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Transmissibility and Compliance comp IFF/DVF/OL

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<title>Stewart Platform - Decentralized Active Damping</title>
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<li><a href="#orgd59c804">1. Inertial Control</a>
<ul>
<li><a href="#org5f749c8">1.1. Identification of the Dynamics</a></li>
<li><a href="#org07e81b1">1.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#org53a0870">1.3. Obtained Damping</a></li>
<li><a href="#org51b20eb">1.4. Conclusion</a></li>
<li><a href="#orgd637197">1.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#orgd895eeb">1.3. Obtained Damping</a></li>
<li><a href="#orgeaf5ef8">1.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org74c7eb4">2. Integral Force Feedback</a>
<ul>
<li><a href="#org9e45139">2.1. Identification of the Dynamics with perfect Joints</a></li>
<li><a href="#org494bb35">2.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#org947ca92">2.3. Obtained Damping</a></li>
<li><a href="#orgade2418">2.4. Conclusion</a></li>
<li><a href="#orgcaa6199">2.1. Identification of the Dynamics with perfect Joints</a></li>
<li><a href="#org1910546">2.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#org9e1f2e2">2.3. Obtained Damping</a></li>
<li><a href="#org405813e">2.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org08917d6">3. Direct Velocity Feedback</a>
<ul>
<li><a href="#orgcaa6199">3.1. Identification of the Dynamics with perfect Joints</a></li>
<li><a href="#orgd637197">3.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#orgd895eeb">3.3. Obtained Damping</a></li>
<li><a href="#orgeaf5ef8">3.4. Conclusion</a></li>
<li><a href="#org7313778">3.1. Identification of the Dynamics with perfect Joints</a></li>
<li><a href="#org3014959">3.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#orga144352">3.3. Obtained Damping</a></li>
<li><a href="#org004b094">3.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org183f3f2">4. Compliance and Transmissibility Comparison</a>
<ul>
<li><a href="#org0ed1499">4.1. Initialization</a></li>
<li><a href="#orgcd64c04">4.2. Identification</a></li>
<li><a href="#orgd30c62d">4.3. Results</a></li>
</ul>
</li>
</ul>
@ -343,7 +328,7 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
</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>);
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
@ -380,8 +365,8 @@ The transfer function from actuator forces to force sensors is shown in Figure <
</div>
</div>
<div id="outline-container-org07e81b1" class="outline-3">
<h3 id="org07e81b1"><span class="section-number-3">1.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
<div id="outline-container-orgd637197" class="outline-3">
<h3 id="orgd637197"><span class="section-number-3">1.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
<div class="outline-text-3" id="text-1-2">
<p>
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
@ -417,8 +402,8 @@ The new dynamics from force actuator to force sensor is shown in Figure <a href=
</div>
</div>
<div id="outline-container-org53a0870" class="outline-3">
<h3 id="org53a0870"><span class="section-number-3">1.3</span> Obtained Damping</h3>
<div id="outline-container-orgd895eeb" class="outline-3">
<h3 id="orgd895eeb"><span class="section-number-3">1.3</span> Obtained Damping</h3>
<div class="outline-text-3" id="text-1-3">
<p>
The control is a performed in a decentralized manner.
@ -443,8 +428,8 @@ The root locus is shown in figure <a href="#org9af9e33">3</a>.
</div>
</div>
<div id="outline-container-org51b20eb" class="outline-3">
<h3 id="org51b20eb"><span class="section-number-3">1.4</span> Conclusion</h3>
<div id="outline-container-orgeaf5ef8" class="outline-3">
<h3 id="orgeaf5ef8"><span class="section-number-3">1.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-4">
<div class="important">
<p>
@ -475,8 +460,8 @@ To run the script, open the Simulink Project, and type <code>run active_damping_
</div>
</div>
<div id="outline-container-org9e45139" class="outline-3">
<h3 id="org9e45139"><span class="section-number-3">2.1</span> Identification of the Dynamics with perfect Joints</h3>
<div id="outline-container-orgcaa6199" class="outline-3">
<h3 id="orgcaa6199"><span class="section-number-3">2.1</span> Identification of the Dynamics with perfect Joints</h3>
<div class="outline-text-3" id="text-2-1">
<p>
We first initialize the Stewart platform without joint stiffness.
@ -497,11 +482,16 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
</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>);
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
<p>
And we identify the dynamics from force actuators to force sensors.
</p>
@ -537,8 +527,8 @@ The transfer function from actuator forces to force sensors is shown in Figure <
</div>
</div>
<div id="outline-container-org494bb35" class="outline-3">
<h3 id="org494bb35"><span class="section-number-3">2.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
<div id="outline-container-org1910546" class="outline-3">
<h3 id="org1910546"><span class="section-number-3">2.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
@ -574,8 +564,8 @@ The new dynamics from force actuator to force sensor is shown in Figure <a href=
</div>
</div>
<div id="outline-container-org947ca92" class="outline-3">
<h3 id="org947ca92"><span class="section-number-3">2.3</span> Obtained Damping</h3>
<div id="outline-container-org9e1f2e2" class="outline-3">
<h3 id="org9e1f2e2"><span class="section-number-3">2.3</span> Obtained Damping</h3>
<div class="outline-text-3" id="text-2-3">
<p>
The control is a performed in a decentralized manner.
@ -607,8 +597,8 @@ The root locus is shown in figure <a href="#orge21bbea">6</a> and the obtained p
</div>
</div>
<div id="outline-container-orgade2418" class="outline-3">
<h3 id="orgade2418"><span class="section-number-3">2.4</span> Conclusion</h3>
<div id="outline-container-org405813e" class="outline-3">
<h3 id="org405813e"><span class="section-number-3">2.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-4">
<div class="important">
<p>
@ -640,8 +630,8 @@ To run the script, open the Simulink Project, and type <code>run active_damping_
</div>
</div>
<div id="outline-container-orgcaa6199" class="outline-3">
<h3 id="orgcaa6199"><span class="section-number-3">3.1</span> Identification of the Dynamics with perfect Joints</h3>
<div id="outline-container-org7313778" class="outline-3">
<h3 id="org7313778"><span class="section-number-3">3.1</span> Identification of the Dynamics with perfect Joints</h3>
<div class="outline-text-3" id="text-3-1">
<p>
We first initialize the Stewart platform without joint stiffness.
@ -662,7 +652,7 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
</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>);
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
@ -703,8 +693,8 @@ The transfer function from actuator forces to relative motion sensors is shown i
</div>
<div id="outline-container-orgd637197" class="outline-3">
<h3 id="orgd637197"><span class="section-number-3">3.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
<div id="outline-container-org3014959" class="outline-3">
<h3 id="org3014959"><span class="section-number-3">3.2</span> Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</h3>
<div class="outline-text-3" id="text-3-2">
<p>
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
@ -740,8 +730,8 @@ The new dynamics from force actuator to relative motion sensor is shown in Figur
</div>
</div>
<div id="outline-container-orgd895eeb" class="outline-3">
<h3 id="orgd895eeb"><span class="section-number-3">3.3</span> Obtained Damping</h3>
<div id="outline-container-orga144352" class="outline-3">
<h3 id="orga144352"><span class="section-number-3">3.3</span> Obtained Damping</h3>
<div class="outline-text-3" id="text-3-3">
<p>
The control is a performed in a decentralized manner.
@ -766,14 +756,115 @@ The root locus is shown in figure <a href="#org277d60d">10</a>.
</div>
</div>
<div id="outline-container-orgeaf5ef8" class="outline-3">
<h3 id="orgeaf5ef8"><span class="section-number-3">3.4</span> Conclusion</h3>
<div id="outline-container-org004b094" class="outline-3">
<h3 id="org004b094"><span class="section-number-3">3.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-4">
<div class="important">
<p>
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org183f3f2" class="outline-2">
<h2 id="org183f3f2"><span class="section-number-2">4</span> Compliance and Transmissibility Comparison</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org0ed1499" class="outline-3">
<h3 id="org0ed1499"><span class="section-number-3">4.1</span> Initialization</h3>
<div class="outline-text-3" id="text-4-1">
<p>
We first initialize the Stewart platform without joint stiffness.
</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>
<p>
The rotation point of the ground is located at the origin of frame \(\{A\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgcd64c04" class="outline-3">
<h3 id="orgcd64c04"><span class="section-number-3">4.2</span> Identification</h3>
<div class="outline-text-3" id="text-4-2">
<p>
Let&rsquo;s first identify the transmissibility and compliance in the open-loop case.
</p>
<div class="org-src-container">
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, freqs] = computeCompliance();
</pre>
</div>
<p>
Now, let&rsquo;s identify the transmissibility and compliance for the Integral Force Feedback architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'iff'</span>);
G_iff = (2e4<span class="org-type">/</span>s)<span class="org-type">*</span>eye(6);
[T_iff, T_norm_iff, <span class="org-type">~</span>] = computeTransmissibility();
[C_iff, C_norm_iff, <span class="org-type">~</span>] = computeCompliance();
</pre>
</div>
<p>
And for the Direct Velocity Feedback.
</p>
<div class="org-src-container">
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'dvf'</span>);
G_dvf = 1e4<span class="org-type">*</span>s<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>5000)<span class="org-type">*</span>eye(6);
[T_dvf, T_norm_dvf, <span class="org-type">~</span>] = computeTransmissibility();
[C_dvf, C_norm_dvf, <span class="org-type">~</span>] = computeCompliance();
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd30c62d" class="outline-3">
<h3 id="orgd30c62d"><span class="section-number-3">4.3</span> Results</h3>
<div class="outline-text-3" id="text-4-3">
<div id="org6691389" class="figure">
<p><img src="figs/transmissibility_iff_dvf.png" alt="transmissibility_iff_dvf.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Obtained transmissibility for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) (<a href="./figs/transmissibility_iff_dvf.png">png</a>, <a href="./figs/transmissibility_iff_dvf.pdf">pdf</a>)</p>
</div>
<div id="orgd29218a" class="figure">
<p><img src="figs/compliance_iff_dvf.png" alt="compliance_iff_dvf.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Obtained compliance for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) (<a href="./figs/compliance_iff_dvf.png">png</a>, <a href="./figs/compliance_iff_dvf.pdf">pdf</a>)</p>
</div>
<div id="org2ee9711" class="figure">
<p><img src="figs/frobenius_norm_T_C_iff_dvf.png" alt="frobenius_norm_T_C_iff_dvf.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Frobenius norm of the Transmissibility and Compliance Matrices (<a href="./figs/frobenius_norm_T_C_iff_dvf.png">png</a>, <a href="./figs/frobenius_norm_T_C_iff_dvf.pdf">pdf</a>)</p>
</div>
</div>
</div>
@ -781,7 +872,7 @@ Joint stiffness does increase the resonance frequencies of the system but does n
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-13 jeu. 16:46</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
</div>
</body>
</html>

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<title>Stewart Platform - Control Study</title>
@ -201,30 +201,7 @@
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<li><a href="#orga9fb0f5">1.5. Controller Design</a></li>
</ul>
</li>
<li><a href="#org1ce6b23">2. Functions</a>
<ul>
<li><a href="#org9b036f8">2.1. <code>initializeController</code>: Initialize the Controller</a>
<ul>
<li><a href="#org89608d1">Function description</a></li>
<li><a href="#orgb457316">Optional Parameters</a></li>
<li><a href="#orgad0bd08">Structure initialization</a></li>
<li><a href="#org05c3878">Add Type</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
@ -387,10 +377,79 @@ Kl = Kl <span class="org-type">*</span> eye(6);
</div>
</div>
</div>
<div id="outline-container-org1ce6b23" class="outline-2">
<h2 id="org1ce6b23"><span class="section-number-2">2</span> Functions</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org9b036f8" class="outline-3">
<h3 id="org9b036f8"><span class="section-number-3">2.1</span> <code>initializeController</code>: Initialize the Controller</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org339969f"></a>
</p>
</div>
<div id="outline-container-org89608d1" class="outline-4">
<h4 id="org89608d1">Function description</h4>
<div class="outline-text-4" id="text-org89608d1">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[controller]</span> = <span class="org-function-name">initializeController</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% initializeController - Initialize the Controller</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [] = initializeController(args)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - args - Can have the following fields:</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb457316" class="outline-4">
<h4 id="orgb457316">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgb457316">
<div class="org-src-container">
<pre class="src src-matlab">arguments
args.type char {mustBeMember(args.type, {<span class="org-string">'open-loop'</span>, <span class="org-string">'iff'</span>, <span class="org-string">'dvf'</span>})} = <span class="org-string">'open-loop'</span>
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgad0bd08" class="outline-4">
<h4 id="orgad0bd08">Structure initialization</h4>
<div class="outline-text-4" id="text-orgad0bd08">
<div class="org-src-container">
<pre class="src src-matlab">controller = struct();
</pre>
</div>
</div>
</div>
<div id="outline-container-org05c3878" class="outline-4">
<h4 id="org05c3878">Add Type</h4>
<div class="outline-text-4" id="text-org05c3878">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">switch</span> <span class="org-constant">args.type</span>
<span class="org-keyword">case</span> <span class="org-string">'open-loop'</span>
controller.type = 0;
<span class="org-keyword">case</span> <span class="org-string">'iff'</span>
controller.type = 1;
<span class="org-keyword">case</span> <span class="org-string">'dvf'</span>
controller.type = 2;
<span class="org-keyword">end</span>
</pre>
</div>
</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. 14:51</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
</div>
</body>
</html>

62
docs/cubic-configuration.html
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<li><a href="#orga88e79a">1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center</a></li>
<li><a href="#orge02ec88">1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center</a></li>
<li><a href="#org43fd7e4">1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center</a></li>
<li><a href="#orgbde7788">1.5. Conclusion</a></li>
<li><a href="#orgd6c60aa">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgd70418b">2. Configuration with the Cube&rsquo;s center above the mobile platform</a>
<ul>
<li><a href="#org8afa645">2.1. Having Cube&rsquo;s center above the top platform</a></li>
<li><a href="#orgc0314ec">2.2. Conclusion</a></li>
<li><a href="#org78f0f9c">2.2. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgcc4ecce">3. Cubic size analysis</a>
<ul>
<li><a href="#org0029d8c">3.1. Analysis</a></li>
<li><a href="#orgb3ca361">3.2. Conclusion</a></li>
<li><a href="#org53a1ab8">3.2. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgf09da67">4. Dynamic Coupling in the Cartesian Frame</a>
<ul>
<li><a href="#org5fe01ec">4.1. Cube&rsquo;s center at the Center of Mass of the mobile platform</a></li>
<li><a href="#org4cb2a36">4.2. Cube&rsquo;s center not coincident with the Mass of the Mobile platform</a></li>
<li><a href="#orge33568e">4.3. Conclusion</a></li>
<li><a href="#orga0d81dc">4.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org8f26dc0">5. Dynamic Coupling between actuators and sensors of each strut</a>
<ul>
<li><a href="#org6e391c9">5.1. Coupling between the actuators and sensors - Cubic Architecture</a></li>
<li><a href="#orgafd808d">5.2. Coupling between the actuators and sensors - Non-Cubic Architecture</a></li>
<li><a href="#org1a90044">5.3. Conclusion</a></li>
<li><a href="#org3e2b41c">5.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org3044455">6. Functions</a>
@ -848,8 +826,8 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
</div>
</div>
<div id="outline-container-orgbde7788" class="outline-3">
<h3 id="orgbde7788"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-orgd6c60aa" class="outline-3">
<h3 id="orgd6c60aa"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@ -1186,8 +1164,8 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
</div>
</div>
<div id="outline-container-orgc0314ec" class="outline-3">
<h3 id="orgc0314ec"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-org78f0f9c" class="outline-3">
<h3 id="org78f0f9c"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important">
<p>
@ -1273,8 +1251,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
</div>
</div>
<div id="outline-container-orgb3ca361" class="outline-3">
<h3 id="orgb3ca361"><span class="section-number-3">3.2</span> Conclusion</h3>
<div id="outline-container-org53a1ab8" class="outline-3">
<h3 id="org53a1ab8"><span class="section-number-3">3.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-2">
<p>
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
@ -1629,8 +1607,8 @@ This was expected as the mass matrix is not diagonal (the Center of Mass of the
</div>
</div>
<div id="outline-container-orge33568e" class="outline-3">
<h3 id="orge33568e"><span class="section-number-3">4.3</span> Conclusion</h3>
<div id="outline-container-orga0d81dc" class="outline-3">
<h3 id="orga0d81dc"><span class="section-number-3">4.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-3">
<div class="important">
<p>
@ -1812,8 +1790,8 @@ And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relati
</div>
</div>
<div id="outline-container-org1a90044" class="outline-3">
<h3 id="org1a90044"><span class="section-number-3">5.3</span> Conclusion</h3>
<div id="outline-container-org3e2b41c" class="outline-3">
<h3 id="org3e2b41c"><span class="section-number-3">5.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-3">
<div class="important">
<p>
@ -1984,7 +1962,7 @@ stewart.platform_M.Mb = Mb;
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-14 ven. 14:11</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
</div>
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49
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<ul>
<li><a href="#org4509b7d">1.1. Comparison with fixed support</a></li>
<li><a href="#org8662186">1.2. Comparison with a flexible support</a></li>
<li><a href="#org1cbdf9a">1.3. Conclusion</a></li>
<li><a href="#org03b2957">1.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org81ab204">2. Comparison of the static transfer function and the Compliance matrix</a>
<ul>
<li><a href="#orge7e7242">2.1. Analysis</a></li>
<li><a href="#org03b2957">2.2. Conclusion</a></li>
<li><a href="#org920d3c4">2.2. Conclusion</a></li>
</ul>
</li>
</ul>
@ -463,8 +441,8 @@ And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly <b>
</div>
<div id="outline-container-org1cbdf9a" class="outline-3">
<h3 id="org1cbdf9a"><span class="section-number-3">1.3</span> Conclusion</h3>
<div id="outline-container-org03b2957" class="outline-3">
<h3 id="org03b2957"><span class="section-number-3">1.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-3">
<div class="important">
<p>
@ -525,11 +503,6 @@ 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>
@ -702,8 +675,8 @@ And now at the Compliance matrix.
</div>
</div>
<div id="outline-container-org03b2957" class="outline-3">
<h3 id="org03b2957"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-org920d3c4" class="outline-3">
<h3 id="org920d3c4"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important">
<p>
@ -717,7 +690,7 @@ The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathc
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-14 ven. 14:11</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
</div>
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<title>Identification of the Stewart Platform using Simscape</title>
@ -201,30 +201,7 @@
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As additional permission under GNU GPL version 3 section 7, you
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<li><a href="#orge7b97c8">1.5. Visualizing the modes</a></li>
</ul>
</li>
<li><a href="#org2891722">2. Transmissibility Analysis</a>
<ul>
<li><a href="#org8c667e9">2.1. Initialize the Stewart platform</a></li>
<li><a href="#org5338f20">2.2. Transmissibility</a></li>
</ul>
</li>
<li><a href="#orgc94edbd">3. Compliance Analysis</a>
<ul>
<li><a href="#orgc8e1f51">3.1. Initialize the Stewart platform</a></li>
<li><a href="#org1177029">3.2. Compliance</a></li>
</ul>
</li>
<li><a href="#org68ca336">4. Functions</a>
<ul>
<li><a href="#org487c4d4">4.1. Compute the Transmissibility</a>
<ul>
<li><a href="#org851f84d">Function description</a></li>
<li><a href="#orgf5e24cd">Optional Parameters</a></li>
<li><a href="#org4629501">Identification of the Transmissibility Matrix</a></li>
<li><a href="#org989379a">Computation of the Frobenius norm</a></li>
</ul>
</li>
<li><a href="#org50e35a6">4.2. Compute the Compliance</a>
<ul>
<li><a href="#org64fc1e2">Function description</a></li>
<li><a href="#org54cab00">Optional Parameters</a></li>
<li><a href="#orgef06b63">Identification of the Compliance Matrix</a></li>
<li><a href="#org6f63d37">Computation of the Frobenius norm</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<p>
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
</p>
<ul class="org-ul">
<li><a href="#org7981e88">1</a></li>
<li><a href="#orga989615">2</a></li>
<li><a href="#org4579374">3</a></li>
</ul>
<div id="outline-container-orgcb2f4c2" class="outline-2">
<h2 id="orgcb2f4c2"><span class="section-number-2">1</span> Modal Analysis of the Stewart Platform</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org7981e88"></a>
</p>
</div>
<div id="outline-container-org66d09e9" class="outline-3">
<h3 id="org66d09e9"><span class="section-number-3">1.1</span> Initialize the Stewart Platform</h3>
@ -577,10 +600,521 @@ Save the movie of the mode shape.
</div>
</div>
</div>
<div id="outline-container-org2891722" class="outline-2">
<h2 id="org2891722"><span class="section-number-2">2</span> Transmissibility Analysis</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orga989615"></a>
</p>
</div>
<div id="outline-container-org8c667e9" class="outline-3">
<h3 id="org8c667e9"><span class="section-number-3">2.1</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-2-1">
<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">'accelerometer'</span>, <span class="org-string">'freq'</span>, 5e3);
</pre>
</div>
<p>
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org5338f20" class="outline-3">
<h3 id="org5338f20"><span class="section-number-3">2.2</span> Transmissibility</h3>
<div class="outline-text-3" id="text-2-2">
<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">'/Disturbances/D_w'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
T = linearize(mdl, io, options);
T.InputName = {<span class="org-string">'Wdx'</span>, <span class="org-string">'Wdy'</span>, <span class="org-string">'Wdz'</span>, <span class="org-string">'Wrx'</span>, <span class="org-string">'Wry'</span>, <span class="org-string">'Wrz'</span>};
T.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">freqs = logspace(1, 4, 1000);
<span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
ylim([1e<span class="org-type">-</span>5, 10]);
xlim([freqs(1), freqs(end)]);
<span class="org-keyword">if</span> ix <span class="org-type">&lt;</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">&gt;</span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
<p>
From <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
</p>
\begin{align*}
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
T_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
T_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<p>
And we normalize by a factor \(\sqrt{6}\) to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gamma = T_norm<span class="org-type">/</span>sqrt(6);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
plot(freqs, Gamma)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgc94edbd" class="outline-2">
<h2 id="orgc94edbd"><span class="section-number-2">3</span> Compliance Analysis</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org4579374"></a>
</p>
</div>
<div id="outline-container-orgc8e1f51" class="outline-3">
<h3 id="orgc8e1f51"><span class="section-number-3">3.1</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-3-1">
<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">'accelerometer'</span>, <span class="org-string">'freq'</span>, 5e3);
</pre>
</div>
<p>
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
</p>
<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">'rigid'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org1177029" class="outline-3">
<h3 id="org1177029"><span class="section-number-3">3.2</span> Compliance</h3>
<div class="outline-text-3" id="text-3-2">
<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">'/Disturbances/F_ext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
C = linearize(mdl, io, options);
C.InputName = {<span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
C.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">freqs = logspace(1, 4, 1000);
<span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
ylim([1e<span class="org-type">-</span>10, 1e<span class="org-type">-</span>3]);
xlim([freqs(1), freqs(end)]);
<span class="org-keyword">if</span> ix <span class="org-type">&lt;</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">&gt;</span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
<p>
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace(1, 4, 1000);
C_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
C_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
plot(freqs, C_norm)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org68ca336" class="outline-2">
<h2 id="org68ca336"><span class="section-number-2">4</span> Functions</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org487c4d4" class="outline-3">
<h3 id="org487c4d4"><span class="section-number-3">4.1</span> Compute the Transmissibility</h3>
<div class="outline-text-3" id="text-4-1">
<p>
<a id="orgbca579c"></a>
</p>
</div>
<div id="outline-container-org851f84d" class="outline-4">
<h4 id="org851f84d">Function description</h4>
<div class="outline-text-4" id="text-org851f84d">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[T, T_norm, freqs]</span> = <span class="org-function-name">computeTransmissibility</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% computeTransmissibility -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [T, T_norm, freqs] = computeTransmissibility(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">% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm</span>
<span class="org-comment">% - freqs [] - Frequency vector to estimate the Frobenius norm</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - T [6x6 ss] - Transmissibility matrix</span>
<span class="org-comment">% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix</span>
<span class="org-comment">% - freqs [length(freqs)x1] - Frequency vector in [Hz]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf5e24cd" class="outline-4">
<h4 id="orgf5e24cd">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgf5e24cd">
<div class="org-src-container">
<pre class="src src-matlab">arguments
args.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
</pre>
</div>
</div>
</div>
<div id="outline-container-org4629501" class="outline-4">
<h4 id="org4629501">Identification of the Transmissibility Matrix</h4>
<div class="outline-text-4" id="text-org4629501">
<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">'/Disturbances/D_w'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
T = linearize(mdl, io, options);
T.InputName = {<span class="org-string">'Wdx'</span>, <span class="org-string">'Wdy'</span>, <span class="org-string">'Wdz'</span>, <span class="org-string">'Wrx'</span>, <span class="org-string">'Wry'</span>, <span class="org-string">'Wrz'</span>};
T.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>
<p>
If wanted, the 6x6 transmissibility matrix is plotted.
</p>
<div class="org-src-container">
<pre class="src src-matlab">p_handle = zeros(6<span class="org-type">*</span>6,1);
<span class="org-keyword">if</span> args.plots
fig = <span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
p_handle((ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy) = subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
<span class="org-keyword">if</span> ix <span class="org-type">&lt;</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">&gt;</span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
linkaxes(p_handle, <span class="org-string">'xy'</span>)
xlim([freqs(1), freqs(end)]);
ylim([1e<span class="org-type">-</span>5, 1e2]);
han = <span class="org-type">axes</span>(fig, <span class="org-string">'visible'</span>, <span class="org-string">'off'</span>);
han.XLabel.Visible = <span class="org-string">'on'</span>;
han.YLabel.Visible = <span class="org-string">'on'</span>;
ylabel(han, <span class="org-string">'Frequency [Hz]'</span>);
xlabel(han, <span class="org-string">'Transmissibility [m/m]'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org989379a" class="outline-4">
<h4 id="org989379a">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-org989379a">
<div class="org-src-container">
<pre class="src src-matlab">T_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
T_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">T_norm = T_norm<span class="org-type">/</span>sqrt(6);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">if</span> args.plots
<span class="org-type">figure</span>;
plot(freqs, T_norm)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
ylabel(<span class="org-string">'Transmissibility - Frobenius Norm'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org50e35a6" class="outline-3">
<h3 id="org50e35a6"><span class="section-number-3">4.2</span> Compute the Compliance</h3>
<div class="outline-text-3" id="text-4-2">
<p>
<a id="org0a73574"></a>
</p>
</div>
<div id="outline-container-org64fc1e2" class="outline-4">
<h4 id="org64fc1e2">Function description</h4>
<div class="outline-text-4" id="text-org64fc1e2">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[C, C_norm, freqs]</span> = <span class="org-function-name">computeCompliance</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% computeCompliance -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [C, C_norm, freqs] = computeCompliance(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">% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm</span>
<span class="org-comment">% - freqs [] - Frequency vector to estimate the Frobenius norm</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - C [6x6 ss] - Compliance matrix</span>
<span class="org-comment">% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix</span>
<span class="org-comment">% - freqs [length(freqs)x1] - Frequency vector in [Hz]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org54cab00" class="outline-4">
<h4 id="org54cab00">Optional Parameters</h4>
<div class="outline-text-4" id="text-org54cab00">
<div class="org-src-container">
<pre class="src src-matlab">arguments
args.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgef06b63" class="outline-4">
<h4 id="orgef06b63">Identification of the Compliance Matrix</h4>
<div class="outline-text-4" id="text-orgef06b63">
<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">'/Disturbances/F_ext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces [N, N*m]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
C = linearize(mdl, io, options);
C.InputName = {<span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
C.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>
<p>
If wanted, the 6x6 transmissibility matrix is plotted.
</p>
<div class="org-src-container">
<pre class="src src-matlab">p_handle = zeros(6<span class="org-type">*</span>6,1);
<span class="org-keyword">if</span> args.plots
fig = <span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
p_handle((ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy) = subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
<span class="org-keyword">if</span> ix <span class="org-type">&lt;</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">&gt;</span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
linkaxes(p_handle, <span class="org-string">'xy'</span>)
xlim([freqs(1), freqs(end)]);
han = <span class="org-type">axes</span>(fig, <span class="org-string">'visible'</span>, <span class="org-string">'off'</span>);
han.XLabel.Visible = <span class="org-string">'on'</span>;
han.YLabel.Visible = <span class="org-string">'on'</span>;
xlabel(han, <span class="org-string">'Frequency [Hz]'</span>);
ylabel(han, <span class="org-string">'Compliance [m/N, rad/(N*m)]'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org6f63d37" class="outline-4">
<h4 id="org6f63d37">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-org6f63d37">
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
C_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
C_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">if</span> args.plots
<span class="org-type">figure</span>;
plot(freqs, C_norm)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
ylabel(<span class="org-string">'Compliance - Frobenius Norm'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</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. 15:44</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
</div>
</body>
</html>

91
docs/simscape-model.html
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@ -4,7 +4,7 @@
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<title>Stewart Platform - Simscape Model</title>
@ -201,30 +201,7 @@
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<script src="./js/readtheorg.js"></script>
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@ -281,18 +259,19 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#org3535b6d">6.1. Payload</a>
<ul>
<li><a href="#org706f994">Function description</a></li>
<li><a href="#orgb15b49f">Optional Parameters</a></li>
<li><a href="#orge7f39a8">Function description</a></li>
<li><a href="#orgb83df72">Optional Parameters</a></li>
<li><a href="#orgeeb8d35">Add Payload Type</a></li>
<li><a href="#org6d52ffc">Add Stiffness, Damping and Mass properties of the Payload</a></li>
</ul>
</li>
<li><a href="#orgaaed406">6.2. Ground</a>
<ul>
<li><a href="#orge7f39a8">Function description</a></li>
<li><a href="#orgb83df72">Optional Parameters</a></li>
<li><a href="#org5d402b9">Function description</a></li>
<li><a href="#orgc0da5ca">Optional Parameters</a></li>
<li><a href="#orgef7035d">Add Ground Type</a></li>
<li><a href="#org95633e8">Add Stiffness and Damping properties of the Ground</a></li>
<li><a href="#org14ff2fc">Rotation Point</a></li>
</ul>
</li>
</ul>
@ -525,9 +504,9 @@ This Matlab function is accessible <a href="../src/initializePayload.m">here</a>
</p>
</div>
<div id="outline-container-org706f994" class="outline-4">
<h4 id="org706f994">Function description</h4>
<div class="outline-text-4" id="text-org706f994">
<div id="outline-container-orge7f39a8" class="outline-4">
<h4 id="orge7f39a8">Function description</h4>
<div class="outline-text-4" id="text-orge7f39a8">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[payload]</span> = <span class="org-function-name">initializePayload</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% initializePayload - Initialize the Payload that can then be used for simulations and analysis</span>
@ -536,7 +515,7 @@ This Matlab function is accessible <a href="../src/initializePayload.m">here</a>
<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">% - type - 'none', 'solid', 'flexible', 'cartesian'</span>
<span class="org-comment">% - type - 'none', 'rigid', 'flexible', 'cartesian'</span>
<span class="org-comment">% - h [1x1] - Height of the CoM of the payload w.r.t {M} [m]</span>
<span class="org-comment">% This also the position where K and C are defined</span>
<span class="org-comment">% - K [6x1] - Stiffness of the Payload [N/m, N/rad]</span>
@ -546,7 +525,7 @@ This Matlab function is accessible <a href="../src/initializePayload.m">here</a>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - payload - Struture with the following properties:</span>
<span class="org-comment">% - type - 1 (none), 2 (solid), 3 (flexible)</span>
<span class="org-comment">% - type - 1 (none), 2 (rigid), 3 (flexible)</span>
<span class="org-comment">% - h [1x1] - Height of the CoM of the payload w.r.t {M} [m]</span>
<span class="org-comment">% - K [6x1] - Stiffness of the Payload [N/m, N/rad]</span>
<span class="org-comment">% - C [6x1] - Stiffness of the Payload [N/(m/s), N/(rad/s)]</span>
@ -557,12 +536,12 @@ This Matlab function is accessible <a href="../src/initializePayload.m">here</a>
</div>
</div>
<div id="outline-container-orgb15b49f" class="outline-4">
<h4 id="orgb15b49f">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgb15b49f">
<div id="outline-container-orgb83df72" class="outline-4">
<h4 id="orgb83df72">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgb83df72">
<div class="org-src-container">
<pre class="src src-matlab">arguments
args.type char {mustBeMember(args.type,{<span class="org-string">'none'</span>, <span class="org-string">'solid'</span>, <span class="org-string">'flexible'</span>, <span class="org-string">'cartesian'</span>})} = <span class="org-string">'none'</span>
args.type char {mustBeMember(args.type,{<span class="org-string">'none'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'flexible'</span>, <span class="org-string">'cartesian'</span>})} = <span class="org-string">'none'</span>
args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e8<span class="org-type">*</span>ones(6,1)
args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1<span class="org-type">*</span>ones(6,1)
args.h (1,1) double {mustBeNumeric, mustBeNonnegative} = 100e<span class="org-type">-</span>3
@ -581,7 +560,7 @@ This Matlab function is accessible <a href="../src/initializePayload.m">here</a>
<pre class="src src-matlab"><span class="org-keyword">switch</span> <span class="org-constant">args.type</span>
<span class="org-keyword">case</span> <span class="org-string">'none'</span>
payload.type = 1;
<span class="org-keyword">case</span> <span class="org-string">'solid'</span>
<span class="org-keyword">case</span> <span class="org-string">'rigid'</span>
payload.type = 2;
<span class="org-keyword">case</span> <span class="org-string">'flexible'</span>
payload.type = 3;
@ -621,9 +600,9 @@ This Matlab function is accessible <a href="../src/initializeGround.m">here</a>.
</p>
</div>
<div id="outline-container-orge7f39a8" class="outline-4">
<h4 id="orge7f39a8">Function description</h4>
<div class="outline-text-4" id="text-orge7f39a8">
<div id="outline-container-org5d402b9" class="outline-4">
<h4 id="org5d402b9">Function description</h4>
<div class="outline-text-4" id="text-org5d402b9">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[ground]</span> = <span class="org-function-name">initializeGround</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% initializeGround - Initialize the Ground that can then be used for simulations and analysis</span>
@ -633,12 +612,13 @@ This Matlab function is accessible <a href="../src/initializeGround.m">here</a>.
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - args - Structure with the following fields:</span>
<span class="org-comment">% - type - 'none', 'solid', 'flexible'</span>
<span class="org-comment">% - rot_point [3x1] - Rotation point for the ground motion [m]</span>
<span class="org-comment">% - K [3x1] - Translation Stiffness of the Ground [N/m]</span>
<span class="org-comment">% - C [3x1] - Translation Damping of the Ground [N/(m/s)]</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - ground - Struture with the following properties:</span>
<span class="org-comment">% - type - 1 (none), 2 (solid), 3 (flexible)</span>
<span class="org-comment">% - type - 1 (none), 2 (rigid), 3 (flexible)</span>
<span class="org-comment">% - K [3x1] - Translation Stiffness of the Ground [N/m]</span>
<span class="org-comment">% - C [3x1] - Translation Damping of the Ground [N/(m/s)]</span>
</pre>
@ -646,12 +626,13 @@ This Matlab function is accessible <a href="../src/initializeGround.m">here</a>.
</div>
</div>
<div id="outline-container-orgb83df72" class="outline-4">
<h4 id="orgb83df72">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgb83df72">
<div id="outline-container-orgc0da5ca" class="outline-4">
<h4 id="orgc0da5ca">Optional Parameters</h4>
<div class="outline-text-4" id="text-orgc0da5ca">
<div class="org-src-container">
<pre class="src src-matlab">arguments
args.type char {mustBeMember(args.type,{<span class="org-string">'none'</span>, <span class="org-string">'solid'</span>, <span class="org-string">'flexible'</span>})} = <span class="org-string">'none'</span>
args.type char {mustBeMember(args.type,{<span class="org-string">'none'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'flexible'</span>})} = <span class="org-string">'none'</span>
args.rot_point (3,1) double {mustBeNumeric} = zeros(3,1)
args.K (3,1) double {mustBeNumeric, mustBeNonnegative} = 1e8<span class="org-type">*</span>ones(3,1)
args.C (3,1) double {mustBeNumeric, mustBeNonnegative} = 1e1<span class="org-type">*</span>ones(3,1)
<span class="org-keyword">end</span>
@ -667,7 +648,7 @@ This Matlab function is accessible <a href="../src/initializeGround.m">here</a>.
<pre class="src src-matlab"><span class="org-keyword">switch</span> <span class="org-constant">args.type</span>
<span class="org-keyword">case</span> <span class="org-string">'none'</span>
ground.type = 1;
<span class="org-keyword">case</span> <span class="org-string">'solid'</span>
<span class="org-keyword">case</span> <span class="org-string">'rigid'</span>
ground.type = 2;
<span class="org-keyword">case</span> <span class="org-string">'flexible'</span>
ground.type = 3;
@ -687,12 +668,22 @@ ground.C = args.C;
</div>
</div>
</div>
<div id="outline-container-org14ff2fc" class="outline-4">
<h4 id="org14ff2fc">Rotation Point</h4>
<div class="outline-text-4" id="text-org14ff2fc">
<div class="org-src-container">
<pre class="src src-matlab">ground.rot_point = args.rot_point;
</pre>
</div>
</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. 15:47</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
</div>
</body>
</html>

BIN
matlab/stewart_platform_model.slx
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177
org/active-damping.org
View File

@ -91,7 +91,7 @@ To run the script, open the Simulink Project, and type =run active_damping_inert
#+end_src
#+begin_src matlab
ground = initializeGround('type', 'none');
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
#+end_src
@ -321,10 +321,14 @@ We first initialize the Stewart platform without joint stiffness.
#+end_src
#+begin_src matlab
ground = initializeGround('type', 'none');
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
#+end_src
#+begin_src matlab
controller = initializeController('type', 'open-loop');
#+end_src
And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
%% Options for Linearized
@ -590,7 +594,7 @@ We first initialize the Stewart platform without joint stiffness.
#+end_src
#+begin_src matlab
ground = initializeGround('type', 'none');
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
#+end_src
@ -773,3 +777,170 @@ The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]].
#+begin_important
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
#+end_important
* Compliance and Transmissibility Comparison
** Initialization
We first initialize the Stewart platform without joint stiffness.
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');
#+end_src
The rotation point of the ground is located at the origin of frame $\{A\}$.
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
#+end_src
** Identification
Let's first identify the transmissibility and compliance in the open-loop case.
#+begin_src matlab
controller = initializeController('type', 'open-loop');
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, freqs] = computeCompliance();
#+end_src
Now, let's identify the transmissibility and compliance for the Integral Force Feedback architecture.
#+begin_src matlab
controller = initializeController('type', 'iff');
G_iff = (2e4/s)*eye(6);
[T_iff, T_norm_iff, ~] = computeTransmissibility();
[C_iff, C_norm_iff, ~] = computeCompliance();
#+end_src
And for the Direct Velocity Feedback.
#+begin_src matlab
controller = initializeController('type', 'dvf');
G_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6);
[T_dvf, T_norm_dvf, ~] = computeTransmissibility();
[C_dvf, C_norm_dvf, ~] = computeCompliance();
#+end_src
** Results
#+begin_src matlab :exports none
p_handle = zeros(6*6,1);
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(T_ol(ix, iy), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(T_iff(ix, iy), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',3);
plot(freqs, abs(squeeze(freqresp(T_dvf(ix, iy), freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Transmissibility');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/transmissibility_iff_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:transmissibility_iff_dvf
#+caption: Obtained transmissibility for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) ([[./figs/transmissibility_iff_dvf.png][png]], [[./figs/transmissibility_iff_dvf.pdf][pdf]])
[[file:figs/transmissibility_iff_dvf.png]]
#+begin_src matlab :exports none
p_handle = zeros(6*6,1);
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(C_ol(ix, iy), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(C_iff(ix, iy), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',3);
plot(freqs, abs(squeeze(freqresp(C_dvf(ix, iy), freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Compliance');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/compliance_iff_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:compliance_iff_dvf
#+caption: Obtained compliance for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) ([[./figs/compliance_iff_dvf.png][png]], [[./figs/compliance_iff_dvf.pdf][pdf]])
[[file:figs/compliance_iff_dvf.png]]
#+begin_src matlab :exports none
figure;
subplot(1,2,1);
hold on;
plot(freqs, T_norm_ol)
plot(freqs, T_norm_iff)
plot(freqs, T_norm_dvf)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Transmissibility - Frobenius Norm');
subplot(1,2,2);
hold on;
plot(freqs, C_norm_ol, 'DisplayName', 'OL')
plot(freqs, C_norm_iff, 'DisplayName', 'IFF')
plot(freqs, C_norm_dvf, 'DisplayName', 'DVF')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Compliance - Frobenius Norm');
legend();
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/frobenius_norm_T_C_iff_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:frobenius_norm_T_C_iff_dvf
#+caption: Frobenius norm of the Transmissibility and Compliance Matrices ([[./figs/frobenius_norm_T_C_iff_dvf.png][png]], [[./figs/frobenius_norm_T_C_iff_dvf.pdf][pdf]])
[[file:figs/frobenius_norm_T_C_iff_dvf.png]]

436
org/control-study.org
View File

@ -259,241 +259,27 @@ A lead is added around the crossover frequency which is set to be around 500Hz.
linkaxes([ax1,ax2],'x');
#+end_src
* Transmissibility Analysis
** Matlab Init :noexport:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'rigid');
#+end_src
** Transmissibility
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
freqs = logspace(1, 4, 1000);
figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-5, 10]);
xlim([freqs(1), freqs(end)]);
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
#+end_src
From cite:preumont07_six_axis_singl_stage_activ, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
\begin{align*}
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
#+begin_src matlab
freqs = logspace(1, 4, 1000);
T_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
#+end_src
And we normalize by a factor $\sqrt{6}$ to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
#+begin_src matlab
Gamma = T_norm/sqrt(6);
#+end_src
#+begin_src matlab
figure;
plot(freqs, Gamma)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src
* Compliance Analysis
** Matlab Init :noexport:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid');
#+end_src
** Compliance
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
freqs = logspace(1, 4, 1000);
figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-10, 1e-3]);
xlim([freqs(1), freqs(end)]);
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
#+end_src
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
#+begin_src matlab
freqs = logspace(1, 4, 1000);
C_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
figure;
plot(freqs, C_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src
* Functions
** Compute the Transmissibility
** =initializeController=: Initialize the Controller
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeTransmissibility.m
:header-args:matlab+: :tangle ../src/initializeController.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeTransmissibility>>
<<sec:initializeController>>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [T, T_norm, freqs] = computeTransmissibility(args)
% computeTransmissibility -
function [controller] = initializeController(args)
% initializeController - Initialize the Controller
%
% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
% Syntax: [] = initializeController(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - T [6x6 ss] - Transmissibility matrix
% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
% - args - Can have the following fields:
#+end_src
*** Optional Parameters
@ -502,215 +288,29 @@ We can try to use the Frobenius norm to obtain a scalar value representing the 6
:END:
#+begin_src matlab
arguments
args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf'})} = 'open-loop'
end
#+end_src
#+begin_src matlab
freqs = args.freqs;
#+end_src
*** Identification of the Transmissibility Matrix
*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
controller = struct();
#+end_src
If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
p_handle = zeros(6*6,1);
if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
ylim([1e-5, 1e2]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Transmissibility [m/m]');
end
#+end_src
*** Computation of the Frobenius norm
*** Add Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
T_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
T_norm = T_norm/sqrt(6);
#+end_src
#+begin_src matlab
if args.plots
figure;
plot(freqs, T_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Transmissibility - Frobenius Norm');
end
#+end_src
** Compute the Compliance
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeCompliance.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeCompliance>>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [C, C_norm, freqs] = computeCompliance(args)
% computeCompliance -
%
% Syntax: [C, C_norm, freqs] = computeCompliance(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - C [6x6 ss] - Compliance matrix
% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
end
#+end_src
#+begin_src matlab
freqs = args.freqs;
#+end_src
*** Identification of the Compliance Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
p_handle = zeros(6*6,1);
if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Compliance [m/N, rad/(N*m)]');
end
#+end_src
*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
freqs = args.freqs;
C_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
if args.plots
figure;
plot(freqs, C_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Compliance - Frobenius Norm');
switch args.type
case 'open-loop'
controller.type = 0;
case 'iff'
controller.type = 1;
case 'dvf'
controller.type = 2;
end
#+end_src

5
org/dynamics-study.org
View File

@ -338,11 +338,6 @@ Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]

466
org/identification.org
View File

@ -39,8 +39,14 @@
:END:
* Introduction :ignore:
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
- [[sec:modal_analysis]]
- [[sec:transmissibility]]
- [[sec:compliance]]
* Modal Analysis of the Stewart Platform
<<sec:modal_analysis>>
** Introduction :ignore:
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -239,3 +245,463 @@ Save the movie of the mode shape.
#+caption: Identified mode - 5
[[file:figs/mode5.gif]]
* Transmissibility Analysis
<<sec:transmissibility>>
** Introduction :ignore:
** Matlab Init :noexport:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'rigid');
#+end_src
** Transmissibility
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
freqs = logspace(1, 4, 1000);
figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-5, 10]);
xlim([freqs(1), freqs(end)]);
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
#+end_src
From cite:preumont07_six_axis_singl_stage_activ, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
\begin{align*}
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
#+begin_src matlab
freqs = logspace(1, 4, 1000);
T_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
#+end_src
And we normalize by a factor $\sqrt{6}$ to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
#+begin_src matlab
Gamma = T_norm/sqrt(6);
#+end_src
#+begin_src matlab
figure;
plot(freqs, Gamma)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src
* Compliance Analysis
<<sec:compliance>>
** Introduction :ignore:
** Matlab Init :noexport:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid');
#+end_src
** Compliance
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
freqs = logspace(1, 4, 1000);
figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-10, 1e-3]);
xlim([freqs(1), freqs(end)]);
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
#+end_src
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
#+begin_src matlab
freqs = logspace(1, 4, 1000);
C_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
figure;
plot(freqs, C_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src
* Functions
** Compute the Transmissibility
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeTransmissibility.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeTransmissibility>>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [T, T_norm, freqs] = computeTransmissibility(args)
% computeTransmissibility -
%
% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - T [6x6 ss] - Transmissibility matrix
% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
end
#+end_src
#+begin_src matlab
freqs = args.freqs;
#+end_src
*** Identification of the Transmissibility Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
p_handle = zeros(6*6,1);
if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
ylim([1e-5, 1e2]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Transmissibility [m/m]');
end
#+end_src
*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
T_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
T_norm = T_norm/sqrt(6);
#+end_src
#+begin_src matlab
if args.plots
figure;
plot(freqs, T_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Transmissibility - Frobenius Norm');
end
#+end_src
** Compute the Compliance
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeCompliance.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeCompliance>>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [C, C_norm, freqs] = computeCompliance(args)
% computeCompliance -
%
% Syntax: [C, C_norm, freqs] = computeCompliance(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - C [6x6 ss] - Compliance matrix
% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
end
#+end_src
#+begin_src matlab
freqs = args.freqs;
#+end_src
*** Identification of the Compliance Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
p_handle = zeros(6*6,1);
if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Compliance [m/N, rad/(N*m)]');
end
#+end_src
*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
freqs = args.freqs;
C_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
if args.plots
figure;
plot(freqs, C_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Compliance - Frobenius Norm');
end
#+end_src

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src/initializeController.m Normal file
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function [controller] = initializeController(args)
% initializeController - Initialize the Controller
%
% Syntax: [] = initializeController(args)
%
% Inputs:
% - args - Can have the following fields:
arguments
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf'})} = 'open-loop'
end
controller = struct();
switch args.type
case 'open-loop'
controller.type = 0;
case 'iff'
controller.type = 1;
case 'dvf'
controller.type = 2;
end