Remove figs, add table for stiffness matrices
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@ -4,7 +4,7 @@
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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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<!-- 2020-02-12 mer. 11:18 -->
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<!-- 2020-02-12 mer. 11:23 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Cubic configuration for the Stewart Platform</title>
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@ -274,26 +274,26 @@ for the JavaScript code in this tag.
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<li><a href="#orga88e79a">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
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<li><a href="#orge02ec88">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org43fd7e4">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
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<li><a href="#orgd35acc0">1.5. Conclusion</a></li>
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<li><a href="#org95af62e">1.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgd70418b">2. Configuration with the Cube’s center above the mobile platform</a>
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<ul>
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<li><a href="#org8afa645">2.1. Having Cube’s center above the top platform</a></li>
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<li><a href="#org4576402">2.2. Conclusion</a></li>
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<li><a href="#org7673551">2.2. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgcc4ecce">3. Cubic size analysis</a>
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<ul>
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<li><a href="#org0029d8c">3.1. Analysis</a></li>
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<li><a href="#org04f1ef6">3.2. Conclusion</a></li>
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<li><a href="#orgcad29fd">3.2. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org5abef15">4. Dynamic Coupling</a>
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<ul>
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<li><a href="#org0d67b92">4.1. Cube’s center at the Center of Mass of the Payload</a></li>
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<li><a href="#org876e05f">4.2. Dynamic decoupling between the actuators and sensors</a></li>
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<li><a href="#org95af62e">4.3. Conclusion</a></li>
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<li><a href="#org213d9b9">4.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org3044455">5. Functions</a>
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@ -388,7 +388,7 @@ We here study what makes the Stiffness matrix diagonal when using a cubic config
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<h3 id="orgf6f7ad2"><span class="section-number-3">1.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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We create a cubic Stewart platform (figure <a href="#org9454f54">1</a>) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
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We create a cubic Stewart platform (figure <a href="#orgaba20c8">1</a>) in such a way that the center of the cube (black star) is located at the center of the Stewart platform (blue dot).
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The Jacobian matrix is estimated at the location of the center of the cube.
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</p>
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@ -412,21 +412,14 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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</div>
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<div id="org9454f54" class="figure">
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<p><img src="figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p>
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</div>
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<div id="orgaba20c8" class="figure">
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<p><img src="figs/cubic_conf_centered_J_center.png" alt="cubic_conf_centered_J_center.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_center.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 1: </span>Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_center.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org4baf591" 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> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -504,7 +497,7 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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<h3 id="orga88e79a"><span class="section-number-3">1.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org9454f54">1</a>).
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We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org47f8142">2</a>).
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The Jacobian matrix is not estimated at the location of the center of the cube.
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</p>
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@ -531,11 +524,11 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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<div id="org47f8142" class="figure">
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<p><img src="figs/cubic_conf_centered_J_not_center.png" alt="cubic_conf_centered_J_not_center.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (<a href="./figs/cubic_conf_centered_J_not_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_not_center.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 2: </span>Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (<a href="./figs/cubic_conf_centered_J_not_center.png">png</a>, <a href="./figs/cubic_conf_centered_J_not_center.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org5cc2020" 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> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -613,7 +606,7 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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<h3 id="orge02ec88"><span class="section-number-3">1.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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Here, the “center” of the Stewart platform is not at the cube center (figure <a href="#org0235d3a">4</a>).
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Here, the “center” of the Stewart platform is not at the cube center (figure <a href="#org0235d3a">3</a>).
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The Jacobian is estimated at the cube center.
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</p>
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@ -640,11 +633,11 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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<div id="org0235d3a" class="figure">
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<p><img src="figs/cubic_conf_not_centered_J_center.png" alt="cubic_conf_not_centered_J_center.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_not_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_center.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 3: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (<a href="./figs/cubic_conf_not_centered_J_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_center.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org6b3d8b1" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 3:</span> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -760,11 +753,11 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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<div id="orgbe766b3" class="figure">
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<p><img src="figs/cubic_conf_not_centered_J_stewart_center.png" alt="cubic_conf_not_centered_J_stewart_center.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (<a href="./figs/cubic_conf_not_centered_J_stewart_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_stewart_center.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 4: </span>Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (<a href="./figs/cubic_conf_not_centered_J_stewart_center.png">png</a>, <a href="./figs/cubic_conf_not_centered_J_stewart_center.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org846d51c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 4:</span> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -838,8 +831,8 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
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</div>
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</div>
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<div id="outline-container-orgd35acc0" class="outline-3">
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<h3 id="orgd35acc0"><span class="section-number-3">1.5</span> Conclusion</h3>
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<div id="outline-container-org95af62e" class="outline-3">
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<h3 id="org95af62e"><span class="section-number-3">1.5</span> Conclusion</h3>
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<div class="outline-text-3" id="text-1-5">
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<div class="important">
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<p>
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@ -892,9 +885,9 @@ We find the several Cubic configuration for the Stewart platform where the cente
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The differences between the configuration are the cube’s size:
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</p>
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<ul class="org-ul">
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<li>Small Cube Size in Figure <a href="#org105635f">6</a></li>
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<li>Medium Cube Size in Figure <a href="#org264ab9c">7</a></li>
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<li>Large Cube Size in Figure <a href="#org52254fe">8</a></li>
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<li>Small Cube Size in Figure <a href="#org105635f">5</a></li>
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<li>Medium Cube Size in Figure <a href="#org264ab9c">6</a></li>
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<li>Large Cube Size in Figure <a href="#org52254fe">7</a></li>
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</ul>
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<p>
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@ -912,11 +905,11 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
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<div id="org105635f" class="figure">
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<p><img src="figs/stewart_cubic_conf_type_1.png" alt="stewart_cubic_conf_type_1.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Cubic Configuration for the Stewart Platform - Small Cube Size (<a href="./figs/stewart_cubic_conf_type_1.png">png</a>, <a href="./figs/stewart_cubic_conf_type_1.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 5: </span>Cubic Configuration for the Stewart Platform - Small Cube Size (<a href="./figs/stewart_cubic_conf_type_1.png">png</a>, <a href="./figs/stewart_cubic_conf_type_1.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org91f89e4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 5:</span> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -998,12 +991,12 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
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<div id="org264ab9c" class="figure">
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<p><img src="figs/stewart_cubic_conf_type_2.png" alt="stewart_cubic_conf_type_2.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Cubic Configuration for the Stewart Platform - Medium Cube Size (<a href="./figs/stewart_cubic_conf_type_2.png">png</a>, <a href="./figs/stewart_cubic_conf_type_2.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 6: </span>Cubic Configuration for the Stewart Platform - Medium Cube Size (<a href="./figs/stewart_cubic_conf_type_2.png">png</a>, <a href="./figs/stewart_cubic_conf_type_2.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orgcf84781" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 6:</span> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -1085,12 +1078,12 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
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<div id="org52254fe" class="figure">
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<p><img src="figs/stewart_cubic_conf_type_3.png" alt="stewart_cubic_conf_type_3.png" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Cubic Configuration for the Stewart Platform - Large Cube Size (<a href="./figs/stewart_cubic_conf_type_3.png">png</a>, <a href="./figs/stewart_cubic_conf_type_3.pdf">pdf</a>)</p>
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<p><span class="figure-number">Figure 7: </span>Cubic Configuration for the Stewart Platform - Large Cube Size (<a href="./figs/stewart_cubic_conf_type_3.png">png</a>, <a href="./figs/stewart_cubic_conf_type_3.pdf">pdf</a>)</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org02f7789" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 7:</span> Stiffness Matrix</caption>
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<colgroup>
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<col class="org-right" />
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@ -1164,8 +1157,8 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
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</div>
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</div>
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<div id="outline-container-org4576402" class="outline-3">
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<h3 id="org4576402"><span class="section-number-3">2.2</span> Conclusion</h3>
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<div id="outline-container-org7673551" class="outline-3">
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<h3 id="org7673551"><span class="section-number-3">2.2</span> Conclusion</h3>
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<div class="outline-text-3" id="text-2-2">
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<div class="important">
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<p>
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@ -1227,20 +1220,20 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
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<p>
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We find that for all the cube’s size, \(k_x = k_y = k_z = k\) where \(k\) is the strut stiffness.
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We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varying with the cube’s size (figure <a href="#orgf5b4a80">9</a>).
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We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varying with the cube’s size (figure <a href="#orgf5b4a80">8</a>).
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</p>
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<div id="orgf5b4a80" class="figure">
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<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
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</p>
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<p><span class="figure-number">Figure 9: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p>
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<p><span class="figure-number">Figure 8: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org04f1ef6" class="outline-3">
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<h3 id="org04f1ef6"><span class="section-number-3">3.2</span> Conclusion</h3>
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<div id="outline-container-orgcad29fd" class="outline-3">
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<h3 id="orgcad29fd"><span class="section-number-3">3.2</span> Conclusion</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
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@ -1271,8 +1264,8 @@ In order to maximize the rotational stiffness of the Stewart platform, the size
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<h3 id="org876e05f"><span class="section-number-3">4.2</span> Dynamic decoupling between the actuators and sensors</h3>
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</div>
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<div id="outline-container-org95af62e" class="outline-3">
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<h3 id="org95af62e"><span class="section-number-3">4.3</span> Conclusion</h3>
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<div id="outline-container-org213d9b9" class="outline-3">
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<h3 id="org213d9b9"><span class="section-number-3">4.3</span> Conclusion</h3>
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</div>
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</div>
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@ -1330,7 +1323,7 @@ This Matlab function is accessible <a href="../src/generateCubicConfiguration.m"
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<div id="org8a7f3d8" class="figure">
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<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
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</p>
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<p><span class="figure-number">Figure 10: </span>Cubic Configuration</p>
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<p><span class="figure-number">Figure 9: </span>Cubic Configuration</p>
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</div>
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</div>
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</div>
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@ -1435,7 +1428,7 @@ stewart.platform_M.Mb = Mb;
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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: 2020-02-12 mer. 11:18</p>
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<p class="date">Created: 2020-02-12 mer. 11:23</p>
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</div>
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</body>
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</html>
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@ -91,7 +91,7 @@ We here study what makes the Stiffness matrix diagonal when using a cubic config
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#+end_src
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** Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
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We create a cubic Stewart platform (figure [[fig:3d-cubic-stewart-aligned]]) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
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We create a cubic Stewart platform (figure [[fig:cubic_conf_centered_J_center]]) in such a way that the center of the cube (black star) is located at the center of the Stewart platform (blue dot).
|
||||
The Jacobian matrix is estimated at the location of the center of the cube.
|
||||
|
||||
#+begin_src matlab
|
||||
@ -111,10 +111,6 @@ The Jacobian matrix is estimated at the location of the center of the cube.
|
||||
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);
|
||||
#+end_src
|
||||
|
||||
#+name: fig:3d-cubic-stewart-aligned
|
||||
#+caption: Centered cubic configuration
|
||||
[[file:figs/3d-cubic-stewart-aligned.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
displayArchitecture(stewart, 'labels', false);
|
||||
scatter3(0, 0, FOc, 200, 'kh');
|
||||
@ -133,6 +129,8 @@ The Jacobian matrix is estimated at the location of the center of the cube.
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:cubic_conf_centered_J_center
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | -2.5e-16 | 0 | 2.1e-17 | 0 |
|
||||
| 0 | 2 | 0 | -7.8e-19 | 0 | 0 |
|
||||
@ -142,7 +140,7 @@ The Jacobian matrix is estimated at the location of the center of the cube.
|
||||
| 6.6e-18 | -3.3e-18 | 0 | 1.7e-18 | 0 | 0.06 |
|
||||
|
||||
** Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
|
||||
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure [[fig:3d-cubic-stewart-aligned]]).
|
||||
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure [[fig:cubic_conf_centered_J_not_center]]).
|
||||
The Jacobian matrix is not estimated at the location of the center of the cube.
|
||||
|
||||
#+begin_src matlab
|
||||
@ -180,6 +178,8 @@ The Jacobian matrix is not estimated at the location of the center of the cube.
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:cubic_conf_centered_J_not_center
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | -2.5e-16 | 0 | -0.14 | 0 |
|
||||
| 0 | 2 | 0 | 0.14 | 0 | 0 |
|
||||
@ -227,6 +227,8 @@ The Jacobian is estimated at the cube center.
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:cubic_conf_not_centered_J_center
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | -1.7e-16 | 0 | 4.9e-17 | 0 |
|
||||
| 0 | 2 | 0 | -2.2e-17 | 0 | 2.8e-17 |
|
||||
@ -281,6 +283,8 @@ The center of the cube from the top platform is at $z = 110 - 175 = -65$.
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:cubic_conf_not_centered_J_stewart_center
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | 1.5e-16 | 0 | 0.02 | 0 |
|
||||
| 0 | 2 | 0 | -0.02 | 0 | 0 |
|
||||
@ -366,6 +370,8 @@ However, the rotational stiffnesses are increasing with the cube's size but the
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:stewart_cubic_conf_type_1
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | -2.8e-16 | 0 | 2.4e-17 | 0 |
|
||||
| 0 | 2 | 0 | -2.3e-17 | 0 | 0 |
|
||||
@ -405,6 +411,8 @@ However, the rotational stiffnesses are increasing with the cube's size but the
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:stewart_cubic_conf_type_2
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | -1.9e-16 | 0 | 5.6e-17 | 0 |
|
||||
| 0 | 2 | 0 | -7.6e-17 | 0 | 0 |
|
||||
@ -444,6 +452,8 @@ However, the rotational stiffnesses are increasing with the cube's size but the
|
||||
data2orgtable(stewart.kinematics.K, {}, {}, ' %.2g ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:stewart_cubic_conf_type_3
|
||||
#+caption: Stiffness Matrix
|
||||
#+RESULTS:
|
||||
| 2 | 0 | -3e-16 | 0 | -8.3e-17 | 0 |
|
||||
| 0 | 2 | 0 | -2.2e-17 | 0 | 5.6e-17 |
|
||||
|
||||
Loading…
Reference in New Issue
Block a user