759 lines
32 KiB
HTML
759 lines
32 KiB
HTML
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<div id="org-div-home-and-up">
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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="#orge42fba6">1. Jacobian Analysis</a>
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<ul>
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<li><a href="#org8938d37">1.1. Jacobian Computation</a></li>
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<li><a href="#orgbcdebed">1.2. Velocity of the struts to the velocity of the mobile platform</a></li>
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<li><a href="#org9482fe8">1.3. Displacement of the struts to the displacement of the mobile platform</a></li>
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<li><a href="#orgc7c6052">1.4. Force Transformation</a></li>
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</ul>
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</li>
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<li><a href="#orgda7fde9">2. Forward and Inverse Kinematics</a>
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<ul>
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<li><a href="#org26477b8">2.1. Inverse Kinematics</a></li>
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<li><a href="#org01066c5">2.2. Forward Kinematics</a></li>
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<li><a href="#org37b3180">2.3. Approximate Forward Kinematics</a></li>
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</ul>
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</li>
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<li><a href="#org5304e0f">3. Stiffness Analysis</a>
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<ul>
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<li><a href="#orgdc3ef4e">3.1. Computation of the Stiffness and Compliance Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org03cb27a">4. Estimated required actuator stroke for specified platform mobility</a>
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<ul>
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<li><a href="#orgfa74621">4.1. Needed Actuator Stroke</a>
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<ul>
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<li><a href="#org79abcb3">4.1.1. Stewart architecture definition</a></li>
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<li><a href="#org5bf59b4">4.1.2. Wanted translations and rotations</a></li>
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<li><a href="#org1dce5e1">4.1.3. Needed stroke for “pure” rotations or translations</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="#org7319607">5. Estimated platform mobility from specified actuator stroke</a></li>
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<li><a href="#org951a228">6. Functions</a>
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<ul>
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<li><a href="#org2387af5">6.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
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<ul>
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<li><a href="#org0734fbe">6.1.1. Function description</a></li>
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<li><a href="#orge2bf995">6.1.2. Compute Jacobian Matrix</a></li>
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<li><a href="#orgc3abc35">6.1.3. Compute Stiffness Matrix</a></li>
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<li><a href="#org5be4e51">6.1.4. Compute Compliance Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org2510ad8">6.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
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<ul>
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<li><a href="#org15abed6">6.2.1. Function description</a></li>
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<li><a href="#orgfa724fa">6.2.2. Optional Parameters</a></li>
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<li><a href="#org9510865">6.2.3. Theory</a></li>
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<li><a href="#orgad46e51">6.2.4. Compute</a></li>
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</ul>
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</li>
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<li><a href="#orgc42ae4c">6.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
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<ul>
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<li><a href="#orgba5a90f">6.3.1. Function description</a></li>
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<li><a href="#org0a3069a">6.3.2. Optional Parameters</a></li>
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<li><a href="#orgf878173">6.3.3. 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-orge42fba6" class="outline-2">
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<h2 id="orge42fba6"><span class="section-number-2">1</span> Jacobian Analysis</h2>
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<h3 id="org8938d37"><span class="section-number-3">1.1</span> Jacobian Computation</h3>
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</div>
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<div id="outline-container-orgbcdebed" class="outline-3">
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<h3 id="orgbcdebed"><span class="section-number-3">1.2</span> Velocity of the struts to the velocity of the mobile platform</h3>
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</div>
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<div id="outline-container-org9482fe8" class="outline-3">
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<h3 id="org9482fe8"><span class="section-number-3">1.3</span> Displacement of the struts to the displacement of the mobile platform</h3>
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</div>
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<div id="outline-container-orgc7c6052" class="outline-3">
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<h3 id="orgc7c6052"><span class="section-number-3">1.4</span> Force Transformation</h3>
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</div>
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<div id="outline-container-orgda7fde9" class="outline-2">
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<h2 id="orgda7fde9"><span class="section-number-2">2</span> Forward and Inverse Kinematics</h2>
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<div class="outline-text-2" id="text-2">
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<div id="outline-container-org26477b8" class="outline-3">
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<h3 id="org26477b8"><span class="section-number-3">2.1</span> Inverse Kinematics</h3>
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</div>
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<h3 id="org01066c5"><span class="section-number-3">2.2</span> Forward Kinematics</h3>
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<h3 id="org37b3180"><span class="section-number-3">2.3</span> Approximate Forward Kinematics</h3>
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<h2 id="org5304e0f"><span class="section-number-2">3</span> Stiffness Analysis</h2>
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<h3 id="orgdc3ef4e"><span class="section-number-3">3.1</span> Computation of the Stiffness and Compliance Matrix</h3>
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<h2 id="org03cb27a"><span class="section-number-2">4</span> Estimated required actuator stroke for specified platform mobility</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
</div>
|
|
<div id="outline-container-orgfa74621" class="outline-3">
|
|
<h3 id="orgfa74621"><span class="section-number-3">4.1</span> Needed Actuator Stroke</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
The goal is to determine the needed stroke of the actuators to obtain wanted translations and rotations.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org79abcb3" class="outline-4">
|
|
<h4 id="org79abcb3"><span class="section-number-4">4.1.1</span> Stewart architecture definition</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
<p>
|
|
We use a cubic architecture.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">opts = struct(...
|
|
<span class="org-string">'H_tot'</span>, 90, ...<span class="org-comment"> % Total height of the Hexapod [mm]</span>
|
|
<span class="org-string">'L'</span>, 200<span class="org-type">/</span>sqrt(3), ...<span class="org-comment"> % Size of the Cube [mm]</span>
|
|
<span class="org-string">'H'</span>, 60, ...<span class="org-comment"> % Height between base joints and platform joints [mm]</span>
|
|
<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>
|
|
);
|
|
stewart = initializeCubicConfiguration(opts);
|
|
opts = struct(...
|
|
<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>
|
|
<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>
|
|
);
|
|
stewart = computeGeometricalProperties(stewart, opts);
|
|
opts = struct(...
|
|
<span class="org-string">'stroke'</span>, 50e<span class="org-type">-</span>6 ...<span class="org-comment"> % Maximum stroke of each actuator [m]</span>
|
|
);
|
|
stewart = initializeMechanicalElements(stewart, opts);
|
|
|
|
save(<span class="org-string">'./mat/stewart.mat'</span>, <span class="org-string">'stewart'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5bf59b4" class="outline-4">
|
|
<h4 id="org5bf59b4"><span class="section-number-4">4.1.2</span> Wanted translations and rotations</h4>
|
|
<div class="outline-text-4" id="text-4-1-2">
|
|
<p>
|
|
We define wanted translations and rotations
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Tx_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
|
|
Ty_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
|
|
Tz_max = 15e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
|
|
Rx_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
|
|
Ry_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1dce5e1" class="outline-4">
|
|
<h4 id="org1dce5e1"><span class="section-number-4">4.1.3</span> Needed stroke for “pure” rotations or translations</h4>
|
|
<div class="outline-text-4" id="text-4-1-3">
|
|
<p>
|
|
First, we estimate the needed actuator stroke for “pure” rotations and translation.
|
|
</p>
|
|
<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>
|
|
</div>
|
|
|
|
<div id="outline-container-org7319607" class="outline-2">
|
|
<h2 id="org7319607"><span class="section-number-2">5</span> Estimated platform mobility from specified actuator stroke</h2>
|
|
</div>
|
|
<div id="outline-container-org951a228" class="outline-2">
|
|
<h2 id="org951a228"><span class="section-number-2">6</span> Functions</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
</div>
|
|
<div id="outline-container-org2387af5" class="outline-3">
|
|
<h3 id="org2387af5"><span class="section-number-3">6.1</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
<p>
|
|
<a id="org2ea0e42"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org0734fbe" class="outline-4">
|
|
<h4 id="org0734fbe"><span class="section-number-4">6.1.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-6-1-1">
|
|
<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-orge2bf995" class="outline-4">
|
|
<h4 id="orge2bf995"><span class="section-number-4">6.1.2</span> Compute Jacobian Matrix</h4>
|
|
<div class="outline-text-4" id="text-6-1-2">
|
|
<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-orgc3abc35" class="outline-4">
|
|
<h4 id="orgc3abc35"><span class="section-number-4">6.1.3</span> Compute Stiffness Matrix</h4>
|
|
<div class="outline-text-4" id="text-6-1-3">
|
|
<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-org5be4e51" class="outline-4">
|
|
<h4 id="org5be4e51"><span class="section-number-4">6.1.4</span> Compute Compliance Matrix</h4>
|
|
<div class="outline-text-4" id="text-6-1-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.C = inv(stewart.K);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2510ad8" class="outline-3">
|
|
<h3 id="org2510ad8"><span class="section-number-3">6.2</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
<p>
|
|
<a id="orgd507362"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org15abed6" class="outline-4">
|
|
<h4 id="org15abed6"><span class="section-number-4">6.2.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-6-2-1">
|
|
<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-orgfa724fa" class="outline-4">
|
|
<h4 id="orgfa724fa"><span class="section-number-4">6.2.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-6-2-2">
|
|
<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-org9510865" class="outline-4">
|
|
<h4 id="org9510865"><span class="section-number-4">6.2.3</span> Theory</h4>
|
|
<div class="outline-text-4" id="text-6-2-3">
|
|
<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-orgad46e51" class="outline-4">
|
|
<h4 id="orgad46e51"><span class="section-number-4">6.2.4</span> Compute</h4>
|
|
<div class="outline-text-4" id="text-6-2-4">
|
|
<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-orgc42ae4c" class="outline-3">
|
|
<h3 id="orgc42ae4c"><span class="section-number-3">6.3</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3>
|
|
<div class="outline-text-3" id="text-6-3">
|
|
<p>
|
|
<a id="org6e7838d"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgba5a90f" class="outline-4">
|
|
<h4 id="orgba5a90f"><span class="section-number-4">6.3.1</span> Function description</h4>
|
|
<div class="outline-text-4" id="text-6-3-1">
|
|
<div class="org-src-container">
|
|
<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>)
|
|
<span class="org-comment">% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using</span>
|
|
<span class="org-comment">% the Jacobian Matrix</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)</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">% - J [6x6] - The Jacobian Matrix</span>
|
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
|
<span class="org-comment">% - dL [6x1] - Displacement of each strut [m]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - P [3x1] - The estimated position of {B} with respect to {A}</span>
|
|
<span class="org-comment">% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0a3069a" class="outline-4">
|
|
<h4 id="org0a3069a"><span class="section-number-4">6.3.2</span> Optional Parameters</h4>
|
|
<div class="outline-text-4" id="text-6-3-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf878173" class="outline-4">
|
|
<h4 id="orgf878173"><span class="section-number-4">6.3.3</span> Computation</h4>
|
|
<div class="outline-text-4" id="text-6-3-3">
|
|
<p>
|
|
From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the
|
|
position and orientation of {B} with respect to {A} using the following formula:
|
|
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
|
|
</p>
|
|
<div class="org-src-container">
|
|
<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: 2020-01-29 mer. 13:29</p>
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</div>
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</body>
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</html>
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