582 lines
16 KiB
Org Mode
582 lines
16 KiB
Org Mode
#+TITLE: HAC-LAC applied on the Simscape Model
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#+SETUPFILE: ./setup/org-setup-file.org
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* Introduction :ignore:
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The position $\bm{\mathcal{X}}$ of the Sample with respect to the granite is measured.
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It is then compare to the wanted position of the Sample $\bm{r}_\mathcal{X}$ in order to obtain the position error $\bm{\epsilon}_\mathcal{X}$ of the Sample with respect to a frame attached to the Stewart top platform.
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#+begin_src latex :file hac_lac_control_schematic.pdf
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\begin{tikzpicture}
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\node[block={3.0cm}{3.0cm}] (G) {Plant};
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% Input and outputs coordinates
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\coordinate[] (outputX) at ($(G.south east)!0.25!(G.north east)$);
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\coordinate[] (outputL) at ($(G.south east)!0.75!(G.north east)$);
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\draw[->] (outputX) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{X}}$};
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\draw[->] (outputL) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{L}}$};
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% Blocs
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\node[addb, left= of G] (addF) {};
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\node[block, left=1.2 of addF] (Kx) {$\bm{K}_\mathcal{X}$};
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\node[block={2cm}{2cm}, align=center, left=1.2 of Kx] (subx) {Computes\\Position\\Error};
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\node[block, above= of addF] (Kl) {$\bm{K}_\mathcal{L}$};
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\node[addb={+}{}{}{-}{}, above= of Kl] (subl) {};
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\node[block, align=center, left=0.8 of subl] (invK) {Inverse\\Kinematics};
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% Connections and labels
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\draw[<-] (subx.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-0.8, 0);
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\draw[->] ($(subx.east) + (0.2, 0)$)node[branch]{} |- (invK.west);
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\draw[->] (invK.east) -- (subl.west) node[above left]{$\bm{r}_\mathcal{L}$};
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\draw[->] (subl.south) -- (Kl.north) node[above right]{$\bm{\epsilon}_\mathcal{L}$};
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\draw[->] (Kl.south) -- (addF.north);
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\draw[->] (subx.east) -- (Kx.west) node[above left]{$\bm{\epsilon}_\mathcal{X}$};
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\draw[->] (Kx.east) node[above right]{$\bm{\tau}^\prime$} -- (addF.west);
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\draw[->] (addF.east) -- (G.west) node[above left]{$\bm{\tau}$};
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\draw[->] ($(outputL.east) + (0.4, 0)$)node[branch](L){} |- (subl.east);
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\draw[->] ($(outputX.east) + (1.2, 0)$)node[branch]{} -- ++(0, -1.6) -| (subx.south);
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\begin{scope}[on background layer]
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\node[fit={(G.south-|Kl.west) (L|-subl.north)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Ktot) {};
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\end{scope}
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\end{tikzpicture}
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#+end_src
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#+name: fig:hac_lac_control_schematic
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#+caption: HAC-LAC Control Architecture used for the Control of the NASS
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#+RESULTS:
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[[file:figs/hac_lac_control_schematic.png]]
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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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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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The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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#+begin_src matlab
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initializeNanoHexapod('actuator', 'piezo');
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initializeSample('mass', 1);
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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', 'Rz_period', 1);
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#+end_src
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#+begin_src matlab
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initializeDisturbances();
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#+end_src
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Open Loop.
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#+begin_src matlab
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initializeController('type', 'open-loop');
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#+end_src
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And we put some gravity.
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#+begin_src matlab
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initializeSimscapeConfiguration('gravity', true);
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#+end_src
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We log the signals.
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#+begin_src matlab
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initializeLoggingConfiguration('log', 'all');
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#+end_src
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* Low Authority Control - Direct Velocity Feedback $\bm{K}_\mathcal{L}$
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** Introduction :ignore:
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The first loop closed corresponds to a direct velocity feedback loop.
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The design of the associated decentralized controller is explained in [[file:control_active_damping.org][this]] file.
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** Identification
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#+begin_src matlab
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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', [], 'Dnlm'); io_i = io_i + 1; % Relative Motion Outputs
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%% Run the linearization
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G_dvf = linearize(mdl, io, 0);
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G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
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#+end_src
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** Plant
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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 = 1:6
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plot(freqs, abs(squeeze(freqresp(G_dvf(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(G_dvf(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$d\\mathcal{L}_%i/\\tau_%i$', i, 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('location', 'northwest');
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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(G_dvf(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(G_dvf(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(G_dvf(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(G_dvf(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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** Root Locus
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#+begin_src matlab :exports none
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gains = logspace(0, 5, 500);
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figure;
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hold on;
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plot(real(pole(G_dvf)), imag(pole(G_dvf)), 'x');
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set(gca,'ColorOrderIndex',1);
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plot(real(tzero(G_dvf)), imag(tzero(G_dvf)), 'o');
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for i = 1:length(gains)
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set(gca,'ColorOrderIndex',1);
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cl_poles = pole(feedback(G_dvf, (gains(i)*s)*eye(6)));
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plot(real(cl_poles), imag(cl_poles), '.');
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end
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ylim([0, 2*pi*500]);
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xlim([-2*pi*500,0]);
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xlabel('Real Part')
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ylabel('Imaginary Part')
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axis square
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#+end_src
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** Controller and Loop Gain
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#+begin_src matlab
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K_dvf = s*15000/(1 + s/2/pi/10000);
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#+end_src
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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, 1, 1);
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hold on;
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(K_dvf*G_dvf(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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ax2 = subplot(2, 1, 2);
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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(K_dvf*G_dvf(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', '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],'x');
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#+end_src
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#+begin_src matlab
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K_dvf = -K_dvf*eye(6);
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#+end_src
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* Uncertainty Improvements thanks to the LAC control
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#+begin_src matlab
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K_dvf_backup = K_dvf;
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initializeController('type', 'hac-dvf');
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#+end_src
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#+begin_src matlab
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masses = [1, 10, 50]; % [kg]
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#+end_src
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#+begin_src matlab
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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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Gm = {zeros(length(masses))};
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K_dvf = tf(zeros(6));
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Kx = tf(zeros(6));
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for i = 1:length(masses)
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initializeSample('mass', masses(i));
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%% Run the linearization
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G = linearize(mdl, io, 0);
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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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Gm(i) = {G};
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end
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#+end_src
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#+begin_src matlab :exports none
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Gm_iff = {zeros(length(masses))};
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K_dvf = K_dvf_backup;
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Kx = tf(zeros(6));
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for i = 1:length(masses)
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initializeSample('mass', masses(i));
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%% Run the linearization
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G = linearize(mdl, io, 0);
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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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Gm_iff(i) = {G};
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end
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#+end_src
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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, 1, 1);
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hold on;
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for i = 1:length(Gm_iff)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz'))), '-');
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set(gca,'ColorOrderIndex',i);
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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 [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 1:length(Gm_iff)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz'))), '-', ...
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'DisplayName', sprintf('$M = %.0f$ [kg]', masses(i)));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))), '--', ...
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'HandleVisibility', 'off');
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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('location', 'southwest');
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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#+end_src
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* High Authority Control - $\bm{K}_\mathcal{X}$
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** Identification of the damped plant
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#+begin_src matlab
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Kx = tf(zeros(6));
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#+end_src
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#+begin_src matlab
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initializeController('type', 'hac-dvf');
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#+end_src
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#+begin_src matlab
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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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%% Run the linearization
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G = linearize(mdl, io, 0);
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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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#+end_src
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The minus sine is put here because there is already a minus sign included due to the computation of the position error.
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#+begin_src matlab
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load('mat/stages.mat', 'nano_hexapod');
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Gx = -G*inv(nano_hexapod.kinematics.J');
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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#+end_src
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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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|
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|
linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
|
|
|
|
** Controller Design
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|
The controller consists of:
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|
- A pure integrator
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|
- A Second integrator up to half the wanted bandwidth
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|
- A Lead around the cross-over frequency
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- A low pass filter with a cut-off equal to two times the wanted bandwidth
|
|
|
|
#+begin_src matlab
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|
wc = 2*pi*15; % Bandwidth Bandwidth [rad/s]
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|
|
h = 1.5; % Lead parameter
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|
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Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/2));
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|
|
|
% Normalization of the gain of have a loop gain of 1 at frequency wc
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|
Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));
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|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
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|
freqs = logspace(0, 3, 1000);
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|
|
|
labels = {'$L_x$', '$L_y$', '$L_z$', '$L_{R_x}$', '$L_{R_y}$', '$L_{R_z}$'};
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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 = 1:6
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|
plot(freqs, abs(squeeze(freqresp(Gx(i, i)*Kx(i,i), freqs, 'Hz'))));
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|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
title('Diagonal elements of the Plant');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i)*Kx(i,i), freqs, 'Hz'))), 'DisplayName', labels{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();
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
isstable(feedback(Gx*Kx, eye(6), -1))
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Kx = inv(nano_hexapod.kinematics.J')*Kx;
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
isstable(feedback(G*Kx, eye(6), 1))
|
|
#+end_src
|
|
|
|
* Simulation
|
|
#+begin_src matlab
|
|
load('mat/conf_simulink.mat');
|
|
set_param(conf_simulink, 'StopTime', '2');
|
|
#+end_src
|
|
|
|
And we simulate the system.
|
|
#+begin_src matlab
|
|
sim('nass_model');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
hac_dvf = simout;
|
|
save('./mat/tomo_exp_hac_lac.mat', 'hac_dvf');
|
|
#+end_src
|
|
|
|
* Results
|
|
Let's load the simulation when no control is applied.
|
|
#+begin_src matlab
|
|
load('./mat/experiment_tomography.mat', 'tomo_align_dist');
|
|
load('./mat/tomo_exp_hac_lac.mat', 'hac_dvf');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 1))
|
|
plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 1))
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dx [m]');
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 2))
|
|
plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 2))
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dy [m]');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 3))
|
|
plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 3))
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dz [m]');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 4))
|
|
plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 4))
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Rx [rad]');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 5))
|
|
plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 5))
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Ry [rad]');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 6), 'DisplayName', '$\mu$-Station')
|
|
plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 6), 'DisplayName', 'HAC-DVF')
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Rz [rad]');
|
|
legend();
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
xlim([0.5, inf]);
|
|
#+end_src
|