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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">Cascade Control applied on the Simscape Model</h1>
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||||
<div id="table-of-contents">
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||||
<h2>Table of Contents</h2>
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||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org143e6e7">1. Initialization</a></li>
|
||||
<li><a href="#orga3bd04c">2. Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</a>
|
||||
<li><a href="#org61c020f">1. Initialization</a></li>
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||||
<li><a href="#orgabcff63">2. Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</a>
|
||||
<ul>
|
||||
<li><a href="#org702c612">2.1. Identification</a></li>
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||||
<li><a href="#org1eda739">2.2. Plant</a></li>
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||||
<li><a href="#orgf33a40f">2.3. Root Locus</a></li>
|
||||
<li><a href="#orgd895b21">2.4. Controller and Loop Gain</a></li>
|
||||
<li><a href="#org340d5c4">2.1. Identification</a></li>
|
||||
<li><a href="#orga3a6bfe">2.2. Plant</a></li>
|
||||
<li><a href="#org08443b2">2.3. Root Locus</a></li>
|
||||
<li><a href="#orgfcc764a">2.4. Controller and Loop Gain</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org101bfbc">3. High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</a>
|
||||
<li><a href="#org5f8f119">3. High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</a>
|
||||
<ul>
|
||||
<li><a href="#org259240d">3.1. Identification of the damped plant</a></li>
|
||||
<li><a href="#orgfc86b4c">3.2. Obtained Plant</a></li>
|
||||
<li><a href="#orgdea5e17">3.3. Controller Design and Loop Gain</a></li>
|
||||
<li><a href="#org4c8cb18">3.1. Identification of the damped plant</a></li>
|
||||
<li><a href="#org8ad1542">3.2. Obtained Plant</a></li>
|
||||
<li><a href="#orge39ae16">3.3. Controller Design and Loop Gain</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org58f9f32">4. Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</a>
|
||||
<li><a href="#org11a22c2">4. Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</a>
|
||||
<ul>
|
||||
<li><a href="#org82ca884">4.1. Identification of the linearized plant</a></li>
|
||||
<li><a href="#org6f5b22e">4.2. Obtained Plant</a></li>
|
||||
<li><a href="#org839cdb3">4.3. Controller Design</a></li>
|
||||
<li><a href="#orgfc45f6f">4.1. Identification of the linearized plant</a></li>
|
||||
<li><a href="#org170c73f">4.2. Obtained Plant</a></li>
|
||||
<li><a href="#orge97b630">4.3. Controller Design</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org3300911">5. Simulation</a></li>
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||||
<li><a href="#org2943496">6. Results</a></li>
|
||||
<li><a href="#org07fbc52">5. Simulation</a></li>
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||||
<li><a href="#org3e9cd70">6. Results</a></li>
|
||||
</ul>
|
||||
</div>
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||||
</div>
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||||
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||||
<p>
|
||||
The control architecture we wish here to study is shown in Figure <a href="#org19741db">1</a>.
|
||||
The control architecture we wish here to study is shown in Figure <a href="#orga0c68cb">1</a>.
|
||||
</p>
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||||
|
||||
|
||||
<div id="org19741db" class="figure">
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||||
<div id="orga0c68cb" class="figure">
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||||
<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>
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||||
@@ -79,28 +83,28 @@ The control architecture we wish here to study is shown in Figure <a href="#org1
|
||||
This cascade control is designed in three steps:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>In section <a href="#orga20c211">2</a>: an active damping controller is designed.
|
||||
<li>In section <a href="#orgbc2e2fd">2</a>: an active damping controller is designed.
|
||||
This is based on the Integral Force Feedback and applied in a decentralized way</li>
|
||||
<li>In section <a href="#org9367d49">3</a>: a decentralized tracking control is designed in the frame of the legs.
|
||||
<li>In section <a href="#orgd16580b">3</a>: a decentralized tracking control is designed in the frame of the legs.
|
||||
This controller is based on the displacement of each of the legs</li>
|
||||
<li>In section <a href="#orgdd883a0">4</a>: a controller is designed in the task space in order to follow the wanted reference path corresponding to the sample position with respect to the granite</li>
|
||||
<li>In section <a href="#org20bc645">4</a>: a controller is designed in the task space in order to follow the wanted reference path corresponding to the sample position with respect to the granite</li>
|
||||
</ul>
|
||||
|
||||
<div id="outline-container-org143e6e7" class="outline-2">
|
||||
<h2 id="org143e6e7"><span class="section-number-2">1</span> Initialization</h2>
|
||||
<div id="outline-container-org61c020f" class="outline-2">
|
||||
<h2 id="org61c020f"><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 class="src src-matlab"> initializeGround();
|
||||
initializeGranite();
|
||||
initializeTy();
|
||||
initializeRy();
|
||||
initializeRz();
|
||||
initializeMicroHexapod();
|
||||
initializeAxisc();
|
||||
initializeMirror();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -108,8 +112,8 @@ initializeMirror();
|
||||
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||||
initializeSample(<span class="org-string">'mass'</span>, 1);
|
||||
<pre class="src src-matlab"> initializeNanoHexapod(<span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||||
initializeSample(<span class="org-string">'mass'</span>, 1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -117,12 +121,12 @@ initializeSample(<span class="org-string">'mass'</span>, 1);
|
||||
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 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();
|
||||
<pre class="src src-matlab"> initializeDisturbances();
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -130,7 +134,7 @@ We set the references that corresponds to a tomography experiment.
|
||||
Open Loop.
|
||||
</p>
|
||||
<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 class="src src-matlab"> initializeController(<span class="org-string">'type'</span>, <span class="org-string">'cascade-hac-lac'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -138,7 +142,7 @@ Open Loop.
|
||||
And we put some gravity.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -146,55 +150,55 @@ And we put some gravity.
|
||||
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 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">Kp = tf(zeros(6));
|
||||
Kl = tf(zeros(6));
|
||||
Kiff = tf(zeros(6));
|
||||
<pre class="src src-matlab"> Kp = tf(zeros(6));
|
||||
Kl = tf(zeros(6));
|
||||
Kiff = tf(zeros(6));
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga3bd04c" class="outline-2">
|
||||
<h2 id="orga3bd04c"><span class="section-number-2">2</span> Low Authority Control - Integral Force Feedback \(\bm{K}_\text{IFF}\)</h2>
|
||||
<div id="outline-container-orgabcff63" class="outline-2">
|
||||
<h2 id="orgabcff63"><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="orga20c211"></a>
|
||||
<a id="orgbc2e2fd"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org702c612" class="outline-3">
|
||||
<h3 id="org702c612"><span class="section-number-3">2.1</span> Identification</h3>
|
||||
<div id="outline-container-org340d5c4" class="outline-3">
|
||||
<h3 id="org340d5c4"><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>;
|
||||
<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">%% 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>};
|
||||
<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-org1eda739" class="outline-3">
|
||||
<h3 id="org1eda739"><span class="section-number-3">2.2</span> Plant</h3>
|
||||
<div id="outline-container-orga3a6bfe" class="outline-3">
|
||||
<h3 id="orga3a6bfe"><span class="section-number-3">2.2</span> Plant</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
|
||||
<div id="org366ff26" class="figure">
|
||||
<div id="orgdfa60cc" class="figure">
|
||||
<p><img src="figs/cascade_iff_plant.png" alt="cascade_iff_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>IFF Plant (<a href="./figs/cascade_iff_plant.png">png</a>, <a href="./figs/cascade_iff_plant.pdf">pdf</a>)</p>
|
||||
@@ -202,11 +206,11 @@ G_iff.OutputName = {<span class="org-string">'Fnlm1'</span>, <span class="org-st
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf33a40f" class="outline-3">
|
||||
<h3 id="orgf33a40f"><span class="section-number-3">2.3</span> Root Locus</h3>
|
||||
<div id="outline-container-org08443b2" class="outline-3">
|
||||
<h3 id="org08443b2"><span class="section-number-3">2.3</span> Root Locus</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
|
||||
<div id="orgdf7a93e" class="figure">
|
||||
<div id="org6320a7a" class="figure">
|
||||
<p><img src="figs/cascade_iff_root_locus.png" alt="cascade_iff_root_locus.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Root Locus for the IFF control (<a href="./figs/cascade_iff_root_locus.png">png</a>, <a href="./figs/cascade_iff_root_locus.pdf">pdf</a>)</p>
|
||||
@@ -218,21 +222,21 @@ The maximum damping is obtained for a control gain of \(\approx 3000\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd895b21" class="outline-3">
|
||||
<h3 id="orgd895b21"><span class="section-number-3">2.4</span> Controller and Loop Gain</h3>
|
||||
<div id="outline-container-orgfcc764a" class="outline-3">
|
||||
<h3 id="orgfcc764a"><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="#org7aaed92">4</a>.
|
||||
The obtained loop gain is shown in Figure <a href="#org59e605f">4</a>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50;
|
||||
Kiff = <span class="org-type">-</span>3000<span class="org-type">/</span>s<span class="org-type">*</span>eye(6);
|
||||
<pre class="src src-matlab"> w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>50;
|
||||
Kiff = <span class="org-type">-</span>3000<span class="org-type">/</span>s<span class="org-type">*</span>eye(6);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7aaed92" class="figure">
|
||||
<div id="org59e605f" class="figure">
|
||||
<p><img src="figs/cascade_iff_loop_gain.png" alt="cascade_iff_loop_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Obtained Loop gain the IFF Control (<a href="./figs/cascade_iff_loop_gain.png">png</a>, <a href="./figs/cascade_iff_loop_gain.pdf">pdf</a>)</p>
|
||||
@@ -241,32 +245,32 @@ Kiff = <span class="org-type">-</span>3000<span class="org-type">/</span>s<span
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org101bfbc" class="outline-2">
|
||||
<h2 id="org101bfbc"><span class="section-number-2">3</span> High Authority Control in the joint space - \(\bm{K}_\mathcal{L}\)</h2>
|
||||
<div id="outline-container-org5f8f119" class="outline-2">
|
||||
<h2 id="org5f8f119"><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="org9367d49"></a>
|
||||
<a id="orgd16580b"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org259240d" class="outline-3">
|
||||
<h3 id="org259240d"><span class="section-number-3">3.1</span> Identification of the damped plant</h3>
|
||||
<div id="outline-container-org4c8cb18" class="outline-3">
|
||||
<h3 id="org4c8cb18"><span class="section-number-3">3.1</span> Identification of the damped plant</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
We now identify the transfer function from \(\tau^\prime\) to \(d\bm{\mathcal{L}}\) as shown in Figure <a href="#org19741db">1</a>.
|
||||
We now identify the transfer function from \(\tau^\prime\) to \(d\bm{\mathcal{L}}\) as shown in Figure <a href="#orga0c68cb">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>;
|
||||
<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">%% 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>};
|
||||
<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>
|
||||
|
||||
@@ -275,19 +279,19 @@ 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 class="src src-matlab"> isstable(Gl)
|
||||
Gl = minreal(Gl);
|
||||
isstable(Gl)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfc86b4c" class="outline-3">
|
||||
<h3 id="orgfc86b4c"><span class="section-number-3">3.2</span> Obtained Plant</h3>
|
||||
<div id="outline-container-org8ad1542" class="outline-3">
|
||||
<h3 id="org8ad1542"><span class="section-number-3">3.2</span> Obtained Plant</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
The obtain plant is shown in Figure <a href="#org455eb07">5</a>.
|
||||
The obtain plant is shown in Figure <a href="#org5030d6d">5</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -295,7 +299,7 @@ We can see that the plant is quite well decoupled.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org455eb07" class="figure">
|
||||
<div id="org5030d6d" class="figure">
|
||||
<p><img src="figs/cascade_hac_joint_plant.png" alt="cascade_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_hac_joint_plant.png">png</a>, <a href="./figs/cascade_hac_joint_plant.pdf">pdf</a>)</p>
|
||||
@@ -304,8 +308,8 @@ We can see that the plant is quite well decoupled.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgdea5e17" class="outline-3">
|
||||
<h3 id="orgdea5e17"><span class="section-number-3">3.3</span> Controller Design and Loop Gain</h3>
|
||||
<div id="outline-container-orge39ae16" class="outline-3">
|
||||
<h3 id="orge39ae16"><span class="section-number-3">3.3</span> Controller Design and Loop Gain</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
The controller consists of:
|
||||
@@ -318,20 +322,20 @@ The controller consists of:
|
||||
</ul>
|
||||
|
||||
<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>400; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
<pre class="src src-matlab"> wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>400; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
|
||||
h = 2; <span class="org-comment">% Lead parameter</span>
|
||||
h = 2; <span class="org-comment">% Lead parameter</span>
|
||||
|
||||
<span class="org-comment">% Kl = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/2));</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> (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s;
|
||||
<span class="org-comment">% Kl = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/2));</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> (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s;
|
||||
|
||||
<span class="org-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))));
|
||||
<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 id="org3dd0142" class="figure">
|
||||
<div id="org3b106c4" class="figure">
|
||||
<p><img src="figs/cascade_hac_joint_loop_gain.png" alt="cascade_hac_joint_loop_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Loop Gain for the High Autority Control in the joint space (<a href="./figs/cascade_hac_joint_loop_gain.png">png</a>, <a href="./figs/cascade_hac_joint_loop_gain.pdf">pdf</a>)</p>
|
||||
@@ -340,32 +344,32 @@ Kl = Kl<span class="org-type">.*</span>diag(1<span class="org-type">./</span>dia
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org58f9f32" class="outline-2">
|
||||
<h2 id="org58f9f32"><span class="section-number-2">4</span> Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</h2>
|
||||
<div id="outline-container-org11a22c2" class="outline-2">
|
||||
<h2 id="org11a22c2"><span class="section-number-2">4</span> Primary Controller in the task space - \(\bm{K}_\mathcal{X}\)</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="orgdd883a0"></a>
|
||||
<a id="org20bc645"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org82ca884" class="outline-3">
|
||||
<h3 id="org82ca884"><span class="section-number-3">4.1</span> Identification of the linearized plant</h3>
|
||||
<div id="outline-container-orgfc45f6f" class="outline-3">
|
||||
<h3 id="orgfc45f6f"><span class="section-number-3">4.1</span> Identification of the linearized plant</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<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>;
|
||||
<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/Kp'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Controller/Cascade-HAC-LAC/Kp'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'output'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Errror</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
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>};
|
||||
<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>
|
||||
|
||||
@@ -373,19 +377,19 @@ Gx.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">
|
||||
As before, we take the minimum realization.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">isstable(Gx)
|
||||
Gx = minreal(Gx);
|
||||
isstable(Gx)
|
||||
<pre class="src src-matlab"> isstable(Gx)
|
||||
Gx = minreal(Gx);
|
||||
isstable(Gx)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6f5b22e" class="outline-3">
|
||||
<h3 id="org6f5b22e"><span class="section-number-3">4.2</span> Obtained Plant</h3>
|
||||
<div id="outline-container-org170c73f" class="outline-3">
|
||||
<h3 id="org170c73f"><span class="section-number-3">4.2</span> Obtained Plant</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
|
||||
<div id="orge364e46" class="figure">
|
||||
<div id="org9c2e85a" class="figure">
|
||||
<p><img src="figs/cascade_primary_plant.png" alt="cascade_primary_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Plant for the Primary Controller (<a href="./figs/cascade_primary_plant.png">png</a>, <a href="./figs/cascade_primary_plant.pdf">pdf</a>)</p>
|
||||
@@ -393,23 +397,23 @@ isstable(Gx)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org839cdb3" class="outline-3">
|
||||
<h3 id="org839cdb3"><span class="section-number-3">4.3</span> Controller Design</h3>
|
||||
<div id="outline-container-orge97b630" class="outline-3">
|
||||
<h3 id="orge97b630"><span class="section-number-3">4.3</span> Controller Design</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<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>10; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
<pre class="src src-matlab"> wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; <span class="org-comment">% Bandwidth Bandwidth [rad/s]</span>
|
||||
|
||||
h = 2; <span class="org-comment">% Lead parameter</span>
|
||||
h = 2; <span class="org-comment">% Lead parameter</span>
|
||||
|
||||
Kp = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> (s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5)<span class="org-type">/</span>s <span class="org-type">*</span> 1<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>20);
|
||||
Kp = (1<span class="org-type">/</span>h) <span class="org-type">*</span> (1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">*</span>h)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc<span class="org-type">/</span>h) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> (s <span class="org-type">+</span> 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>5)<span class="org-type">/</span>s <span class="org-type">*</span> 1<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>20);
|
||||
|
||||
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
|
||||
Kp = Kp<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kp, wc))));
|
||||
<span class="org-comment">% Normalization of the gain of have a loop gain of 1 at frequency wc</span>
|
||||
Kp = Kp<span class="org-type">.*</span>diag(1<span class="org-type">./</span>diag(abs(freqresp(Gx<span class="org-type">*</span>Kp, wc))));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb9d3971" class="figure">
|
||||
<div id="org37a2534" class="figure">
|
||||
<p><img src="figs/cascade_primary_loop_gain.png" alt="cascade_primary_loop_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Loop Gain for the primary controller (outer loop) (<a href="./figs/cascade_primary_loop_gain.png">png</a>, <a href="./figs/cascade_primary_loop_gain.pdf">pdf</a>)</p>
|
||||
@@ -418,12 +422,12 @@ Kp = Kp<span class="org-type">.*</span>diag(1<span class="org-type">./</span>dia
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3300911" class="outline-2">
|
||||
<h2 id="org3300911"><span class="section-number-2">5</span> Simulation</h2>
|
||||
<div id="outline-container-org07fbc52" class="outline-2">
|
||||
<h2 id="org07fbc52"><span class="section-number-2">5</span> Simulation</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<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 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>
|
||||
|
||||
@@ -431,58 +435,58 @@ Kp = Kp<span class="org-type">.*</span>diag(1<span class="org-type">./</span>dia
|
||||
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 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 = simout;
|
||||
save(<span class="org-string">'./mat/cascade_hac_lac.mat'</span>, <span class="org-string">'cascade_hac_lac'</span>);
|
||||
<pre class="src src-matlab"> cascade_hac_lac = simout;
|
||||
save(<span class="org-string">'./mat/cascade_hac_lac.mat'</span>, <span class="org-string">'cascade_hac_lac'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2943496" class="outline-2">
|
||||
<h2 id="org2943496"><span class="section-number-2">6</span> Results</h2>
|
||||
<div id="outline-container-org3e9cd70" class="outline-2">
|
||||
<h2 id="org3e9cd70"><span class="section-number-2">6</span> Results</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/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>);
|
||||
<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>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<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 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);
|
||||
<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_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);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org11aa945" class="figure">
|
||||
<div id="orgb068123" class="figure">
|
||||
<p><img src="figs/cascade_hac_lac_tomography_psd.png" alt="cascade_hac_lac_tomography_psd.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>ASD of the position error (<a href="./figs/cascade_hac_lac_tomography_psd.png">png</a>, <a href="./figs/cascade_hac_lac_tomography_psd.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org845b982" class="figure">
|
||||
<div id="orgb10044c" class="figure">
|
||||
<p><img src="figs/cascade_hac_lac_tomography_cas.png" alt="cascade_hac_lac_tomography_cas.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Cumulative Amplitude Spectrum of the position error (<a href="./figs/cascade_hac_lac_tomography_cas.png">png</a>, <a href="./figs/cascade_hac_lac_tomography_cas.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org0151895" class="figure">
|
||||
<div id="orgd639abb" class="figure">
|
||||
<p><img src="figs/cascade_hac_lac_tomography.png" alt="cascade_hac_lac_tomography.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Results of the Tomography Experiment (<a href="./figs/cascade_hac_lac_tomography.png">png</a>, <a href="./figs/cascade_hac_lac_tomography.pdf">pdf</a>)</p>
|
||||
@@ -492,7 +496,7 @@ Ts = t(2)<span class="org-type">-</span>t(1);
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-04-17 ven. 09:35</p>
|
||||
<p class="date">Created: 2021-02-20 sam. 23:08</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user