
<p><img src="figs/test_bench_principle.png" alt="test_bench_principle.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Test Bench - working principle</p>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
This test-bench will be used to have a first approximation of the bending stiffnesss and stroke of the flexible joints.
|
|
Another test-bench, better engineered will be used to measure the flexible joint’s characteristics with better accuracy.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgc946d23" class="outline-3">
|
|
<h3 id="orgc946d23"><span class="section-number-3">3.1</span> Flexible joint Geometry</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
The flexible joint used for the Nano-Hexapod is shown in Figure <a href="#org41fd7ad">9</a>.
|
|
Its bending stiffness is foreseen to be \(k_{R_y}\approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 25\,mrad\).
|
|
</p>
|
|
|
|
|
|
<div id="org41fd7ad" class="figure">
|
|
<p><img src="figs/flexible_joint_geometry.png" alt="flexible_joint_geometry.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Geometry of the flexible joint</p>
|
|
</div>
|
|
|
|
<p>
|
|
The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 20\,mm\).
|
|
</p>
|
|
|
|
<p>
|
|
Let’s define the parameters on Matlab.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">kRx = 5; <span class="org-comment">% Bending Stiffness [Nm/rad]</span>
|
|
Rxmax = 25e<span class="org-type">-</span>3; <span class="org-comment">% Bending Stroke [rad]</span>
|
|
h = 20e<span class="org-type">-</span>3; <span class="org-comment">% Height [m]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org96fb02e" class="outline-3">
|
|
<h3 id="org96fb02e"><span class="section-number-3">3.2</span> Required external applied force</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:
|
|
</p>
|
|
\begin{equation}
|
|
\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
|
|
\end{equation}
|
|
|
|
<p>
|
|
Therefore, the applied force to test the full range of the flexible joint is:
|
|
</p>
|
|
\begin{equation}
|
|
F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
|
|
\end{equation}
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>Rxmax<span class="org-type">/</span>h; <span class="org-comment">% Force to induce maximum stroke [N]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we obtain:
|
|
</p>
|
|
\begin{equation} F_{x,max} = 6.2\, [N] \end{equation}
|
|
|
|
<p>
|
|
The measurement range of the force sensor should then be higher than \(6.2\,N\).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcd7c170" class="outline-3">
|
|
<h3 id="orgcd7c170"><span class="section-number-3">3.3</span> Required actuator stroke and sensors range</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
The flexible joint is designed to allow a bending motion of \(\pm 25\,mrad\).
|
|
The corresponding stroke at the location of the force sensor is:
|
|
\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">dxmax = h<span class="org-type">*</span>tan(Rxmax);
|
|
</pre>
|
|
</div>
|
|
|
|
\begin{equation} d_{max} = 0.5\, [mm] \end{equation}
|
|
|
|
<p>
|
|
In order to test the full range of the flexible joint, the stroke of the translation stage used to move the force sensor should be higher than \(0.5\,mm\).
|
|
Similarly, the measurement range of the displacement sensor should also be higher than \(0.5\,mm\).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfd51ca6" class="outline-3">
|
|
<h3 id="orgfd51ca6"><span class="section-number-3">3.4</span> Test Bench</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
A CAD view of the measurement bench is shown in Figure <a href="#org1b9c20e">10</a>.
|
|
</p>
|
|
|
|
<div class="note" id="org7979eca">
|
|
<p>
|
|
Here are the different elements used in this bench:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Translation Stage</b>: <a href="doc/V-408-Datasheet.pdf">V-408</a></li>
|
|
<li><b>Load Cells</b>: <a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a></li>
|
|
<li><b>Encoder</b>: <a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">Renishaw Resolute 1nm</a></li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
Both the measured force and displacement are acquired at the same time using a Speedgoat machine.
|
|
</p>
|
|
|
|
|
|
<div id="org1b9c20e" class="figure">
|
|
<p><img src="figs/test_bench_flex_overview.png" alt="test_bench_flex_overview.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
|
|
</div>
|
|
|
|
<p>
|
|
A side view of the bench with the important quantities are shown in Figure <a href="#org97759ae">11</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org97759ae" class="figure">
|
|
<p><img src="figs/test_bench_flex_side.png" alt="test_bench_flex_side.png" width="300px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgdf7ba71" class="outline-2">
|
|
<h2 id="orgdf7ba71"><span class="section-number-2">4</span> Error budget</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="orgbd6228c"></a>
|
|
</p>
|
|
<p>
|
|
Many things can impact the accuracy of the measured bending stiffness such as:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Errors in the force and displacement measurement</li>
|
|
<li>Shear effects</li>
|
|
<li>Deflection of the Force sensor</li>
|
|
<li>Errors in the geometry of the bench</li>
|
|
</ul>
|
|
|
|
<p>
|
|
In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgd3136d0" class="outline-3">
|
|
<h3 id="orgd3136d0"><span class="section-number-3">4.1</span> Finite Element Model</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables <a href="#org32e2bd9">3</a> and <a href="#org8dea15f">4</a>.
|
|
</p>
|
|
|
|
<table id="org32e2bd9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 3:</span> Axial/Shear characteristics</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-right">Stiffness [N/um]</th>
|
|
<th scope="col" class="org-right">Max Force [N]</th>
|
|
<th scope="col" class="org-right">Stroke [um]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Axial</td>
|
|
<td class="org-right">94</td>
|
|
<td class="org-right">469</td>
|
|
<td class="org-right">5</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Shear</td>
|
|
<td class="org-right">13</td>
|
|
<td class="org-right">242</td>
|
|
<td class="org-right">19</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<table id="org8dea15f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 4:</span> Bending/Torsion characteristics</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-right">Stiffness [Nm/rad]</th>
|
|
<th scope="col" class="org-right">Max Torque [Nmm]</th>
|
|
<th scope="col" class="org-right">Stroke [mrad]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Bending</td>
|
|
<td class="org-right">5</td>
|
|
<td class="org-right">118</td>
|
|
<td class="org-right">24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Torsional</td>
|
|
<td class="org-right">260</td>
|
|
<td class="org-right">1508</td>
|
|
<td class="org-right">6</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4ca82ad" class="outline-3">
|
|
<h3 id="org4ca82ad"><span class="section-number-3">4.2</span> Setup</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
The setup is schematically represented in Figure <a href="#org8fdc757">12</a>.
|
|
</p>
|
|
|
|
<p>
|
|
The force is applied on top of the flexible joint with a distance \(h\) with the joint’s center.
|
|
The displacement of the flexible joint is also measured at the same height.
|
|
</p>
|
|
|
|
<p>
|
|
The height between the joint’s center and the force application point is:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = 25e<span class="org-type">-</span>3; <span class="org-comment">% Height [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org8fdc757" class="figure">
|
|
<p><img src="figs/test_bench_flex_side.png" alt="test_bench_flex_side.png" width="300px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org95d9447" class="outline-3">
|
|
<h3 id="org95d9447"><span class="section-number-3">4.3</span> Effect of Bending</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<p>
|
|
The torque applied is:
|
|
</p>
|
|
\begin{equation}
|
|
M_y = F_x \cdot h
|
|
\end{equation}
|
|
|
|
<p>
|
|
The flexible joint is experiencing a rotation \(\theta_y\) due to the torque \(M_y\):
|
|
</p>
|
|
\begin{equation}
|
|
\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x \cdot h}{k_{R_y}}
|
|
\end{equation}
|
|
|
|
<p>
|
|
This rotation is then measured by the displacement sensor.
|
|
The measured displacement is:
|
|
</p>
|
|
\begin{equation}
|
|
D_b = h \tan(\theta_y) = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) \label{eq:bending_stiffness_formula}
|
|
\end{equation}
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org989ae59" class="outline-3">
|
|
<h3 id="org989ae59"><span class="section-number-3">4.4</span> Computation of the bending stiffness</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<p>
|
|
From equation \eqref{eq:bending_stiffness_formula}, we can compute the bending stiffness:
|
|
</p>
|
|
\begin{equation}
|
|
k_{R_y} = \frac{F_x \cdot h}{\tan^{-1}\left( \frac{D_b}{h} \right)}
|
|
\end{equation}
|
|
|
|
<p>
|
|
For small displacement, we have
|
|
</p>
|
|
\begin{equation}
|
|
\boxed{k_{R_y} \approx h^2 \frac{F_x}{d_x}}
|
|
\end{equation}
|
|
|
|
<p>
|
|
And therefore, to precisely measure \(k_{R_y}\), we need to:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>precisely measure the motion \(d_x\)</li>
|
|
<li>precisely measure the applied force \(F_x\)</li>
|
|
<li>precisely now the height of the force application point \(h\)</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org82d111d" class="outline-3">
|
|
<h3 id="org82d111d"><span class="section-number-3">4.5</span> Estimation error due to force and displacement sensors accuracy</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
<p>
|
|
The maximum error on the measured displacement with the encoder is 40 nm.
|
|
This quite negligible compared to the measurement range of 0.5 mm.
|
|
</p>
|
|
|
|
<p>
|
|
The accuracy of the force sensor is around 1% and therefore, we should expect to have an accuracy on the measured stiffness of at most 1%.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org30e4c30" class="outline-3">
|
|
<h3 id="org30e4c30"><span class="section-number-3">4.6</span> Estimation error due to Shear</h3>
|
|
<div class="outline-text-3" id="text-4-6">
|
|
<p>
|
|
The effect of Shear on the measured displacement is simply:
|
|
</p>
|
|
\begin{equation}
|
|
D_s = \frac{F_x}{k_s}
|
|
\end{equation}
|
|
|
|
<p>
|
|
The measured displacement will be the effect of shear + effect of bending
|
|
</p>
|
|
\begin{equation}
|
|
d_x = D_b + D_s = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_s} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_s} \right)
|
|
\end{equation}
|
|
|
|
<p>
|
|
The estimated bending stiffness \(k_{\text{est}}\) will then be:
|
|
</p>
|
|
\begin{equation}
|
|
k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}}
|
|
\end{equation}
|
|
|
|
<pre class="example">
|
|
The measurement error due to Shear is 0.1 %
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8c74234" class="outline-3">
|
|
<h3 id="org8c74234"><span class="section-number-3">4.7</span> Estimation error due to force sensor compression</h3>
|
|
<div class="outline-text-3" id="text-4-7">
|
|
<p>
|
|
The measured displacement is not done directly at the joint’s location.
|
|
The force sensor compression will then induce an error on the joint’s stiffness.
|
|
</p>
|
|
|
|
<p>
|
|
The force sensor stiffness \(k_F\) is estimated to be around:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">kF = 50<span class="org-type">/</span>0.05e<span class="org-type">-</span>3; <span class="org-comment">% [N/m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
k_F = 1.0e+06 [N/m]
|
|
</pre>
|
|
|
|
|
|
<p>
|
|
The measured displacement will be the sum of the displacement induced by the bending and by the compression of the force sensor:
|
|
</p>
|
|
\begin{equation}
|
|
d_x = D_b + \frac{F_x}{k_F} = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_F} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_F} \right)
|
|
\end{equation}
|
|
|
|
<p>
|
|
The estimated bending stiffness \(k_{\text{est}}\) will then be:
|
|
</p>
|
|
\begin{equation}
|
|
k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}}
|
|
\end{equation}
|
|
|
|
<pre class="example">
|
|
The measurement error due to height estimation errors is 0.8 %
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga4639c7" class="outline-3">
|
|
<h3 id="orga4639c7"><span class="section-number-3">4.8</span> Estimation error due to height estimation error</h3>
|
|
<div class="outline-text-3" id="text-4-8">
|
|
<p>
|
|
Let’s consider an error in the estimation of the height from the application of the force to the joint’s center:
|
|
</p>
|
|
\begin{equation}
|
|
h_{\text{est}} = h (1 + \epsilon)
|
|
\end{equation}
|
|
|
|
<p>
|
|
The computed bending stiffness will be:
|
|
</p>
|
|
\begin{equation}
|
|
k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x}
|
|
\end{equation}
|
|
|
|
<p>
|
|
And the stiffness estimation error is:
|
|
</p>
|
|
\begin{equation}
|
|
\frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2
|
|
\end{equation}
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h_err = 0.2e<span class="org-type">-</span>3; <span class="org-comment">% Height estimation error [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
The measurement error due to height estimation errors of 0.2 [mm] is 1.6 %
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfdab5b0" class="outline-3">
|
|
<h3 id="orgfdab5b0"><span class="section-number-3">4.9</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-4-9">
|
|
<p>
|
|
Based on the above analysis, we should expect no better than few percent of accuracy using the current test-bench.
|
|
This is well enough for a first estimation of the bending stiffness of the flexible joints.
|
|
</p>
|
|
|
|
<p>
|
|
Another measurement bench allowing better accuracy will be developed.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgadd76ee" class="outline-2">
|
|
<h2 id="orgadd76ee"><span class="section-number-2">5</span> First Measurements</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orge162618"></a>
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Section <a href="#org8f41c54">5.1</a>:</li>
|
|
<li>Section <a href="#org22b6b03">5.2</a>:</li>
|
|
</ul>
|
|
</div>
|
|
<div id="outline-container-org48ad99b" class="outline-3">
|
|
<h3 id="org48ad99b"><span class="section-number-3">5.1</span> Agreement between the probe and the encoder</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
<a id="org8f41c54"></a>
|
|
</p>
|
|
<div class="note" id="org2980735">
|
|
<ul class="org-ul">
|
|
<li><b>Encoder</b>: <a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">Renishaw Resolute 1nm</a></li>
|
|
<li><b>Displacement Probe</b>: <a href="doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf">Millimar C1216 electronics</a> and <a href="doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf">Millimar 1318 probe</a></li>
|
|
</ul>
|
|
|
|
</div>
|
|
<p>
|
|
The measurement setup is made such that the probe measured the translation table displacement.
|
|
It should then measure the same displacement as the encoder.
|
|
Using this setup, we should be able to compare the probe and the encoder.
|
|
</p>
|
|
<p>
|
|
Let’s load the measurements.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'meas_probe_against_encoder.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'d'</span>, <span class="org-string">'dp'</span>, <span class="org-string">'F'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The time domain measured displacement by the probe and by the encoder is shown in Figure <a href="#org8505712">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org8505712" class="figure">
|
|
<p><img src="figs/comp_encoder_probe_time.png" alt="comp_encoder_probe_time.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Time domain measurement</p>
|
|
</div>
|
|
|
|
<p>
|
|
If we zoom, we see that there is some delay between the encoder and the probe (Figure <a href="#org0ea83ae">14</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org0ea83ae" class="figure">
|
|
<p><img src="figs/comp_encoder_probe_time_zoom.png" alt="comp_encoder_probe_time_zoom.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Time domain measurement (Zoom)</p>
|
|
</div>
|
|
|
|
<p>
|
|
This delay is estimated using the <code>finddelay</code> command.
|
|
</p>
|
|
|
|
<pre class="example">
|
|
The time delay is approximately 15.8 [ms]
|
|
</pre>
|
|
|
|
|
|
<p>
|
|
The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure <a href="#org8c19e48">15</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org8c19e48" class="figure">
|
|
<p><img src="figs/comp_encoder_probe_mismatch.png" alt="comp_encoder_probe_mismatch.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Measurement mismatch, with and without delay compensation</p>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure <a href="#org0313952">16</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org0313952" class="figure">
|
|
<p><img src="figs/comp_encoder_probe_linear_fit.png" alt="comp_encoder_probe_linear_fit.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Measured displacement by the probe as a function of the measured displacement by the encoder</p>
|
|
</div>
|
|
|
|
<div class="important" id="orgf816b27">
|
|
<p>
|
|
From the measurement, it is shown that the probe is well calibrated.
|
|
However, there is some time delay of tens of milliseconds that could induce some measurement errors.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9afc147" class="outline-3">
|
|
<h3 id="org9afc147"><span class="section-number-3">5.2</span> Measurement of the Millimar 1318 probe stiffness</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<p>
|
|
<a id="org22b6b03"></a>
|
|
</p>
|
|
<div class="note" id="org62d5a64">
|
|
<ul class="org-ul">
|
|
<li><b>Translation Stage</b>: <a href="doc/V-408-Datasheet.pdf">V-408</a></li>
|
|
<li><b>Load Cell</b>: <a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a></li>
|
|
<li><b>Encoder</b>: <a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">Renishaw Resolute 1nm</a></li>
|
|
<li><b>Displacement Probe</b>: <a href="doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf">Millimar C1216 electronics</a> and <a href="doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf">Millimar 1318 probe</a></li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
|
|
<div id="orgd0a66d0" class="figure">
|
|
<p><img src="figs/setup_mahr_stiff_meas_side.jpg" alt="setup_mahr_stiff_meas_side.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Setup - Side View</p>
|
|
</div>
|
|
|
|
|
|
<div id="org37fbba6" class="figure">
|
|
<p><img src="figs/setup_mahr_stiff_meas_top.jpg" alt="setup_mahr_stiff_meas_top.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Setup - Top View</p>
|
|
</div>
|
|
<p>
|
|
Let’s load the measurement results.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'meas_stiff_probe.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'d'</span>, <span class="org-string">'dp'</span>, <span class="org-string">'F'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The time domain measured force and displacement are shown in Figure <a href="#orgcf9e77e">19</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgcf9e77e" class="figure">
|
|
<p><img src="figs/mahr_time_domain.png" alt="mahr_time_domain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Time domain measurements</p>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
Now we can estimate the stiffness with a linear fit.
|
|
</p>
|
|
|
|
<p>
|
|
This is very close to the 0.04 [N/mm] written in the <a href="doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf">Millimar 1318 probe datasheet</a>.
|
|
</p>
|
|
|
|
<p>
|
|
And compare the linear fit with the raw measurement data (Figure <a href="#org91cec56">20</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org91cec56" class="figure">
|
|
<p><img src="figs/mahr_stiffness_f_d_plot.png" alt="mahr_stiffness_f_d_plot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Measured displacement as a function of the measured force. Raw data and linear fit</p>
|
|
</div>
|
|
|
|
<div class="summary" id="orgda156e0">
|
|
<p>
|
|
The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org412d759" class="outline-3">
|
|
<h3 id="org412d759"><span class="section-number-3">5.3</span> Force Sensor Calibration</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
<div class="note" id="org55845f0">
|
|
<p>
|
|
<b>Load Cells</b>:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a></li>
|
|
<li><a href="doc/FRE_DS_XFL212R_FR_A3.pdf">XFL212R</a></li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
There are both specified to have \(\pm 1 \%\) of non-linearity over the full range.
|
|
</p>
|
|
|
|
<p>
|
|
The XFL212R has a spherical interface while the FC2231 has a flat surface.
|
|
Therefore, we should have a nice point contact when using the two force sensors as shown in Figure <a href="#orgb68dc52">21</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgb68dc52" class="figure">
|
|
<p><img src="figs/IMG_20210309_145333.jpg" alt="IMG_20210309_145333.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Zoom on the two force sensors in contact</p>
|
|
</div>
|
|
|
|
<p>
|
|
The two force sensors are therefore measuring the exact same force, and we can compare the two measurements.
|
|
</p>
|
|
<p>
|
|
Let’s load the measured force of both sensors.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load measurement data</span></span>
|
|
load(<span class="org-string">'calibration_force_sensor.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'F'</span>, <span class="org-string">'Fc'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We remove any offset such that they are both measuring no force when not in contact.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Remove offset</span></span>
|
|
F = F <span class="org-type">-</span> mean(F( t <span class="org-type">></span> 0.5 <span class="org-type">&</span> t <span class="org-type"><</span> 1.0));
|
|
Fc = Fc <span class="org-type">-</span> mean(Fc(t <span class="org-type">></span> 0.5 <span class="org-type">&</span> t <span class="org-type"><</span> 1.0));
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org2198cb3" class="figure">
|
|
<p><img src="figs/force_calibration_time.png" alt="force_calibration_time.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>Measured force using both sensors as a function of time</p>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s select only the first part from the moment they are in contact until the maximum force is reached.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Only get the first part until maximum force</span></span>
|
|
F = F( t <span class="org-type">></span> 1.55 <span class="org-type">&</span> t <span class="org-type"><</span> 4.65);
|
|
Fc = Fc(t <span class="org-type">></span> 1.55 <span class="org-type">&</span> t <span class="org-type"><</span> 4.65);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, let’s make a linear fit between the two measured forces.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Make a line fit</span></span>
|
|
fit_F = polyfit(Fc, F, 1);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The two forces are plotted against each other as well as the linear fit in Figure <a href="#orgbd6e825">23</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgbd6e825" class="figure">
|
|
<p><img src="figs/calibrated_force_dit.png" alt="calibrated_force_dit.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 23: </span>Measured two forces and linear fit</p>
|
|
</div>
|
|
|
|
<p>
|
|
The measurement error between the two sensors is shown in Figure <a href="#org41621c6">24</a>.
|
|
It is below 0.1N for the full measurement range.
|
|
</p>
|
|
|
|
|
|
<div id="org41621c6" class="figure">
|
|
<p><img src="figs/force_meas_error.png" alt="force_meas_error.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 24: </span>Error in Newtons</p>
|
|
</div>
|
|
|
|
<p>
|
|
The same error is shown in percentage in Figure <a href="#org6fc7087">25</a>.
|
|
The error is less than 1% when the measured force is above 5N.
|
|
</p>
|
|
|
|
|
|
<div id="org6fc7087" class="figure">
|
|
<p><img src="figs/force_meas_error_percentage.png" alt="force_meas_error_percentage.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 25: </span>Error in percentage</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org709b570" class="outline-3">
|
|
<h3 id="org709b570"><span class="section-number-3">5.4</span> Force Sensor Noise</h3>
|
|
<div class="outline-text-3" id="text-5-4">
|
|
<p>
|
|
The objective of this measurement is to estimate the noise of the force sensor <a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a>.
|
|
To do so, we don’t apply any force to the sensor, and we measure its output for 100s.
|
|
</p>
|
|
<p>
|
|
Let’s load the measurement data.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load measurement data</span></span>
|
|
load(<span class="org-string">'force_sensor_noise_meas.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'F'</span>);
|
|
Ts = t(2) <span class="org-type">-</span> t(1);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The measured force is shown in Figure <a href="#orgb4486a9">26</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgb4486a9" class="figure">
|
|
<p><img src="figs/force_noise_time.png" alt="force_noise_time.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 26: </span>Measured force</p>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s now compute the Amplitude Spectral Density of the measured force.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Compute Spectral Density of Measured Force</span></span>
|
|
<span class="org-comment">% Hanning window</span>
|
|
win = hanning(ceil(1<span class="org-type">/</span>Ts));
|
|
|
|
<span class="org-comment">% Power Spectral Density</span>
|
|
[pxx, f] = pwelch(F, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The results is shown in Figure <a href="#orgd71adad">27</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgd71adad" class="figure">
|
|
<p><img src="figs/force_noise_asd.png" alt="force_noise_asd.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 27: </span>Amplitude Spectral Density of the meaured force</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org9cde23b" class="outline-3">
|
|
<h3 id="org9cde23b"><span class="section-number-3">5.5</span> Force Sensor Stiffness</h3>
|
|
<div class="outline-text-3" id="text-5-5">
|
|
<p>
|
|
The objective of this measurement is to estimate the stiffness of the force sensor <a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a>.
|
|
</p>
|
|
|
|
<p>
|
|
To do so, a very stiff element is fixed in front of the force sensor as shown in Figure <a href="#orge139ed1">28</a>.
|
|
</p>
|
|
|
|
<p>
|
|
Then, we apply a force on the stiff element through the force sensor.
|
|
We measure the deflection of the force sensor using an encoder.
|
|
</p>
|
|
|
|
<p>
|
|
Then, having the force and the deflection, we should be able to estimate the stiffness of the force sensor supposing the stiffness of the other elements are much larger.
|
|
</p>
|
|
|
|
|
|
<div id="orge139ed1" class="figure">
|
|
<p><img src="figs/IMG_20210309_145242.jpg" alt="IMG_20210309_145242.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 28: </span>Bench used to measured the stiffness of the force sensor</p>
|
|
</div>
|
|
|
|
<p>
|
|
From the documentation, the deflection of the sensor at the maximum load (50N) is 0.05mm, the stiffness is therefore foreseen to be around \(1\,N/\mu m\).
|
|
</p>
|
|
<p>
|
|
Let’s load the measured force as well as the measured displacement.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load measurement data</span></span>
|
|
load(<span class="org-string">'force_sensor_stiff_meas.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'F'</span>, <span class="org-string">'d'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Select important part of data</span></span>
|
|
F = F( t <span class="org-type">></span> 1.55 <span class="org-type">&</span> t <span class="org-type"><</span> 4.65);
|
|
d = d( t <span class="org-type">></span> 1.55 <span class="org-type">&</span> t <span class="org-type"><</span> 4.65);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Linear fit</span></span>
|
|
fit_k = polyfit(F, d, 1);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Force Sensor Stiffness</span></span>
|
|
fit_k(1)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfddad67" class="outline-2">
|
|
<h2 id="orgfddad67"><span class="section-number-2">6</span> Bending Stiffness Measurement</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="orgcf7ec01"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org366c038" class="outline-3">
|
|
<h3 id="org366c038"><span class="section-number-3">6.1</span> Introduction</h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
<p>
|
|
A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure <a href="#org7f8bbc5">29</a>.
|
|
A closer view on flexible joint is shown in Figure <a href="#orgeee40da">30</a> and a zoom on the force sensor tip is shown in Figure <a href="#org781654e">31</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org7f8bbc5" class="figure">
|
|
<p><img src="figs/picture_bending_x_meas_side_overview.jpg" alt="picture_bending_x_meas_side_overview.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 29: </span>Side view of the flexible joint stiffness bench. X-Bending stiffness is measured.</p>
|
|
</div>
|
|
|
|
|
|
<div id="orgeee40da" class="figure">
|
|
<p><img src="figs/picture_bending_x_meas_side_close.jpg" alt="picture_bending_x_meas_side_close.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 30: </span>Zoom on the flexible joint - Side view</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org781654e" class="figure">
|
|
<p><img src="figs/picture_bending_x_meas_side_zoom.jpg" alt="picture_bending_x_meas_side_zoom.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 31: </span>Zoom on the tip of the force sensor</p>
|
|
</div>
|
|
|
|
<p>
|
|
The same bench used to measure the Y-bending stiffness of the flexible joint is shown in Figure <a href="#org12731d8">32</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org12731d8" class="figure">
|
|
<p><img src="figs/picture_bending_y_meas_side_close.jpg" alt="picture_bending_y_meas_side_close.jpg" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 32: </span>Stiffness measurement bench - Y-d bending measurement</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2174e74" class="outline-3">
|
|
<h3 id="org2174e74"><span class="section-number-3">6.2</span> Analysis of one measurement</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
<p>
|
|
In this section is shown how the data are analysis in order to measured:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>the bending stiffness</li>
|
|
<li>the bending stroke</li>
|
|
<li>the stiffness once the mechanical stops are in contact</li>
|
|
</ul>
|
|
|
|
|
|
<p>
|
|
The height from the flexible joint’s center and the point of application force \(h\) is defined below:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">h = 25e<span class="org-type">-</span>3; <span class="org-comment">% [m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Load Data</span></span>
|
|
load(<span class="org-string">'meas_stiff_flex_1_x.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'F'</span>, <span class="org-string">'d'</span>);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Zero the force</span></span>
|
|
F = F <span class="org-type">-</span> mean(F(t <span class="org-type">></span> 0.1 <span class="org-type">&</span> t <span class="org-type"><</span> 0.3));
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Start measurement at t = 0.2 s</span></span>
|
|
d = d(t <span class="org-type">></span> 0.2);
|
|
F = F(t <span class="org-type">></span> 0.2);
|
|
t = t(t <span class="org-type">></span> 0.2); t = t <span class="org-type">-</span> t(1);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained time domain measurements are shown in Figure <a href="#org2e16681">33</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org2e16681" class="figure">
|
|
<p><img src="figs/flex_joint_meas_example_time_domain.png" alt="flex_joint_meas_example_time_domain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 33: </span>Typical time domain measurements</p>
|
|
</div>
|
|
|
|
<p>
|
|
The displacement as a function of the force is then shown in Figure <a href="#orgb68f8ac">34</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgb68f8ac" class="figure">
|
|
<p><img src="figs/flex_joint_meas_example_F_d.png" alt="flex_joint_meas_example_F_d.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 34: </span>Typical measurement of the diplacement as a function of the applied force</p>
|
|
</div>
|
|
|
|
<p>
|
|
The bending stiffness can be estimated by computing the slope of the curve in Figure <a href="#orgb68f8ac">34</a>.
|
|
The bending stroke and the stiffness when touching the mechanical stop can also be estimated from the same figure.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Determine the linear region and region when touching the mechanical stop</span></span>
|
|
<span class="org-comment">% Find when the force sensor touches the flexible joint</span>
|
|
i_l_start = find(F <span class="org-type">></span> 0.3, 1, <span class="org-string">'first'</span>);
|
|
<span class="org-comment">% Reset the measured diplacement at that point</span>
|
|
d = d <span class="org-type">-</span> d(i_l_start);
|
|
<span class="org-comment">% Find then the maximum force is applied</span>
|
|
[<span class="org-type">~</span>, i_s_stop] = max(F);
|
|
<span class="org-comment">% Linear region stops ~ when 90% of the stroke is reached</span>
|
|
i_l_stop = find(d <span class="org-type">></span> 0.9<span class="org-type">*</span>d(i_s_stop), 1, <span class="org-string">'first'</span>);
|
|
<span class="org-comment">% "Stop" region start ~1N before maximum force is applied</span>
|
|
i_s_start = find(F <span class="org-type">></span> max(F)<span class="org-type">-</span>1, 1, <span class="org-string">'first'</span>);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Define variables for the two regions</span></span>
|
|
F_l = F(i_l_start<span class="org-type">:</span>i_l_stop);
|
|
d_l = d(i_l_start<span class="org-type">:</span>i_l_stop);
|
|
|
|
F_s = F(i_s_start<span class="org-type">:</span>i_s_stop);
|
|
d_s = d(i_s_start<span class="org-type">:</span>i_s_stop);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Fit the best straight line for the two regions</span></span>
|
|
fit_l = polyfit(F_l, d_l, 1);
|
|
fit_s = polyfit(F_s, d_s, 1);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Reset displacement based on fit</span></span>
|
|
d = d <span class="org-type">-</span> fit_l(2);
|
|
fit_s<span class="org-type">(2) </span>= fit_s(2) <span class="org-type">-</span> fit_l(2);
|
|
fit_l<span class="org-type">(2) </span>= 0;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure <a href="#orgafae209">35</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgafae209" class="figure">
|
|
<p><img src="figs/flex_joint_meas_example_F_d_lin_fit.png" alt="flex_joint_meas_example_F_d_lin_fit.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 35: </span>Typical measurement of the diplacement as a function of the applied force with estimated linear fits</p>
|
|
</div>
|
|
|
|
<p>
|
|
Then, the bending stroke is estimated as crossing point between the two fitted lines:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">d_max = fit_l(1)<span class="org-type">*</span>fit_s(2)<span class="org-type">/</span>(fit_l(1) <span class="org-type">-</span> fit_s(1));
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained characteristics are summarized in Table <a href="#orgfece359">5</a>.
|
|
</p>
|
|
|
|
<table id="orgfece359" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 5:</span> Estimated characteristics of the flexible joint number 1 for the X-direction</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Bending Stiffness [Nm/rad]</td>
|
|
<td class="org-right">5.5</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Bending Stiffness @ stop [Nm/rad]</td>
|
|
<td class="org-right">173.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Bending Stroke [mrad]</td>
|
|
<td class="org-right">18.9</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org134bc98" class="outline-3">
|
|
<h3 id="org134bc98"><span class="section-number-3">6.3</span> Bending stiffness and bending stroke of all the flexible joints</h3>
|
|
<div class="outline-text-3" id="text-6-3">
|
|
<p>
|
|
Now, let’s estimate the bending stiffness and stroke for all the flexible joints.
|
|
</p>
|
|
|
|
<p>
|
|
The results are summarized in Table <a href="#orge69a370">6</a> for the X direction and in Table <a href="#orge08146c">7</a> for the Y direction.
|
|
</p>
|
|
|
|
<table id="orge69a370" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 6:</span> Measured characteristics of the flexible joints in the X direction</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-right"> </th>
|
|
<th scope="col" class="org-right">\(R_{R_x}\) [Nm/rad]</th>
|
|
<th scope="col" class="org-right">\(k_{R_x,s}\) [Nm/rad]</th>
|
|
<th scope="col" class="org-right">\(R_{x,\text{max}}\) [mrad]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">5.5</td>
|
|
<td class="org-right">173.6</td>
|
|
<td class="org-right">18.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">6.1</td>
|
|
<td class="org-right">195.0</td>
|
|
<td class="org-right">17.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3</td>
|
|
<td class="org-right">6.1</td>
|
|
<td class="org-right">191.3</td>
|
|
<td class="org-right">17.7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">136.7</td>
|
|
<td class="org-right">18.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">88.9</td>
|
|
<td class="org-right">22.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">183.9</td>
|
|
<td class="org-right">18.7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">157.9</td>
|
|
<td class="org-right">17.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">8</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">166.1</td>
|
|
<td class="org-right">17.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">9</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">159.5</td>
|
|
<td class="org-right">18.2</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">10</td>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">143.6</td>
|
|
<td class="org-right">18.1</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">11</td>
|
|
<td class="org-right">5.0</td>
|
|
<td class="org-right">163.8</td>
|
|
<td class="org-right">17.7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">12</td>
|
|
<td class="org-right">6.1</td>
|
|
<td class="org-right">111.9</td>
|
|
<td class="org-right">17.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">13</td>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">142.0</td>
|
|
<td class="org-right">17.4</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">14</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">130.1</td>
|
|
<td class="org-right">17.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">15</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">170.7</td>
|
|
<td class="org-right">18.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">16</td>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">148.7</td>
|
|
<td class="org-right">17.5</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
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<table id="orge08146c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 7:</span> Measured characteristics of the flexible joints in the Y direction</caption>
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<colgroup>
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-right"> </th>
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<th scope="col" class="org-right">\(R_{R_y}\) [Nm/rad]</th>
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<th scope="col" class="org-right">\(k_{R_y,s}\) [Nm/rad]</th>
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<th scope="col" class="org-right">\(R_{y,\text{may}}\) [mrad]</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-right">1</td>
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<td class="org-right">5.7</td>
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<td class="org-right">323.5</td>
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<td class="org-right">17.9</td>
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</tr>
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<tr>
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<td class="org-right">2</td>
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<td class="org-right">5.9</td>
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<td class="org-right">306.0</td>
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<td class="org-right">17.2</td>
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</tr>
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<tr>
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<td class="org-right">3</td>
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<td class="org-right">6.0</td>
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<td class="org-right">224.4</td>
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<td class="org-right">16.8</td>
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</tr>
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<tr>
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<td class="org-right">4</td>
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<td class="org-right">5.7</td>
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<td class="org-right">247.3</td>
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<td class="org-right">17.8</td>
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</tr>
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<tr>
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<td class="org-right">5</td>
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<td class="org-right">5.8</td>
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<td class="org-right">250.9</td>
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<td class="org-right">13.0</td>
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</tr>
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<tr>
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<td class="org-right">6</td>
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<td class="org-right">5.8</td>
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<td class="org-right">244.5</td>
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<td class="org-right">17.8</td>
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</tr>
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<tr>
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<td class="org-right">7</td>
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<td class="org-right">5.3</td>
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<td class="org-right">214.8</td>
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<td class="org-right">18.1</td>
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</tr>
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<tr>
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<td class="org-right">8</td>
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<td class="org-right">5.8</td>
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<td class="org-right">217.2</td>
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<td class="org-right">17.6</td>
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</tr>
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<tr>
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<td class="org-right">9</td>
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<td class="org-right">5.7</td>
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<td class="org-right">225.0</td>
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<td class="org-right">17.6</td>
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</tr>
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<tr>
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<td class="org-right">10</td>
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<td class="org-right">6.0</td>
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<td class="org-right">254.7</td>
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<td class="org-right">17.3</td>
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</tr>
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<tr>
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<td class="org-right">11</td>
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<td class="org-right">4.9</td>
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<td class="org-right">261.1</td>
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<td class="org-right">18.4</td>
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</tr>
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<tr>
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<td class="org-right">12</td>
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<td class="org-right">5.9</td>
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<td class="org-right">161.5</td>
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<td class="org-right">16.7</td>
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</tr>
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<tr>
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<td class="org-right">13</td>
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<td class="org-right">6.1</td>
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<td class="org-right">227.6</td>
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<td class="org-right">16.8</td>
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</tr>
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<tr>
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<td class="org-right">14</td>
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<td class="org-right">5.9</td>
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<td class="org-right">221.3</td>
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<td class="org-right">17.8</td>
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</tr>
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<tr>
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<td class="org-right">15</td>
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<td class="org-right">5.4</td>
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<td class="org-right">241.5</td>
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<td class="org-right">17.8</td>
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</tr>
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<tr>
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<td class="org-right">16</td>
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<td class="org-right">5.3</td>
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<td class="org-right">291.1</td>
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<td class="org-right">17.7</td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<div id="outline-container-orgbb78147" class="outline-3">
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<h3 id="orgbb78147"><span class="section-number-3">6.4</span> Analysis</h3>
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<div class="outline-text-3" id="text-6-4">
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<p>
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The dispersion of the measured bending stiffness is shown in Figure <a href="#org3a61253">36</a> and of the bending stroke in Figure <a href="#org7429f5c">37</a>.
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</p>
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<div id="org3a61253" class="figure">
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<p><img src="figs/bending_stiffness_histogram.png" alt="bending_stiffness_histogram.png" />
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</p>
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<p><span class="figure-number">Figure 36: </span>Histogram of the measured bending stiffness</p>
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</div>
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<div id="org7429f5c" class="figure">
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<p><img src="figs/bending_stroke_histogram.png" alt="bending_stroke_histogram.png" />
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</p>
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<p><span class="figure-number">Figure 37: </span>Histogram of the measured bending stroke</p>
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</div>
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<p>
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The relation between the measured beam thickness and the measured bending stiffness is shown in Figure <a href="#org94fc4ac">38</a>.
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</p>
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<div id="org94fc4ac" class="figure">
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<p><img src="figs/flex_thickness_vs_bending_stiff.png" alt="flex_thickness_vs_bending_stiff.png" />
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</p>
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<p><span class="figure-number">Figure 38: </span>Measured bending stiffness as a function of the estimated flexible beam thickness</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org1c10279" class="outline-3">
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<h3 id="org1c10279"><span class="section-number-3">6.5</span> Conclusion</h3>
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<div class="outline-text-3" id="text-6-5">
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<div class="important" id="org98be774">
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<p>
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The measured bending stiffness and bending stroke of the flexible joints are very close to the estimated one using a Finite Element Model.
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</p>
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<p>
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The characteristics of all the flexible joints are also quite close to each other.
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This should allow us to model them with unique parameters.
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</p>
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</div>
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</div>
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</div>
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</div>
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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: 2021-03-10 mer. 11:41</p>
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</div>
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</body>
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</html>
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