From 764b1292eb78b09b13a54f6b955389ff13c73fee Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 2 Apr 2020 15:29:38 +0200 Subject: [PATCH] Analysis on dynamics variability to payload --- matlab/nass_model.slx | Bin 204032 -> 204236 bytes org/optimal_stiffness.org | 945 ++++++++++++++++++++++++++++++++++++- src/initializeController.m | 4 +- 3 files changed, 946 insertions(+), 3 deletions(-) diff --git a/matlab/nass_model.slx b/matlab/nass_model.slx index d6c1287488fa3ad31a6272d77e67f28bfecd2b4c..1282eb7b66389d96857824736889f6bc7905f6df 100644 GIT binary patch delta 177911 zcmY&eQ*fY7w2W=rwzIL5%_iB{wz09lSR326v9WF2w(aEp_rBeyo~fFes+m63J$0u2 z+7Kq%5fBw*z#%X|KtNzXMAZZ85GlaG|HB3juozI-{{{+BA$?!Q2@~Kn!0hjS5{%EJ zkY}v6)*|aRCdCHo6r8Ntmk@_#E-VX><;5E8w#F&bTf`}zOs}g$vx>mdKmiM7+E(#D z;tC+~w0fgP;d)KyEF%7ODvnz8urdq&BE59e8VPJgyNVT1|EhW*rhcX;a>}S^trz0p zs|DmzIl`TeWX%O=xh+E40W|N0kUOxXP7FOtzZnO&SZpMvoa+y4{@HXnZ{jf>}BI7K`Tdt$7}xsv{(3TyS6eWz2^w7K(Y^^Q(tU*qwn3vg->(KD6#R4 zbK^Wba$eAPTX+ymCKNLBi)IU58h#yk>UlqWfDjHpoA2Kd|35AiYJRus4gJTA6etJ? 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zHdu*i+Wv>j&IacFlPjgx?2pbwqXgpMB97nm7a*SS7r={+7y)s$F$u&QQpMH3T<2BN zOPC?Hf7WFFx2W?U(Is}U&>sUOcA}-nCsItpCsHWJogGZ|CvX4ig84sFT4FNT!NO$G Jj9?=3zW|H^Jv{&b diff --git a/org/optimal_stiffness.org b/org/optimal_stiffness.org index 9927409..37c1f10 100644 --- a/org/optimal_stiffness.org +++ b/org/optimal_stiffness.org @@ -51,18 +51,21 @@ We wish here to see if we can determine an optimal stiffness of the nano-hexapod - Section [[sec:micro_station_compliance]]: the support compliance dynamics is not much present in the nano-hexapod dynamics - Section [[sec:payload_impedance]]: the change of payload impedance has acceptable effect on the plant dynamics +The overall goal is to design a nano-hexapod that will allow the highest possible control bandwidth. + * Spindle Rotation Speed <> ** 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 @@ -73,7 +76,318 @@ We wish here to see if we can determine an optimal stiffness of the nano-hexapod open('nass_model.slx') #+end_src +** Initialization +We initialize all the stages with the default parameters. +#+begin_src matlab + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); +#+end_src + +The worst case scenario is a rotation speed of 60rpm with a payload mass of 1Kg. +#+begin_src matlab + initializeSample('mass', 10); +#+end_src + +We don't include gravity nor disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics. +#+begin_src matlab + initializeSimscapeConfiguration('gravity', false); + initializeDisturbances('enable', false); +#+end_src + +We set the controller type to Open-Loop, and we do not need to log any signal. +#+begin_src matlab + initializeController('type', 'stabilizing'); + initializeLoggingConfiguration('log', 'none'); +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'nass_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En'); io_i = io_i + 1; +#+end_src + +** Identification when not rotating +We set the range of stiffness that we want to use. +#+begin_src matlab + Ks = logspace(3,9,7) +#+end_src + +#+begin_src matlab + initializeReferences(); +#+end_src + +#+begin_src matlab + Gk_iff = {zeros(length(Ks))}; + Gk_dvf = {zeros(length(Ks))}; + Gk_err = {zeros(length(Ks))}; +#+end_src + +#+begin_src matlab + for i = 1:length(Ks) + initializeNanoHexapod('k', Ks(i)); + + %% Run the linearization + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ... + 'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ... + 'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gk_iff(i) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + Gk_dvf(i) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + + Jinvt = tf(inv(nano_hexapod.J)'); + Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + Gk_err(i) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt}; + end +#+end_src + +** Identification when rotating at maximum speed +#+begin_src matlab + Rz_rpm = 60; + + initializeReferences('Rz_type', 'rotating', ... + 'Rz_period', 60/Rz_rpm, ... % Rotation period [s] + 'Rz_amplitude', -0.2*(2*pi*Rz_rpm/60)); % Angle offset [rad] + + load('mat/nass_references.mat', 'Rz'); % We load the reference for the Spindle + [~, i_end] = min(abs(Rz.signals.values)); % Obtain the indice where the spindle angle is zero + t_sim = Rz.time(i_end); % Simulation time before identification [s] +#+end_src + +#+begin_src matlab + k_sta = -1e8; +#+end_src + +#+begin_src matlab + Gk_wz_iff = {zeros(length(Ks))}; + Gk_wz_dvf = {zeros(length(Ks))}; + Gk_wz_err = {zeros(length(Ks))}; +#+end_src + +#+begin_src matlab + for i = 1:length(Ks) + initializeNanoHexapod('k', Ks(i)); + + %% Run the linearization + G = linearize(mdl, io, t_sim); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ... + 'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ... + 'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gk_wz_iff(i) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + Gk_wz_dvf(i) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + + Jinvt = tf(inv(nano_hexapod.J)'); + Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + Gk_wz_err(i) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt}; + end +#+end_src + +#+begin_src matlab + save('mat/optimal_stiffness_Gk_wz.mat', 'Ks', ... + 'Gk_iff', 'Gk_dvf', 'Gk_err', ... + 'Gk_wz_iff', 'Gk_wz_dvf', 'Gk_wz_err'); +#+end_src + +** Change of dynamics +#+begin_src matlab :exports none + load('mat/optimal_stiffness_Gk_wz.mat'); +#+end_src + +Change of dynamics for decentralized IFF control. +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Gk_iff) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_iff{i}( 'Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_wz_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); + title('Soft Nano-Hexapod'); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Gk_iff) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gk_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gk_wz_iff{i}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +Change of dynamics from $F_x$ to $D_x$. +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Gk_err) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_err{i}( 'Ex', 'Fx'), freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Gk_err) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gk_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gk_wz_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +Change of dynamics from $F_z$ to $D_z$. +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Gk_err) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_err{i}( 'Ez', 'Fz'), freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i}('Ez', 'Fz'), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title('Soft Nano-Hexapod'); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Gk_err) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gk_err{i}('Ez', 'Fz'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gk_wz_err{i}('Ez', 'Fz'), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +** Change of coupling +#+begin_src matlab :exports none + load('mat/optimal_stiffness_Gk_wz.mat'); +#+end_src + +Change of coupling from $F_x$ to $D_y$ when not rotating and when rotating at 60rpm. +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + hold on; + for i = 1:length(Gk_err) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_err{i}( 'Ey', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i}('Ey', 'Fx'), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast'); +#+end_src + +Comparison of the coupling from $F_x$ to $D_y$ when rotating at 60rpm to the direct term $F_x$ to $D_x$. +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + hold on; + for i = 1:length(Gk_err) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gk_wz_err{i}('Ey', 'Fx'), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast'); +#+end_src + ** Conclusion :ignore: +#+begin_important + The leg stiffness should be at higher than $k_i = 10^4\ [N/m]$ such that the main resonance frequency does not shift too much when rotating. + For the coupling, it is more difficult to conclude about the minimum required leg stiffness. +#+end_important + +#+begin_notes + Note that we can use very soft nano-hexapod if we limit the spindle rotating speed. +#+end_notes + * Micro-Station Compliance Effect <> @@ -121,5 +435,632 @@ We wish here to see if we can determine an optimal stiffness of the nano-hexapod open('nass_model.slx') #+end_src +** Initialization +We initialize all the stages with the default parameters. +#+begin_src matlab + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); +#+end_src + +We don't include gravity nor disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics. +#+begin_src matlab + initializeSimscapeConfiguration('gravity', false); + initializeDisturbances('enable', false); +#+end_src + +We set the controller type to Open-Loop, and we do not need to log any signal. +#+begin_src matlab + initializeController('type', 'stabilizing'); + initializeLoggingConfiguration('log', 'none'); +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'nass_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; + io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'En'); io_i = io_i + 1; +#+end_src + +** Change of payload dynamics +- Change of mass: from 1kg to 50kg +- Change of resonance frequency: from 50Hz to 500Hz +- The damping ratio of the payload is fixed to $\xi = 0.2$ + +#+begin_src matlab + initializeReferences(); + Ks = logspace(3,9,7) % [N/m] +#+end_src + +#+begin_src matlab + Ms = [1, 20, 50]; % [Kg] +#+end_src + +#+begin_src matlab + Gm_iff = {zeros(length(Ks), length(Ms))}; + Gm_dvf = {zeros(length(Ks), length(Ms))}; + Gm_err = {zeros(length(Ks), length(Ms))}; +#+end_src + +#+begin_src matlab + for i = 1:length(Ks) + for j = 1:length(Ms) + initializeNanoHexapod('k', Ks(i)); + initializeSample('mass', Ms(j), 'freq', 100*ones(6,1)); + + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ... + 'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ... + 'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gm_iff(i,j) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + Gm_dvf(i,j) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + + Jinvt = tf(inv(nano_hexapod.J)'); + Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + Gm_err(i,j) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt}; + end + end +#+end_src + +#+begin_src matlab + Fs = [50, 200, 500]; % [Hz] +#+end_src + +#+begin_src matlab + Gf_iff = {zeros(length(Ks), length(Fs))}; + Gf_dvf = {zeros(length(Ks), length(Fs))}; + Gf_err = {zeros(length(Ks), length(Fs))}; +#+end_src + +#+begin_src matlab + for i = 1:length(Ks) + for j = 1:length(Fs) + initializeNanoHexapod('k', Ks(i)); + initializeSample('mass', 20, 'freq', Fs(j)*ones(6,1)); + + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ... + 'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ... + 'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gf_iff(i,j) = {minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + Gf_dvf(i,j) = {minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))}; + + Jinvt = tf(inv(nano_hexapod.J)'); + Jinvt.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + Jinvt.OutputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + Gf_err(i,j) = {-minreal(G({'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*Jinvt}; + end + end +#+end_src + +#+begin_src matlab + save('mat/optimal_stiffness_Gm_Gf.mat', 'Ks', 'Ms', 'Fs', ... + 'Gm_iff', 'Gm_dvf', 'Gm_err', ... + 'Gf_iff', 'Gf_dvf', 'Gf_err'); +#+end_src + +** Plots +** Change of optimal gain for decentralized control +For each payload, compute the optimal gain for the IFF control. +The optimal value corresponds to critical damping to *all* the 6 modes of the nano-hexapod. + +#+begin_src matlab + load('mat/optimal_stiffness_Gm_Gf.mat'); +#+end_src + +Change of Mass +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ks) + for j = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gm_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ks) + for j = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + if j == 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i))); + else + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); + end + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +Change of payload resonance frequency +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ks) + for j = 1:length(Fs) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gf_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ks) + for j = 1:length(Fs) + set(gca,'ColorOrderIndex',i); + if j == 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i))); + else + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_iff{i,j}('Fnlm1', 'Fnl1'), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); + end + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +** Change of dynamics for the primary controller +For each stiffness, plot the total spread of dynamics. + +#+begin_src matlab + load('mat/optimal_stiffness_Gm_Gf.mat'); +#+end_src + +*** Frequency variation +Same payload mass, but different stiffness resulting in different resonance frequency. + +All curves +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ks) + for j = 1:length(Fs) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ks) + for j = 1:length(Fs) + set(gca,'ColorOrderIndex',i); + if j == 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i))); + else + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); + end + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +X direction +#+begin_src matlab :exports none + i = 1; + + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + for j = 1:length(Fs) + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title(sprintf('$k = %.0e$ [N/m]', Ks(i))) + + ax2 = subplot(2, 2, 3); + hold on; + for j = 1:length(Fs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + + i = 7; + + ax1 = subplot(2, 2, 2); + hold on; + for j = 1:length(Fs) + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title(sprintf('$k = %.0e$ [N/m]', Ks(i))) + + ax2 = subplot(2, 2, 4); + hold on; + for j = 1:length(Fs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +Z direction: +We can see two mass lines for the soft nano-hexapod: +- The first mass line corresponds to $\frac{1}{(m_n + m_p)s^2}$ where $m_p = 20\ [kg]$ is the mass of the payload and $m_n = 15\ [Kg]$ is the mass of the nano-hexapod top platform and attached mirror +- The second mass line corresponds to $\frac{1}{m_n s^2}$ + +#+begin_src matlab :exports none + i = 1; + + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + for j = 1:length(Fs) + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title(sprintf('$k = %.0e$ [N/m]', Ks(i))) + + ax2 = subplot(2, 2, 3); + hold on; + for j = 1:length(Fs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + + i = 7; + + ax1 = subplot(2, 2, 2); + hold on; + for j = 1:length(Fs) + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title(sprintf('$k = %.0e$ [N/m]', Ks(i))) + + ax2 = subplot(2, 2, 4); + hold on; + for j = 1:length(Fs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$\\omega_0 = %.0f$ [Hz]', Fs(j))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +*** Mass variation +All mixed, X direction +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ks) + for j = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ks) + for j = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + if j == 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i))); + else + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); + end + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +All mixed, Z direction +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ks) + for j = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ks) + for j = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + if j == 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$K = %.0e$ [N/m]', Ks(i))); + else + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); + end + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +Z direction +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + i = 1; + + ax1 = subplot(2, 2, 1); + hold on; + for j = 1:length(Ms) + plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title(sprintf('$k = %.0e$ [N/m]', Ks(i))) + + ax2 = subplot(2, 2, 3); + hold on; + for j = 1:length(Ms) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + + i = 7; + + ax1 = subplot(2, 2, 2); + hold on; + for j = 1:length(Ms) + plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title(sprintf('$k = %.0e$ [N/m]', Ks(i))) + + ax2 = subplot(2, 2, 4); + hold on; + for j = 1:length(Ms) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ez', 'Fz'), freqs, 'Hz'))), '-', ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(j))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +*** Total variation +Total change of dynamics due to change of the payload: +- mass from 1kg to 50kg +- main resonance from 50Hz to 500Hz + +For a soft nano-hexapod +#+begin_src matlab :exports none + i = 1; + + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for j = 1:length(Fs) + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + end + for j = 1:length(Ms) + plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for j = 1:length(Fs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + for j = 1:length(Ms) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +For a stiff nano-hexapod +#+begin_src matlab :exports none + i = 7; + + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for j = 1:length(Fs) + plot(freqs, abs(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + end + for j = 1:length(Ms) + plot(freqs, abs(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for j = 1:length(Fs) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + for j = 1:length(Ms) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_err{i,j}('Ex', 'Fx'), freqs, 'Hz'))), 'k-'); + end + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + + ** Conclusion :ignore: diff --git a/src/initializeController.m b/src/initializeController.m index 2b6c628..33d7201 100644 --- a/src/initializeController.m +++ b/src/initializeController.m @@ -1,7 +1,7 @@ function [] = initializeController(args) arguments - args.type char {mustBeMember(args.type,{'open-loop', 'iff', 'dvf', 'hac-dvf', 'ref-track-L', 'ref-track-iff-L', 'cascade-hac-lac', 'hac-iff'})} = 'open-loop' + args.type char {mustBeMember(args.type,{'open-loop', 'iff', 'dvf', 'hac-dvf', 'ref-track-L', 'ref-track-iff-L', 'cascade-hac-lac', 'hac-iff', 'stabilizing'})} = 'open-loop' end controller = struct(); @@ -23,6 +23,8 @@ switch args.type controller.type = 7; case 'hac-iff' controller.type = 8; + case 'stabilizing' + controller.type = 9; end save('./mat/controller.mat', 'controller');