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Add analysis about cube size

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Thomas Dehaeze
2020-03-12 18:06:56 +01:00
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@@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-03-03 mar. 15:51 -->
<!-- 2020-03-12 jeu. 18:06 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Cubic configuration for the Stewart Platform</title>
@@ -252,33 +252,34 @@
<li><a href="#orga88e79a">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
<li><a href="#orge02ec88">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
<li><a href="#org43fd7e4">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
<li><a href="#org64395b6">1.5. Conclusion</a></li>
<li><a href="#org3356db5">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgd70418b">2. Configuration with the Cube&rsquo;s center above the mobile platform</a>
<ul>
<li><a href="#org8afa645">2.1. Having Cube&rsquo;s center above the top platform</a></li>
<li><a href="#org8e09793">2.2. Conclusion</a></li>
<li><a href="#org25d045b">2.2. Size of the platforms</a></li>
<li><a href="#org9477b7a">2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgcc4ecce">3. Cubic size analysis</a>
<ul>
<li><a href="#org0029d8c">3.1. Analysis</a></li>
<li><a href="#orgc5a2e1f">3.2. Conclusion</a></li>
<li><a href="#org46632b3">3.2. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgf09da67">4. Dynamic Coupling in the Cartesian Frame</a>
<ul>
<li><a href="#org5fe01ec">4.1. Cube&rsquo;s center at the Center of Mass of the mobile platform</a></li>
<li><a href="#org4cb2a36">4.2. Cube&rsquo;s center not coincident with the Mass of the Mobile platform</a></li>
<li><a href="#org24cd25e">4.3. Conclusion</a></li>
<li><a href="#orgc1b6a36">4.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org8f26dc0">5. Dynamic Coupling between actuators and sensors of each strut</a>
<ul>
<li><a href="#org6e391c9">5.1. Coupling between the actuators and sensors - Cubic Architecture</a></li>
<li><a href="#orgafd808d">5.2. Coupling between the actuators and sensors - Non-Cubic Architecture</a></li>
<li><a href="#org3356db5">5.3. Conclusion</a></li>
<li><a href="#org87716af">5.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org3044455">6. Functions</a>
@@ -826,8 +827,8 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
</div>
</div>
<div id="outline-container-org64395b6" class="outline-3">
<h3 id="org64395b6"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org3356db5" class="outline-3">
<h3 id="org3356db5"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@@ -1164,9 +1165,100 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
</div>
</div>
<div id="outline-container-org8e09793" class="outline-3">
<h3 id="org8e09793"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-org25d045b" class="outline-3">
<h3 id="org25d045b"><span class="section-number-3">2.2</span> Size of the platforms</h3>
<div class="outline-text-3" id="text-2-2">
<p>
The minimum size of the platforms depends on the cube&rsquo;s size and the height between the platform and the cube&rsquo;s center.
</p>
<p>
Let&rsquo;s denote:
</p>
<ul class="org-ul">
<li>\(H\) the height between the cube&rsquo;s center and the considered platform</li>
<li>\(D\) the size of the cube&rsquo;s edges</li>
</ul>
<p>
Let&rsquo;s denote by \(a\) and \(b\) the points of both ends of one of the cube&rsquo;s edge.
</p>
<p>
Initially, we have:
</p>
\begin{align}
a &= \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\
b &= \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}
\end{align}
<p>
We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical.
\[ R = \begin{bmatrix}
\frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\
\frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
\frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}}
\end{bmatrix} \]
</p>
<p>
After rotation, the points \(a\) and \(b\) become:
</p>
\begin{align}
a &= \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\
b &= \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix}
\end{align}
<p>
Points \(a\) and \(b\) define a vector \(u = b - a\) that gives the orientation of one of the Stewart platform strut:
\[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]
</p>
<p>
Then we want to find the intersection between the line that defines the strut with the plane defined by the height \(H\) from the cube&rsquo;s center.
To do so, we first find \(g\) such that:
\[ a_z + g u_z = -H \]
We obtain:
</p>
\begin{align}
g &= - \frac{H + a_z}{u_z} \\
&= \sqrt{3} \frac{H}{D} - \frac{1}{2}
\end{align}
<p>
Then, the intersection point \(P\) is given by:
</p>
\begin{align}
P &= a + g u \\
&= \begin{bmatrix}
H \sqrt{2} \\
D \frac{1}{\sqrt{2}} \\
H
\end{bmatrix}
\end{align}
<p>
Finally, the circle can contains the intersection point has a radius \(r\):
</p>
\begin{align}
r &= \sqrt{P_x^2 + P_y^2} \\
&= \sqrt{2 H^2 + \frac{1}{2}D^2}
\end{align}
<p>
By symmetry, we can show that all the other intersection points will also be on the circle with a radius \(r\).
</p>
<p>
For a small cube:
\[ r \approx \sqrt{2} H \]
</p>
</div>
</div>
<div id="outline-container-org9477b7a" class="outline-3">
<h3 id="org9477b7a"><span class="section-number-3">2.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-3">
<div class="important">
<p>
We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform.
@@ -1283,8 +1375,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
</div>
</div>
<div id="outline-container-orgc5a2e1f" class="outline-3">
<h3 id="orgc5a2e1f"><span class="section-number-3">3.2</span> Conclusion</h3>
<div id="outline-container-org46632b3" class="outline-3">
<h3 id="org46632b3"><span class="section-number-3">3.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-2">
<p>
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
@@ -1653,8 +1745,8 @@ This was expected as the mass matrix is not diagonal (the Center of Mass of the
</div>
</div>
<div id="outline-container-org24cd25e" class="outline-3">
<h3 id="org24cd25e"><span class="section-number-3">4.3</span> Conclusion</h3>
<div id="outline-container-orgc1b6a36" class="outline-3">
<h3 id="orgc1b6a36"><span class="section-number-3">4.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-3">
<div class="important">
<p>
@@ -1743,6 +1835,12 @@ controller = initializeController(<span class="org-string">'type'</span>, <span
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">disturbances = initializeDisturbances();
references = initializeReferences(stewart);
</pre>
</div>
<div id="org67d7284" class="figure">
<p><img src="figs/stewart_architecture_coupling_struts_cubic.png" alt="stewart_architecture_coupling_struts_cubic.png" />
@@ -1838,8 +1936,8 @@ And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relati
</div>
</div>
<div id="outline-container-org3356db5" class="outline-3">
<h3 id="org3356db5"><span class="section-number-3">5.3</span> Conclusion</h3>
<div id="outline-container-org87716af" class="outline-3">
<h3 id="org87716af"><span class="section-number-3">5.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-3">
<div class="important">
<p>
@@ -1942,8 +2040,7 @@ H = stewart.geometry.H;
<div class="outline-text-4" id="text-org771c630">
<p>
We define the useful points of the cube with respect to the Cube&rsquo;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}.
\({}^{C}C\) 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">
@@ -2019,7 +2116,7 @@ stewart.platform_M.Mb = Mb;
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-03 mar. 15:51</p>
<p class="date">Created: 2020-03-12 jeu. 18:06</p>
</div>
</body>
</html>