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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 optimal stiffness</h1>
|
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<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org99c5b6d">1. Low Authority Control - Decentralized Direct Velocity Feedback</a>
|
||||
<li><a href="#orgdb6e4f3">1. Low Authority Control - Decentralized Direct Velocity Feedback</a>
|
||||
<ul>
|
||||
<li><a href="#orgf3f8aed">1.1. Initialization</a></li>
|
||||
<li><a href="#orgc5a1e81">1.2. Identification</a></li>
|
||||
<li><a href="#orgfef1a3f">1.3. Controller Design</a></li>
|
||||
<li><a href="#org3c73014">1.4. Effect of the Low Authority Control on the Primary Plant</a></li>
|
||||
<li><a href="#orgee5dbee">1.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
|
||||
<li><a href="#orgdc2eb5a">1.6. Conclusion</a></li>
|
||||
<li><a href="#org9319182">1.1. Initialization</a></li>
|
||||
<li><a href="#org8b02bfa">1.2. Identification</a></li>
|
||||
<li><a href="#org6f820ae">1.3. Controller Design</a></li>
|
||||
<li><a href="#org5434213">1.4. Effect of the Low Authority Control on the Primary Plant</a></li>
|
||||
<li><a href="#orgc187d77">1.5. Effect of the Low Authority Control on the Sensibility to Disturbances</a></li>
|
||||
<li><a href="#org9539bfe">1.6. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org81dc0a8">2. Primary Control in the leg space</a>
|
||||
<li><a href="#org2797edc">2. Primary Control in the leg space</a>
|
||||
<ul>
|
||||
<li><a href="#org1e7a412">2.1. Plant in the leg space</a></li>
|
||||
<li><a href="#orgf39520c">2.2. Control in the leg space</a></li>
|
||||
<li><a href="#org16d192f">2.3. Sensibility to Disturbances and Noise Budget</a></li>
|
||||
<li><a href="#org8f34c09">2.4. Simulations of Tomography Experiment</a></li>
|
||||
<li><a href="#orgcc19864">2.5. Results</a></li>
|
||||
<li><a href="#orgf709759">2.6. Actuator Stroke and Forces</a></li>
|
||||
<li><a href="#orgcf22d67">2.7. Conclusion</a></li>
|
||||
<li><a href="#org8c682ad">2.1. Plant in the leg space</a></li>
|
||||
<li><a href="#org8308914">2.2. Control in the leg space</a></li>
|
||||
<li><a href="#org5a96746">2.3. Sensibility to Disturbances and Noise Budget</a></li>
|
||||
<li><a href="#org370df82">2.4. Simulations of Tomography Experiment</a></li>
|
||||
<li><a href="#orgd19d198">2.5. Results</a></li>
|
||||
<li><a href="#org4bc7412">2.6. Actuator Stroke and Forces</a></li>
|
||||
<li><a href="#orgead2d0d">2.7. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org56b28cd">3. Further More complex simulations</a>
|
||||
<li><a href="#orgbb6ed0e">3. Further More complex simulations</a>
|
||||
<ul>
|
||||
<li><a href="#org6c1ddb5">3.1. Simulation with Micro-Hexapod Offset</a>
|
||||
<li><a href="#orgbe6d807">3.1. Simulation with Micro-Hexapod Offset</a>
|
||||
<ul>
|
||||
<li><a href="#org78cec1a">3.1.1. Simulation</a></li>
|
||||
<li><a href="#org53a553d">3.1.2. Results</a></li>
|
||||
<li><a href="#orgd36f8c4">3.1.1. Simulation</a></li>
|
||||
<li><a href="#org8fe5f58">3.1.2. Results</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org5cb899b">3.2. Simultaneous Translation scans and Spindle’s rotation</a>
|
||||
<li><a href="#org9753678">3.2. Simultaneous Translation scans and Spindle’s rotation</a>
|
||||
<ul>
|
||||
<li><a href="#orgf715899">3.2.1. Simulation</a></li>
|
||||
<li><a href="#org056af12">3.2.2. Results</a></li>
|
||||
<li><a href="#org2619c3f">3.2.1. Simulation</a></li>
|
||||
<li><a href="#org8a2b562">3.2.2. Results</a></li>
|
||||
</ul>
|
||||
</li>
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||||
</ul>
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||||
</li>
|
||||
<li><a href="#org9bd2bf8">4. Primary Control in the task space</a>
|
||||
<li><a href="#org26c4ef6">4. Primary Control in the task space</a>
|
||||
<ul>
|
||||
<li><a href="#org07b4a9d">4.1. Plant in the task space</a></li>
|
||||
<li><a href="#org7d888f9">4.2. Control in the task space</a>
|
||||
<li><a href="#org48bda71">4.1. Plant in the task space</a></li>
|
||||
<li><a href="#org4251f93">4.2. Control in the task space</a>
|
||||
<ul>
|
||||
<li><a href="#orgb28634b">4.2.1. Stability</a></li>
|
||||
<li><a href="#org226215f">4.2.1. Stability</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org8bd1f9d">4.3. Simulation</a></li>
|
||||
<li><a href="#org3cfdfa3">4.4. Conclusion</a></li>
|
||||
<li><a href="#org0aa9f05">4.3. Simulation</a></li>
|
||||
<li><a href="#org4a860c5">4.4. Conclusion</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
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||||
</div>
|
||||
</div>
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||||
|
||||
<div id="outline-container-org99c5b6d" class="outline-2">
|
||||
<h2 id="org99c5b6d"><span class="section-number-2">1</span> Low Authority Control - Decentralized Direct Velocity Feedback</h2>
|
||||
<div id="outline-container-orgdb6e4f3" class="outline-2">
|
||||
<h2 id="orgdb6e4f3"><span class="section-number-2">1</span> Low Authority Control - Decentralized Direct Velocity Feedback</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="orgfec42cb"></a>
|
||||
<a id="org8d1c1a1"></a>
|
||||
</p>
|
||||
|
||||
<div id="org7f11a74" class="figure">
|
||||
<div id="org26ba32b" class="figure">
|
||||
<p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Low Authority Control: Decentralized Direct Velocity Feedback</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgf3f8aed" class="outline-3">
|
||||
<h3 id="orgf3f8aed"><span class="section-number-3">1.1</span> Initialization</h3>
|
||||
<div id="outline-container-org9319182" class="outline-3">
|
||||
<h3 id="org9319182"><span class="section-number-3">1.1</span> Initialization</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<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();
|
||||
|
||||
initializeSimscapeConfiguration();
|
||||
initializeDisturbances('enable', false);
|
||||
initializeLoggingConfiguration('log', 'none');
|
||||
initializeSimscapeConfiguration();
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
|
||||
|
||||
initializeController('type', 'hac-dvf');
|
||||
initializeController(<span class="org-string">'type'</span>, <span class="org-string">'hac-dvf'</span>);
|
||||
</pre>
|
||||
</div>
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||||
|
||||
@@ -125,18 +130,18 @@ initializeController('type', 'hac-dvf');
|
||||
We set the stiffness of the payload fixation:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kp = 1e8; % [N/m]
|
||||
<pre class="src src-matlab"> Kp = 1e8; <span class="org-comment">% [N/m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc5a1e81" class="outline-3">
|
||||
<h3 id="orgc5a1e81"><span class="section-number-3">1.2</span> Identification</h3>
|
||||
<div id="outline-container-org8b02bfa" class="outline-3">
|
||||
<h3 id="org8b02bfa"><span class="section-number-3">1.2</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">K = tf(zeros(6));
|
||||
Kdvf = tf(zeros(6));
|
||||
<pre class="src src-matlab"> K = tf(zeros(6));
|
||||
Kdvf = tf(zeros(6));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -144,7 +149,7 @@ Kdvf = tf(zeros(6));
|
||||
We identify the system for the following payload masses:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Ms = [1, 10, 50];
|
||||
<pre class="src src-matlab"> Ms = [1, 10, 50];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -152,36 +157,36 @@ We identify the system for the following payload masses:
|
||||
The nano-hexapod has the following leg’s stiffness and damping.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeNanoHexapod('k', 1e5, 'c', 2e2);
|
||||
<pre class="src src-matlab"> initializeNanoHexapod(<span class="org-string">'k'</span>, 1e5, <span class="org-string">'c'</span>, 2e2);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfef1a3f" class="outline-3">
|
||||
<h3 id="orgfef1a3f"><span class="section-number-3">1.3</span> Controller Design</h3>
|
||||
<div id="outline-container-org6f820ae" class="outline-3">
|
||||
<h3 id="org6f820ae"><span class="section-number-3">1.3</span> Controller Design</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
The obtain dynamics from actuators forces \(\tau_i\) to the relative motion of the legs \(d\mathcal{L}_i\) is shown in Figure <a href="#orgdb7af3b">2</a> for the three considered payload masses.
|
||||
The obtain dynamics from actuators forces \(\tau_i\) to the relative motion of the legs \(d\mathcal{L}_i\) is shown in Figure <a href="#org3f64a5c">2</a> for the three considered payload masses.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The Root Locus is shown in Figure <a href="#org5814b4f">3</a> and wee see that we have unconditional stability.
|
||||
The Root Locus is shown in Figure <a href="#org4342572">3</a> and wee see that we have unconditional stability.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In order to choose the gain such that we obtain good damping for all the three payload masses, we plot the damping ration of the modes as a function of the gain for all three payload masses in Figure <a href="#orgb16b2c3">4</a>.
|
||||
In order to choose the gain such that we obtain good damping for all the three payload masses, we plot the damping ration of the modes as a function of the gain for all three payload masses in Figure <a href="#org79a7504">4</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgdb7af3b" class="figure">
|
||||
<div id="org3f64a5c" class="figure">
|
||||
<p><img src="figs/opt_stiff_dvf_plant.png" alt="opt_stiff_dvf_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Dynamics for the Direct Velocity Feedback active damping for three payload masses</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5814b4f" class="figure">
|
||||
<div id="org4342572" class="figure">
|
||||
<p><img src="figs/opt_stiff_dvf_root_locus.png" alt="opt_stiff_dvf_root_locus.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Root Locus for the DVF control for three payload masses</p>
|
||||
@@ -191,7 +196,7 @@ In order to choose the gain such that we obtain good damping for all the three p
|
||||
Damping as function of the gain
|
||||
</p>
|
||||
|
||||
<div id="orgb16b2c3" class="figure">
|
||||
<div id="org79a7504" class="figure">
|
||||
<p><img src="figs/opt_stiff_dvf_damping_gain.png" alt="opt_stiff_dvf_damping_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Damping ratio of the poles as a function of the DVF gain</p>
|
||||
@@ -201,14 +206,14 @@ Damping as function of the gain
|
||||
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
|
||||
<pre class="src src-matlab"> Kdvf = 5e3<span class="org-type">*</span>s<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>1e3)<span class="org-type">*</span>eye(6);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3c73014" class="outline-3">
|
||||
<h3 id="org3c73014"><span class="section-number-3">1.4</span> Effect of the Low Authority Control on the Primary Plant</h3>
|
||||
<div id="outline-container-org5434213" class="outline-3">
|
||||
<h3 id="org5434213"><span class="section-number-3">1.4</span> Effect of the Low Authority Control on the Primary Plant</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
Let’s identify the dynamics from actuator forces \(\bm{\tau}\) to displacement as measured by the metrology \(\bm{\mathcal{X}}\):
|
||||
@@ -220,10 +225,10 @@ We do so both when the DVF is applied and when it is not applied.
|
||||
<p>
|
||||
Then, we compute the transfer function from forces applied by the actuators \(\bm{\mathcal{F}}\) to the measured position error in the frame of the nano-hexapod \(\bm{\epsilon}_{\mathcal{X}_n}\):
|
||||
\[ \bm{G}_\mathcal{X}(s) = \frac{\bm{\epsilon}_{\mathcal{X}_n}}{\bm{\mathcal{F}}} = \bm{G}(s) \bm{J}^{-T} \]
|
||||
The obtained dynamics is shown in Figure <a href="#org45c1265">5</a>.
|
||||
The obtained dynamics is shown in Figure <a href="#org4b0b413">5</a>.
|
||||
</p>
|
||||
|
||||
<div class="important">
|
||||
<div class="important" id="orgd787228">
|
||||
<p>
|
||||
A zero with a positive real part is introduced in the transfer function from \(\mathcal{F}_y\) to \(\mathcal{X}_y\) after Decentralized Direct Velocity Feedback is applied.
|
||||
</p>
|
||||
@@ -233,23 +238,23 @@ A zero with a positive real part is introduced in the transfer function from \(\
|
||||
<p>
|
||||
And we compute the transfer function from actuator forces \(\bm{\tau}\) to position error of each leg \(\bm{\epsilon}_\mathcal{L}\):
|
||||
\[ \bm{G}_\mathcal{L} = \frac{\bm{\epsilon}_\mathcal{L}}{\bm{\tau}} = \bm{J} \bm{G}(s) \]
|
||||
The obtained dynamics is shown in Figure <a href="#org069e296">6</a>.
|
||||
The obtained dynamics is shown in Figure <a href="#org0a1130b">6</a>.
|
||||
</p>
|
||||
|
||||
<div id="org45c1265" class="figure">
|
||||
<div id="org4b0b413" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_X.png" alt="opt_stiff_primary_plant_damped_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Primary plant in the task space with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org069e296" class="figure">
|
||||
<div id="org0a1130b" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_L.png" alt="opt_stiff_primary_plant_damped_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
<p>
|
||||
The coupling (off diagonal elements) of \(\bm{G}_\mathcal{X}\) are shown in Figure <a href="#orgbb4e497">7</a> both when DVF is applied and when it is not.
|
||||
The coupling (off diagonal elements) of \(\bm{G}_\mathcal{X}\) are shown in Figure <a href="#org67cb6cd">7</a> both when DVF is applied and when it is not.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -258,25 +263,25 @@ The coupling does not change a lot with DVF.
|
||||
|
||||
|
||||
<p>
|
||||
The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure <a href="#orgc43d759">8</a>.
|
||||
The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure <a href="#org6331a29">8</a>.
|
||||
</p>
|
||||
|
||||
<div class="important">
|
||||
<div class="important" id="orgc16e5dd">
|
||||
<p>
|
||||
The magnitude of the coupling between \(\tau_i\) and \(d\mathcal{L}_j\) (Figure <a href="#orgc43d759">8</a>) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
|
||||
The magnitude of the coupling between \(\tau_i\) and \(d\mathcal{L}_j\) (Figure <a href="#org6331a29">8</a>) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgbb4e497" class="figure">
|
||||
<div id="org67cb6cd" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_coupling_X.png" alt="opt_stiff_primary_plant_damped_coupling_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Coupling in the primary plant in the task with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc43d759" class="figure">
|
||||
<div id="org6331a29" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_damped_coupling_L.png" alt="opt_stiff_primary_plant_damped_coupling_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Coupling in the primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback</p>
|
||||
@@ -284,8 +289,8 @@ The magnitude of the coupling between \(\tau_i\) and \(d\mathcal{L}_j\) (Figure
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgee5dbee" class="outline-3">
|
||||
<h3 id="orgee5dbee"><span class="section-number-3">1.5</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h3>
|
||||
<div id="outline-container-orgc187d77" class="outline-3">
|
||||
<h3 id="orgc187d77"><span class="section-number-3">1.5</span> Effect of the Low Authority Control on the Sensibility to Disturbances</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<p>
|
||||
We may now see how Decentralized Direct Velocity Feedback changes the sensibility to disturbances, namely:
|
||||
@@ -307,15 +312,15 @@ To simplify the analysis, we here only consider the vertical direction, thus, we
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The norm of these transfer functions are shown in Figure <a href="#org199898b">9</a>.
|
||||
The norm of these transfer functions are shown in Figure <a href="#orgc177556">9</a>.
|
||||
</p>
|
||||
|
||||
<div id="org199898b" class="figure">
|
||||
<div id="orgc177556" class="figure">
|
||||
<p><img src="figs/opt_stiff_sensibility_dist_dvf.png" alt="opt_stiff_sensibility_dist_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied</p>
|
||||
</div>
|
||||
<div class="important">
|
||||
<div class="important" id="orgb9e4f65">
|
||||
<p>
|
||||
Decentralized Direct Velocity Feedback is shown to increase the effect of stages vibrations at high frequency and to reduce the effect of ground motion and direct forces at low frequency.
|
||||
</p>
|
||||
@@ -324,10 +329,10 @@ Decentralized Direct Velocity Feedback is shown to increase the effect of stages
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdc2eb5a" class="outline-3">
|
||||
<h3 id="orgdc2eb5a"><span class="section-number-3">1.6</span> Conclusion</h3>
|
||||
<div id="outline-container-org9539bfe" class="outline-3">
|
||||
<h3 id="org9539bfe"><span class="section-number-3">1.6</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
<div class="important">
|
||||
<div class="important" id="orgf2d2cca">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
@@ -337,14 +342,14 @@ Decentralized Direct Velocity Feedback is shown to increase the effect of stages
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org81dc0a8" class="outline-2">
|
||||
<h2 id="org81dc0a8"><span class="section-number-2">2</span> Primary Control in the leg space</h2>
|
||||
<div id="outline-container-org2797edc" class="outline-2">
|
||||
<h2 id="org2797edc"><span class="section-number-2">2</span> Primary Control in the leg space</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="orgd0beb6a"></a>
|
||||
<a id="orge570650"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section we implement the control architecture shown in Figure <a href="#org7d5c8bc">10</a> consisting of:
|
||||
In this section we implement the control architecture shown in Figure <a href="#orga22fd47">10</a> consisting of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>an inner loop with a decentralized direct velocity feedback control</li>
|
||||
@@ -352,25 +357,25 @@ In this section we implement the control architecture shown in Figure <a href="#
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org7d5c8bc" class="figure">
|
||||
<div id="orga22fd47" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_dvf_pos_L.png" alt="control_architecture_hac_dvf_pos_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg’s space</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The controller for decentralized direct velocity feedback is the one designed in Section <a href="#orgfec42cb">1</a>.
|
||||
The controller for decentralized direct velocity feedback is the one designed in Section <a href="#org8d1c1a1">1</a>.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org1e7a412" class="outline-3">
|
||||
<h3 id="org1e7a412"><span class="section-number-3">2.1</span> Plant in the leg space</h3>
|
||||
<div id="outline-container-org8c682ad" class="outline-3">
|
||||
<h3 id="org8c682ad"><span class="section-number-3">2.1</span> Plant in the leg space</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We now look at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses are shown in Figure <a href="#org23d23ae">11</a>.
|
||||
The diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses are shown in Figure <a href="#org15749d9">11</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -378,7 +383,7 @@ The plant dynamics below \(100\ [Hz]\) is only slightly dependent on the payload
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org23d23ae" class="figure">
|
||||
<div id="org15749d9" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_plant_L.png" alt="opt_stiff_primary_plant_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses</p>
|
||||
@@ -387,8 +392,8 @@ The plant dynamics below \(100\ [Hz]\) is only slightly dependent on the payload
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgf39520c" class="outline-3">
|
||||
<h3 id="orgf39520c"><span class="section-number-3">2.2</span> Control in the leg space</h3>
|
||||
<div id="outline-container-org8308914" class="outline-3">
|
||||
<h3 id="org8308914"><span class="section-number-3">2.2</span> Control in the leg space</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
We design a diagonal controller with all the same diagonal elements.
|
||||
@@ -414,21 +419,21 @@ The design controller is as follows:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The loop gain is shown in Figure <a href="#orgbcc0acb">12</a>.
|
||||
The loop gain is shown in Figure <a href="#org05096f1">12</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">h = 2.0;
|
||||
Kl = 2e7 * eye(6) * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
|
||||
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
|
||||
1/(1 + s/2/pi/300);
|
||||
<pre class="src src-matlab"> h = 2.0;
|
||||
Kl = 2e7 <span class="org-type">*</span> eye(6) <span class="org-type">*</span> ...
|
||||
1<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>100<span class="org-type">/</span>h) <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100<span class="org-type">*</span>h) <span class="org-type">+</span> 1) <span class="org-type">*</span> ...
|
||||
1<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>200<span class="org-type">/</span>h) <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>200<span class="org-type">*</span>h) <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-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-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>300);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgbcc0acb" class="figure">
|
||||
<div id="org05096f1" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_loop_gain_L.png" alt="opt_stiff_primary_loop_gain_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Loop gain for the primary plant</p>
|
||||
@@ -438,25 +443,25 @@ Kl = 2e7 * eye(6) * ...
|
||||
Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load('mat/stages.mat', 'nano_hexapod');
|
||||
K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
|
||||
<pre class="src src-matlab"> load(<span class="org-string">'mat/stages.mat'</span>, <span class="org-string">'nano_hexapod'</span>);
|
||||
K = Kl<span class="org-type">*</span>nano_hexapod.kinematics.J<span class="org-type">*</span>diag([1, 1, 1, 1, 1, 0]);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org16d192f" class="outline-3">
|
||||
<h3 id="org16d192f"><span class="section-number-3">2.3</span> Sensibility to Disturbances and Noise Budget</h3>
|
||||
<div id="outline-container-org5a96746" class="outline-3">
|
||||
<h3 id="org5a96746"><span class="section-number-3">2.3</span> Sensibility to Disturbances and Noise Budget</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
||||
</p>
|
||||
<p>
|
||||
We compare the norm of these transfer function for the vertical direction when no control is applied and when HAC-LAC control is applied: Figure <a href="#org9650e03">13</a>.
|
||||
We compare the norm of these transfer function for the vertical direction when no control is applied and when HAC-LAC control is applied: Figure <a href="#orgc3a7b3a">13</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org9650e03" class="figure">
|
||||
<div id="orgc3a7b3a" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_senbility_dist.png" alt="opt_stiff_primary_control_L_senbility_dist.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Sensibility to disturbances when the HAC-LAC control is applied</p>
|
||||
@@ -465,27 +470,27 @@ We compare the norm of these transfer function for the vertical direction when n
|
||||
Then, we load the Power Spectral Density of the perturbations and we look at the obtained PSD of the displacement error in the vertical direction due to the disturbances:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Figure <a href="#org32928e0">14</a>: Amplitude Spectral Density of the vertical position error due to both the vertical ground motion and the vertical vibrations of the spindle</li>
|
||||
<li>Figure <a href="#org7fda8f7">15</a>: Comparison of the Amplitude Spectral Density of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
||||
<li>Figure <a href="#org073608b">16</a>: Comparison of the Cumulative Amplitude Spectrum of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
||||
<li>Figure <a href="#org7ae258c">14</a>: Amplitude Spectral Density of the vertical position error due to both the vertical ground motion and the vertical vibrations of the spindle</li>
|
||||
<li>Figure <a href="#org72bd654">15</a>: Comparison of the Amplitude Spectral Density of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
||||
<li>Figure <a href="#org08b6134">16</a>: Comparison of the Cumulative Amplitude Spectrum of the vertical position error in Open Loop and with the HAC-DVF Control</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org32928e0" class="figure">
|
||||
<div id="org7ae258c" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_psd_dist.png" alt="opt_stiff_primary_control_L_psd_dist.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Amplitude Spectral Density of the vertical position error of the sample when the HAC-DVF control is applied due to both the ground motion and spindle vibrations</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7fda8f7" class="figure">
|
||||
<div id="org72bd654" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_psd_tot.png" alt="opt_stiff_primary_control_L_psd_tot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Amplitude Spectral Density of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org073608b" class="figure">
|
||||
<div id="org08b6134" class="figure">
|
||||
<p><img src="figs/opt_stiff_primary_control_L_cas_tot.png" alt="opt_stiff_primary_control_L_cas_tot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Cumulative Amplitude Spectrum of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied</p>
|
||||
@@ -493,17 +498,17 @@ Then, we load the Power Spectral Density of the perturbations and we look at the
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8f34c09" class="outline-3">
|
||||
<h3 id="org8f34c09"><span class="section-number-3">2.4</span> Simulations of Tomography Experiment</h3>
|
||||
<div id="outline-container-org370df82" class="outline-3">
|
||||
<h3 id="org370df82"><span class="section-number-3">2.4</span> Simulations of Tomography Experiment</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
Let’s now simulate a tomography experiment.
|
||||
To do so, we include all disturbances except vibrations of the translation stage.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances();
|
||||
initializeSimscapeConfiguration('gravity', false);
|
||||
initializeLoggingConfiguration('log', 'all');
|
||||
<pre class="src src-matlab"> initializeDisturbances();
|
||||
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -512,40 +517,40 @@ And we run the simulation for all three payload Masses.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgcc19864" class="outline-3">
|
||||
<h3 id="orgcc19864"><span class="section-number-3">2.5</span> Results</h3>
|
||||
<div id="outline-container-orgd19d198" class="outline-3">
|
||||
<h3 id="orgd19d198"><span class="section-number-3">2.5</span> Results</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<p>
|
||||
Let’s now see how this controller performs.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
First, we compute the Power Spectral Density of the sample’s position error and we compare it with the open loop case in Figure <a href="#org6cab6ef">17</a>.
|
||||
First, we compute the Power Spectral Density of the sample’s position error and we compare it with the open loop case in Figure <a href="#orgeb9afe3">17</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Similarly, the Cumulative Amplitude Spectrum is shown in Figure <a href="#org33e9f1a">18</a>.
|
||||
Similarly, the Cumulative Amplitude Spectrum is shown in Figure <a href="#org6b85652">18</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Finally, the time domain position error signals are shown in Figure <a href="#orgf0f1950">19</a>.
|
||||
Finally, the time domain position error signals are shown in Figure <a href="#orge2cc533">19</a>.
|
||||
</p>
|
||||
|
||||
<div id="org6cab6ef" class="figure">
|
||||
<div id="orgeb9afe3" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_psd_disp_error.png" alt="opt_stiff_hac_dvf_L_psd_disp_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org33e9f1a" class="figure">
|
||||
<div id="org6b85652" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_cas_disp_error.png" alt="opt_stiff_hac_dvf_L_cas_disp_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf0f1950" class="figure">
|
||||
<div id="orge2cc533" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_pos_error.png" alt="opt_stiff_hac_dvf_L_pos_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture</p>
|
||||
@@ -553,18 +558,18 @@ Finally, the time domain position error signals are shown in Figure <a href="#or
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf709759" class="outline-3">
|
||||
<h3 id="orgf709759"><span class="section-number-3">2.6</span> Actuator Stroke and Forces</h3>
|
||||
<div id="outline-container-org4bc7412" class="outline-3">
|
||||
<h3 id="org4bc7412"><span class="section-number-3">2.6</span> Actuator Stroke and Forces</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
|
||||
<div id="orgf9d6367" class="figure">
|
||||
<div id="orgd89d4d1" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_act_force.png" alt="opt_stiff_hac_dvf_L_act_force.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Force applied by the actuator during the simulation</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org11b8730" class="figure">
|
||||
<div id="org78753f1" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_L_act_stroke.png" alt="opt_stiff_hac_dvf_L_act_stroke.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 21: </span>Leg’s stroke during the simulation</p>
|
||||
@@ -572,10 +577,10 @@ Finally, the time domain position error signals are shown in Figure <a href="#or
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcf22d67" class="outline-3">
|
||||
<h3 id="orgcf22d67"><span class="section-number-3">2.7</span> Conclusion</h3>
|
||||
<div id="outline-container-orgead2d0d" class="outline-3">
|
||||
<h3 id="orgead2d0d"><span class="section-number-3">2.7</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-7">
|
||||
<div class="important">
|
||||
<div class="important" id="orgd04ff34">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
@@ -585,16 +590,16 @@ Finally, the time domain position error signals are shown in Figure <a href="#or
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org56b28cd" class="outline-2">
|
||||
<h2 id="org56b28cd"><span class="section-number-2">3</span> Further More complex simulations</h2>
|
||||
<div id="outline-container-orgbb6ed0e" class="outline-2">
|
||||
<h2 id="orgbb6ed0e"><span class="section-number-2">3</span> Further More complex simulations</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
</div>
|
||||
<div id="outline-container-org6c1ddb5" class="outline-3">
|
||||
<h3 id="org6c1ddb5"><span class="section-number-3">3.1</span> Simulation with Micro-Hexapod Offset</h3>
|
||||
<div id="outline-container-orgbe6d807" class="outline-3">
|
||||
<h3 id="orgbe6d807"><span class="section-number-3">3.1</span> Simulation with Micro-Hexapod Offset</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
</div>
|
||||
<div id="outline-container-org78cec1a" class="outline-4">
|
||||
<h4 id="org78cec1a"><span class="section-number-4">3.1.1</span> Simulation</h4>
|
||||
<div id="outline-container-orgd36f8c4" class="outline-4">
|
||||
<h4 id="orgd36f8c4"><span class="section-number-4">3.1.1</span> Simulation</h4>
|
||||
<div class="outline-text-4" id="text-3-1-1">
|
||||
<p>
|
||||
The micro-hexapod is inducing a 10mm offset of the sample center of mass with the rotation axis.
|
||||
@@ -602,43 +607,43 @@ A tomography experiment is then simulated.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances();
|
||||
initializeSimscapeConfiguration('gravity', false);
|
||||
initializeLoggingConfiguration('log', 'all');
|
||||
<pre class="src src-matlab"> initializeDisturbances();
|
||||
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
|
||||
initializeSample('mass', 1, 'freq', 200);
|
||||
initializeMicroHexapod('AP', [10e-3 0 0]);
|
||||
initializeReferences('Rz_type', 'rotating', 'Rz_period', 1, ...
|
||||
'Dh_pos', [10e-3; 0; 0; 0; 0; 0]);
|
||||
initializeSample(<span class="org-string">'mass'</span>, 1, <span class="org-string">'freq'</span>, 200);
|
||||
initializeMicroHexapod(<span class="org-string">'AP'</span>, [10e<span class="org-type">-</span>3 0 0]);
|
||||
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1, ...
|
||||
<span class="org-string">'Dh_pos'</span>, [10e<span class="org-type">-</span>3; 0; 0; 0; 0; 0]);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load('mat/conf_simulink.mat');
|
||||
set_param(conf_simulink, 'StopTime', '3');
|
||||
sim('nass_model');
|
||||
<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">'3'</span>);
|
||||
<span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org53a553d" class="outline-4">
|
||||
<h4 id="org53a553d"><span class="section-number-4">3.1.2</span> Results</h4>
|
||||
<div id="outline-container-org8fe5f58" class="outline-4">
|
||||
<h4 id="org8fe5f58"><span class="section-number-4">3.1.2</span> Results</h4>
|
||||
<div class="outline-text-4" id="text-3-1-2">
|
||||
|
||||
<div id="org6be7e46" class="figure">
|
||||
<div id="org776d205" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_Dh_offset_disp_error.png" alt="opt_stiff_hac_dvf_Dh_offset_disp_error.png" />
|
||||
</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org07fa12d" class="figure">
|
||||
<div id="org26c6131" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_Dh_offset_F.png" alt="opt_stiff_hac_dvf_Dh_offset_F.png" />
|
||||
</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga4d03c5" class="figure">
|
||||
<div id="org176d06f" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_Dh_offset_dL.png" alt="opt_stiff_hac_dvf_Dh_offset_dL.png" />
|
||||
</p>
|
||||
</div>
|
||||
@@ -646,12 +651,12 @@ sim('nass_model');
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5cb899b" class="outline-3">
|
||||
<h3 id="org5cb899b"><span class="section-number-3">3.2</span> Simultaneous Translation scans and Spindle’s rotation</h3>
|
||||
<div id="outline-container-org9753678" class="outline-3">
|
||||
<h3 id="org9753678"><span class="section-number-3">3.2</span> Simultaneous Translation scans and Spindle’s rotation</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
</div>
|
||||
<div id="outline-container-orgf715899" class="outline-4">
|
||||
<h4 id="orgf715899"><span class="section-number-4">3.2.1</span> Simulation</h4>
|
||||
<div id="outline-container-org2619c3f" class="outline-4">
|
||||
<h4 id="org2619c3f"><span class="section-number-4">3.2.1</span> Simulation</h4>
|
||||
<div class="outline-text-4" id="text-3-2-1">
|
||||
<p>
|
||||
A simulation is now performed with translation scans and spindle rotation at the same time.
|
||||
@@ -662,35 +667,35 @@ The sample has a mass of 1kg, the spindle rotation speed is 60rpm and the transl
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeDisturbances();
|
||||
initializeSimscapeConfiguration('gravity', false);
|
||||
initializeLoggingConfiguration('log', 'all');
|
||||
<pre class="src src-matlab"> initializeDisturbances();
|
||||
initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
|
||||
initializeSample('mass', 1, 'freq', 200);
|
||||
initializeReferences('Rz_type', 'rotating', 'Rz_period', 1, ...
|
||||
'Dy_type', 'triangular', 'Dy_amplitude', 5e-3, 'Dy_period', 4);
|
||||
initializeSample(<span class="org-string">'mass'</span>, 1, <span class="org-string">'freq'</span>, 200);
|
||||
initializeReferences(<span class="org-string">'Rz_type'</span>, <span class="org-string">'rotating'</span>, <span class="org-string">'Rz_period'</span>, 1, ...
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'triangular'</span>, <span class="org-string">'Dy_amplitude'</span>, 5e<span class="org-type">-</span>3, <span class="org-string">'Dy_period'</span>, 4);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org056af12" class="outline-4">
|
||||
<h4 id="org056af12"><span class="section-number-4">3.2.2</span> Results</h4>
|
||||
<div id="outline-container-org8a2b562" class="outline-4">
|
||||
<h4 id="org8a2b562"><span class="section-number-4">3.2.2</span> Results</h4>
|
||||
<div class="outline-text-4" id="text-3-2-2">
|
||||
|
||||
<div id="orgbfa1d02" class="figure">
|
||||
<div id="orgc2e9b19" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_Dy_scans_disp_error.png" alt="opt_stiff_hac_dvf_Dy_scans_disp_error.png" />
|
||||
</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org760b96c" class="figure">
|
||||
<div id="org35d233c" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_Dy_scans_F.png" alt="opt_stiff_hac_dvf_Dy_scans_F.png" />
|
||||
</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgae36e3d" class="figure">
|
||||
<div id="org46908f8" class="figure">
|
||||
<p><img src="figs/opt_stiff_hac_dvf_Dy_scans_dL.png" alt="opt_stiff_hac_dvf_Dy_scans_dL.png" />
|
||||
</p>
|
||||
</div>
|
||||
@@ -699,14 +704,14 @@ initializeReferences('Rz_type', 'rotating', 'Rz_period', 1, ...
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9bd2bf8" class="outline-2">
|
||||
<h2 id="org9bd2bf8"><span class="section-number-2">4</span> Primary Control in the task space</h2>
|
||||
<div id="outline-container-org26c4ef6" class="outline-2">
|
||||
<h2 id="org26c4ef6"><span class="section-number-2">4</span> Primary Control in the task space</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="orge9c2f9a"></a>
|
||||
<a id="org06af957"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, the control architecture shown in Figure <a href="#org7e70ccc">28</a> is applied and consists of:
|
||||
In this section, the control architecture shown in Figure <a href="#org9be312d">28</a> is applied and consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>an inner Low Authority Control loop consisting of a decentralized direct velocity control controller</li>
|
||||
@@ -714,14 +719,14 @@ In this section, the control architecture shown in Figure <a href="#org7e70ccc">
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org7e70ccc" class="figure">
|
||||
<div id="org9be312d" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_dvf_pos_X.png" alt="control_architecture_hac_dvf_pos_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 28: </span>HAC-LAC architecture</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org07b4a9d" class="outline-3">
|
||||
<h3 id="org07b4a9d"><span class="section-number-3">4.1</span> Plant in the task space</h3>
|
||||
<div id="outline-container-org48bda71" class="outline-3">
|
||||
<h3 id="org48bda71"><span class="section-number-3">4.1</span> Plant in the task space</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
Let’s look \(\bm{G}_\mathcal{X}(s)\).
|
||||
@@ -729,62 +734,62 @@ Let’s look \(\bm{G}_\mathcal{X}(s)\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7d888f9" class="outline-3">
|
||||
<h3 id="org7d888f9"><span class="section-number-3">4.2</span> Control in the task space</h3>
|
||||
<div id="outline-container-org4251f93" class="outline-3">
|
||||
<h3 id="org4251f93"><span class="section-number-3">4.2</span> Control in the task space</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Kx = tf(zeros(6));
|
||||
<pre class="src src-matlab"> Kx = tf(zeros(6));
|
||||
|
||||
h = 2.5;
|
||||
Kx(1,1) = 3e7 * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1);
|
||||
h = 2.5;
|
||||
Kx<span class="org-type">(1,1) </span>= 3e7 <span class="org-type">*</span> ...
|
||||
1<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>100<span class="org-type">/</span>h) <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100<span class="org-type">*</span>h) <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>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>1);
|
||||
|
||||
Kx(2,2) = Kx(1,1);
|
||||
Kx<span class="org-type">(2,2) </span>= Kx(1,1);
|
||||
|
||||
h = 2.5;
|
||||
Kx(3,3) = 3e7 * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1);
|
||||
h = 2.5;
|
||||
Kx<span class="org-type">(3,3) </span>= 3e7 <span class="org-type">*</span> ...
|
||||
1<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>100<span class="org-type">/</span>h) <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100<span class="org-type">*</span>h) <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>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>1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">h = 1.5;
|
||||
Kx(4,4) = 5e5 * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1);
|
||||
<pre class="src src-matlab"> h = 1.5;
|
||||
Kx<span class="org-type">(4,4) </span>= 5e5 <span class="org-type">*</span> ...
|
||||
1<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>100<span class="org-type">/</span>h) <span class="org-type">+</span> 1)<span class="org-type">/</span>(s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100<span class="org-type">*</span>h) <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>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>1);
|
||||
|
||||
Kx(5,5) = Kx(4,4);
|
||||
Kx<span class="org-type">(5,5) </span>= Kx(4,4);
|
||||
|
||||
h = 1.5;
|
||||
Kx(6,6) = 5e4 * ...
|
||||
1/h*(s/(2*pi*30/h) + 1)/(s/(2*pi*30*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1);
|
||||
h = 1.5;
|
||||
Kx<span class="org-type">(6,6) </span>= 5e4 <span class="org-type">*</span> ...
|
||||
1<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>30<span class="org-type">/</span>h) <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>30<span class="org-type">*</span>h) <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>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>1);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb28634b" class="outline-4">
|
||||
<h4 id="orgb28634b"><span class="section-number-4">4.2.1</span> Stability</h4>
|
||||
<div id="outline-container-org226215f" class="outline-4">
|
||||
<h4 id="org226215f"><span class="section-number-4">4.2.1</span> Stability</h4>
|
||||
<div class="outline-text-4" id="text-4-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">for i = 1:length(Ms)
|
||||
isstable(feedback(Gm_x{i}*Kx, eye(6), -1))
|
||||
end
|
||||
<pre class="src src-matlab"> <span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ms)</span>
|
||||
isstable(feedback(Gm_x{<span class="org-constant">i</span>}<span class="org-type">*</span>Kx, eye(6), <span class="org-type">-</span>1))
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8bd1f9d" class="outline-3">
|
||||
<h3 id="org8bd1f9d"><span class="section-number-3">4.3</span> Simulation</h3>
|
||||
<div id="outline-container-org0aa9f05" class="outline-3">
|
||||
<h3 id="org0aa9f05"><span class="section-number-3">4.3</span> Simulation</h3>
|
||||
</div>
|
||||
<div id="outline-container-org3cfdfa3" class="outline-3">
|
||||
<h3 id="org3cfdfa3"><span class="section-number-3">4.4</span> Conclusion</h3>
|
||||
<div id="outline-container-org4a860c5" class="outline-3">
|
||||
<h3 id="org4a860c5"><span class="section-number-3">4.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="important">
|
||||
<div class="important" id="org4bf4213">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
@@ -796,7 +801,7 @@ end
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-05-20 mer. 16:41</p>
|
||||
<p class="date">Created: 2021-02-20 sam. 23:09</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user