1125 lines
53 KiB
HTML
1125 lines
53 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">Stewart Platform - Simscape Model</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org3273efc">1. Procedure</a></li>
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<li><a href="#org0cfaf56">2. Matlab Code</a>
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<ul>
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<li><a href="#org47e6dec">2.1. Simscape Model</a></li>
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<li><a href="#org80710ea">2.2. Test the functions</a></li>
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</ul>
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</li>
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<li><a href="#org6005875">3. <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</a>
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<ul>
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<li><a href="#orga3d2c9e">3.1. Function description</a></li>
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<li><a href="#org61a1e26">3.2. Documentation</a></li>
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<li><a href="#org713ea9a">3.3. Optional Parameters</a></li>
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<li><a href="#org88e3f87">3.4. Initialize the Stewart structure</a></li>
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<li><a href="#org4a2edf9">3.5. Compute the position of each frame</a></li>
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</ul>
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</li>
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<li><a href="#org8eb2e40">4. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
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<ul>
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<li><a href="#org04bea4c">4.1. Function description</a></li>
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<li><a href="#org924cbfe">4.2. Documentation</a></li>
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<li><a href="#org335de0c">4.3. Optional Parameters</a></li>
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<li><a href="#orgc1a1209">4.4. Position of the Cube</a></li>
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<li><a href="#orga1cfc06">4.5. Compute the pose</a></li>
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</ul>
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</li>
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<li><a href="#orgc137b22">5. <code>computeJointsPose</code>: Compute the Pose of the Joints</a>
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<ul>
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<li><a href="#orge4dc5d9">5.1. Function description</a></li>
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<li><a href="#org5e72282">5.2. Documentation</a></li>
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<li><a href="#orgafa4ad9">5.3. Compute the position of the Joints</a></li>
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<li><a href="#org50bd360">5.4. Compute the strut length and orientation</a></li>
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<li><a href="#org3d0d76f">5.5. Compute the orientation of the Joints</a></li>
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</ul>
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</li>
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<li><a href="#orgc7ae05f">6. <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</a>
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<ul>
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<li><a href="#org234504f">6.1. Function description</a></li>
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<li><a href="#org8edf5d0">6.2. Optional Parameters</a></li>
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<li><a href="#orgf9bb882">6.3. Add Stiffness and Damping properties of each strut</a></li>
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</ul>
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</li>
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<li><a href="#org61b514c">7. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
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<ul>
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<li><a href="#org2983146">7.1. Function description</a></li>
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<li><a href="#org996fbd8">7.2. Compute Jacobian Matrix</a></li>
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<li><a href="#org9e42bbe">7.3. Compute Stiffness Matrix</a></li>
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<li><a href="#org36e44cd">7.4. Compute Compliance Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org76be477">8. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
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<ul>
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<li><a href="#org16e3f5f">8.1. Function description</a></li>
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<li><a href="#org5ccd852">8.2. Optional Parameters</a></li>
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<li><a href="#org8896a32">8.3. Theory</a></li>
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<li><a href="#orgf5481aa">8.4. Compute</a></li>
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</ul>
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</li>
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<li><a href="#org1c1fd0d">9. <code>forwardKinematicsApprox</code>: Compute the Forward Kinematics</a>
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<ul>
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<li><a href="#orgcc0b9b5">9.1. Function description</a></li>
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<li><a href="#orgdbe04db">9.2. Optional Parameters</a></li>
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<li><a href="#orgc0a97e0">9.3. Computation</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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<p>
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|
Stewart platforms are generated in multiple steps.
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</p>
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<p>
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We define 4 important <b>frames</b>:
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</p>
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<ul class="org-ul">
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<li>\(\{F\}\): Frame fixed to the <b>Fixed</b> base and located at the center of its bottom surface.
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This is used to fix the Stewart platform to some support.</li>
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<li>\(\{M\}\): Frame fixed to the <b>Moving</b> platform and located at the center of its top surface.
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This is used to place things on top of the Stewart platform.</li>
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<li>\(\{A\}\): Frame fixed to the fixed base.
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It defined the center of rotation of the moving platform.</li>
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<li>\(\{B\}\): Frame fixed to the moving platform.
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The motion of the moving platforms and forces applied to it are defined with respect to this frame \(\{B\}\).</li>
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</ul>
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<p>
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Then, we define the <b>location of the spherical joints</b>:
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</p>
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<ul class="org-ul">
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<li>\(\bm{a}_{i}\) are the position of the spherical joints fixed to the fixed base</li>
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<li>\(\bm{b}_{i}\) are the position of the spherical joints fixed to the moving platform</li>
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</ul>
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<p>
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We define the <b>rest position</b> of the Stewart platform:
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</p>
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<ul class="org-ul">
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<li>For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.</li>
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<li>Thus, to define the rest position of the Stewart platform, we just have to defined its total height \(H\).
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\(H\) corresponds to the distance from the bottom of the fixed base to the top of the moving platform.</li>
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</ul>
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<p>
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From \(\bm{a}_{i}\) and \(\bm{b}_{i}\), we can determine the <b>length and orientation of each strut</b>:
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</p>
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<ul class="org-ul">
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<li>\(l_{i}\) is the length of the strut</li>
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<li>\({}^{A}\hat{\bm{s}}_{i}\) is the unit vector align with the strut</li>
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</ul>
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<p>
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The position of the Spherical joints can be computed using various methods:
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</p>
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<ul class="org-ul">
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<li>Cubic configuration</li>
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<li>Circular configuration</li>
|
|
<li>Arbitrary position</li>
|
|
<li>These methods should be easily scriptable and corresponds to specific functions that returns \({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\).
|
|
The input of these functions are the parameters corresponding to the wanted geometry.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
For Simscape, we need:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>The position and orientation of each spherical joint fixed to the fixed base: \({}^{F}\bm{a}_{i}\) and \({}^{F}\bm{R}_{a_{i}}\)</li>
|
|
<li>The position and orientation of each spherical joint fixed to the moving platform: \({}^{M}\bm{b}_{i}\) and \({}^{M}\bm{R}_{b_{i}}\)</li>
|
|
<li>The rest length of each strut: \(l_{i}\)</li>
|
|
<li>The stiffness and damping of each actuator: \(k_{i}\) and \(c_{i}\)</li>
|
|
<li>The position of the frame \(\{A\}\) with respect to the frame \(\{F\}\): \({}^{F}\bm{O}_{A}\)</li>
|
|
<li>The position of the frame \(\{B\}\) with respect to the frame \(\{M\}\): \({}^{M}\bm{O}_{B}\)</li>
|
|
</ul>
|
|
|
|
<div id="outline-container-org3273efc" class="outline-2">
|
|
<h2 id="org3273efc"><span class="section-number-2">1</span> Procedure</h2>
|
|
<div class="outline-text-2" id="text-1">
|
|
<p>
|
|
The procedure to define the Stewart platform is the following:
|
|
</p>
|
|
<ol class="org-ol">
|
|
<li>Define the initial position of frames {A}, {B}, {F} and {M}.
|
|
We do that using the <code>initializeFramesPositions</code> function.
|
|
We have to specify the total height of the Stewart platform \(H\) and the position \({}^{M}O_{B}\) of {B} with respect to {M}.</li>
|
|
<li>Compute the positions of joints \({}^{F}a_{i}\) and \({}^{M}b_{i}\).
|
|
We can do that using various methods depending on the wanted architecture:
|
|
<ul class="org-ul">
|
|
<li><code>generateCubicConfiguration</code> permits to generate a cubic configuration</li>
|
|
</ul></li>
|
|
<li>Compute the position and orientation of the joints with respect to the fixed base and the moving platform.
|
|
This is done with the <code>computeJointsPose</code> function.</li>
|
|
<li>Define the dynamical properties of the Stewart platform.
|
|
The output are the stiffness and damping of each strut \(k_{i}\) and \(c_{i}\).
|
|
This can be done we simply choosing directly the stiffness and damping of each strut.
|
|
The stiffness and damping of each actuator can also be determine from the wanted stiffness of the Stewart platform for instance.</li>
|
|
<li>Define the mass and inertia of each element of the Stewart platform.</li>
|
|
</ol>
|
|
|
|
<p>
|
|
By following this procedure, we obtain a Matlab structure <code>stewart</code> that contains all the information for the Simscape model and for further analysis.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org0cfaf56" class="outline-2">
|
|
<h2 id="org0cfaf56"><span class="section-number-2">2</span> Matlab Code</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
</div>
|
|
<div id="outline-container-org47e6dec" class="outline-3">
|
|
<h3 id="org47e6dec"><span class="section-number-3">2.1</span> Simscape Model</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform.slx'</span>)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org80710ea" class="outline-3">
|
|
<h3 id="org80710ea"><span class="section-number-3">2.2</span> Test the functions</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeFramesPositions(<span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
|
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, 60e<span class="org-type">-</span>3, <span class="org-string">'FOc'</span>, 45e<span class="org-type">-</span>3, <span class="org-string">'FHa'</span>, 5e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 5e<span class="org-type">-</span>3);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones(6,1));
|
|
stewart = computeJacobian(stewart);
|
|
[Li, dLi] = inverseKinematics(stewart, <span class="org-string">'AP'</span>, [0;0;0.00001], <span class="org-string">'ARB'</span>, eye(3));
|
|
[P, R] = forwardKinematicsApprox(stewart, <span class="org-string">'dL'</span>, dLi)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6005875" class="outline-2">
|
|
<h2 id="org6005875"><span class="section-number-2">3</span> <code>initializeFramesPositions</code>: Initialize the positions of frames {A}, {B}, {F} and {M}</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org1dfe172"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/initializeFramesPositions.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orga3d2c9e" class="outline-3">
|
|
<h3 id="orga3d2c9e"><span class="section-number-3">3.1</span> Function description</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">[stewart]</span> = <span class="org-function-name">initializeFramesPositions</span>(<span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = initializeFramesPositions(args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
|
<span class="org-comment">% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]</span>
|
|
<span class="org-comment">% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - A structure with the following fields:</span>
|
|
<span class="org-comment">% - H [1x1] - Total Height of the Stewart Platform [m]</span>
|
|
<span class="org-comment">% - FO_M [3x1] - Position of {M} with respect to {F} [m]</span>
|
|
<span class="org-comment">% - MO_B [3x1] - Position of {B} with respect to {M} [m]</span>
|
|
<span class="org-comment">% - FO_A [3x1] - Position of {A} with respect to {F} [m]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org61a1e26" class="outline-3">
|
|
<h3 id="org61a1e26"><span class="section-number-3">3.2</span> Documentation</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
|
|
<div id="org99358c5" class="figure">
|
|
<p><img src="figs/stewart-frames-position.png" alt="stewart-frames-position.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Definition of the position of the frames</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org713ea9a" class="outline-3">
|
|
<h3 id="org713ea9a"><span class="section-number-3">3.3</span> Optional Parameters</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e<span class="org-type">-</span>3
|
|
args.MO_B (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org88e3f87" class="outline-3">
|
|
<h3 id="org88e3f87"><span class="section-number-3">3.4</span> Initialize the Stewart structure</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = struct();
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4a2edf9" class="outline-3">
|
|
<h3 id="org4a2edf9"><span class="section-number-3">3.5</span> Compute the position of each frame</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.H = args.H; <span class="org-comment">% Total Height of the Stewart Platform [m]</span>
|
|
|
|
stewart.FO_M = [0; 0; stewart.H]; <span class="org-comment">% Position of {M} with respect to {F} [m]</span>
|
|
|
|
stewart.MO_B = [0; 0; args.MO_B]; <span class="org-comment">% Position of {B} with respect to {M} [m]</span>
|
|
|
|
stewart.FO_A = stewart.MO_B <span class="org-type">+</span> stewart.FO_M; <span class="org-comment">% Position of {A} with respect to {F} [m]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8eb2e40" class="outline-2">
|
|
<h2 id="org8eb2e40"><span class="section-number-2">4</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org7944b80"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/generateCubicConfiguration.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org04bea4c" class="outline-3">
|
|
<h3 id="org04bea4c"><span class="section-number-3">4.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<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">generateCubicConfiguration</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% generateCubicConfiguration - Generate a Cubic Configuration</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = generateCubicConfiguration(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">% - H [1x1] - Total height of the platform [m]</span>
|
|
<span class="org-comment">% - args - Can have the following fields:</span>
|
|
<span class="org-comment">% - Hc [1x1] - Height of the "useful" part of the cube [m]</span>
|
|
<span class="org-comment">% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]</span>
|
|
<span class="org-comment">% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]</span>
|
|
<span class="org-comment">% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
|
<span class="org-comment">% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org924cbfe" class="outline-3">
|
|
<h3 id="org924cbfe"><span class="section-number-3">4.2</span> Documentation</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
|
|
<div id="org9b7988f" class="figure">
|
|
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Cubic Configuration</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org335de0c" class="outline-3">
|
|
<h3 id="org335de0c"><span class="section-number-3">4.3</span> Optional Parameters</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e<span class="org-type">-</span>3
|
|
args.FOc (1,1) double {mustBeNumeric} = 50e<span class="org-type">-</span>3
|
|
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e<span class="org-type">-</span>3
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc1a1209" class="outline-3">
|
|
<h3 id="orgc1a1209"><span class="section-number-3">4.4</span> Position of the Cube</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<p>
|
|
We define the useful points of the cube with respect to the Cube’s center.
|
|
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
|
|
located at the center of the cube and aligned with {F} and {M}.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">sx = [ 2; <span class="org-type">-</span>1; <span class="org-type">-</span>1];
|
|
sy = [ 0; 1; <span class="org-type">-</span>1];
|
|
sz = [ 1; 1; 1];
|
|
|
|
R = [sx, sy, sz]<span class="org-type">./</span>vecnorm([sx, sy, sz]);
|
|
|
|
L = args.Hc<span class="org-type">*</span>sqrt(3);
|
|
|
|
Cc = R<span class="org-type">'*</span>[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] <span class="org-type">-</span> [0;0;1.5<span class="org-type">*</span>args.Hc];
|
|
|
|
CCf = [Cc(<span class="org-type">:</span>,1), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,3), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,5), Cc(<span class="org-type">:</span>,1)]; <span class="org-comment">% CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg</span>
|
|
CCm = [Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,2), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,4), Cc(<span class="org-type">:</span>,6), Cc(<span class="org-type">:</span>,6)]; <span class="org-comment">% CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga1cfc06" class="outline-3">
|
|
<h3 id="orga1cfc06"><span class="section-number-3">4.5</span> Compute the pose</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
<p>
|
|
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">CSi = (CCm <span class="org-type">-</span> CCf)<span class="org-type">./</span>vecnorm(CCm <span class="org-type">-</span> CCf);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Fa = CCf <span class="org-type">+</span> [0; 0; args.FOc] <span class="org-type">+</span> ((args.FHa<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
|
stewart.Mb = CCf <span class="org-type">+</span> [0; 0; args.FOc<span class="org-type">-</span>stewart.H] <span class="org-type">+</span> ((stewart.H<span class="org-type">-</span>args.MHb<span class="org-type">-</span>(args.FOc<span class="org-type">-</span>args.Hc<span class="org-type">/</span>2))<span class="org-type">./</span>CSi(3,<span class="org-type">:</span>))<span class="org-type">.*</span>CSi;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc137b22" class="outline-2">
|
|
<h2 id="orgc137b22"><span class="section-number-2">5</span> <code>computeJointsPose</code>: Compute the Pose of the Joints</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org566dd7e"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/computeJointsPose.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orge4dc5d9" class="outline-3">
|
|
<h3 id="orge4dc5d9"><span class="section-number-3">5.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<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">computeJointsPose</span>(<span class="org-variable-name">stewart</span>)
|
|
<span class="org-comment">% computeJointsPose -</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = computeJointsPose(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">% - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}</span>
|
|
<span class="org-comment">% - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}</span>
|
|
<span class="org-comment">% - FO_A [3x1] - Position of {A} with respect to {F}</span>
|
|
<span class="org-comment">% - MO_B [3x1] - Position of {B} with respect to {M}</span>
|
|
<span class="org-comment">% - FO_M [3x1] - Position of {M} with respect to {F}</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - A structure with the following added fields</span>
|
|
<span class="org-comment">% - Aa [3x6] - The i'th column is the position of ai with respect to {A}</span>
|
|
<span class="org-comment">% - Ab [3x6] - The i'th column is the position of bi with respect to {A}</span>
|
|
<span class="org-comment">% - Ba [3x6] - The i'th column is the position of ai with respect to {B}</span>
|
|
<span class="org-comment">% - Bb [3x6] - The i'th column is the position of bi with respect to {B}</span>
|
|
<span class="org-comment">% - l [6x1] - The i'th element is the initial length of strut i</span>
|
|
<span class="org-comment">% - As [3x6] - The i'th column is the unit vector of strut i expressed in {A}</span>
|
|
<span class="org-comment">% - Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}</span>
|
|
<span class="org-comment">% - FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}</span>
|
|
<span class="org-comment">% - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5e72282" class="outline-3">
|
|
<h3 id="org5e72282"><span class="section-number-3">5.2</span> Documentation</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
|
|
<div id="org8d57a7b" class="figure">
|
|
<p><img src="figs/stewart-struts.png" alt="stewart-struts.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Position and orientation of the struts</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgafa4ad9" class="outline-3">
|
|
<h3 id="orgafa4ad9"><span class="section-number-3">5.3</span> Compute the position of the Joints</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Aa = stewart.Fa <span class="org-type">-</span> repmat(stewart.FO_A, [1, 6]);
|
|
stewart.Bb = stewart.Mb <span class="org-type">-</span> repmat(stewart.MO_B, [1, 6]);
|
|
|
|
stewart.Ab = stewart.Bb <span class="org-type">-</span> repmat(<span class="org-type">-</span>stewart.MO_B<span class="org-type">-</span>stewart.FO_M<span class="org-type">+</span>stewart.FO_A, [1, 6]);
|
|
stewart.Ba = stewart.Aa <span class="org-type">-</span> repmat( stewart.MO_B<span class="org-type">+</span>stewart.FO_M<span class="org-type">-</span>stewart.FO_A, [1, 6]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org50bd360" class="outline-3">
|
|
<h3 id="org50bd360"><span class="section-number-3">5.4</span> Compute the strut length and orientation</h3>
|
|
<div class="outline-text-3" id="text-5-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.As = (stewart.Ab <span class="org-type">-</span> stewart.Aa)<span class="org-type">./</span>vecnorm(stewart.Ab <span class="org-type">-</span> stewart.Aa); <span class="org-comment">% As_i is the i'th vector of As</span>
|
|
|
|
stewart.l = vecnorm(stewart.Ab <span class="org-type">-</span> stewart.Aa)<span class="org-type">'</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Bs = (stewart.Bb <span class="org-type">-</span> stewart.Ba)<span class="org-type">./</span>vecnorm(stewart.Bb <span class="org-type">-</span> stewart.Ba);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3d0d76f" class="outline-3">
|
|
<h3 id="org3d0d76f"><span class="section-number-3">5.5</span> Compute the orientation of the Joints</h3>
|
|
<div class="outline-text-3" id="text-5-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.FRa = zeros(3,3,6);
|
|
stewart.MRb = zeros(3,3,6);
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:6</span>
|
|
stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = [cross([0;1;0], stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>)) , cross(stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>), cross([0;1;0], stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>))) , stewart.As(<span class="org-type">:</span>,<span class="org-constant">i</span>)];
|
|
stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>)<span class="org-type">./</span>vecnorm(stewart.FRa(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>));
|
|
|
|
stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = [cross([0;1;0], stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>)) , cross(stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>), cross([0;1;0], stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>))) , stewart.Bs(<span class="org-type">:</span>,<span class="org-constant">i</span>)];
|
|
stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>) = stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>)<span class="org-type">./</span>vecnorm(stewart.MRb(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>));
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc7ae05f" class="outline-2">
|
|
<h2 id="orgc7ae05f"><span class="section-number-2">6</span> <code>initializeStrutDynamics</code>: Add Stiffness and Damping properties of each strut</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="orge0b89e3"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/initializeStrutDynamics.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org234504f" class="outline-3">
|
|
<h3 id="org234504f"><span class="section-number-3">6.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
<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">initializeStrutDynamics</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
|
|
<span class="org-comment">% initializeStrutDynamics - Add Stiffness and Damping properties of each strut</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Syntax: [stewart] = initializeStrutDynamics(args)</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Inputs:</span>
|
|
<span class="org-comment">% - args - Structure with the following fields:</span>
|
|
<span class="org-comment">% - Ki [6x1] - Stiffness of each strut [N/m]</span>
|
|
<span class="org-comment">% - Ci [6x1] - Damping of each strut [N/(m/s)]</span>
|
|
<span class="org-comment">%</span>
|
|
<span class="org-comment">% Outputs:</span>
|
|
<span class="org-comment">% - stewart - updated Stewart structure with the added fields:</span>
|
|
<span class="org-comment">% - Ki [6x1] - Stiffness of each strut [N/m]</span>
|
|
<span class="org-comment">% - Ci [6x1] - Damping of each strut [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8edf5d0" class="outline-3">
|
|
<h3 id="org8edf5d0"><span class="section-number-3">6.2</span> Optional Parameters</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">arguments
|
|
stewart
|
|
args.Ki (6,1) double {mustBeNumeric, mustBePositive} = 1e6<span class="org-type">*</span>ones(6,1)
|
|
args.Ci (6,1) double {mustBeNumeric, mustBePositive} = 1e2<span class="org-type">*</span>ones(6,1)
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf9bb882" class="outline-3">
|
|
<h3 id="orgf9bb882"><span class="section-number-3">6.3</span> Add Stiffness and Damping properties of each strut</h3>
|
|
<div class="outline-text-3" id="text-6-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.Ki = args.Ki;
|
|
stewart.Ci = args.Ci;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org61b514c" class="outline-2">
|
|
<h2 id="org61b514c"><span class="section-number-2">7</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
<p>
|
|
<a id="org27d5ba7"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org2983146" class="outline-3">
|
|
<h3 id="org2983146"><span class="section-number-3">7.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-7-1">
|
|
<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-org996fbd8" class="outline-3">
|
|
<h3 id="org996fbd8"><span class="section-number-3">7.2</span> Compute Jacobian Matrix</h3>
|
|
<div class="outline-text-3" id="text-7-2">
|
|
<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-org9e42bbe" class="outline-3">
|
|
<h3 id="org9e42bbe"><span class="section-number-3">7.3</span> Compute Stiffness Matrix</h3>
|
|
<div class="outline-text-3" id="text-7-3">
|
|
<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-org36e44cd" class="outline-3">
|
|
<h3 id="org36e44cd"><span class="section-number-3">7.4</span> Compute Compliance Matrix</h3>
|
|
<div class="outline-text-3" id="text-7-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart.C = inv(stewart.K);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org76be477" class="outline-2">
|
|
<h2 id="org76be477"><span class="section-number-2">8</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h2>
|
|
<div class="outline-text-2" id="text-8">
|
|
<p>
|
|
<a id="orgb8a3565"></a>
|
|
</p>
|
|
|
|
<p>
|
|
This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org16e3f5f" class="outline-3">
|
|
<h3 id="org16e3f5f"><span class="section-number-3">8.1</span> Function description</h3>
|
|
<div class="outline-text-3" id="text-8-1">
|
|
<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-org5ccd852" class="outline-3">
|
|
<h3 id="org5ccd852"><span class="section-number-3">8.2</span> Optional Parameters</h3>
|
|
<div class="outline-text-3" id="text-8-2">
|
|
<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-org8896a32" class="outline-3">
|
|
<h3 id="org8896a32"><span class="section-number-3">8.3</span> Theory</h3>
|
|
<div class="outline-text-3" id="text-8-3">
|
|
<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-orgf5481aa" class="outline-3">
|
|
<h3 id="orgf5481aa"><span class="section-number-3">8.4</span> Compute</h3>
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<div class="outline-text-3" id="text-8-4">
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<div class="org-src-container">
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<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));
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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">dLi = Li<span class="org-type">-</span>stewart.l;
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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 id="outline-container-org1c1fd0d" class="outline-2">
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<h2 id="org1c1fd0d"><span class="section-number-2">9</span> <code>forwardKinematicsApprox</code>: Compute the Forward Kinematics</h2>
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<div class="outline-text-2" id="text-9">
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<p>
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<a id="org55e7dad"></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-orgcc0b9b5" class="outline-3">
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<h3 id="orgcc0b9b5"><span class="section-number-3">9.1</span> Function description</h3>
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<div class="outline-text-3" id="text-9-1">
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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-orgdbe04db" class="outline-3">
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<h3 id="orgdbe04db"><span class="section-number-3">9.2</span> Optional Parameters</h3>
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<div class="outline-text-3" id="text-9-2">
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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-orgc0a97e0" class="outline-3">
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<h3 id="orgc0a97e0"><span class="section-number-3">9.3</span> Computation</h3>
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<div class="outline-text-3" id="text-9-3">
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|
<p>
|
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From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the
|
|
position and orientation of {B} with respect to {A} using the following formula:
|
|
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
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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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|
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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>
|
|
|
|
<p>
|
|
The next 3 elements are the orientation of {B} with respect to {A} expressed
|
|
using the screw axis.
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|
</p>
|
|
<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>
|
|
</pre>
|
|
</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: 2020-01-06 lun. 18:16</p>
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</div>
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</body>
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</html>
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