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<div id="content">
<h1 class="title">Stewart Platforms</h1>
<p>
<a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>
</p>
<div id="outline-container-org9cd44e0" class="outline-2">
<h2 id="org9cd44e0"><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>
<li><a href="identification.html">Identification of the Simscape Model</a></li>
</ul>
</div>
</div>
<div id="outline-container-org7a44762" class="outline-2">
<h2 id="org7a44762"><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>
<li><a href="stiffness-study.html">Stiffness Matrix Study</a></li>
<li>Jacobian Study</li>
<li><a href="cubic-configuration.html">Cubic Architecture</a></li>
</ul>
</div>
</div>
<div id="outline-container-org77767cc" class="outline-2">
<h2 id="org77767cc"><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>
<li>Inertial Control</li>
<li>Decentralized Control</li>
</ul>
</div>
</div>
<div id="outline-container-org9d06c58" class="outline-2">
<h2 id="org9d06c58"><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-orgffe6651" class="outline-3">
<h3 id="orgffe6651"><span class="section-number-3">4.1</span> Jacobian</h3>
<div class="outline-text-3" id="text-4-1">
</div>
<div id="outline-container-org6b92660" class="outline-4">
<h4 id="org6b92660"><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-orgcec2e05" class="outline-4">
<h4 id="orgcec2e05"><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-orgbf33a4e" class="outline-4">
<h4 id="orgbf33a4e"><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-org3710914" class="outline-3">
<h3 id="org3710914"><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-orgd5d2c08" class="outline-3">
<h3 id="orgd5d2c08"><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="orgc118680" 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>
<h1 class='org-ref-bib-h1'>Bibliography</h1>
<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>
</ul>
</p>
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