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Add css and js. Add lots of org mode files.

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Thomas Dehaeze 2019-03-22 12:03:59 +01:00
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<div id="content">
<h1 class="title">Identification of the Stewart Platform using Simscape</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org0c15748">1. Identification</a></li>
<li><a href="#orgb51bebd">2. Cartesian Plot</a></li>
<li><a href="#org8822347">3. From a force to force sensor</a></li>
<li><a href="#orgb3f97c3">4. From a force applied in the leg to the displacement of the leg</a></li>
<li><a href="#org4f7f749">5. Transmissibility</a></li>
<li><a href="#orgc027ff6">6. Compliance</a></li>
<li><a href="#orgeb43267">7. Inertial</a></li>
<li><a href="#org702dc6c">8. identifyPlant</a></li>
</ul>
</div>
</div>
<div id="outline-container-org0c15748" class="outline-2">
<h2 id="org0c15748"><span class="section-number-2">1</span> Identification</h2>
<div class="outline-text-2" id="text-1">
<p>
The hexapod structure and Sample structure are initialized.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeHexapod<span style="color: #DCDCCC;">()</span>;
initializeSample<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G = identifyPlant<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb51bebd" class="outline-2">
<h2 id="orgb51bebd"><span class="section-number-2">2</span> Cartesian Plot</h2>
<div class="outline-text-2" id="text-2">
<p>
From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_cart<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_cart<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
</div>
</div>
<div id="outline-container-org8822347" class="outline-2">
<h2 id="org8822347"><span class="section-number-2">3</span> From a force to force sensor</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">4</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">6</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">4</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_forc<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">6</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb3f97c3" class="outline-2">
<h2 id="orgb3f97c3"><span class="section-number-2">4</span> From a force applied in the leg to the displacement of the leg</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">4</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">6</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">4</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">6</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
</div>
</div>
<div id="outline-container-org4f7f749" class="outline-2">
<h2 id="org4f7f749"><span class="section-number-2">5</span> Transmissibility</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">4</span>, <span style="color: #BFEBBF;">4</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">6</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_tran<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc027ff6" class="outline-2">
<h2 id="orgc027ff6"><span class="section-number-2">6</span> Compliance</h2>
<div class="outline-text-2" id="text-6">
<p>
From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_comp<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_comp<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_comp<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgeb43267" class="outline-2">
<h2 id="orgeb43267"><span class="section-number-2">7</span> Inertial</h2>
<div class="outline-text-2" id="text-7">
<p>
From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
bode<span style="color: #DCDCCC;">(</span>G.G_iner<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_iner<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
bode<span style="color: #DCDCCC;">(</span>G.G_iner<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
hold off;
</pre>
</div>
</div>
</div>
<div id="outline-container-org702dc6c" class="outline-2">
<h2 id="org702dc6c"><span class="section-number-2">8</span> identifyPlant</h2>
<div class="outline-text-2" id="text-8">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">sys</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">identifyPlant</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
</pre>
</div>
<p>
We use this code block to pass optional parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Default values for opts</span>
opts = struct<span style="color: #DCDCCC;">()</span>;
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Populate opts with input parameters</span>
<span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
opts.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
<p>
We defined the options for the <code>linearize</code> command.
Here, we just identify the system at time \(t = 0\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">options = linearizeOptions;
options.SampleTime = <span style="color: #BFEBBF;">0</span>;
</pre>
</div>
<p>
We define the name of the Simulink File used to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">mdl = <span style="color: #CC9393;">'stewart'</span>;
</pre>
</div>
<p>
Then we defined the input/output of the transfer function we want to identify.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Inputs</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>F'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Cartesian forces</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Fl'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Leg forces</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Fd'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Direct forces</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Dw'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Base motion</span>
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Outputs</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Dm'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Relative Motion</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Dlm'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Displacement of each leg</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">7</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Flm'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Force sensor in each leg</span>
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">8</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>Xm'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Absolute motion of platform</span>
</pre>
</div>
<p>
The linearization is run.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<p>
We defined all the Input/Output names of the identified transfer function.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'</span>, <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'F1', 'F2', 'F3', 'F4', 'F5', 'F6'</span>, <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'</span>, <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'</span><span style="color: #DCDCCC;">}</span>;
G.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'</span>, <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'</span>, <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'</span>, <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'</span><span style="color: #DCDCCC;">}</span>;
</pre>
</div>
<p>
We split the transfer function into sub transfer functions and we compute their minimum realization.
</p>
<div class="org-src-container">
<pre class="src src-matlab">sys.G_cart = minreal<span style="color: #DCDCCC;">(</span>G<span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'</span><span style="color: #D0BF8F;">}</span><span style="color: #CC9393;">, </span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
sys.G_forc = minreal<span style="color: #DCDCCC;">(</span>G<span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'</span><span style="color: #D0BF8F;">}</span><span style="color: #CC9393;">, </span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'F1', 'F2', 'F3', 'F4', 'F5', 'F6'</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
sys.G_legs = minreal<span style="color: #DCDCCC;">(</span>G<span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'</span><span style="color: #D0BF8F;">}</span><span style="color: #CC9393;">, </span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'F1', 'F2', 'F3', 'F4', 'F5', 'F6'</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
sys.G_tran = minreal<span style="color: #DCDCCC;">(</span>G<span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'</span><span style="color: #D0BF8F;">}</span><span style="color: #CC9393;">, </span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
sys.G_comp = minreal<span style="color: #DCDCCC;">(</span>G<span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'</span><span style="color: #D0BF8F;">}</span><span style="color: #CC9393;">, </span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
sys.G_iner = minreal<span style="color: #DCDCCC;">(</span>G<span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'</span><span style="color: #D0BF8F;">}</span><span style="color: #CC9393;">, </span><span style="color: #D0BF8F;">{</span><span style="color: #CC9393;">'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7F9F7F;">% sys.G_all = minreal(G);</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-22 ven. 12:03</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>

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#+TITLE: Identification of the Stewart Platform using Simscape
:DRAWER:
#+STARTUP: overview
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
#+HTML_HEAD: <script src="js/jquery.min.js"></script>
#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
#+LATEX_CLASS: cleanreport
#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
#+LaTeX_HEADER: \usepackage{svg}
#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :mkdirp yes
:END:
* Identification
#+begin_src matlab :results none :exports none
<<matlab-init>>
addpath('src');
addpath('library');
#+end_src
#+begin_src matlab :results none :exports none
open stewart
#+end_src
The hexapod structure and Sample structure are initialized.
#+begin_src matlab :results none
initializeHexapod();
initializeSample();
#+end_src
#+begin_src matlab :results none
G = identifyPlant();
#+end_src
* Cartesian Plot
From a force applied in the Cartesian frame to a displacement in the Cartesian frame.
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_cart(1, 1));
bode(G.G_cart(3, 3));
hold off;
#+end_src
* From a force to force sensor
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_forc(1, 1));
bode(G.G_forc(2, 2));
bode(G.G_forc(3, 3));
bode(G.G_forc(4, 4));
bode(G.G_forc(5, 5));
bode(G.G_forc(6, 6));
hold off;
#+end_src
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_forc(1, 1));
bode(G.G_forc(1, 2));
bode(G.G_forc(1, 3));
bode(G.G_forc(1, 4));
bode(G.G_forc(1, 5));
bode(G.G_forc(1, 6));
hold off;
#+end_src
* From a force applied in the leg to the displacement of the leg
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_legs(1, 1));
bode(G.G_legs(2, 2));
bode(G.G_legs(3, 3));
bode(G.G_legs(4, 4));
bode(G.G_legs(5, 5));
bode(G.G_legs(6, 6));
hold off;
#+end_src
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_legs(1, 1));
bode(G.G_legs(1, 2));
bode(G.G_legs(1, 3));
bode(G.G_legs(1, 4));
bode(G.G_legs(1, 5));
bode(G.G_legs(1, 6));
hold off;
#+end_src
* Transmissibility
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_tran(1, 1));
bode(G.G_tran(2, 2));
bode(G.G_tran(3, 3));
hold off;
#+end_src
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_tran(4, 4));
bode(G.G_tran(5, 5));
bode(G.G_tran(6, 6));
hold off;
#+end_src
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_tran(1, 1));
bode(G.G_tran(2, 1));
bode(G.G_tran(3, 1));
hold off;
#+end_src
* Compliance
From a force applied in the Cartesian frame to a relative displacement of the mobile platform with respect to the base.
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_comp(1, 1));
bode(G.G_comp(2, 2));
bode(G.G_comp(3, 3));
hold off;
#+end_src
* Inertial
From a force applied on the Cartesian frame to the absolute displacement of the mobile platform.
#+begin_src matlab :results none
figure;
hold on;
bode(G.G_iner(1, 1));
bode(G.G_iner(2, 2));
bode(G.G_iner(3, 3));
hold off;
#+end_src
* identifyPlant
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments yes
:HEADER-ARGS:matlab+: :eval no
:HEADER-ARGS:matlab+: :tangle src/identifyPlant.m
:END:
#+begin_src matlab
function [sys] = identifyPlant(opts_param)
#+end_src
We use this code block to pass optional parameters.
#+begin_src matlab
%% Default values for opts
opts = struct();
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
#+end_src
We defined the options for the =linearize= command.
Here, we just identify the system at time $t = 0$.
#+begin_src matlab
options = linearizeOptions;
options.SampleTime = 0;
#+end_src
We define the name of the Simulink File used to identification.
#+begin_src matlab
mdl = 'stewart';
#+end_src
Then we defined the input/output of the transfer function we want to identify.
#+begin_src matlab
%% Inputs
io(1) = linio([mdl, '/F'], 1, 'input'); % Cartesian forces
io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion
%% Outputs
io(5) = linio([mdl, '/Dm'], 1, 'output'); % Relative Motion
io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
io(8) = linio([mdl, '/Xm'], 1, 'output'); % Absolute motion of platform
#+end_src
The linearization is run.
#+begin_src matlab
G = linearize(mdl, io, 0);
#+end_src
We defined all the Input/Output names of the identified transfer function.
#+begin_src matlab
G.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};
#+end_src
We split the transfer function into sub transfer functions and we compute their minimum realization.
#+begin_src matlab
sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
sys.G_forc = minreal(G({'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_legs = minreal(G({'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_tran = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
sys.G_comp = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
% sys.G_all = minreal(G);
#+end_src
#+begin_src matlab
end
#+end_src

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<div id="content">
<h1 class="title">Stewart Platform Studies</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org2b3b6a5">1. Simscape Model</a></li>
<li><a href="#org5dc817d">2. Architecture Study</a></li>
<li><a href="#orgccde31a">3. Motion Control</a></li>
</ul>
</div>
</div>
<div id="outline-container-org2b3b6a5" class="outline-2">
<h2 id="org2b3b6a5"><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</a></li>
</ul>
</div>
</div>
<div id="outline-container-org5dc817d" class="outline-2">
<h2 id="org5dc817d"><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>
</ul>
</div>
</div>
<div id="outline-container-orgccde31a" class="outline-2">
<h2 id="orgccde31a"><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>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-22 ven. 12:03</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>

639
index.org
View File

@ -1,4 +1,4 @@
#+TITLE: Stewart Platform with Simscape
#+TITLE: Stewart Platform Studies
:DRAWER:
#+STARTUP: overview
@ -11,6 +11,7 @@
#+LATEX_CLASS: cleanreport
#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
#+LaTeX_HEADER: \usepackage{svg}
#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
@ -23,626 +24,16 @@
#+PROPERTY: header-args:matlab+ :mkdirp yes
:END:
#+begin_src matlab :results none
<<matlab-init>>
addpath('src');
addpath('library');
#+end_src
#+begin_src matlab :results none
open stewart
#+end_src
#+begin_src matlab
hexapod = initializeHexapod();
#+end_src
#+RESULTS:
: org_babel_eoe
#+begin_src matlab
initializeSample();
#+end_src
#+begin_src matlab
G = identifyPlant();
#+end_src
#+RESULTS:
* Functions
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :mkdir yes
:HEADER-ARGS:matlab+: :eval no
:END:
** getMaxPositions
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/getMaxPositions.m
:END:
#+begin_src matlab
function [X, Y, Z] = getMaxPositions(stewart)
Leg = stewart.Leg;
J = stewart.J;
theta = linspace(0, 2*pi, 100);
phi = linspace(-pi/2 , pi/2, 100);
dmax = zeros(length(theta), length(phi));
for i = 1:length(theta)
for j = 1:length(phi)
L = J*[cos(phi(j))*cos(theta(i)) cos(phi(j))*sin(theta(i)) sin(phi(j)) 0 0 0]';
dmax(i, j) = Leg.stroke/max(abs(L));
end
end
X = dmax.*cos(repmat(phi,length(theta),1)).*cos(repmat(theta,length(phi),1))';
Y = dmax.*cos(repmat(phi,length(theta),1)).*sin(repmat(theta,length(phi),1))';
Z = dmax.*sin(repmat(phi,length(theta),1));
end
#+end_src
** getMaxPureDisplacement
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/getMaxPureDisplacement.m
:END:
#+begin_src matlab
function [max_disp] = getMaxPureDisplacement(Leg, J)
max_disp = zeros(6, 1);
max_disp(1) = Leg.stroke/max(abs(J*[1 0 0 0 0 0]'));
max_disp(2) = Leg.stroke/max(abs(J*[0 1 0 0 0 0]'));
max_disp(3) = Leg.stroke/max(abs(J*[0 0 1 0 0 0]'));
max_disp(4) = Leg.stroke/max(abs(J*[0 0 0 1 0 0]'));
max_disp(5) = Leg.stroke/max(abs(J*[0 0 0 0 1 0]'));
max_disp(6) = Leg.stroke/max(abs(J*[0 0 0 0 0 1]'));
end
#+end_src
** getStiffnessMatrix
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/getStiffnessMatrix.m
:END:
#+begin_src matlab
function [K] = getStiffnessMatrix(k, J)
% k - leg stiffness
% J - Jacobian matrix
K = k*(J'*J);
end
#+end_src
** identifyPlant
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/identifyPlant.m
:END:
#+begin_src matlab
function [sys] = identifyPlant(opts_param)
%% Default values for opts
opts = struct();
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_identification';
%% Input/Output definition
io(1) = linio([mdl, '/F'], 1, 'input'); % Cartesian forces
io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion
io(5) = linio([mdl, '/Dm'], 1, 'output'); % Relative Motion
io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
io(8) = linio([mdl, '/Xm'], 1, 'output'); % Absolute motion of platform
%% Run the linearization
G = linearize(mdl, io, 0);
%% Input/Output names
G.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};
%% Cut into sub transfer functions
sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
sys.G_forc = minreal(G({'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_legs = G({'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
sys.G_tran = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
sys.G_comp = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_all = minreal(G);
end
#+end_src
** initializeHexapod
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/initializeHexapod.m
:END:
*** Function description and arguments
The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
#+begin_src matlab
function [stewart] = initializeHexapod(opts_param)
#+end_src
Default values for opts.
#+begin_src matlab
opts = struct(...
'height', 90, ... % Height of the platform [mm]
'density', 8000, ... % Density of the material used for the hexapod [kg/m3]
'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
'c_ax', 100, ... % Damping of each actuator [N/(m/s)]
'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
'name', 'stewart' ... % Name of the file
);
#+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
*** Initialization of the stewart structure
We initialize the Stewart structure
#+begin_src matlab
stewart = struct();
#+end_src
And we defined its total height.
#+begin_src matlab
stewart.H = opts.height; % [mm]
#+end_src
*** Bottom Plate
#+name: fig:stewart_bottom_plate
#+caption: Schematic of the bottom plates with all the parameters
[[file:./figs/stewart_bottom_plate.png]]
The bottom plate structure is initialized.
#+begin_src matlab
BP = struct();
#+end_src
We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
#+begin_src matlab
BP.Rint = 0; % Internal Radius [mm]
BP.Rext = 150; % External Radius [mm]
#+end_src
We define its thickness.
#+begin_src matlab
BP.H = 10; % Thickness of the Bottom Plate [mm]
#+end_src
At which radius legs will be fixed and with that angle offset.
#+begin_src matlab
BP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
BP.alpha = 10; % Angle Offset [deg]
#+end_src
We defined the density of the material of the bottom plate.
#+begin_src matlab
BP.density = opts.density; % Density of the material [kg/m3]
#+end_src
And its color.
#+begin_src matlab
BP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src
Then the profile of the bottom plate is computed and will be used by Simscape
#+begin_src matlab
BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.BP = BP;
#+end_src
*** Top Plate
The top plate structure is initialized.
#+begin_src matlab
TP = struct();
#+end_src
We defined the internal and external radius of the top plate.
#+begin_src matlab
TP.Rint = 0; % [mm]
TP.Rext = 100; % [mm]
#+end_src
The thickness of the top plate.
#+begin_src matlab
TP.H = 10; % [mm]
#+end_src
At which radius and angle are fixed the legs.
#+begin_src matlab
TP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
TP.alpha = 20; % Angle [deg]
TP.dalpha = 0; % Angle Offset from 0 position [deg]
#+end_src
The density of its material.
#+begin_src matlab
TP.density = opts.density; % Density of the material [kg/m3]
#+end_src
Its color.
#+begin_src matlab
TP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src
Then the shape of the top plate is computed
#+begin_src matlab
TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.TP = TP;
#+end_src
*** Legs
#+name: fig:stewart_legs
#+caption: Schematic for the legs of the Stewart platform
[[file:./figs/stewart_legs.png]]
The leg structure is initialized.
#+begin_src matlab
Leg = struct();
#+end_src
The maximum Stroke of each leg is defined.
#+begin_src matlab
Leg.stroke = opts.stroke; % [m]
#+end_src
The stiffness and damping of each leg are defined
#+begin_src matlab
Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
#+end_src
The radius of the legs are defined
#+begin_src matlab
Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
#+end_src
The density of its material.
#+begin_src matlab
Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
#+end_src
Its color.
#+begin_src matlab
Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
#+end_src
The radius of spheres representing the ball joints are defined.
#+begin_src matlab
Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.Leg = Leg;
#+end_src
*** Ball Joints
#+name: fig:stewart_ball_joints
#+caption: Schematic of the support for the ball joints
[[file:./figs/stewart_ball_joints.png]]
=SP= is the structure representing the support for the ball joints at the extremity of each leg.
The =SP= structure is initialized.
#+begin_src matlab
SP = struct();
#+end_src
We can define its rotational stiffness and damping. For now, we use perfect joints.
#+begin_src matlab
SP.k = 0; % [N*m/deg]
SP.c = 0; % [N*m/deg]
#+end_src
Its height is defined
#+begin_src matlab
SP.H = 15; % [mm]
#+end_src
Its radius is based on the radius on the sphere at the end of the legs.
#+begin_src matlab
SP.R = Leg.R; % [mm]
#+end_src
#+begin_src matlab
SP.section = [0 SP.H-SP.R;
0 0;
SP.R 0;
SP.R SP.H];
#+end_src
The density of its material is defined.
#+begin_src matlab
SP.density = opts.density; % [kg/m^3]
#+end_src
Its color is defined.
#+begin_src matlab
SP.color = [0.7 0.7 0.7]; % [RGB]
#+end_src
The structure is added to the Hexapod structure
#+begin_src matlab
stewart.SP = SP;
#+end_src
*** More parameters are initialized
#+begin_src matlab
stewart = initializeParameters(stewart);
#+end_src
*** Save the Stewart Structure
#+begin_src matlab
save('./mat/stewart.mat', 'stewart')
#+end_src
*** initializeParameters Function
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle no
:END:
#+begin_src matlab
function [stewart] = initializeParameters(stewart)
#+end_src
Computation of the position of the connection points on the base and moving platform
We first initialize =pos_base= corresponding to $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and =pos_top= corresponding to $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
stewart.pos_base = zeros(6, 3);
stewart.pos_top = zeros(6, 3);
#+end_src
We estimate the height between the ball joints of the bottom platform and of the top platform.
#+begin_src matlab
height = stewart.H - stewart.BP.H - stewart.TP.H - 2*stewart.SP.H; % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
% base points
angle_m_b = 120*(i-1) - stewart.BP.alpha;
angle_p_b = 120*(i-1) + stewart.BP.alpha;
stewart.pos_base(2*i-1,:) = [stewart.BP.Rleg*cos(angle_m_b), stewart.BP.Rleg*sin(angle_m_b), 0.0];
stewart.pos_base(2*i,:) = [stewart.BP.Rleg*cos(angle_p_b), stewart.BP.Rleg*sin(angle_p_b), 0.0];
% top points
angle_m_t = 120*(i-1) - stewart.TP.alpha + stewart.TP.dalpha;
angle_p_t = 120*(i-1) + stewart.TP.alpha + stewart.TP.dalpha;
stewart.pos_top(2*i-1,:) = [stewart.TP.Rleg*cos(angle_m_t), stewart.TP.Rleg*sin(angle_m_t), height];
stewart.pos_top(2*i,:) = [stewart.TP.Rleg*cos(angle_p_t), stewart.TP.Rleg*sin(angle_p_t), height];
end
% permute pos_top points so that legs are end points of base and top points
stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
#+end_src
leg vectors
#+begin_src matlab
legs = stewart.pos_top - stewart.pos_base;
leg_length = zeros(6, 1);
leg_vectors = zeros(6, 3);
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
stewart.Leg.shape.bot = [0 0; ...
stewart.Leg.rad.bottom 0; ...
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
stewart.Leg.rad.top stewart.Leg.lenght; ...
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
#+end_src
Calculate revolute and cylindrical axes
#+begin_src matlab
rev1 = zeros(6, 3);
rev2 = zeros(6, 3);
cyl1 = zeros(6, 3);
for i = 1:6
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
cyl1(i,:) = leg_vectors(i,:);
end
#+end_src
Coordinate systems
#+begin_src matlab
stewart.lower_leg = struct('rotation', eye(3));
stewart.upper_leg = struct('rotation', eye(3));
for i = 1:6
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
end
#+end_src
Position Matrix
#+begin_src matlab
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.h+stewart.Leg.sphere.top+stewart.SP.h.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
#+end_src
Compute Jacobian Matrix
#+begin_src matlab
% aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.h - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
bb = stewart.pos_top_tranform - (stewart.TP.h + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
stewart.J = getJacobianMatrix(leg_vectors', bb');
stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
end
#+end_src
*** initializeParameters Function - BIS
#+begin_src matlab
function [stewart] = initializeParameters(stewart)
#+end_src
We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
stewart.Aa = zeros(6, 3); % [mm]
stewart.Ab = zeros(6, 3); % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
stewart.BP.H+stewart.SP.H];
stewart.Aa(2*i,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
stewart.BP.H+stewart.SP.H];
stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
stewart.Ab(2*i,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
end
#+end_src
Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
\[ b_i - a_i = l_i \hat{s}_i \]
We initialize $l_i$ and $\hat{s}_i$
#+begin_src matlab
leg_length = zeros(6, 1); % [mm]
leg_vectors = zeros(6, 3);
#+end_src
We compute $b_i - a_i$, and then:
\begin{align*}
l_i &= \left|b_i - a_i\right| \\
\hat{s}_i &= \frac{b_i - a_i}{l_i}
\end{align*}
#+begin_src matlab
legs = stewart.Ab - stewart.Aa;
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
#+end_src
Then the shape of the bottom leg is estimated
#+begin_src matlab
stewart.Leg.lenght = leg_length(1)/1.5;
stewart.Leg.shape.bot = ...
[0 0; ...
stewart.Leg.Rbot 0; ...
stewart.Leg.Rbot stewart.Leg.lenght; ...
stewart.Leg.Rtop stewart.Leg.lenght; ...
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
#+end_src
We compute rotation matrices to have the orientation of the legs.
The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
#+begin_src matlab
stewart.Rm = struct('R', eye(3));
for i = 1:6
sx = cross(leg_vectors(i,:), [1 0 0]);
sx = sx/norm(sx);
sy = -cross(sx, leg_vectors(i,:));
sy = sy/norm(sy);
sz = leg_vectors(i,:);
sz = sz/norm(sz);
stewart.Rm(i).R = [sx', sy', sz'];
end
#+end_src
Compute Jacobian Matrix
#+begin_src matlab
J = zeros(6);
for i = 1:6
J(i, 1:3) = leg_vectors(i, :);
J(i, 4:6) = cross(0.001*(stewart.Ab - stewart.H*[0,0,1]), leg_vectors(i, :));
end
stewart.J = J;
#+end_src
#+begin_src matlab
stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
#+end_src
#+begin_src matlab
end
end
#+end_src
** initializeSample
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/initializeSample.m
:END:
#+begin_src matlab
function [] = initializeSample(opts_param)
%% Default values for opts
sample = struct( ...
'radius', 100, ... % radius of the cylinder [mm]
'height', 300, ... % height of the cylinder [mm]
'mass', 50, ... % mass of the cylinder [kg]
'measheight', 150, ... % measurement point z-offset [mm]
'offset', [0, 0, 0], ... % offset position of the sample [mm]
'color', [0.9 0.1 0.1] ...
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
sample.(opt{1}) = opts_param.(opt{1});
end
end
%% Save
save('./mat/sample.mat', 'sample');
end
#+end_src
* Simscape Model
- [[file:simscape-model.org][Model of the Stewart Platform]]
- [[file:identification.org][Identification]]
* Architecture Study
- [[file:kinematic-study.org][Kinematic Study]]
- [[file:stiffness-study.org][Stiffness Matrix Study]]
- Jacobian Study
* Motion Control
- Active Damping
- Inertial Control
- Decentralized Control

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<h1 class="title">Kinematic Study of the Stewart Platform</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgc1c40d5">1. Functions</a>
<ul>
<li><a href="#org3d6cf9e">1.1. getMaxPositions</a></li>
<li><a href="#orge3ee3ac">1.2. getMaxPureDisplacement</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-orgc1c40d5" class="outline-2">
<h2 id="orgc1c40d5"><span class="section-number-2">1</span> Functions</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org3d6cf9e" class="outline-3">
<h3 id="org3d6cf9e"><span class="section-number-3">1.1</span> getMaxPositions</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">X, Y, Z</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPositions</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>
Leg = stewart.Leg;
J = stewart.J;
theta = linspace<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
phi = linspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> , <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
dmax = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>theta<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>phi<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:length</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">theta</span><span style="color: #DCDCCC;">)</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">j</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:length</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">phi</span><span style="color: #DCDCCC;">)</span>
L = J<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span>cos<span style="color: #BFEBBF;">(</span>phi<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">j</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span>theta<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> cos<span style="color: #BFEBBF;">(</span>phi<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">j</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span>theta<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> sin<span style="color: #BFEBBF;">(</span>phi<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">j</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
dmax<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">j</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>L<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
X = dmax<span style="color: #7CB8BB;">.*</span>cos<span style="color: #DCDCCC;">(</span>repmat<span style="color: #BFEBBF;">(</span>phi,length<span style="color: #D0BF8F;">(</span>theta<span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">.*</span>cos<span style="color: #DCDCCC;">(</span>repmat<span style="color: #BFEBBF;">(</span>theta,length<span style="color: #D0BF8F;">(</span>phi<span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>';
Y = dmax<span style="color: #7CB8BB;">.*</span>cos<span style="color: #DCDCCC;">(</span>repmat<span style="color: #BFEBBF;">(</span>phi,length<span style="color: #D0BF8F;">(</span>theta<span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">.*</span>sin<span style="color: #DCDCCC;">(</span>repmat<span style="color: #BFEBBF;">(</span>theta,length<span style="color: #D0BF8F;">(</span>phi<span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>';
Z = dmax<span style="color: #7CB8BB;">.*</span>sin<span style="color: #DCDCCC;">(</span>repmat<span style="color: #BFEBBF;">(</span>phi,length<span style="color: #D0BF8F;">(</span>theta<span style="color: #D0BF8F;">)</span>,<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orge3ee3ac" class="outline-3">
<h3 id="orge3ee3ac"><span class="section-number-3">1.2</span> getMaxPureDisplacement</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">max_disp</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPureDisplacement</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">Leg</span>, <span style="color: #DFAF8F;">J</span><span style="color: #DCDCCC;">)</span>
max_disp = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
max_disp<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>J<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
max_disp<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>J<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
max_disp<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>J<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
max_disp<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>J<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
max_disp<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>J<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span><span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
max_disp<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span> = Leg.stroke<span style="color: #7CB8BB;">/</span>max<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>J<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
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<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-22 ven. 12:03</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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#+TITLE: Kinematic Study of the Stewart Platform
* Functions
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :mkdir yes
:HEADER-ARGS:matlab+: :eval no
:END:
** getMaxPositions
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/getMaxPositions.m
:END:
#+begin_src matlab
function [X, Y, Z] = getMaxPositions(stewart)
Leg = stewart.Leg;
J = stewart.J;
theta = linspace(0, 2*pi, 100);
phi = linspace(-pi/2 , pi/2, 100);
dmax = zeros(length(theta), length(phi));
for i = 1:length(theta)
for j = 1:length(phi)
L = J*[cos(phi(j))*cos(theta(i)) cos(phi(j))*sin(theta(i)) sin(phi(j)) 0 0 0]';
dmax(i, j) = Leg.stroke/max(abs(L));
end
end
X = dmax.*cos(repmat(phi,length(theta),1)).*cos(repmat(theta,length(phi),1))';
Y = dmax.*cos(repmat(phi,length(theta),1)).*sin(repmat(theta,length(phi),1))';
Z = dmax.*sin(repmat(phi,length(theta),1));
end
#+end_src
** getMaxPureDisplacement
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/getMaxPureDisplacement.m
:END:
#+begin_src matlab
function [max_disp] = getMaxPureDisplacement(Leg, J)
max_disp = zeros(6, 1);
max_disp(1) = Leg.stroke/max(abs(J*[1 0 0 0 0 0]'));
max_disp(2) = Leg.stroke/max(abs(J*[0 1 0 0 0 0]'));
max_disp(3) = Leg.stroke/max(abs(J*[0 0 1 0 0 0]'));
max_disp(4) = Leg.stroke/max(abs(J*[0 0 0 1 0 0]'));
max_disp(5) = Leg.stroke/max(abs(J*[0 0 0 0 1 0]'));
max_disp(6) = Leg.stroke/max(abs(J*[0 0 0 0 0 1]'));
end
#+end_src

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<div id="content">
<h1 class="title">Stewart Platform - Simscape Model</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org9a10766">1. Function description and arguments</a></li>
<li><a href="#orgb6911a1">2. Initialization of the stewart structure</a></li>
<li><a href="#org030aed6">3. Bottom Plate</a></li>
<li><a href="#orged8012a">4. Top Plate</a></li>
<li><a href="#orgc74617a">5. Legs</a></li>
<li><a href="#org7cd2aa5">6. Ball Joints</a></li>
<li><a href="#org1d76ed9">7. More parameters are initialized</a></li>
<li><a href="#orge9faa26">8. Save the Stewart Structure</a></li>
<li><a href="#orga207d03">9. initializeParameters Function</a></li>
<li><a href="#org724c1a1">10. initializeSample</a></li>
</ul>
</div>
</div>
<div id="outline-container-org9a10766" class="outline-2">
<h2 id="org9a10766"><span class="section-number-2">1</span> Function description and arguments</h2>
<div class="outline-text-2" id="text-1">
<p>
The <code>initializeHexapod</code> function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeHexapod</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
</pre>
</div>
<p>
Default values for opts.
</p>
<div class="org-src-container">
<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'height'</span>, <span style="color: #BFEBBF;">90</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height of the platform [mm]</span>
<span style="color: #CC9393;">'density'</span>, <span style="color: #BFEBBF;">8000</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Density of the material used for the hexapod [kg/m3]</span>
<span style="color: #CC9393;">'k_ax'</span>, <span style="color: #BFEBBF;">1e8</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Stiffness of each actuator [N/m]</span>
<span style="color: #CC9393;">'c_ax'</span>, <span style="color: #BFEBBF;">1000</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Damping of each actuator [N/(m/s)]</span>
<span style="color: #CC9393;">'stroke'</span>, <span style="color: #BFEBBF;">50e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Maximum stroke of each actuator [m]</span>
<span style="color: #CC9393;">'name', 'stewart'</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Name of the file</span>
<span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<p>
Populate opts with input parameters
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
opts.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb6911a1" class="outline-2">
<h2 id="orgb6911a1"><span class="section-number-2">2</span> Initialization of the stewart structure</h2>
<div class="outline-text-2" id="text-2">
<p>
We initialize the Stewart structure
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = struct<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
<p>
And we defined its total height.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.H = opts.height; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org030aed6" class="outline-2">
<h2 id="org030aed6"><span class="section-number-2">3</span> Bottom Plate</h2>
<div class="outline-text-2" id="text-3">
<div id="org3d7fe71" class="figure">
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the bottom plates with all the parameters</p>
</div>
<p>
The bottom plate structure is initialized.
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP = struct<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
<p>
We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP.Rint = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% Internal Radius [mm]</span>
BP.Rext = <span style="color: #BFEBBF;">150</span>; <span style="color: #7F9F7F;">% External Radius [mm]</span>
</pre>
</div>
<p>
We define its thickness.
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP.H = <span style="color: #BFEBBF;">10</span>; <span style="color: #7F9F7F;">% Thickness of the Bottom Plate [mm]</span>
</pre>
</div>
<p>
At which radius legs will be fixed and with that angle offset.
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP.Rleg = <span style="color: #BFEBBF;">100</span>; <span style="color: #7F9F7F;">% Radius where the legs articulations are positionned [mm]</span>
BP.alpha = <span style="color: #BFEBBF;">10</span>; <span style="color: #7F9F7F;">% Angle Offset [deg]</span>
</pre>
</div>
<p>
We defined the density of the material of the bottom plate.
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP.density = opts.density; <span style="color: #7F9F7F;">% Density of the material [kg/m3]</span>
</pre>
</div>
<p>
And its color.
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP.color = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span><span style="color: #DCDCCC;">]</span>; <span style="color: #7F9F7F;">% Color [RGB]</span>
</pre>
</div>
<p>
Then the profile of the bottom plate is computed and will be used by Simscape
</p>
<div class="org-src-container">
<pre class="src src-matlab">BP.shape = <span style="color: #DCDCCC;">[</span>BP.Rint BP.H; BP.Rint <span style="color: #BFEBBF;">0</span>; BP.Rext <span style="color: #BFEBBF;">0</span>; BP.Rext BP.H<span style="color: #DCDCCC;">]</span>; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
<p>
The structure is added to the stewart structure
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.BP = BP;
</pre>
</div>
</div>
</div>
<div id="outline-container-orged8012a" class="outline-2">
<h2 id="orged8012a"><span class="section-number-2">4</span> Top Plate</h2>
<div class="outline-text-2" id="text-4">
<p>
The top plate structure is initialized.
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP = struct<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
<p>
We defined the internal and external radius of the top plate.
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP.Rint = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% [mm]</span>
TP.Rext = <span style="color: #BFEBBF;">100</span>; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
<p>
The thickness of the top plate.
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP.H = <span style="color: #BFEBBF;">10</span>; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
<p>
At which radius and angle are fixed the legs.
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP.Rleg = <span style="color: #BFEBBF;">100</span>; <span style="color: #7F9F7F;">% Radius where the legs articulations are positionned [mm]</span>
TP.alpha = <span style="color: #BFEBBF;">20</span>; <span style="color: #7F9F7F;">% Angle [deg]</span>
TP.dalpha = <span style="color: #BFEBBF;">0</span>; % Angle Offset from <span style="color: #BFEBBF;">0</span> position [deg]
</pre>
</div>
<p>
The density of its material.
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP.density = opts.density; <span style="color: #7F9F7F;">% Density of the material [kg/m3]</span>
</pre>
</div>
<p>
Its color.
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP.color = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span><span style="color: #DCDCCC;">]</span>; <span style="color: #7F9F7F;">% Color [RGB]</span>
</pre>
</div>
<p>
Then the shape of the top plate is computed
</p>
<div class="org-src-container">
<pre class="src src-matlab">TP.shape = <span style="color: #DCDCCC;">[</span>TP.Rint TP.H; TP.Rint <span style="color: #BFEBBF;">0</span>; TP.Rext <span style="color: #BFEBBF;">0</span>; TP.Rext TP.H<span style="color: #DCDCCC;">]</span>;
</pre>
</div>
<p>
The structure is added to the stewart structure
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.TP = TP;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc74617a" class="outline-2">
<h2 id="orgc74617a"><span class="section-number-2">5</span> Legs</h2>
<div class="outline-text-2" id="text-5">
<div id="orgc225133" class="figure">
<p><img src="./figs/stewart_legs.png" alt="stewart_legs.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Schematic for the legs of the Stewart platform</p>
</div>
<p>
The leg structure is initialized.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg = struct<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
<p>
The maximum Stroke of each leg is defined.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg.stroke = opts.stroke; <span style="color: #7F9F7F;">% [m]</span>
</pre>
</div>
<p>
The stiffness and damping of each leg are defined
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg.k_ax = opts.k_ax; <span style="color: #7F9F7F;">% Stiffness of each leg [N/m]</span>
Leg.c_ax = opts.c_ax; <span style="color: #7F9F7F;">% Damping of each leg [N/(m/s)]</span>
</pre>
</div>
<p>
The radius of the legs are defined
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg.Rtop = <span style="color: #BFEBBF;">10</span>; <span style="color: #7F9F7F;">% Radius of the cylinder of the top part of the leg[mm]</span>
Leg.Rbot = <span style="color: #BFEBBF;">12</span>; <span style="color: #7F9F7F;">% Radius of the cylinder of the bottom part of the leg [mm]</span>
</pre>
</div>
<p>
The density of its material.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg.density = opts.density; <span style="color: #7F9F7F;">% Density of the material used for the legs [kg/m3]</span>
</pre>
</div>
<p>
Its color.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg.color = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">]</span>; <span style="color: #7F9F7F;">% Color of the top part of the leg [RGB]</span>
</pre>
</div>
<p>
The radius of spheres representing the ball joints are defined.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Leg.R = <span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">*</span>Leg.Rbot; <span style="color: #7F9F7F;">% Size of the sphere at the extremity of the leg [mm]</span>
</pre>
</div>
<p>
The structure is added to the stewart structure
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.Leg = Leg;
</pre>
</div>
</div>
</div>
<div id="outline-container-org7cd2aa5" class="outline-2">
<h2 id="org7cd2aa5"><span class="section-number-2">6</span> Ball Joints</h2>
<div class="outline-text-2" id="text-6">
<div id="org7b92b11" class="figure">
<p><img src="./figs/stewart_ball_joints.png" alt="stewart_ball_joints.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Schematic of the support for the ball joints</p>
</div>
<p>
<code>SP</code> is the structure representing the support for the ball joints at the extremity of each leg.
</p>
<p>
The <code>SP</code> structure is initialized.
</p>
<div class="org-src-container">
<pre class="src src-matlab">SP = struct<span style="color: #DCDCCC;">()</span>;
</pre>
</div>
<p>
We can define its rotational stiffness and damping. For now, we use perfect joints.
</p>
<div class="org-src-container">
<pre class="src src-matlab">SP.k = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% [N*m/deg]</span>
SP.c = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% [N*m/deg]</span>
</pre>
</div>
<p>
Its height is defined
</p>
<div class="org-src-container">
<pre class="src src-matlab">SP.H = <span style="color: #BFEBBF;">15</span>; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
<p>
Its radius is based on the radius on the sphere at the end of the legs.
</p>
<div class="org-src-container">
<pre class="src src-matlab">SP.R = Leg.R; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">SP.section = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> SP.H<span style="color: #7CB8BB;">-</span>SP.R;
<span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span>;
SP.R <span style="color: #BFEBBF;">0</span>;
SP.R SP.H<span style="color: #DCDCCC;">]</span>;
</pre>
</div>
<p>
The density of its material is defined.
</p>
<div class="org-src-container">
<pre class="src src-matlab">SP.density = opts.density; % [kg<span style="color: #7CB8BB;">/</span>m<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">3</span>]
</pre>
</div>
<p>
Its color is defined.
</p>
<div class="org-src-container">
<pre class="src src-matlab">SP.color = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">7</span><span style="color: #DCDCCC;">]</span>; <span style="color: #7F9F7F;">% [RGB]</span>
</pre>
</div>
<p>
The structure is added to the Hexapod structure
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.SP = SP;
</pre>
</div>
</div>
</div>
<div id="outline-container-org1d76ed9" class="outline-2">
<h2 id="org1d76ed9"><span class="section-number-2">7</span> More parameters are initialized</h2>
<div class="outline-text-2" id="text-7">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeParameters<span style="color: #DCDCCC;">(</span>stewart<span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orge9faa26" class="outline-2">
<h2 id="orge9faa26"><span class="section-number-2">8</span> Save the Stewart Structure</h2>
<div class="outline-text-2" id="text-8">
<div class="org-src-container">
<pre class="src src-matlab">save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/stewart.mat', 'stewart'</span><span style="color: #DCDCCC;">)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orga207d03" class="outline-2">
<h2 id="orga207d03"><span class="section-number-2">9</span> initializeParameters Function</h2>
<div class="outline-text-2" id="text-9">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeParameters</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>
</pre>
</div>
<p>
We first compute \([a_1, a_2, a_3, a_4, a_5, a_6]^T\) and \([b_1, b_2, b_3, b_4, b_5, b_6]^T\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.Aa = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
stewart.Ab = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
stewart.Bb = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">3</span>
stewart.Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">-</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">-</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.BP.H<span style="color: #7CB8BB;">+</span>stewart.SP.H<span style="color: #DCDCCC;">]</span>;
stewart.Aa<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.BP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.BP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.BP.H<span style="color: #7CB8BB;">+</span>stewart.SP.H<span style="color: #DCDCCC;">]</span>;
stewart.Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">-</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">-</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.H <span style="color: #7CB8BB;">-</span> stewart.TP.H <span style="color: #7CB8BB;">-</span> stewart.SP.H<span style="color: #DCDCCC;">]</span>;
stewart.Ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = <span style="color: #DCDCCC;">[</span>stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>cos<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">+</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.TP.Rleg<span style="color: #7CB8BB;">*</span>sin<span style="color: #BFEBBF;">(</span> <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">120</span><span style="color: #7CB8BB;">*</span><span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span> <span style="color: #7CB8BB;">+</span> stewart.TP.dalpha <span style="color: #7CB8BB;">+</span> stewart.TP.alpha<span style="color: #D0BF8F;">)</span> <span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span>
stewart.H <span style="color: #7CB8BB;">-</span> stewart.TP.H <span style="color: #7CB8BB;">-</span> stewart.SP.H<span style="color: #DCDCCC;">]</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
stewart.Bb = stewart.Ab <span style="color: #7CB8BB;">-</span> stewart.H<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">]</span>;
</pre>
</div>
<p>
Now, we compute the leg vectors \(\hat{s}_i\) and leg position \(l_i\):
\[ b_i - a_i = l_i \hat{s}_i \]
</p>
<p>
We initialize \(l_i\) and \(\hat{s}_i\)
</p>
<div class="org-src-container">
<pre class="src src-matlab">leg_length = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [mm]</span>
leg_vectors = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<p>
We compute \(b_i - a_i\), and then:
</p>
\begin{align*}
l_i &= \left|b_i - a_i\right| \\
\hat{s}_i &= \frac{b_i - a_i}{l_i}
\end{align*}
<div class="org-src-container">
<pre class="src src-matlab">legs = stewart.Ab <span style="color: #7CB8BB;">-</span> stewart.Aa;
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = norm<span style="color: #DCDCCC;">(</span>legs<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> = legs<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">/</span> leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
<p>
Then the shape of the bottom leg is estimated
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.Leg.lenght = leg_length<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">5</span>;
stewart.Leg.shape.bot = <span style="text-decoration: underline;">...</span>
<span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
stewart.Leg.Rbot <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
stewart.Leg.Rbot stewart.Leg.lenght; <span style="text-decoration: underline;">...</span>
stewart.Leg.Rtop stewart.Leg.lenght; <span style="text-decoration: underline;">...</span>
stewart.Leg.Rtop <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>stewart.Leg.lenght; <span style="text-decoration: underline;">...</span>
<span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>stewart.Leg.lenght<span style="color: #DCDCCC;">]</span>;
</pre>
</div>
<p>
We compute rotation matrices to have the orientation of the legs.
The rotation matrix transforms the \(z\) axis to the axis of the leg. The other axis are not important here.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart.Rm = struct<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'R'</span>, eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
sx = cross<span style="color: #DCDCCC;">(</span>leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
sx = sx<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sx<span style="color: #DCDCCC;">)</span>;
sy = <span style="color: #7CB8BB;">-</span>cross<span style="color: #DCDCCC;">(</span>sx, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
sy = sy<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sy<span style="color: #DCDCCC;">)</span>;
sz = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>,<span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
sz = sz<span style="color: #7CB8BB;">/</span>norm<span style="color: #DCDCCC;">(</span>sz<span style="color: #DCDCCC;">)</span>;
stewart.Rm<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span>.R = <span style="color: #DCDCCC;">[</span>sx', sy', sz'<span style="color: #DCDCCC;">]</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
<p>
Compute Jacobian Matrix
</p>
<div class="org-src-container">
<pre class="src src-matlab">J = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:</span><span style="color: #BFEBBF;">6</span>
J<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = leg_vectors<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>;
J<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #BFEBBF;">4</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">6</span><span style="color: #DCDCCC;">)</span> = cross<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">001</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>stewart.Ab<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">-</span> stewart.H<span style="color: #7CB8BB;">*</span><span style="color: #D0BF8F;">[</span><span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">0</span>,<span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">]</span><span style="color: #BFEBBF;">)</span>, leg_vectors<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">i</span>, <span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
stewart.J = J;
stewart.Jinv = inv<span style="color: #DCDCCC;">(</span>J<span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">stewart.K = stewart.Leg.k_ax<span style="color: #7CB8BB;">*</span>stewart.J'<span style="color: #7CB8BB;">*</span>stewart.J;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org724c1a1" class="outline-2">
<h2 id="org724c1a1"><span class="section-number-2">10</span> initializeSample</h2>
<div class="outline-text-2" id="text-10">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[]</span> = <span style="color: #93E0E3;">initializeSample</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Default values for opts</span>
sample = struct<span style="color: #DCDCCC;">(</span> <span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'radius'</span>, <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% radius of the cylinder [mm]</span>
<span style="color: #CC9393;">'height'</span>, <span style="color: #BFEBBF;">100</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% height of the cylinder [mm]</span>
<span style="color: #CC9393;">'mass'</span>, <span style="color: #BFEBBF;">10</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% mass of the cylinder [kg]</span>
<span style="color: #CC9393;">'measheight'</span>, <span style="color: #BFEBBF;">50</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% measurement point z-offset [mm]</span>
<span style="color: #CC9393;">'offset'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% offset position of the sample [mm]</span>
<span style="color: #CC9393;">'color'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">9</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">]</span> <span style="text-decoration: underline;">...</span>
<span style="color: #DCDCCC;">)</span>;
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Populate opts with input parameters</span>
<span style="color: #F0DFAF; font-weight: bold;">if</span> exist<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'opts_param','var'</span><span style="color: #DCDCCC;">)</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">opt</span> = <span style="color: #BFEBBF;">fieldnames</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">opts_param</span><span style="color: #DCDCCC;">)</span><span style="color: #BFEBBF;">'</span>
sample.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span> = opts_param.<span style="color: #DCDCCC;">(</span>opt<span style="color: #BFEBBF;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Save</span>
save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/sample.mat', 'sample'</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-22 ven. 12:03</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>

503
simscape-model.org Normal file
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#+TITLE: Stewart Platform - Simscape Model
:DRAWER:
#+STARTUP: overview
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
#+HTML_HEAD: <script src="js/jquery.min.js"></script>
#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
#+LATEX_CLASS: cleanreport
#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
#+LaTeX_HEADER: \usepackage{svg}
#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments no
#+PROPERTY: header-args:matlab+ :exports bode
#+PROPERTY: header-args:matlab+ :eval no
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :tangle src/initializeHexapod.m
:END:
* Function description and arguments
The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
#+begin_src matlab
function [stewart] = initializeHexapod(opts_param)
#+end_src
Default values for opts.
#+begin_src matlab
opts = struct(...
'height', 90, ... % Height of the platform [mm]
'density', 8000, ... % Density of the material used for the hexapod [kg/m3]
'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
'c_ax', 1000, ... % Damping of each actuator [N/(m/s)]
'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
'name', 'stewart' ... % Name of the file
);
#+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
* Initialization of the stewart structure
We initialize the Stewart structure
#+begin_src matlab
stewart = struct();
#+end_src
And we defined its total height.
#+begin_src matlab
stewart.H = opts.height; % [mm]
#+end_src
* Bottom Plate
#+name: fig:stewart_bottom_plate
#+caption: Schematic of the bottom plates with all the parameters
[[file:./figs/stewart_bottom_plate.png]]
The bottom plate structure is initialized.
#+begin_src matlab
BP = struct();
#+end_src
We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
#+begin_src matlab
BP.Rint = 0; % Internal Radius [mm]
BP.Rext = 150; % External Radius [mm]
#+end_src
We define its thickness.
#+begin_src matlab
BP.H = 10; % Thickness of the Bottom Plate [mm]
#+end_src
At which radius legs will be fixed and with that angle offset.
#+begin_src matlab
BP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
BP.alpha = 10; % Angle Offset [deg]
#+end_src
We defined the density of the material of the bottom plate.
#+begin_src matlab
BP.density = opts.density; % Density of the material [kg/m3]
#+end_src
And its color.
#+begin_src matlab
BP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src
Then the profile of the bottom plate is computed and will be used by Simscape
#+begin_src matlab
BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.BP = BP;
#+end_src
* Top Plate
The top plate structure is initialized.
#+begin_src matlab
TP = struct();
#+end_src
We defined the internal and external radius of the top plate.
#+begin_src matlab
TP.Rint = 0; % [mm]
TP.Rext = 100; % [mm]
#+end_src
The thickness of the top plate.
#+begin_src matlab
TP.H = 10; % [mm]
#+end_src
At which radius and angle are fixed the legs.
#+begin_src matlab
TP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
TP.alpha = 20; % Angle [deg]
TP.dalpha = 0; % Angle Offset from 0 position [deg]
#+end_src
The density of its material.
#+begin_src matlab
TP.density = opts.density; % Density of the material [kg/m3]
#+end_src
Its color.
#+begin_src matlab
TP.color = [0.7 0.7 0.7]; % Color [RGB]
#+end_src
Then the shape of the top plate is computed
#+begin_src matlab
TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.TP = TP;
#+end_src
* Legs
#+name: fig:stewart_legs
#+caption: Schematic for the legs of the Stewart platform
[[file:./figs/stewart_legs.png]]
The leg structure is initialized.
#+begin_src matlab
Leg = struct();
#+end_src
The maximum Stroke of each leg is defined.
#+begin_src matlab
Leg.stroke = opts.stroke; % [m]
#+end_src
The stiffness and damping of each leg are defined
#+begin_src matlab
Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
#+end_src
The radius of the legs are defined
#+begin_src matlab
Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
#+end_src
The density of its material.
#+begin_src matlab
Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
#+end_src
Its color.
#+begin_src matlab
Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
#+end_src
The radius of spheres representing the ball joints are defined.
#+begin_src matlab
Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
#+end_src
The structure is added to the stewart structure
#+begin_src matlab
stewart.Leg = Leg;
#+end_src
* Ball Joints
#+name: fig:stewart_ball_joints
#+caption: Schematic of the support for the ball joints
[[file:./figs/stewart_ball_joints.png]]
=SP= is the structure representing the support for the ball joints at the extremity of each leg.
The =SP= structure is initialized.
#+begin_src matlab
SP = struct();
#+end_src
We can define its rotational stiffness and damping. For now, we use perfect joints.
#+begin_src matlab
SP.k = 0; % [N*m/deg]
SP.c = 0; % [N*m/deg]
#+end_src
Its height is defined
#+begin_src matlab
SP.H = 15; % [mm]
#+end_src
Its radius is based on the radius on the sphere at the end of the legs.
#+begin_src matlab
SP.R = Leg.R; % [mm]
#+end_src
#+begin_src matlab
SP.section = [0 SP.H-SP.R;
0 0;
SP.R 0;
SP.R SP.H];
#+end_src
The density of its material is defined.
#+begin_src matlab
SP.density = opts.density; % [kg/m^3]
#+end_src
Its color is defined.
#+begin_src matlab
SP.color = [0.7 0.7 0.7]; % [RGB]
#+end_src
The structure is added to the Hexapod structure
#+begin_src matlab
stewart.SP = SP;
#+end_src
* More parameters are initialized
#+begin_src matlab
stewart = initializeParameters(stewart);
#+end_src
* Save the Stewart Structure
#+begin_src matlab
save('./mat/stewart.mat', 'stewart')
#+end_src
* initializeParameters Function :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle no
:END:
#+begin_src matlab
function [stewart] = initializeParameters(stewart)
#+end_src
Computation of the position of the connection points on the base and moving platform
We first initialize =pos_base= corresponding to $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and =pos_top= corresponding to $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
stewart.pos_base = zeros(6, 3);
stewart.pos_top = zeros(6, 3);
#+end_src
We estimate the height between the ball joints of the bottom platform and of the top platform.
#+begin_src matlab
height = stewart.H - stewart.BP.H - stewart.TP.H - 2*stewart.SP.H; % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
% base points
angle_m_b = 120*(i-1) - stewart.BP.alpha;
angle_p_b = 120*(i-1) + stewart.BP.alpha;
stewart.pos_base(2*i-1,:) = [stewart.BP.Rleg*cos(angle_m_b), stewart.BP.Rleg*sin(angle_m_b), 0.0];
stewart.pos_base(2*i,:) = [stewart.BP.Rleg*cos(angle_p_b), stewart.BP.Rleg*sin(angle_p_b), 0.0];
% top points
angle_m_t = 120*(i-1) - stewart.TP.alpha + stewart.TP.dalpha;
angle_p_t = 120*(i-1) + stewart.TP.alpha + stewart.TP.dalpha;
stewart.pos_top(2*i-1,:) = [stewart.TP.Rleg*cos(angle_m_t), stewart.TP.Rleg*sin(angle_m_t), height];
stewart.pos_top(2*i,:) = [stewart.TP.Rleg*cos(angle_p_t), stewart.TP.Rleg*sin(angle_p_t), height];
end
% permute pos_top points so that legs are end points of base and top points
stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom
stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)];
#+end_src
leg vectors
#+begin_src matlab
legs = stewart.pos_top - stewart.pos_base;
leg_length = zeros(6, 1);
leg_vectors = zeros(6, 3);
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
stewart.Leg.lenght = 1000*leg_length(1)/1.5;
stewart.Leg.shape.bot = [0 0; ...
stewart.Leg.rad.bottom 0; ...
stewart.Leg.rad.bottom stewart.Leg.lenght; ...
stewart.Leg.rad.top stewart.Leg.lenght; ...
stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
#+end_src
Calculate revolute and cylindrical axes
#+begin_src matlab
rev1 = zeros(6, 3);
rev2 = zeros(6, 3);
cyl1 = zeros(6, 3);
for i = 1:6
rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]);
rev1(i,:) = rev1(i,:) / norm(rev1(i,:));
rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:));
rev2(i,:) = rev2(i,:) / norm(rev2(i,:));
cyl1(i,:) = leg_vectors(i,:);
end
#+end_src
Coordinate systems
#+begin_src matlab
stewart.lower_leg = struct('rotation', eye(3));
stewart.upper_leg = struct('rotation', eye(3));
for i = 1:6
stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)'];
end
#+end_src
Position Matrix
#+begin_src matlab
stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.h+stewart.Leg.sphere.top+stewart.SP.h.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)];
#+end_src
Compute Jacobian Matrix
#+begin_src matlab
% aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.h - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
bb = stewart.pos_top_tranform - (stewart.TP.h + stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)];
bb = bb - stewart.jacobian*1e-3*[zeros(6, 2),ones(6, 1)];
stewart.J = getJacobianMatrix(leg_vectors', bb');
stewart.K = stewart.Leg.k.ax*stewart.J'*stewart.J;
end
#+end_src
* initializeParameters Function
#+begin_src matlab
function [stewart] = initializeParameters(stewart)
#+end_src
We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
#+begin_src matlab
stewart.Aa = zeros(6, 3); % [mm]
stewart.Ab = zeros(6, 3); % [mm]
stewart.Bb = zeros(6, 3); % [mm]
#+end_src
#+begin_src matlab
for i = 1:3
stewart.Aa(2*i-1,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
stewart.BP.Rleg*sin( pi/180*(120*(i-1) - stewart.BP.alpha) ), ...
stewart.BP.H+stewart.SP.H];
stewart.Aa(2*i,:) = [stewart.BP.Rleg*cos( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
stewart.BP.Rleg*sin( pi/180*(120*(i-1) + stewart.BP.alpha) ), ...
stewart.BP.H+stewart.SP.H];
stewart.Ab(2*i-1,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha - stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
stewart.Ab(2*i,:) = [stewart.TP.Rleg*cos( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
end
stewart.Bb = stewart.Ab - stewart.H*[0,0,1];
#+end_src
Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
\[ b_i - a_i = l_i \hat{s}_i \]
We initialize $l_i$ and $\hat{s}_i$
#+begin_src matlab
leg_length = zeros(6, 1); % [mm]
leg_vectors = zeros(6, 3);
#+end_src
We compute $b_i - a_i$, and then:
\begin{align*}
l_i &= \left|b_i - a_i\right| \\
\hat{s}_i &= \frac{b_i - a_i}{l_i}
\end{align*}
#+begin_src matlab
legs = stewart.Ab - stewart.Aa;
for i = 1:6
leg_length(i) = norm(legs(i,:));
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
#+end_src
Then the shape of the bottom leg is estimated
#+begin_src matlab
stewart.Leg.lenght = leg_length(1)/1.5;
stewart.Leg.shape.bot = ...
[0 0; ...
stewart.Leg.Rbot 0; ...
stewart.Leg.Rbot stewart.Leg.lenght; ...
stewart.Leg.Rtop stewart.Leg.lenght; ...
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
#+end_src
We compute rotation matrices to have the orientation of the legs.
The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
#+begin_src matlab
stewart.Rm = struct('R', eye(3));
for i = 1:6
sx = cross(leg_vectors(i,:), [1 0 0]);
sx = sx/norm(sx);
sy = -cross(sx, leg_vectors(i,:));
sy = sy/norm(sy);
sz = leg_vectors(i,:);
sz = sz/norm(sz);
stewart.Rm(i).R = [sx', sy', sz'];
end
#+end_src
Compute Jacobian Matrix
#+begin_src matlab
J = zeros(6);
for i = 1:6
J(i, 1:3) = leg_vectors(i, :);
J(i, 4:6) = cross(0.001*(stewart.Ab(i, :)- stewart.H*[0,0,1]), leg_vectors(i, :));
end
stewart.J = J;
stewart.Jinv = inv(J);
#+end_src
#+begin_src matlab
stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;
#+end_src
#+begin_src matlab
end
end
#+end_src
* initializeSample
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/initializeSample.m
:END:
#+begin_src matlab
function [] = initializeSample(opts_param)
%% Default values for opts
sample = struct( ...
'radius', 100, ... % radius of the cylinder [mm]
'height', 100, ... % height of the cylinder [mm]
'mass', 10, ... % mass of the cylinder [kg]
'measheight', 50, ... % measurement point z-offset [mm]
'offset', [0, 0, 0], ... % offset position of the sample [mm]
'color', [0.9 0.1 0.1] ...
);
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
sample.(opt{1}) = opts_param.(opt{1});
end
end
%% Save
save('./mat/sample.mat', 'sample');
end
#+end_src

108
src/identifyPlant.m
View File

@ -1,51 +1,67 @@
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:1]]
function [sys] = identifyPlant(opts_param)
% identifyPlant:1 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:2]]
%% Default values for opts
opts = struct();
opts = struct();
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
%% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_identification';
%% Input/Output definition
io(1) = linio([mdl, '/F'], 1, 'input'); % Cartesian forces
io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion
io(5) = linio([mdl, '/Dm'], 1, 'output'); % Relative Motion
io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
io(8) = linio([mdl, '/Xm'], 1, 'output'); % Absolute motion of platform
%% Run the linearization
G = linearize(mdl, io, 0);
%% Input/Output names
G.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};
%% Cut into sub transfer functions
sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
sys.G_forc = minreal(G({'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_legs = G({'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
sys.G_tran = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
sys.G_comp = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_all = minreal(G);
end
% identifyPlant:2 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:3]]
options = linearizeOptions;
options.SampleTime = 0;
% identifyPlant:3 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:4]]
mdl = 'stewart';
% identifyPlant:4 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:5]]
%% Inputs
io(1) = linio([mdl, '/F'], 1, 'input'); % Cartesian forces
io(2) = linio([mdl, '/Fl'], 1, 'input'); % Leg forces
io(3) = linio([mdl, '/Fd'], 1, 'input'); % Direct forces
io(4) = linio([mdl, '/Dw'], 1, 'input'); % Base motion
%% Outputs
io(5) = linio([mdl, '/Dm'], 1, 'output'); % Relative Motion
io(6) = linio([mdl, '/Dlm'], 1, 'output'); % Displacement of each leg
io(7) = linio([mdl, '/Flm'], 1, 'output'); % Force sensor in each leg
io(8) = linio([mdl, '/Xm'], 1, 'output'); % Absolute motion of platform
% identifyPlant:5 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:6]]
G = linearize(mdl, io, 0);
% identifyPlant:6 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:7]]
G.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz', ...
'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'};
G.OutputName = {'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm', ...
'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m', ...
'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m', ...
'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'};
% identifyPlant:7 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:8]]
sys.G_cart = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}));
sys.G_forc = minreal(G({'F1m', 'F2m', 'F3m', 'F4m', 'F5m', 'F6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_legs = minreal(G({'D1m', 'D2m', 'D3m', 'D4m', 'D5m', 'D6m'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}));
sys.G_tran = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}));
sys.G_comp = minreal(G({'Dxm', 'Dym', 'Dzm', 'Rxm', 'Rym', 'Rzm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
sys.G_iner = minreal(G({'Dxtm', 'Dytm', 'Dztm', 'Rxtm', 'Rytm', 'Rztm'}, {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'}));
% sys.G_all = minreal(G);
% identifyPlant:8 ends here
% [[file:~/MEGA/These/Matlab/Simscape/stewart-simscape/identification.org::*identifyPlant][identifyPlant:9]]
end
% identifyPlant:9 ends here

199
src/initializeHexapod.m
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@ -1,86 +1,228 @@
% Function description and arguments
% The =initializeHexapod= function takes one structure that contains configurations for the hexapod and returns one structure representing the hexapod.
function [stewart] = initializeHexapod(opts_param)
% Default values for opts.
opts = struct(...
'height', 90, ... % Height of the platform [mm]
'density', 8000, ... % Density of the material used for the hexapod [kg/m3]
'k_ax', 1e8, ... % Stiffness of each actuator [N/m]
'c_ax', 100, ... % Damping of each actuator [N/(m/s)]
'c_ax', 1000, ... % Damping of each actuator [N/(m/s)]
'stroke', 50e-6, ... % Maximum stroke of each actuator [m]
'name', 'stewart' ... % Name of the file
);
% Populate opts with input parameters
if exist('opts_param','var')
for opt = fieldnames(opts_param)'
opts.(opt{1}) = opts_param.(opt{1});
end
end
% Initialization of the stewart structure
% We initialize the Stewart structure
stewart = struct();
% And we defined its total height.
stewart.H = opts.height; % [mm]
% Bottom Plate
% #+name: fig:stewart_bottom_plate
% #+caption: Schematic of the bottom plates with all the parameters
% [[file:./figs/stewart_bottom_plate.png]]
% The bottom plate structure is initialized.
BP = struct();
% We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
BP.Rint = 0; % Internal Radius [mm]
BP.Rext = 150; % External Radius [mm]
% We define its thickness.
BP.H = 10; % Thickness of the Bottom Plate [mm]
% At which radius legs will be fixed and with that angle offset.
BP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
BP.alpha = 10; % Angle Offset [deg]
% We defined the density of the material of the bottom plate.
BP.density = opts.density; % Density of the material [kg/m3]
% And its color.
BP.color = [0.7 0.7 0.7]; % Color [RGB]
% Then the profile of the bottom plate is computed and will be used by Simscape
BP.shape = [BP.Rint BP.H; BP.Rint 0; BP.Rext 0; BP.Rext BP.H]; % [mm]
% The structure is added to the stewart structure
stewart.BP = BP;
% Top Plate
% The top plate structure is initialized.
TP = struct();
% We defined the internal and external radius of the top plate.
TP.Rint = 0; % [mm]
TP.Rext = 100; % [mm]
% The thickness of the top plate.
TP.H = 10; % [mm]
% At which radius and angle are fixed the legs.
TP.Rleg = 100; % Radius where the legs articulations are positionned [mm]
TP.alpha = 20; % Angle [deg]
TP.dalpha = 0; % Angle Offset from 0 position [deg]
% The density of its material.
TP.density = opts.density; % Density of the material [kg/m3]
% Its color.
TP.color = [0.7 0.7 0.7]; % Color [RGB]
% Then the shape of the top plate is computed
TP.shape = [TP.Rint TP.H; TP.Rint 0; TP.Rext 0; TP.Rext TP.H];
% The structure is added to the stewart structure
stewart.TP = TP;
% Legs
% #+name: fig:stewart_legs
% #+caption: Schematic for the legs of the Stewart platform
% [[file:./figs/stewart_legs.png]]
% The leg structure is initialized.
Leg = struct();
% The maximum Stroke of each leg is defined.
Leg.stroke = opts.stroke; % [m]
% The stiffness and damping of each leg are defined
Leg.k_ax = opts.k_ax; % Stiffness of each leg [N/m]
Leg.c_ax = opts.c_ax; % Damping of each leg [N/(m/s)]
% The radius of the legs are defined
Leg.Rtop = 10; % Radius of the cylinder of the top part of the leg[mm]
Leg.Rbot = 12; % Radius of the cylinder of the bottom part of the leg [mm]
% The density of its material.
Leg.density = opts.density; % Density of the material used for the legs [kg/m3]
% Its color.
Leg.color = [0.5 0.5 0.5]; % Color of the top part of the leg [RGB]
% The radius of spheres representing the ball joints are defined.
Leg.R = 1.3*Leg.Rbot; % Size of the sphere at the extremity of the leg [mm]
% The structure is added to the stewart structure
stewart.Leg = Leg;
% Ball Joints
% #+name: fig:stewart_ball_joints
% #+caption: Schematic of the support for the ball joints
% [[file:./figs/stewart_ball_joints.png]]
% =SP= is the structure representing the support for the ball joints at the extremity of each leg.
% The =SP= structure is initialized.
SP = struct();
% We can define its rotational stiffness and damping. For now, we use perfect joints.
SP.k = 0; % [N*m/deg]
SP.c = 0; % [N*m/deg]
% Its height is defined
SP.H = 15; % [mm]
% Its radius is based on the radius on the sphere at the end of the legs.
SP.R = Leg.R; % [mm]
SP.section = [0 SP.H-SP.R;
@ -88,18 +230,40 @@ SP.section = [0 SP.H-SP.R;
SP.R 0;
SP.R SP.H];
% The density of its material is defined.
SP.density = opts.density; % [kg/m^3]
% Its color is defined.
SP.color = [0.7 0.7 0.7]; % [RGB]
% The structure is added to the Hexapod structure
stewart.SP = SP;
% More parameters are initialized
stewart = initializeParameters(stewart);
% Save the Stewart Structure
save('./mat/stewart.mat', 'stewart')
% initializeParameters Function
function [stewart] = initializeParameters(stewart)
% We first compute $[a_1, a_2, a_3, a_4, a_5, a_6]^T$ and $[b_1, b_2, b_3, b_4, b_5, b_6]^T$.
stewart.Aa = zeros(6, 3); % [mm]
stewart.Ab = zeros(6, 3); % [mm]
stewart.Bb = zeros(6, 3); % [mm]
@ -119,12 +283,27 @@ for i = 1:3
stewart.TP.Rleg*sin( pi/180*(120*(i-1) + stewart.TP.dalpha + stewart.TP.alpha) ), ...
stewart.H - stewart.TP.H - stewart.SP.H];
end
stewart.Bb = stewart.Ab - stewart.H*[0,0,1];
% Now, we compute the leg vectors $\hat{s}_i$ and leg position $l_i$:
% \[ b_i - a_i = l_i \hat{s}_i \]
% We initialize $l_i$ and $\hat{s}_i$
leg_length = zeros(6, 1); % [mm]
leg_vectors = zeros(6, 3);
% We compute $b_i - a_i$, and then:
% \begin{align*}
% l_i &= \left|b_i - a_i\right| \\
% \hat{s}_i &= \frac{b_i - a_i}{l_i}
% \end{align*}
legs = stewart.Ab - stewart.Aa;
for i = 1:6
@ -132,6 +311,10 @@ for i = 1:6
leg_vectors(i,:) = legs(i,:) / leg_length(i);
end
% Then the shape of the bottom leg is estimated
stewart.Leg.lenght = leg_length(1)/1.5;
stewart.Leg.shape.bot = ...
[0 0; ...
@ -141,6 +324,11 @@ stewart.Leg.shape.bot = ...
stewart.Leg.Rtop 0.2*stewart.Leg.lenght; ...
0 0.2*stewart.Leg.lenght];
% We compute rotation matrices to have the orientation of the legs.
% The rotation matrix transforms the $z$ axis to the axis of the leg. The other axis are not important here.
stewart.Rm = struct('R', eye(3));
for i = 1:6
@ -156,14 +344,19 @@ for i = 1:6
stewart.Rm(i).R = [sx', sy', sz'];
end
% Compute Jacobian Matrix
J = zeros(6);
for i = 1:6
J(i, 1:3) = leg_vectors(i, :);
J(i, 4:6) = cross(0.001*stewart.Bb(i, :), leg_vectors(i, :));
J(i, 4:6) = cross(0.001*(stewart.Ab(i, :)- stewart.H*[0,0,1]), leg_vectors(i, :));
end
stewart.J = J;
stewart.Jinv = inv(J);
stewart.K = stewart.Leg.k_ax*stewart.J'*stewart.J;

6
src/initializeSample.m
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@ -2,9 +2,9 @@ function [] = initializeSample(opts_param)
%% Default values for opts
sample = struct( ...
'radius', 100, ... % radius of the cylinder [mm]
'height', 300, ... % height of the cylinder [mm]
'mass', 50, ... % mass of the cylinder [kg]
'measheight', 150, ... % measurement point z-offset [mm]
'height', 100, ... % height of the cylinder [mm]
'mass', 10, ... % mass of the cylinder [kg]
'measheight', 50, ... % measurement point z-offset [mm]
'offset', [0, 0, 0], ... % offset position of the sample [mm]
'color', [0.9 0.1 0.1] ...
);

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stewart.slx
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@ -0,0 +1,284 @@
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<h1 class="title">Stiffness of the Stewart Platform</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org178badc">1. Functions</a>
<ul>
<li><a href="#org31327cd">1.1. getStiffnessMatrix</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org178badc" class="outline-2">
<h2 id="org178badc"><span class="section-number-2">1</span> Functions</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org31327cd" class="outline-3">
<h3 id="org31327cd"><span class="section-number-3">1.1</span> getStiffnessMatrix</h3>
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<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">K</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getStiffnessMatrix</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">k</span>, <span style="color: #DFAF8F;">J</span><span style="color: #DCDCCC;">)</span>
<span style="color: #7F9F7F;">% k - leg stiffness</span>
<span style="color: #7F9F7F;">% J - Jacobian matrix</span>
K = k<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>J'<span style="color: #7CB8BB;">*</span>J<span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
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<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-22 ven. 12:03</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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#+TITLE: Stiffness of the Stewart Platform
* Functions
:PROPERTIES:
:HEADER-ARGS:matlab+: :exports code
:HEADER-ARGS:matlab+: :comments no
:HEADER-ARGS:matlab+: :mkdir yes
:HEADER-ARGS:matlab+: :eval no
:END:
** getStiffnessMatrix
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle src/getStiffnessMatrix.m
:END:
#+begin_src matlab
function [K] = getStiffnessMatrix(k, J)
% k - leg stiffness
% J - Jacobian matrix
K = k*(J'*J);
end
#+end_src