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<title>Control of the NASS with Voice coil actuators</title>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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||||
<h1 class="title">Control of the NASS with Voice coil actuators</h1>
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||||
<div id="table-of-contents">
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||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org6d5d6b2">1. Initialization</a></li>
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||||
<li><a href="#orgf95b045">2. Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</a>
|
||||
<ul>
|
||||
<li><a href="#org75ecf55">2.1. Identification</a></li>
|
||||
<li><a href="#org203d651">2.2. Plant</a></li>
|
||||
<li><a href="#orgccc21d2">2.3. Root Locus</a></li>
|
||||
<li><a href="#org1a8ee8a">2.4. Controller and Loop Gain</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org2a44e66">3. High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</a>
|
||||
<ul>
|
||||
<li><a href="#org989c2e9">3.1. Identification of the damped plant</a></li>
|
||||
<li><a href="#orgd89dc76">3.2. Obtained Plant</a></li>
|
||||
<li><a href="#orgd1632cf">3.3. Controller Design and Loop Gain</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org5e26e70">4. On the usefulness of the High Authority Control loop / Linearization loop</a>
|
||||
<ul>
|
||||
<li><a href="#orge9b2f08">4.1. Identification</a></li>
|
||||
<li><a href="#orgfab8847">4.2. Plant in the Task space</a></li>
|
||||
<li><a href="#org18aeea5">4.3. Plant in the Leg’s space</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org0cd0c63">5. Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</a>
|
||||
<ul>
|
||||
<li><a href="#orga960106">5.1. Identification of the linearized plant</a></li>
|
||||
<li><a href="#org45829b7">5.2. Obtained Plant</a></li>
|
||||
<li><a href="#org16f56fa">5.3. Controller Design</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org0b05a3a">6. Simulation</a></li>
|
||||
<li><a href="#org16024e0">7. Results</a>
|
||||
<ul>
|
||||
<li><a href="#org26b1a39">7.1. Load the simulation results</a></li>
|
||||
<li><a href="#org0f974ff">7.2. Control effort</a></li>
|
||||
<li><a href="#orge126fd7">7.3. Load the simulation results</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The goal here is to study the use of a voice coil based nano-hexapod.
|
||||
That is to say a nano-hexapod with a very small stiffness.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org757d77b" class="figure">
|
||||
<p><img src="figs/cascade_control_architecture.png" alt="cascade_control_architecture.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6d5d6b2" class="outline-2">
|
||||
<h2 id="org6d5d6b2"><span class="section-number-2">1</span> Initialization</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
We initialize all the stages with the default parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The nano-hexapod is a voice coil based hexapod and the sample has a mass of 1kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'lorentz'</span>);
|
||||
initializeSample(<span class="org-string">'mass'</span>, 1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We set the references that corresponds to a tomography experiment.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances();
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||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'cascade-hac-lac'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We log the signals.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kx = tf(zeros(6));
|
||||
Kl = tf(zeros(6));
|
||||
Kiff = tf(zeros(6));
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf95b045" class="outline-2">
|
||||
<h2 id="orgf95b045"><span class="section-number-2">2</span> Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="org224edef"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org75ecf55" class="outline-3">
|
||||
<h3 id="org75ecf55"><span class="section-number-3">2.1</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
Let’s first identify the plant for the IFF controller.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'nass_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Fnlm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Force Sensors</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G_iff = linearize(mdl, io, 0);
|
||||
G_iff.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
|
||||
G_iff.OutputName = {<span class="org-string">'Fnlm1'</span>, <span class="org-string">'Fnlm2'</span>, <span class="org-string">'Fnlm3'</span>, <span class="org-string">'Fnlm4'</span>, <span class="org-string">'Fnlm5'</span>, <span class="org-string">'Fnlm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org203d651" class="outline-3">
|
||||
<h3 id="org203d651"><span class="section-number-3">2.2</span> Plant</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
The obtained plant for IFF is shown in Figure <a href="#orga39f9fa">2</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga39f9fa" class="figure">
|
||||
<p><img src="figs/cascade_vc_iff_plant.png" alt="cascade_vc_iff_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>IFF Plant (<a href="./figs/cascade_vc_iff_plant.png">png</a>, <a href="./figs/cascade_vc_iff_plant.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgccc21d2" class="outline-3">
|
||||
<h3 id="orgccc21d2"><span class="section-number-3">2.3</span> Root Locus</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
As seen in the root locus (Figure <a href="#org528b5f0">3</a>, no damping can be added to modes corresponding to the resonance of the micro-station.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
However, critical damping can be achieve for the resonances of the nano-hexapod as shown in the zoomed part of the root (Figure <a href="#org528b5f0">3</a>, left part).
|
||||
The maximum damping is obtained for a control gain of \(\approx 70\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org528b5f0" class="figure">
|
||||
<p><img src="figs/cascade_vc_iff_root_locus.png" alt="cascade_vc_iff_root_locus.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Root Locus for the IFF control (<a href="./figs/cascade_vc_iff_root_locus.png">png</a>, <a href="./figs/cascade_vc_iff_root_locus.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1a8ee8a" class="outline-3">
|
||||
<h3 id="org1a8ee8a"><span class="section-number-3">2.4</span> Controller and Loop Gain</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
We create the \(6 \times 6\) diagonal Integral Force Feedback controller.
|
||||
The obtained loop gain is shown in Figure <a href="#orgc890275">4</a>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kiff = <span class="org-type">-</span>70<span class="org-type">/</span>s<span class="org-type">*</span>eye(6);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc890275" class="figure">
|
||||
<p><img src="figs/cascade_vc_iff_loop_gain.png" alt="cascade_vc_iff_loop_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Obtained Loop gain the IFF Control (<a href="./figs/cascade_vc_iff_loop_gain.png">png</a>, <a href="./figs/cascade_vc_iff_loop_gain.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2a44e66" class="outline-2">
|
||||
<h2 id="org2a44e66"><span class="section-number-2">3</span> High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org1d54e1b"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org989c2e9" class="outline-3">
|
||||
<h3 id="org989c2e9"><span class="section-number-3">3.1</span> Identification of the damped plant</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
Let’s identify the dynamics from \(\bm{\tau}^\prime\) to \(d\bm{\mathcal{L}}\) as shown in Figure <a href="#org757d77b">1</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'nass_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Inputs</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Micro-Station'</span>], 3, <span class="org-string">'output'</span>, [], <span class="org-string">'Dnlm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Leg Displacement</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
Gl = linearize(mdl, io, 0);
|
||||
Gl.InputName = {<span class="org-string">'Fnl1'</span>, <span class="org-string">'Fnl2'</span>, <span class="org-string">'Fnl3'</span>, <span class="org-string">'Fnl4'</span>, <span class="org-string">'Fnl5'</span>, <span class="org-string">'Fnl6'</span>};
|
||||
Gl.OutputName = {<span class="org-string">'Dnlm1'</span>, <span class="org-string">'Dnlm2'</span>, <span class="org-string">'Dnlm3'</span>, <span class="org-string">'Dnlm4'</span>, <span class="org-string">'Dnlm5'</span>, <span class="org-string">'Dnlm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
There are some unstable poles in the Plant with very small imaginary parts.
|
||||
These unstable poles are probably not physical, and they disappear when taking the minimum realization of the plant.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">isstable(Gl)
|
||||
Gl = minreal(Gl);
|
||||
isstable(Gl)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd89dc76" class="outline-3">
|
||||
<h3 id="orgd89dc76"><span class="section-number-3">3.2</span> Obtained Plant</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
The obtained dynamics is shown in Figure <a href="#orgd1818fd">5</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Few things can be said on the dynamics:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>the dynamics of the diagonal elements are almost all the same</li>
|
||||
<li>the system is well decoupled below the resonances of the nano-hexapod (1Hz)</li>
|
||||
<li>the dynamics of the diagonal elements are almost equivalent to a critically damped mass-spring-system with some spurious resonances above 50Hz corresponding to the resonances of the micro-station</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgd1818fd" class="figure">
|
||||
<p><img src="figs/cascade_vc_hac_joint_plant.png" alt="cascade_vc_hac_joint_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Plant for the High Authority Control in the Joint Space (<a href="./figs/cascade_vc_hac_joint_plant.png">png</a>, <a href="./figs/cascade_vc_hac_joint_plant.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd1632cf" class="outline-3">
|
||||
<h3 id="orgd1632cf"><span class="section-number-3">3.3</span> Controller Design and Loop Gain</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
As the plant is well decoupled, a diagonal plant is designed.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
|
||||
h = 2; <span class="org-comment">% Lead parameter</span>
|
||||
|
||||
Kl = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> ...<span class="org-comment"> % Lead</span>
|
||||
(1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> ...<span class="org-comment"> % Lead</span>
|
||||
(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10)<span class="org-type">/</span>s <span class="org-type">*</span> ...<span class="org-comment"> % Weak Integrator</span>
|
||||
(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1)<span class="org-type">/</span>s <span class="org-type">*</span> ...<span class="org-comment"> % Weak Integrator</span>
|
||||
1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10); <span class="org-comment">% Low pass filter after crossover</span>
|
||||
|
||||
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
|
||||
Kl = Kl<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gl<span class="org-type">*</span>Kl, wc))));
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5e26e70" class="outline-2">
|
||||
<h2 id="org5e26e70"><span class="section-number-2">4</span> On the usefulness of the High Authority Control loop / Linearization loop</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
Let’s see what happens is we omit the HAC loop and we directly try to control the damped plant using the measurement of the sample with respect to the granite \(\bm{\mathcal{X}}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We can do that in two different ways:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>in the task space as shown in Figure <a href="#orge366d0b">6</a></li>
|
||||
<li>in the space of the legs as shown in Figure <a href="#orgd23329e">7</a></li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orge366d0b" class="figure">
|
||||
<p><img src="figs/control_architecture_iff_X.png" alt="control_architecture_iff_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>IFF control + primary controller in the task space</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd23329e" class="figure">
|
||||
<p><img src="figs/control_architecture_iff_L.png" alt="control_architecture_iff_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>HAC-LAC control architecture in the frame of the legs</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge9b2f08" class="outline-3">
|
||||
<h3 id="orge9b2f08"><span class="section-number-3">4.1</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-iff'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'nass_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller/HAC-IFF/Kx'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, 0);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">isstable(G)
|
||||
G = <span class="org-type">-</span>minreal(G);
|
||||
isstable(G)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfab8847" class="outline-3">
|
||||
<h3 id="orgfab8847"><span class="section-number-3">4.2</span> Plant in the Task space</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
The obtained plant is shown in Figure
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gx = G<span class="org-type">*</span>inv(nano_hexapod.J<span class="org-type">'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org18aeea5" class="outline-3">
|
||||
<h3 id="org18aeea5"><span class="section-number-3">4.3</span> Plant in the Leg’s space</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gl = nano_hexapod.J<span class="org-type">*</span>G;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0cd0c63" class="outline-2">
|
||||
<h2 id="org0cd0c63"><span class="section-number-2">5</span> Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="orga738520"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orga960106" class="outline-3">
|
||||
<h3 id="orga960106"><span class="section-number-3">5.1</span> Identification of the linearized plant</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<p>
|
||||
We know identify the dynamics between \(\bm{r}_{\mathcal{X}_n}\) and \(\bm{r}_\mathcal{X}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'nass_model'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller/Cascade-HAC-LAC/Kx'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
Gx = linearize(mdl, io, 0);
|
||||
Gx.InputName = {<span class="org-string">'rL1'</span>, <span class="org-string">'rL2'</span>, <span class="org-string">'rL3'</span>, <span class="org-string">'rL4'</span>, <span class="org-string">'rL5'</span>, <span class="org-string">'rL6'</span>};
|
||||
Gx.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
As before, we take the minimum realization.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">isstable(Gx)
|
||||
Gx = <span class="org-type">-</span>minreal(Gx);
|
||||
isstable(Gx)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org45829b7" class="outline-3">
|
||||
<h3 id="org45829b7"><span class="section-number-3">5.2</span> Obtained Plant</h3>
|
||||
</div>
|
||||
<div id="outline-container-org16f56fa" class="outline-3">
|
||||
<h3 id="org16f56fa"><span class="section-number-3">5.3</span> Controller Design</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>200; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
|
||||
h = 2; <span class="org-comment">% Lead parameter</span>
|
||||
|
||||
Kx = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> ...
|
||||
(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10)<span class="org-type">/</span>s <span class="org-type">*</span> ...
|
||||
(s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100)<span class="org-type">/</span>s <span class="org-type">*</span> ...
|
||||
1<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>500); <span class="org-comment">% For Piezo</span>
|
||||
<span class="org-comment">% Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * (s + 2*pi*10)/s * (s + 2*pi*1)/s ; % For voice coil</span>
|
||||
|
||||
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
|
||||
Kx = Kx<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kx, wc))));
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0b05a3a" class="outline-2">
|
||||
<h2 id="org0b05a3a"><span class="section-number-2">6</span> Simulation</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
||||
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'2'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we simulate the system.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">cascade_hac_lac_lorentz = simout;
|
||||
save(<span class="org-string">'./mat/cascade_hac_lac.mat'</span>, <span class="org-string">'cascade_hac_lac_lorentz'</span>, <span class="org-string">'-append'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org16024e0" class="outline-2">
|
||||
<h2 id="org16024e0"><span class="section-number-2">7</span> Results</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
</div>
|
||||
<div id="outline-container-org26b1a39" class="outline-3">
|
||||
<h3 id="org26b1a39"><span class="section-number-3">7.1</span> Load the simulation results</h3>
|
||||
<div class="outline-text-3" id="text-7-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'./mat/experiment_tomography.mat'</span>, <span class="org-string">'tomo_align_dist'</span>);
|
||||
load(<span class="org-string">'./mat/cascade_hac_lac.mat'</span>, <span class="org-string">'cascade_hac_lac'</span>, <span class="org-string">'cascade_hac_lac_lorentz'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0f974ff" class="outline-3">
|
||||
<h3 id="org0f974ff"><span class="section-number-3">7.2</span> Control effort</h3>
|
||||
</div>
|
||||
<div id="outline-container-orge126fd7" class="outline-3">
|
||||
<h3 id="orge126fd7"><span class="section-number-3">7.3</span> Load the simulation results</h3>
|
||||
<div class="outline-text-3" id="text-7-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n_av = 4;
|
||||
han_win = hanning(ceil(length(cascade_hac_lac.Em.En.Data(<span class="org-type">:</span>,1))<span class="org-type">/</span>n_av));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">t = cascade_hac_lac.Em.En.Time;
|
||||
Ts = t(2)<span class="org-type">-</span>t(1);
|
||||
|
||||
[pxx_ol, f] = pwelch(tomo_align_dist.Em.En.Data, han_win, [], [], 1<span class="org-type">/</span>Ts);
|
||||
[pxx_ca, <span class="org-type">~</span>] = pwelch(cascade_hac_lac.Em.En.Data, han_win, [], [], 1<span class="org-type">/</span>Ts);
|
||||
[pxx_vc, <span class="org-type">~</span>] = pwelch(cascade_hac_lac_lorentz.Em.En.Data, han_win, [], [], 1<span class="org-type">/</span>Ts);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-03-23 lun. 10:05</p>
|
||||
<p class="date">Created: 2020-03-25 mer. 19:22</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user