nass-simscape/active_damping/index.html
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</div><div id="content">
<h1 class="title">Active Damping applied on the Simscape Model</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org5d89cdc">1. Undamped System</a>
<ul>
<li><a href="#org75c59b5">1.1. Identification of the dynamics for Active Damping</a>
<ul>
<li><a href="#org13c1b35">1.1.1. Initialize the Simulation</a></li>
<li><a href="#org93e7893">1.1.2. Identification</a></li>
<li><a href="#orgc283023">1.1.3. Obtained Plants for Active Damping</a></li>
</ul>
</li>
<li><a href="#org05ea47a">1.2. Tomography Experiment</a>
<ul>
<li><a href="#org22b92db">1.2.1. Simulation</a></li>
<li><a href="#org2572ae7">1.2.2. Results</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org7838397">2. Integral Force Feedback</a>
<ul>
<li><a href="#org089ca66">2.1. Control Design</a>
<ul>
<li><a href="#orgdfb8417">2.1.1. Plant</a></li>
<li><a href="#org87cd8ab">2.1.2. Control Design</a></li>
<li><a href="#org82f3b2f">2.1.3. Diagonal Controller</a></li>
<li><a href="#orgb93ab25">2.1.4. IFF with High Pass Filter</a></li>
</ul>
</li>
<li><a href="#org64d8b6e">2.2. Tomography Experiment</a>
<ul>
<li><a href="#orga0d2c49">2.2.1. Simulation with IFF Controller</a></li>
<li><a href="#org73d7e8e">2.2.2. Simulation with IFF Controller with added High Pass Filter</a></li>
<li><a href="#orgf00d1e2">2.2.3. Compare with Undamped system</a></li>
</ul>
</li>
<li><a href="#org1be894d">2.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#orge87f2c6">3. Direct Velocity Feedback</a>
<ul>
<li><a href="#org63708a2">3.1. Control Design</a>
<ul>
<li><a href="#orge75a80d">3.1.1. Plant</a></li>
<li><a href="#org93e6a45">3.1.2. Control Design</a></li>
<li><a href="#orgfe022fd">3.1.3. Diagonal Controller</a></li>
</ul>
</li>
<li><a href="#orgfbc2c6e">3.2. Tomography Experiment</a>
<ul>
<li><a href="#org06c8b9d">3.2.1. Initialize the Simulation</a></li>
<li><a href="#org1172467">3.2.2. Simulation</a></li>
<li><a href="#org712bc79">3.2.3. Compare with Undamped system</a></li>
</ul>
</li>
<li><a href="#orgb7038fc">3.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org189c37a">4. Inertial Control</a>
<ul>
<li><a href="#org0b527cf">4.1. Control Design</a>
<ul>
<li><a href="#org8f10824">4.1.1. Plant</a></li>
<li><a href="#orgbf5fc59">4.1.2. Control Design</a></li>
<li><a href="#org26c423f">4.1.3. Diagonal Controller</a></li>
</ul>
</li>
<li><a href="#org4354127">4.2. Tomography Experiment</a>
<ul>
<li><a href="#org7a7d0fe">4.2.1. Initialize the Simulation</a></li>
<li><a href="#org4e1a6ec">4.2.2. Simulation</a></li>
<li><a href="#org43213af">4.2.3. Compare with Undamped system</a></li>
</ul>
</li>
<li><a href="#org1583f6b">4.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org60a7861">5. Comparison</a>
<ul>
<li><a href="#orgcfb94b6">5.1. Load the plants</a></li>
<li><a href="#org478cd23">5.2. Sensitivity to Disturbance</a></li>
<li><a href="#org0c15520">5.3. Damped Plant</a></li>
<li><a href="#org7cec242">5.4. Tomography Experiment</a>
<ul>
<li><a href="#org9cc6c6a">5.4.1. Load the Simulation Data</a></li>
<li><a href="#org68b07c1">5.4.2. Frequency Domain Analysis</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgd1af630">6. Useful Functions</a>
<ul>
<li><a href="#org4da4ec2">6.1. prepareTomographyExperiment</a>
<ul>
<li><a href="#orgd77a897">Function Description</a></li>
<li><a href="#org8bc4916">Optional Parameters</a></li>
<li><a href="#org2b62851">Initialize the Simulation</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<p>
First, in section <a href="#org977860d">1</a>, we will looked at the undamped system.
</p>
<p>
Then, we will compare three active damping techniques:
</p>
<ul class="org-ul">
<li>In section <a href="#org50f479b">2</a>: the integral force feedback is used</li>
<li>In section <a href="#org41f3a46">3</a>: the direct velocity feedback is used</li>
<li>In section <a href="#org79aba2e">4</a>: inertial control is used</li>
</ul>
<p>
For each of the active damping technique, we will:
</p>
<ul class="org-ul">
<li>Look at the damped plant</li>
<li>Simulate tomography experiments</li>
<li>Compare the sensitivity from disturbances</li>
</ul>
<p>
The disturbances are:
</p>
<ul class="org-ul">
<li>Ground motion</li>
<li>Motion errors of all the stages</li>
</ul>
<div id="outline-container-org5d89cdc" class="outline-2">
<h2 id="org5d89cdc"><span class="section-number-2">1</span> Undamped System</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org977860d"></a>
</p>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/undamped_system.zip">here</a>.
</p>
</div>
<p>
We first look at the undamped system.
The performance of this undamped system will be compared with the damped system using various techniques.
</p>
</div>
<div id="outline-container-org75c59b5" class="outline-3">
<h3 id="org75c59b5"><span class="section-number-3">1.1</span> Identification of the dynamics for Active Damping</h3>
<div class="outline-text-3" id="text-1-1">
</div>
<div id="outline-container-org13c1b35" class="outline-4">
<h4 id="org13c1b35"><span class="section-number-4">1.1.1</span> Initialize the Simulation</h4>
<div class="outline-text-4" id="text-1-1-1">
<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();
initializeAxisc();
initializeMirror();
</pre>
</div>
<p>
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
initializeSample(<span class="org-string">'mass'</span>, 50);
</pre>
</div>
<p>
We set the references to zero.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeReferences();
</pre>
</div>
<p>
And all the controllers are set to 0.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_ine = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>);
K_iff = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org93e7893" class="outline-4">
<h4 id="org93e7893"><span class="section-number-4">1.1.2</span> Identification</h4>
<div class="outline-text-4" id="text-1-1-2">
<p>
First, we identify the dynamics of the system using the <code>linearize</code> function.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nass_active_damping'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Fnl'</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">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Dnlm'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Fnlm'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Vlm'</span>); io_i = io_i <span class="org-type">+</span> 1;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
G.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
G.OutputName = {<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</span>, ...
<span class="org-string">'Fnlm1'</span>, <span class="org-string">'Fnlm2'</span>, <span class="org-string">'Fnlm3'</span>, <span class="org-string">'Fnlm4'</span>, <span class="org-string">'Fnlm5'</span>, <span class="org-string">'Fnlm6'</span>, ...
<span class="org-string">'Vnlm1'</span>, <span class="org-string">'Vnlm2'</span>, <span class="org-string">'Vnlm3'</span>, <span class="org-string">'Vnlm4'</span>, <span class="org-string">'Vnlm5'</span>, <span class="org-string">'Vnlm6'</span>};
</pre>
</div>
<p>
We then create transfer functions corresponding to the active damping plants.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_iff = minreal(G({<span class="org-string">'Fnlm1'</span>, <span class="org-string">'Fnlm2'</span>, <span class="org-string">'Fnlm3'</span>, <span class="org-string">'Fnlm4'</span>, <span class="org-string">'Fnlm5'</span>, <span class="org-string">'Fnlm6'</span>}, {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}));
G_dvf = minreal(G({<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</span>}, {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}));
G_ine = minreal(G({<span class="org-string">'Vnlm1'</span>, <span class="org-string">'Vnlm2'</span>, <span class="org-string">'Vnlm3'</span>, <span class="org-string">'Vnlm4'</span>, <span class="org-string">'Vnlm5'</span>, <span class="org-string">'Vnlm6'</span>}, {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>}));
</pre>
</div>
<p>
And we save them for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'G_ine'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc283023" class="outline-4">
<h4 id="orgc283023"><span class="section-number-4">1.1.3</span> Obtained Plants for Active Damping</h4>
<div class="outline-text-4" id="text-1-1-3">
<div id="orgb06c0f1" class="figure">
<p><img src="figs/nass_active_damping_iff_plant.png" alt="nass_active_damping_iff_plant.png" />
</p>
<p><span class="figure-number">Figure 1: </span><code>G_iff</code>: IFF Plant (<a href="./figs/nass_active_damping_iff_plant.png">png</a>, <a href="./figs/nass_active_damping_iff_plant.pdf">pdf</a>)</p>
</div>
<div id="org90aa5ef" class="figure">
<p><img src="figs/nass_active_damping_ine_plant.png" alt="nass_active_damping_ine_plant.png" />
</p>
<p><span class="figure-number">Figure 2: </span><code>G_dvf</code>: Plant for Direct Velocity Feedback (<a href="./figs/nass_active_damping_dvf_plant.png">png</a>, <a href="./figs/nass_active_damping_dvf_plant.pdf">pdf</a>)</p>
</div>
<div id="orgcbf8f5e" class="figure">
<p><img src="figs/nass_active_damping_inertial_plant.png" alt="nass_active_damping_inertial_plant.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Inertial Feedback Plant (<a href="./figs/nass_active_damping_inertial_plant.png">png</a>, <a href="./figs/nass_active_damping_inertial_plant.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org05ea47a" class="outline-3">
<h3 id="org05ea47a"><span class="section-number-3">1.2</span> Tomography Experiment</h3>
<div class="outline-text-3" id="text-1-2">
</div>
<div id="outline-container-org22b92db" class="outline-4">
<h4 id="org22b92db"><span class="section-number-4">1.2.1</span> Simulation</h4>
<div class="outline-text-4" id="text-1-2-1">
<p>
We initialize elements for the tomography experiment.
</p>
<div class="org-src-container">
<pre class="src src-matlab">prepareTomographyExperiment();
</pre>
</div>
<p>
We change the simulation stop time.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
</pre>
</div>
<p>
And we simulate the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
</pre>
</div>
<p>
Finally, we save the simulation results for further analysis
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'Eg'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org2572ae7" class="outline-4">
<h4 id="org2572ae7"><span class="section-number-4">1.2.2</span> Results</h4>
<div class="outline-text-4" id="text-1-2-2">
<p>
We load the results of tomography experiments.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>);
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
</pre>
</div>
<div id="org603a0d4" class="figure">
<p><img src="figs/nass_act_damp_undamped_sim_tomo_trans.png" alt="nass_act_damp_undamped_sim_tomo_trans.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_undamped_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_undamped_sim_tomo_trans.pdf">pdf</a>)</p>
</div>
<div id="orgdb366a5" class="figure">
<p><img src="figs/nass_act_damp_undamped_sim_tomo_rot.png" alt="nass_act_damp_undamped_sim_tomo_rot.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_undamped_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_undamped_sim_tomo_rot.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
</div>
<div id="outline-container-org7838397" class="outline-2">
<h2 id="org7838397"><span class="section-number-2">2</span> Integral Force Feedback</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org50f479b"></a>
</p>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/iff.zip">here</a>.
</p>
</div>
<p>
Integral Force Feedback is applied on the simscape model.
</p>
</div>
<div id="outline-container-org089ca66" class="outline-3">
<h3 id="org089ca66"><span class="section-number-3">2.1</span> Control Design</h3>
<div class="outline-text-3" id="text-2-1">
</div>
<div id="outline-container-orgdfb8417" class="outline-4">
<h4 id="orgdfb8417"><span class="section-number-4">2.1.1</span> Plant</h4>
<div class="outline-text-4" id="text-2-1-1">
<p>
Let&rsquo;s load the previously indentified undamped plant:
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_iff'</span>);
</pre>
</div>
<p>
Let&rsquo;s look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure <a href="#org709c442">6</a>).
</p>
<div id="org709c442" class="figure">
<p><img src="figs/iff_plant.png" alt="iff_plant.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/iff_plant.png">png</a>, <a href="./figs/iff_plant.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org87cd8ab" class="outline-4">
<h4 id="org87cd8ab"><span class="section-number-4">2.1.2</span> Control Design</h4>
<div class="outline-text-4" id="text-2-1-2">
<p>
The controller for each pair of actuator/sensor is:
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = 1000<span class="org-type">/</span>s;
</pre>
</div>
<p>
The corresponding loop gains are shown in figure <a href="#org86c2154">7</a>.
</p>
<div id="org86c2154" class="figure">
<p><img src="figs/iff_open_loop.png" alt="iff_open_loop.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/iff_open_loop.png">png</a>, <a href="./figs/iff_open_loop.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org82f3b2f" class="outline-4">
<h4 id="org82f3b2f"><span class="section-number-4">2.1.3</span> Diagonal Controller</h4>
<div class="outline-text-4" id="text-2-1-3">
<p>
We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye(6);
</pre>
</div>
<p>
We save the controller for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_iff.mat'</span>, <span class="org-string">'K_iff'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb93ab25" class="outline-4">
<h4 id="orgb93ab25"><span class="section-number-4">2.1.4</span> IFF with High Pass Filter</h4>
<div class="outline-text-4" id="text-2-1-4">
<div class="org-src-container">
<pre class="src src-matlab">w_hpf = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; <span class="org-comment">% Cut-off frequency for the high pass filter [rad/s]</span>
w_lpf = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>200; <span class="org-comment">% Cut-off frequency for the low pass filter [rad/s]</span>
K_iff = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>200<span class="org-type">/</span>s <span class="org-type">*</span> (s<span class="org-type">/</span>w_hpf)<span class="org-type">/</span>(s<span class="org-type">/</span>w_hpf <span class="org-type">+</span> 1) <span class="org-type">*</span> 1<span class="org-type">/</span>(s<span class="org-type">/</span>w_lpf <span class="org-type">+</span> 1);
</pre>
</div>
<p>
The corresponding loop gains are shown in figure <a href="#org483db09">8</a>.
</p>
<div id="org483db09" class="figure">
<p><img src="figs/iff_hpf_open_loop.png" alt="iff_hpf_open_loop.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Loop Gain for the Integral Force Feedback with an High pass filter (<a href="./figs/iff_hpf_open_loop.png">png</a>, <a href="./figs/iff_hpf_open_loop.pdf">pdf</a>)</p>
</div>
<p>
We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = <span class="org-type">-</span>K_iff<span class="org-type">*</span>eye(6);
</pre>
</div>
<p>
We save the controller for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_iff_hpf.mat'</span>, <span class="org-string">'K_iff'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org64d8b6e" class="outline-3">
<h3 id="org64d8b6e"><span class="section-number-3">2.2</span> Tomography Experiment</h3>
<div class="outline-text-3" id="text-2-2">
</div>
<div id="outline-container-orga0d2c49" class="outline-4">
<h4 id="orga0d2c49"><span class="section-number-4">2.2.1</span> Simulation with IFF Controller</h4>
<div class="outline-text-4" id="text-2-2-1">
<p>
We initialize elements for the tomography experiment.
</p>
<div class="org-src-container">
<pre class="src src-matlab">prepareTomographyExperiment();
</pre>
</div>
<p>
We set the IFF controller.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_iff.mat'</span>, <span class="org-string">'K_iff'</span>);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<p>
We change the simulation stop time.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
</pre>
</div>
<p>
And we simulate the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
</pre>
</div>
<p>
Finally, we save the simulation results for further analysis
</p>
<div class="org-src-container">
<pre class="src src-matlab">En_iff = En;
Eg_iff = Eg;
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_iff'</span>, <span class="org-string">'Eg_iff'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org73d7e8e" class="outline-4">
<h4 id="org73d7e8e"><span class="section-number-4">2.2.2</span> Simulation with IFF Controller with added High Pass Filter</h4>
<div class="outline-text-4" id="text-2-2-2">
<p>
We initialize elements for the tomography experiment.
</p>
<div class="org-src-container">
<pre class="src src-matlab">prepareTomographyExperiment();
</pre>
</div>
<p>
We set the IFF controller with the High Pass Filter.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_iff_hpf.mat'</span>, <span class="org-string">'K_iff'</span>);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
<p>
We change the simulation stop time.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
</pre>
</div>
<p>
And we simulate the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
</pre>
</div>
<p>
Finally, we save the simulation results for further analysis
</p>
<div class="org-src-container">
<pre class="src src-matlab">En_iff_hpf = En;
Eg_iff_hpf = Eg;
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_iff_hpf'</span>, <span class="org-string">'Eg_iff_hpf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf00d1e2" class="outline-4">
<h4 id="orgf00d1e2"><span class="section-number-4">2.2.3</span> Compare with Undamped system</h4>
<div class="outline-text-4" id="text-2-2-3">
<p>
We load the results of tomography experiments.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_iff'</span>, <span class="org-string">'En_iff_hpf'</span>);
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
</pre>
</div>
<div id="org40817f4" class="figure">
<p><img src="figs/nass_act_damp_iff_sim_tomo_xy.png" alt="nass_act_damp_iff_sim_tomo_xy.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Position Error during tomography experiment - XY Motion (<a href="./figs/nass_act_damp_iff_sim_tomo_xy.png">png</a>, <a href="./figs/nass_act_damp_iff_sim_tomo_xy.pdf">pdf</a>)</p>
</div>
<div id="org59740ae" class="figure">
<p><img src="figs/nass_act_damp_iff_sim_tomo_trans.png" alt="nass_act_damp_iff_sim_tomo_trans.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_iff_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_iff_sim_tomo_trans.pdf">pdf</a>)</p>
</div>
<div id="orgf2e1614" class="figure">
<p><img src="figs/nass_act_damp_iff_sim_tomo_rot.png" alt="nass_act_damp_iff_sim_tomo_rot.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_iff_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_iff_sim_tomo_rot.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org1be894d" class="outline-3">
<h3 id="org1be894d"><span class="section-number-3">2.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-3">
<div class="important">
<p>
Integral Force Feedback:
</p>
<ul class="org-ul">
<li>Robust (guaranteed stability)</li>
<li>Acceptable Damping</li>
<li>Increase the sensitivity to disturbances at low frequencies</li>
</ul>
</div>
</div>
</div>
</div>
<div id="outline-container-orge87f2c6" class="outline-2">
<h2 id="orge87f2c6"><span class="section-number-2">3</span> Direct Velocity Feedback</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org41f3a46"></a>
</p>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/dvf.zip">here</a>.
</p>
</div>
<p>
In the Direct Velocity Feedback (DVF), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
The actuator displacement can be measured with a capacitive sensor for instance.
</p>
</div>
<div id="outline-container-org63708a2" class="outline-3">
<h3 id="org63708a2"><span class="section-number-3">3.1</span> Control Design</h3>
<div class="outline-text-3" id="text-3-1">
</div>
<div id="outline-container-orge75a80d" class="outline-4">
<h4 id="orge75a80d"><span class="section-number-4">3.1.1</span> Plant</h4>
<div class="outline-text-4" id="text-3-1-1">
<p>
Let&rsquo;s load the undamped plant:
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_dvf'</span>);
</pre>
</div>
<p>
Let&rsquo;s look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure <a href="#org1c4b9c5">12</a>).
</p>
<div id="org1c4b9c5" class="figure">
<p><img src="figs/dvf_plant.png" alt="dvf_plant.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/dvf_plant.png">png</a>, <a href="./figs/dvf_plant.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org93e6a45" class="outline-4">
<h4 id="org93e6a45"><span class="section-number-4">3.1.2</span> Control Design</h4>
<div class="outline-text-4" id="text-3-1-2">
<p>
The Direct Velocity Feedback is defined below.
A Low pass Filter is added to make the controller transfer function proper.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_dvf = s<span class="org-type">*</span>20000<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>10000);
</pre>
</div>
<p>
The obtained loop gains are shown in figure <a href="#orgd5f1986">13</a>.
</p>
<div id="orgd5f1986" class="figure">
<p><img src="figs/dvf_open_loop.png" alt="dvf_open_loop.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/dvf_open_loop.png">png</a>, <a href="./figs/dvf_open_loop.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgfe022fd" class="outline-4">
<h4 id="orgfe022fd"><span class="section-number-4">3.1.3</span> Diagonal Controller</h4>
<div class="outline-text-4" id="text-3-1-3">
<p>
We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_dvf = <span class="org-type">-</span>K_dvf<span class="org-type">*</span>eye(6);
</pre>
</div>
<p>
We save the controller for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_dvf.mat'</span>, <span class="org-string">'K_dvf'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgfbc2c6e" class="outline-3">
<h3 id="orgfbc2c6e"><span class="section-number-3">3.2</span> Tomography Experiment</h3>
<div class="outline-text-3" id="text-3-2">
</div>
<div id="outline-container-org06c8b9d" class="outline-4">
<h4 id="org06c8b9d"><span class="section-number-4">3.2.1</span> Initialize the Simulation</h4>
<div class="outline-text-4" id="text-3-2-1">
<p>
We initialize elements for the tomography experiment.
</p>
<div class="org-src-container">
<pre class="src src-matlab">prepareTomographyExperiment();
</pre>
</div>
<p>
We set the DVF controller.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_dvf.mat'</span>, <span class="org-string">'K_dvf'</span>);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org1172467" class="outline-4">
<h4 id="org1172467"><span class="section-number-4">3.2.2</span> Simulation</h4>
<div class="outline-text-4" id="text-3-2-2">
<p>
We change the simulation stop time.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
</pre>
</div>
<p>
And we simulate the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
</pre>
</div>
<p>
Finally, we save the simulation results for further analysis
</p>
<div class="org-src-container">
<pre class="src src-matlab">En_dvf = En;
Eg_dvf = Eg;
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_dvf'</span>, <span class="org-string">'Eg_dvf'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org712bc79" class="outline-4">
<h4 id="org712bc79"><span class="section-number-4">3.2.3</span> Compare with Undamped system</h4>
<div class="outline-text-4" id="text-3-2-3">
<p>
We load the results of tomography experiments.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_dvf'</span>);
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
</pre>
</div>
<div id="orgcf6c283" class="figure">
<p><img src="figs/nass_act_damp_dvf_sim_tomo_xy.png" alt="nass_act_damp_dvf_sim_tomo_xy.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Position Error during tomography experiment - XY Motion (<a href="./figs/nass_act_damp_dvf_sim_tomo_xy.png">png</a>, <a href="./figs/nass_act_damp_dvf_sim_tomo_xy.pdf">pdf</a>)</p>
</div>
<div id="org03ea40c" class="figure">
<p><img src="figs/nass_act_damp_dvf_sim_tomo_trans.png" alt="nass_act_damp_dvf_sim_tomo_trans.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_dvf_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_dvf_sim_tomo_trans.pdf">pdf</a>)</p>
</div>
<div id="org258bfdb" class="figure">
<p><img src="figs/nass_act_damp_dvf_sim_tomo_rot.png" alt="nass_act_damp_dvf_sim_tomo_rot.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_dvf_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_dvf_sim_tomo_rot.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb7038fc" class="outline-3">
<h3 id="orgb7038fc"><span class="section-number-3">3.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-3">
<div class="important">
<p>
Direct Velocity Feedback:
</p>
<ul class="org-ul">
<li></li>
</ul>
</div>
</div>
</div>
</div>
<div id="outline-container-org189c37a" class="outline-2">
<h2 id="org189c37a"><span class="section-number-2">4</span> Inertial Control</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org79aba2e"></a>
</p>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/ine.zip">here</a>.
</p>
</div>
<p>
In Inertial Control, a feedback is applied between the measured <b>absolute</b> motion (velocity or acceleration) of the platform to the actuator force input.
</p>
</div>
<div id="outline-container-org0b527cf" class="outline-3">
<h3 id="org0b527cf"><span class="section-number-3">4.1</span> Control Design</h3>
<div class="outline-text-3" id="text-4-1">
</div>
<div id="outline-container-org8f10824" class="outline-4">
<h4 id="org8f10824"><span class="section-number-4">4.1.1</span> Plant</h4>
<div class="outline-text-4" id="text-4-1-1">
<p>
Let&rsquo;s load the undamped plant:
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/undamped_plants.mat'</span>, <span class="org-string">'G_ine'</span>);
</pre>
</div>
<p>
Let&rsquo;s look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure <a href="#org524e9d1">17</a>).
</p>
<div id="org524e9d1" class="figure">
<p><img src="figs/ine_plant.png" alt="ine_plant.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/ine_plant.png">png</a>, <a href="./figs/ine_plant.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgbf5fc59" class="outline-4">
<h4 id="orgbf5fc59"><span class="section-number-4">4.1.2</span> Control Design</h4>
<div class="outline-text-4" id="text-4-1-2">
<p>
The controller is defined below and the obtained loop gain is shown in figure <a href="#org9d25b5e">18</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_ine = 1e4<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>100));
</pre>
</div>
<div id="org9d25b5e" class="figure">
<p><img src="figs/ine_open_loop_gain.png" alt="ine_open_loop_gain.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Loop Gain for Inertial Control (<a href="./figs/ine_open_loop_gain.png">png</a>, <a href="./figs/ine_open_loop_gain.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org26c423f" class="outline-4">
<h4 id="org26c423f"><span class="section-number-4">4.1.3</span> Diagonal Controller</h4>
<div class="outline-text-4" id="text-4-1-3">
<p>
We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_ine = <span class="org-type">-</span>K_ine<span class="org-type">*</span>eye(6);
</pre>
</div>
<p>
We save the controller for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'./active_damping/mat/K_ine.mat'</span>, <span class="org-string">'K_ine'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org4354127" class="outline-3">
<h3 id="org4354127"><span class="section-number-3">4.2</span> Tomography Experiment</h3>
<div class="outline-text-3" id="text-4-2">
</div>
<div id="outline-container-org7a7d0fe" class="outline-4">
<h4 id="org7a7d0fe"><span class="section-number-4">4.2.1</span> Initialize the Simulation</h4>
<div class="outline-text-4" id="text-4-2-1">
<p>
We initialize elements for the tomography experiment.
</p>
<div class="org-src-container">
<pre class="src src-matlab">prepareTomographyExperiment();
</pre>
</div>
<p>
We set the Inertial controller.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/K_ine.mat'</span>, <span class="org-string">'K_ine'</span>);
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org4e1a6ec" class="outline-4">
<h4 id="org4e1a6ec"><span class="section-number-4">4.2.2</span> Simulation</h4>
<div class="outline-text-4" id="text-4-2-2">
<p>
We change the simulation stop time.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'3'</span>);
</pre>
</div>
<p>
And we simulate the system.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'sim_nass_active_damping'</span>);
</pre>
</div>
<p>
Finally, we save the simulation results for further analysis
</p>
<div class="org-src-container">
<pre class="src src-matlab">En_ine = En;
Eg_ine = Eg;
save(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En_ine'</span>, <span class="org-string">'Eg_ine'</span>, <span class="org-string">'-append'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org43213af" class="outline-4">
<h4 id="org43213af"><span class="section-number-4">4.2.3</span> Compare with Undamped system</h4>
<div class="outline-text-4" id="text-4-2-3">
<p>
We load the results of tomography experiments.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_ine'</span>);
t = linspace(0, 3, length(En_ine(<span class="org-type">:</span>,1)));
</pre>
</div>
<div id="org0914dbb" class="figure">
<p><img src="figs/nass_act_damp_ine_sim_tomo_xy.png" alt="nass_act_damp_ine_sim_tomo_xy.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Position Error during tomography experiment - XY Motion (<a href="./figs/nass_act_damp_ine_sim_tomo_xy.png">png</a>, <a href="./figs/nass_act_damp_ine_sim_tomo_xy.pdf">pdf</a>)</p>
</div>
<div id="org48f6bf5" class="figure">
<p><img src="figs/nass_act_damp_ine_sim_tomo_trans.png" alt="nass_act_damp_ine_sim_tomo_trans.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Position Error during tomography experiment - Translations (<a href="./figs/nass_act_damp_ine_sim_tomo_trans.png">png</a>, <a href="./figs/nass_act_damp_ine_sim_tomo_trans.pdf">pdf</a>)</p>
</div>
<div id="org89ebbf9" class="figure">
<p><img src="figs/nass_act_damp_ine_sim_tomo_rot.png" alt="nass_act_damp_ine_sim_tomo_rot.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Position Error during tomography experiment - Rotations (<a href="./figs/nass_act_damp_ine_sim_tomo_rot.png">png</a>, <a href="./figs/nass_act_damp_ine_sim_tomo_rot.pdf">pdf</a>)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org1583f6b" class="outline-3">
<h3 id="org1583f6b"><span class="section-number-3">4.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-3">
<div class="important">
<p>
Inertial Control:
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org60a7861" class="outline-2">
<h2 id="org60a7861"><span class="section-number-2">5</span> Comparison</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org7d71743"></a>
</p>
</div>
<div id="outline-container-orgcfb94b6" class="outline-3">
<h3 id="orgcfb94b6"><span class="section-number-3">5.1</span> Load the plants</h3>
<div class="outline-text-3" id="text-5-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_ine'</span>, <span class="org-string">'G_dvf'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org478cd23" class="outline-3">
<h3 id="org478cd23"><span class="section-number-3">5.2</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-5-2">
<div id="org766a5d0" class="figure">
<p><img src="figs/sensitivity_comp_ground_motion_z.png" alt="sensitivity_comp_ground_motion_z.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Sensitivity to ground motion in the Z direction on the Z motion error (<a href="./figs/sensitivity_comp_ground_motion_z.png">png</a>, <a href="./figs/sensitivity_comp_ground_motion_z.pdf">pdf</a>)</p>
</div>
<div id="orgb662906" class="figure">
<p><img src="figs/sensitivity_comp_direct_forces_z.png" alt="sensitivity_comp_direct_forces_z.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Compliance in the Z direction: Sensitivity of direct forces applied on the sample in the Z direction on the Z motion error (<a href="./figs/sensitivity_comp_direct_forces_z.png">png</a>, <a href="./figs/sensitivity_comp_direct_forces_z.pdf">pdf</a>)</p>
</div>
<div id="orgd7e090b" class="figure">
<p><img src="figs/sensitivity_comp_spindle_z.png" alt="sensitivity_comp_spindle_z.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Sensitivity to forces applied in the Z direction by the Spindle on the Z motion error (<a href="./figs/sensitivity_comp_spindle_z.png">png</a>, <a href="./figs/sensitivity_comp_spindle_z.pdf">pdf</a>)</p>
</div>
<div id="orge64e918" class="figure">
<p><img src="figs/sensitivity_comp_ty_z.png" alt="sensitivity_comp_ty_z.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Sensitivity to forces applied in the Z direction by the Y translation stage on the Z motion error (<a href="./figs/sensitivity_comp_ty_z.png">png</a>, <a href="./figs/sensitivity_comp_ty_z.pdf">pdf</a>)</p>
</div>
<div id="org6d663d9" class="figure">
<p><img src="figs/sensitivity_comp_ty_x.png" alt="sensitivity_comp_ty_x.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Sensitivity to forces applied in the X direction by the Y translation stage on the X motion error (<a href="./figs/sensitivity_comp_ty_x.png">png</a>, <a href="./figs/sensitivity_comp_ty_x.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org0c15520" class="outline-3">
<h3 id="org0c15520"><span class="section-number-3">5.3</span> Damped Plant</h3>
<div class="outline-text-3" id="text-5-3">
<div id="org8336dec" class="figure">
<p><img src="figs/plant_comp_damping_z.png" alt="plant_comp_damping_z.png" />
</p>
<p><span class="figure-number">Figure 27: </span>Plant for the \(z\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_z.png">png</a>, <a href="./figs/plant_comp_damping_z.pdf">pdf</a>)</p>
</div>
<div id="orgfe252f1" class="figure">
<p><img src="figs/plant_comp_damping_x.png" alt="plant_comp_damping_x.png" />
</p>
<p><span class="figure-number">Figure 28: </span>Plant for the \(x\) direction for different active damping technique used (<a href="./figs/plant_comp_damping_x.png">png</a>, <a href="./figs/plant_comp_damping_x.pdf">pdf</a>)</p>
</div>
<div id="orgd70da06" class="figure">
<p><img src="figs/plant_comp_damping_coupling.png" alt="plant_comp_damping_coupling.png" />
</p>
<p><span class="figure-number">Figure 29: </span>Comparison of one off-diagonal plant for different damping technique applied (<a href="./figs/plant_comp_damping_coupling.png">png</a>, <a href="./figs/plant_comp_damping_coupling.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org7cec242" class="outline-3">
<h3 id="org7cec242"><span class="section-number-3">5.4</span> Tomography Experiment</h3>
<div class="outline-text-3" id="text-5-4">
</div>
<div id="outline-container-org9cc6c6a" class="outline-4">
<h4 id="org9cc6c6a"><span class="section-number-4">5.4.1</span> Load the Simulation Data</h4>
<div class="outline-text-4" id="text-5-4-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./active_damping/mat/tomo_exp.mat'</span>, <span class="org-string">'En'</span>, <span class="org-string">'En_iff_hpf'</span>, <span class="org-string">'En_dvf'</span>, <span class="org-string">'En_ine'</span>);
En_iff = En_iff_hpf;
t = linspace(0, 3, length(En(<span class="org-type">:</span>,1)));
</pre>
</div>
</div>
</div>
<div id="outline-container-org68b07c1" class="outline-4">
<h4 id="org68b07c1"><span class="section-number-4">5.4.2</span> Frequency Domain Analysis</h4>
<div class="outline-text-4" id="text-5-4-2">
<p>
Window used for <code>pwelch</code> function.
</p>
<div class="org-src-container">
<pre class="src src-matlab">n_av = 8;
han_win = hanning(ceil(length(En(<span class="org-type">:</span>, 1))<span class="org-type">/</span>n_av));
</pre>
</div>
<div id="org520876a" class="figure">
<p><img src="figs/act_damp_tomo_exp_comp_psd_trans.png" alt="act_damp_tomo_exp_comp_psd_trans.png" />
</p>
<p><span class="figure-number">Figure 30: </span>PSD of the translation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_psd_trans.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_psd_trans.pdf">pdf</a>)</p>
</div>
<div id="org23a6962" class="figure">
<p><img src="figs/act_damp_tomo_exp_comp_psd_rot.png" alt="act_damp_tomo_exp_comp_psd_rot.png" />
</p>
<p><span class="figure-number">Figure 31: </span>PSD of the rotation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_psd_rot.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_psd_rot.pdf">pdf</a>)</p>
</div>
<div id="org3b5e07b" class="figure">
<p><img src="figs/act_damp_tomo_exp_comp_cps_trans.png" alt="act_damp_tomo_exp_comp_cps_trans.png" />
</p>
<p><span class="figure-number">Figure 32: </span>CPS of the translation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_cps_trans.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_cps_trans.pdf">pdf</a>)</p>
</div>
<div id="org31dea9b" class="figure">
<p><img src="figs/act_damp_tomo_exp_comp_cps_rot.png" alt="act_damp_tomo_exp_comp_cps_rot.png" />
</p>
<p><span class="figure-number">Figure 33: </span>CPS of the rotation errors for applied Active Damping techniques (<a href="./figs/act_damp_tomo_exp_comp_cps_rot.png">png</a>, <a href="./figs/act_damp_tomo_exp_comp_cps_rot.pdf">pdf</a>)</p>
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<h2 id="orgd1af630"><span class="section-number-2">6</span> Useful Functions</h2>
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<h3 id="org4da4ec2"><span class="section-number-3">6.1</span> prepareTomographyExperiment</h3>
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<a id="orgfe1b669"></a>
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<p>
This Matlab function is accessible <a href="src/prepareTomographyExperiment.m">here</a>.
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<h4 id="orgd77a897">Function Description</h4>
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[]</span> = <span class="org-function-name">prepareTomographyExperiment</span>(<span class="org-variable-name">args</span>)
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<h4 id="org8bc4916">Optional Parameters</h4>
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<pre class="src src-matlab">arguments
args.nass_actuator char {mustBeMember(args.nass_actuator,{<span class="org-string">'piezo'</span>, <span class="org-string">'lorentz'</span>})} = <span class="org-string">'piezo'</span>
args.sample_mass (1,1) double {mustBeNumeric, mustBePositive} = 50
args.Ry_period (1,1) double {mustBeNumeric, mustBePositive} = 1
<span class="org-keyword">end</span>
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<h4 id="org2b62851">Initialize the Simulation</h4>
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<p>
We initialize all the stages with the default parameters.
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<pre class="src src-matlab">initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
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<p>
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, args.nass_actuator);
initializeSample(<span class="org-string">'mass'</span>, args.sample_mass);
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<p>
We set the references to zero.
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<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, args.Ry_period);
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<p>
And all the controllers are set to 0.
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<pre class="src src-matlab">K = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span>);
K_ine = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_ine'</span>, <span class="org-string">'-append'</span>);
K_iff = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span>);
K_dvf = tf(zeros(6));
save(<span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span>);
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-01-20 lun. 17:45</p>
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