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Work on HAC-LAC, Control architectures

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Thomas Dehaeze
2020-02-28 17:35:44 +01:00
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-02-27 jeu. 14:16 -->
<!-- 2020-02-28 ven. 17:33 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Stewart Platform - Decentralized Active Damping</title>
@@ -249,25 +249,25 @@
<li><a href="#orgd59c804">1. Inertial Control</a>
<ul>
<li><a href="#org5f749c8">1.1. Identification of the Dynamics</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>
<li><a href="#org3014959">1.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#orga144352">1.3. Obtained Damping</a></li>
<li><a href="#org004b094">1.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org74c7eb4">2. Integral Force Feedback</a>
<ul>
<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>
<li><a href="#org7313778">2.1. Identification of the Dynamics with perfect Joints</a></li>
<li><a href="#org462c581">2.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#org943bf7b">2.3. Obtained Damping</a></li>
<li><a href="#orga677c7d">2.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org08917d6">3. Direct Velocity Feedback</a>
<ul>
<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>
<li><a href="#orgcd99b62">3.1. Identification of the Dynamics with perfect Joints</a></li>
<li><a href="#orgd0f78f7">3.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics</a></li>
<li><a href="#org3f64d96">3.3. Obtained Damping</a></li>
<li><a href="#org8e1ece7">3.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org183f3f2">4. Compliance and Transmissibility Comparison</a>
@@ -330,6 +330,7 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
<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>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -365,8 +366,8 @@ The transfer function from actuator forces to force sensors is shown in Figure <
</div>
</div>
<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 id="outline-container-org3014959" class="outline-3">
<h3 id="org3014959"><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.
@@ -402,8 +403,8 @@ The new dynamics from force actuator to force sensor is shown in Figure <a href=
</div>
</div>
<div id="outline-container-orgd895eeb" class="outline-3">
<h3 id="orgd895eeb"><span class="section-number-3">1.3</span> Obtained Damping</h3>
<div id="outline-container-orga144352" class="outline-3">
<h3 id="orga144352"><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.
@@ -428,8 +429,8 @@ The root locus is shown in figure <a href="#org9af9e33">3</a>.
</div>
</div>
<div id="outline-container-orgeaf5ef8" class="outline-3">
<h3 id="orgeaf5ef8"><span class="section-number-3">1.4</span> Conclusion</h3>
<div id="outline-container-org004b094" class="outline-3">
<h3 id="org004b094"><span class="section-number-3">1.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-4">
<div class="important">
<p>
@@ -460,8 +461,8 @@ 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">2.1</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">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.
@@ -484,11 +485,7 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
<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 class="org-src-container">
<pre class="src src-matlab">controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -496,11 +493,7 @@ payload = initializePayload(<span class="org-string">'type'</span>, <span class=
And we identify the dynamics from force actuators to force sensors.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
<pre class="src src-matlab"><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>
@@ -509,7 +502,7 @@ io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1,
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'Taum'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Force Sensor Outputs [N]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
G = linearize(mdl, io);
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
</pre>
@@ -527,15 +520,15 @@ The transfer function from actuator forces to force sensors is shown in Figure <
</div>
</div>
<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 id="outline-container-org462c581" class="outline-3">
<h3 id="org462c581"><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.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeJointDynamics(stewart, <span class="org-string">'type_F'</span>, <span class="org-string">'universal'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical'</span>);
Gf = linearize(mdl, io, options);
Gf = linearize(mdl, io);
Gf.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
Gf.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
</pre>
@@ -546,7 +539,7 @@ We now use the amplified actuators and re-identify the dynamics
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeAmplifiedStrutDynamics(stewart);
Ga = linearize(mdl, io, options);
Ga = linearize(mdl, io);
Ga.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
Ga.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
</pre>
@@ -564,8 +557,8 @@ The new dynamics from force actuator to force sensor is shown in Figure <a href=
</div>
</div>
<div id="outline-container-org9e1f2e2" class="outline-3">
<h3 id="org9e1f2e2"><span class="section-number-3">2.3</span> Obtained Damping</h3>
<div id="outline-container-org943bf7b" class="outline-3">
<h3 id="org943bf7b"><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.
@@ -597,8 +590,8 @@ The root locus is shown in figure <a href="#orge21bbea">6</a> and the obtained p
</div>
</div>
<div id="outline-container-org405813e" class="outline-3">
<h3 id="org405813e"><span class="section-number-3">2.4</span> Conclusion</h3>
<div id="outline-container-orga677c7d" class="outline-3">
<h3 id="orga677c7d"><span class="section-number-3">2.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-4">
<div class="important">
<p>
@@ -630,8 +623,8 @@ To run the script, open the Simulink Project, and type <code>run active_damping_
</div>
</div>
<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 id="outline-container-orgcd99b62" class="outline-3">
<h3 id="orgcd99b62"><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.
@@ -654,6 +647,7 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
<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>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -693,8 +687,8 @@ The transfer function from actuator forces to relative motion sensors is shown i
</div>
<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 id="outline-container-orgd0f78f7" class="outline-3">
<h3 id="orgd0f78f7"><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.
@@ -730,8 +724,8 @@ The new dynamics from force actuator to relative motion sensor is shown in Figur
</div>
</div>
<div id="outline-container-orga144352" class="outline-3">
<h3 id="orga144352"><span class="section-number-3">3.3</span> Obtained Damping</h3>
<div id="outline-container-org3f64d96" class="outline-3">
<h3 id="org3f64d96"><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.
@@ -756,8 +750,8 @@ The root locus is shown in figure <a href="#org277d60d">10</a>.
</div>
</div>
<div id="outline-container-org004b094" class="outline-3">
<h3 id="org004b094"><span class="section-number-3">3.4</span> Conclusion</h3>
<div id="outline-container-org8e1ece7" class="outline-3">
<h3 id="org8e1ece7"><span class="section-number-3">3.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-4">
<div class="important">
<p>
@@ -799,6 +793,7 @@ The rotation point of the ground is located at the origin of frame \(\{A\}\).
<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>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
</div>
@@ -822,7 +817,7 @@ Now, let&rsquo;s identify the transmissibility and compliance for the Integral F
</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);
K_iff = (1e4<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();
@@ -834,7 +829,7 @@ 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);
K_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();
@@ -872,7 +867,7 @@ G_dvf = 1e4<span class="org-type">*</span>s<span class="org-type">/</span>(1<spa
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
<p class="date">Created: 2020-02-28 ven. 17:33</p>
</div>
</body>
</html>

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<!-- 2020-02-28 ven. 17:37 -->
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<h1 class="title">Stewart Platform - Tracking Control</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org4c793a2">1. First Control Architecture</a>
<ul>
<li><a href="#org49467e8">1.1. Control Schematic</a></li>
<li><a href="#org67db718">1.2. Initialize the Stewart platform</a></li>
<li><a href="#org641cba6">1.3. Identification of the plant</a></li>
<li><a href="#orgd9d7b44">1.4. Plant Analysis</a></li>
<li><a href="#orgfaf80fa">1.5. Controller Design</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org4c793a2" class="outline-2">
<h2 id="org4c793a2"><span class="section-number-2">1</span> First Control Architecture</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org49467e8" class="outline-3">
<h3 id="org49467e8"><span class="section-number-3">1.1</span> Control Schematic</h3>
<div class="outline-text-3" id="text-1-1">
<div class="figure">
<p><img src="figs/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org67db718" class="outline-3">
<h3 id="org67db718"><span class="section-number-3">1.2</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-1-2">
<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>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org641cba6" class="outline-3">
<h3 id="org641cba6"><span class="section-number-3">1.3</span> Identification of the plant</h3>
<div class="outline-text-3" id="text-1-3">
<p>
Let&rsquo;s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'stewart_platform_model'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Stewart Platform'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'dLm'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [m]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io);
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'L1'</span>, <span class="org-string">'L2'</span>, <span class="org-string">'L3'</span>, <span class="org-string">'L4'</span>, <span class="org-string">'L5'</span>, <span class="org-string">'L6'</span>};
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd9d7b44" class="outline-3">
<h3 id="orgd9d7b44"><span class="section-number-3">1.4</span> Plant Analysis</h3>
<div class="outline-text-3" id="text-1-4">
<p>
Diagonal terms
Compare to off-diagonal terms
</p>
</div>
</div>
<div id="outline-container-orgfaf80fa" class="outline-3">
<h3 id="orgfaf80fa"><span class="section-number-3">1.5</span> Controller Design</h3>
<div class="outline-text-3" id="text-1-5">
<p>
One integrator should be present in the controller.
</p>
<p>
A lead is added around the crossover frequency which is set to be around 500Hz.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% wint = 2*pi*100; % Integrate until [rad]</span>
<span class="org-comment">% wlead = 2*pi*500; % Location of the lead [rad]</span>
<span class="org-comment">% hlead = 2; % Lead strengh</span>
<span class="org-comment">% Kl = 1e6 * ... % Gain</span>
<span class="org-comment">% (s + wint)/(s) * ... % Integrator until 100Hz</span>
<span class="org-comment">% (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead</span>
wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100;
Kl = 1<span class="org-type">/</span>abs(freqresp(G(1,1), wc)) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(3<span class="org-type">*</span>wc));
Kl = Kl <span class="org-type">*</span> eye(6);
</pre>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-28 ven. 17:37</p>
</div>
</body>
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38
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@@ -4,7 +4,7 @@
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<head>
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<!-- 2020-02-28 ven. 17:34 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Cubic configuration for the Stewart Platform</title>
@@ -252,33 +252,33 @@
<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="#orgd6c60aa">1.5. Conclusion</a></li>
<li><a href="#org3e2b41c">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="#org78f0f9c">2.2. Conclusion</a></li>
<li><a href="#orgeeac940">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="#org53a1ab8">3.2. Conclusion</a></li>
<li><a href="#org991d232">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="#orga0d81dc">4.3. Conclusion</a></li>
<li><a href="#orgf0acd1f">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="#org3e2b41c">5.3. Conclusion</a></li>
<li><a href="#org78c4967">5.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org3044455">6. Functions</a>
@@ -826,8 +826,8 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
</div>
</div>
<div id="outline-container-orgd6c60aa" class="outline-3">
<h3 id="orgd6c60aa"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org3e2b41c" class="outline-3">
<h3 id="org3e2b41c"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@@ -1164,8 +1164,8 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
</div>
</div>
<div id="outline-container-org78f0f9c" class="outline-3">
<h3 id="org78f0f9c"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-orgeeac940" class="outline-3">
<h3 id="orgeeac940"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important">
<p>
@@ -1251,8 +1251,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
</div>
</div>
<div id="outline-container-org53a1ab8" class="outline-3">
<h3 id="org53a1ab8"><span class="section-number-3">3.2</span> Conclusion</h3>
<div id="outline-container-org991d232" class="outline-3">
<h3 id="org991d232"><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.
@@ -1391,6 +1391,7 @@ No flexibility below the Stewart platform and no payload.
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -1535,6 +1536,7 @@ No flexibility below the Stewart platform and no payload.
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -1607,8 +1609,8 @@ This was expected as the mass matrix is not diagonal (the Center of Mass of the
</div>
</div>
<div id="outline-container-orga0d81dc" class="outline-3">
<h3 id="orga0d81dc"><span class="section-number-3">4.3</span> Conclusion</h3>
<div id="outline-container-orgf0acd1f" class="outline-3">
<h3 id="orgf0acd1f"><span class="section-number-3">4.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-3">
<div class="important">
<p>
@@ -1693,6 +1695,7 @@ No flexibility below the Stewart platform and no payload.
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -1760,6 +1763,7 @@ No flexibility below the Stewart platform and no payload.
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -1790,8 +1794,8 @@ And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relati
</div>
</div>
<div id="outline-container-org3e2b41c" class="outline-3">
<h3 id="org3e2b41c"><span class="section-number-3">5.3</span> Conclusion</h3>
<div id="outline-container-org78c4967" class="outline-3">
<h3 id="org78c4967"><span class="section-number-3">5.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-3">
<div class="important">
<p>
@@ -1962,7 +1966,7 @@ stewart.platform_M.Mb = Mb;
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
<p class="date">Created: 2020-02-28 ven. 17:34</p>
</div>
</body>
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18
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@@ -4,7 +4,7 @@
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<head>
<!-- 2020-02-27 jeu. 14:16 -->
<!-- 2020-02-28 ven. 17:34 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Stewart Platform - Dynamics Study</title>
@@ -250,13 +250,13 @@
<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="#org03b2957">1.3. Conclusion</a></li>
<li><a href="#org920d3c4">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="#org920d3c4">2.2. Conclusion</a></li>
<li><a href="#orgbb930ae">2.2. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -299,6 +299,7 @@ We also don&rsquo;t put any payload on top of the Stewart platform.
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -441,8 +442,8 @@ And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly <b>
</div>
<div id="outline-container-org03b2957" class="outline-3">
<h3 id="org03b2957"><span class="section-number-3">1.3</span> Conclusion</h3>
<div id="outline-container-org920d3c4" class="outline-3">
<h3 id="org920d3c4"><span class="section-number-3">1.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-3">
<div class="important">
<p>
@@ -489,6 +490,7 @@ No flexibility below the Stewart platform and no payload.
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
@@ -675,8 +677,8 @@ And now at the Compliance matrix.
</div>
</div>
<div id="outline-container-org920d3c4" class="outline-3">
<h3 id="org920d3c4"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-orgbb930ae" class="outline-3">
<h3 id="orgbb930ae"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important">
<p>
@@ -690,7 +692,7 @@ 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: 2020-02-27 jeu. 14:16</p>
<p class="date">Created: 2020-02-28 ven. 17:34</p>
</div>
</body>
</html>

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75
docs/identification.html
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@@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-02-27 jeu. 14:16 -->
<!-- 2020-02-28 ven. 17:34 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Identification of the Stewart Platform using Simscape</title>
@@ -257,13 +257,13 @@
</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="#orgc8e1f51">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="#org55d2544">3.1. Initialize the Stewart platform</a></li>
<li><a href="#org1177029">3.2. Compliance</a></li>
</ul>
</li>
@@ -271,18 +271,18 @@
<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="#org64fc1e2">Function description</a></li>
<li><a href="#org54cab00">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>
<li><a href="#org6f63d37">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="#org3cf1d13">Function description</a></li>
<li><a href="#org726b57d">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>
<li><a href="#org1019eaf">Computation of the Frobenius norm</a></li>
</ul>
</li>
</ul>
@@ -329,6 +329,7 @@ stewart = initializeInertialSensor(stewart);
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
</div>
@@ -608,8 +609,8 @@ Save the movie of the mode shape.
<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 id="outline-container-orgc8e1f51" class="outline-3">
<h3 id="orgc8e1f51"><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();
@@ -632,6 +633,7 @@ We set the rotation point of the ground to be at the same point at frames \(\{A\
<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>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
</div>
@@ -729,8 +731,8 @@ plot(freqs, Gamma)
<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 id="outline-container-org55d2544" class="outline-3">
<h3 id="org55d2544"><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();
@@ -753,6 +755,7 @@ We set the rotation point of the ground to be at the same point at frames \(\{A\
<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>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
</div>
@@ -841,9 +844,9 @@ plot(freqs, C_norm)
</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 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">[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>
@@ -864,9 +867,9 @@ plot(freqs, C_norm)
</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 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>
@@ -896,7 +899,7 @@ 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>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'output'</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);
@@ -935,17 +938,17 @@ If wanted, the 6x6 transmissibility matrix is plotted.
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>);
xlabel(han, <span class="org-string">'Frequency [Hz]'</span>);
ylabel(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 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">T_norm = zeros(length(freqs), 1);
@@ -982,9 +985,9 @@ If wanted, the 6x6 transmissibility matrix is plotted.
</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 id="outline-container-org3cf1d13" class="outline-4">
<h4 id="org3cf1d13">Function description</h4>
<div class="outline-text-4" id="text-org3cf1d13">
<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>
@@ -1005,9 +1008,9 @@ If wanted, the 6x6 transmissibility matrix is plotted.
</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 id="outline-container-org726b57d" class="outline-4">
<h4 id="org726b57d">Optional Parameters</h4>
<div class="outline-text-4" id="text-org726b57d">
<div class="org-src-container">
<pre class="src src-matlab">arguments
args.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
@@ -1037,7 +1040,7 @@ 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>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'output'</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);
@@ -1083,9 +1086,9 @@ If wanted, the 6x6 transmissibility matrix is plotted.
</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 id="outline-container-org1019eaf" class="outline-4">
<h4 id="org1019eaf">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-org1019eaf">
<div class="org-src-container">
<pre class="src src-matlab">freqs = args.freqs;
@@ -1114,7 +1117,7 @@ C_norm = zeros(length(freqs), 1);
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-27 jeu. 14:16</p>
<p class="date">Created: 2020-02-28 ven. 17:34</p>
</div>
</body>
</html>

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org/active-damping.org
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@@ -93,6 +93,7 @@ To run the script, open the Simulink Project, and type =run active_damping_inert
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
#+begin_src matlab
@@ -323,18 +324,11 @@ We first initialize the Stewart platform without joint stiffness.
#+begin_src matlab
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
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
@@ -344,7 +338,7 @@ And we identify the dynamics from force actuators to force sensors.
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensor Outputs [N]
%% Run the linearization
G = linearize(mdl, io, options);
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src
@@ -398,7 +392,7 @@ The transfer function from actuator forces to force sensors is shown in Figure [
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
#+begin_src matlab
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
Gf = linearize(mdl, io, options);
Gf = linearize(mdl, io);
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src
@@ -406,7 +400,7 @@ We add some stiffness and damping in the flexible joints and we re-identify the
We now use the amplified actuators and re-identify the dynamics
#+begin_src matlab
stewart = initializeAmplifiedStrutDynamics(stewart);
Ga = linearize(mdl, io, options);
Ga = linearize(mdl, io);
Ga.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Ga.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src
@@ -596,6 +590,7 @@ We first initialize the Stewart platform without joint stiffness.
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
And we identify the dynamics from force actuators to force sensors.
@@ -778,6 +773,24 @@ The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]].
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
** 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)
<<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
** Initialization
We first initialize the Stewart platform without joint stiffness.
#+begin_src matlab
@@ -798,6 +811,7 @@ 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');
controller = initializeController('type', 'open-loop');
#+end_src
** Identification
@@ -811,7 +825,7 @@ Let's first identify the transmissibility and compliance in the open-loop case.
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);
K_iff = (1e4/s)*eye(6);
[T_iff, T_norm_iff, ~] = computeTransmissibility();
[C_iff, C_norm_iff, ~] = computeCompliance();
@@ -820,7 +834,7 @@ Now, let's identify the transmissibility and compliance for the Integral Force F
And for the Direct Velocity Feedback.
#+begin_src matlab
controller = initializeController('type', 'dvf');
G_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6);
K_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6);
[T_dvf, T_norm_dvf, ~] = computeTransmissibility();
[C_dvf, C_norm_dvf, ~] = computeCompliance();
@@ -857,8 +871,8 @@ And for the Direct Velocity Feedback.
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Transmissibility');
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Transmissibility');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
@@ -900,8 +914,8 @@ And for the Direct Velocity Feedback.
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Compliance');
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Compliance');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes

1186
org/control-study.org
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#+TITLE: Stewart Platform - Tracking Control
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ./index.html
#+HTML_LINK_UP: ./index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* First Control Architecture
** Matlab Init :noexport:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
simulinkproject('../');
#+end_src
** Control Schematic
#+begin_src latex :file control_measure_rotating_2dof.pdf
\begin{tikzpicture}
% Blocs
\node[block] (J) at (0, 0) {$J$};
\node[addb={+}{}{}{}{-}, right=1 of J] (subr) {};
\node[block, right=0.8 of subr] (K) {$K_{L}$};
\node[block, right=1 of K] (G) {$G_{L}$};
% Connections and labels
\draw[<-] (J.west)node[above left]{$\bm{r}_{n}$} -- ++(-1, 0);
\draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{L}$};
\draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{L}$};
\draw[->] (K.east) -- (G.west) node[above left]{$\bm{\tau}$};
\draw[->] (G.east) node[above right]{$\bm{L}$} -| ($(G.east)+(1, -1)$) -| (subr.south);
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/control_measure_rotating_2dof.png]]
** 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
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
#+end_src
** Identification of the plant
Let's identify the transfer function from $\bm{\tau}$ to $\bm{L}$.
#+begin_src matlab
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% 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]
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
%% Run the linearization
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};
#+end_src
** Plant Analysis
Diagonal terms
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
#+end_src
Compare to off-diagonal terms
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
#+end_src
** Controller Design
One integrator should be present in the controller.
A lead is added around the crossover frequency which is set to be around 500Hz.
#+begin_src matlab
% wint = 2*pi*100; % Integrate until [rad]
% wlead = 2*pi*500; % Location of the lead [rad]
% hlead = 2; % Lead strengh
% Kl = 1e6 * ... % Gain
% (s + wint)/(s) * ... % Integrator until 100Hz
% (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead
wc = 2*pi*100;
Kl = 1/abs(freqresp(G(1,1), wc)) * wc/s * 1/(1 + s/(3*wc));
Kl = Kl * eye(6);
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Kl(i,i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
#+end_src

4
org/cubic-configuration.org
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@@ -695,6 +695,7 @@ No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
The obtain geometry is shown in figure [[fig:stewart_cubic_conf_decouple_dynamics]].
@@ -880,6 +881,7 @@ No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]].
@@ -1087,6 +1089,7 @@ No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
#+begin_src matlab :exports none
@@ -1258,6 +1261,7 @@ No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
#+begin_src matlab :exports none

2
org/dynamics-study.org
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@@ -80,6 +80,7 @@ We also don't put any payload on top of the Stewart platform.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
The transfer function from actuator forces $\bm{\tau}$ to the relative displacement of the mobile platform $\mathcal{\bm{X}}$ is extracted.
@@ -327,6 +328,7 @@ No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:

11
org/identification.org
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@@ -83,6 +83,7 @@ In this document, we discuss the various methods to identify the behavior of the
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
** Identification
@@ -284,6 +285,7 @@ We set the rotation point of the ground to be at the same point at frames $\{A\}
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'rigid');
controller = initializeController('type', 'open-loop');
#+end_src
** Transmissibility
@@ -396,6 +398,7 @@ We set the rotation point of the ground to be at the same point at frames $\{A\}
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid');
controller = initializeController('type', 'open-loop');
#+end_src
** Compliance
@@ -517,7 +520,7 @@ We can try to use the Frobenius norm to obtain a scalar value representing the 6
%% 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]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
@@ -553,8 +556,8 @@ If wanted, the 6x6 transmissibility matrix is plotted.
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Transmissibility [m/m]');
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Transmissibility [m/m]');
end
#+end_src
@@ -642,7 +645,7 @@ If wanted, the 6x6 transmissibility matrix is plotted.
%% 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]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);

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simscape_subsystems/stewart_strut.slx
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src/computeCompliance.m
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@@ -30,7 +30,7 @@ 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]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);

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src/computeTransmissibility.m
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@@ -30,7 +30,7 @@ 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]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
@@ -63,8 +63,8 @@ if args.plots
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
ylabel(han, 'Frequency [Hz]');
xlabel(han, 'Transmissibility [m/m]');
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Transmissibility [m/m]');
end
T_norm = zeros(length(freqs), 1);

6
src/initializeController.m
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@@ -7,7 +7,7 @@ function [controller] = initializeController(args)
% - args - Can have the following fields:
arguments
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf'})} = 'open-loop'
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf'})} = 'open-loop'
end
controller = struct();
@@ -19,4 +19,8 @@ switch args.type
controller.type = 1;
case 'dvf'
controller.type = 2;
case 'hac-iff'
controller.type = 3;
case 'hac-dvf'
controller.type = 4;
end