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Thomas Dehaeze
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-04-03 ven. 17:55 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Determination of the optimal nano-hexapod&rsquo;s stiffness</title>
@@ -249,7 +249,7 @@
<ul>
<li><a href="#org84aa9ce">1. Spindle Rotation Speed</a>
<ul>
<li><a href="#org0a41acf">1.1. Initialization</a></li>
<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>
@@ -264,7 +264,7 @@
</li>
<li><a href="#orgd6382d8">3. Payload &ldquo;Impedance&rdquo; Effect</a>
<ul>
<li><a href="#org37d23dd">3.1. Initialization</a></li>
<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>
@@ -321,21 +321,21 @@ The rotation speed will have an effect due to the Coriolis effect.
</p>
</div>
<div id="outline-container-org0a41acf" class="outline-3">
<h3 id="org0a41acf"><span class="section-number-3">1.1</span> Initialization</h3>
<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 class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
</pre>
</div>
@@ -343,7 +343,7 @@ initializeMirror();
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 class="src src-matlab"> initializeSample(<span class="org-string">'mass'</span>, 10);
</pre>
</div>
@@ -352,10 +352,10 @@ We don&rsquo;t include disturbances in this model as it adds complexity to the s
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 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>
@@ -368,7 +368,7 @@ initializeController();
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 class="src src-matlab"> Rz_rpm = linspace(0, 60, 6);
</pre>
</div>
@@ -376,7 +376,7 @@ We identify the dynamics for the following spindle rotation speeds <code>Rz_rpm<
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 class="src src-matlab"> Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
</pre>
</div>
</div>
@@ -389,38 +389,45 @@ And for the following nano-hexapod actuator stiffness <code>Ks</code>:
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">1</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <a href="#org9bfe588">2</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <a href="#org9f887c8">3</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">4</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_y\) (coupling of the centralized positioning plant)</li>
<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 1: </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>
<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 2: </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>
<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 3: </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>
<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 4: </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>
<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>
@@ -462,12 +469,12 @@ Note that we can use very soft nano-hexapod if we limit the spindle rotating spe
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 class="src src-matlab"> initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'compliance'</span>);
</pre>
</div>
@@ -475,23 +482,23 @@ initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-
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 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">5</a>.
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 5: </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>
<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>
@@ -505,7 +512,7 @@ This is equivalent of identifying the dynamics of the nano-hexapod when fixed to
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 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>
@@ -513,7 +520,7 @@ We also choose the sample to be rigid and to have a mass of 10kg.
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 class="src src-matlab"> Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
</pre>
</div>
</div>
@@ -536,38 +543,38 @@ 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">6</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <a href="#orgcea6354">7</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <a href="#org5c6f89c">8</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">9</a>: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)</li>
<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 6: </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_micro_station_iff.png">png</a>, <a href="./figs/opt_stiffness_micro_station_iff.pdf">pdf</a>)</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 7: </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_micro_station_dvf.png">png</a>, <a href="./figs/opt_stiffness_micro_station_dvf.pdf">pdf</a>)</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 8: </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_micro_station_fx_dx.png">png</a>, <a href="./figs/opt_stiffness_micro_station_fx_dx.pdf">pdf</a>)</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 9: </span>Change of dynamics from force \(\mathcal{F}_z\) to displacement \(\mathcal{X}_z\) for a spindle rotation speed from 0rpm to 60rpm (<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>
<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>
@@ -591,15 +598,15 @@ When the nano-hexapod is stiff (\(k>10^7\ [N/m]\)), the compliance of the micro-
</p>
</div>
<div id="outline-container-org37d23dd" class="outline-3">
<h3 id="org37d23dd"><span class="section-number-3">3.1</span> Initialization</h3>
<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&rsquo;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 class="src src-matlab"> initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
</pre>
</div>
@@ -607,10 +614,10 @@ We don&rsquo;t include disturbances in this model as it adds complexity to the s
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 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>
@@ -632,8 +639,8 @@ We make the following change of payload dynamics:
We identify the dynamics for the following payload masses <code>Ms</code> and nano-hexapod leg&rsquo;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 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>
@@ -641,7 +648,7 @@ Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
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 class="src src-matlab"> Fs = [50, 200, 500]; <span class="org-comment">% [Hz]</span>
</pre>
</div>
</div>
@@ -658,12 +665,12 @@ We then identify the dynamics for the following payload resonance frequencies <c
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">10</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">11</a>: same, but for all tested nano-hexapod stiffnesses</li>
<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">10</a>):
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>
@@ -675,14 +682,14 @@ We can see two mass lines for the soft nano-hexapod (Figure <a href="#orgcce8740
<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 10: </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>
<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 11: </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>
<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>
@@ -694,8 +701,8 @@ We can see two mass lines for the soft nano-hexapod (Figure <a href="#orgcce8740
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">12</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">13</a>: same, but for all tested nano-hexapod stiffnesses</li>
<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>
@@ -712,14 +719,14 @@ We can see here that for the soft nano-hexapod:
<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 12: </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>
<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 13: </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>
<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>
@@ -728,7 +735,7 @@ We can see here that for the soft nano-hexapod:
<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 (Figure <a href="#org75c1705">14</a>):
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>
@@ -736,10 +743,17 @@ We now plot the total change of dynamics due to change of the payload (Figure <a
</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 14: </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>
<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>
@@ -771,63 +785,63 @@ We now consider the total change of nano-hexapod dynamics due to:
The obtained dynamics are shown:
</p>
<ul class="org-ul">
<li>Figure <a href="#org045955d">15</a> for a stiffness \(k = 10^3\ [N/m]\)</li>
<li>Figure <a href="#orgc2c0741">16</a> for a stiffness \(k = 10^5\ [N/m]\)</li>
<li>Figure <a href="#org7a8f7c3">17</a> for a stiffness \(k = 10^7\ [N/m]\)</li>
<li>Figure <a href="#org4745a60">18</a> for a stiffness \(k = 10^9\ [N/m]\)</li>
<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">19</a> and <a href="#orge890242">20</a> are shown an animation of the change of dynamics with the nano-hexapod&rsquo;s stiffness.
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&rsquo;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 15: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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>
<p><span class="figure-number">Figure 17: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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 16: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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>
<p><span class="figure-number">Figure 18: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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 17: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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>
<p><span class="figure-number">Figure 19: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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 18: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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>
<p><span class="figure-number">Figure 20: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg&rsquo;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 19: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</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 20: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</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: 2020-04-03 ven. 17:55</p>
<p class="date">Created: 2020-04-05 dim. 19:43</p>
</div>
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