Add noise budget to determine max noise of sensors
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org/noise_budgeting.org
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242
org/noise_budgeting.org
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#+TITLE: Noise Budgeting
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#+SETUPFILE: ./setup/org-setup-file.org
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* Maximum Noise of the Relative Motion Sensors
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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
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simulinkproject('../');
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#+END_SRC
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** Initialization
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#+begin_src matlab
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open('nass_model.slx');
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#+end_src
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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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initializeSimscapeConfiguration();
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initializeDisturbances('enable', false);
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initializeLoggingConfiguration('log', 'none');
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initializeController('type', 'hac-dvf');
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#+end_src
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We set the stiffness of the payload fixation:
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#+begin_src matlab
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Kp = 1e8; % [N/m]
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#+end_src
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#+begin_src matlab
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initializeNanoHexapod('k', 1e5, 'c', 2e2);
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Ms = 50;
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initializeSample('mass', Ms, 'freq', sqrt(Kp/Ms)/2/pi*ones(6,1));
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#+end_src
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#+begin_src matlab
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms);
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#+end_src
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** Control System
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#+begin_src matlab
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Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
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#+end_src
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#+begin_src matlab
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h = 2.0;
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Kl = 2e7 * eye(6) * ...
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1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
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1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
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(s/2/pi/10 + 1)/(s/2/pi/10) * ...
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1/(1 + s/2/pi/300);
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#+end_src
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#+begin_src matlab
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load('mat/stages.mat', 'nano_hexapod');
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K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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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, '/Noises'], 1, 'openinput', [], 'ndL'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1;
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#+end_src
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#+begin_src matlab
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%% Run the linearization
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G = linearize(mdl, io);
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G.InputName = {'ndL1', 'ndL2', 'ndL3', 'ndL4', 'ndL5', 'ndL6'};
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G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
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#+end_src
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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(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G(2, 2), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G(3, 3), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G(4, 4), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G(6, 5), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G(6, 6), freqs, 'Hz'))));
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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/m]'); set(gca, 'XTickLabel',[]);
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#+end_src
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** Maximum induced vibration's ASD
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Required maximum induced ASD of the sample's vibration due to the relative motion sensor noise.
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\[ \bm{\Gamma}_x(\omega) = \begin{bmatrix} \Gamma_x(\omega) & \Gamma_y(\omega) & \Gamma_{R_x}(\omega) & \Gamma_{R_y}(\omega) \end{bmatrix} \]
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#+begin_src matlab
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Gamma_x = [(1e-9)/(1 + s/2/pi/100); % Dx
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(1e-9)/(1 + s/2/pi/100); % Dy
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(1e-9)/(1 + s/2/pi/100); % Dz
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(2e-8)/(1 + s/2/pi/100); % Rx
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(2e-8)/(1 + s/2/pi/100)]; % Ry
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#+end_src
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#+begin_src matlab
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freqs = logspace(0, 3, 1000);
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#+end_src
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Corresponding RMS value in [nm rms, nrad rms]
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([1e9*sqrt(trapz(freqs, (abs(squeeze(freqresp(Gamma_x, freqs, 'Hz')))').^2))]', {'Dx [nm]', 'Dy [nm]', 'Dz [nm]', 'Rx [nrad]', 'Ry [nrad]'}, {'Specifications'}, ' %.1f ');
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#+end_src
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#+RESULTS:
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| | Specifications |
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|-----------+----------------|
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| Dx [nm] | 12.1 |
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| Dy [nm] | 12.1 |
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| Dz [nm] | 12.1 |
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| Rx [nrad] | 241.8 |
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| Ry [nrad] | 241.8 |
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** Computation of the maximum relative motion sensor noise
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Let's note $G$ the transfer function from the 6 sensor noise $n$ to the 5dof pose error $x$.
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We have:
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\[ x_i = \sum_{j=1}^6 G_{ij}(s) n_j, \quad i = 1 \dots 5 \]
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In terms of ASD:
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\[ \Gamma_{x_i}(\omega) = \sqrt{\sum_{j=1}^6 |G_{ij}(j\omega)|^2 \cdot {\Gamma_{n_j}}^2(\omega)}, \quad i = 1 \dots 5 \]
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Let's suppose that the ASD of all the sensor noise are equal:
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\[ \Gamma_{n_j} = \Gamma_{n}, \quad j = 1 \dots 6 \]
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We then have an upper bound of the sensor noise for each of the considered motion errors:
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\[ \Gamma_{n_i, \text{max}}(\omega) = \frac{\Gamma_{x_i}(\omega)}{\sqrt{\sum_{j=1}^6 |G_{ij}(j\omega)|^2}}, \quad i = 1 \dots 5 \]
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#+begin_src matlab
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Gamma_ndL = zeros(5, length(freqs));
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for in = 1:5
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Gamma_ndL(in, :) = abs(squeeze(freqresp(Gamma_x(in), freqs, 'Hz')))./sqrt(sum(abs(squeeze(freqresp(G(in, :), freqs, 'Hz'))).^2))';
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end
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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for in = 1:5
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plot(freqs, Gamma_ndL(in, :), 'k-');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('ASD [$\frac{m}{\sqrt{Hz}}$]');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/noise_budget_ndL_max_asd.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:noise_budget_ndL_max_asd
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#+caption: Maximum estimated ASD of the relative motion sensor noise
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#+RESULTS:
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[[file:figs/noise_budget_ndL_max_asd.png]]
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If the noise ASD of the relative motion sensor is bellow the maximum specified ASD for all the considered motion:
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\[ \Gamma_n < \Gamma_{n_i, \text{max}}, \quad i = 1 \dots 5 \]
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Then, the motion error due to sensor noise should be bellow the one specified.
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#+begin_src matlab
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Gamma_ndL_max = min(Gamma_ndL(1:5, :));
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#+end_src
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Let's take a sensor with a white noise up to 1kHz that is bellow the specified one:
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#+begin_src matlab
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Gamma_ndL_ex = abs(squeeze(freqresp(min(Gamma_ndL_max)/(1 + s/2/pi/1e3), freqs, 'Hz')));
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, Gamma_ndL_max, 'k-', 'DisplayName', 'Specification');
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plot(freqs, Gamma_ndL_ex, 'DisplayName', 'Sensor Example');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('ASD [m/sqrt(Hz)]');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/relative_motion_sensor_noise_ASD_example.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:relative_motion_sensor_noise_ASD_example
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#+caption: Requirement maximum ASD of the sensor noise + example of a sensor validating the requirements
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#+RESULTS:
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[[file:figs/relative_motion_sensor_noise_ASD_example.png]]
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The corresponding RMS value of the sensor noise taken as an example is [nm RMS]:
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#+begin_src matlab :results replace value
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1e9*sqrt(trapz(freqs, Gamma_ndL_max.^2))
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#+end_src
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#+RESULTS:
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: 519.29
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** Verification of the induced motion error
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Verify that by taking the sensor noise, we have to wanted displacement error
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From the sensor noise PSD $\Gamma_n(\omega)$, we can estimate the obtained displacement PSD $\Gamma_x(\omega)$:
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\[ \Gamma_{x,i}(\omega) = \sqrt{ \sum_{j=1}^{6} |G_{ij}|^2(j\omega) \cdot \Gamma_{n,j}^2(\omega) }, \quad i = 1 \dots 5 \]
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#+begin_src matlab
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Gamma_xest = zeros(5, length(freqs));
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for in = 1:5
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Gamma_xest(in, :) = sqrt(sum(abs(squeeze(freqresp(G(in, :), freqs, 'Hz'))).^2.*Gamma_ndL_max.^2));
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end
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([1e9*sqrt(trapz(freqs, (Gamma_xest.^2)')); 1e9*sqrt(trapz(freqs, (abs(squeeze(freqresp(Gamma_x, freqs, 'Hz')))').^2))]', {'Dx [nm]', 'Dy [nm]', 'Dz [nm]', 'Rx [nrad]', 'Ry [nrad]'}, {'Results', 'Specifications'}, ' %.1f ');
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#+end_src
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#+RESULTS:
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| | Results | Specifications |
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|-----------+---------+----------------|
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| Dx [nm] | 8.9 | 12.1 |
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| Dy [nm] | 9.3 | 12.1 |
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| Dz [nm] | 10.2 | 12.1 |
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| Rx [nrad] | 110.2 | 241.8 |
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| Ry [nrad] | 107.8 | 241.8 |
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