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Thomas Dehaeze 2020-02-13 15:37:06 +01:00
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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>
<!-- 2020-02-13 jeu. 15:19 -->
<!-- 2020-02-13 jeu. 15:36 -->
<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>
@ -272,13 +272,13 @@ for the JavaScript code in this tag.
<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="#orgb87f273">1.3. Conclusion</a></li>
<li><a href="#org230655f">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="#org230655f">2.2. Conclusion</a></li>
<li><a href="#org1cbdf9a">2.2. Conclusion</a></li>
</ul>
</li>
</ul>
@ -288,10 +288,17 @@ for the JavaScript code in this tag.
<div id="outline-container-orgc59e712" class="outline-2">
<h2 id="orgc59e712"><span class="section-number-2">1</span> Compare external forces and forces applied by the actuators</h2>
<div class="outline-text-2" id="text-1">
<p>
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
</p>
</div>
<div id="outline-container-org4509b7d" class="outline-3">
<h3 id="org4509b7d"><span class="section-number-3">1.1</span> Comparison with fixed support</h3>
<div class="outline-text-3" id="text-1-1">
<p>
Let&rsquo;s generate a Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
@ -307,6 +314,10 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
</pre>
</div>
<p>
We don&rsquo;t put any flexibility below the Stewart platform such that <b>its base is fixed to an inertial frame</b>.
We also don&rsquo;t put any payload on top of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
@ -314,7 +325,7 @@ payload = initializePayload(<span class="org-string">'type'</span>, <span class=
</div>
<p>
Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):
The transfer function from actuator forces \(\bm{\tau}\) to the relative displacement of the mobile platform \(\mathcal{\bm{X}}\) is extracted.
</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>
@ -336,6 +347,9 @@ G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">
</pre>
</div>
<p>
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame \(\{B\}\) fixed to the mobile platform:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
@ -343,7 +357,7 @@ Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">
</div>
<p>
Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):
We also extract the transfer function from external forces \(\bm{\mathcal{F}}_{\text{ext}}\) on the frame \(\{B\}\) fixed to the mobile platform to the relative displacement \(\mathcal{\bm{X}}\) of \(\{B\}\) with respect to frame \(\{A\}\):
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
@ -357,6 +371,17 @@ Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string"
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
</pre>
</div>
<p>
The comparison of the two transfer functions is shown in Figure <a href="#orgbf9a54a">1</a>.
</p>
<div id="orgbf9a54a" class="figure">
<p><img src="figs/comparison_Fext_F_fixed_base.png" alt="comparison_Fext_F_fixed_base.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_fixed_base.png">png</a>, <a href="./figs/comparison_Fext_F_fixed_base.pdf">pdf</a>)</p>
</div>
</div>
</div>
@ -364,7 +389,7 @@ Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string"
<h3 id="org8662186"><span class="section-number-3">1.2</span> Comparison with a flexible support</h3>
<div class="outline-text-3" id="text-1-2">
<p>
We redo the identification for when the Stewart platform is on a flexible support.
We now add a flexible support under the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'flexible'</span>);
@ -372,17 +397,10 @@ We redo the identification for when the Stewart platform is on a flexible suppor
</div>
<p>
Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):
And we perform again the identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = 0;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'stewart_platform_model'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Controller'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
@ -391,20 +409,11 @@ io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>
G = linearize(mdl, io, options);
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
Gc.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
</pre>
</div>
<p>
Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
@ -415,11 +424,22 @@ Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string"
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
</pre>
</div>
<p>
The comparison between the obtained transfer functions is shown in Figure <a href="#orga2f2bd5">2</a>.
</p>
<div id="orga2f2bd5" class="figure">
<p><img src="figs/comparison_Fext_F_flexible_base.png" alt="comparison_Fext_F_flexible_base.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_flexible_base.png">png</a>, <a href="./figs/comparison_Fext_F_flexible_base.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgb87f273" class="outline-3">
<h3 id="orgb87f273"><span class="section-number-3">1.3</span> Conclusion</h3>
<div id="outline-container-org230655f" class="outline-3">
<h3 id="org230655f"><span class="section-number-3">1.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-3">
<div class="important">
<p>
@ -434,7 +454,11 @@ The transfer function from forces/torques applied by the actuators on the payloa
<div id="outline-container-org81ab204" class="outline-2">
<h2 id="org81ab204"><span class="section-number-2">2</span> Comparison of the static transfer function and the Compliance matrix</h2>
<div class="outline-text-2" id="text-2">
<p>
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
</p>
</div>
<div id="outline-container-orge7e7242" class="outline-3">
<h3 id="orge7e7242"><span class="section-number-3">2.1</span> Analysis</h3>
<div class="outline-text-3" id="text-2-1">
@ -456,6 +480,9 @@ stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</spa
</pre>
</div>
<p>
No flexibility below the Stewart platform and no payload.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
@ -650,8 +677,8 @@ And now at the Compliance matrix.
</div>
</div>
<div id="outline-container-org230655f" class="outline-3">
<h3 id="org230655f"><span class="section-number-3">2.2</span> Conclusion</h3>
<div id="outline-container-org1cbdf9a" class="outline-3">
<h3 id="org1cbdf9a"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important">
<p>
@ -665,7 +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: 2020-02-13 jeu. 15:19</p>
<p class="date">Created: 2020-02-13 jeu. 15:36</p>
</div>
</body>
</html>

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@ -40,6 +40,8 @@
* Compare external forces and forces applied by the actuators
** Introduction :ignore:
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
** 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>>
@ -58,6 +60,7 @@
#+end_src
** Comparison with fixed support
Let's generate a Stewart platform.
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
@ -72,12 +75,14 @@
stewart = initializeInertialSensor(stewart, 'type', 'none');
#+end_src
We don't put any flexibility below the Stewart platform such that *its base is fixed to an inertial frame*.
We also don't put any payload on top of the Stewart platform.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
#+end_src
Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
The transfer function from actuator forces $\bm{\tau}$ to the relative displacement of the mobile platform $\mathcal{\bm{X}}$ is extracted.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
@ -97,12 +102,13 @@ Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame $\{B\}$ fixed to the mobile platform:
#+begin_src matlab
Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src
Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$:
We also extract the transfer function from external forces $\bm{\mathcal{F}}_{\text{ext}}$ on the frame $\{B\}$ fixed to the mobile platform to the relative displacement $\mathcal{\bm{X}}$ of $\{B\}$ with respect to frame $\{A\}$:
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
@ -115,6 +121,8 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
The comparison of the two transfer functions is shown in Figure [[fig:comparison_Fext_F_fixed_base]].
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
@ -126,7 +134,7 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
plot(freqs, abs(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
@ -137,26 +145,28 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend({'$\mathcal{X}_{x}/\mathcal{F}_{x}$', '$\mathcal{X}_{x}/\mathcal{F}_{x,ext}$'});
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/comparison_Fext_F_fixed_base.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:comparison_Fext_F_fixed_base
#+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_fixed_base.png][png]], [[./figs/comparison_Fext_F_fixed_base.pdf][pdf]])
[[file:figs/comparison_Fext_F_fixed_base.png]]
** Comparison with a flexible support
We redo the identification for when the Stewart platform is on a flexible support.
We now add a flexible support under the Stewart platform.
#+begin_src matlab
ground = initializeGround('type', 'flexible');
#+end_src
Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
And we perform again the identification.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
@ -166,15 +176,10 @@ Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src
Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$:
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
@ -186,6 +191,8 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
The comparison between the obtained transfer functions is shown in Figure [[fig:comparison_Fext_F_flexible_base]].
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
@ -197,7 +204,7 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
plot(freqs, abs(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
@ -208,10 +215,20 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend({'$\mathcal{X}_{x}/\mathcal{F}_{x}$', '$\mathcal{X}_{x}/\mathcal{F}_{x,ext}$'});
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/comparison_Fext_F_flexible_base.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:comparison_Fext_F_flexible_base
#+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_flexible_base.png][png]], [[./figs/comparison_Fext_F_flexible_base.pdf][pdf]])
[[file:figs/comparison_Fext_F_flexible_base.png]]
** Conclusion
#+begin_important
The transfer function from forces/torques applied by the actuators on the payload $\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}$ to the pose of the mobile platform $\bm{\mathcal{X}}$ is the same as the transfer function from external forces/torques to $\bm{\mathcal{X}}$ as long as the Stewart platform's base is fixed.
@ -219,6 +236,8 @@ The transfer function from forces/torques applied by the actuators on the payloa
* Comparison of the static transfer function and the Compliance matrix
** Introduction :ignore:
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.
** 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>>
@ -252,6 +271,7 @@ Initialization of the Stewart platform.
stewart = initializeInertialSensor(stewart, 'type', 'none');
#+end_src
No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');