132 lines
4.5 KiB
Matlab
132 lines
4.5 KiB
Matlab
%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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addpath('./mat/');
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% Load Data
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% The measurement data are loaded and the offset are removed using the =detrend= command.
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load('int_enc_comp.mat', 'interferometer', 'encoder', 'u', 't');
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Ts = 1e-4; % Sampling Time [s]
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interferometer = detrend(interferometer, 0);
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encoder = detrend(encoder, 0);
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u = detrend(u, 0);
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% Time Domain Results
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% One period of the displacement of the mass as measured by the encoder and interferometer are shown in Figure [[fig:int_enc_one_cycle]].
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% It consist of the sinusoidal motion at 0.5Hz with an amplitude of approximately $70\mu m$.
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% The frequency of the motion is chosen such that no resonance in the system is excited.
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% This should improve the coherence between the measurements made by the encoder and interferometer.
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figure;
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hold on;
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plot(t, encoder, '-', 'DisplayName', 'Encoder')
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plot(t, interferometer, '--', 'DisplayName', 'Interferometer')
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [m]');
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legend('location', 'southeast');
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xlim([50, 52])
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% #+name: fig:int_enc_one_cycle
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% #+caption: One cycle measurement
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% #+RESULTS:
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% [[file:figs/int_enc_one_cycle.png]]
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% The difference between the two measurements during the same period is shown in Figure [[fig:int_enc_one_cycle_error]].
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figure;
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hold on;
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plot(t, encoder - interferometer, 'DisplayName', 'Difference')
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [m]');
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legend('location', 'northeast');
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xlim([50, 52])
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% Difference between Encoder and Interferometer as a function of time
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% The data is filtered using a second order low pass filter with a cut-off frequency $\omega_0$ as defined below.
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w0 = 2*pi*5; % [rad/s]
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xi = 0.7;
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G_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2);
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% After filtering, the data is "re-shaped" such that we can superimpose all the measured periods as shown in Figure [[fig:int_enc_error_mean_time]].
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% This gives an idea of the measurement error as given by the Attocube during a $70 \mu m$ motion.
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d_err_mean = reshape(lsim(G_lpf, encoder - interferometer, t), [2/Ts floor(Ts/2*length(encoder))]);
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d_err_mean = d_err_mean - mean(d_err_mean);
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figure;
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hold on;
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for i_i = 1:size(d_err_mean, 2)
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plot(t(1:size(d_err_mean, 1)), d_err_mean(:, i_i), 'k-')
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end
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plot(t(1:size(d_err_mean, 1)), mean(d_err_mean, 2), 'r-')
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [m]');
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% Difference between Encoder and Interferometer as a function of position
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% Figure [[fig:int_enc_error_mean_time]] gives the measurement error as a function of time.
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% We here wish the compute this measurement error as a function of the position (as measured by the encoer).
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% To do so, all the attocube measurements corresponding to each position measured by the Encoder (resolution of $1nm$) are averaged.
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% Figure [[fig:int_enc_error_mean_position]] is obtained where we clearly see an error with a period comparable to the motion range and a much smaller period corresponding to the non-linear period errors that we wish the estimate.
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[e_sorted, ~, e_ind] = unique(encoder);
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i_mean = zeros(length(e_sorted), 1);
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for i = 1:length(e_sorted)
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i_mean(i) = mean(interferometer(e_ind == i));
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end
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i_mean_error = (i_mean - e_sorted);
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figure;
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hold on;
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% plot(encoder, interferometer - encoder, 'k.', 'DisplayName', 'Difference')
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plot(1e6*(e_sorted), 1e9*(i_mean_error))
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hold off;
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xlabel('Encoder Measurement [$\mu m$]'); ylabel('Measrement Error [nm]');
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% #+name: fig:int_enc_error_mean_position
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% #+caption: Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles
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% #+RESULTS:
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% [[file:figs/int_enc_error_mean_position.png]]
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% The period of the non-linearity seems to be equal to $765 nm$ which corresponds to half the wavelength of the Laser ($1.53 \mu m$).
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% For the motion range done here, the non-linearity is measured over ~18 periods which permits to do some averaging.
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win_length = 1530/2; % length of the windows (corresponds to 765 nm)
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num_avg = floor(length(e_sorted)/win_length); % number of averaging
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i_init = ceil((length(e_sorted) - win_length*num_avg)/2); % does not start at the extremity
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e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init:i_init+win_length*num_avg-1), [win_length num_avg]), 2);
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% The obtained periodic non-linearity is shown in Figure [[fig:int_non_linearity_period_wavelength]].
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figure;
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hold on;
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plot(1e-3*(0:win_length-1), 1e9*(e_sorted_mean_over_period))
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hold off;
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xlabel('Displacement [$\mu m$]'); ylabel('Measurement Non-Linearity [nm]');
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