785 lines
36 KiB
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785 lines
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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">Kinematic Study of the Stewart Platform</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="#org63c8faa">1. Needed Actuator Stroke</a>
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<ul>
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<li><a href="#orged5be9e">1.1. Stewart architecture definition</a></li>
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<li><a href="#org73e5cf8">1.2. Wanted translations and rotations</a></li>
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<li><a href="#org9825ccf">1.3. Needed stroke for “pure” rotations or translations</a></li>
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<li><a href="#org0440602">1.4. Needed stroke for combined translations and rotations</a></li>
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</ul>
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</li>
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<li><a href="#org092f7f8">2. Maximum Stroke</a></li>
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<li><a href="#org720ba56">3. Functions</a>
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<ul>
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<li><a href="#org8125766">3.1. getMaxPositions</a></li>
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<li><a href="#org91e4101">3.2. getMaxPureDisplacement</a></li>
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<li><a href="#orgf75fefe">3.3. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
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<ul>
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<li><a href="#orgae47616">3.3.1. Function description</a></li>
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<li><a href="#org78705da">3.3.2. Compute Jacobian Matrix</a></li>
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<li><a href="#orgb7dc1d7">3.3.3. Compute Stiffness Matrix</a></li>
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<li><a href="#org7aa6c04">3.3.4. Compute Compliance Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org9c46957">3.4. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
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<ul>
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<li><a href="#org9da7af0">3.4.1. Function description</a></li>
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<li><a href="#orge2cc540">3.4.2. Optional Parameters</a></li>
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<li><a href="#orga1a0cc7">3.4.3. Theory</a></li>
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<li><a href="#org9b86eb9">3.4.4. Compute</a></li>
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</ul>
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</li>
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<li><a href="#org7e6d65c">3.5. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
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<ul>
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<li><a href="#org65e0ce7">3.5.1. Function description</a></li>
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<li><a href="#orgf6a32e1">3.5.2. Optional Parameters</a></li>
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<li><a href="#orgce0b559">3.5.3. Computation</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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<div id="outline-container-org63c8faa" class="outline-2">
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<h2 id="org63c8faa"><span class="section-number-2">1</span> Needed Actuator Stroke</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
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</p>
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</div>
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<div id="outline-container-orged5be9e" class="outline-3">
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<h3 id="orged5be9e"><span class="section-number-3">1.1</span> Stewart architecture definition</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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We use a cubic architecture.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">opts = struct(...
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<span class="org-string">'H_tot'</span>, 90, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
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<span class="org-string">'L'</span>, 200<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
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<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
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<span class="org-string">'H0'</span>, 200<span class="org-type">/</span>2<span class="org-type">-</span>60<span class="org-type">/</span>2 ...<span class="org-comment"> % Height between the corner of the cube and the plane containing the base joints [mm]</span>
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);
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stewart = initializeCubicConfiguration(opts);
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opts = struct(...
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<span class="org-string">'Jd_pos'</span>, [0, 0, 100], ...<span class="org-comment"> % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
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<span class="org-string">'Jf_pos'</span>, [0, 0, <span class="org-type">-</span>50] ...<span class="org-comment"> % Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
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);
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stewart = computeGeometricalProperties(stewart, opts);
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opts = struct(...
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<span class="org-string">'stroke'</span>, 50e<span class="org-type">-</span>6 ...<span class="org-comment"> % Maximum stroke of each actuator [m]</span>
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);
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stewart = initializeMechanicalElements(stewart, opts);
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save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-string">'stewart'</span>);
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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-org73e5cf8" class="outline-3">
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<h3 id="org73e5cf8"><span class="section-number-3">1.2</span> Wanted translations and rotations</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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We define wanted translations and rotations
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Tx_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
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Ty_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
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Tz_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
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Rx_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
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Ry_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
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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-org9825ccf" class="outline-3">
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<h3 id="org9825ccf"><span class="section-number-3">1.3</span> Needed stroke for “pure” rotations or translations</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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First, we estimate the needed actuator stroke for “pure” rotations and translation.
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</p>
|
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<div class="org-src-container">
|
|
<pre class="src src-matlab">LTx = stewart.Jd<span class="org-type">*</span>[Tx_max 0 0 0 0 0]<span class="org-type">'</span>;
|
|
LTy = stewart.Jd<span class="org-type">*</span>[0 Ty_max 0 0 0 0]<span class="org-type">'</span>;
|
|
LTz = stewart.Jd<span class="org-type">*</span>[0 0 Tz_max 0 0 0]<span class="org-type">'</span>;
|
|
LRx = stewart.Jd<span class="org-type">*</span>[0 0 0 Rx_max 0 0]<span class="org-type">'</span>;
|
|
LRy = stewart.Jd<span class="org-type">*</span>[0 0 0 0 Ry_max 0]<span class="org-type">'</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
From -1.2e-05[m] to 1.1e-05[m]: Total stroke = 22.9[um]
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0440602" class="outline-3">
|
|
<h3 id="org0440602"><span class="section-number-3">1.4</span> Needed stroke for combined translations and rotations</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
|
<p>
|
|
Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Lmax = 0;
|
|
Lmin = 0;
|
|
pos = [0, 0, 0, 0, 0];
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">Tx</span> = <span class="org-constant">[-Tx_max</span>,Tx_max]
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">Ty</span> = <span class="org-constant">[-Ty_max</span>,Ty_max]
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">Tz</span> = <span class="org-constant">[-Tz_max</span>,Tz_max]
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">Rx</span> = <span class="org-constant">[-Rx_max</span>,Rx_max]
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">Ry</span> = <span class="org-constant">[-Ry_max</span>,Ry_max]
|
|
lmax = max(stewart.Jd<span class="org-type">*</span>[Tx Ty Tz Rx Ry 0]<span class="org-type">'</span>);
|
|
lmin = min(stewart.Jd<span class="org-type">*</span>[Tx Ty Tz Rx Ry 0]<span class="org-type">'</span>);
|
|
<span class="org-keyword">if</span> lmax <span class="org-type">></span> Lmax
|
|
Lmax = lmax;
|
|
pos = [Tx Ty Tz Rx Ry];
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">if</span> lmin <span class="org-type"><</span> Lmin
|
|
Lmin = lmin;
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We obtain a needed stroke shown below (almost two times the needed stroke for “pure” rotations and translations).
|
|
</p>
|
|
<pre class="example">
|
|
From -3.1e-05[m] to 3.1e-05[m]: Total stroke = 61.5[um]
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org092f7f8" class="outline-2">
|
|
<h2 id="org092f7f8"><span class="section-number-2">2</span> Maximum Stroke</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
<p>
|
|
From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[X, Y, Z] = getMaxPositions(<span class="org-variable-name">stewart</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
plot3(X, Y, Z, <span class="org-string">'k-'</span>)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org720ba56" class="outline-2">
|
|
<h2 id="org720ba56"><span class="section-number-2">3</span> Functions</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
</div>
|
|
<div id="outline-container-org8125766" class="outline-3">
|
|
<h3 id="org8125766"><span class="section-number-3">3.1</span> getMaxPositions</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[X, Y, Z]</span> = <span class="org-function-name">getMaxPositions</span>(<span class="org-variable-name">stewart</span>)
|
|
Leg = stewart.Leg;
|
|
J = stewart.Jd;
|
|
theta = linspace(0, 2<span class="org-type">*</span><span class="org-constant">pi</span>, 100);
|
|
phi = linspace(<span class="org-type">-</span><span class="org-constant">pi</span><span class="org-type">/</span>2 , <span class="org-constant">pi</span><span class="org-type">/</span>2, 100);
|
|
dmax = zeros(length(theta), length(phi));
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(theta)</span>
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(phi)</span>
|
|
L = J<span class="org-type">*</span>[cos(phi(<span class="org-constant">j</span>))<span class="org-type">*</span>cos(theta(<span class="org-constant">i</span>)) cos(phi(<span class="org-constant">j</span>))<span class="org-type">*</span>sin(theta(<span class="org-constant">i</span>)) sin(phi(<span class="org-constant">j</span>)) 0 0 0]<span class="org-type">'</span>;
|
|
dmax(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = Leg.stroke<span class="org-type">/</span>max(abs(L));
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
|
|
X = dmax<span class="org-type">.*</span>cos(repmat(phi,length(theta),1))<span class="org-type">.*</span>cos(repmat(theta,length(phi),1))<span class="org-type">'</span>;
|
|
Y = dmax<span class="org-type">.*</span>cos(repmat(phi,length(theta),1))<span class="org-type">.*</span>sin(repmat(theta,length(phi),1))<span class="org-type">'</span>;
|
|
Z = dmax<span class="org-type">.*</span>sin(repmat(phi,length(theta),1));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org91e4101" class="outline-3">
|
|
<h3 id="org91e4101"><span class="section-number-3">3.2</span> getMaxPureDisplacement</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[max_disp]</span> = <span class="org-function-name">getMaxPureDisplacement</span>(<span class="org-variable-name">Leg</span>, <span class="org-variable-name">J</span>)
|
|
max_disp = zeros(6, 1);
|
|
max_disp<span class="org-type">(1) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[1 0 0 0 0 0]<span class="org-type">'</span>));
|
|
max_disp<span class="org-type">(2) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 1 0 0 0 0]<span class="org-type">'</span>));
|
|
max_disp<span class="org-type">(3) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 1 0 0 0]<span class="org-type">'</span>));
|
|
max_disp<span class="org-type">(4) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 0 1 0 0]<span class="org-type">'</span>));
|
|
max_disp<span class="org-type">(5) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 0 0 1 0]<span class="org-type">'</span>));
|
|
max_disp<span class="org-type">(6) </span>= Leg.stroke<span class="org-type">/</span>max(abs(J<span class="org-type">*</span>[0 0 0 0 0 1]<span class="org-type">'</span>));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgf75fefe" class="outline-3">
|
|
<h3 id="orgf75fefe"><span class="section-number-3">3.3</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
<a id="org02bdbb2"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgae47616" class="outline-4">
|
|
<h4 id="orgae47616"><span class="section-number-4">3.3.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-3-3-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">computeJacobian</span>(<span class="org-variable-name">stewart</span>)
|
|
<span class="org-comment">% computeJacobian -</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = computeJacobian(stewart)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - stewart - With at least the following fields:</span>
|
|
<span class="org-comment">% - As [3x6] - The 6 unit vectors for each strut expressed in {A}</span>
|
|
<span class="org-comment">% - Ab [3x6] - The 6 position of the joints bi expressed in {A}</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - With the 3 added field:</span>
|
|
<span class="org-comment">% - J [6x6] - The Jacobian Matrix</span>
|
|
<span class="org-comment">% - K [6x6] - The Stiffness Matrix</span>
|
|
<span class="org-comment">% - C [6x6] - The Compliance Matrix</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org78705da" class="outline-4">
|
|
<h4 id="org78705da"><span class="section-number-4">3.3.2</span> Compute Jacobian Matrix</h4>
|
|
<div class="outline-text-4" id="text-3-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.J = [stewart.As<span class="org-type">'</span> , cross(stewart.Ab, stewart.As)<span class="org-type">'</span>];
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb7dc1d7" class="outline-4">
|
|
<h4 id="orgb7dc1d7"><span class="section-number-4">3.3.3</span> Compute Stiffness Matrix</h4>
|
|
<div class="outline-text-4" id="text-3-3-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.K = stewart.J<span class="org-type">'*</span>diag(stewart.Ki)<span class="org-type">*</span>stewart.J;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7aa6c04" class="outline-4">
|
|
<h4 id="org7aa6c04"><span class="section-number-4">3.3.4</span> Compute Compliance Matrix</h4>
|
|
<div class="outline-text-4" id="text-3-3-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.C = inv(stewart.K);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9c46957" class="outline-3">
|
|
<h3 id="org9c46957"><span class="section-number-3">3.4</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
<a id="orgab617cc"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org9da7af0" class="outline-4">
|
|
<h4 id="org9da7af0"><span class="section-number-4">3.4.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-3-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[Li, dLi]</span> = <span class="org-function-name">inverseKinematics</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = inverseKinematics(stewart)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - stewart - A structure with the following fields</span>
|
|
<span class="org-comment">% - Aa [3x6] - The positions ai expressed in {A}</span>
|
|
<span class="org-comment">% - Bb [3x6] - The positions bi expressed in {B}</span>
|
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
|
<span class="org-comment">% - AP [3x1] - The wanted position of {B} with respect to {A}</span>
|
|
<span class="org-comment">% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}</span>
|
|
<span class="org-comment">% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge2cc540" class="outline-4">
|
|
<h4 id="orge2cc540"><span class="section-number-4">3.4.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-3-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
|
|
args.ARB (3,3) double {mustBeNumeric} = eye(3)
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga1a0cc7" class="outline-4">
|
|
<h4 id="orga1a0cc7"><span class="section-number-4">3.4.3</span> Theory</h4>
|
|
<div class="outline-text-4" id="text-3-4-3">
|
|
<p>
|
|
For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
|
|
</p>
|
|
|
|
<p>
|
|
From the geometry of the manipulator, the loop closure for each limb, \(i = 1, 2, \dots, 6\) can be written as
|
|
</p>
|
|
\begin{align*}
|
|
l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
|
|
&= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
|
|
\end{align*}
|
|
|
|
<p>
|
|
To obtain the length of each actuator and eliminate \(\hat{\bm{s}}_i\), it is sufficient to dot multiply each side by itself:
|
|
</p>
|
|
\begin{equation}
|
|
l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
|
|
\end{equation}
|
|
|
|
<p>
|
|
Hence, for \(i = 1, 2, \dots, 6\), each limb length can be uniquely determined by:
|
|
</p>
|
|
\begin{equation}
|
|
l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
|
|
\end{equation}
|
|
|
|
<p>
|
|
If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
|
|
Otherwise, when the limbs’ lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9b86eb9" class="outline-4">
|
|
<h4 id="org9b86eb9"><span class="section-number-4">3.4.4</span> Compute</h4>
|
|
<div class="outline-text-4" id="text-3-4-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Li = sqrt(args.AP<span class="org-type">'*</span>args.AP <span class="org-type">+</span> diag(stewart.Bb<span class="org-type">'*</span>stewart.Bb) <span class="org-type">+</span> diag(stewart.Aa<span class="org-type">'*</span>stewart.Aa) <span class="org-type">-</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>stewart.Aa)<span class="org-type">'</span> <span class="org-type">+</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>(args.ARB<span class="org-type">*</span>stewart.Bb))<span class="org-type">'</span> <span class="org-type">-</span> diag(2<span class="org-type">*</span>(args.ARB<span class="org-type">*</span>stewart.Bb)<span class="org-type">'*</span>stewart.Aa));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">dLi = Li<span class="org-type">-</span>stewart.l;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7e6d65c" class="outline-3">
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<h3 id="org7e6d65c"><span class="section-number-3">3.5</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3>
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<div class="outline-text-3" id="text-3-5">
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<p>
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<a id="orgee3cdbf"></a>
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</p>
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<p>
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This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here</a>.
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</p>
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</div>
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<div id="outline-container-org65e0ce7" class="outline-4">
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<h4 id="org65e0ce7"><span class="section-number-4">3.5.1</span> Function description</h4>
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<div class="outline-text-4" id="text-3-5-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[P, R]</span> = <span class="org-function-name">forwardKinematicsApprox</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
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<span class="org-comment">% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using</span>
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<span class="org-comment">% the Jacobian Matrix</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Inputs:</span>
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<span class="org-comment">% - stewart - A structure with the following fields</span>
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<span class="org-comment">% - J [6x6] - The Jacobian Matrix</span>
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<span class="org-comment">% - args - Can have the following fields:</span>
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<span class="org-comment">% - dL [6x1] - Displacement of each strut [m]</span>
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<span class="org-comment">%</span>
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<span class="org-comment">% Outputs:</span>
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<span class="org-comment">% - P [3x1] - The estimated position of {B} with respect to {A}</span>
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<span class="org-comment">% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}</span>
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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-orgf6a32e1" class="outline-4">
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<h4 id="orgf6a32e1"><span class="section-number-4">3.5.2</span> Optional Parameters</h4>
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<div class="outline-text-4" id="text-3-5-2">
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<div class="org-src-container">
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<pre class="src src-matlab">arguments
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stewart
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args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
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<span class="org-keyword">end</span>
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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-orgce0b559" class="outline-4">
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<h4 id="orgce0b559"><span class="section-number-4">3.5.3</span> Computation</h4>
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<div class="outline-text-4" id="text-3-5-3">
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<p>
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From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the
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position and orientation of {B} with respect to {A} using the following formula:
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\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">X = stewart.J<span class="org-type">\</span>args.dL;
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</pre>
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</div>
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<p>
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The position vector corresponds to the first 3 elements.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">P = X(1<span class="org-type">:</span>3);
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</pre>
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</div>
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<p>
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The next 3 elements are the orientation of {B} with respect to {A} expressed
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using the screw axis.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">theta = norm(X(4<span class="org-type">:</span>6));
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s = X(4<span class="org-type">:</span>6)<span class="org-type">/</span>theta;
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</pre>
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</div>
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<p>
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We then compute the corresponding rotation matrix.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">R = [s(1)<span class="org-type">^</span>2<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">+</span> cos(theta) , s(1)<span class="org-type">*</span>s(2)<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">-</span> s(3)<span class="org-type">*</span>sin(theta), s(1)<span class="org-type">*</span>s(3)<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">+</span> s(2)<span class="org-type">*</span>sin(theta);
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s<span class="org-type">(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);</span>
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s<span class="org-type">(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];</span>
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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>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-01-28 mar. 17:38</p>
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</div>
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</body>
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</html>
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