Add analysis for voice coil actuators (uniaxial model)
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figs/uniaxial-cas-iff-vc.png
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figs/uniaxial-sensitivity-vc-disturbances.png
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figs/uniaxial-sensitivity-vc-force-dist.png
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figs/uniaxial-vc-cas-dist.png
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figs/uniaxial-vc-psd-dist.png
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figs/uniaxial_iff_vc_open_loop.png
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@ -47,6 +47,8 @@ The idea is to use the same model as the full Simscape Model but to restrict the
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This is done in order to more easily study the system and evaluate control techniques.
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* Simscape Model
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<<sec:simscape_model>>
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A schematic of the uniaxial model used for simulations is represented in figure [[fig:uniaxial-model-nass-flexible]].
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The perturbations $w$ are:
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@ -623,6 +625,7 @@ Schematics of the active damping techniques are displayed in figure [[fig:uniaxi
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[[file:figs/uniaxial-model-nass-flexible-active-damping.png]]
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* Undamped System
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<<sec:undamped>>
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** Introduction :ignore:
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Let's start by study the undamped system.
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@ -2184,6 +2187,7 @@ And initialize the controllers.
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Direct Velocity Feedback:
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#+end_important
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* With Cedrat Piezo-electric Actuators
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<<sec:cedrat_actuator>>
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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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@ -2749,3 +2753,400 @@ It is important to note that the effect of direct forces applied to the sample a
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| Sensitivity ($F_s$) | - (at low freq) | + | + |
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| Sensitivity ($F_{ty,rz}$) | + | - | + |
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| Overall RMS of $D$ | = | = | = |
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* Voice Coil
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<<sec:voice_coil>>
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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 'simscape/sim_nano_station_uniaxial.slx'
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#+end_src
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** Init
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We initialize all the stages with the default parameters.
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The nano-hexapod is an hexapod with voice coils and the sample has a mass of 50kg.
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#+begin_src matlab :exports none
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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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initializeNanoHexapod(struct('actuator', 'lorentz'));
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initializeSample(struct('mass', 50));
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#+end_src
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All the controllers are set to 0 (Open Loop).
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#+begin_src matlab :exports none
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K = tf(0);
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save('./mat/controllers.mat', 'K', '-append');
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K_iff = tf(0);
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save('./mat/controllers.mat', 'K_iff', '-append');
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K_rmc = tf(0);
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save('./mat/controllers.mat', 'K_rmc', '-append');
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K_dvf = tf(0);
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save('./mat/controllers.mat', 'K_dvf', '-append');
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#+end_src
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** Identification
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We identify the dynamics of the system.
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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 = 'sim_nano_station_uniaxial';
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#+end_src
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The inputs and outputs are defined below and corresponds to the name of simulink blocks.
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#+begin_src matlab
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%% Input/Output definition
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io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
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io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
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io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
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io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
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io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
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io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
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io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
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io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
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io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
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io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
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#+end_src
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Finally, we use the =linearize= Matlab function to extract a state space model from the simscape model.
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#+begin_src matlab
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%% Run the linearization
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G_vc = linearize(mdl, io, options);
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G_vc.InputName = {'Dw', ... % Ground Motion [m]
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'Fs', ... % Force Applied on Sample [N]
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'Fn', ... % Force applied by NASS [N]
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'Fty', ... % Parasitic Force Ty [N]
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'Frz'}; % Parasitic Force Rz [N]
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G_vc.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
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'Fnm', ... % Force Sensor in NASS [N]
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'Dnm', ... % Displacement Sensor in NASS [m]
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'Dgm', ... % Asbolute displacement of Granite [m]
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'Vlm'}; ... % Absolute Velocity of NASS [m/s]
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#+end_src
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Finally, we save the identified system dynamics for further analysis.
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#+begin_src matlab
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save('./uniaxial/mat/plants.mat', 'G_vc', '-append');
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#+end_src
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** Sensitivity to Disturbances
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We load the dynamics when using a piezo-electric nano hexapod to compare the results.
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#+begin_src matlab
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load('./uniaxial/mat/plants.mat', 'G');
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#+end_src
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We show several plots representing the sensitivity to disturbances:
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- in figure [[fig:uniaxial-sensitivity-vc-disturbances]] the transfer functions from ground motion $D_w$ to the sample position $D$ and the transfer function from direct force on the sample $F_s$ to the sample position $D$ are shown
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- in figure [[fig:uniaxial-sensitivity-vc-force-dist]], it is the effect of parasitic forces of the positioning stages ($F_{ty}$ and $F_{rz}$) on the position $D$ of the sample that are shown
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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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subplot(2, 1, 1);
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title('$D_w$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'G - VC');
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plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'G - PZ');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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legend('location', 'northeast');
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ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
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subplot(2, 1, 2);
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title('$F_s$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Fs'), freqs, 'Hz'))), 'k-');
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plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k--');
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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]'); xlabel('Frequency [Hz]');
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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/uniaxial-sensitivity-vc-disturbances.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:uniaxial-sensitivity-vc-disturbances
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#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-vc-disturbances.png][png]], [[./figs/uniaxial-sensitivity-vc-disturbances.pdf][pdf]])
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[[file:figs/uniaxial-sensitivity-vc-disturbances.png]]
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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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subplot(2, 1, 1);
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title('$F_{ty}$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'G - VC');
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plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'G - PZ');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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legend('location', 'northeast');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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subplot(2, 1, 2);
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title('$F_{rz}$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Frz'), freqs, 'Hz'))), 'k-');
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plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k--');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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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/uniaxial-sensitivity-vc-force-dist.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:uniaxial-sensitivity-vc-force-dist
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#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-vc-force-dist.png][png]], [[./figs/uniaxial-sensitivity-vc-force-dist.pdf][pdf]])
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[[file:figs/uniaxial-sensitivity-vc-force-dist.png]]
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** Noise Budget
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We first load the measured PSD of the disturbance.
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#+begin_src matlab
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load('./disturbances/mat/dist_psd.mat', 'dist_f');
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#+end_src
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The effect of these disturbances on the distance $D$ is computed below.
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#+begin_src matlab :exports none
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% Power Spectral Density of the relative Displacement [m^2/Hz]
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psd_vc_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_vc('D', 'Dw'), dist_f.f, 'Hz'))).^2;
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psd_vc_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_vc('D', 'Fty'), dist_f.f, 'Hz'))).^2;
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psd_vc_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_vc('D', 'Frz'), dist_f.f, 'Hz'))).^2;
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#+end_src
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#+begin_src matlab :exports none
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% Power Spectral Density of the relative Displacement [m^2/Hz]
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psd_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
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psd_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
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psd_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
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#+end_src
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The PSD of the obtain distance $D$ due to each of the perturbation is shown in figure [[fig:uniaxial-vc-psd-dist]] and the Cumulative Amplitude Spectrum is shown in figure [[fig:uniaxial-vc-cas-dist]].
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The Root Mean Square value of the obtained displacement $D$ is computed below and can be determined from the figure [[fig:uniaxial-vc-cas-dist]].
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#+begin_src matlab :results value replace :exports results
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cas_tot_d = sqrt(cumtrapz(dist_f.f, psd_vc_rz_d+psd_vc_ty_d+psd_vc_gm_d)); cas_tot_d(end)
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#+end_src
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#+RESULTS:
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: 4.8793e-06
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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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hold on;
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plot(dist_f.f, psd_vc_gm_d, 'DisplayName', '$D_w$');
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plot(dist_f.f, psd_vc_ty_d, 'DisplayName', '$T_y$');
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plot(dist_f.f, psd_vc_rz_d, 'DisplayName', '$R_z$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('CAS of the effect of disturbances on $D$ $\left[\frac{m^2}{Hz}\right]$'); xlabel('Frequency [Hz]');
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legend('location', 'northeast')
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xlim([0.5, 500]);
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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/uniaxial-vc-psd-dist.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:uniaxial-vc-psd-dist
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#+CAPTION: PSD of the displacement $D$ due to disturbances ([[./figs/uniaxial-vc-psd-dist.png][png]], [[./figs/uniaxial-vc-psd-dist.pdf][pdf]])
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[[file:figs/uniaxial-vc-psd-dist.png]]
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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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hold on;
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d)))), 'DisplayName', '$D_w$ - VC');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_ty_d)))), 'DisplayName', '$T_y$ - VC');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_rz_d)))), 'DisplayName', '$R_z$ - VC');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d+psd_vc_ty_d+psd_vc_rz_d)))), 'k-', 'DisplayName', 'tot - VC');
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plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), 'k--', 'DisplayName', 'tot - PZ');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast')
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xlim([0.5, 500]); ylim([1e-12, 5e-6]);
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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/uniaxial-vc-cas-dist.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:uniaxial-vc-cas-dist
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#+CAPTION: CAS of the displacement $D$ due the disturbances ([[./figs/uniaxial-vc-cas-dist.png][png]], [[./figs/uniaxial-vc-cas-dist.pdf][pdf]])
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[[file:figs/uniaxial-vc-cas-dist.png]]
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#+begin_important
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Even though the RMS value of the displacement $D$ is lower when using a piezo-electric actuator, the motion is mainly due to high frequency disturbances which are more difficult to control (an higher control bandwidth is required).
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Thus, it may be desirable to use voice coil actuators.
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#+end_important
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** Integral Force Feedback
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#+begin_src matlab
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K_iff = -20/s;
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(-1, 2, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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plot(freqs, abs(squeeze(freqresp(K_iff*G_vc('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
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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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ax2 = subplot(2, 1, 2);
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plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G_vc('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
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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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#+HEADER: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/uniaxial_iff_vc_open_loop.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:uniaxial_iff_vc_open_loop
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#+CAPTION: Open Loop Transfer Function for IFF control when using a voice coil actuator ([[./figs/uniaxial_iff_vc_open_loop.png][png]], [[./figs/uniaxial_iff_vc_open_loop.pdf][pdf]])
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[[file:figs/uniaxial_iff_vc_open_loop.png]]
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** Identification of the Damped Plant
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Let's initialize the system prior to identification.
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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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initializeNanoHexapod(struct('actuator', 'lorentz'));
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initializeSample(struct('mass', 50));
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#+end_src
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All the controllers are set to 0.
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#+begin_src matlab
|
||||
K = tf(0);
|
||||
save('./mat/controllers.mat', 'K', '-append');
|
||||
K_iff = -K_iff;
|
||||
save('./mat/controllers.mat', 'K_iff', '-append');
|
||||
K_rmc = tf(0);
|
||||
save('./mat/controllers.mat', 'K_rmc', '-append');
|
||||
K_dvf = tf(0);
|
||||
save('./mat/controllers.mat', 'K_dvf', '-append');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'sim_nano_station_uniaxial';
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Input/Output definition
|
||||
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
|
||||
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
|
||||
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
|
||||
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
|
||||
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
|
||||
|
||||
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
|
||||
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
|
||||
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
|
||||
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
|
||||
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Run the linearization
|
||||
G_vc_iff = linearize(mdl, io, options);
|
||||
G_vc_iff.InputName = {'Dw', ... % Ground Motion [m]
|
||||
'Fs', ... % Force Applied on Sample [N]
|
||||
'Fn', ... % Force applied by NASS [N]
|
||||
'Fty', ... % Parasitic Force Ty [N]
|
||||
'Frz'}; % Parasitic Force Rz [N]
|
||||
G_vc_iff.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
|
||||
'Fnm', ... % Force Sensor in NASS [N]
|
||||
'Dnm', ... % Displacement Sensor in NASS [m]
|
||||
'Dgm', ... % Asbolute displacement of Granite [m]
|
||||
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
|
||||
#+end_src
|
||||
|
||||
** Noise Budget
|
||||
We compute the obtain PSD of the displacement $D$ when using IFF.
|
||||
#+begin_src matlab :exports none
|
||||
% Power Spectral Density of the relative Displacement [m^2/Hz]
|
||||
psd_vc_iff_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_vc_iff('D', 'Dw'), dist_f.f, 'Hz'))).^2;
|
||||
psd_vc_iff_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_vc_iff('D', 'Fty'), dist_f.f, 'Hz'))).^2;
|
||||
psd_vc_iff_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_vc_iff('D', 'Frz'), dist_f.f, 'Hz'))).^2;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), '-', 'DisplayName', 'OL - PZ');
|
||||
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d+psd_vc_ty_d+psd_vc_rz_d)))), 'k-', 'DisplayName', 'OL - VC');
|
||||
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_iff_gm_d+psd_vc_iff_ty_d+psd_vc_iff_rz_d)))), 'k--', 'DisplayName', 'IFF - VC');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'northeast')
|
||||
xlim([0.5, 500]); ylim([1e-12, 5e-6]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/uniaxial-cas-iff-vc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:uniaxial-cas-iff-vc
|
||||
#+CAPTION: CAS of the displacement $D$ ([[./figs/uniaxial-cas-iff-vc.png][png]], [[./figs/uniaxial-cas-iff-vc.pdf][pdf]])
|
||||
[[file:figs/uniaxial-cas-iff-vc.png]]
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
The use of voice coil actuators would allow a better disturbance rejection for a fixed bandwidth compared with a piezo-electric hexapod.
|
||||
|
||||
Similarly, it would require much lower bandwidth to attain the same level of disturbance rejection for $D$.
|
||||
#+end_important
|
||||
|
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Reference in New Issue
Block a user