894 lines
29 KiB
HTML
894 lines
29 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="#org8b281d9">1. Flexible Joints</a></li>
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<li><a href="#org95f31e9">2. Measurement Test Bench - Bending Stiffness</a>
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<ul>
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<li><a href="#org49aa26d">2.1. Flexible joint Geometry</a></li>
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<li><a href="#org6176a07">2.2. Required external applied force</a></li>
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<li><a href="#orgd96e919">2.3. Required actuator stroke and sensors range</a></li>
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<li><a href="#org34c590e">2.4. Test Bench</a></li>
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</ul>
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</li>
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<li><a href="#org600c97d">3. Error budget</a>
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<ul>
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<li><a href="#orgc9ee1f1">3.1. Finite Element Model</a></li>
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<li><a href="#orga7f4716">3.2. Setup</a></li>
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<li><a href="#org82790b3">3.3. Effect of Bending</a></li>
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<li><a href="#org36529f3">3.4. Computation of the bending stiffness</a></li>
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<li><a href="#org8b414e4">3.5. Estimation error due to force and displacement sensors accuracy</a></li>
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<li><a href="#org1ab0479">3.6. Estimation error due to Shear</a></li>
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<li><a href="#org6765aeb">3.7. Estimation error due to force sensor compression</a></li>
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<li><a href="#orgdc33e2c">3.8. Estimation error due to height estimation error</a></li>
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<li><a href="#orgbff0a4b">3.9. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orge6a23d4">4. First Measurements</a>
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<ul>
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<li><a href="#orgf3173a7">4.1. Agreement between the probe and the encoder</a></li>
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<li><a href="#org47cc77a">4.2. 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="#org48987bc">5. Bending Stiffness Measurement</a></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="#org1f36cf1">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="#org487b515">2</a>: the measurement bench is presented</li>
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<li>Section <a href="#orgdab23b9">3</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="#org60b7659">4</a>: first measurements are performed</li>
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<li>Section <a href="#org34f6444">5</a>: the bending stiffness of the flexible joints are measured</li>
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</ul>
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<div id="outline-container-org8b281d9" class="outline-2">
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<h2 id="org8b281d9"><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="org1f36cf1"></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="#orgd244b13">1</a>).
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</p>
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<div id="orgd244b13" 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="#org42febf6">1</a>.
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</p>
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<table id="org42febf6" 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="#org904f6bf">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="#org42febf6">1</a>.
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</p>
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<div id="org904f6bf" 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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</div>
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</div>
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<div id="outline-container-org95f31e9" class="outline-2">
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<h2 id="org95f31e9"><span class="section-number-2">2</span> Measurement Test Bench - Bending Stiffness</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org487b515"></a>
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</p>
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<p>
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The most important characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
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</p>
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<p>
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To do so, we have to apply a torque \(T_x\) on the flexible joint and measure its angular deflection \(\theta_x\).
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The stiffness is then
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</p>
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\begin{equation}
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k_{R_x} = \frac{T_x}{\theta_x}
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\end{equation}
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<p>
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As it is quite difficult to apply a pure torque, a force will be applied instead.
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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.
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</p>
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<p>
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The working principle of the bench is schematically shown in Figure <a href="#org5e03b2d">3</a>.
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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.
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The induced rotation is measured with a displacement sensor \(d_x\).
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</p>
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<div id="org5e03b2d" class="figure">
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<p><img src="figs/test_bench_principle.png" alt="test_bench_principle.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Test Bench - working principle</p>
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</div>
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<p>
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This test-bench will be used to have a first approximation of the bending stiffnesss and stroke of the flexible joints.
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Another test-bench, better engineered will be used to measure the flexible joint’s characteristics with better accuracy.
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</p>
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</div>
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<div id="outline-container-org49aa26d" class="outline-3">
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<h3 id="org49aa26d"><span class="section-number-3">2.1</span> Flexible joint Geometry</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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The flexible joint used for the Nano-Hexapod is shown in Figure <a href="#org8f874a3">4</a>.
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Its bending stiffness is foreseen to be \(k_{R_y}\approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 25\,mrad\).
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</p>
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<div id="org8f874a3" class="figure">
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<p><img src="figs/flexible_joint_geometry.png" alt="flexible_joint_geometry.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Geometry of the flexible joint</p>
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</div>
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<p>
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The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 20\,mm\).
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</p>
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<p>
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Let’s define the parameters on Matlab.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">kRx = 5; <span class="org-comment">% Bending Stiffness [Nm/rad]</span>
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Rxmax = 25e<span class="org-type">-</span>3; <span class="org-comment">% Bending Stroke [rad]</span>
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h = 20e<span class="org-type">-</span>3; <span class="org-comment">% Height [m]</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org6176a07" class="outline-3">
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<h3 id="org6176a07"><span class="section-number-3">2.2</span> Required external applied force</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:
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</p>
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\begin{equation}
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\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
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\end{equation}
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<p>
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Therefore, the applied force to test the full range of the flexible joint is:
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</p>
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\begin{equation}
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F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
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\end{equation}
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<div class="org-src-container">
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<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>
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</pre>
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</div>
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<p>
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And we obtain:
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</p>
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\begin{equation} F_{x,max} = 6.2\, [N] \end{equation}
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<p>
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The measurement range of the force sensor should then be higher than \(6.2\,N\).
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</p>
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</div>
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</div>
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<div id="outline-container-orgd96e919" class="outline-3">
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<h3 id="orgd96e919"><span class="section-number-3">2.3</span> Required actuator stroke and sensors range</h3>
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<div class="outline-text-3" id="text-2-3">
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<p>
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The flexible joint is designed to allow a bending motion of \(\pm 25\,mrad\).
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The corresponding stroke at the location of the force sensor is:
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\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">dxmax = h<span class="org-type">*</span>tan(Rxmax);
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</pre>
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</div>
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\begin{equation} d_{max} = 0.5\, [mm] \end{equation}
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<p>
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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 \(0.5\,mm\).
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Similarly, the measurement range of the displacement sensor should also be higher than \(0.5\,mm\).
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</p>
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</div>
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</div>
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<div id="outline-container-org34c590e" class="outline-3">
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<h3 id="org34c590e"><span class="section-number-3">2.4</span> Test Bench</h3>
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<div class="outline-text-3" id="text-2-4">
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<p>
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A CAD view of the measurement bench is shown in Figure <a href="#org8ec9f08">5</a>.
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</p>
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<div class="note" id="org8dbefd9">
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<p>
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Here are the different elements used in this bench:
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</p>
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<ul class="org-ul">
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<li><b>Translation Stage</b>: <a href="doc/V-408-Datasheet.pdf">V-408</a></li>
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<li><b>Load Cells</b>: <a href="doc/A700000007147087.pdf">FC2231-0000-0010-L</a></li>
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<li><b>Encoder</b>: <a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">Renishaw Resolute 1nm</a></li>
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</ul>
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</div>
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<p>
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Both the measured force and displacement are acquired at the same time using a Speedgoat machine.
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</p>
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<div id="org8ec9f08" class="figure">
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<p><img src="figs/test_bench_flex_overview.png" alt="test_bench_flex_overview.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
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</div>
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<p>
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A side view of the bench with the important quantities are shown in Figure <a href="#orge104838">6</a>.
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</p>
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<div id="orge104838" class="figure">
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<p><img src="figs/test_bench_flex_side.png" alt="test_bench_flex_side.png" width="300px" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</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 id="outline-container-org600c97d" class="outline-2">
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<h2 id="org600c97d"><span class="section-number-2">3</span> Error budget</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="orgdab23b9"></a>
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</p>
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<p>
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Many things can impact the accuracy of the measured bending stiffness such as:
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</p>
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<ul class="org-ul">
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<li>Errors in the force and displacement measurement</li>
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<li>Shear effects</li>
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<li>Deflection of the Force sensor</li>
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<li>Errors in the geometry of the bench</li>
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</ul>
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<p>
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In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors.
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</p>
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</div>
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<div id="outline-container-orgc9ee1f1" class="outline-3">
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<h3 id="orgc9ee1f1"><span class="section-number-3">3.1</span> Finite Element Model</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables <a href="#org6032011">2</a> and <a href="#org949868f">3</a>.
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</p>
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<table id="org6032011" 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> Axial/Shear characteristics</caption>
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<colgroup>
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<col class="org-left" />
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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-left"> </th>
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<th scope="col" class="org-right">Stiffness [N/um]</th>
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<th scope="col" class="org-right">Max Force [N]</th>
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|
<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="org949868f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 3:</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-orga7f4716" class="outline-3">
|
|
<h3 id="orga7f4716"><span class="section-number-3">3.2</span> Setup</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The setup is schematically represented in Figure <a href="#org9f57086">7</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="org9f57086" 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 7: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org82790b3" class="outline-3">
|
|
<h3 id="org82790b3"><span class="section-number-3">3.3</span> Effect of Bending</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<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-org36529f3" class="outline-3">
|
|
<h3 id="org36529f3"><span class="section-number-3">3.4</span> Computation of the bending stiffness</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<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-org8b414e4" class="outline-3">
|
|
<h3 id="org8b414e4"><span class="section-number-3">3.5</span> Estimation error due to force and displacement sensors accuracy</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
<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-org1ab0479" class="outline-3">
|
|
<h3 id="org1ab0479"><span class="section-number-3">3.6</span> Estimation error due to Shear</h3>
|
|
<div class="outline-text-3" id="text-3-6">
|
|
<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-org6765aeb" class="outline-3">
|
|
<h3 id="org6765aeb"><span class="section-number-3">3.7</span> Estimation error due to force sensor compression</h3>
|
|
<div class="outline-text-3" id="text-3-7">
|
|
<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-orgdc33e2c" class="outline-3">
|
|
<h3 id="orgdc33e2c"><span class="section-number-3">3.8</span> Estimation error due to height estimation error</h3>
|
|
<div class="outline-text-3" id="text-3-8">
|
|
<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-orgbff0a4b" class="outline-3">
|
|
<h3 id="orgbff0a4b"><span class="section-number-3">3.9</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-3-9">
|
|
<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-orge6a23d4" class="outline-2">
|
|
<h2 id="orge6a23d4"><span class="section-number-2">4</span> First Measurements</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org60b7659"></a>
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Section <a href="#org950c08c">4.1</a>:</li>
|
|
<li>Section <a href="#org08bd7f1">4.2</a>:</li>
|
|
</ul>
|
|
</div>
|
|
<div id="outline-container-orgf3173a7" class="outline-3">
|
|
<h3 id="orgf3173a7"><span class="section-number-3">4.1</span> Agreement between the probe and the encoder</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
<a id="org950c08c"></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="#org582a172">8</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org582a172" class="figure">
|
|
<p><img src="figs/comp_encoder_probe_time.png" alt="comp_encoder_probe_time.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </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="#orgdf303a4">9</a>).
|
|
</p>
|
|
|
|
|
|
<div id="orgdf303a4" 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 9: </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="#org8a6d0a7">10</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org8a6d0a7" class="figure">
|
|
<p><img src="figs/comp_encoder_probe_mismatch.png" alt="comp_encoder_probe_mismatch.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </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="#org7323329">11</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org7323329" 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 11: </span>Measured displacement by the probe as a function of the measured displacement by the encoder</p>
|
|
</div>
|
|
|
|
<div class="important" id="orgd8436d3">
|
|
<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-org47cc77a" class="outline-3">
|
|
<h3 id="org47cc77a"><span class="section-number-3">4.2</span> Measurement of the Millimar 1318 probe stiffness</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
<a id="org08bd7f1"></a>
|
|
</p>
|
|
<div class="note" id="org2dc97c3">
|
|
<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="org5312017" 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 12: </span>Setup - Side View</p>
|
|
</div>
|
|
|
|
|
|
<div id="orge5425ad" 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 13: </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="#org411a5b9">14</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org411a5b9" class="figure">
|
|
<p><img src="figs/mahr_time_domain.png" alt="mahr_time_domain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Time domain measurements</p>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
Now we can estimate the stiffness with a linear fit.
|
|
</p>
|
|
|
|
<pre class="example">
|
|
Stiffness is 0.039 [N/mm]
|
|
</pre>
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<p>
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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>.
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</p>
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<p>
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And compare the linear fit with the raw measurement data (Figure <a href="#org784b37d">15</a>).
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</p>
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<div id="org784b37d" class="figure">
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<p><img src="figs/mahr_stiffness_f_d_plot.png" alt="mahr_stiffness_f_d_plot.png" />
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</p>
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<p><span class="figure-number">Figure 15: </span>Measured displacement as a function of the measured force. Raw data and linear fit</p>
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</div>
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<div class="summary" id="orgbc46442">
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<p>
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The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).
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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 id="outline-container-org48987bc" class="outline-2">
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<h2 id="org48987bc"><span class="section-number-2">5</span> Bending Stiffness Measurement</h2>
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<div class="outline-text-2" id="text-5">
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<p>
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<a id="org34f6444"></a>
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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 id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2021-02-18 jeu. 11:33</p>
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</div>
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</body>
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</html>
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