Add few functions for kinematic study
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Stewart Platforms</title>
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<title>Stewart Platforms</title>
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@ -259,8 +259,8 @@ The goal of this project is to provide a Matlab/Simscape Toolbox to study Stewar
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The project is divided into several section listed below.
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The project is divided into several section listed below.
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<div id="outline-container-org2246378" class="outline-2">
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<div id="outline-container-orge0ffde8" class="outline-2">
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<h2 id="org2246378"><span class="section-number-2">1</span> Simulink Project (<a href="simulink-project.html">link</a>)</h2>
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<h2 id="orge0ffde8"><span class="section-number-2">1</span> Simulink Project (<a href="simulink-project.html">link</a>)</h2>
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<p>
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The project is managed with a <b>Simulink Project</b>.
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The project is managed with a <b>Simulink Project</b>.
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@ -269,8 +269,8 @@ Such project is briefly presented <a href="simulink-project.html">here</a>.
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<div id="outline-container-org213a021" class="outline-2">
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<div id="outline-container-org840f439" class="outline-2">
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<h2 id="org213a021"><span class="section-number-2">2</span> Stewart Platform Architecture Definition (<a href="stewart-architecture.html">link</a>)</h2>
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<h2 id="org840f439"><span class="section-number-2">2</span> Stewart Platform Architecture Definition (<a href="stewart-architecture.html">link</a>)</h2>
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<div class="outline-text-2" id="text-2">
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The way the Stewart Platform is defined <a href="stewart-architecture.html">here</a>.
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The way the Stewart Platform is defined <a href="stewart-architecture.html">here</a>.
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@ -281,8 +281,8 @@ All the geometrical parameters are defined including:
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</p>
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</p>
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<ul class="org-ul">
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<ul class="org-ul">
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<li>Definition of the location of the frames</li>
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<li>Definition of the location of the frames</li>
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<li>Size of the platforms and the limbs</li>
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<li>Location/orientation of the limbs</li>
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<li>Location/orientation of the limbs</li>
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<li>Size/inertia of the platforms and the limbs</li>
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</ul>
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</ul>
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<p>
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@ -290,14 +290,13 @@ Other parameters are also defined such as:
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</p>
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<ul class="org-ul">
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<ul class="org-ul">
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<li>Stiffness and damping of the struts</li>
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<li>Stiffness and damping of the struts</li>
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<li>Inertia of the different elements</li>
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<li>Rest position of the Stewart platform</li>
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<li>Rest position of the Stewart platform</li>
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</ul>
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</ul>
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<div id="outline-container-org43ed1d4" class="outline-2">
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<div id="outline-container-org91daed3" class="outline-2">
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<h2 id="org43ed1d4"><span class="section-number-2">3</span> Simscape Model of the Stewart Platform (<a href="simscape-model.html">link</a>)</h2>
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<h2 id="org91daed3"><span class="section-number-2">3</span> Simscape Model of the Stewart Platform (<a href="simscape-model.html">link</a>)</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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The Stewart Platform is then modeled using <a href="https://www.mathworks.com/products/simscape.html">Simscape</a>.
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The Stewart Platform is then modeled using <a href="https://www.mathworks.com/products/simscape.html">Simscape</a>.
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@ -309,8 +308,8 @@ The way to model is build and works is explained <a href="simscape-model.html">h
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<div id="outline-container-org5851cea" class="outline-2">
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<div id="outline-container-orgf826a60" class="outline-2">
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<h2 id="org5851cea"><span class="section-number-2">4</span> Kinematic Analysis (<a href="kinematic-study.html">link</a>)</h2>
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<h2 id="orgf826a60"><span class="section-number-2">4</span> Kinematic Analysis (<a href="kinematic-study.html">link</a>)</h2>
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<div class="outline-text-2" id="text-4">
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<div class="outline-text-2" id="text-4">
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<p>
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From the defined geometry of the Stewart platform, we can perform static analysis such as:
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From the defined geometry of the Stewart platform, we can perform static analysis such as:
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@ -330,8 +329,8 @@ All these analysis are described <a href="kinematic-study.html">here</a>.
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<div id="outline-container-orgcc5dd70" class="outline-2">
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<div id="outline-container-org34ceac7" class="outline-2">
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<h2 id="orgcc5dd70"><span class="section-number-2">5</span> Identification of the Stewart Dynamics (<a href="identification.html">link</a>)</h2>
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<h2 id="org34ceac7"><span class="section-number-2">5</span> Identification of the Stewart Dynamics (<a href="identification.html">link</a>)</h2>
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<div class="outline-text-2" id="text-5">
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<div class="outline-text-2" id="text-5">
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<p>
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The Dynamics of the Stewart platform can be identified using the Simscape model.
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The Dynamics of the Stewart platform can be identified using the Simscape model.
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@ -352,8 +351,8 @@ The code that is used for identification is explained <a href="identification.ht
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<div id="outline-container-org2c5545b" class="outline-2">
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<h2 id="org58fb0b9"><span class="section-number-2">6</span> Active Damping (<a href="active-damping.html">link</a>)</h2>
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<h2 id="org2c5545b"><span class="section-number-2">6</span> Active Damping (<a href="active-damping.html">link</a>)</h2>
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<div class="outline-text-2" id="text-6">
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The use of different sensors are compared for active damping:
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The use of different sensors are compared for active damping:
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@ -371,8 +370,8 @@ The result of the analysis is accessible <a href="active-damping.html">here</a>.
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<div id="outline-container-org8ab41c3" class="outline-2">
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<h2 id="orgb149881"><span class="section-number-2">7</span> Motion Control of the Stewart Platform (<a href="control-study.html">link</a>)</h2>
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<h2 id="org8ab41c3"><span class="section-number-2">7</span> Motion Control of the Stewart Platform (<a href="control-study.html">link</a>)</h2>
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<div class="outline-text-2" id="text-7">
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<div class="outline-text-2" id="text-7">
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<p>
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Some control architecture for motion control of the Stewart platform are applied on the Simscape model and compared in <a href="control-study.html">this</a> document.
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Some control architecture for motion control of the Stewart platform are applied on the Simscape model and compared in <a href="control-study.html">this</a> document.
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@ -380,8 +379,8 @@ Some control architecture for motion control of the Stewart platform are applied
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</div>
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<div id="outline-container-org6bf9b4a" class="outline-2">
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<div id="outline-container-org46a0310" class="outline-2">
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<h2 id="org6bf9b4a"><span class="section-number-2">8</span> Cubic Configuration (<a href="cubic-configuration.html">link</a>)</h2>
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<h2 id="org46a0310"><span class="section-number-2">8</span> Cubic Configuration (<a href="cubic-configuration.html">link</a>)</h2>
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<div class="outline-text-2" id="text-8">
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<div class="outline-text-2" id="text-8">
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<p>
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<p>
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The cubic configuration is a special class of Stewart platform that has interesting properties.
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The cubic configuration is a special class of Stewart platform that has interesting properties.
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All the geometrical parameters are defined including:
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All the geometrical parameters are defined including:
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- Definition of the location of the frames
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- Definition of the location of the frames
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- Size of the platforms and the limbs
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- Location/orientation of the limbs
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- Location/orientation of the limbs
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- Size/inertia of the platforms and the limbs
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Other parameters are also defined such as:
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Other parameters are also defined such as:
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- Stiffness and damping of the struts
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- Stiffness and damping of the struts
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- Inertia of the different elements
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- Rest position of the Stewart platform
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- Rest position of the Stewart platform
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* Simscape Model of the Stewart Platform ([[file:simscape-model.org][link]])
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* Simscape Model of the Stewart Platform ([[file:simscape-model.org][link]])
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<?xml version="1.0" encoding="utf-8"?>
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Kinematic Study of the Stewart Platform</title>
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<title>Kinematic Study of the Stewart Platform</title>
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<meta name="generator" content="Org mode" />
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<meta name="author" content="Thomas Dehaeze" />
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@licstart The following is the entire license notice for the
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@licstart The following is the entire license notice for the
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JavaScript code in this tag.
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Copyright (C) 2012-2019 Free Software Foundation, Inc.
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Copyright (C) 2012-2020 Free Software Foundation, Inc.
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The JavaScript code in this tag is free software: you can
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The JavaScript code in this tag is free software: you can
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redistribute it and/or modify it under the terms of the GNU
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redistribute it and/or modify it under the terms of the GNU
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<h2>Table of Contents</h2>
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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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<ul>
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<li><a href="#org4a99119">1. Needed Actuator Stroke</a>
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<li><a href="#org63c8faa">1. Needed Actuator Stroke</a>
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<ul>
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<ul>
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<li><a href="#org119868f">1.1. Stewart architecture definition</a></li>
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<li><a href="#orged5be9e">1.1. Stewart architecture definition</a></li>
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<li><a href="#org8ece9f1">1.2. Wanted translations and rotations</a></li>
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<li><a href="#org73e5cf8">1.2. Wanted translations and rotations</a></li>
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<li><a href="#org8cd9dfc">1.3. Needed stroke for "pure" rotations or translations</a></li>
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<li><a href="#org9825ccf">1.3. Needed stroke for “pure” rotations or translations</a></li>
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<li><a href="#org9acbff2">1.4. Needed stroke for combined translations and rotations</a></li>
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<li><a href="#org0440602">1.4. Needed stroke for combined translations and rotations</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org0aef174">2. Maximum Stroke</a></li>
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<li><a href="#org092f7f8">2. Maximum Stroke</a></li>
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<li><a href="#orgff4f3e2">3. Functions</a>
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<li><a href="#org720ba56">3. Functions</a>
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<ul>
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<ul>
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<li><a href="#orgca8f528">3.1. getMaxPositions</a></li>
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<li><a href="#org8125766">3.1. getMaxPositions</a></li>
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<li><a href="#org0ac04ca">3.2. getMaxPureDisplacement</a></li>
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<li><a href="#org91e4101">3.2. getMaxPureDisplacement</a></li>
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<li><a href="#orgf75fefe">3.3. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
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<ul>
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<li><a href="#orgae47616">3.3.1. Function description</a></li>
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<li><a href="#org78705da">3.3.2. Compute Jacobian Matrix</a></li>
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<li><a href="#orgb7dc1d7">3.3.3. Compute Stiffness Matrix</a></li>
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<li><a href="#org7aa6c04">3.3.4. Compute Compliance Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org9c46957">3.4. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
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<ul>
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<li><a href="#org9da7af0">3.4.1. Function description</a></li>
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<li><a href="#orge2cc540">3.4.2. Optional Parameters</a></li>
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<li><a href="#orga1a0cc7">3.4.3. Theory</a></li>
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<li><a href="#org9b86eb9">3.4.4. Compute</a></li>
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</ul>
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</li>
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<li><a href="#org7e6d65c">3.5. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
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<ul>
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<li><a href="#org65e0ce7">3.5.1. Function description</a></li>
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<li><a href="#orgf6a32e1">3.5.2. Optional Parameters</a></li>
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<li><a href="#orgce0b559">3.5.3. Computation</a></li>
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<h2 id="org4a99119"><span class="section-number-2">1</span> Needed Actuator Stroke</h2>
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<h2 id="org63c8faa"><span class="section-number-2">1</span> Needed Actuator Stroke</h2>
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The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
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The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
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<h3 id="org119868f"><span class="section-number-3">1.1</span> Stewart architecture definition</h3>
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<h3 id="orged5be9e"><span class="section-number-3">1.1</span> Stewart architecture definition</h3>
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<p>
|
<p>
|
||||||
We use a cubic architecture.
|
We use a cubic architecture.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">opts = struct<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-underline">...</span>
|
<pre class="src src-matlab">opts = struct(...
|
||||||
<span class="org-string">'H_tot'</span>, <span class="org-highlight-numbers-number">90</span>, <span class="org-underline">...</span> <span class="org-comment">% Total height of the Hexapod [mm]</span>
|
<span class="org-string">'H_tot'</span>, 90, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
||||||
<span class="org-string">'L'</span>, <span class="org-highlight-numbers-number">200</span><span class="org-type">/</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-underline">...</span> <span class="org-comment">% Size of the Cube [mm]</span>
|
<span class="org-string">'L'</span>, 200<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
||||||
<span class="org-string">'H'</span>, <span class="org-highlight-numbers-number">60</span>, <span class="org-underline">...</span> <span class="org-comment">% Height between base joints and platform joints [mm]</span>
|
<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
||||||
<span class="org-string">'H0'</span>, <span class="org-highlight-numbers-number">200</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">-</span><span class="org-highlight-numbers-number">60</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span> <span class="org-underline">...</span> <span class="org-comment">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
<span class="org-string">'H0'</span>, 200<span class="org-type">/</span>2<span class="org-type">-</span>60<span class="org-type">/</span>2 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
|
||||||
<span class="org-rainbow-delimiters-depth-1">)</span>;
|
);
|
||||||
stewart = initializeCubicConfiguration<span class="org-rainbow-delimiters-depth-1">(</span>opts<span class="org-rainbow-delimiters-depth-1">)</span>;
|
stewart = initializeCubicConfiguration(opts);
|
||||||
opts = struct<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-underline">...</span>
|
opts = struct(...
|
||||||
<span class="org-string">'Jd_pos'</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-underline">...</span> <span class="org-comment">% Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
<span class="org-string">'Jd_pos'</span>, [0, 0, 100], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
|
||||||
<span class="org-string">'Jf_pos'</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-type">-</span><span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">]</span> <span class="org-underline">...</span> <span class="org-comment">% Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
<span class="org-string">'Jf_pos'</span>, [0, 0, <span class="org-type">-</span>50] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
|
||||||
<span class="org-rainbow-delimiters-depth-1">)</span>;
|
);
|
||||||
stewart = computeGeometricalProperties<span class="org-rainbow-delimiters-depth-1">(</span>stewart, opts<span class="org-rainbow-delimiters-depth-1">)</span>;
|
stewart = computeGeometricalProperties(stewart, opts);
|
||||||
opts = struct<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-underline">...</span>
|
opts = struct(...
|
||||||
<span class="org-string">'stroke'</span>, <span class="org-highlight-numbers-number">50e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">6</span> <span class="org-underline">...</span> <span class="org-comment">% Maximum stroke of each actuator [m]</span>
|
<span class="org-string">'stroke'</span>, 50e<span class="org-type">-</span>6 ...<span class="org-comment"> % Maximum stroke of each actuator [m]</span>
|
||||||
<span class="org-rainbow-delimiters-depth-1">)</span>;
|
);
|
||||||
stewart = initializeMechanicalElements<span class="org-rainbow-delimiters-depth-1">(</span>stewart, opts<span class="org-rainbow-delimiters-depth-1">)</span>;
|
stewart = initializeMechanicalElements(stewart, opts);
|
||||||
|
|
||||||
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/stewart.mat', 'stewart'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-string">'stewart'</span>);
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org8ece9f1" class="outline-3">
|
<div id="outline-container-org73e5cf8" class="outline-3">
|
||||||
<h3 id="org8ece9f1"><span class="section-number-3">1.2</span> Wanted translations and rotations</h3>
|
<h3 id="org73e5cf8"><span class="section-number-3">1.2</span> Wanted translations and rotations</h3>
|
||||||
<div class="outline-text-3" id="text-1-2">
|
<div class="outline-text-3" id="text-1-2">
|
||||||
<p>
|
<p>
|
||||||
We define wanted translations and rotations
|
We define wanted translations and rotations
|
||||||
</p>
|
</p>
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">Tx_max = <span class="org-highlight-numbers-number">15e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">6</span>; <span class="org-comment">% Translation [m]</span>
|
<pre class="src src-matlab">Tx_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
|
||||||
Ty_max = <span class="org-highlight-numbers-number">15e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">6</span>; <span class="org-comment">% Translation [m]</span>
|
Ty_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
|
||||||
Tz_max = <span class="org-highlight-numbers-number">15e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">6</span>; <span class="org-comment">% Translation [m]</span>
|
Tz_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
|
||||||
Rx_max = <span class="org-highlight-numbers-number">30e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">6</span>; <span class="org-comment">% Rotation [rad]</span>
|
Rx_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
|
||||||
Ry_max = <span class="org-highlight-numbers-number">30e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">6</span>; <span class="org-comment">% Rotation [rad]</span>
|
Ry_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org8cd9dfc" class="outline-3">
|
<div id="outline-container-org9825ccf" class="outline-3">
|
||||||
<h3 id="org8cd9dfc"><span class="section-number-3">1.3</span> Needed stroke for "pure" rotations or translations</h3>
|
<h3 id="org9825ccf"><span class="section-number-3">1.3</span> Needed stroke for “pure” rotations or translations</h3>
|
||||||
<div class="outline-text-3" id="text-1-3">
|
<div class="outline-text-3" id="text-1-3">
|
||||||
<p>
|
<p>
|
||||||
First, we estimate the needed actuator stroke for "pure" rotations and translation.
|
First, we estimate the needed actuator stroke for “pure” rotations and translation.
|
||||||
</p>
|
</p>
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">LTx = stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span>Tx_max <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>';
|
<pre class="src src-matlab">LTx = stewart.Jd<span class="org-type">*</span>[Tx_max 0 0 0 0 0]<span class="org-type">'</span>;
|
||||||
LTy = stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> Ty_max <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>';
|
LTy = stewart.Jd<span class="org-type">*</span>[0 Ty_max 0 0 0 0]<span class="org-type">'</span>;
|
||||||
LTz = stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> Tz_max <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>';
|
LTz = stewart.Jd<span class="org-type">*</span>[0 0 Tz_max 0 0 0]<span class="org-type">'</span>;
|
||||||
LRx = stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> Rx_max <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>';
|
LRx = stewart.Jd<span class="org-type">*</span>[0 0 0 Rx_max 0 0]<span class="org-type">'</span>;
|
||||||
LRy = stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> Ry_max <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>';
|
LRy = stewart.Jd<span class="org-type">*</span>[0 0 0 0 Ry_max 0]<span class="org-type">'</span>;
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
@ -353,26 +402,26 @@ From -1.2e-05[m] to 1.1e-05[m]: Total stroke = 22.9[um]
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org9acbff2" class="outline-3">
|
<div id="outline-container-org0440602" class="outline-3">
|
||||||
<h3 id="org9acbff2"><span class="section-number-3">1.4</span> Needed stroke for combined translations and rotations</h3>
|
<h3 id="org0440602"><span class="section-number-3">1.4</span> Needed stroke for combined translations and rotations</h3>
|
||||||
<div class="outline-text-3" id="text-1-4">
|
<div class="outline-text-3" id="text-1-4">
|
||||||
<p>
|
<p>
|
||||||
Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
|
Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
|
||||||
</p>
|
</p>
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">Lmax = <span class="org-highlight-numbers-number">0</span>;
|
<pre class="src src-matlab">Lmax = 0;
|
||||||
Lmin = <span class="org-highlight-numbers-number">0</span>;
|
Lmin = 0;
|
||||||
pos = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
pos = [0, 0, 0, 0, 0];
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">Tx</span> = <span class="org-constant"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-constant">-Tx_max</span>,Tx_max<span class="org-rainbow-delimiters-depth-1">]</span>
|
<span class="org-keyword">for</span> <span class="org-variable-name">Tx</span> = <span class="org-constant">[-Tx_max</span>,Tx_max]
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">Ty</span> = <span class="org-constant"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-constant">-Ty_max</span>,Ty_max<span class="org-rainbow-delimiters-depth-1">]</span>
|
<span class="org-keyword">for</span> <span class="org-variable-name">Ty</span> = <span class="org-constant">[-Ty_max</span>,Ty_max]
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">Tz</span> = <span class="org-constant"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-constant">-Tz_max</span>,Tz_max<span class="org-rainbow-delimiters-depth-1">]</span>
|
<span class="org-keyword">for</span> <span class="org-variable-name">Tz</span> = <span class="org-constant">[-Tz_max</span>,Tz_max]
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">Rx</span> = <span class="org-constant"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-constant">-Rx_max</span>,Rx_max<span class="org-rainbow-delimiters-depth-1">]</span>
|
<span class="org-keyword">for</span> <span class="org-variable-name">Rx</span> = <span class="org-constant">[-Rx_max</span>,Rx_max]
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">Ry</span> = <span class="org-constant"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-constant">-Ry_max</span>,Ry_max<span class="org-rainbow-delimiters-depth-1">]</span>
|
<span class="org-keyword">for</span> <span class="org-variable-name">Ry</span> = <span class="org-constant">[-Ry_max</span>,Ry_max]
|
||||||
lmax = max<span class="org-rainbow-delimiters-depth-1">(</span>stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">[</span>Tx Ty Tz Rx Ry <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-2">]</span>'<span class="org-rainbow-delimiters-depth-1">)</span>;
|
lmax = max(stewart.Jd<span class="org-type">*</span>[Tx Ty Tz Rx Ry 0]<span class="org-type">'</span>);
|
||||||
lmin = min<span class="org-rainbow-delimiters-depth-1">(</span>stewart.Jd<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">[</span>Tx Ty Tz Rx Ry <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-2">]</span>'<span class="org-rainbow-delimiters-depth-1">)</span>;
|
lmin = min(stewart.Jd<span class="org-type">*</span>[Tx Ty Tz Rx Ry 0]<span class="org-type">'</span>);
|
||||||
<span class="org-keyword">if</span> lmax <span class="org-type">></span> Lmax
|
<span class="org-keyword">if</span> lmax <span class="org-type">></span> Lmax
|
||||||
Lmax = lmax;
|
Lmax = lmax;
|
||||||
pos = <span class="org-rainbow-delimiters-depth-1">[</span>Tx Ty Tz Rx Ry<span class="org-rainbow-delimiters-depth-1">]</span>;
|
pos = [Tx Ty Tz Rx Ry];
|
||||||
<span class="org-keyword">end</span>
|
<span class="org-keyword">end</span>
|
||||||
<span class="org-keyword">if</span> lmin <span class="org-type"><</span> Lmin
|
<span class="org-keyword">if</span> lmin <span class="org-type"><</span> Lmin
|
||||||
Lmin = lmin;
|
Lmin = lmin;
|
||||||
@ -386,7 +435,7 @@ pos = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-high
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
We obtain a needed stroke shown below (almost two times the needed stroke for "pure" rotations and translations).
|
We obtain a needed stroke shown below (almost two times the needed stroke for “pure” rotations and translations).
|
||||||
</p>
|
</p>
|
||||||
<pre class="example">
|
<pre class="example">
|
||||||
From -3.1e-05[m] to 3.1e-05[m]: Total stroke = 61.5[um]
|
From -3.1e-05[m] to 3.1e-05[m]: Total stroke = 61.5[um]
|
||||||
@ -395,80 +444,341 @@ From -3.1e-05[m] to 3.1e-05[m]: Total stroke = 61.5[um]
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0aef174" class="outline-2">
|
<div id="outline-container-org092f7f8" class="outline-2">
|
||||||
<h2 id="org0aef174"><span class="section-number-2">2</span> Maximum Stroke</h2>
|
<h2 id="org092f7f8"><span class="section-number-2">2</span> Maximum Stroke</h2>
|
||||||
<div class="outline-text-2" id="text-2">
|
<div class="outline-text-2" id="text-2">
|
||||||
<p>
|
<p>
|
||||||
From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.
|
From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>X, Y, Z<span class="org-rainbow-delimiters-depth-1">]</span> = getMaxPositions<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">stewart</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
<pre class="src src-matlab">[X, Y, Z] = getMaxPositions(<span class="org-variable-name">stewart</span>);
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
||||||
plot3<span class="org-rainbow-delimiters-depth-1">(</span>X, Y, Z, <span class="org-string">'k-'</span><span class="org-rainbow-delimiters-depth-1">)</span>
|
plot3(X, Y, Z, <span class="org-string">'k-'</span>)
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgff4f3e2" class="outline-2">
|
<div id="outline-container-org720ba56" class="outline-2">
|
||||||
<h2 id="orgff4f3e2"><span class="section-number-2">3</span> Functions</h2>
|
<h2 id="org720ba56"><span class="section-number-2">3</span> Functions</h2>
|
||||||
<div class="outline-text-2" id="text-3">
|
<div class="outline-text-2" id="text-3">
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-orgca8f528" class="outline-3">
|
<div id="outline-container-org8125766" class="outline-3">
|
||||||
<h3 id="orgca8f528"><span class="section-number-3">3.1</span> getMaxPositions</h3>
|
<h3 id="org8125766"><span class="section-number-3">3.1</span> getMaxPositions</h3>
|
||||||
<div class="outline-text-3" id="text-3-1">
|
<div class="outline-text-3" id="text-3-1">
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-variable-name">X, Y, Z</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">getMaxPositions</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">stewart</span><span class="org-rainbow-delimiters-depth-1">)</span>
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[X, Y, Z]</span> = <span class="org-function-name">getMaxPositions</span>(<span class="org-variable-name">stewart</span>)
|
||||||
Leg = stewart.Leg;
|
Leg = stewart.Leg;
|
||||||
J = stewart.Jd;
|
J = stewart.Jd;
|
||||||
theta = linspace<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>, <span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
theta = linspace(0, 2<span class="org-type">*</span><span class="org-constant">pi</span>, 100);
|
||||||
phi = linspace<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">-</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span> , <span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
phi = linspace(<span class="org-type">-</span><span class="org-constant">pi</span><span class="org-type">/</span>2 , <span class="org-constant">pi</span><span class="org-type">/</span>2, 100);
|
||||||
dmax = zeros<span class="org-rainbow-delimiters-depth-1">(</span>length<span class="org-rainbow-delimiters-depth-2">(</span>theta<span class="org-rainbow-delimiters-depth-2">)</span>, length<span class="org-rainbow-delimiters-depth-2">(</span>phi<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
dmax = zeros(length(theta), length(phi));
|
||||||
|
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">theta</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(theta)</span>
|
||||||
<span class="org-keyword">for</span> <span class="org-variable-name">j</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">phi</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(phi)</span>
|
||||||
L = J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">[</span>cos<span class="org-rainbow-delimiters-depth-2">(</span>phi<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">j</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span>cos<span class="org-rainbow-delimiters-depth-2">(</span>theta<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> cos<span class="org-rainbow-delimiters-depth-2">(</span>phi<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">j</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>theta<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> sin<span class="org-rainbow-delimiters-depth-2">(</span>phi<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">j</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>';
|
L = J<span class="org-type">*</span>[cos(phi(<span class="org-constant">j</span>))<span class="org-type">*</span>cos(theta(<span class="org-constant">i</span>)) cos(phi(<span class="org-constant">j</span>))<span class="org-type">*</span>sin(theta(<span class="org-constant">i</span>)) sin(phi(<span class="org-constant">j</span>)) 0 0 0]<span class="org-type">'</span>;
|
||||||
dmax<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-constant">i</span>, <span class="org-constant">j</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>L<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
dmax(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = Leg.stroke<span class="org-type">/</span>max(abs(L));
|
||||||
<span class="org-keyword">end</span>
|
<span class="org-keyword">end</span>
|
||||||
<span class="org-keyword">end</span>
|
<span class="org-keyword">end</span>
|
||||||
|
|
||||||
X = dmax<span class="org-type">.*</span>cos<span class="org-rainbow-delimiters-depth-1">(</span>repmat<span class="org-rainbow-delimiters-depth-2">(</span>phi,length<span class="org-rainbow-delimiters-depth-3">(</span>theta<span class="org-rainbow-delimiters-depth-3">)</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.*</span>cos<span class="org-rainbow-delimiters-depth-1">(</span>repmat<span class="org-rainbow-delimiters-depth-2">(</span>theta,length<span class="org-rainbow-delimiters-depth-3">(</span>phi<span class="org-rainbow-delimiters-depth-3">)</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>';
|
X = dmax<span class="org-type">.*</span>cos(repmat(phi,length(theta),1))<span class="org-type">.*</span>cos(repmat(theta,length(phi),1))<span class="org-type">'</span>;
|
||||||
Y = dmax<span class="org-type">.*</span>cos<span class="org-rainbow-delimiters-depth-1">(</span>repmat<span class="org-rainbow-delimiters-depth-2">(</span>phi,length<span class="org-rainbow-delimiters-depth-3">(</span>theta<span class="org-rainbow-delimiters-depth-3">)</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.*</span>sin<span class="org-rainbow-delimiters-depth-1">(</span>repmat<span class="org-rainbow-delimiters-depth-2">(</span>theta,length<span class="org-rainbow-delimiters-depth-3">(</span>phi<span class="org-rainbow-delimiters-depth-3">)</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>';
|
Y = dmax<span class="org-type">.*</span>cos(repmat(phi,length(theta),1))<span class="org-type">.*</span>sin(repmat(theta,length(phi),1))<span class="org-type">'</span>;
|
||||||
Z = dmax<span class="org-type">.*</span>sin<span class="org-rainbow-delimiters-depth-1">(</span>repmat<span class="org-rainbow-delimiters-depth-2">(</span>phi,length<span class="org-rainbow-delimiters-depth-3">(</span>theta<span class="org-rainbow-delimiters-depth-3">)</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
Z = dmax<span class="org-type">.*</span>sin(repmat(phi,length(theta),1));
|
||||||
<span class="org-keyword">end</span>
|
<span class="org-keyword">end</span>
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0ac04ca" class="outline-3">
|
<div id="outline-container-org91e4101" class="outline-3">
|
||||||
<h3 id="org0ac04ca"><span class="section-number-3">3.2</span> getMaxPureDisplacement</h3>
|
<h3 id="org91e4101"><span class="section-number-3">3.2</span> getMaxPureDisplacement</h3>
|
||||||
<div class="outline-text-3" id="text-3-2">
|
<div class="outline-text-3" id="text-3-2">
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-variable-name">max_disp</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">getMaxPureDisplacement</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">Leg</span>, <span class="org-variable-name">J</span><span class="org-rainbow-delimiters-depth-1">)</span>
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[max_disp]</span> = <span class="org-function-name">getMaxPureDisplacement</span>(<span class="org-variable-name">Leg</span>, <span class="org-variable-name">J</span>)
|
||||||
max_disp = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp = zeros(6, 1);
|
||||||
max_disp<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">[</span><span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp<span class="org-type">(1) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[1 0 0 0 0 0]<span class="org-type">'</span>));
|
||||||
max_disp<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp<span class="org-type">(2) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 1 0 0 0 0]<span class="org-type">'</span>));
|
||||||
max_disp<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp<span class="org-type">(3) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 1 0 0 0]<span class="org-type">'</span>));
|
||||||
max_disp<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp<span class="org-type">(4) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 0 1 0 0]<span class="org-type">'</span>));
|
||||||
max_disp<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp<span class="org-type">(5) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 0 0 1 0]<span class="org-type">'</span>));
|
||||||
max_disp<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = Leg.stroke<span class="org-type">/</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>J<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
max_disp<span class="org-type">(6) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 0 0 0 1]<span class="org-type">'</span>));
|
||||||
<span class="org-keyword">end</span>
|
<span class="org-keyword">end</span>
|
||||||
</pre>
|
</pre>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
<div id="outline-container-orgf75fefe" class="outline-3">
|
||||||
|
<h3 id="orgf75fefe"><span class="section-number-3">3.3</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-3">
|
||||||
|
<p>
|
||||||
|
<a id="org02bdbb2"></a>
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgae47616" class="outline-4">
|
||||||
|
<h4 id="orgae47616"><span class="section-number-4">3.3.1</span> Function description</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-3-1">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">computeJacobian</span>(<span class="org-variable-name">stewart</span>)
|
||||||
|
<span class="org-comment">% computeJacobian -</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Syntax: [stewart] = computeJacobian(stewart)</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Inputs:</span>
|
||||||
|
<span class="org-comment">% - stewart - With at least the following fields:</span>
|
||||||
|
<span class="org-comment">% - As [3x6] - The 6 unit vectors for each strut expressed in {A}</span>
|
||||||
|
<span class="org-comment">% - Ab [3x6] - The 6 position of the joints bi expressed in {A}</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Outputs:</span>
|
||||||
|
<span class="org-comment">% - stewart - With the 3 added field:</span>
|
||||||
|
<span class="org-comment">% - J [6x6] - The Jacobian Matrix</span>
|
||||||
|
<span class="org-comment">% - K [6x6] - The Stiffness Matrix</span>
|
||||||
|
<span class="org-comment">% - C [6x6] - The Compliance Matrix</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org78705da" class="outline-4">
|
||||||
|
<h4 id="org78705da"><span class="section-number-4">3.3.2</span> Compute Jacobian Matrix</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-3-2">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">stewart.J = [stewart.As<span class="org-type">'</span> , cross(stewart.Ab, stewart.As)<span class="org-type">'</span>];
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgb7dc1d7" class="outline-4">
|
||||||
|
<h4 id="orgb7dc1d7"><span class="section-number-4">3.3.3</span> Compute Stiffness Matrix</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-3-3">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">stewart.K = stewart.J<span class="org-type">'*</span>diag(stewart.Ki)<span class="org-type">*</span>stewart.J;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org7aa6c04" class="outline-4">
|
||||||
|
<h4 id="org7aa6c04"><span class="section-number-4">3.3.4</span> Compute Compliance Matrix</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-3-4">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">stewart.C = inv(stewart.K);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org9c46957" class="outline-3">
|
||||||
|
<h3 id="org9c46957"><span class="section-number-3">3.4</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-4">
|
||||||
|
<p>
|
||||||
|
<a id="orgab617cc"></a>
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org9da7af0" class="outline-4">
|
||||||
|
<h4 id="org9da7af0"><span class="section-number-4">3.4.1</span> Function description</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-4-1">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[Li, dLi]</span> = <span class="org-function-name">inverseKinematics</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
||||||
|
<span class="org-comment">% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Syntax: [stewart] = inverseKinematics(stewart)</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Inputs:</span>
|
||||||
|
<span class="org-comment">% - stewart - A structure with the following fields</span>
|
||||||
|
<span class="org-comment">% - Aa [3x6] - The positions ai expressed in {A}</span>
|
||||||
|
<span class="org-comment">% - Bb [3x6] - The positions bi expressed in {B}</span>
|
||||||
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
||||||
|
<span class="org-comment">% - AP [3x1] - The wanted position of {B} with respect to {A}</span>
|
||||||
|
<span class="org-comment">% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Outputs:</span>
|
||||||
|
<span class="org-comment">% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}</span>
|
||||||
|
<span class="org-comment">% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orge2cc540" class="outline-4">
|
||||||
|
<h4 id="orge2cc540"><span class="section-number-4">3.4.2</span> Optional Parameters</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-4-2">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">arguments
|
||||||
|
stewart
|
||||||
|
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
||||||
|
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
||||||
|
<span class="org-keyword">end</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orga1a0cc7" class="outline-4">
|
||||||
|
<h4 id="orga1a0cc7"><span class="section-number-4">3.4.3</span> Theory</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-4-3">
|
||||||
|
<p>
|
||||||
|
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, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
From the geometry of the manipulator, the loop closure for each limb, \(i = 1, 2, \dots, 6\) can be written as
|
||||||
|
</p>
|
||||||
|
\begin{align*}
|
||||||
|
l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
|
||||||
|
&= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
|
||||||
|
\end{align*}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
To obtain the length of each actuator and eliminate \(\hat{\bm{s}}_i\), it is sufficient to dot multiply each side by itself:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Hence, for \(i = 1, 2, \dots, 6\), each limb length can be uniquely determined by:
|
||||||
|
</p>
|
||||||
|
\begin{equation}
|
||||||
|
l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
<p>
|
||||||
|
If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
|
||||||
|
Otherwise, when the limbs’ lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org9b86eb9" class="outline-4">
|
||||||
|
<h4 id="org9b86eb9"><span class="section-number-4">3.4.4</span> Compute</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-4-4">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">Li = sqrt(args.AP<span class="org-type">'*</span>args.AP <span class="org-type">+</span> diag(stewart.Bb<span class="org-type">'*</span>stewart.Bb) <span class="org-type">+</span> diag(stewart.Aa<span class="org-type">'*</span>stewart.Aa) <span class="org-type">-</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>stewart.Aa)<span class="org-type">'</span> <span class="org-type">+</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>(args.ARB<span class="org-type">*</span>stewart.Bb))<span class="org-type">'</span> <span class="org-type">-</span> diag(2<span class="org-type">*</span>(args.ARB<span class="org-type">*</span>stewart.Bb)<span class="org-type">'*</span>stewart.Aa));
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">dLi = Li<span class="org-type">-</span>stewart.l;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org7e6d65c" class="outline-3">
|
||||||
|
<h3 id="org7e6d65c"><span class="section-number-3">3.5</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3>
|
||||||
|
<div class="outline-text-3" id="text-3-5">
|
||||||
|
<p>
|
||||||
|
<a id="orgee3cdbf"></a>
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here</a>.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-org65e0ce7" class="outline-4">
|
||||||
|
<h4 id="org65e0ce7"><span class="section-number-4">3.5.1</span> Function description</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-5-1">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[P, R]</span> = <span class="org-function-name">forwardKinematicsApprox</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
||||||
|
<span class="org-comment">% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using</span>
|
||||||
|
<span class="org-comment">% the Jacobian Matrix</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Inputs:</span>
|
||||||
|
<span class="org-comment">% - stewart - A structure with the following fields</span>
|
||||||
|
<span class="org-comment">% - J [6x6] - The Jacobian Matrix</span>
|
||||||
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
||||||
|
<span class="org-comment">% - dL [6x1] - Displacement of each strut [m]</span>
|
||||||
|
<span class="org-comment">%</span>
|
||||||
|
<span class="org-comment">% Outputs:</span>
|
||||||
|
<span class="org-comment">% - P [3x1] - The estimated position of {B} with respect to {A}</span>
|
||||||
|
<span class="org-comment">% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgf6a32e1" class="outline-4">
|
||||||
|
<h4 id="orgf6a32e1"><span class="section-number-4">3.5.2</span> Optional Parameters</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-5-2">
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">arguments
|
||||||
|
stewart
|
||||||
|
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
|
||||||
|
<span class="org-keyword">end</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<div id="outline-container-orgce0b559" class="outline-4">
|
||||||
|
<h4 id="orgce0b559"><span class="section-number-4">3.5.3</span> Computation</h4>
|
||||||
|
<div class="outline-text-4" id="text-3-5-3">
|
||||||
|
<p>
|
||||||
|
From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the
|
||||||
|
position and orientation of {B} with respect to {A} using the following formula:
|
||||||
|
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">X = stewart.J<span class="org-type">\</span>args.dL;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The position vector corresponds to the first 3 elements.
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">P = X(1<span class="org-type">:</span>3);
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The next 3 elements are the orientation of {B} with respect to {A} expressed
|
||||||
|
using the screw axis.
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">theta = norm(X(4<span class="org-type">:</span>6));
|
||||||
|
s = X(4<span class="org-type">:</span>6)<span class="org-type">/</span>theta;
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
We then compute the corresponding rotation matrix.
|
||||||
|
</p>
|
||||||
|
<div class="org-src-container">
|
||||||
|
<pre class="src src-matlab">R = [s(1)<span class="org-type">^</span>2<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">+</span> cos(theta) , s(1)<span class="org-type">*</span>s(2)<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">-</span> s(3)<span class="org-type">*</span>sin(theta), s(1)<span class="org-type">*</span>s(3)<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">+</span> s(2)<span class="org-type">*</span>sin(theta);
|
||||||
|
s<span class="org-type">(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);</span>
|
||||||
|
s<span class="org-type">(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];</span>
|
||||||
|
</pre>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
<div id="postamble" class="status">
|
<div id="postamble" class="status">
|
||||||
<p class="author">Author: Thomas Dehaeze</p>
|
<p class="author">Author: Dehaeze Thomas</p>
|
||||||
<p class="date">Created: 2019-08-26 lun. 11:55</p>
|
<p class="date">Created: 2020-01-28 mar. 17:38</p>
|
||||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
</html>
|
</html>
|
||||||
|
|||||||
@ -182,3 +182,177 @@ From a specified actuator stroke, we try to estimate the available maneuverabili
|
|||||||
max_disp(6) = Leg.stroke/max(abs(J*[0 0 0 0 0 1]'));
|
max_disp(6) = Leg.stroke/max(abs(J*[0 0 0 0 0 1]'));
|
||||||
end
|
end
|
||||||
#+end_src
|
#+end_src
|
||||||
|
** =computeJacobian=: Compute the Jacobian Matrix
|
||||||
|
:PROPERTIES:
|
||||||
|
:header-args:matlab+: :tangle src/computeJacobian.m
|
||||||
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||||
|
:END:
|
||||||
|
<<sec:computeJacobian>>
|
||||||
|
|
||||||
|
This Matlab function is accessible [[file:src/computeJacobian.m][here]].
|
||||||
|
|
||||||
|
*** Function description
|
||||||
|
#+begin_src matlab
|
||||||
|
function [stewart] = computeJacobian(stewart)
|
||||||
|
% computeJacobian -
|
||||||
|
%
|
||||||
|
% Syntax: [stewart] = computeJacobian(stewart)
|
||||||
|
%
|
||||||
|
% Inputs:
|
||||||
|
% - stewart - With at least the following fields:
|
||||||
|
% - As [3x6] - The 6 unit vectors for each strut expressed in {A}
|
||||||
|
% - Ab [3x6] - The 6 position of the joints bi expressed in {A}
|
||||||
|
%
|
||||||
|
% Outputs:
|
||||||
|
% - stewart - With the 3 added field:
|
||||||
|
% - J [6x6] - The Jacobian Matrix
|
||||||
|
% - K [6x6] - The Stiffness Matrix
|
||||||
|
% - C [6x6] - The Compliance Matrix
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Compute Jacobian Matrix
|
||||||
|
#+begin_src matlab
|
||||||
|
stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Compute Stiffness Matrix
|
||||||
|
#+begin_src matlab
|
||||||
|
stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Compute Compliance Matrix
|
||||||
|
#+begin_src matlab
|
||||||
|
stewart.C = inv(stewart.K);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** =inverseKinematics=: Compute Inverse Kinematics
|
||||||
|
:PROPERTIES:
|
||||||
|
:header-args:matlab+: :tangle src/inverseKinematics.m
|
||||||
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:inverseKinematics>>
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This Matlab function is accessible [[file:src/inverseKinematics.m][here]].
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*** Function description
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#+begin_src matlab
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function [Li, dLi] = inverseKinematics(stewart, args)
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% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
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%
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% Syntax: [stewart] = inverseKinematics(stewart)
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%
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% Inputs:
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% - stewart - A structure with the following fields
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% - Aa [3x6] - The positions ai expressed in {A}
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% - Bb [3x6] - The positions bi expressed in {B}
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% - args - Can have the following fields:
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% - AP [3x1] - The wanted position of {B} with respect to {A}
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% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
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%
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% Outputs:
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% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
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% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
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||||||
|
#+end_src
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|
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||||||
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*** Optional Parameters
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||||||
|
#+begin_src matlab
|
||||||
|
arguments
|
||||||
|
stewart
|
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|
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
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args.ARB (3,3) double {mustBeNumeric} = eye(3)
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||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
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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, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.
|
||||||
|
|
||||||
|
From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as
|
||||||
|
\begin{align*}
|
||||||
|
l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
|
||||||
|
&= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
|
||||||
|
\end{align*}
|
||||||
|
|
||||||
|
To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself:
|
||||||
|
\begin{equation}
|
||||||
|
l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by:
|
||||||
|
\begin{equation}
|
||||||
|
l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
|
||||||
|
Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
|
||||||
|
|
||||||
|
*** Compute
|
||||||
|
#+begin_src matlab
|
||||||
|
Li = sqrt(args.AP'*args.AP + diag(stewart.Bb'*stewart.Bb) + diag(stewart.Aa'*stewart.Aa) - (2*args.AP'*stewart.Aa)' + (2*args.AP'*(args.ARB*stewart.Bb))' - diag(2*(args.ARB*stewart.Bb)'*stewart.Aa));
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
dLi = Li-stewart.l;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** =forwardKinematicsApprox=: Compute the Approximate Forward Kinematics
|
||||||
|
:PROPERTIES:
|
||||||
|
:header-args:matlab+: :tangle src/forwardKinematicsApprox.m
|
||||||
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||||
|
:END:
|
||||||
|
<<sec:forwardKinematicsApprox>>
|
||||||
|
|
||||||
|
This Matlab function is accessible [[file:src/forwardKinematicsApprox.m][here]].
|
||||||
|
|
||||||
|
*** Function description
|
||||||
|
#+begin_src matlab
|
||||||
|
function [P, R] = forwardKinematicsApprox(stewart, args)
|
||||||
|
% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
|
||||||
|
% the Jacobian Matrix
|
||||||
|
%
|
||||||
|
% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
|
||||||
|
%
|
||||||
|
% Inputs:
|
||||||
|
% - stewart - A structure with the following fields
|
||||||
|
% - J [6x6] - The Jacobian Matrix
|
||||||
|
% - args - Can have the following fields:
|
||||||
|
% - dL [6x1] - Displacement of each strut [m]
|
||||||
|
%
|
||||||
|
% Outputs:
|
||||||
|
% - P [3x1] - The estimated position of {B} with respect to {A}
|
||||||
|
% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Optional Parameters
|
||||||
|
#+begin_src matlab
|
||||||
|
arguments
|
||||||
|
stewart
|
||||||
|
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Computation
|
||||||
|
From a small displacement of each strut $d\bm{\mathcal{L}}$, we can compute the
|
||||||
|
position and orientation of {B} with respect to {A} using the following formula:
|
||||||
|
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
|
||||||
|
#+begin_src matlab
|
||||||
|
X = stewart.J\args.dL;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The position vector corresponds to the first 3 elements.
|
||||||
|
#+begin_src matlab
|
||||||
|
P = X(1:3);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The next 3 elements are the orientation of {B} with respect to {A} expressed
|
||||||
|
using the screw axis.
|
||||||
|
#+begin_src matlab
|
||||||
|
theta = norm(X(4:6));
|
||||||
|
s = X(4:6)/theta;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
We then compute the corresponding rotation matrix.
|
||||||
|
#+begin_src matlab
|
||||||
|
R = [s(1)^2*(1-cos(theta)) + cos(theta) , s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta);
|
||||||
|
s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);
|
||||||
|
s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];
|
||||||
|
#+end_src
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user