487 lines
16 KiB
HTML
487 lines
16 KiB
HTML
<?xml version="1.0" encoding="utf-8"?>
|
|
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
|
|
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
|
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
|
<head>
|
|
<!-- 2020-08-05 mer. 13:27 -->
|
|
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
|
<title>Stewart Platform - Dynamics Study</title>
|
|
<meta name="generator" content="Org mode" />
|
|
<meta name="author" content="Dehaeze Thomas" />
|
|
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
|
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
|
<script src="./js/jquery.min.js"></script>
|
|
<script src="./js/bootstrap.min.js"></script>
|
|
<script src="./js/jquery.stickytableheaders.min.js"></script>
|
|
<script src="./js/readtheorg.js"></script>
|
|
<script>MathJax = {
|
|
tex: {
|
|
tags: 'ams',
|
|
macros: {bm: ["\\boldsymbol{#1}",1],}
|
|
}
|
|
};
|
|
</script>
|
|
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
|
</head>
|
|
<body>
|
|
<div id="org-div-home-and-up">
|
|
<a accesskey="h" href="./index.html"> UP </a>
|
|
|
|
|
<a accesskey="H" href="./index.html"> HOME </a>
|
|
</div><div id="content">
|
|
<h1 class="title">Stewart Platform - Dynamics Study</h1>
|
|
<div id="table-of-contents">
|
|
<h2>Table of Contents</h2>
|
|
<div id="text-table-of-contents">
|
|
<ul>
|
|
<li><a href="#orgc59e712">1. Compare external forces and forces applied by the actuators</a>
|
|
<ul>
|
|
<li><a href="#org4509b7d">1.1. Comparison with fixed support</a></li>
|
|
<li><a href="#org8662186">1.2. Comparison with a flexible support</a></li>
|
|
<li><a href="#org55e0dad">1.3. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
<li><a href="#org81ab204">2. Comparison of the static transfer function and the Compliance matrix</a>
|
|
<ul>
|
|
<li><a href="#orge7e7242">2.1. Analysis</a></li>
|
|
<li><a href="#org9ee3939">2.2. Conclusion</a></li>
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc59e712" class="outline-2">
|
|
<h2 id="orgc59e712"><span class="section-number-2">1</span> Compare external forces and forces applied by the actuators</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org4509b7d" class="outline-3">
|
|
<h3 id="org4509b7d"><span class="section-number-3">1.1</span> Comparison with fixed support</h3>
|
|
<div class="outline-text-3" id="text-1-1">
|
|
<p>
|
|
Let’s generate a Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeInertialSensor(stewart, 'type', 'none');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We don’t put any flexibility below the Stewart platform such that <b>its base is fixed to an inertial frame</b>.
|
|
We also don’t put any payload on top of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround('type', 'none');
|
|
payload = initializePayload('type', 'none');
|
|
controller = initializeController('type', 'open-loop');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The transfer function from actuator forces \(\bm{\tau}\) to the relative displacement of the mobile platform \(\mathcal{\bm{X}}\) is extracted.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_platform_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame \(\{B\}\) fixed to the mobile platform:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = minreal(G*inv(stewart.kinematics.J'));
|
|
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We also extract the transfer function from external forces \(\bm{\mathcal{F}}_{\text{ext}}\) on the frame \(\{B\}\) fixed to the mobile platform to the relative displacement \(\mathcal{\bm{X}}\) of \(\{B\}\) with respect to frame \(\{A\}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
|
|
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
|
|
|
|
%% Run the linearization
|
|
Gd = linearize(mdl, io, options);
|
|
Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
|
|
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The comparison of the two transfer functions is shown in Figure <a href="#orgbf9a54a">1</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgbf9a54a" class="figure">
|
|
<p><img src="figs/comparison_Fext_F_fixed_base.png" alt="comparison_Fext_F_fixed_base.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_fixed_base.png">png</a>, <a href="./figs/comparison_Fext_F_fixed_base.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
This can be understood from figure <a href="#org8bd3e63">2</a> where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly the same effect on \(\mathcal{X}_{x}\).
|
|
</p>
|
|
|
|
|
|
<div id="org8bd3e63" class="figure">
|
|
<p><img src="figs/1dof_actuator_external_forces.png" alt="1dof_actuator_external_forces.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Schematic representation of the stewart platform on a rigid support</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8662186" class="outline-3">
|
|
<h3 id="org8662186"><span class="section-number-3">1.2</span> Comparison with a flexible support</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
We now add a flexible support under the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround('type', 'flexible');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And we perform again the identification.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gc = minreal(G*inv(stewart.kinematics.J'));
|
|
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
|
|
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
|
|
|
|
%% Run the linearization
|
|
Gd = linearize(mdl, io, options);
|
|
Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
|
|
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The comparison between the obtained transfer functions is shown in Figure <a href="#orga2f2bd5">3</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orga2f2bd5" class="figure">
|
|
<p><img src="figs/comparison_Fext_F_flexible_base.png" alt="comparison_Fext_F_flexible_base.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_flexible_base.png">png</a>, <a href="./figs/comparison_Fext_F_flexible_base.pdf">pdf</a>)</p>
|
|
</div>
|
|
|
|
<p>
|
|
The addition of a flexible support can be schematically represented in Figure <a href="#orgee3ecbe">4</a>.
|
|
We see that \(\mathcal{F}_{x}\) applies a force both on \(m\) and \(m^{\prime}\) whereas \(\mathcal{F}_{x,\text{ext}}\) only applies a force on \(m\).
|
|
And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly <b>not</b> the same effect on \(\mathcal{X}_{x}\).
|
|
</p>
|
|
|
|
|
|
<div id="orgee3ecbe" class="figure">
|
|
<p><img src="figs/2dof_actuator_external_forces.png" alt="2dof_actuator_external_forces.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Schematic representation of the stewart platform on top of a flexible support</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org55e0dad" class="outline-3">
|
|
<h3 id="org55e0dad"><span class="section-number-3">1.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<div class="important">
|
|
<p>
|
|
The transfer function from forces/torques applied by the actuators on the payload \(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\) to the pose of the mobile platform \(\bm{\mathcal{X}}\) is the same as the transfer function from external forces/torques to \(\bm{\mathcal{X}}\) as long as the Stewart platform’s base is fixed.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org81ab204" class="outline-2">
|
|
<h2 id="org81ab204"><span class="section-number-2">2</span> Comparison of the static transfer function and the Compliance matrix</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orge7e7242" class="outline-3">
|
|
<h3 id="orge7e7242"><span class="section-number-3">2.1</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
Initialization of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeInertialSensor(stewart, 'type', 'none');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
No flexibility below the Stewart platform and no payload.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround('type', 'none');
|
|
payload = initializePayload('type', 'none');
|
|
controller = initializeController('type', 'open-loop');
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_platform_model';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = minreal(G*inv(stewart.kinematics.J'));
|
|
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s first look at the low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-7.2e-19</td>
|
|
<td class="org-right">5.0e-18</td>
|
|
<td class="org-right">-8.9e-18</td>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">9.9e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.7e-18</td>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-5.7e-18</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">-1.6e-17</td>
|
|
<td class="org-right">-1.7e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.3e-18</td>
|
|
<td class="org-right">-6.3e-18</td>
|
|
<td class="org-right">2.1e-08</td>
|
|
<td class="org-right">4.4e-17</td>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">7.4e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3.2e-17</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">6.2e-18</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-3.5e-16</td>
|
|
<td class="org-right">6.3e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">2.7e-17</td>
|
|
<td class="org-right">4.8e-17</td>
|
|
<td class="org-right">-4.5e-16</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-1.2e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.0e-17</td>
|
|
<td class="org-right">-9.5e-17</td>
|
|
<td class="org-right">8.4e-18</td>
|
|
<td class="org-right">4.3e-16</td>
|
|
<td class="org-right">5.8e-16</td>
|
|
<td class="org-right">1.7e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
And now at the Compliance matrix.
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-2.0e-24</td>
|
|
<td class="org-right">7.4e-25</td>
|
|
<td class="org-right">5.9e-23</td>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">5.9e-24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-7.1e-25</td>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">2.9e-25</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">-5.4e-24</td>
|
|
<td class="org-right">-3.3e-23</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.9e-26</td>
|
|
<td class="org-right">-6.4e-25</td>
|
|
<td class="org-right">2.1e-08</td>
|
|
<td class="org-right">1.9e-23</td>
|
|
<td class="org-right">5.3e-25</td>
|
|
<td class="org-right">-6.5e-40</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.4e-23</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">1.3e-23</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">4.9e-22</td>
|
|
<td class="org-right">-3.8e-24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">7.6e-24</td>
|
|
<td class="org-right">1.2e-23</td>
|
|
<td class="org-right">6.9e-22</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-2.6e-22</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.3e-24</td>
|
|
<td class="org-right">-3.2e-23</td>
|
|
<td class="org-right">-1.6e-39</td>
|
|
<td class="org-right">9.9e-23</td>
|
|
<td class="org-right">-3.3e-22</td>
|
|
<td class="org-right">1.7e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9ee3939" class="outline-3">
|
|
<h3 id="org9ee3939"><span class="section-number-3">2.2</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<div class="important">
|
|
<p>
|
|
The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-08-05 mer. 13:27</p>
|
|
</div>
|
|
</body>
|
|
</html>
|