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a/figs/vionic_noise_asd_low_freq.pdf b/figs/vionic_noise_asd_low_freq.pdf new file mode 100644 index 0000000..a5e9ee6 Binary files /dev/null and b/figs/vionic_noise_asd_low_freq.pdf differ diff --git a/figs/vionic_noise_asd_low_freq.png b/figs/vionic_noise_asd_low_freq.png new file mode 100644 index 0000000..1e2308e Binary files /dev/null and b/figs/vionic_noise_asd_low_freq.png differ diff --git a/figs/vionic_noise_asd_model.pdf b/figs/vionic_noise_asd_model.pdf index 4b0ed94..d3c8070 100644 Binary files a/figs/vionic_noise_asd_model.pdf and b/figs/vionic_noise_asd_model.pdf differ diff --git a/test-bench-vionic.html b/test-bench-vionic.html index 3b934bf..99a91be 100644 --- a/test-bench-vionic.html +++ b/test-bench-vionic.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Encoder Renishaw Vionic - Test Bench @@ -39,22 +39,21 @@

Table of Contents

@@ -64,7 +63,7 @@

This report is also available as a pdf.

-
+

You can find below the documentation of:

@@ -89,25 +88,25 @@ In particular, we would like to measure: This document is structured as follow:

    -
  • Section 1: the expected performance of the Vionic encoder system are described
    • -
  • Section 2: a simple model of the encoder is developed
    • -
  • Section 3: the noise of the encoder is measured and a model of the noise is identified
    • -
  • Section 4: the linearity of the sensor is estimated
    • +
  • Section 1: the expected performance of the Vionic encoder system are described
    • +
  • Section 2: a simple model of the encoder is developed
    • +
  • Section 3: the noise of the encoder is measured and a model of the noise is identified
    • +
  • Section 4: the linearity of the sensor is estimated
-
-

1 Expected Performances

+
+

1 Expected Performances

- +

-The Vionic encoder is shown in Figure 1. +The Vionic encoder is shown in Figure 1.

-
+

encoder_vionic.png

Figure 1: Picture of the Vionic Encoder

@@ -135,21 +134,21 @@ Interpolation is within the readhead, with fine resolution versions being furthe

-The expected interpolation errors (non-linearity) is shown in Figure 2. +The expected interpolation errors (non-linearity) is shown in Figure 2.

-
+

vionic_expected_noise.png

Figure 2: Expected interpolation errors for the Vionic Encoder

-The characteristics as advertise in the manual as well as our specifications are shown in Table 1. +The characteristics as advertise in the manual as well as our specifications are shown in Table 1.

- +
@@ -169,7 +168,7 @@ The characteristics as advertise in the manual as well as our specifications are - + @@ -201,11 +200,11 @@ The characteristics as advertise in the manual as well as our specifications are -
-

2 Encoder Model

+
+

2 Encoder Model

- +

@@ -218,38 +217,53 @@ It is also characterized by its measurement noise \(n\) that can be described by

-The model of the encoder is shown in Figure 3. +The model of the encoder is shown in Figure 3.

-
+

encoder-model-schematic.png

Figure 3: Model of the Encoder

-We can also use a transfer function \(G_n(s)\) to shape a noise \(\tilde{n}\) with unity ASD as shown in Figure 2. +We can also use a transfer function \(G_n(s)\) to shape a noise \(\tilde{n}\) with unity ASD as shown in Figure 2.

-
+

encoder-model-schematic-with-asd.png

-
-

3 Noise Measurement

+
+

3 Noise Measurement

- +

+

+This part is structured as follow: +

+
    +
  • Section 3.1: the measurement bench is described
    • +
  • Section 3.2: long measurement is performed to estimate the low frequency drifts in the measurement
    • +
  • Section 3.3: high frequency measurements are performed to estimate the high frequency noise
    • +
  • Section 3.4: the Spectral density of the measurement noise is estimated
    • +
  • Section 3.5: finally, the measured noise is modeled
    • +
-
-

3.1 Test Bench

+ +
+

3.1 Test Bench

+

+ +

+

To measure the noise \(n\) of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured. Then, the measured signal \(y_m\) corresponds to the noise \(n\). @@ -257,62 +271,138 @@ Then, the measured signal \(y_m\) corresponds to the noise \(n\).

-
-

3.2 Thermal drifts

+
+

3.2 Thermal drifts

-
    -
  • [ ] picture of the setup
    • -
  • [ ] long thermal drifts
    • -
  • [ ] once stabilize, look at the noise
    • -
  • [ ] compute low frequency ASD (may still be thermal drifts of the mechanics and not noise)
    • -
+

+ +Measured displacement were recording during approximately 40 hours with a sample frequency of 100Hz. +A first order low pass filter with a corner frequency of 1Hz +

+ +
+
enc_l = load('mat/noise_meas_40h_100Hz_1.mat', 't', 'x');
+
+
+ +

+The measured time domain data are shown in Figure 5. +

+ +
+

vionic_drifts_time.png +

+

Figure 5: Measured thermal drifts

+
+ +

+The measured data seems to experience a constant drift after approximately 20 hour. +Let’s estimate this drift. +

+ +
+The mean drift is approximately 60.9 [nm/hour] or 1.0 [nm/min]
+
+ + +

+Comparison between the data and the linear fit is shown in Figure 6. +

+ +
+

vionic_drifts_linear_fit.png +

+

Figure 6: Measured drift and linear fit

+
+ +

+Let’s now estimate the Power Spectral Density of the measured displacement. +The obtained low frequency ASD is shown in Figure 7. +

+ +
+

vionic_noise_asd_low_freq.png +

+

Figure 7: Amplitude Spectral density of the measured displacement

+
-
-

3.3 Time Domain signals

+
+

3.3 Time Domain signals

-First we load the data. -The raw measured data as well as the low pass filtered data (using a first order low pass filter with a cut-off at 10Hz) are shown in Figure 5. +

-
+

+Then, and for all the 7 encoders, we record the measured motion during 100s with a sampling frequency of 20kHz. +

+ +

+The raw measured data as well as the low pass filtered data (using a first order low pass filter with a cut-off at 10Hz) are shown in Figure 8. +

+ +

vionic_noise_raw_lpf.png

-

Figure 5: Time domain measurement (raw data and low pass filtered data with first order 10Hz LPF)

+

Figure 8: Time domain measurement (raw data and low pass filtered data with first order 10Hz LPF)

-The time domain data for all the encoders are compared in Figure 6. +The time domain data for all the encoders are compared in Figure 9.

-
+

+We can see some drifts that are in the order of few nm to 20nm per minute. +As shown in Section 3.2, these drifts should diminish over time down to 1nm/min. +

+ +

vionic_noise_time.png

-

Figure 6: Comparison of the time domain measurement

+

Figure 9: Comparison of the time domain measurement

-
-

3.4 Noise Spectral Density

+
+

3.4 Noise Spectral Density

-The amplitude spectral density is computed and shown in Figure 7. +

-
+

+The amplitude spectral densities for all the encoder are computed and shown in Figure 10. +

+ +

vionic_noise_asd.png

-

Figure 7: Amplitude Spectral Density of the measured signal

+

Figure 10: Amplitude Spectral Density of the measured signal

+
+ +

+We can combine these measurements with the low frequency noise computed in Section 3.2. +The obtained ASD is shown in Figure 11. +

+ +
+

vionic_noise_asd_combined.png +

+

Figure 11: Combined low frequency and high frequency noise measurements

-
-

3.5 Noise Model

+
+

3.5 Noise Model

+

+ +

+

Let’s create a transfer function that approximate the measured noise of the encoder.

@@ -322,23 +412,18 @@ Let’s create a transfer function that approximate the measured noise of th

-The amplitude of the transfer function and the measured ASD are shown in Figure 8. +The amplitude of the transfer function and the measured ASD are shown in Figure 12.

-
+

vionic_noise_asd_model.png

-

Figure 8: Measured ASD of the noise and modeled one

-
-
+

Figure 12: Measured ASD of the noise and modeled one

-
-

3.6 Validity of the noise model

-

-The cumulative amplitude spectrum is now computed and shown in Figure 9. +The cumulative amplitude spectrum is now computed and shown in Figure 13.

@@ -346,24 +431,24 @@ We can see that the Root Mean Square value of the measurement noise is \(\approx

-
+

vionic_noise_cas_model.png

-

Figure 9: Meassured CAS of the noise and modeled one

+

Figure 13: Meassured CAS of the noise and modeled one

-
-

4 Linearity Measurement

+
+

4 Linearity Measurement

- +

-
-

4.1 Test Bench

+
+

4.1 Test Bench

In order to measure the linearity, we have to compare the measured displacement with a reference sensor with a known linearity. @@ -372,7 +457,7 @@ An actuator should also be there so impose a displacement.

-One idea is to use the test-bench shown in Figure 10. +One idea is to use the test-bench shown in Figure 14.

@@ -385,22 +470,22 @@ As the interferometer has a very large bandwidth, we should be able to estimate

-
+

test_bench_encoder_calibration.png

-

Figure 10: Schematic of the test bench

+

Figure 14: Schematic of the test bench

-
-

4.2 Results

+
+

4.2 Results

Author: Dehaeze Thomas

-

Created: 2021-02-10 mer. 15:14

+

Created: 2021-02-11 jeu. 15:21

diff --git a/test-bench-vionic.org b/test-bench-vionic.org index eb5769a..5d9f3cc 100644 --- a/test-bench-vionic.org +++ b/test-bench-vionic.org @@ -169,7 +169,18 @@ We can also use a transfer function $G_n(s)$ to shape a noise $\tilde{n}$ with u * Noise Measurement <> + +** Introduction :ignore: + +This part is structured as follow: +- Section [[sec:noise_bench]]: the measurement bench is described +- Section [[sec:thermal_drifts]]: long measurement is performed to estimate the low frequency drifts in the measurement +- Section [[sec:vionic_noise_time]]: high frequency measurements are performed to estimate the high frequency noise +- Section [[sec:noise_asd]]: the Spectral density of the measurement noise is estimated +- Section [[sec:vionic_noise_model]]: finally, the measured noise is modeled + ** Test Bench +<> To measure the noise $n$ of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured. Then, the measured signal $y_m$ corresponds to the noise $n$. @@ -192,16 +203,21 @@ addpath('./matlab/'); addpath('./mat/'); #+end_src -** TODO Thermal drifts - -- [ ] picture of the setup -- [ ] long thermal drifts -- [ ] Identification of the drifts (exponential fit) -- [ ] once stabilize, look at the noise -- [ ] compute low frequency ASD (may still be thermal drifts of the mechanics and not noise) +** Thermal drifts +<> +Measured displacement were recording during approximately 40 hours with a sample frequency of 100Hz. +A first order low pass filter with a corner frequency of 1Hz #+begin_src matlab -enc_l = load('mat/noise_meas_40h_200Hz_1.mat', 't', 'x'); +enc_l = load('mat/noise_meas_40h_100Hz_1.mat', 't', 'x'); +#+end_src + +The measured time domain data are shown in Figure [[fig:vionic_drifts_time]]. +#+begin_src matlab :exports none +enc_l.x = enc_l.x(enc_l.t > 5); % Remove first 5 seconds +enc_l.t = enc_l.t(enc_l.t > 5); % Remove first 5 seconds +enc_l.t = enc_l.t - enc_l.t(1); % Start at 0 + enc_l.x = enc_l.x - mean(enc_l.x(enc_l.t < 1)); % Start at zero displacement #+end_src @@ -212,67 +228,94 @@ plot(enc_l.t/3600, 1e9*enc_l.x, '-'); hold off; xlabel('Time [h]'); ylabel('Displacement [nm]'); +xlim([0, 40]); #+end_src -Exponential fit -#+begin_src matlab -f = @(b,x) b(1)*(1 - exp(-x/b(2))); - -y_cur = enc_l.x; -t_cur = end_l.t; - -nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function -B0 = [400e-9, 2*60*60]; % Choose Appropriate Initial Estimates -[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’ +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/vionic_drifts_time.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:vionic_drifts_time +#+caption: Measured thermal drifts +#+RESULTS: +[[file:figs/vionic_drifts_time.png]] + +The measured data seems to experience a constant drift after approximately 20 hour. +Let's estimate this drift. + +#+begin_src matlab :exports none +t0 = 20*3600; % Start time [s] +x_stab = enc_l.x(enc_l.t > t0); +x_stab = x_stab - x_stab(1); +t_stab = enc_l.t(enc_l.t > t0); +t_stab = t_stab - t_stab(1); #+end_src -The corresponding time constant is (in [h]): #+begin_src matlab :results value replace :exports results -B(2)/60/60 +sprintf('The mean drift is approximately %.1f [nm/hour] or %.1f [nm/min]', 3600*1e9*(t_stab\x_stab), 60*1e9*(t_stab\x_stab)) #+end_src -Comparison of the data and exponential fit +#+RESULTS: +: The mean drift is approximately 60.9 [nm/hour] or 1.0 [nm/min] + +Comparison between the data and the linear fit is shown in Figure [[fig:vionic_drifts_linear_fit]]. #+begin_src matlab :exports none figure; hold on; -plot(enc_l.t/60/60, 1e9*enc_l.x); -plot(enc_l.t/60/60, 1e9*f(B, enc_l.t)); +plot(t_stab/3600, 1e9*x_stab, '-'); +plot(t_stab/3600, 1e9*t_stab*(t_stab\x_stab), 'k--'); hold off; -xlim([0, 17.5]) -xlabel('Time [h]'); ylabel('Displacement [nm]'); +xlabel('Time [h]'); +ylabel('Displacement [nm]'); +xlim([0, 20]); #+end_src -Let's get only the data once it is stabilized -#+begin_src matlab -x_stab = enc_l.x(enc_l.t > 20*3600); -x_stab = x_stab - mean(x_stab); -t_stab = enc_l.t(enc_l.t > 20*3600); -x_stab = x_stab - x_stab(1); +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/vionic_drifts_linear_fit.pdf', 'width', 'wide', 'height', 'normal'); #+end_src +#+name: fig:vionic_drifts_linear_fit +#+caption: Measured drift and linear fit +#+RESULTS: +[[file:figs/vionic_drifts_linear_fit.png]] + +Let's now estimate the Power Spectral Density of the measured displacement. +The obtained low frequency ASD is shown in Figure [[fig:vionic_noise_asd_low_freq]]. #+begin_src matlab :exports none % Compute sampling Frequency -Ts = (enc{1}.t(end) - enc{1}.t(1))/(length(enc{1}.t)-1); +Ts = (enc_l.t(end) - enc_l.t(1))/(length(enc_l.t)-1); Fs = 1/Ts; % Hannning Windows -win = hanning(ceil(60/Ts)); +win = hanning(ceil(60*10/Ts)); -[pxx, f] = pwelch(x_stab, win, [], [], Fs); +[pxx_l, f_l] = pwelch(x_stab, win, [], [], Fs); #+end_src #+begin_src matlab :exports none figure; hold on; -plot(f, sqrt(pxx)) +plot(f_l, sqrt(pxx_l)) set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); -% xlim([10, Fs/2]); -% ylim([1e-11, 1e-10]); +xlim([1e-2, 1e0]); +ylim([1e-11, 1e-8]); #+end_src +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/vionic_noise_asd_low_freq.pdf', 'width', 'side', 'height', 'normal'); +#+end_src + +#+name: fig:vionic_noise_asd_low_freq +#+caption: Amplitude Spectral density of the measured displacement +#+RESULTS: +[[file:figs/vionic_noise_asd_low_freq.png]] + ** Time Domain signals -First we load the data. +<> + +Then, and for all the 7 encoders, we record the measured motion during 100s with a sampling frequency of 20kHz. + #+begin_src matlab :exports none %% Load all the measurements enc = {}; @@ -310,6 +353,9 @@ exportFig('figs/vionic_noise_raw_lpf.pdf', 'width', 'wide', 'height', 'normal'); [[file:figs/vionic_noise_raw_lpf.png]] The time domain data for all the encoders are compared in Figure [[fig:vionic_noise_time]]. + +We can see some drifts that are in the order of few nm to 20nm per minute. +As shown in Section [[sec:thermal_drifts]], these drifts should diminish over time down to 1nm/min. #+begin_src matlab :exports none figure; hold on; @@ -333,7 +379,9 @@ exportFig('figs/vionic_noise_time.pdf', 'width', 'wide', 'height', 'normal'); [[file:figs/vionic_noise_time.png]] ** Noise Spectral Density -The amplitude spectral density is computed and shown in Figure [[fig:vionic_noise_asd]]. +<> + +The amplitude spectral densities for all the encoder are computed and shown in Figure [[fig:vionic_noise_asd]]. #+begin_src matlab :exports none % Compute sampling Frequency Ts = (enc{1}.t(end) - enc{1}.t(1))/(length(enc{1}.t)-1); @@ -361,7 +409,7 @@ end set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); xlim([10, Fs/2]); -ylim([1e-11, 1e-10]); +ylim([1e-11, 1e-9]); legend('location', 'northeast'); #+end_src @@ -374,7 +422,33 @@ exportFig('figs/vionic_noise_asd.pdf', 'width', 'wide', 'height', 'normal'); #+RESULTS: [[file:figs/vionic_noise_asd.png]] +We can combine these measurements with the low frequency noise computed in Section [[sec:thermal_drifts]]. +The obtained ASD is shown in Figure [[fig:vionic_noise_asd_combined]]. +#+begin_src matlab :exports none +[pxx_h, f_h] = pwelch(enc{2}.x, hanning(ceil(10/Ts)), [], [], Fs); + +figure; +hold on; +plot(f_h(f_h>0.6), sqrt(pxx_h(f_h>0.6)), 'k-'); +plot(f_l(f_l<1), sqrt(pxx_l(f_l<1)), 'k-') +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); +xlim([1e-2, Fs/2]); +ylim([1e-12, 1e-8]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/vionic_noise_asd_combined.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:vionic_noise_asd_combined +#+caption: Combined low frequency and high frequency noise measurements +#+RESULTS: +[[file:figs/vionic_noise_asd_combined.png]] + ** Noise Model +<> + Let's create a transfer function that approximate the measured noise of the encoder. #+begin_src matlab Gn_e = 1.8e-11/(1 + s/2/pi/1e4); @@ -385,7 +459,7 @@ The amplitude of the transfer function and the measured ASD are shown in Figure #+begin_src matlab :exports none figure; hold on; -plot(f, sqrt(p1), 'color', [0, 0, 0, 0.5], 'DisplayName', '$\Gamma_n(\omega)$'); +plot(f, sqrt(enc{1}.pxx), 'color', [0, 0, 0, 0.5], 'DisplayName', '$\Gamma_n(\omega)$'); for i=2:7 plot(f, sqrt(enc{i}.pxx), 'color', [0, 0, 0, 0.5], ... 'HandleVisibility', 'off'); @@ -408,7 +482,6 @@ exportFig('figs/vionic_noise_asd_model.pdf', 'width', 'wide', 'height', 'normal' #+RESULTS: [[file:figs/vionic_noise_asd_model.png]] -** Validity of the noise model The cumulative amplitude spectrum is now computed and shown in Figure [[fig:vionic_noise_cas_model]]. We can see that the Root Mean Square value of the measurement noise is $\approx 1.6 \, nm$ as advertise in the datasheet. diff --git a/test-bench-vionic.pdf b/test-bench-vionic.pdf index 78451a3..d0f2002 100644 Binary files a/test-bench-vionic.pdf and b/test-bench-vionic.pdf differ
Table 1: Characteristics of the Vionic compared with the specifications
Time Delay < 10 ns < 0.5 ms