Started the stiffness analysis / noise budgeting
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docs/figs/2dof_system_granite_stiffness.pdf
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<title>Simscape Model of the Nano-Active-Stabilization-System</title>
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@ -249,7 +249,7 @@
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<li><a href="#org0605048">10. Effect of support’s compliance on the plant dynamics (link)</a></li>
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<li><a href="#orge777d0f">11. Effect of the payload’s “impedance” on the plant dynamics (link)</a></li>
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<li><a href="#orga323881">12. Effect of Experimental conditions on the plant dynamics (link)</a></li>
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<li><a href="#org1adf7f6">13. Optimal Stiffness of the nano-hexapod (link)</a></li>
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<li><a href="#orge7b9b41">13. Optimal Stiffness of the nano-hexapod to reduce plant uncertainty (link)</a></li>
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<li><a href="#org14a10e8">14. Active Damping Techniques on the full Simscape Model (link)</a></li>
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<li><a href="#orgd818a00">15. Control of the Nano-Active-Stabilization-System (link)</a></li>
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<li><a href="#org361f405">16. Useful Matlab Functions (link)</a></li>
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@ -407,8 +407,8 @@ Conclusion are drawn about what experimental conditions are critical on the vari
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<div id="outline-container-org1adf7f6" class="outline-2">
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<h2 id="org1adf7f6"><span class="section-number-2">13</span> Optimal Stiffness of the nano-hexapod (<a href="optimal_stiffness.html">link</a>)</h2>
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<div id="outline-container-orge7b9b41" class="outline-2">
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<h2 id="orge7b9b41"><span class="section-number-2">13</span> Optimal Stiffness of the nano-hexapod to reduce plant uncertainty (<a href="uncertainty_optimal_stiffness.html">link</a>)</h2>
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-04-01 mer. 17:19</p>
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<p class="date">Created: 2020-04-07 mar. 14:55</p>
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<a accesskey="H" href="index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org72be3da">1. Spectral Density of Disturbances label:sec:psd_disturbances</a>
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<ul>
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<li><a href="#orge84ae10">1.1. Load of the identified disturbances</a></li>
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<li><a href="#org0bec7fc">1.2. Plots</a></li>
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</ul>
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</li>
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<li><a href="#orgc44cf7e">2. Effect of disturbances on the position error</a>
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<ul>
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<li><a href="#org524df41">2.1. Initialization</a></li>
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<li><a href="#orgaf88c9f">2.2. Identification</a></li>
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<li><a href="#org2b8e5e0">2.3. Plots</a></li>
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<li><a href="#org71f73c5">2.4. Save</a></li>
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</ul>
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</li>
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<li><a href="#org6527e58">3. Effect of granite stiffness</a>
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<ul>
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<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a></li>
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<li><a href="#org9215f81">3.2. Soft Granite</a></li>
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<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
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</ul>
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</li>
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<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
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<ul>
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<li><a href="#org2ee8120">4.1. Load of the identified disturbances and transfer functions</a></li>
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<li><a href="#orgb1a3177">4.2. Equations</a></li>
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<li><a href="#org63eade9">4.3. Results</a></li>
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<li><a href="#orgef96b89">4.4. Cumulative Amplitude Spectrum</a></li>
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</ul>
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</li>
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<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
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<ul>
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<li><a href="#orgf2d36a1">5.1. Reduction thanks to feedback - Required bandwidth</a></li>
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</ul>
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</li>
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<li><a href="#orgea74617">6. Conclusion</a></li>
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</ul>
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</div>
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</div>
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<ul class="org-ul">
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<li><a href="#org17d3d6a">1</a></li>
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<li><a href="#orgf9e4300">2</a></li>
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<li><a href="#orgd4105b6">3</a></li>
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<li><a href="#org5d05990">4</a></li>
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<li><a href="#orgd3503fb">5</a></li>
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</ul>
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<ul class="org-ul">
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<li><a href="#sec:psd_disturbances">sec:psd_disturbances</a></li>
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<li><a href="#sec:effect_disturbances">sec:effect_disturbances</a></li>
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<li><a href="#sec:granite_stiffness">sec:granite_stiffness</a></li>
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<li><a href="#sec:open_loop_budget_error">sec:open_loop_budget_error</a></li>
|
||||
<li><a href="#sec:closed_loop_budget_error">sec:closed_loop_budget_error</a></li>
|
||||
</ul>
|
||||
|
||||
<div id="outline-container-org72be3da" class="outline-2">
|
||||
<h2 id="org72be3da"><span class="section-number-2">1</span> Spectral Density of Disturbances <div id="sec:psd_disturbances"></div></h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="org17d3d6a"></a>
|
||||
</p>
|
||||
<p>
|
||||
The level of disturbances has been identified form experiments.
|
||||
This is detailed in <a href="disturbances.html">this</a> document.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orge84ae10" class="outline-3">
|
||||
<h3 id="orge84ae10"><span class="section-number-3">1.1</span> Load of the identified disturbances</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'./mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0bec7fc" class="outline-3">
|
||||
<h3 id="org0bec7fc"><span class="section-number-3">1.2</span> Plots</h3>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgc44cf7e" class="outline-2">
|
||||
<h2 id="orgc44cf7e"><span class="section-number-2">2</span> Effect of disturbances on the position error</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="orgf9e4300"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org524df41" class="outline-3">
|
||||
<h3 id="org524df41"><span class="section-number-3">2.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We use a sample mass of 10kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 10);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeController();
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgaf88c9f" class="outline-3">
|
||||
<h3 id="orgaf88c9f"><span class="section-number-3">2.2</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
Inputs:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><code>Dwx</code>: Ground displacement in the \(x\) direction</li>
|
||||
<li><code>Dwy</code>: Ground displacement in the \(y\) direction</li>
|
||||
<li><code>Dwz</code>: Ground displacement in the \(z\) direction</li>
|
||||
<li><code>Fty_x</code>: Forces applied by the Translation stage in the \(x\) direction</li>
|
||||
<li><code>Fty_z</code>: Forces applied by the Translation stage in the \(z\) direction</li>
|
||||
<li><code>Frz_z</code>: Forces applied by the Spindle in the \(z\) direction</li>
|
||||
<li><code>Fd</code>: Direct forces applied at the center of mass of the Payload</li>
|
||||
</ul>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<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>
|
||||
mdl = <span class="org-string">'nass_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Micro-Hexapod</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Dwx'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% X Ground motion</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Dwy'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Y Ground motion</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Dwz'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Z Ground motion</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Fty_x'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Parasitic force Ty - X</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Fty_z'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Parasitic force Ty - Z</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Frz_z'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Parasitic force Rz - Z</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Fd'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Direct forces</span>
|
||||
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Error</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
initializeNanoHexapod(<span class="org-string">'k'</span>, Ks(<span class="org-constant">i</span>));
|
||||
|
||||
<span class="org-comment">% Run the linearization</span>
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Fty_x'</span>, <span class="org-string">'Fty_z'</span>, <span class="org-string">'Frz_z'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
|
||||
G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
Gd(<span class="org-constant">i</span>) = {minreal(G)};
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2b8e5e0" class="outline-3">
|
||||
<h3 id="org2b8e5e0"><span class="section-number-3">2.3</span> Plots</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
Effect of Stages vibration (Filtering).
|
||||
Effect of Ground motion (Transmissibility).
|
||||
Direct Forces (Compliance).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org71f73c5" class="outline-3">
|
||||
<h3 id="org71f73c5"><span class="section-number-3">2.4</span> Save</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">save(<span class="org-string">'./mat/opt_stiffness_disturbances.mat'</span>, <span class="org-string">'Gd'</span>)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6527e58" class="outline-2">
|
||||
<h2 id="org6527e58"><span class="section-number-2">3</span> Effect of granite stiffness</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="orgd4105b6"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgd3e5fe1" class="outline-3">
|
||||
<h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
|
||||
<div id="org8fb9606" class="figure">
|
||||
<p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Figure caption</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
If we write the equation of motion of the system in Figure <a href="#org8fb9606">1</a>, we obtain:
|
||||
</p>
|
||||
\begin{align}
|
||||
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
|
||||
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
If we note \(d = x^\prime - x\), we obtain:
|
||||
</p>
|
||||
\begin{equation}
|
||||
\label{org4396920}
|
||||
\frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
|
||||
\end{equation}
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9215f81" class="outline-3">
|
||||
<h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
Let’s initialize a soft granite that will act as an isolation stage from ground motion.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGranite(<span class="org-string">'K'</span>, 5e5<span class="org-type">*</span>ones(3,1), <span class="org-string">'C'</span>, 5e3<span class="org-type">*</span>ones(3,1));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
initializeNanoHexapod(<span class="org-string">'k'</span>, Ks(<span class="org-constant">i</span>));
|
||||
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Fty_x'</span>, <span class="org-string">'Fty_z'</span>, <span class="org-string">'Frz_z'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
|
||||
G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
Gdr(<span class="org-constant">i</span>) = {minreal(G)};
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8878556" class="outline-3">
|
||||
<h3 id="org8878556"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org8a88fb0" class="outline-2">
|
||||
<h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org5d05990"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org2ee8120" class="outline-3">
|
||||
<h3 id="org2ee8120"><span class="section-number-3">4.1</span> Load of the identified disturbances and transfer functions</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'./mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>);
|
||||
load(<span class="org-string">'./mat/opt_stiffness_disturbances.mat'</span>, <span class="org-string">'Gd'</span>)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb1a3177" class="outline-3">
|
||||
<h3 id="orgb1a3177"><span class="section-number-3">4.2</span> Equations</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org63eade9" class="outline-3">
|
||||
<h3 id="org63eade9"><span class="section-number-3">4.3</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
Effect of all disturbances
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(dist_f.psd_rz)<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>))));
|
||||
<span class="org-keyword">end</span>
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'ASD $\left[\frac{m}{\sqrt{Hz}}\right]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
||||
xlim([2, 500]);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgef96b89" class="outline-3">
|
||||
<h3 id="orgef96b89"><span class="section-number-3">4.4</span> Cumulative Amplitude Spectrum</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_ty<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Fty_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
||||
xlim([2, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_rz<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
||||
xlim([2, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Ground motion
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Dwz'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
||||
xlim([2, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ex'</span>, <span class="org-string">'Dwx'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'lin'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
||||
xlim([2, 500]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ey'</span>, <span class="org-string">'Dwy'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'lin'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
||||
xlim([2, 500]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Sum of all perturbations
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">psd_tot = zeros(length(freqs), length(Ks));
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>) = dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Dwz'</span> ), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2 <span class="org-type">+</span> ...
|
||||
dist_f.psd_ty<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Fty_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2 <span class="org-type">+</span> ...
|
||||
dist_f.psd_rz<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2;
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_z$ $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
||||
xlim([1, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org34c0f38" class="outline-2">
|
||||
<h2 id="org34c0f38"><span class="section-number-2">5</span> Closed Loop Budget Error</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="orgd3503fb"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgf2d36a1" class="outline-3">
|
||||
<h3 id="orgf2d36a1"><span class="section-number-3">5.1</span> Reduction thanks to feedback - Required bandwidth</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = 1<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
||||
xic = 0.5;
|
||||
|
||||
S = (s<span class="org-type">/</span>wc)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc);
|
||||
|
||||
bodeFig({S}, logspace(<span class="org-type">-</span>1,2,1000))
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = [1, 5, 10, 20, 50, 100, 200];
|
||||
|
||||
S1 = {zeros(length(wc), 1)};
|
||||
S2 = {zeros(length(wc), 1)};
|
||||
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
|
||||
L = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s; <span class="org-comment">% Simple integrator</span>
|
||||
S1{<span class="org-constant">j</span>} = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> L);
|
||||
L = ((2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s)<span class="org-type">^</span>2<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">/</span>2))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">*</span>2));
|
||||
S2{<span class="org-constant">j</span>} = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> L);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-constant">i</span> = 6;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">j</span>);
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(S1{<span class="org-constant">j</span>}, freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$\\omega_c = %.0f$ [Hz]'</span>, wc(<span class="org-constant">j</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'k-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, <span class="org-string">'Open-Loop'</span>);
|
||||
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
||||
xlim([0.5, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = logspace(0, 3, 100);
|
||||
|
||||
Dz1_rms = zeros(length(Ks), length(wc));
|
||||
Dz2_rms = zeros(length(Ks), length(wc));
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
|
||||
L = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s;
|
||||
Dz1_rms(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = sqrt(trapz(freqs, abs(squeeze(freqresp(1<span class="org-type">/</span>(1 <span class="org-type">+</span> L), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>)));
|
||||
|
||||
L = ((2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s)<span class="org-type">^</span>2<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">/</span>2))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">*</span>2));
|
||||
Dz2_rms(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = sqrt(trapz(freqs, abs(squeeze(freqresp(1<span class="org-type">/</span>(1 <span class="org-type">+</span> L), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>)));
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = dist_f.f;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">i</span>);
|
||||
plot(wc, Dz1_rms(<span class="org-constant">i</span>, <span class="org-type">:</span>), <span class="org-string">'-'</span>, ...
|
||||
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)))
|
||||
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">i</span>);
|
||||
plot(wc, Dz2_rms(<span class="org-constant">i</span>, <span class="org-type">:</span>), <span class="org-string">'--'</span>, ...
|
||||
<span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>)
|
||||
<span class="org-keyword">end</span>
|
||||
hold off;
|
||||
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
||||
xlabel(<span class="org-string">'Control Bandwidth [Hz]'</span>); ylabel(<span class="org-string">'$E_z\ [m, rms]$'</span>)
|
||||
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
||||
xlim([1, 500]);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgea74617" class="outline-2">
|
||||
<h2 id="orgea74617"><span class="section-number-2">6</span> Conclusion</h2>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-04-07 mar. 14:57</p>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
|
||||
<a accesskey="H" href="index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<h1 class="title">Determination of the optimal nano-hexapod’s stiffness</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org157c07d">1. Spindle Rotation Speed</a>
|
||||
<ul>
|
||||
<li><a href="#orgb1e5096">1.1. Initialization</a></li>
|
||||
<li><a href="#org687bdef">1.2. Identification when rotating at maximum speed</a></li>
|
||||
<li><a href="#org7dcfddb">1.3. Change of dynamics</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org23ddf26">2. Micro-Station Compliance Effect</a>
|
||||
<ul>
|
||||
<li><a href="#orgdc8aeea">2.1. Identification of the micro-station compliance</a></li>
|
||||
<li><a href="#orga44542b">2.2. Identification of the dynamics with a rigid micro-station</a></li>
|
||||
<li><a href="#org49d6b26">2.3. Identification of the dynamics with a flexible micro-station</a></li>
|
||||
<li><a href="#org4c1ed79">2.4. Obtained Dynamics</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org19559b0">3. Payload “Impedance” Effect</a>
|
||||
<ul>
|
||||
<li><a href="#org67607c3">3.1. Initialization</a></li>
|
||||
<li><a href="#org73f1c6e">3.2. Identification of the dynamics while change the payload dynamics</a></li>
|
||||
<li><a href="#orgd7a519b">3.3. Change of dynamics for the primary controller</a>
|
||||
<ul>
|
||||
<li><a href="#orgb44d421">3.3.1. Frequency variation</a></li>
|
||||
<li><a href="#orgfc270b0">3.3.2. Mass variation</a></li>
|
||||
<li><a href="#org118f0c2">3.3.3. Total variation</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org973d2e3">4. Total Change of dynamics</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
As shown before, many parameters other than the nano-hexapod itself do influence the plant dynamics:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>The micro-station compliance (studied <a href="uncertainty_support.html">here</a>)</li>
|
||||
<li>The payload mass and dynamical properties (studied <a href="uncertainty_payload.html">here</a> and <a href="uncertainty_experiment.html">here</a>)</li>
|
||||
<li>The experimental conditions, mainly the spindle rotation speed (studied <a href="uncertainty_experiment.html">here</a>)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
As seen before, the stiffness of the nano-hexapod greatly influence the effect of such parameters.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We wish here to see if we can determine an optimal stiffness of the nano-hexapod such that:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Section <a href="#org902923f">1</a>: the change of its dynamics due to the spindle rotation speed is acceptable</li>
|
||||
<li>Section <a href="#orgabe2ab2">2</a>: the support compliance dynamics is not much present in the nano-hexapod dynamics</li>
|
||||
<li>Section <a href="#org2bd8390">3</a>: the change of payload impedance has acceptable effect on the plant dynamics</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The overall goal is to design a nano-hexapod that will allow the highest possible control bandwidth.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-org157c07d" class="outline-2">
|
||||
<h2 id="org157c07d"><span class="section-number-2">1</span> Spindle Rotation Speed</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="org902923f"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, we look at the effect of the spindle rotation speed on the plant dynamics.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The rotation speed will have an effect due to the Coriolis effect.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb1e5096" class="outline-3">
|
||||
<h3 id="orgb1e5096"><span class="section-number-3">1.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We use a sample mass of 10kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 10);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We don’t include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics.
|
||||
We however include gravity.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeController();
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org687bdef" class="outline-3">
|
||||
<h3 id="org687bdef"><span class="section-number-3">1.2</span> Identification when rotating at maximum speed</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<p>
|
||||
We identify the dynamics for the following spindle rotation speeds <code>Rz_rpm</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Rz_rpm = linspace(0, 60, 6);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And for the following nano-hexapod actuator stiffness <code>Ks</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7dcfddb" class="outline-3">
|
||||
<h3 id="org7dcfddb"><span class="section-number-3">1.3</span> Change of dynamics</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
We plot the change of dynamics due to the change of the spindle rotation speed (from 0rpm to 60rpm):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#orgfd21b56">2</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
|
||||
<li>Figure <a href="#org2a4cc54">3</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
|
||||
<li>Figure <a href="#orgbf48d68">4</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#org16be775">5</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_y\) (coupling of the centralized positioning plant)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org039ad8e" class="figure">
|
||||
<p><img src="figs/opti_stiffness_iff_root_locus.png" alt="opti_stiffness_iff_root_locus.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Root Locus plot for IFF control when not rotating (in red) and when rotating at 60rpm (in blue) for 4 different nano-hexapod stiffnesses (<a href="./figs/opti_stiffness_iff_root_locus.png">png</a>, <a href="./figs/opti_stiffness_iff_root_locus.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgfd21b56" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_iff.png" alt="opt_stiffness_wz_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_iff.png">png</a>, <a href="./figs/opt_stiffness_wz_iff.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org2a4cc54" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_dvf.png" alt="opt_stiffness_wz_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_dvf.png">png</a>, <a href="./figs/opt_stiffness_wz_dvf.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgbf48d68" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_fx_dx.png" alt="opt_stiffness_wz_fx_dx.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_wz_fx_dx.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org16be775" class="figure">
|
||||
<p><img src="figs/opt_stiffness_wz_coupling.png" alt="opt_stiffness_wz_coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Change of Coupling from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_y\) for a spindle rotation speed from 0rpm to 60rpm (<a href="./figs/opt_stiffness_wz_coupling.png">png</a>, <a href="./figs/opt_stiffness_wz_coupling.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<div class="important">
|
||||
<p>
|
||||
The leg stiffness should be at higher than \(k_i = 10^4\ [N/m]\) such that the main resonance frequency does not shift too much when rotating.
|
||||
For the coupling, it is more difficult to conclude about the minimum required leg stiffness.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="notes">
|
||||
<p>
|
||||
Note that we can use very soft nano-hexapod if we limit the spindle rotating speed.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org23ddf26" class="outline-2">
|
||||
<h2 id="org23ddf26"><span class="section-number-2">2</span> Micro-Station Compliance Effect</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="orgabe2ab2"></a>
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>take the 6dof compliance of the micro-station</li>
|
||||
<li>simple model + uncertainty</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdc8aeea" class="outline-3">
|
||||
<h3 id="orgdc8aeea"><span class="section-number-3">2.1</span> Identification of the micro-station compliance</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'compliance'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We put nothing on top of the micro-hexapod.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.
|
||||
The diagonal element of the identified Micro-Station compliance matrix are shown in Figure <a href="#org6cfb14b">6</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6cfb14b" class="figure">
|
||||
<p><img src="figs/opt_stiff_micro_station_compliance.png" alt="opt_stiff_micro_station_compliance.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Identified Compliance of the Micro-Station (<a href="./figs/opt_stiff_micro_station_compliance.png">png</a>, <a href="./figs/opt_stiff_micro_station_compliance.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga44542b" class="outline-3">
|
||||
<h3 id="orga44542b"><span class="section-number-3">2.2</span> Identification of the dynamics with a rigid micro-station</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
We now identify the dynamics when the micro-station is rigid.
|
||||
This is equivalent of identifying the dynamics of the nano-hexapod when fixed to a rigid ground.
|
||||
We also choose the sample to be rigid and to have a mass of 10kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 10);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
As before, we identify the dynamics for the following actuator stiffnesses:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org49d6b26" class="outline-3">
|
||||
<h3 id="org49d6b26"><span class="section-number-3">2.3</span> Identification of the dynamics with a flexible micro-station</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
We now initialize all the micro-station stages to be flexible.
|
||||
And we identify the dynamics of the nano-hexapod.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org4c1ed79" class="outline-3">
|
||||
<h3 id="org4c1ed79"><span class="section-number-3">2.4</span> Obtained Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
We plot the change of dynamics due to the compliance of the Micro-Station.
|
||||
The solid curves are corresponding to the nano-hexapod without the micro-station, and the dashed curves with the micro-station:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org71f5400">7</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
|
||||
<li>Figure <a href="#org32aef29">8</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
|
||||
<li>Figure <a href="#org8a33fed">9</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
|
||||
<li>Figure <a href="#orge9bd08b">10</a>: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org71f5400" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_iff.png" alt="opt_stiffness_micro_station_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_iff.png">png</a>, <a href="./figs/opt_stiffness_micro_station_iff.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org32aef29" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_dvf.png" alt="opt_stiffness_micro_station_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_dvf.png">png</a>, <a href="./figs/opt_stiffness_micro_station_dvf.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org8a33fed" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_fx_dx.png" alt="opt_stiffness_micro_station_fx_dx.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fx_dx.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge9bd08b" class="figure">
|
||||
<p><img src="figs/opt_stiffness_micro_station_fz_dz.png" alt="opt_stiffness_micro_station_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Change of dynamics from force \(\mathcal{F}_z\) to displacement \(\mathcal{X}_z\) due to the micro-station compliance (<a href="./figs/opt_stiffness_micro_station_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<div class="important">
|
||||
<p>
|
||||
The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when the stiffness of the legs is less than \(10^6\ [N/m]\).
|
||||
When the nano-hexapod is stiff (\(k>10^7\ [N/m]\)), the compliance of the micro-station appears in the primary plant.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org19559b0" class="outline-2">
|
||||
<h2 id="org19559b0"><span class="section-number-2">3</span> Payload “Impedance” Effect</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org2bd8390"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org67607c3" class="outline-3">
|
||||
<h3 id="org67607c3"><span class="section-number-3">3.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
We don’t include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics. :exports none
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We set the controller type to Open-Loop, and we do not need to log any signal.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
initializeController();
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
initializeReferences();
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org73f1c6e" class="outline-3">
|
||||
<h3 id="org73f1c6e"><span class="section-number-3">3.2</span> Identification of the dynamics while change the payload dynamics</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We make the following change of payload dynamics:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Change of mass: from 1kg to 50kg</li>
|
||||
<li>Change of resonance frequency: from 50Hz to 500Hz</li>
|
||||
<li>The damping ratio of the payload is fixed to \(\xi = 0.2\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
We identify the dynamics for the following payload masses <code>Ms</code> and nano-hexapod leg’s stiffnesses <code>Ks</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ms = [1, 20, 50]; <span class="org-comment">% [Kg]</span>
|
||||
Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We then identify the dynamics for the following payload resonance frequencies <code>Fs</code>:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Fs = [50, 200, 500]; <span class="org-comment">% [Hz]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd7a519b" class="outline-3">
|
||||
<h3 id="orgd7a519b"><span class="section-number-3">3.3</span> Change of dynamics for the primary controller</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
</div>
|
||||
<div id="outline-container-orgb44d421" class="outline-4">
|
||||
<h4 id="orgb44d421"><span class="section-number-4">3.3.1</span> Frequency variation</h4>
|
||||
<div class="outline-text-4" id="text-3-3-1">
|
||||
<p>
|
||||
We here compare the dynamics for the same payload mass, but different stiffness resulting in different resonance frequency of the payload:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org00db693">11</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
|
||||
<li>Figure <a href="#org76716ad">12</a>: same, but for all tested nano-hexapod stiffnesses</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
We can see two mass lines for the soft nano-hexapod (Figure <a href="#org00db693">11</a>):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>The first mass line corresponds to \(\frac{1}{(m_n + m_p)s^2}\) where \(m_p = 10\ [kg]\) is the mass of the payload and \(m_n = 15\ [Kg]\) is the mass of the nano-hexapod top platform and attached mirror</li>
|
||||
<li>The second mass line corresponds to \(\frac{1}{m_n s^2}\)</li>
|
||||
<li>The zero corresponds to the resonance of the payload alone (fixed nano-hexapod’s top platform)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org00db693" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_freq_fz_dz.png" alt="opt_stiffness_payload_freq_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_freq_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_freq_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org76716ad" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_freq_all.png" alt="opt_stiffness_payload_freq_all.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency (<a href="./figs/opt_stiffness_payload_freq_all.png">png</a>, <a href="./figs/opt_stiffness_payload_freq_all.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfc270b0" class="outline-4">
|
||||
<h4 id="orgfc270b0"><span class="section-number-4">3.3.2</span> Mass variation</h4>
|
||||
<div class="outline-text-4" id="text-3-3-2">
|
||||
<p>
|
||||
We here compare the dynamics for different payload mass with the same resonance frequency (100Hz):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#orga1343a7">13</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
|
||||
<li>Figure <a href="#org35aebae">14</a>: same, but for all tested nano-hexapod stiffnesses</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
We can see here that for the soft nano-hexapod:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>the first resonance \(\omega_n\) is changing with the mass of the payload as \(\omega_n = \sqrt{\frac{k_n}{m_p + m_n}}\) with \(k_p\) the stiffness of the nano-hexapod, \(m_p\) the payload’s mass and \(m_n\) the mass of the nano-hexapod top platform</li>
|
||||
<li>the first mass line corresponding to \(\frac{1}{(m_p + m_n)s^2}\) is changing with the payload mass</li>
|
||||
<li>the zero at 100Hz is not changing as it corresponds to the resonance of the payload itself</li>
|
||||
<li>the second mass line does not change</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orga1343a7" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_mass_fz_dz.png" alt="opt_stiffness_payload_mass_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_mass_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_mass_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org35aebae" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_mass_all.png" alt="opt_stiffness_payload_mass_all.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass (<a href="./figs/opt_stiffness_payload_mass_all.png">png</a>, <a href="./figs/opt_stiffness_payload_mass_all.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org118f0c2" class="outline-4">
|
||||
<h4 id="org118f0c2"><span class="section-number-4">3.3.3</span> Total variation</h4>
|
||||
<div class="outline-text-4" id="text-3-3-3">
|
||||
<p>
|
||||
We now plot the total change of dynamics due to change of the payload (Figures <a href="#orgf16d005">15</a> and <a href="#org73b8b8a">16</a>):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>mass from 1kg to 50kg</li>
|
||||
<li>main resonance from 50Hz to 500Hz</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgf16d005" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_impedance_all_fz_dz.png" alt="opt_stiffness_payload_impedance_all_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_impedance_all_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_impedance_all_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org73b8b8a" class="figure">
|
||||
<p><img src="figs/opt_stiffness_payload_impedance_fz_dz.png" alt="opt_stiffness_payload_impedance_fz_dz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<a href="./figs/opt_stiffness_payload_impedance_fz_dz.png">png</a>, <a href="./figs/opt_stiffness_payload_impedance_fz_dz.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<div class="important">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org973d2e3" class="outline-2">
|
||||
<h2 id="org973d2e3"><span class="section-number-2">4</span> Total Change of dynamics</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
We now consider the total change of nano-hexapod dynamics due to:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><code>Gk_wz_err</code> - Change of spindle rotation speed</li>
|
||||
<li><code>Gf_err</code> and <code>Gm_err</code> - Change of payload resonance</li>
|
||||
<li><code>Gmf_err</code> and <code>Gmr_err</code> - Micro-Station compliance</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The obtained dynamics are shown:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#orgcf64eb6">17</a> for a stiffness \(k = 10^3\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org175cc57">18</a> for a stiffness \(k = 10^5\ [N/m]\)</li>
|
||||
<li>Figure <a href="#org998cf87">19</a> for a stiffness \(k = 10^7\ [N/m]\)</li>
|
||||
<li>Figure <a href="#orgd3db91c">20</a> for a stiffness \(k = 10^9\ [N/m]\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
And finally, in Figures <a href="#orge05feb5">21</a> and <a href="#org17c5c95">22</a> are shown an animation of the change of dynamics with the nano-hexapod’s stiffness.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgcf64eb6" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e3.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^3\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org175cc57" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e5.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^5\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org998cf87" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e7.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^7\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd3db91c" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.png" alt="opt_stiffness_plant_dynamics_fx_dx_k_1e9.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^9\ [N/m]\) (<a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.png">png</a>, <a href="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge05feb5" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_task_space.gif" alt="opt_stiffness_plant_dynamics_task_space.gif" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 21: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org17c5c95" class="figure">
|
||||
<p><img src="figs/opt_stiffness_plant_dynamics_task_space_colors.gif" alt="opt_stiffness_plant_dynamics_task_space_colors.gif" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-04-07 mar. 14:55</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
BIN
mat/opt_stiffness_disturbances.mat
Normal file
BIN
mat/opt_stiffness_disturbances.mat
Normal file
Binary file not shown.
BIN
mat/stages.mat
BIN
mat/stages.mat
Binary file not shown.
Binary file not shown.
Binary file not shown.
@ -100,7 +100,7 @@ This effect is studied, and conclusions on what characteristics of the isolation
|
||||
In this document, the effect of all the experimental conditions (rotation speed, sample mass, ...) on the plant dynamics are studied.
|
||||
Conclusion are drawn about what experimental conditions are critical on the variability of the plant dynamics.
|
||||
|
||||
* Optimal Stiffness of the nano-hexapod ([[file:optimal_stiffness.org][link]])
|
||||
* Optimal Stiffness of the nano-hexapod to reduce plant uncertainty ([[file:uncertainty_optimal_stiffness.org][link]])
|
||||
|
||||
* Active Damping Techniques on the full Simscape Model ([[file:control_active_damping.org][link]])
|
||||
Active damping techniques are applied to the full Simscape model.
|
||||
|
696
org/optimal_stiffness_disturbances.org
Normal file
696
org/optimal_stiffness_disturbances.org
Normal file
@ -0,0 +1,696 @@
|
||||
#+TITLE: Determination of the optimal nano-hexapod's stiffness for reducing the effect of disturbances
|
||||
:DRAWER:
|
||||
#+STARTUP: overview
|
||||
|
||||
#+LANGUAGE: en
|
||||
#+EMAIL: dehaeze.thomas@gmail.com
|
||||
#+AUTHOR: Dehaeze Thomas
|
||||
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
|
||||
|
||||
#+HTML_MATHJAX: align: center tagside: right font: TeX
|
||||
|
||||
#+PROPERTY: header-args:matlab :session *MATLAB*
|
||||
#+PROPERTY: header-args:matlab+ :comments org
|
||||
#+PROPERTY: header-args:matlab+ :results none
|
||||
#+PROPERTY: header-args:matlab+ :exports both
|
||||
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||
#+PROPERTY: header-args:matlab+ :tangle ../matlab/optimal_stiffness.m
|
||||
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
||||
|
||||
#+PROPERTY: header-args:shell :eval no-export
|
||||
|
||||
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
|
||||
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
||||
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
||||
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
||||
#+PROPERTY: header-args:latex+ :results file raw replace
|
||||
#+PROPERTY: header-args:latex+ :buffer no
|
||||
#+PROPERTY: header-args:latex+ :eval no-export
|
||||
#+PROPERTY: header-args:latex+ :exports results
|
||||
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||
:END:
|
||||
|
||||
* Introduction :ignore:
|
||||
In this document is studied how the stiffness of the nano-hexapod will impact the effect of disturbances on the position error of the sample.
|
||||
|
||||
It is divided in the following sections:
|
||||
- Section [[sec:psd_disturbances]]: the disturbances are listed and their Power Spectral Densities (PSD) are shown
|
||||
- Section [[sec:effect_disturbances]]: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses
|
||||
- Section [[sec:granite_stiffness]]:
|
||||
- Section [[sec:open_loop_budget_error]]: from both the PSD of the disturbances and the transfer function from disturbances to sample's position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)
|
||||
- Section [[sec:closed_loop_budget_error]]: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values
|
||||
|
||||
* Disturbances
|
||||
<<sec:psd_disturbances>>
|
||||
** Introduction :ignore:
|
||||
The main disturbances considered here are:
|
||||
- $D_w$: Ground displacement in the $x$, $y$ and $z$ directions
|
||||
- $F_{ty}$: Forces applied by the Translation stage in the $x$ and $z$ directions
|
||||
- $F_{rz}$: Forces applied by the Spindle in the $z$ direction
|
||||
- $F_d$: Direct forces applied at the center of mass of the Payload
|
||||
|
||||
The level of these disturbances has been identified form experiments which are detailed in [[file:disturbances.org][this]] document.
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no
|
||||
simulinkproject('../');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
load('mat/conf_simulink.mat');
|
||||
|
||||
open('nass_model.slx')
|
||||
#+end_src
|
||||
** Plots :ignore:
|
||||
The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures [[fig:opt_stiff_dist_gm]] and [[fig:opt_stiff_dist_fty_frz]].
|
||||
|
||||
In this study, the expected frequency content of the direct forces applied to the payload is not considered.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
load('./mat/dist_psd.mat', 'dist_f');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(dist_f.f, sqrt(dist_f.psd_gm));
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('$\Gamma_{D_w}$ $\left[\frac{m}{\sqrt{Hz}}\right]$')
|
||||
xlim([1, 500]);
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/opt_stiff_dist_gm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:opt_stiff_dist_gm
|
||||
#+caption: Amplitude Spectral Density of the Ground Displacement ([[./figs/opt_stiff_dist_gm.png][png]], [[./figs/opt_stiff_dist_gm.pdf][pdf]])
|
||||
[[file:figs/opt_stiff_dist_gm.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(dist_f.f, sqrt(dist_f.psd_ty), 'DisplayName', '$F_{T_y}$');
|
||||
plot(dist_f.f, sqrt(dist_f.psd_rz), 'DisplayName', '$F_{R_z}$');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{F}{\sqrt{Hz}}\right]$')
|
||||
legend('Location', 'southwest');
|
||||
xlim([1, 500]);
|
||||
#+end_src
|
||||
|
||||
#+header: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/opt_stiff_dist_fty_frz.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+name: fig:opt_stiff_dist_fty_frz
|
||||
#+caption: Amplitude Spectral Density of the "parasitic" forces comming from the Translation stage and the spindle ([[./figs/opt_stiff_dist_fty_frz.png][png]], [[./figs/opt_stiff_dist_fty_frz.pdf][pdf]])
|
||||
[[file:figs/opt_stiff_dist_fty_frz.png]]
|
||||
|
||||
* Effect of disturbances on the position error
|
||||
<<sec:effect_disturbances>>
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no
|
||||
simulinkproject('../');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
load('mat/conf_simulink.mat');
|
||||
|
||||
open('nass_model.slx')
|
||||
#+end_src
|
||||
** Initialization
|
||||
We initialize all the stages with the default parameters.
|
||||
#+begin_src matlab
|
||||
initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
#+end_src
|
||||
|
||||
We use a sample mass of 10kg.
|
||||
#+begin_src matlab
|
||||
initializeSample('mass', 10);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
initializeSimscapeConfiguration('gravity', true);
|
||||
initializeDisturbances('enable', false);
|
||||
initializeLoggingConfiguration('log', 'none');
|
||||
initializeController();
|
||||
#+end_src
|
||||
|
||||
|
||||
** Identification
|
||||
Inputs:
|
||||
- =Dwx=: Ground displacement in the $x$ direction
|
||||
- =Dwy=: Ground displacement in the $y$ direction
|
||||
- =Dwz=: Ground displacement in the $z$ direction
|
||||
- =Fty_x=: Forces applied by the Translation stage in the $x$ direction
|
||||
- =Fty_z=: Forces applied by the Translation stage in the $z$ direction
|
||||
- =Frz_z=: Forces applied by the Spindle in the $z$ direction
|
||||
- =Fd=: Direct forces applied at the center of mass of the Payload
|
||||
|
||||
#+begin_src matlab
|
||||
Ks = logspace(3,9,7); % [N/m]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
Gd = {zeros(length(Ks), 1)};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'nass_model';
|
||||
|
||||
%% Micro-Hexapod
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwx'); io_i = io_i + 1; % X Ground motion
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwy'); io_i = io_i + 1; % Y Ground motion
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz'); io_i = io_i + 1; % Z Ground motion
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_x'); io_i = io_i + 1; % Parasitic force Ty - X
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_z'); io_i = io_i + 1; % Parasitic force Ty - Z
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Parasitic force Rz - Z
|
||||
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fd'); io_i = io_i + 1; % Direct forces
|
||||
|
||||
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En'); io_i = io_i + 1; % Position Error
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:length(Ks)
|
||||
initializeNanoHexapod('k', Ks(i));
|
||||
|
||||
% Run the linearization
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
||||
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
||||
Gd(i) = {minreal(G)};
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Plots
|
||||
Effect of Stages vibration (Filtering).
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
||||
#+end_src
|
||||
|
||||
Effect of Ground motion (Transmissibility).
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(3, 1, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ex', 'Dwx'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]');
|
||||
|
||||
ax1 = subplot(3, 1, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ey', 'Dwy'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]');
|
||||
|
||||
ax1 = subplot(3, 1, 3);
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
||||
#+end_src
|
||||
|
||||
Direct Forces (Compliance).
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
||||
#+end_src
|
||||
|
||||
** Save
|
||||
#+begin_src matlab
|
||||
save('./mat/opt_stiffness_disturbances.mat', 'Gd')
|
||||
#+end_src
|
||||
|
||||
* Effect of granite stiffness
|
||||
<<sec:granite_stiffness>>
|
||||
** Analytical Analysis
|
||||
#+begin_src latex :file 2dof_system_granite_stiffness.pdf
|
||||
\begin{tikzpicture}
|
||||
% ====================
|
||||
% Parameters
|
||||
% ====================
|
||||
\def\massw{2.2} % Width of the masses
|
||||
\def\massh{0.8} % Height of the masses
|
||||
\def\spaceh{1.2} % Height of the springs/dampers
|
||||
\def\dispw{0.3} % Width of the dashed line for the displacement
|
||||
\def\disph{0.5} % Height of the arrow for the displacements
|
||||
\def\bracs{0.05} % Brace spacing vertically
|
||||
\def\brach{-10pt} % Brace shift horizontaly
|
||||
% ====================
|
||||
|
||||
|
||||
% ====================
|
||||
% Ground
|
||||
% ====================
|
||||
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
|
||||
\draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
|
||||
\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$};
|
||||
|
||||
% ====================
|
||||
% Micro Station
|
||||
% ====================
|
||||
\begin{scope}[shift={(0, 0)}]
|
||||
% Mass
|
||||
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};
|
||||
|
||||
% Spring, Damper, and Actuator
|
||||
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
|
||||
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c$};
|
||||
|
||||
% Displacements
|
||||
\draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
|
||||
\draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x$};
|
||||
|
||||
% Legend
|
||||
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
|
||||
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
|
||||
\end{scope}
|
||||
|
||||
% ====================
|
||||
% Nano Station
|
||||
% ====================
|
||||
\begin{scope}[shift={(0, \spaceh+\massh)}]
|
||||
% Mass
|
||||
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};
|
||||
|
||||
% Spring, Damper, and Actuator
|
||||
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
|
||||
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c^\prime$};
|
||||
|
||||
% Displacements
|
||||
\draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
|
||||
\draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x^\prime$};
|
||||
|
||||
% Legend
|
||||
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
|
||||
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:2dof_system_granite_stiffness
|
||||
#+caption: Figure caption
|
||||
#+RESULTS:
|
||||
[[file:figs/2dof_system_granite_stiffness.png]]
|
||||
|
||||
If we write the equation of motion of the system in Figure [[fig:2dof_system_granite_stiffness]], we obtain:
|
||||
\begin{align}
|
||||
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
|
||||
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
|
||||
\end{align}
|
||||
|
||||
If we note $d = x^\prime - x$, we obtain:
|
||||
#+name: eq:plant_ground_transmissibility
|
||||
\begin{equation}
|
||||
\frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
|
||||
\end{equation}
|
||||
|
||||
** Soft Granite
|
||||
Let's initialize a soft granite that will act as an isolation stage from ground motion.
|
||||
#+begin_src matlab
|
||||
initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
Ks = logspace(3,9,7); % [N/m]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
Gdr = {zeros(length(Ks), 1)};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:length(Ks)
|
||||
initializeNanoHexapod('k', Ks(i));
|
||||
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
||||
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
||||
Gdr(i) = {minreal(G)};
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Effect of the Granite transfer function
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gdr{i}('Ez', 'Dwz'), freqs, 'Hz'))), '--', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gdr{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '--', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
#+end_src
|
||||
|
||||
|
||||
* Open Loop Budget Error
|
||||
<<sec:open_loop_budget_error>>
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no
|
||||
simulinkproject('../');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
load('mat/conf_simulink.mat');
|
||||
|
||||
open('nass_model.slx')
|
||||
#+end_src
|
||||
|
||||
** Load of the identified disturbances and transfer functions
|
||||
#+begin_src matlab
|
||||
load('./mat/dist_psd.mat', 'dist_f');
|
||||
load('./mat/opt_stiffness_disturbances.mat', 'Gd')
|
||||
#+end_src
|
||||
|
||||
** Equations
|
||||
|
||||
** Results
|
||||
Effect of all disturbances
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(dist_f.psd_rz).*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{m}{\sqrt{Hz}}\right]$')
|
||||
legend('Location', 'southwest');
|
||||
xlim([2, 500]);
|
||||
#+end_src
|
||||
|
||||
** Cumulative Amplitude Spectrum
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_ty.*abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))).^2)))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $[m]$')
|
||||
legend('Location', 'southwest');
|
||||
xlim([2, 500]); ylim([1e-10 1e-6]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_rz.*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))).^2)))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $[m]$')
|
||||
legend('Location', 'southwest');
|
||||
xlim([2, 500]); ylim([1e-10 1e-6]);
|
||||
#+end_src
|
||||
|
||||
Ground motion
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))).^2)))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
|
||||
legend('Location', 'northeast');
|
||||
xlim([2, 500]); ylim([1e-10 1e-6]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ex', 'Dwx'), freqs, 'Hz'))).^2)))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
|
||||
legend('Location', 'northeast');
|
||||
xlim([2, 500]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ey', 'Dwy'), freqs, 'Hz'))).^2)))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
|
||||
legend('Location', 'northeast');
|
||||
xlim([2, 500]);
|
||||
#+end_src
|
||||
|
||||
Sum of all perturbations
|
||||
#+begin_src matlab
|
||||
psd_tot = zeros(length(freqs), length(Ks));
|
||||
|
||||
for i = 1:length(Ks)
|
||||
psd_tot(:,i) = dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz' ), freqs, 'Hz'))).^2 + ...
|
||||
dist_f.psd_ty.*abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))).^2 + ...
|
||||
dist_f.psd_rz.*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))).^2;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(psd_tot(:,i))))), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
|
||||
end
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $E_z$ $[m]$')
|
||||
legend('Location', 'northeast');
|
||||
xlim([1, 500]); ylim([1e-10 1e-6]);
|
||||
#+end_src
|
||||
|
||||
* Closed Loop Budget Error
|
||||
<<sec:closed_loop_budget_error>>
|
||||
** Introduction :ignore:
|
||||
** Reduction thanks to feedback - Required bandwidth
|
||||
#+begin_src matlab
|
||||
wc = 1*2*pi; % [rad/s]
|
||||
xic = 0.5;
|
||||
|
||||
S = (s/wc)/(1 + s/wc);
|
||||
|
||||
bodeFig({S}, logspace(-1,2,1000))
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
wc = [1, 5, 10, 20, 50, 100, 200];
|
||||
|
||||
S1 = {zeros(length(wc), 1)};
|
||||
S2 = {zeros(length(wc), 1)};
|
||||
|
||||
for j = 1:length(wc)
|
||||
L = (2*pi*wc(j))/s; % Simple integrator
|
||||
S1{j} = 1/(1 + L);
|
||||
L = ((2*pi*wc(j))/s)^2*(1 + s/(2*pi*wc(j)/2))/(1 + s/(2*pi*wc(j)*2));
|
||||
S2{j} = 1/(1 + L);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
i = 6;
|
||||
for j = 1:length(wc)
|
||||
set(gca,'ColorOrderIndex',j);
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(S1{j}, freqs, 'Hz'))).^2.*psd_tot(:,i))))), '-', ...
|
||||
'DisplayName', sprintf('$\\omega_c = %.0f$ [Hz]', wc(j)));
|
||||
end
|
||||
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(psd_tot(:,i))))), 'k-', ...
|
||||
'DisplayName', 'Open-Loop');
|
||||
plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
|
||||
legend('Location', 'northeast');
|
||||
xlim([0.5, 500]); ylim([1e-10 1e-6]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
wc = logspace(0, 3, 100);
|
||||
|
||||
Dz1_rms = zeros(length(Ks), length(wc));
|
||||
Dz2_rms = zeros(length(Ks), length(wc));
|
||||
for i = 1:length(Ks)
|
||||
for j = 1:length(wc)
|
||||
L = (2*pi*wc(j))/s;
|
||||
Dz1_rms(i, j) = sqrt(trapz(freqs, abs(squeeze(freqresp(1/(1 + L), freqs, 'Hz'))).^2.*psd_tot(:,i)));
|
||||
|
||||
L = ((2*pi*wc(j))/s)^2*(1 + s/(2*pi*wc(j)/2))/(1 + s/(2*pi*wc(j)*2));
|
||||
Dz2_rms(i, j) = sqrt(trapz(freqs, abs(squeeze(freqresp(1/(1 + L), freqs, 'Hz'))).^2.*psd_tot(:,i)));
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = dist_f.f;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(Ks)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(wc, Dz1_rms(i, :), '-', ...
|
||||
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)))
|
||||
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(wc, Dz2_rms(i, :), '--', ...
|
||||
'HandleVisibility', 'off')
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Control Bandwidth [Hz]'); ylabel('$E_z\ [m, rms]$')
|
||||
legend('Location', 'southwest');
|
||||
xlim([1, 500]);
|
||||
#+end_src
|
||||
* Conclusion
|
@ -313,13 +313,14 @@ The output =sample_pos= corresponds to the impact point of the X-ray.
|
||||
args.type char {mustBeMember(args.type,{'rigid', 'flexible', 'none', 'modal-analysis', 'init'})} = 'flexible'
|
||||
args.Foffset logical {mustBeNumericOrLogical} = false
|
||||
args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3]
|
||||
args.K (3,1) double {mustBeNumeric, mustBeNonnegative} = [4e9; 3e8; 8e8] % [N/m]
|
||||
args.C (3,1) double {mustBeNumeric, mustBeNonnegative} = [4.0e5; 1.1e5; 9.0e5] % [N/(m/s)]
|
||||
args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m]
|
||||
args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m]
|
||||
args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m]
|
||||
end
|
||||
#+end_src
|
||||
|
||||
|
||||
** Structure initialization
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
@ -370,8 +371,8 @@ Z-offset for the initial position of the sample with respect to the granite top
|
||||
:END:
|
||||
|
||||
#+begin_src matlab
|
||||
granite.K = [4e9; 3e8; 8e8]; % [N/m]
|
||||
granite.C = [4.0e5; 1.1e5; 9.0e5]; % [N/(m/s)]
|
||||
granite.K = args.K; % [N/m]
|
||||
granite.C = args.C; % [N/(m/s)]
|
||||
#+end_src
|
||||
|
||||
** Equilibrium position of the each joint.
|
||||
|
@ -75,8 +75,8 @@ The rotation speed will have an effect due to the Coriolis effect.
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
% load('mat/conf_simulink.mat');
|
||||
% open('nass_model.slx')
|
||||
load('mat/conf_simulink.mat');
|
||||
open('nass_model.slx')
|
||||
#+end_src
|
||||
|
||||
** Initialization
|
@ -4,6 +4,8 @@ arguments
|
||||
args.type char {mustBeMember(args.type,{'rigid', 'flexible', 'none', 'modal-analysis', 'init'})} = 'flexible'
|
||||
args.Foffset logical {mustBeNumericOrLogical} = false
|
||||
args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3]
|
||||
args.K (3,1) double {mustBeNumeric, mustBeNonnegative} = [4e9; 3e8; 8e8] % [N/m]
|
||||
args.C (3,1) double {mustBeNumeric, mustBeNonnegative} = [4.0e5; 1.1e5; 9.0e5] % [N/(m/s)]
|
||||
args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m]
|
||||
args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m]
|
||||
args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m]
|
||||
@ -29,8 +31,8 @@ granite.STEP = './STEPS/granite/granite.STEP';
|
||||
|
||||
granite.sample_pos = 0.8; % [m]
|
||||
|
||||
granite.K = [4e9; 3e8; 8e8]; % [N/m]
|
||||
granite.C = [4.0e5; 1.1e5; 9.0e5]; % [N/(m/s)]
|
||||
granite.K = args.K; % [N/m]
|
||||
granite.C = args.C; % [N/(m/s)]
|
||||
|
||||
if args.Foffset && ~strcmp(args.type, 'none') && ~strcmp(args.type, 'rigid') && ~strcmp(args.type, 'init')
|
||||
load('mat/Foffset.mat', 'Fgm');
|
||||
|
Loading…
Reference in New Issue
Block a user