Add new file about optimal stiffness / control
This commit is contained in:
779
org/optimal_stiffness_control.org
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779
org/optimal_stiffness_control.org
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#+TITLE: Control of the NASS with optimal stiffness
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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: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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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+ :results none
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:matlab+ :tangle no
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:shell :eval no-export
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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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* Introduction :ignore:
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* Low Authority Control - Decentralized Integral Force Feedback
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** Introduction :ignore:
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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 :tangle no
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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load('mat/conf_simulink.mat');
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open('nass_model.slx')
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#+end_src
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** Initialization
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We initialize all the stages with the default parameters.
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#+begin_src matlab
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initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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#+end_src
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We set the references that corresponds to a tomography experiment.
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#+begin_src matlab
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 1);
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initializeSimscapeConfiguration();
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initializeDisturbances('enable', false);
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initializeLoggingConfiguration('log', 'none');
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#+end_src
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#+begin_src matlab
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initializeController('type', 'hac-iff');
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#+end_src
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** Identification
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#+begin_src matlab
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Kx = tf(zeros(6));
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Kiff = tf(zeros(6));
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#+end_src
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#+begin_src matlab
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Ms = [1, 10, 50];
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Gm_iff = {zeros(length(Ms), 1)};
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#+end_src
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#+begin_src matlab
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initializeNanoHexapod('k', 1e5, 'c', 2e2);
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'nass_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 Inputs
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io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
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#+end_src
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#+begin_src matlab :exports none
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for i = 1:length(Ms)
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initializeSample('mass', Ms(i), 'freq', 200*ones(6,1));
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%% Run the linearization
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G_iff = linearize(mdl, io);
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G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
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Gm_iff(i) = {G_iff};
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end
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#+end_src
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** Controller Design
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#+begin_src matlab :exports none
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freqs = logspace(-1, 3, 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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for i = 1:length(Ms)
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plot(freqs, abs(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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title('Diagonal elements of the Plant');
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 1:length(Ms)
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz')))), ...
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'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
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end
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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([-270, 90]);
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yticks([-360:90:360]);
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legend('location', 'northeast');
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linkaxes([ax1,ax2],'x');
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#+end_src
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Root Locus
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#+begin_src matlab :exports none :post
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figure;
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gains = logspace(0, 3, 300);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(real(pole(Gm_iff{i})), imag(pole(Gm_iff{i})), 'x', ...
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'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
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set(gca,'ColorOrderIndex',i);
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plot(real(tzero(Gm_iff{i})), imag(tzero(Gm_iff{i})), 'o', ...
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'HandleVisibility', 'off');
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for k = 1:length(gains)
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set(gca,'ColorOrderIndex',i);
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cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
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plot(real(cl_poles), imag(cl_poles), '.', ...
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'HandleVisibility', 'off');
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end
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end
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hold off;
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axis square;
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xlim([-140, 10]); ylim([0, 150]);
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xlabel('Real Part'); ylabel('Imaginary Part');
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legend('location', 'northwest');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/opt_stiff_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:opt_stiff_iff_root_locus
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#+caption: Root Locus for the
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#+RESULTS:
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[[file:figs/opt_stiff_iff_root_locus.png]]
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Damping as function of the gain
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#+begin_src matlab :exports none
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c1 = [ 0 0.4470 0.7410]; % Blue
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c2 = [0.8500 0.3250 0.0980]; % Orange
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c3 = [0.9290 0.6940 0.1250]; % Yellow
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c4 = [0.4940 0.1840 0.5560]; % Purple
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c5 = [0.4660 0.6740 0.1880]; % Green
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c6 = [0.3010 0.7450 0.9330]; % Light Blue
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c7 = [0.6350 0.0780 0.1840]; % Red
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colors = [c1; c2; c3; c4; c5; c6; c7];
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figure;
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gains = logspace(0, 3, 100);
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hold on;
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for i = 1:length(Ms)
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for k = 1:length(gains)
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cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
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set(gca,'ColorOrderIndex',i);
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plot(gains(k), sin(-pi/2 + angle(cl_poles)), '.', 'color', colors(i, :));
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end
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end
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hold off;
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xlabel('IFF Gain'); ylabel('Modal Damping');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylim([0, 1]);
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#+end_src
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#+begin_src matlab
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Kiff = -200/s*eye(6);
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#+end_src
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* Primary Control
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** Introduction :ignore:
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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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** Identification
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#+begin_src matlab
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Gm_x = {zeros(length(Ms), 1)};
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Gm_l = {zeros(length(Ms), 1)};
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#+end_src
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#+begin_src matlab
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load('mat/stages.mat', 'nano_hexapod');
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'nass_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, 'input'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
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#+end_src
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#+begin_src matlab :exports none
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for i = 1:length(Ms)
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initializeSample('mass', Ms(i), 'freq', 200*ones(6,1));
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%% Run the linearization
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G = linearize(mdl, io);
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G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
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Gx = -G*inv(nano_hexapod.J');
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gm_x(i) = {Gx};
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Gl = -nano_hexapod.J*G;
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Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
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Gm_l(i) = {Gl};
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end
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#+end_src
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** Controller in the task space
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))));
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end
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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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title('Diagonal elements of the Plant');
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ax2 = subplot(2, 2, 3);
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hold on;
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'DisplayName', labels{i});
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end
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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();
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ax3 = subplot(2, 2, 2);
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hold on;
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for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Gx(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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title('Off-Diagonal elements of the Plant');
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ax4 = subplot(2, 2, 4);
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hold on;
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for i = 1:5
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for j = i+1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(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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linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
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*** Translation
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(Gm_x{i}(1, 1), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(Gm_x{i}(2, 2), freqs, 'Hz'))));
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end
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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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title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
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ax2 = subplot(2, 2, 2);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(Gm_x{i}(3, 3), freqs, 'Hz'))));
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end
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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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title('$\mathcal{X}_z/\mathcal{F}_z$')
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ax3 = subplot(2, 2, 3);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(1, 1), freqs, 'Hz')))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(2, 2), freqs, 'Hz')))));
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end
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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([-270, 90]);
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||||
yticks([-360:90:360]);
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||||
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||||
ax4 = subplot(2, 2, 4);
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hold on;
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for i = 1:length(Ms)
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||||
set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(3, 3), freqs, 'Hz')))), ...
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'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
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||||
end
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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([-270, 90]);
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||||
yticks([-360:90:360]);
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||||
legend('location', 'southwest');
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||||
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||||
linkaxes([ax1,ax2,ax3,ax4],'x');
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||||
#+end_src
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||||
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||||
#+begin_src matlab
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||||
Kx = tf(zeros(6));
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||||
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||||
h = 1.5;
|
||||
Kx(1,1) = 2e6 * ...
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||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||
|
||||
Kx(2,2) = Kx(1,1);
|
||||
|
||||
h = 1.5;
|
||||
Kx(3,3) = 1e7 * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(1, 1)*Kx(1,1), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(2, 2)*Kx(2,2), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
|
||||
|
||||
ax2 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(3, 3)*Kx(3,3), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_z/\mathcal{F}_z$')
|
||||
|
||||
ax3 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(1, 1)*Kx(1,1), freqs, 'Hz')))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(2, 2)*Kx(2,2), freqs, 'Hz')))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(3, 3)*Kx(3,3), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
*** Rotations
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(4, 4), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(5, 5), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_{R_x}/\mathcal{M}_x$, $\mathcal{X}_{R_y}/\mathcal{M}_y$')
|
||||
|
||||
ax2 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(6, 6), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_{R_z}/\mathcal{M}_z$')
|
||||
|
||||
ax3 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(4, 4), freqs, 'Hz')))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(5, 5), freqs, 'Hz')))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(6, 6), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
h = 1.5;
|
||||
Kx(4,4) = 1e5 * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||
|
||||
Kx(5,5) = Kx(4,4);
|
||||
|
||||
h = 1.5;
|
||||
Kx(6,6) = 2e5 * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(4, 4)*Kx(4,4), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(5, 5)*Kx(5,5), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_{R_x}/\mathcal{M}_x$, $\mathcal{X}_{R_y}/\mathcal{M}_y$')
|
||||
|
||||
ax2 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_x{i}(6, 6)*Kx(6,6), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(N m)]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_{R_z}/\mathcal{M}_z$')
|
||||
|
||||
ax3 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(4, 4)*Kx(4,4), freqs, 'Hz')))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(5, 5)*Kx(5,5), freqs, 'Hz')))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_x{i}(6, 6)*Kx(6,6), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
*** Stability
|
||||
#+begin_src matlab
|
||||
for i = 1:length(Ms)
|
||||
isstable(feedback(Gm_x{i}*Kx, eye(6), -1))
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Simulation
|
||||
|
||||
** Control in the leg space
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('Diagonal elements of the Plant');
|
||||
|
||||
ax2 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$d\\mathcal{L}_%i / \\tau_%i$', i, i));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
legend();
|
||||
|
||||
ax3 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('Off-Diagonal elements of the Plant');
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
for j = 1:6
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j), freqs, 'Hz'))));
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('Diagonal elements of the Plant');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
for j = 1:6
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j), freqs, 'Hz'))));
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
h = 1.5;
|
||||
Kl = 5e6 * eye(6) * ...
|
||||
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
||||
(s/2/pi/1 + 1)/(s/2/pi/1) * ...
|
||||
(s/2/pi/10 + 1)/(s/2/pi/10);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:length(Ms)
|
||||
isstable(feedback(Gm_l{i}(1,1)*Kl(1,1), 1, -1))
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
for j = 1:6
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
for j = 1:6
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
|
||||
* Simulations
|
||||
@@ -1240,6 +1240,6 @@ data2orgtable([wb1'; wb2'], {'Required wc with L1 [Hz]', 'Required wc with L2 [H
|
||||
#+begin_important
|
||||
From Figure [[fig:opt_stiff_req_bandwidth_K1_K2]] and Table [[tab:approx_required_wc_10nm]], we can clearly see three different results depending on the nano-hexapod stiffness:
|
||||
- For a soft nano-hexapod ($k < 10^4\ [N/m]$), the required bandwidth is $\omega_c \approx 50-100\ Hz$
|
||||
- For a nano-hexapods with $10^5 < k < 10^6\ [N/m]$), the required bandwidth is $\omega_c \approx 150-300\ Hz$
|
||||
- For a nano-hexapods with $10^5 < k < 10^6\ [N/m]$, the required bandwidth is $\omega_c \approx 150-300\ Hz$
|
||||
- For a stiff nano-hexapods ($k > 10^7\ [N/m]$), the required bandwidth is $\omega_c \approx 250-500\ Hz$
|
||||
#+end_important
|
||||
|
||||
Reference in New Issue
Block a user