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

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');

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]

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

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

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.

Let’s initialize a soft granite and see how the sensitivity to disturbances will change.

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

From Figure 10, we can see that having a “soft” granite suspension greatly lowers the sensitivity to ground motion. The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).

From Figures 11 and 12, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.

Let’s consider Figure 13 there \(G_d(s)\) is the transfer function from a signal \(d\) (the perturbation) to a signal \(y\) (the sample’s position error).

We can compute the Power Spectral Density (PSD) of signal \(y\) from the PSD of \(d\) and the norm of \(G_d(s)\):

If we now consider multiple disturbances \(d_1, \dots, d_n\) as shown in Figure 14, we have that:

Sometimes, we prefer to compute the Amplitude Spectral Density (ASD) which is related to the PSD by: \[ \Gamma_y(\omega) = \sqrt{S_y(\omega)} \]

The Cumulative Power Spectrum (CPS) is here defined as:

And the Cumulative Amplitude Spectrum (CAS):

The CAS evaluation for all frequency corresponds to the rms value of the considered quantity: \[ y_{\text{rms}} = \Psi(\omega = 0) = \sqrt{\int_0^\infty S_y(\nu) d\nu} \]

We compute the effect of perturbations on the motion error thanks to Eq. \eqref{eq:psd_transfer_function}.

The result is shown in:

• Figure 15: PSD of the vertical sample’s motion error due to vertical ground motion
• Figure 16: PSD of the vertical sample’s motion error due to vertical vibrations of the Spindle

We compute the effect of all perturbations on the vertical position error using Eq. \eqref{eq:sum_psd} and the resulting PSD is shown in Figure 17.

Similarly, the Cumulative Amplitude Spectrum of the sample vibrations are shown:

• Figure 18: due to vertical ground motion
• Figure 19: due to vertical vibrations of the Spindle
• Figure 20: due to all considered perturbations

The black dashed line corresponds to the performance objective of a sample vibration equal to \(10\ nm [rms]\).

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([1, 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([1, 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(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]);

Let’s consider Figure 21 where a controller \(K\) is used to reduce the effect of the disturbance \(d\) on the position error \(y\).

The reduction of the impact of \(d\) on \(y\) thanks to feedback is described by the following equation:

The transfer functions corresponding to \(G_d\) are those identified in Section 2.

As a first approximation, we can consider that the controller \(K\) is designed in such a way that the loop gain \(KG\) is a pure integrator: \[ L_1(s) = K_1(s) G(s) = \frac{\omega_c}{s} \] where \(\omega_c\) is the crossover frequency.

We may then consider another controller in such a way that the loop gain corresponds to a double integrator with a lead centered with the crossover frequency \(\omega_c\): \[ L_2(s) = K_2(s) G(s) = \left( \frac{\omega_c}{s} \right)^2 \cdot \frac{1 + \frac{s}{\omega_c/2}}{1 + \frac{s}{2\omega_c}} \]

In the next section, we see how the power spectral density of \(y\) is reduced as a function of the control bandwidth \(\omega_c\). This will help to determine what is the approximate control bandwidth required such that the rms value of \(y\) is below \(10nm\).

Let’s first see how does the Cumulative Amplitude Spectrum of the sample’s motion error is modified by the control.

In Figure 22 is shown the Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses. The controller used in this simulation is \(K_1\). The loop gain is then a pure integrator.

In Figure 23 is shown the expected RMS value of the sample’s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot). As expected, it is shown that \(K_2\) performs better than \(K_1\). This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.

The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample’s motion error less than 10nm rms are gathered in Table 1.