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diff --git a/docs/optimal_stiffness_disturbances.html b/docs/optimal_stiffness_disturbances.html
index f9d812a..a6424bc 100644
--- a/docs/optimal_stiffness_disturbances.html
+++ b/docs/optimal_stiffness_disturbances.html
@@ -4,7 +4,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-04-07 mar. 15:57 -->
+<!-- 2020-04-07 mar. 17:10 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <meta name="viewport" content="width=device-width, initial-scale=1" />
 <title>Determination of the optimal nano-hexapod&rsquo;s stiffness for reducing the effect of disturbances</title>
@@ -227,7 +227,9 @@
 </script>
 <script>
       MathJax = {
-          tex: { macros: {
+          tex: {
+          tags: 'ams',
+          macros: {
                   bm: ["\\boldsymbol{#1}",1],
                   }
               }
@@ -255,14 +257,20 @@
 <li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
 <li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
 <li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
-<li><a href="#orgea74617">2.6. Conclusion</a></li>
+<li><a href="#orge0160c0">2.6. Conclusion</a></li>
 </ul>
 </li>
 <li><a href="#org6527e58">3. Effect of granite stiffness</a>
 <ul>
-<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a></li>
+<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a>
+<ul>
+<li><a href="#orgbc34a65">3.1.1. Simple mass-spring-damper model</a></li>
+<li><a href="#org4ddec32">3.1.2. General Case</a></li>
+</ul>
+</li>
 <li><a href="#org9215f81">3.2. Soft Granite</a></li>
 <li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
+<li><a href="#orgb756362">3.4. Conclusion</a></li>
 </ul>
 </li>
 <li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
@@ -278,7 +286,7 @@
 <li><a href="#orgf2d36a1">5.1. Reduction thanks to feedback - Required bandwidth</a></li>
 </ul>
 </li>
-<li><a href="#org0953c03">6. Conclusion</a></li>
+<li><a href="#orga29f90b">6. Conclusion</a></li>
 </ul>
 </div>
 </div>
@@ -488,12 +496,16 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
 </div>
 </div>
 
-<div id="outline-container-orgea74617" class="outline-3">
-<h3 id="orgea74617"><span class="section-number-3">2.6</span> Conclusion</h3>
+<div id="outline-container-orge0160c0" class="outline-3">
+<h3 id="orge0160c0"><span class="section-number-3">2.6</span> Conclusion</h3>
 <div class="outline-text-3" id="text-2-6">
 <div class="important">
 <p>
+Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
+</p>
 
+<p>
+In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section <a href="#org5d05990">4</a>.
 </p>
 
 </div>
@@ -507,15 +519,28 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
 <p>
 <a id="orgd4105b6"></a>
 </p>
+<p>
+In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
+</p>
 </div>
 <div id="outline-container-orgd3e5fe1" class="outline-3">
 <h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
 <div class="outline-text-3" id="text-3-1">
+</div>
+<div id="outline-container-orgbc34a65" class="outline-4">
+<h4 id="orgbc34a65"><span class="section-number-4">3.1.1</span> Simple mass-spring-damper model</h4>
+<div class="outline-text-4" id="text-3-1-1">
+<p>
+Let&rsquo;s consider the system shown in Figure <a href="#org8fb9606">8</a> consisting of two stacked mass-spring-damper systems.
+The bottom one represents the granite, and the top one all the positioning stages.
+We want the smallest stage &ldquo;deformation&rdquo; \(d = x^\prime - x\) due to ground motion \(w\).
+</p>
+
 
 <div id="org8fb9606" class="figure">
 <p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
 </p>
-<p><span class="figure-number">Figure 8: </span>Figure caption</p>
+<p><span class="figure-number">Figure 8: </span>Mass Spring Damper system consisting of a granite and a positioning stage</p>
 </div>
 
 <p>
@@ -523,53 +548,148 @@ If we write the equation of motion of the system in Figure <a href="#org8fb9606"
 </p>
 \begin{align}
   m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
-  ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
+  ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x)
 \end{align}
 
 <p>
 If we note \(d = x^\prime - x\), we obtain:
 </p>
 \begin{equation}
-\label{org4396920}
-  \frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
+  \frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
 \end{equation}
 </div>
 </div>
 
+<div id="outline-container-org4ddec32" class="outline-4">
+<h4 id="org4ddec32"><span class="section-number-4">3.1.2</span> General Case</h4>
+<div class="outline-text-4" id="text-3-1-2">
+<p>
+Let&rsquo;s now considering a general positioning stage defined by:
+</p>
+<ul class="org-ul">
+<li>\(G^\prime(s) = \frac{F}{x}\): its mechanical &ldquo;impedance&rdquo;</li>
+<li>\(D^\prime(s) = \frac{d}{x}\): its &ldquo;deformation&rdquo; transfer function</li>
+</ul>
+
+
+<div id="org9702e0f" class="figure">
+<p><img src="figs/general_system_granite_stiffness.png" alt="general_system_granite_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 9: </span>Mass Spring Damper representing the granite and a general representation of positioning stages</p>
+</div>
+
+<p>
+The equation of motion are:
+</p>
+\begin{align}
+  ms^2 x &= (cs + k) (x - w) - F \\
+  F &= G^\prime(s) x \\
+  d &= D^\prime(s) x
+\end{align}
+<p>
+where:
+</p>
+<ul class="org-ul">
+<li>\(F\) is the force applied by the position stages on the granite mass</li>
+</ul>
+
+<div class="important">
+<p>
+We can express \(d\) as a function of \(w\):
+</p>
+\begin{equation}
+  \frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
+\end{equation}
+
+<p>
+This is the transfer function that we would like to minimize.
+</p>
+
+</div>
+
+<p>
+Let&rsquo;s verify this formula for a simple mass/spring/damper positioning stage.
+In that case, we have:
+</p>
+\begin{align*}
+  D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\
+  G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
+\end{align*}
+
+<p>
+And finally:
+</p>
+\begin{equation}
+  \frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
+\end{equation}
+<p>
+which is the same as the previously derived equation.
+</p>
+</div>
+</div>
+</div>
+
 <div id="outline-container-org9215f81" class="outline-3">
 <h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
 <div class="outline-text-3" id="text-3-2">
 <p>
-Let&rsquo;s initialize a soft granite that will act as an isolation stage from ground motion.
+Let&rsquo;s initialize a soft granite and see how the sensitivity to disturbances will change.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">initializeGranite(<span class="org-string">'K'</span>, 5e5<span class="org-type">*</span>ones(3,1), <span class="org-string">'C'</span>, 5e3<span class="org-type">*</span>ones(3,1));
 </pre>
 </div>
-
-<div class="org-src-container">
-<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
-</pre>
-</div>
-
-<div class="org-src-container">
-<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
-    initializeNanoHexapod(<span class="org-string">'k'</span>, Ks(<span class="org-constant">i</span>));
-
-    G = linearize(mdl, io);
-    G.InputName  = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Fty_x'</span>, <span class="org-string">'Fty_z'</span>, <span class="org-string">'Frz_z'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
-    G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
-    Gdr(<span class="org-constant">i</span>) = {minreal(G)};
-<span class="org-keyword">end</span>
-</pre>
-</div>
 </div>
 </div>
 
 <div id="outline-container-org8878556" class="outline-3">
 <h3 id="org8878556"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
+<div class="outline-text-3" id="text-3-3">
+<p>
+From Figure <a href="#org38146da">10</a>, we can see that having a &ldquo;soft&rdquo; granite suspension greatly lowers the sensitivity to ground motion.
+The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
+</p>
+
+<p>
+From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
+</p>
+
+
+<div id="org38146da" 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 10: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Dw.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Dw.pdf">pdf</a>)</p>
+</div>
+
+
+<div id="orgc4c14fb" class="figure">
+<p><img src="figs/opt_stiff_soft_granite_Frz.png" alt="opt_stiff_soft_granite_Frz.png" />
+</p>
+<p><span class="figure-number">Figure 11: </span>Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Frz.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Frz.pdf">pdf</a>)</p>
+</div>
+
+
+<div id="org533cc4b" class="figure">
+<p><img src="figs/opt_stiff_soft_granite_Fd.png" alt="opt_stiff_soft_granite_Fd.png" />
+</p>
+<p><span class="figure-number">Figure 12: </span>Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Fd.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Fd.pdf">pdf</a>)</p>
 </div>
 </div>
+</div>
+
+<div id="outline-container-orgb756362" class="outline-3">
+<h3 id="orgb756362"><span class="section-number-3">3.4</span> Conclusion</h3>
+<div class="outline-text-3" id="text-3-4">
+<div class="important">
+<p>
+Having a soft granite suspension could greatly improve the sensitivity the ground motion and thus the level of sample vibration if it is found that ground motion is the limiting factor.
+</p>
+
+</div>
+</div>
+</div>
+</div>
+
 <div id="outline-container-org8a88fb0" class="outline-2">
 <h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
 <div class="outline-text-2" id="text-4">
@@ -846,13 +966,13 @@ xlim([1, 500]);
 </div>
 </div>
 </div>
-<div id="outline-container-org0953c03" class="outline-2">
-<h2 id="org0953c03"><span class="section-number-2">6</span> Conclusion</h2>
+<div id="outline-container-orga29f90b" class="outline-2">
+<h2 id="orga29f90b"><span class="section-number-2">6</span> Conclusion</h2>
 </div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-04-07 mar. 15:57</p>
+<p class="date">Created: 2020-04-07 mar. 17:10</p>
 </div>
 </body>
 </html>
diff --git a/org/optimal_stiffness_disturbances.org b/org/optimal_stiffness_disturbances.org
index fe5a31d..24c2ed7 100644
--- a/org/optimal_stiffness_disturbances.org
+++ b/org/optimal_stiffness_disturbances.org
@@ -389,12 +389,22 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
 
 ** Conclusion
 #+begin_important
+  Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
 
+  In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section [[sec:open_loop_budget_error]].
 #+end_important
 
 * Effect of granite stiffness
 <<sec:granite_stiffness>>
+** Introduction                                                      :ignore:
+In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
+
 ** Analytical Analysis
+*** Simple mass-spring-damper model
+Let's consider the system shown in Figure [[fig:2dof_system_granite_stiffness]] consisting of two stacked mass-spring-damper systems.
+The bottom one represents the granite, and the top one all the positioning stages.
+We want the smallest stage "deformation" $d = x^\prime - x$ due to ground motion $w$.
+
 #+begin_src latex :file 2dof_system_granite_stiffness.pdf
   \begin{tikzpicture}
     % ====================
@@ -415,7 +425,7 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
     % ====================
     \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
     \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
-    \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$};
+    \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
 
     % ====================
     % Micro Station
@@ -425,8 +435,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
       \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};
 
       % Spring, Damper, and Actuator
-      \draw[spring] (-0.4*\massw, 0)   -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
-      \draw[damper] (0, 0)             -- ( 0, \spaceh)          node[midway, left=0.2]{$c$};
+      \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
+      \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};
 
       % Displacements
       \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
@@ -434,8 +444,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
 
       % Legend
       \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
-        (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
-        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
+      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
     \end{scope}
 
     % ====================
@@ -446,8 +456,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
       \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};
 
       % Spring, Damper, and Actuator
-      \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
-      \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c^\prime$};
+      \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
+      \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c^\prime$};
 
       % Displacements
       \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
@@ -455,44 +465,123 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
 
       % Legend
       \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
-        (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
-        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
+      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
     \end{scope}
   \end{tikzpicture}
 #+end_src
 
 #+name: fig:2dof_system_granite_stiffness
-#+caption: Figure caption
+#+caption: Mass Spring Damper system consisting of a granite and a positioning stage
 #+RESULTS:
 [[file:figs/2dof_system_granite_stiffness.png]]
 
 If we write the equation of motion of the system in Figure [[fig:2dof_system_granite_stiffness]], we obtain:
 \begin{align}
   m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
-  ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
+  ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x)
 \end{align}
 
 If we note $d = x^\prime - x$, we obtain:
-#+name: eq:plant_ground_transmissibility
 \begin{equation}
-  \frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
+  \frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
 \end{equation}
 
+*** General Case
+Let's now considering a general positioning stage defined by:
+- $G^\prime(s) = \frac{F}{x}$: its mechanical "impedance"
+- $D^\prime(s) = \frac{d}{x}$: its "deformation" transfer function
+
+#+begin_src latex :file general_system_granite_stiffness.pdf
+  \begin{tikzpicture}
+    \def\massw{2.2}  % Width of the masses
+    \def\massh{0.8}  % Height of the masses
+    \def\spaceh{1.2} % Height of the springs/dampers
+    \def\dispw{0.3}  % Width of the dashed line for the displacement
+    \def\disph{0.5}  % Height of the arrow for the displacements
+    \def\bracs{0.05} % Brace spacing vertically
+    \def\brach{-10pt} % Brace shift horizontaly
+
+    % Mass
+    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle node[left=6pt]{$m$} (0.5*\massw, \spaceh+\massh);
+
+    % Spring, Damper, and Actuator
+    \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
+    \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};
+
+    % Ground
+    \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
+    % Groud Motion
+    \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0);
+    \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
+
+    % Displacements
+    \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(dhigh);
+    \draw[->] (0.5*\massw+1.5*\dispw, \spaceh+\massh) -- ++(0, \disph) node[right]{$x$};
+
+    % Legend
+    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
+      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
+
+    \begin{scope}[shift={(0, \spaceh+\massh)}]
+      \node[piezo={2.2}{1.5}{6}, anchor=south] (piezo) at (0, 0){};
+      \draw[->] (0,0)node[branch]{} -- ++(0, -0.6)node[above right]{$F$}
+
+      \draw[<->] (1.1+0.5*\dispw,0) -- node[midway, right]{$d$} ++(0,1.5);
+
+      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
+        ($(piezo.south west) + (-10pt, 0)$) -- ($(piezo.north west) + (-10pt, 0)$) %
+        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
+    \end{scope}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:general_system_granite_stiffness
+#+caption: Mass Spring Damper representing the granite and a general representation of positioning stages
+#+RESULTS:
+[[file:figs/general_system_granite_stiffness.png]]
+
+The equation of motion are:
+\begin{align}
+  ms^2 x &= (cs + k) (x - w) - F \\
+  F &= G^\prime(s) x \\
+  d &= D^\prime(s) x
+\end{align}
+where:
+- $F$ is the force applied by the position stages on the granite mass
+
+#+begin_important
+We can express $d$ as a function of $w$:
+\begin{equation}
+  \frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
+\end{equation}
+
+This is the transfer function that we would like to minimize.
+#+end_important
+
+Let's verify this formula for a simple mass/spring/damper positioning stage.
+In that case, we have:
+\begin{align*}
+  D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\
+  G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
+\end{align*}
+
+And finally:
+\begin{equation}
+  \frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
+\end{equation}
+which is the same as the previously derived equation.
+
 ** Soft Granite
-Let's initialize a soft granite that will act as an isolation stage from ground motion.
+Let's initialize a soft granite and see how the sensitivity to disturbances will change.
 #+begin_src matlab
   initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));
 #+end_src
 
-#+begin_src matlab
-  Ks = logspace(3,9,7); % [N/m]
-#+end_src
-
 #+begin_src matlab :exports none
   Gdr = {zeros(length(Ks), 1)};
-#+end_src
 
-#+begin_src matlab
   for i = 1:length(Ks)
       initializeNanoHexapod('k', Ks(i));
 
@@ -504,6 +593,11 @@ Let's initialize a soft granite that will act as an isolation stage from ground
 #+end_src
 
 ** Effect of the Granite transfer function
+From Figure [[fig:opt_stiff_soft_granite_Dw]], we can see that having a "soft" granite suspension greatly lowers the sensitivity to ground motion.
+The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
+
+From Figures [[fig:opt_stiff_soft_granite_Frz]] and [[fig:opt_stiff_soft_granite_Fd]], we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
+
 #+begin_src matlab :exports none
   freqs = logspace(0, 3, 1000);
 
@@ -520,9 +614,18 @@ Let's initialize a soft granite that will act as an isolation stage from ground
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
-  legend('location', 'southeast');
+  legend('location', 'southwest');
 #+end_src
 
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Dw.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_soft_granite_Dw
+#+caption: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Dw.png][png]], [[./figs/opt_stiff_soft_granite_Dw.pdf][pdf]])
+[[file:figs/opt_stiff_soft_granite_Dw.png]]
+
 #+begin_src matlab :exports none
   freqs = logspace(0, 3, 1000);
 
@@ -539,9 +642,51 @@ Let's initialize a soft granite that will act as an isolation stage from ground
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
-  legend('location', 'southeast');
+  legend('location', 'southwest');
 #+end_src
 
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_soft_granite_Frz
+#+caption: Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Frz.png][png]], [[./figs/opt_stiff_soft_granite_Frz.pdf][pdf]])
+[[file:figs/opt_stiff_soft_granite_Frz.png]]
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 3, 1000);
+
+  figure;
+  hold on;
+  for i = 1:length(Ks)
+      set(gca,'ColorOrderIndex',i);
+      plot(freqs, abs(squeeze(freqresp(Gd{i}( 'Ez', 'Fdz'), freqs, 'Hz'))), '-', ...
+           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+      set(gca,'ColorOrderIndex',i);
+      plot(freqs, abs(squeeze(freqresp(Gdr{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--', ...
+           'HandleVisibility', 'off');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('$E_{z}/F_{d,z}$ [m/N]'); xlabel('Frequency [Hz]');
+
+  legend('location', 'northeast');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Fd.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:opt_stiff_soft_granite_Fd
+#+caption: Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Fd.png][png]], [[./figs/opt_stiff_soft_granite_Fd.pdf][pdf]])
+[[file:figs/opt_stiff_soft_granite_Fd.png]]
+
+** Conclusion
+#+begin_important
+  Having a soft granite suspension could greatly improve the sensitivity the ground motion and thus the level of sample vibration if it is found that ground motion is the limiting factor.
+#+end_important
 
 * Open Loop Budget Error
 <<sec:open_loop_budget_error>>
