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<a accesskey="h" href="./index.html"> UP </a>
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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">Stewart Platform - Simscape Model</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org002dfa5">1. Parameters used for the Simscape Model</a></li>
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<li><a href="#orgf99cee5">2. Simulation Configuration - Configuration reference</a></li>
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<li><a href="#org7bf96b7">3. Subsystem Reference</a></li>
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<li><a href="#orgae34e57">4. Subsystem - Fixed base and Mobile Platform</a></li>
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<li><a href="#org2e16af3">5. Subsystem - Struts</a>
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<ul>
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<li><a href="#org59382cb">5.1. Strut Configuration</a></li>
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<li><a href="#org5a54286">5.2. Z-Axis Geophone</a>
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<ul>
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<li><a href="#org7f22bd7">5.2.1. Working Principle</a></li>
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<li><a href="#org3956dae">5.2.2. Initialization function</a></li>
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</ul>
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</li>
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<li><a href="#org2561089">5.3. Z-Axis Accelerometer</a>
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<ul>
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<li><a href="#org70a289b">5.3.1. Working Principle</a></li>
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<li><a href="#orgc7763c4">5.3.2. Initialization function</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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</ul>
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</div>
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</div>
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<p>
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In this document is explained how the Simscape model of the Stewart Platform is implemented.
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</p>
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<p>
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It is divided in the following sections:
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</p>
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<ul class="org-ul">
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<li>section <a href="#org2553b43">1</a>: is explained how the parameters of the Stewart platform are set for the Simscape model</li>
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<li>section <a href="#orgc20ead1">2</a>: the Simulink configuration (solver, simulation time, …) is shared among all the Simulink files. It is explain how this is done.</li>
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<li>section <a href="#org374fa44">3</a>: All the elements (platforms, struts, sensors, …) are saved in separate files and imported in Simulink files using “subsystem referenced”.</li>
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<li>section <a href="#org0562d05">4</a>: The simscape model for the fixed base and mobile platform are described in this section.</li>
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<li>section <a href="#org0ff5129">5</a>: The simscape model for the Stewart platform struts is described in this section.</li>
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</ul>
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<div id="outline-container-org002dfa5" class="outline-2">
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<h2 id="org002dfa5"><span class="section-number-2">1</span> Parameters used for the Simscape Model</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org2553b43"></a>
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The Simscape Model of the Stewart Platform is working with the <code>stewart</code> structure generated using the functions described <a href="stewart-architecture.html">here</a>.
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</p>
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<p>
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All the geometry and inertia of the mechanical elements are defined in the <code>stewart</code> structure.
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</p>
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<p>
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By updating the <code>stewart</code> structure in the workspace, the Simscape model will be automatically updated.
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</p>
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<p>
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Thus, nothing should be changed by hand inside the Simscape model.
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</p>
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<p>
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The main advantage to have all the parameters defined in one structure (and not hard-coded in some simulink blocs) it that we can easily change the Stewart architecture/parameters in a Matlab script to perform some parametric study for instance.
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</p>
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</div>
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</div>
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<div id="outline-container-orgf99cee5" class="outline-2">
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<h2 id="orgf99cee5"><span class="section-number-2">2</span> Simulation Configuration - Configuration reference</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orgc20ead1"></a>
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As multiple simulink files will be used for simulation and tests, it is very useful to determine good simulation configuration that will be <b>shared</b> among all the simulink files.
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</p>
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<p>
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This is done using something called “<b>Configuration Reference</b>” (<a href="https://fr.mathworks.com/help/simulink/ug/more-about-configuration-references.html">documentation</a>).
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</p>
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<p>
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Basically, the configuration is stored in a mat file <code>conf_simscape.mat</code> and then loaded in the workspace for it to be accessible to all the simulink models.
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It is automatically loaded when the Simulink project is open. It can be loaded manually with the command:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
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</pre>
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</div>
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<p>
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It is however possible to modify specific parameters just for one simulation using the <code>set_param</code> command:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, 1);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org7bf96b7" class="outline-2">
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<h2 id="org7bf96b7"><span class="section-number-2">3</span> Subsystem Reference</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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<a id="org374fa44"></a>
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Several Stewart platform models are used, for instance one is use to study the dynamics while the other is used to apply active damping techniques.
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</p>
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<p>
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However, all the Simscape models share some subsystems using the <b>Subsystem Reference</b> Simulink block (<a href="https://fr.mathworks.com/help/simulink/ug/referenced-subsystem-1.html">documentation</a>).
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</p>
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<p>
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These shared subsystems are:
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</p>
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<ul class="org-ul">
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<li><code>Fixed_Based.slx</code> - Fixed base of the Stewart Platform</li>
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<li><code>Mobile_Platform.slx</code> - Mobile platform of the Stewart Platform</li>
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<li><code>stewart_strut.slx</code> - One strut containing two spherical/universal joints, the actuator as well as the included sensors. A parameter <code>i</code> is initialized to determine what it the “number” of the strut.</li>
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</ul>
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<p>
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These subsystems are referenced from another subsystem called <code>Stewart_Platform.slx</code> shown in figure <a href="#org7f7ef2b">1</a>, that basically connect them correctly.
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This subsystem is then referenced in other simulink models for various purposes (control, analysis, simulation, …).
|
|
</p>
|
|
|
|
|
|
<div id="org7f7ef2b" class="figure">
|
|
<p><img src="figs/simscape_stewart_platform.png" alt="simscape_stewart_platform.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Simscape Subsystem of the Stewart platform. Encapsulate the Subsystems corresponding to the fixed base, mobile platform and all the struts.</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgae34e57" class="outline-2">
|
|
<h2 id="orgae34e57"><span class="section-number-2">4</span> Subsystem - Fixed base and Mobile Platform</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org0562d05"></a>
|
|
Both the fixed base and the mobile platform simscape models share many similarities.
|
|
</p>
|
|
|
|
<p>
|
|
Their are both composed of:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>a solid body representing the platform</li>
|
|
<li>6 rigid transform blocks to go from the frame \(\{F\}\) (resp. \(\{M\}\)) to the location of the joints.
|
|
These rigid transform are using \({}^F\bm{a}_i\) (resp. \({}^M\bm{b}_i\)) for the position of the joint and \({}^F\bm{R}_{a_i}\) (resp. \({}^M\bm{R}_{b_i}\)) for the orientation of the joint.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
As always, the parameters that define the geometry are taken from the <code>stewart</code> structure.
|
|
</p>
|
|
|
|
|
|
<div id="orga1b9893" class="figure">
|
|
<p><img src="figs/simscape_fixed_base.png" alt="simscape_fixed_base.png" width="1000px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Simscape Model of the Fixed base</p>
|
|
</div>
|
|
|
|
|
|
<div id="org1f71117" class="figure">
|
|
<p><img src="figs/simscape_mobile_platform.png" alt="simscape_mobile_platform.png" width="800px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Simscape Model of the Mobile platform</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2e16af3" class="outline-2">
|
|
<h2 id="org2e16af3"><span class="section-number-2">5</span> Subsystem - Struts</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org0ff5129"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org59382cb" class="outline-3">
|
|
<h3 id="org59382cb"><span class="section-number-3">5.1</span> Strut Configuration</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
For the Stewart platform, the 6 struts are identical.
|
|
Thus, all the struts used in the Stewart platform are referring to the same subsystem called <code>stewart_strut.slx</code> and shown in Figure <a href="#org9ef7b41">4</a>.
|
|
</p>
|
|
|
|
<p>
|
|
This strut as the following structure:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Universal Joint*</b> connected on the Fixed base</li>
|
|
<li><b>Prismatic Joint*</b> for the actuator</li>
|
|
<li><b>Spherical Joint*</b> connected on the Mobile platform</li>
|
|
</ul>
|
|
|
|
<p>
|
|
This configuration is called <b>UPS</b>.
|
|
</p>
|
|
|
|
<p>
|
|
The other common configuration <b>SPS</b> has the disadvantage of having additional passive degrees-of-freedom corresponding to the rotation of the strut around its main axis.
|
|
This is why the <b>UPS</b> configuration is used, but other configuration can be easily implemented.
|
|
</p>
|
|
|
|
|
|
<div id="org9ef7b41" class="figure">
|
|
<p><img src="figs/simscape_strut.png" alt="simscape_strut.png" width="800px" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Simscape model of the Stewart platform’s strut</p>
|
|
</div>
|
|
|
|
<p>
|
|
Several sensors are included in the strut that may or may not be used for control:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Relative Displacement sensor: gives the relative displacement of the strut.</li>
|
|
<li>Force sensor: measure the total force applied by the force actuator, the stiffness and damping forces in the direction of the strut.</li>
|
|
<li>Inertial sensor: measure the absolute motion (velocity) of the top part of the strut in the direction of the strut.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
There is two main types of inertial sensor that can be used to measure the absolute motion of the top part of the strut in the direction of the strut:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>a geophone that measures the absolute velocity above some frequency</li>
|
|
<li>an accelerometer that measures the absolute acceleration below some frequency</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Both inertial sensors are described bellow.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5a54286" class="outline-3">
|
|
<h3 id="org5a54286"><span class="section-number-3">5.2</span> Z-Axis Geophone</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
</div>
|
|
<div id="outline-container-org7f22bd7" class="outline-4">
|
|
<h4 id="org7f22bd7"><span class="section-number-4">5.2.1</span> Working Principle</h4>
|
|
<div class="outline-text-4" id="text-5-2-1">
|
|
<p>
|
|
From the schematic of the Z-axis geophone shown in Figure <a href="#orgbecc7b0">5</a>, we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\):
|
|
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
|
with:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\)</li>
|
|
<li>\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orgbecc7b0" class="figure">
|
|
<p><img src="figs/inertial_sensor.png" alt="inertial_sensor.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Schematic of a Z-Axis geophone</p>
|
|
</div>
|
|
|
|
<p>
|
|
We see that at frequencies above \(\omega_0\):
|
|
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]
|
|
</p>
|
|
|
|
<p>
|
|
And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.
|
|
</p>
|
|
|
|
<p>
|
|
We generally want to have the smallest resonant frequency \(\omega_0\) to measure low frequency absolute velocity, however there is a trade-off between \(\omega_0\) and the mass of the inertial mass.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3956dae" class="outline-4">
|
|
<h4 id="org3956dae"><span class="section-number-4">5.2.2</span> Initialization function</h4>
|
|
<div class="outline-text-4" id="text-5-2-2">
|
|
<p>
|
|
<a id="orgabe7399"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="../src/initializeZAxisGeophone.m">here</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[geophone]</span> = <span class="org-function-name">initializeZAxisGeophone</span>(<span class="org-variable-name">args</span>)
|
|
arguments
|
|
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e<span class="org-type">-</span>3 <span class="org-comment">% [kg]</span>
|
|
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 1 <span class="org-comment">% [Hz]</span>
|
|
<span class="org-keyword">end</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
|
|
geophone.m = args.mass;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% The Stiffness is set to have the damping resonance frequency</span></span>
|
|
geophone.k = geophone.m <span class="org-type">*</span> (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>args.freq)<span class="org-type">^</span>2;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% We set the damping value to have critical damping</span></span>
|
|
geophone.c = 2<span class="org-type">*</span>sqrt(geophone.m <span class="org-type">*</span> geophone.k);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Save</span></span>
|
|
save(<span class="org-string">'./mat/geophone_z_axis.mat'</span>, <span class="org-string">'geophone'</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2561089" class="outline-3">
|
|
<h3 id="org2561089"><span class="section-number-3">5.3</span> Z-Axis Accelerometer</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
</div>
|
|
<div id="outline-container-org70a289b" class="outline-4">
|
|
<h4 id="org70a289b"><span class="section-number-4">5.3.1</span> Working Principle</h4>
|
|
<div class="outline-text-4" id="text-5-3-1">
|
|
<p>
|
|
From the schematic of the Z-axis accelerometer shown in Figure <a href="#orgbacfc60">6</a>, we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\):
|
|
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
|
with:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\)</li>
|
|
<li>\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orgbacfc60" class="figure">
|
|
<p><img src="figs/inertial_sensor.png" alt="inertial_sensor.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Schematic of a Z-Axis geophone</p>
|
|
</div>
|
|
|
|
<p>
|
|
We see that at frequencies below \(\omega_0\):
|
|
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]
|
|
</p>
|
|
|
|
<p>
|
|
And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.
|
|
</p>
|
|
|
|
<p>
|
|
Note that there is trade-off between:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>the highest measurable acceleration \(\omega_0\)</li>
|
|
<li>the sensitivity of the accelerometer which is equal to \(-\frac{1}{{\omega_0}^2}\)</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc7763c4" class="outline-4">
|
|
<h4 id="orgc7763c4"><span class="section-number-4">5.3.2</span> Initialization function</h4>
|
|
<div class="outline-text-4" id="text-5-3-2">
|
|
<p>
|
|
<a id="orge9014b7"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="../src/initializeZAxisAccelerometer.m">here</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[accelerometer]</span> = <span class="org-function-name">initializeZAxisAccelerometer</span>(<span class="org-variable-name">args</span>)
|
|
arguments
|
|
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e<span class="org-type">-</span>3 <span class="org-comment">% [kg]</span>
|
|
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 5e3 <span class="org-comment">% [Hz]</span>
|
|
<span class="org-keyword">end</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
|
|
accelerometer.m = args.mass;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% The Stiffness is set to have the damping resonance frequency</span></span>
|
|
accelerometer.k = accelerometer.m <span class="org-type">*</span> (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>args.freq)<span class="org-type">^</span>2;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% We set the damping value to have critical damping</span></span>
|
|
accelerometer.c = 2<span class="org-type">*</span>sqrt(accelerometer.m <span class="org-type">*</span> accelerometer.k);
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Gain correction of the accelerometer to have a unity gain until the resonance</span></span>
|
|
accelerometer.gain = <span class="org-type">-</span>accelerometer.k<span class="org-type">/</span>accelerometer.m;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Save</span></span>
|
|
save(<span class="org-string">'./mat/accelerometer_z_axis.mat'</span>, <span class="org-string">'accelerometer'</span>);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-01-29 mer. 12:02</p>
|
|
</div>
|
|
</body>
|
|
</html>
|