Change the location of the cubic conf function
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<title>Cubic configuration for the Stewart Platform</title>
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<title>Cubic configuration for the Stewart Platform</title>
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<div id="org-div-home-and-up">
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@ -283,33 +268,46 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<div id="text-table-of-contents">
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<ul>
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<ul>
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<li><a href="#org86c83bf">1. Questions we wish to answer with this analysis</a></li>
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<li><a href="#org43e4755">1. Questions we wish to answer with this analysis</a></li>
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<li><a href="#org0b05973">2. <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</a>
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<li><a href="#org7c85269">2. <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</a>
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<ul>
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<ul>
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<li><a href="#org3f035e8">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="#org7a2e2af">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="#org77ecb36">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="#orgdd082ef">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="#org42ea8ad">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="#org314610d">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="#org38870ce">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="#org460e492">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="#org08c7461">2.5. Conclusion</a></li>
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<li><a href="#orgccb5ef0">2.5. Conclusion</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgc4c2abd">3. <span class="todo TODO">TODO</span> Cubic size analysis</a></li>
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<li><a href="#org29d657d">3. <span class="todo TODO">TODO</span> Cubic size analysis</a></li>
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<li><a href="#org36a27e6">4. <span class="todo TODO">TODO</span> initializeCubicConfiguration</a>
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<li><a href="#orgc12b0fc">4. Functions</a>
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<ul>
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<ul>
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<li><a href="#orgf299c5c">4.1. Function description</a></li>
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<li><a href="#org12a207e">4.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
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<li><a href="#org46c8589">4.2. Optional Parameters</a></li>
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<ul>
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<li><a href="#orgd8d9b14">4.3. Cube Creation</a></li>
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<li><a href="#orgecee38f">Function description</a></li>
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<li><a href="#org181d1d8">4.4. Vectors of each leg</a></li>
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<li><a href="#orgb9948d8">Documentation</a></li>
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<li><a href="#orgb396e98">4.5. Verification of Height of the Stewart Platform</a></li>
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<li><a href="#orgbfb0bd5">Optional Parameters</a></li>
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<li><a href="#orgf38af83">4.6. Determinate the location of the joints</a></li>
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<li><a href="#orgcb22d51">Position of the Cube</a></li>
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<li><a href="#orgdf9e3cf">4.7. Returns Stewart Structure</a></li>
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<li><a href="#org2f09e98">Compute the pose</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgf8fb731">5. <span class="todo TODO">TODO</span> Tests</a>
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</ul>
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</li>
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<li><a href="#org4eaf218">5. <span class="todo TODO">TODO</span> initializeCubicConfiguration</a>
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<ul>
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<ul>
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<li><a href="#org4434fe5">5.1. First attempt to parametrisation</a></li>
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<li><a href="#org4fb2bc6">5.1. Function description</a></li>
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<li><a href="#org723e6eb">5.2. Second attempt</a></li>
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<li><a href="#orgb540658">5.2. Optional Parameters</a></li>
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<li><a href="#orgcc173ac">5.3. Generate the Stewart platform for a Cubic configuration</a></li>
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<li><a href="#org1474f46">5.3. Cube Creation</a></li>
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<li><a href="#org03d2dd7">5.4. Vectors of each leg</a></li>
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<li><a href="#orgfed36b2">5.5. Verification of Height of the Stewart Platform</a></li>
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<li><a href="#orgdb27b02">5.6. Determinate the location of the joints</a></li>
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<li><a href="#org5079890">5.7. Returns Stewart Structure</a></li>
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</ul>
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</li>
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<li><a href="#orgd9f1e20">6. <span class="todo TODO">TODO</span> Tests</a>
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<ul>
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<li><a href="#orgea7297c">6.1. First attempt to parametrisation</a></li>
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<li><a href="#orgd6ed3c3">6.2. Second attempt</a></li>
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<li><a href="#orgf39eafa">6.3. Generate the Stewart platform for a Cubic configuration</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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<p>
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<p>
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The discovery of the Cubic configuration is done in <a class='org-ref-reference' href="#geng94_six_degree_of_freed_activ">geng94_six_degree_of_freed_activ</a>.
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The discovery of the Cubic configuration is done in <a class='org-ref-reference' href="#geng94_six_degree_of_freed_activ">geng94_six_degree_of_freed_activ</a>.
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Further analysis is conducted in <a class='org-ref-reference' href="#jafari03_orthog_gough_stewar_platf_microm">jafari03_orthog_gough_stewar_platf_microm</a>.
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Further analysis is conducted in
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</p>
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</p>
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<p>
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People using orthogonal/cubic configuration: <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>.
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</p>
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<p>
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<p>
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The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
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The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
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</p>
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</p>
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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="#org8b1f609">4</a>).
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The cubic (or orthogonal) configuration of the Stewart platform is now widely used (<a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>,<a class='org-ref-reference' href="#jafari03_orthog_gough_stewar_platf_microm">jafari03_orthog_gough_stewar_platf_microm</a>).
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</p>
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</p>
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<p>
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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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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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</p>
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<div id="outline-container-org86c83bf" class="outline-2">
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<p>
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<h2 id="org86c83bf"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2>
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To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#org83d7db1">5</a>).
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<div id="outline-container-org43e4755" class="outline-2">
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<h2 id="org43e4755"><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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<div class="outline-text-2" id="text-1">
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<p>
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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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The goal is to study the benefits of using a cubic configuration:
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@ -352,20 +349,20 @@ 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-org0b05973" class="outline-2">
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<div id="outline-container-org7c85269" class="outline-2">
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<h2 id="org0b05973"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</h2>
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<h2 id="org7c85269"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</h2>
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<div class="outline-text-2" id="text-2">
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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-org3f035e8" class="outline-3">
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<div id="outline-container-org7a2e2af" class="outline-3">
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<h3 id="org3f035e8"><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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<h3 id="org7a2e2af"><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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<div class="outline-text-3" id="text-2-1">
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<p>
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<p>
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We create a cubic Stewart platform (figure <a href="#org1effc0f">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="#org01dbe25">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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The Jacobian matrix is estimated at the location of the center of the cube.
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</p>
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</p>
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<div id="org1effc0f" class="figure">
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<div id="org01dbe25" 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><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" />
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</p>
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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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<p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p>
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@ -469,11 +466,11 @@ save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-strin
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<div id="outline-container-org77ecb36" class="outline-3">
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<div id="outline-container-orgdd082ef" class="outline-3">
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<h3 id="org77ecb36"><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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<h3 id="orgdd082ef"><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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<div class="outline-text-3" id="text-2-2">
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<p>
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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="#org1effc0f">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="#org01dbe25">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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The Jacobian matrix is not estimated at the location of the center of the cube.
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</p>
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</p>
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@ -573,16 +570,16 @@ stewart = computeGeometricalProperties(stewart, opts);
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<div id="outline-container-org42ea8ad" class="outline-3">
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<div id="outline-container-org314610d" class="outline-3">
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<h3 id="org42ea8ad"><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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<h3 id="org314610d"><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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<div class="outline-text-3" id="text-2-3">
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<p>
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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="#org3f10bc2">2</a>).
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Here, the “center” of the Stewart platform is not at the cube center (figure <a href="#org4aa7b60">2</a>).
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The Jacobian is estimated at the cube center.
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The Jacobian is estimated at the cube center.
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</p>
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</p>
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<div id="org3f10bc2" class="figure">
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<div id="org4aa7b60" 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><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
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</p>
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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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<p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p>
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@ -695,8 +692,8 @@ 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-org38870ce" class="outline-3">
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<div id="outline-container-org460e492" class="outline-3">
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<h3 id="org38870ce"><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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<h3 id="org460e492"><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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<div class="outline-text-3" id="text-2-4">
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<p>
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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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Here, the “center” of the Stewart platform is not at the cube center.
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@ -810,8 +807,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
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<div id="outline-container-orgccb5ef0" class="outline-3">
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<h3 id="org08c7461"><span class="section-number-3">2.5</span> Conclusion</h3>
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<h3 id="orgccb5ef0"><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="outline-text-3" id="text-2-5">
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<div class="important">
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<div class="important">
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<ul class="org-ul">
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<ul class="org-ul">
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@ -824,8 +821,8 @@ 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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<h2 id="orgc4c2abd"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Cubic size analysis</h2>
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<h2 id="org29d657d"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Cubic size analysis</h2>
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<div class="outline-text-2" id="text-3">
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<div class="outline-text-2" id="text-3">
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<p>
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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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We here study the effect of the size of the cube used for the Stewart configuration.
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||||||
@ -900,7 +897,7 @@ xlabel(<span class="org-string">'Cube Size [mm]'</span>); ylabel(<span class="or
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="org659a01f" class="figure">
|
<div id="orgeec8e66" class="figure">
|
||||||
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
|
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
|
||||||
</p>
|
</p>
|
||||||
<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>
|
<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>
|
||||||
@ -921,17 +918,143 @@ In that case, the legs will the further separated. Size of the cube is then limi
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org36a27e6" class="outline-2">
|
<div id="outline-container-orgc12b0fc" class="outline-2">
|
||||||
<h2 id="org36a27e6"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> initializeCubicConfiguration</h2>
|
<h2 id="orgc12b0fc"><span class="section-number-2">4</span> Functions</h2>
|
||||||
<div class="outline-text-2" id="text-4">
|
<div class="outline-text-2" id="text-4">
|
||||||
<p>
|
<p>
|
||||||
<a id="org8b1f609"></a>
|
<a id="orgee0330a"></a>
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf299c5c" class="outline-3">
|
<div id="outline-container-org12a207e" class="outline-3">
|
||||||
<h3 id="orgf299c5c"><span class="section-number-3">4.1</span> Function description</h3>
|
<h3 id="org12a207e"><span class="section-number-3">4.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
|
||||||
<div class="outline-text-3" id="text-4-1">
|
<div class="outline-text-3" id="text-4-1">
|
||||||
|
<p>
|
||||||
|
<a id="org0a684d8"></a>
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
This Matlab function is accessible <a href="src/generateCubicConfiguration.m">here</a>.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgecee38f" class="outline-4">
|
||||||
|
<h4 id="orgecee38f">Function description</h4>
|
||||||
|
<div class="outline-text-4" id="text-orgecee38f">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
||||||
|
<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(stewart, args)</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Inputs:</span>
|
||||||
|
<span class="org-comment">% - stewart - A structure with the following fields</span>
|
||||||
|
<span class="org-comment">% - H [1x1] - Total height of the platform [m]</span>
|
||||||
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
||||||
|
<span class="org-comment">% - Hc [1x1] - Height of the "useful" part of the cube [m]</span>
|
||||||
|
<span class="org-comment">% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]</span>
|
||||||
|
<span class="org-comment">% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]</span>
|
||||||
|
<span class="org-comment">% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Outputs:</span>
|
||||||
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
||||||
|
<span class="org-comment">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
||||||
|
<span class="org-comment">% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgb9948d8" class="outline-4">
|
||||||
|
<h4 id="orgb9948d8">Documentation</h4>
|
||||||
|
<div class="outline-text-4" id="text-orgb9948d8">
|
||||||
|
|
||||||
|
<div id="orgff1f403" class="figure">
|
||||||
|
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
|
||||||
|
</p>
|
||||||
|
<p><span class="figure-number">Figure 4: </span>Cubic Configuration</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgbfb0bd5" class="outline-4">
|
||||||
|
<h4 id="orgbfb0bd5">Optional Parameters</h4>
|
||||||
|
<div class="outline-text-4" id="text-orgbfb0bd5">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">arguments
|
||||||
|
stewart
|
||||||
|
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e<span class="org-type">-</span>3
|
||||||
|
args.FOc (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
||||||
|
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
||||||
|
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
||||||
|
<span class="org-keyword">end</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgcb22d51" class="outline-4">
|
||||||
|
<h4 id="orgcb22d51">Position of the Cube</h4>
|
||||||
|
<div class="outline-text-4" id="text-orgcb22d51">
|
||||||
|
<p>
|
||||||
|
We define the useful points of the cube with respect to the Cube’s center.
|
||||||
|
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
|
||||||
|
located at the center of the cube and aligned with {F} and {M}.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">sx = [ 2; <span class="org-type">-</span>1; <span class="org-type">-</span>1];
|
||||||
|
sy = [ 0; 1; <span class="org-type">-</span>1];
|
||||||
|
sz = [ 1; 1; 1];
|
||||||
|
|
||||||
|
R = [sx, sy, sz]<span class="org-type">./</span>vecnorm([sx, sy, sz]);
|
||||||
|
|
||||||
|
L = args.Hc<span class="org-type">*</span>sqrt(3);
|
||||||
|
|
||||||
|
Cc = R<span class="org-type">'*</span>[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] <span class="org-type">-</span> [0;0;1.5<span class="org-type">*</span>args.Hc];
|
||||||
|
|
||||||
|
CCf = [Cc(<span class="org-type">:</span>,1), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,1)]; <span class="org-comment">% CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg</span>
|
||||||
|
CCm = [Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,6), Cc(<span class="org-type">:</span>,6)]; <span class="org-comment">% CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org2f09e98" class="outline-4">
|
||||||
|
<h4 id="org2f09e98">Compute the pose</h4>
|
||||||
|
<div class="outline-text-4" id="text-org2f09e98">
|
||||||
|
<p>
|
||||||
|
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">CSi = (CCm <span class="org-type">-</span> CCf)<span class="org-type">./</span>vecnorm(CCm <span class="org-type">-</span> CCf);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">stewart.Fa = CCf <span class="org-type">+</span> [0; 0; args.FOc] <span class="org-type">+</span> ((args.FHa<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
||||||
|
stewart.Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>stewart.H] <span class="org-type">+</span> ((stewart.H<span class="org-type">-</span>args.MHb<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org4eaf218" class="outline-2">
|
||||||
|
<h2 id="org4eaf218"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> initializeCubicConfiguration</h2>
|
||||||
|
<div class="outline-text-2" id="text-5">
|
||||||
|
<p>
|
||||||
|
<a id="org83d7db1"></a>
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org4fb2bc6" class="outline-3">
|
||||||
|
<h3 id="org4fb2bc6"><span class="section-number-3">5.1</span> Function description</h3>
|
||||||
|
<div class="outline-text-3" id="text-5-1">
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">initializeCubicConfiguration</span>(<span class="org-variable-name">opts_param</span>)
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">initializeCubicConfiguration</span>(<span class="org-variable-name">opts_param</span>)
|
||||||
</pre>
|
</pre>
|
||||||
@ -939,9 +1062,9 @@ In that case, the legs will the further separated. Size of the cube is then limi
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org46c8589" class="outline-3">
|
<div id="outline-container-orgb540658" class="outline-3">
|
||||||
<h3 id="org46c8589"><span class="section-number-3">4.2</span> Optional Parameters</h3>
|
<h3 id="orgb540658"><span class="section-number-3">5.2</span> Optional Parameters</h3>
|
||||||
<div class="outline-text-3" id="text-4-2">
|
<div class="outline-text-3" id="text-5-2">
|
||||||
<p>
|
<p>
|
||||||
Default values for opts.
|
Default values for opts.
|
||||||
</p>
|
</p>
|
||||||
@ -969,9 +1092,9 @@ Populate opts with input parameters
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd8d9b14" class="outline-3">
|
<div id="outline-container-org1474f46" class="outline-3">
|
||||||
<h3 id="orgd8d9b14"><span class="section-number-3">4.3</span> Cube Creation</h3>
|
<h3 id="org1474f46"><span class="section-number-3">5.3</span> Cube Creation</h3>
|
||||||
<div class="outline-text-3" id="text-4-3">
|
<div class="outline-text-3" id="text-5-3">
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">points = [0, 0, 0; ...
|
<pre class="src src-matlab">points = [0, 0, 0; ...
|
||||||
0, 0, 1; ...
|
0, 0, 1; ...
|
||||||
@ -1015,9 +1138,9 @@ We use to rotation matrix to rotate the cube
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org181d1d8" class="outline-3">
|
<div id="outline-container-org03d2dd7" class="outline-3">
|
||||||
<h3 id="org181d1d8"><span class="section-number-3">4.4</span> Vectors of each leg</h3>
|
<h3 id="org03d2dd7"><span class="section-number-3">5.4</span> Vectors of each leg</h3>
|
||||||
<div class="outline-text-3" id="text-4-4">
|
<div class="outline-text-3" id="text-5-4">
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">leg_indices = [3, 4; ...
|
<pre class="src src-matlab">leg_indices = [3, 4; ...
|
||||||
2, 4; ...
|
2, 4; ...
|
||||||
@ -1044,9 +1167,9 @@ legs_start = zeros(6, 3);
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgb396e98" class="outline-3">
|
<div id="outline-container-orgfed36b2" class="outline-3">
|
||||||
<h3 id="orgb396e98"><span class="section-number-3">4.5</span> Verification of Height of the Stewart Platform</h3>
|
<h3 id="orgfed36b2"><span class="section-number-3">5.5</span> Verification of Height of the Stewart Platform</h3>
|
||||||
<div class="outline-text-3" id="text-4-5">
|
<div class="outline-text-3" id="text-5-5">
|
||||||
<p>
|
<p>
|
||||||
If the Stewart platform is not contained in the cube, throw an error.
|
If the Stewart platform is not contained in the cube, throw an error.
|
||||||
</p>
|
</p>
|
||||||
@ -1064,9 +1187,9 @@ If the Stewart platform is not contained in the cube, throw an error.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf38af83" class="outline-3">
|
<div id="outline-container-orgdb27b02" class="outline-3">
|
||||||
<h3 id="orgf38af83"><span class="section-number-3">4.6</span> Determinate the location of the joints</h3>
|
<h3 id="orgdb27b02"><span class="section-number-3">5.6</span> Determinate the location of the joints</h3>
|
||||||
<div class="outline-text-3" id="text-4-6">
|
<div class="outline-text-3" id="text-5-6">
|
||||||
<p>
|
<p>
|
||||||
We now determine the location of the joints on the fixed platform w.r.t the fixed frame \(\{A\}\).
|
We now determine the location of the joints on the fixed platform w.r.t the fixed frame \(\{A\}\).
|
||||||
\(\{A\}\) is fixed to the bottom of the base.
|
\(\{A\}\) is fixed to the bottom of the base.
|
||||||
@ -1110,9 +1233,9 @@ Ab = Ab <span class="org-type">-</span> h<span class="org-type">*</span>[0, 0, 1
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgdf9e3cf" class="outline-3">
|
<div id="outline-container-org5079890" class="outline-3">
|
||||||
<h3 id="orgdf9e3cf"><span class="section-number-3">4.7</span> Returns Stewart Structure</h3>
|
<h3 id="org5079890"><span class="section-number-3">5.7</span> Returns Stewart Structure</h3>
|
||||||
<div class="outline-text-3" id="text-4-7">
|
<div class="outline-text-3" id="text-5-7">
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab"> stewart = struct();
|
<pre class="src src-matlab"> stewart = struct();
|
||||||
stewart.Aa = Aa;
|
stewart.Aa = Aa;
|
||||||
@ -1126,18 +1249,18 @@ Ab = Ab <span class="org-type">-</span> h<span class="org-type">*</span>[0, 0, 1
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf8fb731" class="outline-2">
|
<div id="outline-container-orgd9f1e20" class="outline-2">
|
||||||
<h2 id="orgf8fb731"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> Tests</h2>
|
<h2 id="orgd9f1e20"><span class="section-number-2">6</span> <span class="todo TODO">TODO</span> Tests</h2>
|
||||||
<div class="outline-text-2" id="text-5">
|
<div class="outline-text-2" id="text-6">
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-org4434fe5" class="outline-3">
|
<div id="outline-container-orgea7297c" class="outline-3">
|
||||||
<h3 id="org4434fe5"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3>
|
<h3 id="orgea7297c"><span class="section-number-3">6.1</span> First attempt to parametrisation</h3>
|
||||||
<div class="outline-text-3" id="text-5-1">
|
<div class="outline-text-3" id="text-6-1">
|
||||||
|
|
||||||
<div id="org8dfcb96" class="figure">
|
<div id="org65e66e5" class="figure">
|
||||||
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
|
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 4: </span>Schematic of the bottom plates with all the parameters</p>
|
<p><span class="figure-number">Figure 5: </span>Schematic of the bottom plates with all the parameters</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -1169,9 +1292,9 @@ Lets express \(a_i\), \(b_i\) and \(a_j\):
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org723e6eb" class="outline-3">
|
<div id="outline-container-orgd6ed3c3" class="outline-3">
|
||||||
<h3 id="org723e6eb"><span class="section-number-3">5.2</span> Second attempt</h3>
|
<h3 id="orgd6ed3c3"><span class="section-number-3">6.2</span> Second attempt</h3>
|
||||||
<div class="outline-text-3" id="text-5-2">
|
<div class="outline-text-3" id="text-6-2">
|
||||||
<p>
|
<p>
|
||||||
We start with the point of a cube in space:
|
We start with the point of a cube in space:
|
||||||
</p>
|
</p>
|
||||||
@ -1297,9 +1420,9 @@ Let’s then estimate the middle position of the platform
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgcc173ac" class="outline-3">
|
<div id="outline-container-orgf39eafa" class="outline-3">
|
||||||
<h3 id="orgcc173ac"><span class="section-number-3">5.3</span> Generate the Stewart platform for a Cubic configuration</h3>
|
<h3 id="orgf39eafa"><span class="section-number-3">6.3</span> Generate the Stewart platform for a Cubic configuration</h3>
|
||||||
<div class="outline-text-3" id="text-5-3">
|
<div class="outline-text-3" id="text-6-3">
|
||||||
<p>
|
<p>
|
||||||
First we defined the height of the Hexapod.
|
First we defined the height of the Hexapod.
|
||||||
</p>
|
</p>
|
||||||
@ -1360,15 +1483,15 @@ zlim([0, 2]);
|
|||||||
<p>
|
<p>
|
||||||
|
|
||||||
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
||||||
<ul class='org-ref-bib'><li><a id="geng94_six_degree_of_freed_activ">[geng94_six_degree_of_freed_activ]</a> <a name="geng94_six_degree_of_freed_activ"></a>Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, <i>IEEE Transactions on Control Systems Technology</i>, <b>2(1)</b>, 45-53 (1994). <a href="https://doi.org/10.1109/87.273110">link</a>. <a href="http://dx.doi.org/10.1109/87.273110">doi</a>.</li>
|
<ul class='org-ref-bib'><li><a id="geng94_six_degree_of_freed_activ">[geng94_six_degree_of_freed_activ]</a> <a name="geng94_six_degree_of_freed_activ"></a>Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, <i>IEEE Transactions on Control Systems Technology</i>, <b>2(1)</b>, 45-53 (1994). <a href="https://doi.org/10.1109/87.273110">link</a>. <a href="http://dx.doi.org/10.1109/87.273110">doi</a>.</li>
|
||||||
|
<li><a id="preumont07_six_axis_singl_stage_activ">[preumont07_six_axis_singl_stage_activ]</a> <a name="preumont07_six_axis_singl_stage_activ"></a>Preumont, Horodinca, Romanescu, de Marneffe, Avraam, Deraemaeker, Bossens & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, <i>Journal of Sound and Vibration</i>, <b>300(3-5)</b>, 644-661 (2007). <a href="https://doi.org/10.1016/j.jsv.2006.07.050">link</a>. <a href="http://dx.doi.org/10.1016/j.jsv.2006.07.050">doi</a>.</li>
|
||||||
<li><a id="jafari03_orthog_gough_stewar_platf_microm">[jafari03_orthog_gough_stewar_platf_microm]</a> <a name="jafari03_orthog_gough_stewar_platf_microm"></a>Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, <i>IEEE Transactions on Robotics and Automation</i>, <b>19(4)</b>, 595-603 (2003). <a href="https://doi.org/10.1109/tra.2003.814506">link</a>. <a href="http://dx.doi.org/10.1109/tra.2003.814506">doi</a>.</li>
|
<li><a id="jafari03_orthog_gough_stewar_platf_microm">[jafari03_orthog_gough_stewar_platf_microm]</a> <a name="jafari03_orthog_gough_stewar_platf_microm"></a>Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, <i>IEEE Transactions on Robotics and Automation</i>, <b>19(4)</b>, 595-603 (2003). <a href="https://doi.org/10.1109/tra.2003.814506">link</a>. <a href="http://dx.doi.org/10.1109/tra.2003.814506">doi</a>.</li>
|
||||||
<li><a id="preumont07_six_axis_singl_stage_activ">[preumont07_six_axis_singl_stage_activ]</a> <a name="preumont07_six_axis_singl_stage_activ"></a>Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, <i>Journal of Sound and Vibration</i>, <b>300(3-5)</b>, 644-661 (2007). <a href="https://doi.org/10.1016/j.jsv.2006.07.050">link</a>. <a href="http://dx.doi.org/10.1016/j.jsv.2006.07.050">doi</a>.</li>
|
|
||||||
</ul>
|
</ul>
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
<div id="postamble" class="status">
|
<div id="postamble" class="status">
|
||||||
<p class="author">Author: Dehaeze Thomas</p>
|
<p class="author">Author: Dehaeze Thomas</p>
|
||||||
<p class="date">Created: 2020-01-27 lun. 17:41</p>
|
<p class="date">Created: 2020-02-06 jeu. 17:25</p>
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
</html>
|
</html>
|
||||||
|
@ -22,19 +22,18 @@
|
|||||||
:END:
|
:END:
|
||||||
|
|
||||||
* Introduction :ignore:
|
* Introduction :ignore:
|
||||||
The discovery of the Cubic configuration is done in citenum:geng94_six_degree_of_freed_activ.
|
The discovery of the Cubic configuration is done in cite:geng94_six_degree_of_freed_activ.
|
||||||
Further analysis is conducted in cite:jafari03_orthog_gough_stewar_platf_microm.
|
Further analysis is conducted in
|
||||||
|
|
||||||
People using orthogonal/cubic configuration: cite:preumont07_six_axis_singl_stage_activ.
|
|
||||||
|
|
||||||
|
|
||||||
The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
|
The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
|
||||||
|
|
||||||
To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]).
|
The cubic (or orthogonal) configuration of the Stewart platform is now widely used (cite:preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm).
|
||||||
|
|
||||||
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).
|
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:
|
To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]).
|
||||||
|
|
||||||
|
* Matlab Init :noexport:ignore:
|
||||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
<<matlab-dir>>
|
<<matlab-dir>>
|
||||||
#+end_src
|
#+end_src
|
||||||
@ -304,6 +303,103 @@ We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linear
|
|||||||
In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
|
In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
|
||||||
#+end_important
|
#+end_important
|
||||||
|
|
||||||
|
* Functions
|
||||||
|
<<sec:functions>>
|
||||||
|
|
||||||
|
** =generateCubicConfiguration=: Generate a Cubic Configuration
|
||||||
|
:PROPERTIES:
|
||||||
|
:header-args:matlab+: :tangle src/generateCubicConfiguration.m
|
||||||
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||||
|
:END:
|
||||||
|
<<sec:generateCubicConfiguration>>
|
||||||
|
|
||||||
|
This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here]].
|
||||||
|
|
||||||
|
*** Function description
|
||||||
|
:PROPERTIES:
|
||||||
|
:UNNUMBERED: t
|
||||||
|
:END:
|
||||||
|
#+begin_src matlab
|
||||||
|
function [stewart] = generateCubicConfiguration(stewart, args)
|
||||||
|
% generateCubicConfiguration - Generate a Cubic Configuration
|
||||||
|
%
|
||||||
|
% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
|
||||||
|
%
|
||||||
|
% Inputs:
|
||||||
|
% - stewart - A structure with the following fields
|
||||||
|
% - H [1x1] - Total height of the platform [m]
|
||||||
|
% - args - Can have the following fields:
|
||||||
|
% - Hc [1x1] - Height of the "useful" part of the cube [m]
|
||||||
|
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
|
||||||
|
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
|
||||||
|
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
|
||||||
|
%
|
||||||
|
% Outputs:
|
||||||
|
% - stewart - updated Stewart structure with the added fields:
|
||||||
|
% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
|
||||||
|
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Documentation
|
||||||
|
:PROPERTIES:
|
||||||
|
:UNNUMBERED: t
|
||||||
|
:END:
|
||||||
|
#+name: fig:cubic-configuration-definition
|
||||||
|
#+caption: Cubic Configuration
|
||||||
|
[[file:figs/cubic-configuration-definition.png]]
|
||||||
|
|
||||||
|
*** Optional Parameters
|
||||||
|
:PROPERTIES:
|
||||||
|
:UNNUMBERED: t
|
||||||
|
:END:
|
||||||
|
#+begin_src matlab
|
||||||
|
arguments
|
||||||
|
stewart
|
||||||
|
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
|
||||||
|
args.FOc (1,1) double {mustBeNumeric} = 50e-3
|
||||||
|
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
|
||||||
|
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Position of the Cube
|
||||||
|
:PROPERTIES:
|
||||||
|
:UNNUMBERED: t
|
||||||
|
:END:
|
||||||
|
We define the useful points of the cube with respect to the Cube's center.
|
||||||
|
${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is
|
||||||
|
located at the center of the cube and aligned with {F} and {M}.
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
sx = [ 2; -1; -1];
|
||||||
|
sy = [ 0; 1; -1];
|
||||||
|
sz = [ 1; 1; 1];
|
||||||
|
|
||||||
|
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
|
||||||
|
|
||||||
|
L = args.Hc*sqrt(3);
|
||||||
|
|
||||||
|
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
|
||||||
|
|
||||||
|
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
|
||||||
|
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Compute the pose
|
||||||
|
:PROPERTIES:
|
||||||
|
:UNNUMBERED: t
|
||||||
|
:END:
|
||||||
|
We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).
|
||||||
|
#+begin_src matlab
|
||||||
|
CSi = (CCm - CCf)./vecnorm(CCm - CCf);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
|
||||||
|
#+begin_src matlab
|
||||||
|
stewart.Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
||||||
|
stewart.Mb = CCf + [0; 0; args.FOc-stewart.H] + ((stewart.H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
* TODO initializeCubicConfiguration
|
* TODO initializeCubicConfiguration
|
||||||
:PROPERTIES:
|
:PROPERTIES:
|
||||||
:HEADER-ARGS:matlab+: :exports code
|
:HEADER-ARGS:matlab+: :exports code
|
||||||
@ -615,5 +711,5 @@ And we plot the legs.
|
|||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
* Bibliography :ignore:
|
* Bibliography :ignore:
|
||||||
bibliographystyle:unsrt
|
bibliographystyle:unsrtnat
|
||||||
bibliography:ref.bib
|
bibliography:ref.bib
|
||||||
|
File diff suppressed because it is too large
Load Diff
@ -18,6 +18,18 @@
|
|||||||
#+PROPERTY: header-args:matlab+ :noweb yes
|
#+PROPERTY: header-args:matlab+ :noweb yes
|
||||||
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
||||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||||
|
|
||||||
|
#+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 file raw replace
|
||||||
|
#+PROPERTY: header-args:latex+ :buffer no
|
||||||
|
#+PROPERTY: header-args:latex+ :eval no-export
|
||||||
|
#+PROPERTY: header-args:latex+ :exports results
|
||||||
|
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||||
|
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||||
|
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||||
:END:
|
:END:
|
||||||
|
|
||||||
* Introduction :ignore:
|
* Introduction :ignore:
|
||||||
@ -82,10 +94,12 @@ Then, we define the *location of the spherical joints* (see Figure [[fig:joint_l
|
|||||||
The location of the joints will define the Geometry of the Stewart platform.
|
The location of the joints will define the Geometry of the Stewart platform.
|
||||||
Many characteristics of the platform depend on the location of the joints.
|
Many characteristics of the platform depend on the location of the joints.
|
||||||
|
|
||||||
The location of the joints can be set to arbitrary positions (function =generateGeneralConfiguration= described [[sec:generateGeneralConfiguration][here]]) or can be computed to obtain specific configurations such as:
|
The location of the joints can be set to arbitrary positions or it can be computed to obtain specific configurations such as:
|
||||||
- A cubic configuration: function =generateCubicConfiguration= ([[sec:generateCubicConfiguration][link]]).
|
- A cubic configuration: function =generateCubicConfiguration= (described in [[file:cubic-configuration.org][this]] file)
|
||||||
- A symmetrical configuration
|
- A symmetrical configuration
|
||||||
|
|
||||||
|
A function (=generateGeneralConfiguration=) to set the position of the joints on a circle is described [[sec:generateGeneralConfiguration][here]].
|
||||||
|
|
||||||
The location of the spherical joints are then given by ${}^{F}\bm{a}_{i}$ and ${}^{M}\bm{b}_{i}$.
|
The location of the spherical joints are then given by ${}^{F}\bm{a}_{i}$ and ${}^{M}\bm{b}_{i}$.
|
||||||
|
|
||||||
#+name: fig:joint_location
|
#+name: fig:joint_location
|
||||||
@ -421,7 +435,55 @@ This Matlab function is accessible [[file:src/generateGeneralConfiguration.m][he
|
|||||||
:UNNUMBERED: t
|
:UNNUMBERED: t
|
||||||
:END:
|
:END:
|
||||||
Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}.
|
Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}.
|
||||||
The radius of the circles can be chosen as well as the angles where the joints are located.
|
The radius of the circles can be chosen as well as the angles where the joints are located (see Figure [[fig:joint_position_general]]).
|
||||||
|
|
||||||
|
#+begin_src latex :file stewart_bottom_plate.pdf :exports results
|
||||||
|
\begin{tikzpicture}
|
||||||
|
% Internal and external limit
|
||||||
|
\draw[fill=white!80!black] (0, 0) circle [radius=3];
|
||||||
|
% Circle where the joints are located
|
||||||
|
\draw[dashed] (0, 0) circle [radius=2.5];
|
||||||
|
|
||||||
|
% Bullets for the positions of the joints
|
||||||
|
\node[] (J1) at ( 80:2.5){$\bullet$};
|
||||||
|
\node[] (J2) at (100:2.5){$\bullet$};
|
||||||
|
\node[] (J3) at (200:2.5){$\bullet$};
|
||||||
|
\node[] (J4) at (220:2.5){$\bullet$};
|
||||||
|
\node[] (J5) at (320:2.5){$\bullet$};
|
||||||
|
\node[] (J6) at (340:2.5){$\bullet$};
|
||||||
|
|
||||||
|
% Name of the points
|
||||||
|
\node[above right] at (J1) {$a_{1}$};
|
||||||
|
\node[above left] at (J2) {$a_{2}$};
|
||||||
|
\node[above left] at (J3) {$a_{3}$};
|
||||||
|
\node[right ] at (J4) {$a_{4}$};
|
||||||
|
\node[left ] at (J5) {$a_{5}$};
|
||||||
|
\node[above right] at (J6) {$a_{6}$};
|
||||||
|
|
||||||
|
% First 2 angles
|
||||||
|
\draw[dashed, ->] (0:1) arc [start angle=0, end angle=80, radius=1] node[below right]{$\theta_{1}$};
|
||||||
|
\draw[dashed, ->] (0:1.5) arc [start angle=0, end angle=100, radius=1.5] node[left ]{$\theta_{2}$};
|
||||||
|
|
||||||
|
% Division of 360 degrees by 3
|
||||||
|
\draw[dashed] (0, 0) -- ( 80:3.2);
|
||||||
|
\draw[dashed] (0, 0) -- (100:3.2);
|
||||||
|
\draw[dashed] (0, 0) -- (200:3.2);
|
||||||
|
\draw[dashed] (0, 0) -- (220:3.2);
|
||||||
|
\draw[dashed] (0, 0) -- (320:3.2);
|
||||||
|
\draw[dashed] (0, 0) -- (340:3.2);
|
||||||
|
|
||||||
|
% Radius for the position of the joints
|
||||||
|
\draw[<->] (0, 0) --node[near end, above]{$R$} (180:2.5);
|
||||||
|
|
||||||
|
\draw[->] (0, 0) -- ++(3.4, 0) node[above]{$x$};
|
||||||
|
\draw[->] (0, 0) -- ++(0, 3.4) node[left]{$y$};
|
||||||
|
\end{tikzpicture}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:joint_position_general
|
||||||
|
#+caption: Position of the joints
|
||||||
|
#+RESULTS:
|
||||||
|
[[file:figs/stewart_bottom_plate.png]]
|
||||||
|
|
||||||
*** Optional Parameters
|
*** Optional Parameters
|
||||||
:PROPERTIES:
|
:PROPERTIES:
|
||||||
|
Loading…
Reference in New Issue
Block a user