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<table>
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<tr valign="top">
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<td align="right" class="bibtexnumber">
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[<a name="smith99_scien_engin_guide_digit_signal">1</a>]
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</td>
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<td class="bibtexitem">
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Steven W. Smith.
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<em>The Scientist and Engineer's Guide to Digital Signal Processing
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- Second Edition</em>.
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California Technical Publishing, 1999.
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</td>
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</tr>
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</table>
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@ -3,7 +3,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2019-01-24 jeu. 14:05 -->
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<!-- 2019-01-24 jeu. 15:17 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Control in a rotating frame</title>
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@ -275,86 +275,86 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgf151bb1">1. Introduction</a></li>
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<li><a href="#org268da4c">2. System Description and Analysis</a>
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<li><a href="#org35986a6">1. Introduction</a></li>
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<li><a href="#org2cfc65e">2. System Description and Analysis</a>
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<ul>
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<li><a href="#orge7664a5">2.1. System description</a></li>
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<li><a href="#org60bfb2d">2.2. Equations</a></li>
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<li><a href="#orgf397277">2.3. Numerical Values for the NASS</a></li>
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<li><a href="#org7aa02cb">2.4. Euler and Coriolis forces - Numerical Result</a></li>
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<li><a href="#org9672480">2.5. Negative Spring Effect - Numerical Result</a></li>
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<li><a href="#org57a38eb">2.6. Limitations due to coupling</a>
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<li><a href="#org52d1b39">2.1. System description</a></li>
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<li><a href="#org56f1c8e">2.2. Equations</a></li>
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<li><a href="#org23e861a">2.3. Numerical Values for the NASS</a></li>
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<li><a href="#org8834a4b">2.4. Euler and Coriolis forces - Numerical Result</a></li>
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<li><a href="#org3fc75f8">2.5. Negative Spring Effect - Numerical Result</a></li>
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<li><a href="#orgca44f56">2.6. Limitations due to coupling</a>
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<ul>
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<li><a href="#org6274181">2.6.1. Numerical Analysis</a></li>
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<li><a href="#org972ba28">2.6.1. Numerical Analysis</a></li>
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</ul>
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</li>
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<li><a href="#org4030106">2.7. Limitations due to negative stiffness effect</a></li>
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<li><a href="#org7049dc3">2.8. Effect of rotation speed on the plant</a>
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<li><a href="#org24a2547">2.7. Limitations due to negative stiffness effect</a></li>
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<li><a href="#org90bd4c5">2.8. Effect of rotation speed on the plant</a>
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<ul>
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<li><a href="#org755ed06">2.8.1. Voice coil actuator</a></li>
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<li><a href="#org53fec97">2.8.2. Piezoelectric actuator</a></li>
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<li><a href="#orgf6be1a4">2.8.3. Analysis</a></li>
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<li><a href="#org2262aaa">2.8.4. Campbell diagram</a></li>
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<li><a href="#orgb2a8b4a">2.8.1. Voice coil actuator</a></li>
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<li><a href="#org34e6778">2.8.2. Piezoelectric actuator</a></li>
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<li><a href="#org36cd742">2.8.3. Analysis</a></li>
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<li><a href="#org23ea4ed">2.8.4. Campbell diagram</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orga07d0dd">3. Control Strategies</a>
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<li><a href="#org89b80ab">3. Control Strategies</a>
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<ul>
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<li><a href="#orgeb25ab0">3.1. Measurement in the fixed reference frame</a></li>
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<li><a href="#org9456905">3.2. Measurement in the rotating frame</a></li>
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<li><a href="#orgbdd9948">3.1. Measurement in the fixed reference frame</a></li>
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<li><a href="#org724b218">3.2. Measurement in the rotating frame</a></li>
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</ul>
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</li>
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<li><a href="#orgab7ac9c">4. Multi Body Model - Simscape</a>
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<li><a href="#org30fbee8">4. Multi Body Model - Simscape</a>
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<ul>
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<li><a href="#org123b2ae">4.1. Initialize</a></li>
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<li><a href="#orgc1906bb">4.2. Parameter for the Simscape simulations</a></li>
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<li><a href="#org255159f">4.3. Identification in the rotating referenced frame</a></li>
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<li><a href="#org5f1926d">4.4. Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</a></li>
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<li><a href="#orge5f2b9f">4.5. Plant Control</a>
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<li><a href="#orge1f000c">4.1. Initialize</a></li>
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<li><a href="#org8b4df15">4.2. Parameter for the Simscape simulations</a></li>
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<li><a href="#orga3ac610">4.3. Identification in the rotating referenced frame</a></li>
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<li><a href="#orga381ded">4.4. Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</a></li>
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<li><a href="#org6b388ff">4.5. Plant Control</a>
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<ul>
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<li><a href="#orgb9cef97">4.5.1. Low rotation speed and High rotation speed</a></li>
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<li><a href="#orgdb709bf">4.5.1. Low rotation speed and High rotation speed</a></li>
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</ul>
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</li>
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<li><a href="#org09ff6ca">4.6. Identification in the fixed frame</a></li>
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<li><a href="#org588dae5">4.7. Identification from actuator forces to displacement in the fixed frame</a></li>
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<li><a href="#org53255e3">4.8. Effect of the rotating Speed</a>
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<li><a href="#org5822ce2">4.6. Identification in the fixed frame</a></li>
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<li><a href="#orgfa9ed99">4.7. Identification from actuator forces to displacement in the fixed frame</a></li>
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<li><a href="#orgbc833bb">4.8. Effect of the rotating Speed</a>
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<ul>
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<li><a href="#org14c5fe5">4.8.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
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<li><a href="#org5347efa">4.8.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
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<li><a href="#orgaf21bf8">4.8.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
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<li><a href="#orgdd964cc">4.8.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
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</ul>
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</li>
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<li><a href="#orgd2cb6ed">4.9. Effect of the X-Y stage stiffness</a>
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<li><a href="#orgc30bae9">4.9. Effect of the X-Y stage stiffness</a>
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<ul>
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<li><a href="#org177c370">4.9.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
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<li><a href="#org3a4478a">4.9.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org4965ab2">5. Control Implementation</a>
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<li><a href="#org12e1d75">5. Control Implementation</a>
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<ul>
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<li><a href="#org9f42bc5">5.1. Measurement in the fixed reference frame</a></li>
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<li><a href="#org70652b4">5.1. Measurement in the fixed reference frame</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgf151bb1" class="outline-2">
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<h2 id="orgf151bb1"><span class="section-number-2">1</span> Introduction</h2>
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<div id="outline-container-org35986a6" class="outline-2">
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<h2 id="org35986a6"><span class="section-number-2">1</span> Introduction</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The objective of this note it to highlight some control problems that arises when controlling the position of an object using actuators that are rotating with respect to a fixed reference frame.
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</p>
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<p>
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In section <a href="#org3669843">2</a>, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
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In section <a href="#org0986a46">2</a>, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
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The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system.
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This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.
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</p>
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<p>
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Then, in section <a href="#org3747048">3</a>, two different control approach are compared where:
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Then, in section <a href="#org786bfb0">3</a>, two different control approach are compared where:
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</p>
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<ul class="org-ul">
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<li>the measurement is made in the fixed frame</li>
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@ -362,27 +362,31 @@ Then, in section <a href="#org3747048">3</a>, two different control approach are
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</ul>
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<p>
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In section <a href="#org149db50">4</a>, the analytical study will be validated using a multi body model of the studied system.
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In section <a href="#orgfce2ea4">4</a>, the analytical study will be validated using a multi body model of the studied system.
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</p>
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<p>
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Finally, in section <a href="#org9e7daf4">5</a>, the control strategies are implemented using Simulink and Simscape and compared.
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Finally, in section <a href="#org4a3b8a3">5</a>, the control strategies are implemented using Simulink and Simscape and compared.
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</p>
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<p>
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Test citation: [<a href="#smith99_scien_engin_guide_digit_signal">1</a>].
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</p>
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</div>
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</div>
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<div id="outline-container-org268da4c" class="outline-2">
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<h2 id="org268da4c"><span class="section-number-2">2</span> System Description and Analysis</h2>
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<div id="outline-container-org2cfc65e" class="outline-2">
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<h2 id="org2cfc65e"><span class="section-number-2">2</span> System Description and Analysis</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org3669843"></a>
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<a id="org0986a46"></a>
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</p>
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</div>
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<div id="outline-container-orge7664a5" class="outline-3">
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<h3 id="orge7664a5"><span class="section-number-3">2.1</span> System description</h3>
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<div id="outline-container-org52d1b39" class="outline-3">
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<h3 id="org52d1b39"><span class="section-number-3">2.1</span> System description</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org5ddd11b">1</a>).
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The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org455bae8">1</a>).
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</p>
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<p>
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@ -395,7 +399,7 @@ The measurement is either the \(x-y\) displacement of the object located on top
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</p>
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<div id="org5ddd11b" class="figure">
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<div id="org455bae8" class="figure">
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<p><img src="./Figures/rotating_frame_2dof.png" alt="rotating_frame_2dof.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of the mecanical system</p>
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@ -429,19 +433,19 @@ Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (
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</div>
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</div>
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<div id="outline-container-org60bfb2d" class="outline-3">
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<h3 id="org60bfb2d"><span class="section-number-3">2.2</span> Equations</h3>
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<div id="outline-container-org56f1c8e" class="outline-3">
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<h3 id="org56f1c8e"><span class="section-number-3">2.2</span> Equations</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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<a id="orgc4fe841"></a>
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Based on the figure <a href="#org5ddd11b">1</a>, we can write the equations of motion of the system.
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<a id="org8074d39"></a>
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Based on the figure <a href="#org455bae8">1</a>, we can write the equations of motion of the system.
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</p>
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<p>
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Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
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</p>
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\begin{align}
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\label{org7c77780}
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\label{org9b4a615}
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T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
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V & = \frac{1}{2} k \left( x^2 + y^2 \right)
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\end{align}
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@ -450,7 +454,7 @@ V & = \frac{1}{2} k \left( x^2 + y^2 \right)
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The Lagrangian is the kinetic energy minus the potential energy.
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</p>
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\begin{equation}
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\label{orgc4495ac}
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\label{org81b342f}
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L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
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\end{equation}
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@ -459,7 +463,7 @@ L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \le
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The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:
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</p>
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\begin{align*}
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\label{org5e103d6}
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\label{orgf5d2cb1}
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\frac{\partial L}{\partial x} & = -kx \\
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\frac{\partial L}{\partial y} & = -ky \\
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\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
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@ -529,11 +533,11 @@ We can then subtract and add the previous equations to obtain the following equa
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</p>
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<div class="important">
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\begin{equation}
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\label{orgb342505}
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\label{orgb43453a}
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m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta}
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\end{equation}
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\begin{equation}
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\label{org97a2349}
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\label{org01f818e}
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m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta}
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\end{equation}
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@ -559,8 +563,8 @@ The resulting effect of those forces should then be higher when using softer act
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</div>
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</div>
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<div id="outline-container-orgf397277" class="outline-3">
|
||||
<h3 id="orgf397277"><span class="section-number-3">2.3</span> Numerical Values for the NASS</h3>
|
||||
<div id="outline-container-org23e861a" class="outline-3">
|
||||
<h3 id="org23e861a"><span class="section-number-3">2.3</span> Numerical Values for the NASS</h3>
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||||
<div class="outline-text-3" id="text-2-3">
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||||
<p>
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||||
Let's define the parameters for the NASS.
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@ -623,8 +627,8 @@ Let's define the parameters for the NASS.
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||||
</div>
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||||
</div>
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||||
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<div id="outline-container-org7aa02cb" class="outline-3">
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||||
<h3 id="org7aa02cb"><span class="section-number-3">2.4</span> Euler and Coriolis forces - Numerical Result</h3>
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||||
<div id="outline-container-org8834a4b" class="outline-3">
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||||
<h3 id="org8834a4b"><span class="section-number-3">2.4</span> Euler and Coriolis forces - Numerical Result</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
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||||
<p>
|
||||
First we will determine the value for Euler and Coriolis forces during regular experiment.
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||||
@ -635,10 +639,10 @@ First we will determine the value for Euler and Coriolis forces during regular e
|
||||
</ul>
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||||
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||||
<p>
|
||||
The obtained values are displayed in table <a href="#org3e40f1c">1</a>.
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||||
The obtained values are displayed in table <a href="#orgdbd5160">1</a>.
|
||||
</p>
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||||
|
||||
<table id="org3e40f1c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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||||
<table id="orgdbd5160" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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||||
<caption class="t-above"><span class="table-number">Table 1:</span> Euler and Coriolis forces for the NASS</caption>
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||||
<colgroup>
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||||
@ -672,22 +676,22 @@ The obtained values are displayed in table <a href="#org3e40f1c">1</a>.
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||||
</div>
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||||
</div>
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||||
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||||
<div id="outline-container-org9672480" class="outline-3">
|
||||
<h3 id="org9672480"><span class="section-number-3">2.5</span> Negative Spring Effect - Numerical Result</h3>
|
||||
<div id="outline-container-org3fc75f8" class="outline-3">
|
||||
<h3 id="org3fc75f8"><span class="section-number-3">2.5</span> Negative Spring Effect - Numerical Result</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<p>
|
||||
The negative stiffness due to the rotation is equal to \(-m{\omega_0}^2\).
|
||||
</p>
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||||
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||||
<p>
|
||||
The values for the negative spring effect are displayed in table <a href="#org7b1aaf6">2</a>.
|
||||
The values for the negative spring effect are displayed in table <a href="#org7c845ef">2</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
||||
</p>
|
||||
|
||||
<table id="org7b1aaf6" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="org7c845ef" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 2:</span> Negative Spring effect</caption>
|
||||
|
||||
<colgroup>
|
||||
@ -715,15 +719,15 @@ This is definitely negligible when using piezoelectric actuators. It may not be
|
||||
</div>
|
||||
</div>
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||||
|
||||
<div id="outline-container-org57a38eb" class="outline-3">
|
||||
<h3 id="org57a38eb"><span class="section-number-3">2.6</span> Limitations due to coupling</h3>
|
||||
<div id="outline-container-orgca44f56" class="outline-3">
|
||||
<h3 id="orgca44f56"><span class="section-number-3">2.6</span> Limitations due to coupling</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<p>
|
||||
To simplify, we consider a constant rotating speed \(\dot{\theta} = {\omega_0}\) and thus \(\ddot{\theta} = 0\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From equations \eqref{orgb342505} and \eqref{org97a2349}, we obtain:
|
||||
From equations \eqref{orgb43453a} and \eqref{org01f818e}, we obtain:
|
||||
</p>
|
||||
\begin{align*}
|
||||
(m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\
|
||||
@ -761,7 +765,7 @@ The two previous equations can be written in a matrix form:
|
||||
</p>
|
||||
<div class="important">
|
||||
\begin{equation}
|
||||
\label{orga4820eb}
|
||||
\label{org2b23e3b}
|
||||
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
|
||||
\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
|
||||
\begin{bmatrix}
|
||||
@ -778,26 +782,26 @@ Then, coupling is negligible if \(|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6274181" class="outline-4">
|
||||
<h4 id="org6274181"><span class="section-number-4">2.6.1</span> Numerical Analysis</h4>
|
||||
<div id="outline-container-org972ba28" class="outline-4">
|
||||
<h4 id="org972ba28"><span class="section-number-4">2.6.1</span> Numerical Analysis</h4>
|
||||
<div class="outline-text-4" id="text-2-6-1">
|
||||
<p>
|
||||
We plot on the same graph \(\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \omega_0 \omega|}\) for the voice coil and the piezo:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>with the light sample (figure <a href="#orgeb8c982">2</a>).</li>
|
||||
<li>with the heavy sample (figure <a href="#orga3125c6">3</a>).</li>
|
||||
<li>with the light sample (figure <a href="#org2eaf004">2</a>).</li>
|
||||
<li>with the heavy sample (figure <a href="#orge6601b9">3</a>).</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgeb8c982" class="figure">
|
||||
<div id="org2eaf004" class="figure">
|
||||
<p><img src="Figures/coupling_light.png" alt="coupling_light.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Relative Coupling for light mass and high rotation speed</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga3125c6" class="figure">
|
||||
<div id="orge6601b9" class="figure">
|
||||
<p><img src="Figures/coupling_heavy.png" alt="coupling_heavy.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Relative Coupling for heavy mass and low rotation speed</p>
|
||||
@ -813,17 +817,17 @@ Coupling is higher for actuators with small stiffness.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4030106" class="outline-3">
|
||||
<h3 id="org4030106"><span class="section-number-3">2.7</span> Limitations due to negative stiffness effect</h3>
|
||||
<div id="outline-container-org24a2547" class="outline-3">
|
||||
<h3 id="org24a2547"><span class="section-number-3">2.7</span> Limitations due to negative stiffness effect</h3>
|
||||
<div class="outline-text-3" id="text-2-7">
|
||||
<p>
|
||||
If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table <a href="#org7eddfba">3</a>.
|
||||
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table <a href="#orge84ae0f">3</a>.
|
||||
</p>
|
||||
<table id="org7eddfba" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="orge84ae0f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 3:</span> Maximum rotation speed at which negative stiffness is negligible (\(0.1\sqrt{\frac{k}{m}}\))</caption>
|
||||
|
||||
<colgroup>
|
||||
@ -872,10 +876,10 @@ The system can even goes unstable when \(m \omega^2 > k\), that is when the cent
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table <a href="#org63d2716">4</a>)
|
||||
From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table <a href="#org94d23e2">4</a>)
|
||||
</p>
|
||||
|
||||
<table id="org63d2716" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="org94d23e2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 4:</span> Minimum possible stiffness</caption>
|
||||
|
||||
<colgroup>
|
||||
@ -903,11 +907,11 @@ From this analysis, we can determine the lowest practical stiffness that is poss
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7049dc3" class="outline-3">
|
||||
<h3 id="org7049dc3"><span class="section-number-3">2.8</span> Effect of rotation speed on the plant</h3>
|
||||
<div id="outline-container-org90bd4c5" class="outline-3">
|
||||
<h3 id="org90bd4c5"><span class="section-number-3">2.8</span> Effect of rotation speed on the plant</h3>
|
||||
<div class="outline-text-3" id="text-2-8">
|
||||
<p>
|
||||
As shown in equation \eqref{orga4820eb}, the plant changes with the rotation speed \(\omega_0\).
|
||||
As shown in equation \eqref{org2b23e3b}, the plant changes with the rotation speed \(\omega_0\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -919,18 +923,18 @@ Then we compare the result between voice coil and piezoelectric actuators.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org755ed06" class="outline-4">
|
||||
<h4 id="org755ed06"><span class="section-number-4">2.8.1</span> Voice coil actuator</h4>
|
||||
<div id="outline-container-orgb2a8b4a" class="outline-4">
|
||||
<h4 id="orgb2a8b4a"><span class="section-number-4">2.8.1</span> Voice coil actuator</h4>
|
||||
<div class="outline-text-4" id="text-2-8-1">
|
||||
|
||||
<div id="org4eafd24" class="figure">
|
||||
<div id="org0f9ed57" class="figure">
|
||||
<p><img src="Figures/G_ws_vc.png" alt="G_ws_vc.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Voice coil</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9ef750d" class="figure">
|
||||
<div id="orgb82c1d1" class="figure">
|
||||
<p><img src="Figures/Gc_ws_vc.png" alt="Gc_ws_vc.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Voice coil</p>
|
||||
@ -938,18 +942,18 @@ Then we compare the result between voice coil and piezoelectric actuators.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org53fec97" class="outline-4">
|
||||
<h4 id="org53fec97"><span class="section-number-4">2.8.2</span> Piezoelectric actuator</h4>
|
||||
<div id="outline-container-org34e6778" class="outline-4">
|
||||
<h4 id="org34e6778"><span class="section-number-4">2.8.2</span> Piezoelectric actuator</h4>
|
||||
<div class="outline-text-4" id="text-2-8-2">
|
||||
|
||||
<div id="orgc028298" class="figure">
|
||||
<div id="org359d5f5" class="figure">
|
||||
<p><img src="Figures/G_ws_pz.png" alt="G_ws_pz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Piezoelectric actuator</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org991a168" class="figure">
|
||||
<div id="org4f616e4" class="figure">
|
||||
<p><img src="Figures/Gc_ws_pz.png" alt="Gc_ws_pz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Piezoelectric actuator</p>
|
||||
@ -957,8 +961,8 @@ Then we compare the result between voice coil and piezoelectric actuators.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf6be1a4" class="outline-4">
|
||||
<h4 id="orgf6be1a4"><span class="section-number-4">2.8.3</span> Analysis</h4>
|
||||
<div id="outline-container-org36cd742" class="outline-4">
|
||||
<h4 id="org36cd742"><span class="section-number-4">2.8.3</span> Analysis</h4>
|
||||
<div class="outline-text-4" id="text-2-8-3">
|
||||
<p>
|
||||
When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system.
|
||||
@ -982,11 +986,11 @@ As shown in the previous figures, the system with voice coil is much more sensit
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2262aaa" class="outline-4">
|
||||
<h4 id="org2262aaa"><span class="section-number-4">2.8.4</span> Campbell diagram</h4>
|
||||
<div id="outline-container-org23ea4ed" class="outline-4">
|
||||
<h4 id="org23ea4ed"><span class="section-number-4">2.8.4</span> Campbell diagram</h4>
|
||||
<div class="outline-text-4" id="text-2-8-4">
|
||||
<p>
|
||||
The poles of the system are computed for multiple values of the rotation frequency.
|
||||
The poles of the system are computed for multiple values of the rotation frequency. To simplify the computation of the poles, we add some damping to the system.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -1022,7 +1026,7 @@ polespz = zeros<span style="color: #707183;">(</span><span style="color: #D0372D
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures <a href="#org2e762b4">8</a> and <a href="#orgf969e06">9</a>).
|
||||
We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures <a href="#org0f74744">8</a> and <a href="#orgab3524b">9</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1030,11 +1034,11 @@ When the real part of one pole becomes positive, the system goes unstable.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
For the voice coil (figure <a href="#org2e762b4">8</a>), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure <a href="#orgf969e06">9</a>).
|
||||
For the voice coil (figure <a href="#org0f74744">8</a>), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure <a href="#orgab3524b">9</a>).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org2e762b4" class="figure">
|
||||
<div id="org0f74744" class="figure">
|
||||
<p><img src="Figures/poles_w_vc.png" alt="poles_w_vc.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample</p>
|
||||
@ -1042,10 +1046,10 @@ For the voice coil (figure <a href="#org2e762b4">8</a>), the system is unstable
|
||||
|
||||
|
||||
|
||||
<div id="orgf969e06" class="figure">
|
||||
<div id="orgab3524b" class="figure">
|
||||
<p><img src="Figures/poles_w_pz.png" alt="poles_w_pz.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Piezoelectric actuator and light sample</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -1053,15 +1057,15 @@ For the voice coil (figure <a href="#org2e762b4">8</a>), the system is unstable
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orga07d0dd" class="outline-2">
|
||||
<h2 id="orga07d0dd"><span class="section-number-2">3</span> Control Strategies</h2>
|
||||
<div id="outline-container-org89b80ab" class="outline-2">
|
||||
<h2 id="org89b80ab"><span class="section-number-2">3</span> Control Strategies</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org3747048"></a>
|
||||
<a id="org786bfb0"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgeb25ab0" class="outline-3">
|
||||
<h3 id="orgeb25ab0"><span class="section-number-3">3.1</span> Measurement in the fixed reference frame</h3>
|
||||
<div id="outline-container-orgbdd9948" class="outline-3">
|
||||
<h3 id="orgbdd9948"><span class="section-number-3">3.1</span> Measurement in the fixed reference frame</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
First, let's consider a measurement in the fixed referenced frame.
|
||||
@ -1084,11 +1088,11 @@ Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The block diagram is shown on figure <a href="#orgf3c5c1c">10</a>.
|
||||
The block diagram is shown on figure <a href="#org4a8c2aa">10</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf3c5c1c" class="figure">
|
||||
<div id="org4a8c2aa" class="figure">
|
||||
<p><img src="./Figures/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Control with a measure from fixed frame</p>
|
||||
@ -1104,19 +1108,19 @@ One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\the
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9456905" class="outline-3">
|
||||
<h3 id="org9456905"><span class="section-number-3">3.2</span> Measurement in the rotating frame</h3>
|
||||
<div id="outline-container-org724b218" class="outline-3">
|
||||
<h3 id="org724b218"><span class="section-number-3">3.2</span> Measurement in the rotating frame</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
Let's consider that the measurement is made in the rotating reference frame.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The corresponding block diagram is shown figure <a href="#org6cffd33">11</a>
|
||||
The corresponding block diagram is shown figure <a href="#orge83e07d">11</a>
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6cffd33" class="figure">
|
||||
<div id="orge83e07d" class="figure">
|
||||
<p><img src="./Figures/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Control with a measure from rotating frame</p>
|
||||
@ -1129,19 +1133,19 @@ The loop gain is \(L = G K\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgab7ac9c" class="outline-2">
|
||||
<h2 id="orgab7ac9c"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
|
||||
<div id="outline-container-org30fbee8" class="outline-2">
|
||||
<h2 id="org30fbee8"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org149db50"></a>
|
||||
<a id="orgfce2ea4"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org123b2ae" class="outline-3">
|
||||
<h3 id="org123b2ae"><span class="section-number-3">4.1</span> Initialize</h3>
|
||||
<div id="outline-container-orge1f000c" class="outline-3">
|
||||
<h3 id="orge1f000c"><span class="section-number-3">4.1</span> Initialize</h3>
|
||||
</div>
|
||||
<div id="outline-container-orgc1906bb" class="outline-3">
|
||||
<h3 id="orgc1906bb"><span class="section-number-3">4.2</span> Parameter for the Simscape simulations</h3>
|
||||
<div id="outline-container-org8b4df15" class="outline-3">
|
||||
<h3 id="org8b4df15"><span class="section-number-3">4.2</span> Parameter for the Simscape simulations</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
First we define the parameters that must be defined in order to run the Simscape simulation.
|
||||
@ -1180,8 +1184,8 @@ freqs = logspace<span style="color: #707183;">(</span><span style="color: #6434A
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org255159f" class="outline-3">
|
||||
<h3 id="org255159f"><span class="section-number-3">4.3</span> Identification in the rotating referenced frame</h3>
|
||||
<div id="outline-container-orga3ac610" class="outline-3">
|
||||
<h3 id="orga3ac610"><span class="section-number-3">4.3</span> Identification in the rotating referenced frame</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
We initialize the inputs and outputs of the system to identify:
|
||||
@ -1248,23 +1252,23 @@ Gvc_heavy.OutputName = <span style="color: #707183;">{</span><span style="color:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5f1926d" class="outline-3">
|
||||
<h3 id="org5f1926d"><span class="section-number-3">4.4</span> Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</h3>
|
||||
<div id="outline-container-orga381ded" class="outline-3">
|
||||
<h3 id="orga381ded"><span class="section-number-3">4.4</span> Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<p>
|
||||
From the previous identification, we plot the coupling ratio in both case (figure <a href="#orgcd55860">12</a>).
|
||||
We obtain the same result than the analytical case (figures <a href="#orgeb8c982">2</a> and <a href="#orga3125c6">3</a>).
|
||||
From the previous identification, we plot the coupling ratio in both case (figure <a href="#org1359930">12</a>).
|
||||
We obtain the same result than the analytical case (figures <a href="#org2eaf004">2</a> and <a href="#orge6601b9">3</a>).
|
||||
</p>
|
||||
|
||||
<div id="orgcd55860" class="figure">
|
||||
<div id="org1359930" class="figure">
|
||||
<p><img src="Figures/coupling_ration_light_heavy.png" alt="coupling_ration_light_heavy.png" />
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge5f2b9f" class="outline-3">
|
||||
<h3 id="orge5f2b9f"><span class="section-number-3">4.5</span> Plant Control</h3>
|
||||
<div id="outline-container-org6b388ff" class="outline-3">
|
||||
<h3 id="org6b388ff"><span class="section-number-3">4.5</span> Plant Control</h3>
|
||||
<div class="outline-text-3" id="text-4-5">
|
||||
<p>
|
||||
The goal is the study control problems due to the coupling that appears because of the rotation.
|
||||
@ -1329,8 +1333,8 @@ Plot the ratio between the main transfer function and the coupling term:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb9cef97" class="outline-4">
|
||||
<h4 id="orgb9cef97"><span class="section-number-4">4.5.1</span> Low rotation speed and High rotation speed</h4>
|
||||
<div id="outline-container-orgdb709bf" class="outline-4">
|
||||
<h4 id="orgdb709bf"><span class="section-number-4">4.5.1</span> Low rotation speed and High rotation speed</h4>
|
||||
<div class="outline-text-4" id="text-4-5-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">60</span>; angle_e = <span style="color: #D0372D;">0</span>;
|
||||
@ -1355,8 +1359,8 @@ bode<span style="color: #707183;">(</span>G_low, G_high<span style="color: #7071
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org09ff6ca" class="outline-3">
|
||||
<h3 id="org09ff6ca"><span class="section-number-3">4.6</span> Identification in the fixed frame</h3>
|
||||
<div id="outline-container-org5822ce2" class="outline-3">
|
||||
<h3 id="org5822ce2"><span class="section-number-3">4.6</span> Identification in the fixed frame</h3>
|
||||
<div class="outline-text-3" id="text-4-6">
|
||||
<p>
|
||||
Let's define some options as well as the inputs and outputs for linearization.
|
||||
@ -1437,8 +1441,8 @@ bode<span style="color: #707183;">(</span>Ge<span style="color: #707183;">)</spa
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org588dae5" class="outline-3">
|
||||
<h3 id="org588dae5"><span class="section-number-3">4.7</span> Identification from actuator forces to displacement in the fixed frame</h3>
|
||||
<div id="outline-container-orgfa9ed99" class="outline-3">
|
||||
<h3 id="orgfa9ed99"><span class="section-number-3">4.7</span> Identification from actuator forces to displacement in the fixed frame</h3>
|
||||
<div class="outline-text-3" id="text-4-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
||||
@ -1496,48 +1500,67 @@ exportFig<span style="color: #707183;">(</span><span style="color: #008000;">'G_
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org53255e3" class="outline-3">
|
||||
<h3 id="org53255e3"><span class="section-number-3">4.8</span> Effect of the rotating Speed</h3>
|
||||
<div id="outline-container-orgbc833bb" class="outline-3">
|
||||
<h3 id="orgbc833bb"><span class="section-number-3">4.8</span> Effect of the rotating Speed</h3>
|
||||
<div class="outline-text-3" id="text-4-8">
|
||||
<p>
|
||||
<a id="org09b2961"></a>
|
||||
<a id="org45bb7b1"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org14c5fe5" class="outline-4">
|
||||
<h4 id="org14c5fe5"><span class="section-number-4">4.8.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h4>
|
||||
<div id="outline-container-orgaf21bf8" class="outline-4">
|
||||
<h4 id="orgaf21bf8"><span class="section-number-4">4.8.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h4>
|
||||
</div>
|
||||
<div id="outline-container-org5347efa" class="outline-4">
|
||||
<h4 id="org5347efa"><span class="section-number-4">4.8.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h4>
|
||||
<div id="outline-container-orgdd964cc" class="outline-4">
|
||||
<h4 id="orgdd964cc"><span class="section-number-4">4.8.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h4>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgd2cb6ed" class="outline-3">
|
||||
<h3 id="orgd2cb6ed"><span class="section-number-3">4.9</span> Effect of the X-Y stage stiffness</h3>
|
||||
<div id="outline-container-orgc30bae9" class="outline-3">
|
||||
<h3 id="orgc30bae9"><span class="section-number-3">4.9</span> Effect of the X-Y stage stiffness</h3>
|
||||
<div class="outline-text-3" id="text-4-9">
|
||||
<p>
|
||||
<a id="org2bcac98"></a>
|
||||
<a id="orge951cc4"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org177c370" class="outline-4">
|
||||
<h4 id="org177c370"><span class="section-number-4">4.9.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
|
||||
<div id="outline-container-org3a4478a" class="outline-4">
|
||||
<h4 id="org3a4478a"><span class="section-number-4">4.9.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org4965ab2" class="outline-2">
|
||||
<h2 id="org4965ab2"><span class="section-number-2">5</span> Control Implementation</h2>
|
||||
<div id="outline-container-org12e1d75" class="outline-2">
|
||||
<h2 id="org12e1d75"><span class="section-number-2">5</span> Control Implementation</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="org9e7daf4"></a>
|
||||
<a id="org4a3b8a3"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org9f42bc5" class="outline-3">
|
||||
<h3 id="org9f42bc5"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
|
||||
<div id="outline-container-org70652b4" class="outline-3">
|
||||
<h3 id="org70652b4"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
|
||||
</div>
|
||||
</div>
|
||||
<div id="bibliography">
|
||||
<h2>References</h2>
|
||||
|
||||
<table>
|
||||
|
||||
<tr valign="top">
|
||||
<td align="right" class="bibtexnumber">
|
||||
[<a name="smith99_scien_engin_guide_digit_signal">1</a>]
|
||||
</td>
|
||||
<td class="bibtexitem">
|
||||
Steven W. Smith.
|
||||
<em>The Scientist and Engineer's Guide to Digital Signal Processing
|
||||
- Second Edition</em>.
|
||||
California Technical Publishing, 1999.
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
</table>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2019-01-24 jeu. 14:05</p>
|
||||
<p class="date">Created: 2019-01-24 jeu. 15:17</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
@ -626,7 +626,7 @@ As shown in the previous figures, the system with voice coil is much more sensit
|
||||
|
||||
*** Campbell diagram
|
||||
|
||||
The poles of the system are computed for multiple values of the rotation frequency.
|
||||
The poles of the system are computed for multiple values of the rotation frequency. To simplify the computation of the poles, we add some damping to the system.
|
||||
|
||||
#+begin_src matlab :results silent :exports code
|
||||
m = mlight;
|
||||
@ -739,7 +739,6 @@ For the voice coil (figure [[fig:poles_w_vc]]), the system is unstable when the
|
||||
#+RESULTS:
|
||||
[[file:Figures/poles_w_pz.png]]
|
||||
|
||||
|
||||
* Control Strategies
|
||||
<<sec:control_strategies>>
|
||||
** Measurement in the fixed reference frame
|
||||
@ -1254,3 +1253,5 @@ Finally, we run the linearization.
|
||||
* Control Implementation
|
||||
<<sec:control>>
|
||||
** Measurement in the fixed reference frame
|
||||
* Bibliography :ignore:
|
||||
#+BIBLIOGRAPHY: /home/tdehaeze/MEGA/These/Ressources/references.bib plain option:-a option:-noabstract option:-nokeywords option:-noheader option:-nofooter option:-nobibsource limit:t
|
||||
|
||||
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Reference in New Issue
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