Encoder - Test Bench
+Table of Contents
+1 Experimental Setup
++The experimental Setup is schematically represented in Figure 1. +
+ + +
+
Figure 1: Schematic of the Experiment
+
+
Figure 2: Side View of the encoder
+
+
Figure 3: Front View of the encoder
+2 Huddle Test
++The goal in this section is the estimate the noise of both the encoder and the intereferometer. +
+2.1 Load Data
+load('mat/int_enc_huddle_test.mat', 'interferometer', 'encoder', 't'); ++
interferometer = detrend(interferometer, 0); +encoder = detrend(encoder, 0); ++
2.2 Time Domain Results
+
+
Figure 4: Huddle test - Time domain signals
+G_lpf = 1/(1 + s/2/pi/10); ++
+
Figure 5: Huddle test - Time domain signals filtered with a LPF at 10Hz
+2.3 Frequency Domain Noise
+Ts = 1e-4; +win = hann(ceil(10/Ts)); + +[p_i, f] = pwelch(interferometer, win, [], [], 1/Ts); +[p_e, ~] = pwelch(encoder, win, [], [], 1/Ts); ++
+
Figure 6: Amplitude Spectral Density of the signals during the Huddle test
+3 Comparison Interferometer / Encoder
++The goal here is to make sure that the interferometer and encoder measurements are coherent. +We may see non-linearity in the interferometric measurement. +
+3.1 Load Data
+load('mat/int_enc_comp.mat', 'interferometer', 'encoder', 'u', 't'); ++
interferometer = detrend(interferometer, 0); +encoder = detrend(encoder, 0); +u = detrend(u, 0); ++
3.2 Time Domain Results
+
+
Figure 7: One cycle measurement
+
+
Figure 8: Difference between the Encoder and the interferometer during one cycle
+3.3 Difference between Encoder and Interferometer as a function of time
+Ts = 1e-4; +d_i_mean = reshape(interferometer, [2/Ts floor(Ts/2*length(interferometer))]); +d_e_mean = reshape(encoder, [2/Ts floor(Ts/2*length(encoder))]); ++
w0 = 2*pi*5; % [rad/s] +xi = 0.7; + +G_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2); + +d_err_mean = reshape(lsim(G_lpf, encoder - interferometer, t), [2/Ts floor(Ts/2*length(encoder))]); +d_err_mean = d_err_mean - mean(d_err_mean); ++
+
Figure 9: Difference between the two measurement in the time domain, averaged for all the cycles
+3.4 Difference between Encoder and Interferometer as a function of position
++Compute the mean of the interferometer measurement corresponding to each of the encoder measurement. +
+ +[e_sorted, ~, e_ind] = unique(encoder); + +i_mean = zeros(length(e_sorted), 1); +for i = 1:length(e_sorted) + i_mean(i) = mean(interferometer(e_ind == i)); +end + +i_mean_error = (i_mean - e_sorted); ++
+
Figure 10: Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles
++The period of the non-linearity seems to be \(1.53 \mu m\) which corresponds to the wavelength of the Laser. +
+ +win_length = 1530; % length of the windows (corresponds to 1.53 um) +num_avg = floor(length(e_sorted)/win_length); % number of averaging + +i_init = ceil((length(e_sorted) - win_length*num_avg)/2); % does not start at the extremity + +e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init:i_init+win_length*num_avg-1), [win_length num_avg]), 2); ++
+
Figure 11: Non-Linearity of the Interferometer over the period of the wavelength
+4 Identification
+4.1 Load Data
+load('mat/int_enc_id_noise_bis.mat', 'interferometer', 'encoder', 'u', 't'); ++
interferometer = detrend(interferometer, 0); +encoder = detrend(encoder, 0); +u = detrend(u, 0); ++
4.2 Identification
+Ts = 1e-4; % Sampling Time [s] +win = hann(ceil(10/Ts)); ++
[tf_i_est, f] = tfestimate(u, interferometer, win, [], [], 1/Ts); +[co_i_est, ~] = mscohere(u, interferometer, win, [], [], 1/Ts); + +[tf_e_est, ~] = tfestimate(u, encoder, win, [], [], 1/Ts); +[co_e_est, ~] = mscohere(u, encoder, win, [], [], 1/Ts); ++
+
+