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<title>Finite Element Model with Simscape</title>
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<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org3ded9a3">1. Amplified Piezoelectric Actuator - 3D elements</a>
<li><a href="#org5a554a0">1. Amplified Piezoelectric Actuator - 3D elements</a>
<ul>
<li><a href="#org7436688">1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org4ad8afa">1.2. Output parameters</a></li>
<li><a href="#org1477fec">1.3. Piezoelectric parameters</a></li>
<li><a href="#orgdd0f3d7">1.4. Identification of the Dynamics</a></li>
<li><a href="#org740df84">1.5. Comparison with Ansys</a></li>
<li><a href="#org1555a0d">1.6. Force Sensor</a></li>
<li><a href="#org71b73d0">1.7. Distributed Actuator</a></li>
<li><a href="#org023858d">1.8. Distributed Actuator and Force Sensor</a></li>
<li><a href="#org91149a1">1.9. Dynamics from input voltage to displacement</a></li>
<li><a href="#orgc531f2d">1.10. Dynamics from input voltage to output voltage</a></li>
<li><a href="#org527cbaa">1.11. Identification for a simpler model</a></li>
<li><a href="#org29056ab">1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#orgccb0a56">1.2. Output parameters</a></li>
<li><a href="#orgdd7c7ba">1.3. Piezoelectric parameters</a></li>
<li><a href="#orgb2959d5">1.4. Identification of the Dynamics</a></li>
<li><a href="#org1e29135">1.5. Comparison with Ansys</a></li>
<li><a href="#org3f1fc2e">1.6. Force Sensor</a></li>
<li><a href="#org4f1753f">1.7. Distributed Actuator</a></li>
<li><a href="#org21246e5">1.8. Distributed Actuator and Force Sensor</a></li>
<li><a href="#orgd0b2aaa">1.9. Dynamics from input voltage to displacement</a></li>
<li><a href="#org5452b7e">1.10. Dynamics from input voltage to output voltage</a></li>
<li><a href="#orgaff4afc">1.11. Identification for a simpler model</a></li>
</ul>
</li>
<li><a href="#org5e5f531">2. APA300ML</a>
<li><a href="#org3dc5d8d">2. APA300ML</a>
<ul>
<li><a href="#org9691c9e">2.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org34676bd">2.2. Output parameters</a></li>
<li><a href="#orgf6ad2fe">2.3. Piezoelectric parameters</a></li>
<li><a href="#orgfcc3b27">2.4. Identification of the APA Characteristics</a>
<li><a href="#org3eaf978">2.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org160ca26">2.2. Output parameters</a></li>
<li><a href="#org7932f3c">2.3. Piezoelectric parameters</a></li>
<li><a href="#org18316cd">2.4. Identification of the APA Characteristics</a>
<ul>
<li><a href="#org7c141d1">2.4.1. Stiffness</a></li>
<li><a href="#org6336a4d">2.4.2. Resonance Frequency</a></li>
<li><a href="#org7adcbea">2.4.3. Amplification factor</a></li>
<li><a href="#org924ba9a">2.4.4. Stroke</a></li>
<li><a href="#orgc0281f1">2.4.1. Stiffness</a></li>
<li><a href="#orgcebe0f9">2.4.2. Resonance Frequency</a></li>
<li><a href="#orgda4f233">2.4.3. Amplification factor</a></li>
<li><a href="#org59829b6">2.4.4. Stroke</a></li>
</ul>
</li>
<li><a href="#org0334d98">2.5. Identification of the Dynamics</a></li>
<li><a href="#org889c8e8">2.6. IFF</a></li>
<li><a href="#org6f11c82">2.7. DVF</a></li>
<li><a href="#org1c376b5">2.8. Identification for a simpler model</a></li>
<li><a href="#org30bc4bf">2.9. Identification of the stiffness properties</a>
<li><a href="#org1cbc8a6">2.5. Identification of the Dynamics</a></li>
<li><a href="#org44a32d5">2.6. IFF</a></li>
<li><a href="#org3e558c6">2.7. DVF</a></li>
<li><a href="#orgad3fdd9">2.8. Identification for a simpler model</a></li>
<li><a href="#orge0b9f5a">2.9. Identification of the stiffness properties</a>
<ul>
<li><a href="#orge89f3f8">2.9.1. APA Alone</a></li>
<li><a href="#org4651c6e">2.9.2. See how the global stiffness is changing with the flexible joints</a></li>
<li><a href="#org52ddecb">2.9.1. APA Alone</a></li>
<li><a href="#org02b6855">2.9.2. See how the global stiffness is changing with the flexible joints</a></li>
</ul>
</li>
<li><a href="#orgbb1e485">2.10. Effect of APA300ML in the flexibility of the leg</a></li>
<li><a href="#org34de703">2.10. Effect of APA300ML in the flexibility of the leg</a></li>
</ul>
</li>
<li><a href="#org71e2995">3. Flexible Joint</a>
<li><a href="#orgb6c0ee0">3. Flexible Joint</a>
<ul>
<li><a href="#org4609327">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org222b467">3.2. Output parameters</a></li>
<li><a href="#orgace43b0">3.3. Flexible Joint Characteristics</a></li>
<li><a href="#orgc60e392">3.4. Identification of the parameters using Simscape</a></li>
<li><a href="#org43c8aa7">3.5. Simpler Model</a></li>
<li><a href="#orgd7b1d5f">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org9778a32">3.2. Output parameters</a></li>
<li><a href="#orgcb9bad1">3.3. Flexible Joint Characteristics</a></li>
<li><a href="#org4dadc02">3.4. Identification of the parameters using Simscape</a></li>
<li><a href="#org30336a6">3.5. Simpler Model</a></li>
</ul>
</li>
<li><a href="#org5d2c10d">4. Optimal Flexible Joint</a>
<li><a href="#orgd9d5aff">4. Optimal Flexible Joint</a>
<ul>
<li><a href="#orgfec12e9">4.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org51a4b8d">4.2. Output parameters</a></li>
<li><a href="#org9df419b">4.3. Flexible Joint Characteristics</a></li>
<li><a href="#org4ea4053">4.4. Identification of the parameters using Simscape</a></li>
<li><a href="#org070daa9">4.5. Simpler Model</a></li>
<li><a href="#org83c1679">4.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#orgbee4a84">4.2. Output parameters</a></li>
<li><a href="#org7609951">4.3. Flexible Joint Characteristics</a></li>
<li><a href="#org8bf4f56">4.4. Identification of the parameters using Simscape</a></li>
<li><a href="#orgd8cb8ff">4.5. Simpler Model</a></li>
</ul>
</li>
<li><a href="#org72ebb5c">5. Integral Force Feedback with Amplified Piezo</a>
<li><a href="#org7f2d76d">5. Integral Force Feedback with Amplified Piezo</a>
<ul>
<li><a href="#orgffa90de">5.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org4ac5a6e">5.2. IFF Plant</a></li>
<li><a href="#orgdc46434">5.3. IFF controller</a></li>
<li><a href="#orgc9d8168">5.4. Closed Loop System</a></li>
<li><a href="#orgd9dc7be">5.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org3671fca">5.2. IFF Plant</a></li>
<li><a href="#org1cbfb66">5.3. IFF controller</a></li>
<li><a href="#org7b29313">5.4. Closed Loop System</a></li>
</ul>
</li>
<li><a href="#orge46f2bf">6. Complete Strut with Encoder</a>
<li><a href="#org1272d3f">6. Complete Strut with Encoder</a>
<ul>
<li><a href="#org9c8b2a0">6.1. Introduction</a></li>
<li><a href="#org6b21925">6.2. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org40668e1">6.3. Output parameters</a></li>
<li><a href="#org6d5c440">6.4. Piezoelectric parameters</a></li>
<li><a href="#org2521017">6.5. Identification of the Dynamics</a></li>
<li><a href="#orgddf8d43">6.1. Introduction</a></li>
<li><a href="#org4742c38">6.2. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org332b172">6.3. Output parameters</a></li>
<li><a href="#orgadca4a4">6.4. Piezoelectric parameters</a></li>
<li><a href="#org226d3f3">6.5. Identification of the Dynamics</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org3ded9a3" class="outline-2">
<h2 id="org3ded9a3"><span class="section-number-2">1</span> Amplified Piezoelectric Actuator - 3D elements</h2>
<div id="outline-container-org5a554a0" class="outline-2">
<h2 id="org5a554a0"><span class="section-number-2">1</span> Amplified Piezoelectric Actuator - 3D elements</h2>
<div class="outline-text-2" id="text-1">
<p>
The idea here is to:
@@ -129,8 +125,8 @@ The idea here is to:
</ul>
</div>
<div id="outline-container-org7436688" class="outline-3">
<h3 id="org7436688"><span class="section-number-3">1.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div id="outline-container-org29056ab" class="outline-3">
<h3 id="org29056ab"><span class="section-number-3">1.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-1-1">
<p>
We first extract the stiffness and mass matrices.
@@ -156,8 +152,8 @@ Then, we extract the coordinates of the interface nodes.
</div>
</div>
<div id="outline-container-org4ad8afa" class="outline-3">
<h3 id="org4ad8afa"><span class="section-number-3">1.2</span> Output parameters</h3>
<div id="outline-container-orgccb0a56" class="outline-3">
<h3 id="orgccb0a56"><span class="section-number-3">1.2</span> Output parameters</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/piezo_amplified_3d.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>);
@@ -196,7 +192,7 @@ Then, we extract the coordinates of the interface nodes.
</table>
<div id="org52ce3d2" class="figure">
<div id="orgf427fec" class="figure">
<p><img src="figs/amplified_piezo_interface_nodes.png" alt="amplified_piezo_interface_nodes.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Interface Nodes for the Amplified Piezo Actuator</p>
@@ -654,8 +650,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div>
<div id="outline-container-org1477fec" class="outline-3">
<h3 id="org1477fec"><span class="section-number-3">1.3</span> Piezoelectric parameters</h3>
<div id="outline-container-orgdd7c7ba" class="outline-3">
<h3 id="orgdd7c7ba"><span class="section-number-3">1.3</span> Piezoelectric parameters</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Parameters for the APA95ML:
@@ -716,8 +712,8 @@ where:
</div>
</div>
<div id="outline-container-orgdd0f3d7" class="outline-3">
<h3 id="orgdd0f3d7"><span class="section-number-3">1.4</span> Identification of the Dynamics</h3>
<div id="outline-container-orgb2959d5" class="outline-3">
<h3 id="orgb2959d5"><span class="section-number-3">1.4</span> Identification of the Dynamics</h3>
<div class="outline-text-3" id="text-1-4">
<p>
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block.
@@ -769,7 +765,7 @@ And the dynamics is identified.
</p>
<p>
The two identified dynamics are compared in Figure <a href="#org49f8567">2</a>.
The two identified dynamics are compared in Figure <a href="#orgd90f204">2</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
@@ -785,7 +781,7 @@ Ghm = <span class="org-type">-</span>linearize(mdl, io);
</div>
<div id="org49f8567" class="figure">
<div id="orgd90f204" class="figure">
<p><img src="figs/dynamics_act_disp_comp_mass.png" alt="dynamics_act_disp_comp_mass.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Dynamics from \(F\) to \(d\) without a payload and with a 10kg payload</p>
@@ -793,8 +789,8 @@ Ghm = <span class="org-type">-</span>linearize(mdl, io);
</div>
</div>
<div id="outline-container-org740df84" class="outline-3">
<h3 id="org740df84"><span class="section-number-3">1.5</span> Comparison with Ansys</h3>
<div id="outline-container-org1e29135" class="outline-3">
<h3 id="org1e29135"><span class="section-number-3">1.5</span> Comparison with Ansys</h3>
<div class="outline-text-3" id="text-1-5">
<p>
Let&rsquo;s import the results from an Harmonic response analysis in Ansys.
@@ -806,11 +802,11 @@ Gresp10 = readtable(<span class="org-string">'FEA_HarmResponse_10kg.txt'</span>)
</div>
<p>
The obtained dynamics from the Simscape model and from the Ansys analysis are compare in Figure <a href="#orga47bfac">3</a>.
The obtained dynamics from the Simscape model and from the Ansys analysis are compare in Figure <a href="#org32de39c">3</a>.
</p>
<div id="orga47bfac" class="figure">
<div id="org32de39c" class="figure">
<p><img src="figs/dynamics_force_disp_comp_anasys.png" alt="dynamics_force_disp_comp_anasys.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Comparison of the obtained dynamics using Simscape with the harmonic response analysis using Ansys</p>
@@ -818,15 +814,15 @@ The obtained dynamics from the Simscape model and from the Ansys analysis are co
</div>
</div>
<div id="outline-container-org1555a0d" class="outline-3">
<h3 id="org1555a0d"><span class="section-number-3">1.6</span> Force Sensor</h3>
<div id="outline-container-org3f1fc2e" class="outline-3">
<h3 id="org3f1fc2e"><span class="section-number-3">1.6</span> Force Sensor</h3>
<div class="outline-text-3" id="text-1-6">
<p>
The dynamics is identified from internal forces applied between nodes 3 and 11 to the relative displacement of nodes 11 and 13.
</p>
<p>
The obtained dynamics is shown in Figure <a href="#orgb045fc0">4</a>.
The obtained dynamics is shown in Figure <a href="#org5ab3306">4</a>.
</p>
<div class="org-src-container">
@@ -866,7 +862,7 @@ Gfm = linearize(mdl, io);
</div>
<div id="orgb045fc0" class="figure">
<div id="org5ab3306" class="figure">
<p><img src="figs/dynamics_force_force_sensor_comp_mass.png" alt="dynamics_force_force_sensor_comp_mass.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)</p>
@@ -874,8 +870,8 @@ Gfm = linearize(mdl, io);
</div>
</div>
<div id="outline-container-org71b73d0" class="outline-3">
<h3 id="org71b73d0"><span class="section-number-3">1.7</span> Distributed Actuator</h3>
<div id="outline-container-org4f1753f" class="outline-3">
<h3 id="org4f1753f"><span class="section-number-3">1.7</span> Distributed Actuator</h3>
<div class="outline-text-3" id="text-1-7">
<div class="org-src-container">
<pre class="src src-matlab">m = 0;
@@ -924,8 +920,8 @@ Gdm = linearize(mdl, io);
</div>
</div>
<div id="outline-container-org023858d" class="outline-3">
<h3 id="org023858d"><span class="section-number-3">1.8</span> Distributed Actuator and Force Sensor</h3>
<div id="outline-container-org21246e5" class="outline-3">
<h3 id="org21246e5"><span class="section-number-3">1.8</span> Distributed Actuator and Force Sensor</h3>
<div class="outline-text-3" id="text-1-8">
<div class="org-src-container">
<pre class="src src-matlab">m = 0;
@@ -965,8 +961,8 @@ Gfdm = linearize(mdl, io);
</div>
</div>
<div id="outline-container-org91149a1" class="outline-3">
<h3 id="org91149a1"><span class="section-number-3">1.9</span> Dynamics from input voltage to displacement</h3>
<div id="outline-container-orgd0b2aaa" class="outline-3">
<h3 id="orgd0b2aaa"><span class="section-number-3">1.9</span> Dynamics from input voltage to displacement</h3>
<div class="outline-text-3" id="text-1-9">
<div class="org-src-container">
<pre class="src src-matlab">m = 5;
@@ -978,7 +974,7 @@ And the dynamics is identified.
</p>
<p>
The two identified dynamics are compared in Figure <a href="#org49f8567">2</a>.
The two identified dynamics are compared in Figure <a href="#orgd90f204">2</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
@@ -1000,8 +996,8 @@ G = <span class="org-type">-</span>linearize(mdl, io);
</div>
</div>
<div id="outline-container-orgc531f2d" class="outline-3">
<h3 id="orgc531f2d"><span class="section-number-3">1.10</span> Dynamics from input voltage to output voltage</h3>
<div id="outline-container-org5452b7e" class="outline-3">
<h3 id="org5452b7e"><span class="section-number-3">1.10</span> Dynamics from input voltage to output voltage</h3>
<div class="outline-text-3" id="text-1-10">
<div class="org-src-container">
<pre class="src src-matlab">m = 5;
@@ -1023,8 +1019,8 @@ G = <span class="org-type">-</span>linearize(mdl, io);
</div>
</div>
<div id="outline-container-org527cbaa" class="outline-3">
<h3 id="org527cbaa"><span class="section-number-3">1.11</span> Identification for a simpler model</h3>
<div id="outline-container-orgaff4afc" class="outline-3">
<h3 id="orgaff4afc"><span class="section-number-3">1.11</span> Identification for a simpler model</h3>
<div class="outline-text-3" id="text-1-11">
<p>
The goal in this section is to identify the parameters of a simple APA model from the FEM.
@@ -1036,12 +1032,12 @@ The presented model is based on (<a href="#citeproc_bib_item_2">Souleille et al.
</p>
<p>
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#orgb1100b8">5</a>).
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#org73ab5e6">5</a>).
The parameters are shown in the table below.
</p>
<div id="orgb1100b8" class="figure">
<div id="org73ab5e6" class="figure">
<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
@@ -1190,7 +1186,7 @@ Adjust the DC gain for the force sensor:
</div>
<div id="org6fbe971" class="figure">
<div id="org629ff2d" class="figure">
<p><img src="figs/apa95ml_comp_simpler_model.png" alt="apa95ml_comp_simpler_model.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Comparison of the Dynamics between the FEM model and the simplified one</p>
@@ -1207,19 +1203,19 @@ We save the parameters of the simplified model for the APA95ML:
</div>
</div>
<div id="outline-container-org5e5f531" class="outline-2">
<h2 id="org5e5f531"><span class="section-number-2">2</span> APA300ML</h2>
<div id="outline-container-org3dc5d8d" class="outline-2">
<h2 id="org3dc5d8d"><span class="section-number-2">2</span> APA300ML</h2>
<div class="outline-text-2" id="text-2">
<div id="orgbd02022" class="figure">
<div id="org60aa4c9" class="figure">
<p><img src="figs/apa300ml_ansys.jpg" alt="apa300ml_ansys.jpg" />
</p>
<p><span class="figure-number">Figure 7: </span>Ansys FEM of the APA300ML</p>
</div>
</div>
<div id="outline-container-org9691c9e" class="outline-3">
<h3 id="org9691c9e"><span class="section-number-3">2.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div id="outline-container-org3eaf978" class="outline-3">
<h3 id="org3eaf978"><span class="section-number-3">2.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-2-1">
<p>
We first extract the stiffness and mass matrices.
@@ -1245,8 +1241,8 @@ Then, we extract the coordinates of the interface nodes.
</div>
</div>
<div id="outline-container-org34676bd" class="outline-3">
<h3 id="org34676bd"><span class="section-number-3">2.2</span> Output parameters</h3>
<div id="outline-container-org160ca26" class="outline-3">
<h3 id="org160ca26"><span class="section-number-3">2.2</span> Output parameters</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/APA300ML.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>);
@@ -1687,8 +1683,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div>
</div>
<div id="outline-container-orgf6ad2fe" class="outline-3">
<h3 id="orgf6ad2fe"><span class="section-number-3">2.3</span> Piezoelectric parameters</h3>
<div id="outline-container-org7932f3c" class="outline-3">
<h3 id="org7932f3c"><span class="section-number-3">2.3</span> Piezoelectric parameters</h3>
<div class="outline-text-3" id="text-2-3">
<p>
Parameters for the APA300ML:
@@ -1749,12 +1745,12 @@ where:
</div>
</div>
<div id="outline-container-orgfcc3b27" class="outline-3">
<h3 id="orgfcc3b27"><span class="section-number-3">2.4</span> Identification of the APA Characteristics</h3>
<div id="outline-container-org18316cd" class="outline-3">
<h3 id="org18316cd"><span class="section-number-3">2.4</span> Identification of the APA Characteristics</h3>
<div class="outline-text-3" id="text-2-4">
</div>
<div id="outline-container-org7c141d1" class="outline-4">
<h4 id="org7c141d1"><span class="section-number-4">2.4.1</span> Stiffness</h4>
<div id="outline-container-orgc0281f1" class="outline-4">
<h4 id="orgc0281f1"><span class="section-number-4">2.4.1</span> Stiffness</h4>
<div class="outline-text-4" id="text-2-4-1">
<p>
The transfer function from vertical external force to the relative vertical displacement is identified.
@@ -1779,16 +1775,16 @@ The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
</div>
</div>
<div id="outline-container-org6336a4d" class="outline-4">
<h4 id="org6336a4d"><span class="section-number-4">2.4.2</span> Resonance Frequency</h4>
<div id="outline-container-orgcebe0f9" class="outline-4">
<h4 id="orgcebe0f9"><span class="section-number-4">2.4.2</span> Resonance Frequency</h4>
<div class="outline-text-4" id="text-2-4-2">
<p>
The resonance frequency is specified to be between 650Hz and 840Hz.
This is also the case for the FEM model (Figure <a href="#org2f62cd6">8</a>).
This is also the case for the FEM model (Figure <a href="#orgbb4a26e">8</a>).
</p>
<div id="org2f62cd6" class="figure">
<div id="orgbb4a26e" class="figure">
<p><img src="figs/apa300ml_resonance.png" alt="apa300ml_resonance.png" />
</p>
<p><span class="figure-number">Figure 8: </span>First resonance is around 800Hz</p>
@@ -1796,8 +1792,8 @@ This is also the case for the FEM model (Figure <a href="#org2f62cd6">8</a>).
</div>
</div>
<div id="outline-container-org7adcbea" class="outline-4">
<h4 id="org7adcbea"><span class="section-number-4">2.4.3</span> Amplification factor</h4>
<div id="outline-container-orgda4f233" class="outline-4">
<h4 id="orgda4f233"><span class="section-number-4">2.4.3</span> Amplification factor</h4>
<div class="outline-text-4" id="text-2-4-3">
<p>
The amplification factor is the ratio of the axial displacement to the stack displacement.
@@ -1830,8 +1826,8 @@ If we take the ratio of the piezo height and length (approximation of the amplif
</div>
</div>
<div id="outline-container-org924ba9a" class="outline-4">
<h4 id="org924ba9a"><span class="section-number-4">2.4.4</span> Stroke</h4>
<div id="outline-container-org59829b6" class="outline-4">
<h4 id="org59829b6"><span class="section-number-4">2.4.4</span> Stroke</h4>
<div class="outline-text-4" id="text-2-4-4">
<p>
Estimation of the actuator stroke:
@@ -1862,8 +1858,8 @@ This is exactly the specified stroke in the data-sheet.
</div>
</div>
<div id="outline-container-org0334d98" class="outline-3">
<h3 id="org0334d98"><span class="section-number-3">2.5</span> Identification of the Dynamics</h3>
<div id="outline-container-org1cbc8a6" class="outline-3">
<h3 id="org1cbc8a6"><span class="section-number-3">2.5</span> Identification of the Dynamics</h3>
<div class="outline-text-3" id="text-2-5">
<p>
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block.
@@ -1889,7 +1885,7 @@ The same dynamics is identified for a payload mass of 10Kg.
</div>
<div id="org452a3a7" class="figure">
<div id="orgb07566c" class="figure">
<p><img src="figs/apa300ml_plant_dynamics.png" alt="apa300ml_plant_dynamics.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Transfer function from forces applied by the stack to the axial displacement of the APA</p>
@@ -1897,28 +1893,28 @@ The same dynamics is identified for a payload mass of 10Kg.
</div>
</div>
<div id="outline-container-org889c8e8" class="outline-3">
<h3 id="org889c8e8"><span class="section-number-3">2.6</span> IFF</h3>
<div id="outline-container-org44a32d5" class="outline-3">
<h3 id="org44a32d5"><span class="section-number-3">2.6</span> IFF</h3>
<div class="outline-text-3" id="text-2-6">
<p>
Let&rsquo;s use 2 stacks as actuators and 1 stack as force sensor.
</p>
<p>
The transfer function from actuator to sensors is identified and shown in Figure <a href="#orge704515">10</a>.
The transfer function from actuator to sensors is identified and shown in Figure <a href="#org0cb4e3c">10</a>.
</p>
<div id="orge704515" class="figure">
<div id="org0cb4e3c" class="figure">
<p><img src="figs/apa300ml_iff_plant.png" alt="apa300ml_iff_plant.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Transfer function from actuator to force sensor</p>
</div>
<p>
For root locus corresponding to IFF is shown in Figure <a href="#org4d28155">11</a>.
For root locus corresponding to IFF is shown in Figure <a href="#org57c3b0d">11</a>.
</p>
<div id="org4d28155" class="figure">
<div id="org57c3b0d" class="figure">
<p><img src="figs/apa300ml_iff_root_locus.png" alt="apa300ml_iff_root_locus.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Root Locus for IFF</p>
@@ -1926,25 +1922,25 @@ For root locus corresponding to IFF is shown in Figure <a href="#org4d28155">11<
</div>
</div>
<div id="outline-container-org6f11c82" class="outline-3">
<h3 id="org6f11c82"><span class="section-number-3">2.7</span> DVF</h3>
<div id="outline-container-org3e558c6" class="outline-3">
<h3 id="org3e558c6"><span class="section-number-3">2.7</span> DVF</h3>
<div class="outline-text-3" id="text-2-7">
<p>
Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure <a href="#org84dd7d9">12</a>.
Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure <a href="#org828d315">12</a>.
</p>
<div id="org84dd7d9" class="figure">
<div id="org828d315" class="figure">
<p><img src="figs/apa300ml_dvf_plant.png" alt="apa300ml_dvf_plant.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Transfer function from stack actuator to relative motion sensor</p>
</div>
<p>
The root locus for DVF is shown in Figure <a href="#org362161f">13</a>.
The root locus for DVF is shown in Figure <a href="#orgf2f0551">13</a>.
</p>
<div id="org362161f" class="figure">
<div id="orgf2f0551" class="figure">
<p><img src="figs/apa300ml_dvf_root_locus.png" alt="apa300ml_dvf_root_locus.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Root Locus for Direct Velocity Feedback</p>
@@ -1952,8 +1948,8 @@ The root locus for DVF is shown in Figure <a href="#org362161f">13</a>.
</div>
</div>
<div id="outline-container-org1c376b5" class="outline-3">
<h3 id="org1c376b5"><span class="section-number-3">2.8</span> Identification for a simpler model</h3>
<div id="outline-container-orgad3fdd9" class="outline-3">
<h3 id="orgad3fdd9"><span class="section-number-3">2.8</span> Identification for a simpler model</h3>
<div class="outline-text-3" id="text-2-8">
<p>
The goal in this section is to identify the parameters of a simple APA model from the FEM.
@@ -1965,12 +1961,12 @@ The presented model is based on (<a href="#citeproc_bib_item_2">Souleille et al.
</p>
<p>
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#orgb1100b8">5</a>).
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#org73ab5e6">5</a>).
The parameters are shown in the table below.
</p>
<div id="orgd4eeaf2" class="figure">
<div id="org2d53ab0" class="figure">
<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
@@ -2119,7 +2115,7 @@ Adjust the DC gain for the force sensor:
</div>
<div id="org36826f6" class="figure">
<div id="org6c57210" class="figure">
<p><img src="figs/apa300ml_comp_simpler_model.png" alt="apa300ml_comp_simpler_model.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Comparison of the Dynamics between the FEM model and the simplified one</p>
@@ -2129,7 +2125,7 @@ Adjust the DC gain for the force sensor:
We now compare the FEM model with the simplified simscape model.
</p>
<div id="orge85d5a8" class="figure">
<div id="orgdec5f8c" class="figure">
<p><img src="figs/apa300ml_comp_simpler_simscape.png" alt="apa300ml_comp_simpler_simscape.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Comparison of the Dynamics between the FEM model and the simplified simscape model</p>
@@ -2145,12 +2141,12 @@ We save the parameters of the simplified model for the APA300ML:
</div>
</div>
<div id="outline-container-org30bc4bf" class="outline-3">
<h3 id="org30bc4bf"><span class="section-number-3">2.9</span> Identification of the stiffness properties</h3>
<div id="outline-container-orge0b9f5a" class="outline-3">
<h3 id="orge0b9f5a"><span class="section-number-3">2.9</span> Identification of the stiffness properties</h3>
<div class="outline-text-3" id="text-2-9">
</div>
<div id="outline-container-orge89f3f8" class="outline-4">
<h4 id="orge89f3f8"><span class="section-number-4">2.9.1</span> APA Alone</h4>
<div id="outline-container-org52ddecb" class="outline-4">
<h4 id="org52ddecb"><span class="section-number-4">2.9.1</span> APA Alone</h4>
<div class="outline-text-4" id="text-2-9-1">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@@ -2201,8 +2197,8 @@ We save the parameters of the simplified model for the APA300ML:
</div>
</div>
<div id="outline-container-org4651c6e" class="outline-4">
<h4 id="org4651c6e"><span class="section-number-4">2.9.2</span> See how the global stiffness is changing with the flexible joints</h4>
<div id="outline-container-org02b6855" class="outline-4">
<h4 id="org02b6855"><span class="section-number-4">2.9.2</span> See how the global stiffness is changing with the flexible joints</h4>
<div class="outline-text-4" id="text-2-9-2">
<div class="org-src-container">
<pre class="src src-matlab">flex = load(<span class="org-string">'./mat/flexor_ID16.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>);
@@ -2290,8 +2286,8 @@ legend(<span class="org-string">'location'</span>, <span class="org-string">'nor
</div>
</div>
<div id="outline-container-orgbb1e485" class="outline-3">
<h3 id="orgbb1e485"><span class="section-number-3">2.10</span> Effect of APA300ML in the flexibility of the leg</h3>
<div id="outline-container-org34de703" class="outline-3">
<h3 id="org34de703"><span class="section-number-3">2.10</span> Effect of APA300ML in the flexibility of the leg</h3>
<div class="outline-text-3" id="text-2-10">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@@ -2352,11 +2348,11 @@ legend(<span class="org-string">'location'</span>, <span class="org-string">'nor
</div>
</div>
<div id="outline-container-org71e2995" class="outline-2">
<h2 id="org71e2995"><span class="section-number-2">3</span> Flexible Joint</h2>
<div id="outline-container-orgb6c0ee0" class="outline-2">
<h2 id="orgb6c0ee0"><span class="section-number-2">3</span> Flexible Joint</h2>
<div class="outline-text-2" id="text-3">
<p>
The studied flexor is shown in Figure <a href="#org0b718d7">17</a>.
The studied flexor is shown in Figure <a href="#orgdff9d67">17</a>.
</p>
<p>
@@ -2369,15 +2365,15 @@ A simplified model of the flexor is then developped.
</p>
<div id="org0b718d7" class="figure">
<div id="orgdff9d67" class="figure">
<p><img src="figs/flexor_id16_screenshot.png" alt="flexor_id16_screenshot.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Flexor studied</p>
</div>
</div>
<div id="outline-container-org4609327" class="outline-3">
<h3 id="org4609327"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div id="outline-container-orgd7b1d5f" class="outline-3">
<h3 id="orgd7b1d5f"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-3-1">
<p>
We first extract the stiffness and mass matrices.
@@ -2403,8 +2399,8 @@ Then, we extract the coordinates of the interface nodes.
</div>
</div>
<div id="outline-container-org222b467" class="outline-3">
<h3 id="org222b467"><span class="section-number-3">3.2</span> Output parameters</h3>
<div id="outline-container-org9778a32" class="outline-3">
<h3 id="org9778a32"><span class="section-number-3">3.2</span> Output parameters</h3>
<div class="outline-text-3" id="text-3-2">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/flexor_ID16.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>);
@@ -2805,8 +2801,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div>
</div>
<div id="outline-container-orgace43b0" class="outline-3">
<h3 id="orgace43b0"><span class="section-number-3">3.3</span> Flexible Joint Characteristics</h3>
<div id="outline-container-orgcb9bad1" class="outline-3">
<h3 id="orgcb9bad1"><span class="section-number-3">3.3</span> Flexible Joint Characteristics</h3>
<div class="outline-text-3" id="text-3-3">
<p>
The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
@@ -2864,8 +2860,8 @@ The most important parameters of the flexible joint can be directly estimated fr
</div>
</div>
<div id="outline-container-orgc60e392" class="outline-3">
<h3 id="orgc60e392"><span class="section-number-3">3.4</span> Identification of the parameters using Simscape</h3>
<div id="outline-container-org4dadc02" class="outline-3">
<h3 id="org4dadc02"><span class="section-number-3">3.4</span> Identification of the parameters using Simscape</h3>
<div class="outline-text-3" id="text-3-4">
<p>
The flexor is now imported into Simscape and its parameters are estimated using an identification.
@@ -2922,15 +2918,15 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
</div>
</div>
<div id="outline-container-org43c8aa7" class="outline-3">
<h3 id="org43c8aa7"><span class="section-number-3">3.5</span> Simpler Model</h3>
<div id="outline-container-org30336a6" class="outline-3">
<h3 id="org30336a6"><span class="section-number-3">3.5</span> Simpler Model</h3>
<div class="outline-text-3" id="text-3-5">
<p>
Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#orga4765e3">18</a>.
Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#org9c5b090">18</a>.
</p>
<div id="orga4765e3" class="figure">
<div id="org9c5b090" class="figure">
<p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Bushing Joint used to model the flexible joint</p>
@@ -2955,7 +2951,7 @@ The two obtained dynamics are compared in Figure
</p>
<div id="org54ce633" class="figure">
<div id="org6baee4c" class="figure">
<p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p>
@@ -2964,19 +2960,19 @@ The two obtained dynamics are compared in Figure
</div>
</div>
<div id="outline-container-org5d2c10d" class="outline-2">
<h2 id="org5d2c10d"><span class="section-number-2">4</span> Optimal Flexible Joint</h2>
<div id="outline-container-orgd9d5aff" class="outline-2">
<h2 id="orgd9d5aff"><span class="section-number-2">4</span> Optimal Flexible Joint</h2>
<div class="outline-text-2" id="text-4">
<div id="orgc598a8a" class="figure">
<div id="org47739fa" class="figure">
<p><img src="data/flexor_circ_025/CS.jpg" alt="CS.jpg" />
</p>
<p><span class="figure-number">Figure 20: </span>Flexor studied</p>
</div>
</div>
<div id="outline-container-orgfec12e9" class="outline-3">
<h3 id="orgfec12e9"><span class="section-number-3">4.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div id="outline-container-org83c1679" class="outline-3">
<h3 id="org83c1679"><span class="section-number-3">4.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-4-1">
<p>
We first extract the stiffness and mass matrices.
@@ -3002,8 +2998,8 @@ Then, we extract the coordinates of the interface nodes.
</div>
</div>
<div id="outline-container-org51a4b8d" class="outline-3">
<h3 id="org51a4b8d"><span class="section-number-3">4.2</span> Output parameters</h3>
<div id="outline-container-orgbee4a84" class="outline-3">
<h3 id="orgbee4a84"><span class="section-number-3">4.2</span> Output parameters</h3>
<div class="outline-text-3" id="text-4-2">
<div class="org-src-container">
<pre class="src src-matlab">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>);
@@ -3404,8 +3400,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div>
</div>
<div id="outline-container-org9df419b" class="outline-3">
<h3 id="org9df419b"><span class="section-number-3">4.3</span> Flexible Joint Characteristics</h3>
<div id="outline-container-org7609951" class="outline-3">
<h3 id="org7609951"><span class="section-number-3">4.3</span> Flexible Joint Characteristics</h3>
<div class="outline-text-3" id="text-4-3">
<p>
The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
@@ -3455,8 +3451,8 @@ The most important parameters of the flexible joint can be directly estimated fr
</div>
</div>
<div id="outline-container-org4ea4053" class="outline-3">
<h3 id="org4ea4053"><span class="section-number-3">4.4</span> Identification of the parameters using Simscape</h3>
<div id="outline-container-org8bf4f56" class="outline-3">
<h3 id="org8bf4f56"><span class="section-number-3">4.4</span> Identification of the parameters using Simscape</h3>
<div class="outline-text-3" id="text-4-4">
<p>
The flexor is now imported into Simscape and its parameters are estimated using an identification.
@@ -3513,15 +3509,15 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
</div>
</div>
<div id="outline-container-org070daa9" class="outline-3">
<h3 id="org070daa9"><span class="section-number-3">4.5</span> Simpler Model</h3>
<div id="outline-container-orgd8cb8ff" class="outline-3">
<h3 id="orgd8cb8ff"><span class="section-number-3">4.5</span> Simpler Model</h3>
<div class="outline-text-3" id="text-4-5">
<p>
Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#orga4765e3">18</a>.
Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#org9c5b090">18</a>.
</p>
<div id="orgaded736" class="figure">
<div id="orgc5e5982" class="figure">
<p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Bushing Joint used to model the flexible joint</p>
@@ -3546,7 +3542,7 @@ The two obtained dynamics are compared in Figure
</p>
<div id="orga26c578" class="figure">
<div id="org2936555" class="figure">
<p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p>
@@ -3555,16 +3551,16 @@ The two obtained dynamics are compared in Figure
</div>
</div>
<div id="outline-container-org72ebb5c" class="outline-2">
<h2 id="org72ebb5c"><span class="section-number-2">5</span> Integral Force Feedback with Amplified Piezo</h2>
<div id="outline-container-org7f2d76d" class="outline-2">
<h2 id="org7f2d76d"><span class="section-number-2">5</span> Integral Force Feedback with Amplified Piezo</h2>
<div class="outline-text-2" id="text-5">
<p>
In this section, we try to replicate the results obtained in (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>).
</p>
</div>
<div id="outline-container-orgffa90de" class="outline-3">
<h3 id="orgffa90de"><span class="section-number-3">5.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div id="outline-container-orgd9dc7be" class="outline-3">
<h3 id="orgd9dc7be"><span class="section-number-3">5.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-5-1">
<p>
We first extract the stiffness and mass matrices.
@@ -3585,11 +3581,11 @@ Then, we extract the coordinates of the interface nodes.
</div>
</div>
<div id="outline-container-org4ac5a6e" class="outline-3">
<h3 id="org4ac5a6e"><span class="section-number-3">5.2</span> IFF Plant</h3>
<div id="outline-container-org3671fca" class="outline-3">
<h3 id="org3671fca"><span class="section-number-3">5.2</span> IFF Plant</h3>
<div class="outline-text-3" id="text-5-2">
<p>
The transfer function from the force actuator to the force sensor is identified and shown in Figure <a href="#org4390f0c">23</a>.
The transfer function from the force actuator to the force sensor is identified and shown in Figure <a href="#org294cdfb">23</a>.
</p>
<div class="org-src-container">
@@ -3626,7 +3622,7 @@ Gf = linearize(mdl, io);
</div>
<div id="org4390f0c" class="figure">
<div id="org294cdfb" class="figure">
<p><img src="figs/piezo_amplified_iff_plant.png" alt="piezo_amplified_iff_plant.png" />
</p>
<p><span class="figure-number">Figure 23: </span>IFF Plant</p>
@@ -3634,11 +3630,11 @@ Gf = linearize(mdl, io);
</div>
</div>
<div id="outline-container-orgdc46434" class="outline-3">
<h3 id="orgdc46434"><span class="section-number-3">5.3</span> IFF controller</h3>
<div id="outline-container-org1cbfb66" class="outline-3">
<h3 id="org1cbfb66"><span class="section-number-3">5.3</span> IFF controller</h3>
<div class="outline-text-3" id="text-5-3">
<p>
The controller is defined and the loop gain is shown in Figure <a href="#orgc28c610">24</a>.
The controller is defined and the loop gain is shown in Figure <a href="#orgbcc66e9">24</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Kiff = <span class="org-type">-</span>1e12<span class="org-type">/</span>s;
@@ -3646,7 +3642,7 @@ The controller is defined and the loop gain is shown in Figure <a href="#orgc28c
</div>
<div id="orgc28c610" class="figure">
<div id="orgbcc66e9" class="figure">
<p><img src="figs/piezo_amplified_iff_loop_gain.png" alt="piezo_amplified_iff_loop_gain.png" />
</p>
<p><span class="figure-number">Figure 24: </span>IFF Loop Gain</p>
@@ -3654,8 +3650,8 @@ The controller is defined and the loop gain is shown in Figure <a href="#orgc28c
</div>
</div>
<div id="outline-container-orgc9d8168" class="outline-3">
<h3 id="orgc9d8168"><span class="section-number-3">5.4</span> Closed Loop System</h3>
<div id="outline-container-org7b29313" class="outline-3">
<h3 id="org7b29313"><span class="section-number-3">5.4</span> Closed Loop System</h3>
<div class="outline-text-3" id="text-5-4">
<div class="org-src-container">
<pre class="src src-matlab">m = 10;
@@ -3698,7 +3694,7 @@ G.OutputName = {<span class="org-string">'x1'</span>, <span class="org-string">
</div>
<div id="orgcc44982" class="figure">
<div id="org9317d23" class="figure">
<p><img src="figs/piezo_amplified_iff_comp.png" alt="piezo_amplified_iff_comp.png" />
</p>
<p><span class="figure-number">Figure 25: </span>OL and CL transfer functions</p>
@@ -3706,7 +3702,7 @@ G.OutputName = {<span class="org-string">'x1'</span>, <span class="org-string">
<div id="org9e21ddc" class="figure">
<div id="org7f216a9" class="figure">
<p><img src="figs/souleille18_results.png" alt="souleille18_results.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Results obtained in <a class='org-ref-reference' href="#souleille18_concep_activ_mount_space_applic">souleille18_concep_activ_mount_space_applic</a></p>
@@ -3715,15 +3711,15 @@ G.OutputName = {<span class="org-string">'x1'</span>, <span class="org-string">
</div>
</div>
<div id="outline-container-orge46f2bf" class="outline-2">
<h2 id="orge46f2bf"><span class="section-number-2">6</span> Complete Strut with Encoder</h2>
<div id="outline-container-org1272d3f" class="outline-2">
<h2 id="org1272d3f"><span class="section-number-2">6</span> Complete Strut with Encoder</h2>
<div class="outline-text-2" id="text-6">
</div>
<div id="outline-container-org9c8b2a0" class="outline-3">
<h3 id="org9c8b2a0"><span class="section-number-3">6.1</span> Introduction</h3>
<div id="outline-container-orgddf8d43" class="outline-3">
<h3 id="orgddf8d43"><span class="section-number-3">6.1</span> Introduction</h3>
<div class="outline-text-3" id="text-6-1">
<div id="org169745c" class="figure">
<div id="org1af2e05" class="figure">
<p><img src="data/strut_encoder/points3.jpg" alt="points3.jpg" />
</p>
<p><span class="figure-number">Figure 27: </span>Interface points</p>
@@ -3735,8 +3731,8 @@ Flexible joints have 0.25mm width.
</div>
</div>
<div id="outline-container-org6b21925" class="outline-3">
<h3 id="org6b21925"><span class="section-number-3">6.2</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div id="outline-container-org4742c38" class="outline-3">
<h3 id="org4742c38"><span class="section-number-3">6.2</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-6-2">
<p>
We first extract the stiffness and mass matrices.
@@ -3762,8 +3758,8 @@ Then, we extract the coordinates of the interface nodes.
</div>
</div>
<div id="outline-container-org40668e1" class="outline-3">
<h3 id="org40668e1"><span class="section-number-3">6.3</span> Output parameters</h3>
<div id="outline-container-org332b172" class="outline-3">
<h3 id="org332b172"><span class="section-number-3">6.3</span> Output parameters</h3>
<div class="outline-text-3" id="text-6-3">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/strut_encoder.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>);
@@ -4215,8 +4211,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
<div id="outline-container-org6d5c440" class="outline-3">
<h3 id="org6d5c440"><span class="section-number-3">6.4</span> Piezoelectric parameters</h3>
<div id="outline-container-orgadca4a4" class="outline-3">
<h3 id="orgadca4a4"><span class="section-number-3">6.4</span> Piezoelectric parameters</h3>
<div class="outline-text-3" id="text-6-4">
<p>
Parameters for the APA300ML:
@@ -4240,8 +4236,8 @@ ns = 1; <span class="org-comment">% Number of stacks used as force sensor</span>
</div>
</div>
<div id="outline-container-org2521017" class="outline-3">
<h3 id="org2521017"><span class="section-number-3">6.5</span> Identification of the Dynamics</h3>
<div id="outline-container-org226d3f3" class="outline-3">
<h3 id="org226d3f3"><span class="section-number-3">6.5</span> Identification of the Dynamics</h3>
<div class="outline-text-3" id="text-6-5">
<p>
The dynamics is identified from the applied force to the measured relative displacement.
@@ -4267,7 +4263,7 @@ The same dynamics is identified for a payload mass of 10Kg.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-10-29 jeu. 10:08</p>
<p class="date">Created: 2020-11-12 jeu. 10:34</p>
</div>
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