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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 16:10 -->
<!-- 2025-12-02 Tue 16:28 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Delta Robot</title>
<meta name="author" content="Dehaeze Thomas" />
@@ -38,60 +38,61 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#org4739c6a">1. The Delta Robot Kinematics</a>
<li><a href="#org041b2c8">1. The Delta Robot Kinematics</a>
<ul>
<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>
<li><a href="#org5a562dd">1.1. Studied Geometry</a></li>
<li><a href="#orgc8f560d">1.2. Kinematics: Jacobian Matrix and Mobility</a></li>
<li><a href="#orge719dd6">1.3. Kinematics: Degrees of Freedom</a></li>
<li><a href="#org46aafd7">1.4. Kinematics: Number of modes</a></li>
</ul>
</li>
<li><a href="#orga49dc0d">2. Flexible Joint Design</a>
<li><a href="#orgb8b8b90">2. Flexible Joint Design</a>
<ul>
<li><a href="#orgb0e63d4">2.1. Studied Geometry</a></li>
<li><a href="#org79fe25a">2.2. Bending Stiffness</a>
<li><a href="#org5fe6551">2.1. Studied Geometry</a></li>
<li><a href="#org5079ccf">2.2. Bending Stiffness</a>
<ul>
<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>
<li><a href="#org43ab0fc">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>
</ul>
</li>
<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>
<li><a href="#orga5a0755">2.3. Axial Stiffness</a></li>
<li><a href="#org68cef68">2.4. Torsional Stiffness</a></li>
<li><a href="#orgb87d3e3">2.5. Shear Stiffness</a></li>
<li><a href="#org49a4ae9">2.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgcae1947">3. Effect of the Geometry</a>
<li><a href="#org44befb2">3. Effect of the Geometry</a>
<ul>
<li><a href="#orgd98514d">3.1. Effect of cube&rsquo;s size</a>
<li><a href="#orgc1397b5">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>
<li><a href="#org650af84">3.1.1. Obtained geometries</a></li>
<li><a href="#orgbcb5a2c">3.1.2. Effect on the plant dynamics</a></li>
<li><a href="#orgf9cf4f5">3.1.3. Effect on the compliance</a></li>
</ul>
</li>
<li><a href="#org226e341">3.2. Effect of the strut length</a>
<li><a href="#org49cdf64">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>
<li><a href="#orgd2961b9">3.2.1. Effect on the compliance</a></li>
<li><a href="#org1103628">3.2.2. Effect on the plant dynamics</a></li>
</ul>
</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>
<li><a href="#org947becc">3.3. Having the Center of Mass at the cube&rsquo;s center</a></li>
<li><a href="#org118ef08">3.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgacdb08e">4. Conclusion</a></li>
<li><a href="#org9bd757a">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 id="outline-container-org041b2c8" class="outline-2">
<h2 id="org041b2c8"><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>
<div id="outline-container-orga5f1cfb" class="outline-3">
<h3 id="orga5f1cfb"><span class="section-number-3">1.1.</span> Studied Geometry</h3>
<div id="outline-container-org5a562dd" class="outline-3">
<h3 id="org5a562dd"><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>.
@@ -102,7 +103,7 @@ The geometry is fully defined by three parameters:
</p>
<ul class="org-ul">
<li><code>d</code>: Cube&rsquo;s size (i.e., the length of the cube edge)</li>
<li><code>a</code>: Distance from cube&rsquo;s vertex to top flexible joint</li>
<li><code>b</code>: Distance from cube&rsquo;s vertex to top flexible joint</li>
<li><code>L</code>: Distance between two flexible joints (i.e., the length of the struts)</li>
</ul>
@@ -187,8 +188,8 @@ Let&rsquo;s initialize a Delta Robot architecture, and plot the obtained geometr
</div>
</div>
</div>
<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 id="outline-container-orgc8f560d" class="outline-3">
<h3 id="orgc8f560d"><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 \(\hat{s}_3\);
@@ -261,8 +262,8 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
</div>
</div>
</div>
<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 id="outline-container-orge719dd6" class="outline-3">
<h3 id="orge719dd6"><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.
@@ -332,8 +333,8 @@ Therefore, to model some compliance of the top platform in rotation, both the ax
</p>
</div>
</div>
<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 id="outline-container-org46aafd7" class="outline-3">
<h3 id="org46aafd7"><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).
@@ -350,8 +351,8 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
</div>
</div>
</div>
<div id="outline-container-orga49dc0d" class="outline-2">
<h2 id="orga49dc0d"><span class="section-number-2">2.</span> Flexible Joint Design</h2>
<div id="outline-container-orgb8b8b90" class="outline-2">
<h2 id="orgb8b8b90"><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>
@@ -371,8 +372,8 @@ 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>
</ul>
</div>
<div id="outline-container-orgb0e63d4" class="outline-3">
<h3 id="orgb0e63d4"><span class="section-number-3">2.1.</span> Studied Geometry</h3>
<div id="outline-container-org5fe6551" class="outline-3">
<h3 id="org5fe6551"><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>
@@ -384,7 +385,7 @@ The actuator stiffness is \(1\,N/\mu m\).
</p>
<p>
The obtained geometry is shown in Figure [].
The obtained geometry is shown in Figure <a href="#fig:delta_robot_studied_geometry">7</a>.
</p>
@@ -409,15 +410,15 @@ The dynamics is shown in Figure <a href="#fig:delta_robot_dynamics_perfect">8</a
</div>
</div>
</div>
<div id="outline-container-org79fe25a" class="outline-3">
<h3 id="org79fe25a"><span class="section-number-3">2.2.</span> Bending Stiffness</h3>
<div id="outline-container-org5079ccf" class="outline-3">
<h3 id="org5079ccf"><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-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 id="outline-container-org43ab0fc" 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>
<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.
@@ -469,8 +470,8 @@ It is not critical from a dynamical point of view, it just decreases the achieva
</div>
</div>
</div>
<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 id="outline-container-org48688f9" class="outline-4">
<h4 id="org48688f9"><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:
@@ -500,8 +501,8 @@ Therefore, the bending stiffness of the flexible joints should be minimized (10N
</div>
</div>
</div>
<div id="outline-container-orgf8008e6" class="outline-3">
<h3 id="orgf8008e6"><span class="section-number-3">2.3.</span> Axial Stiffness</h3>
<div id="outline-container-orga5a0755" class="outline-3">
<h3 id="orga5a0755"><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>
@@ -523,8 +524,8 @@ Therefore, we should aim at \(k_a > 100\,N/\mu m\).
</div>
</div>
</div>
<div id="outline-container-orga1548a3" class="outline-3">
<h3 id="orga1548a3"><span class="section-number-3">2.4.</span> Torsional Stiffness</h3>
<div id="outline-container-org68cef68" class="outline-3">
<h3 id="org68cef68"><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>
@@ -561,8 +562,8 @@ Therefore, the torsional stiffness is not a super important metric for the desig
</p>
</div>
</div>
<div id="outline-container-orge057443" class="outline-3">
<h3 id="orge057443"><span class="section-number-3">2.5.</span> Shear Stiffness</h3>
<div id="outline-container-orgb87d3e3" class="outline-3">
<h3 id="orgb87d3e3"><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>
@@ -585,8 +586,8 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div>
</div>
</div>
<div id="outline-container-org834006b" class="outline-3">
<h3 id="org834006b"><span class="section-number-3">2.6.</span> Conclusion</h3>
<div id="outline-container-org49a4ae9" class="outline-3">
<h3 id="org49a4ae9"><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>
@@ -634,8 +635,8 @@ A value of \(100\,N/\mu m\) seems reasonable.
</div>
</div>
</div>
<div id="outline-container-orgcae1947" class="outline-2">
<h2 id="orgcae1947"><span class="section-number-2">3.</span> Effect of the Geometry</h2>
<div id="outline-container-org44befb2" class="outline-2">
<h2 id="org44befb2"><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>
@@ -644,12 +645,9 @@ A value of \(100\,N/\mu m\) seems reasonable.
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-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:
Reasonable values for the flexible joints are taken:
</p>
<ul class="org-ul">
<li>Bending stiffness of 50Nm/rad</li>
@@ -659,21 +657,51 @@ Let&rsquo;s choose reasonable values for the flexible joints:
</ul>
<p>
And we see the effect of changing the cube&rsquo;s size.
The effect of the following geometrical features are studied:
</p>
<ul class="org-ul">
<li>The cube&rsquo;s size in Section <a href="#ssec:delta_robot_flexible_geometry_cube_size">3.1</a></li>
<li>The strut length in Section <a href="#ssec:delta_robot_flexible_geometry_strut_length">3.2</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>
</div>
<div id="outline-container-orgc1397b5" class="outline-3">
<h3 id="orgc1397b5"><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>
<a id="ssec:delta_robot_flexible_geometry_cube_size"></a>
</p>
</div>
<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 id="outline-container-org650af84" class="outline-4">
<h4 id="org650af84"><span class="section-number-4">3.1.1.</span> Obtained geometries</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>
<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">16</a>:
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.
</p>
<div id="fig:delta_robot_cube_size_small" class="figure">
<p><img src="figs/delta_robot_cube_size_small.png" alt="delta_robot_cube_size_small.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Obtained Delta Robot for a cube&rsquo;s size of 10mm</p>
</div>
<div id="fig:delta_robot_cube_size_large" class="figure">
<p><img src="figs/delta_robot_cube_size_large.png" alt="delta_robot_cube_size_large.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Obtained Delta Robot for a cube&rsquo;s size of 100mm</p>
</div>
</div>
</div>
<div id="outline-container-orgbcb5a2c" class="outline-4">
<h4 id="orgbcb5a2c"><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">
<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>:
</p>
<ul class="org-ul">
<li>coupling decreases with the cube&rsquo;s size</li>
<li>coupling decreases with the cube&rsquo;s size (probably because of the reduced effect of the flexible joints&rsquo; bending stiffness)</li>
<li>one resonance frequency increases with the cube&rsquo;s size (resonances in rotation), which may be beneficial from a control point of view</li>
<li>coupling at the main resonance varies with the cube&rsquo;s size, but it may also depend on the relative position between the CoM and the cube&rsquo;s center</li>
</ul>
@@ -682,34 +710,38 @@ The effect of the cube&rsquo;s size on the plant dynamics is shown in Figure <a
<div id="fig:delta_robot_cube_size_plant_dynamics" class="figure">
<p><img src="figs/delta_robot_cube_size_plant_dynamics.png" alt="delta_robot_cube_size_plant_dynamics.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Effect of the cube&rsquo;s size on the plant dynamics</p>
<p><span class="figure-number">Figure 18: </span>Effect of the cube&rsquo;s size on the plant dynamics</p>
</div>
</div>
</div>
<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">
<div id="outline-container-orgf9cf4f5" class="outline-4">
<h4 id="orgf9cf4f5"><span class="section-number-4">3.1.3.</span> Effect on the compliance</h4>
<div class="outline-text-4" id="text-3-1-3">
<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.
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.
</p>
<p>
Therefore, if possible the cube&rsquo;s size should be increased.
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
</p>
<div id="fig:delta_robot_cube_size_compliance_rotation" class="figure">
<p><img src="figs/delta_robot_cube_size_compliance_rotation.png" alt="delta_robot_cube_size_compliance_rotation.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Effect of the cube&rsquo;s size on the rotational compliance of the top platform</p>
</div>
<p>
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
</p>
<p><span class="figure-number">Figure 19: </span>Effect of the cube&rsquo;s size on the rotational compliance of the top platform</p>
</div>
</div>
</div>
<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>
<div id="outline-container-org49cdf64" class="outline-3">
<h3 id="org49cdf64"><span class="section-number-3">3.2.</span> Effect of the strut length</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="ssec:delta_robot_flexible_geometry_strut_length"></a>
</p>
<p>
Let&rsquo;s choose reasonable values for the flexible joints:
</p>
<ul class="org-ul">
@@ -722,11 +754,11 @@ 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-org891c97e" class="outline-4">
<h4 id="org891c97e"><span class="section-number-4">3.2.1.</span> Effect on the compliance</h4>
<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>
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">20</a>, the strut length has an effect on the system stiffness in translation (left plot) but almost not in rotation (right plot).
</p>
<p>
@@ -742,15 +774,15 @@ Indeed, the stiffness in rotation is a combination of:
<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>
<p><span class="figure-number">Figure 18: </span>Effect of the cube&rsquo;s size on the rotational compliance of the top platform</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>
</div>
</div>
</div>
<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 id="outline-container-org1103628" class="outline-4">
<h4 id="org1103628"><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:
As shown in Figure <a href="#fig:delta_robot_strut_length_plant_dynamics">21</a>, having longer struts:
</p>
<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.
@@ -759,8 +791,7 @@ This is reasonable as the &ldquo;lever&rdquo; arm is getting larger, so the bend
Probably: when pushing with one actuator, it induces some rotation of the struts corresponding to the other two actuators.
This rotation is proportional to the strut length.
Then, this rotation, combined with the limited compliance in bending of the flexible joints induces some force applied on the other actuators, hence the coupling.
This is similar to what was observed when varying the bending stiffness of the flexible joints: the coupling was increased with an increased of the bending stiffness (See Figure <a href="#fig:delta_robot_bending_stiffness_couplign">11</a>)
<b>So we should also observed a decrease of the coupling when decreasing the bending stiffness of the actuators</b></li>
This is similar to what was observed when varying the bending stiffness of the flexible joints: the coupling was increased with an increased of the bending stiffness (See Figure <a href="#fig:delta_robot_bending_stiffness_couplign">11</a>)</li>
</ul>
<p>
@@ -772,20 +803,24 @@ 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">
<p><img src="figs/delta_robot_strut_length_plant_dynamics.png" alt="delta_robot_strut_length_plant_dynamics.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Effect of the Strut length on the plant dynamics</p>
<p><span class="figure-number">Figure 21: </span>Effect of the Strut length on the plant dynamics</p>
</div>
</div>
</div>
</div>
<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 id="outline-container-org947becc" 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>
<div class="outline-text-3" id="text-3-3">
<p>
<a id="ssec:delta_robot_flexible_geometry_com"></a>
</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.
</p>
<p>
As shown in Figure <a href="#fig:delta_robot_CoM_pos_effect_plant">20</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">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).
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.
</p>
@@ -794,21 +829,54 @@ 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">
<p><img src="figs/delta_robot_CoM_pos_effect_plant.png" alt="delta_robot_CoM_pos_effect_plant.png" />
</p>
<p><span class="figure-number">Figure 20: </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 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>
</div>
</div>
</div>
<div id="outline-container-orgd45c7b6" class="outline-3">
<h3 id="orgd45c7b6"><span class="section-number-3">3.4.</span> Conclusion</h3>
<div id="outline-container-org118ef08" class="outline-3">
<h3 id="org118ef08"><span class="section-number-3">3.4.</span> Conclusion</h3>
<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">
<caption class="t-above"><span class="table-number">Table 2:</span> Recommendations for the Delta Robot Geometry</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Geometrical feature</th>
<th scope="col" class="org-left">Effect</th>
<th scope="col" class="org-left">Recommendation</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Cube&rsquo;s size <code>d</code></td>
<td class="org-left">Increasing the cube&rsquo;s size increases the rotational stiffness</td>
<td class="org-left">Should be make as large as possible</td>
</tr>
<tr>
<td class="org-left">Strut length <code>L</code></td>
<td class="org-left">Changes the stiffness and coupling of the system (by changing the effect of the flexible joint bending stiffness)</td>
<td class="org-left">Trade-off between higher stiffness and lower coupling</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgacdb08e" class="outline-2">
<h2 id="orgacdb08e"><span class="section-number-2">4.</span> Conclusion</h2>
</div>
<div id="outline-container-org9bd757a" class="outline-2">
<h2 id="org9bd757a"><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 16:10</p>
<p class="date">Created: 2025-12-02 Tue 16:28</p>
</div>
</body>
</html>
delta-robot.org
+99 -18
View File
@@ -121,7 +121,7 @@ The Delta Robot geometry is defined as shown in Figure [[fig:delta_robot_schemat
The geometry is fully defined by three parameters:
- =d=: Cube's size (i.e., the length of the cube edge)
- =a=: Distance from cube's vertex to top flexible joint
- =b=: Distance from cube's vertex to top flexible joint
- =L=: Distance between two flexible joints (i.e., the length of the struts)
#+name: fig:delta_robot_schematic
@@ -622,7 +622,7 @@ The goal is to extract specifications for the flexible joints of the six struts,
The cube's edge length is equal to 50mm, the distance between cube'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 actuator stiffness is $1\,N/\mu m$.
The obtained geometry is shown in Figure [].
The obtained geometry is shown in Figure [[fig:delta_robot_studied_geometry]].
#+begin_src matlab
%% Geometry
@@ -1192,6 +1192,17 @@ exportFig('figs/delta_robot_shear_stiffness_compliance.pdf', 'width', 'full', 'h
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.
Reasonable values for the flexible joints are taken:
- Bending stiffness of 50Nm/rad
- Torsional stiffness of 500Nm/rad
- Axial stiffness of 100N/um
- Shear stiffness of 100N/um
The effect of the following geometrical features are studied:
- The cube's size in Section [[ssec:delta_robot_flexible_geometry_cube_size]]
- The strut length in Section [[ssec:delta_robot_flexible_geometry_strut_length]]
- The location of the payload's Center of Mass with respect to the cube's center in Section [[ssec:delta_robot_flexible_geometry_com]]
** 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>>
@@ -1217,23 +1228,85 @@ The goal is to better understand the different trade-offs, and to extract specif
<<m-init-other>>
#+end_src
#+begin_src matlab
%% Geometry
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]
%% Actuator
k = 1e6; % [N/m]
%% Sample on top
sample = initializeSample('type', 'cylindrical', 'm', 1, 'H', 50e-3, 'R', 20e-3);
#+end_src
** Effect of cube's size
*** Introduction :ignore:
<<ssec:delta_robot_flexible_geometry_cube_size>>
*** Obtained geometries
Let's choose reasonable values for the flexible joints:
- Bending stiffness of 50Nm/rad
- Torsional stiffness of 500Nm/rad
- Axial stiffness of 100N/um
- Shear stiffness of 100N/um
The cube size is varied from 10mm (Figure [[fig:delta_robot_cube_size_small]]) to 100mm (Figure [[fig:delta_robot_cube_size_large]]) to study the effect on the system dynamics.
And we see the effect of changing the cube's size.
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
delta_robot = initializeStewartPlatform();
delta_robot = generateDeltaRobot(delta_robot, 'd', 10e-3, 'b', b, 'L', L);
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', 10e-3/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_cube_size_small.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:delta_robot_cube_size_small
#+caption: Obtained Delta Robot for a cube's size of 10mm
#+RESULTS:
[[file:figs/delta_robot_cube_size_small.png]]
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
delta_robot = initializeStewartPlatform();
delta_robot = generateDeltaRobot(delta_robot, 'd', 100e-3, 'b', b, 'L', L);
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', 100e-3/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_cube_size_large.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:delta_robot_cube_size_large
#+caption: Obtained Delta Robot for a cube's size of 100mm
#+RESULTS:
[[file:figs/delta_robot_cube_size_large.png]]
*** Effect on the plant dynamics
- [ ] *Understand why such different dynamics between 3dof_a joints and 6dof joints with very high shear stiffnesses*
The effect of the cube's size on the plant dynamics is shown in Figure [[fig:delta_robot_cube_size_plant_dynamics]]:
- coupling decreases with the cube's size
- coupling decreases with the cube's size (probably because of the reduced effect of the flexible joints' bending stiffness)
- one resonance frequency increases with the cube's size (resonances in rotation), which may be beneficial from a control point of view
- coupling at the main resonance varies with the cube's size, but it may also depend on the relative position between the CoM and the cube's center
@@ -1257,7 +1330,6 @@ for i = 1:length(cube_sizes)
delta_robot = generateDeltaRobot(delta_robot, 'd', cube_sizes(i), 'b', b, 'L', L);
delta_robot = computeJointsPose(delta_robot);
delta_robot = initializeActuatorDynamics(delta_robot, 'type', '1dof', 'k', k);
% delta_robot = initializeJointDynamics(delta_robot, 'type_F', '3dof_a', 'type_M', '3dof_a', 'Ka', joint_axial, 'Kf', joint_bending, 'Kt', joint_torsion, 'Ks', joint_shear);
delta_robot = initializeJointDynamics(delta_robot, 'type_F', '6dof', 'type_M', '6dof', 'Ka', joint_axial, 'Kf', joint_bending, 'Kt', joint_torsion, 'Ks', joint_shear);
delta_robot = initializeCylindricalStruts(delta_robot);
delta_robot = computeJacobian(delta_robot);
@@ -1294,7 +1366,7 @@ end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
% xlim([1, 1e4]);
ylim([1e-12, 1e-4])
@@ -1313,6 +1385,9 @@ exportFig('figs/delta_robot_cube_size_plant_dynamics.pdf', 'width', 'wide', 'hei
As shown in Figure [[fig:delta_robot_cube_size_compliance_rotation]], the stiffness of the delta robot in rotation increases with the cube's size.
Therefore, if possible the cube's size should be increased.
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
#+begin_src matlab
%% Effect of torsional stiffness on the plant dynamics
joint_axial = 100e6; % [N/m]
@@ -1359,7 +1434,7 @@ for i = 1:length(cube_sizes)
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/Nm]');
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
% xlim([1, 1e4]);
@@ -1375,9 +1450,8 @@ exportFig('figs/delta_robot_cube_size_compliance_rotation.pdf', 'width', 'wide',
#+RESULTS:
[[file:figs/delta_robot_cube_size_compliance_rotation.png]]
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
** Effect of the strut length
<<ssec:delta_robot_flexible_geometry_strut_length>>
*** Introduction :ignore:
Let's choose reasonable values for the flexible joints:
- Bending stiffness of 50Nm/rad
@@ -1485,7 +1559,6 @@ As shown in Figure [[fig:delta_robot_strut_length_plant_dynamics]], having longe
This rotation is proportional to the strut length.
Then, this rotation, combined with the limited compliance in bending of the flexible joints induces some force applied on the other actuators, hence the coupling.
This is similar to what was observed when varying the bending stiffness of the flexible joints: the coupling was increased with an increased of the bending stiffness (See Figure [[fig:delta_robot_bending_stiffness_couplign]])
*So we should also observed a decrease of the coupling when decreasing the bending stiffness of the actuators*
But even with relatively short struts (20mm and above), the low frequency decoupling is already around two orders of magnitude, which is enough from a control point of view.
So, the struts length can be optimized to not decrease too much the stiffness of the platform while still getting good low frequency decoupling.
@@ -1565,6 +1638,7 @@ exportFig('figs/delta_robot_strut_length_plant_dynamics.pdf', 'width', 'wide', '
[[file:figs/delta_robot_strut_length_plant_dynamics.png]]
** Having the Center of Mass at the cube's center
<<ssec:delta_robot_flexible_geometry_com>>
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.
@@ -1666,6 +1740,13 @@ exportFig('figs/delta_robot_CoM_pos_effect_plant.pdf', 'width', 'wide', 'height'
** Conclusion
#+name: tab:delta_robot_geometry_recommendations
#+caption: Recommendations for the Delta Robot Geometry
| Geometrical feature | Effect | Recommendation |
|---------------------+-------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------|
| Cube's size =d= | Increasing the cube's size increases the rotational stiffness | Should be make as large as possible |
| Strut length =L= | Changes the stiffness and coupling of the system (by changing the effect of the flexible joint bending stiffness) | Trade-off between higher stiffness and lower coupling |
* Conclusion
* Bibliography :ignore:
delta-robot.pdf
BIN
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delta-robot.tex
+51 -17
View File
@@ -1,4 +1,4 @@
% Created 2025-12-02 Tue 16:08
% Created 2025-12-02 Tue 16:28
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -22,14 +22,15 @@
\tableofcontents
\clearpage
\chapter{The Delta Robot Kinematics}
\label{sec:delta_robot_kinematics}
\chapter{Studied Geometry}
\section{Studied Geometry}
The Delta Robot geometry is defined as shown in Figure \ref{fig:delta_robot_schematic}.
The geometry is fully defined by three parameters:
\begin{itemize}
\item \texttt{d}: Cube's size (i.e., the length of the cube edge)
\item \texttt{a}: Distance from cube's vertex to top flexible joint
\item \texttt{b}: Distance from cube's vertex to top flexible joint
\item \texttt{L}: Distance between two flexible joints (i.e., the length of the struts)
\end{itemize}
@@ -98,7 +99,7 @@ L = 50e-3; % Length of the struts [m]
\includegraphics[scale=1]{figs/delta_robot_architecture_top.png}
\caption{\label{fig:delta_robot_architecture_top}Delta Robot Architecture - Top View}
\end{figure}
\chapter{Kinematics: Jacobian Matrix and Mobility}
\section{Kinematics: Jacobian Matrix and Mobility}
There are three actuators in the following directions \(\hat{s}_1\), \(\hat{s}_2\) and \(\hat{s}_3\);
@@ -156,7 +157,7 @@ Maximum YZ mobility for an angle of 270 degrees, square with edge size of 117 um
\includegraphics[scale=1]{figs/delta_robot_2d_workspace_optimal.png}
\caption{\label{fig:delta_robot_2d_workspace_optimal}2D mobility for the optimal Rz angle}
\end{figure}
\chapter{Kinematics: Degrees of Freedom}
\section{Kinematics: Degrees of Freedom}
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 order to have some compliance in rotation, the flexible joints need to have some compliance in torsion \textbf{and} in the axial direction.
@@ -203,7 +204,7 @@ In that case we get some compliance in rotation.
So it is a combination of axial and torsion stiffness that gives some rotational stiffness of the top platform.
Therefore, to model some compliance of the top platform in rotation, both the axial compliance and the torsional compliance of the flexible joints should be considered.
\chapter{Kinematics: Number of modes}
\section{Kinematics: Number of modes}
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).
@@ -233,7 +234,7 @@ The goal is to extract specifications for the flexible joints of the six struts,
The cube's edge length is equal to 50mm, the distance between cube'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 actuator stiffness is \(1\,N/\mu m\).
The obtained geometry is shown in Figure [].
The obtained geometry is shown in Figure \ref{fig:delta_robot_studied_geometry}.
\begin{figure}[htbp]
\centering
@@ -372,8 +373,10 @@ Shear & Can limit the stiffness of the system & As high as possible (less import
\end{table}
\chapter{Effect of the Geometry}
\label{sec:delta_robot_flexible_geometry}
\section{Effect of cube's size}
Let's choose reasonable values for the flexible joints:
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.
Reasonable values for the flexible joints are taken:
\begin{itemize}
\item Bending stiffness of 50Nm/rad
\item Torsional stiffness of 500Nm/rad
@@ -381,16 +384,34 @@ Let's choose reasonable values for the flexible joints:
\item Shear stiffness of 100N/um
\end{itemize}
And we see the effect of changing the cube's size.
\subsection{Effect on the plant dynamics}
The effect of the following geometrical features are studied:
\begin{itemize}
\item[{$\square$}] \textbf{Understand why such different dynamics between 3dof\_a joints and 6dof joints with very high shear stiffnesses}
\item The cube's size in Section \ref{ssec:delta_robot_flexible_geometry_cube_size}
\item The strut length in Section \ref{ssec:delta_robot_flexible_geometry_strut_length}
\item The location of the payload's Center of Mass with respect to the cube's center in Section \ref{ssec:delta_robot_flexible_geometry_com}
\end{itemize}
\section{Effect of cube's size}
\label{ssec:delta_robot_flexible_geometry_cube_size}
\subsection{Obtained geometries}
The cube size is varied from 10mm (Figure \ref{fig:delta_robot_cube_size_small}) to 100mm (Figure \ref{fig:delta_robot_cube_size_large}) to study the effect on the system dynamics.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/delta_robot_cube_size_small.png}
\caption{\label{fig:delta_robot_cube_size_small}Obtained Delta Robot for a cube's size of 10mm}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/delta_robot_cube_size_large.png}
\caption{\label{fig:delta_robot_cube_size_large}Obtained Delta Robot for a cube's size of 100mm}
\end{figure}
\subsection{Effect on the plant dynamics}
The effect of the cube's size on the plant dynamics is shown in Figure \ref{fig:delta_robot_cube_size_plant_dynamics}:
\begin{itemize}
\item coupling decreases with the cube's size
\item coupling decreases with the cube's size (probably because of the reduced effect of the flexible joints' bending stiffness)
\item one resonance frequency increases with the cube's size (resonances in rotation), which may be beneficial from a control point of view
\item coupling at the main resonance varies with the cube's size, but it may also depend on the relative position between the CoM and the cube's center
\end{itemize}
@@ -404,14 +425,16 @@ The effect of the cube's size on the plant dynamics is shown in Figure \ref{fig:
As shown in Figure \ref{fig:delta_robot_cube_size_compliance_rotation}, the stiffness of the delta robot in rotation increases with the cube's size.
Therefore, if possible the cube's size should be increased.
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/delta_robot_cube_size_compliance_rotation.png}
\caption{\label{fig:delta_robot_cube_size_compliance_rotation}Effect of the cube's size on the rotational compliance of the top platform}
\end{figure}
With a cube size of 50mm, the resonance frequency is already above 1kHz with seems reasonable.
\section{Effect of the strut length}
\label{ssec:delta_robot_flexible_geometry_strut_length}
Let's choose reasonable values for the flexible joints:
\begin{itemize}
\item Bending stiffness of 50Nm/rad
@@ -447,7 +470,6 @@ Probably: when pushing with one actuator, it induces some rotation of the struts
This rotation is proportional to the strut length.
Then, this rotation, combined with the limited compliance in bending of the flexible joints induces some force applied on the other actuators, hence the coupling.
This is similar to what was observed when varying the bending stiffness of the flexible joints: the coupling was increased with an increased of the bending stiffness (See Figure \ref{fig:delta_robot_bending_stiffness_couplign})
\textbf{So we should also observed a decrease of the coupling when decreasing the bending stiffness of the actuators}
\end{itemize}
But even with relatively short struts (20mm and above), the low frequency decoupling is already around two orders of magnitude, which is enough from a control point of view.
@@ -459,6 +481,7 @@ So, the struts length can be optimized to not decrease too much the stiffness of
\caption{\label{fig:delta_robot_strut_length_plant_dynamics}Effect of the Strut length on the plant dynamics}
\end{figure}
\section{Having the Center of Mass at the cube's center}
\label{ssec:delta_robot_flexible_geometry_com}
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.
@@ -472,6 +495,17 @@ But how sensitive this decoupling is to the exact position of the CoM still need
\caption{\label{fig:delta_robot_CoM_pos_effect_plant}Effect of the payload's Center of Mass position with respect to the cube's size on the plant dynamics}
\end{figure}
\section{Conclusion}
\begin{table}[htbp]
\caption{\label{tab:delta_robot_geometry_recommendations}Recommendations for the Delta Robot Geometry}
\centering
\begin{tabular}{lll}
Geometrical feature & Effect & Recommendation\\
\hline
Cube's size \texttt{d} & Increasing the cube's size increases the rotational stiffness & Should be make as large as possible\\
Strut length \texttt{L} & Changes the stiffness and coupling of the system (by changing the effect of the flexible joint bending stiffness) & Trade-off between higher stiffness and lower coupling\\
\end{tabular}
\end{table}
\chapter{Conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]
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