From 16e3878dbbdfd74b627bdd925573c7d8a7039a20 Mon Sep 17 00:00:00 2001
From: Thomas Dehaeze <dehaeze.thomas@gmail.com>
Date: Wed, 29 Apr 2020 19:02:03 +0200
Subject: [PATCH] Re-read section about multi-body model

---
 figs/simscape_picture.png | Bin 88965 -> 106643 bytes
 index.html                | 751 ++++++++++++++++++++------------------
 index.org                 |  81 ++--
 3 files changed, 440 insertions(+), 392 deletions(-)

diff --git a/figs/simscape_picture.png b/figs/simscape_picture.png
index a77fe07f4df8c68b60e5fb7936cecbc83015d67a..63c7ded07836f5c34ce672db97901c90e6985122 100644
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diff --git a/index.html b/index.html
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 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-04-29 mer. 18:17 -->
+<!-- 2020-04-29 mer. 19:02 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <title>Design of the Nano-Hexapod and associated Control Architectures - Summary</title>
 <meta name="generator" content="Org mode" />
@@ -35,94 +35,94 @@
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org466c91c">1. Introduction to Feedback Systems and Noise budgeting</a>
+<li><a href="#org5ebe35e">1. Introduction to Feedback Systems and Noise budgeting</a>
 <ul>
-<li><a href="#orgad80709">1.1. Feedback System</a>
+<li><a href="#org8eabb73">1.1. Feedback System</a>
 <ul>
-<li><a href="#orge99f149">1.1.1. Simplified Feedback Control Diagram for the NASS</a></li>
-<li><a href="#orgfef2aa4">1.1.2. How does the feedback loop is modifying the system behavior?</a></li>
-<li><a href="#org7e92a2e">1.1.3. Trade off: Disturbance Reduction / Noise Injection</a></li>
-<li><a href="#orgfbb078e">1.1.4. Trade off: Robustness / Performance</a></li>
+<li><a href="#orgea180e3">1.1.1. Simplified Feedback Control Diagram for the NASS</a></li>
+<li><a href="#orgee4b57a">1.1.2. How does the feedback loop is modifying the system behavior?</a></li>
+<li><a href="#org672b56a">1.1.3. Trade off: Disturbance Reduction / Noise Injection</a></li>
+<li><a href="#org24cd2f0">1.1.4. Trade off: Robustness / Performance</a></li>
 </ul>
 </li>
-<li><a href="#orgb6996cc">1.2. Dynamic error budgeting</a>
+<li><a href="#orgcd11903">1.2. Dynamic error budgeting</a>
 <ul>
-<li><a href="#orga061b8d">1.2.1. Power Spectral Density</a></li>
-<li><a href="#orgd87885d">1.2.2. Cumulative Power Spectrum</a></li>
-<li><a href="#org30dd763">1.2.3. Modification of a signal&rsquo;s PSD when going through a dynamical system</a></li>
-<li><a href="#org2273db0">1.2.4. PSD of combined signals</a></li>
-<li><a href="#orga3546fb">1.2.5. Dynamic Noise Budgeting</a></li>
+<li><a href="#org3320434">1.2.1. Power Spectral Density</a></li>
+<li><a href="#org129133e">1.2.2. Cumulative Power Spectrum</a></li>
+<li><a href="#org5472bba">1.2.3. Modification of a signal&rsquo;s PSD when going through a dynamical system</a></li>
+<li><a href="#orgbbc41e7">1.2.4. PSD of combined signals</a></li>
+<li><a href="#org454c2e9">1.2.5. Dynamic Noise Budgeting</a></li>
 </ul>
 </li>
 </ul>
 </li>
-<li><a href="#org4920a6e">2. Identification of the Micro-Station Dynamics</a>
+<li><a href="#org98142af">2. Identification of the Micro-Station Dynamics</a>
 <ul>
-<li><a href="#org4edd614">2.1. Experimental Setup</a></li>
-<li><a href="#orgf6e05b4">2.2. Results</a></li>
-<li><a href="#orge3cb290">2.3. Conclusion</a></li>
+<li><a href="#org8c825c9">2.1. Experimental Setup</a></li>
+<li><a href="#orgd31b11c">2.2. Results</a></li>
+<li><a href="#org72bfef9">2.3. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org8c43d49">3. Identification of the Disturbances</a>
+<li><a href="#org726da55">3. Identification of the Disturbances</a>
 <ul>
-<li><a href="#orgb1af52d">3.1. Ground Motion</a></li>
-<li><a href="#org9284950">3.2. Stage Vibration - Effect of Control systems</a></li>
-<li><a href="#org482cc38">3.3. Stage Vibration - Effect of Motion</a>
+<li><a href="#org0887051">3.1. Ground Motion</a></li>
+<li><a href="#org4359dc1">3.2. Stage Vibration - Effect of Control systems</a></li>
+<li><a href="#orgf5d9e59">3.3. Stage Vibration - Effect of Motion</a>
 <ul>
-<li><a href="#org1af435f">Spindle and Slip-Ring</a></li>
-<li><a href="#org98f5b12">Translation Stage</a></li>
+<li><a href="#orgc61ac0b">Spindle and Slip-Ring</a></li>
+<li><a href="#org2401436">Translation Stage</a></li>
 </ul>
 </li>
-<li><a href="#org561160c">3.4. Open Loop noise budgeting</a></li>
-<li><a href="#org669e909">3.5. Better estimation of the disturbances</a></li>
-<li><a href="#orgb70e152">3.6. Conclusion</a></li>
+<li><a href="#orgc564be1">3.4. Open Loop noise budgeting</a></li>
+<li><a href="#orge4dfb1c">3.5. Better estimation of the disturbances</a></li>
+<li><a href="#org2c65932">3.6. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org02f1102">4. Multi Body Model</a>
+<li><a href="#orgde0231c">4. Multi Body Model</a>
 <ul>
-<li><a href="#org517071d">4.1. Multi-Body model</a></li>
-<li><a href="#orgbaa6f2e">4.2. Validity of the model&rsquo;s dynamics</a></li>
-<li><a href="#org0057f81">4.3. Wanted position of the sample and position error</a></li>
-<li><a href="#org68f9b14">4.4. Simulation of Experiments</a></li>
-<li><a href="#org1945128">4.5. Conclusion</a></li>
+<li><a href="#org67778e4">4.1. Multi-Body model</a></li>
+<li><a href="#orgcc3b970">4.2. Validity of the model&rsquo;s dynamics</a></li>
+<li><a href="#orgeaffc11">4.3. Wanted position of the sample and position error</a></li>
+<li><a href="#orgba6084c">4.4. Simulation of Experiments</a></li>
+<li><a href="#orgc7d8ccb">4.5. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org30cddff">5. Optimal Nano-Hexapod Design</a>
+<li><a href="#org80c8d2b">5. Optimal Nano-Hexapod Design</a>
 <ul>
-<li><a href="#orgb0f8fc2">5.1. Optimal Stiffness to reduce the effect of disturbances</a>
+<li><a href="#org448fd3a">5.1. Optimal Stiffness to reduce the effect of disturbances</a>
 <ul>
-<li><a href="#org43274b2">Sensibility to stage vibrations</a></li>
-<li><a href="#org0f8cf88">Sensibility to ground motion</a></li>
-<li><a href="#org8248b99">Dynamic Noise Budgeting considering all the disturbances</a></li>
+<li><a href="#org32fd6fe">Sensibility to stage vibrations</a></li>
+<li><a href="#org323b4b9">Sensibility to ground motion</a></li>
+<li><a href="#org31bc407">Dynamic Noise Budgeting considering all the disturbances</a></li>
 </ul>
 </li>
-<li><a href="#org8de3012">5.2. Optimal Stiffness to reduce the plant uncertainty</a>
+<li><a href="#orga1f43fd">5.2. Optimal Stiffness to reduce the plant uncertainty</a>
 <ul>
-<li><a href="#orgda82a97">Effect of Payload</a></li>
-<li><a href="#org897b055">Effect of Micro-Station Compliance</a></li>
-<li><a href="#orgfbd7e41">Effect of Spindle Rotating Speed</a></li>
-<li><a href="#orge317967">Total Plant Uncertainty</a></li>
+<li><a href="#org4ea8b43">Effect of Payload</a></li>
+<li><a href="#orga6762b8">Effect of Micro-Station Compliance</a></li>
+<li><a href="#orga003084">Effect of Spindle Rotating Speed</a></li>
+<li><a href="#org63b95dd">Total Plant Uncertainty</a></li>
 </ul>
 </li>
-<li><a href="#org668965e">5.3. Conclusion</a></li>
+<li><a href="#orgcaa22c1">5.3. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org3b2affe">6. Robust Control Architecture</a>
+<li><a href="#org1c328ea">6. Robust Control Architecture</a>
 <ul>
-<li><a href="#org01e1894">6.1. High Authority Control / Low Authority Control Architecture</a></li>
-<li><a href="#orgc9b0f9a">6.2. Active Damping and Sensors to be included in the nano-hexapod</a></li>
-<li><a href="#org3e166d9">6.3. High Authority Control</a></li>
-<li><a href="#org10bf44e">6.4. Simulation of Tomography Experiments</a></li>
-<li><a href="#orgadf6472">6.5. Conclusion</a></li>
+<li><a href="#org45f8ce7">6.1. High Authority Control / Low Authority Control Architecture</a></li>
+<li><a href="#org6e08f63">6.2. Active Damping and Sensors to be included in the nano-hexapod</a></li>
+<li><a href="#orgb499508">6.3. High Authority Control</a></li>
+<li><a href="#org9085983">6.4. Simulation of Tomography Experiments</a></li>
+<li><a href="#org4b72da9">6.5. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org6d31a2a">7. General Conclusion and Further notes</a>
+<li><a href="#org6dfaa6a">7. General Conclusion and Further notes</a>
 <ul>
-<li><a href="#org3c26e13">7.1. General Conclusion</a></li>
-<li><a href="#orgf2dd801">7.2. Sensor Noise introduced by the Metrology</a></li>
-<li><a href="#orga032112">7.3. Further Work</a></li>
-<li><a href="#orgc79208c">7.4. Using soft mounts for the</a></li>
-<li><a href="#org2ca3a7d">7.5. Others</a></li>
+<li><a href="#org8a29acd">7.1. General Conclusion</a></li>
+<li><a href="#orgf76caba">7.2. Sensor Noise introduced by the Metrology</a></li>
+<li><a href="#org34c5e7a">7.3. Further Work</a></li>
+<li><a href="#org8b69a8a">7.4. Using soft mounts for the</a></li>
+<li><a href="#org71917ed">7.5. Others</a></li>
 </ul>
 </li>
 </ul>
@@ -136,7 +136,7 @@ This consists of a nano-hexapod and an associated control architecture that are
 
 
 <p>
-To understand the design challenges of such system, a short introduction to Feedback control is provided in Section <a href="#orga77de9c">1</a>.
+To understand the design challenges of such system, a short introduction to Feedback control is provided in Section <a href="#orgd4cf90d">1</a>.
 The mathematical tools (Power Spectral Density, Noise Budgeting, &#x2026;) that will be used throughout this study are also introduced.
 </p>
 
@@ -145,51 +145,51 @@ The mathematical tools (Power Spectral Density, Noise Budgeting, &#x2026;) that
 To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of:
 </p>
 <ul class="org-ul">
-<li>the micro-station dynamics (Section <a href="#org2385fed">2</a>)</li>
-<li>the frequency content of the sources of disturbances such as vibrations induced by the micro-station&rsquo;s stages and ground motion (Section <a href="#org1f22d62">3</a>)</li>
+<li>the micro-station dynamics (Section <a href="#org49cefbd">2</a>)</li>
+<li>the frequency content of the sources of disturbances such as vibrations induced by the micro-station&rsquo;s stages and ground motion (Section <a href="#org471db68">3</a>)</li>
 </ul>
 
 
 <p>
-A model of the micro-station is then developed and tuned using the previous estimations (Section <a href="#orgcbc55a2">4</a>).
+A model of the micro-station is then developed and tuned using the previous estimations (Section <a href="#org754fc97">4</a>).
 The nano-hexapod is further included in the model.
 </p>
 
 
 <p>
 The effects of the nano-hexapod characteristics on the system dynamics are then studied.
-Based on that, an optimal choice of the nano-hexapod stiffness is made (Section <a href="#org6fa5be0">5</a>).
+Based on that, an optimal choice of the nano-hexapod stiffness is made (Section <a href="#orgc2045d1">5</a>).
 </p>
 
 
 <p>
 Finally, using the optimally designed nano-hexapod, a robust control architecture is developed.
-Simulations are performed to show that this design gives acceptable performance and the required robustness (Section <a href="#orgf46ed74">6</a>).
+Simulations are performed to show that this design gives acceptable performance and the required robustness (Section <a href="#org22e89f5">6</a>).
 </p>
 
-<div id="outline-container-org466c91c" class="outline-2">
-<h2 id="org466c91c"><span class="section-number-2">1</span> Introduction to Feedback Systems and Noise budgeting</h2>
+<div id="outline-container-org5ebe35e" class="outline-2">
+<h2 id="org5ebe35e"><span class="section-number-2">1</span> Introduction to Feedback Systems and Noise budgeting</h2>
 <div class="outline-text-2" id="text-1">
 <p>
-<a id="orga77de9c"></a>
+<a id="orgd4cf90d"></a>
 </p>
 <p>
-In this section, some basics of <b>feedback systems</b> are first introduced (Section <a href="#org151dd27">1.1</a>).
+In this section, some basics of <b>feedback systems</b> are first introduced (Section <a href="#org9ba858e">1.1</a>).
 This should highlight the challenges of the required combined performance and robustness.
 </p>
 
 
 <p>
-In Section <a href="#orgf8c3e7b">1.2</a> is introduced the <b>dynamic error budgeting</b> which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources.
+In Section <a href="#org54b0e5a">1.2</a> is introduced the <b>dynamic error budgeting</b> which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources.
 This tool will be widely used throughout this study to both predict the performances and identify the effects that do limit the performances.
 </p>
 </div>
 
-<div id="outline-container-orgad80709" class="outline-3">
-<h3 id="orgad80709"><span class="section-number-3">1.1</span> Feedback System</h3>
+<div id="outline-container-org8eabb73" class="outline-3">
+<h3 id="org8eabb73"><span class="section-number-3">1.1</span> Feedback System</h3>
 <div class="outline-text-3" id="text-1-1">
 <p>
-<a id="org151dd27"></a>
+<a id="org9ba858e"></a>
 </p>
 <p>
 The use of Feedback control in a motion system required to use some sensors to monitor the actual status of the system and actuators to modifies this status.
@@ -224,11 +224,11 @@ Thus the <i>robustness</i> properties of the feedback system must be carefully g
 </ul>
 </div>
 
-<div id="outline-container-orge99f149" class="outline-4">
-<h4 id="orge99f149"><span class="section-number-4">1.1.1</span> Simplified Feedback Control Diagram for the NASS</h4>
+<div id="outline-container-orgea180e3" class="outline-4">
+<h4 id="orgea180e3"><span class="section-number-4">1.1.1</span> Simplified Feedback Control Diagram for the NASS</h4>
 <div class="outline-text-4" id="text-1-1-1">
 <p>
-Let&rsquo;s consider the block diagram shown in Figure <a href="#org7afb345">1</a> where the signals are:
+Let&rsquo;s consider the block diagram shown in Figure <a href="#org6391cc0">1</a> where the signals are:
 </p>
 <ul class="org-ul">
 <li>\(y\): the relative position of the sample with respect to the granite (the quantity to be controlled)</li>
@@ -248,7 +248,7 @@ The <i>dynamical</i> blocks are:
 </ul>
 
 
-<div id="org7afb345" class="figure">
+<div id="org6391cc0" class="figure">
 <p><img src="figs/classical_feedback_small.png" alt="classical_feedback_small.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>Block Diagram of a simple feedback system</p>
@@ -270,11 +270,11 @@ In the next section, we see how the use of the feedback system permits to lower
 </div>
 </div>
 
-<div id="outline-container-orgfef2aa4" class="outline-4">
-<h4 id="orgfef2aa4"><span class="section-number-4">1.1.2</span> How does the feedback loop is modifying the system behavior?</h4>
+<div id="outline-container-orgee4b57a" class="outline-4">
+<h4 id="orgee4b57a"><span class="section-number-4">1.1.2</span> How does the feedback loop is modifying the system behavior?</h4>
 <div class="outline-text-4" id="text-1-1-2">
 <p>
-If we write down the position error signal \(\epsilon = r - y\) as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\) (using the feedback diagram in Figure <a href="#org7afb345">1</a>), we obtain:
+If we write down the position error signal \(\epsilon = r - y\) as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\) (using the feedback diagram in Figure <a href="#org6391cc0">1</a>), we obtain:
 \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
 </p>
 
@@ -315,8 +315,8 @@ Ideally, we would like to design the controller \(K\) such that:
 </div>
 </div>
 
-<div id="outline-container-org7e92a2e" class="outline-4">
-<h4 id="org7e92a2e"><span class="section-number-4">1.1.3</span> Trade off: Disturbance Reduction / Noise Injection</h4>
+<div id="outline-container-org672b56a" class="outline-4">
+<h4 id="org672b56a"><span class="section-number-4">1.1.3</span> Trade off: Disturbance Reduction / Noise Injection</h4>
 <div class="outline-text-4" id="text-1-1-3">
 <p>
 We have from the definition of \(S\) and \(T\) that:
@@ -334,7 +334,7 @@ There is therefore a <b>trade-off between the disturbance rejection and the meas
 
 
 <p>
-Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure <a href="#orgcd9f2be">2</a>.
+Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure <a href="#orgf21ee4d">2</a>.
 We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on the frequency band:
 </p>
 
@@ -360,7 +360,7 @@ We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on
 </ul>
 
 
-<div id="orgcd9f2be" class="figure">
+<div id="orgf21ee4d" class="figure">
 <p><img src="figs/h-infinity-2-blocs-constrains.png" alt="h-infinity-2-blocs-constrains.png" />
 </p>
 <p><span class="figure-number">Figure 2: </span>Typical shapes and constrain of the Sensibility and Transmibility closed-loop transfer functions</p>
@@ -368,11 +368,11 @@ We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on
 </div>
 </div>
 
-<div id="outline-container-orgfbb078e" class="outline-4">
-<h4 id="orgfbb078e"><span class="section-number-4">1.1.4</span> Trade off: Robustness / Performance</h4>
+<div id="outline-container-org24cd2f0" class="outline-4">
+<h4 id="org24cd2f0"><span class="section-number-4">1.1.4</span> Trade off: Robustness / Performance</h4>
 <div class="outline-text-4" id="text-1-1-4">
 <p>
-<a id="org8f6201f"></a>
+<a id="org4392511"></a>
 </p>
 
 <p>
@@ -393,11 +393,11 @@ The main issue it that for stability reasons, <b>the system dynamics must be kno
 </p>
 
 <p>
-For mechanical systems, this generally means that the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure <a href="#orgb229d23">3</a>).
+For mechanical systems, this generally means that the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure <a href="#org789fe82">3</a>).
 </p>
 
 
-<div id="orgb229d23" class="figure">
+<div id="org789fe82" class="figure">
 <p><img src="figs/oomen18_next_gen_loop_gain.png" alt="oomen18_next_gen_loop_gain.png" />
 </p>
 <p><span class="figure-number">Figure 3: </span>Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth. <a class='org-ref-reference' href="#oomen18_advan_motion_contr_precis_mechat">oomen18_advan_motion_contr_precis_mechat</a></p>
@@ -427,11 +427,11 @@ This problem of <b>robustness</b> represent one of the main challenge for the de
 </div>
 </div>
 
-<div id="outline-container-orgb6996cc" class="outline-3">
-<h3 id="orgb6996cc"><span class="section-number-3">1.2</span> Dynamic error budgeting</h3>
+<div id="outline-container-orgcd11903" class="outline-3">
+<h3 id="orgcd11903"><span class="section-number-3">1.2</span> Dynamic error budgeting</h3>
 <div class="outline-text-3" id="text-1-2">
 <p>
-<a id="orgf8c3e7b"></a>
+<a id="org54b0e5a"></a>
 </p>
 <p>
 The dynamic error budgeting is a powerful tool to study the effects of multiple error sources (i.e. disturbances and measurement noise) and to predict how much these effects are reduced by a feedback system.
@@ -442,19 +442,19 @@ The dynamic error budgeting uses two important mathematical functions: the <b>Po
 </p>
 
 <p>
-After these two functions are introduced (in Sections <a href="#orga211787">1.2.1</a> and <a href="#org45892a4">1.2.2</a>), is shown how do multiple error sources are combined and modified by dynamical systems (in Section <a href="#org56d9109">1.2.3</a> and <a href="#org8958979">1.2.4</a>).
+After these two functions are introduced (in Sections <a href="#orgb370b52">1.2.1</a> and <a href="#org5065e5f">1.2.2</a>), is shown how do multiple error sources are combined and modified by dynamical systems (in Section <a href="#org6726f61">1.2.3</a> and <a href="#org6d22498">1.2.4</a>).
 </p>
 
 <p>
-Finally, the dynamic noise budgeting for the NASS is derived in Section <a href="#org57a0176">1.2.5</a>.
+Finally, the dynamic noise budgeting for the NASS is derived in Section <a href="#org11ab089">1.2.5</a>.
 </p>
 </div>
 
-<div id="outline-container-orga061b8d" class="outline-4">
-<h4 id="orga061b8d"><span class="section-number-4">1.2.1</span> Power Spectral Density</h4>
+<div id="outline-container-org3320434" class="outline-4">
+<h4 id="org3320434"><span class="section-number-4">1.2.1</span> Power Spectral Density</h4>
 <div class="outline-text-4" id="text-1-2-1">
 <p>
-<a id="orga211787"></a>
+<a id="orgb370b52"></a>
 </p>
 
 <p>
@@ -483,11 +483,11 @@ One can also integrate the infinitesimal power \(S_{xx}(\omega)d\omega\) over a
 </div>
 </div>
 
-<div id="outline-container-orgd87885d" class="outline-4">
-<h4 id="orgd87885d"><span class="section-number-4">1.2.2</span> Cumulative Power Spectrum</h4>
+<div id="outline-container-org129133e" class="outline-4">
+<h4 id="org129133e"><span class="section-number-4">1.2.2</span> Cumulative Power Spectrum</h4>
 <div class="outline-text-4" id="text-1-2-2">
 <p>
-<a id="org45892a4"></a>
+<a id="org5065e5f"></a>
 </p>
 
 <p>
@@ -521,19 +521,19 @@ The Cumulative Power Spectrum is generally shown as a function of frequency, and
 </div>
 </div>
 
-<div id="outline-container-org30dd763" class="outline-4">
-<h4 id="org30dd763"><span class="section-number-4">1.2.3</span> Modification of a signal&rsquo;s PSD when going through a dynamical system</h4>
+<div id="outline-container-org5472bba" class="outline-4">
+<h4 id="org5472bba"><span class="section-number-4">1.2.3</span> Modification of a signal&rsquo;s PSD when going through a dynamical system</h4>
 <div class="outline-text-4" id="text-1-2-3">
 <p>
-<a id="org56d9109"></a>
+<a id="org6726f61"></a>
 </p>
 
 <p>
-Let&rsquo;s consider a signal \(u\) with a PSD \(S_{uu}\) going through a LTI system \(G(s)\) that outputs a signal \(y\) with a PSD (Figure <a href="#orgf820166">4</a>).
+Let&rsquo;s consider a signal \(u\) with a PSD \(S_{uu}\) going through a LTI system \(G(s)\) that outputs a signal \(y\) with a PSD (Figure <a href="#org03d1473">4</a>).
 </p>
 
 
-<div id="orgf820166" class="figure">
+<div id="org03d1473" class="figure">
 <p><img src="figs/psd_lti_system.png" alt="psd_lti_system.png" />
 </p>
 <p><span class="figure-number">Figure 4: </span>LTI dynamical system \(G(s)\) with input signal \(u\) and output signal \(y\)</p>
@@ -548,15 +548,15 @@ The Power Spectral Density of the output signal \(y\) can be computed using:
 </div>
 </div>
 
-<div id="outline-container-org2273db0" class="outline-4">
-<h4 id="org2273db0"><span class="section-number-4">1.2.4</span> PSD of combined signals</h4>
+<div id="outline-container-orgbbc41e7" class="outline-4">
+<h4 id="orgbbc41e7"><span class="section-number-4">1.2.4</span> PSD of combined signals</h4>
 <div class="outline-text-4" id="text-1-2-4">
 <p>
-<a id="org8958979"></a>
+<a id="org6d22498"></a>
 </p>
 
 <p>
-Let&rsquo;s consider a signal \(y\) that is the sum of two <b>uncorrelated</b> signals \(u\) and \(v\) (Figure <a href="#org09cf203">5</a>).
+Let&rsquo;s consider a signal \(y\) that is the sum of two <b>uncorrelated</b> signals \(u\) and \(v\) (Figure <a href="#orgfcba80c">5</a>).
 </p>
 
 <p>
@@ -565,7 +565,7 @@ We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD o
 </p>
 
 
-<div id="org09cf203" class="figure">
+<div id="orgfcba80c" class="figure">
 <p><img src="figs/psd_sum.png" alt="psd_sum.png" />
 </p>
 <p><span class="figure-number">Figure 5: </span>\(y\) as the sum of two signals \(u\) and \(v\)</p>
@@ -573,15 +573,15 @@ We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD o
 </div>
 </div>
 
-<div id="outline-container-orga3546fb" class="outline-4">
-<h4 id="orga3546fb"><span class="section-number-4">1.2.5</span> Dynamic Noise Budgeting</h4>
+<div id="outline-container-org454c2e9" class="outline-4">
+<h4 id="org454c2e9"><span class="section-number-4">1.2.5</span> Dynamic Noise Budgeting</h4>
 <div class="outline-text-4" id="text-1-2-5">
 <p>
-<a id="org57a0176"></a>
+<a id="org11ab089"></a>
 </p>
 
 <p>
-Let&rsquo;s consider the Feedback architecture in Figure <a href="#org7afb345">1</a> where the position error \(\epsilon\) is equal to:
+Let&rsquo;s consider the Feedback architecture in Figure <a href="#org6391cc0">1</a> where the position error \(\epsilon\) is equal to:
 \[ \epsilon = S r + T n - G_d S d \]
 </p>
 
@@ -605,25 +605,25 @@ To estimate the PSD of the position error \(\epsilon\) and thus the RMS residual
 <ul class="org-ul">
 <li>The Power Spectral Densities of the signals affecting the system:
 <ul class="org-ul">
-<li>The disturbances \(S_{dd}\): this will be done in Section <a href="#org1f22d62">3</a></li>
+<li>The disturbances \(S_{dd}\): this will be done in Section <a href="#org471db68">3</a></li>
 <li>The sensor noise \(S_{nn}\): this can be estimated from the sensor data-sheet</li>
 <li>The wanted sample&rsquo;s motion \(S_{rr}\): this is a deterministic signal that we choose.
 For a simple tomography experiment, we can consider that it is equal to \(0\) as we only want to compensate all the sample&rsquo;s vibrations</li>
 </ul></li>
 <li>The dynamics of the complete system comprising the micro-station and the nano-hexapod: \(G\), \(G_d\).
-To do so, we need to identify the dynamics of the micro-station (Section <a href="#org2385fed">2</a>), include this dynamics in a model (Section <a href="#orgcbc55a2">4</a>) and add a model of the nano-hexapod to the model (Section <a href="#org6fa5be0">5</a>)</li>
-<li>The controller \(K\) that will be designed in Section <a href="#orgf46ed74">6</a></li>
+To do so, we need to identify the dynamics of the micro-station (Section <a href="#org49cefbd">2</a>), include this dynamics in a model (Section <a href="#org754fc97">4</a>) and add a model of the nano-hexapod to the model (Section <a href="#orgc2045d1">5</a>)</li>
+<li>The controller \(K\) that will be designed in Section <a href="#org22e89f5">6</a></li>
 </ul>
 </div>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org4920a6e" class="outline-2">
-<h2 id="org4920a6e"><span class="section-number-2">2</span> Identification of the Micro-Station Dynamics</h2>
+<div id="outline-container-org98142af" class="outline-2">
+<h2 id="org98142af"><span class="section-number-2">2</span> Identification of the Micro-Station Dynamics</h2>
 <div class="outline-text-2" id="text-2">
 <p>
-<a id="org2385fed"></a>
+<a id="org49cefbd"></a>
 </p>
 <p>
 As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be used:
@@ -641,7 +641,7 @@ All the measurements performed on the micro-station are detailed in <a href="htt
 
 
 <p>
-The general procedure to identify the dynamics of the micro-station is shown in Figure <a href="#org61b1b58">6</a>.
+The general procedure to identify the dynamics of the micro-station is shown in Figure <a href="#org48f93a0">6</a>.
 The steps are:
 </p>
 <ol class="org-ol">
@@ -651,7 +651,7 @@ The steps are:
 </ol>
 
 
-<div id="org61b1b58" class="figure">
+<div id="org48f93a0" class="figure">
 <p><img src="figs/vibration_analysis_procedure.png" alt="vibration_analysis_procedure.png" />
 </p>
 <p><span class="figure-number">Figure 6: </span>Vibration Analysis Procedure</p>
@@ -663,11 +663,11 @@ Instead, the model will be tuned using both the modal model and the response mod
 </p>
 </div>
 
-<div id="outline-container-org4edd614" class="outline-3">
-<h3 id="org4edd614"><span class="section-number-3">2.1</span> Experimental Setup</h3>
+<div id="outline-container-org8c825c9" class="outline-3">
+<h3 id="org8c825c9"><span class="section-number-3">2.1</span> Experimental Setup</h3>
 <div class="outline-text-3" id="text-2-1">
 <p>
-<a id="org60ca63d"></a>
+<a id="org6648e76"></a>
 </p>
 
 <p>
@@ -693,13 +693,13 @@ In order to perform the modal analysis, the following devices were used:
 The measurement consists of:
 </p>
 <ul class="org-ul">
-<li>Exciting the structure at the same location with the instrumented hammer (Figure <a href="#orgae94e60">7</a>)</li>
+<li>Exciting the structure at the same location with the instrumented hammer (Figure <a href="#orge1bae07">7</a>)</li>
 <li>Fix the accelerometers on each of the stages to measure all the DOF of the structure.
 The position of the accelerometers are:
 <ul class="org-ul">
 <li>4 on the first granite</li>
 <li>4 on the second granite</li>
-<li>4 on top of the translation stage (Figure <a href="#org22f46a3">8</a>)</li>
+<li>4 on top of the translation stage (Figure <a href="#org8b8f9bd">8</a>)</li>
 <li>4 on top of the tilt stage</li>
 <li>3 on top of the spindle</li>
 <li>4 on top of the hexapod</li>
@@ -715,14 +715,14 @@ It was chosen to have some redundancy in the measurement to be able to verify th
 </p>
 
 
-<div id="orgae94e60" class="figure">
+<div id="orge1bae07" class="figure">
 <p><img src="figs/hammer_z.gif" alt="hammer_z.gif" />
 </p>
 <p><span class="figure-number">Figure 7: </span>Example of one hammer impact</p>
 </div>
 
 
-<div id="org22f46a3" class="figure">
+<div id="org8b8f9bd" class="figure">
 <p><img src="figs/accelerometers_ty_overview.jpg" alt="accelerometers_ty_overview.jpg" />
 </p>
 <p><span class="figure-number">Figure 8: </span>3 tri axis accelerometers fixed to the translation stage</p>
@@ -730,11 +730,11 @@ It was chosen to have some redundancy in the measurement to be able to verify th
 </div>
 </div>
 
-<div id="outline-container-orgf6e05b4" class="outline-3">
-<h3 id="orgf6e05b4"><span class="section-number-3">2.2</span> Results</h3>
+<div id="outline-container-orgd31b11c" class="outline-3">
+<h3 id="orgd31b11c"><span class="section-number-3">2.2</span> Results</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
-<a id="orgf4283df"></a>
+<a id="orgfd29a2a"></a>
 </p>
 
 <p>
@@ -743,18 +743,18 @@ From the measurements are extracted all the transfer functions from forces appli
 
 <p>
 Modal shapes and natural frequencies are then computed.
-Example of the obtained micro-station&rsquo;s mode shapes are shown in Figures <a href="#org5f8825e">9</a> and <a href="#orgd8eea98">10</a>.
+Example of the obtained micro-station&rsquo;s mode shapes are shown in Figures <a href="#orgc8f5c69">9</a> and <a href="#orgf201e60">10</a>.
 </p>
 
 
-<div id="org5f8825e" class="figure">
+<div id="orgc8f5c69" class="figure">
 <p><img src="figs/mode1.gif" alt="mode1.gif" />
 </p>
 <p><span class="figure-number">Figure 9: </span>First mode that shows a suspension mode, probably due to bad leveling of one Airloc</p>
 </div>
 
 
-<div id="orgd8eea98" class="figure">
+<div id="orgf201e60" class="figure">
 <p><img src="figs/mode6.gif" alt="mode6.gif" />
 </p>
 <p><span class="figure-number">Figure 10: </span>Sixth mode</p>
@@ -782,12 +782,12 @@ This thus means that <b>a multi-body model can be used to correctly represent th
 
 <p>
 Many Frequency Response Functions (FRF) are obtained from the measurements.
-Examples of FRF are shown in Figure <a href="#orge6a4117">11</a>.
+Examples of FRF are shown in Figure <a href="#orgfefaa82">11</a>.
 These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.
 </p>
 
 
-<div id="orge6a4117" class="figure">
+<div id="orgfefaa82" class="figure">
 <p><img src="figs/frf_all_bodies_one_direction.png" alt="frf_all_bodies_one_direction.png" />
 </p>
 <p><span class="figure-number">Figure 11: </span>Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction</p>
@@ -795,8 +795,8 @@ These FRF will be used to compare the dynamics of the multi-body model with the
 </div>
 </div>
 
-<div id="outline-container-orge3cb290" class="outline-3">
-<h3 id="orge3cb290"><span class="section-number-3">2.3</span> Conclusion</h3>
+<div id="outline-container-org72bfef9" class="outline-3">
+<h3 id="org72bfef9"><span class="section-number-3">2.3</span> Conclusion</h3>
 <div class="outline-text-3" id="text-2-3">
 <div class="important">
 <p>
@@ -804,7 +804,7 @@ The dynamical measurements made on the micro-station confirmed the fact that a m
 </p>
 
 <p>
-In Section <a href="#orgcbc55a2">4</a>, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
+In Section <a href="#org754fc97">4</a>, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
 </p>
 
 </div>
@@ -812,11 +812,11 @@ In Section <a href="#orgcbc55a2">4</a>, the obtained Frequency Response Function
 </div>
 </div>
 
-<div id="outline-container-org8c43d49" class="outline-2">
-<h2 id="org8c43d49"><span class="section-number-2">3</span> Identification of the Disturbances</h2>
+<div id="outline-container-org726da55" class="outline-2">
+<h2 id="org726da55"><span class="section-number-2">3</span> Identification of the Disturbances</h2>
 <div class="outline-text-2" id="text-3">
 <p>
-<a id="org1f22d62"></a>
+<a id="org471db68"></a>
 </p>
 <p>
 In this section, all the disturbances affecting the system are identified and quantified.
@@ -830,13 +830,13 @@ Note that the low frequency disturbances such as static guiding errors and therm
 The main challenge is to reduce the disturbances containing high frequencies, and thus efforts are made to identify these high frequency disturbances such as:
 </p>
 <ul class="org-ul">
-<li>Ground motion (Section <a href="#org8c2cb84">3.1</a>)</li>
-<li>Vibration introduced by control systems (Section <a href="#org6645da4">3.2</a>)</li>
-<li>Vibration introduced by the motion of the spindle and of the translation stage (Section <a href="#orgdba1aaa">3.3</a>)</li>
+<li>Ground motion (Section <a href="#orgaa9b6aa">3.1</a>)</li>
+<li>Vibration introduced by control systems (Section <a href="#org46dcf85">3.2</a>)</li>
+<li>Vibration introduced by the motion of the spindle and of the translation stage (Section <a href="#org59015f5">3.3</a>)</li>
 </ul>
 
 <p>
-A noise budgeting is performed in Section <a href="#orgf262191">3.4</a>, the vibrations induced by the disturbances are compared and the required control bandwidth is estimated.
+A noise budgeting is performed in Section <a href="#org1cf32f4">3.4</a>, the vibrations induced by the disturbances are compared and the required control bandwidth is estimated.
 </p>
 
 <p>
@@ -844,11 +844,11 @@ The measurements are presented in more detail in <a href="https://tdehaeze.githu
 </p>
 </div>
 
-<div id="outline-container-orgb1af52d" class="outline-3">
-<h3 id="orgb1af52d"><span class="section-number-3">3.1</span> Ground Motion</h3>
+<div id="outline-container-org0887051" class="outline-3">
+<h3 id="org0887051"><span class="section-number-3">3.1</span> Ground Motion</h3>
 <div class="outline-text-3" id="text-3-1">
 <p>
-<a id="org8c2cb84"></a>
+<a id="orgaa9b6aa"></a>
 </p>
 
 <p>
@@ -856,12 +856,12 @@ Ground motion can easily be estimated using an inertial sensor with sufficient s
 </p>
 
 <p>
-To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure <a href="#org1235686">12</a>).
+To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure <a href="#orge70b12c">12</a>).
 The details of the Huddle Test can be found <a href="https://tdehaeze.github.io/meas-analysis/huddle-test-geophones/index.html">here</a>.
 </p>
 
 
-<div id="org1235686" class="figure">
+<div id="orge70b12c" class="figure">
 <p><img src="figs/geophones.jpg" alt="geophones.jpg" />
 </p>
 <p><span class="figure-number">Figure 12: </span>Huddle Test Setup</p>
@@ -873,7 +873,7 @@ The low frequency differences between the ground motion at ID31 and ID09 is just
 </p>
 
 
-<div id="org7376b94" class="figure">
+<div id="org17c776b" class="figure">
 <p><img src="figs/ground_motion_compare.png" alt="ground_motion_compare.png" />
 </p>
 <p><span class="figure-number">Figure 13: </span>Comparison of the PSD of the ground motion measured at different location</p>
@@ -881,11 +881,11 @@ The low frequency differences between the ground motion at ID31 and ID09 is just
 </div>
 </div>
 
-<div id="outline-container-org9284950" class="outline-3">
-<h3 id="org9284950"><span class="section-number-3">3.2</span> Stage Vibration - Effect of Control systems</h3>
+<div id="outline-container-org4359dc1" class="outline-3">
+<h3 id="org4359dc1"><span class="section-number-3">3.2</span> Stage Vibration - Effect of Control systems</h3>
 <div class="outline-text-3" id="text-3-2">
 <p>
-<a id="org6645da4"></a>
+<a id="org46dcf85"></a>
 </p>
 
 <p>
@@ -908,11 +908,11 @@ Complete reports on these measurements are accessible <a href="https://tdehaeze.
 </div>
 </div>
 
-<div id="outline-container-org482cc38" class="outline-3">
-<h3 id="org482cc38"><span class="section-number-3">3.3</span> Stage Vibration - Effect of Motion</h3>
+<div id="outline-container-orgf5d9e59" class="outline-3">
+<h3 id="orgf5d9e59"><span class="section-number-3">3.3</span> Stage Vibration - Effect of Motion</h3>
 <div class="outline-text-3" id="text-3-3">
 <p>
-<a id="orgdba1aaa"></a>
+<a id="org59015f5"></a>
 </p>
 <p>
 We consider here the vibrations induced by <b>scans of the translation stage</b> and <b>rotation of the spindle</b>.
@@ -923,15 +923,15 @@ Details reports are accessible <a href="https://tdehaeze.github.io/meas-analysis
 </p>
 </div>
 
-<div id="outline-container-org1af435f" class="outline-4">
-<h4 id="org1af435f">Spindle and Slip-Ring</h4>
-<div class="outline-text-4" id="text-org1af435f">
+<div id="outline-container-orgc61ac0b" class="outline-4">
+<h4 id="orgc61ac0b">Spindle and Slip-Ring</h4>
+<div class="outline-text-4" id="text-orgc61ac0b">
 <p>
-The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure <a href="#org73f4a87">14</a>.
+The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure <a href="#orgcb1088b">14</a>.
 </p>
 
 
-<div id="org73f4a87" class="figure">
+<div id="orgcb1088b" class="figure">
 <p><img src="figs/rz_meas_errors.gif" alt="rz_meas_errors.gif" />
 </p>
 <p><span class="figure-number">Figure 14: </span>Measurement of the sample&rsquo;s vertical motion when rotating at 6rpm</p>
@@ -947,7 +947,7 @@ A geophone is fixed at the location of the sample and the motion is measured:
 </ul>
 
 <p>
-The obtained Power Spectral Densities of the sample&rsquo;s absolute velocity are shown in Figure <a href="#org03b427f">15</a>.
+The obtained Power Spectral Densities of the sample&rsquo;s absolute velocity are shown in Figure <a href="#org6630afb">15</a>.
 </p>
 
 <p>
@@ -964,7 +964,7 @@ Its cause has not been identified yet</li>
 </ul>
 
 
-<div id="org03b427f" class="figure">
+<div id="org6630afb" class="figure">
 <p><img src="figs/sr_sp_psd_sample_compare.png" alt="sr_sp_psd_sample_compare.png" />
 </p>
 <p><span class="figure-number">Figure 15: </span>Comparison of the ASD of the measured voltage from the Geophone at the sample location</p>
@@ -979,19 +979,19 @@ Some investigation should be performed to determine where does this 23Hz motion
 </div>
 </div>
 
-<div id="outline-container-org98f5b12" class="outline-4">
-<h4 id="org98f5b12">Translation Stage</h4>
-<div class="outline-text-4" id="text-org98f5b12">
+<div id="outline-container-org2401436" class="outline-4">
+<h4 id="org2401436">Translation Stage</h4>
+<div class="outline-text-4" id="text-org2401436">
 <p>
 The same setup is used: a geophone is located at the sample&rsquo;s location and another on the granite.
 </p>
 
 <p>
-A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translation stage (Figure <a href="#orge4e7a2d">16</a>), and the absolute velocities of the sample and the granite are measured.
+A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translation stage (Figure <a href="#orgdfc7637">16</a>), and the absolute velocities of the sample and the granite are measured.
 </p>
 
 
-<div id="orge4e7a2d" class="figure">
+<div id="orgdfc7637" class="figure">
 <p><img src="figs/ty_position_time.png" alt="ty_position_time.png" />
 </p>
 <p><span class="figure-number">Figure 16: </span>Y position of the translation stage measured by the encoders</p>
@@ -999,20 +999,20 @@ A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translat
 
 
 <p>
-The time domain absolute vertical velocity of the sample and granite are shown in Figure <a href="#org37494a8">17</a>.
+The time domain absolute vertical velocity of the sample and granite are shown in Figure <a href="#org0b60a4d">17</a>.
 It is shown that quite large motion of the granite is induced by the translation stage scans.
 This could be a problem if this is shown to excite the metrology frame of the nano-focusing lens position stage.
 </p>
 
 
-<div id="org37494a8" class="figure">
+<div id="org0b60a4d" class="figure">
 <p><img src="figs/ty_z_time.png" alt="ty_z_time.png" />
 </p>
 <p><span class="figure-number">Figure 17: </span>Vertical velocity of the sample and marble when scanning with the translation stage</p>
 </div>
 
 <p>
-The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure <a href="#orge750ee1">18</a>.
+The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure <a href="#orgdf17aa7">18</a>.
 The ASD contains any peaks starting from 1Hz showing the large spectral content of the motion which is probably due to the triangular reference of the translation stage.
 </p>
 
@@ -1029,7 +1029,7 @@ Thus, if the detector is only used in between the triangular peaks, the vibratio
 </div>
 
 
-<div id="orge750ee1" class="figure">
+<div id="orgdf17aa7" class="figure">
 <p><img src="figs/asd_z_direction.png" alt="asd_z_direction.png" />
 </p>
 <p><span class="figure-number">Figure 18: </span>Amplitude spectral density of the measure velocity corresponding to the geophone in the vertical direction located on the granite and at the sample location when the translation stage is scanning at 1Hz</p>
@@ -1038,11 +1038,11 @@ Thus, if the detector is only used in between the triangular peaks, the vibratio
 </div>
 </div>
 
-<div id="outline-container-org561160c" class="outline-3">
-<h3 id="org561160c"><span class="section-number-3">3.4</span> Open Loop noise budgeting</h3>
+<div id="outline-container-orgc564be1" class="outline-3">
+<h3 id="orgc564be1"><span class="section-number-3">3.4</span> Open Loop noise budgeting</h3>
 <div class="outline-text-3" id="text-3-4">
 <p>
-<a id="orgf262191"></a>
+<a id="org1cf32f4"></a>
 </p>
 
 <p>
@@ -1050,7 +1050,7 @@ We can now compare the effect of all the disturbance sources on the position err
 </p>
 
 <p>
-The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure <a href="#orgb03d2e4">19</a>.
+The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure <a href="#org2ab0c8e">19</a>.
 </p>
 
 <p>
@@ -1058,26 +1058,26 @@ We can see that the ground motion is quite small compare to the translation stag
 </p>
 
 
-<div id="orgb03d2e4" class="figure">
+<div id="org2ab0c8e" class="figure">
 <p><img src="figs/dist_effect_relative_motion.png" alt="dist_effect_relative_motion.png" />
 </p>
 <p><span class="figure-number">Figure 19: </span>Amplitude Spectral Density fo the motion error due to disturbances</p>
 </div>
 
 <p>
-The Cumulative Amplitude Spectrum is shown in Figure <a href="#org3ebc44b">20</a>.
+The Cumulative Amplitude Spectrum is shown in Figure <a href="#orgaa03026">20</a>.
 It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range for frequencies above 1Hz.
 </p>
 
 
-<div id="org3ebc44b" class="figure">
+<div id="orgaa03026" class="figure">
 <p><img src="figs/dist_effect_relative_motion_cas.png" alt="dist_effect_relative_motion_cas.png" />
 </p>
 <p><span class="figure-number">Figure 20: </span>Cumulative Amplitude Spectrum of the motion error due to disturbances</p>
 </div>
 
 <p>
-From Figure <a href="#org3ebc44b">20</a>, required bandwidth can be estimated by seeing that \(10\ nm [rms]\) motion is induced by the perturbations above 100Hz.
+From Figure <a href="#orgaa03026">20</a>, required bandwidth can be estimated by seeing that \(10\ nm [rms]\) motion is induced by the perturbations above 100Hz.
 </p>
 
 <p>
@@ -1090,8 +1090,8 @@ From that, it can be concluded that control bandwidth will have to be around 100
 </div>
 </div>
 
-<div id="outline-container-org669e909" class="outline-3">
-<h3 id="org669e909"><span class="section-number-3">3.5</span> Better estimation of the disturbances</h3>
+<div id="outline-container-orge4dfb1c" class="outline-3">
+<h3 id="orge4dfb1c"><span class="section-number-3">3.5</span> Better estimation of the disturbances</h3>
 <div class="outline-text-3" id="text-3-5">
 <p>
 All the disturbance measurements were made with inertial sensors, and to obtain the relative motion sample/granite, two inertial sensors were used and the signals were subtracted.
@@ -1111,8 +1111,8 @@ The detector requirement would need to have a sample frequency above \(400Hz\) a
 </div>
 </div>
 
-<div id="outline-container-orgb70e152" class="outline-3">
-<h3 id="orgb70e152"><span class="section-number-3">3.6</span> Conclusion</h3>
+<div id="outline-container-org2c65932" class="outline-3">
+<h3 id="org2c65932"><span class="section-number-3">3.6</span> Conclusion</h3>
 <div class="outline-text-3" id="text-3-6">
 <div class="important">
 <p>
@@ -1137,44 +1137,59 @@ This should however not change the conclusion of this study nor significantly ch
 </div>
 </div>
 
-<div id="outline-container-org02f1102" class="outline-2">
-<h2 id="org02f1102"><span class="section-number-2">4</span> Multi Body Model</h2>
+<div id="outline-container-orgde0231c" class="outline-2">
+<h2 id="orgde0231c"><span class="section-number-2">4</span> Multi Body Model</h2>
 <div class="outline-text-2" id="text-4">
 <p>
-<a id="orgcbc55a2"></a>
+<a id="org754fc97"></a>
 </p>
 <p>
-As was shown during the modal analysis (Section <a href="#org2385fed">2</a>), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies.
+As was shown during the modal analysis (Section <a href="#org49cefbd">2</a>), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) connected with some discrete flexibility (stiffnesses and dampers).
 </p>
 
 <p>
-Thus, a <b>multi-body model</b> is perfect to represent such dynamics.
+Thus, <b>a multi-body model is perfectly adapted to represent the dynamics of the micro-station</b>.
 </p>
 
 <p>
-To do so, we use the Matlab&rsquo;s <a href="https://www.mathworks.com/products/simscape.html">Simscape</a> toolbox.
+The Matlab&rsquo;s <a href="https://www.mathworks.com/products/simscape.html">Simscape</a> toolbox is used to develop the multi-body model.
 A small summary of the multi-body Simscape is available <a href="https://tdehaeze.github.io/nass-simscape/simscape.html">here</a> and each of the modeled stage is described <a href="https://tdehaeze.github.io/nass-simscape/simscape_subsystems.html">here</a>.
 </p>
 </div>
 
-<div id="outline-container-org517071d" class="outline-3">
-<h3 id="org517071d"><span class="section-number-3">4.1</span> Multi-Body model</h3>
+<div id="outline-container-org67778e4" class="outline-3">
+<h3 id="org67778e4"><span class="section-number-3">4.1</span> Multi-Body model</h3>
 <div class="outline-text-3" id="text-4-1">
+<p>
+<a id="org4240743"></a>
+</p>
+
+<p>
+The parameters to tune the dynamics of the multi body are:
+</p>
+<ul class="org-ul">
+<li>the mass/moment of inertia of each of the solid bodies</li>
+<li>the 6 stiffnesses and 6 damping properties representing each of the the mechanical guiding between two solid bodies</li>
+</ul>
+
+
 <p>
 The mass/inertia of each stage is automatically computed from the imported geometry and the material&rsquo;s density.
 </p>
 
-<p>
-The (6dof) stiffness between two solid bodies is first guessed from either measurements of data-sheets.
-Then, the values of the stiffness and damping of each joint is manually tuned until the obtained dynamics is sufficiently close to the measured dynamics.
-</p>
 
 <p>
-The 3D representation of the simscape model is shown in Figure <a href="#org6d479c1">21</a>.
+The stiffnesses between two solid bodies is first guessed from either measurements of data-sheets.
+Then, the values of the stiffnesses and damping properties of each joint is manually tuned until the obtained dynamics is sufficiently close to the measured dynamics.
 </p>
 
 
-<div id="org6d479c1" class="figure">
+<p>
+The 3D representation of the simscape model is shown in Figure <a href="#org9adaa7f">21</a>.
+</p>
+
+
+<div id="org9adaa7f" class="figure">
 <p><img src="figs/simscape_picture.png" alt="simscape_picture.png" />
 </p>
 <p><span class="figure-number">Figure 21: </span>3D representation of the simscape model</p>
@@ -1182,15 +1197,19 @@ The 3D representation of the simscape model is shown in Figure <a href="#org6d47
 </div>
 </div>
 
-<div id="outline-container-orgbaa6f2e" class="outline-3">
-<h3 id="orgbaa6f2e"><span class="section-number-3">4.2</span> Validity of the model&rsquo;s dynamics</h3>
+<div id="outline-container-orgcc3b970" class="outline-3">
+<h3 id="orgcc3b970"><span class="section-number-3">4.2</span> Validity of the model&rsquo;s dynamics</h3>
 <div class="outline-text-3" id="text-4-2">
 <p>
-It is very difficult the tune the dynamics of such model as there are more than 50 parameters and many curves to compare between the model and the measurements.
+<a id="orgf4ffbea"></a>
 </p>
 
 <p>
-The comparison of three of the Frequency Response Functions are shown in Figure <a href="#org752342e">22</a>.
+Tuning the dynamics of such model is very difficult as there are more than 50 parameters to tune and many different dynamics to compare between the model and the measurements.
+</p>
+
+<p>
+The comparison of three of the Frequency Response Functions are shown in Figure <a href="#org21ba1fd">22</a>.
 </p>
 
 <p>
@@ -1202,7 +1221,7 @@ We believe that the model is representing the micro-station dynamics with suffic
 </p>
 
 
-<div id="org752342e" class="figure">
+<div id="org21ba1fd" class="figure">
 <p><img src="figs/identification_comp_top_stages.png" alt="identification_comp_top_stages.png" />
 </p>
 <p><span class="figure-number">Figure 22: </span>Frequency Response function from Hammer force in the X,Y and Z directions to the X,Y and Z displacements of the micro-hexapod&rsquo;s top platform. The measurements are shown in blue and the Model in red.</p>
@@ -1229,40 +1248,39 @@ Then, using the model, we can
 <ul class="org-ul">
 <li>perform simulation of experiments in presence of disturbances</li>
 <li>measure the motion of the solid-bodies</li>
-<li>identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod and control design</li>
-<li>include a multi-body model of the nano-hexapod and closed-loop simulations</li>
+<li>identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod design and the control synthesis</li>
+<li>include a multi-body model of the nano-hexapod and perform closed-loop simulations</li>
 </ul>
 </div>
 </div>
 
-<div id="outline-container-org0057f81" class="outline-3">
-<h3 id="org0057f81"><span class="section-number-3">4.3</span> Wanted position of the sample and position error</h3>
+<div id="outline-container-orgeaffc11" class="outline-3">
+<h3 id="orgeaffc11"><span class="section-number-3">4.3</span> Wanted position of the sample and position error</h3>
 <div class="outline-text-3" id="text-4-3">
 <p>
-<a id="org82c65f0"></a>
+<a id="org92c9b7d"></a>
 </p>
 
 <p>
-For the control of the nano-hexapod, we need to now the sample position error (the motion to be compensated) in the frame of the nano-hexapod.
+For the control of the nano-hexapod, the sample position error (the motion to be compensated) in the frame of the nano-hexapod needs to be computed.
 </p>
 
 <p>
-To do so, we need to perform several computations (summarized in Figure <a href="#orgb875b49">23</a>):
+To do so, several computations are performed (summarized in Figure <a href="#org39b35d8">23</a>):
 </p>
 <ul class="org-ul">
-<li>First, we need to determine the actual <b>wanted pose</b> (3 translations and 3 rotations) of the sample with respect to the granite.
-This is determined from the wanted motion of each micro-station stage.
-Each wanted stage motion is represented by an homogeneous transformation matrix (explain <a href="http://planning.cs.uiuc.edu/node111.html">here</a>), then these matrices are combined to give to total wanted motion of the sample with respect to the granite.</li>
-<li>Then, we need to determine the <b>actual pose</b> of the sample with respect to the granite.
-This will be performed by several interferometers and several computation will be required to compute the pose of the sample from the interferometers measurements.
-However we here directly measure the 3 translations and 3 rotations of the sample using a special simscape block.</li>
-<li>Finally, we need to compare the wanted pose with the measured pose to compute the position error of the sample.
+<li>First, the <b>wanted pose</b> (3 translations and 3 rotations) of the sample with respect to the granite is computed.
+This is determined from the wanted motion of each micro-station stage: each wanted stage motion is represented by a homogeneous transformation matrix that are combined to give to total wanted motion of the sample with respect to the granite</li>
+<li>Then, the <b>actual pose</b> of the sample with respect to the granite is computed.
+For the real system, this will require the use of several interferometers and computations to obtain the sample&rsquo;s pose from the individual measurements.
+However, the pose of the sample with respect to the granite is directly measured using a special simscape block</li>
+<li>Finally, the wanted pose is compared with the measured pose to compute the <b>position error of the sample</b>.
 This position error can be expressed in the frame of the granite, or in the frame of the (rotating) nano-hexapod.
-Both computation are performed.</li>
+Both computation are performed</li>
 </ul>
 
 
-<div id="orgb875b49" class="figure">
+<div id="org39b35d8" class="figure">
 <p><img src="figs/control-schematic-nass.png" alt="control-schematic-nass.png" />
 </p>
 <p><span class="figure-number">Figure 23: </span>Figure caption</p>
@@ -1274,34 +1292,43 @@ More details about these computations are accessible <a href="https://tdehaeze.g
 </div>
 </div>
 
-<div id="outline-container-org68f9b14" class="outline-3">
-<h3 id="org68f9b14"><span class="section-number-3">4.4</span> Simulation of Experiments</h3>
+<div id="outline-container-orgba6084c" class="outline-3">
+<h3 id="orgba6084c"><span class="section-number-3">4.4</span> Simulation of Experiments</h3>
 <div class="outline-text-3" id="text-4-4">
 <p>
-<a id="orgd794530"></a>
-Now that the dynamics of the model is tuned and the disturbances included in the model, we can perform simulation of experiments.
+<a id="orgc54105a"></a>
 </p>
 
 <p>
-We first do a simulation where the nano-hexapod is considered to be a solid-body to estimate the sample&rsquo;s motion that we have without an control.
+Now that the dynamics of the model is tuned and the disturbances included in the model, simulations of experiments can be performed.
 </p>
 
 <p>
-An animation of the obtained motion is shown in Figure <a href="#org8337027">24</a>.
-A zoom in the micro-meter ranger on the sample&rsquo;s location is shown in Figure <a href="#orga41a823">25</a>.
+A first simulation is done with the nano-hexapod modeled as a rigid-body.
+This does represent the system without the NASS and permits to estimate the sample&rsquo;s vibrations using the micro-station alone.
+The results of this simulation will be compared to simulations using the NASS in Section <a href="#org9af06ea">6.4</a>.
+</p>
+
+
+<p>
+An 3D animation of the simulation is shown in Figure <a href="#org03b70e6">24</a>.
 </p>
 
 <p>
-Two frames are displayed:
+A zoom in the micro-meter ranger on the sample&rsquo;s location is shown in Figure <a href="#org97382ea">25</a> with two frames:
 </p>
 <ul class="org-ul">
-<li>a non-rotating frame that corresponds to the wanted position of the sample.
-Note that here this frame is moving with the granite.</li>
+<li>a non-rotating frame corresponding to the wanted position of the sample.
+Note that this frame is moving with the granite.</li>
 <li>a rotating frame that corresponds to the actual pose of the sample</li>
 </ul>
 
+<p>
+The motion of the sample follows the wanted motion but with vibrations in the micro-meter range as was expected.
+</p>
 
-<div id="org8337027" class="figure">
+
+<div id="org03b70e6" class="figure">
 <p><img src="figs/open_loop_sim.gif" alt="open_loop_sim.gif" />
 </p>
 <p><span class="figure-number">Figure 24: </span>Tomography Experiment using the Simscape Model</p>
@@ -1309,15 +1336,15 @@ Note that here this frame is moving with the granite.</li>
 
 
 
-<div id="orga41a823" class="figure">
+<div id="org97382ea" class="figure">
 <p><img src="figs/open_loop_sim_zoom.gif" alt="open_loop_sim_zoom.gif" />
 </p>
 <p><span class="figure-number">Figure 25: </span>Tomography Experiment using the Simscape Model - Zoom on the sample&rsquo;s position (the full vertical scale is \(\approx 10 \mu m\))</p>
 </div>
 
 <p>
-The position error of the sample with respect to the granite are shown in Figure <a href="#orga1e10e4">26</a>.
-It is shown that the X-Y-Z position errors are in the micro-meter range.
+The position error of the sample with respect to the granite are shown in Figure <a href="#org57622b5">26</a>.
+It is confirmed that the X-Y-Z position errors are in the micro-meter range.
 </p>
 
 <p>
@@ -1326,11 +1353,15 @@ This is explained by the fact that no torque disturbances is considered in the m
 </p>
 
 <p>
-For the vertical rotation, this is due to the fact that we suppose perfect rotation of the Spindle, and anyway, no measurement of the sample with respect to the granite is made by the interferometers.
+The vertical rotation error is meaningless for two reasons:
 </p>
+<ul class="org-ul">
+<li>the rotation of the Spindle is considered to be perfect</li>
+<li>no measurement of the sample&rsquo;s vertical rotation with respect to the granite is made by the interferometers</li>
+</ul>
 
 
-<div id="orga1e10e4" class="figure">
+<div id="org57622b5" class="figure">
 <p><img src="figs/exp_scans_rz_dist.png" alt="exp_scans_rz_dist.png" />
 </p>
 <p><span class="figure-number">Figure 26: </span>Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances</p>
@@ -1338,25 +1369,25 @@ For the vertical rotation, this is due to the fact that we suppose perfect rotat
 </div>
 </div>
 
-<div id="outline-container-org1945128" class="outline-3">
-<h3 id="org1945128"><span class="section-number-3">4.5</span> Conclusion</h3>
+<div id="outline-container-orgc7d8ccb" class="outline-3">
+<h3 id="orgc7d8ccb"><span class="section-number-3">4.5</span> Conclusion</h3>
 <div class="outline-text-3" id="text-4-5">
 <div class="important">
 <p>
-The multi-body model developed using Simscape is shown to be sufficiently close to the micro-station dynamics.
+The multi-body model has been tuned to represents the micro-station dynamics and includes disturbances such as ground motion and stages vibrations.
 </p>
 
 <p>
-It makes possible to:
+It can be used to:
 </p>
 <ul class="org-ul">
 <li>study many effects such as the change of dynamics due to the rotation, the sample mass, etc.</li>
-<li>extract transfer function like \(G\) and \(G_d\)</li>
-<li>simulate experiments to validate performance</li>
+<li>extract transfer function like plant dynamics \(G\) and sensibility to disturbances \(G_d\)</li>
+<li>simulate experiments</li>
 </ul>
 
 <p>
-This model will be used in the next sections to help the design of the nano-hexapod, to develop the robust control architecture and to perform simulation in order to validate.
+In the next sections, it will allows to optimally design the nano-hexapod, to develop a robust control architecture and to perform simulations to estimate the system&rsquo;s performances.
 </p>
 
 </div>
@@ -1364,11 +1395,11 @@ This model will be used in the next sections to help the design of the nano-hexa
 </div>
 </div>
 
-<div id="outline-container-org30cddff" class="outline-2">
-<h2 id="org30cddff"><span class="section-number-2">5</span> Optimal Nano-Hexapod Design</h2>
+<div id="outline-container-org80c8d2b" class="outline-2">
+<h2 id="org80c8d2b"><span class="section-number-2">5</span> Optimal Nano-Hexapod Design</h2>
 <div class="outline-text-2" id="text-5">
 <p>
-<a id="org6fa5be0"></a>
+<a id="orgc2045d1"></a>
 </p>
 <p>
 As explain before, the nano-hexapod properties (mass, stiffness, architecture, &#x2026;) will influence:
@@ -1382,8 +1413,8 @@ As explain before, the nano-hexapod properties (mass, stiffness, architecture, &
 Thus, we here wish to find the optimal nano-hexapod properties such that:
 </p>
 <ul class="org-ul">
-<li>the effect of disturbances is minimized (Section <a href="#orgeaacd92">5.1</a>)</li>
-<li>the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section <a href="#orgca94eaa">5.2</a>)</li>
+<li>the effect of disturbances is minimized (Section <a href="#orga0317ee">5.1</a>)</li>
+<li>the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section <a href="#org700daec">5.2</a>)</li>
 </ul>
 
 <p>
@@ -1395,11 +1426,11 @@ The study presented here only consider changes in the nano-hexapod <b>stiffness<
 </ul>
 </div>
 
-<div id="outline-container-orgb0f8fc2" class="outline-3">
-<h3 id="orgb0f8fc2"><span class="section-number-3">5.1</span> Optimal Stiffness to reduce the effect of disturbances</h3>
+<div id="outline-container-org448fd3a" class="outline-3">
+<h3 id="org448fd3a"><span class="section-number-3">5.1</span> Optimal Stiffness to reduce the effect of disturbances</h3>
 <div class="outline-text-3" id="text-5-1">
 <p>
-<a id="orgeaacd92"></a>
+<a id="orga0317ee"></a>
 </p>
 <p>
 As will be seen, the nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of \(G_d\)).
@@ -1411,11 +1442,11 @@ A complete study of the optimal nano-hexapod stiffness for the minimization of d
 </p>
 </div>
 
-<div id="outline-container-org43274b2" class="outline-4">
-<h4 id="org43274b2">Sensibility to stage vibrations</h4>
-<div class="outline-text-4" id="text-org43274b2">
+<div id="outline-container-org32fd6fe" class="outline-4">
+<h4 id="org32fd6fe">Sensibility to stage vibrations</h4>
+<div class="outline-text-4" id="text-org32fd6fe">
 <p>
-The sensibility to the spindle&rsquo;s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure <a href="#orgf8691c0">27</a>.
+The sensibility to the spindle&rsquo;s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure <a href="#orgdac74f1">27</a>.
 It is shown that a softer nano-hexapod it better to filter out vertical vibrations of the spindle.
 More precisely, is start to filters the vibration at the first suspension mode of the payload on top of the nano-hexapod.
 </p>
@@ -1425,7 +1456,7 @@ The same conclusion is made for vibrations of the translation stage.
 </p>
 
 
-<div id="orgf8691c0" class="figure">
+<div id="orgdac74f1" class="figure">
 <p><img src="figs/opt_stiff_sensitivity_Frz.png" alt="opt_stiff_sensitivity_Frz.png" />
 </p>
 <p><span class="figure-number">Figure 27: </span>Sensitivity to Spindle vertical motion error to the vertical error position of the sample</p>
@@ -1433,21 +1464,21 @@ The same conclusion is made for vibrations of the translation stage.
 </div>
 </div>
 
-<div id="outline-container-org0f8cf88" class="outline-4">
-<h4 id="org0f8cf88">Sensibility to ground motion</h4>
-<div class="outline-text-4" id="text-org0f8cf88">
+<div id="outline-container-org323b4b9" class="outline-4">
+<h4 id="org323b4b9">Sensibility to ground motion</h4>
+<div class="outline-text-4" id="text-org323b4b9">
 <p>
-The sensibilities to ground motion in the Y and Z directions are shown in Figure <a href="#org2279a42">28</a>.
+The sensibilities to ground motion in the Y and Z directions are shown in Figure <a href="#orgbdd0e7f">28</a>.
 We can see that above the suspension mode of the nano-hexapod, the norm of the transmissibility is close to one until the suspension mode of the granite.
 Thus, a stiff nano-hexapod is better for reducing the effect of ground motion at low frequency.
 </p>
 
 <p>
-It will be suggested in Section <a href="#org48f1970">7.4</a> that using soft mounts for the granite can greatly lower the sensibility to ground motion.
+It will be suggested in Section <a href="#org4327c13">7.4</a> that using soft mounts for the granite can greatly lower the sensibility to ground motion.
 </p>
 
 
-<div id="org2279a42" class="figure">
+<div id="orgbdd0e7f" class="figure">
 <p><img src="figs/opt_stiff_sensitivity_Dw.png" alt="opt_stiff_sensitivity_Dw.png" />
 </p>
 <p><span class="figure-number">Figure 28: </span>Sensitivity to Ground motion to the position error of the sample</p>
@@ -1455,13 +1486,13 @@ It will be suggested in Section <a href="#org48f1970">7.4</a> that using soft mo
 </div>
 </div>
 
-<div id="outline-container-org8248b99" class="outline-4">
-<h4 id="org8248b99">Dynamic Noise Budgeting considering all the disturbances</h4>
-<div class="outline-text-4" id="text-org8248b99">
+<div id="outline-container-org31bc407" class="outline-4">
+<h4 id="org31bc407">Dynamic Noise Budgeting considering all the disturbances</h4>
+<div class="outline-text-4" id="text-org31bc407">
 <p>
 However, lowering the sensibility to some disturbance at a frequency where its effect is already small compare to the other disturbances sources is not really interesting.
 What is more important than comparing the sensitivity to disturbances, is thus to compare the obtain power spectral density of the sample&rsquo;s position error.
-From the Power Spectral Density of all the sources of disturbances identified in Section <a href="#org1f22d62">3</a>, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure <a href="#orgddfccca">29</a>).
+From the Power Spectral Density of all the sources of disturbances identified in Section <a href="#org471db68">3</a>, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure <a href="#orgb647b64">29</a>).
 </p>
 
 <p>
@@ -1469,7 +1500,7 @@ We can see that the most important change is in the frequency range 30Hz to 300H
 </p>
 
 
-<div id="orgddfccca" class="figure">
+<div id="orgb647b64" class="figure">
 <p><img src="figs/opt_stiff_psd_dz_tot.png" alt="opt_stiff_psd_dz_tot.png" />
 </p>
 <p><span class="figure-number">Figure 29: </span>Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses</p>
@@ -1477,13 +1508,13 @@ We can see that the most important change is in the frequency range 30Hz to 300H
 
 <div class="important">
 <p>
-If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure <a href="#orgb8c75a7">30</a>, we can observe that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance.
+If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure <a href="#orge7570d8">30</a>, we can observe that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance.
 </p>
 
 </div>
 
 
-<div id="orgb8c75a7" class="figure">
+<div id="orge7570d8" class="figure">
 <p><img src="figs/opt_stiff_cas_dz_tot.png" alt="opt_stiff_cas_dz_tot.png" />
 </p>
 <p><span class="figure-number">Figure 30: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses</p>
@@ -1492,11 +1523,11 @@ If we look at the Cumulative amplitude spectrum of the vertical error motion in
 </div>
 </div>
 
-<div id="outline-container-org8de3012" class="outline-3">
-<h3 id="org8de3012"><span class="section-number-3">5.2</span> Optimal Stiffness to reduce the plant uncertainty</h3>
+<div id="outline-container-orga1f43fd" class="outline-3">
+<h3 id="orga1f43fd"><span class="section-number-3">5.2</span> Optimal Stiffness to reduce the plant uncertainty</h3>
 <div class="outline-text-3" id="text-5-2">
 <p>
-<a id="orgca94eaa"></a>
+<a id="org700daec"></a>
 </p>
 <p>
 One of the most important design goal is to obtain a system that is <b>robust</b> to all changes in the system.
@@ -1528,15 +1559,15 @@ However, the dynamics from forces to sensors located in the nano-hexapod legs, s
 </p>
 </div>
 
-<div id="outline-container-orgda82a97" class="outline-4">
-<h4 id="orgda82a97">Effect of Payload</h4>
-<div class="outline-text-4" id="text-orgda82a97">
+<div id="outline-container-org4ea8b43" class="outline-4">
+<h4 id="org4ea8b43">Effect of Payload</h4>
+<div class="outline-text-4" id="text-org4ea8b43">
 <p>
 The most obvious change in the system is the change of payload.
 </p>
 
 <p>
-In Figure <a href="#org3dc6283">31</a> the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg.
+In Figure <a href="#orgd7aa778">31</a> the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg.
 On the left side, the change of dynamics is computed for a very soft nano-hexapod, while on the right side, it is computed for a very stiff nano-hexapod.
 </p>
 
@@ -1554,14 +1585,14 @@ For the stiff-nano-hexapod, the change of payload mass has very little effect (t
 </p>
 
 
-<div id="org3dc6283" class="figure">
+<div id="orgd7aa778" 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 31: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod (left) and a stiff nano-hexapod (right)</p>
 </div>
 
 <p>
-In Figure <a href="#org447fd2f">32</a> is shown the effect of a change of payload dynamics.
+In Figure <a href="#orga399143">32</a> is shown the effect of a change of payload dynamics.
 The mass of the payload is fixed and its resonance frequency is changing from 50Hz to 500Hz.
 </p>
 
@@ -1570,7 +1601,7 @@ We can see (more easily for the soft nano-hexapod), that resonance of the payloa
 </p>
 
 
-<div id="org447fd2f" class="figure">
+<div id="orga399143" 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 32: </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</p>
@@ -1578,7 +1609,7 @@ We can see (more easily for the soft nano-hexapod), that resonance of the payloa
 
 
 <p>
-The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure <a href="#org0a8d3b6">33</a>.
+The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure <a href="#org64f6864">33</a>.
 </p>
 
 <p>
@@ -1599,7 +1630,7 @@ For nano-hexapod stiffnesses above \(10^7\,[N/m]\):
 </ul>
 
 
-<div id="org0a8d3b6" class="figure">
+<div id="org64f6864" 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 33: </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</p>
@@ -1628,11 +1659,11 @@ Heavy samples with low first resonance mode will be very problematic.
 </div>
 </div>
 
-<div id="outline-container-org897b055" class="outline-4">
-<h4 id="org897b055">Effect of Micro-Station Compliance</h4>
-<div class="outline-text-4" id="text-org897b055">
+<div id="outline-container-orga6762b8" class="outline-4">
+<h4 id="orga6762b8">Effect of Micro-Station Compliance</h4>
+<div class="outline-text-4" id="text-orga6762b8">
 <p>
-The micro-station dynamics is quite complex as was shown in Section <a href="#org2385fed">2</a>, moreover, its dynamics can change due to:
+The micro-station dynamics is quite complex as was shown in Section <a href="#org49cefbd">2</a>, moreover, its dynamics can change due to:
 </p>
 <ul class="org-ul">
 <li>a change in some mechanical elements</li>
@@ -1653,7 +1684,7 @@ This as several other advantages:
 
 
 <p>
-To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure <a href="#org3f9aa88">34</a>):
+To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure <a href="#orga08e2f0">34</a>):
 </p>
 <ul class="org-ul">
 <li>without the micro-station (solid curves)</li>
@@ -1669,7 +1700,7 @@ For nano-hexapod stiffnesses above \(10^7\,[N/m]\), the micro-station compliance
 </p>
 
 
-<div id="org3f9aa88" class="figure">
+<div id="orga08e2f0" 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 34: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance</p>
@@ -1689,15 +1720,15 @@ If a stiff nano-hexapod is used, the control bandwidth should probably be limite
 </div>
 </div>
 
-<div id="outline-container-orgfbd7e41" class="outline-4">
-<h4 id="orgfbd7e41">Effect of Spindle Rotating Speed</h4>
-<div class="outline-text-4" id="text-orgfbd7e41">
+<div id="outline-container-orga003084" class="outline-4">
+<h4 id="orga003084">Effect of Spindle Rotating Speed</h4>
+<div class="outline-text-4" id="text-orga003084">
 <p>
 Let&rsquo;s now consider the rotation of the Spindle.
 </p>
 
 <p>
-The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure <a href="#org7d7e4dc">35</a>.
+The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure <a href="#orgf2b6792">35</a>.
 </p>
 
 <p>
@@ -1709,7 +1740,7 @@ For very soft nano-hexapods, the main resonance is split into two resonances and
 </p>
 
 
-<div id="org7d7e4dc" class="figure">
+<div id="orgf2b6792" 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 35: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm</p>
@@ -1728,16 +1759,16 @@ A very soft (\(k < 10^4\,[N/m]\)) nano-hexapod should not be used due to the eff
 </div>
 </div>
 
-<div id="outline-container-orge317967" class="outline-4">
-<h4 id="orge317967">Total Plant Uncertainty</h4>
-<div class="outline-text-4" id="text-orge317967">
+<div id="outline-container-org63b95dd" class="outline-4">
+<h4 id="org63b95dd">Total Plant Uncertainty</h4>
+<div class="outline-text-4" id="text-org63b95dd">
 <p>
-Finally, let&rsquo;s combined all the uncertainties and display the plant dynamics &ldquo;spread&rdquo; for all the nano-hexapod stiffnesses (Figure <a href="#org56845f5">36</a>).
+Finally, let&rsquo;s combined all the uncertainties and display the plant dynamics &ldquo;spread&rdquo; for all the nano-hexapod stiffnesses (Figure <a href="#orga618628">36</a>).
 This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics.
 </p>
 
 
-<div id="org56845f5" class="figure">
+<div id="orga618628" 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 36: </span>Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness</p>
@@ -1770,22 +1801,22 @@ In such case, the main limitation will be heavy samples with small stiffnesses.
 </div>
 </div>
 
-<div id="outline-container-org668965e" class="outline-3">
-<h3 id="org668965e"><span class="section-number-3">5.3</span> Conclusion</h3>
+<div id="outline-container-orgcaa22c1" class="outline-3">
+<h3 id="orgcaa22c1"><span class="section-number-3">5.3</span> Conclusion</h3>
 <div class="outline-text-3" id="text-5-3">
 <div class="important">
 <p>
-In Section <a href="#orgeaacd92">5.1</a>, it has been concluded that a nano-hexapod stiffness below \(10^5-10^6\,[N/m]\) helps reducing the high frequency vibrations induced by all sources of disturbances considered.
+In Section <a href="#orga0317ee">5.1</a>, it has been concluded that a nano-hexapod stiffness below \(10^5-10^6\,[N/m]\) helps reducing the high frequency vibrations induced by all sources of disturbances considered.
 As the high frequency vibrations are the most difficult to compensate for when using feedback control, a soft hexapod will most certainly helps improving the performances.
 </p>
 
 <p>
-In Section <a href="#orgca94eaa">5.2</a>, we concluded that a nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\) is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed.
+In Section <a href="#org700daec">5.2</a>, we concluded that a nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\) is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed.
 Provided that the samples used have a first mode that is sufficiently high in frequency, the total plant dynamic uncertainty should be manageable.
 </p>
 
 <p>
-Thus, a stiffness of \(10^5\,[N/m]\) will be used in Section <a href="#orgf46ed74">6</a> to develop the robust control architecture and to perform simulations.
+Thus, a stiffness of \(10^5\,[N/m]\) will be used in Section <a href="#org22e89f5">6</a> to develop the robust control architecture and to perform simulations.
 </p>
 
 <p>
@@ -1797,11 +1828,11 @@ A more detailed study of the determination of the optimal stiffness based on all
 </div>
 </div>
 
-<div id="outline-container-org3b2affe" class="outline-2">
-<h2 id="org3b2affe"><span class="section-number-2">6</span> Robust Control Architecture</h2>
+<div id="outline-container-org1c328ea" class="outline-2">
+<h2 id="org1c328ea"><span class="section-number-2">6</span> Robust Control Architecture</h2>
 <div class="outline-text-2" id="text-6">
 <p>
-<a id="orgf46ed74"></a>
+<a id="org22e89f5"></a>
 </p>
 <p>
 Before designing the control system, let&rsquo;s summarize what has been done:
@@ -1828,8 +1859,8 @@ This would require to measure the mass/inertia of each used payload and manually
 </p>
 </div>
 
-<div id="outline-container-org01e1894" class="outline-3">
-<h3 id="org01e1894"><span class="section-number-3">6.1</span> High Authority Control / Low Authority Control Architecture</h3>
+<div id="outline-container-org45f8ce7" class="outline-3">
+<h3 id="org45f8ce7"><span class="section-number-3">6.1</span> High Authority Control / Low Authority Control Architecture</h3>
 <div class="outline-text-3" id="text-6-1">
 <p>
 Many control architecture could be used for the control of the nano-hexapod.
@@ -1845,7 +1876,7 @@ Some properties of the HAC-LAC architecture are explained below (taken from <a c
 </p>
 <blockquote>
 <p>
-The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <a href="#org41decdd">37</a>.
+The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <a href="#org0d4a50a">37</a>.
 The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
 This approach has the following advantages:
 </p>
@@ -1857,7 +1888,7 @@ This approach has the following advantages:
 </blockquote>
 
 
-<div id="org41decdd" class="figure">
+<div id="org0d4a50a" class="figure">
 <p><img src="figs/control_architecture_hac_lac_one_input.png" alt="control_architecture_hac_lac_one_input.png" />
 </p>
 <p><span class="figure-number">Figure 37: </span>HAC-LAC Architecture with a system having only one input</p>
@@ -1867,17 +1898,17 @@ This approach has the following advantages:
 The HAC-LAC architecture thus consisted of two cascade controllers:
 </p>
 <ul class="org-ul">
-<li>a Low Authority Controller that is used to damp the system (Section <a href="#orgd035d37">6.2</a>)</li>
-<li>a High Authority Controller used to suppress the sample&rsquo;s vibration in a wide frequency range (Section <a href="#orgb8a5785">6.3</a>)</li>
+<li>a Low Authority Controller that is used to damp the system (Section <a href="#org34db031">6.2</a>)</li>
+<li>a High Authority Controller used to suppress the sample&rsquo;s vibration in a wide frequency range (Section <a href="#orga9b9d4f">6.3</a>)</li>
 </ul>
 </div>
 </div>
 
-<div id="outline-container-orgc9b0f9a" class="outline-3">
-<h3 id="orgc9b0f9a"><span class="section-number-3">6.2</span> Active Damping and Sensors to be included in the nano-hexapod</h3>
+<div id="outline-container-org6e08f63" class="outline-3">
+<h3 id="org6e08f63"><span class="section-number-3">6.2</span> Active Damping and Sensors to be included in the nano-hexapod</h3>
 <div class="outline-text-3" id="text-6-2">
 <p>
-<a id="orgd035d37"></a>
+<a id="org34db031"></a>
 </p>
 
 <p>
@@ -1905,11 +1936,11 @@ It would also be difficult to apply in a robust way due to the non-collocation w
 
 
 <p>
-<b>Relative motion sensors</b> are then included in each of the nano-hexapod&rsquo;s leg and a decentralized direct velocity feedback control architecture is applied (Figure <a href="#orgc202b21">38</a>).
+<b>Relative motion sensors</b> are then included in each of the nano-hexapod&rsquo;s leg and a decentralized direct velocity feedback control architecture is applied (Figure <a href="#org0928e5a">38</a>).
 </p>
 
 <p>
-The signals shown in Figure <a href="#orgc202b21">38</a> are:
+The signals shown in Figure <a href="#org0928e5a">38</a> are:
 </p>
 <ul class="org-ul">
 <li>\(\bm{\tau}\): Actuator forces applied in each leg</li>
@@ -1925,7 +1956,7 @@ The force applied in each leg being proportional to the relative velocity of the
 </p>
 
 
-<div id="orgc202b21" class="figure">
+<div id="org0928e5a" class="figure">
 <p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
 </p>
 <p><span class="figure-number">Figure 38: </span>Low Authority Control: Decentralized Direct Velocity Feedback</p>
@@ -1938,20 +1969,20 @@ This may not be the optimal choice as will be further explained.
 
 
 <p>
-The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure <a href="#org2975a46">39</a>.
+The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure <a href="#org7d01e00">39</a>.
 It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz up too 100Hz.
 This will make the primary controller more robust and easier to develop.
 </p>
 
 
-<div id="org2975a46" class="figure">
+<div id="org7d01e00" class="figure">
 <p><img src="figs/opt_stiff_primary_plant_damped_L.png" alt="opt_stiff_primary_plant_damped_L.png" />
 </p>
 <p><span class="figure-number">Figure 39: </span>Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
 </div>
 
 <p>
-The change of sensibility to disturbances with the use of DVF is shown in Figure <a href="#org3b5d6d3">40</a>.
+The change of sensibility to disturbances with the use of DVF is shown in Figure <a href="#org7e0ea6d">40</a>.
 It is shown that the DVF control lowers the sensibility to disturbances in the vicinity of the nano-hexapod resonance but increases the sensibility at higher frequencies.
 </p>
 
@@ -1960,7 +1991,7 @@ This is probably not the optimal gain that could have been used, and further ana
 </p>
 
 
-<div id="org3b5d6d3" class="figure">
+<div id="org7e0ea6d" class="figure">
 <p><img src="figs/opt_stiff_sensibility_dist_dvf.png" alt="opt_stiff_sensibility_dist_dvf.png" />
 </p>
 <p><span class="figure-number">Figure 40: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied</p>
@@ -1968,20 +1999,20 @@ This is probably not the optimal gain that could have been used, and further ana
 </div>
 </div>
 
-<div id="outline-container-org3e166d9" class="outline-3">
-<h3 id="org3e166d9"><span class="section-number-3">6.3</span> High Authority Control</h3>
+<div id="outline-container-orgb499508" class="outline-3">
+<h3 id="orgb499508"><span class="section-number-3">6.3</span> High Authority Control</h3>
 <div class="outline-text-3" id="text-6-3">
 <p>
-<a id="orgb8a5785"></a>
+<a id="orga9b9d4f"></a>
 </p>
 
 <p>
-The complete HAC-LAC architecture is shown in Figure <a href="#org20d6928">41</a> where an outer loop is added to the decentralized direct velocity feedback loop.
+The complete HAC-LAC architecture is shown in Figure <a href="#orgb60004f">41</a> where an outer loop is added to the decentralized direct velocity feedback loop.
 </p>
 
 <p>
 The block <code>Compute Position Error</code> is used to compute the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) of the sample with respect to the nano-hexapod&rsquo;s base platform from the actual measurement of the sample&rsquo;s pose \(\bm{\mathcal{X}}\) and the wanted pose \(\bm{r}_\mathcal{X}\).
-The computation done in such block was briefly explained in Section <a href="#org82c65f0">4.3</a>.
+The computation done in such block was briefly explained in Section <a href="#org92c9b7d">4.3</a>.
 </p>
 
 
@@ -1994,7 +2025,7 @@ Then, a diagonal controller \(\bm{K}_\mathcal{L}\) generates the required force
 </p>
 
 
-<div id="org20d6928" class="figure">
+<div id="orgb60004f" class="figure">
 <p><img src="figs/control_architecture_hac_dvf_pos_L.png" alt="control_architecture_hac_dvf_pos_L.png" />
 </p>
 <p><span class="figure-number">Figure 41: </span>Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg&rsquo;s space</p>
@@ -2005,12 +2036,12 @@ Some alternative to this control architecture have been studied, but this is the
 </p>
 
 <p>
-The plant dynamics for each of the six legs and for the three payload&rsquo;s masses is shown in Figure <a href="#orgd72aeb2">42</a>.
+The plant dynamics for each of the six legs and for the three payload&rsquo;s masses is shown in Figure <a href="#orgb1f37dc">42</a>.
 The dynamical spread is kept reasonably small thanks to both the optimal nano-hexapod design and the Low Authority Controller.
 </p>
 
 
-<div id="orgd72aeb2" class="figure">
+<div id="orgb1f37dc" class="figure">
 <p><img src="figs/opt_stiff_primary_plant_L.png" alt="opt_stiff_primary_plant_L.png" />
 </p>
 <p><span class="figure-number">Figure 42: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses</p>
@@ -2019,18 +2050,18 @@ The dynamical spread is kept reasonably small thanks to both the optimal nano-he
 
 <p>
 The diagonal controller \(\bm{K}_\mathcal{L}\) is then tuned in such a way that the control bandwidth is around 100Hz and such that enough stability margins are obtained for all the payload&rsquo;s masses used.
-The obtained loop gain is shown in Figure <a href="#orgb542f54">43</a>.
+The obtained loop gain is shown in Figure <a href="#orgb4188b9">43</a>.
 </p>
 
 
-<div id="orgb542f54" class="figure">
+<div id="orgb4188b9" class="figure">
 <p><img src="figs/opt_stiff_primary_loop_gain_L.png" alt="opt_stiff_primary_loop_gain_L.png" />
 </p>
 <p><span class="figure-number">Figure 43: </span>Loop gain for the primary plant</p>
 </div>
 
 <p>
-The sensibility to disturbance after the use of HAC-LAC control is shown in Figure <a href="#org32f7c3a">44</a>.
+The sensibility to disturbance after the use of HAC-LAC control is shown in Figure <a href="#org3bcef30">44</a>.
 The change of sensibility is very typical for feedback system:
 </p>
 <ul class="org-ul">
@@ -2048,7 +2079,7 @@ This should gives slightly better performance and robustness, but should not cha
 </p>
 
 
-<div id="org32f7c3a" class="figure">
+<div id="org3bcef30" class="figure">
 <p><img src="figs/opt_stiff_primary_control_L_senbility_dist.png" alt="opt_stiff_primary_control_L_senbility_dist.png" />
 </p>
 <p><span class="figure-number">Figure 44: </span>Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)</p>
@@ -2056,22 +2087,22 @@ This should gives slightly better performance and robustness, but should not cha
 </div>
 </div>
 
-<div id="outline-container-org10bf44e" class="outline-3">
-<h3 id="org10bf44e"><span class="section-number-3">6.4</span> Simulation of Tomography Experiments</h3>
+<div id="outline-container-org9085983" class="outline-3">
+<h3 id="org9085983"><span class="section-number-3">6.4</span> Simulation of Tomography Experiments</h3>
 <div class="outline-text-3" id="text-6-4">
 <p>
-<a id="org7b6bb0b"></a>
+<a id="org9af06ea"></a>
 </p>
 
 <p>
 A new simulation of a tomography is performed with the optimal nano-hexapod and the HAC-LAC architecture implemented in the model.
-The results of this simulation will be compare to the simulation performed in Section <a href="#orgd794530">4.4</a> without the nano-hexapod.
+The results of this simulation will be compare to the simulation performed in Section <a href="#orgc54105a">4.4</a> without the nano-hexapod.
 All the disturbances are included such as ground motion, spindle and translation stage vibrations.
 </p>
 
 
 <p>
-The Power Spectral Density of the sample&rsquo;s position error is plotted in Figure <a href="#org20cf815">45</a> and the Cumulative Amplitude Spectrum is shown in Figure <a href="#org7a6bc29">46</a>.
+The Power Spectral Density of the sample&rsquo;s position error is plotted in Figure <a href="#orgef93fb6">45</a> and the Cumulative Amplitude Spectrum is shown in Figure <a href="#orga7f9385">46</a>.
 The top three plots corresponds to the X, Y and Z translations and the bottom three plots corresponds to the X,Y and Z rotations.
 </p>
 
@@ -2100,7 +2131,7 @@ This increase in rotation is still very small and is not foreseen to be a proble
 </ul>
 
 
-<div id="org20cf815" class="figure">
+<div id="orgef93fb6" class="figure">
 <p><img src="figs/opt_stiff_hac_dvf_L_psd_disp_error.png" alt="opt_stiff_hac_dvf_L_psd_disp_error.png" />
 </p>
 <p><span class="figure-number">Figure 45: </span>Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller</p>
@@ -2108,7 +2139,7 @@ This increase in rotation is still very small and is not foreseen to be a proble
 
 
 
-<div id="org7a6bc29" class="figure">
+<div id="orga7f9385" class="figure">
 <p><img src="figs/opt_stiff_hac_dvf_L_cas_disp_error.png" alt="opt_stiff_hac_dvf_L_cas_disp_error.png" />
 </p>
 <p><span class="figure-number">Figure 46: </span>Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller</p>
@@ -2116,23 +2147,23 @@ This increase in rotation is still very small and is not foreseen to be a proble
 
 
 <p>
-The time domain sample&rsquo;s vibrations are shown in Figure <a href="#org9f0ea07">47</a>.
+The time domain sample&rsquo;s vibrations are shown in Figure <a href="#orgdd9e627">47</a>.
 The use of the nano-hexapod combined with the HAC-LAC architecture is shown to considerably reduce the sample&rsquo;s vibrations.
 </p>
 
 <p>
-An animation of the experiment is shown in Figure <a href="#org13a1d6e">48</a> and we can see that the actual sample&rsquo;s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure <a href="#orga41a823">25</a> (same scale was used for both animations).
+An animation of the experiment is shown in Figure <a href="#org338d327">48</a> and we can see that the actual sample&rsquo;s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure <a href="#org97382ea">25</a> (same scale was used for both animations).
 </p>
 
 
-<div id="org9f0ea07" class="figure">
+<div id="orgdd9e627" class="figure">
 <p><img src="figs/opt_stiff_hac_dvf_L_pos_error.png" alt="opt_stiff_hac_dvf_L_pos_error.png" />
 </p>
 <p><span class="figure-number">Figure 47: </span>Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture</p>
 </div>
 
 
-<div id="org13a1d6e" class="figure">
+<div id="org338d327" class="figure">
 <p><img src="figs/closed_loop_sim_zoom.gif" alt="closed_loop_sim_zoom.gif" />
 </p>
 <p><span class="figure-number">Figure 48: </span>Tomography Experiment using the Simscape Model in Closed Loop with the HAC-LAC Control - Zoom on the sample&rsquo;s position (the full vertical scale is \(\approx 10 \mu m\))</p>
@@ -2140,8 +2171,8 @@ An animation of the experiment is shown in Figure <a href="#org13a1d6e">48</a> a
 </div>
 </div>
 
-<div id="outline-container-orgadf6472" class="outline-3">
-<h3 id="orgadf6472"><span class="section-number-3">6.5</span> Conclusion</h3>
+<div id="outline-container-org4b72da9" class="outline-3">
+<h3 id="org4b72da9"><span class="section-number-3">6.5</span> Conclusion</h3>
 <div class="outline-text-3" id="text-6-5">
 <div class="important">
 <p>
@@ -2176,21 +2207,21 @@ A more complete study of the control of the NASS is performed <a href="https://t
 </div>
 </div>
 
-<div id="outline-container-org6d31a2a" class="outline-2">
-<h2 id="org6d31a2a"><span class="section-number-2">7</span> General Conclusion and Further notes</h2>
+<div id="outline-container-org6dfaa6a" class="outline-2">
+<h2 id="org6dfaa6a"><span class="section-number-2">7</span> General Conclusion and Further notes</h2>
 <div class="outline-text-2" id="text-7">
 <p>
-<a id="org2e38497"></a>
+<a id="org126e968"></a>
 </p>
 </div>
 
-<div id="outline-container-org3c26e13" class="outline-3">
-<h3 id="org3c26e13"><span class="section-number-3">7.1</span> General Conclusion</h3>
+<div id="outline-container-org8a29acd" class="outline-3">
+<h3 id="org8a29acd"><span class="section-number-3">7.1</span> General Conclusion</h3>
 </div>
 
 
-<div id="outline-container-orgf2dd801" class="outline-3">
-<h3 id="orgf2dd801"><span class="section-number-3">7.2</span> Sensor Noise introduced by the Metrology</h3>
+<div id="outline-container-orgf76caba" class="outline-3">
+<h3 id="orgf76caba"><span class="section-number-3">7.2</span> Sensor Noise introduced by the Metrology</h3>
 <div class="outline-text-3" id="text-7-2">
 <p>
 Say that is will introduce noise inside the bandwidth (100Hz)
@@ -2200,20 +2231,20 @@ This should not be significant.
 </div>
 
 
-<div id="outline-container-orga032112" class="outline-3">
-<h3 id="orga032112"><span class="section-number-3">7.3</span> Further Work</h3>
+<div id="outline-container-org34c5e7a" class="outline-3">
+<h3 id="org34c5e7a"><span class="section-number-3">7.3</span> Further Work</h3>
 </div>
 
 
-<div id="outline-container-orgc79208c" class="outline-3">
-<h3 id="orgc79208c"><span class="section-number-3">7.4</span> Using soft mounts for the</h3>
+<div id="outline-container-org8b69a8a" class="outline-3">
+<h3 id="org8b69a8a"><span class="section-number-3">7.4</span> Using soft mounts for the</h3>
 <div class="outline-text-3" id="text-7-4">
 <p>
-<a id="org48f1970"></a>
+<a id="org4327c13"></a>
 </p>
 
 
-<div id="org055128d" class="figure">
+<div id="org64f4d32" class="figure">
 <p><img src="figs/opt_stiff_soft_granite_Dw.png" alt="opt_stiff_soft_granite_Dw.png" />
 </p>
 <p><span class="figure-number">Figure 49: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves)</p>
@@ -2229,8 +2260,8 @@ Sensible to detector motion?
 </div>
 </div>
 
-<div id="outline-container-org2ca3a7d" class="outline-3">
-<h3 id="org2ca3a7d"><span class="section-number-3">7.5</span> Others</h3>
+<div id="outline-container-org71917ed" class="outline-3">
+<h3 id="org71917ed"><span class="section-number-3">7.5</span> Others</h3>
 <div class="outline-text-3" id="text-7-5">
 <p>
 Common metrology frame for the nano-focusing optics and the measurement of the sample position?
@@ -2259,7 +2290,7 @@ Slip-Ring noise?
 <div id="postamble" class="status">
 <p class="date">Date: 05-2020</p>
 <p class="author">Author: Thomas Dehaeze</p>
-<p class="date">Created: 2020-04-29 mer. 18:17</p>
+<p class="date">Created: 2020-04-29 mer. 19:02</p>
 </div>
 </body>
 </html>
diff --git a/index.org b/index.org
index 4243b35..42842e5 100644
--- a/index.org
+++ b/index.org
@@ -666,18 +666,27 @@ This should however not change the conclusion of this study nor significantly ch
 <<sec:multi_body_model>>
 
 ** Introduction                                                      :ignore:
-As was shown during the modal analysis (Section [[sec:micro_station_dynamics]]), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies.
+As was shown during the modal analysis (Section [[sec:micro_station_dynamics]]), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) connected with some discrete flexibility (stiffnesses and dampers).
 
-Thus, a *multi-body model* is perfect to represent such dynamics.
+Thus, *a multi-body model is perfectly adapted to represent the dynamics of the micro-station*.
 
-To do so, we use the Matlab's [[https://www.mathworks.com/products/simscape.html][Simscape]] toolbox.
+The Matlab's [[https://www.mathworks.com/products/simscape.html][Simscape]] toolbox is used to develop the multi-body model.
 A small summary of the multi-body Simscape is available [[https://tdehaeze.github.io/nass-simscape/simscape.html][here]] and each of the modeled stage is described [[https://tdehaeze.github.io/nass-simscape/simscape_subsystems.html][here]].
 
 ** Multi-Body model
+<<sec:multi_body_model_introduction>>
+
+The parameters to tune the dynamics of the multi body are:
+- the mass/moment of inertia of each of the solid bodies
+- the 6 stiffnesses and 6 damping properties representing each of the the mechanical guiding between two solid bodies
+
+ 
 The mass/inertia of each stage is automatically computed from the imported geometry and the material's density.
 
-The (6dof) stiffness between two solid bodies is first guessed from either measurements of data-sheets.
-Then, the values of the stiffness and damping of each joint is manually tuned until the obtained dynamics is sufficiently close to the measured dynamics.
+
+The stiffnesses between two solid bodies is first guessed from either measurements of data-sheets.
+Then, the values of the stiffnesses and damping properties of each joint is manually tuned until the obtained dynamics is sufficiently close to the measured dynamics.
+
 
 The 3D representation of the simscape model is shown in Figure [[fig:simscape_picture]].
 
@@ -686,7 +695,9 @@ The 3D representation of the simscape model is shown in Figure [[fig:simscape_pi
 [[file:figs/simscape_picture.png]]
 
 ** Validity of the model's dynamics
-It is very difficult the tune the dynamics of such model as there are more than 50 parameters and many curves to compare between the model and the measurements.
+<<sec:model_validity>>
+
+Tuning the dynamics of such model is very difficult as there are more than 50 parameters to tune and many different dynamics to compare between the model and the measurements.
 
 The comparison of three of the Frequency Response Functions are shown in Figure [[fig:identification_comp_top_stages]].
 
@@ -710,24 +721,23 @@ Now that the multi-body model dynamics as been tuned, the following elements are
 Then, using the model, we can
 - perform simulation of experiments in presence of disturbances
 - measure the motion of the solid-bodies
-- identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod and control design
-- include a multi-body model of the nano-hexapod and closed-loop simulations
+- identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod design and the control synthesis
+- include a multi-body model of the nano-hexapod and perform closed-loop simulations
 
 ** Wanted position of the sample and position error
 <<sec:pos_error_nass>>
 
-For the control of the nano-hexapod, we need to now the sample position error (the motion to be compensated) in the frame of the nano-hexapod.
+For the control of the nano-hexapod, the sample position error (the motion to be compensated) in the frame of the nano-hexapod needs to be computed.
 
-To do so, we need to perform several computations (summarized in Figure [[fig:control-schematic-nass]]):
-- First, we need to determine the actual *wanted pose* (3 translations and 3 rotations) of the sample with respect to the granite.
-  This is determined from the wanted motion of each micro-station stage.
-  Each wanted stage motion is represented by an homogeneous transformation matrix (explain [[http://planning.cs.uiuc.edu/node111.html][here]]), then these matrices are combined to give to total wanted motion of the sample with respect to the granite.
-- Then, we need to determine the *actual pose* of the sample with respect to the granite.
-  This will be performed by several interferometers and several computation will be required to compute the pose of the sample from the interferometers measurements.
-  However we here directly measure the 3 translations and 3 rotations of the sample using a special simscape block.
-- Finally, we need to compare the wanted pose with the measured pose to compute the position error of the sample.
+To do so, several computations are performed (summarized in Figure [[fig:control-schematic-nass]]):
+- First, the *wanted pose* (3 translations and 3 rotations) of the sample with respect to the granite is computed.
+  This is determined from the wanted motion of each micro-station stage: each wanted stage motion is represented by a homogeneous transformation matrix that are combined to give to total wanted motion of the sample with respect to the granite
+- Then, the *actual pose* of the sample with respect to the granite is computed.
+  For the real system, this will require the use of several interferometers and computations to obtain the sample's pose from the individual measurements.
+  However, the pose of the sample with respect to the granite is directly measured using a special simscape block
+- Finally, the wanted pose is compared with the measured pose to compute the *position error of the sample*.
   This position error can be expressed in the frame of the granite, or in the frame of the (rotating) nano-hexapod.
-  Both computation are performed.
+  Both computation are performed
 
 #+name: fig:control-schematic-nass
 #+caption: Figure caption
@@ -737,18 +747,23 @@ More details about these computations are accessible [[https://tdehaeze.github.i
 
 ** Simulation of Experiments
 <<sec:micro_station_simulation>>
-Now that the dynamics of the model is tuned and the disturbances included in the model, we can perform simulation of experiments.
 
-We first do a simulation where the nano-hexapod is considered to be a solid-body to estimate the sample's motion that we have without an control.
+Now that the dynamics of the model is tuned and the disturbances included in the model, simulations of experiments can be performed.
 
-An animation of the obtained motion is shown in Figure [[fig:open_loop_sim]].
-A zoom in the micro-meter ranger on the sample's location is shown in Figure [[fig:open_loop_sim_zoom]].
+A first simulation is done with the nano-hexapod modeled as a rigid-body.
+This does represent the system without the NASS and permits to estimate the sample's vibrations using the micro-station alone.
+The results of this simulation will be compared to simulations using the NASS in Section [[sec:tomography_experiment]].
 
-Two frames are displayed:
-- a non-rotating frame that corresponds to the wanted position of the sample.
-  Note that here this frame is moving with the granite.
+
+An 3D animation of the simulation is shown in Figure [[fig:open_loop_sim]].
+
+A zoom in the micro-meter ranger on the sample's location is shown in Figure [[fig:open_loop_sim_zoom]] with two frames:
+- a non-rotating frame corresponding to the wanted position of the sample.
+  Note that this frame is moving with the granite.
 - a rotating frame that corresponds to the actual pose of the sample
 
+The motion of the sample follows the wanted motion but with vibrations in the micro-meter range as was expected.
+
 #+name: fig:open_loop_sim
 #+caption: Tomography Experiment using the Simscape Model
 [[file:figs/open_loop_sim.gif]]
@@ -759,12 +774,14 @@ Two frames are displayed:
 [[file:figs/open_loop_sim_zoom.gif]]
 
 The position error of the sample with respect to the granite are shown in Figure [[fig:exp_scans_rz_dist]].
-It is shown that the X-Y-Z position errors are in the micro-meter range.
+It is confirmed that the X-Y-Z position errors are in the micro-meter range.
 
 For the rotation around X and Y, the errors are quite small.
 This is explained by the fact that no torque disturbances is considered in the model.
 
-For the vertical rotation, this is due to the fact that we suppose perfect rotation of the Spindle, and anyway, no measurement of the sample with respect to the granite is made by the interferometers.
+The vertical rotation error is meaningless for two reasons:
+- the rotation of the Spindle is considered to be perfect
+- no measurement of the sample's vertical rotation with respect to the granite is made by the interferometers
 
 #+name: fig:exp_scans_rz_dist
 #+caption: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances
@@ -772,14 +789,14 @@ For the vertical rotation, this is due to the fact that we suppose perfect rotat
 
 ** Conclusion
 #+begin_important
-The multi-body model developed using Simscape is shown to be sufficiently close to the micro-station dynamics.
+The multi-body model has been tuned to represents the micro-station dynamics and includes disturbances such as ground motion and stages vibrations.
 
-It makes possible to:
+It can be used to:
 - study many effects such as the change of dynamics due to the rotation, the sample mass, etc.
-- extract transfer function like $G$ and $G_d$
-- simulate experiments to validate performance
+- extract transfer function like plant dynamics $G$ and sensibility to disturbances $G_d$
+- simulate experiments
 
-This model will be used in the next sections to help the design of the nano-hexapod, to develop the robust control architecture and to perform simulation in order to validate.
+In the next sections, it will allows to optimally design the nano-hexapod, to develop a robust control architecture and to perform simulations to estimate the system's performances.
 #+end_important
 
 * Optimal Nano-Hexapod Design