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<title>Stewart Platforms</title>
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<div id="content">
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<h1 class="title">Stewart Platforms</h1>
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<div id="outline-container-orge672724" class="outline-2">
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<h2 id="orge672724"><span class="section-number-2">1</span> Simscape Model</h2>
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<p>
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<a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>
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<div id="outline-container-org9cd44e0" class="outline-2">
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<h2 id="org9cd44e0"><span class="section-number-2">1</span> Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
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<ul class="org-ul">
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<li><a href="simscape-model.html">Model of the Stewart Platform</a></li>
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<div id="outline-container-orgfce4cb7" class="outline-2">
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<h2 id="orgfce4cb7"><span class="section-number-2">2</span> Architecture Study</h2>
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<div id="outline-container-org7a44762" class="outline-2">
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<h2 id="org7a44762"><span class="section-number-2">2</span> Architecture Study</h2>
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<div class="outline-text-2" id="text-2">
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<li><a href="kinematic-study.html">Kinematic Study</a></li>
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<h2 id="org92e9216"><span class="section-number-2">3</span> Motion Control</h2>
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<h2 id="org77767cc"><span class="section-number-2">3</span> Motion Control</h2>
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<div class="outline-text-2" id="text-3">
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<li>Active Damping</li>
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<div id="outline-container-org5ab21e2" class="outline-2">
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<h2 id="org5ab21e2"><span class="section-number-2">4</span> Notes about Stewart platforms</h2>
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<div id="outline-container-org9d06c58" class="outline-2">
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<h2 id="org9d06c58"><span class="section-number-2">4</span> Notes about Stewart platforms</h2>
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<div class="outline-text-2" id="text-4">
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<div id="outline-container-orgf0627f0" class="outline-3">
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<h3 id="orgf0627f0"><span class="section-number-3">4.1</span> Jacobian</h3>
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<div id="outline-container-orgffe6651" class="outline-3">
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<h3 id="orgffe6651"><span class="section-number-3">4.1</span> Jacobian</h3>
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<div class="outline-text-3" id="text-4-1">
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<div id="outline-container-orge3fb927" class="outline-4">
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<h4 id="orge3fb927"><span class="section-number-4">4.1.1</span> Relation to platform parameters</h4>
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<div id="outline-container-org6b92660" class="outline-4">
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<h4 id="org6b92660"><span class="section-number-4">4.1.1</span> Relation to platform parameters</h4>
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<div class="outline-text-4" id="text-4-1-1">
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<p>
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A Jacobian is defined by:
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@@ -329,8 +336,8 @@ Then, the choice of \(O_B\) changes the Jacobian.
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<div id="outline-container-org99049d5" class="outline-4">
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<h4 id="org99049d5"><span class="section-number-4">4.1.2</span> Jacobian for displacement</h4>
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<div id="outline-container-orgcec2e05" class="outline-4">
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<h4 id="orgcec2e05"><span class="section-number-4">4.1.2</span> Jacobian for displacement</h4>
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<div class="outline-text-4" id="text-4-1-2">
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<p>
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\[ \dot{q} = J \dot{X} \]
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@@ -347,8 +354,8 @@ For very small displacements \(\delta q\) and \(\delta X\), we have \(\delta q =
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<div id="outline-container-orgb7963ed" class="outline-4">
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<h4 id="orgb7963ed"><span class="section-number-4">4.1.3</span> Jacobian for forces</h4>
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<div id="outline-container-orgbf33a4e" class="outline-4">
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<h4 id="orgbf33a4e"><span class="section-number-4">4.1.3</span> Jacobian for forces</h4>
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<div class="outline-text-4" id="text-4-1-3">
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<p>
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\[ F = J^T \tau \]
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@@ -362,8 +369,8 @@ With:
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<h3 id="org9fcd675"><span class="section-number-3">4.2</span> Stiffness matrix \(K\)</h3>
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<div id="outline-container-org3710914" class="outline-3">
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<h3 id="org3710914"><span class="section-number-3">4.2</span> Stiffness matrix \(K\)</h3>
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<div class="outline-text-3" id="text-4-2">
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<p>
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\[ K = J^T \text{diag}(k_i) J \]
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@@ -396,8 +403,8 @@ The compliance element \(C_{ij}\) is then the stiffness
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<div id="outline-container-orge5eb09a" class="outline-3">
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<h3 id="orge5eb09a"><span class="section-number-3">4.3</span> Coupling</h3>
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<div id="outline-container-orgd5d2c08" class="outline-3">
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<h3 id="orgd5d2c08"><span class="section-number-3">4.3</span> Coupling</h3>
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<div class="outline-text-3" id="text-4-3">
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<p>
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What causes the coupling from \(F_i\) to \(X_i\) ?
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@@ -418,7 +425,7 @@ What causes the coupling from \(F_i\) to \(X_i\) ?
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</div>
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<div id="org064c4c6" class="figure">
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<div id="orgc118680" class="figure">
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<p><img src="figs/coupling.png" alt="coupling.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Block diagram to control an hexapod</p>
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@@ -441,7 +448,9 @@ Thus, the system is uncoupled if \(G\) and \(K\) are diagonal.
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<p>
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<a href="references.bib">references.bib</a>
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<h1 class='org-ref-bib-h1'>Bibliography</h1>
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<ul class='org-ref-bib'><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>
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