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Update strut length

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Thomas Dehaeze
2025-12-02 16:32:26 +01:00
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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>
<!-- 2025-12-02 Tue 16:28 --> <!-- 2025-12-02 Tue 16:32 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Delta Robot</title> <title>Delta Robot</title>
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
@@ -38,61 +38,62 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc"> <div id="text-table-of-contents" role="doc-toc">
<ul> <ul>
<li><a href="#org041b2c8">1. The Delta Robot Kinematics</a> <li><a href="#org246e757">1. The Delta Robot Kinematics</a>
<ul> <ul>
<li><a href="#org5a562dd">1.1. Studied Geometry</a></li> <li><a href="#orgd65a25c">1.1. Studied Geometry</a></li>
<li><a href="#orgc8f560d">1.2. Kinematics: Jacobian Matrix and Mobility</a></li> <li><a href="#orgae1dbdf">1.2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#orge719dd6">1.3. Kinematics: Degrees of Freedom</a></li> <li><a href="#orgeecfe67">1.3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#org46aafd7">1.4. Kinematics: Number of modes</a></li> <li><a href="#orgbbed25f">1.4. Kinematics: Number of modes</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgb8b8b90">2. Flexible Joint Design</a> <li><a href="#org8d5cefc">2. Flexible Joint Design</a>
<ul> <ul>
<li><a href="#org5fe6551">2.1. Studied Geometry</a></li> <li><a href="#orgea43cfb">2.1. Studied Geometry</a></li>
<li><a href="#org5079ccf">2.2. Bending Stiffness</a> <li><a href="#org813b934">2.2. Bending Stiffness</a>
<ul> <ul>
<li><a href="#org43ab0fc">2.2.1. Stiffness seen by the actuator, and decrease of the achievable stroke</a></li> <li><a href="#orgcf384e6">2.2.1. Stiffness seen by the actuator, and decrease of the achievable stroke</a></li>
<li><a href="#org48688f9">2.2.2. Effect on the coupling</a></li> <li><a href="#org095ff34">2.2.2. Effect on the coupling</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orga5a0755">2.3. Axial Stiffness</a></li> <li><a href="#orga674e80">2.3. Axial Stiffness</a></li>
<li><a href="#org68cef68">2.4. Torsional Stiffness</a></li> <li><a href="#orgeb122f2">2.4. Torsional Stiffness</a></li>
<li><a href="#orgb87d3e3">2.5. Shear Stiffness</a></li> <li><a href="#org87d23ca">2.5. Shear Stiffness</a></li>
<li><a href="#org49a4ae9">2.6. Conclusion</a></li> <li><a href="#orga9cc0e4">2.6. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org44befb2">3. Effect of the Geometry</a> <li><a href="#org59e46a3">3. Effect of the Geometry</a>
<ul> <ul>
<li><a href="#orgc1397b5">3.1. Effect of cube&rsquo;s size</a> <li><a href="#org824382a">3.1. Effect of cube&rsquo;s size</a>
<ul> <ul>
<li><a href="#org650af84">3.1.1. Obtained geometries</a></li> <li><a href="#org9653578">3.1.1. Obtained geometries</a></li>
<li><a href="#orgbcb5a2c">3.1.2. Effect on the plant dynamics</a></li> <li><a href="#org882d5ef">3.1.2. Effect on the plant dynamics</a></li>
<li><a href="#orgf9cf4f5">3.1.3. Effect on the compliance</a></li> <li><a href="#orgdf09522">3.1.3. Effect on the compliance</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org49cdf64">3.2. Effect of the strut length</a> <li><a href="#orgdbfba1e">3.2. Effect of the strut length</a>
<ul> <ul>
<li><a href="#orgd2961b9">3.2.1. Effect on the compliance</a></li> <li><a href="#orgaec6022">3.2.1. Obtained geometries</a></li>
<li><a href="#org1103628">3.2.2. Effect on the plant dynamics</a></li> <li><a href="#orgfc3a953">3.2.2. Effect on the compliance</a></li>
<li><a href="#orgb286aae">3.2.3. Effect on the plant dynamics</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org947becc">3.3. Having the Center of Mass at the cube&rsquo;s center</a></li> <li><a href="#orgf9dc819">3.3. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#org118ef08">3.4. Conclusion</a></li> <li><a href="#orgb3b8684">3.4. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org9bd757a">4. Conclusion</a></li> <li><a href="#org47c28e2">4. Conclusion</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org041b2c8" class="outline-2"> <div id="outline-container-org246e757" class="outline-2">
<h2 id="org041b2c8"><span class="section-number-2">1.</span> The Delta Robot Kinematics</h2> <h2 id="org246e757"><span class="section-number-2">1.</span> The Delta Robot Kinematics</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="sec:delta_robot_kinematics"></a> <a id="sec:delta_robot_kinematics"></a>
</p> </p>
</div> </div>
<div id="outline-container-org5a562dd" class="outline-3"> <div id="outline-container-orgd65a25c" class="outline-3">
<h3 id="org5a562dd"><span class="section-number-3">1.1.</span> Studied Geometry</h3> <h3 id="orgd65a25c"><span class="section-number-3">1.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
The Delta Robot geometry is defined as shown in Figure <a href="#fig:delta_robot_schematic">1</a>. The Delta Robot geometry is defined as shown in Figure <a href="#fig:delta_robot_schematic">1</a>.
@@ -188,8 +189,8 @@ Let&rsquo;s initialize a Delta Robot architecture, and plot the obtained geometr
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgc8f560d" class="outline-3"> <div id="outline-container-orgae1dbdf" class="outline-3">
<h3 id="orgc8f560d"><span class="section-number-3">1.2.</span> Kinematics: Jacobian Matrix and Mobility</h3> <h3 id="orgae1dbdf"><span class="section-number-3">1.2.</span> Kinematics: Jacobian Matrix and Mobility</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\); There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);
@@ -262,8 +263,8 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orge719dd6" class="outline-3"> <div id="outline-container-orgeecfe67" class="outline-3">
<h3 id="orge719dd6"><span class="section-number-3">1.3.</span> Kinematics: Degrees of Freedom</h3> <h3 id="orgeecfe67"><span class="section-number-3">1.3.</span> Kinematics: Degrees of Freedom</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
In the perfect case (flexible joints having no stiffness in bending, and infinite stiffness in torsion and in the axial direction), the top platform is allowed to move only in the X, Y and Z directions while the three rotations are fixed. In the perfect case (flexible joints having no stiffness in bending, and infinite stiffness in torsion and in the axial direction), the top platform is allowed to move only in the X, Y and Z directions while the three rotations are fixed.
@@ -333,8 +334,8 @@ Therefore, to model some compliance of the top platform in rotation, both the ax
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org46aafd7" class="outline-3"> <div id="outline-container-orgbbed25f" class="outline-3">
<h3 id="org46aafd7"><span class="section-number-3">1.4.</span> Kinematics: Number of modes</h3> <h3 id="orgbbed25f"><span class="section-number-3">1.4.</span> Kinematics: Number of modes</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
<p> <p>
In the perfect condition (i.e. infinite stiffness in torsion and in compression of the flexible joints), the system has 6 states (i.e. 3 modes, one for each DoF: X, Y and Z). In the perfect condition (i.e. infinite stiffness in torsion and in compression of the flexible joints), the system has 6 states (i.e. 3 modes, one for each DoF: X, Y and Z).
@@ -351,8 +352,8 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgb8b8b90" class="outline-2"> <div id="outline-container-org8d5cefc" class="outline-2">
<h2 id="orgb8b8b90"><span class="section-number-2">2.</span> Flexible Joint Design</h2> <h2 id="org8d5cefc"><span class="section-number-2">2.</span> Flexible Joint Design</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="sec:delta_robot_flexible_joints"></a> <a id="sec:delta_robot_flexible_joints"></a>
@@ -372,8 +373,8 @@ The goal is to extract specifications for the flexible joints of the six struts,
<li>shear stiffness (Section <a href="#ssec:delta_robot_flexible_joints_shear">2.5</a>)</li> <li>shear stiffness (Section <a href="#ssec:delta_robot_flexible_joints_shear">2.5</a>)</li>
</ul> </ul>
</div> </div>
<div id="outline-container-org5fe6551" class="outline-3"> <div id="outline-container-orgea43cfb" class="outline-3">
<h3 id="org5fe6551"><span class="section-number-3">2.1.</span> Studied Geometry</h3> <h3 id="orgea43cfb"><span class="section-number-3">2.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
<a id="ssec:delta_robot_flexible_joints_geometry"></a> <a id="ssec:delta_robot_flexible_joints_geometry"></a>
@@ -410,15 +411,15 @@ The dynamics is shown in Figure <a href="#fig:delta_robot_dynamics_perfect">8</a
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org5079ccf" class="outline-3"> <div id="outline-container-org813b934" class="outline-3">
<h3 id="org5079ccf"><span class="section-number-3">2.2.</span> Bending Stiffness</h3> <h3 id="org813b934"><span class="section-number-3">2.2.</span> Bending Stiffness</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
<a id="ssec:delta_robot_flexible_joints_bending"></a> <a id="ssec:delta_robot_flexible_joints_bending"></a>
</p> </p>
</div> </div>
<div id="outline-container-org43ab0fc" class="outline-4"> <div id="outline-container-orgcf384e6" class="outline-4">
<h4 id="org43ab0fc"><span class="section-number-4">2.2.1.</span> Stiffness seen by the actuator, and decrease of the achievable stroke</h4> <h4 id="orgcf384e6"><span class="section-number-4">2.2.1.</span> Stiffness seen by the actuator, and decrease of the achievable stroke</h4>
<div class="outline-text-4" id="text-2-2-1"> <div class="outline-text-4" id="text-2-2-1">
<p> <p>
Because the flexible joints will have some bending stiffness, the actuator in one direction will &ldquo;see&rdquo; some stiffness due to the struts in the other directions. Because the flexible joints will have some bending stiffness, the actuator in one direction will &ldquo;see&rdquo; some stiffness due to the struts in the other directions.
@@ -470,8 +471,8 @@ It is not critical from a dynamical point of view, it just decreases the achieva
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org48688f9" class="outline-4"> <div id="outline-container-org095ff34" class="outline-4">
<h4 id="org48688f9"><span class="section-number-4">2.2.2.</span> Effect on the coupling</h4> <h4 id="org095ff34"><span class="section-number-4">2.2.2.</span> Effect on the coupling</h4>
<div class="outline-text-4" id="text-2-2-2"> <div class="outline-text-4" id="text-2-2-2">
<p> <p>
Here, reasonable values for the flexible joints (modelled as a 6DoF joint) stiffness are taken: Here, reasonable values for the flexible joints (modelled as a 6DoF joint) stiffness are taken:
@@ -501,8 +502,8 @@ Therefore, the bending stiffness of the flexible joints should be minimized (10N
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orga5a0755" class="outline-3"> <div id="outline-container-orga674e80" class="outline-3">
<h3 id="orga5a0755"><span class="section-number-3">2.3.</span> Axial Stiffness</h3> <h3 id="orga674e80"><span class="section-number-3">2.3.</span> Axial Stiffness</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
<a id="ssec:delta_robot_flexible_joints_axial"></a> <a id="ssec:delta_robot_flexible_joints_axial"></a>
@@ -524,8 +525,8 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org68cef68" class="outline-3"> <div id="outline-container-orgeb122f2" class="outline-3">
<h3 id="org68cef68"><span class="section-number-3">2.4.</span> Torsional Stiffness</h3> <h3 id="orgeb122f2"><span class="section-number-3">2.4.</span> Torsional Stiffness</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
<a id="ssec:delta_robot_flexible_joints_torsion"></a> <a id="ssec:delta_robot_flexible_joints_torsion"></a>
@@ -562,8 +563,8 @@ Therefore, the torsional stiffness is not a super important metric for the desig
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-orgb87d3e3" class="outline-3"> <div id="outline-container-org87d23ca" class="outline-3">
<h3 id="orgb87d3e3"><span class="section-number-3">2.5.</span> Shear Stiffness</h3> <h3 id="org87d23ca"><span class="section-number-3">2.5.</span> Shear Stiffness</h3>
<div class="outline-text-3" id="text-2-5"> <div class="outline-text-3" id="text-2-5">
<p> <p>
<a id="ssec:delta_robot_flexible_joints_shear"></a> <a id="ssec:delta_robot_flexible_joints_shear"></a>
@@ -586,8 +587,8 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org49a4ae9" class="outline-3"> <div id="outline-container-orga9cc0e4" class="outline-3">
<h3 id="org49a4ae9"><span class="section-number-3">2.6.</span> Conclusion</h3> <h3 id="orga9cc0e4"><span class="section-number-3">2.6.</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-6"> <div class="outline-text-3" id="text-2-6">
<table id="tab:delta_robot_flexible_joints_recommendations" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="tab:delta_robot_flexible_joints_recommendations" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Recommendations for the flexible joints</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Recommendations for the flexible joints</caption>
@@ -635,8 +636,8 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org44befb2" class="outline-2"> <div id="outline-container-org59e46a3" class="outline-2">
<h2 id="org44befb2"><span class="section-number-2">3.</span> Effect of the Geometry</h2> <h2 id="org59e46a3"><span class="section-number-2">3.</span> Effect of the Geometry</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="sec:delta_robot_flexible_geometry"></a> <a id="sec:delta_robot_flexible_geometry"></a>
@@ -665,15 +666,15 @@ The effect of the following geometrical features are studied:
<li>The location of the payload&rsquo;s Center of Mass with respect to the cube&rsquo;s center in Section <a href="#ssec:delta_robot_flexible_geometry_com">3.3</a></li> <li>The location of the payload&rsquo;s Center of Mass with respect to the cube&rsquo;s center in Section <a href="#ssec:delta_robot_flexible_geometry_com">3.3</a></li>
</ul> </ul>
</div> </div>
<div id="outline-container-orgc1397b5" class="outline-3"> <div id="outline-container-org824382a" class="outline-3">
<h3 id="orgc1397b5"><span class="section-number-3">3.1.</span> Effect of cube&rsquo;s size</h3> <h3 id="org824382a"><span class="section-number-3">3.1.</span> Effect of cube&rsquo;s size</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
<a id="ssec:delta_robot_flexible_geometry_cube_size"></a> <a id="ssec:delta_robot_flexible_geometry_cube_size"></a>
</p> </p>
</div> </div>
<div id="outline-container-org650af84" class="outline-4"> <div id="outline-container-org9653578" class="outline-4">
<h4 id="org650af84"><span class="section-number-4">3.1.1.</span> Obtained geometries</h4> <h4 id="org9653578"><span class="section-number-4">3.1.1.</span> Obtained geometries</h4>
<div class="outline-text-4" id="text-3-1-1"> <div class="outline-text-4" id="text-3-1-1">
<p> <p>
The cube size is varied from 10mm (Figure <a href="#fig:delta_robot_cube_size_small">16</a>) to 100mm (Figure <a href="#fig:delta_robot_cube_size_large">17</a>) to study the effect on the system dynamics. The cube size is varied from 10mm (Figure <a href="#fig:delta_robot_cube_size_small">16</a>) to 100mm (Figure <a href="#fig:delta_robot_cube_size_large">17</a>) to study the effect on the system dynamics.
@@ -694,8 +695,8 @@ The cube size is varied from 10mm (Figure <a href="#fig:delta_robot_cube_size_sm
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgbcb5a2c" class="outline-4"> <div id="outline-container-org882d5ef" class="outline-4">
<h4 id="orgbcb5a2c"><span class="section-number-4">3.1.2.</span> Effect on the plant dynamics</h4> <h4 id="org882d5ef"><span class="section-number-4">3.1.2.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-3-1-2"> <div class="outline-text-4" id="text-3-1-2">
<p> <p>
The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a href="#fig:delta_robot_cube_size_plant_dynamics">18</a>: The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a href="#fig:delta_robot_cube_size_plant_dynamics">18</a>:
@@ -714,8 +715,8 @@ The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgf9cf4f5" class="outline-4"> <div id="outline-container-orgdf09522" class="outline-4">
<h4 id="orgf9cf4f5"><span class="section-number-4">3.1.3.</span> Effect on the compliance</h4> <h4 id="orgdf09522"><span class="section-number-4">3.1.3.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-3-1-3"> <div class="outline-text-4" id="text-3-1-3">
<p> <p>
As shown in Figure <a href="#fig:delta_robot_cube_size_compliance_rotation">19</a>, the stiffness of the delta robot in rotation increases with the cube&rsquo;s size. As shown in Figure <a href="#fig:delta_robot_cube_size_compliance_rotation">19</a>, the stiffness of the delta robot in rotation increases with the cube&rsquo;s size.
@@ -735,12 +736,16 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org49cdf64" class="outline-3"> <div id="outline-container-orgdbfba1e" class="outline-3">
<h3 id="org49cdf64"><span class="section-number-3">3.2.</span> Effect of the strut length</h3> <h3 id="orgdbfba1e"><span class="section-number-3">3.2.</span> Effect of the strut length</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
<a id="ssec:delta_robot_flexible_geometry_strut_length"></a> <a id="ssec:delta_robot_flexible_geometry_strut_length"></a>
</p> </p>
</div>
<div id="outline-container-orgaec6022" class="outline-4">
<h4 id="orgaec6022"><span class="section-number-4">3.2.1.</span> Obtained geometries</h4>
<div class="outline-text-4" id="text-3-2-1">
<p> <p>
Let&rsquo;s choose reasonable values for the flexible joints: Let&rsquo;s choose reasonable values for the flexible joints:
</p> </p>
@@ -753,12 +758,31 @@ Let&rsquo;s choose reasonable values for the flexible joints:
<p> <p>
And we see the effect of changing the strut length. And we see the effect of changing the strut length.
</p> </p>
</div>
<div id="outline-container-orgd2961b9" class="outline-4">
<h4 id="orgd2961b9"><span class="section-number-4">3.2.1.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-3-2-1">
<p> <p>
As shown in Figure <a href="#fig:delta_robot_strut_length_compliance_rotation">20</a>, the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot). The strut length is varied from 10mm (Figure <a href="#fig:delta_robot_strut_small">20</a>) to 100mm (Figure <a href="#fig:delta_robot_strut_large">21</a>) to study the effect on the system dynamics.
</p>
<div id="fig:delta_robot_strut_small" class="figure">
<p><img src="figs/delta_robot_strut_small.png" alt="delta_robot_strut_small.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Obtained Delta Robot for a cube&rsquo;s size of 10mm</p>
</div>
<div id="fig:delta_robot_strut_large" class="figure">
<p><img src="figs/delta_robot_strut_large.png" alt="delta_robot_strut_large.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Obtained Delta Robot for a cube&rsquo;s size of 100mm</p>
</div>
</div>
</div>
<div id="outline-container-orgfc3a953" class="outline-4">
<h4 id="orgfc3a953"><span class="section-number-4">3.2.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-3-2-2">
<p>
As shown in Figure <a href="#fig:delta_robot_strut_length_compliance_rotation">22</a>, the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot).
</p> </p>
<p> <p>
@@ -774,15 +798,15 @@ Indeed, the stiffness in rotation is a combination of:
<div id="fig:delta_robot_strut_length_compliance_rotation" class="figure"> <div id="fig:delta_robot_strut_length_compliance_rotation" class="figure">
<p><img src="figs/delta_robot_strut_length_compliance_rotation.png" alt="delta_robot_strut_length_compliance_rotation.png" /> <p><img src="figs/delta_robot_strut_length_compliance_rotation.png" alt="delta_robot_strut_length_compliance_rotation.png" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Effect of the cube&rsquo;s size on the rotational compliance of the top platform</p> <p><span class="figure-number">Figure 22: </span>Effect of the cube&rsquo;s size on the rotational compliance of the top platform</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org1103628" class="outline-4"> <div id="outline-container-orgb286aae" class="outline-4">
<h4 id="org1103628"><span class="section-number-4">3.2.2.</span> Effect on the plant dynamics</h4> <h4 id="orgb286aae"><span class="section-number-4">3.2.3.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-3-2-2"> <div class="outline-text-4" id="text-3-2-3">
<p> <p>
As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">21</a>, having longer struts: As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">23</a>, having longer struts:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>decreases the main resonance frequency: this means that the stiffness in the X,Y and Z directions is decreased when the length of the strut is longer. <li>decreases the main resonance frequency: this means that the stiffness in the X,Y and Z directions is decreased when the length of the strut is longer.
@@ -803,13 +827,13 @@ So, the struts length can be optimized to not decrease too much the stiffness of
<div id="fig:delta_robot_strut_length_plant_dynamics" class="figure"> <div id="fig:delta_robot_strut_length_plant_dynamics" class="figure">
<p><img src="figs/delta_robot_strut_length_plant_dynamics.png" alt="delta_robot_strut_length_plant_dynamics.png" /> <p><img src="figs/delta_robot_strut_length_plant_dynamics.png" alt="delta_robot_strut_length_plant_dynamics.png" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Effect of the Strut length on the plant dynamics</p> <p><span class="figure-number">Figure 23: </span>Effect of the Strut length on the plant dynamics</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org947becc" class="outline-3"> <div id="outline-container-orgf9dc819" class="outline-3">
<h3 id="org947becc"><span class="section-number-3">3.3.</span> Having the Center of Mass at the cube&rsquo;s center</h3> <h3 id="orgf9dc819"><span class="section-number-3">3.3.</span> Having the Center of Mass at the cube&rsquo;s center</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
<a id="ssec:delta_robot_flexible_geometry_com"></a> <a id="ssec:delta_robot_flexible_geometry_com"></a>
@@ -820,7 +844,7 @@ To make things easier, we take a top platform with no mass, mass-less struts, an
</p> </p>
<p> <p>
As shown in Figure <a href="#fig:delta_robot_CoM_pos_effect_plant">22</a>, having the CoM of the payload at the cube&rsquo;s center allow to have better decoupling properties above the suspension mode of the system (i.e. above the first mode). As shown in Figure <a href="#fig:delta_robot_CoM_pos_effect_plant">24</a>, having the CoM of the payload at the cube&rsquo;s center allow to have better decoupling properties above the suspension mode of the system (i.e. above the first mode).
This could allow to have a bandwidth exceeding the frequency of the first mode. This could allow to have a bandwidth exceeding the frequency of the first mode.
But how sensitive this decoupling is to the exact position of the CoM still need to be studied. But how sensitive this decoupling is to the exact position of the CoM still need to be studied.
</p> </p>
@@ -829,12 +853,12 @@ But how sensitive this decoupling is to the exact position of the CoM still need
<div id="fig:delta_robot_CoM_pos_effect_plant" class="figure"> <div id="fig:delta_robot_CoM_pos_effect_plant" class="figure">
<p><img src="figs/delta_robot_CoM_pos_effect_plant.png" alt="delta_robot_CoM_pos_effect_plant.png" /> <p><img src="figs/delta_robot_CoM_pos_effect_plant.png" alt="delta_robot_CoM_pos_effect_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Effect of the payload&rsquo;s Center of Mass position with respect to the cube&rsquo;s size on the plant dynamics</p> <p><span class="figure-number">Figure 24: </span>Effect of the payload&rsquo;s Center of Mass position with respect to the cube&rsquo;s size on the plant dynamics</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org118ef08" class="outline-3"> <div id="outline-container-orgb3b8684" class="outline-3">
<h3 id="org118ef08"><span class="section-number-3">3.4.</span> Conclusion</h3> <h3 id="orgb3b8684"><span class="section-number-3">3.4.</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<table id="tab:delta_robot_geometry_recommendations" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="tab:delta_robot_geometry_recommendations" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Recommendations for the Delta Robot Geometry</caption> <caption class="t-above"><span class="table-number">Table 2:</span> Recommendations for the Delta Robot Geometry</caption>
@@ -870,13 +894,13 @@ But how sensitive this decoupling is to the exact position of the CoM still need
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org9bd757a" class="outline-2"> <div id="outline-container-org47c28e2" class="outline-2">
<h2 id="org9bd757a"><span class="section-number-2">4.</span> Conclusion</h2> <h2 id="org47c28e2"><span class="section-number-2">4.</span> Conclusion</h2>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2025-12-02 Tue 16:28</p> <p class="date">Created: 2025-12-02 Tue 16:32</p>
</div> </div>
</body> </body>
</html> </html>

67
delta-robot.org
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@@ -1452,7 +1452,7 @@ exportFig('figs/delta_robot_cube_size_compliance_rotation.pdf', 'width', 'wide',
** Effect of the strut length ** Effect of the strut length
<<ssec:delta_robot_flexible_geometry_strut_length>> <<ssec:delta_robot_flexible_geometry_strut_length>>
*** Introduction :ignore: *** Obtained geometries
Let's choose reasonable values for the flexible joints: Let's choose reasonable values for the flexible joints:
- Bending stiffness of 50Nm/rad - Bending stiffness of 50Nm/rad
- Torsional stiffness of 500Nm/rad - Torsional stiffness of 500Nm/rad
@@ -1460,6 +1460,71 @@ Let's choose reasonable values for the flexible joints:
And we see the effect of changing the strut length. And we see the effect of changing the strut length.
The strut length is varied from 10mm (Figure [[fig:delta_robot_strut_small]]) to 100mm (Figure [[fig:delta_robot_strut_large]]) to study the effect on the system dynamics.
#+begin_src matlab
d = 50e-3; % Cube's edge length [m]
b = 20e-3; % Distance between cube's vertices and top joints [m]
L = 50e-3; % Length of the struts [m]
#+end_src
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
delta_robot = initializeStewartPlatform();
delta_robot = generateDeltaRobot(delta_robot, 'd', d, 'b', b, 'L', 10e-3);
delta_robot = computeJointsPose(delta_robot);
delta_robot = initializeActuatorDynamics(delta_robot);
delta_robot = initializeJointDynamics(delta_robot);
delta_robot = initializeCylindricalStruts(delta_robot);
delta_robot = computeJacobian(delta_robot);
delta_robot = initializeStewartPose(delta_robot);
#+end_src
#+begin_src matlab :exports none :results none
%% Delta Robot Architecture
displayArchitecture(delta_robot, 'labels', false, 'frames', false);
plotCube(delta_robot, 'Hc', d/sqrt(3), 'FOc', delta_robot.geometry.H+delta_robot.platform_M.MO_B(3), 'color', [0,0,0,0.2], 'link_to_struts', false);
view([70, 30]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/delta_robot_strut_small.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:delta_robot_strut_small
#+caption: Obtained Delta Robot for a cube's size of 10mm
#+RESULTS:
[[file:figs/delta_robot_strut_small.png]]
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
delta_robot = initializeStewartPlatform();
delta_robot = generateDeltaRobot(delta_robot, 'd', d, 'b', b, 'L', 100e-3);
delta_robot = computeJointsPose(delta_robot);
delta_robot = initializeActuatorDynamics(delta_robot);
delta_robot = initializeJointDynamics(delta_robot);
delta_robot = initializeCylindricalStruts(delta_robot);
delta_robot = computeJacobian(delta_robot);
delta_robot = initializeStewartPose(delta_robot);
#+end_src
#+begin_src matlab :exports none :results none
%% Delta Robot Architecture
displayArchitecture(delta_robot, 'labels', false, 'frames', false);
plotCube(delta_robot, 'Hc', d/sqrt(3), 'FOc', delta_robot.geometry.H+delta_robot.platform_M.MO_B(3), 'color', [0,0,0,0.2], 'link_to_struts', false);
view([70, 30]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/delta_robot_strut_large.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:delta_robot_strut_large
#+caption: Obtained Delta Robot for a cube's size of 100mm
#+RESULTS:
[[file:figs/delta_robot_strut_large.png]]
*** Effect on the compliance *** Effect on the compliance
As shown in Figure [[fig:delta_robot_strut_length_compliance_rotation]], the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot). As shown in Figure [[fig:delta_robot_strut_length_compliance_rotation]], the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot).

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