Update conclusion about flexible joints
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-05-05 mar. 11:26 -->
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<!-- 2020-05-05 mar. 11:50 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Study of the Flexible Joints</title>
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<meta name="generator" content="Org mode" />
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@ -37,43 +37,43 @@
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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="#org8fdef7f">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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<li><a href="#orgdb214f9">1.2.1. Direct Velocity Feedback</a></li>
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<li><a href="#org4069e58">1.2.2. Primary Plant</a></li>
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<li><a href="#orga32adf0">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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<li><a href="#org4adf147">1.3.1. Direct Velocity Feedback</a></li>
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<li><a href="#org53e5f08">1.3.2. Primary Control</a></li>
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<li><a href="#orgc45ccb0">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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<li><a href="#orgdaf7b6c">2. Axial 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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<li><a href="#orge82a7c2">2.1.1. Initialization</a></li>
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<li><a href="#org44f67b8">2.1.2. Direct Velocity Feedback</a></li>
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<li><a href="#orgd5fd59b">2.1.3. Primary Plant</a></li>
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<li><a href="#org552093a">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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<li><a href="#orge13b41c">2.2.1. Direct Velocity Feedback</a></li>
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<li><a href="#org15c2c08">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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<li><a href="#orgce1052e">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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<li><a href="#org865157e">3. Conclusion</a></li>
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</ul>
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</div>
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</div>
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@ -106,8 +106,8 @@ In this section, we wish to study the effect of the rotation flexibility of the
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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 id="outline-container-org8fdef7f" class="outline-3">
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<h3 id="org8fdef7f"><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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@ -168,8 +168,8 @@ 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 id="outline-container-orgdb214f9" class="outline-4">
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<h4 id="orgdb214f9"><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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@ -189,8 +189,8 @@ It is shown that the adding of stiffness for the flexible joints does increase a
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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 id="outline-container-org4069e58" class="outline-4">
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<h4 id="org4069e58"><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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@ -210,8 +210,8 @@ The plant dynamics is not found to be changing significantly.
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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 id="outline-container-orga32adf0" class="outline-4">
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<h4 id="orga32adf0"><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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@ -248,8 +248,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
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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 id="outline-container-org4adf147" class="outline-4">
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<h4 id="org4adf147"><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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@ -279,8 +279,8 @@ It is shown that the bending stiffness of the flexible joints does indeed change
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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 id="outline-container-org53e5f08" class="outline-4">
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<h4 id="org53e5f08"><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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@ -299,8 +299,8 @@ It is shown that the bending stiffness of the flexible joints have very little i
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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 id="outline-container-orgc45ccb0" class="outline-4">
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<h4 id="orgc45ccb0"><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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@ -313,8 +313,8 @@ The bending stiffness of the flexible joint does not significantly change the dy
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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 id="outline-container-orgdaf7b6c" class="outline-2">
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<h2 id="orgdaf7b6c"><span class="section-number-2">2</span> Axial 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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@ -341,8 +341,8 @@ Cz_M = 1*ones(6,1); % [N/(m/s)]
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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 id="outline-container-orge82a7c2" class="outline-4">
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<h4 id="orge82a7c2"><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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@ -370,8 +370,8 @@ initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
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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 id="outline-container-org44f67b8" class="outline-4">
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<h4 id="org44f67b8"><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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@ -390,8 +390,8 @@ The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
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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 id="outline-container-orgd5fd59b" class="outline-4">
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<h4 id="orgd5fd59b"><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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@ -415,8 +415,8 @@ The dynamics is compare with and without the joint flexibility in Figure <a href
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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 id="outline-container-org552093a" class="outline-4">
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<h4 id="org552093a"><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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@ -448,8 +448,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
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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>
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<div id="outline-container-orge13b41c" class="outline-4">
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<h4 id="orge13b41c"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
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<div class="outline-text-4" id="text-2-2-1">
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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="#orgab9ab86">8</a>.
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@ -491,8 +491,8 @@ It can be seen that very little active damping can be achieve for axial stiffnes
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</div>
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</div>
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<div id="outline-container-org53e5f08" class="outline-4">
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<h4 id="org53e5f08"><span class="section-number-4">2.2.2</span> Primary Control</h4>
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<div id="outline-container-org15c2c08" class="outline-4">
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<h4 id="org15c2c08"><span class="section-number-4">2.2.2</span> Primary Control</h4>
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<div class="outline-text-4" id="text-2-2-2">
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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="#org6070692">11</a>.
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@ -508,8 +508,8 @@ The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for
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</div>
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</div>
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<div id="outline-container-org1ddd8bf" class="outline-3">
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<h3 id="org1ddd8bf"><span class="section-number-3">2.3</span> Conclusion</h3>
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<div id="outline-container-orgce1052e" class="outline-3">
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<h3 id="orgce1052e"><span class="section-number-3">2.3</span> Conclusion</h3>
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<div class="outline-text-3" id="text-2-3">
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<div class="important">
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<p>
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@ -533,30 +533,52 @@ We may interpolate the results and say that the axial joint stiffness should be
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</div>
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</div>
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<div id="outline-container-orga32adf0" class="outline-2">
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<h2 id="orga32adf0"><span class="section-number-2">3</span> Conclusion</h2>
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<div id="outline-container-org865157e" class="outline-2">
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<h2 id="org865157e"><span class="section-number-2">3</span> Conclusion</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="org6614f42"></a>
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</p>
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<div class="important">
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<p>
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In this study we considered the bending, torsional and axial stiffnesses of the flexible joints used for the nano-hexapod.
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</p>
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<p>
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The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the legs.
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It is not found to be much problematic for the considered control architecture (it is however, if Integral Force Feedback is to be used).
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As a consequence of the added stiffness, it could increase a little bit the required actuator force.
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</p>
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<p>
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The axial stiffness of the flexible joints can be very problematic for control.
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Small values of the axial stiffness are shown to limit the achievable damping with Direct Velocity Feedback.
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The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod.
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</p>
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<p>
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For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:
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</p>
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<ul class="org-ul">
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<li>Axial Stiffness: \(K_a > 10^7\,[N/m]\)</li>
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<li>Bending Stiffness: \(K_b < 50\,[Nm/rad]\)</li>
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<li>Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)</li>
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<li>Axial Stiffness: \(K_a > 10^7\,[N/m]\)</li>
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</ul>
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<p>
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As there is generally a trade-off between bending stiffness and axial stiffness, it should be highlighted that the <b>axial</b> stiffness is the most important property of the flexible joints.
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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="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-05-05 mar. 11:26</p>
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<p class="date">Created: 2020-05-05 mar. 11:50</p>
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</div>
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</body>
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</html>
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@ -504,7 +504,7 @@ It is shown that the bending stiffness of the flexible joints have very little i
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The bending stiffness of the flexible joint does not significantly change the dynamics.
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#+end_important
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* Translation Stiffness
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* Axial Stiffness
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<<sec:trans_stiffness>>
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** Introduction :ignore:
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@ -993,8 +993,22 @@ The dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the
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<<sec:conclusion>>
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#+begin_important
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For the identified optimal actuator stiffness $k = 10^5\,[N/m]$, the flexible joint should have the following stiffness properties:
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- Bending Stiffness: $K_b < 50\,[Nm/rad]$
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- Torsion Stiffness: $K_t < 50\,[Nm/rad]$
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- Axial Stiffness: $K_a > 10^7\,[N/m]$
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In this study we considered the bending, torsional and axial stiffnesses of the flexible joints used for the nano-hexapod.
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The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the legs.
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It is not found to be much problematic for the considered control architecture (it is however, if Integral Force Feedback is to be used).
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As a consequence of the added stiffness, it could increase a little bit the required actuator force.
|
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|
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The axial stiffness of the flexible joints can be very problematic for control.
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Small values of the axial stiffness are shown to limit the achievable damping with Direct Velocity Feedback.
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The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod.
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For the identified optimal actuator stiffness $k = 10^5\,[N/m]$, the flexible joint should have the following stiffness properties:
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- Axial Stiffness: $K_a > 10^7\,[N/m]$
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- Bending Stiffness: $K_b < 50\,[Nm/rad]$
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- Torsion Stiffness: $K_t < 50\,[Nm/rad]$
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As there is generally a trade-off between bending stiffness and axial stiffness, it should be highlighted that the *axial* stiffness is the most important property of the flexible joints.
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#+end_important
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|
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