Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances
Table of Contents
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.
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 \(x\) directionDwy
: Ground displacement in the \(y\) directionDwz
: Ground displacement in the \(z\) directionFty_x
: Forces applied by the Translation stage in the \(x\) directionFty_z
: Forces applied by the Translation stage in the \(z\) directionFrz_z
: Forces applied by the Spindle in the \(z\) directionFd
: 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)
2.4 Effect of Ground motion (Transmissibility).
2.5 Direct Forces (Compliance).
2.6 Conclusion
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section 4.
3 Effect of granite stiffness
In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
3.1 Analytical Analysis
3.1.1 Simple mass-spring-damper model
Let’s consider the system shown in Figure 8 consisting of two stacked mass-spring-damper systems. The bottom one represents the granite, and the top one all the positioning stages. We want the smallest stage “deformation” \(d = x^\prime - x\) due to ground motion \(w\).
Figure 8: Mass Spring Damper system consisting of a granite and a positioning stage
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) (w - x) \end{align}If we note \(d = x^\prime - x\), we obtain:
\begin{equation} \frac{d}{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}3.1.2 General Case
Let’s now considering a general positioning stage defined by:
- \(G^\prime(s) = \frac{F}{x}\): its mechanical “impedance”
- \(D^\prime(s) = \frac{d}{x}\): its “deformation” transfer function
Figure 9: Mass Spring Damper representing the granite and a general representation of positioning stages
The equation of motion are:
\begin{align} ms^2 x &= (cs + k) (x - w) - F \\ F &= G^\prime(s) x \\ d &= D^\prime(s) x \end{align}where:
- \(F\) is the force applied by the position stages on the granite mass
We can express \(d\) as a function of \(w\):
\begin{equation} \frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)} \end{equation}This is the transfer function that we would like to minimize.
Let’s verify this formula for a simple mass/spring/damper positioning stage. In that case, we have:
\begin{align*} D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\ G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \end{align*}And finally:
\begin{equation} \frac{d}{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}which is the same as the previously derived equation.
3.2 Soft Granite
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));
3.3 Effect of the Granite transfer function
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.
Figure 10: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (png, pdf)
3.4 Conclusion
Having a soft granite suspension could greatly improve the sensitivity the ground motion and thus the level of sample vibration if it is found that ground motion is the limiting factor.
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]);