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