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<title>Active Damping of Rotating Platforms using Integral Force Feedback - Matlab Computation</title>
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<h1 class="title">Active Damping of Rotating Platforms using Integral Force Feedback - Matlab Computation</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org06be299">1. System Description and Analysis</a>
<ul>
<li><a href="#org4600b5b">1.1. System description</a></li>
<li><a href="#org2d46257">1.2. Equations</a></li>
<li><a href="#org1698c67">1.3. Numerical Values</a></li>
<li><a href="#org12f3c6c">1.4. Campbell Diagram</a></li>
<li><a href="#org7a2ea7e">1.5. Simscape Model</a></li>
<li><a href="#orgd90ebb8">1.6. Effect of the rotation speed</a></li>
</ul>
</li>
<li><a href="#org59dec10">2. Problem with pure Integral Force Feedback</a>
<ul>
<li><a href="#orge6cb2c2">2.1. Plant Parameters</a></li>
<li><a href="#org86cebac">2.2. Equations</a></li>
<li><a href="#org5733bdd">2.3. Comparison of the Analytical Model and the Simscape Model</a></li>
<li><a href="#org25f390d">2.4. Effect of the rotation speed</a></li>
<li><a href="#orgb92b242">2.5. Decentralized Integral Force Feedback</a></li>
</ul>
</li>
<li><a href="#org2e3307b">3. Integral Force Feedback with an High Pass Filter</a>
<ul>
<li><a href="#orgd87bec0">3.1. Plant Parameters</a></li>
<li><a href="#org38459ba">3.2. Modified Integral Force Feedback Controller</a></li>
<li><a href="#org1fe7230">3.3. Root Locus</a></li>
<li><a href="#orgbe65c44">3.4. What is the optimal \(\omega_i\) and \(g\)?</a></li>
</ul>
</li>
<li><a href="#orgd7e3817">4. IFF with a stiffness in parallel with the force sensor</a>
<ul>
<li><a href="#org84e074b">4.1. Schematic</a></li>
<li><a href="#orgafb020a">4.2. Equations</a></li>
<li><a href="#org218ccb0">4.3. Plant Parameters</a></li>
<li><a href="#org92b8dc6">4.4. Comparison of the Analytical Model and the Simscape Model</a></li>
<li><a href="#orgffb6014">4.5. Effect of the parallel stiffness on the IFF plant</a></li>
<li><a href="#orgccd60cd">4.6. IFF when adding a spring in parallel</a></li>
<li><a href="#org62a8e82">4.7. Effect of \(k_p\) on the attainable damping</a></li>
</ul>
</li>
<li><a href="#org57a9404">5. Comparison</a>
<ul>
<li><a href="#org10a2ed1">5.1. Plant Parameters</a></li>
<li><a href="#orgc78a96c">5.2. Root Locus</a></li>
<li><a href="#org2018523">5.3. Controllers - Optimal Gains</a></li>
<li><a href="#orgfe35652">5.4. Passive Damping - Critical Damping</a></li>
<li><a href="#org8aebf94">5.5. Transmissibility And Compliance</a>
<ul>
<li><a href="#org53602c4">5.5.1. Passive Damping</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org6c5b405">6. Notations</a></li>
</ul>
</div>
</div>
<hr>
<p>This report is also available as a <a href="./index.pdf">pdf</a>.</p>
<hr>
<p>
This document gathers the Matlab code used to for the conference paper (<a href="#citeproc_bib_item_1">Dehaeze and Collette 2020</a>) and the journal paper (<a href="#citeproc_bib_item_3">Dehaeze and Collette 2021</a>).
</p>
<p>
It is structured in several sections:
</p>
<ul class="org-ul">
<li>Section <a href="#org33d481f">1</a>: presents a simple model of a rotating suspended platform that will be used throughout this study.</li>
<li>Section <a href="#org4004ae5">2</a>: explains how the unconditional stability of IFF is lost due to Gyroscopic effects induced by the rotation.</li>
<li>Section <a href="#org6148a93">3</a>: suggests a simple modification of the control law such that damping can be added to the suspension modes in a robust way.</li>
<li>Section <a href="#orgb35716d">4</a>: proposes to add springs in parallel with the force sensors to regain the unconditional stability of IFF.</li>
<li>Section <a href="#org4387152">5</a>: compares both proposed modifications to the classical IFF in terms of damping authority and closed-loop system behavior.</li>
<li>Section <a href="#org19b2219">6</a>: contains the notations used for both the Matlab code and the paper</li>
</ul>
<p>
The matlab code is accessible on <a href="https://zenodo.org/record/3894343">Zonodo</a> and <a href="https://github.com/tdehaeze/dehaeze20_contr_stewa_platf">Github</a> (<a href="#citeproc_bib_item_2">Dehaeze 2020</a>). It can also be download as a <code>.zip</code> file <a href="matlab.zip">here</a>.
</p>
<p>
To run the Matlab code, go in the <code>matlab</code> directory and run the following Matlab files corresponding to each section.
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Paper&rsquo;s sections and corresponding Matlab files</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Sections</th>
<th scope="col" class="org-left">Matlab File</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Section <a href="#org33d481f">1</a></td>
<td class="org-left"><code>s1_system_description.m</code></td>
</tr>
<tr>
<td class="org-left">Section <a href="#org4004ae5">2</a></td>
<td class="org-left"><code>s2_iff_pure_int.m</code></td>
</tr>
<tr>
<td class="org-left">Section <a href="#org6148a93">3</a></td>
<td class="org-left"><code>s3_iff_hpf.m</code></td>
</tr>
<tr>
<td class="org-left">Section <a href="#orgb35716d">4</a></td>
<td class="org-left"><code>s4_iff_kp.m</code></td>
</tr>
<tr>
<td class="org-left">Section <a href="#org4387152">5</a></td>
<td class="org-left"><code>s5_act_damp_comparison.m</code></td>
</tr>
</tbody>
</table>
<div id="outline-container-org06be299" class="outline-2">
<h2 id="org06be299"><span class="section-number-2">1</span> System Description and Analysis</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org33d481f"></a>
</p>
</div>
<div id="outline-container-org4600b5b" class="outline-3">
<h3 id="org4600b5b"><span class="section-number-3">1.1</span> System description</h3>
<div class="outline-text-3" id="text-1-1">
<p>
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org06f9bae">1</a>).
</p>
<div id="org06f9bae" class="figure">
<p><img src="figs-paper/system.png" alt="system.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the studied system</p>
</div>
<p>
The control inputs are the forces applied by the actuators of the translation stage (\(F_u\) and \(F_v\)).
As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(\vec{i}_u\) and \(\vec{i}_v\).
</p>
</div>
</div>
<div id="outline-container-org2d46257" class="outline-3">
<h3 id="org2d46257"><span class="section-number-3">1.2</span> Equations</h3>
<div class="outline-text-3" id="text-1-2">
<p>
Based on the Figure <a href="#org06f9bae">1</a>, the equations of motions are:
</p>
<div class="important" id="orge2d1a3a">
\begin{equation}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
\bm{G}_d
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
<p>
Where \(\bm{G}_d\) is a \(2 \times 2\) transfer function matrix.
</p>
\begin{equation}
\bm{G}_d = \frac{1}{k} \frac{1}{G_{dp}}
\begin{bmatrix}
G_{dz} & G_{dc} \\
-G_{dc} & G_{dz}
\end{bmatrix}
\end{equation}
<p>
With:
</p>
\begin{align}
G_{dp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{dz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
G_{dc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
\end{align}
</div>
</div>
</div>
<div id="outline-container-org1698c67" class="outline-3">
<h3 id="org1698c67"><span class="section-number-3">1.3</span> Numerical Values</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Let&rsquo;s define initial values for the model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> k = 1; <span class="org-comment">% Actuator Stiffness [N/m]</span>
c = 0.05; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
m = 1; <span class="org-comment">% Payload mass [kg]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> xi = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m));
w0 = sqrt(k<span class="org-type">/</span>m); <span class="org-comment">% [rad/s]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org12f3c6c" class="outline-3">
<h3 id="org12f3c6c"><span class="section-number-3">1.4</span> Campbell Diagram</h3>
<div class="outline-text-3" id="text-1-4">
<p>
The Campbell Diagram displays the evolution of the real and imaginary parts of the system as a function of the rotating speed.
</p>
<p>
It is shown in Figures <a href="#org3e5c429">2</a> and <a href="#org3bb19b1">3</a>, and one can see that the system becomes unstable for \(\Omega > \omega_0\) (the real part of one of the poles becomes positive).
</p>
<div id="org3e5c429" class="figure">
<p><img src="figs/campbell_diagram_real.png" alt="campbell_diagram_real.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Campbell Diagram - Real Part</p>
</div>
<div id="org3bb19b1" class="figure">
<p><img src="figs/campbell_diagram_imag.png" alt="campbell_diagram_imag.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Campbell Diagram - Imaginary Part</p>
</div>
</div>
</div>
<div id="outline-container-org7a2ea7e" class="outline-3">
<h3 id="org7a2ea7e"><span class="section-number-3">1.5</span> Simscape Model</h3>
<div class="outline-text-3" id="text-1-5">
<p>
In order to validate all the equations of motion, a Simscape model of the same system has been developed.
The dynamics of the system can be identified from the Simscape model and compare with the analytical model.
</p>
<p>
The rotating speed for the Simscape Model is defined.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> W = 0.1; <span class="org-comment">% Rotation Speed [rad/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> open(<span class="org-string">'rotating_frame.slx'</span>);
</pre>
</div>
<p>
The transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) is identified from the Simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'rotating_frame'</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">'/K'</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">'/G'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> G = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
G.InputName = {<span class="org-string">'Fu'</span>, <span class="org-string">'Fv'</span>};
G.OutputName = {<span class="org-string">'du'</span>, <span class="org-string">'dv'</span>};
</pre>
</div>
<p>
The same transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) is written down from the analytical model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gth = (1<span class="org-type">/</span>k)<span class="org-type">/</span>(((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2) <span class="org-type">*</span> ...
[(s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2), 2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2) ; ...
<span class="org-type">-</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2), (s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2)];
</pre>
</div>
<p>
Both transfer functions are compared in Figure <a href="#orgfc7990e">4</a> and are found to perfectly match.
</p>
<div id="orgfc7990e" class="figure">
<p><img src="figs/plant_simscape_analytical.png" alt="plant_simscape_analytical.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Bode plot of the transfer function from \([F_u, F_v]\) to \([d_u, d_v]\) as identified from the Simscape model and from an analytical model</p>
</div>
</div>
</div>
<div id="outline-container-orgd90ebb8" class="outline-3">
<h3 id="orgd90ebb8"><span class="section-number-3">1.6</span> Effect of the rotation speed</h3>
<div class="outline-text-3" id="text-1-6">
<p>
The transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) are identified for the following rotating speeds.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Ws = [0, 0.2, 0.7, 1.1]<span class="org-type">*</span>w0; <span class="org-comment">% Rotating Speeds [rad/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> Gs = {zeros(2, 2, length(Ws))};
<span class="org-keyword">for</span> <span class="org-variable-name">W_i</span> = <span class="org-constant">1:length(Ws)</span>
W = Ws(W_i);
Gs(<span class="org-type">:</span>, <span class="org-type">:</span>, W_i) = {(1<span class="org-type">/</span>k)<span class="org-type">/</span>(((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2) <span class="org-type">*</span> ...
[(s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2), 2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2) ; ...
<span class="org-type">-</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2), (s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2)]};
<span class="org-keyword">end</span>
</pre>
</div>
<p>
They are compared in Figures <a href="#org032061b">5</a> and <a href="#orgec75241">6</a>.
</p>
<div id="org032061b" class="figure">
<p><img src="figs/plant_compare_rotating_speed_direct.png" alt="plant_compare_rotating_speed_direct.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Comparison of the transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) for several rotating speed - Direct Terms</p>
</div>
<div id="orgec75241" class="figure">
<p><img src="figs/plant_compare_rotating_speed_coupling.png" alt="plant_compare_rotating_speed_coupling.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Comparison of the transfer functions from \([F_u, F_v]\) to \([d_u, d_v]\) for several rotating speed - Coupling Terms</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org59dec10" class="outline-2">
<h2 id="org59dec10"><span class="section-number-2">2</span> Problem with pure Integral Force Feedback</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org4004ae5"></a>
</p>
<p>
Force sensors are added in series with the two actuators (Figure <a href="#org368ee29">7</a>).
</p>
<p>
Two identical controllers \(K_F\) are used to feedback each of the sensed force to its associated actuator.
</p>
<div id="org368ee29" class="figure">
<p><img src="figs-paper/system_iff.png" alt="system_iff.png" />
</p>
<p><span class="figure-number">Figure 7: </span>System with added Force Sensor in series with the actuators</p>
</div>
</div>
<div id="outline-container-orge6cb2c2" class="outline-3">
<h3 id="orge6cb2c2"><span class="section-number-3">2.1</span> Plant Parameters</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Let&rsquo;s define initial values for the model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> k = 1; <span class="org-comment">% Actuator Stiffness [N/m]</span>
c = 0.05; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
m = 1; <span class="org-comment">% Payload mass [kg]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> xi = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m));
w0 = sqrt(k<span class="org-type">/</span>m); <span class="org-comment">% [rad/s]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org86cebac" class="outline-3">
<h3 id="org86cebac"><span class="section-number-3">2.2</span> Equations</h3>
<div class="outline-text-3" id="text-2-2">
<p>
The sensed forces are equal to:
</p>
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k)
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation}
<p>
Which then gives:
</p>
<div class="important" id="org31425e1">
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\bm{G}_{f}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\frac{1}{G_{fp}}
\begin{bmatrix}
G_{fz} & -G_{fc} \\
G_{fc} & G_{fz}
\end{bmatrix}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{align}
G_{fp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{fz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{fc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)
\end{align}
</div>
</div>
</div>
<div id="outline-container-org5733bdd" class="outline-3">
<h3 id="org5733bdd"><span class="section-number-3">2.3</span> Comparison of the Analytical Model and the Simscape Model</h3>
<div class="outline-text-3" id="text-2-3">
<p>
The rotation speed is set to \(\Omega = 0.1 \omega_0\).
</p>
<div class="org-src-container">
<pre class="src src-matlab"> W = 0.1<span class="org-type">*</span>w0; <span class="org-comment">% [rad/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> open(<span class="org-string">'rotating_frame.slx'</span>);
</pre>
</div>
<p>
And the transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is identified using the Simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'rotating_frame'</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">'/K'</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">'/G'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> Giff = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
Giff.InputName = {<span class="org-string">'Fu'</span>, <span class="org-string">'Fv'</span>};
Giff.OutputName = {<span class="org-string">'fu'</span>, <span class="org-string">'fv'</span>};
</pre>
</div>
<p>
The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down from the analytical model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Giff_th = 1<span class="org-type">/</span>(((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2) <span class="org-type">*</span> ...
[(s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2)) <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2, <span class="org-type">-</span> (2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2) ; ...
(2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2];
</pre>
</div>
<p>
The two are compared in Figure <a href="#org93df5f4">8</a> and found to perfectly match.
</p>
<div id="org93df5f4" class="figure">
<p><img src="figs/plant_iff_comp_simscape_analytical.png" alt="plant_iff_comp_simscape_analytical.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) between the Simscape model and the analytical one</p>
</div>
</div>
</div>
<div id="outline-container-org25f390d" class="outline-3">
<h3 id="org25f390d"><span class="section-number-3">2.4</span> Effect of the rotation speed</h3>
<div class="outline-text-3" id="text-2-4">
<p>
The transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) are identified for the following rotating speeds.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Ws = [0, 0.2, 0.7]<span class="org-type">*</span>w0; <span class="org-comment">% Rotating Speeds [rad/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> Gsiff = {zeros(2, 2, length(Ws))};
<span class="org-keyword">for</span> <span class="org-variable-name">W_i</span> = <span class="org-constant">1:length(Ws)</span>
W = Ws(W_i);
Gsiff(<span class="org-type">:</span>, <span class="org-type">:</span>, W_i) = {1<span class="org-type">/</span>(((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2) <span class="org-type">*</span> ...
[(s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2)) <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2, <span class="org-type">-</span> (2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2) ; ...
(2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2]};
<span class="org-keyword">end</span>
</pre>
</div>
<p>
The obtained transfer functions are shown in Figure <a href="#org0fcab55">9</a>.
</p>
<div id="org0fcab55" class="figure">
<p><img src="figs/plant_iff_compare_rotating_speed.png" alt="plant_iff_compare_rotating_speed.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) for several rotating speed</p>
</div>
</div>
</div>
<div id="outline-container-orgb92b242" class="outline-3">
<h3 id="orgb92b242"><span class="section-number-3">2.5</span> Decentralized Integral Force Feedback</h3>
<div class="outline-text-3" id="text-2-5">
<p>
The decentralized IFF controller consists of pure integrators:
</p>
\begin{equation}
\bm{K}_{\text{IFF}}(s) = \frac{g}{s} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{equation}
<p>
The Root Locus (evolution of the poles of the closed loop system in the complex plane as a function of \(g\)) is shown in Figure <a href="#orgd204056">10</a>.
It is shown that for non-null rotating speed, one pole is bound to the right-half plane, and thus the closed loop system is unstable.
</p>
<div id="orgd204056" class="figure">
<p><img src="figs/root_locus_pure_iff.png" alt="root_locus_pure_iff.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown.</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org2e3307b" class="outline-2">
<h2 id="org2e3307b"><span class="section-number-2">3</span> Integral Force Feedback with an High Pass Filter</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org6148a93"></a>
</p>
</div>
<div id="outline-container-orgd87bec0" class="outline-3">
<h3 id="orgd87bec0"><span class="section-number-3">3.1</span> Plant Parameters</h3>
<div class="outline-text-3" id="text-3-1">
<p>
Let&rsquo;s define initial values for the model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> k = 1; <span class="org-comment">% Actuator Stiffness [N/m]</span>
c = 0.05; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
m = 1; <span class="org-comment">% Payload mass [kg]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> xi = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m));
w0 = sqrt(k<span class="org-type">/</span>m); <span class="org-comment">% [rad/s]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org38459ba" class="outline-3">
<h3 id="org38459ba"><span class="section-number-3">3.2</span> Modified Integral Force Feedback Controller</h3>
<div class="outline-text-3" id="text-3-2">
<p>
Let&rsquo;s modify the initial Integral Force Feedback Controller ; instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
</p>
\begin{equation}
K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{equation}
<p>
where \(\omega_i\) characterize down to which frequency the signal is integrated.
</p>
<p>
Let&rsquo;s arbitrary choose the following control parameters:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> g = 2;
wi = 0.1<span class="org-type">*</span>w0;
</pre>
</div>
<p>
And the following rotating speed.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Giff = 1<span class="org-type">/</span>(((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2) <span class="org-type">*</span> ...
[(s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2)) <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2, <span class="org-type">-</span> (2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2) ; ...
(2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2];
</pre>
</div>
<p>
The obtained Loop Gain is shown in Figure <a href="#org2511f91">11</a>.
</p>
<div id="org2511f91" class="figure">
<p><img src="figs/loop_gain_modified_iff.png" alt="loop_gain_modified_iff.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Loop Gain for the modified IFF controller</p>
</div>
</div>
</div>
<div id="outline-container-org1fe7230" class="outline-3">
<h3 id="org1fe7230"><span class="section-number-3">3.3</span> Root Locus</h3>
<div class="outline-text-3" id="text-3-3">
<p>
As shown in the Root Locus plot (Figure <a href="#org65c72ad">12</a>), for some value of the gain, the system remains stable.
</p>
<div id="org65c72ad" class="figure">
<p><img src="figs/root_locus_modified_iff.png" alt="root_locus_modified_iff.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Root Locus for the modified IFF controller</p>
</div>
<div id="orgd6abc3e" class="figure">
<p><img src="figs/root_locus_modified_iff_zoom.png" alt="root_locus_modified_iff_zoom.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Root Locus for the modified IFF controller - Zoom</p>
</div>
</div>
</div>
<div id="outline-container-orgbe65c44" class="outline-3">
<h3 id="orgbe65c44"><span class="section-number-3">3.4</span> What is the optimal \(\omega_i\) and \(g\)?</h3>
<div class="outline-text-3" id="text-3-4">
<p>
In order to visualize the effect of \(\omega_i\) on the attainable damping, the Root Locus is displayed in Figure <a href="#org7f8fd53">14</a> for the following \(\omega_i\):
</p>
<div class="org-src-container">
<pre class="src src-matlab"> wis = [0.01, 0.1, 0.5, 1]<span class="org-type">*</span>w0; <span class="org-comment">% [rad/s]</span>
</pre>
</div>
<div id="org7f8fd53" class="figure">
<p><img src="figs/root_locus_wi_modified_iff.png" alt="root_locus_wi_modified_iff.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Root Locus for the modified IFF controller (zoomed plot on the left)</p>
</div>
<div id="org7fe57c8" class="figure">
<p><img src="figs/root_locus_wi_modified_iff_zoom.png" alt="root_locus_wi_modified_iff_zoom.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Root Locus for the modified IFF controller (zoomed plot on the left)</p>
</div>
<p>
For the controller
</p>
\begin{equation}
K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{equation}
<p>
The gain at which the system becomes unstable is
</p>
\begin{equation}
g_\text{max} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right) \label{eq:iff_gmax}
\end{equation}
<p>
While it seems that small \(\omega_i\) do allow more damping to be added to the system (Figure <a href="#org7f8fd53">14</a>), the control gains may be limited to small values due to \eqref{eq:iff_gmax} thus reducing the attainable damping.
</p>
<p>
There must be an optimum for \(\omega_i\).
To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of \(\omega_i\) (Figure <a href="#org5abda38">16</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab"> wis = logspace(<span class="org-type">-</span>2, 1, 100)<span class="org-type">*</span>w0; <span class="org-comment">% [rad/s]</span>
opt_xi = zeros(1, length(wis)); <span class="org-comment">% Optimal simultaneous damping</span>
opt_gain = zeros(1, length(wis)); <span class="org-comment">% Corresponding optimal gain</span>
<span class="org-keyword">for</span> <span class="org-variable-name">wi_i</span> = <span class="org-constant">1:length(wis)</span>
wi = wis(wi_i);
Kiff = 1<span class="org-type">/</span>(s <span class="org-type">+</span> wi)<span class="org-type">*</span>eye(2);
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
[g_opt, xi_opt] = fminsearch(fun, 0.5<span class="org-type">*</span>wi<span class="org-type">*</span>((w0<span class="org-type">/</span>W)<span class="org-type">^</span>2 <span class="org-type">-</span> 1));
opt_xi(wi_i) = 1<span class="org-type">/</span>xi_opt;
opt_gain(wi_i) = g_opt;
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org5abda38" class="figure">
<p><img src="figs/mod_iff_damping_wi.png" alt="mod_iff_damping_wi.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Simultaneous attainable damping of the closed loop poles as a function of \(\omega_i\)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgd7e3817" class="outline-2">
<h2 id="orgd7e3817"><span class="section-number-2">4</span> IFF with a stiffness in parallel with the force sensor</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="orgb35716d"></a>
</p>
</div>
<div id="outline-container-org84e074b" class="outline-3">
<h3 id="org84e074b"><span class="section-number-3">4.1</span> Schematic</h3>
<div class="outline-text-3" id="text-4-1">
<p>
In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the rotation.
</p>
<div id="orge3677aa" class="figure">
<p><img src="figs-paper/system_parallel_springs.png" alt="system_parallel_springs.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Studied system with additional springs in parallel with the actuators and force sensors</p>
</div>
<p>
In order to keep the overall stiffness \(k = k_a + k_p\) constant, a scalar parameter \(\alpha\) (\(0 \le \alpha < 1\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor
</p>
\begin{equation}
k_p = \alpha k, \quad k_a = (1 - \alpha) k
\end{equation}
</div>
</div>
<div id="outline-container-orgafb020a" class="outline-3">
<h3 id="orgafb020a"><span class="section-number-3">4.2</span> Equations</h3>
<div class="outline-text-3" id="text-4-2">
<div class="important" id="org5c8ee21">
\begin{equation}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
\bm{G}_k
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
\frac{1}{G_{kp}}
\begin{bmatrix}
G_{kz} & -G_{kc} \\
G_{kc} & G_{kz}
\end{bmatrix}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
<p>
With:
</p>
\begin{align}
G_{kp} &= \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{{\omega_0}^2} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\
G_{kz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{{\omega_0}^2} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\
G_{kc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)
\end{align}
</div>
<p>
If we compare \(G_{kz}\) and \(G_{fz}\), we see that the spring in parallel adds a term \(\alpha\).
In order to have two complex conjugate zeros (instead of real zeros):
</p>
\begin{equation}
\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2
\end{equation}
</div>
</div>
<div id="outline-container-org218ccb0" class="outline-3">
<h3 id="org218ccb0"><span class="section-number-3">4.3</span> Plant Parameters</h3>
<div class="outline-text-3" id="text-4-3">
<p>
Let&rsquo;s define initial values for the model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> k = 1; <span class="org-comment">% Actuator Stiffness [N/m]</span>
c = 0.05; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
m = 1; <span class="org-comment">% Payload mass [kg]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> xi = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m));
w0 = sqrt(k<span class="org-type">/</span>m); <span class="org-comment">% [rad/s]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org92b8dc6" class="outline-3">
<h3 id="org92b8dc6"><span class="section-number-3">4.4</span> Comparison of the Analytical Model and the Simscape Model</h3>
<div class="outline-text-3" id="text-4-4">
<p>
The same transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) is written down from the analytical model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> W = 0.1<span class="org-type">*</span>w0; <span class="org-comment">% [rad/s]</span>
kp = 1.5<span class="org-type">*</span>m<span class="org-type">*</span>W<span class="org-type">^</span>2;
cp = 0;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> open(<span class="org-string">'rotating_frame.slx'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'rotating_frame'</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">'/K'</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">'/G'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
Giff = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
Giff.InputName = {<span class="org-string">'Fu'</span>, <span class="org-string">'Fv'</span>};
Giff.OutputName = {<span class="org-string">'fu'</span>, <span class="org-string">'fv'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> w0p = sqrt((k <span class="org-type">+</span> kp)<span class="org-type">/</span>m);
xip = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt((k<span class="org-type">+</span>kp)<span class="org-type">*</span>m));
Giff_th = 1<span class="org-type">/</span>( (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ) <span class="org-type">*</span> [ ...
(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2, <span class="org-type">-</span>(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p));
(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p)), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ];
Giff_th.InputName = {<span class="org-string">'Fu'</span>, <span class="org-string">'Fv'</span>};
Giff_th.OutputName = {<span class="org-string">'fu'</span>, <span class="org-string">'fv'</span>};
</pre>
</div>
<div id="orge74dc78" class="figure">
<p><img src="figs/plant_iff_kp_comp_simscape_analytical.png" alt="plant_iff_kp_comp_simscape_analytical.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Comparison of the transfer functions from \([F_u, F_v]\) to \([f_u, f_v]\) between the Simscape model and the analytical one</p>
</div>
</div>
</div>
<div id="outline-container-orgffb6014" class="outline-3">
<h3 id="orgffb6014"><span class="section-number-3">4.5</span> Effect of the parallel stiffness on the IFF plant</h3>
<div class="outline-text-3" id="text-4-5">
<p>
The rotation speed is set to \(\Omega = 0.1 \omega_0\).
</p>
<div class="org-src-container">
<pre class="src src-matlab"> W = 0.1<span class="org-type">*</span>w0; <span class="org-comment">% [rad/s]</span>
</pre>
</div>
<p>
And the IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified in three different cases:
</p>
<ul class="org-ul">
<li>without parallel stiffness</li>
<li>with a small parallel stiffness \(k_p < m \Omega^2\)</li>
<li>with a large parallel stiffness \(k_p > m \Omega^2\)</li>
</ul>
<p>
The results are shown in Figure <a href="#org30a4559">19</a>.
</p>
<p>
One can see that for \(k_p > m \Omega^2\), the systems shows alternating complex conjugate poles and zeros.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> kp = 0;
w0p = sqrt((k <span class="org-type">+</span> kp)<span class="org-type">/</span>m);
xip = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt((k<span class="org-type">+</span>kp)<span class="org-type">*</span>m));
Giff = 1<span class="org-type">/</span>( (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ) <span class="org-type">*</span> [ ...
(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2, <span class="org-type">-</span>(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p));
(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p)), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> kp = 0.5<span class="org-type">*</span>m<span class="org-type">*</span>W<span class="org-type">^</span>2;
k = 1 <span class="org-type">-</span> kp;
w0p = sqrt((k <span class="org-type">+</span> kp)<span class="org-type">/</span>m);
xip = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt((k<span class="org-type">+</span>kp)<span class="org-type">*</span>m));
Giff_s = 1<span class="org-type">/</span>( (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ) <span class="org-type">*</span> [ ...
(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2, <span class="org-type">-</span>(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p));
(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p)), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> kp = 1.5<span class="org-type">*</span>m<span class="org-type">*</span>W<span class="org-type">^</span>2;
k = 1 <span class="org-type">-</span> kp;
w0p = sqrt((k <span class="org-type">+</span> kp)<span class="org-type">/</span>m);
xip = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt((k<span class="org-type">+</span>kp)<span class="org-type">*</span>m));
Giff_l = 1<span class="org-type">/</span>( (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ) <span class="org-type">*</span> [ ...
(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2, <span class="org-type">-</span>(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p));
(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p)), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2];
</pre>
</div>
<div id="org30a4559" class="figure">
<p><img src="figs/plant_iff_kp.png" alt="plant_iff_kp.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Transfer function from \([F_u, F_v]\) to \([f_u, f_v]\) for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\)</p>
</div>
</div>
</div>
<div id="outline-container-orgccd60cd" class="outline-3">
<h3 id="orgccd60cd"><span class="section-number-3">4.6</span> IFF when adding a spring in parallel</h3>
<div class="outline-text-3" id="text-4-6">
<p>
In Figure <a href="#org6cd3905">20</a> is displayed the Root Locus in the three considered cases with
</p>
\begin{equation}
K_{\text{IFF}} = \frac{g}{s} \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{equation}
<p>
One can see that for \(k_p > m \Omega^2\), the root locus stays in the left half of the complex plane and thus the control system is unconditionally stable.
</p>
<p>
Thus, decentralized IFF controller with pure integrators can be used if:
</p>
\begin{equation}
k_{p} > m \Omega^2
\end{equation}
<div id="org6cd3905" class="figure">
<p><img src="figs/root_locus_iff_kp.png" alt="root_locus_iff_kp.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Root Locus</p>
</div>
<div id="org79a6aef" class="figure">
<p><img src="figs/root_locus_iff_kp_zoom.png" alt="root_locus_iff_kp_zoom.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Root Locus</p>
</div>
</div>
</div>
<div id="outline-container-org62a8e82" class="outline-3">
<h3 id="org62a8e82"><span class="section-number-3">4.7</span> Effect of \(k_p\) on the attainable damping</h3>
<div class="outline-text-3" id="text-4-7">
<p>
However, having large values of \(k_p\) may decrease the attainable damping.
</p>
<p>
To study the second point, Root Locus plots for the following values of \(k_p\) are shown in Figure <a href="#orgfab2d58">22</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> kps = [2, 20, 40]<span class="org-type">*</span>m<span class="org-type">*</span>W<span class="org-type">^</span>2;
</pre>
</div>
<p>
It is shown that large values of \(k_p\) decreases the attainable damping.
</p>
<div id="orgfab2d58" class="figure">
<p><img src="figs/root_locus_iff_kps.png" alt="root_locus_iff_kps.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Root Locus plot</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> alphas = logspace(<span class="org-type">-</span>2, 0, 100);
opt_xi = zeros(1, length(alphas)); <span class="org-comment">% Optimal simultaneous damping</span>
opt_gain = zeros(1, length(alphas)); <span class="org-comment">% Corresponding optimal gain</span>
Kiff = 1<span class="org-type">/</span>s<span class="org-type">*</span>eye(2);
<span class="org-keyword">for</span> <span class="org-variable-name">alpha_i</span> = <span class="org-constant">1:length(alphas)</span>
kp = alphas(alpha_i);
k = 1 <span class="org-type">-</span> alphas(alpha_i);
w0p = sqrt((k <span class="org-type">+</span> kp)<span class="org-type">/</span>m);
xip = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt((k<span class="org-type">+</span>kp)<span class="org-type">*</span>m));
Giff = 1<span class="org-type">/</span>( (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ) <span class="org-type">*</span> [ ...
(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2, <span class="org-type">-</span>(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p));
(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p)), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2];
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
[g_opt, xi_opt] = fminsearch(fun, 2);
opt_xi(alpha_i) = 1<span class="org-type">/</span>xi_opt;
opt_gain(alpha_i) = g_opt;
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org11fecca" class="figure">
<p><img src="figs/opt_damp_alpha.png" alt="opt_damp_alpha.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Attainable damping ratio and corresponding controller gain for different parameter \(\alpha\)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org57a9404" class="outline-2">
<h2 id="org57a9404"><span class="section-number-2">5</span> Comparison</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org4387152"></a>
</p>
<p>
Two modifications to adapt the IFF control strategy to rotating platforms have been proposed.
These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
</p>
</div>
<div id="outline-container-org10a2ed1" class="outline-3">
<h3 id="org10a2ed1"><span class="section-number-3">5.1</span> Plant Parameters</h3>
<div class="outline-text-3" id="text-5-1">
<p>
Let&rsquo;s define initial values for the model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> k = 1; <span class="org-comment">% Actuator Stiffness [N/m]</span>
c = 0.05; <span class="org-comment">% Actuator Damping [N/(m/s)]</span>
m = 1; <span class="org-comment">% Payload mass [kg]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> xi = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m));
w0 = sqrt(k<span class="org-type">/</span>m); <span class="org-comment">% [rad/s]</span>
</pre>
</div>
<p>
The rotating speed is set to \(\Omega = 0.1 \omega_0\).
</p>
<div class="org-src-container">
<pre class="src src-matlab"> W = 0.1<span class="org-type">*</span>w0;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc78a96c" class="outline-3">
<h3 id="orgc78a96c"><span class="section-number-3">5.2</span> Root Locus</h3>
<div class="outline-text-3" id="text-5-2">
<p>
IFF with High Pass Filter
</p>
<div class="org-src-container">
<pre class="src src-matlab"> wi = 0.1<span class="org-type">*</span>w0; <span class="org-comment">% [rad/s]</span>
Giff = 1<span class="org-type">/</span>(((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2) <span class="org-type">*</span> ...
[(s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2)) <span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2, <span class="org-type">-</span> (2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2) ; ...
(2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1)<span class="org-type">*</span>2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2)<span class="org-type">*</span>((s<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2) <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1 <span class="org-type">-</span> (W<span class="org-type">^</span>2)<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">+</span> (2<span class="org-type">*</span>W<span class="org-type">*</span>s<span class="org-type">/</span>(w0<span class="org-type">^</span>2))<span class="org-type">^</span>2];
</pre>
</div>
<p>
IFF With parallel Stiffness
</p>
<div class="org-src-container">
<pre class="src src-matlab"> kp = 5<span class="org-type">*</span>m<span class="org-type">*</span>W<span class="org-type">^</span>2;
k = k <span class="org-type">-</span> kp;
w0p = sqrt((k <span class="org-type">+</span> kp)<span class="org-type">/</span>m);
xip = c<span class="org-type">/</span>(2<span class="org-type">*</span>sqrt((k<span class="org-type">+</span>kp)<span class="org-type">*</span>m));
Giff_kp = 1<span class="org-type">/</span>( (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">^</span>2 <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ) <span class="org-type">*</span> [ ...
(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2, <span class="org-type">-</span>(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p));
(2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> k<span class="org-type">/</span>(k <span class="org-type">+</span> kp))<span class="org-type">*</span>(2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p)), (s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> kp<span class="org-type">/</span>(k <span class="org-type">+</span> kp) <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2)<span class="org-type">*</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xip<span class="org-type">*</span>s<span class="org-type">/</span>w0p <span class="org-type">+</span> 1 <span class="org-type">-</span> W<span class="org-type">^</span>2<span class="org-type">/</span>w0p<span class="org-type">^</span>2) <span class="org-type">+</span> (2<span class="org-type">*</span>(s<span class="org-type">/</span>w0p)<span class="org-type">*</span>(W<span class="org-type">/</span>w0p))<span class="org-type">^</span>2 ];
k = k <span class="org-type">+</span> kp;
</pre>
</div>
<div id="org75bbc2a" class="figure">
<p><img src="figs/comp_root_locus.png" alt="comp_root_locus.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness</p>
</div>
</div>
</div>
<div id="outline-container-org2018523" class="outline-3">
<h3 id="org2018523"><span class="section-number-3">5.3</span> Controllers - Optimal Gains</h3>
<div class="outline-text-3" id="text-5-3">
<p>
In order to compare to three considered Active Damping techniques, gains that yield maximum damping of all the modes are computed for each case.
</p>
<p>
The obtained damping ratio and control are shown below.
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">Obtained \(\xi\)</th>
<th scope="col" class="org-right">Control Gain</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Modified IFF</td>
<td class="org-right">0.83</td>
<td class="org-right">1.99</td>
</tr>
<tr>
<td class="org-left">IFF with \(k_p\)</td>
<td class="org-right">0.83</td>
<td class="org-right">2.02</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgfe35652" class="outline-3">
<h3 id="orgfe35652"><span class="section-number-3">5.4</span> Passive Damping - Critical Damping</h3>
<div class="outline-text-3" id="text-5-4">
\begin{equation}
\xi = \frac{c}{2 \sqrt{km}}
\end{equation}
<p>
Critical Damping corresponds to to \(\xi = 1\), and thus:
</p>
\begin{equation}
c_{\text{crit}} = 2 \sqrt{km}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"> c_opt = 2<span class="org-type">*</span>sqrt(k<span class="org-type">*</span>m);
</pre>
</div>
</div>
</div>
<div id="outline-container-org8aebf94" class="outline-3">
<h3 id="org8aebf94"><span class="section-number-3">5.5</span> Transmissibility And Compliance</h3>
<div class="outline-text-3" id="text-5-5">
<p>
<a id="org6ee546c"></a>
</p>
<div class="org-src-container">
<pre class="src src-matlab"> open(<span class="org-string">'rotating_frame.slx'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'rotating_frame'</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">'/dw'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/fd'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Meas'</span>], 1, <span class="org-string">'output'</span>); io_i = io_i <span class="org-type">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> G_ol = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
G_ol.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>};
G_ol.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>};
</pre>
</div>
</div>
<div id="outline-container-org53602c4" class="outline-4">
<h4 id="org53602c4"><span class="section-number-4">5.5.1</span> Passive Damping</h4>
<div class="outline-text-4" id="text-5-5-1">
<div class="org-src-container">
<pre class="src src-matlab"> kp = 0;
cp = 0;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> c_old = c;
c = c_opt;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> G_pas = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
G_pas.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>};
G_pas.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> c = c_old;
</pre>
</div>
</div>
</div>
<div class="outline-text-3" id="text-5-5">
<div class="org-src-container">
<pre class="src src-matlab"> Kiff = opt_gain_iff<span class="org-type">/</span>(wi <span class="org-type">+</span> s)<span class="org-type">*</span>tf(eye(2));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> G_iff = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
G_iff.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>};
G_iff.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>};
</pre>
</div>
</div>
<div class="outline-text-3" id="text-5-5">
<div class="org-src-container">
<pre class="src src-matlab"> kp = 5<span class="org-type">*</span>m<span class="org-type">*</span>W<span class="org-type">^</span>2;
cp = 0.01;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> Kiff = opt_gain_kp<span class="org-type">/</span>s<span class="org-type">*</span>tf(eye(2));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> G_kp = linearize(mdl, io, 0);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
G_kp.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>};
G_kp.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>};
</pre>
</div>
</div>
<div class="outline-text-3" id="text-5-5">
<div id="orgb246369" class="figure">
<p><img src="figs/comp_transmissibility.png" alt="comp_transmissibility.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Comparison of the transmissibility</p>
</div>
</div>
<div class="outline-text-3" id="text-5-5">
<div id="org098f670" class="figure">
<p><img src="figs/comp_compliance.png" alt="comp_compliance.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Comparison of the obtained Compliance</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org6c5b405" class="outline-2">
<h2 id="org6c5b405"><span class="section-number-2">6</span> Notations</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="org19b2219"></a>
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Mathematical Notation</th>
<th scope="col" class="org-left">Matlab</th>
<th scope="col" class="org-left">Unit</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Actuator Stiffness</td>
<td class="org-left">\(k\)</td>
<td class="org-left"><code>k</code></td>
<td class="org-left">N/m</td>
</tr>
<tr>
<td class="org-left">Actuator Damping</td>
<td class="org-left">\(c\)</td>
<td class="org-left"><code>c</code></td>
<td class="org-left">N/(m/s)</td>
</tr>
<tr>
<td class="org-left">Payload Mass</td>
<td class="org-left">\(m\)</td>
<td class="org-left"><code>m</code></td>
<td class="org-left">kg</td>
</tr>
<tr>
<td class="org-left">Damping Ratio</td>
<td class="org-left">\(\xi = \frac{c}{2\sqrt{km}}\)</td>
<td class="org-left"><code>xi</code></td>
<td class="org-left">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Actuator Force</td>
<td class="org-left">\(\bm{F}, F_u, F_v\)</td>
<td class="org-left"><code>F</code> <code>Fu</code> <code>Fv</code></td>
<td class="org-left">N</td>
</tr>
<tr>
<td class="org-left">Force Sensor signal</td>
<td class="org-left">\(\bm{f}, f_u, f_v\)</td>
<td class="org-left"><code>f</code> <code>fu</code> <code>fv</code></td>
<td class="org-left">N</td>
</tr>
<tr>
<td class="org-left">Relative Displacement</td>
<td class="org-left">\(\bm{d}, d_u, d_v\)</td>
<td class="org-left"><code>d</code> <code>du</code> <code>dv</code></td>
<td class="org-left">m</td>
</tr>
<tr>
<td class="org-left">Resonance freq. when \(\Omega = 0\)</td>
<td class="org-left">\(\omega_0\)</td>
<td class="org-left"><code>w0</code></td>
<td class="org-left">rad/s</td>
</tr>
<tr>
<td class="org-left">Rotation Speed</td>
<td class="org-left">\(\Omega = \dot{\theta}\)</td>
<td class="org-left"><code>W</code></td>
<td class="org-left">rad/s</td>
</tr>
<tr>
<td class="org-left">Low Pass Filter corner frequency</td>
<td class="org-left">\(\omega_i\)</td>
<td class="org-left"><code>wi</code></td>
<td class="org-left">rad/s</td>
</tr>
</tbody>
</table>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Mathematical Notation</th>
<th scope="col" class="org-left">Matlab</th>
<th scope="col" class="org-left">Unit</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Laplace variable</td>
<td class="org-left">\(s\)</td>
<td class="org-left"><code>s</code></td>
<td class="org-left">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Complex number</td>
<td class="org-left">\(j\)</td>
<td class="org-left"><code>j</code></td>
<td class="org-left">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Frequency</td>
<td class="org-left">\(\omega\)</td>
<td class="org-left"><code>w</code></td>
<td class="org-left">[rad/s]</td>
</tr>
</tbody>
</table>
</div>
</div>
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Dehaeze, T., and C. Collette. 2020. “Active Damping of Rotating Platforms Using Integral Force Feedback.” In <i>Proceedings of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA)</i>.</div>
<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Dehaeze, Thomas. 2020. “Active Damping of Rotating Positioning Platforms.” <a href="https://doi.org/10.5281/zenodo.3894342">https://doi.org/10.5281/zenodo.3894342</a>.</div>
<div class="csl-entry"><a name="citeproc_bib_item_3"></a>Dehaeze, Thomas, and Christophe Collette. 2021. “Active Damping of Rotating Platforms Using Integral Force Feedback.” <i>Engineering Research Express</i>. <a href="http://iopscience.iop.org/article/10.1088/2631-8695/abe803">http://iopscience.iop.org/article/10.1088/2631-8695/abe803</a>.</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2021-02-20 sam. 14:44</p>
</div>
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