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

1. Disturbances
2. Effect of disturbances on the position error
3. Effect of granite stiffness
4. Open Loop Budget Error
5. Closed Loop Budget Error
- 5.1. Reduction thanks to feedback - Required bandwidth
6. Conclusion

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 1: the disturbances are listed and their Power Spectral Densities (PSD) are shown
Section 2: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses
Section 3:
Section 4: 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 5: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values

1 Disturbances

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.

The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures 1 and 2.

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

Figure 1: Amplitude Spectral Density of the Ground Displacement (png, pdf)

Figure 2: Amplitude Spectral Density of the “parasitic” forces comming from the Translation stage and the spindle (png, pdf)

2 Effect of disturbances on the position error

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.

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);

We include gravity, and we use no controller.

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

2.2 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]

2.3 Sensitivity to Stages vibration (Filtering)

The sensitivity the stage vibrations are displayed:

Figure 3: sensitivity to vertical spindle vibrations
Figure 4: sensitivity to vertical translation stage vibrations
Figure 5: sensitivity to horizontal (x) translation stage vibrations

Figure 3: Sensitivity to Spindle vertical motion error (\(F_{rz}\)) to the vertical error position of the sample (\(E_z\)) (png, pdf)

Figure 4: Sensitivity to Translation stage vertical motion error (\(F_{ty,z}\)) to the vertical error position of the sample (\(E_z\)) (png, pdf)

Figure 5: 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)

2.4 Effect of Ground motion (Transmissibility).

The effect of Ground motion on the position error of the sample is shown in Figure 6.

Figure 6: Sensitivity to Ground motion (\(D_{w}\)) to the position error of the sample (\(E_y\) and \(E_z\)) (png, pdf)

2.5 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 7.

Figure 7: Sensitivity to Direct forces and torques applied to the sample (\(F_d\), \(M_d\)) to the position error of the sample (png, pdf)

2.6 Conclusion

3 Effect of granite stiffness

3.1 Analytical Analysis

Figure 8: Figure caption

If we write the equation of motion of the system in Figure 8, 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]);