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<title>Finite Element Model with Simscape</title> <title>NASS - Finite Element Models with Simscape</title>
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<a accesskey="H" href="../index.html"> HOME </a> <a accesskey="H" href="../index.html"> HOME </a>
</div><div id="content"> </div><div id="content">
<h1 class="title">Finite Element Model with Simscape</h1> <h1 class="title">NASS - Finite Element Models with Simscape</h1>
<div id="table-of-contents"> <div id="table-of-contents">
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org18d2db8">1. APA300ML</a> <li><a href="#orgb231366">1. APA300ML</a>
<ul> <ul>
<li><a href="#org5c45df4">1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li> <li><a href="#orga4e3f9c">1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#orga9811cf">1.2. Piezoelectric parameters</a></li> <li><a href="#org4f3db59">1.2. Piezoelectric parameters</a></li>
<li><a href="#orgf60bf13">1.3. Simscape Model</a></li> <li><a href="#org364e184">1.3. Simscape Model</a></li>
<li><a href="#org7589ca6">1.4. Identification of the APA Characteristics</a> <li><a href="#org8bf66af">1.4. Identification of the APA Characteristics</a>
<ul> <ul>
<li><a href="#org839ecc0">1.4.1. Stiffness</a></li> <li><a href="#orgc2b9be5">1.4.1. Stiffness</a></li>
<li><a href="#org9b9cedd">1.4.2. Resonance Frequency</a></li> <li><a href="#orgd55eeff">1.4.2. Resonance Frequency</a></li>
<li><a href="#org87cbe72">1.4.3. Amplification factor</a></li> <li><a href="#org59f7b55">1.4.3. Amplification factor</a></li>
<li><a href="#org0071048">1.4.4. Stroke</a></li> <li><a href="#orga970d47">1.4.4. Stroke</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4fdb600">1.5. Identification of the Dynamics from actuator to replace displacement</a></li> <li><a href="#org875f674">1.5. Identification of the Dynamics from actuator to replace displacement</a></li>
<li><a href="#orga2f4fd6">1.6. Identification of the Dynamics from actuator to force sensor</a></li> <li><a href="#org926378e">1.6. Identification of the Dynamics from actuator to force sensor</a></li>
<li><a href="#org8ece2ce">1.7. Identification for a simpler model</a></li> <li><a href="#org0b533cc">1.7. Identification for a simpler model</a></li>
<li><a href="#org43ae9e5">1.8. Integral Force Feedback</a></li> <li><a href="#orgd7e3154">1.8. Integral Force Feedback</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgc203c93">2. First Flexible Joint Geometry</a> <li><a href="#orge12e432">2. First Flexible Joint Geometry</a>
<ul> <ul>
<li><a href="#orga48c65c">2.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li> <li><a href="#org91559c3">2.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org5de18c6">2.2. Identification of the parameters using Simscape and looking at the Stiffness Matrix</a></li> <li><a href="#org0c0ae39">2.2. Identification of the parameters using Simscape and looking at the Stiffness Matrix</a></li>
<li><a href="#orgb2d0259">2.3. Simpler Model</a></li> <li><a href="#orgb1eeb49">2.3. Simpler Model</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4dadb53">3. Optimized Flexible Joint</a> <li><a href="#org6fa0f81">3. Optimized Flexible Joint</a>
<ul> <ul>
<li><a href="#orgb1d1e31">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li> <li><a href="#orgadfaeb7">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#orgc7107db">3.2. Identification of the parameters using Simscape</a></li> <li><a href="#org1a74e71">3.2. Identification of the parameters using Simscape</a></li>
<li><a href="#org96d2775">3.3. Simpler Model</a></li> <li><a href="#org3ba1fee">3.3. Simpler Model</a></li>
<li><a href="#org2916e5d">3.4. Comparison with a stiffer Flexible Joint</a></li> <li><a href="#orgec51432">3.4. Comparison with a stiffer Flexible Joint</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org79ad15f">4. Complete Strut with Encoder</a> <li><a href="#org91975b5">4. Complete Strut with Encoder</a>
<ul> <ul>
<li><a href="#org26e8494">4.1. Introduction</a></li> <li><a href="#orgd829824">4.1. Introduction</a></li>
<li><a href="#orgd702a5f">4.2. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li> <li><a href="#orgd7f754c">4.2. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
<li><a href="#org42c2461">4.3. Piezoelectric parameters</a></li> <li><a href="#org5019141">4.3. Piezoelectric parameters</a></li>
<li><a href="#orga5ba630">4.4. Identification of the Dynamics</a></li> <li><a href="#org72bb8f1">4.4. Identification of the Dynamics</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -80,23 +80,26 @@
In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis. In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.
</p> </p>
<p>
It is divided in the following sections:
</p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org6804357">1</a>: <li>Section <a href="#org31bfe65">1</a>:
A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape. A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape.
The static and dynamical properties of the APA300ML are then estimated using the Simscape model.</li> The static and dynamical properties of the APA300ML are then estimated using the Simscape model.</li>
<li>Section <a href="#orgb1abf80">2</a>: <li>Section <a href="#orga0ece29">2</a>:
A first geometry of a Flexible joint is modelled and its characteristics are identified from the Stiffness matrix as well as from the Simscape model.</li> A first geometry of a Flexible joint is modelled and its characteristics are identified from the Stiffness matrix as well as from the Simscape model.</li>
<li>Section <a href="#org146537a">3</a>: <li>Section <a href="#org513c349">3</a>:
An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.</li> An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.</li>
<li>Section <a href="#orgc4b9146">4</a>: <li>Section <a href="#orgcff61d6">4</a>:
A super element of a complete strut is exported.</li> A super element of a complete strut is studied.</li>
</ul> </ul>
<div id="outline-container-org18d2db8" class="outline-2"> <div id="outline-container-orgb231366" class="outline-2">
<h2 id="org18d2db8"><span class="section-number-2">1</span> APA300ML</h2> <h2 id="orgb231366"><span class="section-number-2">1</span> APA300ML</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="org6804357"></a> <a id="org31bfe65"></a>
</p> </p>
<p> <p>
In this section, the Amplified Piezoelectric Actuator APA300ML (<a href="doc/APA300ML.pdf">doc</a>) is modeled using a Finite Element Software. In this section, the Amplified Piezoelectric Actuator APA300ML (<a href="doc/APA300ML.pdf">doc</a>) is modeled using a Finite Element Software.
@ -104,19 +107,19 @@ Then a <i>super element</i> is exported and imported in Simscape where its dynam
</p> </p>
<p> <p>
A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure <a href="#org7a29cfe">1</a>. A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure <a href="#orgfaefa60">1</a>.
The remote point used are also shown in this figure. The remote point used are also shown in this figure.
</p> </p>
<div id="org7a29cfe" class="figure"> <div id="orgfaefa60" class="figure">
<p><img src="figs/apa300ml_ansys.jpg" alt="apa300ml_ansys.jpg" /> <p><img src="figs/apa300ml_ansys.jpg" alt="apa300ml_ansys.jpg" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Ansys FEM of the APA300ML</p> <p><span class="figure-number">Figure 1: </span>Ansys FEM of the APA300ML</p>
</div> </div>
</div> </div>
<div id="outline-container-org5c45df4" class="outline-3"> <div id="outline-container-orga4e3f9c" class="outline-3">
<h3 id="org5c45df4"><span class="section-number-3">1.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3> <h3 id="orga4e3f9c"><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"> <div class="outline-text-3" id="text-1-1">
<p> <p>
We first extract the stiffness and mass matrices. We first extract the stiffness and mass matrices.
@ -570,8 +573,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can now use th
</div> </div>
</div> </div>
<div id="outline-container-orga9811cf" class="outline-3"> <div id="outline-container-org4f3db59" class="outline-3">
<h3 id="orga9811cf"><span class="section-number-3">1.2</span> Piezoelectric parameters</h3> <h3 id="org4f3db59"><span class="section-number-3">1.2</span> Piezoelectric parameters</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material: In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
@ -590,7 +593,7 @@ C = 5e<span class="org-type">-</span>6; <span class="org-comment">% Stack c
The ratio of the developed force to applied voltage is: The ratio of the developed force to applied voltage is:
</p> </p>
\begin{equation} \begin{equation}
\label{org0bf4f86} \label{org26cf049}
F_a = g_a V_a, \quad g_a = d_{33} n k_a F_a = g_a V_a, \quad g_a = d_{33} n k_a
\end{equation} \end{equation}
<p> <p>
@ -621,7 +624,7 @@ If we take the numerical values, we obtain:
From (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>) (page 123), the relation between relative displacement of the sensor stack and generated voltage is: From (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>) (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
</p> </p>
\begin{equation} \begin{equation}
\label{org0478126} \label{orgd71c6e4}
V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
\end{equation} \end{equation}
<p> <p>
@ -650,8 +653,8 @@ If we take the numerical values, we obtain:
</div> </div>
</div> </div>
<div id="outline-container-orgf60bf13" class="outline-3"> <div id="outline-container-org364e184" class="outline-3">
<h3 id="orgf60bf13"><span class="section-number-3">1.3</span> Simscape Model</h3> <h3 id="org364e184"><span class="section-number-3">1.3</span> Simscape Model</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block. The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block.
@ -666,7 +669,7 @@ Let&rsquo;s say we use two stacks as a force sensor and one stack as an actuator
</ul> </ul>
<p> <p>
The interface nodes are shown in Figure <a href="#org7a29cfe">1</a>. The interface nodes are shown in Figure <a href="#orgfaefa60">1</a>.
</p> </p>
<p> <p>
@ -675,12 +678,12 @@ One mass is fixed at one end of the piezo-electric stack actuator (remove point
</div> </div>
</div> </div>
<div id="outline-container-org7589ca6" class="outline-3"> <div id="outline-container-org8bf66af" class="outline-3">
<h3 id="org7589ca6"><span class="section-number-3">1.4</span> Identification of the APA Characteristics</h3> <h3 id="org8bf66af"><span class="section-number-3">1.4</span> Identification of the APA Characteristics</h3>
<div class="outline-text-3" id="text-1-4"> <div class="outline-text-3" id="text-1-4">
</div> </div>
<div id="outline-container-org839ecc0" class="outline-4"> <div id="outline-container-orgc2b9be5" class="outline-4">
<h4 id="org839ecc0"><span class="section-number-4">1.4.1</span> Stiffness</h4> <h4 id="orgc2b9be5"><span class="section-number-4">1.4.1</span> Stiffness</h4>
<div class="outline-text-4" id="text-1-4-1"> <div class="outline-text-4" id="text-1-4-1">
<p> <p>
The transfer function from vertical external force to the relative vertical displacement is identified. The transfer function from vertical external force to the relative vertical displacement is identified.
@ -705,16 +708,16 @@ The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
</div> </div>
</div> </div>
<div id="outline-container-org9b9cedd" class="outline-4"> <div id="outline-container-orgd55eeff" class="outline-4">
<h4 id="org9b9cedd"><span class="section-number-4">1.4.2</span> Resonance Frequency</h4> <h4 id="orgd55eeff"><span class="section-number-4">1.4.2</span> Resonance Frequency</h4>
<div class="outline-text-4" id="text-1-4-2"> <div class="outline-text-4" id="text-1-4-2">
<p> <p>
The resonance frequency is specified to be between 650Hz and 840Hz. The resonance frequency is specified to be between 650Hz and 840Hz.
This is also the case for the FEM model (Figure <a href="#org2662235">2</a>). This is also the case for the FEM model (Figure <a href="#org5a0e1d6">2</a>).
</p> </p>
<div id="org2662235" class="figure"> <div id="org5a0e1d6" class="figure">
<p><img src="figs/apa300ml_resonance.png" alt="apa300ml_resonance.png" /> <p><img src="figs/apa300ml_resonance.png" alt="apa300ml_resonance.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>First resonance is around 800Hz</p> <p><span class="figure-number">Figure 2: </span>First resonance is around 800Hz</p>
@ -722,8 +725,8 @@ This is also the case for the FEM model (Figure <a href="#org2662235">2</a>).
</div> </div>
</div> </div>
<div id="outline-container-org87cbe72" class="outline-4"> <div id="outline-container-org59f7b55" class="outline-4">
<h4 id="org87cbe72"><span class="section-number-4">1.4.3</span> Amplification factor</h4> <h4 id="org59f7b55"><span class="section-number-4">1.4.3</span> Amplification factor</h4>
<div class="outline-text-4" id="text-1-4-3"> <div class="outline-text-4" id="text-1-4-3">
<p> <p>
The amplification factor is the ratio of the vertical displacement to the stack displacement. The amplification factor is the ratio of the vertical displacement to the stack displacement.
@ -756,8 +759,8 @@ This is actually correct and approximately corresponds to the ratio of the piezo
</div> </div>
</div> </div>
<div id="outline-container-org0071048" class="outline-4"> <div id="outline-container-orga970d47" class="outline-4">
<h4 id="org0071048"><span class="section-number-4">1.4.4</span> Stroke</h4> <h4 id="orga970d47"><span class="section-number-4">1.4.4</span> Stroke</h4>
<div class="outline-text-4" id="text-1-4-4"> <div class="outline-text-4" id="text-1-4-4">
<p> <p>
Estimation of the actuator stroke: Estimation of the actuator stroke:
@ -788,8 +791,8 @@ This is exactly the specified stroke in the data-sheet.
</div> </div>
</div> </div>
<div id="outline-container-org4fdb600" class="outline-3"> <div id="outline-container-org875f674" class="outline-3">
<h3 id="org4fdb600"><span class="section-number-3">1.5</span> Identification of the Dynamics from actuator to replace displacement</h3> <h3 id="org875f674"><span class="section-number-3">1.5</span> Identification of the Dynamics from actuator to replace displacement</h3>
<div class="outline-text-3" id="text-1-5"> <div class="outline-text-3" id="text-1-5">
<p> <p>
We first set the mass to be approximately zero. We first set the mass to be approximately zero.
@ -802,17 +805,17 @@ The same dynamics is identified for a payload mass of 10Kg.
</div> </div>
<div id="orgb9beb05" class="figure"> <div id="org0bf96a7" class="figure">
<p><img src="figs/apa300ml_plant_dynamics.png" alt="apa300ml_plant_dynamics.png" /> <p><img src="figs/apa300ml_plant_dynamics.png" alt="apa300ml_plant_dynamics.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Transfer function from forces applied by the stack to the axial displacement of the APA</p> <p><span class="figure-number">Figure 3: </span>Transfer function from forces applied by the stack to the axial displacement of the APA</p>
</div> </div>
<p> <p>
The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure <a href="#orgcdc7a42">4</a>. The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure <a href="#orgf443cba">4</a>.
</p> </p>
<div id="orgcdc7a42" class="figure"> <div id="orgf443cba" class="figure">
<p><img src="figs/apa300ml_dvf_root_locus.png" alt="apa300ml_dvf_root_locus.png" /> <p><img src="figs/apa300ml_dvf_root_locus.png" alt="apa300ml_dvf_root_locus.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Root Locus for Direct Velocity Feedback</p> <p><span class="figure-number">Figure 4: </span>Root Locus for Direct Velocity Feedback</p>
@ -820,28 +823,28 @@ The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is
</div> </div>
</div> </div>
<div id="outline-container-orga2f4fd6" class="outline-3"> <div id="outline-container-org926378e" class="outline-3">
<h3 id="orga2f4fd6"><span class="section-number-3">1.6</span> Identification of the Dynamics from actuator to force sensor</h3> <h3 id="org926378e"><span class="section-number-3">1.6</span> Identification of the Dynamics from actuator to force sensor</h3>
<div class="outline-text-3" id="text-1-6"> <div class="outline-text-3" id="text-1-6">
<p> <p>
Let&rsquo;s use 2 stacks as a force sensor and 1 stack as force actuator. Let&rsquo;s use 2 stacks as a force sensor and 1 stack as force actuator.
</p> </p>
<p> <p>
The transfer function from actuator voltage to sensor voltage is identified and shown in Figure <a href="#org838a459">5</a>. The transfer function from actuator voltage to sensor voltage is identified and shown in Figure <a href="#org0571899">5</a>.
</p> </p>
<div id="org838a459" class="figure"> <div id="org0571899" class="figure">
<p><img src="figs/apa300ml_iff_plant.png" alt="apa300ml_iff_plant.png" /> <p><img src="figs/apa300ml_iff_plant.png" alt="apa300ml_iff_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Transfer function from actuator to force sensor</p> <p><span class="figure-number">Figure 5: </span>Transfer function from actuator to force sensor</p>
</div> </div>
<p> <p>
For root locus corresponding to IFF is shown in Figure <a href="#org6a92e46">6</a>. For root locus corresponding to IFF is shown in Figure <a href="#org4c7369c">6</a>.
</p> </p>
<div id="org6a92e46" class="figure"> <div id="org4c7369c" class="figure">
<p><img src="figs/apa300ml_iff_root_locus.png" alt="apa300ml_iff_root_locus.png" /> <p><img src="figs/apa300ml_iff_root_locus.png" alt="apa300ml_iff_root_locus.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Root Locus for IFF</p> <p><span class="figure-number">Figure 6: </span>Root Locus for IFF</p>
@ -849,8 +852,8 @@ For root locus corresponding to IFF is shown in Figure <a href="#org6a92e46">6</
</div> </div>
</div> </div>
<div id="outline-container-org8ece2ce" class="outline-3"> <div id="outline-container-org0b533cc" class="outline-3">
<h3 id="org8ece2ce"><span class="section-number-3">1.7</span> Identification for a simpler model</h3> <h3 id="org0b533cc"><span class="section-number-3">1.7</span> Identification for a simpler model</h3>
<div class="outline-text-3" id="text-1-7"> <div class="outline-text-3" id="text-1-7">
<p> <p>
The goal in this section is to identify the parameters of a simple APA model from the FEM. The goal in this section is to identify the parameters of a simple APA model from the FEM.
@ -862,12 +865,12 @@ The presented model is based on (<a href="#citeproc_bib_item_2">Souleille et al.
</p> </p>
<p> <p>
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#org6551a86">7</a>). The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure <a href="#orgdda4959">7</a>).
The parameters are shown in the table below. The parameters are shown in the table below.
</p> </p>
<div id="org6551a86" class="figure"> <div id="orgdda4959" class="figure">
<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" /> <p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p> <p><span class="figure-number">Figure 7: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
@ -1016,11 +1019,11 @@ And the DC gain is adjusted for the force sensor:
</div> </div>
<p> <p>
The dynamics of the FEM model and the simpler model are compared in Figure <a href="#orgbd0f182">8</a>. The dynamics of the FEM model and the simpler model are compared in Figure <a href="#org25d35cd">8</a>.
</p> </p>
<div id="orgbd0f182" class="figure"> <div id="org25d35cd" class="figure">
<p><img src="figs/apa300ml_comp_simpler_model.png" alt="apa300ml_comp_simpler_model.png" /> <p><img src="figs/apa300ml_comp_simpler_model.png" alt="apa300ml_comp_simpler_model.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Comparison of the Dynamics between the FEM model and the simplified one</p> <p><span class="figure-number">Figure 8: </span>Comparison of the Dynamics between the FEM model and the simplified one</p>
@ -1031,10 +1034,10 @@ The simplified model has also been implemented in Simscape.
</p> </p>
<p> <p>
The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure <a href="#orgc032d03">9</a>. The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure <a href="#org3ca18e2">9</a>.
</p> </p>
<div id="orgc032d03" class="figure"> <div id="org3ca18e2" class="figure">
<p><img src="figs/apa300ml_comp_simpler_simscape.png" alt="apa300ml_comp_simpler_simscape.png" /> <p><img src="figs/apa300ml_comp_simpler_simscape.png" alt="apa300ml_comp_simpler_simscape.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Comparison of the Dynamics between the FEM model and the simplified simscape model</p> <p><span class="figure-number">Figure 9: </span>Comparison of the Dynamics between the FEM model and the simplified simscape model</p>
@ -1042,8 +1045,8 @@ The dynamics of the Simscape simplified model is identified and compared with th
</div> </div>
</div> </div>
<div id="outline-container-org43ae9e5" class="outline-3"> <div id="outline-container-orgd7e3154" class="outline-3">
<h3 id="org43ae9e5"><span class="section-number-3">1.8</span> Integral Force Feedback</h3> <h3 id="orgd7e3154"><span class="section-number-3">1.8</span> Integral Force Feedback</h3>
<div class="outline-text-3" id="text-1-8"> <div class="outline-text-3" id="text-1-8">
<p> <p>
In this section, Integral Force Feedback control architecture is applied on the APA300ML. In this section, Integral Force Feedback control architecture is applied on the APA300ML.
@ -1059,18 +1062,18 @@ The payload mass is set to 10kg.
</div> </div>
<p> <p>
The obtained dynamics is shown in Figure <a href="#orga6e0dfc">10</a>. The obtained dynamics is shown in Figure <a href="#org41e4933">10</a>.
</p> </p>
<div id="orga6e0dfc" class="figure"> <div id="org41e4933" class="figure">
<p><img src="figs/piezo_amplified_iff_plant.png" alt="piezo_amplified_iff_plant.png" /> <p><img src="figs/piezo_amplified_iff_plant.png" alt="piezo_amplified_iff_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>IFF Plant</p> <p><span class="figure-number">Figure 10: </span>IFF Plant</p>
</div> </div>
<p> <p>
The controller is defined below and the loop gain is shown in Figure <a href="#orgeb0376d">11</a>. The controller is defined below and the loop gain is shown in Figure <a href="#org8791595">11</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Kiff = <span class="org-type">-</span>1e3<span class="org-type">/</span>s; <pre class="src src-matlab">Kiff = <span class="org-type">-</span>1e3<span class="org-type">/</span>s;
@ -1078,29 +1081,29 @@ The controller is defined below and the loop gain is shown in Figure <a href="#o
</div> </div>
<div id="orgeb0376d" class="figure"> <div id="org8791595" class="figure">
<p><img src="figs/piezo_amplified_iff_loop_gain.png" alt="piezo_amplified_iff_loop_gain.png" /> <p><img src="figs/piezo_amplified_iff_loop_gain.png" alt="piezo_amplified_iff_loop_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>IFF Loop Gain</p> <p><span class="figure-number">Figure 11: </span>IFF Loop Gain</p>
</div> </div>
<p> <p>
Now the closed-loop system is identified again and compare with the open loop system in Figure <a href="#org51f452b">12</a>. Now the closed-loop system is identified again and compare with the open loop system in Figure <a href="#org9002d80">12</a>.
</p> </p>
<p> <p>
It is the expected behavior as shown in the Figure <a href="#orged35295">13</a> (from (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>)). It is the expected behavior as shown in the Figure <a href="#orgf085b71">13</a> (from (<a href="#citeproc_bib_item_2">Souleille et al. 2018</a>)).
</p> </p>
<div id="org51f452b" class="figure"> <div id="org9002d80" class="figure">
<p><img src="figs/piezo_amplified_iff_comp.png" alt="piezo_amplified_iff_comp.png" /> <p><img src="figs/piezo_amplified_iff_comp.png" alt="piezo_amplified_iff_comp.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>OL and CL transfer functions</p> <p><span class="figure-number">Figure 12: </span>OL and CL transfer functions</p>
</div> </div>
<div id="orged35295" class="figure"> <div id="orgf085b71" class="figure">
<p><img src="figs/souleille18_results.png" alt="souleille18_results.png" /> <p><img src="figs/souleille18_results.png" alt="souleille18_results.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Results obtained in <a class='org-ref-reference' href="#souleille18_concep_activ_mount_space_applic">souleille18_concep_activ_mount_space_applic</a></p> <p><span class="figure-number">Figure 13: </span>Results obtained in <a class='org-ref-reference' href="#souleille18_concep_activ_mount_space_applic">souleille18_concep_activ_mount_space_applic</a></p>
@ -1110,14 +1113,14 @@ It is the expected behavior as shown in the Figure <a href="#orged35295">13</a>
</div> </div>
<div id="outline-container-orgc203c93" class="outline-2"> <div id="outline-container-orge12e432" class="outline-2">
<h2 id="orgc203c93"><span class="section-number-2">2</span> First Flexible Joint Geometry</h2> <h2 id="orge12e432"><span class="section-number-2">2</span> First Flexible Joint Geometry</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="orgb1abf80"></a> <a id="orga0ece29"></a>
</p> </p>
<p> <p>
The studied flexor is shown in Figure <a href="#orgd3ad3b1">14</a>. The studied flexor is shown in Figure <a href="#orgcd75ab8">14</a>.
</p> </p>
<p> <p>
@ -1130,14 +1133,14 @@ A simplified model of the flexor is then developped.
</p> </p>
<div id="orgd3ad3b1" class="figure"> <div id="orgcd75ab8" class="figure">
<p><img src="figs/flexor_id16_screenshot.png" alt="flexor_id16_screenshot.png" /> <p><img src="figs/flexor_id16_screenshot.png" alt="flexor_id16_screenshot.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Flexor studied</p> <p><span class="figure-number">Figure 14: </span>Flexor studied</p>
</div> </div>
</div> </div>
<div id="outline-container-orga48c65c" class="outline-3"> <div id="outline-container-org91559c3" class="outline-3">
<h3 id="orga48c65c"><span class="section-number-3">2.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3> <h3 id="org91559c3"><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"> <div class="outline-text-3" id="text-2-1">
<p> <p>
We first extract the stiffness and mass matrices. We first extract the stiffness and mass matrices.
@ -1549,8 +1552,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div> </div>
</div> </div>
<div id="outline-container-org5de18c6" class="outline-3"> <div id="outline-container-org0c0ae39" class="outline-3">
<h3 id="org5de18c6"><span class="section-number-3">2.2</span> Identification of the parameters using Simscape and looking at the Stiffness Matrix</h3> <h3 id="org0c0ae39"><span class="section-number-3">2.2</span> Identification of the parameters using Simscape and looking at the Stiffness Matrix</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
The flexor is now imported into Simscape and its parameters are estimated using an identification. The flexor is now imported into Simscape and its parameters are estimated using an identification.
@ -1607,15 +1610,15 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
</div> </div>
</div> </div>
<div id="outline-container-orgb2d0259" class="outline-3"> <div id="outline-container-orgb1eeb49" class="outline-3">
<h3 id="orgb2d0259"><span class="section-number-3">2.3</span> Simpler Model</h3> <h3 id="orgb1eeb49"><span class="section-number-3">2.3</span> Simpler Model</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#org8f309d8">15</a>. Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#orgc8a4dd1">15</a>.
</p> </p>
<div id="org8f309d8" class="figure"> <div id="orgc8a4dd1" class="figure">
<p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" /> <p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Bushing Joint used to model the flexible joint</p> <p><span class="figure-number">Figure 15: </span>Bushing Joint used to model the flexible joint</p>
@ -1640,7 +1643,7 @@ The two obtained dynamics are compared in Figure
</p> </p>
<div id="orge65cd1b" class="figure"> <div id="org168dbda" class="figure">
<p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" /> <p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p> <p><span class="figure-number">Figure 16: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p>
@ -1649,29 +1652,29 @@ The two obtained dynamics are compared in Figure
</div> </div>
</div> </div>
<div id="outline-container-org4dadb53" class="outline-2"> <div id="outline-container-org6fa0f81" class="outline-2">
<h2 id="org4dadb53"><span class="section-number-2">3</span> Optimized Flexible Joint</h2> <h2 id="org6fa0f81"><span class="section-number-2">3</span> Optimized Flexible Joint</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="org146537a"></a> <a id="org513c349"></a>
</p> </p>
<p> <p>
The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness. The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness.
</p> </p>
<p> <p>
The obtained geometry is shown in Figure <a href="#org89612a5">17</a>. The obtained geometry is shown in Figure <a href="#orge1d8231">17</a>.
</p> </p>
<div id="org89612a5" class="figure"> <div id="orge1d8231" class="figure">
<p><img src="figs/flexor_025_MDoF.jpg" alt="flexor_025_MDoF.jpg" /> <p><img src="figs/flexor_025_MDoF.jpg" alt="flexor_025_MDoF.jpg" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Flexor studied</p> <p><span class="figure-number">Figure 17: </span>Flexor studied</p>
</div> </div>
</div> </div>
<div id="outline-container-orgb1d1e31" class="outline-3"> <div id="outline-container-orgadfaeb7" class="outline-3">
<h3 id="orgb1d1e31"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3> <h3 id="orgadfaeb7"><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"> <div class="outline-text-3" id="text-3-1">
<p> <p>
We first extract the stiffness and mass matrices. We first extract the stiffness and mass matrices.
@ -2085,8 +2088,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div> </div>
</div> </div>
<div id="outline-container-orgc7107db" class="outline-3"> <div id="outline-container-org1a74e71" class="outline-3">
<h3 id="orgc7107db"><span class="section-number-3">3.2</span> Identification of the parameters using Simscape</h3> <h3 id="org1a74e71"><span class="section-number-3">3.2</span> Identification of the parameters using Simscape</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
The flexor is now imported into Simscape and its parameters are estimated using an identification. The flexor is now imported into Simscape and its parameters are estimated using an identification.
@ -2143,15 +2146,15 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
</div> </div>
</div> </div>
<div id="outline-container-org96d2775" class="outline-3"> <div id="outline-container-org3ba1fee" class="outline-3">
<h3 id="org96d2775"><span class="section-number-3">3.3</span> Simpler Model</h3> <h3 id="org3ba1fee"><span class="section-number-3">3.3</span> Simpler Model</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#org8f309d8">15</a>. Let&rsquo;s now model the flexible joint with a &ldquo;perfect&rdquo; Bushing joint as shown in Figure <a href="#orgc8a4dd1">15</a>.
</p> </p>
<div id="org9fc5457" class="figure"> <div id="org1f2487e" class="figure">
<p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" /> <p><img src="figs/flexible_joint_simscape.png" alt="flexible_joint_simscape.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Bushing Joint used to model the flexible joint</p> <p><span class="figure-number">Figure 18: </span>Bushing Joint used to model the flexible joint</p>
@ -2176,7 +2179,7 @@ The two obtained dynamics are compared in Figure
</p> </p>
<div id="org3f0ca2d" class="figure"> <div id="org520525f" class="figure">
<p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" /> <p><img src="figs/flexor_ID16_compare_bushing_joint.png" alt="flexor_ID16_compare_bushing_joint.png" />
</p> </p>
<p><span class="figure-number">Figure 19: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p> <p><span class="figure-number">Figure 19: </span>Comparison of the Joint compliance between the FEM model and the simpler model</p>
@ -2184,8 +2187,8 @@ The two obtained dynamics are compared in Figure
</div> </div>
</div> </div>
<div id="outline-container-org2916e5d" class="outline-3"> <div id="outline-container-orgec51432" class="outline-3">
<h3 id="org2916e5d"><span class="section-number-3">3.4</span> Comparison with a stiffer Flexible Joint</h3> <h3 id="orgec51432"><span class="section-number-3">3.4</span> Comparison with a stiffer Flexible Joint</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
The stiffness matrix with the flexible joint with a &ldquo;hinge&rdquo; size of 0.50mm is loaded. The stiffness matrix with the flexible joint with a &ldquo;hinge&rdquo; size of 0.50mm is loaded.
@ -2252,37 +2255,37 @@ Its parameters are compared with the Flexible Joint with a size of 0.25mm in the
</div> </div>
</div> </div>
<div id="outline-container-org79ad15f" class="outline-2"> <div id="outline-container-org91975b5" class="outline-2">
<h2 id="org79ad15f"><span class="section-number-2">4</span> Complete Strut with Encoder</h2> <h2 id="org91975b5"><span class="section-number-2">4</span> Complete Strut with Encoder</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="orgc4b9146"></a> <a id="orgcff61d6"></a>
</p> </p>
</div> </div>
<div id="outline-container-org26e8494" class="outline-3"> <div id="outline-container-orgd829824" class="outline-3">
<h3 id="org26e8494"><span class="section-number-3">4.1</span> Introduction</h3> <h3 id="orgd829824"><span class="section-number-3">4.1</span> Introduction</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
Now, the full nano-hexapod strut is modelled using Ansys. Now, the full nano-hexapod strut is modelled using Ansys.
</p> </p>
<p> <p>
The 3D as well as the interface nodes are shown in Figure <a href="#org8a816bb">20</a>. The 3D as well as the interface nodes are shown in Figure <a href="#org9f2a66d">20</a>.
</p> </p>
<div id="org8a816bb" class="figure"> <div id="org9f2a66d" class="figure">
<p><img src="figs/strut_encoder_nodes.jpg" alt="strut_encoder_nodes.jpg" /> <p><img src="figs/strut_encoder_nodes.jpg" alt="strut_encoder_nodes.jpg" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Interface points</p> <p><span class="figure-number">Figure 20: </span>Interface points</p>
</div> </div>
<p> <p>
A side view is shown in Figure <a href="#org4c74f6e">21</a>. A side view is shown in Figure <a href="#org3437ed1">21</a>.
</p> </p>
<div id="org4c74f6e" class="figure"> <div id="org3437ed1" class="figure">
<p><img src="figs/strut_encoder_nodes_side.jpg" alt="strut_encoder_nodes_side.jpg" /> <p><img src="figs/strut_encoder_nodes_side.jpg" alt="strut_encoder_nodes_side.jpg" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Interface points - Side view</p> <p><span class="figure-number">Figure 21: </span>Interface points - Side view</p>
@ -2294,8 +2297,8 @@ The flexible joints used have a 0.25mm width size.
</div> </div>
</div> </div>
<div id="outline-container-orgd702a5f" class="outline-3"> <div id="outline-container-orgd7f754c" class="outline-3">
<h3 id="orgd702a5f"><span class="section-number-3">4.2</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3> <h3 id="orgd7f754c"><span class="section-number-3">4.2</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
We first extract the stiffness and mass matrices. We first extract the stiffness and mass matrices.
@ -2757,8 +2760,8 @@ Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <c
</div> </div>
</div> </div>
<div id="outline-container-org42c2461" class="outline-3"> <div id="outline-container-org5019141" class="outline-3">
<h3 id="org42c2461"><span class="section-number-3">4.3</span> Piezoelectric parameters</h3> <h3 id="org5019141"><span class="section-number-3">4.3</span> Piezoelectric parameters</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<p> <p>
Parameters for the APA300ML: Parameters for the APA300ML:
@ -2782,8 +2785,8 @@ ns = 1; <span class="org-comment">% Number of stacks used as force sensor</span>
</div> </div>
</div> </div>
<div id="outline-container-orga5ba630" class="outline-3"> <div id="outline-container-org72bb8f1" class="outline-3">
<h3 id="orga5ba630"><span class="section-number-3">4.4</span> Identification of the Dynamics</h3> <h3 id="org72bb8f1"><span class="section-number-3">4.4</span> Identification of the Dynamics</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<p> <p>
The dynamics is identified from the applied force to the measured relative displacement. The dynamics is identified from the applied force to the measured relative displacement.
@ -2795,7 +2798,7 @@ The same dynamics is identified for a payload mass of 10Kg.
</div> </div>
<div id="org4ba34f1" class="figure"> <div id="orgda90142" class="figure">
<p><img src="figs/dynamics_encoder_full_strut.png" alt="dynamics_encoder_full_strut.png" /> <p><img src="figs/dynamics_encoder_full_strut.png" alt="dynamics_encoder_full_strut.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Dynamics from the force actuator to the measured motion by the encoder</p> <p><span class="figure-number">Figure 22: </span>Dynamics from the force actuator to the measured motion by the encoder</p>
@ -2816,7 +2819,7 @@ The same dynamics is identified for a payload mass of 10Kg.
</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: 2020-11-12 jeu. 18:48</p> <p class="date">Created: 2020-11-13 ven. 08:56</p>
</div> </div>
</body> </body>
</html> </html>

5
index.org
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@ -1,4 +1,4 @@
#+TITLE: Finite Element Model with Simscape #+TITLE: NASS - Finite Element Models with Simscape
:DRAWER: :DRAWER:
#+STARTUP: overview #+STARTUP: overview
@ -38,6 +38,7 @@
In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis. In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.
It is divided in the following sections:
- Section [[sec:APA300ML]]: - Section [[sec:APA300ML]]:
A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape. A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape.
The static and dynamical properties of the APA300ML are then estimated using the Simscape model. The static and dynamical properties of the APA300ML are then estimated using the Simscape model.
@ -46,7 +47,7 @@ In this document, Finite Element Models (FEM) of parts of the Nano-Hexapod are d
- Section [[sec:flexor_025]]: - Section [[sec:flexor_025]]:
An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model. An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.
- Section [[sec:strut_encoder]]: - Section [[sec:strut_encoder]]:
A super element of a complete strut is exported. A super element of a complete strut is studied.
* APA300ML * APA300ML
:PROPERTIES: :PROPERTIES: