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Estimate Stewart actuator stroke / Stewart maneuverability relationship

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<title>Cubic configuration for the Stewart Platform</title> <title>Cubic configuration for the Stewart Platform</title>
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<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgec4f5e2">1. Questions we wish to answer with this analysis</a></li> <li><a href="#org43e3f4a">1. Questions we wish to answer with this analysis</a></li>
<li><a href="#orgef11581">2. Configuration Analysis - Stiffness Matrix</a> <li><a href="#org9890470">2. Configuration Analysis - Stiffness Matrix</a>
<ul> <ul>
<li><a href="#org4203cad">2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li> <li><a href="#org5d80bd3">2.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
<li><a href="#org3344772">2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li> <li><a href="#orga5ac347">2.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
<li><a href="#org52de20d">2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li> <li><a href="#org47743ef">2.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
<li><a href="#orgd7e1449">2.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li> <li><a href="#orgd0daf23">2.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
<li><a href="#orgf16b788">2.5. Conclusion</a></li> <li><a href="#org6729b53">2.5. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orga2cd408">3. Cubic size analysis</a></li> <li><a href="#org0969b06">3. Cubic size analysis</a></li>
<li><a href="#org9220275">4. initializeCubicConfiguration</a> <li><a href="#org0f56064">4. initializeCubicConfiguration</a>
<ul> <ul>
<li><a href="#orgdee5436">4.1. Function description</a></li> <li><a href="#orge3f31b8">4.1. Function description</a></li>
<li><a href="#org68794ca">4.2. Optional Parameters</a></li> <li><a href="#org2580fde">4.2. Optional Parameters</a></li>
<li><a href="#org93d8028">4.3. Cube Creation</a></li> <li><a href="#org230a253">4.3. Cube Creation</a></li>
<li><a href="#org00e16e1">4.4. Vectors of each leg</a></li> <li><a href="#orgd1a04ef">4.4. Vectors of each leg</a></li>
<li><a href="#orgd7df0cc">4.5. Verification of Height of the Stewart Platform</a></li> <li><a href="#orgfec0778">4.5. Verification of Height of the Stewart Platform</a></li>
<li><a href="#orgc4b765a">4.6. Determinate the location of the joints</a></li> <li><a href="#orgc3cd1d1">4.6. Determinate the location of the joints</a></li>
<li><a href="#org0a84b4d">4.7. Returns Stewart Structure</a></li> <li><a href="#org9063896">4.7. Returns Stewart Structure</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgcb21f88">5. Tests</a> <li><a href="#org5807740">5. Tests</a>
<ul> <ul>
<li><a href="#orgc47f87d">5.1. First attempt to parametrisation</a></li> <li><a href="#org8809c47">5.1. First attempt to parametrisation</a></li>
<li><a href="#orgff4f69c">5.2. Second attempt</a></li> <li><a href="#org71c64bc">5.2. Second attempt</a></li>
<li><a href="#org011ab94">5.3. Generate the Stewart platform for a Cubic configuration</a></li> <li><a href="#orgc132128">5.3. Generate the Stewart platform for a Cubic configuration</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
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</div> </div>
<p> <p>
The discovery of the Cubic configuration is done in. 1 The discovery of the Cubic configuration is done in <a class='org-ref-reference' href="#geng94_six_degree_of_freed_activ">geng94_six_degree_of_freed_activ</a>.
Further analysis is conducted in <a class='org-ref-reference' href="#jafari03_orthog_gough_stewar_platf_microm">jafari03_orthog_gough_stewar_platf_microm</a>.
</p>
<p>
People using orthogonal/cubic configuration: <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>.
</p>
<p>
The specificity of the Cubic configuration is that each actuator is orthogonal with the others. The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
</p> </p>
<p> <p>
To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#org8876664">4</a>). To generate and study the Cubic configuration, <code>initializeCubicConfiguration</code> is used (description in section <a href="#org7e73a4b">4</a>).
</p> </p>
<div id="outline-container-orgec4f5e2" class="outline-2"> <div id="outline-container-org43e3f4a" class="outline-2">
<h2 id="orgec4f5e2"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2> <h2 id="org43e3f4a"><span class="section-number-2">1</span> Questions we wish to answer with this analysis</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
The goal is to study the benefits of using a cubic configuration: The goal is to study the benefits of using a cubic configuration:
@ -331,20 +340,20 @@ The goal is to study the benefits of using a cubic configuration:
</div> </div>
</div> </div>
<div id="outline-container-orgef11581" class="outline-2"> <div id="outline-container-org9890470" class="outline-2">
<h2 id="orgef11581"><span class="section-number-2">2</span> Configuration Analysis - Stiffness Matrix</h2> <h2 id="org9890470"><span class="section-number-2">2</span> Configuration Analysis - Stiffness Matrix</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
</div> </div>
<div id="outline-container-org4203cad" class="outline-3"> <div id="outline-container-org5d80bd3" class="outline-3">
<h3 id="org4203cad"><span class="section-number-3">2.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3> <h3 id="org5d80bd3"><span class="section-number-3">2.1</span> Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
We create a cubic Stewart platform (figure <a href="#org620d9b9">1</a>) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot). We create a cubic Stewart platform (figure <a href="#org813c22d">1</a>) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
The Jacobian matrix is estimated at the location of the center of the cube. The Jacobian matrix is estimated at the location of the center of the cube.
</p> </p>
<div id="org620d9b9" class="figure"> <div id="org813c22d" class="figure">
<p><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" /> <p><img src="./figs/3d-cubic-stewart-aligned.png" alt="3d-cubic-stewart-aligned.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p> <p><span class="figure-number">Figure 1: </span>Centered cubic configuration</p>
@ -448,11 +457,11 @@ save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/s
</div> </div>
</div> </div>
<div id="outline-container-org3344772" class="outline-3"> <div id="outline-container-orga5ac347" class="outline-3">
<h3 id="org3344772"><span class="section-number-3">2.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3> <h3 id="orga5ac347"><span class="section-number-3">2.2</span> Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org620d9b9">1</a>). We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure <a href="#org813c22d">1</a>).
The Jacobian matrix is not estimated at the location of the center of the cube. The Jacobian matrix is not estimated at the location of the center of the cube.
</p> </p>
@ -552,16 +561,16 @@ stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stew
</div> </div>
</div> </div>
<div id="outline-container-org52de20d" class="outline-3"> <div id="outline-container-org47743ef" class="outline-3">
<h3 id="org52de20d"><span class="section-number-3">2.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3> <h3 id="org47743ef"><span class="section-number-3">2.3</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
Here, the "center" of the Stewart platform is not at the cube center (figure <a href="#org283dc40">2</a>). Here, the "center" of the Stewart platform is not at the cube center (figure <a href="#org8a2bd4c">2</a>).
The Jacobian is estimated at the cube center. The Jacobian is estimated at the cube center.
</p> </p>
<div id="org283dc40" class="figure"> <div id="org8a2bd4c" class="figure">
<p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" /> <p><img src="./figs/3d-cubic-stewart-misaligned.png" alt="3d-cubic-stewart-misaligned.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p> <p><span class="figure-number">Figure 2: </span>Not centered cubic configuration</p>
@ -674,8 +683,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
</div> </div>
</div> </div>
<div id="outline-container-orgd7e1449" class="outline-3"> <div id="outline-container-orgd0daf23" class="outline-3">
<h3 id="orgd7e1449"><span class="section-number-3">2.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3> <h3 id="orgd0daf23"><span class="section-number-3">2.4</span> Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
Here, the "center" of the Stewart platform is not at the cube center. Here, the "center" of the Stewart platform is not at the cube center.
@ -789,8 +798,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
</div> </div>
</div> </div>
<div id="outline-container-orgf16b788" class="outline-3"> <div id="outline-container-org6729b53" class="outline-3">
<h3 id="orgf16b788"><span class="section-number-3">2.5</span> Conclusion</h3> <h3 id="org6729b53"><span class="section-number-3">2.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-5"> <div class="outline-text-3" id="text-2-5">
<div class="important"> <div class="important">
<ul class="org-ul"> <ul class="org-ul">
@ -803,8 +812,8 @@ We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiff
</div> </div>
</div> </div>
<div id="outline-container-orga2cd408" class="outline-2"> <div id="outline-container-org0969b06" class="outline-2">
<h2 id="orga2cd408"><span class="section-number-2">3</span> Cubic size analysis</h2> <h2 id="org0969b06"><span class="section-number-2">3</span> Cubic size analysis</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
We here study the effect of the size of the cube used for the Stewart configuration. We here study the effect of the size of the cube used for the Stewart configuration.
@ -879,7 +888,7 @@ xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Cube
</div> </div>
<div id="org859b371" class="figure"> <div id="orgad7fc21" class="figure">
<p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" /> <p><img src="figs/stiffness_cube_size.png" alt="stiffness_cube_size.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p> <p><span class="figure-number">Figure 3: </span>\(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube</p>
@ -900,16 +909,16 @@ In that case, the legs will the further separated. Size of the cube is then limi
</div> </div>
</div> </div>
<div id="outline-container-org9220275" class="outline-2"> <div id="outline-container-org0f56064" class="outline-2">
<h2 id="org9220275"><span class="section-number-2">4</span> initializeCubicConfiguration</h2> <h2 id="org0f56064"><span class="section-number-2">4</span> initializeCubicConfiguration</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="org8876664"></a> <a id="org7e73a4b"></a>
</p> </p>
</div> </div>
<div id="outline-container-orgdee5436" class="outline-3"> <div id="outline-container-orge3f31b8" class="outline-3">
<h3 id="orgdee5436"><span class="section-number-3">4.1</span> Function description</h3> <h3 id="orge3f31b8"><span class="section-number-3">4.1</span> Function description</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeCubicConfiguration</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span> <pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">initializeCubicConfiguration</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">opts_param</span><span style="color: #DCDCCC;">)</span>
@ -918,8 +927,8 @@ In that case, the legs will the further separated. Size of the cube is then limi
</div> </div>
</div> </div>
<div id="outline-container-org68794ca" class="outline-3"> <div id="outline-container-org2580fde" class="outline-3">
<h3 id="org68794ca"><span class="section-number-3">4.2</span> Optional Parameters</h3> <h3 id="org2580fde"><span class="section-number-3">4.2</span> Optional Parameters</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
Default values for opts. Default values for opts.
@ -948,8 +957,8 @@ Populate opts with input parameters
</div> </div>
</div> </div>
<div id="outline-container-org93d8028" class="outline-3"> <div id="outline-container-org230a253" class="outline-3">
<h3 id="org93d8028"><span class="section-number-3">4.3</span> Cube Creation</h3> <h3 id="org230a253"><span class="section-number-3">4.3</span> Cube Creation</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">points = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span> <pre class="src src-matlab">points = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>; <span style="text-decoration: underline;">...</span>
@ -994,8 +1003,8 @@ We use to rotation matrix to rotate the cube
</div> </div>
</div> </div>
<div id="outline-container-org00e16e1" class="outline-3"> <div id="outline-container-orgd1a04ef" class="outline-3">
<h3 id="org00e16e1"><span class="section-number-3">4.4</span> Vectors of each leg</h3> <h3 id="orgd1a04ef"><span class="section-number-3">4.4</span> Vectors of each leg</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">leg_indices = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">4</span>; <span style="text-decoration: underline;">...</span> <pre class="src src-matlab">leg_indices = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">4</span>; <span style="text-decoration: underline;">...</span>
@ -1023,8 +1032,8 @@ legs_start = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFE
</div> </div>
</div> </div>
<div id="outline-container-orgd7df0cc" class="outline-3"> <div id="outline-container-orgfec0778" class="outline-3">
<h3 id="orgd7df0cc"><span class="section-number-3">4.5</span> Verification of Height of the Stewart Platform</h3> <h3 id="orgfec0778"><span class="section-number-3">4.5</span> Verification of Height of the Stewart Platform</h3>
<div class="outline-text-3" id="text-4-5"> <div class="outline-text-3" id="text-4-5">
<p> <p>
If the Stewart platform is not contained in the cube, throw an error. If the Stewart platform is not contained in the cube, throw an error.
@ -1043,8 +1052,8 @@ If the Stewart platform is not contained in the cube, throw an error.
</div> </div>
</div> </div>
<div id="outline-container-orgc4b765a" class="outline-3"> <div id="outline-container-orgc3cd1d1" class="outline-3">
<h3 id="orgc4b765a"><span class="section-number-3">4.6</span> Determinate the location of the joints</h3> <h3 id="orgc3cd1d1"><span class="section-number-3">4.6</span> Determinate the location of the joints</h3>
<div class="outline-text-3" id="text-4-6"> <div class="outline-text-3" id="text-4-6">
<p> <p>
We now determine the location of the joints on the fixed platform w.r.t the fixed frame \(\{A\}\). We now determine the location of the joints on the fixed platform w.r.t the fixed frame \(\{A\}\).
@ -1089,8 +1098,8 @@ Ab = Ab <span style="color: #7CB8BB;">-</span> h<span style="color: #7CB8BB;">*<
</div> </div>
</div> </div>
<div id="outline-container-org0a84b4d" class="outline-3"> <div id="outline-container-org9063896" class="outline-3">
<h3 id="org0a84b4d"><span class="section-number-3">4.7</span> Returns Stewart Structure</h3> <h3 id="org9063896"><span class="section-number-3">4.7</span> Returns Stewart Structure</h3>
<div class="outline-text-3" id="text-4-7"> <div class="outline-text-3" id="text-4-7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"> stewart = struct<span style="color: #DCDCCC;">()</span>; <pre class="src src-matlab"> stewart = struct<span style="color: #DCDCCC;">()</span>;
@ -1105,15 +1114,15 @@ Ab = Ab <span style="color: #7CB8BB;">-</span> h<span style="color: #7CB8BB;">*<
</div> </div>
</div> </div>
<div id="outline-container-orgcb21f88" class="outline-2"> <div id="outline-container-org5807740" class="outline-2">
<h2 id="orgcb21f88"><span class="section-number-2">5</span> Tests</h2> <h2 id="org5807740"><span class="section-number-2">5</span> Tests</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
</div> </div>
<div id="outline-container-orgc47f87d" class="outline-3"> <div id="outline-container-org8809c47" class="outline-3">
<h3 id="orgc47f87d"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3> <h3 id="org8809c47"><span class="section-number-3">5.1</span> First attempt to parametrisation</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<div id="orgb15dddd" class="figure"> <div id="org520c5d6" class="figure">
<p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" /> <p><img src="./figs/stewart_bottom_plate.png" alt="stewart_bottom_plate.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Schematic of the bottom plates with all the parameters</p> <p><span class="figure-number">Figure 4: </span>Schematic of the bottom plates with all the parameters</p>
@ -1148,8 +1157,8 @@ Lets express \(a_i\), \(b_i\) and \(a_j\):
</div> </div>
</div> </div>
<div id="outline-container-orgff4f69c" class="outline-3"> <div id="outline-container-org71c64bc" class="outline-3">
<h3 id="orgff4f69c"><span class="section-number-3">5.2</span> Second attempt</h3> <h3 id="org71c64bc"><span class="section-number-3">5.2</span> Second attempt</h3>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-3" id="text-5-2">
<p> <p>
We start with the point of a cube in space: We start with the point of a cube in space:
@ -1276,8 +1285,8 @@ Let's then estimate the middle position of the platform
</div> </div>
</div> </div>
<div id="outline-container-org011ab94" class="outline-3"> <div id="outline-container-orgc132128" class="outline-3">
<h3 id="org011ab94"><span class="section-number-3">5.3</span> Generate the Stewart platform for a Cubic configuration</h3> <h3 id="orgc132128"><span class="section-number-3">5.3</span> Generate the Stewart platform for a Cubic configuration</h3>
<div class="outline-text-3" id="text-5-3"> <div class="outline-text-3" id="text-5-3">
<p> <p>
First we defined the height of the Hexapod. First we defined the height of the Hexapod.
@ -1336,13 +1345,18 @@ zlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>
</div> </div>
</div> </div>
<ol class="org-ol"> <p>
<li>Z.J. Geng, and L.S. Haynes, , <i>Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms</i>, IEEE Transactions on Control Systems Technology, 2<b>(1)</b>, pp. 45-53 (1994). <a href="http://dx.doi.org/10.1109/87.273110">http://dx.doi.org/10.1109/87.273110</a>.</li>
</ol> <h1 class='org-ref-bib-h1'>Bibliography</h1>
<ul class='org-ref-bib'><li><a id="geng94_six_degree_of_freed_activ">[geng94_six_degree_of_freed_activ]</a> <a name="geng94_six_degree_of_freed_activ"></a>Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, <i>IEEE Transactions on Control Systems Technology</i>, <b>2(1)</b>, 45-53 (1994). <a href="https://doi.org/10.1109/87.273110">link</a>. <a href="http://dx.doi.org/10.1109/87.273110">doi</a>.</li>
<li><a id="jafari03_orthog_gough_stewar_platf_microm">[jafari03_orthog_gough_stewar_platf_microm]</a> <a name="jafari03_orthog_gough_stewar_platf_microm"></a>Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, <i>IEEE Transactions on Robotics and Automation</i>, <b>19(4)</b>, 595-603 (2003). <a href="https://doi.org/10.1109/tra.2003.814506">link</a>. <a href="http://dx.doi.org/10.1109/tra.2003.814506">doi</a>.</li>
<li><a id="preumont07_six_axis_singl_stage_activ">[preumont07_six_axis_singl_stage_activ]</a> <a name="preumont07_six_axis_singl_stage_activ"></a>Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, <i>Journal of Sound and Vibration</i>, <b>300(3-5)</b>, 644-661 (2007). <a href="https://doi.org/10.1016/j.jsv.2006.07.050">link</a>. <a href="http://dx.doi.org/10.1016/j.jsv.2006.07.050">doi</a>.</li>
</ul>
</p>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p> <p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-25 lun. 18:11</p> <p class="date">Created: 2019-03-26 mar. 08:47</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

13
cubic-configuration.org
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@ -31,6 +31,11 @@
#+end_src #+end_src
The discovery of the Cubic configuration is done in citenum:geng94_six_degree_of_freed_activ. The discovery of the Cubic configuration is done in citenum:geng94_six_degree_of_freed_activ.
Further analysis is conducted in cite:jafari03_orthog_gough_stewar_platf_microm.
People using orthogonal/cubic configuration: cite:preumont07_six_axis_singl_stage_activ.
The specificity of the Cubic configuration is that each actuator is orthogonal with the others. The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]). To generate and study the Cubic configuration, =initializeCubicConfiguration= is used (description in section [[sec:initializeCubicConfiguration]]).
@ -68,7 +73,7 @@ The Jacobian matrix is estimated at the location of the center of the cube.
#+end_src #+end_src
#+begin_src matlab :results none :exports code #+begin_src matlab :results none :exports code
K = stewart.Jd'*stewart.Jd; K = stewart.Jf'*stewart.Jf;
#+end_src #+end_src
#+begin_src matlab :results value table :exports results #+begin_src matlab :results value table :exports results
@ -104,7 +109,7 @@ The Jacobian matrix is not estimated at the location of the center of the cube.
#+end_src #+end_src
#+begin_src matlab :results none :exports code #+begin_src matlab :results none :exports code
K = stewart.Jd'*stewart.Jd; K = stewart.Jf'*stewart.Jf;
#+end_src #+end_src
#+begin_src matlab :results value table :exports results #+begin_src matlab :results value table :exports results
@ -149,7 +154,7 @@ The center of the cube from the top platform is at $z = 110 - 175 = -65$.
#+end_src #+end_src
#+begin_src matlab :results none :exports code #+begin_src matlab :results none :exports code
K = stewart.Jd'*stewart.Jd; K = stewart.Jf'*stewart.Jf;
#+end_src #+end_src
#+begin_src matlab :results value table :exports results #+begin_src matlab :results value table :exports results
@ -192,7 +197,7 @@ The center of the cube from the top platform is at $z = 110 - 175 = -65$.
#+end_src #+end_src
#+begin_src matlab :results none :exports code #+begin_src matlab :results none :exports code
K = stewart.Jd'*stewart.Jd; K = stewart.Jf'*stewart.Jf;
#+end_src #+end_src
#+begin_src matlab :results value table :exports results #+begin_src matlab :results value table :exports results

175
kinematic-study.html
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2019-03-22 ven. 12:03 --> <!-- 2019-03-26 mar. 09:24 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Kinematic Study of the Stewart Platform</title> <title>Kinematic Study of the Stewart Platform</title>
@ -193,6 +193,12 @@
.org-svg { width: 90%; } .org-svg { width: 90%; }
/*]]>*/--> /*]]>*/-->
</style> </style>
<link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
<link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
<script src="js/jquery.min.js"></script>
<script src="js/bootstrap.min.js"></script>
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<script type="text/javascript" src="js/readtheorg.js"></script>
<script type="text/javascript"> <script type="text/javascript">
/* /*
@licstart The following is the entire license notice for the @licstart The following is the entire license notice for the
@ -247,27 +253,170 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgc1c40d5">1. Functions</a> <li><a href="#org2e1bd58">1. Needed Actuator Stroke</a>
<ul> <ul>
<li><a href="#org3d6cf9e">1.1. getMaxPositions</a></li> <li><a href="#org16d1370">1.1. Stewart architecture definition</a></li>
<li><a href="#orge3ee3ac">1.2. getMaxPureDisplacement</a></li> <li><a href="#orgaf07b82">1.2. Wanted translations and rotations</a></li>
<li><a href="#org920b62b">1.3. Needed stroke for "pure" rotations or translations</a></li>
<li><a href="#org27bf97e">1.4. Needed stroke for combined translations and rotations</a></li>
</ul>
</li>
<li><a href="#orgaebf111">2. Maximum Stroke</a></li>
<li><a href="#orgfb8a1e7">3. Functions</a>
<ul>
<li><a href="#org465746a">3.1. getMaxPositions</a></li>
<li><a href="#org527f7ca">3.2. getMaxPureDisplacement</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orgc1c40d5" class="outline-2"> <div id="outline-container-org2e1bd58" class="outline-2">
<h2 id="orgc1c40d5"><span class="section-number-2">1</span> Functions</h2> <h2 id="org2e1bd58"><span class="section-number-2">1</span> Needed Actuator Stroke</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p>
The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
</p>
</div> </div>
<div id="outline-container-org3d6cf9e" class="outline-3">
<h3 id="org3d6cf9e"><span class="section-number-3">1.1</span> getMaxPositions</h3> <div id="outline-container-org16d1370" class="outline-3">
<h3 id="org16d1370"><span class="section-number-3">1.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p>
We use a cubic architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'H_tot'</span>, <span style="color: #BFEBBF;">90</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">60</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">60</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
<span style="color: #DCDCCC;">)</span>;
stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'Jd_pos'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">100</span><span style="color: #BFEBBF;">]</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
<span style="color: #CC9393;">'Jf_pos'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">50</span><span style="color: #BFEBBF;">]</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
<span style="color: #DCDCCC;">)</span>;
stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'stroke'</span>, <span style="color: #BFEBBF;">50e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Maximum stroke of each actuator [m]</span>
<span style="color: #DCDCCC;">)</span>;
stewart = initializeMechanicalElements<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/stewart.mat', 'stewart'</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgaf07b82" class="outline-3">
<h3 id="orgaf07b82"><span class="section-number-3">1.2</span> Wanted translations and rotations</h3>
<div class="outline-text-3" id="text-1-2">
<p>
We define wanted translations and rotations
</p>
<div class="org-src-container">
<pre class="src src-matlab">Tx_max = <span style="color: #BFEBBF;">15e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Translation [m]</span>
Ty_max = <span style="color: #BFEBBF;">15e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Translation [m]</span>
Tz_max = <span style="color: #BFEBBF;">15e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Translation [m]</span>
Rx_max = <span style="color: #BFEBBF;">30e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Rotation [rad]</span>
Ry_max = <span style="color: #BFEBBF;">30e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Rotation [rad]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org920b62b" class="outline-3">
<h3 id="org920b62b"><span class="section-number-3">1.3</span> Needed stroke for "pure" rotations or translations</h3>
<div class="outline-text-3" id="text-1-3">
<p>
First, we estimate the needed actuator stroke for "pure" rotations and translation.
</p>
<div class="org-src-container">
<pre class="src src-matlab">LTx = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span>Tx_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
LTy = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> Ty_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
LTz = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> Tz_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
LRx = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> Rx_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
LRy = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> Ry_max <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
</pre>
</div>
<pre class="example">
1.0607e-05
</pre>
</div>
</div>
<div id="outline-container-org27bf97e" class="outline-3">
<h3 id="org27bf97e"><span class="section-number-3">1.4</span> Needed stroke for combined translations and rotations</h3>
<div class="outline-text-3" id="text-1-4">
<p>
Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Lmax = <span style="color: #BFEBBF;">0</span>;
pos = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>;
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Tx</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Tx_max</span>,Tx_max<span style="color: #DCDCCC;">]</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Ty</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Ty_max</span>,Ty_max<span style="color: #DCDCCC;">]</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Tz</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Tz_max</span>,Tz_max<span style="color: #DCDCCC;">]</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Rx</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Rx_max</span>,Rx_max<span style="color: #DCDCCC;">]</span>
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Ry</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Ry_max</span>,Ry_max<span style="color: #DCDCCC;">]</span>
L = max<span style="color: #DCDCCC;">(</span>stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">[</span>Tx Ty Tz Rx Ry <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">if</span> L <span style="color: #7CB8BB;">&gt;</span> Lmax
Lmax = L;
pos = <span style="color: #DCDCCC;">[</span>Tx Ty Tz Rx Ry<span style="color: #DCDCCC;">]</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
<p>
We obtain a needed stroke shown below (almost two times the needed stroke for "pure" rotations and translations).
</p>
<pre class="example">
3.0927e-05
</pre>
</div>
</div>
</div>
<div id="outline-container-orgaebf111" class="outline-2">
<h2 id="orgaebf111"><span class="section-number-2">2</span> Maximum Stroke</h2>
<div class="outline-text-2" id="text-2">
<p>
From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #DCDCCC;">[</span>X, Y, Z<span style="color: #DCDCCC;">]</span> = getMaxPositions<span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
plot3<span style="color: #DCDCCC;">(</span>X, Y, Z, <span style="color: #CC9393;">'k-'</span><span style="color: #DCDCCC;">)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfb8a1e7" class="outline-2">
<h2 id="orgfb8a1e7"><span class="section-number-2">3</span> Functions</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-org465746a" class="outline-3">
<h3 id="org465746a"><span class="section-number-3">3.1</span> getMaxPositions</h3>
<div class="outline-text-3" id="text-3-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">X, Y, Z</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPositions</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span> <pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">X, Y, Z</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPositions</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>
Leg = stewart.Leg; Leg = stewart.Leg;
J = stewart.J; J = stewart.Jd;
theta = linspace<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>; theta = linspace<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
phi = linspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> , <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>; phi = linspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> , <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
dmax = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>theta<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>phi<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>; dmax = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>theta<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>phi<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
@ -288,9 +437,9 @@ for the JavaScript code in this tag.
</div> </div>
</div> </div>
<div id="outline-container-orge3ee3ac" class="outline-3"> <div id="outline-container-org527f7ca" class="outline-3">
<h3 id="orge3ee3ac"><span class="section-number-3">1.2</span> getMaxPureDisplacement</h3> <h3 id="org527f7ca"><span class="section-number-3">3.2</span> getMaxPureDisplacement</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-3-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">max_disp</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPureDisplacement</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">Leg</span>, <span style="color: #DFAF8F;">J</span><span style="color: #DCDCCC;">)</span> <pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">max_disp</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPureDisplacement</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">Leg</span>, <span style="color: #DFAF8F;">J</span><span style="color: #DCDCCC;">)</span>
max_disp = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>; max_disp = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
@ -309,7 +458,7 @@ for the JavaScript code in this tag.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p> <p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-22 ven. 12:03</p> <p class="date">Created: 2019-03-26 mar. 09:24</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

104
kinematic-study.org
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@ -24,6 +24,108 @@
#+PROPERTY: header-args:matlab+ :mkdirp yes #+PROPERTY: header-args:matlab+ :mkdirp yes
:END: :END:
#+begin_src matlab :results none :exports none :noweb yes
<<matlab-init>>
addpath('src');
addpath('library');
#+end_src
* Needed Actuator Stroke
The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
** Stewart architecture definition
We use a cubic architecture.
#+begin_src matlab :results silent
opts = struct(...
'H_tot', 90, ... % Total height of the Hexapod [mm]
'L', 180/sqrt(3), ... % Size of the Cube [mm]
'H', 60, ... % Height between base joints and platform joints [mm]
'H0', 180/2-60/2 ... % Height between the corner of the cube and the plane containing the base joints [mm]
);
stewart = initializeCubicConfiguration(opts);
opts = struct(...
'Jd_pos', [0, 0, 100], ... % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]
'Jf_pos', [0, 0, -50] ... % Position of the Jacobian for force location from the top of the mobile platform [mm]
);
stewart = computeGeometricalProperties(stewart, opts);
opts = struct(...
'stroke', 50e-6 ... % Maximum stroke of each actuator [m]
);
stewart = initializeMechanicalElements(stewart, opts);
save('./mat/stewart.mat', 'stewart');
#+end_src
** Wanted translations and rotations
We define wanted translations and rotations
#+begin_src matlab :results silent
Tx_max = 15e-6; % Translation [m]
Ty_max = 15e-6; % Translation [m]
Tz_max = 15e-6; % Translation [m]
Rx_max = 30e-6; % Rotation [rad]
Ry_max = 30e-6; % Rotation [rad]
#+end_src
** Needed stroke for "pure" rotations or translations
First, we estimate the needed actuator stroke for "pure" rotations and translation.
#+begin_src matlab :results silent
LTx = stewart.Jd*[Tx_max 0 0 0 0 0]';
LTy = stewart.Jd*[0 Ty_max 0 0 0 0]';
LTz = stewart.Jd*[0 0 Tz_max 0 0 0]';
LRx = stewart.Jd*[0 0 0 Rx_max 0 0]';
LRy = stewart.Jd*[0 0 0 0 Ry_max 0]';
#+end_src
#+begin_src matlab :results value :exports results
ans = max(max([LTx, LTy, LTz, LRx, LRy]))
#+end_src
#+RESULTS:
: 1.0607e-05
** Needed stroke for combined translations and rotations
Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
#+begin_src matlab :results none
Lmax = 0;
pos = [0, 0, 0, 0, 0];
for Tx = [-Tx_max,Tx_max]
for Ty = [-Ty_max,Ty_max]
for Tz = [-Tz_max,Tz_max]
for Rx = [-Rx_max,Rx_max]
for Ry = [-Ry_max,Ry_max]
L = max(stewart.Jd*[Tx Ty Tz Rx Ry 0]');
if L > Lmax
Lmax = L;
pos = [Tx Ty Tz Rx Ry];
end
end
end
end
end
end
#+end_src
We obtain a needed stroke shown below (almost two times the needed stroke for "pure" rotations and translations).
#+begin_src matlab :results value :exports results
ans = Lmax
#+end_src
#+RESULTS:
: 3.0927e-05
* Maximum Stroke
From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.
#+begin_src matlab :results silent
[X, Y, Z] = getMaxPositions(stewart);
#+end_src
#+begin_src matlab :results silent
figure;
plot3(X, Y, Z, 'k-')
#+end_src
* Functions * Functions
:PROPERTIES: :PROPERTIES:
:HEADER-ARGS:matlab+: :exports code :HEADER-ARGS:matlab+: :exports code
@ -38,7 +140,7 @@
#+begin_src matlab #+begin_src matlab
function [X, Y, Z] = getMaxPositions(stewart) function [X, Y, Z] = getMaxPositions(stewart)
Leg = stewart.Leg; Leg = stewart.Leg;
J = stewart.J; J = stewart.Jd;
theta = linspace(0, 2*pi, 100); theta = linspace(0, 2*pi, 100);
phi = linspace(-pi/2 , pi/2, 100); phi = linspace(-pi/2 , pi/2, 100);
dmax = zeros(length(theta), length(phi)); dmax = zeros(length(theta), length(phi));

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32
references.bib
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@ -34,4 +34,36 @@
doi = {10.1109/87.273110}, doi = {10.1109/87.273110},
url = {https://doi.org/10.1109/87.273110}, url = {https://doi.org/10.1109/87.273110},
keywords = {}, keywords = {},
}
@article{jafari03_orthog_gough_stewar_platf_microm,
author = {Jafari, F. and McInroy, J.E.},
title = {Orthogonal Gough-Stewart Platforms for Micromanipulation},
journal = {IEEE Transactions on Robotics and Automation},
volume = 19,
number = 4,
pages = {595-603},
year = 2003,
doi = {10.1109/tra.2003.814506},
url = {https://doi.org/10.1109/tra.2003.814506},
issn = {1042-296X},
month = {Aug},
publisher = {Institute of Electrical and Electronics Engineers (IEEE)},
keywords = {},
}
@article{preumont07_six_axis_singl_stage_activ,
author = {A. Preumont and M. Horodinca and I. Romanescu and B. de
Marneffe and M. Avraam and A. Deraemaeker and F. Bossens and
A. Abu Hanieh},
title = {A Six-Axis Single-Stage Active Vibration Isolator Based on
Stewart Platform},
journal = {Journal of Sound and Vibration},
volume = 300,
number = {3-5},
pages = {644-661},
year = 2007,
doi = {10.1016/j.jsv.2006.07.050},
url = {https://doi.org/10.1016/j.jsv.2006.07.050},
keywords = {},
} }

2
src/getMaxPositions.m
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@ -1,6 +1,6 @@
function [X, Y, Z] = getMaxPositions(stewart) function [X, Y, Z] = getMaxPositions(stewart)
Leg = stewart.Leg; Leg = stewart.Leg;
J = stewart.J; J = stewart.Jd;
theta = linspace(0, 2*pi, 100); theta = linspace(0, 2*pi, 100);
phi = linspace(-pi/2 , pi/2, 100); phi = linspace(-pi/2 , pi/2, 100);
dmax = zeros(length(theta), length(phi)); dmax = zeros(length(theta), length(phi));

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