Identification of the disturbances

The following Simscape model of the micro-station is the same model used for the dynamical analysis. However, we here constrain all the stage to only move in the vertical direction.

We add disturbances forces in the vertical direction for the Translation Stage and the Spindle. Also, we measure the absolute displacement of the granite and of the top platform of the Hexapod.

We load the configuration and we set a small StopTime.

  load('mat/conf_simulink.mat');
  set_param(conf_simulink, 'StopTime', '0.5');

We initialize all the stages without the sample nor the nano-hexapod. The obtained system corresponds to the status micro-station when the vibration measurements were conducted.

  initializeGround();
  initializeGranite('type', 'modal-analysis');
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod('type', 'modal-analysis');
  initializeAxisc('type', 'none');
  initializeMirror('type', 'none');
  initializeNanoHexapod('type', 'none');
  initializeSample('type', 'none');

Open Loop Control.

  initializeController('type', 'open-loop');

We don’t need gravity here.

  initializeSimscapeConfiguration('gravity', false);

We log the signals.

  initializeLoggingConfiguration('log', 'all');

The transfer functions from the disturbance forces to the relative velocity of the hexapod with respect to the granite are computed using the Simscape Model representing the experimental setup with the code below.

  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Micro-Hexapod
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz'); io_i = io_i + 1; % Vertical Ground Motion
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_z'); io_i = io_i + 1; % Parasitic force Ty
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Parasitic force Rz

  io(io_i) = linio([mdl, '/Micro-Station/Granite/Modal Analysis/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; % Absolute motion - Granite
  io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Modal Analysis/accelerometer'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion - Hexapod

  % Run the linearization
  G = linearize(mdl, io, 0);

We Take only the outputs corresponding to the vertical acceleration.

  G = G([3,9], :);

  % Input/Output names
  G.InputName  = {'Dw', 'Fty', 'Frz'};
  G.OutputName = {'Agm', 'Ahm'};

We integrate 1 time the output to have the velocity and we substract the absolute velocities to have the relative velocity.

  G = (1/s)*tf([-1, 1])*G;

  % Input/Output names
  G.InputName  = {'Dw', 'Fty', 'Frz'};
  G.OutputName = {'Vm'};

The obtained sensitivity to disturbances are shown bellow:

• The transfer function from vertical ground motion \(D_w\) to the vertical relative displacement from the micro-hexapod to the granite \(D\) is shown in Figure 2
• The sensitive from vertical forces applied in the Translation stage is shown in Figure 3

The Power Spectral Densities of the relative velocity between the hexapod and the marble in \([(m/s)^2/Hz]\) are loaded for the following sources of disturbance:

• Slip Ring Rotation (\(F_{r_z}\))
• Scan of the translation stage (effect in the vertical direction and in the horizontal direction)

Also, the Ground Motion is measured.

  gm  = load('./mat/psd_gm.mat', 'f', 'psd_gm');
  rz  = load('./mat/pxsp_r.mat', 'f', 'pxsp_r');
  tyz = load('./mat/pxz_ty_r.mat', 'f', 'pxz_ty_r');
  tyx = load('./mat/pxe_ty_r.mat', 'f', 'pxe_ty_r');

Because some 50Hz and harmonics were present in the ground motion measurement, we remove these peaks with the following code:

  f0s = [50, 100, 150, 200, 250, 350, 450];
  for f0 = f0s
      i = find(gm.f > f0-0.5 & gm.f < f0+0.5);
      gm.psd_gm(i) = linspace(gm.psd_gm(i(1)), gm.psd_gm(i(end)), length(i));
  end

We now compute the relative velocity between the hexapod and the granite due to ground motion.

  gm.psd_rv = gm.psd_gm.*abs(squeeze(freqresp(G('Vm', 'Dw'), gm.f, 'Hz'))).^2;

The Power Spectral Density of the relative motion and velocity of the hexapod with respect to the granite are shown in figures 5 and 6.

The Cumulative Amplitude Spectrum of the relative motion is shown in figure 7.

Using the extracted transfer functions from the disturbance force to the relative motion of the hexapod with respect to the granite (section 3) and using the measured PSD of the relative motion (section 4), we can compute the PSD of the disturbance force.

This is done below.

  rz.psd_f  = rz.pxsp_r./abs(squeeze(freqresp(G('Vm', 'Frz'), rz.f, 'Hz'))).^2;
  tyz.psd_f = tyz.pxz_ty_r./abs(squeeze(freqresp(G('Vm', 'Fty'), tyz.f, 'Hz'))).^2;

The obtained amplitude spectral densities of the disturbance forces are shown in Figure 8.

From the obtained spectral density of the disturbance sources, we can compute the resulting relative motion of the Hexapod with respect to the granite using the model.

This is equivalent as doing the inverse that was done in the previous section. This is done in order to verify that this is coherent with the measurements.

The power spectral density of the relative motion is computed below and the result is shown in Figure 9. We can see that this is exactly the same as the Figure 6.

  psd_gm_d = gm.psd_gm.*abs(squeeze(freqresp(G('Vm', 'Dw')/s, gm.f, 'Hz'))).^2;
  psd_ty_d = tyz.psd_f.*abs(squeeze(freqresp(G('Vm', 'Fty')/s, tyz.f, 'Hz'))).^2;
  psd_rz_d = rz.psd_f.*abs(squeeze(freqresp(G('Vm', 'Frz')/s, rz.f, 'Hz'))).^2;

The PSD of the disturbance force are now saved for further analysis.

  dist_f = struct();

  dist_f.f = gm.f; % Frequency Vector [Hz]

  dist_f.psd_gm = gm.psd_gm; % Power Spectral Density of the Ground Motion [m^2/Hz]
  dist_f.psd_ty = tyz.psd_f; % Power Spectral Density of the force induced by the Ty stage in the Z direction [N^2/Hz]
  dist_f.psd_rz = rz.psd_f; % Power Spectral Density of the force induced by the Rz stage in the Z direction [N^2/Hz]

  save('./mat/dist_psd.mat', 'dist_f');

Let’s initialize the time domain disturbances and load them.

  initializeDisturbances();
  dist = load('nass_disturbances.mat');

The time domain disturbance signals are shown in Figure 10.