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<h1 class="title">Measurement Analysis</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org8a6aabc">1. Importation of the data</a></li>
<li><a href="#orgad6ba6a">2. Variables for analysis</a></li>
<li><a href="#org1edb14b">3. Measurement 1 - Effect of Ty stage</a></li>
<li><a href="#org7de0db0">4. Measurement 2 - Effect of Ry stage</a></li>
<li><a href="#orgb549f7e">5. Measurement 3 - Effect of the Hexapod</a></li>
<li><a href="#org8c5be93">6. Measurement 4 - Effect of the Splip-Ring and Spindle</a></li>
<li><a href="#org9be1dbe">7. Measurement 5 - Transmission from ground to marble</a></li>
</ul>
</div>
</div>
<div id="outline-container-org8a6aabc" class="outline-2">
<h2 id="org8a6aabc"><span class="section-number-2">1</span> Importation of the data</h2>
<div class="outline-text-2" id="text-1">
<p>
First, load all the measurement files:
</p>
<div class="org-src-container">
<pre class="src src-matlab">meas = <span style="color: #707183;">{}</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span> = load<span style="color: #707183;">(</span><span style="color: #008000;">'./mat/Measurement1.mat'</span><span style="color: #707183;">)</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span> = load<span style="color: #707183;">(</span><span style="color: #008000;">'./mat/Measurement2.mat'</span><span style="color: #707183;">)</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">3</span><span style="color: #707183;">}</span> = load<span style="color: #707183;">(</span><span style="color: #008000;">'./mat/Measurement3.mat'</span><span style="color: #707183;">)</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">4</span><span style="color: #707183;">}</span> = load<span style="color: #707183;">(</span><span style="color: #008000;">'./mat/Measurement4.mat'</span><span style="color: #707183;">)</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">5</span><span style="color: #707183;">}</span> = load<span style="color: #707183;">(</span><span style="color: #008000;">'./mat/Measurement5.mat'</span><span style="color: #707183;">)</span>;
</pre>
</div>
<p>
Change the track name for measurements 3 and 4.
</p>
<div class="org-src-container">
<pre class="src src-matlab">meas<span style="color: #707183;">{</span><span style="color: #D0372D;">3</span><span style="color: #707183;">}</span>.Track1_Name = 'Input <span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span> Hexa Z';
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">4</span><span style="color: #707183;">}</span>.Track1_Name = 'Input <span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span> Hexa Z';
</pre>
</div>
<p>
For the measurements 1 to 4, the measurement channels are shown table <a href="#org9c4944a">1</a>.
</p>
<table id="org9c4944a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Channels for measurements 1 to 4</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left">Channel 1</th>
<th scope="col" class="org-left">Channel 2</th>
<th scope="col" class="org-left">Channel 3</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Meas. 1</td>
<td class="org-left">Input 1: tilt1 Z</td>
<td class="org-left">Input 2: tilt2 Z</td>
<td class="org-left">Input 3: Ty Y</td>
</tr>
<tr>
<td class="org-left">Meas. 2</td>
<td class="org-left">Input 1: tilt1 Z</td>
<td class="org-left">Input 2: tilt2 Z</td>
<td class="org-left">Input 3: Ty Y</td>
</tr>
<tr>
<td class="org-left">Meas. 3</td>
<td class="org-left">Input 1: Hexa Z</td>
<td class="org-left">Input 2: tilt2 Z</td>
<td class="org-left">Input 3: Ty Y</td>
</tr>
<tr>
<td class="org-left">Meas. 4</td>
<td class="org-left">Input 1: Hexa Z</td>
<td class="org-left">Input 2: tilt2 Z</td>
<td class="org-left">Input 3: Ty Y</td>
</tr>
</tbody>
</table>
<p>
For the measurement 5, the channels are shown table <a href="#org3301908">2</a>.
</p>
<table id="org3301908" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Channels for measurement 5</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"> </th>
<th scope="col" class="org-left">Channel 1</th>
<th scope="col" class="org-left">Channel 2</th>
<th scope="col" class="org-left">Channel 3</th>
<th scope="col" class="org-left">Channel 4</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Meas. 5</td>
<td class="org-left">Input 1: Floor Z</td>
<td class="org-left">Input 2: Marble Z</td>
<td class="org-left">Input 3: Floor Y</td>
<td class="org-left">Input 4: Marble Y</td>
</tr>
</tbody>
</table>
<p>
When using two geophone sensors on the same tilt stage (measurements 1 and 2), we post-process the data to obtain the z displacement and the rotation of the tilt stage:
</p>
<div class="org-src-container">
<pre class="src src-matlab">meas1_z = <span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #6434A3;">+</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #707183;">)</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">2</span>;
meas1_tilt = <span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #6434A3;">-</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #707183;">)</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">2</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span>.Track1 = meas1_z;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span>.Track1_Y_Magnitude = <span style="color: #008000;">'Meter / second'</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span>.Track1_Name = <span style="color: #008000;">'Ry Z'</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span>.Track2 = meas1_tilt;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span>.Track2_Y_Magnitude = <span style="color: #008000;">'Rad / second'</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">1</span><span style="color: #707183;">}</span>.Track2_Name = <span style="color: #008000;">'Ry Tilt'</span>;
meas2_z = <span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #6434A3;">+</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #707183;">)</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">2</span>;
meas2_tilt = <span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #6434A3;">-</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #707183;">)</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">2</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span>.Track1 = meas2_z;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span>.Track1_Y_Magnitude = <span style="color: #008000;">'Meter / second'</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span>.Track1_Name = <span style="color: #008000;">'Ry Z'</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span>.Track2 = meas2_tilt;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span>.Track2_Y_Magnitude = <span style="color: #008000;">'Rad / second'</span>;
meas<span style="color: #707183;">{</span><span style="color: #D0372D;">2</span><span style="color: #707183;">}</span>.Track2_Name = <span style="color: #008000;">'Ry Tilt'</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgad6ba6a" class="outline-2">
<h2 id="orgad6ba6a"><span class="section-number-2">2</span> Variables for analysis</h2>
<div class="outline-text-2" id="text-2">
<p>
We define the sampling frequency and the time vectors for the plots.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Fs = <span style="color: #D0372D;">256</span>; <span style="color: #8D8D84; font-style: italic;">% [Hz]</span>
dt = <span style="color: #D0372D;">1</span><span style="color: #6434A3;">/</span><span style="color: #707183;">(</span>Fs<span style="color: #707183;">)</span>;
t1 = dt<span style="color: #6434A3;">*</span><span style="color: #707183;">[</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>length<span style="color: #7388D6;">(</span>meas<span style="color: #909183;">{</span><span style="color: #D0372D;">1</span><span style="color: #909183;">}</span>.Track1<span style="color: #7388D6;">)</span><span style="color: #707183;">]</span>;
t2 = dt<span style="color: #6434A3;">*</span><span style="color: #707183;">[</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>length<span style="color: #7388D6;">(</span>meas<span style="color: #909183;">{</span><span style="color: #D0372D;">2</span><span style="color: #909183;">}</span>.Track1<span style="color: #7388D6;">)</span><span style="color: #707183;">]</span>;
t3 = dt<span style="color: #6434A3;">*</span><span style="color: #707183;">[</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>length<span style="color: #7388D6;">(</span>meas<span style="color: #909183;">{</span><span style="color: #D0372D;">3</span><span style="color: #909183;">}</span>.Track1<span style="color: #7388D6;">)</span><span style="color: #707183;">]</span>;
t4 = dt<span style="color: #6434A3;">*</span><span style="color: #707183;">[</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>length<span style="color: #7388D6;">(</span>meas<span style="color: #909183;">{</span><span style="color: #D0372D;">4</span><span style="color: #909183;">}</span>.Track1<span style="color: #7388D6;">)</span><span style="color: #707183;">]</span>;
t5 = dt<span style="color: #6434A3;">*</span><span style="color: #707183;">[</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>length<span style="color: #7388D6;">(</span>meas<span style="color: #909183;">{</span><span style="color: #D0372D;">5</span><span style="color: #909183;">}</span>.Track1<span style="color: #7388D6;">)</span><span style="color: #707183;">]</span>;
</pre>
</div>
<p>
For the frequency analysis, we define the frequency limits for the plot.
</p>
<div class="org-src-container">
<pre class="src src-matlab">fmin = <span style="color: #D0372D;">1</span>; <span style="color: #8D8D84; font-style: italic;">% [Hz]</span>
fmax = <span style="color: #D0372D;">100</span>; <span style="color: #8D8D84; font-style: italic;">% [Hz]</span>
</pre>
</div>
<p>
Then we define the windows that will be used to average the results.
</p>
<div class="org-src-container">
<pre class="src src-matlab">psd_window = hanning<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span>fmin<span style="color: #6434A3;">/</span>dt<span style="color: #707183;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org1edb14b" class="outline-2">
<h2 id="org1edb14b"><span class="section-number-2">3</span> Measurement 1 - Effect of Ty stage</h2>
<div class="outline-text-2" id="text-3">
<p>
The configuration for this measurement is shown table <a href="#org91518a8">3</a>.
</p>
<table id="org91518a8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Stages configuration - Measurement 1</caption>
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Time</th>
<th scope="col" class="org-right">0-309</th>
<th scope="col" class="org-left">309-end</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Ty</td>
<td class="org-right">OFF</td>
<td class="org-left">ON</td>
</tr>
</tbody>
</table>
<p>
We then plot the measurements in time domain (figure <a href="#orgf668224">1</a>).
</p>
<div class="important">
<p>
We observe strange behavior when the Ty stage is turned on.
How can we explain that?
</p>
</div>
<div id="orgf668224" class="figure">
<p><img src="Figures/meas1.png" alt="meas1.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Time domain - measurement 1</p>
</div>
<p>
To understand what is going on, instead of looking at the velocity, we can look at the displacement by integrating the data. The displacement is computed by integrating the velocity using <code>cumtrapz</code> function.
</p>
<div class="org-src-container">
<pre class="src src-matlab">tdisp = t1<span style="color: #707183;">(</span>ceil<span style="color: #7388D6;">(</span><span style="color: #D0372D;">300</span><span style="color: #6434A3;">/</span>dt<span style="color: #7388D6;">)</span><span style="color: #6434A3;">:</span>ceil<span style="color: #7388D6;">(</span><span style="color: #D0372D;">340</span><span style="color: #6434A3;">/</span>dt<span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
xdisp = cumtrapz<span style="color: #707183;">(</span>tdisp, meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">300</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">340</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span><span style="color: #707183;">)</span>;
</pre>
</div>
<p>
Then we plot the position with respect to time (figure <a href="#org162b20d">2</a>).
</p>
<div id="org162b20d" class="figure">
<p><img src="Figures/meas1_disp.png" alt="meas1_disp.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Y displacement of the Ty stage</p>
</div>
<p>
We when compute the power spectral density of each measurement before and after turning on the stage.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #707183;">[</span>pxx111, f111<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">300</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx112, f112<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">350</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx121, f121<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">300</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx122, f122<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">350</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx131, f131<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">300</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx132, f132<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">1</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">350</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
</pre>
</div>
<p>
We finally plot the power spectral density of each track (figures <a href="#orgc8c714a">3</a>, <a href="#org9301433">4</a>, <a href="#org958556b">5</a>).
</p>
<div id="orgc8c714a" class="figure">
<p><img src="Figures/meas1_ry_z_psd.png" alt="meas1_ry_z_psd.png" />
</p>
<p><span class="figure-number">Figure 3: </span>PSD of the Z velocity of Ry stage - measurement 1</p>
</div>
<div id="org9301433" class="figure">
<p><img src="Figures/meas1_ry_tilt_psd.png" alt="meas1_ry_tilt_psd.png" />
</p>
<p><span class="figure-number">Figure 4: </span>PSD of the Rotation of Ry Stage - measurement 1</p>
</div>
<div id="org958556b" class="figure">
<p><img src="Figures/meas1_ty_y_psd.png" alt="meas1_ty_y_psd.png" />
</p>
<p><span class="figure-number">Figure 5: </span>PSD of the Ty velocity in the Y direction - measurement 1</p>
</div>
<div class="important">
<p>
Turning on the Y-translation stage increases the velocity of the Ty stage in the Y direction and the rotation motion of the tilt stage:
</p>
<ul class="org-ul">
<li>at 20Hz</li>
<li>at 40Hz</li>
<li>between 80Hz and 90Hz</li>
</ul>
<p>
It does not seems to have any effect on the Z motion of the tilt stage.
</p>
</div>
</div>
</div>
<div id="outline-container-org7de0db0" class="outline-2">
<h2 id="org7de0db0"><span class="section-number-2">4</span> Measurement 2 - Effect of Ry stage</h2>
<div class="outline-text-2" id="text-4">
<p>
The tilt stage is turned ON at around 326 seconds (table <a href="#org296416d">4</a>).
</p>
<table id="org296416d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Stages configuration - Measurement 2</caption>
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Time</th>
<th scope="col" class="org-right">0-326</th>
<th scope="col" class="org-left">326-end</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Tilt</td>
<td class="org-right">OFF</td>
<td class="org-left">ON</td>
</tr>
</tbody>
</table>
<p>
We plot the time domain (figure <a href="#org074f8c1">6</a>) and we don't observe anything special in the time domain.
</p>
<div id="org074f8c1" class="figure">
<p><img src="Figures/meas2.png" alt="meas2.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Time domain - measurement 2</p>
</div>
<p>
We compute the PSD of each track and we plot them (figures <a href="#org2cfdba7">7</a>, <a href="#orgc7b7dca">8</a> and <a href="#orged46ee5">9</a> ).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #707183;">[</span>pxx211, f211<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">326</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx212, f212<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">326</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx221, f221<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">326</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx222, f222<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">326</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx231, f231<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">326</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx232, f232<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">2</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">326</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
</pre>
</div>
<div id="org2cfdba7" class="figure">
<p><img src="Figures/meas2_ry_z_psd.png" alt="meas2_ry_z_psd.png" />
</p>
<p><span class="figure-number">Figure 7: </span>PSD of the Z velocity of Ry Stage - measurement 2</p>
</div>
<div id="orgc7b7dca" class="figure">
<p><img src="Figures/meas2_ry_tilt_psd.png" alt="meas2_ry_tilt_psd.png" />
</p>
<p><span class="figure-number">Figure 8: </span>PSD of the Rotation motion of Ry Stage - measurement 2</p>
</div>
<div id="orged46ee5" class="figure">
<p><img src="Figures/meas2_ty_y_psd.png" alt="meas2_ty_y_psd.png" />
</p>
<p><span class="figure-number">Figure 9: </span>PSD of the Ty velocity in the Y direction - measurement 2</p>
</div>
<div class="important">
<p>
We observe no noticeable difference when the Tilt-stage is turned ON expect a small decrease of the Z motion of the tilt stage around 10Hz.
</p>
</div>
</div>
</div>
<div id="outline-container-orgb549f7e" class="outline-2">
<h2 id="orgb549f7e"><span class="section-number-2">5</span> Measurement 3 - Effect of the Hexapod</h2>
<div class="outline-text-2" id="text-5">
<p>
The hexapod is turned off after 406 seconds (table <a href="#org5597972">5</a>).
</p>
<table id="org5597972" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> Stages configuration - Measurement 3</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Time</th>
<th scope="col" class="org-left">0-406</th>
<th scope="col" class="org-left">406-end</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Tilt</td>
<td class="org-left">ON</td>
<td class="org-left">ON</td>
</tr>
<tr>
<td class="org-left">Hexa</td>
<td class="org-left">ON</td>
<td class="org-left">OFF</td>
</tr>
</tbody>
</table>
<p>
The time domain result is shown figure <a href="#org986a794">10</a>.
</p>
<div id="org986a794" class="figure">
<p><img src="Figures/meas3.png" alt="meas3.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Time domain - measurement 3</p>
</div>
<p>
We then compute the PSD of each track before and after turning off the hexapod and plot the results in the figures <a href="#orgbb8631e">11</a>, <a href="#org1306ed1">12</a> and <a href="#orgd453e30">13</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #707183;">[</span>pxx311, f311<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">3</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">400</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx312, f312<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">3</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">420</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx321, f321<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">3</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">400</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx322, f322<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">3</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">420</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx331, f331<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">3</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span><span style="color: #D0372D;">1</span><span style="color: #6434A3;">:</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">400</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx332, f332<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">3</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span>ceil<span style="color: #909183;">(</span><span style="color: #D0372D;">420</span><span style="color: #6434A3;">/</span>dt<span style="color: #909183;">)</span><span style="color: #6434A3;">:</span>end<span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
</pre>
</div>
<div id="orgbb8631e" class="figure">
<p><img src="Figures/meas3_hexa_z_psd.png" alt="meas3_hexa_z_psd.png" />
</p>
<p><span class="figure-number">Figure 11: </span>PSD of the Z velocity of the Hexapod - measurement 3</p>
</div>
<div id="org1306ed1" class="figure">
<p><img src="Figures/meas3_ry_z_psd.png" alt="meas3_ry_z_psd.png" />
</p>
<p><span class="figure-number">Figure 12: </span>PSD of the Z velocity of the Ry stage - measurement 3</p>
</div>
<div id="orgd453e30" class="figure">
<p><img src="Figures/meas3_ty_y_psd.png" alt="meas3_ty_y_psd.png" />
</p>
<p><span class="figure-number">Figure 13: </span>PSD of the Ty velocity in the Y direction - measurement 3</p>
</div>
<div class="important">
<p>
Turning ON induces some motion on the hexapod in the z direction (figure <a href="#orgbb8631e">11</a>), on the tilt stage in the z direction (figure <a href="#org1306ed1">12</a>) and on the y motion of the Ty stage (figure <a href="#orgd453e30">13</a>):
</p>
<ul class="org-ul">
<li>at 17Hz</li>
<li>at 34Hz</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-org8c5be93" class="outline-2">
<h2 id="org8c5be93"><span class="section-number-2">6</span> Measurement 4 - Effect of the Splip-Ring and Spindle</h2>
<div class="outline-text-2" id="text-6">
<p>
The slip ring is turned on at 300s, then the spindle is turned on at 620s (table <a href="#orgdce5060">6</a>). The time domain signals are shown figure <a href="#orge41f920">14</a>.
</p>
<table id="orgdce5060" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 6:</span> Stages configuration - Measurement 4</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Time</th>
<th scope="col" class="org-left">0-300</th>
<th scope="col" class="org-left">300-620</th>
<th scope="col" class="org-left">620-end</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">SlipRing</td>
<td class="org-left">OFF</td>
<td class="org-left">ON</td>
<td class="org-left">ON</td>
</tr>
<tr>
<td class="org-left">Hexa</td>
<td class="org-left">OFF</td>
<td class="org-left">OFF</td>
<td class="org-left">OFF</td>
</tr>
<tr>
<td class="org-left">Spindle</td>
<td class="org-left">OFF</td>
<td class="org-left">OFF</td>
<td class="org-left">ON</td>
</tr>
</tbody>
</table>
<div id="orge41f920" class="figure">
<p><img src="Figures/meas4.png" alt="meas4.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Time domain - measurement 4</p>
</div>
<p>
The PSD of each track are computed using the code below.
</p>
<div id="org823e771" class="figure">
<p><img src="Figures/meas4_hexa_z_psd.png" alt="meas4_hexa_z_psd.png" />
</p>
<p><span class="figure-number">Figure 15: </span>PSD of the Z velocity of the Hexapod - measurement 4</p>
</div>
<div id="orgc34be34" class="figure">
<p><img src="Figures/meas4_ry_z_psd.png" alt="meas4_ry_z_psd.png" />
</p>
<p><span class="figure-number">Figure 16: </span>PSD of the Ry rotation in the Y direction - measurement 4</p>
</div>
<div id="org6693a0d" class="figure">
<p><img src="Figures/meas4_ty_y_psd.png" alt="meas4_ty_y_psd.png" />
</p>
<p><span class="figure-number">Figure 17: </span>PSD of the Ty velocity in the Y direction - measurement 4</p>
</div>
<div class="important">
<p>
Turning ON the splipring seems to not add motions on the stages measured.
It even seems to lower the motion of the Ty stage (figure <a href="#org6693a0d">17</a>): does that make any sense?
</p>
<p>
Turning ON the spindle induces motions:
</p>
<ul class="org-ul">
<li>at 5Hz on each motion measured</li>
<li>at 22.5Hz on the Z motion of the Hexapod. Can this is due to some 50Hz?</li>
<li>at 62Hz on each motion measured</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-org9be1dbe" class="outline-2">
<h2 id="org9be1dbe"><span class="section-number-2">7</span> Measurement 5 - Transmission from ground to marble</h2>
<div class="outline-text-2" id="text-7">
<p>
This measurement just consists of measurement of Y-Z motion of the ground and the marble.
</p>
<p>
The time domain signals are shown on figure <a href="#org18a5544">18</a>.
</p>
<div id="org18a5544" class="figure">
<p><img src="Figures/meas5.png" alt="meas5.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Time domain - measurement 5</p>
</div>
<p>
We compute the PSD of each track and we plot the PSD of the Z motion for the ground and marble on figure <a href="#org8d852a7">19</a> and for the Y motion on figure <a href="#orgbd725c3">20</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #707183;">[</span>pxx51, f51<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx52, f52<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx53, f53<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>pxx54, f54<span style="color: #707183;">]</span> = pwelch<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track4<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
</pre>
</div>
<div id="org8d852a7" class="figure">
<p><img src="Figures/meas5_z_psd.png" alt="meas5_z_psd.png" />
</p>
<p><span class="figure-number">Figure 19: </span>PSD of the ground and marble in the Z direction</p>
</div>
<div id="orgbd725c3" class="figure">
<p><img src="Figures/meas5_y_psd.png" alt="meas5_y_psd.png" />
</p>
<p><span class="figure-number">Figure 20: </span>PSD of the ground and marble in the Y direction</p>
</div>
<p>
Then, instead of looking at the Power Spectral Density, we can try to estimate the transfer function from a ground motion to the motion of the marble.
The transfer functions are shown on figure <a href="#org966e909">21</a> and the coherence on figure <a href="#orgca2d629">22</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #707183;">[</span>tfz, fz<span style="color: #707183;">]</span> = tfestimate<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>tfy, fy<span style="color: #707183;">]</span> = tfestimate<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track4<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
</pre>
</div>
<div id="org966e909" class="figure">
<p><img src="Figures/meas5_tf.png" alt="meas5_tf.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Transfer function estimation - measurement 5</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #707183;">[</span>cohz, fz<span style="color: #707183;">]</span> = mscohere<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track1<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track2<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
<span style="color: #707183;">[</span>cohy, fy<span style="color: #707183;">]</span> = mscohere<span style="color: #707183;">(</span>meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track3<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, meas<span style="color: #7388D6;">{</span><span style="color: #D0372D;">5</span><span style="color: #7388D6;">}</span>.Track4<span style="color: #7388D6;">(</span><span style="color: #6434A3;">:</span><span style="color: #7388D6;">)</span>, psd_window, <span style="color: #7388D6;">[]</span>, <span style="color: #7388D6;">[]</span>, Fs<span style="color: #707183;">)</span>;
</pre>
</div>
<div id="orgca2d629" class="figure">
<p><img src="Figures/meas5_coh.png" alt="meas5_coh.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Coherence - measurement 5</p>
</div>
<div class="important">
<p>
The marble seems to have a resonance at around 20Hz on the Y direction.
But the coherence is not good above 20Hz, so it is difficult to estimate resonances.
</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-14 jeu. 11:55</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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