849 lines
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849 lines
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<a accesskey="h" href="index.html"> UP </a>
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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</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="#org157c07d">1. Spindle Rotation Speed</a>
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<ul>
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<li><a href="#orgb1e5096">1.1. Initialization</a></li>
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<li><a href="#org687bdef">1.2. Identification when rotating at maximum speed</a></li>
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<li><a href="#org7dcfddb">1.3. Change of dynamics</a></li>
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</ul>
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</li>
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<li><a href="#org23ddf26">2. Micro-Station Compliance Effect</a>
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<ul>
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<li><a href="#orgdc8aeea">2.1. Identification of the micro-station compliance</a></li>
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<li><a href="#orga44542b">2.2. Identification of the dynamics with a rigid micro-station</a></li>
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<li><a href="#org49d6b26">2.3. Identification of the dynamics with a flexible micro-station</a></li>
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<li><a href="#org4c1ed79">2.4. Obtained Dynamics</a></li>
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</ul>
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</li>
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<li><a href="#org19559b0">3. Payload “Impedance” Effect</a>
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<ul>
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<li><a href="#org67607c3">3.1. Initialization</a></li>
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<li><a href="#org73f1c6e">3.2. Identification of the dynamics while change the payload dynamics</a></li>
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<li><a href="#orgd7a519b">3.3. Change of dynamics for the primary controller</a>
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<ul>
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<li><a href="#orgb44d421">3.3.1. Frequency variation</a></li>
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<li><a href="#orgfc270b0">3.3.2. Mass variation</a></li>
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<li><a href="#org118f0c2">3.3.3. Total variation</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org973d2e3">4. Total Change of dynamics</a></li>
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</ul>
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</div>
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</div>
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<p>
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As shown before, many parameters other than the nano-hexapod itself do influence the plant dynamics:
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</p>
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<ul class="org-ul">
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<li>The micro-station compliance (studied <a href="uncertainty_support.html">here</a>)</li>
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<li>The payload mass and dynamical properties (studied <a href="uncertainty_payload.html">here</a> and <a href="uncertainty_experiment.html">here</a>)</li>
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<li>The experimental conditions, mainly the spindle rotation speed (studied <a href="uncertainty_experiment.html">here</a>)</li>
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</ul>
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<p>
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As seen before, the stiffness of the nano-hexapod greatly influence the effect of such parameters.
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</p>
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<p>
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We wish here to see if we can determine an optimal stiffness of the nano-hexapod such that:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org902923f">1</a>: the change of its dynamics due to the spindle rotation speed is acceptable</li>
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<li>Section <a href="#orgabe2ab2">2</a>: the support compliance dynamics is not much present in the nano-hexapod dynamics</li>
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<li>Section <a href="#org2bd8390">3</a>: the change of payload impedance has acceptable effect on the plant dynamics</li>
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</ul>
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<p>
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||
|
|
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>
|