2.6 Conclusion
+
+
2.6 Conclusion
-
-
3.4 Conclusion
+
+
-
3.4 Conclusion
@@ -696,27 +697,130 @@ Having a soft granite suspension could greatly improve the sensitivity the groun
++Now that the frequency content of disturbances have been estimated (Section 1) and the transfer functions from disturbances to the position error of the sample have been identified (Section 2), we can compute the level of sample vibration due to the disturbances. +
+ ++We then can conclude and the nano-hexapod stiffness that will lower the sample position error. +
-
4.1 Load of the identified disturbances and transfer functions
+
+
4.1 Noise Budgeting - Theory
-
+
+
-
+load('./mat/dist_psd.mat', 'dist_f'); -load('./mat/opt_stiffness_disturbances.mat', 'Gd') -+
+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). +
+ + +
+
+
+
+
Figure 13: Figure caption
++We can compute the Power Spectral Density (PSD) of signal \(y\) from the PSD of \(d\) and the norm of \(G_d(s)\): +
+\begin{equation} + S_{y}(\omega) = \left|G_d(j\omega)\right|^2 S_{d}(\omega) \label{eq:psd_transfer_function} +\end{equation} + ++If we now consider multiple disturbances \(d_1, \dots, d_n\) as shown in Figure 14, we have that: +
+\begin{equation} + S_{y}(\omega) = \left|G_{d_1}(j\omega)\right|^2 S_{d_1}(\omega) + \dots + \left|G_{d_n}(j\omega)\right|^2 S_{d_n}(\omega) \label{eq:sum_psd} +\end{equation} + ++Sometimes, we prefer to compute the Amplitude Spectral Density (ASD) which is related to the PSD by: +\[ \Gamma_y(\omega) = \sqrt{S_y(\omega)} \] +
+ + +
+
+
+
+
Figure 14: Figure caption
++The Cumulative Power Spectrum (CPS) is here defined as: +
+\begin{equation} + \Phi_y(\omega) = \int_\omega^\infty S_y(\nu) d\nu +\end{equation} + ++And the Cumulative Amplitude Spectrum (CAS): +
+\begin{equation} + \Psi(\omega) = \sqrt{\Phi(\omega)} = \sqrt{\int_\omega^\infty S_y(\nu) d\nu} +\end{equation} + ++The CAS evaluation for all frequency corresponds to the rms value of the considered quantity: +\[ y_{\text{rms}} = \Psi(\omega = 0) = \sqrt{\int_0^\infty S_y(\nu) d\nu} \] +
+
+
-4.2 Power Spectral Densities
+
+
+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 +
+
+
+
+
+
Figure 15: Amplitude Spectral Density of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (png, pdf)
+
+
+
+
+
Figure 16: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (png, pdf)
++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. +
+ +
+
-
-
-4.2 Equations
-
-
-4.3 Results
+
+
-
-
-4.3 Cumulative Amplitude Spectrum
-Effect of all disturbances +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 \(10\ nm [rms]\).
@@ -725,39 +829,25 @@ Effect of all disturbances
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]);
-
-
-
-
@@ -875,9 +920,45 @@ xlim([1, 500]); ylim([1e-10 1e
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)))), '-', ... + 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 $[m]$') -legend('Location', 'southwest'); -xlim([2, 500]); ylim([1e-10 1e-6]); +xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$') +legend('Location', 'northeast'); +xlim([1, 500]); ylim([1e-10 1e-6]);
+
+
+
Figure 18: Cumulative Amplitude Spectrum of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (png, pdf)
+freqs = dist_f.f; @@ -772,79 +862,15 @@ 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]); +xlim([1, 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 -+
Figure 19: Cumulative Amplitude Spectrum of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (png, pdf)
@@ -864,6 +890,25 @@ legend('Location', 'nor
xlim([1, 500]); ylim([1e-10 1e-6]);
+
+
+
+
+
+
4.4 Conclusion
+
+
+
+From Figure 20, we can see that a soft nano-hexapod \(k<10^6\ [N/m]\) significantly reduces the effect of perturbations from 20Hz to 300Hz. +
+ +
->
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+ <>
+#+end_src
+
+#+begin_src matlab :tangle no
+ simulinkproject('../');
+#+end_src
+
** Analytical Analysis
*** Simple mass-spring-damper model
Let's consider the system shown in Figure [[fig:2dof_system_granite_stiffness]] consisting of two stacked mass-spring-damper systems.
@@ -691,6 +699,10 @@ From Figures [[fig:opt_stiff_soft_granite_Frz]] and [[fig:opt_stiff_soft_granite
* Open Loop Budget Error
<>
** Introduction :ignore:
+Now that the frequency content of disturbances have been estimated (Section [[sec:psd_disturbances]]) and the transfer functions from disturbances to the position error of the sample have been identified (Section [[sec:effect_disturbances]]), we can compute the level of sample vibration due to the disturbances.
+
+We then can conclude and the nano-hexapod stiffness that will lower the sample position error.
+
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
@@ -704,30 +716,120 @@ From Figures [[fig:opt_stiff_soft_granite_Frz]] and [[fig:opt_stiff_soft_granite
simulinkproject('../');
#+end_src
-#+begin_src matlab
- load('mat/conf_simulink.mat');
+** Noise Budgeting - Theory
+Let's consider Figure [[fig:psd_change_tf]] there $G_d(s)$ is the transfer function from a signal $d$ (the perturbation) to a signal $y$ (the sample's position error).
- open('nass_model.slx')
+#+begin_src latex :file psd_change_tf.pdf
+ \begin{tikzpicture}
+ \node[block] (G) at (0, 0) {$G_d(s)$};
+
+ \draw[<-] (G.west) -- ++(-1, 0) node[above right]{$d$};
+ \draw[->] (G.east) -- ++( 1, 0) node[above left ]{$y$};
+ \end{tikzpicture}
#+end_src
-** Load of the identified disturbances and transfer functions
-#+begin_src matlab
+#+name: fig:psd_change_tf
+#+caption: Figure caption
+#+RESULTS:
+[[file:figs/psd_change_tf.png]]
+
+We can compute the Power Spectral Density (PSD) of signal $y$ from the PSD of $d$ and the norm of $G_d(s)$:
+\begin{equation}
+ S_{y}(\omega) = \left|G_d(j\omega)\right|^2 S_{d}(\omega) \label{eq:psd_transfer_function}
+\end{equation}
+
+If we now consider multiple disturbances $d_1, \dots, d_n$ as shown in Figure [[fig:psd_change_tf_multiple_pert]], we have that:
+\begin{equation}
+ S_{y}(\omega) = \left|G_{d_1}(j\omega)\right|^2 S_{d_1}(\omega) + \dots + \left|G_{d_n}(j\omega)\right|^2 S_{d_n}(\omega) \label{eq:sum_psd}
+\end{equation}
+
+Sometimes, we prefer to compute the *Amplitude* Spectral Density (ASD) which is related to the PSD by:
+\[ \Gamma_y(\omega) = \sqrt{S_y(\omega)} \]
+
+#+begin_src latex :file psd_change_tf_multiple_pert.pdf
+ \begin{tikzpicture}
+ \node[block] (Gm) at (0, 0) {$\dots$};
+ \draw[<-] (Gm.west) -- ++(-1, 0);
+
+ \node[block, above=0.5 of Gm] (G1) {$G_{d_1}(s)$};
+ \draw[<-] (G1.west) -- ++(-1, 0) node[above right]{$d_1$};
+
+ \node[block, below=0.5 of Gm] (Gn) {$G_{d_n}(s)$};
+ \draw[<-] (Gn.west) -- ++(-1, 0) node[above right]{$d_n$};
+
+ \node[addb, right= of Gm] (add) {};
+
+ \draw[->] (G1.east) -| (add.north);
+ \draw[->] (Gm.east) -- (add.west);
+ \draw[->] (Gn.east) -| (add.south);
+
+ \draw[->] (add) -- ++( 1, 0) node[above left]{$y$};
+ \end{tikzpicture}
+#+end_src
+
+#+name: fig:psd_change_tf_multiple_pert
+#+caption: Figure caption
+#+RESULTS:
+[[file:figs/psd_change_tf_multiple_pert.png]]
+
+The Cumulative Power Spectrum (CPS) is here defined as:
+\begin{equation}
+ \Phi_y(\omega) = \int_\omega^\infty S_y(\nu) d\nu
+\end{equation}
+
+And the Cumulative Amplitude Spectrum (CAS):
+\begin{equation}
+ \Psi(\omega) = \sqrt{\Phi(\omega)} = \sqrt{\int_\omega^\infty S_y(\nu) d\nu}
+\end{equation}
+
+The CAS evaluation for all frequency corresponds to the rms value of the considered quantity:
+\[ y_{\text{rms}} = \Psi(\omega = 0) = \sqrt{\int_0^\infty S_y(\nu) d\nu} \]
+
+** Power Spectral Densities
+We compute the effect of perturbations on the motion error thanks to Eq. eqref:eq:psd_transfer_function.
+
+The result is shown in:
+- Figure [[fig:opt_stiff_psd_dz_gm]]: PSD of the vertical sample's motion error due to vertical ground motion
+- Figure [[fig:opt_stiff_psd_dz_rz]]: PSD of the vertical sample's motion error due to vertical vibrations of the Spindle
+
+#+begin_src matlab :exports none
load('./mat/dist_psd.mat', 'dist_f');
load('./mat/opt_stiffness_disturbances.mat', 'Gd')
#+end_src
-** Equations
-
-** Results
-Effect of all disturbances
-
-#+begin_src matlab
+#+begin_src matlab :exports none
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'))));
+ plot(freqs, sqrt(dist_f.psd_gm).*abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), '-', ...
+ 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+ 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([1, 500]);
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_psd_dz_gm.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+name: fig:opt_stiff_psd_dz_gm
+#+caption: Amplitude Spectral Density of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses ([[./figs/opt_stiff_psd_dz_gm.png][png]], [[./figs/opt_stiff_psd_dz_gm.pdf][pdf]])
+[[file:figs/opt_stiff_psd_dz_gm.png]]
+
+#+begin_src matlab :exports none
+ 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'))), '-', ...
+ 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
end
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
@@ -736,24 +838,85 @@ Effect of all disturbances
xlim([2, 500]);
#+end_src
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_psd_dz_rz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+name: fig:opt_stiff_psd_dz_rz
+#+caption: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses ([[./figs/opt_stiff_psd_dz_rz.png][png]], [[./figs/opt_stiff_psd_dz_rz.pdf][pdf]])
+[[file:figs/opt_stiff_psd_dz_rz.png]]
+
+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 [[fig:opt_stiff_psd_dz_tot]].
+#+begin_src matlab :exports none
+ 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
+#+end_src
+
+#+begin_src matlab :exports none
+ freqs = dist_f.f;
+
+ figure;
+ hold on;
+ for i = 1:length(Ks)
+ plot(freqs, sqrt(psd_tot(:,i)), '-', ...
+ 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
+ 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([1, 500]);
+#+end_src
+
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_psd_dz_tot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+name: fig:opt_stiff_psd_dz_tot
+#+caption: Amplitude Spectral Density of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses ([[./figs/opt_stiff_psd_dz_tot.png][png]], [[./figs/opt_stiff_psd_dz_tot.pdf][pdf]])
+[[file:figs/opt_stiff_psd_dz_tot.png]]
+
** Cumulative Amplitude Spectrum
+Similarly, the Cumulative Amplitude Spectrum of the sample vibrations are shown:
+- Figure [[fig:opt_stiff_cas_dz_gm]]: due to vertical ground motion
+- Figure [[fig:opt_stiff_cas_dz_rz]]: due to vertical vibrations of the Spindle
+- Figure [[fig:opt_stiff_cas_dz_tot]]: due to all considered perturbations
+
+The black dashed line corresponds to the performance objective of a sample vibration equal to $10\ nm [rms]$.
+
#+begin_src matlab
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)))), '-', ...
+ 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 $[m]$')
- legend('Location', 'southwest');
- xlim([2, 500]); ylim([1e-10 1e-6]);
+ xlabel('Frequency [Hz]'); ylabel('CAS $E_y$ $[m]$')
+ legend('Location', 'northeast');
+ xlim([1, 500]); ylim([1e-10 1e-6]);
#+end_src
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_cas_dz_gm.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+name: fig:opt_stiff_cas_dz_gm
+#+caption: Cumulative Amplitude Spectrum of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses ([[./figs/opt_stiff_cas_dz_gm.png][png]], [[./figs/opt_stiff_cas_dz_gm.pdf][pdf]])
+[[file:figs/opt_stiff_cas_dz_gm.png]]
+
#+begin_src matlab
freqs = dist_f.f;
@@ -768,71 +931,17 @@ Effect of all disturbances
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]);
+ xlim([1, 500]); ylim([1e-10 1e-6]);
#+end_src
-Ground motion
-#+begin_src matlab
- 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]);
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_cas_dz_rz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
#+end_src
-#+begin_src matlab
- 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]);
-#+end_src
-
-#+begin_src matlab
- 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]);
-#+end_src
-
-Sum of all perturbations
-#+begin_src matlab
- 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
-#+end_src
+#+name: fig:opt_stiff_cas_dz_rz
+#+caption: Cumulative Amplitude Spectrum of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses ([[./figs/opt_stiff_cas_dz_rz.png][png]], [[./figs/opt_stiff_cas_dz_rz.pdf][pdf]])
+[[file:figs/opt_stiff_cas_dz_rz.png]]
#+begin_src matlab
freqs = dist_f.f;
@@ -851,9 +960,77 @@ Sum of all perturbations
xlim([1, 500]); ylim([1e-10 1e-6]);
#+end_src
+#+header: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/opt_stiff_cas_dz_tot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+<>
+#+end_src
+
+#+name: fig:opt_stiff_cas_dz_tot
+#+caption: Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses ([[./figs/opt_stiff_cas_dz_tot.png][png]], [[./figs/opt_stiff_cas_dz_tot.pdf][pdf]])
+[[file:figs/opt_stiff_cas_dz_tot.png]]
+
+** Conclusion
+#+begin_important
+ From Figure [[fig:opt_stiff_cas_dz_tot]], we can see that a soft nano-hexapod $k<10^6\ [N/m]$ significantly reduces the effect of perturbations from 20Hz to 300Hz.
+#+end_important
+
* Closed Loop Budget Error
<>
** Introduction :ignore:
+** Matlab Init :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+ <>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+ <>
+#+end_src
+
+#+begin_src matlab :tangle no
+ simulinkproject('../');
+#+end_src
+
+** Approximation of the effect of feedback on the motion error
+Let's consider Figure [[fig:effect_feedback_disturbance_diagram]] where a controller $K$ is used to reduce the effect of the disturbance $d$ on the position error $y$.
+
+#+begin_src latex :file effect_feedback_disturbance_diagram.pdf
+ \begin{tikzpicture}
+ \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
+ \node[block, right=0.6 of addfb] (K){$K$};
+ \node[block, right=0.6 of K] (G){$G$};
+ \node[addb={+}{}{}{}{}, right=0.6 of G] (adddy){};
+ \node[block, above=0.6 of adddy] (Gd){$G_d$};
+
+ \draw[<-] (addfb.west) -- ++(-0.6, 0) node[above right]{$r$};
+ \draw[->] (addfb.east) -- (K.west);
+ \draw[->] (K.east) -- (G.west) node[above left]{$u$};
+ \draw[->] (G.east) -- (adddy.west);
+ \draw[->] (adddy.east) -- ++(1, 0) node[above left]{$y$};
+ \draw[->] ($(adddy.east)+(0.6, 0)$) node[branch]{} -- ++(0, -1) -| (addfb.south);
+ \draw[<-] (Gd.north) -- ++(0, 0.6) node[below right]{$d$};
+ \draw[->] (Gd.south) -- (adddy.north);
+ \end{tikzpicture}
+#+end_src
+
+#+name: fig:effect_feedback_disturbance_diagram
+#+caption: Feedback System
+#+RESULTS:
+[[file:figs/effect_feedback_disturbance_diagram.png]]
+
+The reduction of the impact of $d$ on $y$ thanks to feedback is described by the following equation:
+\begin{equation}
+ \frac{y}{d} = \frac{G_d}{1 + KG}
+\end{equation}
+
+
+As a first approximation, we can consider that the controller 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 $\omega_c$:
+\[ L_2(s) = K_2(s) G(s) = \left( \frac{\omega_c}{s} \right)^2 \cdot \frac{1 + \frac{s}{\omega_c/2}}{1 + \frac{s}{2\omega_c}} \]
+
** Reduction thanks to feedback - Required bandwidth
#+begin_src matlab
wc = 1*2*pi; % [rad/s]
5.1 Reduction thanks to feedback - Required bandwidth
+
+
+
+5.1 Approximation of the effect of feedback on the motion error
+
++Let’s consider Figure 21 where a controller \(K\) is used to reduce the effect of the disturbance \(d\) on the position error \(y\). +
+ + +
+
+
+
+
Figure 21: Feedback System
++The reduction of the impact of \(d\) on \(y\) thanks to feedback is described by the following equation: +
+\begin{equation} + \frac{y}{d} = \frac{G_d}{1 + KG} +\end{equation} + + ++As a first approximation, we can consider that the controller 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 \(\omega_c\): +\[ L_2(s) = K_2(s) G(s) = \left( \frac{\omega_c}{s} \right)^2 \cdot \frac{1 + \frac{s}{\omega_c/2}}{1 + \frac{s}{2\omega_c}} \] +
+
+
-5.2 Reduction thanks to feedback - Required bandwidth
+wc = 1*2*pi; % [rad/s] xic = 0.5; @@ -966,13 +1047,13 @@ xlim([1, 500]);
-
6 Conclusion
+
+
6 Conclusion
-
diff --git a/mat/stages.mat b/mat/stages.mat
index 12eb802..6fc3d68 100644
Binary files a/mat/stages.mat and b/mat/stages.mat differ
diff --git a/org/optimal_stiffness_disturbances.org b/org/optimal_stiffness_disturbances.org
index 24c2ed7..81e33bc 100644
--- a/org/optimal_stiffness_disturbances.org
+++ b/org/optimal_stiffness_disturbances.org
@@ -74,11 +74,6 @@ The level of these disturbances has been identified form experiments which are d
simulinkproject('../');
#+end_src
-#+begin_src matlab
- load('mat/conf_simulink.mat');
-
- open('nass_model.slx')
-#+end_src
** 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]].
@@ -149,9 +144,9 @@ We do that for a wide range of nano-hexapod stiffnesses and we compare the obtai
#+begin_src matlab
load('mat/conf_simulink.mat');
-
open('nass_model.slx')
#+end_src
+
** Initialization
We initialize all the stages with the default parameters.
#+begin_src matlab
@@ -399,6 +394,19 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
** Introduction :ignore:
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.
+** Matlab Init :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+ <Created: 2020-04-07 mar. 17:10
+Created: 2020-04-07 mar. 19:33