Estimate Stewart actuator stroke / Stewart maneuverability relationship
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<!-- 2019-03-22 ven. 12:03 -->
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<!-- 2019-03-26 mar. 09:24 -->
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<title>Kinematic Study of the Stewart Platform</title>
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/*
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@licstart The following is the entire license notice for the
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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="#orgc1c40d5">1. Functions</a>
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<li><a href="#org2e1bd58">1. Needed Actuator Stroke</a>
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<ul>
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<li><a href="#org3d6cf9e">1.1. getMaxPositions</a></li>
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<li><a href="#orge3ee3ac">1.2. getMaxPureDisplacement</a></li>
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<li><a href="#org16d1370">1.1. Stewart architecture definition</a></li>
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<li><a href="#orgaf07b82">1.2. Wanted translations and rotations</a></li>
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<li><a href="#org920b62b">1.3. Needed stroke for "pure" rotations or translations</a></li>
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<li><a href="#org27bf97e">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="#orgaebf111">2. Maximum Stroke</a></li>
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<li><a href="#orgfb8a1e7">3. Functions</a>
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<ul>
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<li><a href="#org465746a">3.1. getMaxPositions</a></li>
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<li><a href="#org527f7ca">3.2. getMaxPureDisplacement</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orgc1c40d5" class="outline-2">
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<h2 id="orgc1c40d5"><span class="section-number-2">1</span> Functions</h2>
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<div id="outline-container-org2e1bd58" class="outline-2">
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<h2 id="org2e1bd58"><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-org3d6cf9e" class="outline-3">
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<h3 id="org3d6cf9e"><span class="section-number-3">1.1</span> getMaxPositions</h3>
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<div id="outline-container-org16d1370" class="outline-3">
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<h3 id="org16d1370"><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<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
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<span style="color: #CC9393;">'H_tot'</span>, <span style="color: #BFEBBF;">90</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Total height of the Hexapod [mm]</span>
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<span style="color: #CC9393;">'L'</span>, <span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">/</span>sqrt<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #BFEBBF;">)</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Size of the Cube [mm]</span>
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<span style="color: #CC9393;">'H'</span>, <span style="color: #BFEBBF;">60</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between base joints and platform joints [mm]</span>
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<span style="color: #CC9393;">'H0'</span>, <span style="color: #BFEBBF;">180</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">60</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Height between the corner of the cube and the plane containing the base joints [mm]</span>
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<span style="color: #DCDCCC;">)</span>;
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stewart = initializeCubicConfiguration<span style="color: #DCDCCC;">(</span>opts<span style="color: #DCDCCC;">)</span>;
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opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
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<span style="color: #CC9393;">'Jd_pos'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">100</span><span style="color: #BFEBBF;">]</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]</span>
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<span style="color: #CC9393;">'Jf_pos'</span>, <span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">50</span><span style="color: #BFEBBF;">]</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Position of the Jacobian for force location from the top of the mobile platform [mm]</span>
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<span style="color: #DCDCCC;">)</span>;
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stewart = computeGeometricalProperties<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
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opts = struct<span style="color: #DCDCCC;">(</span><span style="text-decoration: underline;">...</span>
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<span style="color: #CC9393;">'stroke'</span>, <span style="color: #BFEBBF;">50e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Maximum stroke of each actuator [m]</span>
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<span style="color: #DCDCCC;">)</span>;
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stewart = initializeMechanicalElements<span style="color: #DCDCCC;">(</span>stewart, opts<span style="color: #DCDCCC;">)</span>;
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save<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/stewart.mat', 'stewart'</span><span style="color: #DCDCCC;">)</span>;
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</pre>
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</div>
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<div id="outline-container-orgaf07b82" class="outline-3">
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<h3 id="orgaf07b82"><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 = <span style="color: #BFEBBF;">15e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Translation [m]</span>
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Ty_max = <span style="color: #BFEBBF;">15e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Translation [m]</span>
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Tz_max = <span style="color: #BFEBBF;">15e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Translation [m]</span>
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Rx_max = <span style="color: #BFEBBF;">30e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% Rotation [rad]</span>
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Ry_max = <span style="color: #BFEBBF;">30e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% 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-org920b62b" class="outline-3">
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<h3 id="org920b62b"><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">
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<pre class="src src-matlab">LTx = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span>Tx_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
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LTy = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> Ty_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
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LTz = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> Tz_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
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LRx = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> Rx_max <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
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LRy = stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> <span style="color: #BFEBBF;">0</span> Ry_max <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>';
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</pre>
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</div>
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<pre class="example">
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1.0607e-05
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</pre>
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</div>
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</div>
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<div id="outline-container-org27bf97e" class="outline-3">
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<h3 id="org27bf97e"><span class="section-number-3">1.4</span> Needed stroke for combined translations and rotations</h3>
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<div class="outline-text-3" id="text-1-4">
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<p>
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Now, we combine translations and rotations, and we try to find the worst case (that we suppose to happen at the border).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Lmax = <span style="color: #BFEBBF;">0</span>;
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pos = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">]</span>;
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<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Tx</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Tx_max</span>,Tx_max<span style="color: #DCDCCC;">]</span>
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<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Ty</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Ty_max</span>,Ty_max<span style="color: #DCDCCC;">]</span>
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<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Tz</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Tz_max</span>,Tz_max<span style="color: #DCDCCC;">]</span>
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<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Rx</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Rx_max</span>,Rx_max<span style="color: #DCDCCC;">]</span>
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<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">Ry</span> = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">-Ry_max</span>,Ry_max<span style="color: #DCDCCC;">]</span>
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L = max<span style="color: #DCDCCC;">(</span>stewart.Jd<span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">[</span>Tx Ty Tz Rx Ry <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
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<span style="color: #F0DFAF; font-weight: bold;">if</span> L <span style="color: #7CB8BB;">></span> Lmax
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Lmax = L;
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pos = <span style="color: #DCDCCC;">[</span>Tx Ty Tz Rx Ry<span style="color: #DCDCCC;">]</span>;
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<span style="color: #F0DFAF; font-weight: bold;">end</span>
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<span style="color: #F0DFAF; font-weight: bold;">end</span>
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<span style="color: #F0DFAF; font-weight: bold;">end</span>
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<span style="color: #F0DFAF; font-weight: bold;">end</span>
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<span style="color: #F0DFAF; font-weight: bold;">end</span>
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<span style="color: #F0DFAF; font-weight: bold;">end</span>
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</pre>
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</div>
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<p>
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We obtain a needed stroke shown below (almost two times the needed stroke for "pure" rotations and translations).
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</p>
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<pre class="example">
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3.0927e-05
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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-orgaebf111" class="outline-2">
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<h2 id="orgaebf111"><span class="section-number-2">2</span> Maximum Stroke</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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From a specified actuator stroke, we try to estimate the available maneuverability of the Stewart platform.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span style="color: #DCDCCC;">[</span>X, Y, Z<span style="color: #DCDCCC;">]</span> = getMaxPositions<span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>;
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
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plot3<span style="color: #DCDCCC;">(</span>X, Y, Z, <span style="color: #CC9393;">'k-'</span><span style="color: #DCDCCC;">)</span>
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</pre>
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</div>
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<div id="outline-container-orgfb8a1e7" class="outline-2">
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<h2 id="orgfb8a1e7"><span class="section-number-2">3</span> Functions</h2>
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<div class="outline-text-2" id="text-3">
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</div>
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<div id="outline-container-org465746a" class="outline-3">
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<h3 id="org465746a"><span class="section-number-3">3.1</span> getMaxPositions</h3>
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<div class="outline-text-3" id="text-3-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">X, Y, Z</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPositions</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">stewart</span><span style="color: #DCDCCC;">)</span>
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Leg = stewart.Leg;
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J = stewart.J;
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J = stewart.Jd;
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theta = linspace<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
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phi = linspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> , <span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
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dmax = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>theta<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>phi<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
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@@ -288,9 +437,9 @@ for the JavaScript code in this tag.
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<div id="outline-container-orge3ee3ac" class="outline-3">
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<h3 id="orge3ee3ac"><span class="section-number-3">1.2</span> getMaxPureDisplacement</h3>
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<div class="outline-text-3" id="text-1-2">
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<div id="outline-container-org527f7ca" class="outline-3">
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<h3 id="org527f7ca"><span class="section-number-3">3.2</span> getMaxPureDisplacement</h3>
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<div class="outline-text-3" id="text-3-2">
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<div class="org-src-container">
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<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">max_disp</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getMaxPureDisplacement</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">Leg</span>, <span style="color: #DFAF8F;">J</span><span style="color: #DCDCCC;">)</span>
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max_disp = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">6</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Thomas Dehaeze</p>
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<p class="date">Created: 2019-03-22 ven. 12:03</p>
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<p class="date">Created: 2019-03-26 mar. 09:24</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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Reference in New Issue
Block a user