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<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="#org84aa9ce">1. Spindle Rotation Speed</a>
<ul>
<li><a href="#org37d23dd">1.1. Initialization</a></li>
<li><a href="#org75fea48">1.2. Identification when rotating at maximum speed</a></li>
<li><a href="#orgf19758e">1.3. Change of dynamics</a></li>
</ul>
</li>
<li><a href="#org57a0e57">2. Micro-Station Compliance Effect</a>
<ul>
<li><a href="#orgca3001c">2.1. Identification of the micro-station compliance</a></li>
<li><a href="#orgffe89ca">2.2. Identification of the dynamics with a rigid micro-station</a></li>
<li><a href="#org21c0a38">2.3. Identification of the dynamics with a flexible micro-station</a></li>
<li><a href="#orge9344e2">2.4. Obtained Dynamics</a></li>
</ul>
</li>
<li><a href="#orgd6382d8">3. Payload “Impedance” Effect</a>
<ul>
<li><a href="#orgc1416f3">3.1. Initialization</a></li>
<li><a href="#orgc8cd70d">3.2. Identification of the dynamics while change the payload dynamics</a></li>
<li><a href="#org4da36db">3.3. Change of dynamics for the primary controller</a>
<ul>
<li><a href="#org913b8cb">3.3.1. Frequency variation</a></li>
<li><a href="#org906dcba">3.3.2. Mass variation</a></li>
<li><a href="#org632af10">3.3.3. Total variation</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgf0a88c5">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="#org1f375bb">1</a>: the change of its dynamics due to the spindle rotation speed is acceptable</li>
<li>Section <a href="#orgcf5798c">2</a>: the support compliance dynamics is not much present in the nano-hexapod dynamics</li>
<li>Section <a href="#org1a9d09c">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-org84aa9ce" class="outline-2">
<h2 id="org84aa9ce"><span class="section-number-2">1</span> Spindle Rotation Speed</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org1f375bb"></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-org37d23dd" class="outline-3">
<h3 id="org37d23dd"><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-org75fea48" class="outline-3">
<h3 id="org75fea48"><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-orgf19758e" class="outline-3">
<h3 id="orgf19758e"><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="#org2b01fbe">2</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <a href="#org9bfe588">3</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <a href="#org9f887c8">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="#org5926aca">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="org3c83afc" 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="org2b01fbe" 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="org9bfe588" 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="org9f887c8" 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="org5926aca" 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-org57a0e57" class="outline-2">
<h2 id="org57a0e57"><span class="section-number-2">2</span> Micro-Station Compliance Effect</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orgcf5798c"></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-orgca3001c" class="outline-3">
<h3 id="orgca3001c"><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="#org15a14d9">6</a>.
</p>
<div id="org15a14d9" 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-orgffe89ca" class="outline-3">
<h3 id="orgffe89ca"><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-org21c0a38" class="outline-3">
<h3 id="org21c0a38"><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-orge9344e2" class="outline-3">
<h3 id="orge9344e2"><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="#org6257db9">7</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <a href="#orgcea6354">8</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <a href="#org5c6f89c">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="#org7f2360c">10</a>: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)</li>
</ul>
<div id="org6257db9" 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="orgcea6354" 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="org5c6f89c" 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="org7f2360c" 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-orgd6382d8" class="outline-2">
<h2 id="orgd6382d8"><span class="section-number-2">3</span> Payload “Impedance” Effect</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org1a9d09c"></a>
</p>
</div>
<div id="outline-container-orgc1416f3" class="outline-3">
<h3 id="orgc1416f3"><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-orgc8cd70d" class="outline-3">
<h3 id="orgc8cd70d"><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-org4da36db" class="outline-3">
<h3 id="org4da36db"><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-org913b8cb" class="outline-4">
<h4 id="org913b8cb"><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="#orgcce8740">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="#org87ecfc8">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="#orgcce8740">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="orgcce8740" 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="org87ecfc8" 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-org906dcba" class="outline-4">
<h4 id="org906dcba"><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="#org4eb6bfc">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="#org6e013a7">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="org4eb6bfc" 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="org6e013a7" 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-org632af10" class="outline-4">
<h4 id="org632af10"><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="#orgf33beff">15</a> and <a href="#org75c1705">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="orgf33beff" 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="org75c1705" 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-orgf0a88c5" class="outline-2">
<h2 id="orgf0a88c5"><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="#org045955d">17</a> for a stiffness \(k = 10^3\ [N/m]\)</li>
<li>Figure <a href="#orgc2c0741">18</a> for a stiffness \(k = 10^5\ [N/m]\)</li>
<li>Figure <a href="#org7a8f7c3">19</a> for a stiffness \(k = 10^7\ [N/m]\)</li>
<li>Figure <a href="#org4745a60">20</a> for a stiffness \(k = 10^9\ [N/m]\)</li>
</ul>
<p>
And finally, in Figures <a href="#org0a4e875">21</a> and <a href="#orge890242">22</a> are shown an animation of the change of dynamics with the nano-hexapod’s stiffness.
</p>
<div id="org045955d" 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="orgc2c0741" 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="org7a8f7c3" 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="org4745a60" 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="org0a4e875" 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="orge890242" 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: 2020-04-05 dim. 19:43</p>
</div>
</body>
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