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

Table of Contents

1 Spectral Density of Disturbances

The level of disturbances has been identified form experiments. This is detailed in this document.

1.1 Load of the identified disturbances

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

1.2 Plots

2 Effect of disturbances on the position error

2.1 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();

2.2 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

2.3 Plots

Effect of Stages vibration (Filtering). Effect of Ground motion (Transmissibility). Direct Forces (Compliance).

2.4 Save

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

3 Effect of granite stiffness

3.1 Analytical Analysis

2dof_system_granite_stiffness.png

Figure 1: Figure caption

If we write the equation of motion of the system in Figure 1, 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} \label{org4396920} \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}

3.2 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

3.3 Effect of the Granite transfer function

4 Open Loop Budget Error

4.1 Load of the identified disturbances and transfer functions

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

4.2 Equations

4.3 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]);

4.4 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]);

5 Closed Loop Budget Error

5.1 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]);

6 Conclusion

Author: Dehaeze Thomas

Created: 2020-04-07 mar. 14:57