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Important analysis of the cubic configuration

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Thomas Dehaeze 2020-02-12 18:27:31 +01:00
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Cubic configuration for the Stewart Platform</title>
@ -268,37 +268,44 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org8c6677e">1. Configuration Analysis - Stiffness Matrix</a>
<li><a href="#org3d18192">1. Stiffness Matrix for the Cubic configuration</a>
<ul>
<li><a href="#orgf6f7ad2">1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center</a></li>
<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="#org95af62e">1.5. Conclusion</a></li>
<li><a href="#org778aab3">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="#org7673551">2.2. Conclusion</a></li>
<li><a href="#org8e940b8">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="#orgcad29fd">3.2. Conclusion</a></li>
<li><a href="#org9d42e84">3.2. Conclusion</a></li>
</ul>
</li>
<li><a href="#org5abef15">4. Dynamic Coupling</a>
<li><a href="#orgf09da67">4. Dynamic Coupling in the Cartesian Frame</a>
<ul>
<li><a href="#org0d67b92">4.1. Cube&rsquo;s center at the Center of Mass of the Payload</a></li>
<li><a href="#org876e05f">4.2. Dynamic decoupling between the actuators and sensors</a></li>
<li><a href="#org213d9b9">4.3. Conclusion</a></li>
<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="#org5214dec">4.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org3044455">5. Functions</a>
<li><a href="#org8f26dc0">5. Dynamic Coupling between actuators and sensors of each strut</a>
<ul>
<li><a href="#org56504f1">5.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<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="#org0936a8a">5.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org3044455">6. Functions</a>
<ul>
<li><a href="#org56504f1">6.1. <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</a>
<ul>
<li><a href="#orga5a9ba8">Function description</a></li>
<li><a href="#org3253792">Documentation</a></li>
@ -337,15 +344,13 @@ In this document, the cubic architecture is analyzed:
<li>In section <a href="#orgda0ee50">1</a>, we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix</li>
<li>In section <a href="#orgb73265d">2</a>, we study cubic configurations where the cube&rsquo;s center is located above the mobile platform</li>
<li>In section <a href="#org348ec7d">3</a>, we study the effect of the cube&rsquo;s size on the Stewart platform properties</li>
<li>In section <a href="#orgc379ec8">4</a>, we study the dynamic coupling of the cubic configuration</li>
<li>In section <a href="#org00d3816">4</a>, we study the dynamic coupling of the cubic configuration in the cartesian frame</li>
<li>In section <a href="#org5b5c8a9">5</a>, we study the dynamic coupling of the cubic configuration from actuators to sensors of each strut</li>
<li>In section <a href="#org28ba607">6</a>, function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function <code>generateCubicConfiguration</code> is used (described <a href="#orga8311d3">here</a>).</li>
</ul>
<p>
To generate and study the Stewart platform with a Cubic configuration, the Matlab function <code>generateCubicConfiguration</code> is used (described <a href="#orga8311d3">here</a>).
</p>
<div id="outline-container-org8c6677e" class="outline-2">
<h2 id="org8c6677e"><span class="section-number-2">1</span> Configuration Analysis - Stiffness Matrix</h2>
<div id="outline-container-org3d18192" class="outline-2">
<h2 id="org3d18192"><span class="section-number-2">1</span> Stiffness Matrix for the Cubic configuration</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgda0ee50"></a>
@ -831,8 +836,8 @@ stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'
</div>
</div>
<div id="outline-container-org95af62e" class="outline-3">
<h3 id="org95af62e"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-org778aab3" class="outline-3">
<h3 id="org778aab3"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@ -1157,8 +1162,8 @@ FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Cente
</div>
</div>
<div id="outline-container-org7673551" class="outline-3">
<h3 id="org7673551"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-org8e940b8" class="outline-3">
<h3 id="org8e940b8"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important">
<p>
@ -1232,8 +1237,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi
</div>
</div>
<div id="outline-container-orgcad29fd" class="outline-3">
<h3 id="orgcad29fd"><span class="section-number-3">3.2</span> Conclusion</h3>
<div id="outline-container-org9d42e84" class="outline-3">
<h3 id="org9d42e84"><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.
@ -1249,37 +1254,506 @@ In order to maximize the rotational stiffness of the Stewart platform, the size
</div>
</div>
<div id="outline-container-org5abef15" class="outline-2">
<h2 id="org5abef15"><span class="section-number-2">4</span> Dynamic Coupling</h2>
<div id="outline-container-orgf09da67" class="outline-2">
<h2 id="orgf09da67"><span class="section-number-2">4</span> Dynamic Coupling in the Cartesian Frame</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="orgc379ec8"></a>
<a id="org00d3816"></a>
</p>
<p>
In this section, we study the dynamics of the platform in the cartesian frame.
</p>
<p>
We here suppose that there is one relative motion sensor in each strut (\(\delta\bm{\mathcal{L}}\) is measured) and we would like to control the position of the top platform pose \(\delta \bm{\mathcal{X}}\).
</p>
<p>
Thanks to the Jacobian matrix, we can use the &ldquo;architecture&rdquo; shown in Figure <a href="#org76f24a0">9</a> to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.
</p>
<div class="org-src-container">
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate">\node</span>[block] (Jt) at (0, 0) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{J}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-T}$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right= of Jt] (G) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{G}$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right= of G] (J) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{J}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(Jt.west)+(-0.8, 0)$</span>) -- (Jt.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{F}}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Jt.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tau</span></span><span class="org-font-latex-math">}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G.east) -- (J.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\delta\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{L}}$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (J.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\delta\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{X}}$</span>};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="org76f24a0" class="figure">
<p><img src="figs/local_to_cartesian_coordinates.png" alt="local_to_cartesian_coordinates.png" />
</p>
<p><span class="figure-number">Figure 9: </span>From Strut coordinate to Cartesian coordinate using the Jacobian matrix</p>
</div>
<p>
We here study the dynamics from \(\bm{\mathcal{F}}\) to \(\delta\bm{\mathcal{X}}\).
</p>
<p>
One has to note that when considering the static behavior:
\[ \bm{G}(s = 0) = \begin{bmatrix}
1/k_1 & & 0 \\
& \ddots & 0 \\
0 & & 1/k_6
\end{bmatrix}\]
</p>
<p>
And thus:
\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]
</p>
<p>
We conclude that the <b>static</b> behavior of the platform depends on the stiffness matrix.
For the cubic configuration, we have a diagonal stiffness matrix is the frames \(\{A\}\) and \(\{B\}\) are coincident with the cube&rsquo;s center.
</p>
</div>
<div id="outline-container-org0d67b92" class="outline-3">
<h3 id="org0d67b92"><span class="section-number-3">4.1</span> Cube&rsquo;s center at the Center of Mass of the Payload</h3>
<div id="outline-container-org5fe01ec" class="outline-3">
<h3 id="org5fe01ec"><span class="section-number-3">4.1</span> Cube&rsquo;s center at the Center of Mass of the mobile platform</h3>
<div class="outline-text-3" id="text-4-1">
<p>
Let&rsquo;s create a Cubic Stewart Platform where the <b>Center of Mass of the mobile platform is located at the center of the cube</b>.
</p>
<p>
We define the size of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
</pre>
</div>
<div id="outline-container-org876e05f" class="outline-3">
<h3 id="org876e05f"><span class="section-number-3">4.2</span> Dynamic decoupling between the actuators and sensors</h3>
<p>
Now, we set the cube&rsquo;s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
</pre>
</div>
<div id="outline-container-org213d9b9" class="outline-3">
<h3 id="org213d9b9"><span class="section-number-3">4.3</span> Conclusion</h3>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
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>);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
</pre>
</div>
<p>
Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
<span class="org-string">'Mpm'</span>, 10, ...
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
</pre>
</div>
<p>
And we set small mass for the struts.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
stewart = initializeInertialSensor(stewart);
</pre>
</div>
<p>
The obtain geometry is shown in figure <a href="#orgc92a65b">10</a>.
</p>
<div id="orgc92a65b" class="figure">
<p><img src="figs/stewart_cubic_conf_decouple_dynamics.png" alt="stewart_cubic_conf_decouple_dynamics.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Geometry used for the simulations - The cube&rsquo;s center, the frames \(\{A\}\) and \(\{B\}\) and the Center of mass of the mobile platform are coincident (<a href="./figs/stewart_cubic_conf_decouple_dynamics.png">png</a>, <a href="./figs/stewart_cubic_conf_decouple_dynamics.pdf">pdf</a>)</p>
</div>
<p>
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'simulink/stewart_active_damping.slx'</span>)
<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_active_damping'</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; <span class="org-comment">% Actuator Force Inputs [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Dm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Displacement of each leg [m]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
</pre>
</div>
<p>
Now, thanks to the Jacobian (Figure <a href="#org76f24a0">9</a>), we compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
</pre>
</div>
<p>
The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure <a href="#orgcb3ac4d">11</a>.
</p>
<div id="orgcb3ac4d" class="figure">
<p><img src="figs/stewart_cubic_decoupled_dynamics_cartesian.png" alt="stewart_cubic_decoupled_dynamics_cartesian.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (<a href="./figs/stewart_cubic_decoupled_dynamics_cartesian.png">png</a>, <a href="./figs/stewart_cubic_decoupled_dynamics_cartesian.pdf">pdf</a>)</p>
</div>
<div class="important">
<p>
The dynamics is well decoupled at all frequencies.
</p>
<p>
We have the same dynamics for:
</p>
<ul class="org-ul">
<li>\(D_x/F_x\), \(D_y/F_y\) and \(D_z/F_z\)</li>
<li>\(R_x/M_x\) and \(D_y/F_y\)</li>
</ul>
<p>
The Dynamics from \(F_i\) to \(D_i\) is just a 1-dof mass-spring-damper system.
</p>
<p>
This is because the Mass, Damping and Stiffness matrices are all diagonal.
</p>
</div>
</div>
</div>
<div id="outline-container-org4cb2a36" class="outline-3">
<h3 id="org4cb2a36"><span class="section-number-3">4.2</span> Cube&rsquo;s center not coincident with the Mass of the Mobile platform</h3>
<div class="outline-text-3" id="text-4-2">
<p>
Let&rsquo;s create a Stewart platform with a cubic architecture where the cube&rsquo;s center is at the center of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
MO_B = <span class="org-type">-</span>100e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
</pre>
</div>
<p>
Now, we set the cube&rsquo;s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
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>);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
</pre>
</div>
<p>
However, the Center of Mass of the mobile platform is <b>not</b> located at the cube&rsquo;s center.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
<span class="org-string">'Mpm'</span>, 10, ...
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
</pre>
</div>
<p>
And we set small mass for the struts.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
stewart = initializeInertialSensor(stewart);
</pre>
</div>
<p>
The obtain geometry is shown in figure <a href="#orgfce7805">12</a>.
</p>
<div id="orgfce7805" class="figure">
<p><img src="figs/stewart_cubic_conf_mass_above.png" alt="stewart_cubic_conf_mass_above.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Geometry used for the simulations - The cube&rsquo;s center is coincident with the frames \(\{A\}\) and \(\{B\}\) but not with the Center of mass of the mobile platform (<a href="./figs/stewart_cubic_conf_mass_above.png">png</a>, <a href="./figs/stewart_cubic_conf_mass_above.pdf">pdf</a>)</p>
</div>
<p>
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">open(<span class="org-string">'simulink/stewart_active_damping.slx'</span>)
<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_active_damping'</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; <span class="org-comment">% Actuator Force Inputs [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Dm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Displacement of each leg [m]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, options);
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
</pre>
</div>
<p>
And we use the Jacobian to compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gc = inv(stewart.kinematics.J)<span class="org-type">*</span>G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>);
Gc.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
Gc.OutputName = {<span class="org-string">'Dx'</span>, <span class="org-string">'Dy'</span>, <span class="org-string">'Dz'</span>, <span class="org-string">'Rx'</span>, <span class="org-string">'Ry'</span>, <span class="org-string">'Rz'</span>};
</pre>
</div>
<p>
The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure <a href="#org7a04d45">13</a>.
</p>
<div id="org7a04d45" class="figure">
<p><img src="figs/stewart_conf_coupling_mass_matrix.png" alt="stewart_conf_coupling_mass_matrix.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Obtained Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (<a href="./figs/stewart_conf_coupling_mass_matrix.png">png</a>, <a href="./figs/stewart_conf_coupling_mass_matrix.pdf">pdf</a>)</p>
</div>
<div class="important">
<p>
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is <b>not</b> decoupled at all frequencies.
</p>
<p>
This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame \(\{B\}\)).
</p>
</div>
</div>
</div>
<div id="outline-container-org5214dec" class="outline-3">
<h3 id="org5214dec"><span class="section-number-3">4.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-3">
<div class="important">
<p>
Some conclusions can be drawn from the above analysis:
</p>
<ul class="org-ul">
<li>Static Decoupling &lt;=&gt; Diagonal Stiffness matrix &lt;=&gt; {A} and {B} at the cube&rsquo;s center</li>
<li>Dynamic Decoupling &lt;=&gt; Static Decoupling + CoM of mobile platform coincident with {A} and {B}.</li>
</ul>
</div>
</div>
</div>
</div>
<div id="outline-container-org8f26dc0" class="outline-2">
<h2 id="org8f26dc0"><span class="section-number-2">5</span> Dynamic Coupling between actuators and sensors of each strut</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org5b5c8a9"></a>
</p>
<p>
From <a class='org-ref-reference' href="#preumont07_six_axis_singl_stage_activ">preumont07_six_axis_singl_stage_activ</a>, the cubic configuration &ldquo;<i>minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)</i>&rdquo;.
</p>
<p>
In this section, we wish to study such properties of the cubic architecture.
</p>
<p>
We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).
</p>
</div>
<div id="outline-container-org6e391c9" class="outline-3">
<h3 id="org6e391c9"><span class="section-number-3">5.1</span> Coupling between the actuators and sensors - Cubic Architecture</h3>
<div class="outline-text-3" id="text-5-1">
<p>
Let&rsquo;s generate a Cubic architecture where the cube&rsquo;s center and the frames \(\{A\}\) and \(\{B\}\) are coincident.
</p>
<div class="org-src-container">
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
Hc = 2.5<span class="org-type">*</span>H; <span class="org-comment">% Size of the useful part of the cube [m]</span>
FOc = H <span class="org-type">+</span> MO_B; <span class="org-comment">% Center of the cube with respect to {F}</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
stewart = generateCubicConfiguration(stewart, <span class="org-string">'Hc'</span>, Hc, <span class="org-string">'FOc'</span>, FOc, <span class="org-string">'FHa'</span>, 25e<span class="org-type">-</span>3, <span class="org-string">'MHb'</span>, 25e<span class="org-type">-</span>3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
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>);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
<span class="org-string">'Mpm'</span>, 10, ...
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
stewart = initializeInertialSensor(stewart);
</pre>
</div>
<div id="org67d7284" class="figure">
<p><img src="figs/stewart_architecture_coupling_struts_cubic.png" alt="stewart_architecture_coupling_struts_cubic.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Geometry of the generated Stewart platform (<a href="./figs/stewart_architecture_coupling_struts_cubic.png">png</a>, <a href="./figs/stewart_architecture_coupling_struts_cubic.pdf">pdf</a>)</p>
</div>
<p>
And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure <a href="#orga20cd7d">15</a>) and to the force sensors \(\tau_{m,i}\) (Figure <a href="#org645e6c3">16</a>).
</p>
<div id="orga20cd7d" class="figure">
<p><img src="figs/coupling_struts_relative_sensor_cubic.png" alt="coupling_struts_relative_sensor_cubic.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Dynamics from the force actuators to the relative motion sensors (<a href="./figs/coupling_struts_relative_sensor_cubic.png">png</a>, <a href="./figs/coupling_struts_relative_sensor_cubic.pdf">pdf</a>)</p>
</div>
<div id="org645e6c3" class="figure">
<p><img src="figs/coupling_struts_force_sensor_cubic.png" alt="coupling_struts_force_sensor_cubic.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Dynamics from the force actuators to the force sensors (<a href="./figs/coupling_struts_force_sensor_cubic.png">png</a>, <a href="./figs/coupling_struts_force_sensor_cubic.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgafd808d" class="outline-3">
<h3 id="orgafd808d"><span class="section-number-3">5.2</span> Coupling between the actuators and sensors - Non-Cubic Architecture</h3>
<div class="outline-text-3" id="text-5-2">
<p>
Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">H = 200e<span class="org-type">-</span>3; <span class="org-comment">% height of the Stewart platform [m]</span>
MO_B = <span class="org-type">-</span>10e<span class="org-type">-</span>3; <span class="org-comment">% Position {B} with respect to {M} [m]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, H, <span class="org-string">'MO_B'</span>, MO_B);
stewart = generateGeneralConfiguration(stewart, <span class="org-string">'FR'</span>, 250e<span class="org-type">-</span>3, <span class="org-string">'MR'</span>, 150e<span class="org-type">-</span>3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'K'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'C'</span>, 1e1<span class="org-type">*</span>ones(6,1));
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>);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, <span class="org-string">'Fpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_F.Fa)), ...
<span class="org-string">'Mpm'</span>, 10, ...
<span class="org-string">'Mph'</span>, 20e<span class="org-type">-</span>3, ...
<span class="org-string">'Mpr'</span>, 1.2<span class="org-type">*</span>max(vecnorm(stewart.platform_M.Mb)));
stewart = initializeCylindricalStruts(stewart, <span class="org-string">'Fsm'</span>, 1e<span class="org-type">-</span>3, <span class="org-string">'Msm'</span>, 1e<span class="org-type">-</span>3);
stewart = initializeInertialSensor(stewart);
</pre>
</div>
<div id="org14d3492" class="figure">
<p><img src="figs/stewart_architecture_coupling_struts_non_cubic.png" alt="stewart_architecture_coupling_struts_non_cubic.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Geometry of the generated Stewart platform (<a href="./figs/stewart_architecture_coupling_struts_non_cubic.png">png</a>, <a href="./figs/stewart_architecture_coupling_struts_non_cubic.pdf">pdf</a>)</p>
</div>
<p>
And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure <a href="#orgff23a38">18</a>) and to the force sensors \(\tau_{m,i}\) (Figure <a href="#orgd802951">19</a>).
</p>
<div id="orgff23a38" class="figure">
<p><img src="figs/coupling_struts_relative_sensor_non_cubic.png" alt="coupling_struts_relative_sensor_non_cubic.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Dynamics from the force actuators to the relative motion sensors (<a href="./figs/coupling_struts_relative_sensor_non_cubic.png">png</a>, <a href="./figs/coupling_struts_relative_sensor_non_cubic.pdf">pdf</a>)</p>
</div>
<div id="orgd802951" class="figure">
<p><img src="figs/coupling_struts_force_sensor_non_cubic.png" alt="coupling_struts_force_sensor_non_cubic.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Dynamics from the force actuators to the force sensors (<a href="./figs/coupling_struts_force_sensor_non_cubic.png">png</a>, <a href="./figs/coupling_struts_force_sensor_non_cubic.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org0936a8a" class="outline-3">
<h3 id="org0936a8a"><span class="section-number-3">5.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-3">
<div class="important">
<p>
The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org3044455" class="outline-2">
<h2 id="org3044455"><span class="section-number-2">5</span> Functions</h2>
<div class="outline-text-2" id="text-5">
<h2 id="org3044455"><span class="section-number-2">6</span> Functions</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="org28ba607"></a>
</p>
</div>
<div id="outline-container-org56504f1" class="outline-3">
<h3 id="org56504f1"><span class="section-number-3">5.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
<div class="outline-text-3" id="text-5-1">
<h3 id="org56504f1"><span class="section-number-3">6.1</span> <code>generateCubicConfiguration</code>: Generate a Cubic Configuration</h3>
<div class="outline-text-3" id="text-6-1">
<p>
<a id="orga8311d3"></a>
</p>
@ -1323,7 +1797,7 @@ This Matlab function is accessible <a href="../src/generateCubicConfiguration.m"
<div id="org8a7f3d8" class="figure">
<p><img src="figs/cubic-configuration-definition.png" alt="cubic-configuration-definition.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Cubic Configuration</p>
<p><span class="figure-number">Figure 20: </span>Cubic Configuration</p>
</div>
</div>
</div>
@ -1428,7 +1902,7 @@ stewart.platform_M.Mb = Mb;
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-12 mer. 11:23</p>
<p class="date">Created: 2020-02-12 mer. 18:26</p>
</div>
</body>
</html>

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@ -52,11 +52,11 @@ In this document, the cubic architecture is analyzed:
- In section [[sec:cubic_conf_stiffness]], we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix
- In section [[sec:cubic_conf_above_platform]], we study cubic configurations where the cube's center is located above the mobile platform
- In section [[sec:cubic_conf_size_analysis]], we study the effect of the cube's size on the Stewart platform properties
- In section [[sec:cubic_conf_coupling]], we study the dynamic coupling of the cubic configuration
- In section [[sec:cubic_conf_coupling_cartesian]], we study the dynamic coupling of the cubic configuration in the cartesian frame
- In section [[sec:cubic_conf_coupling_struts]], we study the dynamic coupling of the cubic configuration from actuators to sensors of each strut
- In section [[sec:functions]], function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]).
To generate and study the Stewart platform with a Cubic configuration, the Matlab function =generateCubicConfiguration= is used (described [[sec:generateCubicConfiguration][here]]).
* Configuration Analysis - Stiffness Matrix
* Stiffness Matrix for the Cubic configuration
<<sec:cubic_conf_stiffness>>
** Introduction :ignore:
First, we have to understand what is the physical meaning of the Stiffness matrix $\bm{K}$.
@ -552,9 +552,47 @@ We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linear
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
#+end_important
* Dynamic Coupling
<<sec:cubic_conf_coupling>>
* Dynamic Coupling in the Cartesian Frame
<<sec:cubic_conf_coupling_cartesian>>
** Introduction :ignore:
In this section, we study the dynamics of the platform in the cartesian frame.
We here suppose that there is one relative motion sensor in each strut ($\delta\bm{\mathcal{L}}$ is measured) and we would like to control the position of the top platform pose $\delta \bm{\mathcal{X}}$.
Thanks to the Jacobian matrix, we can use the "architecture" shown in Figure [[fig:local_to_cartesian_coordinates]] to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.
#+begin_src latex :file local_to_cartesian_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports both
\begin{tikzpicture}
\node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$};
\node[block, right= of Jt] (G) {$\bm{G}$};
\node[block, right= of G] (J) {$\bm{J}^{-1}$};
\draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$};
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
\draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$};
\draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$};
\end{tikzpicture}
#+end_src
#+name: fig:local_to_cartesian_coordinates
#+caption: From Strut coordinate to Cartesian coordinate using the Jacobian matrix
#+RESULTS:
[[file:figs/local_to_cartesian_coordinates.png]]
We here study the dynamics from $\bm{\mathcal{F}}$ to $\delta\bm{\mathcal{X}}$.
One has to note that when considering the static behavior:
\[ \bm{G}(s = 0) = \begin{bmatrix}
1/k_1 & & 0 \\
& \ddots & 0 \\
0 & & 1/k_6
\end{bmatrix}\]
And thus:
\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]
We conclude that the *static* behavior of the platform depends on the stiffness matrix.
For the cubic configuration, we have a diagonal stiffness matrix is the frames $\{A\}$ and $\{B\}$ are coincident with the cube's center.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -569,13 +607,724 @@ We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linear
simulinkproject('../');
#+end_src
** Cube's center at the Center of Mass of the mobile platform
Let's create a Cubic Stewart Platform where the *Center of Mass of the mobile platform is located at the center of the cube*.
We define the size of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
H = 200e-3; % height of the Stewart platform [m]
MO_B = -10e-3; % Position {B} with respect to {M} [m]
#+end_src
** Cube's center at the Center of Mass of the Payload
Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
#+begin_src matlab
Hc = 2.5*H; % Size of the useful part of the cube [m]
FOc = H + MO_B; % Center of the cube with respect to {F}
#+end_src
** Dynamic decoupling between the actuators and sensors
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
#+end_src
Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with $\{A\}$ and $\{B\}$.
#+begin_src matlab
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
'Mpm', 10, ...
'Mph', 20e-3, ...
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
#+end_src
And we set small mass for the struts.
#+begin_src matlab
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
stewart = initializeInertialSensor(stewart);
#+end_src
The obtain geometry is shown in figure [[fig:stewart_cubic_conf_decouple_dynamics]].
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false, 'view', 'all');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_decouple_dynamics.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:stewart_cubic_conf_decouple_dynamics
#+caption: Geometry used for the simulations - The cube's center, the frames $\{A\}$ and $\{B\}$ and the Center of mass of the mobile platform are coincident ([[./figs/stewart_cubic_conf_decouple_dynamics.png][png]], [[./figs/stewart_cubic_conf_decouple_dynamics.pdf][pdf]])
[[file:figs/stewart_cubic_conf_decouple_dynamics.png]]
We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
#+begin_src matlab
open('simulink/stewart_active_damping.slx')
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_active_damping';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src
Now, thanks to the Jacobian (Figure [[fig:local_to_cartesian_coordinates]]), we compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
#+begin_src matlab
Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
#+end_src
The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_cubic_decoupled_dynamics_cartesian]].
#+begin_src matlab :exports none
freqs = logspace(1, 3, 500);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax3 = subplot(2, 2, 3);
hold on;
for i = 1:6
for j = i+1:6
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), 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]);
legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
ax2 = subplot(2, 2, 2);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:6
for j = i+1:6
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), 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]);
legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/stewart_cubic_decoupled_dynamics_cartesian.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:stewart_cubic_decoupled_dynamics_cartesian
#+caption: Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/stewart_cubic_decoupled_dynamics_cartesian.png][png]], [[./figs/stewart_cubic_decoupled_dynamics_cartesian.pdf][pdf]])
[[file:figs/stewart_cubic_decoupled_dynamics_cartesian.png]]
#+begin_important
The dynamics is well decoupled at all frequencies.
We have the same dynamics for:
- $D_x/F_x$, $D_y/F_y$ and $D_z/F_z$
- $R_x/M_x$ and $D_y/F_y$
The Dynamics from $F_i$ to $D_i$ is just a 1-dof mass-spring-damper system.
This is because the Mass, Damping and Stiffness matrices are all diagonal.
#+end_important
** Cube's center not coincident with the Mass of the Mobile platform
Let's create a Stewart platform with a cubic architecture where the cube's center is at the center of the Stewart platform.
#+begin_src matlab
H = 200e-3; % height of the Stewart platform [m]
MO_B = -100e-3; % Position {B} with respect to {M} [m]
#+end_src
Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
#+begin_src matlab
Hc = 2.5*H; % Size of the useful part of the cube [m]
FOc = H + MO_B; % Center of the cube with respect to {F}
#+end_src
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
#+end_src
However, the Center of Mass of the mobile platform is *not* located at the cube's center.
#+begin_src matlab
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
'Mpm', 10, ...
'Mph', 20e-3, ...
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
#+end_src
And we set small mass for the struts.
#+begin_src matlab
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
stewart = initializeInertialSensor(stewart);
#+end_src
The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]].
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false, 'view', 'all');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/stewart_cubic_conf_mass_above.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:stewart_cubic_conf_mass_above
#+caption: Geometry used for the simulations - The cube's center is coincident with the frames $\{A\}$ and $\{B\}$ but not with the Center of mass of the mobile platform ([[./figs/stewart_cubic_conf_mass_above.png][png]], [[./figs/stewart_cubic_conf_mass_above.pdf][pdf]])
[[file:figs/stewart_cubic_conf_mass_above.png]]
We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
#+begin_src matlab
open('simulink/stewart_active_damping.slx')
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_active_damping';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src
And we use the Jacobian to compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
#+begin_src matlab
Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
#+end_src
The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_conf_coupling_mass_matrix]].
#+begin_src matlab :exports none
freqs = logspace(1, 3, 500);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax3 = subplot(2, 2, 3);
hold on;
for i = 1:6
for j = i+1:6
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), 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]);
legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
ax2 = subplot(2, 2, 2);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:6
for j = i+1:6
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), 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]);
legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/stewart_conf_coupling_mass_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:stewart_conf_coupling_mass_matrix
#+caption: Obtained Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/stewart_conf_coupling_mass_matrix.png][png]], [[./figs/stewart_conf_coupling_mass_matrix.pdf][pdf]])
[[file:figs/stewart_conf_coupling_mass_matrix.png]]
#+begin_important
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is *not* decoupled at all frequencies.
This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame $\{B\}$).
#+end_important
** Conclusion
#+begin_important
Some conclusions can be drawn from the above analysis:
- Static Decoupling <=> Diagonal Stiffness matrix <=> {A} and {B} at the cube's center
- Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with {A} and {B}.
#+end_important
* Dynamic Coupling between actuators and sensors of each strut
<<sec:cubic_conf_coupling_struts>>
** Introduction :ignore:
From cite:preumont07_six_axis_singl_stage_activ, the cubic configuration "/minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)/".
In this section, we wish to study such properties of the cubic architecture.
We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).
** 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 :results none :exports none
simulinkproject('../');
#+end_src
** Coupling between the actuators and sensors - Cubic Architecture
Let's generate a Cubic architecture where the cube's center and the frames $\{A\}$ and $\{B\}$ are coincident.
#+begin_src matlab
H = 200e-3; % height of the Stewart platform [m]
MO_B = -10e-3; % Position {B} with respect to {M} [m]
Hc = 2.5*H; % Size of the useful part of the cube [m]
FOc = H + MO_B; % Center of the cube with respect to {F}
#+end_src
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
'Mpm', 10, ...
'Mph', 20e-3, ...
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
stewart = initializeInertialSensor(stewart);
#+end_src
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false, 'view', 'all');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/stewart_architecture_coupling_struts_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:stewart_architecture_coupling_struts_cubic
#+caption: Geometry of the generated Stewart platform ([[./figs/stewart_architecture_coupling_struts_cubic.png][png]], [[./figs/stewart_architecture_coupling_struts_cubic.pdf][pdf]])
[[file:figs/stewart_architecture_coupling_struts_cubic.png]]
And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure [[fig:coupling_struts_relative_sensor_cubic]]) and to the force sensors $\tau_{m,i}$ (Figure [[fig:coupling_struts_force_sensor_cubic]]).
#+begin_src matlab :exports none
open('simulink/stewart_active_damping.slx')
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_active_damping';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
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',[]);
ax3 = subplot(2, 1, 2);
hold on;
for i = 1:6
for j = i+1:6
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = 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]);
legend([p1, p2], {'$L_i/\tau_i$', '$L_i/\tau_j$'})
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/coupling_struts_relative_sensor_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:coupling_struts_relative_sensor_cubic
#+caption: Dynamics from the force actuators to the relative motion sensors ([[./figs/coupling_struts_relative_sensor_cubic.png][png]], [[./figs/coupling_struts_relative_sensor_cubic.pdf][pdf]])
[[file:figs/coupling_struts_relative_sensor_cubic.png]]
#+begin_src matlab :exports none
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 500);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
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 [N/N]'); set(gca, 'XTickLabel',[]);
ax3 = subplot(2, 1, 2);
hold on;
for i = 1:6
for j = i+1:6
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = 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]);
legend([p1, p2], {'$F_{m,i}/\tau_i$', '$F_{m,i}/\tau_j$'})
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/coupling_struts_force_sensor_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:coupling_struts_force_sensor_cubic
#+caption: Dynamics from the force actuators to the force sensors ([[./figs/coupling_struts_force_sensor_cubic.png][png]], [[./figs/coupling_struts_force_sensor_cubic.pdf][pdf]])
[[file:figs/coupling_struts_force_sensor_cubic.png]]
** Coupling between the actuators and sensors - Non-Cubic Architecture
Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.
#+begin_src matlab
H = 200e-3; % height of the Stewart platform [m]
MO_B = -10e-3; % Position {B} with respect to {M} [m]
#+end_src
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateGeneralConfiguration(stewart, 'FR', 250e-3, 'MR', 150e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
'Mpm', 10, ...
'Mph', 20e-3, ...
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
stewart = initializeInertialSensor(stewart);
#+end_src
#+begin_src matlab :exports none
displayArchitecture(stewart, 'labels', false, 'view', 'all');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/stewart_architecture_coupling_struts_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:stewart_architecture_coupling_struts_non_cubic
#+caption: Geometry of the generated Stewart platform ([[./figs/stewart_architecture_coupling_struts_non_cubic.png][png]], [[./figs/stewart_architecture_coupling_struts_non_cubic.pdf][pdf]])
[[file:figs/stewart_architecture_coupling_struts_non_cubic.png]]
And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure [[fig:coupling_struts_relative_sensor_non_cubic]]) and to the force sensors $\tau_{m,i}$ (Figure [[fig:coupling_struts_force_sensor_non_cubic]]).
#+begin_src matlab :exports none
open('simulink/stewart_active_damping.slx')
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_active_damping';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
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',[]);
ax3 = subplot(2, 1, 2);
hold on;
for i = 1:6
for j = i+1:6
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = 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]);
legend([p1, p2], {'$L_i/\tau_i$', '$L_i/\tau_j$'})
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/coupling_struts_relative_sensor_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:coupling_struts_relative_sensor_non_cubic
#+caption: Dynamics from the force actuators to the relative motion sensors ([[./figs/coupling_struts_relative_sensor_non_cubic.png][png]], [[./figs/coupling_struts_relative_sensor_non_cubic.pdf][pdf]])
[[file:figs/coupling_struts_relative_sensor_non_cubic.png]]
#+begin_src matlab :exports none
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Displacement of each leg [m]
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 500);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
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 [N/N]'); set(gca, 'XTickLabel',[]);
ax3 = subplot(2, 1, 2);
hold on;
for i = 1:6
for j = i+1:6
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'k-');
end
end
set(gca,'ColorOrderIndex',1);
p1 = 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]);
legend([p1, p2], {'$F_{m,i}/\tau_i$', '$F_{m,i}/\tau_j$'})
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/coupling_struts_force_sensor_non_cubic.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:coupling_struts_force_sensor_non_cubic
#+caption: Dynamics from the force actuators to the force sensors ([[./figs/coupling_struts_force_sensor_non_cubic.png][png]], [[./figs/coupling_struts_force_sensor_non_cubic.pdf][pdf]])
[[file:figs/coupling_struts_force_sensor_non_cubic.png]]
** Conclusion
#+begin_important
The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut.
#+end_important
* Functions
<<sec:functions>>

2
src/displayArchitecture.m
View File

@ -235,4 +235,6 @@ if strcmp(args.views, 'all')
axis equal;
axis off;
title('Side')
close(f);
end