nass-simscape/org/optimal_stiffness_disturbances.org

Determination of the optimal nano-hexapod's stiffness for reducing the effect of disturbances

• Introduction
• Disturbances
  • Introduction
  • Plots
• Effect of disturbances on the position error
  • Introduction
  • Initialization
  • Identification
  • Sensitivity to Stages vibration (Filtering)
  • Effect of Ground motion (Transmissibility).
  • Direct Forces (Compliance).
  • Conclusion
• Effect of granite stiffness
  • Analytical Analysis
  • Soft Granite
  • Effect of the Granite transfer function
• Open Loop Budget Error
  • Introduction
  • Load of the identified disturbances and transfer functions
  • Equations
  • Results
  • Cumulative Amplitude Spectrum
• Closed Loop Budget Error
  • Introduction
  • Reduction thanks to feedback - Required bandwidth
• Conclusion

Introduction   ignore

In this document is studied how the stiffness of the nano-hexapod will impact the effect of disturbances on the position error of the sample.

It is divided in the following sections:

• Section sec:psd_disturbances: the disturbances are listed and their Power Spectral Densities (PSD) are shown
• Section sec:effect_disturbances: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses
• Section sec:granite_stiffness:
• Section sec:open_loop_budget_error: from both the PSD of the disturbances and the transfer function from disturbances to sample's position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)
• Section sec:closed_loop_budget_error: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values

Disturbances

<<sec:psd_disturbances>>

Introduction   ignore

The main disturbances considered here are:

• $D_w$: Ground displacement in the $x$, $y$ and $z$ directions
• $F_{ty}$: Forces applied by the Translation stage in the $x$ and $z$ directions
• $F_{rz}$: Forces applied by the Spindle in the $z$ direction
• $F_d$: Direct forces applied at the center of mass of the Payload

The level of these disturbances has been identified form experiments which are detailed in this document.

Plots   ignore

The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures fig:opt_stiff_dist_gm and fig:opt_stiff_dist_fty_frz.

In this study, the expected frequency content of the direct forces applied to the payload is not considered.

<<plt-matlab>>

Amplitude Spectral Density of the Ground Displacement (png, pdf)

<<plt-matlab>>

Amplitude Spectral Density of the "parasitic" forces comming from the Translation stage and the spindle (png, pdf)

Effect of disturbances on the position error

<<sec:effect_disturbances>>

Introduction   ignore

In this section, we use the Simscape model to identify the transfer function from disturbances to the position error of the sample. We do that for a wide range of nano-hexapod stiffnesses and we compare the obtained results.

Initialization

We initialize all the stages with the default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

We use a sample mass of 10kg.

  initializeSample('mass', 10);

We include gravity, and we use no controller.

  initializeSimscapeConfiguration('gravity', true);
  initializeController();
  initializeDisturbances('enable', false);
  initializeLoggingConfiguration('log', 'none');

Identification

The considered inputs are:

• Dwx: Ground displacement in the $x$ direction
• Dwy: Ground displacement in the $y$ direction
• Dwz: Ground displacement in the $z$ direction
• Fty_x: Forces applied by the Translation stage in the $x$ direction
• Fty_z: Forces applied by the Translation stage in the $z$ direction
• Frz_z: Forces applied by the Spindle in the $z$ direction
• Fd: Direct forces applied at the center of mass of the Payload

The outputs are Ex, Ey, Ez, Erx, Ery, Erz which are the 3 positions and 3 orientations errors of the sample.

We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.

  Ks = logspace(3,9,7); % [N/m]

Sensitivity to Stages vibration (Filtering)

The sensitivity the stage vibrations are displayed:

• Figure fig:opt_stiff_sensitivity_Frz: sensitivity to vertical spindle vibrations
• Figure fig:opt_stiff_sensitivity_Fty_z: sensitivity to vertical translation stage vibrations
• Figure fig:opt_stiff_sensitivity_Fty_x: sensitivity to horizontal (x) translation stage vibrations

<<plt-matlab>>

Sensitivity to Spindle vertical motion error ($F_{rz}$) to the vertical error position of the sample ($E_z$) (png, pdf)

<<plt-matlab>>

Sensitivity to Translation stage vertical motion error ($F_{ty,z}$) to the vertical error position of the sample ($E_z$) (png, pdf)

<<plt-matlab>>

Sensitivity to Translation stage $x$ motion error ($F_{ty,x}$) to the error position of the sample in the $x$ direction ($E_x$) (png, pdf)

Effect of Ground motion (Transmissibility).

The effect of Ground motion on the position error of the sample is shown in Figure fig:opt_stiff_sensitivity_Dw.

<<plt-matlab>>

Sensitivity to Ground motion ($D_{w}$) to the position error of the sample ($E_y$ and $E_z$) (png, pdf)

Direct Forces (Compliance).

The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure fig:opt_stiff_sensitivity_Fd.

<<plt-matlab>>

Sensitivity to Direct forces and torques applied to the sample ($F_d$, $M_d$) to the position error of the sample (png, pdf)

Conclusion

Effect of granite stiffness

<<sec:granite_stiffness>>

Analytical Analysis

  \begin{tikzpicture}
    % ====================
    % Parameters
    % ====================
    \def\massw{2.2}  % Width of the masses
    \def\massh{0.8}  % Height of the masses
    \def\spaceh{1.2} % Height of the springs/dampers
    \def\dispw{0.3}  % Width of the dashed line for the displacement
    \def\disph{0.5}  % Height of the arrow for the displacements
    \def\bracs{0.05} % Brace spacing vertically
    \def\brach{-10pt} % Brace shift horizontaly
    % ====================


    % ====================
    % Ground
    % ====================
    \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
    \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
    \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$};

    % ====================
    % Micro Station
    % ====================
    \begin{scope}[shift={(0, 0)}]
      % Mass
      \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};

      % Spring, Damper, and Actuator
      \draw[spring] (-0.4*\massw, 0)   -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
      \draw[damper] (0, 0)             -- ( 0, \spaceh)          node[midway, left=0.2]{$c$};

      % Displacements
      \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
      \draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x$};

      % Legend
      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
        (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
    \end{scope}

    % ====================
    % Nano Station
    % ====================
    \begin{scope}[shift={(0, \spaceh+\massh)}]
      % Mass
      \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};

      % Spring, Damper, and Actuator
      \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
      \draw[damper] (0, 0)           -- ( 0, \spaceh)          node[midway, left=0.2]{$c^\prime$};

      % Displacements
      \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
      \draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x^\prime$};

      % Legend
      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
        (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
    \end{scope}
  \end{tikzpicture}

Figure caption

If we write the equation of motion of the system in Figure fig:2dof_system_granite_stiffness, we obtain:

\begin{align} m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\ ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x) \end{align}

If we note $d = x^\prime - x$, we obtain:

\begin{equation} \frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)} \end{equation}

Soft Granite

Let's initialize a soft granite that will act as an isolation stage from ground motion.

  initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));

  Ks = logspace(3,9,7); % [N/m]

  for i = 1:length(Ks)
      initializeNanoHexapod('k', Ks(i));

      G = linearize(mdl, io);
      G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
      G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
      Gdr(i) = {minreal(G)};
  end

Effect of the Granite transfer function

Open Loop Budget Error

<<sec:open_loop_budget_error>>

Introduction   ignore

Load of the identified disturbances and transfer functions

  load('./mat/dist_psd.mat', 'dist_f');
  load('./mat/opt_stiffness_disturbances.mat', 'Gd')

Equations

Results

Effect of all disturbances

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(dist_f.psd_rz).*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{m}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);

Cumulative Amplitude Spectrum

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_ty.*abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))).^2)))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
  end
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS $[m]$')
  legend('Location', 'southwest');
  xlim([2, 500]); ylim([1e-10 1e-6]);

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_rz.*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))).^2)))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
  end
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS $[m]$')
  legend('Location', 'southwest');
  xlim([2, 500]); ylim([1e-10 1e-6]);

Ground motion

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))).^2)))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
  end
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
  legend('Location', 'northeast');
  xlim([2, 500]); ylim([1e-10 1e-6]);

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ex', 'Dwx'), freqs, 'Hz'))).^2)))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
  end
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'lin');
  xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
  legend('Location', 'northeast');
  xlim([2, 500]);

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ey', 'Dwy'), freqs, 'Hz'))).^2)))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
  end
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'lin');
  xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
  legend('Location', 'northeast');
  xlim([2, 500]);

Sum of all perturbations

  psd_tot = zeros(length(freqs), length(Ks));

  for i = 1:length(Ks)
      psd_tot(:,i) = dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'  ), freqs, 'Hz'))).^2 + ...
          dist_f.psd_ty.*abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))).^2 + ...
          dist_f.psd_rz.*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))).^2;
  end

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(psd_tot(:,i))))), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
  end
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS $E_z$ $[m]$')
  legend('Location', 'northeast');
  xlim([1, 500]); ylim([1e-10 1e-6]);

Closed Loop Budget Error

<<sec:closed_loop_budget_error>>

Introduction   ignore

Reduction thanks to feedback - Required bandwidth

  wc = 1*2*pi; % [rad/s]
  xic = 0.5;

  S = (s/wc)/(1 + s/wc);

  bodeFig({S}, logspace(-1,2,1000))

  wc = [1, 5, 10, 20, 50, 100, 200];

  S1 = {zeros(length(wc), 1)};
  S2 = {zeros(length(wc), 1)};

  for j = 1:length(wc)
      L = (2*pi*wc(j))/s; % Simple integrator
      S1{j} = 1/(1 + L);
      L = ((2*pi*wc(j))/s)^2*(1 + s/(2*pi*wc(j)/2))/(1 + s/(2*pi*wc(j)*2));
      S2{j} = 1/(1 + L);
  end

  freqs = dist_f.f;

  figure;
  hold on;
  i = 6;
  for j = 1:length(wc)
      set(gca,'ColorOrderIndex',j);
      plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(S1{j}, freqs, 'Hz'))).^2.*psd_tot(:,i))))), '-', ...
           'DisplayName', sprintf('$\\omega_c = %.0f$ [Hz]', wc(j)));
  end
  plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(psd_tot(:,i))))), 'k-', ...
       'DisplayName', 'Open-Loop');
  plot([freqs(1) freqs(end)], [10e-9 10e-9], 'k--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
  legend('Location', 'northeast');
  xlim([0.5, 500]); ylim([1e-10 1e-6]);

  wc = logspace(0, 3, 100);

  Dz1_rms = zeros(length(Ks), length(wc));
  Dz2_rms = zeros(length(Ks), length(wc));
  for i = 1:length(Ks)
      for j = 1:length(wc)
          L = (2*pi*wc(j))/s;
          Dz1_rms(i, j) = sqrt(trapz(freqs, abs(squeeze(freqresp(1/(1 + L), freqs, 'Hz'))).^2.*psd_tot(:,i)));

          L = ((2*pi*wc(j))/s)^2*(1 + s/(2*pi*wc(j)/2))/(1 + s/(2*pi*wc(j)*2));
          Dz2_rms(i, j) = sqrt(trapz(freqs, abs(squeeze(freqresp(1/(1 + L), freqs, 'Hz'))).^2.*psd_tot(:,i)));
      end
  end

  freqs = dist_f.f;

  figure;
  hold on;
  for i = 1:length(Ks)
    set(gca,'ColorOrderIndex',i);
    plot(wc, Dz1_rms(i, :), '-', ...
           'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)))

    set(gca,'ColorOrderIndex',i);
    plot(wc, Dz2_rms(i, :), '--', ...
           'HandleVisibility', 'off')
  end
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Control Bandwidth [Hz]'); ylabel('$E_z\ [m, rms]$')
  legend('Location', 'southwest');
  xlim([1, 500]);

Conclusion