563 lines
19 KiB
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563 lines
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<title>Study of the Flexible Joints</title>
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<div id="org-div-home-and-up">
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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">Study of the Flexible Joints</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="#orge032d30">1. Bending and Torsional Stiffness</a>
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<ul>
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<li><a href="#org14d57c4">1.1. Initialization</a></li>
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<li><a href="#orgde60939">1.2. Realistic Bending Stiffness Values</a>
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<ul>
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<li><a href="#org5ed48b8">1.2.1. Direct Velocity Feedback</a></li>
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<li><a href="#orgddae25e">1.2.2. Primary Plant</a></li>
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<li><a href="#orgb8a9692">1.2.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org8ad3f34">1.3. Parametric Study</a>
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<ul>
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<li><a href="#org44ccdbe">1.3.1. Direct Velocity Feedback</a></li>
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<li><a href="#org5d9965b">1.3.2. Primary Control</a></li>
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<li><a href="#org0f9f990">1.3.3. Conclusion</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org81f1d95">2. Translation Stiffness</a>
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<ul>
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<li><a href="#org969d9e7">2.1. Realistic Translation Stiffness Values</a>
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<ul>
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<li><a href="#org8fdef7f">2.1.1. Initialization</a></li>
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<li><a href="#orgc087bb9">2.1.2. Direct Velocity Feedback</a></li>
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<li><a href="#org4069e58">2.1.3. Primary Plant</a></li>
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<li><a href="#org3d8a1a7">2.1.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org0275632">2.2. Parametric study</a>
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<ul>
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<li><a href="#orgdb214f9">2.2.1. Direct Velocity Feedback</a></li>
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<li><a href="#org53e5f08">2.2.2. Primary Control</a></li>
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</ul>
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</li>
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<li><a href="#org1ddd8bf">2.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orga32adf0">3. Conclusion</a></li>
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</ul>
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</div>
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</div>
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<p>
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In this document is studied the effect of the mechanical behavior of the flexible joints that are located the extremities of each nano-hexapod’s legs.
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</p>
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<p>
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Ideally, we want the x and y rotations to be free and all the translations to be blocked.
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However, this is never the case and be have to consider:
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</p>
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<ul class="org-ul">
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<li>Finite bending stiffnesses (Section <a href="#org3eb4121">1</a>)</li>
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<li>Axial stiffness in the direction of the legs (Section <a href="#org8f4d83b">2</a>)</li>
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</ul>
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<p>
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This may impose some limitations, also, the goal is to specify the required joints stiffnesses (summarized in Section <a href="#org6614f42">3</a>).
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</p>
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<div id="outline-container-orge032d30" class="outline-2">
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<h2 id="orge032d30"><span class="section-number-2">1</span> Bending and Torsional Stiffness</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org3eb4121"></a>
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</p>
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<p>
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In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.
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</p>
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</div>
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<div id="outline-container-org14d57c4" class="outline-3">
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<h3 id="org14d57c4"><span class="section-number-3">1.1</span> Initialization</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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Let’s initialize all the stages with default parameters.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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</pre>
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</div>
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<p>
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Let’s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSample('mass', 50, 'freq', 200*ones(6,1));
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
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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-orgde60939" class="outline-3">
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<h3 id="orgde60939"><span class="section-number-3">1.2</span> Realistic Bending Stiffness Values</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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Let’s compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:
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</p>
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<ul class="org-ul">
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<li>\(K_{\theta, \phi} = 15\,[Nm/rad]\) stiffness in flexion</li>
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<li>\(K_{\psi} = 20\,[Nm/rad]\) stiffness in torsion</li>
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</ul>
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<div class="org-src-container">
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<pre class="src src-matlab">Kf_M = 15*ones(6,1);
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Kf_F = 15*ones(6,1);
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Kt_M = 20*ones(6,1);
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Kt_F = 20*ones(6,1);
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</pre>
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</div>
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<p>
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The stiffness and damping of the nano-hexapod’s legs are:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">k_opt = 1e5; % [N/m]
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c_opt = 2e2; % [N/(m/s)]
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</pre>
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</div>
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<p>
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This corresponds to the optimal identified stiffness.
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</p>
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</div>
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<div id="outline-container-org5ed48b8" class="outline-4">
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<h4 id="org5ed48b8"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
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<div class="outline-text-4" id="text-1-2-1">
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<p>
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We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness.
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</p>
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<p>
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The obtained dynamics are shown in Figure <a href="#org656fd1c">1</a>.
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It is shown that the adding of stiffness for the flexible joints does increase a little bit the frequencies of the mass suspension modes. It stiffen the structure.
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</p>
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<div id="org656fd1c" class="figure">
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<p><img src="figs/flex_joint_rot_dvf.png" alt="flex_joint_rot_dvf.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint stiffness</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgddae25e" class="outline-4">
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<h4 id="orgddae25e"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
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<div class="outline-text-4" id="text-1-2-2">
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<p>
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Let’s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
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</p>
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<p>
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The dynamics is compare with and without the joint flexibility in Figure <a href="#org4322feb">2</a>.
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The plant dynamics is not found to be changing significantly.
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</p>
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<div id="org4322feb" class="figure">
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<p><img src="figs/flex_joints_rot_primary_plant_L.png" alt="flex_joints_rot_primary_plant_L.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints and with 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-orgb8a9692" class="outline-4">
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<h4 id="orgb8a9692"><span class="section-number-4">1.2.3</span> Conclusion</h4>
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<div class="outline-text-4" id="text-1-2-3">
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<div class="important">
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<p>
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Considering realistic flexible joint bending stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.
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</p>
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<p>
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It only increases a little bit the suspension modes of the sample on top of the nano-hexapod.
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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-org8ad3f34" class="outline-3">
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<h3 id="org8ad3f34"><span class="section-number-3">1.3</span> Parametric Study</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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We wish now to see what is the impact of the rotation stiffness of the flexible joints on the dynamics.
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This will help to determine the requirements on the joint’s stiffness.
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</p>
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<p>
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Let’s consider the following bending stiffness of the flexible joints:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Ks = [1, 5, 10, 50, 100]; % [Nm/rad]
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</pre>
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</div>
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<p>
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We also consider here a nano-hexapod with the identified optimal actuator stiffness.
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</p>
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</div>
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<div id="outline-container-org44ccdbe" class="outline-4">
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<h4 id="org44ccdbe"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
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<div class="outline-text-4" id="text-1-3-1">
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<p>
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The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#org8fbbf9d">3</a>.
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</p>
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<p>
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The corresponding Root Locus plot is shown in Figure <a href="#orgb9f3389">4</a>.
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</p>
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<p>
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It is shown that the bending stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback.
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</p>
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<div id="org8fbbf9d" class="figure">
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<p><img src="figs/flex_joints_rot_study_dvf.png" alt="flex_joints_rot_study_dvf.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered Rotation Stiffnesses</p>
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</div>
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<div id="orgb9f3389" class="figure">
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<p><img src="figs/flex_joints_rot_study_dvf_root_locus.png" alt="flex_joints_rot_study_dvf_root_locus.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Root Locus for all the considered Rotation Stiffnesses</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org5d9965b" class="outline-4">
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<h4 id="org5d9965b"><span class="section-number-4">1.3.2</span> Primary Control</h4>
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<div class="outline-text-4" id="text-1-3-2">
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<p>
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The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#orgb739560">5</a>.
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</p>
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<p>
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It is shown that the bending stiffness of the flexible joints have very little impact on the dynamics.
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</p>
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<div id="orgb739560" class="figure">
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<p><img src="figs/flex_joints_rot_study_primary_plant.png" alt="flex_joints_rot_study_primary_plant.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered bending stiffnesses</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org0f9f990" class="outline-4">
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<h4 id="org0f9f990"><span class="section-number-4">1.3.3</span> Conclusion</h4>
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<div class="outline-text-4" id="text-1-3-3">
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<div class="important">
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<p>
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The bending stiffness of the flexible joint does not significantly change the dynamics.
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</p>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org81f1d95" class="outline-2">
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<h2 id="org81f1d95"><span class="section-number-2">2</span> Translation 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="org8f4d83b"></a>
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</p>
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<p>
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Let’s know consider a flexibility in translation of the flexible joint, in the axis of the legs.
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</p>
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</div>
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<div id="outline-container-org969d9e7" class="outline-3">
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<h3 id="org969d9e7"><span class="section-number-3">2.1</span> Realistic Translation Stiffness Values</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We choose realistic values of the axial stiffness of the joints:
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\[ K_a = 60\,[N/\mu m] \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Kz_F = 60e6*ones(6,1); % [N/m]
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Kz_M = 60e6*ones(6,1); % [N/m]
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Cz_F = 1*ones(6,1); % [N/(m/s)]
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Cz_M = 1*ones(6,1); % [N/(m/s)]
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</pre>
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</div>
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</div>
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<div id="outline-container-org8fdef7f" class="outline-4">
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<h4 id="org8fdef7f"><span class="section-number-4">2.1.1</span> Initialization</h4>
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<div class="outline-text-4" id="text-2-1-1">
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<p>
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Let’s initialize all the stages with default parameters.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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</pre>
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</div>
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<p>
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Let’s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSample('mass', 50, 'freq', 200*ones(6,1));
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
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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-orgc087bb9" class="outline-4">
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<h4 id="orgc087bb9"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
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<div class="outline-text-4" id="text-2-1-2">
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<p>
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The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified.
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</p>
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<p>
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The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
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</p>
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<div id="org78dd87a" class="figure">
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<p><img src="figs/flex_joint_trans_dvf.png" alt="flex_joint_trans_dvf.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint axis stiffness</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org4069e58" class="outline-4">
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<h4 id="org4069e58"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
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<div class="outline-text-4" id="text-2-1-3">
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<div class="org-src-container">
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<pre class="src src-matlab">Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
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</pre>
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</div>
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<p>
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Let’s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
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</p>
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<p>
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The dynamics is compare with and without the joint flexibility in Figure <a href="#org9bd0791">7</a>.
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</p>
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<div id="org9bd0791" class="figure">
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<p><img src="figs/flex_joints_trans_primary_plant_L.png" alt="flex_joints_trans_primary_plant_L.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org3d8a1a7" class="outline-4">
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<h4 id="org3d8a1a7"><span class="section-number-4">2.1.4</span> Conclusion</h4>
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<div class="outline-text-4" id="text-2-1-4">
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<div class="important">
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<p>
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For the realistic value of the flexible joint axial stiffness, the dynamics is not much impact, and this should not be a problem for control.
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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-org0275632" class="outline-3">
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<h3 id="org0275632"><span class="section-number-3">2.2</span> Parametric study</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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We wish now to see what is the impact of the <b>axial</b> stiffness of the flexible joints on the dynamics.
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</p>
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<p>
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Let’s consider the following values for the axial stiffness:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Kzs = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; % [N/m]
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</pre>
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</div>
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<p>
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We also consider here a nano-hexapod with the identified optimal actuator stiffness (\(k = 10^5\,[N/m]\)).
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</p>
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</div>
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<div id="outline-container-orgdb214f9" class="outline-4">
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<h4 id="orgdb214f9"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
|
|
<div class="outline-text-4" id="text-2-2-1">
|
|
<p>
|
|
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#orgab9ab86">8</a>.
|
|
</p>
|
|
|
|
<p>
|
|
It is shown that the axial stiffness of the flexible joints does have a huge impact on the dynamics.
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|
</p>
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|
|
|
<p>
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|
If the axial stiffness of the flexible joints is \(K_a > 10^7\,[N/m]\) (here \(100\) times higher than the actuator stiffness), then the change of dynamics stays reasonably small.
|
|
</p>
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|
|
|
<p>
|
|
This is more clear by looking at the root locus (Figures <a href="#org9d43966">9</a> and <a href="#org987d98e">10</a>).
|
|
It can be seen that very little active damping can be achieve for axial stiffnesses below \(10^7\,[N/m]\).
|
|
</p>
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|
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|
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|
<div id="orgab9ab86" class="figure">
|
|
<p><img src="figs/flex_joints_trans_study_dvf.png" alt="flex_joints_trans_study_dvf.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered axis Stiffnesses</p>
|
|
</div>
|
|
|
|
|
|
<div id="org9d43966" class="figure">
|
|
<p><img src="figs/flex_joints_trans_study_dvf_root_locus.png" alt="flex_joints_trans_study_dvf_root_locus.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Root Locus for all the considered axial Stiffnesses</p>
|
|
</div>
|
|
|
|
|
|
<div id="org987d98e" class="figure">
|
|
<p><img src="figs/flex_joints_trans_study_root_locus_unzoom.png" alt="flex_joints_trans_study_root_locus_unzoom.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Root Locus (unzoom) for all the considered axial Stiffnesses</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org53e5f08" class="outline-4">
|
|
<h4 id="org53e5f08"><span class="section-number-4">2.2.2</span> Primary Control</h4>
|
|
<div class="outline-text-4" id="text-2-2-2">
|
|
<p>
|
|
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#org6070692">11</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org6070692" class="figure">
|
|
<p><img src="figs/flex_joints_trans_study_primary_plant.png" alt="flex_joints_trans_study_primary_plant.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered axial stiffnesses</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1ddd8bf" class="outline-3">
|
|
<h3 id="org1ddd8bf"><span class="section-number-3">2.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<div class="important">
|
|
<p>
|
|
The axial stiffness of the flexible joints should be maximized.
|
|
</p>
|
|
|
|
<p>
|
|
For the considered actuator stiffness \(k = 10^5\,[N/m]\), the axial stiffness of the flexible joints should ideally be above \(10^7\,[N/m]\).
|
|
</p>
|
|
|
|
<p>
|
|
This is a reasonable stiffness value for such joints.
|
|
</p>
|
|
|
|
<p>
|
|
We may interpolate the results and say that the axial joint stiffness should be 100 times higher than the actuator stiffness, but this should be confirmed with further analysis.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga32adf0" class="outline-2">
|
|
<h2 id="orga32adf0"><span class="section-number-2">3</span> Conclusion</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org6614f42"></a>
|
|
</p>
|
|
|
|
<div class="important">
|
|
<p>
|
|
For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Bending Stiffness: \(K_b < 50\,[Nm/rad]\)</li>
|
|
<li>Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)</li>
|
|
<li>Axial Stiffness: \(K_a > 10^7\,[N/m]\)</li>
|
|
</ul>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-05-05 mar. 11:26</p>
|
|
</div>
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</body>
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</html>
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