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Add schematic to explain the dynamics

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Thomas Dehaeze 2020-02-14 14:11:57 +01:00
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@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-02-13 jeu. 15:36 --> <!-- 2020-02-14 ven. 14:11 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Stewart Platform - Dynamics Study</title> <title>Stewart Platform - Dynamics Study</title>
@ -272,13 +272,13 @@ for the JavaScript code in this tag.
<ul> <ul>
<li><a href="#org4509b7d">1.1. Comparison with fixed support</a></li> <li><a href="#org4509b7d">1.1. Comparison with fixed support</a></li>
<li><a href="#org8662186">1.2. Comparison with a flexible support</a></li> <li><a href="#org8662186">1.2. Comparison with a flexible support</a></li>
<li><a href="#org230655f">1.3. Conclusion</a></li> <li><a href="#org1cbdf9a">1.3. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org81ab204">2. Comparison of the static transfer function and the Compliance matrix</a> <li><a href="#org81ab204">2. Comparison of the static transfer function and the Compliance matrix</a>
<ul> <ul>
<li><a href="#orge7e7242">2.1. Analysis</a></li> <li><a href="#orge7e7242">2.1. Analysis</a></li>
<li><a href="#org1cbdf9a">2.2. Conclusion</a></li> <li><a href="#org03b2957">2.2. Conclusion</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -382,6 +382,17 @@ The comparison of the two transfer functions is shown in Figure <a href="#orgbf9
</p> </p>
<p><span class="figure-number">Figure 1: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_fixed_base.png">png</a>, <a href="./figs/comparison_Fext_F_fixed_base.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 1: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_fixed_base.png">png</a>, <a href="./figs/comparison_Fext_F_fixed_base.pdf">pdf</a>)</p>
</div> </div>
<p>
This can be understood from figure <a href="#org8bd3e63">2</a> where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly the same effect on \(\mathcal{X}_{x}\).
</p>
<div id="org8bd3e63" class="figure">
<p><img src="figs/1dof_actuator_external_forces.png" alt="1dof_actuator_external_forces.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Schematic representation of the stewart platform on a rigid support</p>
</div>
</div> </div>
</div> </div>
@ -426,20 +437,34 @@ Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string"
</div> </div>
<p> <p>
The comparison between the obtained transfer functions is shown in Figure <a href="#orga2f2bd5">2</a>. The comparison between the obtained transfer functions is shown in Figure <a href="#orga2f2bd5">3</a>.
</p> </p>
<div id="orga2f2bd5" class="figure"> <div id="orga2f2bd5" class="figure">
<p><img src="figs/comparison_Fext_F_flexible_base.png" alt="comparison_Fext_F_flexible_base.png" /> <p><img src="figs/comparison_Fext_F_flexible_base.png" alt="comparison_Fext_F_flexible_base.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_flexible_base.png">png</a>, <a href="./figs/comparison_Fext_F_flexible_base.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 3: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_flexible_base.png">png</a>, <a href="./figs/comparison_Fext_F_flexible_base.pdf">pdf</a>)</p>
</div>
<p>
The addition of a flexible support can be schematically represented in Figure <a href="#orgee3ecbe">4</a>.
We see that \(\mathcal{F}_{x}\) applies a force both on \(m\) and \(m^{\prime}\) whereas \(\mathcal{F}_{x,\text{ext}}\) only applies a force on \(m\).
And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly <b>not</b> the same effect on \(\mathcal{X}_{x}\).
</p>
<div id="orgee3ecbe" class="figure">
<p><img src="figs/2dof_actuator_external_forces.png" alt="2dof_actuator_external_forces.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Schematic representation of the stewart platform on top of a flexible support</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org230655f" class="outline-3">
<h3 id="org230655f"><span class="section-number-3">1.3</span> Conclusion</h3> <div id="outline-container-org1cbdf9a" class="outline-3">
<h3 id="org1cbdf9a"><span class="section-number-3">1.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<div class="important"> <div class="important">
<p> <p>
@ -677,8 +702,8 @@ And now at the Compliance matrix.
</div> </div>
</div> </div>
<div id="outline-container-org1cbdf9a" class="outline-3"> <div id="outline-container-org03b2957" class="outline-3">
<h3 id="org1cbdf9a"><span class="section-number-3">2.2</span> Conclusion</h3> <h3 id="org03b2957"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<div class="important"> <div class="important">
<p> <p>
@ -692,7 +717,7 @@ The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathc
</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: 2020-02-13 jeu. 15:36</p> <p class="date">Created: 2020-02-14 ven. 14:11</p>
</div> </div>
</body> </body>
</html> </html>

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@ -159,6 +159,29 @@ The comparison of the two transfer functions is shown in Figure [[fig:comparison
#+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_fixed_base.png][png]], [[./figs/comparison_Fext_F_fixed_base.pdf][pdf]]) #+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_fixed_base.png][png]], [[./figs/comparison_Fext_F_fixed_base.pdf][pdf]])
[[file:figs/comparison_Fext_F_fixed_base.png]] [[file:figs/comparison_Fext_F_fixed_base.png]]
This can be understood from figure [[fig:1dof_actuator_external_forces]] where $\mathcal{F}_{x}$ and $\mathcal{F}_{x,\text{ext}}$ have clearly the same effect on $\mathcal{X}_{x}$.
#+begin_src latex :file 1dof_actuator_external_forces.pdf
\begin{tikzpicture}
\draw[ground] (-1, 0) -- (1, 0);
\draw[spring] (-0.6, 0) -- (-0.6, 1.5) node[midway, left=0.1]{$k$};
\draw[actuator] ( 0.6, 0) -- ( 0.6, 1.5) node[midway, left=0.1](F){$\mathcal{F}_{x}$};
\draw[fill=white] (-1, 1.5) rectangle (1, 2) node[pos=0.5]{$m$};
\draw[dashed] (1, 2) -- ++(0.5, 0);
\draw[->] (1.5, 2) -- ++(0, 0.5) node[right]{$\mathcal{X}_{x}$};
\draw[->] (0, 2) node[]{$\bullet$} -- (0, 2.8) node[below right]{$\mathcal{F}_{x,\text{ext}}$};
\end{tikzpicture}
#+end_src
#+name: fig:1dof_actuator_external_forces
#+caption: Schematic representation of the stewart platform on a rigid support
#+RESULTS:
[[file:figs/1dof_actuator_external_forces.png]]
** Comparison with a flexible support ** Comparison with a flexible support
We now add a flexible support under the Stewart platform. We now add a flexible support under the Stewart platform.
#+begin_src matlab #+begin_src matlab
@ -229,6 +252,35 @@ The comparison between the obtained transfer functions is shown in Figure [[fig:
#+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_flexible_base.png][png]], [[./figs/comparison_Fext_F_flexible_base.pdf][pdf]]) #+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_flexible_base.png][png]], [[./figs/comparison_Fext_F_flexible_base.pdf][pdf]])
[[file:figs/comparison_Fext_F_flexible_base.png]] [[file:figs/comparison_Fext_F_flexible_base.png]]
The addition of a flexible support can be schematically represented in Figure [[fig:2dof_actuator_external_forces]].
We see that $\mathcal{F}_{x}$ applies a force both on $m$ and $m^{\prime}$ whereas $\mathcal{F}_{x,\text{ext}}$ only applies a force on $m$.
And thus $\mathcal{F}_{x}$ and $\mathcal{F}_{x,\text{ext}}$ have clearly *not* the same effect on $\mathcal{X}_{x}$.
#+begin_src latex :file 2dof_actuator_external_forces.pdf
\begin{tikzpicture}
\draw[ground] (-1, 0) -- (1, 0);
\draw[spring] (0, 0) -- (0, 1.5) node[midway, left=0.1]{$k^{\prime}$};
\draw[fill=white] (-1, 1.5) rectangle (1, 2) node[pos=0.5]{$m^{\prime}$};
\draw[spring] (-0.6, 2) -- (-0.6, 3.5) node[midway, left=0.1]{$k$};
\draw[actuator] ( 0.6, 2) -- ( 0.6, 3.5) node[midway, left=0.1](F){$\mathcal{F}_{x}$};
\draw[fill=white] (-1, 3.5) rectangle (1, 4) node[pos=0.5]{$m$};
\draw[dashed] (1, 4) -- ++(0.5, 0);
\draw[->] (1.5, 4) -- ++(0, 0.5) node[right]{$\mathcal{X}_{x}$};
\draw[->] (0, 4) node[]{$\bullet$} -- (0, 4.8) node[below right]{$\mathcal{F}_{x,\text{ext}}$};
\end{tikzpicture}
#+end_src
#+name: fig:2dof_actuator_external_forces
#+caption: Schematic representation of the stewart platform on top of a flexible support
#+RESULTS:
[[file:figs/2dof_actuator_external_forces.png]]
** Conclusion ** Conclusion
#+begin_important #+begin_important
The transfer function from forces/torques applied by the actuators on the payload $\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}$ to the pose of the mobile platform $\bm{\mathcal{X}}$ is the same as the transfer function from external forces/torques to $\bm{\mathcal{X}}$ as long as the Stewart platform's base is fixed. The transfer function from forces/torques applied by the actuators on the payload $\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}$ to the pose of the mobile platform $\bm{\mathcal{X}}$ is the same as the transfer function from external forces/torques to $\bm{\mathcal{X}}$ as long as the Stewart platform's base is fixed.