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2025-12-02 15:33:28 +01:00
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<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" />
@@ -11,31 +11,21 @@
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<div id="org-div-home-and-up"> <div id="org-div-home-and-up">
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<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="#org53f469a">1. Geometry</a></li> <li><a href="#org27506b9">1. Geometry</a></li>
<li><a href="#org149f9ab">2. Kinematics: Jacobian Matrix and Mobility</a></li> <li><a href="#org1adb3ee">2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#org5895290">3. Kinematics: Degrees of Freedom</a></li> <li><a href="#org603ccc6">3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#org028031c">4. Kinematics: Number of modes</a></li> <li><a href="#org88264b0">4. Kinematics: Number of modes</a></li>
<li><a href="#orgee25d76">5. Flexible Joint Design</a> <li><a href="#org7f469de">5. Flexible Joint Design</a>
<ul> <ul>
<li><a href="#orgb6c609b">5.1. Studied Geometry</a></li> <li><a href="#org56c40c7">5.1. Studied Geometry</a></li>
<li><a href="#org348dd75">5.2. Stiffness seen by the actuator</a></li> <li><a href="#org8cef7a8">5.2. Stiffness seen by the actuator</a></li>
<li><a href="#org31a5262">5.3. Bending Stiffness</a></li> <li><a href="#orgaf5193b">5.3. Bending Stiffness</a></li>
<li><a href="#orgc8c79de">5.4. Axial Stiffness</a></li> <li><a href="#org1d8d21a">5.4. Axial Stiffness</a></li>
<li><a href="#org466dfab">5.5. Torsional Stiffness</a></li> <li><a href="#org439f6c2">5.5. Torsional Stiffness</a></li>
<li><a href="#org82410c1">5.6. Shear Stiffness</a></li> <li><a href="#org8c09ae7">5.6. Shear Stiffness</a></li>
<li><a href="#org3f56f9f">5.7. Effect of cube&rsquo;s size</a> <li><a href="#org5f740e7">5.7. Effect of cube&rsquo;s size</a>
<ul> <ul>
<li><a href="#org26cb099">5.7.1. Effect on the plant dynamics</a></li> <li><a href="#orga3a8b34">5.7.1. Effect on the plant dynamics</a></li>
<li><a href="#org3877b8c">5.7.2. Effect on the compliance</a></li> <li><a href="#org18443f4">5.7.2. Effect on the compliance</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org65c6b51">5.8. Effect of the strut length ?</a> <li><a href="#org7e1cd7b">5.8. Effect of the strut length ?</a>
<ul> <ul>
<li><a href="#orgde6fcac">5.8.1. Effect on the plant dynamics</a></li> <li><a href="#org945fe3a">5.8.1. Effect on the plant dynamics</a></li>
<li><a href="#org1c3f6b3">5.8.2. Effect on the compliance</a></li> <li><a href="#orgb2de208">5.8.2. Effect on the compliance</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgc930dc6">5.9. Having the Center of Mass at the cube&rsquo;s center</a></li> <li><a href="#org4999aef">5.9. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#org735360a">5.10. Conclusion</a></li> <li><a href="#orgc5d83a6">5.10. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org1efe995">6. Conclusion</a></li> <li><a href="#org7840f5a">6. Conclusion</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<p> <p>
<a id="sec:delta_robot_introduction"></a> <a id="sec:delta_robot_introduction"></a>
</p> </p>
<div id="outline-container-org53f469a" class="outline-2"> <div id="outline-container-org27506b9" class="outline-2">
<h2 id="org53f469a"><span class="section-number-2">1.</span> Geometry</h2> <h2 id="org27506b9"><span class="section-number-2">1.</span> Geometry</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-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>.
@@ -172,8 +162,8 @@ Let&rsquo;s initialize a Delta Robot architecture, and plot the obtained geometr
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</div> </div>
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<div id="outline-container-org149f9ab" class="outline-2"> <div id="outline-container-org1adb3ee" class="outline-2">
<h2 id="org149f9ab"><span class="section-number-2">2.</span> Kinematics: Jacobian Matrix and Mobility</h2> <h2 id="org1adb3ee"><span class="section-number-2">2.</span> Kinematics: Jacobian Matrix and Mobility</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-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 \(\hat{s}_3\);
@@ -246,8 +236,8 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
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</div> </div>
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<div id="outline-container-org5895290" class="outline-2"> <div id="outline-container-org603ccc6" class="outline-2">
<h2 id="org5895290"><span class="section-number-2">3.</span> Kinematics: Degrees of Freedom</h2> <h2 id="org603ccc6"><span class="section-number-2">3.</span> Kinematics: Degrees of Freedom</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-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.
@@ -537,8 +527,8 @@ Therefore, to model some compliance of the top platform in rotation, both the ax
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org028031c" class="outline-2"> <div id="outline-container-org88264b0" class="outline-2">
<h2 id="org028031c"><span class="section-number-2">4.</span> Kinematics: Number of modes</h2> <h2 id="org88264b0"><span class="section-number-2">4.</span> Kinematics: Number of modes</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-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).
@@ -554,8 +544,8 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
</pre> </pre>
</div> </div>
</div> </div>
<div id="outline-container-orgee25d76" class="outline-2"> <div id="outline-container-org7f469de" class="outline-2">
<h2 id="orgee25d76"><span class="section-number-2">5.</span> Flexible Joint Design</h2> <h2 id="org7f469de"><span class="section-number-2">5.</span> Flexible Joint Design</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
<a id="sec:delta_robot_flexible_joints"></a> <a id="sec:delta_robot_flexible_joints"></a>
@@ -584,8 +574,8 @@ 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. Then, the impact of the flexible joint&rsquo;s imperfections will be studied.
</p> </p>
</div> </div>
<div id="outline-container-orgb6c609b" class="outline-3"> <div id="outline-container-org56c40c7" class="outline-3">
<h3 id="orgb6c609b"><span class="section-number-3">5.1.</span> Studied Geometry</h3> <h3 id="org56c40c7"><span class="section-number-3">5.1.</span> Studied Geometry</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<p> <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. 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.
@@ -618,8 +608,8 @@ 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-org348dd75" class="outline-3"> <div id="outline-container-org8cef7a8" class="outline-3">
<h3 id="org348dd75"><span class="section-number-3">5.2.</span> Stiffness seen by the actuator</h3> <h3 id="org8cef7a8"><span class="section-number-3">5.2.</span> Stiffness seen by the actuator</h3>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-3" id="text-5-2">
<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.
@@ -653,8 +643,8 @@ This should be validated with the final geometry.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org31a5262" class="outline-3"> <div id="outline-container-orgaf5193b" class="outline-3">
<h3 id="org31a5262"><span class="section-number-3">5.3.</span> Bending Stiffness</h3> <h3 id="orgaf5193b"><span class="section-number-3">5.3.</span> Bending Stiffness</h3>
<div class="outline-text-3" id="text-5-3"> <div class="outline-text-3" id="text-5-3">
<p> <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>. 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>.
@@ -674,8 +664,8 @@ It is not critical from a dynamical point of view, it just decreases the achieva
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</div> </div>
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<div id="outline-container-orgc8c79de" class="outline-3"> <div id="outline-container-org1d8d21a" class="outline-3">
<h3 id="orgc8c79de"><span class="section-number-3">5.4.</span> Axial Stiffness</h3> <h3 id="org1d8d21a"><span class="section-number-3">5.4.</span> Axial Stiffness</h3>
<div class="outline-text-3" id="text-5-4"> <div class="outline-text-3" id="text-5-4">
<p> <p>
Now, the effect of the axial stiffness on the dynamics is studied (Figure <a href="#fig:delta_robot_axial_stiffness_dynamics">11</a>). Now, the effect of the axial stiffness on the dynamics is studied (Figure <a href="#fig:delta_robot_axial_stiffness_dynamics">11</a>).
@@ -693,8 +683,8 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org466dfab" class="outline-3"> <div id="outline-container-org439f6c2" class="outline-3">
<h3 id="org466dfab"><span class="section-number-3">5.5.</span> Torsional Stiffness</h3> <h3 id="org439f6c2"><span class="section-number-3">5.5.</span> Torsional Stiffness</h3>
<div class="outline-text-3" id="text-5-5"> <div class="outline-text-3" id="text-5-5">
<p> <p>
Now the compliance in torsion of the flexible joints is considered. Now the compliance in torsion of the flexible joints is considered.
@@ -727,8 +717,8 @@ Therefore, the torsional stiffness is not a super important metric for the desig
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org82410c1" class="outline-3"> <div id="outline-container-org8c09ae7" class="outline-3">
<h3 id="org82410c1"><span class="section-number-3">5.6.</span> Shear Stiffness</h3> <h3 id="org8c09ae7"><span class="section-number-3">5.6.</span> Shear Stiffness</h3>
<div class="outline-text-3" id="text-5-6"> <div class="outline-text-3" id="text-5-6">
<p> <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. 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.
@@ -747,8 +737,8 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div> </div>
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<div id="outline-container-org3f56f9f" class="outline-3"> <div id="outline-container-org5f740e7" class="outline-3">
<h3 id="org3f56f9f"><span class="section-number-3">5.7.</span> Effect of cube&rsquo;s size</h3> <h3 id="org5f740e7"><span class="section-number-3">5.7.</span> Effect of cube&rsquo;s size</h3>
<div class="outline-text-3" id="text-5-7"> <div class="outline-text-3" id="text-5-7">
<p> <p>
Let&rsquo;s choose reasonable values for the flexible joints: Let&rsquo;s choose reasonable values for the flexible joints:
@@ -764,8 +754,8 @@ Let&rsquo;s choose reasonable values for the flexible joints:
And we see the effect of changing the cube&rsquo;s size. And we see the effect of changing the cube&rsquo;s size.
</p> </p>
</div> </div>
<div id="outline-container-org26cb099" class="outline-4"> <div id="outline-container-orga3a8b34" class="outline-4">
<h4 id="org26cb099"><span class="section-number-4">5.7.1.</span> Effect on the plant dynamics</h4> <h4 id="orga3a8b34"><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"> <div class="outline-text-4" id="text-5-7-1">
<ul class="org-ul"> <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> <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>
@@ -788,8 +778,8 @@ The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a
</div> </div>
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<div id="outline-container-org3877b8c" class="outline-4"> <div id="outline-container-org18443f4" class="outline-4">
<h4 id="org3877b8c"><span class="section-number-4">5.7.2.</span> Effect on the compliance</h4> <h4 id="org18443f4"><span class="section-number-4">5.7.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-5-7-2"> <div class="outline-text-4" id="text-5-7-2">
<p> <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. 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.
@@ -808,8 +798,8 @@ With a cube size of 50mm, the resonance frequency is already above 1kHz with see
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<div id="outline-container-org65c6b51" class="outline-3"> <div id="outline-container-org7e1cd7b" class="outline-3">
<h3 id="org65c6b51"><span class="section-number-3">5.8.</span> Effect of the strut length ?</h3> <h3 id="org7e1cd7b"><span class="section-number-3">5.8.</span> Effect of the strut length ?</h3>
<div class="outline-text-3" id="text-5-8"> <div class="outline-text-3" id="text-5-8">
<p> <p>
Let&rsquo;s choose reasonable values for the flexible joints: Let&rsquo;s choose reasonable values for the flexible joints:
@@ -824,8 +814,8 @@ Let&rsquo;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.
</p> </p>
</div> </div>
<div id="outline-container-orgde6fcac" class="outline-4"> <div id="outline-container-org945fe3a" class="outline-4">
<h4 id="orgde6fcac"><span class="section-number-4">5.8.1.</span> Effect on the plant dynamics</h4> <h4 id="org945fe3a"><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"> <div class="outline-text-4" id="text-5-8-1">
<p> <p>
As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">17</a>, having longer struts: As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">17</a>, having longer struts:
@@ -849,8 +839,8 @@ So, the struts length can be optimized to not decrease too much the stiffness of
</div> </div>
</div> </div>
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<div id="outline-container-org1c3f6b3" class="outline-4"> <div id="outline-container-orgb2de208" class="outline-4">
<h4 id="org1c3f6b3"><span class="section-number-4">5.8.2.</span> Effect on the compliance</h4> <h4 id="orgb2de208"><span class="section-number-4">5.8.2.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-5-8-2"> <div class="outline-text-4" id="text-5-8-2">
<p> <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). 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).
@@ -865,8 +855,8 @@ As shown in Figure <a href="#fig:delta_robot_strut_length_compliance_rotation">1
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<div id="outline-container-orgc930dc6" class="outline-3"> <div id="outline-container-org4999aef" 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> <h3 id="org4999aef"><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"> <div class="outline-text-3" id="text-5-9">
<p> <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. 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.
@@ -886,17 +876,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need
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<div id="outline-container-org735360a" class="outline-3"> <div id="outline-container-orgc5d83a6" class="outline-3">
<h3 id="org735360a"><span class="section-number-3">5.10.</span> Conclusion</h3> <h3 id="orgc5d83a6"><span class="section-number-3">5.10.</span> Conclusion</h3>
</div> </div>
</div> </div>
<div id="outline-container-org1efe995" class="outline-2"> <div id="outline-container-org7840f5a" class="outline-2">
<h2 id="org1efe995"><span class="section-number-2">6.</span> Conclusion</h2> <h2 id="org7840f5a"><span class="section-number-2">6.</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 15:31</p> <p class="date">Created: 2025-12-02 Tue 15:33</p>
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