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"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 16:08 -->
<!-- 2025-12-02 Tue 16:10 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Delta Robot</title>
<meta name="author" content="Dehaeze Thomas" />
@@ -38,53 +38,61 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org3e81dc8">1. Studied Geometry</a></li>
<li><a href="#orgaa65a8a">2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#org9f53811">3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#org534ad74">4. Kinematics: Number of modes</a></li>
<li><a href="#org97c3c13">5. Flexible Joint Design</a>
<li><a href="#org4739c6a">1. The Delta Robot Kinematics</a>
<ul>
<li><a href="#orgd837bf2">5.1. Studied Geometry</a></li>
<li><a href="#orgad7f9e4">5.2. Bending Stiffness</a>
<ul>
<li><a href="#orgea15e0a">5.2.1. Stiffness seen by the actuator, and decrease of the achievable stroke</a></li>
<li><a href="#org2f22a71">5.2.2. Effect on the coupling</a></li>
<li><a href="#orga5f1cfb">1.1. Studied Geometry</a></li>
<li><a href="#org5398db8">1.2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#org10f74fc">1.3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#orgdb828b8">1.4. Kinematics: Number of modes</a></li>
</ul>
</li>
<li><a href="#org712e050">5.3. Axial Stiffness</a></li>
<li><a href="#org716a92b">5.4. Torsional Stiffness</a></li>
<li><a href="#org031c26f">5.5. Shear Stiffness</a></li>
<li><a href="#orgd32bf80">5.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#org76b25f1">6. Effect of the Geometry</a>
<li><a href="#orga49dc0d">2. Flexible Joint Design</a>
<ul>
<li><a href="#org51e9a9b">6.1. Effect of cube&rsquo;s size</a>
<li><a href="#orgb0e63d4">2.1. Studied Geometry</a></li>
<li><a href="#org79fe25a">2.2. Bending Stiffness</a>
<ul>
<li><a href="#org8bb4129">6.1.1. Effect on the plant dynamics</a></li>
<li><a href="#orge21d460">6.1.2. Effect on the compliance</a></li>
<li><a href="#org283df00">2.2.1. Stiffness seen by the actuator, and decrease of the achievable stroke</a></li>
<li><a href="#orgee8e0bf">2.2.2. Effect on the coupling</a></li>
</ul>
</li>
<li><a href="#orgdeb7742">6.2. Effect of the strut length</a>
<li><a href="#orgf8008e6">2.3. Axial Stiffness</a></li>
<li><a href="#orga1548a3">2.4. Torsional Stiffness</a></li>
<li><a href="#orge057443">2.5. Shear Stiffness</a></li>
<li><a href="#org834006b">2.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgcae1947">3. Effect of the Geometry</a>
<ul>
<li><a href="#orgc8740b1">6.2.1. Effect on the compliance</a></li>
<li><a href="#org924ba25">6.2.2. Effect on the plant dynamics</a></li>
<li><a href="#orgd98514d">3.1. Effect of cube&rsquo;s size</a>
<ul>
<li><a href="#org4c31a39">3.1.1. Effect on the plant dynamics</a></li>
<li><a href="#orgb677c76">3.1.2. Effect on the compliance</a></li>
</ul>
</li>
<li><a href="#org24827ae">6.3. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#orgea55578">6.4. Conclusion</a></li>
<li><a href="#org226e341">3.2. Effect of the strut length</a>
<ul>
<li><a href="#org891c97e">3.2.1. Effect on the compliance</a></li>
<li><a href="#orgb031680">3.2.2. Effect on the plant dynamics</a></li>
</ul>
</li>
<li><a href="#org0ecfbfc">7. Conclusion</a></li>
<li><a href="#orgedf64a9">3.3. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#orgd45c7b6">3.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgacdb08e">4. Conclusion</a></li>
</ul>
</div>
</div>
<div id="outline-container-org4739c6a" class="outline-2">
<h2 id="org4739c6a"><span class="section-number-2">1.</span> The Delta Robot Kinematics</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="sec:delta_robot_kinematics"></a>
</p>
<div id="outline-container-org3e81dc8" class="outline-2">
<h2 id="org3e81dc8"><span class="section-number-2">1.</span> Studied Geometry</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-orga5f1cfb" class="outline-3">
<h3 id="orga5f1cfb"><span class="section-number-3">1.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-1-1">
<p>
The Delta Robot geometry is defined as shown in Figure <a href="#fig:delta_robot_schematic">1</a>.
</p>
@@ -179,9 +187,9 @@ Let&rsquo;s initialize a Delta Robot architecture, and plot the obtained geometr
</div>
</div>
</div>
<div id="outline-container-orgaa65a8a" class="outline-2">
<h2 id="orgaa65a8a"><span class="section-number-2">2.</span> Kinematics: Jacobian Matrix and Mobility</h2>
<div class="outline-text-2" id="text-2">
<div id="outline-container-org5398db8" class="outline-3">
<h3 id="org5398db8"><span class="section-number-3">1.2.</span> Kinematics: Jacobian Matrix and Mobility</h3>
<div class="outline-text-3" id="text-1-2">
<p>
There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);
</p>
@@ -253,9 +261,9 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
</div>
</div>
</div>
<div id="outline-container-org9f53811" class="outline-2">
<h2 id="org9f53811"><span class="section-number-2">3.</span> Kinematics: Degrees of Freedom</h2>
<div class="outline-text-2" id="text-3">
<div id="outline-container-org10f74fc" class="outline-3">
<h3 id="org10f74fc"><span class="section-number-3">1.3.</span> Kinematics: Degrees of Freedom</h3>
<div class="outline-text-3" id="text-1-3">
<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.
</p>
@@ -324,9 +332,9 @@ Therefore, to model some compliance of the top platform in rotation, both the ax
</p>
</div>
</div>
<div id="outline-container-org534ad74" class="outline-2">
<h2 id="org534ad74"><span class="section-number-2">4.</span> Kinematics: Number of modes</h2>
<div class="outline-text-2" id="text-4">
<div id="outline-container-orgdb828b8" class="outline-3">
<h3 id="orgdb828b8"><span class="section-number-3">1.4.</span> Kinematics: Number of modes</h3>
<div class="outline-text-3" id="text-1-4">
<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).
</p>
@@ -341,14 +349,15 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
</pre>
</div>
</div>
<div id="outline-container-org97c3c13" class="outline-2">
<h2 id="org97c3c13"><span class="section-number-2">5.</span> Flexible Joint Design</h2>
<div class="outline-text-2" id="text-5">
</div>
<div id="outline-container-orga49dc0d" class="outline-2">
<h2 id="orga49dc0d"><span class="section-number-2">2.</span> Flexible Joint Design</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="sec:delta_robot_flexible_joints"></a>
</p>
<p>
First, in Section <a href="#ssec:delta_robot_flexible_joints_geometry">5.1</a>, the dynamics of a &ldquo;perfect&rdquo; Delta-Robot is identified (i.e. with perfect 2DoF rotational joints).
First, in Section <a href="#ssec:delta_robot_flexible_joints_geometry">2.1</a>, the dynamics of a &ldquo;perfect&rdquo; Delta-Robot is identified (i.e. with perfect 2DoF rotational joints).
</p>
<p>
@@ -356,15 +365,15 @@ Then, the impact of the flexible joint&rsquo;s imperfections will be studied.
The goal is to extract specifications for the flexible joints of the six struts, in terms of:
</p>
<ul class="org-ul">
<li>bending stiffness (Section <a href="#ssec:delta_robot_flexible_joints_bending">5.2</a>)</li>
<li>axial stiffness (Section <a href="#ssec:delta_robot_flexible_joints_axial">5.3</a>)</li>
<li>torsional stiffness (Section <a href="#ssec:delta_robot_flexible_joints_torsion">5.4</a>)</li>
<li>shear stiffness (Section <a href="#ssec:delta_robot_flexible_joints_shear">5.5</a>)</li>
<li>bending stiffness (Section <a href="#ssec:delta_robot_flexible_joints_bending">2.2</a>)</li>
<li>axial stiffness (Section <a href="#ssec:delta_robot_flexible_joints_axial">2.3</a>)</li>
<li>torsional stiffness (Section <a href="#ssec:delta_robot_flexible_joints_torsion">2.4</a>)</li>
<li>shear stiffness (Section <a href="#ssec:delta_robot_flexible_joints_shear">2.5</a>)</li>
</ul>
</div>
<div id="outline-container-orgd837bf2" class="outline-3">
<h3 id="orgd837bf2"><span class="section-number-3">5.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-5-1">
<div id="outline-container-orgb0e63d4" class="outline-3">
<h3 id="orgb0e63d4"><span class="section-number-3">2.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="ssec:delta_robot_flexible_joints_geometry"></a>
</p>
@@ -400,16 +409,16 @@ The dynamics is shown in Figure <a href="#fig:delta_robot_dynamics_perfect">8</a
</div>
</div>
</div>
<div id="outline-container-orgad7f9e4" class="outline-3">
<h3 id="orgad7f9e4"><span class="section-number-3">5.2.</span> Bending Stiffness</h3>
<div class="outline-text-3" id="text-5-2">
<div id="outline-container-org79fe25a" class="outline-3">
<h3 id="org79fe25a"><span class="section-number-3">2.2.</span> Bending Stiffness</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="ssec:delta_robot_flexible_joints_bending"></a>
</p>
</div>
<div id="outline-container-orgea15e0a" class="outline-4">
<h4 id="orgea15e0a"><span class="section-number-4">5.2.1.</span> Stiffness seen by the actuator, and decrease of the achievable stroke</h4>
<div class="outline-text-4" id="text-5-2-1">
<div id="outline-container-org283df00" class="outline-4">
<h4 id="org283df00"><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">
<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.
This will limit its effective stroke.
@@ -460,9 +469,9 @@ It is not critical from a dynamical point of view, it just decreases the achieva
</div>
</div>
</div>
<div id="outline-container-org2f22a71" class="outline-4">
<h4 id="org2f22a71"><span class="section-number-4">5.2.2.</span> Effect on the coupling</h4>
<div class="outline-text-4" id="text-5-2-2">
<div id="outline-container-orgee8e0bf" class="outline-4">
<h4 id="orgee8e0bf"><span class="section-number-4">2.2.2.</span> Effect on the coupling</h4>
<div class="outline-text-4" id="text-2-2-2">
<p>
Here, reasonable values for the flexible joints (modelled as a 6DoF joint) stiffness are taken:
</p>
@@ -491,9 +500,9 @@ Therefore, the bending stiffness of the flexible joints should be minimized (10N
</div>
</div>
</div>
<div id="outline-container-org712e050" class="outline-3">
<h3 id="org712e050"><span class="section-number-3">5.3.</span> Axial Stiffness</h3>
<div class="outline-text-3" id="text-5-3">
<div id="outline-container-orgf8008e6" class="outline-3">
<h3 id="orgf8008e6"><span class="section-number-3">2.3.</span> Axial Stiffness</h3>
<div class="outline-text-3" id="text-2-3">
<p>
<a id="ssec:delta_robot_flexible_joints_axial"></a>
</p>
@@ -514,9 +523,9 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
</div>
</div>
</div>
<div id="outline-container-org716a92b" class="outline-3">
<h3 id="org716a92b"><span class="section-number-3">5.4.</span> Torsional Stiffness</h3>
<div class="outline-text-3" id="text-5-4">
<div id="outline-container-orga1548a3" class="outline-3">
<h3 id="orga1548a3"><span class="section-number-3">2.4.</span> Torsional Stiffness</h3>
<div class="outline-text-3" id="text-2-4">
<p>
<a id="ssec:delta_robot_flexible_joints_torsion"></a>
</p>
@@ -552,9 +561,9 @@ Therefore, the torsional stiffness is not a super important metric for the desig
</p>
</div>
</div>
<div id="outline-container-org031c26f" class="outline-3">
<h3 id="org031c26f"><span class="section-number-3">5.5.</span> Shear Stiffness</h3>
<div class="outline-text-3" id="text-5-5">
<div id="outline-container-orge057443" class="outline-3">
<h3 id="orge057443"><span class="section-number-3">2.5.</span> Shear Stiffness</h3>
<div class="outline-text-3" id="text-2-5">
<p>
<a id="ssec:delta_robot_flexible_joints_shear"></a>
</p>
@@ -576,9 +585,9 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div>
</div>
</div>
<div id="outline-container-orgd32bf80" class="outline-3">
<h3 id="orgd32bf80"><span class="section-number-3">5.6.</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-6">
<div id="outline-container-org834006b" class="outline-3">
<h3 id="org834006b"><span class="section-number-3">2.6.</span> Conclusion</h3>
<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">
<caption class="t-above"><span class="table-number">Table 1:</span> Recommendations for the flexible joints</caption>
@@ -625,16 +634,20 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div>
</div>
</div>
<div id="outline-container-org76b25f1" class="outline-2">
<h2 id="org76b25f1"><span class="section-number-2">6.</span> Effect of the Geometry</h2>
<div class="outline-text-2" id="text-6">
<div id="outline-container-orgcae1947" class="outline-2">
<h2 id="orgcae1947"><span class="section-number-2">3.</span> Effect of the Geometry</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="sec:delta_robot_flexible_geometry"></a>
</p>
<p>
In this section, the effect of the geometry on the system properties are studied.
The goal is to better understand the different trade-offs, and to extract specifications in terms of the Delta Robot geometry.
</p>
</div>
<div id="outline-container-org51e9a9b" class="outline-3">
<h3 id="org51e9a9b"><span class="section-number-3">6.1.</span> Effect of cube&rsquo;s size</h3>
<div class="outline-text-3" id="text-6-1">
<div id="outline-container-orgd98514d" class="outline-3">
<h3 id="orgd98514d"><span class="section-number-3">3.1.</span> Effect of cube&rsquo;s size</h3>
<div class="outline-text-3" id="text-3-1">
<p>
Let&rsquo;s choose reasonable values for the flexible joints:
</p>
@@ -649,9 +662,9 @@ 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-org8bb4129" class="outline-4">
<h4 id="org8bb4129"><span class="section-number-4">6.1.1.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-6-1-1">
<div id="outline-container-org4c31a39" class="outline-4">
<h4 id="org4c31a39"><span class="section-number-4">3.1.1.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-3-1-1">
<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>
</ul>
@@ -673,9 +686,9 @@ The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a
</div>
</div>
</div>
<div id="outline-container-orge21d460" class="outline-4">
<h4 id="orge21d460"><span class="section-number-4">6.1.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-6-1-2">
<div id="outline-container-orgb677c76" class="outline-4">
<h4 id="orgb677c76"><span class="section-number-4">3.1.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-3-1-2">
<p>
As shown in Figure <a href="#fig:delta_robot_cube_size_compliance_rotation">17</a>, the stiffness of the delta robot in rotation increases with the cube&rsquo;s size.
</p>
@@ -693,9 +706,9 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see
</div>
</div>
</div>
<div id="outline-container-orgdeb7742" class="outline-3">
<h3 id="orgdeb7742"><span class="section-number-3">6.2.</span> Effect of the strut length</h3>
<div class="outline-text-3" id="text-6-2">
<div id="outline-container-org226e341" class="outline-3">
<h3 id="org226e341"><span class="section-number-3">3.2.</span> Effect of the strut length</h3>
<div class="outline-text-3" id="text-3-2">
<p>
Let&rsquo;s choose reasonable values for the flexible joints:
</p>
@@ -709,9 +722,9 @@ 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-orgc8740b1" class="outline-4">
<h4 id="orgc8740b1"><span class="section-number-4">6.2.1.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-6-2-1">
<div id="outline-container-org891c97e" class="outline-4">
<h4 id="org891c97e"><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>
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).
</p>
@@ -733,9 +746,9 @@ Indeed, the stiffness in rotation is a combination of:
</div>
</div>
</div>
<div id="outline-container-org924ba25" class="outline-4">
<h4 id="org924ba25"><span class="section-number-4">6.2.2.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-6-2-2">
<div id="outline-container-orgb031680" class="outline-4">
<h4 id="orgb031680"><span class="section-number-4">3.2.2.</span> Effect on the plant dynamics</h4>
<div class="outline-text-4" id="text-3-2-2">
<p>
As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">19</a>, having longer struts:
</p>
@@ -764,9 +777,9 @@ So, the struts length can be optimized to not decrease too much the stiffness of
</div>
</div>
</div>
<div id="outline-container-org24827ae" class="outline-3">
<h3 id="org24827ae"><span class="section-number-3">6.3.</span> Having the Center of Mass at the cube&rsquo;s center</h3>
<div class="outline-text-3" id="text-6-3">
<div id="outline-container-orgedf64a9" class="outline-3">
<h3 id="orgedf64a9"><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">
<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.
</p>
@@ -785,17 +798,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need
</div>
</div>
</div>
<div id="outline-container-orgea55578" class="outline-3">
<h3 id="orgea55578"><span class="section-number-3">6.4.</span> Conclusion</h3>
<div id="outline-container-orgd45c7b6" class="outline-3">
<h3 id="orgd45c7b6"><span class="section-number-3">3.4.</span> Conclusion</h3>
</div>
</div>
<div id="outline-container-org0ecfbfc" class="outline-2">
<h2 id="org0ecfbfc"><span class="section-number-2">7.</span> Conclusion</h2>
<div id="outline-container-orgacdb08e" class="outline-2">
<h2 id="orgacdb08e"><span class="section-number-2">4.</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 16:08</p>
<p class="date">Created: 2025-12-02 Tue 16:10</p>
</div>
</body>
</html>

5
delta-robot.org
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@@ -88,7 +88,7 @@
org-ref-acronyms-before-parsing))
#+END_SRC
* The Delta Robot Kinematics :ignore:
* The Delta Robot Kinematics
<<sec:delta_robot_kinematics>>
** Introduction :ignore:
** Matlab Init :noexport:ignore:
@@ -1189,6 +1189,9 @@ exportFig('figs/delta_robot_shear_stiffness_compliance.pdf', 'width', 'full', 'h
* Effect of the Geometry
<<sec:delta_robot_flexible_geometry>>
** Introduction :ignore:
In this section, the effect of the geometry on the system properties are studied.
The goal is to better understand the different trade-offs, and to extract specifications in terms of the Delta Robot geometry.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>