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< title > Identification of the disturbances< / title >
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< h1 class = "title" > Identification of the disturbances< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#Simscape-Model" > 1. Simscape Model< / a > < / li >
< li > < a href = "#Identification" > 2. Identification< / a > < / li >
< li > < a href = "#Sensitivity-to-Disturbances" > 3. Sensitivity to Disturbances< / a > < / li >
< li > < a href = "#Power-Spectral-Density-of-the-effect-of-the-disturbances" > 4. Power Spectral Density of the effect of the disturbances< / a > < / li >
< li > < a href = "#Compute-the-Power-Spectral-Density-of-the-disturbance-force" > 5. Compute the Power Spectral Density of the disturbance force< / a > < / li >
< li > < a href = "#Noise-Budget" > 6. Noise Budget< / a > < / li >
< li > < a href = "#Save" > 7. Save< / a > < / li >
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< li > < a href = "#orgd564a58" > 8. Time Domain Disturbances< / a > < / li >
< li > < a href = "#org8212830" > 9. Time Domain Effect of Disturbances< / a >
< ul >
< li > < a href = "#org76c96f4" > 9.1. Initialization of the Experiment< / a > < / li >
< li > < a href = "#orgf9c1ad9" > 9.2. Simulations< / a > < / li >
< li > < a href = "#org8107e0b" > 9.3. Comparison< / a > < / li >
< / ul >
< / li >
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< / ul >
< / div >
< / div >
< p >
The goal here is to extract the Power Spectral Density of the sources of perturbation.
< / p >
< p >
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The sources of perturbations are (schematically shown in figure < a href = "#org3b8025c" > 1< / a > ):
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< / p >
< ul class = "org-ul" >
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< li > \(D_w\): Ground Motion< / li >
< li > Parasitic forces applied in the system when scanning with the Translation Stage and the Spindle (\(F_{rz}\) and \(F_{ty}\)).
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These forces can be due to imperfect guiding for instance.< / li >
< / ul >
< p >
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Because we cannot measure directly the perturbation forces, we have the measure the effect of those perturbations on the system (in terms of velocity for instance using geophones, \(D\) on figure < a href = "#org3b8025c" > 1< / a > ) and then, using a model, compute the forces that induced such velocity.
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< / p >
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< div id = "org3b8025c" class = "figure" >
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< p > < img src = "figs/uniaxial-model-micro-station.png" alt = "uniaxial-model-micro-station.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Schematic of the Micro Station and the sources of disturbance< / p >
< / div >
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< p >
This file is divided in the following sections:
< / p >
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< ul class = "org-ul" >
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< li > Section < a href = "#org7cdab70" > 1< / a > : the simscape model used here is presented< / li >
< li > Section < a href = "#org43ec76b" > 2< / a > : transfer functions from the disturbance forces to the relative velocity of the hexapod with respect to the granite are computed using the Simscape Model representing the experimental setup< / li >
< li > Section < a href = "#org5636fee" > 3< / a > : the bode plot of those transfer functions are shown< / li >
< li > Section < a href = "#org40a7e4e" > 4< / a > : the measured PSD of the effect of the disturbances are shown< / li >
< li > Section < a href = "#orgd113ba5" > 5< / a > : from the model and the measured PSD, the PSD of the disturbance forces are computed< / li >
< li > Section < a href = "#org71da6bd" > 6< / a > : with the computed PSD, the noise budget of the system is done< / li >
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< / ul >
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< div id = "outline-container-Simscape-Model" class = "outline-2" >
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< h2 id = "Simscape-Model" > < span class = "section-number-2" > 1< / span > Simscape Model< / h2 >
< div class = "outline-text-2" id = "text-Simscape-Model" >
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< p >
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< a id = "org7cdab70" > < / a >
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< / p >
< p >
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The following Simscape model of the micro-station is the same model used for the dynamical analysis.
However, we here constrain all the stage to only move in the vertical direction.
< / p >
< p >
We add disturbances forces in the vertical direction for the Translation Stage and the Spindle.
Also, we measure the absolute displacement of the granite and of the top platform of the Hexapod.
< / p >
< p >
We load the configuration and we set a small < code > StopTime< / code > .
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load('mat/conf_simulink.mat');
set_param(conf_simulink, 'StopTime', '0.5');
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< / pre >
< / div >
< p >
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We initialize all the stages without the sample nor the nano-hexapod.
The obtained system corresponds to the status micro-station when the vibration measurements were conducted.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeGround();
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initializeGranite('type', 'modal-analysis');
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initializeTy();
initializeRy();
initializeRz();
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initializeMicroHexapod('type', 'modal-analysis');
initializeAxisc('type', 'none');
initializeMirror('type', 'none');
initializeNanoHexapod('type', 'none');
initializeSample('type', 'none');
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< / pre >
< / div >
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< p >
Open Loop Control.
< / p >
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< pre class = "src src-matlab" > initializeController('type', 'open-loop');
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< / pre >
< / div >
< p >
We don’ t need gravity here.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeSimscapeConfiguration('gravity', false);
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< / pre >
< / div >
< p >
We log the signals.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeLoggingConfiguration('log', 'all');
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< / pre >
< / div >
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< / div >
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< div id = "outline-container-Identification" class = "outline-2" >
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< h2 id = "Identification" > < span class = "section-number-2" > 2< / span > Identification< / h2 >
< div class = "outline-text-2" id = "text-Identification" >
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< p >
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< a id = "org43ec76b" > < / a >
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The transfer functions from the disturbance forces to the relative velocity of the hexapod with respect to the granite are computed using the Simscape Model representing the experimental setup with the code below.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'nass_model';
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%% Micro-Hexapod
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz'); io_i = io_i + 1; % Vertical Ground Motion
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_z'); io_i = io_i + 1; % Parasitic force Ty
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Parasitic force Rz
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io(io_i) = linio([mdl, '/Micro-Station/Granite/Modal Analysis/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; % Absolute motion - Granite
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Modal Analysis/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion - Hexapod
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% Run the linearization
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G = linearize(mdl, io, 0);
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< / pre >
< / div >
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< p >
We Take only the outputs corresponding to the vertical acceleration.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > G = G([3,9], :);
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% Input/Output names
G.InputName = {'Dw', 'Fty', 'Frz'};
G.OutputName = {'Agm', 'Ahm'};
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< / pre >
< / div >
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< p >
We integrate 1 time the output to have the velocity and we substract the absolute velocities to have the relative velocity.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > G = (1/s)*tf([-1, 1])*G;
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% Input/Output names
G.InputName = {'Dw', 'Fty', 'Frz'};
G.OutputName = {'Vm'};
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-Sensitivity-to-Disturbances" class = "outline-2" >
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< h2 id = "Sensitivity-to-Disturbances" > < span class = "section-number-2" > 3< / span > Sensitivity to Disturbances< / h2 >
< div class = "outline-text-2" id = "text-Sensitivity-to-Disturbances" >
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< p >
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< a id = "org5636fee" > < / a >
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< / p >
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< p >
The obtained sensitivity to disturbances are shown bellow:
< / p >
< ul class = "org-ul" >
< li > The transfer function from vertical ground motion \(D_w\) to the vertical relative displacement from the micro-hexapod to the granite \(D\) is shown in Figure < a href = "#org7fe296c" > 2< / a > < / li >
< li > The sensitive from vertical forces applied in the Translation stage is shown in Figure < a href = "#orgce9715c" > 3< / a > < / li >
< / ul >
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< div id = "org7fe296c" class = "figure" >
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< p > < img src = "figs/sensitivity_dist_gm.png" alt = "sensitivity_dist_gm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 2: < / span > Sensitivity to Ground Motion (< a href = "./figs/sensitivity_dist_gm.png" > png< / a > , < a href = "./figs/sensitivity_dist_gm.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orgce9715c" class = "figure" >
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< p > < img src = "figs/sensitivity_dist_fty.png" alt = "sensitivity_dist_fty.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Sensitivity to vertical forces applied by the Ty stage (< a href = "./figs/sensitivity_dist_fty.png" > png< / a > , < a href = "./figs/sensitivity_dist_fty.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orga3e6fbc" class = "figure" >
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< p > < img src = "figs/sensitivity_dist_frz.png" alt = "sensitivity_dist_frz.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Sensitivity to vertical forces applied by the Rz stage (< a href = "./figs/sensitivity_dist_frz.png" > png< / a > , < a href = "./figs/sensitivity_dist_frz.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-Power-Spectral-Density-of-the-effect-of-the-disturbances" class = "outline-2" >
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< h2 id = "Power-Spectral-Density-of-the-effect-of-the-disturbances" > < span class = "section-number-2" > 4< / span > Power Spectral Density of the effect of the disturbances< / h2 >
< div class = "outline-text-2" id = "text-Power-Spectral-Density-of-the-effect-of-the-disturbances" >
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< p >
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< a id = "org40a7e4e" > < / a >
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< / p >
< p >
The Power Spectral Densities of the relative velocity between the hexapod and the marble in \([(m/s)^2/Hz]\) are loaded for the following sources of disturbance:
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< / p >
< ul class = "org-ul" >
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< li > Slip Ring Rotation (\(F_{r_z}\))< / li >
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< li > Scan of the translation stage (effect in the vertical direction and in the horizontal direction)< / li >
< / ul >
< p >
Also, the Ground Motion is measured.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > gm = load('./mat/psd_gm.mat', 'f', 'psd_gm');
rz = load('./mat/pxsp_r.mat', 'f', 'pxsp_r');
tyz = load('./mat/pxz_ty_r.mat', 'f', 'pxz_ty_r');
tyx = load('./mat/pxe_ty_r.mat', 'f', 'pxe_ty_r');
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< / pre >
< / div >
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< p >
Because some 50Hz and harmonics were present in the ground motion measurement, we remove these peaks with the following code:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > f0s = [50, 100, 150, 200, 250, 350, 450];
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for f0 = f0s
i = find(gm.f > f0-0.5 & gm.f < f0+0.5);
gm.psd_gm(i) = linspace(gm.psd_gm(i(1)), gm.psd_gm(i(end)), length(i));
end
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< / pre >
< / div >
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< p >
We now compute the relative velocity between the hexapod and the granite due to ground motion.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > gm.psd_rv = gm.psd_gm.*abs(squeeze(freqresp(G('Vm', 'Dw'), gm.f, 'Hz'))).^2;
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< / pre >
< / div >
< p >
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The Power Spectral Density of the relative motion and velocity of the hexapod with respect to the granite are shown in figures < a href = "#orgb815ef2" > 5< / a > and < a href = "#org4c2c215" > 6< / a > .
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< / p >
< p >
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The Cumulative Amplitude Spectrum of the relative motion is shown in figure < a href = "#orgaf910f1" > 7< / a > .
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< / p >
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< div id = "orgb815ef2" class = "figure" >
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< p > < img src = "figs/dist_effect_relative_velocity.png" alt = "dist_effect_relative_velocity.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Amplitude Spectral Density of the relative velocity of the hexapod with respect to the granite due to different sources of perturbation (< a href = "./figs/dist_effect_relative_velocity.png" > png< / a > , < a href = "./figs/dist_effect_relative_velocity.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org4c2c215" class = "figure" >
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< p > < img src = "figs/dist_effect_relative_motion.png" alt = "dist_effect_relative_motion.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Amplitude Spectral Density of the relative displacement of the hexapod with respect to the granite due to different sources of perturbation (< a href = "./figs/dist_effect_relative_motion.png" > png< / a > , < a href = "./figs/dist_effect_relative_motion.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orgaf910f1" class = "figure" >
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< p > < img src = "figs/dist_effect_relative_motion_cas.png" alt = "dist_effect_relative_motion_cas.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Cumulative Amplitude Spectrum of the relative motion due to different sources of perturbation (< a href = "./figs/dist_effect_relative_motion_cas.png" > png< / a > , < a href = "./figs/dist_effect_relative_motion_cas.pdf" > pdf< / a > )< / p >
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< / div >
< div class = "important" >
< p >
From Figure < a href = "#orgaf910f1" > 7< / a > , we can see that the translation stage and the rotation stage have almost the same effect on the position error.
Also, the ground motion has a relatively negligible effect on the position error.
< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-Compute-the-Power-Spectral-Density-of-the-disturbance-force" class = "outline-2" >
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< h2 id = "Compute-the-Power-Spectral-Density-of-the-disturbance-force" > < span class = "section-number-2" > 5< / span > Compute the Power Spectral Density of the disturbance force< / h2 >
< div class = "outline-text-2" id = "text-Compute-the-Power-Spectral-Density-of-the-disturbance-force" >
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< p >
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< a id = "orgd113ba5" > < / a >
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< / p >
< p >
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Using the extracted transfer functions from the disturbance force to the relative motion of the hexapod with respect to the granite (section < a href = "#org5636fee" > 3< / a > ) and using the measured PSD of the relative motion (section < a href = "#org40a7e4e" > 4< / a > ), we can compute the PSD of the disturbance force.
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< / p >
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< p >
This is done below.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > rz.psd_f = rz.pxsp_r./abs(squeeze(freqresp(G('Vm', 'Frz'), rz.f, 'Hz'))).^2;
tyz.psd_f = tyz.pxz_ty_r./abs(squeeze(freqresp(G('Vm', 'Fty'), tyz.f, 'Hz'))).^2;
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< p >
The obtained amplitude spectral densities of the disturbance forces are shown in Figure < a href = "#org45cf35c" > 8< / a > .
< / p >
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< div id = "org45cf35c" class = "figure" >
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< p > < img src = "figs/dist_force_psd.png" alt = "dist_force_psd.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Amplitude Spectral Density of the disturbance force (< a href = "./figs/dist_force_psd.png" > png< / a > , < a href = "./figs/dist_force_psd.pdf" > pdf< / a > )< / p >
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< h2 id = "Noise-Budget" > < span class = "section-number-2" > 6< / span > Noise Budget< / h2 >
< div class = "outline-text-2" id = "text-Noise-Budget" >
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< p >
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< a id = "org71da6bd" > < / a >
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< / p >
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< p >
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From the obtained spectral density of the disturbance sources, we can compute the resulting relative motion of the Hexapod with respect to the granite using the model.
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< / p >
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< p >
This is equivalent as doing the inverse that was done in the previous section.
This is done in order to verify that this is coherent with the measurements.
< / p >
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< p >
The power spectral density of the relative motion is computed below and the result is shown in Figure < a href = "#orgcacf809" > 9< / a > .
We can see that this is exactly the same as the Figure < a href = "#org4c2c215" > 6< / a > .
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< / p >
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< pre class = "src src-matlab" > psd_gm_d = gm.psd_gm.*abs(squeeze(freqresp(G('Vm', 'Dw')/s, gm.f, 'Hz'))).^2;
psd_ty_d = tyz.psd_f.*abs(squeeze(freqresp(G('Vm', 'Fty')/s, tyz.f, 'Hz'))).^2;
psd_rz_d = rz.psd_f.*abs(squeeze(freqresp(G('Vm', 'Frz')/s, rz.f, 'Hz'))).^2;
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< div id = "orgcacf809" class = "figure" >
< p > < img src = "figs/psd_effect_dist_verif.png" alt = "psd_effect_dist_verif.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Computed Effect of the disturbances on the relative displacement hexapod/granite (< a href = "./figs/psd_effect_dist_verif.png" > png< / a > , < a href = "./figs/psd_effect_dist_verif.pdf" > pdf< / a > )< / p >
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< div id = "outline-container-Save" class = "outline-2" >
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< h2 id = "Save" > < span class = "section-number-2" > 7< / span > Save< / h2 >
< div class = "outline-text-2" id = "text-Save" >
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< p >
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The PSD of the disturbance force are now saved for further analysis.
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< pre class = "src src-matlab" > dist_f = struct();
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dist_f.f = gm.f; % Frequency Vector [Hz]
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dist_f.psd_gm = gm.psd_gm; % Power Spectral Density of the Ground Motion [m^2/Hz]
dist_f.psd_ty = tyz.psd_f; % Power Spectral Density of the force induced by the Ty stage in the Z direction [N^2/Hz]
dist_f.psd_rz = rz.psd_f; % Power Spectral Density of the force induced by the Rz stage in the Z direction [N^2/Hz]
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save('./mat/dist_psd.mat', 'dist_f');
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< h2 id = "orgd564a58" > < span class = "section-number-2" > 8< / span > Time Domain Disturbances< / h2 >
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< p >
Let’ s initialize the time domain disturbances and load them.
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< pre class = "src src-matlab" > initializeDisturbances();
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dist = load('nass_disturbances.mat');
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< p >
The time domain disturbance signals are shown in Figure < a href = "#org3ff859b" > 10< / a > .
< / p >
< div id = "org3ff859b" class = "figure" >
< p > < img src = "figs/disturbances_time_domain.png" alt = "disturbances_time_domain.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Disturbances in the Time Domain (< a href = "./figs/disturbances_time_domain.png" > png< / a > , < a href = "./figs/disturbances_time_domain.pdf" > pdf< / a > )< / p >
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< h2 id = "org8212830" > < span class = "section-number-2" > 9< / span > Time Domain Effect of Disturbances< / h2 >
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< h3 id = "org76c96f4" > < span class = "section-number-3" > 9.1< / span > Initialization of the Experiment< / h3 >
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< p >
We initialize all the stages with the default parameters.
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< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
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< p >
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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< pre class = "src src-matlab" > initializeNanoHexapod('type', 'rigid');
initializeSample('mass', 1);
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< pre class = "src src-matlab" > initializeReferences();
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initializeController('type', 'open-loop');
initializeSimscapeConfiguration('gravity', false);
initializeLoggingConfiguration('log', 'all');
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< pre class = "src src-matlab" > load('mat/conf_simulink.mat');
set_param(conf_simulink, 'StopTime', '2');
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< h3 id = "orgf9c1ad9" > < span class = "section-number-3" > 9.2< / span > Simulations< / h3 >
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< p >
No disturbances:
< / p >
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< pre class = "src src-matlab" > initializeDisturbances('enable', false);
sim('nass_model');
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sim_no = simout;
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< p >
Ground Motion:
< / p >
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< pre class = "src src-matlab" > initializeDisturbances('Fty_x', false, 'Fty_z', false, 'Frz_z', false);
sim('nass_model');
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sim_gm = simout;
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< p >
Translation Stage Vibrations:
< / p >
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< pre class = "src src-matlab" > initializeDisturbances('Dwx', false, 'Dwy', false, 'Dwz', false, 'Frz_z', false);
sim('nass_model');
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sim_ty = simout;
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< p >
Rotation Stage Vibrations:
< / p >
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< pre class = "src src-matlab" > initializeDisturbances('Dwx', false, 'Dwy', false, 'Dwz', false, 'Fty_x', false, 'Fty_z', false);
sim('nass_model');
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sim_rz = simout;
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< h3 id = "org8107e0b" > < span class = "section-number-3" > 9.3< / span > Comparison< / h3 >
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< p >
Let’ s now compare the effect of those perturbations on the position error of the sample (Figure < a href = "#orgee898d4" > 11< / a > )
< / p >
< div id = "orgee898d4" class = "figure" >
< p > < img src = "figs/effect_disturbances_position_error.png" alt = "effect_disturbances_position_error.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Effect of Perturbations on the position error (< a href = "./figs/effect_disturbances_position_error.png" > png< / a > , < a href = "./figs/effect_disturbances_position_error.pdf" > pdf< / a > )< / p >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-09-01 mar. 13:48< / p >
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