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figs/*.svg
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figs/*.pdf
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figures/
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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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<!-- 2019-08-26 lun. 11:58 -->
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<!-- 2019-10-09 mer. 11:08 -->
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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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@ -279,33 +279,33 @@ for the JavaScript code in this tag.
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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="#org1bafc26">1. Questions we wish to answer with this analysis</a></li>
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<li><a href="#orga5361fc">2. Configuration Analysis - Stiffness Matrix</a>
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<li><a href="#orgc57423d">1. Questions we wish to answer with this analysis</a></li>
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<li><a href="#org5539c71">2. Configuration Analysis - Stiffness Matrix</a>
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<ul>
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<li><a href="#orge00f8a7">2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org575d55b">2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
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<li><a href="#orgcda0ff4">2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org06f7f99">2.4. 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="#org42ac83d">2.5. Conclusion</a></li>
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<li><a href="#orga0e5e7a">2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org2b14a19">2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
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<li><a href="#orgdd2c3a5">2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org2c1dada">2.4. 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="#org6305043">2.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgb37e81b">3. Cubic size analysis</a></li>
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<li><a href="#org2b0a41e">4. initializeCubicConfiguration</a>
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<li><a href="#org00efd87">3. Cubic size analysis</a></li>
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<li><a href="#org3841131">4. initializeCubicConfiguration</a>
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<ul>
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<li><a href="#orgfb743ea">4.1. Function description</a></li>
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<li><a href="#orgcc92353">4.2. Optional Parameters</a></li>
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<li><a href="#org384ec97">4.3. Cube Creation</a></li>
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<li><a href="#orgefa2328">4.4. Vectors of each leg</a></li>
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<li><a href="#orgf6960ff">4.5. Verification of Height of the Stewart Platform</a></li>
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<li><a href="#orgd7e65db">4.6. Determinate the location of the joints</a></li>
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<li><a href="#org38f602f">4.7. Returns Stewart Structure</a></li>
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<li><a href="#orgff95f33">4.1. Function description</a></li>
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<li><a href="#org3163673">4.2. Optional Parameters</a></li>
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<li><a href="#orgda7067a">4.3. Cube Creation</a></li>
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<li><a href="#org2c8b79d">4.4. Vectors of each leg</a></li>
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<li><a href="#org2f2eeb2">4.5. Verification of Height of the Stewart Platform</a></li>
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<li><a href="#org7c5ca24">4.6. Determinate the location of the joints</a></li>
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<li><a href="#org723d8e6">4.7. Returns Stewart Structure</a></li>
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</ul>
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</li>
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<li><a href="#org243b392">5. Tests</a>
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<li><a href="#org1963ce8">5. Tests</a>
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<ul>
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<li><a href="#org86fb4aa">5.1. First attempt to parametrisation</a></li>
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<li><a href="#orgcc2eaf4">5.2. Second attempt</a></li>
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<li><a href="#org7824b39">5.3. Generate the Stewart platform for a Cubic configuration</a></li>
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<li><a href="#org546f291">5.1. First attempt to parametrisation</a></li>
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<li><a href="#org2231886">5.2. Second attempt</a></li>
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<li><a href="#org736f58d">5.3. Generate the Stewart platform for a Cubic configuration</a></li>
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</ul>
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</li>
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</ul>
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@ -327,11 +327,15 @@ The specificity of the Cubic configuration is that each actuator is orthogonal w
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</p>
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<p>
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To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#org7faef27">4</a>).
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To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#orga589e9f">4</a>).
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</p>
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<div id="outline-container-org1bafc26" class="outline-2">
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<h2 id="org1bafc26"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2>
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<p>
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According to <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>, the cubic configuration provides a uniform stiffness in all directions and <b>minimizes the crosscoupling</b> from actuator to sensor of different legs (being orthogonal to each other).
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</p>
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<div id="outline-container-orgc57423d" class="outline-2">
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<h2 id="orgc57423d"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The goal is to study the benefits of using a cubic configuration:
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@ -344,20 +348,20 @@ The goal is to study the benefits of using a cubic configuration:
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</div>
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</div>
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<div id="outline-container-orga5361fc" class="outline-2">
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<h2 id="orga5361fc"><span class="section-number-2">2</span> Configuration Analysis - Stiffness Matrix</h2>
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<div id="outline-container-org5539c71" class="outline-2">
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<h2 id="org5539c71"><span class="section-number-2">2</span> Configuration Analysis - Stiffness Matrix</h2>
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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-orge00f8a7" class="outline-3">
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<h3 id="orge00f8a7"><span class="section-number-3">2.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3>
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<div id="outline-container-orga0e5e7a" class="outline-3">
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<h3 id="orga0e5e7a"><span class="section-number-3">2.1</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-2-1">
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<p>
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We create a cubic Stewart platform (figure <a href="#orgb095247">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="#org1d5da43">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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The Jacobian matrix is estimated at the location of the center of the cube.
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</p>
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<div id="orgb095247" class="figure">
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<div id="org1d5da43" 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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@ -461,11 +465,11 @@ save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string
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</div>
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</div>
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<div id="outline-container-org575d55b" class="outline-3">
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<h3 id="org575d55b"><span class="section-number-3">2.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
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<div id="outline-container-org2b14a19" class="outline-3">
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<h3 id="org2b14a19"><span class="section-number-3">2.2</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-2-2">
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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="#orgb095247">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="#org1d5da43">1</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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@ -565,16 +569,16 @@ stewart = computeGeometricalProperties<span class="org-rainbow-delimiters-depth-
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</div>
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</div>
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<div id="outline-container-orgcda0ff4" class="outline-3">
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<h3 id="orgcda0ff4"><span class="section-number-3">2.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
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<div id="outline-container-orgdd2c3a5" class="outline-3">
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<h3 id="orgdd2c3a5"><span class="section-number-3">2.3</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-2-3">
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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="#org741ffe8">2</a>).
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Here, the "center" of the Stewart platform is not at the cube center (figure <a href="#org95caad9">2</a>).
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The Jacobian is estimated at the cube center.
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</p>
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<div id="org741ffe8" class="figure">
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<div id="org95caad9" class="figure">
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<p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p>
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@ -687,8 +691,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
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</div>
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</div>
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<div id="outline-container-org06f7f99" class="outline-3">
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<h3 id="org06f7f99"><span class="section-number-3">2.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3>
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<div id="outline-container-org2c1dada" class="outline-3">
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<h3 id="org2c1dada"><span class="section-number-3">2.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3>
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<div class="outline-text-3" id="text-2-4">
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<p>
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Here, the "center" of the Stewart platform is not at the cube center.
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@ -802,8 +806,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
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</div>
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</div>
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<div id="outline-container-org42ac83d" class="outline-3">
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<h3 id="org42ac83d"><span class="section-number-3">2.5</span> Conclusion</h3>
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<div id="outline-container-org6305043" class="outline-3">
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<h3 id="org6305043"><span class="section-number-3">2.5</span> Conclusion</h3>
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<div class="outline-text-3" id="text-2-5">
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<div class="important">
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<ul class="org-ul">
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@ -816,8 +820,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
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</div>
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</div>
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<div id="outline-container-orgb37e81b" class="outline-2">
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<h2 id="orgb37e81b"><span class="section-number-2">3</span> Cubic size analysis</h2>
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<div id="outline-container-org00efd87" class="outline-2">
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<h2 id="org00efd87"><span class="section-number-2">3</span> Cubic size analysis</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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We here study the effect of the size of the cube used for the Stewart configuration.
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@ -892,7 +896,7 @@ xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-stri
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</div>
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<div id="org5647f41" class="figure">
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<div id="org5211ce6" 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 3: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p>
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@ -913,16 +917,16 @@ In that case, the legs will the further separated. Size of the cube is then limi
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</div>
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</div>
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<div id="outline-container-org2b0a41e" class="outline-2">
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<h2 id="org2b0a41e"><span class="section-number-2">4</span> initializeCubicConfiguration</h2>
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<div id="outline-container-org3841131" class="outline-2">
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<h2 id="org3841131"><span class="section-number-2">4</span> initializeCubicConfiguration</h2>
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<div class="outline-text-2" id="text-4">
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<p>
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<a id="org7faef27"></a>
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<a id="orga589e9f"></a>
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</p>
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</div>
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<div id="outline-container-orgfb743ea" class="outline-3">
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<h3 id="orgfb743ea"><span class="section-number-3">4.1</span> Function description</h3>
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<div id="outline-container-orgff95f33" class="outline-3">
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<h3 id="orgff95f33"><span class="section-number-3">4.1</span> Function description</h3>
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<div class="outline-text-3" id="text-4-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-variable-name">stewart</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">initializeCubicConfiguration</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">opts_param</span><span class="org-rainbow-delimiters-depth-1">)</span>
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@ -931,8 +935,8 @@ In that case, the legs will the further separated. Size of the cube is then limi
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</div>
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</div>
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<div id="outline-container-orgcc92353" class="outline-3">
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<h3 id="orgcc92353"><span class="section-number-3">4.2</span> Optional Parameters</h3>
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<div id="outline-container-org3163673" class="outline-3">
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<h3 id="org3163673"><span class="section-number-3">4.2</span> Optional Parameters</h3>
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<div class="outline-text-3" id="text-4-2">
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<p>
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Default values for opts.
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@ -961,8 +965,8 @@ Populate opts with input parameters
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</div>
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</div>
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<div id="outline-container-org384ec97" class="outline-3">
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<h3 id="org384ec97"><span class="section-number-3">4.3</span> Cube Creation</h3>
|
||||
<div id="outline-container-orgda7067a" class="outline-3">
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||||
<h3 id="orgda7067a"><span class="section-number-3">4.3</span> Cube Creation</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">points = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>; <span class="org-underline">...</span>
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||||
@ -1007,8 +1011,8 @@ We use to rotation matrix to rotate the cube
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||||
</div>
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||||
</div>
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||||
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<div id="outline-container-orgefa2328" class="outline-3">
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||||
<h3 id="orgefa2328"><span class="section-number-3">4.4</span> Vectors of each leg</h3>
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<div id="outline-container-org2c8b79d" class="outline-3">
|
||||
<h3 id="org2c8b79d"><span class="section-number-3">4.4</span> Vectors of each leg</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">leg_indices = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">4</span>; <span class="org-underline">...</span>
|
||||
@ -1036,8 +1040,8 @@ legs_start = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span cla
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||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf6960ff" class="outline-3">
|
||||
<h3 id="orgf6960ff"><span class="section-number-3">4.5</span> Verification of Height of the Stewart Platform</h3>
|
||||
<div id="outline-container-org2f2eeb2" class="outline-3">
|
||||
<h3 id="org2f2eeb2"><span class="section-number-3">4.5</span> Verification of Height of the Stewart Platform</h3>
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||||
<div class="outline-text-3" id="text-4-5">
|
||||
<p>
|
||||
If the Stewart platform is not contained in the cube, throw an error.
|
||||
@ -1056,8 +1060,8 @@ If the Stewart platform is not contained in the cube, throw an error.
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||||
</div>
|
||||
</div>
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||||
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||||
<div id="outline-container-orgd7e65db" class="outline-3">
|
||||
<h3 id="orgd7e65db"><span class="section-number-3">4.6</span> Determinate the location of the joints</h3>
|
||||
<div id="outline-container-org7c5ca24" class="outline-3">
|
||||
<h3 id="org7c5ca24"><span class="section-number-3">4.6</span> Determinate the location of the joints</h3>
|
||||
<div class="outline-text-3" id="text-4-6">
|
||||
<p>
|
||||
We now determine the location of the joints on the fixed platform w.r.t the fixed frame \(\{A\}\).
|
||||
@ -1102,8 +1106,8 @@ Ab = Ab <span class="org-type">-</span> h<span class="org-type">*</span><span cl
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org38f602f" class="outline-3">
|
||||
<h3 id="org38f602f"><span class="section-number-3">4.7</span> Returns Stewart Structure</h3>
|
||||
<div id="outline-container-org723d8e6" class="outline-3">
|
||||
<h3 id="org723d8e6"><span class="section-number-3">4.7</span> Returns Stewart Structure</h3>
|
||||
<div class="outline-text-3" id="text-4-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = struct<span class="org-rainbow-delimiters-depth-1">()</span>;
|
||||
@ -1118,15 +1122,15 @@ Ab = Ab <span class="org-type">-</span> h<span class="org-type">*</span><span cl
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org243b392" class="outline-2">
|
||||
<h2 id="org243b392"><span class="section-number-2">5</span> Tests</h2>
|
||||
<div id="outline-container-org1963ce8" class="outline-2">
|
||||
<h2 id="org1963ce8"><span class="section-number-2">5</span> Tests</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
</div>
|
||||
<div id="outline-container-org86fb4aa" class="outline-3">
|
||||
<h3 id="org86fb4aa"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3>
|
||||
<div id="outline-container-org546f291" class="outline-3">
|
||||
<h3 id="org546f291"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
|
||||
<div id="org10fe09c" class="figure">
|
||||
<div id="org16ba25a" class="figure">
|
||||
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Schematic of the bottom plates with all the parameters</p>
|
||||
@ -1161,8 +1165,8 @@ Lets express \(a_i\), \(b_i\) and \(a_j\):
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcc2eaf4" class="outline-3">
|
||||
<h3 id="orgcc2eaf4"><span class="section-number-3">5.2</span> Second attempt</h3>
|
||||
<div id="outline-container-org2231886" class="outline-3">
|
||||
<h3 id="org2231886"><span class="section-number-3">5.2</span> Second attempt</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
We start with the point of a cube in space:
|
||||
@ -1289,8 +1293,8 @@ Let's then estimate the middle position of the platform
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7824b39" class="outline-3">
|
||||
<h3 id="org7824b39"><span class="section-number-3">5.3</span> Generate the Stewart platform for a Cubic configuration</h3>
|
||||
<div id="outline-container-org736f58d" class="outline-3">
|
||||
<h3 id="org736f58d"><span class="section-number-3">5.3</span> Generate the Stewart platform for a Cubic configuration</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<p>
|
||||
First we defined the height of the Hexapod.
|
||||
@ -1360,7 +1364,7 @@ zlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2019-08-26 lun. 11:58</p>
|
||||
<p class="date">Created: 2019-10-09 mer. 11:08</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
@ -21,12 +21,7 @@
|
||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||
:END:
|
||||
|
||||
#+begin_src matlab :results none :exports none :noweb yes
|
||||
<<matlab-init>>
|
||||
addpath('src');
|
||||
addpath('library');
|
||||
#+end_src
|
||||
|
||||
* Introduction :ignore:
|
||||
The discovery of the Cubic configuration is done in citenum:geng94_six_degree_of_freed_activ.
|
||||
Further analysis is conducted in cite:jafari03_orthog_gough_stewar_platf_microm.
|
||||
|
||||
@ -37,6 +32,22 @@ The specificity of the Cubic configuration is that each actuator is orthogonal w
|
||||
|
||||
To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]).
|
||||
|
||||
According to cite:preumont07_six_axis_singl_stage_activ, the cubic configuration provides a uniform stiffness in all directions and *minimizes the crosscoupling* from actuator to sensor of different legs (being orthogonal to each other).
|
||||
|
||||
* Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none :exports none
|
||||
addpath('src');
|
||||
addpath('library');
|
||||
#+end_src
|
||||
|
||||
* Questions we wish to answer with this analysis
|
||||
The goal is to study the benefits of using a cubic configuration:
|
||||
- Equal stiffness in all the degrees of freedom?
|
||||
|
||||
BIN
figs/coupling.png
Normal file
BIN
figs/coupling.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 3.8 KiB |
175
index.html
175
index.html
@ -3,7 +3,7 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2019-08-26 lun. 11:56 -->
|
||||
<!-- 2019-10-09 mer. 11:07 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>Stewart Platforms</title>
|
||||
@ -245,13 +245,35 @@ for the JavaScript code in this tag.
|
||||
}
|
||||
/*]]>*///-->
|
||||
</script>
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
displayAlign: "center",
|
||||
displayIndent: "0em",
|
||||
|
||||
"HTML-CSS": { scale: 100,
|
||||
linebreaks: { automatic: "false" },
|
||||
webFont: "TeX"
|
||||
},
|
||||
SVG: {scale: 100,
|
||||
linebreaks: { automatic: "false" },
|
||||
font: "TeX"},
|
||||
NativeMML: {scale: 100},
|
||||
TeX: { equationNumbers: {autoNumber: "AMS"},
|
||||
MultLineWidth: "85%",
|
||||
TagSide: "right",
|
||||
TagIndent: ".8em"
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript"
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
</head>
|
||||
<body>
|
||||
<div id="content">
|
||||
<h1 class="title">Stewart Platforms</h1>
|
||||
|
||||
<div id="outline-container-org6a8d5e6" class="outline-2">
|
||||
<h2 id="org6a8d5e6"><span class="section-number-2">1</span> Simscape Model</h2>
|
||||
<div id="outline-container-orge672724" class="outline-2">
|
||||
<h2 id="orge672724"><span class="section-number-2">1</span> Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<ul class="org-ul">
|
||||
<li><a href="simscape-model.html">Model of the Stewart Platform</a></li>
|
||||
@ -260,8 +282,8 @@ for the JavaScript code in this tag.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org28395cb" class="outline-2">
|
||||
<h2 id="org28395cb"><span class="section-number-2">2</span> Architecture Study</h2>
|
||||
<div id="outline-container-orgfce4cb7" class="outline-2">
|
||||
<h2 id="orgfce4cb7"><span class="section-number-2">2</span> Architecture Study</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<ul class="org-ul">
|
||||
<li><a href="kinematic-study.html">Kinematic Study</a></li>
|
||||
@ -272,8 +294,8 @@ for the JavaScript code in this tag.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6738e47" class="outline-2">
|
||||
<h2 id="org6738e47"><span class="section-number-2">3</span> Motion Control</h2>
|
||||
<div id="outline-container-org92e9216" class="outline-2">
|
||||
<h2 id="org92e9216"><span class="section-number-2">3</span> Motion Control</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<ul class="org-ul">
|
||||
<li>Active Damping</li>
|
||||
@ -282,6 +304,145 @@ for the JavaScript code in this tag.
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org5ab21e2" class="outline-2">
|
||||
<h2 id="org5ab21e2"><span class="section-number-2">4</span> Notes about Stewart platforms</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
</div>
|
||||
<div id="outline-container-orgf0627f0" class="outline-3">
|
||||
<h3 id="orgf0627f0"><span class="section-number-3">4.1</span> Jacobian</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
</div>
|
||||
<div id="outline-container-orge3fb927" class="outline-4">
|
||||
<h4 id="orge3fb927"><span class="section-number-4">4.1.1</span> Relation to platform parameters</h4>
|
||||
<div class="outline-text-4" id="text-4-1-1">
|
||||
<p>
|
||||
A Jacobian is defined by:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>the orientations of the struts \(\hat{s}_i\) expressed in a frame \(\{A\}\) linked to the fixed platform.</li>
|
||||
<li>the vectors from \(O_B\) to \(b_i\) expressed in the frame \(\{A\}\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Then, the choice of \(O_B\) changes the Jacobian.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org99049d5" class="outline-4">
|
||||
<h4 id="org99049d5"><span class="section-number-4">4.1.2</span> Jacobian for displacement</h4>
|
||||
<div class="outline-text-4" id="text-4-1-2">
|
||||
<p>
|
||||
\[ \dot{q} = J \dot{X} \]
|
||||
With:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(q = [q_1\ q_2\ q_3\ q_4\ q_5\ q_6]\) vector of linear displacement of actuated joints</li>
|
||||
<li>\(X = [x\ y\ z\ \theta_x\ \theta_y\ \theta_z]\) position and orientation of \(O_B\) expressed in the frame \(\{A\}\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
For very small displacements \(\delta q\) and \(\delta X\), we have \(\delta q = J \delta X\).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb7963ed" class="outline-4">
|
||||
<h4 id="orgb7963ed"><span class="section-number-4">4.1.3</span> Jacobian for forces</h4>
|
||||
<div class="outline-text-4" id="text-4-1-3">
|
||||
<p>
|
||||
\[ F = J^T \tau \]
|
||||
With:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(\tau = [\tau_1\ \tau_2\ \tau_3\ \tau_4\ \tau_5\ \tau_6]\) vector of actuator forces</li>
|
||||
<li>\(F = [f_x\ f_y\ f_z\ n_x\ n_y\ n_z]\) force and torque acting on point \(O_B\)</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9fcd675" class="outline-3">
|
||||
<h3 id="org9fcd675"><span class="section-number-3">4.2</span> Stiffness matrix \(K\)</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
\[ K = J^T \text{diag}(k_i) J \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
If all the struts have the same stiffness \(k\), then \(K = k J^T J\)
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\(K\) only depends of the geometry of the stewart platform: it depends on the Jacobian, that is on the orientations of the struts, position of the joints and choice of frame \(\{B\}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ F = K X \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
With \(F\) forces and torques applied to the moving platform at the origin of \(\{B\}\) and \(X\) the translations and rotations of \(\{B\}\) with respect to \(\{A\}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ C = K^{-1} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The compliance element \(C_{ij}\) is then the stiffness
|
||||
\[ X_i = C_{ij} F_j \]
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge5eb09a" class="outline-3">
|
||||
<h3 id="orge5eb09a"><span class="section-number-3">4.3</span> Coupling</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
What causes the coupling from \(F_i\) to \(X_i\) ?
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (Jt) at (0, 0) {<span class="org-font-latex-math">$J</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^{-T}</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right= of Jt] (G) {<span class="org-font-latex-math">$G$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right= of G] (J) {<span class="org-font-latex-math">$J</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^{-1}</span></span><span class="org-font-latex-math">$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(Jt.west)+(-0.8, 0)$</span>) -- (Jt.west) node[above left]{<span class="org-font-latex-math">$F</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_i</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Jt.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tau</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_i</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G.east) -- (J.west) node[above left]{<span class="org-font-latex-math">$q</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_i</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (J.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$X</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_i</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org064c4c6" class="figure">
|
||||
<p><img src="figs/coupling.png" alt="coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Block diagram to control an hexapod</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
There is no coupling from \(F_i\) to \(X_j\) if \(J^{-1} G J^{-T}\) is diagonal.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
If \(G\) is diagonal (cubic configuration), then \(J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}\)
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Thus, the system is uncoupled if \(G\) and \(K\) are diagonal.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
|
||||
<a href="references.bib">references.bib</a>
|
||||
</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
80
index.org
80
index.org
@ -9,6 +9,16 @@
|
||||
#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
|
||||
#+HTML_HEAD: <script src="js/jquery.stickytableheaders.min.js"></script>
|
||||
#+HTML_HEAD: <script src="js/readtheorg.js"></script>
|
||||
|
||||
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
|
||||
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
||||
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
||||
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
||||
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
|
||||
#+PROPERTY: header-args:latex+ :eval no-export
|
||||
#+PROPERTY: header-args:latex+ :exports both
|
||||
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||
:END:
|
||||
|
||||
* Simscape Model
|
||||
@ -25,3 +35,73 @@
|
||||
- Active Damping
|
||||
- Inertial Control
|
||||
- Decentralized Control
|
||||
* Notes about Stewart platforms
|
||||
** Jacobian
|
||||
*** Relation to platform parameters
|
||||
A Jacobian is defined by:
|
||||
- the orientations of the struts $\hat{s}_i$ expressed in a frame $\{A\}$ linked to the fixed platform.
|
||||
- the vectors from $O_B$ to $b_i$ expressed in the frame $\{A\}$
|
||||
|
||||
Then, the choice of $O_B$ changes the Jacobian.
|
||||
|
||||
*** Jacobian for displacement
|
||||
\[ \dot{q} = J \dot{X} \]
|
||||
With:
|
||||
- $q = [q_1\ q_2\ q_3\ q_4\ q_5\ q_6]$ vector of linear displacement of actuated joints
|
||||
- $X = [x\ y\ z\ \theta_x\ \theta_y\ \theta_z]$ position and orientation of $O_B$ expressed in the frame $\{A\}$
|
||||
|
||||
For very small displacements $\delta q$ and $\delta X$, we have $\delta q = J \delta X$.
|
||||
|
||||
*** Jacobian for forces
|
||||
\[ F = J^T \tau \]
|
||||
With:
|
||||
- $\tau = [\tau_1\ \tau_2\ \tau_3\ \tau_4\ \tau_5\ \tau_6]$ vector of actuator forces
|
||||
- $F = [f_x\ f_y\ f_z\ n_x\ n_y\ n_z]$ force and torque acting on point $O_B$
|
||||
|
||||
** Stiffness matrix $K$
|
||||
|
||||
\[ K = J^T \text{diag}(k_i) J \]
|
||||
|
||||
If all the struts have the same stiffness $k$, then $K = k J^T J$
|
||||
|
||||
$K$ only depends of the geometry of the stewart platform: it depends on the Jacobian, that is on the orientations of the struts, position of the joints and choice of frame $\{B\}$.
|
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|
||||
\[ F = K X \]
|
||||
|
||||
With $F$ forces and torques applied to the moving platform at the origin of $\{B\}$ and $X$ the translations and rotations of $\{B\}$ with respect to $\{A\}$.
|
||||
|
||||
\[ C = K^{-1} \]
|
||||
|
||||
The compliance element $C_{ij}$ is then the stiffness
|
||||
\[ X_i = C_{ij} F_j \]
|
||||
|
||||
** Coupling
|
||||
What causes the coupling from $F_i$ to $X_i$ ?
|
||||
|
||||
#+begin_src latex :file coupling.pdf :post pdf2svg(file=*this*, ext="png") :exports both
|
||||
\begin{tikzpicture}
|
||||
\node[block] (Jt) at (0, 0) {$J^{-T}$};
|
||||
\node[block, right= of Jt] (G) {$G$};
|
||||
\node[block, right= of G] (J) {$J^{-1}$};
|
||||
|
||||
\draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$F_i$};
|
||||
\draw[->] (Jt.east) -- (G.west) node[above left]{$\tau_i$};
|
||||
\draw[->] (G.east) -- (J.west) node[above left]{$q_i$};
|
||||
\draw[->] (J.east) -- ++(0.8, 0) node[above left]{$X_i$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:block_diag_coupling
|
||||
#+caption: Block diagram to control an hexapod
|
||||
#+RESULTS:
|
||||
[[file:figs/coupling.png]]
|
||||
|
||||
There is no coupling from $F_i$ to $X_j$ if $J^{-1} G J^{-T}$ is diagonal.
|
||||
|
||||
If $G$ is diagonal (cubic configuration), then $J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}$
|
||||
|
||||
Thus, the system is uncoupled if $G$ and $K$ are diagonal.
|
||||
|
||||
* Bibliography :ignore:
|
||||
bibliographystyle:unsrt
|
||||
bibliography:references.bib
|
||||
|
||||
@ -20,6 +20,7 @@
|
||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||
:END:
|
||||
|
||||
* Introduction :ignore:
|
||||
Stewart platforms are generated in multiple steps.
|
||||
|
||||
First, geometrical parameters are defined:
|
||||
|
||||
Loading…
Reference in New Issue
Block a user