Add analysis about virtual mass addition
This commit is contained in:
707
org/control_virtual_mass.org
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707
org/control_virtual_mass.org
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@@ -0,0 +1,707 @@
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#+TITLE: Decentralize control to add virtual mass
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#+SETUPFILE: ./setup/org-setup-file.org
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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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load('mat/conf_simulink.mat');
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open('nass_model.slx')
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#+end_src
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* Initialization
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#+begin_src matlab
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initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeSimscapeConfiguration();
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initializeDisturbances('enable', false);
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initializeLoggingConfiguration('log', 'none');
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initializeController('type', 'hac-dvf');
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#+end_src
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We set the stiffness of the payload fixation:
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#+begin_src matlab
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Kp = 1e8; % [N/m]
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#+end_src
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* Identification
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** Identification of the transfer function from $\tau$ to $d\mathcal{L}$
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#+begin_src matlab
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K = tf(zeros(6));
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Kdvf = tf(zeros(6));
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#+end_src
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We identify the system for the following payload masses:
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#+begin_src matlab
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Ms = [1, 10, 50];
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#+end_src
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#+begin_src matlab :exports none
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Gm = {zeros(length(Ms), 1)};
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#+end_src
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The nano-hexapod has the following leg's stiffness and damping.
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#+begin_src matlab
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initializeNanoHexapod('k', 1e5, 'c', 2e2);
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'nass_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Force Sensors
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#+end_src
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#+begin_src matlab :exports none
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for i = 1:length(Ms)
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initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));
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%% Run the linearization
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G_dvf = linearize(mdl, io);
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G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
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Gm(i) = {G_dvf};
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end
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#+end_src
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** Identification of the Primary plant without virtual add of mass
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#+begin_src matlab :exports none
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G_x = {zeros(length(Ms), 1)};
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G_l = {zeros(length(Ms), 1)};
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#+end_src
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#+begin_src matlab :exports none
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for i = 1:length(Ms)
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initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));
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%% Run the linearization
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G = linearize(mdl, io);
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G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
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Gx = -G*inv(nano_hexapod.J');
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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G_x(i) = {Gx};
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Gl = -nano_hexapod.J*G;
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Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
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G_l(i) = {Gl};
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end
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#+end_src
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* Adding Virtual Mass in the Leg's Space
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** Plant
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#+begin_src matlab :exports none
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freqs = logspace(-1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i = 1:length(Ms)
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plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 1:length(Ms)
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz')))), ...
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'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-270, 90]);
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yticks([-360:90:360]);
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legend('location', 'northeast');
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/virtual_mass_plant_L.pdf', 'width', 'full', 'height', 'full')
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#+end_src
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#+name: fig:virtual_mass_plant_L
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#+caption: Transfer function from $\tau_i$ to $d\mathcal{L}_i$ for three payload masses
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#+RESULTS:
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[[file:figs/virtual_mass_plant_L.png]]
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** Controller Design
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#+begin_src matlab
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Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6);
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#+end_src
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#+begin_src matlab :exports none
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for i = 1:length(Ms)
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isstable(feedback(Gm{i}*Kdvf, eye(6), -1))
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end
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(-1, 4, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i = 1:length(Ms)
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plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 1:length(Ms)
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz')))), ...
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'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-360:90:360]);
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legend('location', 'northeast');
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/virtual_mass_loop_gain_L.pdf', 'width', 'full', 'height', 'full')
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#+end_src
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#+name: fig:virtual_mass_loop_gain_L
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#+caption: Loop Gain for the addition of virtual mass in the leg's space
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#+RESULTS:
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[[file:figs/virtual_mass_loop_gain_L.png]]
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** Identification of the Primary Plant
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'nass_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
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load('mat/stages.mat', 'nano_hexapod');
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GmL_x = {zeros(length(Ms), 1)};
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GmL_l = {zeros(length(Ms), 1)};
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for i = 1:length(Ms)
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initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
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initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));
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%% Run the linearization
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G = linearize(mdl, io);
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G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
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G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
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Gx = -G*inv(nano_hexapod.J');
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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GmL_x(i) = {Gx};
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Gl = -nano_hexapod.J*G;
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Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
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GmL_l(i) = {Gl};
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end
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 5000);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(GmL_x{i}(1, 1), freqs, 'Hz'))), '--');
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(GmL_x{i}(2, 2), freqs, 'Hz'))), '--');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
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ax2 = subplot(2, 2, 2);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(GmL_x{i}(3, 3), freqs, 'Hz'))), '--');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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title('$\mathcal{X}_z/\mathcal{F}_z$')
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ax3 = subplot(2, 2, 3);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(1, 1), freqs, 'Hz')))), '--');
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(2, 2), freqs, 'Hz')))), '--');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-270, 90]);
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yticks([-360:90:360]);
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ax4 = subplot(2, 2, 4);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ...
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'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(3, 3), freqs, 'Hz')))), '--', ...
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'HandleVisibility', 'off');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-270, 90]);
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yticks([-360:90:360]);
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legend('location', 'southwest');
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linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/virtual_mass_L_primary_plant_X.pdf', 'width', 'full', 'height', 'full')
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#+end_src
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#+name: fig:virtual_mass_L_primary_plant_X
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#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the leg's space
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#+RESULTS:
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[[file:figs/virtual_mass_L_primary_plant_X.png]]
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 5000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(GmL_l{i}(1, 1), freqs, 'Hz'))), '--');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ...
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'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_l{i}(1, 1), freqs, 'Hz')))), '--', ...
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'HandleVisibility', 'off');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-270, 90]);
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yticks([-360:90:360]);
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legend('location', 'southwest');
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/virtual_mass_L_primary_plant_L.pdf', 'width', 'full', 'height', 'full')
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#+end_src
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#+name: fig:virtual_mass_L_primary_plant_L
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#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the leg's space
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#+RESULTS:
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[[file:figs/virtual_mass_L_primary_plant_L.png]]
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* Adding Virtual Mass in the Task Space
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** Plant
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Let's look at the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$:
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\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]
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#+begin_src matlab :exports none
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load('mat/stages.mat', 'nano_hexapod');
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GmX = {zeros(length(Ms), 1)};
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for i = 1:length(Ms)
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GmX(i) = {inv(nano_hexapod.J) * Gm{i} * inv(nano_hexapod.J')};
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end
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(-1, 3, 1000);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',i);
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plot(freqs, abs(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 2, 3);
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hold on;
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for i = 1:length(Ms)
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set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz')))), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax1 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/virtual_mass_plant_X.pdf', 'width', 'full', 'height', 'full')
|
||||
#+end_src
|
||||
|
||||
#+name: fig:virtual_mass_plant_X
|
||||
#+caption: Dynamics from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ used for virtual mass addition in the task space
|
||||
#+RESULTS:
|
||||
[[file:figs/virtual_mass_plant_X.png]]
|
||||
|
||||
** Controller Design
|
||||
#+begin_src matlab
|
||||
KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0]));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
for i = 1:length(Ms)
|
||||
isstable(feedback(GmX{i}*KmX, eye(6), -1))
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
LmX = GmX{i}*KmX;
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(LmX(1, 1), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(LmX(2, 2), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
LmX = GmX{i}*KmX;
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(1, 1), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(2, 2), freqs, 'Hz')))), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax1 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
LmX = GmX{i}*KmX;
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(LmX(3, 3), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
LmX = GmX{i}*KmX;
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(3, 3), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/virtual_mass_loop_gain_X.pdf', 'width', 'full', 'height', 'full')
|
||||
#+end_src
|
||||
|
||||
#+name: fig:virtual_mass_loop_gain_X
|
||||
#+caption: Loop gain for virtual mass addition in the task space
|
||||
#+RESULTS:
|
||||
[[file:figs/virtual_mass_loop_gain_X.png]]
|
||||
|
||||
#+begin_src matlab
|
||||
Kdvf = inv(nano_hexapod.J')*KmX*inv(nano_hexapod.J);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
for i = 1:length(Ms)
|
||||
isstable(feedback(Gm{i}*Kdvf, eye(6), -1))
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Identification of the Primary Plant
|
||||
#+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, 'input'); io_i = io_i + 1; % Actuator Inputs
|
||||
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
||||
|
||||
load('mat/stages.mat', 'nano_hexapod');
|
||||
|
||||
GmX_x = {zeros(length(Ms), 1)};
|
||||
GmX_l = {zeros(length(Ms), 1)};
|
||||
|
||||
for i = 1:length(Ms)
|
||||
initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
|
||||
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i));
|
||||
|
||||
%% Run the linearization
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
||||
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
||||
|
||||
Gx = -G*inv(nano_hexapod.J');
|
||||
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
GmX_x(i) = {Gx};
|
||||
|
||||
Gl = -nano_hexapod.J*G;
|
||||
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
||||
GmX_l(i) = {Gl};
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 5000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(GmX_x{i}(1, 1), freqs, 'Hz'))), '--');
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(GmX_x{i}(2, 2), freqs, 'Hz'))), '--');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$')
|
||||
|
||||
ax2 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(GmX_x{i}(3, 3), freqs, 'Hz'))), '--');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
title('$\mathcal{X}_z/\mathcal{F}_z$')
|
||||
|
||||
ax3 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(1, 1), freqs, 'Hz')))), '--');
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(2, 2), freqs, 'Hz')))), '--');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(3, 3), freqs, 'Hz')))), '--', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/virtual_mass_X_primary_plant_X.pdf', 'width', 'full', 'height', 'full')
|
||||
#+end_src
|
||||
|
||||
#+name: fig:virtual_mass_X_primary_plant_X
|
||||
#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the task space
|
||||
#+RESULTS:
|
||||
[[file:figs/virtual_mass_X_primary_plant_X.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 5000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:length(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, abs(squeeze(freqresp(GmX_l{i}(1, 1), 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(Ms)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ...
|
||||
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_l{i}(1, 1), freqs, 'Hz')))), '--', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
yticks([-360:90:360]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/virtual_mass_X_primary_plant_L.pdf', 'width', 'full', 'height', 'full')
|
||||
#+end_src
|
||||
|
||||
#+name: fig:virtual_mass_X_primary_plant_L
|
||||
#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the task space
|
||||
#+RESULTS:
|
||||
[[file:figs/virtual_mass_X_primary_plant_L.png]]
|
||||
@@ -233,6 +233,9 @@ Then, we compute the transfer function from forces applied by the actuators $\bm
|
||||
\[ \bm{G}_\mathcal{X}(s) = \frac{\bm{\epsilon}_{\mathcal{X}_n}}{\bm{\mathcal{F}}} = \bm{G}(s) \bm{J}^{-T} \]
|
||||
The obtained dynamics is shown in Figure [[fig:opt_stiff_primary_plant_damped_X]].
|
||||
|
||||
#+begin_important
|
||||
A zero with a positive real part is introduced in the transfer function from $\mathcal{F}_y$ to $\mathcal{X}_y$ after Decentralized Direct Velocity Feedback is applied.
|
||||
#+end_important
|
||||
|
||||
And we compute the transfer function from actuator forces $\bm{\tau}$ to position error of each leg $\bm{\epsilon}_\mathcal{L}$:
|
||||
\[ \bm{G}_\mathcal{L} = \frac{\bm{\epsilon}_\mathcal{L}}{\bm{\tau}} = \bm{J} \bm{G}(s) \]
|
||||
@@ -449,7 +452,10 @@ The coupling does not change a lot with DVF.
|
||||
|
||||
|
||||
The coupling in the space of the legs $\bm{G}_\mathcal{L}$ are shown in Figure [[fig:opt_stiff_primary_plant_damped_coupling_L]].
|
||||
The magnitude of the coupling around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
|
||||
|
||||
#+begin_important
|
||||
The magnitude of the coupling between $\tau_i$ and $d\mathcal{L}_j$ (Figure [[fig:opt_stiff_primary_plant_damped_coupling_L]]) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.
|
||||
#+end_important
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(0, 3, 1000);
|
||||
@@ -653,6 +659,11 @@ exportFig('figs/opt_stiff_sensibility_dist_dvf.pdf', 'width', 'full', 'height',
|
||||
#+RESULTS:
|
||||
[[file:figs/opt_stiff_sensibility_dist_dvf.png]]
|
||||
|
||||
*** Conclusion :ignore:
|
||||
#+begin_important
|
||||
Decentralized Direct Velocity Feedback is shown to increase the effect of stages vibrations at high frequency and to reduce the effect of ground motion and direct forces at low frequency.
|
||||
#+end_important
|
||||
|
||||
** Conclusion
|
||||
#+begin_important
|
||||
|
||||
@@ -672,7 +683,7 @@ In this section we implement the control architecture shown in Figure [[fig:cont
|
||||
The controller for decentralized direct velocity feedback is the one designed in Section [[sec:lac_dvf]].
|
||||
|
||||
** Plant in the leg space
|
||||
We now loop at the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the design of $\bm{K}_\mathcal{L}$.
|
||||
We now look at the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the design of $\bm{K}_\mathcal{L}$.
|
||||
|
||||
The diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the three considered masses are shown in Figure [[fig:opt_stiff_primary_plant_L]].
|
||||
|
||||
@@ -793,7 +804,7 @@ The loop gain is shown in Figure [[fig:opt_stiff_primary_loop_gain_L]].
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
@@ -821,12 +832,12 @@ exportFig('figs/opt_stiff_primary_loop_gain_L.pdf', 'width', 'full', 'height', '
|
||||
#+RESULTS:
|
||||
[[file:figs/opt_stiff_primary_loop_gain_L.png]]
|
||||
|
||||
Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system.
|
||||
#+begin_src matlab
|
||||
load('mat/stages.mat', 'nano_hexapod');
|
||||
K = Kl*nano_hexapod.J*diag([1, 1, 1, 1, 1, 0]);
|
||||
#+end_src
|
||||
|
||||
Check the MIMO stability
|
||||
#+begin_src matlab :exports none
|
||||
for i = 1:length(Ms)
|
||||
isstable(feedback(nano_hexapod.J\Gm_l{i}*K, eye(6), -1))
|
||||
|
||||
Reference in New Issue
Block a user