nass-simscape/org/disturbances.org

Identification of the disturbances

• Introduction
• Simscape Model
• Identification
• Sensitivity to Disturbances
• Power Spectral Density of the effect of the disturbances
• Compute the Power Spectral Density of the disturbance force
• Noise Budget
• Save

Introduction   ignore

The goal here is to extract the Power Spectral Density of the sources of perturbation.

The sources of perturbations are (schematically shown in figure fig:uniaxial-model-micro-station):

• $D_w$: Ground Motion
• Parasitic forces applied in the system when scanning with the Translation Stage and the Spindle ($F_{rz}$ and $F_{ty}$). These forces can be due to imperfect guiding for instance.

Because we cannot measure directly the perturbation forces, we have the measure the effect of those perturbations on the system (in terms of velocity for instance using geophones, $D$ on figure fig:uniaxial-model-micro-station) and then, using a model, compute the forces that induced such velocity.

Schematic of the Micro Station and the sources of disturbance

This file is divided in the following sections:

• Section sec:simscape_model: the simscape model used here is presented
• Section sec:identification: 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
• Section sec:sensitivity_disturbances: the bode plot of those transfer functions are shown
• Section sec:psd_dist: the measured PSD of the effect of the disturbances are shown
• Section sec:psd_force_dist: from the model and the measured PSD, the PSD of the disturbance forces are computed
• Section sec:noise_budget: with the computed PSD, the noise budget of the system is done

Simscape Model

<<sec:simscape_model>>

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');

Identification

<<sec:identification>> 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
  % io(io_i) = linio([mdl, '/Vm'],   1, 'openoutput'); io_i = io_i + 1; % Relative Velocity hexapod/granite

  % 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'};

Sensitivity to Disturbances

<<sec:sensitivity_disturbances>>

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 fig:sensitivity_dist_gm
• The sensitive from vertical forces applied in the Translation stage is shown in Figure fig:sensitivity_dist_fty

  <<plt-matlab>>

Sensitivity to Ground Motion (png, pdf)

  <<plt-matlab>>

Sensitivity to vertical forces applied by the Ty stage (png, pdf)

  <<plt-matlab>>

Sensitivity to vertical forces applied by the Rz stage (png, pdf)

Power Spectral Density of the effect of the disturbances

<<sec:psd_dist>>

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');

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 fig:dist_effect_relative_velocity and fig:dist_effect_relative_motion.

The Cumulative Amplitude Spectrum of the relative motion is shown in figure fig:dist_effect_relative_motion_cas.

  <<plt-matlab>>

Amplitude Spectral Density of the relative velocity of the hexapod with respect to the granite due to different sources of perturbation (png, pdf)

  <<plt-matlab>>

Amplitude Spectral Density of the relative displacement of the hexapod with respect to the granite due to different sources of perturbation (png, pdf)

  <<plt-matlab>>

Cumulative Amplitude Spectrum of the relative motion due to different sources of perturbation (png, pdf)

From Figure fig:dist_effect_relative_motion_cas, we can see that the translation stage and the rotation stage have almost the same effect on the position error. Also, the ground motion has a relatively negligible effect on the position error.

Compute the Power Spectral Density of the disturbance force

<<sec:psd_force_dist>>

Using the extracted transfer functions from the disturbance force to the relative motion of the hexapod with respect to the granite (section sec:sensitivity_disturbances) and using the measured PSD of the relative motion (section sec:psd_dist), 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 fig:dist_force_psd.

  <<plt-matlab>>

Amplitude Spectral Density of the disturbance force (png, pdf)

Noise Budget

<<sec:noise_budget>>

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 fig:psd_effect_dist_verif. We can see that this is exactly the same as the Figure fig:dist_effect_relative_motion.

  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;

  <<plt-matlab>>

Computed Effect of the disturbances on the relative displacement hexapod/granite (png, pdf)

Save

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');