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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-01-08 ven. 15:30 -->
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<!-- 2021-01-08 ven. 15:52 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Cubic configuration for the Stewart Platform</title>
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<meta name="generator" content="Org mode" />
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@@ -45,34 +45,34 @@
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<li><a href="#org6359f2f">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
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<li><a href="#org5c37be2">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
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<li><a href="#org32ac59a">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
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<li><a href="#org4e88cdb">1.5. Conclusion</a></li>
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<li><a href="#orgeb8ae82">1.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org312b7d4">2. Configuration with the Cube’s center above the mobile platform</a>
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<ul>
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<li><a href="#org4983654">2.1. Having Cube’s center above the top platform</a></li>
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<li><a href="#orge53b7f6">2.2. Size of the platforms</a></li>
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<li><a href="#org561a2bc">2.3. Conclusion</a></li>
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<li><a href="#org52825e8">2.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org2387b96">3. Cubic size analysis</a>
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<ul>
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||||
<li><a href="#org3647f9f">3.1. Analysis</a></li>
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<li><a href="#org948a425">3.2. Conclusion</a></li>
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<li><a href="#org701701b">3.2. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org174af3a">4. Dynamic Coupling in the Cartesian Frame</a>
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<ul>
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<li><a href="#orgdb33aa6">4.1. Cube’s center at the Center of Mass of the mobile platform</a></li>
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<li><a href="#org49b330b">4.2. Cube’s center not coincident with the Mass of the Mobile platform</a></li>
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<li><a href="#org7d50eae">4.3. Conclusion</a></li>
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<li><a href="#orgf407e4d">4.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org7831cff">5. Dynamic Coupling between actuators and sensors of each strut</a>
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<ul>
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<li><a href="#org38e9e8f">5.1. Coupling between the actuators and sensors - Cubic Architecture</a></li>
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<li><a href="#org21d40d3">5.2. Coupling between the actuators and sensors - Non-Cubic Architecture</a></li>
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<li><a href="#orgeb8ae82">5.3. Conclusion</a></li>
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<li><a href="#org0348380">5.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org3ce1c89">6. Functions</a>
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@@ -128,7 +128,7 @@ In this document, the cubic architecture is analyzed:
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<a id="org6bc5f56"></a>
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</p>
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<div class="note" id="org6a03293">
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||||
<div class="note" id="org783c5d6">
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||||
<p>
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||||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_stiffnessl.m">here</a>.
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||||
</p>
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||||
@@ -182,21 +182,21 @@ The Jacobian matrix is estimated at the location of the center of the cube.
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</p>
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||||
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab"> H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
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||||
MO_B = <span class="org-type">-</span>H<span class="org-type">/</span>2; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
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||||
Hc = H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>H<span class="org-type">/</span>2; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
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||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
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||||
</pre>
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||||
</div>
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||||
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
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||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
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||||
stewart = computeJointsPose(stewart);
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||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
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||||
stewart = computeJacobian(stewart);
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||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
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||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
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||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
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||||
stewart = computeJointsPose(stewart);
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||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
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||||
stewart = computeJacobian(stewart);
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||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
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||||
</pre>
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||||
</div>
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||||
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@@ -291,21 +291,21 @@ The Jacobian matrix is not estimated at the location of the center of the cube.
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||||
</p>
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||||
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||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = 20e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H<span class="org-type">/</span>2; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = 20e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H<span class="org-type">/</span>2; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
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||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
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||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
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||||
stewart = computeJointsPose(stewart);
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||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
||||
</pre>
|
||||
</div>
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||||
|
||||
@@ -400,21 +400,21 @@ The Jacobian is estimated at the cube center.
|
||||
</p>
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||||
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||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 80e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>30e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = 100e<span class="org-type">-</span>3; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">H = 80e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>30e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = 100e<span class="org-type">-</span>3; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 175e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 150e<span class="org-type">-</span>3);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -520,21 +520,21 @@ The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>H<span class="org-type">/</span>2; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = 1.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H<span class="org-type">/</span>2 <span class="org-type">+</span> 10e<span class="org-type">-</span>3; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>H<span class="org-type">/</span>2; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = 1.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H<span class="org-type">/</span>2 <span class="org-type">+</span> 10e<span class="org-type">-</span>3; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 215e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 195e<span class="org-type">-</span>3);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 0, <span class="org-string">'MHb'</span>, 0);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, ones(6,1));
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 215e<span class="org-type">-</span>3, <span class="org-string">'Mpr'</span>, 195e<span class="org-type">-</span>3);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -620,10 +620,10 @@ The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4e88cdb" class="outline-3">
|
||||
<h3 id="org4e88cdb"><span class="section-number-3">1.5</span> Conclusion</h3>
|
||||
<div id="outline-container-orgeb8ae82" class="outline-3">
|
||||
<h3 id="orgeb8ae82"><span class="section-number-3">1.5</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<div class="important" id="org2fc62c4">
|
||||
<div class="important" id="orgb449c4a">
|
||||
<p>
|
||||
Here are the conclusion about the Stiffness matrix for the Cubic configuration:
|
||||
</p>
|
||||
@@ -644,7 +644,7 @@ Here are the conclusion about the Stiffness matrix for the Cubic configuration:
|
||||
<a id="org419cdb0"></a>
|
||||
</p>
|
||||
|
||||
<div class="note" id="orgd5c0d4d">
|
||||
<div class="note" id="orge405fbc">
|
||||
<p>
|
||||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_above_platforml.m">here</a>.
|
||||
</p>
|
||||
@@ -676,8 +676,8 @@ Thus, we want the cube’s center to be located above the top center.
|
||||
Let’s fix the Height of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = 20e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = 20e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -697,8 +697,8 @@ However, the rotational stiffnesses are increasing with the cube’s size bu
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Hc = 0.4<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">Hc = 0.4<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -783,8 +783,8 @@ However, the rotational stiffnesses are increasing with the cube’s size bu
|
||||
</table>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Hc = 1.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">Hc = 1.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -870,8 +870,8 @@ However, the rotational stiffnesses are increasing with the cube’s size bu
|
||||
</table>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1049,10 +1049,10 @@ For a small cube:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org561a2bc" class="outline-3">
|
||||
<h3 id="org561a2bc"><span class="section-number-3">2.3</span> Conclusion</h3>
|
||||
<div id="outline-container-org52825e8" class="outline-3">
|
||||
<h3 id="org52825e8"><span class="section-number-3">2.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<div class="important" id="orga24e443">
|
||||
<div class="important" id="orgc3fb4db">
|
||||
<p>
|
||||
We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform.
|
||||
Depending on the cube’s size, we obtain 3 different configurations.
|
||||
@@ -1102,7 +1102,7 @@ Depending on the cube’s size, we obtain 3 different configurations.
|
||||
<a id="org53ade24"></a>
|
||||
</p>
|
||||
|
||||
<div class="note" id="orgfd32e5a">
|
||||
<div class="note" id="org6ff8a60">
|
||||
<p>
|
||||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_size_analysisl.m">here</a>.
|
||||
</p>
|
||||
@@ -1132,8 +1132,8 @@ We only vary the size of the cube.
|
||||
We initialize the wanted cube’s size.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Hcs = 1e<span class="org-type">-</span>3<span class="org-type">*</span>[250<span class="org-type">:</span>20<span class="org-type">:</span>350]; <span class="org-comment">% Heights for the Cube [m]</span>
|
||||
Ks = zeros(6, 6, length(Hcs));
|
||||
<pre class="src src-matlab">Hcs = 1e<span class="org-type">-</span>3<span class="org-type">*</span>[250<span class="org-type">:</span>20<span class="org-type">:</span>350]; <span class="org-comment">% Heights for the Cube [m]</span>
|
||||
Ks = zeros(6, 6, length(Hcs));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1141,7 +1141,7 @@ We initialize the wanted cube’s size.
|
||||
The height of the Stewart platform is fixed:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
<pre class="src src-matlab">H = 100e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1149,8 +1149,8 @@ The height of the Stewart platform is fixed:
|
||||
The frames \(\{A\}\) and \(\{B\}\) are positioned at the Stewart platform center as well as the cube’s center:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> MO_B = <span class="org-type">-</span>50e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">MO_B = <span class="org-type">-</span>50e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1168,14 +1168,14 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org948a425" class="outline-3">
|
||||
<h3 id="org948a425"><span class="section-number-3">3.2</span> Conclusion</h3>
|
||||
<div id="outline-container-org701701b" class="outline-3">
|
||||
<h3 id="org701701b"><span class="section-number-3">3.2</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
|
||||
</p>
|
||||
|
||||
<div class="important" id="org60cc507">
|
||||
<div class="important" id="org93b8347">
|
||||
<p>
|
||||
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
|
||||
</p>
|
||||
@@ -1192,7 +1192,7 @@ In order to maximize the rotational stiffness of the Stewart platform, the size
|
||||
<a id="org3507b2b"></a>
|
||||
</p>
|
||||
|
||||
<div class="note" id="org5934a9c">
|
||||
<div class="note" id="org265afc7">
|
||||
<p>
|
||||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_coupling_cartesianl.m">here</a>.
|
||||
</p>
|
||||
@@ -1256,8 +1256,8 @@ Let’s create a Cubic Stewart Platform where the <b>Center of Mass of the m
|
||||
We define the size of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1265,20 +1265,20 @@ We define the size of the Stewart platform and the position of frames \(\{A\}\)
|
||||
Now, we set the cube’s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1286,10 +1286,10 @@ Now, we set the cube’s parameters such that the center of the cube is coin
|
||||
Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with \(\{A\}\) and \(\{B\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
<pre class="src src-matlab">stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1297,8 +1297,8 @@ Now we set the geometry and mass of the mobile platform such that its center of
|
||||
And we set small mass for the struts.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
<pre class="src src-matlab">stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1306,9 +1306,9 @@ And we set small mass for the struts.
|
||||
No flexibility below the Stewart platform and no payload.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1327,24 +1327,24 @@ The obtain geometry is shown in figure <a href="#orgb6b060a">10</a>.
|
||||
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> open(<span class="org-string">'stewart_platform_model.slx'</span>)
|
||||
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform_model.slx'</span>)
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1352,10 +1352,10 @@ We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to
|
||||
Now, thanks to the Jacobian (Figure <a href="#org2137f5a">9</a>), we compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
||||
Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>stewart.kinematics.J;
|
||||
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
||||
<pre class="src src-matlab">Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
||||
Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>stewart.kinematics.J;
|
||||
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1381,7 +1381,7 @@ It is interesting to note here that the system shown in Figure <a href="#org9d84
|
||||
<p><span class="figure-number">Figure 12: </span>Alternative way to decouple the system</p>
|
||||
</div>
|
||||
|
||||
<div class="important" id="org489ed3b">
|
||||
<div class="important" id="orgd31482e">
|
||||
<p>
|
||||
The dynamics is well decoupled at all frequencies.
|
||||
</p>
|
||||
@@ -1413,8 +1413,8 @@ This is because the Mass, Damping and Stiffness matrices are all diagonal.
|
||||
Let’s create a Stewart platform with a cubic architecture where the cube’s center is at the center of the Stewart platform.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>100e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>100e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1422,20 +1422,20 @@ Let’s create a Stewart platform with a cubic architecture where the cube&r
|
||||
Now, we set the cube’s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1443,10 +1443,10 @@ Now, we set the cube’s parameters such that the center of the cube is coin
|
||||
However, the Center of Mass of the mobile platform is <b>not</b> located at the cube’s center.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
<pre class="src src-matlab">stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1454,8 +1454,8 @@ However, the Center of Mass of the mobile platform is <b>not</b> located at the
|
||||
And we set small mass for the struts.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
<pre class="src src-matlab">stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1463,9 +1463,9 @@ And we set small mass for the struts.
|
||||
No flexibility below the Stewart platform and no payload.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1483,24 +1483,24 @@ The obtain geometry is shown in figure <a href="#orgc57dcd2">13</a>.
|
||||
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> open(<span class="org-string">'stewart_platform_model.slx'</span>)
|
||||
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform_model.slx'</span>)
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1508,9 +1508,9 @@ We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to
|
||||
And we use the Jacobian to compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
||||
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
||||
<pre class="src src-matlab">Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
|
||||
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1525,7 +1525,7 @@ The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is sho
|
||||
<p><span class="figure-number">Figure 14: </span>Obtained Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (<a href="./figs/stewart_conf_coupling_mass_matrix.png">png</a>, <a href="./figs/stewart_conf_coupling_mass_matrix.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<div class="important" id="org798e5c6">
|
||||
<div class="important" id="orgc60cb20">
|
||||
<p>
|
||||
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is <b>not</b> decoupled at all frequencies.
|
||||
</p>
|
||||
@@ -1538,10 +1538,10 @@ This was expected as the mass matrix is not diagonal (the Center of Mass of the
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7d50eae" class="outline-3">
|
||||
<h3 id="org7d50eae"><span class="section-number-3">4.3</span> Conclusion</h3>
|
||||
<div id="outline-container-orgf407e4d" class="outline-3">
|
||||
<h3 id="orgf407e4d"><span class="section-number-3">4.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div class="important" id="org9b1be89">
|
||||
<div class="important" id="org982344b">
|
||||
<p>
|
||||
Some conclusions can be drawn from the above analysis:
|
||||
</p>
|
||||
@@ -1562,7 +1562,7 @@ Some conclusions can be drawn from the above analysis:
|
||||
<a id="org7b3ed31"></a>
|
||||
</p>
|
||||
|
||||
<div class="note" id="org7a9b096">
|
||||
<div class="note" id="org96fba24">
|
||||
<p>
|
||||
The Matlab script corresponding to this section is accessible <a href="../matlab/cubic_conf_coupling_strutsl.m">here</a>.
|
||||
</p>
|
||||
@@ -1593,28 +1593,28 @@ Let’s generate a Cubic architecture where the cube’s center and the
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
|
||||
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1622,15 +1622,15 @@ Let’s generate a Cubic architecture where the cube’s center and the
|
||||
No flexibility below the Stewart platform and no payload.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> disturbances = initializeDisturbances();
|
||||
references = initializeReferences(stewart);
|
||||
<pre class="src src-matlab">disturbances = initializeDisturbances();
|
||||
references = initializeReferences(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1669,26 +1669,26 @@ Now we generate a Stewart platform which is not cubic but with approximately the
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
|
||||
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateGeneralConfiguration(stewart, <span class="org-string">'FR'</span>, 250e<span class="org-type">-</span>3, <span class="org-string">'MR'</span>, 150e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
|
||||
stewart = generateGeneralConfiguration(stewart, <span class="org-string">'FR'</span>, 250e<span class="org-type">-</span>3, <span class="org-string">'MR'</span>, 150e<span class="org-type">-</span>3);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
|
||||
<span class="org-string">'Mpm'</span>, 10, ...
|
||||
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
|
||||
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
|
||||
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
|
||||
stewart = initializeInertialSensor(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1696,9 +1696,9 @@ Now we generate a Stewart platform which is not cubic but with approximately the
|
||||
No flexibility below the Stewart platform and no payload.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -1729,10 +1729,10 @@ And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relati
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgeb8ae82" class="outline-3">
|
||||
<h3 id="orgeb8ae82"><span class="section-number-3">5.3</span> Conclusion</h3>
|
||||
<div id="outline-container-org0348380" class="outline-3">
|
||||
<h3 id="org0348380"><span class="section-number-3">5.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<div class="important" id="org74729c6">
|
||||
<div class="important" id="orgd92f0ac">
|
||||
<p>
|
||||
The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control.
|
||||
</p>
|
||||
@@ -1766,24 +1766,24 @@ This Matlab function is accessible <a href="../src/generateCubicConfiguration.m"
|
||||
<h4 id="org2cafc68">Function description</h4>
|
||||
<div class="outline-text-4" id="text-org2cafc68">
|
||||
<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">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, 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">% - geometry.H [1x1] - Total height of the platform [m]</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 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>
|
||||
<span class="org-comment">% Outputs:</span>
|
||||
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
||||
<span class="org-comment">% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
||||
<span class="org-comment">% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</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, 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">% - geometry.H [1x1] - Total height of the platform [m]</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 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>
|
||||
<span class="org-comment">% Outputs:</span>
|
||||
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
||||
<span class="org-comment">% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
||||
<span class="org-comment">% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1805,13 +1805,13 @@ This Matlab function is accessible <a href="../src/generateCubicConfiguration.m"
|
||||
<h4 id="org4fd2c96">Optional Parameters</h4>
|
||||
<div class="outline-text-4" id="text-org4fd2c96">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> <span class="org-keyword">arguments</span>
|
||||
<span class="org-variable-name">stewart</span>
|
||||
<span class="org-variable-name">args</span>.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e<span class="org-type">-</span>3
|
||||
<span class="org-variable-name">args</span>.FOc (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
||||
<span class="org-variable-name">args</span>.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e<span class="org-type">-</span>3
|
||||
<span class="org-variable-name">args</span>.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e<span class="org-type">-</span>3
|
||||
<span class="org-keyword">end</span>
|
||||
<pre class="src src-matlab"><span class="org-keyword">arguments</span>
|
||||
<span class="org-variable-name">stewart</span>
|
||||
<span class="org-variable-name">args</span>.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e<span class="org-type">-</span>3
|
||||
<span class="org-variable-name">args</span>.FOc (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
||||
<span class="org-variable-name">args</span>.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e<span class="org-type">-</span>3
|
||||
<span class="org-variable-name">args</span>.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e<span class="org-type">-</span>3
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1821,8 +1821,8 @@ This Matlab function is accessible <a href="../src/generateCubicConfiguration.m"
|
||||
<h4 id="orgac26a8b">Check the <code>stewart</code> structure elements</h4>
|
||||
<div class="outline-text-4" id="text-orgac26a8b">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> assert(isfield(stewart.geometry, <span class="org-string">'H'</span>), <span class="org-string">'stewart.geometry should have attribute H'</span>)
|
||||
H = stewart.geometry.H;
|
||||
<pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'H'</span>), <span class="org-string">'stewart.geometry should have attribute H'</span>)
|
||||
H = stewart.geometry.H;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1837,18 +1837,18 @@ We define the useful points of the cube with respect to the Cube’s center.
|
||||
</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];
|
||||
<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]);
|
||||
R = [sx, sy, sz]<span class="org-type">./</span>vecnorm([sx, sy, sz]);
|
||||
|
||||
L = args.Hc<span class="org-type">*</span>sqrt(3);
|
||||
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];
|
||||
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>
|
||||
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>
|
||||
@@ -1861,7 +1861,7 @@ We define the useful points of the cube with respect to the Cube’s center.
|
||||
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> CSi = (CCm <span class="org-type">-</span> CCf)<span class="org-type">./</span>vecnorm(CCm <span class="org-type">-</span> CCf);
|
||||
<pre 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>
|
||||
|
||||
@@ -1869,8 +1869,8 @@ 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"> 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;
|
||||
Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>H] <span class="org-type">+</span> ((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 class="src src-matlab">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;
|
||||
Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>H] <span class="org-type">+</span> ((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>
|
||||
@@ -1880,8 +1880,8 @@ We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
|
||||
<h4 id="org153763b">Populate the <code>stewart</code> structure</h4>
|
||||
<div class="outline-text-4" id="text-org153763b">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"> stewart.platform_F.Fa = Fa;
|
||||
stewart.platform_M.Mb = Mb;
|
||||
<pre class="src src-matlab">stewart.platform_F.Fa = Fa;
|
||||
stewart.platform_M.Mb = Mb;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1905,7 +1905,7 @@ We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2021-01-08 ven. 15:30</p>
|
||||
<p class="date">Created: 2021-01-08 ven. 15:52</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user