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</div><div id="content">
<h1 class="title">Compensating the gravity forces to start at steady state</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org356e04b">1. Initialization of the Experimental Conditions</a></li>
<li><a href="#org263e5db">2. Without compensation</a></li>
<li><a href="#orgfd6d9e9">3. Simulation to compute the required force in each joint</a></li>
<li><a href="#org44bde7a">4. New simulation with compensation of gravity forces</a></li>
<li><a href="#orgb714922">5. Conclusion</a></li>
</ul>
</div>
</div>
<p>
In this file is shown a technique used to compensate the gravity forces at t=0.
</p>
<p>
The problem is that in presence of gravity, the system does not start at steady state and experience a transient phase (section <a href="#org4bc70b2">2</a>).
</p>
<p>
In order to start the simulation at steady state in presence of gravity:
</p>
<ul class="org-ul">
<li>section <a href="#org2de8ac1">3</a>: first the stages are initialize in such a way that they are rigid, and the forces/torques applied at the location of their joints is measured</li>
<li>section <a href="#org5364ea3">4</a>: Then, the equilibrium position of each joint is modified in such a way that at t=0, the forces in each joints exactly compensate the forces due to gravity forces</li>
</ul>
<div id="outline-container-org356e04b" class="outline-2">
<h2 id="org356e04b"><span class="section-number-2">1</span> Initialization of the Experimental Conditions</h2>
<div class="outline-text-2" id="text-1">
<p>
We don&rsquo;t inject any perturbations and no reference tracking.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeReferences();
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
initializeController();
</pre>
</div>
<p>
We include the gravity and log all the signals to display.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org263e5db" class="outline-2">
<h2 id="org263e5db"><span class="section-number-2">2</span> Without compensation</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org4bc70b2"></a>
Let&rsquo;s simulate the system without any compensation of gravity forces.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod();
initializeSample();
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
sim_no_compensation = simout;
</pre>
</div>
<p>
And we can observe on Figure <a href="#org0568928">1</a> that there are some motion in the system.
</p>
<div id="org0568928" class="figure">
<p><img src="figs/transient_phase_gravity_no_compensation.png" alt="transient_phase_gravity_no_compensation.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Motion of the sample at the start of the simulation in presence of gravity (<a href="./figs/transient_phase_gravity_no_compensation.png">png</a>, <a href="./figs/transient_phase_gravity_no_compensation.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgfd6d9e9" class="outline-2">
<h2 id="orgfd6d9e9"><span class="section-number-2">3</span> Simulation to compute the required force in each joint</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org2de8ac1"></a>
We here wish to simulate the system in order to compute the required force in each joint to compensate the gravity forces.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'init'</span>);
</pre>
</div>
<p>
We simulate for a short time period (all the bodies are solid, so nothing should move).
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.1'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
</pre>
</div>
<p>
Verification that nothing is moving by looking at the maximum displacement of the sample:
</p>
<div class="org-src-container">
<pre class="src src-matlab">max(max(simout.Em.En.Data))
</pre>
</div>
<pre class="example">
1.0681e-15
</pre>
<p>
We here show the measured total force/torque applied at the location of each joint.
</p>
<div class="org-src-container">
<pre class="src src-matlab">data2orgtable([Fgm 0 0 0; Ftym; Fym; Fsm], {<span class="org-string">'Granite'</span>, <span class="org-string">'Translation Stage'</span>, <span class="org-string">'Tilt Stage'</span>, <span class="org-string">'Sample'</span>}, {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>}, <span class="org-string">' %.1e '</span>);
</pre>
</div>
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<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<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">Fx</th>
<th scope="col" class="org-right">Fy</th>
<th scope="col" class="org-right">Fz</th>
<th scope="col" class="org-right">Mx</th>
<th scope="col" class="org-right">My</th>
<th scope="col" class="org-right">Mz</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Granite</td>
<td class="org-right">-7.6e-12</td>
<td class="org-right">1.2e-11</td>
<td class="org-right">-34000.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.0</td>
</tr>
<tr>
<td class="org-left">Translation Stage</td>
<td class="org-right">-7.6e-12</td>
<td class="org-right">1.2e-11</td>
<td class="org-right">-12000.0</td>
<td class="org-right">31.0</td>
<td class="org-right">2.5</td>
<td class="org-right">6.6e-13</td>
</tr>
<tr>
<td class="org-left">Tilt Stage</td>
<td class="org-right">-7.6e-12</td>
<td class="org-right">1.2e-11</td>
<td class="org-right">-8800.0</td>
<td class="org-right">33.0</td>
<td class="org-right">-0.52</td>
<td class="org-right">6.6e-13</td>
</tr>
<tr>
<td class="org-left">Sample</td>
<td class="org-right">-5.7e-12</td>
<td class="org-right">1.3e-11</td>
<td class="org-right">-490.0</td>
<td class="org-right">-2.5e-12</td>
<td class="org-right">-8.1e-13</td>
<td class="org-right">2.7e-13</td>
</tr>
</tbody>
</table>
<div class="org-src-container">
<pre class="src src-matlab">data2orgtable([Fhm; Fnm], {<span class="org-string">'Micro-Hexapod'</span>, <span class="org-string">'Nano-Hexapod'</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>}, <span class="org-string">' %.1e '</span>);
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<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">F1</th>
<th scope="col" class="org-right">F2</th>
<th scope="col" class="org-right">F3</th>
<th scope="col" class="org-right">F4</th>
<th scope="col" class="org-right">F5</th>
<th scope="col" class="org-right">F6</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Micro-Hexapod</td>
<td class="org-right">-180.0</td>
<td class="org-right">-180.0</td>
<td class="org-right">-180.0</td>
<td class="org-right">-180.0</td>
<td class="org-right">-180.0</td>
<td class="org-right">-180.0</td>
</tr>
<tr>
<td class="org-left">Nano-Hexapod</td>
<td class="org-right">-160.0</td>
<td class="org-right">-160.0</td>
<td class="org-right">-160.0</td>
<td class="org-right">-160.0</td>
<td class="org-right">-160.0</td>
<td class="org-right">-160.0</td>
</tr>
</tbody>
</table>
<p>
We save these forces in <code>Foffset.mat</code>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'mat/Foffset.mat'</span>, <span class="org-string">'Fgm'</span>, <span class="org-string">'Ftym'</span>, <span class="org-string">'Fym'</span>, <span class="org-string">'Fzm'</span>, <span class="org-string">'Fhm'</span>, <span class="org-string">'Fnm'</span>, <span class="org-string">'Fsm'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org44bde7a" class="outline-2">
<h2 id="org44bde7a"><span class="section-number-2">4</span> New simulation with compensation of gravity forces</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org5364ea3"></a>
We now initialize the stages with the option <code>Foffset</code>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
initializeTy(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
initializeRy(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
initializeRz(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
initializeMicroHexapod(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
initializeAxisc();
initializeMirror();
initializeNanoHexapod(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
initializeSample(<span class="org-string">'Foffset'</span>, <span class="org-constant">true</span>);
</pre>
</div>
<p>
And we simulate the system for 0.5 seconds.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
sim_compensation = simout;
</pre>
</div>
<p>
Verification that nothing is moving
</p>
<div id="orgc677acf" class="figure">
<p><img src="figs/transient_phase_gravity_compensation.png" alt="transient_phase_gravity_compensation.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Motion of the sample at the start of the simulation in presence of gravity when compensating the gravity forces (<a href="./figs/transient_phase_gravity_compensation.png">png</a>, <a href="./figs/transient_phase_gravity_compensation.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgb714922" class="outline-2">
<h2 id="orgb714922"><span class="section-number-2">5</span> Conclusion</h2>
<div class="outline-text-2" id="text-5">
<p>
This initialization technique permits to compute the required forces/torques to be applied in each joint in order to compensate for gravity forces.
This initialization should be redone for each configuration (change of sample mass, change of tilt angle), but not when changing the stiffness of joints, for instant when changing from lorentz based nano-hexapod or piezo based.
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p>
</div>
</body>
</html>

96
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@ -4,7 +4,7 @@
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<title>Control of the Nano-Active-Stabilization-System</title>
@ -202,50 +202,28 @@
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@ -286,7 +264,8 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#orgd9c84f0">4.1. HAC-LAC using IFF and Tracking control in the frame of the Legs</a></li>
<li><a href="#orgeb80da1">4.2. HAC-LAC using IFF and Tracking control in the Cartesian frame</a></li>
<li><a href="#org6054c1a">4.3. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite</a></li>
<li><a href="#org8b2b21e">4.3. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the task space</a></li>
<li><a href="#org1c04b26">4.4. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</a></li>
</ul>
</li>
<li><a href="#orgab73896">5. Cascade Architectures (link)</a>
@ -620,8 +599,8 @@ Usually, the Low Authority Controller is first design, and then the High Authori
</div>
</div>
<div id="outline-container-org6054c1a" class="outline-3">
<h3 id="org6054c1a"><span class="section-number-3">4.3</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite</h3>
<div id="outline-container-org8b2b21e" class="outline-3">
<h3 id="org8b2b21e"><span class="section-number-3">4.3</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the task space</h3>
<div class="outline-text-3" id="text-4-3">
<div class="figure">
@ -630,6 +609,17 @@ Usually, the Low Authority Controller is first design, and then the High Authori
</div>
</div>
</div>
<div id="outline-container-org1c04b26" class="outline-3">
<h3 id="org1c04b26"><span class="section-number-3">4.4</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</h3>
<div class="outline-text-3" id="text-4-4">
<div class="figure">
<p><img src="figs/control_architecture_hac_iff_pos_L.png" alt="control_architecture_hac_iff_pos_L.png" />
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgab73896" class="outline-2">
@ -639,13 +629,13 @@ Usually, the Low Authority Controller is first design, and then the High Authori
<a id="orga038762"></a>
</p>
<p>
The principle of Cascade control is shown in Figure <a href="#org03ef231">12</a> and explained as follow:
The principle of Cascade control is shown in Figure <a href="#org03ef231">13</a> and explained as follow:
</p>
<blockquote>
<p>
To follow <b>two objectives</b> with different properties in one control system, usually a <b>hierarchy</b> of two feedback loops is used in practice.
This kind of control topology is called <b>cascade control</b>, which is used when there are <b>several measurements and one prime control variable</b>.
Cascade control is implemented by <b>nesting</b> the control loops, as shown in Figure <a href="#org03ef231">12</a>.
Cascade control is implemented by <b>nesting</b> the control loops, as shown in Figure <a href="#org03ef231">13</a>.
The output control loop is called the <b>primary loop</b>, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system. &#x2013; <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>
</p>
</blockquote>
@ -654,7 +644,7 @@ The output control loop is called the <b>primary loop</b>, while the inner loop
<div id="org03ef231" class="figure">
<p><img src="figs/control_architecture_cascade_control.png" alt="control_architecture_cascade_control.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Cascade Control Architecture</p>
<p><span class="figure-number">Figure 13: </span>Cascade Control Architecture</p>
</div>
<p>
@ -679,7 +669,7 @@ The inner loop can be composed of the system controlled with the HAC-LAC topolog
<div id="orgff7dfc6" class="figure">
<p><img src="figs/control_architecture_cascade_L.png" alt="control_architecture_cascade_L.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
<p><span class="figure-number">Figure 14: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
</div>
</div>
</div>
@ -691,7 +681,7 @@ The inner loop can be composed of the system controlled with the HAC-LAC topolog
<div id="org4bc4c4c" class="figure">
<p><img src="figs/control_architecture_cascade_X.png" alt="control_architecture_cascade_X.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the Cartesian Frame</p>
<p><span class="figure-number">Figure 15: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the Cartesian Frame</p>
</div>
</div>
</div>
@ -728,7 +718,7 @@ Signals:
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-26 jeu. 17:25</p>
<p class="date">Created: 2020-04-01 mer. 16:16</p>
</div>
</body>
</html>

88
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@ -4,7 +4,7 @@
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<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Cascade Control applied on the Simscape Model</title>
@ -202,50 +202,28 @@
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@ -281,14 +259,14 @@ for the JavaScript code in this tag.
<li><a href="#org101bfbc">3. High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</a>
<ul>
<li><a href="#org259240d">3.1. Identification of the damped plant</a></li>
<li><a href="#org878cff7">3.2. Obtained Plant</a></li>
<li><a href="#orgb574395">3.2. Obtained Plant</a></li>
<li><a href="#orgdea5e17">3.3. Controller Design and Loop Gain</a></li>
</ul>
</li>
<li><a href="#org58f9f32">4. Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</a>
<ul>
<li><a href="#org82ca884">4.1. Identification of the linearized plant</a></li>
<li><a href="#orgb574395">4.2. Obtained Plant</a></li>
<li><a href="#org3b0d2c2">4.2. Obtained Plant</a></li>
<li><a href="#org839cdb3">4.3. Controller Design</a></li>
</ul>
</li>
@ -385,7 +363,7 @@ We log the signals.
</div>
<div class="org-src-container">
<pre class="src src-matlab">Kx = tf(zeros(6));
<pre class="src src-matlab">Kp = tf(zeros(6));
Kl = tf(zeros(6));
Kiff = tf(zeros(6));
</pre>
@ -517,8 +495,8 @@ isstable(Gl)
</div>
</div>
<div id="outline-container-org878cff7" class="outline-3">
<h3 id="org878cff7"><span class="section-number-3">3.2</span> Obtained Plant</h3>
<div id="outline-container-orgb574395" class="outline-3">
<h3 id="orgb574395"><span class="section-number-3">3.2</span> Obtained Plant</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The obtain plant is shown in Figure <a href="#org455eb07">5</a>.
@ -593,7 +571,7 @@ mdl = <span class="org-string">'nass_model'</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">'/Controller/Cascade-HAC-LAC/Kx'</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">'/Controller/Cascade-HAC-LAC/Kp'</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">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
@ -615,8 +593,8 @@ isstable(Gx)
</div>
</div>
<div id="outline-container-orgb574395" class="outline-3">
<h3 id="orgb574395"><span class="section-number-3">4.2</span> Obtained Plant</h3>
<div id="outline-container-org3b0d2c2" class="outline-3">
<h3 id="org3b0d2c2"><span class="section-number-3">4.2</span> Obtained Plant</h3>
<div class="outline-text-3" id="text-4-2">
<div id="orge364e46" class="figure">
@ -635,10 +613,10 @@ isstable(Gx)
h = 2; <span class="org-comment">% Lead parameter</span>
Kx = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> (s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5)<span class="org-type">/</span>s <span class="org-type">*</span> 1<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>20);
Kp = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> (s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5)<span class="org-type">/</span>s <span class="org-type">*</span> 1<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>20);
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
Kx = Kx<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kx, wc))));
Kp = Kp<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kp, wc))));
</pre>
</div>
@ -726,7 +704,7 @@ Ts = t(2)<span class="org-type">-</span>t(1);
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-25 mer. 19:23</p>
<p class="date">Created: 2020-04-01 mer. 16:16</p>
</div>
</body>
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@ -4,7 +4,7 @@
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<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Identification</title>
@ -202,7 +202,7 @@
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@ -257,7 +257,12 @@
<li><a href="#org0c6ab2b">2.5. Compare with measurements</a></li>
</ul>
</li>
<li><a href="#orga1de7a7">3. Conclusion</a></li>
<li><a href="#org4704f01">3. Obtained Compliance of the Micro-Station</a>
<ul>
<li><a href="#orge0c91d3">3.1. Initialization</a></li>
</ul>
</li>
<li><a href="#orga1de7a7">4. Conclusion</a></li>
</ul>
</div>
</div>
@ -312,11 +317,6 @@ Some of the springs and dampers values can be estimated from the joints/stages s
<div id="outline-container-orgcfb741d" class="outline-3">
<h3 id="orgcfb741d"><span class="section-number-3">2.1</span> Prepare the Simulation</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'nass_model.slx'</span>)
</pre>
</div>
<p>
We load the configuration.
</p>
@ -728,9 +728,82 @@ We then compare the measurements with the identified transfer functions using th
</div>
<div id="outline-container-orga1de7a7" class="outline-2">
<h2 id="orga1de7a7"><span class="section-number-2">3</span> Conclusion</h2>
<div id="outline-container-org4704f01" class="outline-2">
<h2 id="org4704f01"><span class="section-number-2">3</span> Obtained Compliance of the Micro-Station</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orge0c91d3" class="outline-3">
<h3 id="orge0c91d3"><span class="section-number-3">3.1</span> Initialization</h3>
<div class="outline-text-3" id="text-3-1">
<p>
We initialize all the stages with the default parameters.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'compliance'</span>);
</pre>
</div>
<p>
We put nothing on top of the micro-hexapod.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">initializeReferences();
initializeDisturbances();
initializeController();
initializeSimscapeConfiguration();
initializeLoggingConfiguration();
</pre>
</div>
<p>
And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.
</p>
<p>
The obtained compliance is shown in Figure <a href="#org67dfd1a">4</a>.
</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">'nass_model'</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">'/Micro-Station/Micro Hexapod/Compliance/Fm'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Direct Forces/Torques applied on the micro-hexapod top platform</span>
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station/Micro Hexapod/Compliance/Dm'</span>], 1, <span class="org-string">'output'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute displacement of the top platform</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
Gm = linearize(mdl, io, 0);
Gm.InputName = {<span class="org-string">'Fmx'</span>, <span class="org-string">'Fmy'</span>, <span class="org-string">'Fmz'</span>, <span class="org-string">'Mmx'</span>, <span class="org-string">'Mmy'</span>, <span class="org-string">'Mmz'</span>};
Gm.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Drx'</span>, <span class="org-string">'Dry'</span>, <span class="org-string">'Drz'</span>};
</pre>
</div>
<div id="org67dfd1a" class="figure">
<p><img src="figs/compliance_micro_station.png" alt="compliance_micro_station.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Obtained compliance of the Micro-Station (<a href="./figs/compliance_micro_station.png">png</a>, <a href="./figs/compliance_micro_station.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orga1de7a7" class="outline-2">
<h2 id="orga1de7a7"><span class="section-number-2">4</span> Conclusion</h2>
<div class="outline-text-2" id="text-4">
<div class="important">
<p>
For such a complex system, we believe that the Simscape Model represents the dynamics of the system with enough fidelity.
@ -742,7 +815,7 @@ For such a complex system, we believe that the Simscape Model represents the dyn
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-13 ven. 17:39</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p>
</div>
</body>
</html>

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@ -1,11 +1,10 @@
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<!-- 2020-03-26 jeu. 17:25 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Effect of Uncertainty on the payload&rsquo;s dynamics on the isolation platform dynamics</title>
@ -203,50 +202,28 @@
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@ -272,22 +249,22 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#orgcc5f0ec">1. Simple Introductory Example</a>
<ul>
<li><a href="#org00d3412">1.1. Equations of motion</a></li>
<li><a href="#org6264842">1.1. Equations of motion</a></li>
<li><a href="#org4efccbf">1.2. Initialization of the payload dynamics</a></li>
<li><a href="#orgb400ca3">1.3. Initialization of the isolation platform</a></li>
<li><a href="#orgd0dd88b">1.4. Comparison</a></li>
<li><a href="#org3bef024">1.5. Conclusion</a></li>
<li><a href="#org1b051ce">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org1f8e63e">2. Generalization to arbitrary dynamics</a>
<ul>
<li><a href="#orgc4fa63e">2.1. Introduction</a></li>
<li><a href="#org6264842">2.2. Equations of motion</a></li>
<li><a href="#org35ac80d">2.2. Equations of motion</a></li>
<li><a href="#orge217a33">2.3. Impedance \(G^\prime(s)\) of a mass-spring-damper payload</a></li>
<li><a href="#org0ee44da">2.4. First Analytical analysis</a></li>
<li><a href="#orgfe81c1c">2.5. Impedance of the Payload and Dynamical Uncertainty</a></li>
<li><a href="#orgcc0c290">2.6. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org5e0366d">2.7. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#org5e0366d">2.6. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orgcc0c290">2.7. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org1466bd9">2.8. Reduce the Uncertainty on the plant</a>
<ul>
<li><a href="#org4be463f">2.8.1. Effect of the platform&rsquo;s stiffness \(k\)</a></li>
@ -295,7 +272,7 @@ for the JavaScript code in this tag.
<li><a href="#org9086831">2.8.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#org1b051ce">2.9. Conclusion</a></li>
<li><a href="#org43f33dc">2.9. Conclusion</a></li>
</ul>
</li>
</ul>
@ -349,8 +326,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payloa
</div>
</div>
<div id="outline-container-org00d3412" class="outline-3">
<h3 id="org00d3412"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org6264842" class="outline-3">
<h3 id="org6264842"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div class="outline-text-3" id="text-1-1">
<p>
If we write the equation of motion of the system in Figure <a href="#orgaa77a57">1</a>, we obtain:
@ -418,6 +395,7 @@ One can see that the payload has a resonance frequency of \(\omega_0^\prime = 25
</div>
</div>
</div>
<div id="outline-container-orgb400ca3" class="outline-3">
<h3 id="orgb400ca3"><span class="section-number-3">1.3</span> Initialization of the isolation platform</h3>
<div class="outline-text-3" id="text-1-3">
@ -456,8 +434,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-org3bef024" class="outline-3">
<h3 id="org3bef024"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org1b051ce" class="outline-3">
<h3 id="org1b051ce"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@ -509,8 +487,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org6264842" class="outline-3">
<h3 id="org6264842"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org35ac80d" class="outline-3">
<h3 id="org35ac80d"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We have to following equations of motion:
@ -757,10 +735,47 @@ A set of uncertainty payload&rsquo;s impedance transfer functions is shown in Fi
</div>
</div>
<div id="outline-container-orgcc0c290" class="outline-3">
<h3 id="orgcc0c290"><span class="section-number-3">2.6</span> Effect of the Isolation platform Stiffness</h3>
<div id="outline-container-org5e0366d" class="outline-3">
<h3 id="org5e0366d"><span class="section-number-3">2.6</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-6">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + G_0^\prime(s)} \cdot \frac{1}{1 + \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#org90e7eef">9</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="org90e7eef" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Inverse Multiplicative Uncertainty</p>
</div>
</div>
</div>
<div id="outline-container-orgcc0c290" class="outline-3">
<h3 id="orgcc0c290"><span class="section-number-3">2.7</span> Effect of the Isolation platform Stiffness</h3>
<div class="outline-text-3" id="text-2-7">
<p>
Let&rsquo;s first fix the mass of the isolation platform:
</p>
<div class="org-src-container">
@ -785,6 +800,8 @@ Soft Isolation Platform:
c_soft = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_soft);
G_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> Gp);
G0_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> Gp0);
wiI_soft = Gp0<span class="org-type">*</span>G0_soft<span class="org-type">*</span>wI;
</pre>
</div>
@ -796,6 +813,8 @@ Mid Isolation Platform
c_mid = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_mid);
G_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> Gp);
G0_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> Gp0);
wiI_mid = Gp0<span class="org-type">*</span>G0_mid<span class="org-type">*</span>wI;
</pre>
</div>
@ -807,55 +826,20 @@ Stiff Isolation Platform
c_stiff = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_stiff);
G_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> Gp);
G0_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> Gp0);
wiI_stiff = Gp0<span class="org-type">*</span>G0_stiff<span class="org-type">*</span>wI;
</pre>
</div>
<p>
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org864ba65">9</a>.
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org864ba65">10</a>.
</p>
<div id="org864ba65" class="figure">
<p><img src="figs/plant_uncertainty_impedance_payload.png" alt="plant_uncertainty_impedance_payload.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_impedance_payload.png">png</a>, <a href="./figs/plant_uncertainty_impedance_payload.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org5e0366d" class="outline-3">
<h3 id="org5e0366d"><span class="section-number-3">2.7</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-7">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + G_0^\prime(s)} \cdot \frac{1}{1 + \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#org90e7eef">10</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="org90e7eef" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Inverse Multiplicative Uncertainty</p>
<p><span class="figure-number">Figure 10: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_impedance_payload.png">png</a>, <a href="./figs/plant_uncertainty_impedance_payload.pdf">pdf</a>)</p>
</div>
</div>
</div>
@ -961,8 +945,8 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
</div>
</div>
<div id="outline-container-org1b051ce" class="outline-3">
<h3 id="org1b051ce"><span class="section-number-3">2.9</span> Conclusion</h3>
<div id="outline-container-org43f33dc" class="outline-3">
<h3 id="org43f33dc"><span class="section-number-3">2.9</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-9">
<div class="important">
<p>
@ -986,7 +970,7 @@ In that case, maximizing the stiffness of the payload is a good idea.
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-26 jeu. 17:25</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p>
</div>
</body>
</html>

188
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@ -4,7 +4,7 @@
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<!-- 2020-03-26 jeu. 17:24 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Effect of Uncertainty on the support&rsquo;s dynamics on the isolation platform dynamics</title>
@ -202,50 +202,28 @@
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@ -271,20 +249,20 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#orgbe6e0b8">1. Simple Introductory Example</a>
<ul>
<li><a href="#org440a84d">1.1. Equations of motion</a></li>
<li><a href="#org283e8c3">1.1. Equations of motion</a></li>
<li><a href="#org8bd2a4a">1.2. Initialization of the support dynamics</a></li>
<li><a href="#orgefb9b71">1.3. Initialization of the isolation platform</a></li>
<li><a href="#org3bc4ad1">1.4. Comparison</a></li>
<li><a href="#orgc2af076">1.5. Conclusion</a></li>
<li><a href="#orgd28ebd8">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orge1d3484">2. Generalization to arbitrary dynamics</a>
<ul>
<li><a href="#org3948d1f">2.1. Introduction</a></li>
<li><a href="#org283e8c3">2.2. Equations of motion</a></li>
<li><a href="#org28c48e3">2.2. Equations of motion</a></li>
<li><a href="#orgc20cabb">2.3. Compliance of the Support</a></li>
<li><a href="#orgf1c8c33">2.4. Effect of the Isolation platform Stiffness.</a></li>
<li><a href="#org67810a4">2.5. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#org67810a4">2.4. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orge950395">2.5. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org6967854">2.6. Reduce the Uncertainty on the plant</a>
<ul>
<li><a href="#orgafebadd">2.6.1. Effect of the platform&rsquo;s stiffness \(k\)</a></li>
@ -292,7 +270,7 @@ for the JavaScript code in this tag.
<li><a href="#orgd2fc303">2.6.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#orgd28ebd8">2.7. Conclusion</a></li>
<li><a href="#org718f0f1">2.7. Conclusion</a></li>
</ul>
</li>
</ul>
@ -346,8 +324,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the suppor
</div>
</div>
<div id="outline-container-org440a84d" class="outline-3">
<h3 id="org440a84d"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org283e8c3" class="outline-3">
<h3 id="org283e8c3"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div class="outline-text-3" id="text-1-1">
<p>
If we write the equation of motion of the system in Figure <a href="#org41bc770">1</a>, we obtain:
@ -451,8 +429,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-orgc2af076" class="outline-3">
<h3 id="orgc2af076"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-orgd28ebd8" class="outline-3">
<h3 id="orgd28ebd8"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@ -497,8 +475,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org283e8c3" class="outline-3">
<h3 id="org283e8c3"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org28c48e3" class="outline-3">
<h3 id="org28c48e3"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We have to following equations of motion:
@ -581,7 +559,7 @@ The parameters are defined below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">r0 = 0.5;
tau = 1<span class="org-type">/</span>(50<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
tau = 1<span class="org-type">/</span>(100<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
rinf = 10;
wI = (tau<span class="org-type">*</span>s <span class="org-type">+</span> r0)<span class="org-type">/</span>((tau<span class="org-type">/</span>rinf)<span class="org-type">*</span>s <span class="org-type">+</span> 1);
@ -617,10 +595,48 @@ A set of uncertainty support&rsquo;s compliance transfer functions is shown in F
</div>
</div>
<div id="outline-container-orgf1c8c33" class="outline-3">
<h3 id="orgf1c8c33"><span class="section-number-3">2.4</span> Effect of the Isolation platform Stiffness.</h3>
<div id="outline-container-org67810a4" class="outline-3">
<h3 id="org67810a4"><span class="section-number-3">2.4</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-4">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s) + ms^2(cs + k)G_0^\prime(s) w_I(s)\Delta(s)}\\
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#orge738173">8</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} = G_0(s) ms^2(cs + k)G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="orge738173" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Inverse Multiplicative Uncertainty</p>
</div>
</div>
</div>
<div id="outline-container-orge950395" class="outline-3">
<h3 id="orge950395"><span class="section-number-3">2.5</span> Effect of the Isolation platform Stiffness</h3>
<div class="outline-text-3" id="text-2-5">
<p>
Let&rsquo;s first fix the mass of the payload to be isolated:
</p>
<div class="org-src-container">
@ -645,6 +661,8 @@ Soft Isolation Platform:
c_soft = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_soft);
G_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp);
G0_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp0);
wiI_soft = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>G0_soft<span class="org-type">*</span>wI;
</pre>
</div>
@ -656,6 +674,8 @@ Mid Isolation Platform
c_mid = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_mid);
G_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp);
G0_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp0);
wiI_mid = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>G0_mid<span class="org-type">*</span>wI;
</pre>
</div>
@ -667,18 +687,20 @@ Stiff Isolation Platform
c_stiff = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_stiff);
G_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp);
G0_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp0);
wiI_stiff = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>G0_stiff<span class="org-type">*</span>wI;
</pre>
</div>
<p>
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org89aa89f">8</a>.
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org89aa89f">9</a>.
</p>
<div id="org89aa89f" class="figure">
<p><img src="figs/plant_uncertainty_stiffness_isolator.png" alt="plant_uncertainty_stiffness_isolator.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_stiffness_isolator.png">png</a>, <a href="./figs/plant_uncertainty_stiffness_isolator.pdf">pdf</a>)</p>
<p><span class="figure-number">Figure 9: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_stiffness_isolator.png">png</a>, <a href="./figs/plant_uncertainty_stiffness_isolator.pdf">pdf</a>)</p>
</div>
<p>
@ -688,44 +710,6 @@ This is due to the fact that with the current model, at high frequency, the supp
</div>
</div>
<div id="outline-container-org67810a4" class="outline-3">
<h3 id="org67810a4"><span class="section-number-3">2.5</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-5">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s) + ms^2(cs + k)G_0^\prime(s) w_I(s)\Delta(s)}\\
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#orge738173">9</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} = G_0(s) ms^2(cs + k)G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="orge738173" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Inverse Multiplicative Uncertainty</p>
</div>
</div>
</div>
<div id="outline-container-org6967854" class="outline-3">
<h3 id="org6967854"><span class="section-number-3">2.6</span> Reduce the Uncertainty on the plant</h3>
<div class="outline-text-3" id="text-2-6">
@ -828,8 +812,8 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
</div>
</div>
<div id="outline-container-orgd28ebd8" class="outline-3">
<h3 id="orgd28ebd8"><span class="section-number-3">2.7</span> Conclusion</h3>
<div id="outline-container-org718f0f1" class="outline-3">
<h3 id="org718f0f1"><span class="section-number-3">2.7</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-7">
<div class="important">
<p>
@ -857,7 +841,7 @@ Thus, if a stiff isolation platform is used, the recommendation is to have the l
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-26 jeu. 17:24</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p>
</div>
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