389 lines
16 KiB
Org Mode
389 lines
16 KiB
Org Mode
#+TITLE: Stewart Platform - Dynamics Study
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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* Compare external forces and forces applied by the actuators
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** Introduction :ignore:
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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.
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
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** Comparison with fixed support
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Let's generate a Stewart platform.
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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#+end_src
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We don't put any flexibility below the Stewart platform such that *its base is fixed to an inertial frame*.
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We also don't put any payload on top of the Stewart platform.
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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controller = initializeController('type', 'open-loop');
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#+end_src
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The transfer function from actuator forces $\bm{\tau}$ to the relative displacement of the mobile platform $\mathcal{\bm{X}}$ is extracted.
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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#+end_src
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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:
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#+begin_src matlab
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Gc = minreal(G*inv(stewart.kinematics.J'));
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Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
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#+end_src
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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 $\{A\}$:
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#+begin_src matlab
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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Gd = linearize(mdl, io, options);
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Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
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Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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#+end_src
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The comparison of the two transfer functions is shown in Figure [[fig:comparison_Fext_F_fixed_base]].
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#+begin_src matlab :exports none
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freqs = logspace(1, 4, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(Gc(1,1), freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1,1), freqs, 'Hz'))), '-');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend({'$\mathcal{X}_{x}/\mathcal{F}_{x}$', '$\mathcal{X}_{x}/\mathcal{F}_{x,ext}$'});
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/comparison_Fext_F_fixed_base.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:comparison_Fext_F_fixed_base
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#+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]])
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[[file:figs/comparison_Fext_F_fixed_base.png]]
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This can be understood from figure [[fig:1dof_actuator_external_forces]] where $\mathcal{F}_{x}$ and $\mathcal{F}_{x,\text{ext}}$ have clearly the same effect on $\mathcal{X}_{x}$.
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#+begin_src latex :file 1dof_actuator_external_forces.pdf
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\begin{tikzpicture}
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\draw[ground] (-1, 0) -- (1, 0);
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\draw[spring] (-0.6, 0) -- (-0.6, 1.5) node[midway, left=0.1]{$k$};
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\draw[actuator] ( 0.6, 0) -- ( 0.6, 1.5) node[midway, left=0.1](F){$\mathcal{F}_{x}$};
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\draw[fill=white] (-1, 1.5) rectangle (1, 2) node[pos=0.5]{$m$};
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\draw[dashed] (1, 2) -- ++(0.5, 0);
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\draw[->] (1.5, 2) -- ++(0, 0.5) node[right]{$\mathcal{X}_{x}$};
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\draw[->] (0, 2) node[]{$\bullet$} -- (0, 2.8) node[below right]{$\mathcal{F}_{x,\text{ext}}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:1dof_actuator_external_forces
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#+caption: Schematic representation of the stewart platform on a rigid support
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#+RESULTS:
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[[file:figs/1dof_actuator_external_forces.png]]
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** Comparison with a flexible support
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We now add a flexible support under the Stewart platform.
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#+begin_src matlab
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ground = initializeGround('type', 'flexible');
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#+end_src
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And we perform again the identification.
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#+begin_src matlab
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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Gc = minreal(G*inv(stewart.kinematics.J'));
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Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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Gd = linearize(mdl, io, options);
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Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
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Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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#+end_src
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The comparison between the obtained transfer functions is shown in Figure [[fig:comparison_Fext_F_flexible_base]].
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#+begin_src matlab :exports none
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freqs = logspace(1, 4, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(Gc(1,1), freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1,1), freqs, 'Hz'))), '-');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend({'$\mathcal{X}_{x}/\mathcal{F}_{x}$', '$\mathcal{X}_{x}/\mathcal{F}_{x,ext}$'});
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/comparison_Fext_F_flexible_base.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:comparison_Fext_F_flexible_base
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#+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]])
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[[file:figs/comparison_Fext_F_flexible_base.png]]
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The addition of a flexible support can be schematically represented in Figure [[fig:2dof_actuator_external_forces]].
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We see that $\mathcal{F}_{x}$ applies a force both on $m$ and $m^{\prime}$ whereas $\mathcal{F}_{x,\text{ext}}$ only applies a force on $m$.
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And thus $\mathcal{F}_{x}$ and $\mathcal{F}_{x,\text{ext}}$ have clearly *not* the same effect on $\mathcal{X}_{x}$.
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#+begin_src latex :file 2dof_actuator_external_forces.pdf
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\begin{tikzpicture}
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\draw[ground] (-1, 0) -- (1, 0);
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\draw[spring] (0, 0) -- (0, 1.5) node[midway, left=0.1]{$k^{\prime}$};
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\draw[fill=white] (-1, 1.5) rectangle (1, 2) node[pos=0.5]{$m^{\prime}$};
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\draw[spring] (-0.6, 2) -- (-0.6, 3.5) node[midway, left=0.1]{$k$};
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\draw[actuator] ( 0.6, 2) -- ( 0.6, 3.5) node[midway, left=0.1](F){$\mathcal{F}_{x}$};
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\draw[fill=white] (-1, 3.5) rectangle (1, 4) node[pos=0.5]{$m$};
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\draw[dashed] (1, 4) -- ++(0.5, 0);
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\draw[->] (1.5, 4) -- ++(0, 0.5) node[right]{$\mathcal{X}_{x}$};
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\draw[->] (0, 4) node[]{$\bullet$} -- (0, 4.8) node[below right]{$\mathcal{F}_{x,\text{ext}}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:2dof_actuator_external_forces
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#+caption: Schematic representation of the stewart platform on top of a flexible support
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#+RESULTS:
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[[file:figs/2dof_actuator_external_forces.png]]
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** Conclusion
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#+begin_important
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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.
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#+end_important
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* Comparison of the static transfer function and the Compliance matrix
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** Introduction :ignore:
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In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
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** Analysis
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Initialization of the Stewart platform.
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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#+end_src
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No flexibility below the Stewart platform and no payload.
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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controller = initializeController('type', 'open-loop');
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#+end_src
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Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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#+end_src
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#+begin_src matlab
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Gc = minreal(G*inv(stewart.kinematics.J'));
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Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
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#+end_src
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Let's first look at the low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(real(freqresp(Gd, 0.1)), {}, {}, ' %.1e ');
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#+end_src
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#+RESULTS:
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| 4.7e-08 | -7.2e-19 | 5.0e-18 | -8.9e-18 | 3.2e-07 | 9.9e-18 |
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| 4.7e-18 | 4.7e-08 | -5.7e-18 | -3.2e-07 | -1.6e-17 | -1.7e-17 |
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| 3.3e-18 | -6.3e-18 | 2.1e-08 | 4.4e-17 | 6.6e-18 | 7.4e-18 |
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| -3.2e-17 | -3.2e-07 | 6.2e-18 | 5.2e-06 | -3.5e-16 | 6.3e-17 |
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| 3.2e-07 | 2.7e-17 | 4.8e-17 | -4.5e-16 | 5.2e-06 | -1.2e-19 |
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| 4.0e-17 | -9.5e-17 | 8.4e-18 | 4.3e-16 | 5.8e-16 | 1.7e-06 |
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And now at the Compliance matrix.
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(stewart.kinematics.C, {}, {}, ' %.1e ');
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#+end_src
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#+RESULTS:
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| 4.7e-08 | -2.0e-24 | 7.4e-25 | 5.9e-23 | 3.2e-07 | 5.9e-24 |
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| -7.1e-25 | 4.7e-08 | 2.9e-25 | -3.2e-07 | -5.4e-24 | -3.3e-23 |
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| 7.9e-26 | -6.4e-25 | 2.1e-08 | 1.9e-23 | 5.3e-25 | -6.5e-40 |
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| 1.4e-23 | -3.2e-07 | 1.3e-23 | 5.2e-06 | 4.9e-22 | -3.8e-24 |
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| 3.2e-07 | 7.6e-24 | 1.2e-23 | 6.9e-22 | 5.2e-06 | -2.6e-22 |
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| 7.3e-24 | -3.2e-23 | -1.6e-39 | 9.9e-23 | -3.3e-22 | 1.7e-06 |
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** Conclusion
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#+begin_important
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The low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$ corresponds to the compliance matrix of the Stewart platform.
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#+end_important
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