672 lines
29 KiB
HTML
672 lines
29 KiB
HTML
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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 - Dynamics Study</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="#orgc59e712">1. Compare external forces and forces applied by the actuators</a>
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<ul>
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<li><a href="#org4509b7d">1.1. Comparison with fixed support</a></li>
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<li><a href="#org8662186">1.2. Comparison with a flexible support</a></li>
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<li><a href="#orgb87f273">1.3. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org81ab204">2. Comparison of the static transfer function and the Compliance matrix</a>
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<ul>
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<li><a href="#orge7e7242">2.1. Analysis</a></li>
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<li><a href="#org230655f">2.2. Conclusion</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-orgc59e712" class="outline-2">
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<h2 id="orgc59e712"><span class="section-number-2">1</span> Compare external forces and forces applied by the actuators</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org4509b7d" class="outline-3">
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<h3 id="org4509b7d"><span class="section-number-3">1.1</span> Comparison with fixed support</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab">stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, <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);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'none'</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">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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</pre>
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</div>
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<p>
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Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
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options = linearizeOptions;
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options.SampleTime = 0;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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mdl = <span class="org-string">'stewart_platform_model'</span>;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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clear io; io_i = 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
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io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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G = linearize(mdl, io, options);
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G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
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G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</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">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
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Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
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</pre>
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</div>
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<p>
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Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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clear io; io_i = 1;
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io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
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io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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Gd = linearize(mdl, io, options);
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Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string">'Fey'</span>, <span class="org-string">'Fez'</span>, <span class="org-string">'Mex'</span>, <span class="org-string">'Mey'</span>, <span class="org-string">'Mez'</span>};
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Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</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-org8662186" class="outline-3">
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<h3 id="org8662186"><span class="section-number-3">1.2</span> Comparison with a flexible support</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
We redo the identification for when the Stewart platform is on a flexible support.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'flexible'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
|
|
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
Gd = linearize(mdl, io, options);
|
|
Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string">'Fey'</span>, <span class="org-string">'Fez'</span>, <span class="org-string">'Mex'</span>, <span class="org-string">'Mey'</span>, <span class="org-string">'Mez'</span>};
|
|
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb87f273" class="outline-3">
|
|
<h3 id="orgb87f273"><span class="section-number-3">1.3</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<div class="important">
|
|
<p>
|
|
The transfer function from forces/torques applied by the actuators on the payload \(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\) to the pose of the mobile platform \(\bm{\mathcal{X}}\) is the same as the transfer function from external forces/torques to \(\bm{\mathcal{X}}\) as long as the Stewart platform’s base is fixed.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org81ab204" class="outline-2">
|
|
<h2 id="org81ab204"><span class="section-number-2">2</span> Comparison of the static transfer function and the Compliance matrix</h2>
|
|
<div class="outline-text-2" id="text-2">
|
|
</div>
|
|
<div id="outline-container-orge7e7242" class="outline-3">
|
|
<h3 id="orge7e7242"><span class="section-number-3">2.1</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
<p>
|
|
Initialization of the Stewart platform.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
|
stewart = initializeFramesPositions(stewart, <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 = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'stewart_platform_model'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
|
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
|
|
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Let’s first look at the low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-7.2e-19</td>
|
|
<td class="org-right">5.0e-18</td>
|
|
<td class="org-right">-8.9e-18</td>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">9.9e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.7e-18</td>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-5.7e-18</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">-1.6e-17</td>
|
|
<td class="org-right">-1.7e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.3e-18</td>
|
|
<td class="org-right">-6.3e-18</td>
|
|
<td class="org-right">2.1e-08</td>
|
|
<td class="org-right">4.4e-17</td>
|
|
<td class="org-right">6.6e-18</td>
|
|
<td class="org-right">7.4e-18</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3.2e-17</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">6.2e-18</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-3.5e-16</td>
|
|
<td class="org-right">6.3e-17</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">2.7e-17</td>
|
|
<td class="org-right">4.8e-17</td>
|
|
<td class="org-right">-4.5e-16</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-1.2e-19</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.0e-17</td>
|
|
<td class="org-right">-9.5e-17</td>
|
|
<td class="org-right">8.4e-18</td>
|
|
<td class="org-right">4.3e-16</td>
|
|
<td class="org-right">5.8e-16</td>
|
|
<td class="org-right">1.7e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
And now at the Compliance matrix.
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">-2.0e-24</td>
|
|
<td class="org-right">7.4e-25</td>
|
|
<td class="org-right">5.9e-23</td>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">5.9e-24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-7.1e-25</td>
|
|
<td class="org-right">4.7e-08</td>
|
|
<td class="org-right">2.9e-25</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">-5.4e-24</td>
|
|
<td class="org-right">-3.3e-23</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.9e-26</td>
|
|
<td class="org-right">-6.4e-25</td>
|
|
<td class="org-right">2.1e-08</td>
|
|
<td class="org-right">1.9e-23</td>
|
|
<td class="org-right">5.3e-25</td>
|
|
<td class="org-right">-6.5e-40</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1.4e-23</td>
|
|
<td class="org-right">-3.2e-07</td>
|
|
<td class="org-right">1.3e-23</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">4.9e-22</td>
|
|
<td class="org-right">-3.8e-24</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.2e-07</td>
|
|
<td class="org-right">7.6e-24</td>
|
|
<td class="org-right">1.2e-23</td>
|
|
<td class="org-right">6.9e-22</td>
|
|
<td class="org-right">5.2e-06</td>
|
|
<td class="org-right">-2.6e-22</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.3e-24</td>
|
|
<td class="org-right">-3.2e-23</td>
|
|
<td class="org-right">-1.6e-39</td>
|
|
<td class="org-right">9.9e-23</td>
|
|
<td class="org-right">-3.3e-22</td>
|
|
<td class="org-right">1.7e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org230655f" class="outline-3">
|
|
<h3 id="org230655f"><span class="section-number-3">2.2</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<div class="important">
|
|
<p>
|
|
The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-02-13 jeu. 15:19</p>
|
|
</div>
|
|
</body>
|
|
</html>
|