Delete old function to make cubic configurations
This commit is contained in:
parent
d462bc2481
commit
bfc62cac82
@ -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>
|
||||
<!-- 2020-02-06 jeu. 17:25 -->
|
||||
<!-- 2020-02-06 jeu. 17:29 -->
|
||||
<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>
|
||||
@ -268,48 +268,29 @@ for the JavaScript code in this tag.
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org43e4755">1. Questions we wish to answer with this analysis</a></li>
|
||||
<li><a href="#org7c85269">2. <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</a>
|
||||
<li><a href="#org75c6951">1. <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</a>
|
||||
<ul>
|
||||
<li><a href="#org7a2e2af">2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
|
||||
<li><a href="#orgdd082ef">2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
|
||||
<li><a href="#org314610d">2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
|
||||
<li><a href="#org460e492">2.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
|
||||
<li><a href="#orgccb5ef0">2.5. Conclusion</a></li>
|
||||
<li><a href="#orga823f72">1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
|
||||
<li><a href="#org4261310">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
|
||||
<li><a href="#orgf297eb8">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
|
||||
<li><a href="#orgfeaf9c1">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
|
||||
<li><a href="#org24bdf29">1.5. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org29d657d">3. <span class="todo TODO">TODO</span> Cubic size analysis</a></li>
|
||||
<li><a href="#orgc12b0fc">4. Functions</a>
|
||||
<li><a href="#org2cb2ab0">2. <span class="todo TODO">TODO</span> Cubic size analysis</a></li>
|
||||
<li><a href="#orgeec7b47">3. Functions</a>
|
||||
<ul>
|
||||
<li><a href="#org12a207e">4.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
|
||||
<li><a href="#org92224ef">3.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
|
||||
<ul>
|
||||
<li><a href="#orgecee38f">Function description</a></li>
|
||||
<li><a href="#orgb9948d8">Documentation</a></li>
|
||||
<li><a href="#orgbfb0bd5">Optional Parameters</a></li>
|
||||
<li><a href="#orgcb22d51">Position of the Cube</a></li>
|
||||
<li><a href="#org2f09e98">Compute the pose</a></li>
|
||||
<li><a href="#org715472d">Function description</a></li>
|
||||
<li><a href="#orgbab37f8">Documentation</a></li>
|
||||
<li><a href="#orgddbe42e">Optional Parameters</a></li>
|
||||
<li><a href="#org66dd074">Position of the Cube</a></li>
|
||||
<li><a href="#org388f35d">Compute the pose</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org4eaf218">5. <span class="todo TODO">TODO</span> initializeCubicConfiguration</a>
|
||||
<ul>
|
||||
<li><a href="#org4fb2bc6">5.1. Function description</a></li>
|
||||
<li><a href="#orgb540658">5.2. Optional Parameters</a></li>
|
||||
<li><a href="#org1474f46">5.3. Cube Creation</a></li>
|
||||
<li><a href="#org03d2dd7">5.4. Vectors of each leg</a></li>
|
||||
<li><a href="#orgfed36b2">5.5. Verification of Height of the Stewart Platform</a></li>
|
||||
<li><a href="#orgdb27b02">5.6. Determinate the location of the joints</a></li>
|
||||
<li><a href="#org5079890">5.7. Returns Stewart Structure</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgd9f1e20">6. <span class="todo TODO">TODO</span> Tests</a>
|
||||
<ul>
|
||||
<li><a href="#orgea7297c">6.1. First attempt to parametrisation</a></li>
|
||||
<li><a href="#orgd6ed3c3">6.2. Second attempt</a></li>
|
||||
<li><a href="#orgf39eafa">6.3. Generate the Stewart platform for a Cubic configuration</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
@ -332,13 +313,7 @@ According to <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage
|
||||
</p>
|
||||
|
||||
<p>
|
||||
To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#org83d7db1">5</a>).
|
||||
</p>
|
||||
|
||||
<div id="outline-container-org43e4755" class="outline-2">
|
||||
<h2 id="org43e4755"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
To generate and study the Cubic configuration, <code>generateCubicConfiguration</code> is used (description in section <a href="#orgb0ae4eb">3.1</a>).
|
||||
The goal is to study the benefits of using a cubic configuration:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
@ -346,23 +321,21 @@ The goal is to study the benefits of using a cubic configuration:
|
||||
<li>No coupling between the actuators?</li>
|
||||
<li>Is the center of the cube an important point?</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7c85269" class="outline-2">
|
||||
<h2 id="org7c85269"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<div id="outline-container-org75c6951" class="outline-2">
|
||||
<h2 id="org75c6951"><span class="section-number-2">1</span> <span class="todo TODO">TODO</span> Configuration Analysis - Stiffness Matrix</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
</div>
|
||||
<div id="outline-container-org7a2e2af" class="outline-3">
|
||||
<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>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<div id="outline-container-orga823f72" class="outline-3">
|
||||
<h3 id="orga823f72"><span class="section-number-3">1.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
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).
|
||||
We create a cubic Stewart platform (figure <a href="#org66ade8d">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).
|
||||
The Jacobian matrix is estimated at the location of the center of the cube.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org01dbe25" class="figure">
|
||||
<div id="org66ade8d" class="figure">
|
||||
<p><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p>
|
||||
@ -466,11 +439,11 @@ save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-strin
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdd082ef" class="outline-3">
|
||||
<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>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<div id="outline-container-org4261310" class="outline-3">
|
||||
<h3 id="org4261310"><span class="section-number-3">1.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<p>
|
||||
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>).
|
||||
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org66ade8d">1</a>).
|
||||
The Jacobian matrix is not estimated at the location of the center of the cube.
|
||||
</p>
|
||||
|
||||
@ -570,16 +543,16 @@ stewart = computeGeometricalProperties(stewart, opts);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org314610d" class="outline-3">
|
||||
<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>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<div id="outline-container-orgf297eb8" class="outline-3">
|
||||
<h3 id="orgf297eb8"><span class="section-number-3">1.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
Here, the “center” of the Stewart platform is not at the cube center (figure <a href="#org4aa7b60">2</a>).
|
||||
Here, the “center” of the Stewart platform is not at the cube center (figure <a href="#org4492663">2</a>).
|
||||
The Jacobian is estimated at the cube center.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org4aa7b60" class="figure">
|
||||
<div id="org4492663" class="figure">
|
||||
<p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p>
|
||||
@ -692,9 +665,9 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org460e492" class="outline-3">
|
||||
<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>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<div id="outline-container-orgfeaf9c1" class="outline-3">
|
||||
<h3 id="orgfeaf9c1"><span class="section-number-3">1.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
Here, the “center” of the Stewart platform is not at the cube center.
|
||||
The Jacobian is estimated at the center of the Stewart platform.
|
||||
@ -807,9 +780,9 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgccb5ef0" class="outline-3">
|
||||
<h3 id="orgccb5ef0"><span class="section-number-3">2.5</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<div id="outline-container-org24bdf29" class="outline-3">
|
||||
<h3 id="org24bdf29"><span class="section-number-3">1.5</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<div class="important">
|
||||
<ul class="org-ul">
|
||||
<li>The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta\x} = k_{\theta_y}\)</li>
|
||||
@ -821,9 +794,9 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org29d657d" class="outline-2">
|
||||
<h2 id="org29d657d"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Cubic size analysis</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<div id="outline-container-org2cb2ab0" class="outline-2">
|
||||
<h2 id="org2cb2ab0"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Cubic size analysis</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
We here study the effect of the size of the cube used for the Stewart configuration.
|
||||
</p>
|
||||
@ -897,7 +870,7 @@ xlabel(<span class="org-string">'Cube Size [mm]'</span>); ylabel(<span class="or
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgeec8e66" class="figure">
|
||||
<div id="org009489e" class="figure">
|
||||
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
|
||||
</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>
|
||||
@ -918,19 +891,19 @@ In that case, the legs will the further separated. Size of the cube is then limi
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc12b0fc" class="outline-2">
|
||||
<h2 id="orgc12b0fc"><span class="section-number-2">4</span> Functions</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<div id="outline-container-orgeec7b47" class="outline-2">
|
||||
<h2 id="orgeec7b47"><span class="section-number-2">3</span> Functions</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="orgee0330a"></a>
|
||||
<a id="orgb108018"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org12a207e" class="outline-3">
|
||||
<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 id="outline-container-org92224ef" class="outline-3">
|
||||
<h3 id="org92224ef"><span class="section-number-3">3.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
<a id="org0a684d8"></a>
|
||||
<a id="orgb0ae4eb"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -938,9 +911,9 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
|
||||
</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 id="outline-container-org715472d" class="outline-4">
|
||||
<h4 id="org715472d">Function description</h4>
|
||||
<div class="outline-text-4" id="text-org715472d">
|
||||
<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>
|
||||
@ -965,11 +938,11 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
|
||||
</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="outline-container-orgbab37f8" class="outline-4">
|
||||
<h4 id="orgbab37f8">Documentation</h4>
|
||||
<div class="outline-text-4" id="text-orgbab37f8">
|
||||
|
||||
<div id="orgff1f403" class="figure">
|
||||
<div id="org946a873" 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>
|
||||
@ -977,9 +950,9 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
|
||||
</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 id="outline-container-orgddbe42e" class="outline-4">
|
||||
<h4 id="orgddbe42e">Optional Parameters</h4>
|
||||
<div class="outline-text-4" id="text-orgddbe42e">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">arguments
|
||||
stewart
|
||||
@ -993,9 +966,9 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
|
||||
</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">
|
||||
<div id="outline-container-org66dd074" class="outline-4">
|
||||
<h4 id="org66dd074">Position of the Cube</h4>
|
||||
<div class="outline-text-4" id="text-org66dd074">
|
||||
<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
|
||||
@ -1020,9 +993,9 @@ CCm = [Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>
|
||||
</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">
|
||||
<div id="outline-container-org388f35d" class="outline-4">
|
||||
<h4 id="org388f35d">Compute the pose</h4>
|
||||
<div class="outline-text-4" id="text-org388f35d">
|
||||
<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>
|
||||
@ -1044,442 +1017,6 @@ stewart.Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org
|
||||
</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">
|
||||
<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>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb540658" class="outline-3">
|
||||
<h3 id="orgb540658"><span class="section-number-3">5.2</span> Optional Parameters</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
Default values for opts.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">opts = struct(...
|
||||
<span class="org-string">'H_tot'</span>, 90, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
||||
<span class="org-string">'L'</span>, 110, ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
||||
<span class="org-string">'H'</span>, 40, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
||||
<span class="org-string">'H0'</span>, 75 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
||||
);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Populate opts with input parameters
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">if</span> exist(<span class="org-string">'opts_param'</span>,<span class="org-string">'var'</span>)
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">opt</span> = <span class="org-constant">fieldnames(opts_param)'</span>
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1474f46" class="outline-3">
|
||||
<h3 id="org1474f46"><span class="section-number-3">5.3</span> Cube Creation</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">points = [0, 0, 0; ...
|
||||
0, 0, 1; ...
|
||||
0, 1, 0; ...
|
||||
0, 1, 1; ...
|
||||
1, 0, 0; ...
|
||||
1, 0, 1; ...
|
||||
1, 1, 0; ...
|
||||
1, 1, 1];
|
||||
points = opts.L<span class="org-type">*</span>points;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We create the rotation matrix to rotate the cube
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">sx = cross([1, 1, 1], [1 0 0]);
|
||||
sx = sx<span class="org-type">/</span>norm(sx);
|
||||
|
||||
sy = <span class="org-type">-</span>cross(sx, [1, 1, 1]);
|
||||
sy = sy<span class="org-type">/</span>norm(sy);
|
||||
|
||||
sz = [1, 1, 1];
|
||||
sz = sz<span class="org-type">/</span>norm(sz);
|
||||
|
||||
R = [sx<span class="org-type">'</span>, sy<span class="org-type">'</span>, sz<span class="org-type">'</span>]<span class="org-type">'</span>;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We use to rotation matrix to rotate the cube
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">cube = zeros(size(points));
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:size(points, 1)</span>
|
||||
cube(<span class="org-constant">i</span>, <span class="org-type">:</span>) = R <span class="org-type">*</span> points(<span class="org-constant">i</span>, <span class="org-type">:</span>)<span class="org-type">'</span>;
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org03d2dd7" class="outline-3">
|
||||
<h3 id="org03d2dd7"><span class="section-number-3">5.4</span> Vectors of each leg</h3>
|
||||
<div class="outline-text-3" id="text-5-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">leg_indices = [3, 4; ...
|
||||
2, 4; ...
|
||||
2, 6; ...
|
||||
5, 6; ...
|
||||
5, 7; ...
|
||||
3, 7];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Vectors are:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">legs = zeros(6, 3);
|
||||
legs_start = zeros(6, 3);
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
legs(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 2), <span class="org-type">:</span>) <span class="org-type">-</span> cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>);
|
||||
legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfed36b2" class="outline-3">
|
||||
<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-5-5">
|
||||
<p>
|
||||
If the Stewart platform is not contained in the cube, throw an error.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Hmax = cube(4, 3) <span class="org-type">-</span> cube(2, 3);
|
||||
<span class="org-keyword">if</span> opts.H0 <span class="org-type"><</span> cube(2, 3)
|
||||
error(sprintf(<span class="org-string">'H0 is not high enought. Minimum H0 = %.1f'</span>, cube(2, 3)));
|
||||
<span class="org-keyword">else</span> <span class="org-keyword">if</span> opts.H0 <span class="org-type">+</span> opts.H <span class="org-type">></span> cube(4, 3)
|
||||
error(sprintf(<span class="org-string">'H0+H is too high. Maximum H0+H = %.1f'</span>, cube(4, 3)));
|
||||
error(<span class="org-string">'H0+H is too high'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdb27b02" class="outline-3">
|
||||
<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-5-6">
|
||||
<p>
|
||||
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.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Aa = zeros(6, 3);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
t = (opts.H0<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
||||
Aa(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And the location of the joints on the mobile platform with respect to \(\{A\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ab = zeros(6, 3);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
t = (opts.H0<span class="org-type">+</span>opts.H<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
||||
Ab(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And the location of the joints on the mobile platform with respect to \(\{B\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Bb = zeros(6, 3);
|
||||
Bb = Ab <span class="org-type">-</span> (opts.H0 <span class="org-type">+</span> opts.H_tot<span class="org-type">/</span>2 <span class="org-type">+</span> opts.H<span class="org-type">/</span>2)<span class="org-type">*</span>[0, 0, 1];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">h = opts.H0 <span class="org-type">+</span> opts.H<span class="org-type">/</span>2 <span class="org-type">-</span> opts.H_tot<span class="org-type">/</span>2;
|
||||
Aa = Aa <span class="org-type">-</span> h<span class="org-type">*</span>[0, 0, 1];
|
||||
Ab = Ab <span class="org-type">-</span> h<span class="org-type">*</span>[0, 0, 1];
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5079890" class="outline-3">
|
||||
<h3 id="org5079890"><span class="section-number-3">5.7</span> Returns Stewart Structure</h3>
|
||||
<div class="outline-text-3" id="text-5-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = struct();
|
||||
stewart.Aa = Aa;
|
||||
stewart.Ab = Ab;
|
||||
stewart.Bb = Bb;
|
||||
stewart.H_tot = opts.H_tot;
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd9f1e20" class="outline-2">
|
||||
<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-6">
|
||||
</div>
|
||||
<div id="outline-container-orgea7297c" class="outline-3">
|
||||
<h3 id="orgea7297c"><span class="section-number-3">6.1</span> First attempt to parametrisation</h3>
|
||||
<div class="outline-text-3" id="text-6-1">
|
||||
|
||||
<div id="org65e66e5" class="figure">
|
||||
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Schematic of the bottom plates with all the parameters</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The goal is to choose \(\alpha\), \(\beta\), \(R_\text{leg, t}\) and \(R_\text{leg, b}\) in such a way that the configuration is cubic.
|
||||
</p>
|
||||
|
||||
|
||||
<p>
|
||||
The configuration is cubic if:
|
||||
\[ \overrightarrow{a_i b_i} \cdot \overrightarrow{a_j b_j} = 0, \ \forall i, j = [1, \hdots, 6], i \ne j \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Lets express \(a_i\), \(b_i\) and \(a_j\):
|
||||
</p>
|
||||
\begin{equation*}
|
||||
a_1 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 - \alpha) \\ R_{\text{leg,b}} \cos(120 - \alpha) \\ 0\end{bmatrix} ; \quad
|
||||
a_2 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 + \alpha) \\ R_{\text{leg,b}} \cos(120 + \alpha) \\ 0\end{bmatrix} ; \quad
|
||||
\end{equation*}
|
||||
|
||||
\begin{equation*}
|
||||
b_1 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 - \beta) \\ R_{\text{leg,t}} \cos(120 - \beta\\ H\end{bmatrix} ; \quad
|
||||
b_2 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 + \beta) \\ R_{\text{leg,t}} \cos(120 + \beta\\ H\end{bmatrix} ; \quad
|
||||
\end{equation*}
|
||||
|
||||
<p>
|
||||
\[ \overrightarrow{a_1 b_1} = b_1 - a_1 = \begin{bmatrix}R_{\text{leg}} \cos(120 - \alpha) \\ R_{\text{leg}} \cos(120 - \alpha) \\ 0\end{bmatrix}\]
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd6ed3c3" class="outline-3">
|
||||
<h3 id="orgd6ed3c3"><span class="section-number-3">6.2</span> Second attempt</h3>
|
||||
<div class="outline-text-3" id="text-6-2">
|
||||
<p>
|
||||
We start with the point of a cube in space:
|
||||
</p>
|
||||
\begin{align*}
|
||||
[0, 0, 0] ; \ [0, 0, 1]; \ ...
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
We also want the cube to point upward:
|
||||
\[ [1, 1, 1] \Rightarrow [0, 0, 1] \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Then we have the direction of all the vectors expressed in the frame of the hexapod.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">points = [0, 0, 0; ...
|
||||
0, 0, 1; ...
|
||||
0, 1, 0; ...
|
||||
0, 1, 1; ...
|
||||
1, 0, 0; ...
|
||||
1, 0, 1; ...
|
||||
1, 1, 0; ...
|
||||
1, 1, 1];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
plot3(points(<span class="org-type">:</span>,1), points(<span class="org-type">:</span>,2), points(<span class="org-type">:</span>,3), <span class="org-string">'ko'</span>)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">sx = cross([1, 1, 1], [1 0 0]);
|
||||
sx = sx<span class="org-type">/</span>norm(sx);
|
||||
|
||||
sy = <span class="org-type">-</span>cross(sx, [1, 1, 1]);
|
||||
sy = sy<span class="org-type">/</span>norm(sy);
|
||||
|
||||
sz = [1, 1, 1];
|
||||
sz = sz<span class="org-type">/</span>norm(sz);
|
||||
|
||||
R = [sx<span class="org-type">'</span>, sy<span class="org-type">'</span>, sz<span class="org-type">'</span>]<span class="org-type">'</span>;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">cube = zeros(size(points));
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:size(points, 1)</span>
|
||||
cube(<span class="org-constant">i</span>, <span class="org-type">:</span>) = R <span class="org-type">*</span> points(<span class="org-constant">i</span>, <span class="org-type">:</span>)<span class="org-type">'</span>;
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
hold on;
|
||||
plot3(points(<span class="org-type">:</span>,1), points(<span class="org-type">:</span>,2), points(<span class="org-type">:</span>,3), <span class="org-string">'ko'</span>);
|
||||
plot3(cube(<span class="org-type">:</span>,1), cube(<span class="org-type">:</span>,2), cube(<span class="org-type">:</span>,3), <span class="org-string">'ro'</span>);
|
||||
hold off;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Now we plot the legs of the hexapod.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">leg_indices = [3, 4; ...
|
||||
2, 4; ...
|
||||
2, 6; ...
|
||||
5, 6; ...
|
||||
5, 7; ...
|
||||
3, 7]
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
plot3(cube(leg_indices(<span class="org-constant">i</span>, <span class="org-type">:</span>),1), cube(leg_indices(<span class="org-constant">i</span>, <span class="org-type">:</span>),2), cube(leg_indices(<span class="org-constant">i</span>, <span class="org-type">:</span>),3), <span class="org-string">'-'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
hold off;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Vectors are:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">legs = zeros(6, 3);
|
||||
legs_start = zeros(6, 3);
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
legs(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 2), <span class="org-type">:</span>) <span class="org-type">-</span> cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>);
|
||||
legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) = cube(leg_indices(<span class="org-constant">i</span>, 1), <span class="org-type">:</span>)
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We now have the orientation of each leg.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We here want to see if the position of the “slice” changes something.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let’s first estimate the maximum height of the Stewart platform.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Hmax = cube(4, 3) <span class="org-type">-</span> cube(2, 3);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Let’s then estimate the middle position of the platform
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Hmid = cube(8, 3)<span class="org-type">/</span>2;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf39eafa" class="outline-3">
|
||||
<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-6-3">
|
||||
<p>
|
||||
First we defined the height of the Hexapod.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">H = Hmax<span class="org-type">/</span>2;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Zs = 1.2<span class="org-type">*</span>cube(2, 3); <span class="org-comment">% Height of the fixed platform</span>
|
||||
Ze = Zs <span class="org-type">+</span> H; <span class="org-comment">% Height of the mobile platform</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We now determine the location of the joints on the fixed platform.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Aa = zeros(6, 3);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
t = (Zs<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
||||
Aa(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And the location of the joints on the mobile platform
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ab = zeros(6, 3);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
t = (Ze<span class="org-type">-</span>legs_start(<span class="org-constant">i</span>, 3))<span class="org-type">/</span>(legs(<span class="org-constant">i</span>, 3));
|
||||
Ab(<span class="org-constant">i</span>, <span class="org-type">:</span>) = legs_start(<span class="org-constant">i</span>, <span class="org-type">:</span>) <span class="org-type">+</span> t<span class="org-type">*</span>legs(<span class="org-constant">i</span>, <span class="org-type">:</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we plot the legs.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
||||
plot3([Ab(<span class="org-constant">i</span>, 1),Aa(<span class="org-constant">i</span>, 1)], [Ab(<span class="org-constant">i</span>, 2),Aa(<span class="org-constant">i</span>, 2)], [Ab(<span class="org-constant">i</span>, 3),Aa(<span class="org-constant">i</span>, 3)], <span class="org-string">'k-'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
hold off;
|
||||
xlim([<span class="org-type">-</span>1, 1]);
|
||||
ylim([<span class="org-type">-</span>1, 1]);
|
||||
zlim([0, 2]);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
|
||||
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
||||
@ -1491,7 +1028,7 @@ zlim([0, 2]);
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-02-06 jeu. 17:25</p>
|
||||
<p class="date">Created: 2020-02-06 jeu. 17:29</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
@ -31,7 +31,11 @@ The cubic (or orthogonal) configuration of the Stewart platform is now widely us
|
||||
|
||||
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).
|
||||
|
||||
To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]).
|
||||
To generate and study the Cubic configuration, =generateCubicConfiguration= is used (description in section [[sec:generateCubicConfiguration]]).
|
||||
The goal is to study the benefits of using a cubic configuration:
|
||||
- Equal stiffness in all the degrees of freedom?
|
||||
- No coupling between the actuators?
|
||||
- Is the center of the cube an important point?
|
||||
|
||||
* Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
@ -46,12 +50,6 @@ To generate and study the Cubic configuration, =initializeCubicConfiguration= is
|
||||
simulinkproject('./');
|
||||
#+end_src
|
||||
|
||||
* Questions we wish to answer with this analysis
|
||||
The goal is to study the benefits of using a cubic configuration:
|
||||
- Equal stiffness in all the degrees of freedom?
|
||||
- No coupling between the actuators?
|
||||
- Is the center of the cube an important point?
|
||||
|
||||
* TODO Configuration Analysis - Stiffness Matrix
|
||||
** Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
|
||||
We create a cubic Stewart platform (figure [[fig:3d-cubic-stewart-aligned]]) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
|
||||
@ -400,316 +398,6 @@ We now which to compute the position of the joints $a_{i}$ and $b_{i}$.
|
||||
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
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :exports code
|
||||
:HEADER-ARGS:matlab+: :comments no
|
||||
:HEADER-ARGS:matlab+: :eval no
|
||||
:HEADER-ARGS:matlab+: :tangle src/initializeCubicConfiguration.m
|
||||
:END:
|
||||
<<sec:initializeCubicConfiguration>>
|
||||
|
||||
** Function description
|
||||
#+begin_src matlab
|
||||
function [stewart] = initializeCubicConfiguration(opts_param)
|
||||
#+end_src
|
||||
|
||||
** Optional Parameters
|
||||
Default values for opts.
|
||||
#+begin_src matlab
|
||||
opts = struct(...
|
||||
'H_tot', 90, ... % Total height of the Hexapod [mm]
|
||||
'L', 110, ... % Size of the Cube [mm]
|
||||
'H', 40, ... % Height between base joints and platform joints [mm]
|
||||
'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm]
|
||||
);
|
||||
#+end_src
|
||||
|
||||
Populate opts with input parameters
|
||||
#+begin_src matlab
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Cube Creation
|
||||
#+begin_src matlab :results none
|
||||
points = [0, 0, 0; ...
|
||||
0, 0, 1; ...
|
||||
0, 1, 0; ...
|
||||
0, 1, 1; ...
|
||||
1, 0, 0; ...
|
||||
1, 0, 1; ...
|
||||
1, 1, 0; ...
|
||||
1, 1, 1];
|
||||
points = opts.L*points;
|
||||
#+end_src
|
||||
|
||||
We create the rotation matrix to rotate the cube
|
||||
#+begin_src matlab :results none
|
||||
sx = cross([1, 1, 1], [1 0 0]);
|
||||
sx = sx/norm(sx);
|
||||
|
||||
sy = -cross(sx, [1, 1, 1]);
|
||||
sy = sy/norm(sy);
|
||||
|
||||
sz = [1, 1, 1];
|
||||
sz = sz/norm(sz);
|
||||
|
||||
R = [sx', sy', sz']';
|
||||
#+end_src
|
||||
|
||||
We use to rotation matrix to rotate the cube
|
||||
#+begin_src matlab :results none
|
||||
cube = zeros(size(points));
|
||||
for i = 1:size(points, 1)
|
||||
cube(i, :) = R * points(i, :)';
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Vectors of each leg
|
||||
#+begin_src matlab :results none
|
||||
leg_indices = [3, 4; ...
|
||||
2, 4; ...
|
||||
2, 6; ...
|
||||
5, 6; ...
|
||||
5, 7; ...
|
||||
3, 7];
|
||||
#+end_src
|
||||
|
||||
Vectors are:
|
||||
#+begin_src matlab :results none
|
||||
legs = zeros(6, 3);
|
||||
legs_start = zeros(6, 3);
|
||||
|
||||
for i = 1:6
|
||||
legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
|
||||
legs_start(i, :) = cube(leg_indices(i, 1), :);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Verification of Height of the Stewart Platform
|
||||
If the Stewart platform is not contained in the cube, throw an error.
|
||||
|
||||
#+begin_src matlab :results none
|
||||
Hmax = cube(4, 3) - cube(2, 3);
|
||||
if opts.H0 < cube(2, 3)
|
||||
error(sprintf('H0 is not high enought. Minimum H0 = %.1f', cube(2, 3)));
|
||||
else if opts.H0 + opts.H > cube(4, 3)
|
||||
error(sprintf('H0+H is too high. Maximum H0+H = %.1f', cube(4, 3)));
|
||||
error('H0+H is too high');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Determinate the location of the joints
|
||||
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.
|
||||
#+begin_src matlab :results none
|
||||
Aa = zeros(6, 3);
|
||||
for i = 1:6
|
||||
t = (opts.H0-legs_start(i, 3))/(legs(i, 3));
|
||||
Aa(i, :) = legs_start(i, :) + t*legs(i, :);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
And the location of the joints on the mobile platform with respect to $\{A\}$.
|
||||
#+begin_src matlab :results none
|
||||
Ab = zeros(6, 3);
|
||||
for i = 1:6
|
||||
t = (opts.H0+opts.H-legs_start(i, 3))/(legs(i, 3));
|
||||
Ab(i, :) = legs_start(i, :) + t*legs(i, :);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
And the location of the joints on the mobile platform with respect to $\{B\}$.
|
||||
#+begin_src matlab :results none
|
||||
Bb = zeros(6, 3);
|
||||
Bb = Ab - (opts.H0 + opts.H_tot/2 + opts.H/2)*[0, 0, 1];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none
|
||||
h = opts.H0 + opts.H/2 - opts.H_tot/2;
|
||||
Aa = Aa - h*[0, 0, 1];
|
||||
Ab = Ab - h*[0, 0, 1];
|
||||
#+end_src
|
||||
|
||||
** Returns Stewart Structure
|
||||
#+begin_src matlab :results none
|
||||
stewart = struct();
|
||||
stewart.Aa = Aa;
|
||||
stewart.Ab = Ab;
|
||||
stewart.Bb = Bb;
|
||||
stewart.H_tot = opts.H_tot;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
* TODO Tests
|
||||
** First attempt to parametrisation
|
||||
#+name: fig:stewart_bottom_plate
|
||||
#+caption: Schematic of the bottom plates with all the parameters
|
||||
[[file:./figs/stewart_bottom_plate.png]]
|
||||
|
||||
The goal is to choose $\alpha$, $\beta$, $R_\text{leg, t}$ and $R_\text{leg, b}$ in such a way that the configuration is cubic.
|
||||
|
||||
|
||||
The configuration is cubic if:
|
||||
\[ \overrightarrow{a_i b_i} \cdot \overrightarrow{a_j b_j} = 0, \ \forall i, j = [1, \hdots, 6], i \ne j \]
|
||||
|
||||
Lets express $a_i$, $b_i$ and $a_j$:
|
||||
\begin{equation*}
|
||||
a_1 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 - \alpha) \\ R_{\text{leg,b}} \cos(120 - \alpha) \\ 0\end{bmatrix} ; \quad
|
||||
a_2 = \begin{bmatrix}R_{\text{leg,b}} \cos(120 + \alpha) \\ R_{\text{leg,b}} \cos(120 + \alpha) \\ 0\end{bmatrix} ; \quad
|
||||
\end{equation*}
|
||||
|
||||
\begin{equation*}
|
||||
b_1 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 - \beta) \\ R_{\text{leg,t}} \cos(120 - \beta\\ H\end{bmatrix} ; \quad
|
||||
b_2 = \begin{bmatrix}R_{\text{leg,t}} \cos(120 + \beta) \\ R_{\text{leg,t}} \cos(120 + \beta\\ H\end{bmatrix} ; \quad
|
||||
\end{equation*}
|
||||
|
||||
\[ \overrightarrow{a_1 b_1} = b_1 - a_1 = \begin{bmatrix}R_{\text{leg}} \cos(120 - \alpha) \\ R_{\text{leg}} \cos(120 - \alpha) \\ 0\end{bmatrix}\]
|
||||
|
||||
** Second attempt
|
||||
We start with the point of a cube in space:
|
||||
\begin{align*}
|
||||
[0, 0, 0] ; \ [0, 0, 1]; \ ...
|
||||
\end{align*}
|
||||
|
||||
We also want the cube to point upward:
|
||||
\[ [1, 1, 1] \Rightarrow [0, 0, 1] \]
|
||||
|
||||
Then we have the direction of all the vectors expressed in the frame of the hexapod.
|
||||
|
||||
#+begin_src matlab :results none
|
||||
points = [0, 0, 0; ...
|
||||
0, 0, 1; ...
|
||||
0, 1, 0; ...
|
||||
0, 1, 1; ...
|
||||
1, 0, 0; ...
|
||||
1, 0, 1; ...
|
||||
1, 1, 0; ...
|
||||
1, 1, 1];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none
|
||||
figure;
|
||||
plot3(points(:,1), points(:,2), points(:,3), 'ko')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none
|
||||
sx = cross([1, 1, 1], [1 0 0]);
|
||||
sx = sx/norm(sx);
|
||||
|
||||
sy = -cross(sx, [1, 1, 1]);
|
||||
sy = sy/norm(sy);
|
||||
|
||||
sz = [1, 1, 1];
|
||||
sz = sz/norm(sz);
|
||||
|
||||
R = [sx', sy', sz']';
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none
|
||||
cube = zeros(size(points));
|
||||
for i = 1:size(points, 1)
|
||||
cube(i, :) = R * points(i, :)';
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none
|
||||
figure;
|
||||
hold on;
|
||||
plot3(points(:,1), points(:,2), points(:,3), 'ko');
|
||||
plot3(cube(:,1), cube(:,2), cube(:,3), 'ro');
|
||||
hold off;
|
||||
#+end_src
|
||||
|
||||
Now we plot the legs of the hexapod.
|
||||
#+begin_src matlab :results none
|
||||
leg_indices = [3, 4; ...
|
||||
2, 4; ...
|
||||
2, 6; ...
|
||||
5, 6; ...
|
||||
5, 7; ...
|
||||
3, 7]
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot3(cube(leg_indices(i, :),1), cube(leg_indices(i, :),2), cube(leg_indices(i, :),3), '-');
|
||||
end
|
||||
hold off;
|
||||
#+end_src
|
||||
|
||||
Vectors are:
|
||||
#+begin_src matlab :results none
|
||||
legs = zeros(6, 3);
|
||||
legs_start = zeros(6, 3);
|
||||
|
||||
for i = 1:6
|
||||
legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
|
||||
legs_start(i, :) = cube(leg_indices(i, 1), :)
|
||||
end
|
||||
#+end_src
|
||||
|
||||
We now have the orientation of each leg.
|
||||
|
||||
We here want to see if the position of the "slice" changes something.
|
||||
|
||||
Let's first estimate the maximum height of the Stewart platform.
|
||||
#+begin_src matlab :results none
|
||||
Hmax = cube(4, 3) - cube(2, 3);
|
||||
#+end_src
|
||||
|
||||
Let's then estimate the middle position of the platform
|
||||
#+begin_src matlab :results none
|
||||
Hmid = cube(8, 3)/2;
|
||||
#+end_src
|
||||
|
||||
** Generate the Stewart platform for a Cubic configuration
|
||||
|
||||
First we defined the height of the Hexapod.
|
||||
#+begin_src matlab :results none
|
||||
H = Hmax/2;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results none
|
||||
Zs = 1.2*cube(2, 3); % Height of the fixed platform
|
||||
Ze = Zs + H; % Height of the mobile platform
|
||||
#+end_src
|
||||
|
||||
We now determine the location of the joints on the fixed platform.
|
||||
#+begin_src matlab :results none
|
||||
Aa = zeros(6, 3);
|
||||
for i = 1:6
|
||||
t = (Zs-legs_start(i, 3))/(legs(i, 3));
|
||||
Aa(i, :) = legs_start(i, :) + t*legs(i, :);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
And the location of the joints on the mobile platform
|
||||
#+begin_src matlab :results none
|
||||
Ab = zeros(6, 3);
|
||||
for i = 1:6
|
||||
t = (Ze-legs_start(i, 3))/(legs(i, 3));
|
||||
Ab(i, :) = legs_start(i, :) + t*legs(i, :);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
And we plot the legs.
|
||||
#+begin_src matlab :results none
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot3([Ab(i, 1),Aa(i, 1)], [Ab(i, 2),Aa(i, 2)], [Ab(i, 3),Aa(i, 3)], 'k-');
|
||||
end
|
||||
hold off;
|
||||
xlim([-1, 1]);
|
||||
ylim([-1, 1]);
|
||||
zlim([0, 2]);
|
||||
#+end_src
|
||||
|
||||
* Bibliography :ignore:
|
||||
bibliographystyle:unsrtnat
|
||||
bibliography:ref.bib
|
||||
|
@ -1,89 +0,0 @@
|
||||
function [stewart] = initializeCubicConfiguration(opts_param)
|
||||
|
||||
opts = struct(...
|
||||
'H_tot', 90, ... % Total height of the Hexapod [mm]
|
||||
'L', 110, ... % Size of the Cube [mm]
|
||||
'H', 40, ... % Height between base joints and platform joints [mm]
|
||||
'H0', 75 ... % Height between the corner of the cube and the plane containing the base joints [mm]
|
||||
);
|
||||
|
||||
if exist('opts_param','var')
|
||||
for opt = fieldnames(opts_param)'
|
||||
opts.(opt{1}) = opts_param.(opt{1});
|
||||
end
|
||||
end
|
||||
|
||||
points = [0, 0, 0; ...
|
||||
0, 0, 1; ...
|
||||
0, 1, 0; ...
|
||||
0, 1, 1; ...
|
||||
1, 0, 0; ...
|
||||
1, 0, 1; ...
|
||||
1, 1, 0; ...
|
||||
1, 1, 1];
|
||||
points = opts.L*points;
|
||||
|
||||
sx = cross([1, 1, 1], [1 0 0]);
|
||||
sx = sx/norm(sx);
|
||||
|
||||
sy = -cross(sx, [1, 1, 1]);
|
||||
sy = sy/norm(sy);
|
||||
|
||||
sz = [1, 1, 1];
|
||||
sz = sz/norm(sz);
|
||||
|
||||
R = [sx', sy', sz']';
|
||||
|
||||
cube = zeros(size(points));
|
||||
for i = 1:size(points, 1)
|
||||
cube(i, :) = R * points(i, :)';
|
||||
end
|
||||
|
||||
leg_indices = [3, 4; ...
|
||||
2, 4; ...
|
||||
2, 6; ...
|
||||
5, 6; ...
|
||||
5, 7; ...
|
||||
3, 7];
|
||||
|
||||
legs = zeros(6, 3);
|
||||
legs_start = zeros(6, 3);
|
||||
|
||||
for i = 1:6
|
||||
legs(i, :) = cube(leg_indices(i, 2), :) - cube(leg_indices(i, 1), :);
|
||||
legs_start(i, :) = cube(leg_indices(i, 1), :);
|
||||
end
|
||||
|
||||
Hmax = cube(4, 3) - cube(2, 3);
|
||||
if opts.H0 < cube(2, 3)
|
||||
error(sprintf('H0 is not high enought. Minimum H0 = %.1f', cube(2, 3)));
|
||||
else if opts.H0 + opts.H > cube(4, 3)
|
||||
error(sprintf('H0+H is too high. Maximum H0+H = %.1f', cube(4, 3)));
|
||||
error('H0+H is too high');
|
||||
end
|
||||
|
||||
Aa = zeros(6, 3);
|
||||
for i = 1:6
|
||||
t = (opts.H0-legs_start(i, 3))/(legs(i, 3));
|
||||
Aa(i, :) = legs_start(i, :) + t*legs(i, :);
|
||||
end
|
||||
|
||||
Ab = zeros(6, 3);
|
||||
for i = 1:6
|
||||
t = (opts.H0+opts.H-legs_start(i, 3))/(legs(i, 3));
|
||||
Ab(i, :) = legs_start(i, :) + t*legs(i, :);
|
||||
end
|
||||
|
||||
Bb = zeros(6, 3);
|
||||
Bb = Ab - (opts.H0 + opts.H_tot/2 + opts.H/2)*[0, 0, 1];
|
||||
|
||||
h = opts.H0 + opts.H/2 - opts.H_tot/2;
|
||||
Aa = Aa - h*[0, 0, 1];
|
||||
Ab = Ab - h*[0, 0, 1];
|
||||
|
||||
stewart = struct();
|
||||
stewart.Aa = Aa;
|
||||
stewart.Ab = Ab;
|
||||
stewart.Bb = Bb;
|
||||
stewart.H_tot = opts.H_tot;
|
||||
end
|
Loading…
Reference in New Issue
Block a user