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

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>>
figs/opt_stiff_dist_gm.png
Amplitude Spectral Density of the Ground Displacement (png, pdf) 
<<plt-matlab>>
figs/opt_stiff_dist_fty_frz.png
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

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

Identification

Inputs:

  • 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
  Ks = logspace(3,9,7); % [N/m]
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Micro-Hexapod
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwx');   io_i = io_i + 1; % X Ground motion
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwy');   io_i = io_i + 1; % Y Ground motion
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz');   io_i = io_i + 1; % Z Ground motion
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_x'); io_i = io_i + 1; % Parasitic force Ty - X
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fty_z'); io_i = io_i + 1; % Parasitic force Ty - Z
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Parasitic force Rz - Z
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fd');    io_i = io_i + 1; % Direct forces

  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En');   io_i = io_i + 1; % Position Error
  for i = 1:length(Ks)
      initializeNanoHexapod('k', Ks(i));

      % Run the linearization
      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'};
      Gd(i) = {minreal(G)};
  end

Plots

Effect of Stages vibration (Filtering).

Effect of Ground motion (Transmissibility).

Direct Forces (Compliance).

Save 

  save('./mat/opt_stiffness_disturbances.mat', 'Gd')

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}

figs/2dof_system_granite_stiffness.png

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