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<title>Study of the Flexible Joints</title>
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</div><div id="content">
<h1 class="title">Study of the Flexible Joints</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orge032d30">1. Bending and Torsional Stiffness</a>
<li><a href="#org157c72f">1. Bending and Torsional Stiffness</a>
<ul>
<li><a href="#orge82a7c2">1.1. Initialization</a></li>
<li><a href="#orgde60939">1.2. Realistic Bending Stiffness Values</a>
<li><a href="#org19fa00c">1.1. Initialization</a></li>
<li><a href="#orgc66d5ef">1.2. Realistic Bending Stiffness Values</a>
<ul>
<li><a href="#orge13b41c">1.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orgd5fd59b">1.2.2. Primary Plant</a></li>
<li><a href="#org865157e">1.2.3. Conclusion</a></li>
<li><a href="#org7f86aa6">1.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orgd763ab8">1.2.2. Primary Plant</a></li>
<li><a href="#org34d48fd">1.2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org8ad3f34">1.3. Parametric Study</a>
<li><a href="#org2383b0a">1.3. Parametric Study</a>
<ul>
<li><a href="#orgc98ee7c">1.3.1. Direct Velocity Feedback</a></li>
<li><a href="#org15c2c08">1.3.2. Primary Control</a></li>
<li><a href="#org5322ecd">1.3.3. Conclusion</a></li>
<li><a href="#org0b2ea81">1.3.1. Direct Velocity Feedback</a></li>
<li><a href="#org4bcbd52">1.3.2. Primary Control</a></li>
<li><a href="#orgd481cbd">1.3.3. Conclusion</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgdaf7b6c">2. Axial Stiffness</a>
<li><a href="#org122af58">2. Axial Stiffness</a>
<ul>
<li><a href="#org969d9e7">2.1. Realistic Translation Stiffness Values</a>
<li><a href="#orgbd421fd">2.1. Realistic Translation Stiffness Values</a>
<ul>
<li><a href="#org7dd21d5">2.1.1. Initialization</a></li>
<li><a href="#org47be52b">2.1.2. Direct Velocity Feedback</a></li>
<li><a href="#org15105f5">2.1.3. Primary Plant</a></li>
<li><a href="#org2098f1e">2.1.4. Conclusion</a></li>
<li><a href="#org7e915de">2.1.1. Initialization</a></li>
<li><a href="#org484b32d">2.1.2. Direct Velocity Feedback</a></li>
<li><a href="#org4c02123">2.1.3. Primary Plant</a></li>
<li><a href="#org8aa50f5">2.1.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org0275632">2.2. Parametric study</a>
<li><a href="#orge0a9c34">2.2. Parametric study</a>
<ul>
<li><a href="#orgd87b94b">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orge5d1c12">2.2.2. Primary Control</a></li>
<li><a href="#org564cc95">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orgc1721b8">2.2.2. Primary Control</a></li>
</ul>
</li>
<li><a href="#org382b3cb">2.3. Conclusion</a></li>
<li><a href="#org13f48c9">2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgb6f6c0a">3. Conclusion</a></li>
<li><a href="#orgdf2870d">4. Designed Flexible Joints</a>
<li><a href="#org5519f74">3. Conclusion</a></li>
<li><a href="#org358d063">4. Designed Flexible Joints</a>
<ul>
<li><a href="#orgd355fcb">4.1. Initialization</a></li>
<li><a href="#org43c7d3c">4.2. Direct Velocity Feedback</a></li>
<li><a href="#org056a1de">4.3. Integral Force Feedback</a></li>
<li><a href="#org1148778">4.1. Initialization</a></li>
<li><a href="#orgd889755">4.2. Direct Velocity Feedback</a></li>
<li><a href="#org12972b2">4.3. Integral Force Feedback</a></li>
</ul>
</li>
</ul>
@@ -93,40 +98,40 @@ Ideally, we want the x and y rotations to be free and all the translations to be
However, this is never the case and be have to consider:
</p>
<ul class="org-ul">
<li>Finite bending stiffnesses (Section <a href="#org3eb4121">1</a>)</li>
<li>Axial stiffness in the direction of the legs (Section <a href="#org8f4d83b">2</a>)</li>
<li>Finite bending stiffnesses (Section <a href="#org27b9411">1</a>)</li>
<li>Axial stiffness in the direction of the legs (Section <a href="#org7789cf6">2</a>)</li>
</ul>
<p>
This may impose some limitations, also, the goal is to specify the required joints stiffnesses (summarized in Section <a href="#org6614f42">3</a>).
This may impose some limitations, also, the goal is to specify the required joints stiffnesses (summarized in Section <a href="#org8e0b118">3</a>).
</p>
<div id="outline-container-orge032d30" class="outline-2">
<h2 id="orge032d30"><span class="section-number-2">1</span> Bending and Torsional Stiffness</h2>
<div id="outline-container-org157c72f" class="outline-2">
<h2 id="org157c72f"><span class="section-number-2">1</span> Bending and Torsional Stiffness</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org3eb4121"></a>
<a id="org27b9411"></a>
</p>
<p>
In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.
</p>
</div>
<div id="outline-container-orge82a7c2" class="outline-3">
<h3 id="orge82a7c2"><span class="section-number-3">1.1</span> Initialization</h3>
<div id="outline-container-org19fa00c" class="outline-3">
<h3 id="org19fa00c"><span class="section-number-3">1.1</span> Initialization</h3>
<div class="outline-text-3" id="text-1-1">
<p>
Let&rsquo;s initialize all the stages with default parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
<pre class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
</pre>
</div>
@@ -134,15 +139,15 @@ initializeMirror();
Let&rsquo;s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 50, <span class="org-string">'freq'</span>, 200<span class="org-type">*</span>ones(6,1));
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 60);
<pre class="src src-matlab"> initializeSample(<span class="org-string">'mass'</span>, 50, <span class="org-string">'freq'</span>, 200<span class="org-type">*</span>ones(6,1));
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 60);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgde60939" class="outline-3">
<h3 id="orgde60939"><span class="section-number-3">1.2</span> Realistic Bending Stiffness Values</h3>
<div id="outline-container-orgc66d5ef" class="outline-3">
<h3 id="orgc66d5ef"><span class="section-number-3">1.2</span> Realistic Bending Stiffness Values</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Let&rsquo;s compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:
@@ -153,10 +158,10 @@ Let&rsquo;s compare the ideal case (zero stiffness in rotation and infinite stif
</ul>
<div class="org-src-container">
<pre class="src src-matlab">Kf_M = 15<span class="org-type">*</span>ones(6,1);
Kf_F = 15<span class="org-type">*</span>ones(6,1);
Kt_M = 20<span class="org-type">*</span>ones(6,1);
Kt_F = 20<span class="org-type">*</span>ones(6,1);
<pre class="src src-matlab"> Kf_M = 15<span class="org-type">*</span>ones(6,1);
Kf_F = 15<span class="org-type">*</span>ones(6,1);
Kt_M = 20<span class="org-type">*</span>ones(6,1);
Kt_F = 20<span class="org-type">*</span>ones(6,1);
</pre>
</div>
@@ -164,8 +169,8 @@ Kt_F = 20<span class="org-type">*</span>ones(6,1);
The stiffness and damping of the nano-hexapod&rsquo;s legs are:
</p>
<div class="org-src-container">
<pre class="src src-matlab">k_opt = 1e5; <span class="org-comment">% [N/m]</span>
c_opt = 2e2; <span class="org-comment">% [N/(m/s)]</span>
<pre class="src src-matlab"> k_opt = 1e5; <span class="org-comment">% [N/m]</span>
c_opt = 2e2; <span class="org-comment">% [N/(m/s)]</span>
</pre>
</div>
@@ -174,20 +179,20 @@ This corresponds to the optimal identified stiffness.
</p>
</div>
<div id="outline-container-orge13b41c" class="outline-4">
<h4 id="orge13b41c"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
<div id="outline-container-org7f86aa6" class="outline-4">
<h4 id="org7f86aa6"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-1-2-1">
<p>
We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness.
</p>
<p>
The obtained dynamics are shown in Figure <a href="#org656fd1c">1</a>.
The obtained dynamics are shown in Figure <a href="#orgc127d0f">1</a>.
It is shown that the adding of stiffness for the flexible joints does increase a little bit the frequencies of the mass suspension modes. It stiffen the structure.
</p>
<div id="org656fd1c" class="figure">
<div id="orgc127d0f" class="figure">
<p><img src="figs/flex_joint_rot_dvf.png" alt="flex_joint_rot_dvf.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint stiffness</p>
@@ -195,20 +200,20 @@ It is shown that the adding of stiffness for the flexible joints does increase a
</div>
</div>
<div id="outline-container-orgd5fd59b" class="outline-4">
<h4 id="orgd5fd59b"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
<div id="outline-container-orgd763ab8" class="outline-4">
<h4 id="orgd763ab8"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
<div class="outline-text-4" id="text-1-2-2">
<p>
Let&rsquo;s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
</p>
<p>
The dynamics is compare with and without the joint flexibility in Figure <a href="#org4322feb">2</a>.
The dynamics is compare with and without the joint flexibility in Figure <a href="#orgbca1f59">2</a>.
The plant dynamics is not found to be changing significantly.
</p>
<div id="org4322feb" class="figure">
<div id="orgbca1f59" class="figure">
<p><img src="figs/flex_joints_rot_primary_plant_L.png" alt="flex_joints_rot_primary_plant_L.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints and with flexible joints</p>
@@ -216,10 +221,10 @@ The plant dynamics is not found to be changing significantly.
</div>
</div>
<div id="outline-container-org865157e" class="outline-4">
<h4 id="org865157e"><span class="section-number-4">1.2.3</span> Conclusion</h4>
<div id="outline-container-org34d48fd" class="outline-4">
<h4 id="org34d48fd"><span class="section-number-4">1.2.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-1-2-3">
<div class="important" id="org69f9617">
<div class="important" id="org5365e9f">
<p>
Considering realistic flexible joint bending stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.
</p>
@@ -233,8 +238,8 @@ It only increases a little bit the suspension modes of the sample on top of the
</div>
</div>
<div id="outline-container-org8ad3f34" class="outline-3">
<h3 id="org8ad3f34"><span class="section-number-3">1.3</span> Parametric Study</h3>
<div id="outline-container-org2383b0a" class="outline-3">
<h3 id="org2383b0a"><span class="section-number-3">1.3</span> Parametric Study</h3>
<div class="outline-text-3" id="text-1-3">
<p>
We wish now to see what is the impact of the rotation stiffness of the flexible joints on the dynamics.
@@ -245,7 +250,7 @@ This will help to determine the requirements on the joint&rsquo;s stiffness.
Let&rsquo;s consider the following bending stiffness of the flexible joints:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ks = [1, 5, 10, 50, 100]; <span class="org-comment">% [Nm/rad]</span>
<pre class="src src-matlab"> Ks = [1, 5, 10, 50, 100]; <span class="org-comment">% [Nm/rad]</span>
</pre>
</div>
@@ -254,15 +259,15 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p>
</div>
<div id="outline-container-orgc98ee7c" class="outline-4">
<h4 id="orgc98ee7c"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
<div id="outline-container-org0b2ea81" class="outline-4">
<h4 id="org0b2ea81"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-1-3-1">
<p>
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#org8fbbf9d">3</a>.
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#org849b1d6">3</a>.
</p>
<p>
The corresponding Root Locus plot is shown in Figure <a href="#orgb9f3389">4</a>.
The corresponding Root Locus plot is shown in Figure <a href="#org311fafc">4</a>.
</p>
<p>
@@ -270,14 +275,14 @@ It is shown that the bending stiffness of the flexible joints does indeed change
</p>
<div id="org8fbbf9d" class="figure">
<div id="org849b1d6" class="figure">
<p><img src="figs/flex_joints_rot_study_dvf.png" alt="flex_joints_rot_study_dvf.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered Rotation Stiffnesses</p>
</div>
<div id="orgb9f3389" class="figure">
<div id="org311fafc" class="figure">
<p><img src="figs/flex_joints_rot_study_dvf_root_locus.png" alt="flex_joints_rot_study_dvf_root_locus.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Root Locus for all the considered Rotation Stiffnesses</p>
@@ -285,11 +290,11 @@ It is shown that the bending stiffness of the flexible joints does indeed change
</div>
</div>
<div id="outline-container-org15c2c08" class="outline-4">
<h4 id="org15c2c08"><span class="section-number-4">1.3.2</span> Primary Control</h4>
<div id="outline-container-org4bcbd52" class="outline-4">
<h4 id="org4bcbd52"><span class="section-number-4">1.3.2</span> Primary Control</h4>
<div class="outline-text-4" id="text-1-3-2">
<p>
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#orgb739560">5</a>.
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#orgaed64b5">5</a>.
</p>
<p>
@@ -297,7 +302,7 @@ It is shown that the bending stiffness of the flexible joints have very little i
</p>
<div id="orgb739560" class="figure">
<div id="orgaed64b5" class="figure">
<p><img src="figs/flex_joints_rot_study_primary_plant.png" alt="flex_joints_rot_study_primary_plant.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered bending stiffnesses</p>
@@ -305,10 +310,10 @@ It is shown that the bending stiffness of the flexible joints have very little i
</div>
</div>
<div id="outline-container-org5322ecd" class="outline-4">
<h4 id="org5322ecd"><span class="section-number-4">1.3.3</span> Conclusion</h4>
<div id="outline-container-orgd481cbd" class="outline-4">
<h4 id="orgd481cbd"><span class="section-number-4">1.3.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-1-3-3">
<div class="important" id="orga223c1a">
<div class="important" id="org98dc84f">
<p>
The bending stiffness of the flexible joint does not significantly change the dynamics.
</p>
@@ -319,19 +324,19 @@ The bending stiffness of the flexible joint does not significantly change the dy
</div>
</div>
<div id="outline-container-orgdaf7b6c" class="outline-2">
<h2 id="orgdaf7b6c"><span class="section-number-2">2</span> Axial Stiffness</h2>
<div id="outline-container-org122af58" class="outline-2">
<h2 id="org122af58"><span class="section-number-2">2</span> Axial Stiffness</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org8f4d83b"></a>
<a id="org7789cf6"></a>
</p>
<p>
Let&rsquo;s know consider a flexibility in translation of the flexible joint, in the axis of the legs.
</p>
</div>
<div id="outline-container-org969d9e7" class="outline-3">
<h3 id="org969d9e7"><span class="section-number-3">2.1</span> Realistic Translation Stiffness Values</h3>
<div id="outline-container-orgbd421fd" class="outline-3">
<h3 id="orgbd421fd"><span class="section-number-3">2.1</span> Realistic Translation Stiffness Values</h3>
<div class="outline-text-3" id="text-2-1">
<p>
We choose realistic values of the axial stiffness of the joints:
@@ -339,29 +344,29 @@ We choose realistic values of the axial stiffness of the joints:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ka_F = 60e6<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/m]</span>
Ka_M = 60e6<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/m]</span>
Ca_F = 1<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/(m/s)]</span>
Ca_M = 1<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/(m/s)]</span>
<pre class="src src-matlab"> Ka_F = 60e6<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/m]</span>
Ka_M = 60e6<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/m]</span>
Ca_F = 1<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/(m/s)]</span>
Ca_M = 1<span class="org-type">*</span>ones(6,1); <span class="org-comment">% [N/(m/s)]</span>
</pre>
</div>
</div>
<div id="outline-container-org7dd21d5" class="outline-4">
<h4 id="org7dd21d5"><span class="section-number-4">2.1.1</span> Initialization</h4>
<div id="outline-container-org7e915de" class="outline-4">
<h4 id="org7e915de"><span class="section-number-4">2.1.1</span> Initialization</h4>
<div class="outline-text-4" id="text-2-1-1">
<p>
Let&rsquo;s initialize all the stages with default parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
<pre class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
</pre>
</div>
@@ -369,26 +374,26 @@ initializeMirror();
Let&rsquo;s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 50, <span class="org-string">'freq'</span>, 200<span class="org-type">*</span>ones(6,1));
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 60);
<pre class="src src-matlab"> initializeSample(<span class="org-string">'mass'</span>, 50, <span class="org-string">'freq'</span>, 200<span class="org-type">*</span>ones(6,1));
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 60);
</pre>
</div>
</div>
</div>
<div id="outline-container-org47be52b" class="outline-4">
<h4 id="org47be52b"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
<div id="outline-container-org484b32d" class="outline-4">
<h4 id="org484b32d"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-2-1-2">
<p>
The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified.
</p>
<p>
The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
The obtained dynamics are shown in Figure <a href="#org90e79ad">6</a>.
</p>
<div id="org78dd87a" class="figure">
<div id="org90e79ad" class="figure">
<p><img src="figs/flex_joint_trans_dvf.png" alt="flex_joint_trans_dvf.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint axis stiffness</p>
@@ -396,11 +401,11 @@ The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
</div>
</div>
<div id="outline-container-org15105f5" class="outline-4">
<h4 id="org15105f5"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
<div id="outline-container-org4c02123" class="outline-4">
<h4 id="org4c02123"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
<div class="outline-text-4" id="text-2-1-3">
<div class="org-src-container">
<pre class="src src-matlab">Kdvf = 5e3<span class="org-type">*</span>s<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1e3)<span class="org-type">*</span>eye(6);
<pre class="src src-matlab"> Kdvf = 5e3<span class="org-type">*</span>s<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1e3)<span class="org-type">*</span>eye(6);
</pre>
</div>
@@ -409,11 +414,11 @@ Let&rsquo;s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilo
</p>
<p>
The dynamics is compare with and without the joint flexibility in Figure <a href="#org9bd0791">7</a>.
The dynamics is compare with and without the joint flexibility in Figure <a href="#org98bb072">7</a>.
</p>
<div id="org9bd0791" class="figure">
<div id="org98bb072" class="figure">
<p><img src="figs/flex_joints_trans_primary_plant_L.png" alt="flex_joints_trans_primary_plant_L.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed)</p>
@@ -421,10 +426,10 @@ The dynamics is compare with and without the joint flexibility in Figure <a href
</div>
</div>
<div id="outline-container-org2098f1e" class="outline-4">
<h4 id="org2098f1e"><span class="section-number-4">2.1.4</span> Conclusion</h4>
<div id="outline-container-org8aa50f5" class="outline-4">
<h4 id="org8aa50f5"><span class="section-number-4">2.1.4</span> Conclusion</h4>
<div class="outline-text-4" id="text-2-1-4">
<div class="important" id="org3a7d9f4">
<div class="important" id="org2e8394a">
<p>
For the realistic value of the flexible joint axial stiffness, the dynamics is not much impact, and this should not be a problem for control.
</p>
@@ -434,8 +439,8 @@ For the realistic value of the flexible joint axial stiffness, the dynamics is n
</div>
</div>
<div id="outline-container-org0275632" class="outline-3">
<h3 id="org0275632"><span class="section-number-3">2.2</span> Parametric study</h3>
<div id="outline-container-orge0a9c34" class="outline-3">
<h3 id="orge0a9c34"><span class="section-number-3">2.2</span> Parametric study</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We wish now to see what is the impact of the <b>axial</b> stiffness of the flexible joints on the dynamics.
@@ -445,7 +450,7 @@ We wish now to see what is the impact of the <b>axial</b> stiffness of the flexi
Let&rsquo;s consider the following values for the axial stiffness:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kas = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; <span class="org-comment">% [N/m]</span>
<pre class="src src-matlab"> Kas = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; <span class="org-comment">% [N/m]</span>
</pre>
</div>
@@ -454,11 +459,11 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p>
</div>
<div id="outline-container-orgd87b94b" class="outline-4">
<h4 id="orgd87b94b"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
<div id="outline-container-org564cc95" class="outline-4">
<h4 id="org564cc95"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-2-2-1">
<p>
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#orgab9ab86">8</a>.
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#orgeed898e">8</a>.
</p>
<p>
@@ -470,26 +475,26 @@ If the axial stiffness of the flexible joints is \(K_a > 10^7\,[N/m]\) (here \(1
</p>
<p>
This is more clear by looking at the root locus (Figures <a href="#org9d43966">9</a> and <a href="#org987d98e">10</a>).
This is more clear by looking at the root locus (Figures <a href="#org399589b">9</a> and <a href="#org5154002">10</a>).
It can be seen that very little active damping can be achieve for axial stiffnesses below \(10^7\,[N/m]\).
</p>
<div id="orgab9ab86" class="figure">
<div id="orgeed898e" class="figure">
<p><img src="figs/flex_joints_trans_study_dvf.png" alt="flex_joints_trans_study_dvf.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered axis Stiffnesses</p>
</div>
<div id="org9d43966" class="figure">
<div id="org399589b" class="figure">
<p><img src="figs/flex_joints_trans_study_dvf_root_locus.png" alt="flex_joints_trans_study_dvf_root_locus.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Root Locus for all the considered axial Stiffnesses</p>
</div>
<div id="org987d98e" class="figure">
<div id="org5154002" class="figure">
<p><img src="figs/flex_joints_trans_study_root_locus_unzoom.png" alt="flex_joints_trans_study_root_locus_unzoom.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Root Locus (unzoom) for all the considered axial Stiffnesses</p>
@@ -497,15 +502,15 @@ It can be seen that very little active damping can be achieve for axial stiffnes
</div>
</div>
<div id="outline-container-orge5d1c12" class="outline-4">
<h4 id="orge5d1c12"><span class="section-number-4">2.2.2</span> Primary Control</h4>
<div id="outline-container-orgc1721b8" class="outline-4">
<h4 id="orgc1721b8"><span class="section-number-4">2.2.2</span> Primary Control</h4>
<div class="outline-text-4" id="text-2-2-2">
<p>
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#org6070692">11</a>.
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#orgcab6267">11</a>.
</p>
<div id="org6070692" class="figure">
<div id="orgcab6267" class="figure">
<p><img src="figs/flex_joints_trans_study_primary_plant.png" alt="flex_joints_trans_study_primary_plant.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered axial stiffnesses</p>
@@ -514,10 +519,10 @@ The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for
</div>
</div>
<div id="outline-container-org382b3cb" class="outline-3">
<h3 id="org382b3cb"><span class="section-number-3">2.3</span> Conclusion</h3>
<div id="outline-container-org13f48c9" class="outline-3">
<h3 id="org13f48c9"><span class="section-number-3">2.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-3">
<div class="important" id="org422e802">
<div class="important" id="org4a497ad">
<p>
The axial stiffness of the flexible joints should be maximized.
</p>
@@ -539,14 +544,14 @@ We may interpolate the results and say that the axial joint stiffness should be
</div>
</div>
<div id="outline-container-orgb6f6c0a" class="outline-2">
<h2 id="orgb6f6c0a"><span class="section-number-2">3</span> Conclusion</h2>
<div id="outline-container-org5519f74" class="outline-2">
<h2 id="org5519f74"><span class="section-number-2">3</span> Conclusion</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org6614f42"></a>
<a id="org8e0b118"></a>
</p>
<div class="important" id="org3cbf243">
<div class="important" id="org3aeae4e">
<p>
In this study we considered the bending, torsional and axial stiffnesses of the flexible joints used for the nano-hexapod.
</p>
@@ -583,26 +588,26 @@ As there is generally a trade-off between bending stiffness and axial stiffness,
</div>
<div id="outline-container-orgdf2870d" class="outline-2">
<h2 id="orgdf2870d"><span class="section-number-2">4</span> Designed Flexible Joints</h2>
<div id="outline-container-org358d063" class="outline-2">
<h2 id="org358d063"><span class="section-number-2">4</span> Designed Flexible Joints</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-orgd355fcb" class="outline-3">
<h3 id="orgd355fcb"><span class="section-number-3">4.1</span> Initialization</h3>
<div id="outline-container-org1148778" class="outline-3">
<h3 id="org1148778"><span class="section-number-3">4.1</span> Initialization</h3>
<div class="outline-text-3" id="text-4-1">
<p>
Let&rsquo;s initialize all the stages with default parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror();
<pre class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror();
</pre>
</div>
@@ -610,51 +615,51 @@ initializeMirror();
Let&rsquo;s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 50, <span class="org-string">'freq'</span>, 200<span class="org-type">*</span>ones(6,1));
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 60);
<pre class="src src-matlab"> initializeSample(<span class="org-string">'mass'</span>, 50, <span class="org-string">'freq'</span>, 200<span class="org-type">*</span>ones(6,1));
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating-not-filtered'</span>, <span class="org-string">'Rz_period'</span>, 60);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">flex_joint = load(<span class="org-string">'./mat/flexor_025.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
apa = load(<span class="org-string">'./mat/APA300ML_simplified_model.mat'</span>);
<pre class="src src-matlab"> flex_joint = load(<span class="org-string">'./mat/flexor_025.mat'</span>, <span class="org-string">'int_xyz'</span>, <span class="org-string">'int_i'</span>, <span class="org-string">'n_xyz'</span>, <span class="org-string">'n_i'</span>, <span class="org-string">'nodes'</span>, <span class="org-string">'M'</span>, <span class="org-string">'K'</span>);
apa = load(<span class="org-string">'./mat/APA300ML_simplified_model.mat'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'amplified'</span>, ...
<span class="org-string">'ke'</span>, apa.ke, <span class="org-string">'ka'</span>, apa.ka, <span class="org-string">'k1'</span>, apa.k1, <span class="org-string">'c1'</span>, apa.c1, <span class="org-string">'F_gain'</span>, apa.F_gain, ...
<span class="org-string">'type_M'</span>, <span class="org-string">'spherical_3dof'</span>, ...
<span class="org-string">'Kr_M'</span>, flex_joint.K(1,1)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Ka_M'</span>, flex_joint.K(3,3)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kf_M'</span>, flex_joint.K(4,4)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kt_M'</span>, flex_joint.K(6,6)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'type_F'</span>, <span class="org-string">'spherical_3dof'</span>, ...
<span class="org-string">'Kr_F'</span>, flex_joint.K(1,1)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Ka_F'</span>, flex_joint.K(3,3)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kf_F'</span>, flex_joint.K(4,4)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kt_F'</span>, flex_joint.K(6,6)<span class="org-type">*</span>ones(6,1));
<pre class="src src-matlab"> initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'amplified'</span>, ...
<span class="org-string">'ke'</span>, apa.ke, <span class="org-string">'ka'</span>, apa.ka, <span class="org-string">'k1'</span>, apa.k1, <span class="org-string">'c1'</span>, apa.c1, <span class="org-string">'F_gain'</span>, apa.F_gain, ...
<span class="org-string">'type_M'</span>, <span class="org-string">'spherical_3dof'</span>, ...
<span class="org-string">'Kr_M'</span>, flex_joint.K(1,1)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Ka_M'</span>, flex_joint.K(3,3)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kf_M'</span>, flex_joint.K(4,4)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kt_M'</span>, flex_joint.K(6,6)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'type_F'</span>, <span class="org-string">'spherical_3dof'</span>, ...
<span class="org-string">'Kr_F'</span>, flex_joint.K(1,1)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Ka_F'</span>, flex_joint.K(3,3)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kf_F'</span>, flex_joint.K(4,4)<span class="org-type">*</span>ones(6,1), ...
<span class="org-string">'Kt_F'</span>, flex_joint.K(6,6)<span class="org-type">*</span>ones(6,1));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeNanoHexapod();
<pre class="src src-matlab"> initializeNanoHexapod();
</pre>
</div>
</div>
</div>
<div id="outline-container-org43c7d3c" class="outline-3">
<h3 id="org43c7d3c"><span class="section-number-3">4.2</span> Direct Velocity Feedback</h3>
<div id="outline-container-orgd889755" class="outline-3">
<h3 id="orgd889755"><span class="section-number-3">4.2</span> Direct Velocity Feedback</h3>
</div>
<div id="outline-container-org056a1de" class="outline-3">
<h3 id="org056a1de"><span class="section-number-3">4.3</span> Integral Force Feedback</h3>
<div id="outline-container-org12972b2" class="outline-3">
<h3 id="org12972b2"><span class="section-number-3">4.3</span> Integral Force Feedback</h3>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-03 mar. 09:45</p>
<p class="date">Created: 2021-02-20 sam. 23:09</p>
</div>
</body>
</html>