1144 lines
61 KiB
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1144 lines
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<a accesskey="h" href="./index.html"> UP </a>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Stewart Platform - Simscape Model</h1>
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<div id="table-of-contents">
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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="#orgac25f89">1. <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</a>
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<ul>
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<li><a href="#org5cdd36d">1.1. Function description</a></li>
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<li><a href="#orge837700">1.2. Documentation</a></li>
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<li><a href="#org312e9a7">1.3. Optional Parameters</a></li>
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<li><a href="#org3622825">1.4. Initialize the Stewart structure</a></li>
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<li><a href="#org7d50d54">1.5. Compute the position of each frame</a></li>
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</ul>
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</li>
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<li><a href="#org4310d42">2. Initialize the position of the Joints</a>
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<ul>
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<li><a href="#org65fc289">2.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
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<ul>
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<li><a href="#org5a5b987">2.1.1. Function description</a></li>
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<li><a href="#org40ea23a">2.1.2. Documentation</a></li>
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<li><a href="#org37651aa">2.1.3. Optional Parameters</a></li>
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<li><a href="#orge94a885">2.1.4. Position of the Cube</a></li>
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<li><a href="#org789a28b">2.1.5. Compute the pose</a></li>
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</ul>
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</li>
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<li><a href="#orgccb31c6">2.2. <code>generateGeneralConfiguration</code>: Generate a Very General Configuration</a>
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<ul>
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<li><a href="#org0ba3fa4">2.2.1. Function description</a></li>
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<li><a href="#org839cae7">2.2.2. Documentation</a></li>
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<li><a href="#org2e06fba">2.2.3. Optional Parameters</a></li>
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<li><a href="#orgfac23f3">2.2.4. Compute the pose</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org9944c04">3. <code>computeJointsPose</code>: Compute the Pose of the Joints</a>
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<ul>
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<li><a href="#org465ebb2">3.1. Function description</a></li>
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<li><a href="#orgc11609b">3.2. Documentation</a></li>
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<li><a href="#orge87b302">3.3. Compute the position of the Joints</a></li>
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<li><a href="#org3a7e3c5">3.4. Compute the strut length and orientation</a></li>
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<li><a href="#org9e1258f">3.5. Compute the orientation of the Joints</a></li>
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</ul>
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</li>
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<li><a href="#orgae8d0dc">4. <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</a>
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<ul>
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<li><a href="#org9af8ca0">4.1. Function description</a></li>
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<li><a href="#org335e959">4.2. Optional Parameters</a></li>
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<li><a href="#org3c2e550">4.3. Add Stiffness and Damping properties of each strut</a></li>
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</ul>
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</li>
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<li><a href="#org9e8cbfa">5. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
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<ul>
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<li><a href="#org64e795c">5.1. Function description</a></li>
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<li><a href="#org9bcd9b9">5.2. Compute Jacobian Matrix</a></li>
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<li><a href="#orgf08eda6">5.3. Compute Stiffness Matrix</a></li>
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<li><a href="#orgd164132">5.4. Compute Compliance Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org53d13cc">6. Initialize the Geometry of the Mechanical Elements</a>
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<ul>
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<li><a href="#org4674203">6.1. <code>initializeCylindricalPlatforms</code>: Initialize the geometry of the Fixed and Mobile Platforms</a>
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<ul>
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<li><a href="#org1af6832">6.1.1. Function description</a></li>
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<li><a href="#org12a2c2b">6.1.2. Optional Parameters</a></li>
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<li><a href="#org2965fe8">6.1.3. Create the <code>platforms</code> struct</a></li>
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<li><a href="#org7e1a4d4">6.1.4. Save the <code>platforms</code> struct</a></li>
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</ul>
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</li>
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<li><a href="#org6b4f1ca">6.2. <code>initializeCylindricalStruts</code>: Define the mass and moment of inertia of cylindrical struts</a>
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<ul>
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<li><a href="#org407cf8c">6.2.1. Function description</a></li>
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<li><a href="#org420f14e">6.2.2. Optional Parameters</a></li>
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<li><a href="#org3f9f0ef">6.2.3. Create the <code>struts</code> structure</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgb4dc92b">7. Utility Functions</a>
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<ul>
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<li><a href="#org03168fc">7.1. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
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<ul>
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<li><a href="#orgb7853bd">7.1.1. Function description</a></li>
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<li><a href="#orge9021eb">7.1.2. Optional Parameters</a></li>
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<li><a href="#orgbdc5fb1">7.1.3. Theory</a></li>
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<li><a href="#org8b70a76">7.1.4. Compute</a></li>
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</ul>
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</li>
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<li><a href="#orgf4ebfa3">7.2. <code>forwardKinematicsApprox</code>: Compute the Forward Kinematics</a>
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<ul>
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<li><a href="#org37e6edf">7.2.1. Function description</a></li>
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<li><a href="#org6441308">7.2.2. Optional Parameters</a></li>
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<li><a href="#orgf17cab9">7.2.3. Computation</a></li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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Stewart platforms are generated in multiple steps.
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</p>
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<p>
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We define 4 important <b>frames</b>:
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</p>
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<ul class="org-ul">
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<li>\(\{F\}\): Frame fixed to the <b>Fixed</b> base and located at the center of its bottom surface.
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This is used to fix the Stewart platform to some support.</li>
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<li>\(\{M\}\): Frame fixed to the <b>Moving</b> platform and located at the center of its top surface.
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This is used to place things on top of the Stewart platform.</li>
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<li>\(\{A\}\): Frame fixed to the fixed base.
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It defined the center of rotation of the moving platform.</li>
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<li>\(\{B\}\): Frame fixed to the moving platform.
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The motion of the moving platforms and forces applied to it are defined with respect to this frame \(\{B\}\).</li>
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</ul>
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<p>
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Then, we define the <b>location of the spherical joints</b>:
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</p>
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<ul class="org-ul">
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<li>\(\bm{a}_{i}\) are the position of the spherical joints fixed to the fixed base</li>
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<li>\(\bm{b}_{i}\) are the position of the spherical joints fixed to the moving platform</li>
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</ul>
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<p>
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We define the <b>rest position</b> of the Stewart platform:
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</p>
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<ul class="org-ul">
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<li>For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.</li>
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<li>Thus, to define the rest position of the Stewart platform, we just have to defined its total height \(H\).
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\(H\) corresponds to the distance from the bottom of the fixed base to the top of the moving platform.</li>
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</ul>
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<p>
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From \(\bm{a}_{i}\) and \(\bm{b}_{i}\), we can determine the <b>length and orientation of each strut</b>:
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</p>
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<ul class="org-ul">
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<li>\(l_{i}\) is the length of the strut</li>
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<li>\({}^{A}\hat{\bm{s}}_{i}\) is the unit vector align with the strut</li>
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</ul>
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<p>
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The position of the Spherical joints can be computed using various methods:
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</p>
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<ul class="org-ul">
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<li>Cubic configuration</li>
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<li>Circular configuration</li>
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<li>Arbitrary position</li>
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<li>These methods should be easily scriptable and corresponds to specific functions that returns \({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\).
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The input of these functions are the parameters corresponding to the wanted geometry.</li>
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</ul>
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<p>
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For Simscape, we need:
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</p>
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<ul class="org-ul">
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<li>The position and orientation of each spherical joint fixed to the fixed base: \({}^{F}\bm{a}_{i}\) and \({}^{F}\bm{R}_{a_{i}}\)</li>
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<li>The position and orientation of each spherical joint fixed to the moving platform: \({}^{M}\bm{b}_{i}\) and \({}^{M}\bm{R}_{b_{i}}\)</li>
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<li>The rest length of each strut: \(l_{i}\)</li>
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<li>The stiffness and damping of each actuator: \(k_{i}\) and \(c_{i}\)</li>
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<li>The position of the frame \(\{A\}\) with respect to the frame \(\{F\}\): \({}^{F}\bm{O}_{A}\)</li>
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<li>The position of the frame \(\{B\}\) with respect to the frame \(\{M\}\): \({}^{M}\bm{O}_{B}\)</li>
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</ul>
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<div id="outline-container-orgac25f89" class="outline-2">
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<h2 id="orgac25f89"><span class="section-number-2">1</span> <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="orga56a5c6"></a>
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</p>
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<p>
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This Matlab function is accessible <a href="src/initializeFramesPositions.m">here</a>.
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</p>
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</div>
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<div id="outline-container-org5cdd36d" class="outline-3">
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<h3 id="org5cdd36d"><span class="section-number-3">1.1</span> Function description</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">initializeFramesPositions</span>(<span class="org-variable-name">args</span>)
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<span class="org-comment">% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Syntax: [stewart] = initializeFramesPositions(args)</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Inputs:</span>
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<span class="org-comment">% - args - Can have the following fields:</span>
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<span class="org-comment">% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]</span>
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<span class="org-comment">% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Outputs:</span>
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<span class="org-comment">% - stewart - A structure with the following fields:</span>
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<span class="org-comment">% - H [1x1] - Total Height of the Stewart Platform [m]</span>
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<span class="org-comment">% - FO_M [3x1] - Position of {M} with respect to {F} [m]</span>
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<span class="org-comment">% - MO_B [3x1] - Position of {B} with respect to {M} [m]</span>
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<span class="org-comment">% - FO_A [3x1] - Position of {A} with respect to {F} [m]</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orge837700" class="outline-3">
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<h3 id="orge837700"><span class="section-number-3">1.2</span> Documentation</h3>
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<div class="outline-text-3" id="text-1-2">
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<div id="orgebd0f49" class="figure">
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<p><img src="figs/stewart-frames-position.png" alt="stewart-frames-position.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Definition of the position of the frames</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org312e9a7" class="outline-3">
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<h3 id="org312e9a7"><span class="section-number-3">1.3</span> Optional Parameters</h3>
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<div class="outline-text-3" id="text-1-3">
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<div class="org-src-container">
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<pre class="src src-matlab">arguments
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args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e<span class="org-type">-</span>3
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args.MO_B (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
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<span class="org-keyword">end</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org3622825" class="outline-3">
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<h3 id="org3622825"><span class="section-number-3">1.4</span> Initialize the Stewart structure</h3>
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<div class="outline-text-3" id="text-1-4">
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<div class="org-src-container">
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<pre class="src src-matlab">stewart = struct();
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org7d50d54" class="outline-3">
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<h3 id="org7d50d54"><span class="section-number-3">1.5</span> Compute the position of each frame</h3>
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<div class="outline-text-3" id="text-1-5">
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<div class="org-src-container">
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<pre class="src src-matlab">stewart.H = args.H; <span class="org-comment">% Total Height of the Stewart Platform [m]</span>
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stewart.FO_M = [0; 0; stewart.H]; <span class="org-comment">% Position of {M} with respect to {F} [m]</span>
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stewart.MO_B = [0; 0; args.MO_B]; <span class="org-comment">% Position of {B} with respect to {M} [m]</span>
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stewart.FO_A = stewart.MO_B <span class="org-type">+</span> stewart.FO_M; <span class="org-comment">% Position of {A} with respect to {F} [m]</span>
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</pre>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org4310d42" class="outline-2">
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<h2 id="org4310d42"><span class="section-number-2">2</span> Initialize the position of the Joints</h2>
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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-org65fc289" class="outline-3">
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<h3 id="org65fc289"><span class="section-number-3">2.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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<a id="org677ea95"></a>
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</p>
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<p>
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This Matlab function is accessible <a href="src/generateCubicConfiguration.m">here</a>.
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</p>
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</div>
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<div id="outline-container-org5a5b987" class="outline-4">
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<h4 id="org5a5b987"><span class="section-number-4">2.1.1</span> Function description</h4>
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<div class="outline-text-4" id="text-2-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
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<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(stewart, args)</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Inputs:</span>
|
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<span class="org-comment">% - stewart - A structure with the following fields</span>
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<span class="org-comment">% - H [1x1] - Total height of the platform [m]</span>
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<span class="org-comment">% - args - Can have the following fields:</span>
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<span class="org-comment">% - Hc [1x1] - Height of the "useful" part of the cube [m]</span>
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<span class="org-comment">% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]</span>
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<span class="org-comment">% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]</span>
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<span class="org-comment">% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Outputs:</span>
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<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
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<span class="org-comment">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
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<span class="org-comment">% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org40ea23a" class="outline-4">
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<h4 id="org40ea23a"><span class="section-number-4">2.1.2</span> Documentation</h4>
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<div class="outline-text-4" id="text-2-1-2">
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<div id="org70070f0" class="figure">
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<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Cubic Configuration</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org37651aa" class="outline-4">
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<h4 id="org37651aa"><span class="section-number-4">2.1.3</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-2-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e<span class="org-type">-</span>3
|
|
args.FOc (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
|
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge94a885" class="outline-4">
|
|
<h4 id="orge94a885"><span class="section-number-4">2.1.4</span> Position of the Cube</h4>
|
|
<div class="outline-text-4" id="text-2-1-4">
|
|
<p>
|
|
We define the useful points of the cube with respect to the Cube’s center.
|
|
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
|
|
located at the center of the cube and aligned with {F} and {M}.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sx = [ 2; <span class="org-type">-</span>1; <span class="org-type">-</span>1];
|
|
sy = [ 0; 1; <span class="org-type">-</span>1];
|
|
sz = [ 1; 1; 1];
|
|
|
|
R = [sx, sy, sz]<span class="org-type">./</span>vecnorm([sx, sy, sz]);
|
|
|
|
L = args.Hc<span class="org-type">*</span>sqrt(3);
|
|
|
|
Cc = R<span class="org-type">'*</span>[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] <span class="org-type">-</span> [0;0;1.5<span class="org-type">*</span>args.Hc];
|
|
|
|
CCf = [Cc(<span class="org-type">:</span>,1), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,1)]; <span class="org-comment">% CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg</span>
|
|
CCm = [Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,6), Cc(<span class="org-type">:</span>,6)]; <span class="org-comment">% CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org789a28b" class="outline-4">
|
|
<h4 id="org789a28b"><span class="section-number-4">2.1.5</span> Compute the pose</h4>
|
|
<div class="outline-text-4" id="text-2-1-5">
|
|
<p>
|
|
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">CSi = (CCm <span class="org-type">-</span> CCf)<span class="org-type">./</span>vecnorm(CCm <span class="org-type">-</span> CCf);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Fa = CCf <span class="org-type">+</span> [0; 0; args.FOc] <span class="org-type">+</span> ((args.FHa<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
|
stewart.Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>stewart.H] <span class="org-type">+</span> ((stewart.H<span class="org-type">-</span>args.MHb<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgccb31c6" class="outline-3">
|
|
<h3 id="orgccb31c6"><span class="section-number-3">2.2</span> <code>generateGeneralConfiguration</code>: Generate a Very General Configuration</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
<a id="org32105b0"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/generateGeneralConfiguration.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org0ba3fa4" class="outline-4">
|
|
<h4 id="org0ba3fa4"><span class="section-number-4">2.2.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-2-2-1">
|
|
<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">generateGeneralConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% generateGeneralConfiguration - Generate a Very General Configuration</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
|
<span class="org-comment">% - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]</span>
|
|
<span class="org-comment">% - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m]</span>
|
|
<span class="org-comment">% - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]</span>
|
|
<span class="org-comment">% - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]</span>
|
|
<span class="org-comment">% - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m]</span>
|
|
<span class="org-comment">% - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
|
<span class="org-comment">% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org839cae7" class="outline-4">
|
|
<h4 id="org839cae7"><span class="section-number-4">2.2.2</span> Documentation</h4>
|
|
<div class="outline-text-4" id="text-2-2-2">
|
|
<p>
|
|
Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}.
|
|
The radius of the circles can be chosen as well as the angles where the joints are located.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2e06fba" class="outline-4">
|
|
<h4 id="org2e06fba"><span class="section-number-4">2.2.3</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-2-2-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 90e<span class="org-type">-</span>3;
|
|
args.FTh (6,1) double {mustBeNumeric} = [<span class="org-type">-</span>10, 10, 120<span class="org-type">-</span>10, 120<span class="org-type">+</span>10, 240<span class="org-type">-</span>10, 240<span class="org-type">+</span>10]<span class="org-type">*</span>(<span class="org-constant">pi</span><span class="org-type">/</span>180);
|
|
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 70e<span class="org-type">-</span>3;
|
|
args.MTh (6,1) double {mustBeNumeric} = [<span class="org-type">-</span>60<span class="org-type">+</span>10, 60<span class="org-type">-</span>10, 60<span class="org-type">+</span>10, 180<span class="org-type">-</span>10, 180<span class="org-type">+</span>10, <span class="org-type">-</span>60<span class="org-type">-</span>10]<span class="org-type">*</span>(<span class="org-constant">pi</span><span class="org-type">/</span>180);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfac23f3" class="outline-4">
|
|
<h4 id="orgfac23f3"><span class="section-number-4">2.2.4</span> Compute the pose</h4>
|
|
<div class="outline-text-4" id="text-2-2-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Fa = zeros(3,6);
|
|
stewart.Mb = zeros(3,6);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
stewart.Fa(<span class="org-type">:</span>,<span class="org-constant">i</span>) = [args.FR<span class="org-type">*</span>cos(args.FTh(<span class="org-constant">i</span>)); args.FR<span class="org-type">*</span>sin(args.FTh(<span class="org-constant">i</span>)); args.FH];
|
|
stewart.Mb(<span class="org-type">:</span>,<span class="org-constant">i</span>) = [args.MR<span class="org-type">*</span>cos(args.MTh(<span class="org-constant">i</span>)); args.MR<span class="org-type">*</span>sin(args.MTh(<span class="org-constant">i</span>)); <span class="org-type">-</span>args.MH];
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9944c04" class="outline-2">
|
|
<h2 id="org9944c04"><span class="section-number-2">3</span> <code>computeJointsPose</code>: Compute the Pose of the Joints</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgd0bee51"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/computeJointsPose.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org465ebb2" class="outline-3">
|
|
<h3 id="org465ebb2"><span class="section-number-3">3.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-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">computeJointsPose</span>(<span class="org-variable-name">stewart</span>)
|
|
<span class="org-comment">% computeJointsPose -</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = computeJointsPose(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">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
|
<span class="org-comment">% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
|
<span class="org-comment">% - FO_A [3x1] - Position of {A} with respect to {F}</span>
|
|
<span class="org-comment">% - MO_B [3x1] - Position of {B} with respect to {M}</span>
|
|
<span class="org-comment">% - FO_M [3x1] - Position of {M} with respect to {F}</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - A structure with the following added fields</span>
|
|
<span class="org-comment">% - Aa [3x6] - The i'th column is the position of ai with respect to {A}</span>
|
|
<span class="org-comment">% - Ab [3x6] - The i'th column is the position of bi with respect to {A}</span>
|
|
<span class="org-comment">% - Ba [3x6] - The i'th column is the position of ai with respect to {B}</span>
|
|
<span class="org-comment">% - Bb [3x6] - The i'th column is the position of bi with respect to {B}</span>
|
|
<span class="org-comment">% - l [6x1] - The i'th element is the initial length of strut i</span>
|
|
<span class="org-comment">% - As [3x6] - The i'th column is the unit vector of strut i expressed in {A}</span>
|
|
<span class="org-comment">% - Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}</span>
|
|
<span class="org-comment">% - FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}</span>
|
|
<span class="org-comment">% - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc11609b" class="outline-3">
|
|
<h3 id="orgc11609b"><span class="section-number-3">3.2</span> Documentation</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
|
|
<div id="org20f7106" class="figure">
|
|
<p><img src="figs/stewart-struts.png" alt="stewart-struts.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Position and orientation of the struts</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge87b302" class="outline-3">
|
|
<h3 id="orge87b302"><span class="section-number-3">3.3</span> Compute the position of the Joints</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Aa = stewart.Fa <span class="org-type">-</span> repmat(stewart.FO_A, [1, 6]);
|
|
stewart.Bb = stewart.Mb <span class="org-type">-</span> repmat(stewart.MO_B, [1, 6]);
|
|
|
|
stewart.Ab = stewart.Bb <span class="org-type">-</span> repmat(<span class="org-type">-</span>stewart.MO_B<span class="org-type">-</span>stewart.FO_M<span class="org-type">+</span>stewart.FO_A, [1, 6]);
|
|
stewart.Ba = stewart.Aa <span class="org-type">-</span> repmat( stewart.MO_B<span class="org-type">+</span>stewart.FO_M<span class="org-type">-</span>stewart.FO_A, [1, 6]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3a7e3c5" class="outline-3">
|
|
<h3 id="org3a7e3c5"><span class="section-number-3">3.4</span> Compute the strut length and orientation</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.As = (stewart.Ab <span class="org-type">-</span> stewart.Aa)<span class="org-type">./</span>vecnorm(stewart.Ab <span class="org-type">-</span> stewart.Aa); <span class="org-comment">% As_i is the i'th vector of As</span>
|
|
|
|
stewart.l = vecnorm(stewart.Ab <span class="org-type">-</span> stewart.Aa)<span class="org-type">'</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Bs = (stewart.Bb <span class="org-type">-</span> stewart.Ba)<span class="org-type">./</span>vecnorm(stewart.Bb <span class="org-type">-</span> stewart.Ba);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9e1258f" class="outline-3">
|
|
<h3 id="org9e1258f"><span class="section-number-3">3.5</span> Compute the orientation of the Joints</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.FRa = zeros(3,3,6);
|
|
stewart.MRb = zeros(3,3,6);
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = [cross([0;1;0], stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>)) , cross(stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>), cross([0;1;0], stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>))) , stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>)];
|
|
stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>)<span class="org-type">./</span>vecnorm(stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>));
|
|
|
|
stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = [cross([0;1;0], stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>)) , cross(stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>), cross([0;1;0], stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>))) , stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>)];
|
|
stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>)<span class="org-type">./</span>vecnorm(stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgae8d0dc" class="outline-2">
|
|
<h2 id="orgae8d0dc"><span class="section-number-2">4</span> <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="orgd55e892"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/initializeStrutDynamics.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org9af8ca0" class="outline-3">
|
|
<h3 id="org9af8ca0"><span class="section-number-3">4.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<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">initializeStrutDynamics</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% initializeStrutDynamics - Add Stiffness and Damping properties of each strut</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = initializeStrutDynamics(args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - args - Structure with the following fields:</span>
|
|
<span class="org-comment">% - Ki [6x1] - Stiffness of each strut [N/m]</span>
|
|
<span class="org-comment">% - Ci [6x1] - Damping of each strut [N/(m/s)]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - Ki [6x1] - Stiffness of each strut [N/m]</span>
|
|
<span class="org-comment">% - Ci [6x1] - Damping of each strut [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org335e959" class="outline-3">
|
|
<h3 id="org335e959"><span class="section-number-3">4.2</span> Optional Parameters</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Ki (6,1) double {mustBeNumeric, mustBePositive} = 1e6<span class="org-type">*</span>ones(6,1)
|
|
args.Ci (6,1) double {mustBeNumeric, mustBePositive} = 1e3<span class="org-type">*</span>ones(6,1)
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3c2e550" class="outline-3">
|
|
<h3 id="org3c2e550"><span class="section-number-3">4.3</span> Add Stiffness and Damping properties of each strut</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Ki = args.Ki;
|
|
stewart.Ci = args.Ci;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9e8cbfa" class="outline-2">
|
|
<h2 id="org9e8cbfa"><span class="section-number-2">5</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orgfb113f5"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org64e795c" class="outline-3">
|
|
<h3 id="org64e795c"><span class="section-number-3">5.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">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-org9bcd9b9" class="outline-3">
|
|
<h3 id="org9bcd9b9"><span class="section-number-3">5.2</span> Compute Jacobian Matrix</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<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-orgf08eda6" class="outline-3">
|
|
<h3 id="orgf08eda6"><span class="section-number-3">5.3</span> Compute Stiffness Matrix</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
<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-orgd164132" class="outline-3">
|
|
<h3 id="orgd164132"><span class="section-number-3">5.4</span> Compute Compliance Matrix</h3>
|
|
<div class="outline-text-3" id="text-5-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.C = inv(stewart.K);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org53d13cc" class="outline-2">
|
|
<h2 id="org53d13cc"><span class="section-number-2">6</span> Initialize the Geometry of the Mechanical Elements</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
</div>
|
|
<div id="outline-container-org4674203" class="outline-3">
|
|
<h3 id="org4674203"><span class="section-number-3">6.1</span> <code>initializeCylindricalPlatforms</code>: Initialize the geometry of the Fixed and Mobile Platforms</h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
<p>
|
|
<a id="orgac77ed6"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/initializeCylindricalPlatforms.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org1af6832" class="outline-4">
|
|
<h4 id="org1af6832"><span class="section-number-4">6.1.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-6-1-1">
|
|
<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">initializeCylindricalPlatforms</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = initializeCylindricalPlatforms(args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - args - Structure with the following fields:</span>
|
|
<span class="org-comment">% - Fpm [1x1] - Fixed Platform Mass [kg]</span>
|
|
<span class="org-comment">% - Fph [1x1] - Fixed Platform Height [m]</span>
|
|
<span class="org-comment">% - Fpr [1x1] - Fixed Platform Radius [m]</span>
|
|
<span class="org-comment">% - Mpm [1x1] - Mobile Platform Mass [kg]</span>
|
|
<span class="org-comment">% - Mph [1x1] - Mobile Platform Height [m]</span>
|
|
<span class="org-comment">% - Mpr [1x1] - Mobile Platform Radius [m]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - platforms [struct] - structure with the following fields:</span>
|
|
<span class="org-comment">% - Fpm [1x1] - Fixed Platform Mass [kg]</span>
|
|
<span class="org-comment">% - Msi [3x3] - Mobile Platform Inertia matrix [kg*m^2]</span>
|
|
<span class="org-comment">% - Fph [1x1] - Fixed Platform Height [m]</span>
|
|
<span class="org-comment">% - Fpr [1x1] - Fixed Platform Radius [m]</span>
|
|
<span class="org-comment">% - Mpm [1x1] - Mobile Platform Mass [kg]</span>
|
|
<span class="org-comment">% - Fsi [3x3] - Fixed Platform Inertia matrix [kg*m^2]</span>
|
|
<span class="org-comment">% - Mph [1x1] - Mobile Platform Height [m]</span>
|
|
<span class="org-comment">% - Mpr [1x1] - Mobile Platform Radius [m]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org12a2c2b" class="outline-4">
|
|
<h4 id="org12a2c2b"><span class="section-number-4">6.1.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-6-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
|
|
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e<span class="org-type">-</span>3
|
|
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e<span class="org-type">-</span>3
|
|
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
|
|
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e<span class="org-type">-</span>3
|
|
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2965fe8" class="outline-4">
|
|
<h4 id="org2965fe8"><span class="section-number-4">6.1.3</span> Create the <code>platforms</code> struct</h4>
|
|
<div class="outline-text-4" id="text-6-1-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">platforms = struct();
|
|
|
|
platforms.Fpm = args.Fpm;
|
|
platforms.Fph = args.Fph;
|
|
platforms.Fpr = args.Fpr;
|
|
platforms.Fpi = diag([1<span class="org-type">/</span>12 <span class="org-type">*</span> platforms.Fpm <span class="org-type">*</span> (3<span class="org-type">*</span>platforms.Fpr<span class="org-type">^</span>2 <span class="org-type">+</span> platforms.Fph<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>12 <span class="org-type">*</span> platforms.Fpm <span class="org-type">*</span> (3<span class="org-type">*</span>platforms.Fpr<span class="org-type">^</span>2 <span class="org-type">+</span> platforms.Fph<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>2 <span class="org-type">*</span> platforms.Fpm <span class="org-type">*</span> platforms.Fpr<span class="org-type">^</span>2]);
|
|
|
|
platforms.Mpm = args.Mpm;
|
|
platforms.Mph = args.Mph;
|
|
platforms.Mpr = args.Mpr;
|
|
platforms.Mpi = diag([1<span class="org-type">/</span>12 <span class="org-type">*</span> platforms.Mpm <span class="org-type">*</span> (3<span class="org-type">*</span>platforms.Mpr<span class="org-type">^</span>2 <span class="org-type">+</span> platforms.Mph<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>12 <span class="org-type">*</span> platforms.Mpm <span class="org-type">*</span> (3<span class="org-type">*</span>platforms.Mpr<span class="org-type">^</span>2 <span class="org-type">+</span> platforms.Mph<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>2 <span class="org-type">*</span> platforms.Mpm <span class="org-type">*</span> platforms.Mpr<span class="org-type">^</span>2]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7e1a4d4" class="outline-4">
|
|
<h4 id="org7e1a4d4"><span class="section-number-4">6.1.4</span> Save the <code>platforms</code> struct</h4>
|
|
<div class="outline-text-4" id="text-6-1-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.platforms = platforms;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6b4f1ca" class="outline-3">
|
|
<h3 id="org6b4f1ca"><span class="section-number-3">6.2</span> <code>initializeCylindricalStruts</code>: Define the mass and moment of inertia of cylindrical struts</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
<p>
|
|
<a id="orgfd38bf8"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/initializeCylindricalStruts.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org407cf8c" class="outline-4">
|
|
<h4 id="org407cf8c"><span class="section-number-4">6.2.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-6-2-1">
|
|
<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">initializeCylindricalStruts</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = initializeCylindricalStruts(args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - args - Structure with the following fields:</span>
|
|
<span class="org-comment">% - Fsm [1x1] - Mass of the Fixed part of the struts [kg]</span>
|
|
<span class="org-comment">% - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]</span>
|
|
<span class="org-comment">% - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]</span>
|
|
<span class="org-comment">% - Msm [1x1] - Mass of the Mobile part of the struts [kg]</span>
|
|
<span class="org-comment">% - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]</span>
|
|
<span class="org-comment">% - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - struts [struct] - structure with the following fields:</span>
|
|
<span class="org-comment">% - Fsm [6x1] - Mass of the Fixed part of the struts [kg]</span>
|
|
<span class="org-comment">% - Fsi [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]</span>
|
|
<span class="org-comment">% - Msm [6x1] - Mass of the Mobile part of the struts [kg]</span>
|
|
<span class="org-comment">% - Msi [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]</span>
|
|
<span class="org-comment">% - Fsh [6x1] - Height of cylinder for the Fixed part of the struts [m]</span>
|
|
<span class="org-comment">% - Fsr [6x1] - Radius of cylinder for the Fixed part of the struts [m]</span>
|
|
<span class="org-comment">% - Msh [6x1] - Height of cylinder for the Mobile part of the struts [m]</span>
|
|
<span class="org-comment">% - Msr [6x1] - Radius of cylinder for the Mobile part of the struts [m]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org420f14e" class="outline-4">
|
|
<h4 id="org420f14e"><span class="section-number-4">6.2.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-6-2-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
|
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e<span class="org-type">-</span>3
|
|
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e<span class="org-type">-</span>3
|
|
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
|
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e<span class="org-type">-</span>3
|
|
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3f9f0ef" class="outline-4">
|
|
<h4 id="org3f9f0ef"><span class="section-number-4">6.2.3</span> Create the <code>struts</code> structure</h4>
|
|
<div class="outline-text-4" id="text-6-2-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">struts = struct();
|
|
|
|
struts.Fsm = ones(6,1)<span class="org-type">.*</span>args.Fsm;
|
|
struts.Msm = ones(6,1)<span class="org-type">.*</span>args.Msm;
|
|
|
|
struts.Fsh = ones(6,1)<span class="org-type">.*</span>args.Fsh;
|
|
struts.Fsr = ones(6,1)<span class="org-type">.*</span>args.Fsr;
|
|
struts.Msh = ones(6,1)<span class="org-type">.*</span>args.Msh;
|
|
struts.Msr = ones(6,1)<span class="org-type">.*</span>args.Msr;
|
|
|
|
struts.Fsi = zeros(3, 3, 6);
|
|
struts.Msi = zeros(3, 3, 6);
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
struts.Fsi(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = diag([1<span class="org-type">/</span>12 <span class="org-type">*</span> struts.Fsm(<span class="org-constant">i</span>) <span class="org-type">*</span> (3<span class="org-type">*</span>struts.Fsr(<span class="org-constant">i</span>)<span class="org-type">^</span>2 <span class="org-type">+</span> struts.Fsh(<span class="org-constant">i</span>)<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>12 <span class="org-type">*</span> struts.Fsm(<span class="org-constant">i</span>) <span class="org-type">*</span> (3<span class="org-type">*</span>struts.Fsr(<span class="org-constant">i</span>)<span class="org-type">^</span>2 <span class="org-type">+</span> struts.Fsh(<span class="org-constant">i</span>)<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>2 <span class="org-type">*</span> struts.Fsm(<span class="org-constant">i</span>) <span class="org-type">*</span> struts.Fsr(<span class="org-constant">i</span>)<span class="org-type">^</span>2]);
|
|
struts.Msi(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = diag([1<span class="org-type">/</span>12 <span class="org-type">*</span> struts.Msm(<span class="org-constant">i</span>) <span class="org-type">*</span> (3<span class="org-type">*</span>struts.Msr(<span class="org-constant">i</span>)<span class="org-type">^</span>2 <span class="org-type">+</span> struts.Msh(<span class="org-constant">i</span>)<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>12 <span class="org-type">*</span> struts.Msm(<span class="org-constant">i</span>) <span class="org-type">*</span> (3<span class="org-type">*</span>struts.Msr(<span class="org-constant">i</span>)<span class="org-type">^</span>2 <span class="org-type">+</span> struts.Msh(<span class="org-constant">i</span>)<span class="org-type">^</span>2), ...
|
|
1<span class="org-type">/</span>2 <span class="org-type">*</span> struts.Msm(<span class="org-constant">i</span>) <span class="org-type">*</span> struts.Msr(<span class="org-constant">i</span>)<span class="org-type">^</span>2]);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.struts = struts;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
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|
|
|
<div id="outline-container-orgb4dc92b" class="outline-2">
|
|
<h2 id="orgb4dc92b"><span class="section-number-2">7</span> Utility Functions</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
</div>
|
|
<div id="outline-container-org03168fc" class="outline-3">
|
|
<h3 id="org03168fc"><span class="section-number-3">7.1</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
|
|
<div class="outline-text-3" id="text-7-1">
|
|
<p>
|
|
<a id="org681bcb5"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb7853bd" class="outline-4">
|
|
<h4 id="orgb7853bd"><span class="section-number-4">7.1.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-7-1-1">
|
|
<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-orge9021eb" class="outline-4">
|
|
<h4 id="orge9021eb"><span class="section-number-4">7.1.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-7-1-2">
|
|
<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-orgbdc5fb1" class="outline-4">
|
|
<h4 id="orgbdc5fb1"><span class="section-number-4">7.1.3</span> Theory</h4>
|
|
<div class="outline-text-4" id="text-7-1-3">
|
|
<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-org8b70a76" class="outline-4">
|
|
<h4 id="org8b70a76"><span class="section-number-4">7.1.4</span> Compute</h4>
|
|
<div class="outline-text-4" id="text-7-1-4">
|
|
<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-orgf4ebfa3" class="outline-3">
|
|
<h3 id="orgf4ebfa3"><span class="section-number-3">7.2</span> <code>forwardKinematicsApprox</code>: Compute the Forward Kinematics</h3>
|
|
<div class="outline-text-3" id="text-7-2">
|
|
<p>
|
|
<a id="org5b15db4"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org37e6edf" class="outline-4">
|
|
<h4 id="org37e6edf"><span class="section-number-4">7.2.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-7-2-1">
|
|
<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-org6441308" class="outline-4">
|
|
<h4 id="org6441308"><span class="section-number-4">7.2.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-7-2-2">
|
|
<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-orgf17cab9" class="outline-4">
|
|
<h4 id="orgf17cab9"><span class="section-number-4">7.2.3</span> Computation</h4>
|
|
<div class="outline-text-4" id="text-7-2-3">
|
|
<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 id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-04-17 ven. 09:36</p>
|
|
</div>
|
|
</body>
|
|
</html>
|