Noise Budgeting
+Table of Contents
+ +1 Maximum Noise of the Relative Motion Sensors
+1.1 Initialization
+open('nass_model.slx');
+
+initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeAxisc();
+initializeMirror();
+
+initializeSimscapeConfiguration();
+initializeDisturbances('enable', false);
+initializeLoggingConfiguration('log', 'none');
+
+initializeController('type', 'hac-dvf');
+
++We set the stiffness of the payload fixation: +
+Kp = 1e8; % [N/m] ++
initializeNanoHexapod('k', 1e5, 'c', 2e2);
+
+Ms = 50;
+initializeSample('mass', Ms, 'freq', sqrt(Kp/Ms)/2/pi*ones(6,1));
+
+initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms);
+
+1.2 Control System
+Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6); ++
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); ++
load('mat/stages.mat', 'nano_hexapod');
+K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
+
+%% Run the linearization
+G = linearize(mdl, io);
+G.InputName = {'ndL1', 'ndL2', 'ndL3', 'ndL4', 'ndL5', 'ndL6'};
+G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
+
+1.3 Maximum induced vibration’s ASD
++Required maximum induced ASD of the sample’s vibration due to the relative motion sensor noise. +\[ \bm{\Gamma}_x(\omega) = \begin{bmatrix} \Gamma_x(\omega) & \Gamma_y(\omega) & \Gamma_{R_x}(\omega) & \Gamma_{R_y}(\omega) \end{bmatrix} \] +
+ +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 ++
freqs = logspace(0, 3, 1000); ++
+Corresponding RMS value in [nm rms, nrad rms] +
+1e9*sqrt(trapz(freqs, (abs(squeeze(freqresp(Gamma_x, freqs, 'Hz')))').^2)) ++
| + | Specifications | +
|---|---|
| Dx [nm] | +12.1 | +
| Dy [nm] | +12.1 | +
| Dz [nm] | +12.1 | +
| Rx [nrad] | +241.8 | +
| Ry [nrad] | +241.8 | +
1.4 Computation of the maximum relative motion sensor noise
++Let’s note \(G\) the transfer function from the 6 sensor noise \(n\) to the 6dof 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: +\[ \Gamma_{x_i}(\omega) = \sum_{j=1}^6 |G_{ij}(j\omega)|^2 \Gamma_{n_j}(\omega), \quad i = 1 \dots 5 \] +
+ ++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: +\[ \Gamma_{n_i, \text{max}}(\omega) = \frac{\Gamma_{n_i}(\omega)}{\sum_{j=1}^6 |G_{ij}(j\omega)|^2}, \quad i = 1 \dots 5 \] +
+ +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 ++
+
Figure 1: Maximum estimated ASD of the relative motion sensor noise
++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. +
+ +Gamma_ndL_max = min(Gamma_ndL(1:5, :)); ++
+Let’s take a sensor with a white noise up to 1kHz that is bellow the specified one: +
+Gamma_ndL_ex = abs(squeeze(freqresp(min(Gamma_ndL_max)/(1 + s/2/pi/1e3), freqs, 'Hz'))); ++
+
Figure 2: Requirement maximum ASD of the sensor noise + example of a sensor validating the requirements
++The corresponding RMS value of the sensor noise taken as an example is [nm RMS]: +
+1e9*sqrt(trapz(freqs, Gamma_ndL_max.^2)) ++
+519.29 ++
1.5 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 \(\Gamma_x(\omega)\): +\[ \Gamma_{x,i}(\omega) = \sqrt{ \sum_{j=1}^{6} |G_{ij}|^2(j\omega) \Gamma_{n,j}^2(\omega) }, \quad i = 1 \dots 5 \] +
+ +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 ++
| + | 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 | +