Update simulation of kinematics
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2019-12-10 mar. 18:03 -->
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<!-- 2019-12-11 mer. 14:47 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Kinematics of the station</title>
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@ -283,16 +283,16 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgc0809a7">1. Micro Hexapod</a>
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<li><a href="#orgf49d055">1. Micro Hexapod</a>
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<ul>
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<li><a href="#org5ed8144">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li>
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<li><a href="#orge1f9456">1.2. Control of the Micro-Hexapod using Simscape</a>
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<li><a href="#orgb024fa1">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li>
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<li><a href="#org34abe0f">1.2. Control of the Micro-Hexapod using Simscape</a>
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<ul>
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<li><a href="#orgaec6c7a">1.2.1. Using Bushing Joint</a></li>
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<li><a href="#org964676d">1.2.2. Using Inverse Kinematics and Leg Actuators</a>
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<li><a href="#org118cdf5">1.2.1. Using Bushing Joint</a></li>
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<li><a href="#org37b4bdd">1.2.2. Using Inverse Kinematics and Leg Actuators</a>
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<ul>
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<li><a href="#org3fcf5f8">1.2.2.1. Theory</a></li>
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<li><a href="#orgc39da76">1.2.2.2. Matlab Implementation</a></li>
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<li><a href="#org9839e83">1.2.2.1. Theory</a></li>
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<li><a href="#org78fd3cf">1.2.2.2. Matlab Implementation</a></li>
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</ul>
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</li>
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</ul>
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@ -307,12 +307,12 @@ for the JavaScript code in this tag.
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In this document, we discuss the way the motion of each stage is defined.
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</p>
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<div id="outline-container-orgc0809a7" class="outline-2">
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<h2 id="orgc0809a7"><span class="section-number-2">1</span> Micro Hexapod</h2>
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<div id="outline-container-orgf49d055" class="outline-2">
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<h2 id="orgf49d055"><span class="section-number-2">1</span> Micro Hexapod</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org5ed8144" class="outline-3">
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<h3 id="org5ed8144"><span class="section-number-3">1.1</span> How the Symetrie Hexapod is controlled on the micro station</h3>
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<div id="outline-container-orgb024fa1" class="outline-3">
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<h3 id="orgb024fa1"><span class="section-number-3">1.1</span> How the Symetrie Hexapod is controlled on the micro station</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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For the Micro-Hexapod, the convention for the angles are defined in <code>MAN_A_Software API_4.0.150918_EN.pdf</code> on page 13 (section 2.4 - Rotation Vectors):
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@ -360,8 +360,8 @@ Thus, it does the translations and then the rotation around the new translated f
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</div>
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</div>
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<div id="outline-container-orge1f9456" class="outline-3">
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<h3 id="orge1f9456"><span class="section-number-3">1.2</span> Control of the Micro-Hexapod using Simscape</h3>
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<div id="outline-container-org34abe0f" class="outline-3">
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<h3 id="org34abe0f"><span class="section-number-3">1.2</span> Control of the Micro-Hexapod using Simscape</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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We can think of two main ways to position the Micro-Hexapod using Simscape.
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@ -378,15 +378,15 @@ This require a little bit more of mathematical derivations but this is the chose
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</p>
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</div>
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<div id="outline-container-orgaec6c7a" class="outline-4">
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<h4 id="orgaec6c7a"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4>
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<div id="outline-container-org118cdf5" class="outline-4">
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<h4 id="org118cdf5"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4>
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<div class="outline-text-4" id="text-1-2-1">
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<p>
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In the documentation of the Bushing Joint (<code>doc "Bushing Joint"</code>) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure <a href="#orgaa7f4a2">1</a>.
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In the documentation of the Bushing Joint (<code>doc "Bushing Joint"</code>) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure <a href="#orgbf74afe">1</a>.
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</p>
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<div id="orgaa7f4a2" class="figure">
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<div id="orgbf74afe" class="figure">
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<p><img src="figs/bushing_joint_transform.png" alt="bushing_joint_transform.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Joint Transformation Sequence for the Bushing Joint</p>
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@ -404,8 +404,8 @@ However, the Bushing Joint makes rotations around mobiles axes (X, Y' and then Z
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</div>
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</div>
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<div id="outline-container-org964676d" class="outline-4">
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<h4 id="org964676d"><span class="section-number-4">1.2.2</span> Using Inverse Kinematics and Leg Actuators</h4>
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<div id="outline-container-org37b4bdd" class="outline-4">
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<h4 id="org37b4bdd"><span class="section-number-4">1.2.2</span> Using Inverse Kinematics and Leg Actuators</h4>
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<div class="outline-text-4" id="text-1-2-2">
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<p>
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Here, we can use the Inverse Kinematic of the Hexapod to determine the length of each leg in order to obtain some defined translation and rotation of the mobile platform.
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@ -418,10 +418,21 @@ The advantages are:
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<li>we can position the Hexapod as we want by specifying a rotation matrix</li>
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<li>the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position</li>
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</ul>
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<p>
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However:
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</p>
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<ul class="org-ul">
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<li>even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity</li>
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</ul>
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<p>
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Thus, for this simulation, we <b>remove the gravity</b>.
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</p>
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</div>
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<div id="outline-container-org3fcf5f8" class="outline-5">
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<h5 id="org3fcf5f8"><span class="section-number-5">1.2.2.1</span> Theory</h5>
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<div id="outline-container-org9839e83" class="outline-5">
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<h5 id="org9839e83"><span class="section-number-5">1.2.2.1</span> Theory</h5>
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<div class="outline-text-5" id="text-1-2-2-1">
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<p>
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For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
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@ -456,20 +467,29 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
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</div>
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</div>
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<div id="outline-container-orgc39da76" class="outline-5">
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<h5 id="orgc39da76"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5>
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<div id="outline-container-org78fd3cf" class="outline-5">
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<h5 id="org78fd3cf"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5>
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<div class="outline-text-5" id="text-1-2-2-2">
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<p>
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We open the Simulink file.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">open <span class="org-string">'simscape/hexapod_tests.slx'</span>
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</pre>
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</div>
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<p>
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We load the configuration and set a small <code>StopTime</code>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'simscape/conf_simscape.mat'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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<span class="org-matlab-simulink-keyword">set_param</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, '<span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">5</span><span class="org-type">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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</div>
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<p>
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We define the wanted position/orientation of the Hexapod under study.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">tx = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [rad]</span>
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ty = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% [rad]</span>
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@ -494,6 +514,9 @@ hexapod = initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">(</
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</pre>
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</div>
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<p>
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We run the simulation.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'simscape/hexapod_tests.slx'</span><span class="org-rainbow-delimiters-depth-1">)</span>
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</pre>
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@ -515,15 +538,15 @@ And we verify that we indeed succeed to go to the wanted position.
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</colgroup>
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<tbody>
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<tr>
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<td class="org-right">-2.12e-06</td>
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<td class="org-right">1.611e-10</td>
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</tr>
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<tr>
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<td class="org-right">2.9787e-06</td>
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<td class="org-right">-1.3748e-10</td>
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</tr>
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<tr>
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<td class="org-right">-4.4341e-06</td>
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<td class="org-right">8.4879e-11</td>
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</tr>
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</tbody>
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</table>
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@ -545,21 +568,21 @@ And we verify that we indeed succeed to go to the wanted position.
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</colgroup>
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<tbody>
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<tr>
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<td class="org-right">-1.5714e-06</td>
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<td class="org-right">1.4513e-06</td>
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<td class="org-right">7.8133e-06</td>
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<td class="org-right">-1.2659e-10</td>
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<td class="org-right">6.5603e-11</td>
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<td class="org-right">6.2183e-10</td>
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</tr>
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<tr>
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<td class="org-right">8.4113e-07</td>
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<td class="org-right">-7.1485e-07</td>
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<td class="org-right">-7.4572e-06</td>
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<td class="org-right">1.0354e-10</td>
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<td class="org-right">-5.2439e-11</td>
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<td class="org-right">-5.2425e-10</td>
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</tr>
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<tr>
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<td class="org-right">-7.5348e-06</td>
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<td class="org-right">7.7112e-06</td>
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<td class="org-right">-2.3088e-06</td>
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<td class="org-right">-5.9816e-10</td>
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<td class="org-right">5.532e-10</td>
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<td class="org-right">-1.7737e-10</td>
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</tr>
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</tbody>
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</table>
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@ -571,7 +594,7 @@ And we verify that we indeed succeed to go to the wanted position.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2019-12-10 mar. 18:03</p>
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<p class="date">Created: 2019-12-11 mer. 14:47</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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@ -104,6 +104,11 @@ The advantages are:
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- we can position the Hexapod as we want by specifying a rotation matrix
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- the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position
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However:
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- even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity
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Thus, for this simulation, we *remove the gravity*.
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**** Theory
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For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.
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@ -140,15 +145,18 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
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#+end_src
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**** Matlab Implementation
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We open the Simulink file.
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#+begin_src matlab
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open 'simscape/hexapod_tests.slx'
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#+end_src
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We load the configuration and set a small =StopTime=.
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#+begin_src matlab
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load('simscape/conf_simscape.mat');
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set_param(conf_simscape, 'StopTime', '0.5');
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#+end_src
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We define the wanted position/orientation of the Hexapod under study.
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#+begin_src matlab
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tx = 0.1; % [rad]
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ty = 0.2; % [rad]
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@ -172,6 +180,7 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
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hexapod = initializeMicroHexapod(struct('AP', AP, 'ARB', ARB));
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#+end_src
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We run the simulation.
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#+begin_src matlab
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sim('simscape/hexapod_tests.slx')
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#+end_src
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@ -182,15 +191,15 @@ And we verify that we indeed succeed to go to the wanted position.
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#+end_src
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#+RESULTS:
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| -2.12e-06 |
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| 2.9787e-06 |
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| -4.4341e-06 |
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| 1.611e-10 |
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| -1.3748e-10 |
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| 8.4879e-11 |
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#+begin_src matlab :results table replace
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simout.R.Data(:, :, end)-ARB
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#+end_src
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#+RESULTS:
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| -1.5714e-06 | 1.4513e-06 | 7.8133e-06 |
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| 8.4113e-07 | -7.1485e-07 | -7.4572e-06 |
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| -7.5348e-06 | 7.7112e-06 | -2.3088e-06 |
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| -1.2659e-10 | 6.5603e-11 | 6.2183e-10 |
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| 1.0354e-10 | -5.2439e-11 | -5.2425e-10 |
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| -5.9816e-10 | 5.532e-10 | -1.7737e-10 |
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