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Add few functions and function documentation

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Thomas Dehaeze 2020-01-06 18:17:22 +01:00
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@ -1,14 +1,15 @@
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2019-12-20 ven. 17:49 -->
<!-- 2020-01-06 lun. 18:16 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Stewart Platform - Simscape Model</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Thomas Dehaeze" />
<meta name="author" content="Dehaeze Thomas" />
<style type="text/css">
<!--/*--><![CDATA[/*><!--*/
.title { text-align: center;
@ -282,42 +283,68 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org363c6fe">1. Procedure</a></li>
<li><a href="#orgbb97f5b">2. Matlab Code</a>
<li><a href="#org3273efc">1. Procedure</a></li>
<li><a href="#org0cfaf56">2. Matlab Code</a>
<ul>
<li><a href="#org26c49f2">2.1. Simscape Model</a></li>
<li><a href="#org0aee86d">2.2. Test the functions</a></li>
<li><a href="#org47e6dec">2.1. Simscape Model</a></li>
<li><a href="#org80710ea">2.2. Test the functions</a></li>
</ul>
</li>
<li><a href="#org6b23b15">3. <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</a>
<li><a href="#org6005875">3. <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</a>
<ul>
<li><a href="#orgacce02d">3.1. Function description</a></li>
<li><a href="#orgbb19223">3.2. Optional Parameters</a></li>
<li><a href="#org830651c">3.3. Initialize the Stewart structure</a></li>
<li><a href="#org302ef83">3.4. Compute the position of each frame</a></li>
<li><a href="#orga3d2c9e">3.1. Function description</a></li>
<li><a href="#org61a1e26">3.2. Documentation</a></li>
<li><a href="#org713ea9a">3.3. Optional Parameters</a></li>
<li><a href="#org88e3f87">3.4. Initialize the Stewart structure</a></li>
<li><a href="#org4a2edf9">3.5. Compute the position of each frame</a></li>
</ul>
</li>
<li><a href="#orgefabb7d">4. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<li><a href="#org8eb2e40">4. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<ul>
<li><a href="#orgaa4ebd9">4.1. Function description</a></li>
<li><a href="#org59cf450">4.2. Optional Parameters</a></li>
<li><a href="#orgc3b07d9">4.3. Position of the Cube</a></li>
<li><a href="#org4ab3f4c">4.4. Compute the pose</a></li>
<li><a href="#org04bea4c">4.1. Function description</a></li>
<li><a href="#org924cbfe">4.2. Documentation</a></li>
<li><a href="#org335de0c">4.3. Optional Parameters</a></li>
<li><a href="#orgc1a1209">4.4. Position of the Cube</a></li>
<li><a href="#orga1cfc06">4.5. Compute the pose</a></li>
</ul>
</li>
<li><a href="#org1a92b53">5. <code>computeJointsPose</code>: Compute the Pose of the Joints</a>
<li><a href="#orgc137b22">5. <code>computeJointsPose</code>: Compute the Pose of the Joints</a>
<ul>
<li><a href="#org8d2b6ea">5.1. Function description</a></li>
<li><a href="#org663872a">5.2. Compute the position of the Joints</a></li>
<li><a href="#org08b9460">5.3. Compute the strut length and orientation</a></li>
<li><a href="#orgde1237d">5.4. Compute the orientation of the Joints</a></li>
<li><a href="#orge4dc5d9">5.1. Function description</a></li>
<li><a href="#org5e72282">5.2. Documentation</a></li>
<li><a href="#orgafa4ad9">5.3. Compute the position of the Joints</a></li>
<li><a href="#org50bd360">5.4. Compute the strut length and orientation</a></li>
<li><a href="#org3d0d76f">5.5. Compute the orientation of the Joints</a></li>
</ul>
</li>
<li><a href="#org23b3bc2">6. <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</a>
<li><a href="#orgc7ae05f">6. <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</a>
<ul>
<li><a href="#orgfc3e940">6.1. Function description</a></li>
<li><a href="#org33a48bd">6.2. Optional Parameters</a></li>
<li><a href="#org458bd27">6.3. Add Stiffness and Damping properties of each strut</a></li>
<li><a href="#org234504f">6.1. Function description</a></li>
<li><a href="#org8edf5d0">6.2. Optional Parameters</a></li>
<li><a href="#orgf9bb882">6.3. Add Stiffness and Damping properties of each strut</a></li>
</ul>
</li>
<li><a href="#org61b514c">7. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
<ul>
<li><a href="#org2983146">7.1. Function description</a></li>
<li><a href="#org996fbd8">7.2. Compute Jacobian Matrix</a></li>
<li><a href="#org9e42bbe">7.3. Compute Stiffness Matrix</a></li>
<li><a href="#org36e44cd">7.4. Compute Compliance Matrix</a></li>
</ul>
</li>
<li><a href="#org76be477">8. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
<ul>
<li><a href="#org16e3f5f">8.1. Function description</a></li>
<li><a href="#org5ccd852">8.2. Optional Parameters</a></li>
<li><a href="#org8896a32">8.3. Theory</a></li>
<li><a href="#orgf5481aa">8.4. Compute</a></li>
</ul>
</li>
<li><a href="#org1c1fd0d">9. <code>forwardKinematicsApprox</code>: Compute the Forward Kinematics</a>
<ul>
<li><a href="#orgcc0b9b5">9.1. Function description</a></li>
<li><a href="#orgdbe04db">9.2. Optional Parameters</a></li>
<li><a href="#orgc0a97e0">9.3. Computation</a></li>
</ul>
</li>
</ul>
@ -390,8 +417,8 @@ For Simscape, we need:
<li>The position of the frame \(\{B\}\) with respect to the frame \(\{M\}\): \({}^{M}\bm{O}_{B}\)</li>
</ul>
<div id="outline-container-org363c6fe" class="outline-2">
<h2 id="org363c6fe"><span class="section-number-2">1</span> Procedure</h2>
<div id="outline-container-org3273efc" class="outline-2">
<h2 id="org3273efc"><span class="section-number-2">1</span> Procedure</h2>
<div class="outline-text-2" id="text-1">
<p>
The procedure to define the Stewart platform is the following:
@ -420,39 +447,42 @@ By following this procedure, we obtain a Matlab structure <code>stewart</code> t
</div>
</div>
<div id="outline-container-orgbb97f5b" class="outline-2">
<h2 id="orgbb97f5b"><span class="section-number-2">2</span> Matlab Code</h2>
<div id="outline-container-org0cfaf56" class="outline-2">
<h2 id="org0cfaf56"><span class="section-number-2">2</span> Matlab Code</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org26c49f2" class="outline-3">
<h3 id="org26c49f2"><span class="section-number-3">2.1</span> Simscape Model</h3>
<div id="outline-container-org47e6dec" class="outline-3">
<h3 id="org47e6dec"><span class="section-number-3">2.1</span> Simscape Model</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">open<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'stewart_platform.slx'</span><span class="org-rainbow-delimiters-depth-1">)</span>
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform.slx'</span>)
</pre>
</div>
</div>
</div>
<div id="outline-container-org0aee86d" class="outline-3">
<h3 id="org0aee86d"><span class="section-number-3">2.2</span> Test the functions</h3>
<div id="outline-container-org80710ea" class="outline-3">
<h3 id="org80710ea"><span class="section-number-3">2.2</span> Test the functions</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeFramesPositions<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 50e<span class="org-type">-</span>3<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
stewart = generateCubicConfiguration<span class="org-rainbow-delimiters-depth-1">(</span>stewart, struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'Hc'</span>, 60e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 15e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 15e<span class="org-type">-</span>3<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
stewart = computeJointsPose<span class="org-rainbow-delimiters-depth-1">(</span>stewart<span class="org-rainbow-delimiters-depth-1">)</span>;
stewart = initializeStrutDynamics<span class="org-rainbow-delimiters-depth-1">(</span>stewart, struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones<span class="org-rainbow-delimiters-depth-3">(</span>6,1<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones<span class="org-rainbow-delimiters-depth-3">(</span>6,1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 60e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 45e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 5e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 5e<span class="org-type">-</span>3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones(6,1));
stewart = computeJacobian(stewart);
[Li, dLi] = inverseKinematics(stewart, <span class="org-string">'AP'</span>, [0;0;0.00001], <span class="org-string">'ARB'</span>, eye(3));
[P, R] = forwardKinematicsApprox(stewart, <span class="org-string">'dL'</span>, dLi)
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org6b23b15" class="outline-2">
<h2 id="org6b23b15"><span class="section-number-2">3</span> <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</h2>
<div id="outline-container-org6005875" class="outline-2">
<h2 id="org6005875"><span class="section-number-2">3</span> <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org23d747c"></a>
<a id="org1dfe172"></a>
</p>
<p>
@ -460,18 +490,18 @@ This Matlab function is accessible <a href="src/initializeFramesPositions.m">her
</p>
</div>
<div id="outline-container-orgacce02d" class="outline-3">
<h3 id="orgacce02d"><span class="section-number-3">3.1</span> Function description</h3>
<div id="outline-container-orga3d2c9e" class="outline-3">
<h3 id="orga3d2c9e"><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"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-variable-name">stewart</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">initializeFramesPositions</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">opts_param</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">[stewart]</span> = <span class="org-function-name">initializeFramesPositions</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [stewart] = initializeFramesPositions(H, MO_B)</span>
<span class="org-comment">% Syntax: [stewart] = initializeFramesPositions(args)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - opts_param - Structure with the following fields:</span>
<span class="org-comment">% - H [1x1] - Total Height of the Stewart Platform [m]</span>
<span class="org-comment">% - args - Can have the following fields:</span>
<span class="org-comment">% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]</span>
<span class="org-comment">% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
@ -485,53 +515,50 @@ This Matlab function is accessible <a href="src/initializeFramesPositions.m">her
</div>
</div>
<div id="outline-container-orgbb19223" class="outline-3">
<h3 id="orgbb19223"><span class="section-number-3">3.2</span> Optional Parameters</h3>
<div id="outline-container-org61a1e26" class="outline-3">
<h3 id="org61a1e26"><span class="section-number-3">3.2</span> Documentation</h3>
<div class="outline-text-3" id="text-3-2">
<p>
Default values for opts.
</p>
<div class="org-src-container">
<pre class="src src-matlab">opts = struct<span class="org-rainbow-delimiters-depth-1">(</span> ...
<span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, ...<span class="org-comment"> % [m]</span>
<span class="org-string">'MO_B'</span>, 50e<span class="org-type">-</span>3 ...<span class="org-comment"> % [m]</span>
<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Populate opts with input parameters
<div id="org99358c5" class="figure">
<p><img src="figs/stewart-frames-position.png" alt="stewart-frames-position.png" />
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">if</span> exist<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'opts_param'</span>,<span class="org-string">'var'</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-keyword">for</span> <span class="org-variable-name">opt</span> = <span class="org-constant">fieldnames</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">opts_param</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-constant">'</span>
opts.<span class="org-rainbow-delimiters-depth-1">(</span>opt<span class="org-rainbow-delimiters-depth-2">{</span>1<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span> = opts_param.<span class="org-rainbow-delimiters-depth-1">(</span>opt<span class="org-rainbow-delimiters-depth-2">{</span>1<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
<p><span class="figure-number">Figure 1: </span>Definition of the position of the frames</p>
</div>
</div>
</div>
<div id="outline-container-org830651c" class="outline-3">
<h3 id="org830651c"><span class="section-number-3">3.3</span> Initialize the Stewart structure</h3>
<div id="outline-container-org713ea9a" class="outline-3">
<h3 id="org713ea9a"><span class="section-number-3">3.3</span> Optional Parameters</h3>
<div class="outline-text-3" id="text-3-3">
<div class="org-src-container">
<pre class="src src-matlab">stewart = struct<span class="org-rainbow-delimiters-depth-1">()</span>;
<pre class="src src-matlab">arguments
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e<span class="org-type">-</span>3
args.MO_B (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org302ef83" class="outline-3">
<h3 id="org302ef83"><span class="section-number-3">3.4</span> Compute the position of each frame</h3>
<div id="outline-container-org88e3f87" class="outline-3">
<h3 id="org88e3f87"><span class="section-number-3">3.4</span> Initialize the Stewart structure</h3>
<div class="outline-text-3" id="text-3-4">
<div class="org-src-container">
<pre class="src src-matlab">stewart.H = opts.H; <span class="org-comment">% Total Height of the Stewart Platform [m]</span>
<pre class="src src-matlab">stewart = struct();
</pre>
</div>
</div>
</div>
stewart.FO_M = <span class="org-rainbow-delimiters-depth-1">[</span>0; 0; stewart.H<span class="org-rainbow-delimiters-depth-1">]</span>; <span class="org-comment">% Position of {M} with respect to {F} [m]</span>
<div id="outline-container-org4a2edf9" class="outline-3">
<h3 id="org4a2edf9"><span class="section-number-3">3.5</span> Compute the position of each frame</h3>
<div class="outline-text-3" id="text-3-5">
<div class="org-src-container">
<pre class="src src-matlab">stewart.H = args.H; <span class="org-comment">% Total Height of the Stewart Platform [m]</span>
stewart.MO_B = <span class="org-rainbow-delimiters-depth-1">[</span>0; 0; opts.MO_B<span class="org-rainbow-delimiters-depth-1">]</span>; <span class="org-comment">% Position of {B} with respect to {M} [m]</span>
stewart.FO_M = [0; 0; stewart.H]; <span class="org-comment">% Position of {M} with respect to {F} [m]</span>
stewart.MO_B = [0; 0; args.MO_B]; <span class="org-comment">% Position of {B} with respect to {M} [m]</span>
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>
</pre>
@ -540,11 +567,11 @@ stewart.FO_A = stewart.MO_B <span class="org-type">+</span> stewart.FO_M; <span
</div>
</div>
<div id="outline-container-orgefabb7d" class="outline-2">
<h2 id="orgefabb7d"><span class="section-number-2">4</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h2>
<div id="outline-container-org8eb2e40" class="outline-2">
<h2 id="org8eb2e40"><span class="section-number-2">4</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org90fb821"></a>
<a id="org7944b80"></a>
</p>
<p>
@ -552,21 +579,21 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
</p>
</div>
<div id="outline-container-orgaa4ebd9" class="outline-3">
<h3 id="orgaa4ebd9"><span class="section-number-3">4.1</span> Function description</h3>
<div id="outline-container-org04bea4c" class="outline-3">
<h3 id="org04bea4c"><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"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-variable-name">stewart</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">generateCubicConfiguration</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">stewart</span>, <span class="org-variable-name">opts_param</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">[stewart]</span> = <span class="org-function-name">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(stewart, opts_param)</span>
<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(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">% - H [1x1] - Total height of the platform [m]</span>
<span class="org-comment">% - opts_param - Structure with the following fields:</span>
<span class="org-comment">% - args - Can have the following fields:</span>
<span class="org-comment">% - Hc [1x1] - Height of the "useful" part of the cube [m]</span>
<span class="org-comment">% - FOc [1x1] - Height of the center of the cute with respect to {F} [m]</span>
<span class="org-comment">% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]</span>
<span class="org-comment">% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]</span>
<span class="org-comment">% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]</span>
<span class="org-comment">%</span>
@ -579,69 +606,69 @@ This Matlab function is accessible <a href="src/generateCubicConfiguration.m">he
</div>
</div>
<div id="outline-container-org59cf450" class="outline-3">
<h3 id="org59cf450"><span class="section-number-3">4.2</span> Optional Parameters</h3>
<div id="outline-container-org924cbfe" class="outline-3">
<h3 id="org924cbfe"><span class="section-number-3">4.2</span> Documentation</h3>
<div class="outline-text-3" id="text-4-2">
<p>
Default values for opts.
</p>
<div class="org-src-container">
<pre class="src src-matlab">opts = struct<span class="org-rainbow-delimiters-depth-1">(</span> ...
<span class="org-string">'Hc'</span>, 60e<span class="org-type">-</span>3, ...<span class="org-comment"> % [m]</span>
<span class="org-string">'FOc'</span>, 50e<span class="org-type">-</span>3, ...<span class="org-comment"> % [m]</span>
<span class="org-string">'FHa'</span>, 15e<span class="org-type">-</span>3, ...<span class="org-comment"> % [m]</span>
<span class="org-string">'MHb'</span>, 15e<span class="org-type">-</span>3 ...<span class="org-comment"> % [m]</span>
<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Populate opts with input parameters
<div id="org9b7988f" class="figure">
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">if</span> exist<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'opts_param'</span>,<span class="org-string">'var'</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-keyword">for</span> <span class="org-variable-name">opt</span> = <span class="org-constant">fieldnames</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">opts_param</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-constant">'</span>
opts.<span class="org-rainbow-delimiters-depth-1">(</span>opt<span class="org-rainbow-delimiters-depth-2">{</span>1<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span> = opts_param.<span class="org-rainbow-delimiters-depth-1">(</span>opt<span class="org-rainbow-delimiters-depth-2">{</span>1<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
<p><span class="figure-number">Figure 2: </span>Cubic Configuration</p>
</div>
</div>
</div>
<div id="outline-container-orgc3b07d9" class="outline-3">
<h3 id="orgc3b07d9"><span class="section-number-3">4.3</span> Position of the Cube</h3>
<div id="outline-container-org335de0c" class="outline-3">
<h3 id="org335de0c"><span class="section-number-3">4.3</span> Optional Parameters</h3>
<div class="outline-text-3" id="text-4-3">
<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 = <span class="org-rainbow-delimiters-depth-1">[</span> 2; <span class="org-type">-</span>1; <span class="org-type">-</span>1<span class="org-rainbow-delimiters-depth-1">]</span>;
sy = <span class="org-rainbow-delimiters-depth-1">[</span> 0; 1; <span class="org-type">-</span>1<span class="org-rainbow-delimiters-depth-1">]</span>;
sz = <span class="org-rainbow-delimiters-depth-1">[</span> 1; 1; 1<span class="org-rainbow-delimiters-depth-1">]</span>;
R = <span class="org-rainbow-delimiters-depth-1">[</span>sx, sy, sz<span class="org-rainbow-delimiters-depth-1">]</span><span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>sx, sy, sz<span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
L = opts.Hc<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-1">(</span>3<span class="org-rainbow-delimiters-depth-1">)</span>;
Cc = R<span class="org-type">'*</span><span class="org-rainbow-delimiters-depth-1">[</span><span class="org-rainbow-delimiters-depth-2">[</span>0;0;L<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-rainbow-delimiters-depth-2">[</span>L;0;L<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-rainbow-delimiters-depth-2">[</span>L;0;0<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-rainbow-delimiters-depth-2">[</span>L;L;0<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-rainbow-delimiters-depth-2">[</span>0;L;0<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-rainbow-delimiters-depth-2">[</span>0;L;L<span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-type">-</span> <span class="org-rainbow-delimiters-depth-1">[</span>0;0;1.5<span class="org-type">*</span>opts.Hc<span class="org-rainbow-delimiters-depth-1">]</span>;
CCf = <span class="org-rainbow-delimiters-depth-1">[</span>Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,1<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,3<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,3<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,5<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,5<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,1<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>; <span class="org-comment">% CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg</span>
CCm = <span class="org-rainbow-delimiters-depth-1">[</span>Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,2<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,2<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,4<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,4<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,6<span class="org-rainbow-delimiters-depth-2">)</span>, Cc<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,6<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>; <span class="org-comment">% CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg</span>
<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-org4ab3f4c" class="outline-3">
<h3 id="org4ab3f4c"><span class="section-number-3">4.4</span> Compute the pose</h3>
<div id="outline-container-orgc1a1209" class="outline-3">
<h3 id="orgc1a1209"><span class="section-number-3">4.4</span> Position of the Cube</h3>
<div class="outline-text-3" id="text-4-4">
<p>
We define the useful points of the cube with respect to the Cube&rsquo;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-orga1cfc06" class="outline-3">
<h3 id="orga1cfc06"><span class="section-number-3">4.5</span> Compute the pose</h3>
<div class="outline-text-3" id="text-4-5">
<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 = <span class="org-rainbow-delimiters-depth-1">(</span>CCm <span class="org-type">-</span> CCf<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>CCm <span class="org-type">-</span> CCf<span class="org-rainbow-delimiters-depth-1">)</span>;
<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>
@ -649,19 +676,19 @@ We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector fr
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> <span class="org-rainbow-delimiters-depth-1">[</span>0; 0; opts.FOc<span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>opts.FHa<span class="org-type">-</span><span class="org-rainbow-delimiters-depth-3">(</span>opts.FOc<span class="org-type">-</span>opts.Hc<span class="org-type">/</span>2<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span>CSi<span class="org-rainbow-delimiters-depth-2">(</span>3,<span class="org-type">:</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.*</span>CSi;
stewart.Mb = CCf <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-1">[</span>0; 0; opts.FOc<span class="org-type">-</span>stewart.H<span class="org-rainbow-delimiters-depth-1">]</span> <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>stewart.H<span class="org-type">-</span>opts.MHb<span class="org-type">-</span><span class="org-rainbow-delimiters-depth-3">(</span>opts.FOc<span class="org-type">-</span>opts.Hc<span class="org-type">/</span>2<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span>CSi<span class="org-rainbow-delimiters-depth-2">(</span>3,<span class="org-type">:</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">.*</span>CSi;
<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-org1a92b53" class="outline-2">
<h2 id="org1a92b53"><span class="section-number-2">5</span> <code>computeJointsPose</code>: Compute the Pose of the Joints</h2>
<div id="outline-container-orgc137b22" class="outline-2">
<h2 id="orgc137b22"><span class="section-number-2">5</span> <code>computeJointsPose</code>: Compute the Pose of the Joints</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="orgcb2a95c"></a>
<a id="org566dd7e"></a>
</p>
<p>
@ -669,17 +696,19 @@ This Matlab function is accessible <a href="src/computeJointsPose.m">here</a>.
</p>
</div>
<div id="outline-container-org8d2b6ea" class="outline-3">
<h3 id="org8d2b6ea"><span class="section-number-3">5.1</span> Function description</h3>
<div id="outline-container-orge4dc5d9" class="outline-3">
<h3 id="orge4dc5d9"><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"><span class="org-rainbow-delimiters-depth-1">[</span></span><span class="org-variable-name">stewart</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">computeJointsPose</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">[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, opts_param)</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>
@ -700,50 +729,62 @@ This Matlab function is accessible <a href="src/computeJointsPose.m">here</a>.
</div>
</div>
<div id="outline-container-org663872a" class="outline-3">
<h3 id="org663872a"><span class="section-number-3">5.2</span> Compute the position of the Joints</h3>
<div id="outline-container-org5e72282" class="outline-3">
<h3 id="org5e72282"><span class="section-number-3">5.2</span> Documentation</h3>
<div class="outline-text-3" id="text-5-2">
<div class="org-src-container">
<pre class="src src-matlab">stewart.Aa = stewart.Fa <span class="org-type">-</span> repmat<span class="org-rainbow-delimiters-depth-1">(</span>stewart.FO_A, <span class="org-rainbow-delimiters-depth-2">[</span>1, 6<span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
stewart.Bb = stewart.Mb <span class="org-type">-</span> repmat<span class="org-rainbow-delimiters-depth-1">(</span>stewart.MO_B, <span class="org-rainbow-delimiters-depth-2">[</span>1, 6<span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
stewart.Ab = stewart.Bb <span class="org-type">-</span> repmat<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">-</span>stewart.MO_B<span class="org-type">-</span>stewart.FO_M<span class="org-type">+</span>stewart.FO_A, <span class="org-rainbow-delimiters-depth-2">[</span>1, 6<span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
stewart.Ba = stewart.Aa <span class="org-type">-</span> repmat<span class="org-rainbow-delimiters-depth-1">(</span> stewart.MO_B<span class="org-type">+</span>stewart.FO_M<span class="org-type">-</span>stewart.FO_A, <span class="org-rainbow-delimiters-depth-2">[</span>1, 6<span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
<div id="org8d57a7b" 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-org08b9460" class="outline-3">
<h3 id="org08b9460"><span class="section-number-3">5.3</span> Compute the strut length and orientation</h3>
<div id="outline-container-orgafa4ad9" class="outline-3">
<h3 id="orgafa4ad9"><span class="section-number-3">5.3</span> Compute the position of the Joints</h3>
<div class="outline-text-3" id="text-5-3">
<div class="org-src-container">
<pre class="src src-matlab">stewart.As = <span class="org-rainbow-delimiters-depth-1">(</span>stewart.Ab <span class="org-type">-</span> stewart.Aa<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>stewart.Ab <span class="org-type">-</span> stewart.Aa<span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% As_i is the i'th vector of As</span>
<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.l = vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>stewart.Ab <span class="org-type">-</span> stewart.Aa<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">stewart.Bs = <span class="org-rainbow-delimiters-depth-1">(</span>stewart.Bb <span class="org-type">-</span> stewart.Ba<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>stewart.Bb <span class="org-type">-</span> stewart.Ba<span class="org-rainbow-delimiters-depth-1">)</span>;
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-orgde1237d" class="outline-3">
<h3 id="orgde1237d"><span class="section-number-3">5.4</span> Compute the orientation of the Joints</h3>
<div id="outline-container-org50bd360" class="outline-3">
<h3 id="org50bd360"><span class="section-number-3">5.4</span> Compute the strut length and orientation</h3>
<div class="outline-text-3" id="text-5-4">
<div class="org-src-container">
<pre class="src src-matlab">stewart.FRa = zeros<span class="org-rainbow-delimiters-depth-1">(</span>3,3,6<span class="org-rainbow-delimiters-depth-1">)</span>;
stewart.MRb = zeros<span class="org-rainbow-delimiters-depth-1">(</span>3,3,6<span class="org-rainbow-delimiters-depth-1">)</span>;
<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-org3d0d76f" class="outline-3">
<h3 id="org3d0d76f"><span class="section-number-3">5.5</span> Compute the orientation of the Joints</h3>
<div class="outline-text-3" id="text-5-5">
<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-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span> = <span class="org-rainbow-delimiters-depth-1">[</span>cross<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">[</span>0;1;0<span class="org-rainbow-delimiters-depth-3">]</span>, stewart.As<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> , cross<span class="org-rainbow-delimiters-depth-2">(</span>stewart.As<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span>, cross<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span>0;1;0<span class="org-rainbow-delimiters-depth-4">]</span>, stewart.As<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> , stewart.As<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>;
stewart.FRa<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span> = stewart.FRa<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>stewart.FRa<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</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-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span> = <span class="org-rainbow-delimiters-depth-1">[</span>cross<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">[</span>0;1;0<span class="org-rainbow-delimiters-depth-3">]</span>, stewart.Bs<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> , cross<span class="org-rainbow-delimiters-depth-2">(</span>stewart.Bs<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span>, cross<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span>0;1;0<span class="org-rainbow-delimiters-depth-4">]</span>, stewart.Bs<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span> , stewart.Bs<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">]</span>;
stewart.MRb<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span> = stewart.MRb<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>stewart.MRb<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</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>
@ -751,11 +792,11 @@ stewart.MRb = zeros<span class="org-rainbow-delimiters-depth-1">(</span>3,3,6<sp
</div>
</div>
<div id="outline-container-org23b3bc2" class="outline-2">
<h2 id="org23b3bc2"><span class="section-number-2">6</span> <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</h2>
<div id="outline-container-orgc7ae05f" class="outline-2">
<h2 id="orgc7ae05f"><span class="section-number-2">6</span> <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="org15d65d8"></a>
<a id="orge0b89e3"></a>
</p>
<p>
@ -763,17 +804,17 @@ This Matlab function is accessible <a href="src/initializeStrutDynamics.m">here<
</p>
</div>
<div id="outline-container-orgfc3e940" class="outline-3">
<h3 id="orgfc3e940"><span class="section-number-3">6.1</span> Function description</h3>
<div id="outline-container-org234504f" class="outline-3">
<h3 id="org234504f"><span class="section-number-3">6.1</span> Function description</h3>
<div class="outline-text-3" id="text-6-1">
<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">stewart</span><span class="org-variable-name"><span class="org-rainbow-delimiters-depth-1">]</span></span> = <span class="org-function-name">initializeStrutDynamics</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">stewart</span>, <span class="org-variable-name">opts_param</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">[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(opts_param)</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">% - opts_param - Structure with the following fields:</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>
@ -786,40 +827,289 @@ This Matlab function is accessible <a href="src/initializeStrutDynamics.m">here<
</div>
</div>
<div id="outline-container-org33a48bd" class="outline-3">
<h3 id="org33a48bd"><span class="section-number-3">6.2</span> Optional Parameters</h3>
<div id="outline-container-org8edf5d0" class="outline-3">
<h3 id="org8edf5d0"><span class="section-number-3">6.2</span> Optional Parameters</h3>
<div class="outline-text-3" id="text-6-2">
<p>
Default values for opts.
</p>
<div class="org-src-container">
<pre class="src src-matlab">opts = struct<span class="org-rainbow-delimiters-depth-1">(</span> ...
<span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones<span class="org-rainbow-delimiters-depth-2">(</span>6,1<span class="org-rainbow-delimiters-depth-2">)</span>, ...<span class="org-comment"> % [N/m]</span>
<span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones<span class="org-rainbow-delimiters-depth-2">(</span>6,1<span class="org-rainbow-delimiters-depth-2">)</span> ...<span class="org-comment"> % [N/(m/s)]</span>
<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Populate opts with input parameters
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">if</span> exist<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'opts_param'</span>,<span class="org-string">'var'</span><span class="org-rainbow-delimiters-depth-1">)</span>
<span class="org-keyword">for</span> <span class="org-variable-name">opt</span> = <span class="org-constant">fieldnames</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">opts_param</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-constant">'</span>
opts.<span class="org-rainbow-delimiters-depth-1">(</span>opt<span class="org-rainbow-delimiters-depth-2">{</span>1<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span> = opts_param.<span class="org-rainbow-delimiters-depth-1">(</span>opt<span class="org-rainbow-delimiters-depth-2">{</span>1<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-keyword">end</span>
<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} = 1e2<span class="org-type">*</span>ones(6,1)
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org458bd27" class="outline-3">
<h3 id="org458bd27"><span class="section-number-3">6.3</span> Add Stiffness and Damping properties of each strut</h3>
<div id="outline-container-orgf9bb882" class="outline-3">
<h3 id="orgf9bb882"><span class="section-number-3">6.3</span> Add Stiffness and Damping properties of each strut</h3>
<div class="outline-text-3" id="text-6-3">
<div class="org-src-container">
<pre class="src src-matlab">stewart.Ki = opts.Ki;
stewart.Ci = opts.Ci;
<pre class="src src-matlab">stewart.Ki = args.Ki;
stewart.Ci = args.Ci;
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org61b514c" class="outline-2">
<h2 id="org61b514c"><span class="section-number-2">7</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h2>
<div class="outline-text-2" id="text-7">
<p>
<a id="org27d5ba7"></a>
</p>
<p>
This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
</p>
</div>
<div id="outline-container-org2983146" class="outline-3">
<h3 id="org2983146"><span class="section-number-3">7.1</span> Function description</h3>
<div class="outline-text-3" id="text-7-1">
<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-org996fbd8" class="outline-3">
<h3 id="org996fbd8"><span class="section-number-3">7.2</span> Compute Jacobian Matrix</h3>
<div class="outline-text-3" id="text-7-2">
<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-org9e42bbe" class="outline-3">
<h3 id="org9e42bbe"><span class="section-number-3">7.3</span> Compute Stiffness Matrix</h3>
<div class="outline-text-3" id="text-7-3">
<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-org36e44cd" class="outline-3">
<h3 id="org36e44cd"><span class="section-number-3">7.4</span> Compute Compliance Matrix</h3>
<div class="outline-text-3" id="text-7-4">
<div class="org-src-container">
<pre class="src src-matlab">stewart.C = inv(stewart.K);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org76be477" class="outline-2">
<h2 id="org76be477"><span class="section-number-2">8</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h2>
<div class="outline-text-2" id="text-8">
<p>
<a id="orgb8a3565"></a>
</p>
<p>
This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
</p>
</div>
<div id="outline-container-org16e3f5f" class="outline-3">
<h3 id="org16e3f5f"><span class="section-number-3">8.1</span> Function description</h3>
<div class="outline-text-3" id="text-8-1">
<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-org5ccd852" class="outline-3">
<h3 id="org5ccd852"><span class="section-number-3">8.2</span> Optional Parameters</h3>
<div class="outline-text-3" id="text-8-2">
<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-org8896a32" class="outline-3">
<h3 id="org8896a32"><span class="section-number-3">8.3</span> Theory</h3>
<div class="outline-text-3" id="text-8-3">
<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&rsquo; lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
</p>
</div>
</div>
<div id="outline-container-orgf5481aa" class="outline-3">
<h3 id="orgf5481aa"><span class="section-number-3">8.4</span> Compute</h3>
<div class="outline-text-3" id="text-8-4">
<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-org1c1fd0d" class="outline-2">
<h2 id="org1c1fd0d"><span class="section-number-2">9</span> <code>forwardKinematicsApprox</code>: Compute the Forward Kinematics</h2>
<div class="outline-text-2" id="text-9">
<p>
<a id="org55e7dad"></a>
</p>
<p>
This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here</a>.
</p>
</div>
<div id="outline-container-orgcc0b9b5" class="outline-3">
<h3 id="orgcc0b9b5"><span class="section-number-3">9.1</span> Function description</h3>
<div class="outline-text-3" id="text-9-1">
<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-orgdbe04db" class="outline-3">
<h3 id="orgdbe04db"><span class="section-number-3">9.2</span> Optional Parameters</h3>
<div class="outline-text-3" id="text-9-2">
<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-orgc0a97e0" class="outline-3">
<h3 id="orgc0a97e0"><span class="section-number-3">9.3</span> Computation</h3>
<div class="outline-text-3" id="text-9-3">
<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>
@ -827,9 +1117,8 @@ stewart.Ci = opts.Ci;
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-12-20 ven. 17:49</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-01-06 lun. 18:16</p>
</div>
</body>
</html>

214
simscape-model.org
View File

@ -100,10 +100,13 @@ By following this procedure, we obtain a Matlab structure =stewart= that contain
** Test the functions
#+begin_src matlab
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 50e-3);
stewart = generateCubicConfiguration(stewart, 'Hc', 60e-3, 'FOc', 50e-3, 'FHa', 15e-3, 'MHb', 15e-3);
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
stewart = generateCubicConfiguration(stewart, 'Hc', 60e-3, 'FOc', 45e-3, 'FHa', 5e-3, 'MHb', 5e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1));
stewart = computeJacobian(stewart);
[Li, dLi] = inverseKinematics(stewart, 'AP', [0;0;0.00001], 'ARB', eye(3));
[P, R] = forwardKinematicsApprox(stewart, 'dL', dLi)
#+end_src
* =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M}
@ -124,7 +127,7 @@ This Matlab function is accessible [[file:src/initializeFramesPositions.m][here]
%
% Inputs:
% - args - Can have the following fields:
% - H [1x1] - Total Height of the Stewart Platform [m]
% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
%
% Outputs:
@ -135,11 +138,17 @@ This Matlab function is accessible [[file:src/initializeFramesPositions.m][here]
% - FO_A [3x1] - Position of {A} with respect to {F} [m]
#+end_src
** Documentation
#+name: fig:stewart-frames-position
#+caption: Definition of the position of the frames
[[file:figs/stewart-frames-position.png]]
** Optional Parameters
#+begin_src matlab
arguments
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
args.MO_B (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.MO_B (1,1) double {mustBeNumeric} = 50e-3
end
#+end_src
@ -180,7 +189,7 @@ This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here
% - H [1x1] - Total height of the platform [m]
% - args - Can have the following fields:
% - Hc [1x1] - Height of the "useful" part of the cube [m]
% - FOc [1x1] - Height of the center of the cute with respect to {F} [m]
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
%
@ -190,12 +199,17 @@ This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
#+end_src
** Documentation
#+name: fig:cubic-configuration-definition
#+caption: Cubic Configuration
[[file:figs/cubic-configuration-definition.png]]
** Optional Parameters
#+begin_src matlab
arguments
stewart
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
args.FOc (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.FOc (1,1) double {mustBeNumeric} = 50e-3
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
end
@ -203,7 +217,9 @@ This Matlab function is accessible [[file:src/generateCubicConfiguration.m][here
** Position of the Cube
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}.
${}^{C}C$ 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}.
#+begin_src matlab
sx = [ 2; -1; -1];
sy = [ 0; 1; -1];
@ -245,10 +261,12 @@ This Matlab function is accessible [[file:src/computeJointsPose.m][here]].
function [stewart] = computeJointsPose(stewart)
% computeJointsPose -
%
% Syntax: [stewart] = computeJointsPose(stewart, opts_param)
% Syntax: [stewart] = computeJointsPose(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
% - FO_A [3x1] - Position of {A} with respect to {F}
% - MO_B [3x1] - Position of {B} with respect to {M}
% - FO_M [3x1] - Position of {M} with respect to {F}
@ -266,6 +284,11 @@ This Matlab function is accessible [[file:src/computeJointsPose.m][here]].
% - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
#+end_src
** Documentation
#+name: fig:stewart-struts
#+caption: Position and orientation of the struts
[[file:figs/stewart-struts.png]]
** Compute the position of the Joints
#+begin_src matlab
stewart.Aa = stewart.Fa - repmat(stewart.FO_A, [1, 6]);
@ -342,6 +365,181 @@ This Matlab function is accessible [[file:src/initializeStrutDynamics.m][here]].
stewart.Ci = args.Ci;
#+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
:END:
<<sec:inverseKinematics>>
This Matlab function is accessible [[file:src/inverseKinematics.m][here]].
** Function description
#+begin_src matlab
function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - Aa [3x6] - The positions ai expressed in {A}
% - Bb [3x6] - The positions bi expressed in {B}
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
% - 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}
#+end_src
** Optional Parameters
#+begin_src matlab
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
#+end_src
** Theory
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 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
* OLD :noexport:
** Define the Height of the Platform :noexport:
#+begin_src matlab

21
src/computeJacobian.m Normal file
View File

@ -0,0 +1,21 @@
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
stewart.J = [stewart.As' , cross(stewart.Ab, stewart.As)'];
stewart.K = stewart.J'*diag(stewart.Ki)*stewart.J;
stewart.C = inv(stewart.K);

30
src/forwardKinematics.m Normal file
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@ -0,0 +1,30 @@
function [P, R] = forwardKinematics(stewart, args)
% forwardKinematics - Computed the pose of {B} with respect to {A} from the length of each strut
%
% Syntax: [in_data] = forwardKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - J [6x6] - The Jacobian Matrix
% - args - Can have the following fields:
% - L [6x1] - Length 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}
arguments
stewart
args.L (6,1) double {mustBeNumeric} = zeros(6,1)
end
X = stewart.J\args.L;
P = X(1:3);
theta = norm(X(4:6));
s = X(4:6)/theta;
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)];

4
src/generateCubicConfiguration.m
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@ -8,7 +8,7 @@ function [stewart] = generateCubicConfiguration(stewart, args)
% - H [1x1] - Total height of the platform [m]
% - args - Can have the following fields:
% - Hc [1x1] - Height of the "useful" part of the cube [m]
% - FOc [1x1] - Height of the center of the cute with respect to {F} [m]
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
%
@ -20,7 +20,7 @@ function [stewart] = generateCubicConfiguration(stewart, args)
arguments
stewart
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
args.FOc (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.FOc (1,1) double {mustBeNumeric} = 50e-3
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
end

4
src/initializeFramesPositions.m
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@ -5,7 +5,7 @@ function [stewart] = initializeFramesPositions(args)
%
% Inputs:
% - args - Can have the following fields:
% - H [1x1] - Total Height of the Stewart Platform [m]
% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
%
% Outputs:
@ -17,7 +17,7 @@ function [stewart] = initializeFramesPositions(args)
arguments
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
args.MO_B (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.MO_B (1,1) double {mustBeNumeric} = 50e-3
end
stewart = struct();

26
src/inverseKinematics.m Normal file
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@ -0,0 +1,26 @@
function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - Aa [3x6] - The positions ai expressed in {A}
% - Bb [3x6] - The positions bi expressed in {B}
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
% - 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}
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
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));
dLi = Li-stewart.l;