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Thomas Dehaeze
2020-03-03 15:53:02 +01:00
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<!-- 2020-02-13 jeu. 16:46 -->
<!-- 2020-03-02 lun. 17:57 -->
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<title>Kinematic Study of the Stewart Platform</title>
@@ -201,50 +201,28 @@
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</li>
<li><a href="#org86b4b35">4. Estimation of the range validity of the approximate inverse kinematics</a>
<ul>
<li><a href="#org4c04fb5">4.1. Stewart architecture definition</a></li>
<li><a href="#org7423428">4.1. Stewart architecture definition</a></li>
<li><a href="#orgd83ccf3">4.2. Comparison for &ldquo;pure&rdquo; translations</a></li>
<li><a href="#org4871c83">4.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org63255f9">5. Estimated required actuator stroke from specified platform mobility</a>
<ul>
<li><a href="#org62824e9">5.1. Stewart architecture definition</a></li>
<li><a href="#org3e1d400">5.1. Stewart architecture definition</a></li>
<li><a href="#orgde50dd3">5.2. Wanted translations and rotations</a></li>
<li><a href="#org24e45ca">5.3. Needed stroke for &ldquo;pure&rdquo; rotations or translations</a></li>
<li><a href="#orgf6ba90c">5.4. Needed stroke for &ldquo;combined&rdquo; rotations or translations</a></li>
@@ -304,7 +282,7 @@ for the JavaScript code in this tag.
</li>
<li><a href="#orgbbbf7b3">6. Estimated platform mobility from specified actuator stroke</a>
<ul>
<li><a href="#org7423428">6.1. Stewart architecture definition</a></li>
<li><a href="#org53d6532">6.1. Stewart architecture definition</a></li>
<li><a href="#org2c6819e">6.2. Pure translations</a></li>
</ul>
</li>
@@ -312,8 +290,8 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#org26e8b28">7.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
<ul>
<li><a href="#orgd29f673">Function description</a></li>
<li><a href="#org4bcb03f">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orgd0d007d">Function description</a></li>
<li><a href="#orge1b5b04">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org0cd57b5">Compute Jacobian Matrix</a></li>
<li><a href="#orge21dcfc">Compute Stiffness Matrix</a></li>
<li><a href="#orgae76071">Compute Compliance Matrix</a></li>
@@ -323,17 +301,17 @@ for the JavaScript code in this tag.
<li><a href="#orgb82066f">7.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
<ul>
<li><a href="#org89930b7">Theory</a></li>
<li><a href="#org9c9b2ba">Function description</a></li>
<li><a href="#orgd8e2476">Optional Parameters</a></li>
<li><a href="#org8d85541">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org755b2ae">Function description</a></li>
<li><a href="#org867b3a0">Optional Parameters</a></li>
<li><a href="#org318eb5f">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org0d64c23">Compute</a></li>
</ul>
</li>
<li><a href="#orgf5d8f0b">7.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
<ul>
<li><a href="#orgd0d007d">Function description</a></li>
<li><a href="#org867b3a0">Optional Parameters</a></li>
<li><a href="#orge1b5b04">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orgba3bc64">Function description</a></li>
<li><a href="#org7af7974">Optional Parameters</a></li>
<li><a href="#org2ba5e64">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orge5ade24">Computation</a></li>
</ul>
</li>
@@ -679,8 +657,8 @@ This will also gives us the range for which the approximate forward kinematic is
</p>
</div>
<div id="outline-container-org4c04fb5" class="outline-3">
<h3 id="org4c04fb5"><span class="section-number-3">4.1</span> Stewart architecture definition</h3>
<div id="outline-container-org7423428" class="outline-3">
<h3 id="org7423428"><span class="section-number-3">4.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-4-1">
<p>
We first define some general Stewart architecture.
@@ -780,8 +758,8 @@ This is what is analyzed in this section.
</p>
</div>
<div id="outline-container-org62824e9" class="outline-3">
<h3 id="org62824e9"><span class="section-number-3">5.1</span> Stewart architecture definition</h3>
<div id="outline-container-org3e1d400" class="outline-3">
<h3 id="org3e1d400"><span class="section-number-3">5.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-5-1">
<p>
Let&rsquo;s first define the Stewart platform architecture that we want to study.
@@ -1200,8 +1178,8 @@ However, for small displacements, we can use the Jacobian as an approximate solu
</p>
</div>
<div id="outline-container-org7423428" class="outline-3">
<h3 id="org7423428"><span class="section-number-3">6.1</span> Stewart architecture definition</h3>
<div id="outline-container-org53d6532" class="outline-3">
<h3 id="org53d6532"><span class="section-number-3">6.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-6-1">
<p>
Let&rsquo;s first define the Stewart platform architecture that we want to study.
@@ -1332,9 +1310,9 @@ This Matlab function is accessible <a href="../src/computeJacobian.m">here</a>.
</p>
</div>
<div id="outline-container-orgd29f673" class="outline-4">
<h4 id="orgd29f673">Function description</h4>
<div class="outline-text-4" id="text-orgd29f673">
<div id="outline-container-orgd0d007d" class="outline-4">
<h4 id="orgd0d007d">Function description</h4>
<div class="outline-text-4" id="text-orgd0d007d">
<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>
@@ -1357,9 +1335,9 @@ This Matlab function is accessible <a href="../src/computeJacobian.m">here</a>.
</div>
</div>
<div id="outline-container-org4bcb03f" class="outline-4">
<h4 id="org4bcb03f">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org4bcb03f">
<div id="outline-container-orge1b5b04" class="outline-4">
<h4 id="orge1b5b04">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-orge1b5b04">
<div class="org-src-container">
<pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'As'</span>), <span class="org-string">'stewart.geometry should have attribute As'</span>)
As = stewart.geometry.As;
@@ -1467,9 +1445,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div>
</div>
<div id="outline-container-org9c9b2ba" class="outline-4">
<h4 id="org9c9b2ba">Function description</h4>
<div class="outline-text-4" id="text-org9c9b2ba">
<div id="outline-container-org755b2ae" class="outline-4">
<h4 id="org755b2ae">Function description</h4>
<div class="outline-text-4" id="text-org755b2ae">
<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>
@@ -1493,9 +1471,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div>
</div>
<div id="outline-container-orgd8e2476" class="outline-4">
<h4 id="orgd8e2476">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgd8e2476">
<div id="outline-container-org867b3a0" class="outline-4">
<h4 id="org867b3a0">Optional Parameters</h4>
<div class="outline-text-4" id="text-org867b3a0">
<div class="org-src-container">
<pre class="src src-matlab">arguments
stewart
@@ -1507,9 +1485,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div>
</div>
<div id="outline-container-org8d85541" class="outline-4">
<h4 id="org8d85541">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org8d85541">
<div id="outline-container-org318eb5f" class="outline-4">
<h4 id="org318eb5f">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org318eb5f">
<div class="org-src-container">
<pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'Aa'</span>), <span class="org-string">'stewart.geometry should have attribute Aa'</span>)
Aa = stewart.geometry.Aa;
@@ -1553,9 +1531,9 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</p>
</div>
<div id="outline-container-orgd0d007d" class="outline-4">
<h4 id="orgd0d007d">Function description</h4>
<div class="outline-text-4" id="text-orgd0d007d">
<div id="outline-container-orgba3bc64" class="outline-4">
<h4 id="orgba3bc64">Function description</h4>
<div class="outline-text-4" id="text-orgba3bc64">
<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>
@@ -1577,9 +1555,9 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</div>
</div>
<div id="outline-container-org867b3a0" class="outline-4">
<h4 id="org867b3a0">Optional Parameters</h4>
<div class="outline-text-4" id="text-org867b3a0">
<div id="outline-container-org7af7974" class="outline-4">
<h4 id="org7af7974">Optional Parameters</h4>
<div class="outline-text-4" id="text-org7af7974">
<div class="org-src-container">
<pre class="src src-matlab">arguments
stewart
@@ -1590,9 +1568,9 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</div>
</div>
<div id="outline-container-orge1b5b04" class="outline-4">
<h4 id="orge1b5b04">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-orge1b5b04">
<div id="outline-container-org2ba5e64" class="outline-4">
<h4 id="org2ba5e64">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org2ba5e64">
<div class="org-src-container">
<pre class="src src-matlab">assert(isfield(stewart.kinematics, <span class="org-string">'J'</span>), <span class="org-string">'stewart.kinematics should have attribute J'</span>)
J = stewart.kinematics.J;
@@ -1655,7 +1633,7 @@ We then compute the corresponding rotation matrix.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-13 jeu. 16:46</p>
<p class="date">Created: 2020-03-02 lun. 17:57</p>
</div>
</body>
</html>