diff --git a/.SimulinkProject/Root.type.ProjectPath/1ca9b462-4cfc-4efa-a446-402b77a1e6fd.type.Reference.xml b/.SimulinkProject/Root.type.ProjectPath/1ca9b462-4cfc-4efa-a446-402b77a1e6fd.type.Reference.xml
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@@ -0,0 +1,2 @@
+<?xml version='1.0' encoding='UTF-8'?>
+<Info Ref="matlab" Type="Relative" />
\ No newline at end of file
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+++ b/active-damping.html
@@ -1,14 +1,13 @@
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-<?xml version="1.0" encoding="utf-8"?>
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 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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 <head>
-<!-- 2020-01-22 mer. 16:31 -->
+<!-- 2020-02-06 jeu. 15:39 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <meta name="viewport" content="width=device-width, initial-scale=1" />
-<title>Stewart Platform - Active Damping</title>
+<title>Stewart Platform - Decentralized Active Damping</title>
 <meta name="generator" content="Org mode" />
 <meta name="author" content="Dehaeze Thomas" />
 <style type="text/css">
@@ -247,6 +246,16 @@ for the JavaScript code in this tag.
  }
 /*]]>*///-->
 </script>
+<script>
+      MathJax = {
+          tex: { macros: {
+                  bm: ["\\boldsymbol{#1}",1],
+                  }
+              }
+          };
+          </script>
+          <script type="text/javascript"
+          src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
 </head>
 <body>
 <div id="org-div-home-and-up">
@@ -254,55 +263,72 @@ for the JavaScript code in this tag.
  |
  <a accesskey="H" href="./index.html"> HOME </a>
 </div><div id="content">
-<h1 class="title">Stewart Platform - Active Damping</h1>
+<h1 class="title">Stewart Platform - Decentralized Active Damping</h1>
 <div id="table-of-contents">
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#orgbab9da7">1. Inertial Control</a>
+<li><a href="#orgfba33d4">1. Inertial Control</a>
 <ul>
-<li><a href="#orgd9e8d20">1.1. Simscape Model</a></li>
-<li><a href="#orge6285ea">1.2. Initialize the Stewart model</a></li>
-<li><a href="#org471f4bb">1.3. Identification of the Dynamics</a></li>
-<li><a href="#org30b2e11">1.4. Analysis</a></li>
-<li><a href="#org7f07dca">1.5. Conclusion</a></li>
+<li><a href="#org0ea4bd4">1.1. Identification of the Dynamics</a></li>
+<li><a href="#org5a29480">1.2. Effect of the Flexible Joint stiffness on the Dynamics</a></li>
+<li><a href="#orga92be75">1.3. Obtained Damping</a></li>
+<li><a href="#orgb29f377">1.4. Conclusion</a></li>
+</ul>
+</li>
+<li><a href="#org5fde56d">2. Integral Force Feedback</a>
+<ul>
+<li><a href="#org8823e64">2.1. Identification of the Dynamics with perfect Joints</a></li>
+<li><a href="#org2aff899">2.2. Effect of the Flexible Joint stiffness on the Dynamics</a></li>
+<li><a href="#org40dffdd">2.3. Obtained Damping</a></li>
+<li><a href="#org2ae5aaf">2.4. Conclusion</a></li>
+</ul>
+</li>
+<li><a href="#org9425768">3. Direct Velocity Feedback</a>
+<ul>
+<li><a href="#org61043ac">3.1. Identification of the Dynamics with perfect Joints</a></li>
+<li><a href="#org8f71141">3.2. Effect of the Flexible Joint stiffness on the Dynamics</a></li>
+<li><a href="#org87c6911">3.3. Obtained Damping</a></li>
+<li><a href="#org516fed1">3.4. Conclusion</a></li>
 </ul>
 </li>
 </ul>
 </div>
 </div>
 
-<div id="outline-container-orgbab9da7" class="outline-2">
-<h2 id="orgbab9da7"><span class="section-number-2">1</span> Inertial Control</h2>
+<p>
+The following decentralized active damping techniques are briefly studied:
+</p>
+<ul class="org-ul">
+<li>Inertial Control (proportional feedback of the absolute velocity): Section <a href="#org3c68d9e">1</a></li>
+<li>Integral Force Feedback: Section <a href="#org62cd19c">2</a></li>
+<li>Direct feedback of the relative velocity of each strut: Section <a href="#org587277a">3</a></li>
+</ul>
+
+<div id="outline-container-orgfba33d4" class="outline-2">
+<h2 id="orgfba33d4"><span class="section-number-2">1</span> Inertial Control</h2>
 <div class="outline-text-2" id="text-1">
+<p>
+<a id="org3c68d9e"></a>
+</p>
 </div>
-<div id="outline-container-orgd9e8d20" class="outline-3">
-<h3 id="orgd9e8d20"><span class="section-number-3">1.1</span> Simscape Model</h3>
+
+<div id="outline-container-org0ea4bd4" class="outline-3">
+<h3 id="org0ea4bd4"><span class="section-number-3">1.1</span> Identification of the Dynamics</h3>
 <div class="outline-text-3" id="text-1-1">
 <div class="org-src-container">
-<pre class="src src-matlab">open(<span class="org-string">'simulink/stewart_active_damping.slx'</span>)
-</pre>
-</div>
-</div>
-</div>
-
-<div id="outline-container-orge6285ea" class="outline-3">
-<h3 id="orge6285ea"><span class="section-number-3">1.2</span> Initialize the Stewart model</h3>
-<div class="outline-text-3" id="text-1-2">
-<div class="org-src-container">
 <pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
-stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 40e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 45e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 5e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 5e<span class="org-type">-</span>3);
+stewart = generateGeneralConfiguration(stewart);
 stewart = computeJointsPose(stewart);
-stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones(6,1));
+stewart = initializeStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
 stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
 </pre>
 </div>
-</div>
-</div>
 
-<div id="outline-container-org471f4bb" class="outline-3">
-<h3 id="org471f4bb"><span class="section-number-3">1.3</span> Identification of the Dynamics</h3>
-<div class="outline-text-3" id="text-1-3">
 <div class="org-src-container">
 <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
 options = linearizeOptions;
@@ -313,59 +339,373 @@ mdl = <span class="org-string">'stewart_active_damping'</span>;
 
 <span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
 clear io; io_i = 1;
-io(io_i) = linio([mdl, <span class="org-string">'/F'</span>],   1, <span class="org-string">'openinput'</span>);  io_i = io_i <span class="org-type">+</span> 1;
-io(io_i) = linio([mdl, <span class="org-string">'/WVB'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
-io(io_i) = linio([mdl, <span class="org-string">'/Dm'</span>],  1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
+io(io_i) = linio([mdl, <span class="org-string">'/F'</span>],   1, <span class="org-string">'openinput'</span>);  io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
+io(io_i) = linio([mdl, <span class="org-string">'/Vm'</span>],  1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute velocity of each leg [m/s]</span>
 
 <span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
 G = linearize(mdl, io, options);
 G.InputName  = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
-G.OutputName = {<span class="org-string">'Vx'</span>, <span class="org-string">'Vy'</span>, <span class="org-string">'Vz'</span>, <span class="org-string">'Wx'</span>, <span class="org-string">'Wy'</span>, <span class="org-string">'Wz'</span>, ...
-                <span class="org-string">'D1'</span>, <span class="org-string">'D2'</span>, <span class="org-string">'D3'</span>, <span class="org-string">'D4'</span>, <span class="org-string">'D5'</span>, <span class="org-string">'D6'</span>};
-</pre>
-</div>
-</div>
-</div>
-
-<div id="outline-container-org30b2e11" class="outline-3">
-<h3 id="org30b2e11"><span class="section-number-3">1.4</span> Analysis</h3>
-<div class="outline-text-3" id="text-1-4">
-<div class="org-src-container">
-<pre class="src src-matlab">freqs = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>logspace(1, 4, 1000);
-
-<span class="org-type">figure</span>;
-bode(G({<span class="org-string">'D1'</span>, <span class="org-string">'D2'</span>, <span class="org-string">'D3'</span>, <span class="org-string">'D4'</span>, <span class="org-string">'D5'</span>, <span class="org-string">'D6'</span>}, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}), freqs)
-
-<span class="org-type">figure</span>;
-bode(stewart.J<span class="org-type">*</span>G({<span class="org-string">'Vx'</span>, <span class="org-string">'Vy'</span>, <span class="org-string">'Vz'</span>, <span class="org-string">'Wx'</span>, <span class="org-string">'Wy'</span>, <span class="org-string">'Wz'</span>}, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}), freqs)
+G.OutputName = {<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>};
 </pre>
 </div>
 
-<div class="org-src-container">
-<pre class="src src-matlab"><span class="org-type">figure</span>;
-bode(G({<span class="org-string">'D1'</span>, <span class="org-string">'D2'</span>, <span class="org-string">'D3'</span>, <span class="org-string">'D4'</span>, <span class="org-string">'D5'</span>, <span class="org-string">'D6'</span>}, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}), stewart.J<span class="org-type">*</span>G({<span class="org-string">'Vx'</span>, <span class="org-string">'Vy'</span>, <span class="org-string">'Vz'</span>, <span class="org-string">'Wx'</span>, <span class="org-string">'Wy'</span>, <span class="org-string">'Wz'</span>}, {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>}), freqs)
-</pre>
-</div>
-</div>
-</div>
-
-<div id="outline-container-org7f07dca" class="outline-3">
-<h3 id="org7f07dca"><span class="section-number-3">1.5</span> Conclusion</h3>
-<div class="outline-text-3" id="text-1-5">
 <p>
-It is similar to use:
+The transfer function from actuator forces to force sensors is shown in Figure <a href="#orgfc5367b">1</a>.
 </p>
-<ul class="org-ul">
-<li>one 6dof inertial sensor and the Jacobian the have the velocity of each lim</li>
-<li>6 1dof inertial sensor in each top part of the limbs</li>
-</ul>
+
+<div id="orgfc5367b" class="figure">
+<p><img src="figs/inertial_plant_coupling.png" alt="inertial_plant_coupling.png" />
+</p>
+<p><span class="figure-number">Figure 1: </span>Transfer function from the Actuator force \(F_{i}\) to the absolute velocity of the same leg \(v_{m,i}\) and to the absolute velocity of the other legs \(v_{m,j}\) with \(i \neq j\) in grey (<a href="./figs/inertial_plant_coupling.png">png</a>, <a href="./figs/inertial_plant_coupling.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org5a29480" class="outline-3">
+<h3 id="org5a29480"><span class="section-number-3">1.2</span> Effect of the Flexible Joint stiffness on the Dynamics</h3>
+<div class="outline-text-3" id="text-1-2">
+<p>
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
+Gf = linearize(mdl, io, options);
+Gf.InputName  = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
+Gf.OutputName = {<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>};
+</pre>
+</div>
+
+<p>
+The new dynamics from force actuator to force sensor is shown in Figure <a href="#org2ee5d65">2</a>.
+</p>
+
+<div id="org2ee5d65" class="figure">
+<p><img src="figs/inertial_plant_flexible_joint_decentralized.png" alt="inertial_plant_flexible_joint_decentralized.png" />
+</p>
+<p><span class="figure-number">Figure 2: </span>Transfer function from the Actuator force \(F_{i}\) to the absolute velocity sensor \(v_{m,i}\) (<a href="./figs/inertial_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/inertial_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orga92be75" class="outline-3">
+<h3 id="orga92be75"><span class="section-number-3">1.3</span> Obtained Damping</h3>
+<div class="outline-text-3" id="text-1-3">
+<p>
+The control is a performed in a decentralized manner.
+The \(6 \times 6\) control is a diagonal matrix with pure proportional action on the diagonal:
+\[ K(s) = g
+  \begin{bmatrix}
+    1 & & 0 \\
+    & \ddots & \\
+    0 & & 1
+  \end{bmatrix} \]
+</p>
+
+<p>
+The root locus is shown in figure <a href="#org78a599c">3</a> and the obtained pole damping function of the control gain is shown in figure <a href="#org0b6bb28">4</a>.
+</p>
+
+<div id="org78a599c" class="figure">
+<p><img src="figs/root_locus_inertial_rot_stiffness.png" alt="root_locus_inertial_rot_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 3: </span>Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints (<a href="./figs/root_locus_inertial_rot_stiffness.png">png</a>, <a href="./figs/root_locus_inertial_rot_stiffness.pdf">pdf</a>)</p>
+</div>
+
+
+<div id="org0b6bb28" class="figure">
+<p><img src="figs/pole_damping_gain_inertial_rot_stiffness.png" alt="pole_damping_gain_inertial_rot_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 4: </span>Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints (<a href="./figs/pole_damping_gain_inertial_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_inertial_rot_stiffness.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgb29f377" class="outline-3">
+<h3 id="orgb29f377"><span class="section-number-3">1.4</span> Conclusion</h3>
+<div class="outline-text-3" id="text-1-4">
+<div class="important">
+<p>
+Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
+</p>
+
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org5fde56d" class="outline-2">
+<h2 id="org5fde56d"><span class="section-number-2">2</span> Integral Force Feedback</h2>
+<div class="outline-text-2" id="text-2">
+<p>
+<a id="org62cd19c"></a>
+</p>
+</div>
+
+<div id="outline-container-org8823e64" class="outline-3">
+<h3 id="org8823e64"><span class="section-number-3">2.1</span> Identification of the Dynamics with perfect Joints</h3>
+<div class="outline-text-3" id="text-2-1">
+<p>
+We first initialize the Stewart platform without joint stiffness.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+</pre>
+</div>
+
+<p>
+And we identify the dynamics from force actuators to force sensors.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
+options = linearizeOptions;
+options.SampleTime = 0;
+
+<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
+mdl = <span class="org-string">'stewart_active_damping'</span>;
+
+<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
+clear io; io_i = 1;
+io(io_i) = linio([mdl, <span class="org-string">'/F'</span>],   1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
+io(io_i) = linio([mdl, <span class="org-string">'/Fm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Force Sensor Outputs [N]</span>
+
+<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
+G = linearize(mdl, io, options);
+G.InputName  = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
+G.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
+</pre>
+</div>
+
+<p>
+The transfer function from actuator forces to force sensors is shown in Figure <a href="#orgae4e327">5</a>.
+</p>
+
+<div id="orgae4e327" class="figure">
+<p><img src="figs/iff_plant_coupling.png" alt="iff_plant_coupling.png" />
+</p>
+<p><span class="figure-number">Figure 5: </span>Transfer function from the Actuator force \(F_{i}\) to the Force sensor of the same leg \(F_{m,i}\) and to the force sensor of the other legs \(F_{m,j}\) with \(i \neq j\) in grey (<a href="./figs/iff_plant_coupling.png">png</a>, <a href="./figs/iff_plant_coupling.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org2aff899" class="outline-3">
+<h3 id="org2aff899"><span class="section-number-3">2.2</span> Effect of the Flexible Joint stiffness on the Dynamics</h3>
+<div class="outline-text-3" id="text-2-2">
+<p>
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
+Gf = linearize(mdl, io, options);
+Gf.InputName  = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
+Gf.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
+</pre>
+</div>
+
+<p>
+The new dynamics from force actuator to force sensor is shown in Figure <a href="#orgd21a8a8">6</a>.
+</p>
+
+<div id="orgd21a8a8" class="figure">
+<p><img src="figs/iff_plant_flexible_joint_decentralized.png" alt="iff_plant_flexible_joint_decentralized.png" />
+</p>
+<p><span class="figure-number">Figure 6: </span>Transfer function from the Actuator force \(F_{i}\) to the force sensor \(F_{m,i}\) (<a href="./figs/iff_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/iff_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org40dffdd" class="outline-3">
+<h3 id="org40dffdd"><span class="section-number-3">2.3</span> Obtained Damping</h3>
+<div class="outline-text-3" id="text-2-3">
+<p>
+The control is a performed in a decentralized manner.
+The \(6 \times 6\) control is a diagonal matrix with pure integration action on the diagonal:
+\[ K(s) = g
+  \begin{bmatrix}
+    \frac{1}{s} & & 0 \\
+    & \ddots & \\
+    0 & & \frac{1}{s}
+  \end{bmatrix} \]
+</p>
+
+<p>
+The root locus is shown in figure <a href="#org2cdbf69">7</a> and the obtained pole damping function of the control gain is shown in figure <a href="#orge344229">8</a>.
+</p>
+
+<div id="org2cdbf69" class="figure">
+<p><img src="figs/root_locus_iff_rot_stiffness.png" alt="root_locus_iff_rot_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 7: </span>Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints (<a href="./figs/root_locus_iff_rot_stiffness.png">png</a>, <a href="./figs/root_locus_iff_rot_stiffness.pdf">pdf</a>)</p>
+</div>
+
+
+<div id="orge344229" class="figure">
+<p><img src="figs/pole_damping_gain_iff_rot_stiffness.png" alt="pole_damping_gain_iff_rot_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 8: </span>Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints (<a href="./figs/pole_damping_gain_iff_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_iff_rot_stiffness.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org2ae5aaf" class="outline-3">
+<h3 id="org2ae5aaf"><span class="section-number-3">2.4</span> Conclusion</h3>
+<div class="outline-text-3" id="text-2-4">
+<div class="important">
+<p>
+The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
+Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
+</p>
+
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org9425768" class="outline-2">
+<h2 id="org9425768"><span class="section-number-2">3</span> Direct Velocity Feedback</h2>
+<div class="outline-text-2" id="text-3">
+<p>
+<a id="org587277a"></a>
+</p>
+</div>
+
+<div id="outline-container-org61043ac" class="outline-3">
+<h3 id="org61043ac"><span class="section-number-3">3.1</span> Identification of the Dynamics with perfect Joints</h3>
+<div class="outline-text-3" id="text-3-1">
+<p>
+We first initialize the Stewart platform without joint stiffness.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+</pre>
+</div>
+
+<p>
+And we identify the dynamics from force actuators to force sensors.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
+options = linearizeOptions;
+options.SampleTime = 0;
+
+<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
+mdl = <span class="org-string">'stewart_active_damping'</span>;
+
+<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
+clear io; io_i = 1;
+io(io_i) = linio([mdl, <span class="org-string">'/F'</span>],   1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
+io(io_i) = linio([mdl, <span class="org-string">'/Dm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [N]</span>
+
+<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
+G = linearize(mdl, io, options);
+G.InputName  = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
+G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
+</pre>
+</div>
+
+<p>
+The transfer function from actuator forces to relative motion sensors is shown in Figure <a href="#orgd8d51db">9</a>.
+</p>
+
+<div id="orgd8d51db" class="figure">
+<p><img src="figs/dvf_plant_coupling.png" alt="dvf_plant_coupling.png" />
+</p>
+<p><span class="figure-number">Figure 9: </span>Transfer function from the Actuator force \(F_{i}\) to the Relative Motion Sensor \(D_{m,j}\) with \(i \neq j\) (<a href="./figs/dvf_plant_coupling.png">png</a>, <a href="./figs/dvf_plant_coupling.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+
+<div id="outline-container-org8f71141" class="outline-3">
+<h3 id="org8f71141"><span class="section-number-3">3.2</span> Effect of the Flexible Joint stiffness on the Dynamics</h3>
+<div class="outline-text-3" id="text-3-2">
+<p>
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
+Gf = linearize(mdl, io, options);
+Gf.InputName  = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
+Gf.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
+</pre>
+</div>
+
+<p>
+The new dynamics from force actuator to relative motion sensor is shown in Figure <a href="#orgb18f950">10</a>.
+</p>
+
+<div id="orgb18f950" class="figure">
+<p><img src="figs/dvf_plant_flexible_joint_decentralized.png" alt="dvf_plant_flexible_joint_decentralized.png" />
+</p>
+<p><span class="figure-number">Figure 10: </span>Transfer function from the Actuator force \(F_{i}\) to the relative displacement sensor \(D_{m,i}\) (<a href="./figs/dvf_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/dvf_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org87c6911" class="outline-3">
+<h3 id="org87c6911"><span class="section-number-3">3.3</span> Obtained Damping</h3>
+<div class="outline-text-3" id="text-3-3">
+<p>
+The control is a performed in a decentralized manner.
+The \(6 \times 6\) control is a diagonal matrix with pure derivative action on the diagonal:
+\[ K(s) = g
+  \begin{bmatrix}
+    s & & \\
+    & \ddots & \\
+    & & s
+  \end{bmatrix} \]
+</p>
+
+<p>
+The root locus is shown in figure <a href="#org5cb31c8">11</a> and the obtained pole damping function of the control gain is shown in figure <a href="#org4618492">12</a>.
+</p>
+
+<div id="org5cb31c8" class="figure">
+<p><img src="figs/root_locus_dvf_rot_stiffness.png" alt="root_locus_dvf_rot_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 11: </span>Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints (<a href="./figs/root_locus_dvf_rot_stiffness.png">png</a>, <a href="./figs/root_locus_dvf_rot_stiffness.pdf">pdf</a>)</p>
+</div>
+
+
+<div id="org4618492" class="figure">
+<p><img src="figs/pole_damping_gain_dvf_rot_stiffness.png" alt="pole_damping_gain_dvf_rot_stiffness.png" />
+</p>
+<p><span class="figure-number">Figure 12: </span>Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints (<a href="./figs/pole_damping_gain_dvf_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_dvf_rot_stiffness.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org516fed1" class="outline-3">
+<h3 id="org516fed1"><span class="section-number-3">3.4</span> Conclusion</h3>
+<div class="outline-text-3" id="text-3-4">
+<div class="important">
+<p>
+Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
+</p>
+
+</div>
 </div>
 </div>
 </div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-01-22 mer. 16:31</p>
+<p class="date">Created: 2020-02-06 jeu. 15:39</p>
 </div>
 </body>
 </html>
diff --git a/active-damping.org b/active-damping.org
index 24c18ac..027790a 100644
--- a/active-damping.org
+++ b/active-damping.org
@@ -1,4 +1,4 @@
-#+TITLE: Stewart Platform - Active Damping
+#+TITLE: Stewart Platform - Decentralized Active Damping
 :DRAWER:
 #+HTML_LINK_HOME: ./index.html
 #+HTML_LINK_UP: ./index.html
@@ -20,7 +20,249 @@
 #+PROPERTY: header-args:matlab+ :output-dir figs
 :END:
 
+* Introduction                                                        :ignore:
+
+The following decentralized active damping techniques are briefly studied:
+- Inertial Control (proportional feedback of the absolute velocity): Section [[sec:active_damping_inertial]]
+- Integral Force Feedback: Section [[sec:active_damping_iff]]
+- Direct feedback of the relative velocity of each strut: Section [[sec:active_damping_dvf]]
+
 * Inertial Control
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/active_damping_inertial.m
+:header-args:matlab+: :comments org :mkdirp yes
+:END:
+<<sec:active_damping_inertial>>
+
+** Introduction                                                      :ignore:
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+<<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+<<matlab-init>>
+#+end_src
+
+#+begin_src matlab
+  simulinkproject('./');
+#+end_src
+
+#+begin_src matlab
+  open('simulink/stewart_active_damping.slx')
+#+end_src
+
+** Identification of the Dynamics
+#+begin_src matlab
+  stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+  stewart = initializeStrutDynamics(stewart);
+  stewart = initializeJointDynamics(stewart, 'disable', true);
+  stewart = initializeCylindricalPlatforms(stewart);
+  stewart = initializeCylindricalStruts(stewart);
+  stewart = computeJacobian(stewart);
+  stewart = initializeStewartPose(stewart);
+#+end_src
+
+#+begin_src matlab
+  %% Options for Linearized
+  options = linearizeOptions;
+  options.SampleTime = 0;
+
+  %% Name of the Simulink File
+  mdl = 'stewart_active_damping';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/F'],   1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
+  io(io_i) = linio([mdl, '/Vm'],  1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]
+
+  %% Run the linearization
+  G = linearize(mdl, io, options);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
+#+end_src
+
+The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  for i = 2:6
+    set(gca,'ColorOrderIndex',2);
+    plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 2:6
+    set(gca,'ColorOrderIndex',2);
+    p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
+  end
+  set(gca,'ColorOrderIndex',1);
+  p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([-180, 180]);
+  yticks([-180, -90, 0, 90, 180]);
+  legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
+
+  linkaxes([ax1,ax2],'x');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/inertial_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:inertial_plant_coupling
+#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity of the same leg $v_{m,i}$ and to the absolute velocity of the other legs $v_{m,j}$ with $i \neq j$ in grey ([[./figs/inertial_plant_coupling.png][png]], [[./figs/inertial_plant_coupling.pdf][pdf]])
+[[file:figs/inertial_plant_coupling.png]]
+
+** Effect of the Flexible Joint stiffness on the Dynamics
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+#+begin_src matlab
+  stewart = initializeJointDynamics(stewart);
+  Gf = linearize(mdl, io, options);
+  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
+#+end_src
+
+The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([-180, 180]);
+  yticks([-180, -90, 0, 90, 180]);
+  legend('location', 'southwest')
+
+  linkaxes([ax1,ax2],'x');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/inertial_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:inertial_plant_flexible_joint_decentralized
+#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity sensor $v_{m,i}$ ([[./figs/inertial_plant_flexible_joint_decentralized.png][png]], [[./figs/inertial_plant_flexible_joint_decentralized.pdf][pdf]])
+[[file:figs/inertial_plant_flexible_joint_decentralized.png]]
+
+** Obtained Damping
+The control is a performed in a decentralized manner.
+The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
+\[ K(s) = g
+  \begin{bmatrix}
+    1 & & 0 \\
+    & \ddots & \\
+    0 & & 1
+  \end{bmatrix} \]
+
+The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
+#+begin_src matlab :exports none
+  gains = logspace(0, 5, 1000);
+
+  figure;
+  hold on;
+  plot(real(pole(G)),  imag(pole(G)),  'x');
+  plot(real(pole(Gf)), imag(pole(Gf)), 'x');
+  set(gca,'ColorOrderIndex',1);
+  plot(real(tzero(G)),  imag(tzero(G)),  'o');
+  plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
+  for i = 1:length(gains)
+    cl_poles = pole(feedback(G, gains(i)*eye(6)));
+    set(gca,'ColorOrderIndex',1);
+    plot(real(cl_poles), imag(cl_poles), '.');
+    cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
+    set(gca,'ColorOrderIndex',2);
+    plot(real(cl_poles), imag(cl_poles), '.');
+  end
+  ylim([0,2000]);
+  xlim([-2000,0]);
+  xlabel('Real Part')
+  ylabel('Imaginary Part')
+  axis square
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/root_locus_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:root_locus_inertial_rot_stiffness
+#+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
+[[file:figs/root_locus_inertial_rot_stiffness.png]]
+
+#+begin_src matlab :exports none
+  gains = logspace(0, 5, 1000);
+
+  figure;
+  hold on;
+  for i = 1:length(gains)
+    set(gca,'ColorOrderIndex',1);
+    cl_poles = pole(feedback(G, gains(i)*eye(6)));
+    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+    set(gca,'ColorOrderIndex',2);
+    cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
+    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+  end
+  xlabel('Control Gain');
+  ylabel('Damping of the Poles');
+  set(gca, 'XScale', 'log');
+  ylim([0,pi/2]);
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/pole_damping_gain_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:pole_damping_gain_inertial_rot_stiffness
+#+caption: Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_inertial_rot_stiffness.png][png]], [[./figs/pole_damping_gain_inertial_rot_stiffness.pdf][pdf]])
+[[file:figs/pole_damping_gain_inertial_rot_stiffness.png]]
+
+** Conclusion
+#+begin_important
+  Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
+#+end_important
+
+* Integral Force Feedback
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/active_damping_iff.m
+:header-args:matlab+: :comments org :mkdirp yes
+:END:
+<<sec:active_damping_iff>>
+
+** Introduction                                                      :ignore:
 ** Matlab Init                                             :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
@@ -34,24 +276,25 @@
   simulinkproject('./');
 #+end_src
 
-** Simscape Model
 #+begin_src matlab
   open('simulink/stewart_active_damping.slx')
 #+end_src
 
-** Initialize the Stewart model
+** Identification of the Dynamics with perfect Joints
+We first initialize the Stewart platform without joint stiffness.
 #+begin_src matlab
   stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
-  stewart = generateCubicConfiguration(stewart, 'Hc', 40e-3, 'FOc', 45e-3, 'FHa', 5e-3, 'MHb', 5e-3);
+  stewart = generateGeneralConfiguration(stewart);
   stewart = computeJointsPose(stewart);
-  stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1));
+  stewart = initializeStrutDynamics(stewart);
+  stewart = initializeJointDynamics(stewart, 'disable', true);
   stewart = initializeCylindricalPlatforms(stewart);
   stewart = initializeCylindricalStruts(stewart);
   stewart = computeJacobian(stewart);
   stewart = initializeStewartPose(stewart);
 #+end_src
 
-** Identification of the Dynamics
+And we identify the dynamics from force actuators to force sensors.
 #+begin_src matlab
   %% Options for Linearized
   options = linearizeOptions;
@@ -62,34 +305,414 @@
 
   %% Input/Output definition
   clear io; io_i = 1;
-  io(io_i) = linio([mdl, '/F'],   1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/WVB'], 1, 'openoutput'); io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Dm'],  1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/F'],   1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
+  io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]
 
   %% Run the linearization
   G = linearize(mdl, io, options);
   G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
-  G.OutputName = {'Vx', 'Vy', 'Vz', 'Wx', 'Wy', 'Wz', ...
-                  'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
+  G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
 #+end_src
 
-** Analysis
+The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  for i = 2:6
+    set(gca,'ColorOrderIndex',2);
+    plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 2:6
+    set(gca,'ColorOrderIndex',2);
+    p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
+  end
+  set(gca,'ColorOrderIndex',1);
+  p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([-180, 180]);
+  yticks([-180, -90, 0, 90, 180]);
+  legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
+
+  linkaxes([ax1,ax2],'x');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/iff_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:iff_plant_coupling
+#+caption: Transfer function from the Actuator force $F_{i}$ to the Force sensor of the same leg $F_{m,i}$ and to the force sensor of the other legs $F_{m,j}$ with $i \neq j$ in grey ([[./figs/iff_plant_coupling.png][png]], [[./figs/iff_plant_coupling.pdf][pdf]])
+[[file:figs/iff_plant_coupling.png]]
+
+** Effect of the Flexible Joint stiffness on the Dynamics
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
 #+begin_src matlab
-  freqs = 2*pi*logspace(1, 4, 1000);
-
-  figure;
-  bode(G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), freqs)
-
-  figure;
-  bode(stewart.J*G({'Vx', 'Vy', 'Vz', 'Wx', 'Wy', 'Wz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), freqs)
+  stewart = initializeJointDynamics(stewart);
+  Gf = linearize(mdl, io, options);
+  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
 #+end_src
 
-#+begin_src matlab
+The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
   figure;
-  bode(G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), stewart.J*G({'Vx', 'Vy', 'Vz', 'Wx', 'Wy', 'Wz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), freqs)
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([-180, 180]);
+  yticks([-180, -90, 0, 90, 180]);
+  legend('location', 'southwest')
+
+  linkaxes([ax1,ax2],'x');
 #+end_src
 
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/iff_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:iff_plant_flexible_joint_decentralized
+#+caption: Transfer function from the Actuator force $F_{i}$ to the force sensor $F_{m,i}$ ([[./figs/iff_plant_flexible_joint_decentralized.png][png]], [[./figs/iff_plant_flexible_joint_decentralized.pdf][pdf]])
+[[file:figs/iff_plant_flexible_joint_decentralized.png]]
+
+** Obtained Damping
+The control is a performed in a decentralized manner.
+The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
+\[ K(s) = g
+  \begin{bmatrix}
+    \frac{1}{s} & & 0 \\
+    & \ddots & \\
+    0 & & \frac{1}{s}
+  \end{bmatrix} \]
+
+The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
+#+begin_src matlab :exports none
+  gains = logspace(0, 5, 1000);
+
+  figure;
+  hold on;
+  plot(real(pole(G)),  imag(pole(G)),  'x');
+  plot(real(pole(Gf)), imag(pole(Gf)), 'x');
+  set(gca,'ColorOrderIndex',1);
+  plot(real(tzero(G)),  imag(tzero(G)),  'o');
+  plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
+  for i = 1:length(gains)
+    cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
+    set(gca,'ColorOrderIndex',1);
+    plot(real(cl_poles), imag(cl_poles), '.');
+    cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
+    set(gca,'ColorOrderIndex',2);
+    plot(real(cl_poles), imag(cl_poles), '.');
+  end
+  ylim([0,inf]);
+  xlim([-3000,0]);
+  xlabel('Real Part')
+  ylabel('Imaginary Part')
+  axis square
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/root_locus_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:root_locus_iff_rot_stiffness
+#+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
+[[file:figs/root_locus_iff_rot_stiffness.png]]
+
+#+begin_src matlab :exports none
+  gains = logspace(0, 5, 1000);
+
+  figure;
+  hold on;
+  for i = 1:length(gains)
+    set(gca,'ColorOrderIndex',1);
+    cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
+    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+    set(gca,'ColorOrderIndex',2);
+    cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
+    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+  end
+  xlabel('Control Gain');
+  ylabel('Damping of the Poles');
+  set(gca, 'XScale', 'log');
+  ylim([0,pi/2]);
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/pole_damping_gain_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:pole_damping_gain_iff_rot_stiffness
+#+caption: Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_iff_rot_stiffness.png][png]], [[./figs/pole_damping_gain_iff_rot_stiffness.pdf][pdf]])
+[[file:figs/pole_damping_gain_iff_rot_stiffness.png]]
+
 ** Conclusion
-It is similar to use:
-- one 6dof inertial sensor and the Jacobian the have the velocity of each lim
-- 6 1dof inertial sensor in each top part of the limbs
+#+begin_important
+  The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
+  Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
+#+end_important
+
+* Direct Velocity Feedback
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/active_damping_dvf.m
+:header-args:matlab+: :comments org :mkdirp yes
+:END:
+<<sec:active_damping_dvf>>
+
+** Introduction                                                      :ignore:
+** Matlab Init                                             :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+<<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+<<matlab-init>>
+#+end_src
+
+#+begin_src matlab
+  simulinkproject('./');
+#+end_src
+
+#+begin_src matlab
+  open('simulink/stewart_active_damping.slx')
+#+end_src
+
+** Identification of the Dynamics with perfect Joints
+We first initialize the Stewart platform without joint stiffness.
+#+begin_src matlab
+  stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+  stewart = initializeStrutDynamics(stewart);
+  stewart = initializeJointDynamics(stewart, 'disable', true);
+  stewart = initializeCylindricalPlatforms(stewart);
+  stewart = initializeCylindricalStruts(stewart);
+  stewart = computeJacobian(stewart);
+  stewart = initializeStewartPose(stewart);
+#+end_src
+
+And we identify the dynamics from force actuators to force sensors.
+#+begin_src matlab
+  %% Options for Linearized
+  options = linearizeOptions;
+  options.SampleTime = 0;
+
+  %% Name of the Simulink File
+  mdl = 'stewart_active_damping';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/F'],   1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
+  io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]
+
+  %% Run the linearization
+  G = linearize(mdl, io, options);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
+#+end_src
+
+The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  for i = 2:6
+    set(gca,'ColorOrderIndex',2);
+    plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 2:6
+    set(gca,'ColorOrderIndex',2);
+    p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
+  end
+  set(gca,'ColorOrderIndex',1);
+  p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([-180, 180]);
+  yticks([-180, -90, 0, 90, 180]);
+  legend([p1, p2], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})
+
+  linkaxes([ax1,ax2],'x');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/dvf_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:dvf_plant_coupling
+#+caption: Transfer function from the Actuator force $F_{i}$ to the Relative Motion Sensor $D_{m,j}$ with $i \neq j$ ([[./figs/dvf_plant_coupling.png][png]], [[./figs/dvf_plant_coupling.pdf][pdf]])
+[[file:figs/dvf_plant_coupling.png]]
+
+
+** Effect of the Flexible Joint stiffness on the Dynamics
+We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+#+begin_src matlab
+  stewart = initializeJointDynamics(stewart);
+  Gf = linearize(mdl, io, options);
+  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
+#+end_src
+
+The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+  ylim([-180, 180]);
+  yticks([-180, -90, 0, 90, 180]);
+  legend('location', 'northeast');
+
+  linkaxes([ax1,ax2],'x');
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/dvf_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:dvf_plant_flexible_joint_decentralized
+#+caption: Transfer function from the Actuator force $F_{i}$ to the relative displacement sensor $D_{m,i}$ ([[./figs/dvf_plant_flexible_joint_decentralized.png][png]], [[./figs/dvf_plant_flexible_joint_decentralized.pdf][pdf]])
+[[file:figs/dvf_plant_flexible_joint_decentralized.png]]
+
+** Obtained Damping
+The control is a performed in a decentralized manner.
+The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
+\[ K(s) = g
+  \begin{bmatrix}
+    s & & \\
+    & \ddots & \\
+    & & s
+  \end{bmatrix} \]
+
+The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
+#+begin_src matlab :exports none
+  gains = logspace(0, 5, 1000);
+
+  figure;
+  hold on;
+  plot(real(pole(G)),  imag(pole(G)),  'x');
+  plot(real(pole(Gf)), imag(pole(Gf)), 'x');
+  set(gca,'ColorOrderIndex',1);
+  plot(real(tzero(G)),  imag(tzero(G)),  'o');
+  plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
+  for i = 1:length(gains)
+    cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
+    set(gca,'ColorOrderIndex',1);
+    plot(real(cl_poles), imag(cl_poles), '.');
+    cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
+    set(gca,'ColorOrderIndex',2);
+    plot(real(cl_poles), imag(cl_poles), '.');
+  end
+  ylim([0,inf]);
+  xlim([-3000,0]);
+  xlabel('Real Part')
+  ylabel('Imaginary Part')
+  axis square
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/root_locus_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:root_locus_dvf_rot_stiffness
+#+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
+[[file:figs/root_locus_dvf_rot_stiffness.png]]
+
+#+begin_src matlab :exports none
+  gains = logspace(0, 5, 1000);
+
+  figure;
+  hold on;
+  for i = 1:length(gains)
+    set(gca,'ColorOrderIndex',1);
+    cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
+    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+    set(gca,'ColorOrderIndex',2);
+    cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
+    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+  end
+  xlabel('Control Gain');
+  ylabel('Damping of the Poles');
+  set(gca, 'XScale', 'log');
+  ylim([0,pi/2]);
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/pole_damping_gain_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+<<plt-matlab>>
+#+end_src
+
+#+name: fig:pole_damping_gain_dvf_rot_stiffness
+#+caption: Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/pole_damping_gain_dvf_rot_stiffness.png][png]], [[./figs/pole_damping_gain_dvf_rot_stiffness.pdf][pdf]])
+[[file:figs/pole_damping_gain_dvf_rot_stiffness.png]]
+
+** Conclusion
+#+begin_important
+  Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
+#+end_important
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diff --git a/matlab/active_damping_dvf.m b/matlab/active_damping_dvf.m
new file mode 100644
index 0000000..474f4b9
--- /dev/null
+++ b/matlab/active_damping_dvf.m
@@ -0,0 +1,178 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+simulinkproject('./');
+
+open('simulink/stewart_active_damping.slx')
+
+% Identification of the Dynamics with perfect Joints
+% We first initialize the Stewart platform without joint stiffness.
+
+stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart, 'disable', true);
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+
+
+
+% And we identify the dynamics from force actuators to force sensors.
+
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'stewart_active_damping';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F'],   1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
+io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]
+
+%% Run the linearization
+G = linearize(mdl, io, options);
+G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
+
+
+
+% The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
+
+freqs = logspace(1, 3, 1000);
+
+figure;
+
+ax1 = subplot(2, 1, 1);
+hold on;
+for i = 2:6
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+
+ax2 = subplot(2, 1, 2);
+hold on;
+for i = 2:6
+  set(gca,'ColorOrderIndex',2);
+  p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
+end
+set(gca,'ColorOrderIndex',1);
+p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+legend([p1, p2], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})
+
+linkaxes([ax1,ax2],'x');
+
+% Effect of the Flexible Joint stiffness on the Dynamics
+% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+
+stewart = initializeJointDynamics(stewart);
+Gf = linearize(mdl, io, options);
+Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
+
+
+
+% The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
+
+freqs = logspace(1, 3, 1000);
+
+figure;
+
+ax1 = subplot(2, 1, 1);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+
+ax2 = subplot(2, 1, 2);
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+legend('location', 'northeast');
+
+linkaxes([ax1,ax2],'x');
+
+% Obtained Damping
+% The control is a performed in a decentralized manner.
+% The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
+% \[ K(s) = g
+%   \begin{bmatrix}
+%     s & & \\
+%     & \ddots & \\
+%     & & s
+%   \end{bmatrix} \]
+
+% The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+plot(real(pole(G)),  imag(pole(G)),  'x');
+plot(real(pole(Gf)), imag(pole(Gf)), 'x');
+set(gca,'ColorOrderIndex',1);
+plot(real(tzero(G)),  imag(tzero(G)),  'o');
+plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
+for i = 1:length(gains)
+  cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
+  set(gca,'ColorOrderIndex',1);
+  plot(real(cl_poles), imag(cl_poles), '.');
+  cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
+  set(gca,'ColorOrderIndex',2);
+  plot(real(cl_poles), imag(cl_poles), '.');
+end
+ylim([0,inf]);
+xlim([-3000,0]);
+xlabel('Real Part')
+ylabel('Imaginary Part')
+axis square
+
+
+
+% #+name: fig:root_locus_dvf_rot_stiffness
+% #+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
+% [[file:figs/root_locus_dvf_rot_stiffness.png]]
+
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+for i = 1:length(gains)
+  set(gca,'ColorOrderIndex',1);
+  cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
+  poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+  plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+  set(gca,'ColorOrderIndex',2);
+  cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
+  poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+  plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+end
+xlabel('Control Gain');
+ylabel('Damping of the Poles');
+set(gca, 'XScale', 'log');
+ylim([0,pi/2]);
diff --git a/matlab/active_damping_iff.m b/matlab/active_damping_iff.m
new file mode 100644
index 0000000..dbfaa90
--- /dev/null
+++ b/matlab/active_damping_iff.m
@@ -0,0 +1,178 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+simulinkproject('./');
+
+open('simulink/stewart_active_damping.slx')
+
+% Identification of the Dynamics with perfect Joints
+% We first initialize the Stewart platform without joint stiffness.
+
+stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart, 'disable', true);
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+
+
+
+% And we identify the dynamics from force actuators to force sensors.
+
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'stewart_active_damping';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F'],   1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
+io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]
+
+%% Run the linearization
+G = linearize(mdl, io, options);
+G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+
+
+% The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
+
+freqs = logspace(1, 3, 1000);
+
+figure;
+
+ax1 = subplot(2, 1, 1);
+hold on;
+for i = 2:6
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+
+ax2 = subplot(2, 1, 2);
+hold on;
+for i = 2:6
+  set(gca,'ColorOrderIndex',2);
+  p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
+end
+set(gca,'ColorOrderIndex',1);
+p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
+
+linkaxes([ax1,ax2],'x');
+
+% Effect of the Flexible Joint stiffness on the Dynamics
+% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+
+stewart = initializeJointDynamics(stewart);
+Gf = linearize(mdl, io, options);
+Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+
+
+% The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
+
+freqs = logspace(1, 3, 1000);
+
+figure;
+
+ax1 = subplot(2, 1, 1);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
+
+ax2 = subplot(2, 1, 2);
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+legend('location', 'southwest')
+
+linkaxes([ax1,ax2],'x');
+
+% Obtained Damping
+% The control is a performed in a decentralized manner.
+% The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
+% \[ K(s) = g
+%   \begin{bmatrix}
+%     \frac{1}{s} & & 0 \\
+%     & \ddots & \\
+%     0 & & \frac{1}{s}
+%   \end{bmatrix} \]
+
+% The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+plot(real(pole(G)),  imag(pole(G)),  'x');
+plot(real(pole(Gf)), imag(pole(Gf)), 'x');
+set(gca,'ColorOrderIndex',1);
+plot(real(tzero(G)),  imag(tzero(G)),  'o');
+plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
+for i = 1:length(gains)
+  cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
+  set(gca,'ColorOrderIndex',1);
+  plot(real(cl_poles), imag(cl_poles), '.');
+  cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
+  set(gca,'ColorOrderIndex',2);
+  plot(real(cl_poles), imag(cl_poles), '.');
+end
+ylim([0,inf]);
+xlim([-3000,0]);
+xlabel('Real Part')
+ylabel('Imaginary Part')
+axis square
+
+
+
+% #+name: fig:root_locus_iff_rot_stiffness
+% #+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
+% [[file:figs/root_locus_iff_rot_stiffness.png]]
+
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+for i = 1:length(gains)
+  set(gca,'ColorOrderIndex',1);
+  cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
+  poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+  plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+  set(gca,'ColorOrderIndex',2);
+  cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
+  poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+  plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+end
+xlabel('Control Gain');
+ylabel('Damping of the Poles');
+set(gca, 'XScale', 'log');
+ylim([0,pi/2]);
diff --git a/matlab/active_damping_inertial.m b/matlab/active_damping_inertial.m
new file mode 100644
index 0000000..9d6f03f
--- /dev/null
+++ b/matlab/active_damping_inertial.m
@@ -0,0 +1,173 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+simulinkproject('./');
+
+open('simulink/stewart_active_damping.slx')
+
+% Identification of the Dynamics
+
+stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart, 'disable', true);
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'stewart_active_damping';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F'],   1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
+io(io_i) = linio([mdl, '/Vm'],  1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]
+
+%% Run the linearization
+G = linearize(mdl, io, options);
+G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
+
+
+
+% The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
+
+freqs = logspace(1, 3, 1000);
+
+figure;
+
+ax1 = subplot(2, 1, 1);
+hold on;
+for i = 2:6
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
+end
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
+
+ax2 = subplot(2, 1, 2);
+hold on;
+for i = 2:6
+  set(gca,'ColorOrderIndex',2);
+  p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
+end
+set(gca,'ColorOrderIndex',1);
+p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
+
+linkaxes([ax1,ax2],'x');
+
+% Effect of the Flexible Joint stiffness on the Dynamics
+% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
+
+stewart = initializeJointDynamics(stewart);
+Gf = linearize(mdl, io, options);
+Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
+
+
+
+% The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
+
+freqs = logspace(1, 3, 1000);
+
+figure;
+
+ax1 = subplot(2, 1, 1);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
+
+ax2 = subplot(2, 1, 2);
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180, -90, 0, 90, 180]);
+legend('location', 'southwest')
+
+linkaxes([ax1,ax2],'x');
+
+% Obtained Damping
+% The control is a performed in a decentralized manner.
+% The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
+% \[ K(s) = g
+%   \begin{bmatrix}
+%     1 & & 0 \\
+%     & \ddots & \\
+%     0 & & 1
+%   \end{bmatrix} \]
+
+% The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+plot(real(pole(G)),  imag(pole(G)),  'x');
+plot(real(pole(Gf)), imag(pole(Gf)), 'x');
+set(gca,'ColorOrderIndex',1);
+plot(real(tzero(G)),  imag(tzero(G)),  'o');
+plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
+for i = 1:length(gains)
+  cl_poles = pole(feedback(G, gains(i)*eye(6)));
+  set(gca,'ColorOrderIndex',1);
+  plot(real(cl_poles), imag(cl_poles), '.');
+  cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
+  set(gca,'ColorOrderIndex',2);
+  plot(real(cl_poles), imag(cl_poles), '.');
+end
+ylim([0,2000]);
+xlim([-2000,0]);
+xlabel('Real Part')
+ylabel('Imaginary Part')
+axis square
+
+
+
+% #+name: fig:root_locus_inertial_rot_stiffness
+% #+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
+% [[file:figs/root_locus_inertial_rot_stiffness.png]]
+
+
+gains = logspace(0, 5, 1000);
+
+figure;
+hold on;
+for i = 1:length(gains)
+  set(gca,'ColorOrderIndex',1);
+  cl_poles = pole(feedback(G, gains(i)*eye(6)));
+  poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+  plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+  set(gca,'ColorOrderIndex',2);
+  cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
+  poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
+  plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
+end
+xlabel('Control Gain');
+ylabel('Damping of the Poles');
+set(gca, 'XScale', 'log');
+ylim([0,pi/2]);
diff --git a/simscape_subsystems/Stewart_Platform.slx b/simscape_subsystems/Stewart_Platform.slx
index 0cf4090..a9ccecb 100644
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diff --git a/simscape_subsystems/stewart_strut.slx b/simscape_subsystems/stewart_strut.slx
index 26727d0..172fc5b 100644
Binary files a/simscape_subsystems/stewart_strut.slx and b/simscape_subsystems/stewart_strut.slx differ
diff --git a/simulink/stewart_active_damping.slx b/simulink/stewart_active_damping.slx
index 49f1b3f..d0183c1 100644
Binary files a/simulink/stewart_active_damping.slx and b/simulink/stewart_active_damping.slx differ
