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<head> <head>
<!-- 2021-04-23 ven. 15:50 --> <!-- 2021-04-23 ven. 17:35 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Nano-Hexapod</title> <title>Nano-Hexapod</title>
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
@ -76,26 +76,26 @@
</li> </li>
<li><a href="#org2fc54dd">2. Active Damping using Integral Force Feedback</a> <li><a href="#org2fc54dd">2. Active Damping using Integral Force Feedback</a>
<ul> <ul>
<li><a href="#org18c0a7a">2.1. Plant Identification</a></li> <li><a href="#org1565813">2.1. Plant Identification</a></li>
<li><a href="#org12ae962">2.2. Root Locus</a></li> <li><a href="#org9a946f0">2.2. Root Locus</a></li>
<li><a href="#org78de4b8">2.3. Effect of IFF on the plant</a></li> <li><a href="#org78de4b8">2.3. Effect of IFF on the plant</a></li>
<li><a href="#org8db31cb">2.4. Effect of IFF on the compliance</a></li> <li><a href="#org8db31cb">2.4. Effect of IFF on the compliance</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org605df2e">3. Active Damping using Direct Velocity Feedback - Encoders on the struts</a> <li><a href="#org605df2e">3. Active Damping using Direct Velocity Feedback - Encoders on the struts</a>
<ul> <ul>
<li><a href="#orge9801cd">3.1. Plant Identification</a></li> <li><a href="#orgbb8eda6">3.1. Plant Identification</a></li>
<li><a href="#org0e2a9f0">3.2. Root Locus</a></li> <li><a href="#org7ce0b94">3.2. Root Locus</a></li>
<li><a href="#org06b118d">3.3. Effect of DVF on the plant</a></li> <li><a href="#org949e780">3.3. Effect of DVF on the plant</a></li>
<li><a href="#org4a7a21f">3.4. Effect of DVF on the compliance</a></li> <li><a href="#org755376c">3.4. Effect of DVF on the compliance</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgac5a0fc">4. Active Damping using Direct Velocity Feedback - Encoders on the plates</a> <li><a href="#orgac5a0fc">4. Active Damping using Direct Velocity Feedback - Encoders on the plates</a>
<ul> <ul>
<li><a href="#org79dc366">4.1. Plant Identification</a></li> <li><a href="#org5344b4a">4.1. Plant Identification</a></li>
<li><a href="#org3088061">4.2. Root Locus</a></li> <li><a href="#orge23105c">4.2. Root Locus</a></li>
<li><a href="#orged4d86f">4.3. Effect of DVF on the plant</a></li> <li><a href="#org4e3c598">4.3. Effect of DVF on the plant</a></li>
<li><a href="#orgc5c32c0">4.4. Effect of DVF on the compliance</a></li> <li><a href="#org276122b">4.4. Effect of DVF on the compliance</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8862f6b">5. Function - Initialize Nano Hexapod</a> <li><a href="#org8862f6b">5. Function - Initialize Nano Hexapod</a>
@ -129,31 +129,31 @@ In this document, a Simscape model of the nano-hexapod is developed and studied.
It is structured as follows: It is structured as follows:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org5550b38">1</a>: the simscape model of the nano-hexapod is presented. Few of its elements can be configured as wanted. The effect of the configuration on the obtained dynamics is studied.</li> <li>Section <a href="#org3121227">1</a>: the simscape model of the nano-hexapod is presented. Few of its elements can be configured as wanted. The effect of the configuration on the obtained dynamics is studied.</li>
<li>Section <a href="#orgda93dd4">2</a>: Direct Velocity Feedback is applied and the obtained damping is derived.</li> <li>Section <a href="#orgf5bb4a7">2</a>: Direct Velocity Feedback is applied and the obtained damping is derived.</li>
<li>Section <a href="#orge64a27c">3</a>: the encoders are fixed to the struts, and Integral Force Feedback is applied. The obtained damping is computed.</li> <li>Section <a href="#orgd06ebe5">3</a>: the encoders are fixed to the struts, and Integral Force Feedback is applied. The obtained damping is computed.</li>
<li>Section <a href="#org6e295ec">4</a>: the same is done when the encoders are fixed on the plates</li> <li>Section <a href="#orgb09098a">4</a>: the same is done when the encoders are fixed on the plates</li>
</ul> </ul>
<div id="outline-container-org3cc44d5" class="outline-2"> <div id="outline-container-org3cc44d5" class="outline-2">
<h2 id="org3cc44d5"><span class="section-number-2">1</span> Nano-Hexapod</h2> <h2 id="org3cc44d5"><span class="section-number-2">1</span> Nano-Hexapod</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="org5550b38"></a> <a id="org3121227"></a>
</p> </p>
</div> </div>
<div id="outline-container-org17a9502" class="outline-3"> <div id="outline-container-org17a9502" class="outline-3">
<h3 id="org17a9502"><span class="section-number-3">1.1</span> Nano Hexapod - Configuration</h3> <h3 id="org17a9502"><span class="section-number-3">1.1</span> Nano Hexapod - Configuration</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
<a id="org876957e"></a> <a id="orgb783e3d"></a>
</p> </p>
<p> <p>
The nano-hexapod can be initialized and configured using the <code>initializeNanoHexapodFinal</code> function (<a href="#org32d0912">link</a>). The nano-hexapod can be initialized and configured using the <code>initializeNanoHexapodFinal</code> function (<a href="#org877b216">link</a>).
</p> </p>
<p> <p>
The following code would produce the model shown in Figure <a href="#org4ec5fab">1</a>. The following code would produce the model shown in Figure <a href="#orgb0efd5b">1</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ... <pre class="src src-matlab">n_hexapod = initializeNanoHexapodFinal(<span class="org-string">'flex_bot_type'</span>, <span class="org-string">'4dof'</span>, ...
@ -165,7 +165,7 @@ The following code would produce the model shown in Figure <a href="#org4ec5fab"
</div> </div>
<div id="org4ec5fab" class="figure"> <div id="orgb0efd5b" class="figure">
<p><img src="figs/nano_hexapod_simscape_encoder_struts.png" alt="nano_hexapod_simscape_encoder_struts.png" /> <p><img src="figs/nano_hexapod_simscape_encoder_struts.png" alt="nano_hexapod_simscape_encoder_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>3D view of the Sismcape model for the Nano-Hexapod</p> <p><span class="figure-number">Figure 1: </span>3D view of the Sismcape model for the Nano-Hexapod</p>
@ -175,21 +175,21 @@ The following code would produce the model shown in Figure <a href="#org4ec5fab"
Several elements on the nano-hexapod can be configured: Several elements on the nano-hexapod can be configured:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>The flexible joints (Section <a href="#org9bbaee3">1.1.1</a>)</li> <li>The flexible joints (Section <a href="#org6815cd6">1.1.1</a>)</li>
<li>The amplified piezoelectric actuators (Section <a href="#org4a142ba">1.1.2</a>)</li> <li>The amplified piezoelectric actuators (Section <a href="#orgf7bc82d">1.1.2</a>)</li>
<li>The encoders (Section <a href="#orgef9e0ec">1.1.3</a>)</li> <li>The encoders (Section <a href="#orgb099f97">1.1.3</a>)</li>
<li>The Jacobian matrices (Section <a href="#orge0269f6">1.1.4</a>)</li> <li>The Jacobian matrices (Section <a href="#org5d30139">1.1.4</a>)</li>
</ul> </ul>
</div> </div>
<div id="outline-container-orgd406621" class="outline-4"> <div id="outline-container-orgd406621" class="outline-4">
<h4 id="orgd406621"><span class="section-number-4">1.1.1</span> Flexible Joints</h4> <h4 id="orgd406621"><span class="section-number-4">1.1.1</span> Flexible Joints</h4>
<div class="outline-text-4" id="text-1-1-1"> <div class="outline-text-4" id="text-1-1-1">
<p> <p>
<a id="org9bbaee3"></a> <a id="org6815cd6"></a>
</p> </p>
<p> <p>
The model of the flexible joint is composed of 3 solid bodies as shown in Figure <a href="#org3a7d26c">2</a> which are connected by joints representing the flexibility of the joint. The model of the flexible joint is composed of 3 solid bodies as shown in Figure <a href="#org6afa01e">2</a> which are connected by joints representing the flexibility of the joint.
</p> </p>
<p> <p>
@ -202,10 +202,10 @@ We can represent:
</ul> </ul>
<p> <p>
The configurations and the represented flexibilities are summarized in Table <a href="#orgb044203">1</a>. The configurations and the represented flexibilities are summarized in Table <a href="#org933a214">1</a>.
</p> </p>
<table id="orgb044203" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org933a214" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Flexible joint&rsquo;s configuration and associated represented flexibility</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Flexible joint&rsquo;s configuration and associated represented flexibility</caption>
<colgroup> <colgroup>
@ -254,7 +254,7 @@ Of course, adding more DoF for the flexible joint will induce an addition of man
</p> </p>
<div id="org3a7d26c" class="figure"> <div id="org6afa01e" class="figure">
<p><img src="figs/simscape_model_flexible_joint.png" alt="simscape_model_flexible_joint.png" /> <p><img src="figs/simscape_model_flexible_joint.png" alt="simscape_model_flexible_joint.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>3D view of the Sismcape model for the Flexible joint (4DoF configuration)</p> <p><span class="figure-number">Figure 2: </span>3D view of the Sismcape model for the Flexible joint (4DoF configuration)</p>
@ -266,7 +266,7 @@ Of course, adding more DoF for the flexible joint will induce an addition of man
<h4 id="org38f71cc"><span class="section-number-4">1.1.2</span> Amplified Piezoelectric Actuators</h4> <h4 id="org38f71cc"><span class="section-number-4">1.1.2</span> Amplified Piezoelectric Actuators</h4>
<div class="outline-text-4" id="text-1-1-2"> <div class="outline-text-4" id="text-1-1-2">
<p> <p>
<a id="org4a142ba"></a> <a id="orgf7bc82d"></a>
</p> </p>
<p> <p>
@ -278,11 +278,11 @@ The APA can be modeled in different ways which can be configured with the <code>
</p> </p>
<p> <p>
The simplest model is a 2-DoF system shown in Figure <a href="#orgee648ad">3</a>. The simplest model is a 2-DoF system shown in Figure <a href="#orgb2afdea">3</a>.
</p> </p>
<div id="orgee648ad" class="figure"> <div id="orgb2afdea" class="figure">
<p><img src="figs/2dof_apa_model.png" alt="2dof_apa_model.png" /> <p><img src="figs/2dof_apa_model.png" alt="2dof_apa_model.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Schematic of the 2DoF model for the Amplified Piezoelectric Actuator</p> <p><span class="figure-number">Figure 3: </span>Schematic of the 2DoF model for the Amplified Piezoelectric Actuator</p>
@ -298,11 +298,11 @@ Then, a more complex model based on a Finite Element Model can be used.
<h4 id="orgad18643"><span class="section-number-4">1.1.3</span> Encoders</h4> <h4 id="orgad18643"><span class="section-number-4">1.1.3</span> Encoders</h4>
<div class="outline-text-4" id="text-1-1-3"> <div class="outline-text-4" id="text-1-1-3">
<p> <p>
<a id="orgef9e0ec"></a> <a id="orgb099f97"></a>
</p> </p>
<p> <p>
The encoders can be either fixed directly on the struts (Figure <a href="#orga44ae3d">4</a>) or on the two plates (Figure <a href="#org0d3f6e9">5</a>). The encoders can be either fixed directly on the struts (Figure <a href="#org63b5540">4</a>) or on the two plates (Figure <a href="#org9165423">5</a>).
</p> </p>
<p> <p>
@ -310,32 +310,32 @@ This can be configured with the <code>motion_sensor_type</code> parameters which
</p> </p>
<div id="orga44ae3d" class="figure"> <div id="org63b5540" class="figure">
<p><img src="figs/encoder_struts.png" alt="encoder_struts.png" /> <p><img src="figs/encoder_struts.png" alt="encoder_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>3D view of the Encoders fixed on the struts</p> <p><span class="figure-number">Figure 4: </span>3D view of the Encoders fixed on the struts</p>
</div> </div>
<div id="org0d3f6e9" class="figure"> <div id="org9165423" class="figure">
<p><img src="figs/encoders_plates_with_apa.png" alt="encoders_plates_with_apa.png" /> <p><img src="figs/encoders_plates_with_apa.png" alt="encoders_plates_with_apa.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>3D view of the Encoders fixed on the plates</p> <p><span class="figure-number">Figure 5: </span>3D view of the Encoders fixed on the plates</p>
</div> </div>
<p> <p>
A complete view of the nano-hexapod with encoders fixed to the struts is shown in Figure <a href="#org4ec5fab">1</a> while it is shown in Figure <a href="#org8f430b9">6</a> when the encoders are fixed to the plates. A complete view of the nano-hexapod with encoders fixed to the struts is shown in Figure <a href="#orgb0efd5b">1</a> while it is shown in Figure <a href="#org96bb17b">6</a> when the encoders are fixed to the plates.
</p> </p>
<div id="org8f430b9" class="figure"> <div id="org96bb17b" class="figure">
<p><img src="figs/nano_hexapod_simscape_encoder_plates.png" alt="nano_hexapod_simscape_encoder_plates.png" /> <p><img src="figs/nano_hexapod_simscape_encoder_plates.png" alt="nano_hexapod_simscape_encoder_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Nano-Hexapod with encoders fixed to the plates</p> <p><span class="figure-number">Figure 6: </span>Nano-Hexapod with encoders fixed to the plates</p>
</div> </div>
<p> <p>
The encoder model is schematically represented in Figure <a href="#org5b503cd">7</a>: The encoder model is schematically represented in Figure <a href="#orgc57bc17">7</a>:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>a frame {B}, fixed to the ruler is positioned on its top surface</li> <li>a frame {B}, fixed to the ruler is positioned on its top surface</li>
@ -347,18 +347,18 @@ The output measurement is then the x displacement of the origin of the frame {F}
</p> </p>
<div id="org5b503cd" class="figure"> <div id="orgc57bc17" class="figure">
<p><img src="figs/simscape_encoder_model.png" alt="simscape_encoder_model.png" /> <p><img src="figs/simscape_encoder_model.png" alt="simscape_encoder_model.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Schematic of the encoder model</p> <p><span class="figure-number">Figure 7: </span>Schematic of the encoder model</p>
</div> </div>
<p> <p>
If the encoder is experiencing some tilt, it is then &ldquo;converted&rdquo; into a measured displacement as shown in Figure <a href="#org78384b8">8</a>. If the encoder is experiencing some tilt, it is then &ldquo;converted&rdquo; into a measured displacement as shown in Figure <a href="#org684088d">8</a>.
</p> </p>
<div id="org78384b8" class="figure"> <div id="org684088d" class="figure">
<p><img src="figs/simscape_encoder_model_disp.png" alt="simscape_encoder_model_disp.png" /> <p><img src="figs/simscape_encoder_model_disp.png" alt="simscape_encoder_model_disp.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Schematic of the encoder model</p> <p><span class="figure-number">Figure 8: </span>Schematic of the encoder model</p>
@ -370,7 +370,7 @@ If the encoder is experiencing some tilt, it is then &ldquo;converted&rdquo; int
<h4 id="org37ecf7a"><span class="section-number-4">1.1.4</span> Jacobians</h4> <h4 id="org37ecf7a"><span class="section-number-4">1.1.4</span> Jacobians</h4>
<div class="outline-text-4" id="text-1-1-4"> <div class="outline-text-4" id="text-1-1-4">
<p> <p>
<a id="orge0269f6"></a> <a id="org5d30139"></a>
</p> </p>
<p> <p>
@ -390,7 +390,7 @@ Same thing can be done for the measured encoder displacements.
<h3 id="orgf279891"><span class="section-number-3">1.2</span> Effect of encoders on the decentralized plant</h3> <h3 id="orgf279891"><span class="section-number-3">1.2</span> Effect of encoders on the decentralized plant</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
<a id="org1a9e872"></a> <a id="org0d966f9"></a>
</p> </p>
<p> <p>
@ -446,18 +446,19 @@ Gp.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string">
</div> </div>
<p> <p>
The obtained plants are compared in Figure <a href="#org97d47f2">9</a>. The obtained plants are compared in Figure <a href="#orga82a8d6">9</a>.
</p> </p>
<div id="org97d47f2" class="figure"> <div id="orga82a8d6" class="figure">
<p><img src="figs/nano_hexapod_effect_encoder.png" alt="nano_hexapod_effect_encoder.png" /> <p><img src="figs/nano_hexapod_effect_encoder.png" alt="nano_hexapod_effect_encoder.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Comparison of the plants from actuator to associated encoder when the encoders are either fixed to the struts or to the plates</p> <p><span class="figure-number">Figure 9: </span>Comparison of the plants from actuator to associated encoder when the encoders are either fixed to the struts or to the plates</p>
</div> </div>
<div class="question" id="orgf2977e4"> <div class="important" id="org19a2b42">
<p> <p>
Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the struts? The zeros at 400Hz and 800Hz should corresponds to resonances of the system when one of the APA is blocked.
It is linked to the axial stiffness of the flexible joints: increasing the axial stiffness of the joints will increase the frequency of the zeros.
</p> </p>
</div> </div>
@ -468,7 +469,7 @@ Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the strut
<h3 id="orgbea8fbc"><span class="section-number-3">1.3</span> Effect of APA flexibility</h3> <h3 id="orgbea8fbc"><span class="section-number-3">1.3</span> Effect of APA flexibility</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
<a id="orga454a3a"></a> <a id="org447e5c9"></a>
</p> </p>
<p> <p>
@ -503,15 +504,15 @@ Gf.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string">
</div> </div>
<div id="orge805a6a" class="figure"> <div id="org152edd8" class="figure">
<p><img src="figs/nano_hexapod_effect_flexible_apa.png" alt="nano_hexapod_effect_flexible_apa.png" /> <p><img src="figs/nano_hexapod_effect_flexible_apa.png" alt="nano_hexapod_effect_flexible_apa.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Comparison of the plants from actuator to associated strut encoder when the APA are modelled with a 2DoF system of with a flexible one</p> <p><span class="figure-number">Figure 10: </span>Comparison of the plants from actuator to associated strut encoder when the APA are modelled with a 2DoF system of with a flexible one</p>
</div> </div>
<div class="question" id="orgaab5d4b"> <div class="question" id="org7df512f">
<p> <p>
The first resonance is strange when using the flexible APA model (Figure <a href="#orge805a6a">10</a>). The first resonance is strange when using the flexible APA model (Figure <a href="#org152edd8">10</a>).
Moreover the system is unstable. Moreover the system is unstable.
Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity. Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity.
</p> </p>
@ -524,7 +525,7 @@ Otherwise, the 2DoF model matches quite well the flexible model considering its
<h3 id="orgb2e1d3b"><span class="section-number-3">1.4</span> Nano Hexapod - Number of DoF</h3> <h3 id="orgb2e1d3b"><span class="section-number-3">1.4</span> Nano Hexapod - Number of DoF</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
<p> <p>
<a id="org9a2a201"></a> <a id="org106ba36"></a>
</p> </p>
<p> <p>
@ -549,10 +550,10 @@ There are 24 states.
<p> <p>
These states are summarized on table <a href="#orgb13a7cc">2</a>. These states are summarized on table <a href="#org12d70d0">2</a>.
</p> </p>
<table id="orgb13a7cc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org12d70d0" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Number of states for the minimalist model</caption> <caption class="t-above"><span class="table-number">Table 2:</span> Number of states for the minimalist model</caption>
<colgroup> <colgroup>
@ -636,7 +637,7 @@ There are 60 states.
</pre> </pre>
<div class="important" id="orgce83bfe"> <div class="important" id="orgda52f21">
<p> <p>
Obtained number of states is very comprehensible. Obtained number of states is very comprehensible.
Depending on the physical effects we want to model, we therefore know how many states are added when configuring the model. Depending on the physical effects we want to model, we therefore know how many states are added when configuring the model.
@ -650,7 +651,7 @@ Depending on the physical effects we want to model, we therefore know how many s
<h3 id="orgcd30976"><span class="section-number-3">1.5</span> Direct Velocity Feedback Plant</h3> <h3 id="orgcd30976"><span class="section-number-3">1.5</span> Direct Velocity Feedback Plant</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
<p> <p>
<a id="org97e76e4"></a> <a id="orgcc85320"></a>
</p> </p>
<p> <p>
@ -692,9 +693,9 @@ DCgain = 1.87e-08 [m/N]
<p> <p>
Let&rsquo;s verify that by looking at the DC gain of the \(6 \times 6\) DVF plant in Table <a href="#org87ee4dc">3</a>. Let&rsquo;s verify that by looking at the DC gain of the \(6 \times 6\) DVF plant in Table <a href="#org24f45a8">3</a>.
</p> </p>
<table id="org87ee4dc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org24f45a8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> DC gain of the DVF plant</caption> <caption class="t-above"><span class="table-number">Table 3:</span> DC gain of the DVF plant</caption>
<colgroup> <colgroup>
@ -768,10 +769,10 @@ Let&rsquo;s verify that by looking at the DC gain of the \(6 \times 6\) DVF plan
</table> </table>
<p> <p>
And the bode plot of the DVF plant is shown in Figure <a href="#org7fd5fe3">11</a>. And the bode plot of the DVF plant is shown in Figure <a href="#org9ec592f">11</a>.
</p> </p>
<div id="org7fd5fe3" class="figure"> <div id="org9ec592f" class="figure">
<p><img src="figs/nano_hexapod_struts_2dof_dvf_plant.png" alt="nano_hexapod_struts_2dof_dvf_plant.png" /> <p><img src="figs/nano_hexapod_struts_2dof_dvf_plant.png" alt="nano_hexapod_struts_2dof_dvf_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Bode plot of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors attached to the struts \(\mathcal{L}_i\). Diagonal terms are shown in blue, and off-diagonal terms in black.</p> <p><span class="figure-number">Figure 11: </span>Bode plot of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors attached to the struts \(\mathcal{L}_i\). Diagonal terms are shown in blue, and off-diagonal terms in black.</p>
@ -783,7 +784,7 @@ And the bode plot of the DVF plant is shown in Figure <a href="#org7fd5fe3">11</
<h3 id="orgdefb22f"><span class="section-number-3">1.6</span> Integral Force Feedback Plant</h3> <h3 id="orgdefb22f"><span class="section-number-3">1.6</span> Integral Force Feedback Plant</h3>
<div class="outline-text-3" id="text-1-6"> <div class="outline-text-3" id="text-1-6">
<p> <p>
<a id="org065b8d6"></a> <a id="org74c9981"></a>
</p> </p>
<p> <p>
@ -816,10 +817,10 @@ This is corresponding to the dynamics for the Integral Force Feedback (IFF) cont
</p> </p>
<p> <p>
The bode plot is shown in Figure <a href="#org5d594c5">12</a>. The bode plot is shown in Figure <a href="#org53859fa">12</a>.
</p> </p>
<div id="org5d594c5" class="figure"> <div id="org53859fa" class="figure">
<p><img src="figs/nano_hexapod_struts_2dof_iff_plant.png" alt="nano_hexapod_struts_2dof_iff_plant.png" /> <p><img src="figs/nano_hexapod_struts_2dof_iff_plant.png" alt="nano_hexapod_struts_2dof_iff_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Bode plot of the transfer functions from actuator forces \(\tau_i\) to force sensors \(F_{m,i}\). Diagonal terms are shown in blue, and off-diagonal terms in black.</p> <p><span class="figure-number">Figure 12: </span>Bode plot of the transfer functions from actuator forces \(\tau_i\) to force sensors \(F_{m,i}\). Diagonal terms are shown in blue, and off-diagonal terms in black.</p>
@ -831,10 +832,10 @@ The bode plot is shown in Figure <a href="#org5d594c5">12</a>.
<h3 id="orgde9c23b"><span class="section-number-3">1.7</span> Decentralized Plant - Cartesian coordinates</h3> <h3 id="orgde9c23b"><span class="section-number-3">1.7</span> Decentralized Plant - Cartesian coordinates</h3>
<div class="outline-text-3" id="text-1-7"> <div class="outline-text-3" id="text-1-7">
<p> <p>
<a id="org960b2db"></a> <a id="org684f012"></a>
</p> </p>
<p> <p>
Consider the plant shown in Figure <a href="#orgb1d0313">13</a> with: Consider the plant shown in Figure <a href="#org2ec5b6e">13</a> with:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>\(\tau\) the 6 input forces (APA)</li> <li>\(\tau\) the 6 input forces (APA)</li>
@ -844,7 +845,7 @@ Consider the plant shown in Figure <a href="#orgb1d0313">13</a> with:
</ul> </ul>
<div id="orgb1d0313" class="figure"> <div id="org2ec5b6e" class="figure">
<p><img src="figs/nano_hexapod_decentralized_schematic.png" alt="nano_hexapod_decentralized_schematic.png" /> <p><img src="figs/nano_hexapod_decentralized_schematic.png" alt="nano_hexapod_decentralized_schematic.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Plant in the cartesian Frame</p> <p><span class="figure-number">Figure 13: </span>Plant in the cartesian Frame</p>
@ -898,10 +899,10 @@ Gsp = <span class="org-type">-</span>Gs({<span class="org-string">'Dx'</span>, <
</div> </div>
<p> <p>
The diagonal elements of the plant are shown in Figure <a href="#org6440a27">14</a>. The diagonal elements of the plant are shown in Figure <a href="#org178ee96">14</a>.
</p> </p>
<div id="org6440a27" class="figure"> <div id="org178ee96" class="figure">
<p><img src="figs/nano_hexapod_comp_cartesian_plants_struts.png" alt="nano_hexapod_comp_cartesian_plants_struts.png" /> <p><img src="figs/nano_hexapod_comp_cartesian_plants_struts.png" alt="nano_hexapod_comp_cartesian_plants_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using &ldquo;perfect&rdquo; 6dof sensor (dashed). The encoders are fixed on the struts.</p> <p><span class="figure-number">Figure 14: </span>Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using &ldquo;perfect&rdquo; 6dof sensor (dashed). The encoders are fixed on the struts.</p>
@ -929,16 +930,16 @@ Gpp = <span class="org-type">-</span>Gp({<span class="org-string">'Dx'</span>, <
</div> </div>
<p> <p>
The obtained bode plots are shown in Figure <a href="#org8fcd838">15</a>. The obtained bode plots are shown in Figure <a href="#orgc69bfd5">15</a>.
</p> </p>
<div id="org8fcd838" class="figure"> <div id="orgc69bfd5" class="figure">
<p><img src="figs/nano_hexapod_comp_cartesian_plants_plates.png" alt="nano_hexapod_comp_cartesian_plants_plates.png" /> <p><img src="figs/nano_hexapod_comp_cartesian_plants_plates.png" alt="nano_hexapod_comp_cartesian_plants_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using &ldquo;perfect&rdquo; 6dof sensor (dashed). The encoders are fixed on the plates.</p> <p><span class="figure-number">Figure 15: </span>Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using &ldquo;perfect&rdquo; 6dof sensor (dashed). The encoders are fixed on the plates.</p>
</div> </div>
<div class="important" id="org48ed4cf"> <div class="important" id="orga22af1a">
<p> <p>
The Jacobian for the encoders is working properly both when the encoders are fixed to the plates or to the struts. The Jacobian for the encoders is working properly both when the encoders are fixed to the plates or to the struts.
</p> </p>
@ -955,10 +956,10 @@ However, then the encoders are fixed to the struts, there is a mismatch between
<h4 id="org664eab9"><span class="section-number-4">1.7.2</span> Comparison of the decentralized plants</h4> <h4 id="org664eab9"><span class="section-number-4">1.7.2</span> Comparison of the decentralized plants</h4>
<div class="outline-text-4" id="text-1-7-2"> <div class="outline-text-4" id="text-1-7-2">
<p> <p>
The decentralized plants are now compared whether the encoders are fixed on the struts or on the plates in Figure <a href="#org82fcbd0">16</a>. The decentralized plants are now compared whether the encoders are fixed on the struts or on the plates in Figure <a href="#org1b4b9c9">16</a>.
</p> </p>
<div id="org82fcbd0" class="figure"> <div id="org1b4b9c9" class="figure">
<p><img src="figs/nano_hexapod_cartesian_plant_encoder_comp.png" alt="nano_hexapod_cartesian_plant_encoder_comp.png" /> <p><img src="figs/nano_hexapod_cartesian_plant_encoder_comp.png" alt="nano_hexapod_cartesian_plant_encoder_comp.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Bode plot of the &ldquo;cartesian&rdquo; plant (transfer function from \(\mathcal{F}\) to \(d\mathcal{X}\)) when the encoders are fixed on the struts (solid) and on the plates (dashed)</p> <p><span class="figure-number">Figure 16: </span>Bode plot of the &ldquo;cartesian&rdquo; plant (transfer function from \(\mathcal{F}\) to \(d\mathcal{X}\)) when the encoders are fixed on the struts (solid) and on the plates (dashed)</p>
@ -971,7 +972,7 @@ The decentralized plants are now compared whether the encoders are fixed on the
<h3 id="org6cf8089"><span class="section-number-3">1.8</span> Decentralized Plant - Decoupling at the Center of Stiffness</h3> <h3 id="org6cf8089"><span class="section-number-3">1.8</span> Decentralized Plant - Decoupling at the Center of Stiffness</h3>
<div class="outline-text-3" id="text-1-8"> <div class="outline-text-3" id="text-1-8">
<p> <p>
<a id="org04363e6"></a> <a id="org73c58d9"></a>
</p> </p>
</div> </div>
@ -979,7 +980,7 @@ The decentralized plants are now compared whether the encoders are fixed on the
<h4 id="org8c0186d"><span class="section-number-4">1.8.1</span> Center of Stiffness</h4> <h4 id="org8c0186d"><span class="section-number-4">1.8.1</span> Center of Stiffness</h4>
<div class="outline-text-4" id="text-1-8-1"> <div class="outline-text-4" id="text-1-8-1">
<p> <p>
<a id="org8ca00ea"></a> <a id="orgd86c328"></a>
</p> </p>
<p> <p>
@ -1085,7 +1086,7 @@ And the (normalized) stiffness matrix is computed as follows:
</pre> </pre>
</div> </div>
<table id="org2d3bf33" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgfd93441" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Normalized Stiffness Matrix - Center of Stiffness</caption> <caption class="t-above"><span class="table-number">Table 4:</span> Normalized Stiffness Matrix - Center of Stiffness</caption>
<colgroup> <colgroup>
@ -1202,9 +1203,9 @@ Then use the Jacobian matrices to obtain the &ldquo;cartesian&rdquo; centralized
</div> </div>
<p> <p>
The DC gain of the obtained plant is shown in Table <a href="#orgb273443">5</a>. The DC gain of the obtained plant is shown in Table <a href="#orgda45738">5</a>.
</p> </p>
<table id="orgb273443" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgda45738" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> DC gain of the centralized plant at the center of stiffness</caption> <caption class="t-above"><span class="table-number">Table 5:</span> DC gain of the centralized plant at the center of stiffness</caption>
<colgroup> <colgroup>
@ -1286,16 +1287,16 @@ As the rotations and translations have very different gains, we normalize each m
</div> </div>
<p> <p>
The diagonal and off-diagonal elements are shown in Figure <a href="#org2a79433">17</a>, and we can see good decoupling at low frequency. The diagonal and off-diagonal elements are shown in Figure <a href="#org17fc09d">17</a>, and we can see good decoupling at low frequency.
</p> </p>
<div id="org2a79433" class="figure"> <div id="org17fc09d" class="figure">
<p><img src="figs/nano_hexapod_diagonal_plant_cok.png" alt="nano_hexapod_diagonal_plant_cok.png" /> <p><img src="figs/nano_hexapod_diagonal_plant_cok.png" alt="nano_hexapod_diagonal_plant_cok.png" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Diagonal and off-diagonal elements of the (normalized) decentralized plant with the Jacobians estimated at the &ldquo;center of stiffness&rdquo;</p> <p><span class="figure-number">Figure 17: </span>Diagonal and off-diagonal elements of the (normalized) decentralized plant with the Jacobians estimated at the &ldquo;center of stiffness&rdquo;</p>
</div> </div>
<div class="important" id="org7ca3e61"> <div class="important" id="orgf6e29ed">
<p> <p>
The Jacobian matrices can be used to decoupled the plant at low frequency. The Jacobian matrices can be used to decoupled the plant at low frequency.
</p> </p>
@ -1309,7 +1310,7 @@ The Jacobian matrices can be used to decoupled the plant at low frequency.
<h3 id="orgd60e2b2"><span class="section-number-3">1.9</span> Stiffness matrix</h3> <h3 id="orgd60e2b2"><span class="section-number-3">1.9</span> Stiffness matrix</h3>
<div class="outline-text-3" id="text-1-9"> <div class="outline-text-3" id="text-1-9">
<p> <p>
<a id="org3c2f2a0"></a> <a id="orgc465638"></a>
</p> </p>
<p> <p>
The stiffness matrix of the nano-hexapod describes its induced static displacement/rotation when a force/torque is applied on its top platform. The stiffness matrix of the nano-hexapod describes its induced static displacement/rotation when a force/torque is applied on its top platform.
@ -1385,7 +1386,7 @@ ks = 1.737e+06 [N/m]
</pre> </pre>
<div class="important" id="org265b2d6"> <div class="important" id="org3758637">
<p> <p>
We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts. We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts.
</p> </p>
@ -1409,9 +1410,9 @@ And the compliance matrix can be computed as the inverse of the stiffness matrix
</div> </div>
<p> <p>
The obtained compliance matrix is shown in Table <a href="#org904bc46">6</a>. The obtained compliance matrix is shown in Table <a href="#org2bb39c8">6</a>.
</p> </p>
<table id="org904bc46" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org2bb39c8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 6:</span> Compliance Matrix - Perfect Joints</caption> <caption class="t-above"><span class="table-number">Table 6:</span> Compliance Matrix - Perfect Joints</caption>
<colgroup> <colgroup>
@ -1515,10 +1516,10 @@ It takes into account the bending and torsional stiffness of the flexible joints
</p> </p>
<p> <p>
The obtained compliance matrix is shown in Table <a href="#orgd28838f">7</a>. The obtained compliance matrix is shown in Table <a href="#org61a591a">7</a>.
</p> </p>
<table id="orgd28838f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org61a591a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 7:</span> Compliance Matrix - Estimated from Simscape</caption> <caption class="t-above"><span class="table-number">Table 7:</span> Compliance Matrix - Estimated from Simscape</caption>
<colgroup> <colgroup>
@ -1591,10 +1592,10 @@ The obtained compliance matrix is shown in Table <a href="#orgd28838f">7</a>.
</tbody> </tbody>
</table> </table>
<div class="important" id="org75979d5"> <div class="important" id="org5872054">
<p> <p>
The bending and torsional stiffness of the flexible joints induces a lot of coupling between forces/torques applied to the to platform to its displacement/rotation. The bending and torsional stiffness of the flexible joints induces a lot of coupling between forces/torques applied to the to platform to its displacement/rotation.
It can be seen by comparison the compliance matrices in Tables <a href="#org904bc46">6</a> and <a href="#orgd28838f">7</a>. It can be seen by comparison the compliance matrices in Tables <a href="#org2bb39c8">6</a> and <a href="#org61a591a">7</a>.
</p> </p>
</div> </div>
@ -1607,7 +1608,7 @@ It can be seen by comparison the compliance matrices in Tables <a href="#org904b
<h2 id="org2fc54dd"><span class="section-number-2">2</span> Active Damping using Integral Force Feedback</h2> <h2 id="org2fc54dd"><span class="section-number-2">2</span> Active Damping using Integral Force Feedback</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="orgda93dd4"></a> <a id="orgf5bb4a7"></a>
</p> </p>
<p> <p>
In this section <b>Integral Force Feedback</b> (IFF) strategy is used to damp the nano-hexapod resonances. In this section <b>Integral Force Feedback</b> (IFF) strategy is used to damp the nano-hexapod resonances.
@ -1617,17 +1618,17 @@ In this section <b>Integral Force Feedback</b> (IFF) strategy is used to damp th
It is structured as follows: It is structured as follows:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org85c3163">2.1</a>: the IFF plant is identified</li> <li>Section <a href="#orgc0b4ccd">2.1</a>: the IFF plant is identified</li>
<li>Section <a href="#org81a418a">2.2</a>: the optimal control gain is identified using the Root Locus plot</li> <li>Section <a href="#org12040da">2.2</a>: the optimal control gain is identified using the Root Locus plot</li>
<li>Section <a href="#org9fa9be2">2.3</a>: the IFF is applied, and the effect on the damped plant is identified and compared with the un-damped one</li> <li>Section <a href="#org52b7e12">2.3</a>: the IFF is applied, and the effect on the damped plant is identified and compared with the un-damped one</li>
<li>Section <a href="#org3ea0a63">2.4</a>: the IFF is applied, and the effect on the compliance is identified</li> <li>Section <a href="#org02ed765">2.4</a>: the IFF is applied, and the effect on the compliance is identified</li>
</ul> </ul>
</div> </div>
<div id="outline-container-org18c0a7a" class="outline-3"> <div id="outline-container-org1565813" class="outline-3">
<h3 id="org18c0a7a"><span class="section-number-3">2.1</span> Plant Identification</h3> <h3 id="org1565813"><span class="section-number-3">2.1</span> Plant Identification</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
<a id="org85c3163"></a> <a id="orgc0b4ccd"></a>
</p> </p>
<p> <p>
@ -1667,7 +1668,7 @@ Giff.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-strin
Its bode plot is shown in Figure Its bode plot is shown in Figure
</p> </p>
<div id="org484e5ff" class="figure"> <div id="orgcd7ed31" class="figure">
<p><img src="figs/nano_hexapod_iff_plant_bode_plot.png" alt="nano_hexapod_iff_plant_bode_plot.png" /> <p><img src="figs/nano_hexapod_iff_plant_bode_plot.png" alt="nano_hexapod_iff_plant_bode_plot.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Integral Force Feedback plant</p> <p><span class="figure-number">Figure 18: </span>Integral Force Feedback plant</p>
@ -1675,11 +1676,11 @@ Its bode plot is shown in Figure
</div> </div>
</div> </div>
<div id="outline-container-org12ae962" class="outline-3"> <div id="outline-container-org9a946f0" class="outline-3">
<h3 id="org12ae962"><span class="section-number-3">2.2</span> Root Locus</h3> <h3 id="org9a946f0"><span class="section-number-3">2.2</span> Root Locus</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
<a id="org81a418a"></a> <a id="org12040da"></a>
</p> </p>
<p> <p>
@ -1702,11 +1703,11 @@ It is here chosen to have quite a large \(\omega_c\) in order to not modify the
</div> </div>
<p> <p>
The obtained Root Locus is shown in Figure <a href="#org50ddd84">19</a>. The obtained Root Locus is shown in Figure <a href="#orgf46b7b8">19</a>.
The control gain chosen for future plots is shown by the red crosses. The control gain chosen for future plots is shown by the red crosses.
</p> </p>
<div id="org50ddd84" class="figure"> <div id="orgf46b7b8" class="figure">
<p><img src="figs/nano_hexapod_iff_root_locus.png" alt="nano_hexapod_iff_root_locus.png" /> <p><img src="figs/nano_hexapod_iff_root_locus.png" alt="nano_hexapod_iff_root_locus.png" />
</p> </p>
<p><span class="figure-number">Figure 19: </span>Root locus for the decentralized IFF control strategy</p> <p><span class="figure-number">Figure 19: </span>Root locus for the decentralized IFF control strategy</p>
@ -1722,11 +1723,11 @@ The obtained controller is then:
</div> </div>
<p> <p>
The corresponding loop gain of the diagonal terms are shown in Figure <a href="#org83f2f21">20</a>. The corresponding loop gain of the diagonal terms are shown in Figure <a href="#org900a4ff">20</a>.
It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of \(\omega_c\). It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of \(\omega_c\).
</p> </p>
<div id="org83f2f21" class="figure"> <div id="org900a4ff" class="figure">
<p><img src="figs/nano_hexapod_iff_loop_gain.png" alt="nano_hexapod_iff_loop_gain.png" /> <p><img src="figs/nano_hexapod_iff_loop_gain.png" alt="nano_hexapod_iff_loop_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{IFF}}(i,i)\)</p> <p><span class="figure-number">Figure 20: </span>Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{IFF}}(i,i)\)</p>
@ -1738,7 +1739,7 @@ It is shown that the loop gain is quite large around resonances (which allows to
<h3 id="org78de4b8"><span class="section-number-3">2.3</span> Effect of IFF on the plant</h3> <h3 id="org78de4b8"><span class="section-number-3">2.3</span> Effect of IFF on the plant</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
<a id="org9fa9be2"></a> <a id="org52b7e12"></a>
</p> </p>
<p> <p>
@ -1775,16 +1776,16 @@ Giff.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string
</div> </div>
<p> <p>
The obtained plants are compared in Figure <a href="#org5668fd9">21</a>. The obtained plants are compared in Figure <a href="#org3f71930">21</a>.
</p> </p>
<div id="org5668fd9" class="figure"> <div id="org3f71930" class="figure">
<p><img src="figs/nano_hexapod_effect_iff_plant.png" alt="nano_hexapod_effect_iff_plant.png" /> <p><img src="figs/nano_hexapod_effect_iff_plant.png" alt="nano_hexapod_effect_iff_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the IFF controller.</p> <p><span class="figure-number">Figure 21: </span>Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the IFF controller.</p>
</div> </div>
<div class="important" id="orgbdce0ed"> <div class="important" id="orgc707f7a">
<p> <p>
The Integral Force Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod. The Integral Force Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod.
</p> </p>
@ -1797,7 +1798,7 @@ The Integral Force Feedback Strategy is very effective to damp the 6 suspension
<h3 id="org8db31cb"><span class="section-number-3">2.4</span> Effect of IFF on the compliance</h3> <h3 id="org8db31cb"><span class="section-number-3">2.4</span> Effect of IFF on the compliance</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
<a id="org3ea0a63"></a> <a id="org02ed765"></a>
</p> </p>
<p> <p>
@ -1806,13 +1807,13 @@ Let&rsquo;s quantify that for the nano-hexapod.
The obtained compliances are compared in Figure The obtained compliances are compared in Figure
</p> </p>
<div id="org702b440" class="figure"> <div id="org0330998" class="figure">
<p><img src="figs/nano_hexapod_iff_compare_compliance.png" alt="nano_hexapod_iff_compare_compliance.png" /> <p><img src="figs/nano_hexapod_iff_compare_compliance.png" alt="nano_hexapod_iff_compare_compliance.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Comparison of the compliances in Open Loop and with Integral Force Feedback controller</p> <p><span class="figure-number">Figure 22: </span>Comparison of the compliances in Open Loop and with Integral Force Feedback controller</p>
</div> </div>
<div class="important" id="org8548e42"> <div class="important" id="orga17dabf">
<p> <p>
The use of IFF induces a degradation of the compliance. The use of IFF induces a degradation of the compliance.
This degradation is limited due to the use of a pseudo integrator (instead of a pure integrator). This degradation is limited due to the use of a pseudo integrator (instead of a pure integrator).
@ -1828,7 +1829,7 @@ Also, it should not be a major problem for the NASS, as no direct forces should
<h2 id="org605df2e"><span class="section-number-2">3</span> Active Damping using Direct Velocity Feedback - Encoders on the struts</h2> <h2 id="org605df2e"><span class="section-number-2">3</span> Active Damping using Direct Velocity Feedback - Encoders on the struts</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="orge64a27c"></a> <a id="orgd06ebe5"></a>
</p> </p>
<p> <p>
In this section, the <b>Direct Velocity Feedback</b> (DVF) strategy is used to damp the nano-hexapod resonances. In this section, the <b>Direct Velocity Feedback</b> (DVF) strategy is used to damp the nano-hexapod resonances.
@ -1838,17 +1839,17 @@ In this section, the <b>Direct Velocity Feedback</b> (DVF) strategy is used to d
It is structured as follows: It is structured as follows:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org3c7fe40">3.1</a>: the DVF plant is identified</li> <li>Section <a href="#orgbb1c58e">3.1</a>: the DVF plant is identified</li>
<li>Section <a href="#orgf211f53">3.2</a>: the optimal control gain is identified using the Root Locus plot</li> <li>Section <a href="#org4d9253b">3.2</a>: the optimal control gain is identified using the Root Locus plot</li>
<li>Section <a href="#orgaddaa31">3.3</a>: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one</li> <li>Section <a href="#org59677b3">3.3</a>: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one</li>
<li>Section <a href="#org7a4a83e">3.4</a>: the DVF is applied, and the effect on the compliance is identified</li> <li>Section <a href="#org4e34121">3.4</a>: the DVF is applied, and the effect on the compliance is identified</li>
</ul> </ul>
</div> </div>
<div id="outline-container-orge9801cd" class="outline-3"> <div id="outline-container-orgbb8eda6" class="outline-3">
<h3 id="orge9801cd"><span class="section-number-3">3.1</span> Plant Identification</h3> <h3 id="orgbb8eda6"><span class="section-number-3">3.1</span> Plant Identification</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
<a id="org3c7fe40"></a> <a id="orgbb1c58e"></a>
</p> </p>
<p> <p>
@ -1885,10 +1886,10 @@ Gdvf.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string
</div> </div>
<p> <p>
Its bode plot is shown in Figure <a href="#org4cd0969">23</a>. Its bode plot is shown in Figure <a href="#org69acb98">23</a>.
</p> </p>
<div id="org4cd0969" class="figure"> <div id="org69acb98" class="figure">
<p><img src="figs/nano_hexapod_dvf_plant_bode_plot_struts.png" alt="nano_hexapod_dvf_plant_bode_plot_struts.png" /> <p><img src="figs/nano_hexapod_dvf_plant_bode_plot_struts.png" alt="nano_hexapod_dvf_plant_bode_plot_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 23: </span>Direct Velocity Feedback plant</p> <p><span class="figure-number">Figure 23: </span>Direct Velocity Feedback plant</p>
@ -1896,11 +1897,11 @@ Its bode plot is shown in Figure <a href="#org4cd0969">23</a>.
</div> </div>
</div> </div>
<div id="outline-container-org0e2a9f0" class="outline-3"> <div id="outline-container-org7ce0b94" class="outline-3">
<h3 id="org0e2a9f0"><span class="section-number-3">3.2</span> Root Locus</h3> <h3 id="org7ce0b94"><span class="section-number-3">3.2</span> Root Locus</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
<a id="orgf211f53"></a> <a id="org4d9253b"></a>
</p> </p>
<p> <p>
@ -1920,11 +1921,11 @@ The value of \(\omega_d\) sets the frequency above high the derivative action is
</div> </div>
<p> <p>
The obtained Root Locus is shown in Figure <a href="#org3fb3734">24</a>. The obtained Root Locus is shown in Figure <a href="#org9369c44">24</a>.
The control gain chosen for future plots is shown by the red crosses. The control gain chosen for future plots is shown by the red crosses.
</p> </p>
<div id="org3fb3734" class="figure"> <div id="org9369c44" class="figure">
<p><img src="figs/nano_hexapod_dvf_root_locus_struts.png" alt="nano_hexapod_dvf_root_locus_struts.png" /> <p><img src="figs/nano_hexapod_dvf_root_locus_struts.png" alt="nano_hexapod_dvf_root_locus_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 24: </span>Root locus for the decentralized DVF control strategy</p> <p><span class="figure-number">Figure 24: </span>Root locus for the decentralized DVF control strategy</p>
@ -1940,11 +1941,11 @@ The obtained controller is then:
</div> </div>
<p> <p>
The corresponding loop gain of the diagonal terms are shown in Figure <a href="#org154eff7">25</a>. The corresponding loop gain of the diagonal terms are shown in Figure <a href="#org69421ce">25</a>.
It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of \(\omega_c\). It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of \(\omega_c\).
</p> </p>
<div id="org154eff7" class="figure"> <div id="org69421ce" class="figure">
<p><img src="figs/nano_hexapod_dvf_loop_gain_struts.png" alt="nano_hexapod_dvf_loop_gain_struts.png" /> <p><img src="figs/nano_hexapod_dvf_loop_gain_struts.png" alt="nano_hexapod_dvf_loop_gain_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 25: </span>Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)</p> <p><span class="figure-number">Figure 25: </span>Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)</p>
@ -1952,11 +1953,11 @@ It is shown that the loop gain is quite large around resonances (which allows to
</div> </div>
</div> </div>
<div id="outline-container-org06b118d" class="outline-3"> <div id="outline-container-org949e780" class="outline-3">
<h3 id="org06b118d"><span class="section-number-3">3.3</span> Effect of DVF on the plant</h3> <h3 id="org949e780"><span class="section-number-3">3.3</span> Effect of DVF on the plant</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
<a id="orgaddaa31"></a> <a id="org59677b3"></a>
</p> </p>
<p> <p>
@ -1993,16 +1994,16 @@ Gdvf.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string
</div> </div>
<p> <p>
The obtained plants are compared in Figure <a href="#orgdea4079">26</a>. The obtained plants are compared in Figure <a href="#org21711b9">26</a>.
</p> </p>
<div id="orgdea4079" class="figure"> <div id="org21711b9" class="figure">
<p><img src="figs/nano_hexapod_effect_dvf_plant_struts.png" alt="nano_hexapod_effect_dvf_plant_struts.png" /> <p><img src="figs/nano_hexapod_effect_dvf_plant_struts.png" alt="nano_hexapod_effect_dvf_plant_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 26: </span>Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the DVF controller.</p> <p><span class="figure-number">Figure 26: </span>Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the DVF controller.</p>
</div> </div>
<div class="important" id="org17e22c5"> <div class="important" id="orgabfd5ca">
<p> <p>
The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod. The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod.
</p> </p>
@ -2011,20 +2012,20 @@ The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension
</div> </div>
</div> </div>
<div id="outline-container-org4a7a21f" class="outline-3"> <div id="outline-container-org755376c" class="outline-3">
<h3 id="org4a7a21f"><span class="section-number-3">3.4</span> Effect of DVF on the compliance</h3> <h3 id="org755376c"><span class="section-number-3">3.4</span> Effect of DVF on the compliance</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
<a id="org7a4a83e"></a> <a id="org4e34121"></a>
</p> </p>
<p> <p>
The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large.
Let&rsquo;s quantify that for the nano-hexapod. Let&rsquo;s quantify that for the nano-hexapod.
The obtained compliances are compared in Figure <a href="#org999cd3a">27</a>. The obtained compliances are compared in Figure <a href="#org6a80fec">27</a>.
</p> </p>
<div id="org999cd3a" class="figure"> <div id="org6a80fec" class="figure">
<p><img src="figs/nano_hexapod_dvf_compare_compliance_struts.png" alt="nano_hexapod_dvf_compare_compliance_struts.png" /> <p><img src="figs/nano_hexapod_dvf_compare_compliance_struts.png" alt="nano_hexapod_dvf_compare_compliance_struts.png" />
</p> </p>
<p><span class="figure-number">Figure 27: </span>Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller</p> <p><span class="figure-number">Figure 27: </span>Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller</p>
@ -2037,7 +2038,7 @@ The obtained compliances are compared in Figure <a href="#org999cd3a">27</a>.
<h2 id="orgac5a0fc"><span class="section-number-2">4</span> Active Damping using Direct Velocity Feedback - Encoders on the plates</h2> <h2 id="orgac5a0fc"><span class="section-number-2">4</span> Active Damping using Direct Velocity Feedback - Encoders on the plates</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="org6e295ec"></a> <a id="orgb09098a"></a>
</p> </p>
<p> <p>
In this section, the <b>Direct Velocity Feedback</b> (DVF) strategy is used to damp the nano-hexapod resonances. In this section, the <b>Direct Velocity Feedback</b> (DVF) strategy is used to damp the nano-hexapod resonances.
@ -2047,17 +2048,17 @@ In this section, the <b>Direct Velocity Feedback</b> (DVF) strategy is used to d
It is structured as follows: It is structured as follows:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org123c758">4.1</a>: the DVF plant is identified</li> <li>Section <a href="#orgdb20ff7">4.1</a>: the DVF plant is identified</li>
<li>Section <a href="#org5ca4e46">4.2</a>: the optimal control gain is identified using the Root Locus plot</li> <li>Section <a href="#org0c92a60">4.2</a>: the optimal control gain is identified using the Root Locus plot</li>
<li>Section <a href="#org11682e7">4.3</a>: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one</li> <li>Section <a href="#org5e5dc9a">4.3</a>: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one</li>
<li>Section <a href="#org9eb6cd8">4.4</a>: the DVF is applied, and the effect on the compliance is identified</li> <li>Section <a href="#org33bc3f9">4.4</a>: the DVF is applied, and the effect on the compliance is identified</li>
</ul> </ul>
</div> </div>
<div id="outline-container-org79dc366" class="outline-3"> <div id="outline-container-org5344b4a" class="outline-3">
<h3 id="org79dc366"><span class="section-number-3">4.1</span> Plant Identification</h3> <h3 id="org5344b4a"><span class="section-number-3">4.1</span> Plant Identification</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
<a id="org123c758"></a> <a id="orgdb20ff7"></a>
</p> </p>
<p> <p>
@ -2094,10 +2095,10 @@ Gdvf.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string
</div> </div>
<p> <p>
Its bode plot is shown in Figure <a href="#org756f99a">28</a>. Its bode plot is shown in Figure <a href="#orgb87ebad">28</a>.
</p> </p>
<div id="org756f99a" class="figure"> <div id="orgb87ebad" class="figure">
<p><img src="figs/nano_hexapod_dvf_plant_bode_plot_plates.png" alt="nano_hexapod_dvf_plant_bode_plot_plates.png" /> <p><img src="figs/nano_hexapod_dvf_plant_bode_plot_plates.png" alt="nano_hexapod_dvf_plant_bode_plot_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 28: </span>Direct Velocity Feedback plant</p> <p><span class="figure-number">Figure 28: </span>Direct Velocity Feedback plant</p>
@ -2105,11 +2106,11 @@ Its bode plot is shown in Figure <a href="#org756f99a">28</a>.
</div> </div>
</div> </div>
<div id="outline-container-org3088061" class="outline-3"> <div id="outline-container-orge23105c" class="outline-3">
<h3 id="org3088061"><span class="section-number-3">4.2</span> Root Locus</h3> <h3 id="orge23105c"><span class="section-number-3">4.2</span> Root Locus</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
<a id="org5ca4e46"></a> <a id="org0c92a60"></a>
</p> </p>
<p> <p>
@ -2129,11 +2130,11 @@ The value of \(\omega_d\) sets the frequency above high the derivative action is
</div> </div>
<p> <p>
The obtained Root Locus is shown in Figure <a href="#orgad1c8a8">29</a>. The obtained Root Locus is shown in Figure <a href="#orgcc25050">29</a>.
The control gain chosen for future plots is shown by the red crosses. The control gain chosen for future plots is shown by the red crosses.
</p> </p>
<div id="orgad1c8a8" class="figure"> <div id="orgcc25050" class="figure">
<p><img src="figs/nano_hexapod_dvf_root_locus_plates.png" alt="nano_hexapod_dvf_root_locus_plates.png" /> <p><img src="figs/nano_hexapod_dvf_root_locus_plates.png" alt="nano_hexapod_dvf_root_locus_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 29: </span>Root locus for the decentralized DVF control strategy</p> <p><span class="figure-number">Figure 29: </span>Root locus for the decentralized DVF control strategy</p>
@ -2149,11 +2150,11 @@ The obtained controller is then:
</div> </div>
<p> <p>
The corresponding loop gain of the diagonal terms are shown in Figure <a href="#org588851a">30</a>. The corresponding loop gain of the diagonal terms are shown in Figure <a href="#org0799113">30</a>.
It is shown that the loop gain is quite large around resonances (which allows to add lots of damping). It is shown that the loop gain is quite large around resonances (which allows to add lots of damping).
</p> </p>
<div id="org588851a" class="figure"> <div id="org0799113" class="figure">
<p><img src="figs/nano_hexapod_dvf_loop_gain_plates.png" alt="nano_hexapod_dvf_loop_gain_plates.png" /> <p><img src="figs/nano_hexapod_dvf_loop_gain_plates.png" alt="nano_hexapod_dvf_loop_gain_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 30: </span>Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)</p> <p><span class="figure-number">Figure 30: </span>Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)</p>
@ -2161,11 +2162,11 @@ It is shown that the loop gain is quite large around resonances (which allows to
</div> </div>
</div> </div>
<div id="outline-container-orged4d86f" class="outline-3"> <div id="outline-container-org4e3c598" class="outline-3">
<h3 id="orged4d86f"><span class="section-number-3">4.3</span> Effect of DVF on the plant</h3> <h3 id="org4e3c598"><span class="section-number-3">4.3</span> Effect of DVF on the plant</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<p> <p>
<a id="org11682e7"></a> <a id="org5e5dc9a"></a>
</p> </p>
<p> <p>
@ -2202,16 +2203,16 @@ Gdvf.OutputName = {<span class="org-string">'D1'</span>, <span class="org-string
</div> </div>
<p> <p>
The obtained plants are compared in Figure <a href="#org6513a41">31</a>. The obtained plants are compared in Figure <a href="#org0f206fb">31</a>.
</p> </p>
<div id="org6513a41" class="figure"> <div id="org0f206fb" class="figure">
<p><img src="figs/nano_hexapod_effect_dvf_plant_plates.png" alt="nano_hexapod_effect_dvf_plant_plates.png" /> <p><img src="figs/nano_hexapod_effect_dvf_plant_plates.png" alt="nano_hexapod_effect_dvf_plant_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 31: </span>Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the DVF controller.</p> <p><span class="figure-number">Figure 31: </span>Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the DVF controller.</p>
</div> </div>
<div class="important" id="org1727d6f"> <div class="important" id="orgd8f88d8">
<p> <p>
The Direct Velocity Feedback Strategy is very effective in damping the 6 suspension modes of the nano-hexapod. The Direct Velocity Feedback Strategy is very effective in damping the 6 suspension modes of the nano-hexapod.
</p> </p>
@ -2220,20 +2221,20 @@ The Direct Velocity Feedback Strategy is very effective in damping the 6 suspens
</div> </div>
</div> </div>
<div id="outline-container-orgc5c32c0" class="outline-3"> <div id="outline-container-org276122b" class="outline-3">
<h3 id="orgc5c32c0"><span class="section-number-3">4.4</span> Effect of DVF on the compliance</h3> <h3 id="org276122b"><span class="section-number-3">4.4</span> Effect of DVF on the compliance</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<p> <p>
<a id="org9eb6cd8"></a> <a id="org33bc3f9"></a>
</p> </p>
<p> <p>
The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large.
Let&rsquo;s quantify that for the nano-hexapod. Let&rsquo;s quantify that for the nano-hexapod.
The obtained compliances are compared in Figure <a href="#orgbd53db3">32</a>. The obtained compliances are compared in Figure <a href="#org907b6e3">32</a>.
</p> </p>
<div id="orgbd53db3" class="figure"> <div id="org907b6e3" class="figure">
<p><img src="figs/nano_hexapod_dvf_compare_compliance_plates.png" alt="nano_hexapod_dvf_compare_compliance_plates.png" /> <p><img src="figs/nano_hexapod_dvf_compare_compliance_plates.png" alt="nano_hexapod_dvf_compare_compliance_plates.png" />
</p> </p>
<p><span class="figure-number">Figure 32: </span>Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller</p> <p><span class="figure-number">Figure 32: </span>Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller</p>
@ -2246,7 +2247,7 @@ The obtained compliances are compared in Figure <a href="#orgbd53db3">32</a>.
<h2 id="org8862f6b"><span class="section-number-2">5</span> Function - Initialize Nano Hexapod</h2> <h2 id="org8862f6b"><span class="section-number-2">5</span> Function - Initialize Nano Hexapod</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
<a id="org32d0912"></a> <a id="org877b216"></a>
</p> </p>
</div> </div>
@ -2604,7 +2605,7 @@ nano_hexapod.geometry.J = [nano_hexapod.geometry.si<span class="org-type">'</spa
</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: 2021-04-23 ven. 15:50</p> <p class="date">Created: 2021-04-23 ven. 17:35</p>
</div> </div>
</body> </body>
</html> </html>

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@ -82,6 +82,8 @@ Active damping techniques are applied to the full Simscape model.
In this file are gathered all studies about the control the Nano-Active-Stabilization-System. In this file are gathered all studies about the control the Nano-Active-Stabilization-System.
* Nano-Hexapod Simscape Model ([[file:nano_hexapod.org][link]]) * Nano-Hexapod Simscape Model ([[file:nano_hexapod.org][link]])
The nano-hexapod simscape model is described and used for simulations.
* Useful Matlab Functions ([[./functions.org][link]]) * Useful Matlab Functions ([[./functions.org][link]])
Many matlab functions are shared among all the files of the projects. Many matlab functions are shared among all the files of the projects.

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@ -264,9 +264,10 @@ exportFig('figs/nano_hexapod_effect_encoder.pdf', 'width', 'full', 'height', 'ta
#+RESULTS: #+RESULTS:
[[file:figs/nano_hexapod_effect_encoder.png]] [[file:figs/nano_hexapod_effect_encoder.png]]
#+begin_question #+begin_important
Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the struts? The zeros at 400Hz and 800Hz should corresponds to resonances of the system when one of the APA is blocked.
#+end_question It is linked to the axial stiffness of the flexible joints: increasing the axial stiffness of the joints will increase the frequency of the zeros.
#+end_important
** Effect of APA flexibility ** Effect of APA flexibility
<<sec:effect_apa_flexibility>> <<sec:effect_apa_flexibility>>
@ -2446,6 +2447,46 @@ exportFig('figs/nano_hexapod_dvf_compare_compliance_plates.pdf', 'width', 'wide'
[[file:figs/nano_hexapod_dvf_compare_compliance_plates.png]] [[file:figs/nano_hexapod_dvf_compare_compliance_plates.png]]
* To-order :noexport: * To-order :noexport:
** Why Zero when encoder on the struts
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
'flex_top_type', '4dof', ...
'motion_sensor_type', 'struts', ...
'actuator_type', '2dof');
#+end_src
The transfer function from actuator inputs to force sensors outputs is identified.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'test_apa300ml';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dLs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/dLp'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
G = linearize(mdl, io, 0.0, options);
G.InputName = {'F'};
G.OutputName = {'dLs', 'dLp'};
bodeFig({G(1), G(2)}, logspace(1,4,1000))
#+end_src
The zero seems to be linked to the axial flexibility of the top joint.
In this mode, the APA does not experience any motion (hence the zero).
The resonance frequency then corresponds to the top mass on top of the axial stiffness of the two joints in series.
For the nano-hexapod, it corresponds to the resonance of the top mass when all (or *just one*?) of the APA is blocked.
#+begin_src matlab
sqrt((n_hexapod.flex_bot.kz(1)/2)/(55/3))/2/pi
#+end_src
** Verify why unstable strut ** Verify why unstable strut
#+begin_src matlab :results value replace #+begin_src matlab :results value replace