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Add uniaxial folder with analysis of active damping techniques
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Add noise budget to compare active damping techniques
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Add uniaxial folder with analysis of active damping techniques
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<h1 class="title">Simscape Uniaxial Model</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
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<li><a href="#orgac9343f">1. Simscape Model</a></li>
<li><a href="#org3890417">2. Undamped System</a>
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<ul>
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<li><a href="#orgbc01ced">2.1. Init</a></li>
<li><a href="#org699bc2c">2.2. Identification</a></li>
<li><a href="#org0c59d84">2.3. Sensitivity to Disturbances</a></li>
<li><a href="#orgb191841">2.4. Noise Budget</a></li>
<li><a href="#org1b2f77b">2.5. Plant</a></li>
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</ul>
</li>
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<li><a href="#org27aabe3">3. Integral Force Feedback</a>
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<ul>
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<li><a href="#org04b5ef2">3.1. Control Design</a></li>
<li><a href="#orgd976043">3.2. Identification</a></li>
<li><a href="#orgb785fd5">3.3. Sensitivity to Disturbance</a></li>
<li><a href="#org7f2e353">3.4. Damped Plant</a></li>
<li><a href="#org46695ba">3.5. Conclusion</a></li>
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</ul>
</li>
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<li><a href="#org8f75c3f">4. Relative Motion Control</a>
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<ul>
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<li><a href="#org7291ab1">4.1. Control Design</a></li>
<li><a href="#org72b4f0a">4.2. Identification</a></li>
<li><a href="#org7669eee">4.3. Sensitivity to Disturbance</a></li>
<li><a href="#orgab2a3d1">4.4. Damped Plant</a></li>
<li><a href="#orgbc19b27">4.5. Conclusion</a></li>
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</ul>
</li>
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<li><a href="#org264e55c">5. Direct Velocity Feedback</a>
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<ul>
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<li><a href="#orge1aa5cd">5.1. Control Design</a></li>
<li><a href="#org2a880b1">5.2. Identification</a></li>
<li><a href="#orgd7e4638">5.3. Sensitivity to Disturbance</a></li>
<li><a href="#org9524bf4">5.4. Damped Plant</a></li>
<li><a href="#org329f7b9">5.5. Conclusion</a></li>
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</ul>
</li>
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<li><a href="#org8ffadeb">6. With Cedrat Piezo-electric Actuators</a>
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<ul>
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<li><a href="#org8f92e1b">6.1. Identification</a></li>
<li><a href="#org9183f23">6.2. Control Design</a></li>
<li><a href="#orge8484c9">6.3. Identification</a></li>
<li><a href="#org35ee201">6.4. Sensitivity to Disturbance</a></li>
<li><a href="#orgfdfe26c">6.5. Damped Plant</a></li>
<li><a href="#org7b0ae1d">6.6. Conclusion</a></li>
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</ul>
</li>
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<li><a href="#org51051c1">7. Comparison of Active Damping Techniques</a>
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<ul>
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<li><a href="#orga925887">7.1. Load the plants</a></li>
<li><a href="#orga01b1a9">7.2. Sensitivity to Disturbance</a></li>
<li><a href="#orga4fdd66">7.3. Noise Budget</a></li>
<li><a href="#orgbb2291d">7.4. Damped Plant</a></li>
<li><a href="#org40f7a4d">7.5. Conclusion</a></li>
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</ul>
</li>
</ul>
</div>
</div>
<p>
The idea is to use the same model as the full Simscape Model but to restrict the motion only in the vertical direction.
</p>
<p>
This is done in order to more easily study the system and evaluate control techniques.
</p>
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<div id="outline-container-orgac9343f" class="outline-2">
<h2 id="orgac9343f"><span class="section-number-2">1</span> Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
<p>
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A schematic of the uniaxial model used for simulations is represented in figure <a href="#org7ac1a00">1</a>.
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</p>
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<p>
The perturbations \(w\) are:
</p>
<ul class="org-ul">
<li>\(F_s\): direct forces applied to the sample such as inertia forces and cable forces</li>
<li>\(F_{rz}\): parasitic forces due to the rotation of the spindle</li>
<li>\(F_{ty}\): parasitic forces due to scans with the translation stage</li>
<li>\(D_w\): ground motion</li>
</ul>
<p>
The quantity to \(z\) to control is:
</p>
<ul class="org-ul">
<li>\(D\): the position of the sample with respect to the granite</li>
</ul>
<p>
The measured quantities \(v\) are:
</p>
<ul class="org-ul">
<li>\(D\): the position of the sample with respect to the granite</li>
</ul>
<p>
We study the use of an additional sensor:
</p>
<ul class="org-ul">
<li>\(F_n\): a force sensor located in the nano-hexapod</li>
<li>\(v_n\): an absolute velocity sensor located on the top platform of the nano-hexapod</li>
<li>\(d_r\): a relative motion sensor located in the nano-hexapod</li>
</ul>
<p>
The control signal \(u\) is:
</p>
<ul class="org-ul">
<li>\(F\) the force applied by the nano-hexapod actuator</li>
</ul>
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<div id="org7ac1a00" class="figure">
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<p><img src="figs/uniaxial-model-nass-flexible.png" alt="uniaxial-model-nass-flexible.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the uniaxial model used</p>
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</div>
Update Analysis
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<p>
Update Analysis
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Few active damping techniques will be compared in order to decide which sensor is to be included in the system.
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Schematics of the active damping techniques are displayed in figure <a href="#orgb379a31">2</a>.
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</p>
Update Analysis
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<div id="orgb379a31" class="figure">
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<p><img src="figs/uniaxial-model-nass-flexible-active-damping.png" alt="uniaxial-model-nass-flexible-active-damping.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Comparison of used active damping techniques</p>
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</div>
</div>
</div>
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<div id="outline-container-org3890417" class="outline-2">
<h2 id="org3890417"><span class="section-number-2">2</span> Undamped System</h2>
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<div class="outline-text-2" id="text-2">
<p>
Let's start by study the undamped system.
</p>
</div>
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<div id="outline-container-orgbc01ced" class="outline-3">
<h3 id="orgbc01ced"><span class="section-number-3">2.1</span> Init</h3>
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<div class="outline-text-3" id="text-2-1">
<p>
We initialize all the stages with the default parameters.
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
</p>
<p>
All the controllers are set to 0 (Open Loop).
</p>
</div>
</div>
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<div id="outline-container-org699bc2c" class="outline-3">
<h3 id="org699bc2c"><span class="section-number-3">2.2</span> Identification</h3>
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<div class="outline-text-3" id="text-2-2">
<p>
We identify the dynamics of the system.
</p>
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<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 = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
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<p>
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
</p>
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<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fs'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied on the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnl'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdrz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
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io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dsm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">7</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">9</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
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<p>
Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model.
</p>
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<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <span class="org-comment">% Parasitic Force Rz [N]</span>
G.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span><span class="org-rainbow-delimiters-depth-1">}</span>; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
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<p>
Finally, we save the identified system dynamics for further analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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</div>
</div>
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<div id="outline-container-org0c59d84" class="outline-3">
<h3 id="org0c59d84"><span class="section-number-3">2.3</span> Sensitivity to Disturbances</h3>
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<div class="outline-text-3" id="text-2-3">
<p>
We show several plots representing the sensitivity to disturbances:
</p>
<ul class="org-ul">
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<li>in figure <a href="#orgc6b4646">3</a> the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown</li>
<li>in figure <a href="#orgbcfd91e">4</a>, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown</li>
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</ul>
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<div id="orgc6b4646" class="figure">
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<p><img src="figs/uniaxial-sensitivity-disturbances.png" alt="uniaxial-sensitivity-disturbances.png" />
</p>
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<p><span class="figure-number">Figure 3: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-disturbances.pdf">pdf</a>)</p>
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</div>
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<div id="orgbcfd91e" class="figure">
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<p><img src="figs/uniaxial-sensitivity-force-dist.png" alt="uniaxial-sensitivity-force-dist.png" />
</p>
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<p><span class="figure-number">Figure 4: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-force-dist.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orgb191841" class="outline-3">
<h3 id="orgb191841"><span class="section-number-3">2.4</span> Noise Budget</h3>
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<div class="outline-text-3" id="text-2-4">
<p>
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We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
The effect of these disturbances on the distance \(D\) is computed below.
The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure <a href="#orgb199386">5</a> and the Cumulative Amplitude Spectrum is shown in figure <a href="#org3989b84">6</a>.
</p>
<p>
The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure <a href="#org3989b84">6</a>.
</p>
<pre class="example">
3.3793e-06
</pre>
<div id="orgb199386" class="figure">
<p><img src="figs/uniaxial-psd-dist.png" alt="uniaxial-psd-dist.png" />
</p>
<p><span class="figure-number">Figure 5: </span>caption (<a href="./figs/uniaxial-psd-dist.png">png</a>, <a href="./figs/uniaxial-psd-dist.pdf">pdf</a>)</p>
</div>
<div id="org3989b84" class="figure">
<p><img src="figs/uniaxial-cas-dist.png" alt="uniaxial-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 6: </span>caption (<a href="./figs/uniaxial-cas-dist.png">png</a>, <a href="./figs/uniaxial-cas-dist.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org1b2f77b" class="outline-3">
<h3 id="org1b2f77b"><span class="section-number-3">2.5</span> Plant</h3>
<div class="outline-text-3" id="text-2-5">
<p>
The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure <a href="#org63b704d">7</a>.
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It corresponds to the plant to control.
</p>
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<div id="org63b704d" class="figure">
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<p><img src="figs/uniaxial-plant.png" alt="uniaxial-plant.png" />
</p>
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<p><span class="figure-number">Figure 7: </span>Bode plot of the Plant (<a href="./figs/uniaxial-plant.png">png</a>, <a href="./figs/uniaxial-plant.pdf">pdf</a>)</p>
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</div>
</div>
</div>
</div>
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<div id="outline-container-org27aabe3" class="outline-2">
<h2 id="org27aabe3"><span class="section-number-2">3</span> Integral Force Feedback</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="orgcfb1bdd"></a>
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</p>
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<div id="org2faff5f" class="figure">
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<p><img src="figs/uniaxial-model-nass-flexible-iff.png" alt="uniaxial-model-nass-flexible-iff.png" />
</p>
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<p><span class="figure-number">Figure 8: </span>Uniaxial IFF Control Schematic</p>
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</div>
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</div>
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<div id="outline-container-org04b5ef2" class="outline-3">
<h3 id="org04b5ef2"><span class="section-number-3">3.1</span> Control Design</h3>
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<div class="outline-text-3" id="text-3-1">
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<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Let'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.
</p>
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<div id="org7cefef0" class="figure">
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<p><img src="figs/uniaxial_iff_plant.png" alt="uniaxial_iff_plant.png" />
</p>
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<p><span class="figure-number">Figure 9: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_iff_plant.png">png</a>, <a href="./figs/uniaxial_iff_plant.pdf">pdf</a>)</p>
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</div>
<p>
The controller for each pair of actuator/sensor is:
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = <span class="org-type">-</span><span class="org-highlight-numbers-number">1000</span><span class="org-type">/</span>s;
</pre>
</div>
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<div id="org2c6c9a0" class="figure">
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<p><img src="figs/uniaxial_iff_open_loop.png" alt="uniaxial_iff_open_loop.png" />
</p>
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<p><span class="figure-number">Figure 10: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_iff_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_open_loop.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orgd976043" class="outline-3">
<h3 id="orgd976043"><span class="section-number-3">3.2</span> Identification</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
Let's initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
All the controllers are set to 0.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_iff = <span class="org-type">-</span>K_iff;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<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 = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fs'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied on the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnl'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdrz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dsm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">7</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">9</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_iff = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G_iff.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_iff.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span><span class="org-rainbow-delimiters-depth-1">}</span>; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
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<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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</div>
</div>
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<div id="outline-container-orgb785fd5" class="outline-3">
<h3 id="orgb785fd5"><span class="section-number-3">3.3</span> Sensitivity to Disturbance</h3>
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<div class="outline-text-3" id="text-3-3">
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<div id="org9085a3e" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_iff.png" alt="uniaxial_sensitivity_dist_iff.png" />
</p>
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<p><span class="figure-number">Figure 11: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_iff.pdf">pdf</a>)</p>
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</div>
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<div id="org087512a" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_stages_iff.png" alt="uniaxial_sensitivity_dist_stages_iff.png" />
</p>
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<p><span class="figure-number">Figure 12: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_iff.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org7f2e353" class="outline-3">
<h3 id="org7f2e353"><span class="section-number-3">3.4</span> Damped Plant</h3>
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<div class="outline-text-3" id="text-3-4">
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<div id="orge8dfaf6" class="figure">
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<p><img src="figs/uniaxial_plant_iff_damped.png" alt="uniaxial_plant_iff_damped.png" />
</p>
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<p><span class="figure-number">Figure 13: </span>Damped Plant after IFF is applied (<a href="./figs/uniaxial_plant_iff_damped.png">png</a>, <a href="./figs/uniaxial_plant_iff_damped.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org46695ba" class="outline-3">
<h3 id="org46695ba"><span class="section-number-3">3.5</span> Conclusion</h3>
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<div class="outline-text-3" id="text-3-5">
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<div class="important">
<p>
Integral Force Feedback:
</p>
</div>
</div>
</div>
</div>
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<div id="outline-container-org8f75c3f" class="outline-2">
<h2 id="org8f75c3f"><span class="section-number-2">4</span> Relative Motion Control</h2>
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<div class="outline-text-2" id="text-4">
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<p>
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<a id="org14dacd3"></a>
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</p>
<p>
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
</p>
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<div id="orgcb12d53" class="figure">
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<p><img src="figs/uniaxial-model-nass-flexible-rmc.png" alt="uniaxial-model-nass-flexible-rmc.png" />
</p>
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<p><span class="figure-number">Figure 14: </span>Uniaxial RMC Control Schematic</p>
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</div>
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<div id="outline-container-org7291ab1" class="outline-3">
<h3 id="org7291ab1"><span class="section-number-3">4.1</span> Control Design</h3>
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<div class="outline-text-3" id="text-4-1">
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<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Let'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.
</p>
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<div id="org7c9427d" class="figure">
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<p><img src="figs/uniaxial_rmc_plant.png" alt="uniaxial_rmc_plant.png" />
</p>
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<p><span class="figure-number">Figure 15: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/uniaxial_rmc_plant.png">png</a>, <a href="./figs/uniaxial_rmc_plant.pdf">pdf</a>)</p>
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</div>
<p>
The Relative Motion Controller 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_rmc = s<span class="org-type">*</span><span class="org-highlight-numbers-number">50000</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">10000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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<div id="org782296b" class="figure">
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<p><img src="figs/uniaxial_rmc_open_loop.png" alt="uniaxial_rmc_open_loop.png" />
</p>
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<p><span class="figure-number">Figure 16: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_rmc_open_loop.png">png</a>, <a href="./figs/uniaxial_rmc_open_loop.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org72b4f0a" class="outline-3">
<h3 id="org72b4f0a"><span class="section-number-3">4.2</span> Identification</h3>
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<div class="outline-text-3" id="text-4-2">
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<p>
Let's initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
And initialize the controllers.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_rmc = <span class="org-type">-</span>K_rmc;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<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 = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fs'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied on the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnl'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdrz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dsm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">7</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">9</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_rmc = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G_rmc.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_rmc.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span><span class="org-rainbow-delimiters-depth-1">}</span>; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
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<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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</div>
</div>
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<div id="outline-container-org7669eee" class="outline-3">
<h3 id="org7669eee"><span class="section-number-3">4.3</span> Sensitivity to Disturbance</h3>
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<div class="outline-text-3" id="text-4-3">
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<div id="orga53e45b" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_rmc.png" alt="uniaxial_sensitivity_dist_rmc.png" />
</p>
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<p><span class="figure-number">Figure 17: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_rmc.pdf">pdf</a>)</p>
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</div>
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<div id="orgb21d169" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_stages_rmc.png" alt="uniaxial_sensitivity_dist_stages_rmc.png" />
</p>
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<p><span class="figure-number">Figure 18: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_rmc.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orgab2a3d1" class="outline-3">
<h3 id="orgab2a3d1"><span class="section-number-3">4.4</span> Damped Plant</h3>
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<div class="outline-text-3" id="text-4-4">
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<div id="orgc8a382a" class="figure">
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<p><img src="figs/uniaxial_plant_rmc_damped.png" alt="uniaxial_plant_rmc_damped.png" />
</p>
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<p><span class="figure-number">Figure 19: </span>Damped Plant after RMC is applied (<a href="./figs/uniaxial_plant_rmc_damped.png">png</a>, <a href="./figs/uniaxial_plant_rmc_damped.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orgbc19b27" class="outline-3">
<h3 id="orgbc19b27"><span class="section-number-3">4.5</span> Conclusion</h3>
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<div class="outline-text-3" id="text-4-5">
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<div class="important">
<p>
Relative Motion Control:
</p>
</div>
</div>
</div>
</div>
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<div id="outline-container-org264e55c" class="outline-2">
<h2 id="org264e55c"><span class="section-number-2">5</span> Direct Velocity Feedback</h2>
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<div class="outline-text-2" id="text-5">
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<p>
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<a id="org96d3a12"></a>
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</p>
<p>
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
</p>
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<div id="org0856bca" class="figure">
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<p><img src="figs/uniaxial-model-nass-flexible-dvf.png" alt="uniaxial-model-nass-flexible-dvf.png" />
</p>
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<p><span class="figure-number">Figure 20: </span>Uniaxial DVF Control Schematic</p>
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</div>
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<div id="outline-container-orge1aa5cd" class="outline-3">
<h3 id="orge1aa5cd"><span class="section-number-3">5.1</span> Control Design</h3>
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<div class="outline-text-3" id="text-5-1">
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<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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<div id="orga963d24" class="figure">
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<p><img src="figs/uniaxial_dvf_plant.png" alt="uniaxial_dvf_plant.png" />
</p>
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<p><span class="figure-number">Figure 21: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_plant.png">png</a>, <a href="./figs/uniaxial_dvf_plant.pdf">pdf</a>)</p>
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</div>
<div class="org-src-container">
<pre class="src src-matlab">K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5e4</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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<div id="org84510c2" class="figure">
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<p><img src="figs/uniaxial_dvf_loop_gain.png" alt="uniaxial_dvf_loop_gain.png" />
</p>
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<p><span class="figure-number">Figure 22: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_loop_gain.png">png</a>, <a href="./figs/uniaxial_dvf_loop_gain.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org2a880b1" class="outline-3">
<h3 id="org2a880b1"><span class="section-number-3">5.2</span> Identification</h3>
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<div class="outline-text-3" id="text-5-2">
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<p>
Let's initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
And initialize the controllers.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_dvf = <span class="org-type">-</span>K_dvf;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<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 = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fs'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied on the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnl'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdrz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dsm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">7</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">9</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_dvf = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G_dvf.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_dvf.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span><span class="org-rainbow-delimiters-depth-1">}</span>; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
Update Analysis
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<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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</div>
</div>
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<div id="outline-container-orgd7e4638" class="outline-3">
<h3 id="orgd7e4638"><span class="section-number-3">5.3</span> Sensitivity to Disturbance</h3>
Update Analysis
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<div class="outline-text-3" id="text-5-3">
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<div id="org5a094e1" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_dvf.png" alt="uniaxial_sensitivity_dist_dvf.png" />
</p>
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<p><span class="figure-number">Figure 23: </span>Sensitivity to disturbance once the DVF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_dvf.pdf">pdf</a>)</p>
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</div>
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<div id="orga67a694" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_stages_dvf.png" alt="uniaxial_sensitivity_dist_stages_dvf.png" />
</p>
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<p><span class="figure-number">Figure 24: </span>Sensitivity to force disturbances in various stages when DVF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_dvf.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org9524bf4" class="outline-3">
<h3 id="org9524bf4"><span class="section-number-3">5.4</span> Damped Plant</h3>
Update Analysis
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<div class="outline-text-3" id="text-5-4">
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<div id="org6b78506" class="figure">
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<p><img src="figs/uniaxial_plant_dvf_damped.png" alt="uniaxial_plant_dvf_damped.png" />
</p>
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<p><span class="figure-number">Figure 25: </span>Damped Plant after DVF is applied (<a href="./figs/uniaxial_plant_dvf_damped.png">png</a>, <a href="./figs/uniaxial_plant_dvf_damped.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org329f7b9" class="outline-3">
<h3 id="org329f7b9"><span class="section-number-3">5.5</span> Conclusion</h3>
Update Analysis
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<div class="outline-text-3" id="text-5-5">
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<div class="important">
<p>
Direct Velocity Feedback:
</p>
</div>
</div>
</div>
</div>
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<div id="outline-container-org8ffadeb" class="outline-2">
<h2 id="org8ffadeb"><span class="section-number-2">6</span> With Cedrat Piezo-electric Actuators</h2>
Update Analysis
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<div class="outline-text-2" id="text-6">
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</div>
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<div id="outline-container-org8f92e1b" class="outline-3">
<h3 id="org8f92e1b"><span class="section-number-3">6.1</span> Identification</h3>
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<div class="outline-text-3" id="text-6-1">
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<p>
Add Cedrat Actuator
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We identify the dynamics of the system.
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</p>
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<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 = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial_cedrat'</span>;
</pre>
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</div>
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<p>
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fs'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied on the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnl'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdrz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dsm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">7</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">9</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<p>
Finally, we use the <code>linearize</code> Matlab function to extract a state space model from the simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <span class="org-comment">% Parasitic Force Rz [N]</span>
G.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span><span class="org-rainbow-delimiters-depth-1">}</span>; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
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</pre>
</div>
</div>
</div>
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<div id="outline-container-org9183f23" class="outline-3">
<h3 id="org9183f23"><span class="section-number-3">6.2</span> Control Design</h3>
Update Analysis
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<div class="outline-text-3" id="text-6-2">
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<p>
Let'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.
</p>
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<div id="org8d3fc99" class="figure">
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<p><img src="figs/uniaxial_cedrat_plant.png" alt="uniaxial_cedrat_plant.png" />
</p>
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<p><span class="figure-number">Figure 26: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_cedrat_plant.png">png</a>, <a href="./figs/uniaxial_cedrat_plant.pdf">pdf</a>)</p>
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</div>
<p>
The controller for each pair of actuator/sensor is:
</p>
<div class="org-src-container">
<pre class="src src-matlab">K_cedrat = <span class="org-highlight-numbers-number">1000</span><span class="org-type">/</span>s;
</pre>
</div>
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<div id="orgc5810ab" class="figure">
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<p><img src="figs/uniaxial_cedrat_open_loop.png" alt="uniaxial_cedrat_open_loop.png" />
</p>
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<p><span class="figure-number">Figure 27: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_cedrat_open_loop.png">png</a>, <a href="./figs/uniaxial_cedrat_open_loop.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orge8484c9" class="outline-3">
<h3 id="orge8484c9"><span class="section-number-3">6.3</span> Identification</h3>
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<div class="outline-text-3" id="text-6-3">
<p>
Let's initialize the system prior to identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGround<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeGranite<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeTy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRy<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeRz<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMicroHexapod<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeAxisc<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeMirror<span class="org-rainbow-delimiters-depth-1">()</span>;
initializeNanoHexapod<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
initializeSample<span class="org-rainbow-delimiters-depth-1">(</span>struct<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mass'</span>, <span class="org-highlight-numbers-number">50</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
All the controllers are set to 0.
</p>
<div class="org-src-container">
<pre class="src src-matlab">K = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_iff = <span class="org-type">-</span>K_cedrat;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_iff'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_rmc = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_rmc'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
K_dvf = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/controllers.mat'</span>, <span class="org-string">'K_dvf'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<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 = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'sim_nano_station_uniaxial_cedrat'</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Ground Motion</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fs'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied on the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnl'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force applied by the NASS</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdty'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Ty</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fdrz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Parasitic force Rz</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dsm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of the sample</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">7</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Force sensor in NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dnlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Displacement of NASS's legs</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">9</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dgm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Absolute displacement of the granite</span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Vlm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Measured absolute velocity of the top NASS platform</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_cedrat = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G_cedrat.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dw'</span>, ...<span class="org-comment"> % Ground Motion [m]</span>
<span class="org-string">'Fs'</span>, ...<span class="org-comment"> % Force Applied on Sample [N]</span>
<span class="org-string">'Fn'</span>, ...<span class="org-comment"> % Force applied by NASS [N]</span>
<span class="org-string">'Fty'</span>, ...<span class="org-comment"> % Parasitic Force Ty [N]</span>
<span class="org-string">'Frz'</span><span class="org-rainbow-delimiters-depth-1">}</span>; <span class="org-comment">% Parasitic Force Rz [N]</span>
G_cedrat.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'D'</span>, ...<span class="org-comment"> % Measured sample displacement x.r.t. granite [m]</span>
<span class="org-string">'Fnm'</span>, ...<span class="org-comment"> % Force Sensor in NASS [N]</span>
<span class="org-string">'Dnm'</span>, ...<span class="org-comment"> % Displacement Sensor in NASS [m]</span>
<span class="org-string">'Dgm'</span>, ...<span class="org-comment"> % Asbolute displacement of Granite [m]</span>
<span class="org-string">'Vlm'</span><span class="org-rainbow-delimiters-depth-1">}</span>; ...<span class="org-comment"> % Absolute Velocity of NASS [m/s]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G_cedrat'</span>, <span class="org-string">'-append'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
</div>
</div>
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<div id="outline-container-org35ee201" class="outline-3">
<h3 id="org35ee201"><span class="section-number-3">6.4</span> Sensitivity to Disturbance</h3>
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<div class="outline-text-3" id="text-6-4">
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<div id="org25c9462" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_cedrat.png" alt="uniaxial_sensitivity_dist_cedrat.png" />
</p>
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<p><span class="figure-number">Figure 28: </span>Sensitivity to disturbance once the CEDRAT controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_cedrat.pdf">pdf</a>)</p>
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</div>
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<div id="org401a0e9" class="figure">
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<p><img src="figs/uniaxial_sensitivity_dist_stages_cedrat.png" alt="uniaxial_sensitivity_dist_stages_cedrat.png" />
</p>
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<p><span class="figure-number">Figure 29: </span>Sensitivity to force disturbances in various stages when CEDRAT is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orgfdfe26c" class="outline-3">
<h3 id="orgfdfe26c"><span class="section-number-3">6.5</span> Damped Plant</h3>
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<div class="outline-text-3" id="text-6-5">
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<div id="org96e840c" class="figure">
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<p><img src="figs/uniaxial_plant_cedrat_damped.png" alt="uniaxial_plant_cedrat_damped.png" />
</p>
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<p><span class="figure-number">Figure 30: </span>Damped Plant after CEDRAT is applied (<a href="./figs/uniaxial_plant_cedrat_damped.png">png</a>, <a href="./figs/uniaxial_plant_cedrat_damped.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-org7b0ae1d" class="outline-3">
<h3 id="org7b0ae1d"><span class="section-number-3">6.6</span> Conclusion</h3>
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<div class="outline-text-3" id="text-6-6">
<div class="important">
<p>
This gives similar results than with a classical force sensor.
</p>
</div>
</div>
</div>
</div>
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<div id="outline-container-org51051c1" class="outline-2">
<h2 id="org51051c1"><span class="section-number-2">7</span> Comparison of Active Damping Techniques</h2>
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<div class="outline-text-2" id="text-7">
<p>
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<a id="orgf321de8"></a>
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</p>
</div>
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<div id="outline-container-orga925887" class="outline-3">
<h3 id="orga925887"><span class="section-number-3">7.1</span> Load the plants</h3>
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<div class="outline-text-3" id="text-7-1">
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'G_dvf'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
</div>
</div>
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<div id="outline-container-orga01b1a9" class="outline-3">
<h3 id="orga01b1a9"><span class="section-number-3">7.2</span> Sensitivity to Disturbance</h3>
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<div class="outline-text-3" id="text-7-2">
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<div id="orgafd9b97" class="figure">
Update Analysis
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<p><img src="figs/uniaxial_sensitivity_ground_motion.png" alt="uniaxial_sensitivity_ground_motion.png" />
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</p>
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<p><span class="figure-number">Figure 31: </span>Sensitivity to Ground Motion - Comparison (<a href="./figs/uniaxial_sensitivity_ground_motion.png">png</a>, <a href="./figs/uniaxial_sensitivity_ground_motion.pdf">pdf</a>)</p>
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</div>
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<div id="org3efc30e" class="figure">
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<p><img src="figs/uniaxial_sensitivity_direct_force.png" alt="uniaxial_sensitivity_direct_force.png" />
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</p>
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<p><span class="figure-number">Figure 32: </span>Sensitivity to disturbance - Comparison (<a href="./figs/uniaxial_sensitivity_direct_force.png">png</a>, <a href="./figs/uniaxial_sensitivity_direct_force.pdf">pdf</a>)</p>
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</div>
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<div id="orgb329c3d" class="figure">
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<p><img src="figs/uniaxial_sensitivity_fty.png" alt="uniaxial_sensitivity_fty.png" />
</p>
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<p><span class="figure-number">Figure 33: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_fty.png">png</a>, <a href="./figs/uniaxial_sensitivity_fty.pdf">pdf</a>)</p>
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<div id="orgc3f8a25" class="figure">
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<p><img src="figs/uniaxial_sensitivity_frz.png" alt="uniaxial_sensitivity_frz.png" />
</p>
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<p><span class="figure-number">Figure 34: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_frz.png">png</a>, <a href="./figs/uniaxial_sensitivity_frz.pdf">pdf</a>)</p>
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</div>
</div>
</div>
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<div id="outline-container-orga4fdd66" class="outline-3">
<h3 id="orga4fdd66"><span class="section-number-3">7.3</span> Noise Budget</h3>
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<div class="outline-text-3" id="text-7-3">
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<p>
We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
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<p>
The effect of these disturbances on the distance \(D\) is computed for all active damping techniques.
We then compute the Cumulative Amplitude Spectrum (figure <a href="#orge43e5e4">35</a>).
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</p>
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<div id="orge43e5e4" class="figure">
<p><img src="figs/uniaxial-comp-cas-dist.png" alt="uniaxial-comp-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 35: </span>Comparison of the Cumulative Amplitude Spectrum of \(D\) for different active damping techniques (<a href="./figs/uniaxial-comp-cas-dist.png">png</a>, <a href="./figs/uniaxial-comp-cas-dist.pdf">pdf</a>)</p>
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<p>
The obtained Root Mean Square Value for each active damping technique is shown below.
</p>
<table id="org3b74f43" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Obtain Root Mean Square value of \(D\) for each Active Damping Technique applied</caption>
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">D [m rms]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">OL</td>
<td class="org-right">3.38e-06</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-right">3.40e-06</td>
</tr>
<tr>
<td class="org-left">RMC</td>
<td class="org-right">3.37e-06</td>
</tr>
<tr>
<td class="org-left">DVF</td>
<td class="org-right">3.38e-06</td>
</tr>
</tbody>
</table>
<p>
It is important to note that the effect of direct forces applied to the sample are not taken into account here.
</p>
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</div>
</div>
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<div id="outline-container-orgbb2291d" class="outline-3">
<h3 id="orgbb2291d"><span class="section-number-3">7.4</span> Damped Plant</h3>
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<div class="outline-text-3" id="text-7-4">
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<div id="org3213591" class="figure">
<p><img src="figs/uniaxial_plant_damped_comp.png" alt="uniaxial_plant_damped_comp.png" />
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<p><span class="figure-number">Figure 36: </span>Damped Plant - Comparison (<a href="./figs/uniaxial_plant_damped_comp.png">png</a>, <a href="./figs/uniaxial_plant_damped_comp.pdf">pdf</a>)</p>
</div>
</div>
</div>
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<h3 id="org40f7a4d"><span class="section-number-3">7.5</span> Conclusion</h3>
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<caption class="t-above"><span class="table-number">Table 2:</span> Comparison of proposed active damping techniques</caption>
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<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">IFF</th>
<th scope="col" class="org-left">RMC</th>
<th scope="col" class="org-left">DVF</th>
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</thead>
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<tr>
<td class="org-left">Sensor Type</td>
<td class="org-left">Force sensor</td>
<td class="org-left">Relative Motion</td>
<td class="org-left">Inertial</td>
</tr>
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<td class="org-left">Guaranteed Stability</td>
<td class="org-left">+</td>
<td class="org-left">+</td>
<td class="org-left">-</td>
</tr>
<tr>
<td class="org-left">Sensitivity (\(D_w\))</td>
<td class="org-left">-</td>
<td class="org-left">+</td>
<td class="org-left">-</td>
</tr>
<tr>
<td class="org-left">Sensitivity (\(F_s\))</td>
<td class="org-left">- (at low freq)</td>
<td class="org-left">+</td>
<td class="org-left">+</td>
</tr>
<tr>
<td class="org-left">Sensitivity (\(F_{ty,rz}\))</td>
<td class="org-left">+</td>
<td class="org-left">-</td>
<td class="org-left">+</td>
</tr>
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<td class="org-left">Overall RMS of \(D\)</td>
<td class="org-left">=</td>
<td class="org-left">=</td>
<td class="org-left">=</td>
</tr>
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
Add noise budget to compare active damping techniques
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<p class="date">Created: 2019-11-04 lun. 17:33</p>
Add uniaxial folder with analysis of active damping techniques
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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