From ca42338fe323c5fd3044d7640095ec10c832ae1d Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 7 Apr 2020 15:58:26 +0200 Subject: [PATCH] Dist. sensitivity / optimal stiffness analysis --- docs/figs/opt_stiff_sensitivity_Dw.pdf | Bin 0 -> 133091 bytes docs/figs/opt_stiff_sensitivity_Dw.png | Bin 0 -> 225615 bytes docs/figs/opt_stiff_sensitivity_Fd.pdf | Bin 0 -> 165026 bytes docs/figs/opt_stiff_sensitivity_Fd.png | Bin 0 -> 226145 bytes docs/figs/opt_stiff_sensitivity_Frz.pdf | Bin 0 -> 105370 bytes docs/figs/opt_stiff_sensitivity_Frz.png | Bin 0 -> 184196 bytes docs/figs/opt_stiff_sensitivity_Fty_x.pdf | Bin 0 -> 101946 bytes docs/figs/opt_stiff_sensitivity_Fty_x.png | Bin 0 -> 177086 bytes docs/figs/opt_stiff_sensitivity_Fty_z.pdf | Bin 0 -> 100975 bytes docs/figs/opt_stiff_sensitivity_Fty_z.png | Bin 0 -> 170497 bytes docs/optimal_stiffness_disturbances.html | 228 +++++++++++++--------- org/optimal_stiffness_disturbances.org | 159 ++++++++++++--- 12 files changed, 269 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b/docs/optimal_stiffness_disturbances.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances @@ -247,18 +247,15 @@

    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: +

      -
    • 1
    • -
    • 2
    • -
    • 3
    • -
    • 4
    • -
    • 5
    • +
    • 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 Spectral Density of Disturbances

    +
    +

    1 Disturbances

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

    + + +
    +

    opt_stiff_dist_gm.png +

    +

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

    -
    -

    1.1 Load of the identified disturbances

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

    opt_stiff_dist_fty_frz.png +

    +

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

    -
    -

    1.2 Plots

    -
    -

    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

    @@ -361,22 +378,24 @@ We use a sample mass of 10kg.
    +

    +We include gravity, and we use no controller. +

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

    2.2 Identification

    -Inputs: +The considered inputs are:

    • Dwx: Ground displacement in the \(x\) direction
    • @@ -388,60 +407,95 @@ Inputs:
    • 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]
     
    - -
    -
    %% 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

    +
    +

    2.3 Sensitivity to Stages vibration (Filtering)

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

    opt_stiff_sensitivity_Frz.png +

    +

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

    +
    + + +
    +

    opt_stiff_sensitivity_Fty_z.png +

    +

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

    +
    + + +
    +

    opt_stiff_sensitivity_Fty_x.png +

    +

    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 Save

    +
    + +
    +

    2.4 Effect of Ground motion (Transmissibility).

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

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

    + + +
    +

    opt_stiff_sensitivity_Dw.png +

    +

    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. +

    + + +
    +

    opt_stiff_sensitivity_Fd.png +

    +

    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

    +
    +
    +

    + +

    +
    @@ -461,11 +515,11 @@ Direct Forces (Compliance).

    2dof_system_granite_stiffness.png

    -

    Figure 1: Figure caption

    +

    Figure 8: Figure caption

    -If we write the equation of motion of the system in Figure 1, we obtain: +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) \\ @@ -792,13 +846,13 @@ xlim([1, 500]);
    -
    -

    6 Conclusion

    +
    +

    6 Conclusion

    Author: Dehaeze Thomas

    -

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

    +

    Created: 2020-04-07 mar. 15:57

    diff --git a/org/optimal_stiffness_disturbances.org b/org/optimal_stiffness_disturbances.org index 60dc62f..fe5a31d 100644 --- a/org/optimal_stiffness_disturbances.org +++ b/org/optimal_stiffness_disturbances.org @@ -131,6 +131,9 @@ In this study, the expected frequency content of the direct forces applied to th * Effect of disturbances on the position error <> ** Introduction :ignore: +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. + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -167,16 +170,16 @@ We use a sample mass of 10kg. initializeSample('mass', 10); #+end_src +We include gravity, and we use no controller. #+begin_src matlab initializeSimscapeConfiguration('gravity', true); + initializeController(); initializeDisturbances('enable', false); initializeLoggingConfiguration('log', 'none'); - initializeController(); #+end_src - ** Identification -Inputs: +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 @@ -185,15 +188,14 @@ Inputs: - =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. #+begin_src matlab Ks = logspace(3,9,7); % [N/m] #+end_src #+begin_src matlab :exports none - Gd = {zeros(length(Ks), 1)}; -#+end_src - -#+begin_src matlab %% Name of the Simulink File mdl = 'nass_model'; @@ -210,7 +212,9 @@ Inputs: io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En'); io_i = io_i + 1; % Position Error #+end_src -#+begin_src matlab +#+begin_src matlab :exports none + Gd = {zeros(length(Ks), 1)}; + for i = 1:length(Ks) initializeNanoHexapod('k', Ks(i)); @@ -222,8 +226,12 @@ Inputs: end #+end_src -** Plots -Effect of Stages vibration (Filtering). +** Sensitivity to Stages vibration (Filtering) +The sensitivity the stage vibrations are displayed: +- Figure [[fig:opt_stiff_sensitivity_Frz]]: sensitivity to vertical spindle vibrations +- Figure [[fig:opt_stiff_sensitivity_Fty_z]]: sensitivity to vertical translation stage vibrations +- Figure [[fig:opt_stiff_sensitivity_Fty_x]]: sensitivity to horizontal (x) translation stage vibrations + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -235,26 +243,76 @@ Effect of Stages vibration (Filtering). end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + ylabel('Effect of $F_{rz}$ on $E_z$ [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'southwest'); #+end_src -Effect of Ground motion (Transmissibility). +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:opt_stiff_sensitivity_Frz +#+caption: Sensitivity to Spindle vertical motion error ($F_{rz}$) to the vertical error position of the sample ($E_z$) ([[./figs/opt_stiff_sensitivity_Frz.png][png]], [[./figs/opt_stiff_sensitivity_Frz.pdf][pdf]]) +[[file:figs/opt_stiff_sensitivity_Frz.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + hold on; + for i = 1:length(Ks) + plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Effect of $F_{ty}$ on $E_z$ [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'southwest'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Fty_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:opt_stiff_sensitivity_Fty_z +#+caption: Sensitivity to Translation stage vertical motion error ($F_{ty,z}$) to the vertical error position of the sample ($E_z$) ([[./figs/opt_stiff_sensitivity_Fty_z.png][png]], [[./figs/opt_stiff_sensitivity_Fty_z.pdf][pdf]]) +[[file:figs/opt_stiff_sensitivity_Fty_z.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + hold on; + for i = 1:length(Ks) + plot(freqs, abs(squeeze(freqresp(Gd{i}('Ex', 'Fty_x'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Effect of $F_{ty}$ on $E_x$ [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Fty_x.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:opt_stiff_sensitivity_Fty_x +#+caption: Sensitivity to Translation stage $x$ motion error ($F_{ty,x}$) to the error position of the sample in the $x$ direction ($E_x$) ([[./figs/opt_stiff_sensitivity_Fty_x.png][png]], [[./figs/opt_stiff_sensitivity_Fty_x.pdf][pdf]]) +[[file:figs/opt_stiff_sensitivity_Fty_x.png]] + +** Effect of Ground motion (Transmissibility). +The effect of Ground motion on the position error of the sample is shown in Figure [[fig:opt_stiff_sensitivity_Dw]]. + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); figure; - ax1 = subplot(3, 1, 1); - hold on; - for i = 1:length(Ks) - plot(freqs, abs(squeeze(freqresp(Gd{i}('Ex', 'Dwx'), freqs, 'Hz'))), '-', ... - 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); - end - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/m]'); - - ax1 = subplot(3, 1, 2); + ax1 = subplot(1, 2, 1); hold on; for i = 1:length(Ks) plot(freqs, abs(squeeze(freqresp(Gd{i}('Ey', 'Dwy'), freqs, 'Hz'))), '-', ... @@ -262,9 +320,9 @@ Effect of Ground motion (Transmissibility). end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/m]'); + ylabel('$E_y/D_{wy}$ [m/m]'); xlabel('Frequency [Hz]'); - ax1 = subplot(3, 1, 3); + ax2 = subplot(1, 2, 2); hold on; for i = 1:length(Ks) plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), '-', ... @@ -272,14 +330,36 @@ Effect of Ground motion (Transmissibility). end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); + ylabel('$E_z/D_{wz}$ [m/m]'); xlabel('Frequency [Hz]'); #+end_src -Direct Forces (Compliance). +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Dw.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:opt_stiff_sensitivity_Dw +#+caption: Sensitivity to Ground motion ($D_{w}$) to the position error of the sample ($E_y$ and $E_z$) ([[./figs/opt_stiff_sensitivity_Dw.png][png]], [[./figs/opt_stiff_sensitivity_Dw.pdf][pdf]]) +[[file:figs/opt_stiff_sensitivity_Dw.png]] + +** 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 [[fig:opt_stiff_sensitivity_Fd]]. + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); figure; + + ax1 = subplot(1, 2, 1); + hold on; + for i = 1:length(Ks) + plot(freqs, abs(squeeze(freqresp(Gd{i}('Ery', 'Mdy'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('$E_{ry}/M_{d,y}\ \left[\frac{rad}{N m}\right]$'); xlabel('Frequency [Hz]'); + + ax2 = subplot(1, 2, 2); hold on; for i = 1:length(Ks) plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))), '-', ... @@ -287,14 +367,31 @@ Direct Forces (Compliance). end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + ylabel('$E_{z}/F_{d,z}$ [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); + + linkaxes([ax1 ax2], 'xy') #+end_src -** Save -#+begin_src matlab - save('./mat/opt_stiffness_disturbances.mat', 'Gd') +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/opt_stiff_sensitivity_Fd.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> #+end_src +#+name: fig:opt_stiff_sensitivity_Fd +#+caption: Sensitivity to Direct forces and torques applied to the sample ($F_d$, $M_d$) to the position error of the sample ([[./figs/opt_stiff_sensitivity_Fd.png][png]], [[./figs/opt_stiff_sensitivity_Fd.pdf][pdf]]) +[[file:figs/opt_stiff_sensitivity_Fd.png]] + +** Save :noexport: +#+begin_src matlab + save('./mat/opt_stiffness_disturbances.mat', 'Ks', 'Gd') +#+end_src + +** Conclusion +#+begin_important + +#+end_important + * Effect of granite stiffness <> ** Analytical Analysis

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