1673 lines
51 KiB
HTML
1673 lines
51 KiB
HTML
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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="#org8806767">1. Flexible Joints</a></li>
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<li><a href="#org3c5ad35">2. Dimensional Measurements</a>
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<ul>
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<li><a href="#org614274e">2.1. Measurement Bench</a></li>
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<li><a href="#orgf1efb87">2.2. Measurement Results</a></li>
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</ul>
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</li>
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<li><a href="#org2e0bb14">3. Measurement Test Bench - Bending Stiffness</a>
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<ul>
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<li><a href="#orgc6ff781">3.1. Flexible joint Geometry</a></li>
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<li><a href="#org8a54a82">3.2. Required external applied force</a></li>
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<li><a href="#org59014dc">3.3. Required actuator stroke and sensors range</a></li>
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<li><a href="#org4e83202">3.4. Test Bench</a></li>
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</ul>
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</li>
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<li><a href="#org9ebdf1e">4. Error budget</a>
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<ul>
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<li><a href="#orgd997606">4.1. Finite Element Model</a></li>
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<li><a href="#org52173fb">4.2. Setup</a></li>
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<li><a href="#org1cf25e3">4.3. Effect of Bending</a></li>
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<li><a href="#org4d23125">4.4. Computation of the bending stiffness</a></li>
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<li><a href="#org5080a7a">4.5. Estimation error due to force and displacement sensors accuracy</a></li>
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<li><a href="#orgc30a0a9">4.6. Estimation error due to Shear</a></li>
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<li><a href="#org36df1ef">4.7. Estimation error due to force sensor compression</a></li>
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<li><a href="#orge1e499b">4.8. Estimation error due to height estimation error</a></li>
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<li><a href="#orgbd1e7ce">4.9. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgde0870c">5. First Measurements</a>
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<ul>
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<li><a href="#org240b2a8">5.1. Agreement between the probe and the encoder</a></li>
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<li><a href="#orgaf2ffc2">5.2. Measurement of the Millimar 1318 probe stiffness</a></li>
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</ul>
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</li>
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<li><a href="#orge80a8c7">6. Bending Stiffness Measurement</a>
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<ul>
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<li><a href="#org72961af">6.1. Introduction</a></li>
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<li><a href="#org1b652ba">6.2. Analysis of one measurement</a></li>
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<li><a href="#org40e8841">6.3. Bending stiffness and bending stroke of all the flexible joints</a></li>
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<li><a href="#orga8e6591">6.4. Analysis</a></li>
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<li><a href="#org2a5f73f">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="#orgad7a6ac">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="#org9f44f20">2</a>: each flexible joint is measured using a profile projector</li>
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<li>Section <a href="#org6525f52">3</a>: the stiffness measurement bench is presented</li>
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<li>Section <a href="#org86b41ab">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="#org1c1b934">5</a>: first measurements are performed</li>
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<li>Section <a href="#org6a94bb1">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-org8806767" class="outline-2">
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<h2 id="org8806767"><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="orgad7a6ac"></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="#orgfbc0e5f">1</a>).
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</p>
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<div id="orgfbc0e5f" 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="#org97363c2">1</a>.
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</p>
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<table id="org97363c2" 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="#orgac842e5">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="#org97363c2">1</a>.
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</p>
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<div id="orgac842e5" 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="#orgbe13c96">3</a>.
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</p>
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<div id="orgbe13c96" 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-org3c5ad35" class="outline-2">
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<h2 id="org3c5ad35"><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="org9f44f20"></a>
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</p>
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</div>
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<div id="outline-container-org614274e" class="outline-3">
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<h3 id="org614274e"><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="#orga554165">4</a>.
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</p>
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<div id="orga554165" 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="#org9d90b87">5</a>.
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</p>
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<div id="org9d90b87" 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="#orgce58b21">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="orgce58b21" 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-orgf1efb87" class="outline-3">
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<h3 id="orgf1efb87"><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="#org0d281b0">2</a>.
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</p>
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<table id="org0d281b0" 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" />
|
|
|
|
<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="#orgf8b68bc">7</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgf8b68bc" 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-org2e0bb14" class="outline-2">
|
|
<h2 id="org2e0bb14"><span class="section-number-2">3</span> Measurement Test Bench - Bending Stiffness</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org6525f52"></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="#org34009d6">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="org34009d6" 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-orgc6ff781" class="outline-3">
|
|
<h3 id="orgc6ff781"><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="#org2ffa562">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="org2ffa562" 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-org8a54a82" class="outline-3">
|
|
<h3 id="org8a54a82"><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-org59014dc" class="outline-3">
|
|
<h3 id="org59014dc"><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-org4e83202" class="outline-3">
|
|
<h3 id="org4e83202"><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="#orgb823e3b">10</a>.
|
|
</p>
|
|
|
|
<div class="note" id="orge86a86e">
|
|
<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="orgb823e3b" 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="#org0e9f540">11</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org0e9f540" 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-org9ebdf1e" class="outline-2">
|
|
<h2 id="org9ebdf1e"><span class="section-number-2">4</span> Error budget</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org86b41ab"></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-orgd997606" class="outline-3">
|
|
<h3 id="orgd997606"><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="#org501f635">3</a> and <a href="#org828c921">4</a>.
|
|
</p>
|
|
|
|
<table id="org501f635" 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="org828c921" 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-org52173fb" class="outline-3">
|
|
<h3 id="org52173fb"><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="#orgebb925e">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="orgebb925e" 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-org1cf25e3" class="outline-3">
|
|
<h3 id="org1cf25e3"><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-org4d23125" class="outline-3">
|
|
<h3 id="org4d23125"><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-org5080a7a" class="outline-3">
|
|
<h3 id="org5080a7a"><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-orgc30a0a9" class="outline-3">
|
|
<h3 id="orgc30a0a9"><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-org36df1ef" class="outline-3">
|
|
<h3 id="org36df1ef"><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-orge1e499b" class="outline-3">
|
|
<h3 id="orge1e499b"><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-orgbd1e7ce" class="outline-3">
|
|
<h3 id="orgbd1e7ce"><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-orgde0870c" class="outline-2">
|
|
<h2 id="orgde0870c"><span class="section-number-2">5</span> First Measurements</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org1c1b934"></a>
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Section <a href="#org747ca5f">5.1</a>:</li>
|
|
<li>Section <a href="#org3cf8ecc">5.2</a>:</li>
|
|
</ul>
|
|
</div>
|
|
<div id="outline-container-org240b2a8" class="outline-3">
|
|
<h3 id="org240b2a8"><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="org747ca5f"></a>
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Load Cells</b>: <a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a> (and <a href="doc/FRE_DS_XFL212R_FR_A3.pdf">XFL212R</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>
|
|
<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="#org2feaa4d">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org2feaa4d" 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="#org711cb25">14</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org711cb25" 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="#org4a8e321">15</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org4a8e321" 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="#orgd70cc1a">16</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgd70cc1a" 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="org50df7ec">
|
|
<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-orgaf2ffc2" class="outline-3">
|
|
<h3 id="orgaf2ffc2"><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="org3cf8ecc"></a>
|
|
</p>
|
|
<div class="note" id="orga33d652">
|
|
<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="org5c8165d" 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="org49f8121" 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="#org863b08a">19</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org863b08a" 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="#org0b938db">20</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org0b938db" 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="org9bef6d3">
|
|
<p>
|
|
The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge80a8c7" class="outline-2">
|
|
<h2 id="orge80a8c7"><span class="section-number-2">6</span> Bending Stiffness Measurement</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="org6a94bb1"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org72961af" class="outline-3">
|
|
<h3 id="org72961af"><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="#org8b09393">21</a>.
|
|
A closer view on flexible joint is shown in Figure <a href="#org6938e6b">22</a> and a zoom on the force sensor tip is shown in Figure <a href="#orgf7e7e7d">23</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org8b09393" 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 21: </span>Side view of the flexible joint stiffness bench. X-Bending stiffness is measured.</p>
|
|
</div>
|
|
|
|
|
|
<div id="org6938e6b" 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 22: </span>Zoom on the flexible joint - Side view</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orgf7e7e7d" 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 23: </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="#org59f600b">24</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org59f600b" 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 24: </span>Stiffness measurement bench - Y-d bending measurement</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1b652ba" class="outline-3">
|
|
<h3 id="org1b652ba"><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="#org06b65db">25</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org06b65db" 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 25: </span>Typical time domain measurements</p>
|
|
</div>
|
|
|
|
<p>
|
|
The displacement as a function of the force is then shown in Figure <a href="#org6b4a964">26</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org6b4a964" 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 26: </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="#org6b4a964">26</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="#org3fa2b9f">27</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org3fa2b9f" 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 27: </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="#orga883164">5</a>.
|
|
</p>
|
|
|
|
<table id="orga883164" 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-org40e8841" class="outline-3">
|
|
<h3 id="org40e8841"><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="#org2399169">6</a> for the X direction and in Table <a href="#org250e0c3">7</a> for the Y direction.
|
|
</p>
|
|
|
|
<table id="org2399169" 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"><b>Bending Stiff X [Nm/rad]</b></th>
|
|
<th scope="col" class="org-right"><b>Stop Stiff X [Nm/rad]</b></th>
|
|
<th scope="col" class="org-right"><b>Bending Stroke X [mrad]</b></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>
|
|
|
|
<table id="org250e0c3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 7:</span> Measured characteristics of the flexible joints in the Y 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"><b>Bending Stiff Y [Nm/rad]</b></th>
|
|
<th scope="col" class="org-right"><b>Stop Stiff Y [Nm/rad]</b></th>
|
|
<th scope="col" class="org-right"><b>Bending Stroke Y [mrad]</b></th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">323.5</td>
|
|
<td class="org-right">17.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">5.9</td>
|
|
<td class="org-right">306.0</td>
|
|
<td class="org-right">17.2</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3</td>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">224.4</td>
|
|
<td class="org-right">16.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">247.3</td>
|
|
<td class="org-right">17.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">250.9</td>
|
|
<td class="org-right">13.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">244.5</td>
|
|
<td class="org-right">17.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7</td>
|
|
<td class="org-right">5.3</td>
|
|
<td class="org-right">214.8</td>
|
|
<td class="org-right">18.1</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">8</td>
|
|
<td class="org-right">5.8</td>
|
|
<td class="org-right">217.2</td>
|
|
<td class="org-right">17.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">9</td>
|
|
<td class="org-right">5.7</td>
|
|
<td class="org-right">225.0</td>
|
|
<td class="org-right">17.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">10</td>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">254.7</td>
|
|
<td class="org-right">17.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">11</td>
|
|
<td class="org-right">4.9</td>
|
|
<td class="org-right">261.1</td>
|
|
<td class="org-right">18.4</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">12</td>
|
|
<td class="org-right">5.9</td>
|
|
<td class="org-right">161.5</td>
|
|
<td class="org-right">16.7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">13</td>
|
|
<td class="org-right">6.1</td>
|
|
<td class="org-right">227.6</td>
|
|
<td class="org-right">16.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">14</td>
|
|
<td class="org-right">5.9</td>
|
|
<td class="org-right">221.3</td>
|
|
<td class="org-right">17.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">15</td>
|
|
<td class="org-right">5.4</td>
|
|
<td class="org-right">241.5</td>
|
|
<td class="org-right">17.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">16</td>
|
|
<td class="org-right">5.3</td>
|
|
<td class="org-right">291.1</td>
|
|
<td class="org-right">17.7</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga8e6591" class="outline-3">
|
|
<h3 id="orga8e6591"><span class="section-number-3">6.4</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-6-4">
|
|
<p>
|
|
The dispersion of the measured bending stiffness is shown in Figure <a href="#org518e311">28</a> and of the bending stroke in Figure <a href="#org20c82ae">29</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org518e311" class="figure">
|
|
<p><img src="figs/bending_stiffness_histogram.png" alt="bending_stiffness_histogram.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 28: </span>Histogram of the measured bending stiffness</p>
|
|
</div>
|
|
|
|
|
|
<div id="org20c82ae" class="figure">
|
|
<p><img src="figs/bending_stroke_histogram.png" alt="bending_stroke_histogram.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 29: </span>Histogram of the measured bending stroke</p>
|
|
</div>
|
|
|
|
<p>
|
|
The relation between the measured beam thickness and the measured bending stiffness is shown in Figure <a href="#org2afa71c">30</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org2afa71c" class="figure">
|
|
<p><img src="figs/flex_thickness_vs_bending_stiff.png" alt="flex_thickness_vs_bending_stiff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 30: </span>Measured bending stiffness as a function of the estimated flexible beam thickness</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2a5f73f" class="outline-3">
|
|
<h3 id="org2a5f73f"><span class="section-number-3">6.5</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-6-5">
|
|
<div class="important" id="org3490684">
|
|
<p>
|
|
The measured bending stiffness and bending stroke of the flexible joints are very close to the estimated one using a Finite Element Model.
|
|
</p>
|
|
|
|
<p>
|
|
The characteristics of all the flexible joints are also quite close to each other.
|
|
This should allow us to model them with unique parameters.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2021-03-04 jeu. 17:20</p>
|
|
</div>
|
|
</body>
|
|
</html>
|