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Thomas Dehaeze
2025-12-02 15:31:07 +01:00
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@@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2025-12-02 Tue 15:22 -->
<!-- 2025-12-02 Tue 15:31 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Delta Robot</title>
<meta name="author" content="Dehaeze Thomas" />
@@ -11,22 +11,31 @@
<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
<script>
MathJax = {
svg: {
scale: 1,
fontCache: "global"
},
tex: {
tags: "ams", // or "all" or "none"
multlineWidth: "85%", // Set this to a real value
tagSide: "right",
tagIndent: ".8em",
macros: {bm: ["\\boldsymbol{#1}",1]}
}
};
</script>
<script id="MathJax-script" async
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js"></script>
MathJax = {
loader: {load: ['[tex]/tagformat']},
tex: {
packages: {'[+]': ['tagformat']},
tags: "ams",
multlineWidth: "85%",
tagSide: "right",
tagIndent: ".8em",
macros: {
bm: ["\\boldsymbol{#1}", 1]
},
tagformat: {
tag: (tag) => '(' + tag + ')',
ref: '', // use same format for qref and ef
number: (n) => n.toString(),
id: (id) => 'mjx-eqn:' + id.replace(/\s/g, '_'),
url: (id, base) => base + '#' + encodeURIComponent(id),
}
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js"></script>
</head>
<body>
<div id="org-div-home-and-up">
@@ -39,43 +48,43 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org63d3ede">1. Geometry</a></li>
<li><a href="#org878cbcc">2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#org2f57217">3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#org11a603f">4. Kinematics: Number of modes</a></li>
<li><a href="#org934acc6">5. Flexible Joint Design</a>
<li><a href="#org53f469a">1. Geometry</a></li>
<li><a href="#org149f9ab">2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#org5895290">3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#org028031c">4. Kinematics: Number of modes</a></li>
<li><a href="#orgee25d76">5. Flexible Joint Design</a>
<ul>
<li><a href="#org73fff96">5.1. Studied Geometry</a></li>
<li><a href="#org6b727af">5.2. Stiffness seen by the actuator</a></li>
<li><a href="#orgdacfd2a">5.3. Bending Stiffness</a></li>
<li><a href="#org57a369f">5.4. Axial Stiffness</a></li>
<li><a href="#org3668433">5.5. Torsional Stiffness</a></li>
<li><a href="#org146f0e6">5.6. Shear Stiffness</a></li>
<li><a href="#org19f08de">5.7. Effect of cube&rsquo;s size</a>
<li><a href="#orgb6c609b">5.1. Studied Geometry</a></li>
<li><a href="#org348dd75">5.2. Stiffness seen by the actuator</a></li>
<li><a href="#org31a5262">5.3. Bending Stiffness</a></li>
<li><a href="#orgc8c79de">5.4. Axial Stiffness</a></li>
<li><a href="#org466dfab">5.5. Torsional Stiffness</a></li>
<li><a href="#org82410c1">5.6. Shear Stiffness</a></li>
<li><a href="#org3f56f9f">5.7. Effect of cube&rsquo;s size</a>
<ul>
<li><a href="#org6140687">5.7.1. Effect on the plant dynamics</a></li>
<li><a href="#org06767fd">5.7.2. Effect on the compliance</a></li>
<li><a href="#org26cb099">5.7.1. Effect on the plant dynamics</a></li>
<li><a href="#org3877b8c">5.7.2. Effect on the compliance</a></li>
</ul>
</li>
<li><a href="#orgee7695d">5.8. Effect of the strut length ?</a>
<li><a href="#org65c6b51">5.8. Effect of the strut length ?</a>
<ul>
<li><a href="#org83589ea">5.8.1. Effect on the plant dynamics</a></li>
<li><a href="#orgaf684cf">5.8.2. Effect on the compliance</a></li>
<li><a href="#orgde6fcac">5.8.1. Effect on the plant dynamics</a></li>
<li><a href="#org1c3f6b3">5.8.2. Effect on the compliance</a></li>
</ul>
</li>
<li><a href="#orgbbc65af">5.9. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#orgd8665fa">5.10. Conclusion</a></li>
<li><a href="#orgc930dc6">5.9. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#org735360a">5.10. Conclusion</a></li>
</ul>
</li>
<li><a href="#orga614bfb">6. Conclusion</a></li>
<li><a href="#org1efe995">6. Conclusion</a></li>
</ul>
</div>
</div>
<p>
<a id="sec:delta_robot_introduction"></a>
</p>
<div id="outline-container-org63d3ede" class="outline-2">
<h2 id="org63d3ede"><span class="section-number-2">1.</span> Geometry</h2>
<div id="outline-container-org53f469a" class="outline-2">
<h2 id="org53f469a"><span class="section-number-2">1.</span> Geometry</h2>
<div class="outline-text-2" id="text-1">
<p>
The Delta Robot geometry is defined as shown in Figure <a href="#fig:delta_robot_schematic">1</a>.
@@ -163,8 +172,8 @@ Let&rsquo;s initialize a Delta Robot architecture, and plot the obtained geometr
</div>
</div>
</div>
<div id="outline-container-org878cbcc" class="outline-2">
<h2 id="org878cbcc"><span class="section-number-2">2.</span> Kinematics: Jacobian Matrix and Mobility</h2>
<div id="outline-container-org149f9ab" class="outline-2">
<h2 id="org149f9ab"><span class="section-number-2">2.</span> Kinematics: Jacobian Matrix and Mobility</h2>
<div class="outline-text-2" id="text-2">
<p>
There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);
@@ -237,8 +246,8 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
</div>
</div>
</div>
<div id="outline-container-org2f57217" class="outline-2">
<h2 id="org2f57217"><span class="section-number-2">3.</span> Kinematics: Degrees of Freedom</h2>
<div id="outline-container-org5895290" class="outline-2">
<h2 id="org5895290"><span class="section-number-2">3.</span> Kinematics: Degrees of Freedom</h2>
<div class="outline-text-2" id="text-3">
<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.
@@ -528,8 +537,8 @@ Therefore, to model some compliance of the top platform in rotation, both the ax
</p>
</div>
</div>
<div id="outline-container-org11a603f" class="outline-2">
<h2 id="org11a603f"><span class="section-number-2">4.</span> Kinematics: Number of modes</h2>
<div id="outline-container-org028031c" class="outline-2">
<h2 id="org028031c"><span class="section-number-2">4.</span> Kinematics: Number of modes</h2>
<div class="outline-text-2" id="text-4">
<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).
@@ -545,8 +554,8 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
</pre>
</div>
</div>
<div id="outline-container-org934acc6" class="outline-2">
<h2 id="org934acc6"><span class="section-number-2">5.</span> Flexible Joint Design</h2>
<div id="outline-container-orgee25d76" class="outline-2">
<h2 id="orgee25d76"><span class="section-number-2">5.</span> Flexible Joint Design</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="sec:delta_robot_flexible_joints"></a>
@@ -575,8 +584,8 @@ First, the dynamics of a &ldquo;perfect&rdquo; Delta-Robot is identified (i.e. w
Then, the impact of the flexible joint&rsquo;s imperfections will be studied.
</p>
</div>
<div id="outline-container-org73fff96" class="outline-3">
<h3 id="org73fff96"><span class="section-number-3">5.1.</span> Studied Geometry</h3>
<div id="outline-container-orgb6c609b" class="outline-3">
<h3 id="orgb6c609b"><span class="section-number-3">5.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-5-1">
<p>
The cube&rsquo;s edge length is equal to 50mm, the distance between cube&rsquo;s vertices and top joints is 20mm and the length of the struts (i.e. the distance between the two flexible joints of the same strut) is 50mm.
@@ -609,8 +618,8 @@ The dynamics is shown in Figure <a href="#fig:delta_robot_dynamics_perfect">8</a
</div>
</div>
</div>
<div id="outline-container-org6b727af" class="outline-3">
<h3 id="org6b727af"><span class="section-number-3">5.2.</span> Stiffness seen by the actuator</h3>
<div id="outline-container-org348dd75" class="outline-3">
<h3 id="org348dd75"><span class="section-number-3">5.2.</span> Stiffness seen by the actuator</h3>
<div class="outline-text-3" id="text-5-2">
<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.
@@ -644,8 +653,8 @@ This should be validated with the final geometry.
</p>
</div>
</div>
<div id="outline-container-orgdacfd2a" class="outline-3">
<h3 id="orgdacfd2a"><span class="section-number-3">5.3.</span> Bending Stiffness</h3>
<div id="outline-container-org31a5262" class="outline-3">
<h3 id="org31a5262"><span class="section-number-3">5.3.</span> Bending Stiffness</h3>
<div class="outline-text-3" id="text-5-3">
<p>
Then, the dynamics is identified for a bending Stiffness of \(50\,Nm/\text{rad}\) and compared with a Delta robot with no bending stiffness in Figure <a href="#fig:delta_robot_bending_stiffness_dynamics">10</a>.
@@ -665,8 +674,8 @@ It is not critical from a dynamical point of view, it just decreases the achieva
</div>
</div>
</div>
<div id="outline-container-org57a369f" class="outline-3">
<h3 id="org57a369f"><span class="section-number-3">5.4.</span> Axial Stiffness</h3>
<div id="outline-container-orgc8c79de" class="outline-3">
<h3 id="orgc8c79de"><span class="section-number-3">5.4.</span> Axial Stiffness</h3>
<div class="outline-text-3" id="text-5-4">
<p>
Now, the effect of the axial stiffness on the dynamics is studied (Figure <a href="#fig:delta_robot_axial_stiffness_dynamics">11</a>).
@@ -684,8 +693,8 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
</div>
</div>
</div>
<div id="outline-container-org3668433" class="outline-3">
<h3 id="org3668433"><span class="section-number-3">5.5.</span> Torsional Stiffness</h3>
<div id="outline-container-org466dfab" class="outline-3">
<h3 id="org466dfab"><span class="section-number-3">5.5.</span> Torsional Stiffness</h3>
<div class="outline-text-3" id="text-5-5">
<p>
Now the compliance in torsion of the flexible joints is considered.
@@ -718,8 +727,8 @@ Therefore, the torsional stiffness is not a super important metric for the desig
</p>
</div>
</div>
<div id="outline-container-org146f0e6" class="outline-3">
<h3 id="org146f0e6"><span class="section-number-3">5.6.</span> Shear Stiffness</h3>
<div id="outline-container-org82410c1" class="outline-3">
<h3 id="org82410c1"><span class="section-number-3">5.6.</span> Shear Stiffness</h3>
<div class="outline-text-3" id="text-5-6">
<p>
As shown in Figure <a href="#fig:delta_robot_shear_stiffness_compliance">14</a>, the shear stiffness of the flexible joints has some effect on the compliance in translation and almost no effect on the compliance in rotation.
@@ -738,8 +747,8 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div>
</div>
</div>
<div id="outline-container-org19f08de" class="outline-3">
<h3 id="org19f08de"><span class="section-number-3">5.7.</span> Effect of cube&rsquo;s size</h3>
<div id="outline-container-org3f56f9f" class="outline-3">
<h3 id="org3f56f9f"><span class="section-number-3">5.7.</span> Effect of cube&rsquo;s size</h3>
<div class="outline-text-3" id="text-5-7">
<p>
Let&rsquo;s choose reasonable values for the flexible joints:
@@ -755,8 +764,8 @@ Let&rsquo;s choose reasonable values for the flexible joints:
And we see the effect of changing the cube&rsquo;s size.
</p>
</div>
<div id="outline-container-org6140687" class="outline-4">
<h4 id="org6140687"><span class="section-number-4">5.7.1.</span> Effect on the plant dynamics</h4>
<div id="outline-container-org26cb099" class="outline-4">
<h4 id="org26cb099"><span class="section-number-4">5.7.1.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-5-7-1">
<ul class="org-ul">
<li class="off"><code>[&#xa0;]</code> <b>Understand why such different dynamics between 3dof_a joints and 6dof joints with very high shear stiffnesses</b></li>
@@ -779,8 +788,8 @@ The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a
</div>
</div>
</div>
<div id="outline-container-org06767fd" class="outline-4">
<h4 id="org06767fd"><span class="section-number-4">5.7.2.</span> Effect on the compliance</h4>
<div id="outline-container-org3877b8c" class="outline-4">
<h4 id="org3877b8c"><span class="section-number-4">5.7.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-5-7-2">
<p>
As shown in Figure <a href="#fig:delta_robot_cube_size_compliance_rotation">16</a>, the stiffness of the delta robot in rotation increases with the cube&rsquo;s size.
@@ -799,8 +808,8 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see
</div>
</div>
</div>
<div id="outline-container-orgee7695d" class="outline-3">
<h3 id="orgee7695d"><span class="section-number-3">5.8.</span> Effect of the strut length ?</h3>
<div id="outline-container-org65c6b51" class="outline-3">
<h3 id="org65c6b51"><span class="section-number-3">5.8.</span> Effect of the strut length ?</h3>
<div class="outline-text-3" id="text-5-8">
<p>
Let&rsquo;s choose reasonable values for the flexible joints:
@@ -815,8 +824,8 @@ Let&rsquo;s choose reasonable values for the flexible joints:
And we see the effect of changing the strut length.
</p>
</div>
<div id="outline-container-org83589ea" class="outline-4">
<h4 id="org83589ea"><span class="section-number-4">5.8.1.</span> Effect on the plant dynamics</h4>
<div id="outline-container-orgde6fcac" class="outline-4">
<h4 id="orgde6fcac"><span class="section-number-4">5.8.1.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-5-8-1">
<p>
As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">17</a>, having longer struts:
@@ -840,8 +849,8 @@ So, the struts length can be optimized to not decrease too much the stiffness of
</div>
</div>
</div>
<div id="outline-container-orgaf684cf" class="outline-4">
<h4 id="orgaf684cf"><span class="section-number-4">5.8.2.</span> Effect on the compliance</h4>
<div id="outline-container-org1c3f6b3" class="outline-4">
<h4 id="org1c3f6b3"><span class="section-number-4">5.8.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-5-8-2">
<p>
As shown in Figure <a href="#fig:delta_robot_strut_length_compliance_rotation">18</a>, the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot).
@@ -856,8 +865,8 @@ As shown in Figure <a href="#fig:delta_robot_strut_length_compliance_rotation">1
</div>
</div>
</div>
<div id="outline-container-orgbbc65af" class="outline-3">
<h3 id="orgbbc65af"><span class="section-number-3">5.9.</span> Having the Center of Mass at the cube&rsquo;s center</h3>
<div id="outline-container-orgc930dc6" class="outline-3">
<h3 id="orgc930dc6"><span class="section-number-3">5.9.</span> Having the Center of Mass at the cube&rsquo;s center</h3>
<div class="outline-text-3" id="text-5-9">
<p>
To make things easier, we take a top platform with no mass, mass-less struts, and we put a payload on top of the platform.
@@ -877,17 +886,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need
</div>
</div>
</div>
<div id="outline-container-orgd8665fa" class="outline-3">
<h3 id="orgd8665fa"><span class="section-number-3">5.10.</span> Conclusion</h3>
<div id="outline-container-org735360a" class="outline-3">
<h3 id="org735360a"><span class="section-number-3">5.10.</span> Conclusion</h3>
</div>
</div>
<div id="outline-container-orga614bfb" class="outline-2">
<h2 id="orga614bfb"><span class="section-number-2">6.</span> Conclusion</h2>
<div id="outline-container-org1efe995" class="outline-2">
<h2 id="org1efe995"><span class="section-number-2">6.</span> Conclusion</h2>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2025-12-02 Tue 15:22</p>
<p class="date">Created: 2025-12-02 Tue 15:31</p>
</div>
</body>
</html>