3192 lines
100 KiB
Org Mode
3192 lines
100 KiB
Org Mode
#+TITLE: Amplified Piezoelectric Stack Actuator
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#+SETUPFILE: ./setup/org-setup-file.org
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* Introduction :ignore:
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The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
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The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
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The parameters are shown in the table below.
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#+name: fig:souleille18_model_piezo
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#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
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[[file:./figs/souleille18_model_piezo.png]]
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#+caption: Parameters used for the model of the APA 100M
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| | Value | Meaning |
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|-------+-------------------+----------------------------------------------------------------|
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| $m$ | $1\,[kg]$ | Payload mass |
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| $k_e$ | $4.8\,[N/\mu m]$ | Stiffness used to adjust the pole of the isolator |
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| $k_1$ | $0.96\,[N/\mu m]$ | Stiffness of the metallic suspension when the stack is removed |
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| $k_a$ | $65\,[N/\mu m]$ | Stiffness of the actuator |
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| $c_1$ | $10\,[N/(m/s)]$ | Added viscous damping |
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* Simplified Model
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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
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simulinkproject('../');
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#+END_SRC
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#+begin_src matlab
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open 'amplified_piezo_model.slx'
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#+end_src
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** Parameters
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#+begin_src matlab
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m = 1; % [kg]
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ke = 4.8e6; % [N/m]
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ce = 5; % [N/(m/s)]
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me = 0.001; % [kg]
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k1 = 0.96e6; % [N/m]
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c1 = 10; % [N/(m/s)]
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ka = 65e6; % [N/m]
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ca = 5; % [N/(m/s)]
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ma = 0.001; % [kg]
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h = 0.2; % [m]
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#+end_src
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IFF Controller:
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#+begin_src matlab
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Kiff = -8000/s;
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#+end_src
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** Identification
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Identification in open-loop.
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'amplified_piezo_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, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
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io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
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io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
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G = linearize(mdl, io, 0);
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G.InputName = {'w', 'f', 'F'};
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G.OutputName = {'Fs', 'x1'};
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#+end_src
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Identification in closed-loop.
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'amplified_piezo_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, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
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io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
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io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
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Giff = linearize(mdl, io, 0);
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Giff.InputName = {'w', 'f', 'F'};
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Giff.OutputName = {'Fs', 'x1'};
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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, 3, 1);
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title('$\displaystyle \frac{x_1}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
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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/m]');xlabel('Frequency [Hz]');
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ax2 = subplot(2, 3, 2);
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title('$\displaystyle \frac{x_1}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
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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]');xlabel('Frequency [Hz]');
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ax3 = subplot(2, 3, 3);
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title('$\displaystyle \frac{x_1}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
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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]');xlabel('Frequency [Hz]');
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ax4 = subplot(2, 3, 4);
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title('$\displaystyle \frac{F_s}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
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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/m]');xlabel('Frequency [Hz]');
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ax5 = subplot(2, 3, 5);
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title('$\displaystyle \frac{F_s}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
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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]');xlabel('Frequency [Hz]');
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ax6 = subplot(2, 3, 6);
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title('$\displaystyle \frac{F_s}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
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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]'); xlabel('Frequency [Hz]');
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linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'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/amplified_piezo_tf_ol_and_cl.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:amplified_piezo_tf_ol_and_cl
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#+caption: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)
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#+RESULTS:
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[[file:figs/amplified_piezo_tf_ol_and_cl.png]]
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** Root Locus
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#+begin_src matlab :exports none :post
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figure;
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gains = logspace(1, 6, 500);
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hold on;
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plot(real(pole(G('Fs', 'f'))), imag(pole(G('Fs', 'f'))), 'kx');
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plot(real(tzero(G('Fs', 'f'))), imag(tzero(G('Fs', 'f'))), 'ko');
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for k = 1:length(gains)
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cl_poles = pole(feedback(G('Fs', 'f'), -gains(k)/s));
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plot(real(cl_poles), imag(cl_poles), 'k.');
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end
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hold off;
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axis square;
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xlim([-2500, 100]); ylim([0, 2600]);
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xlabel('Real Part'); ylabel('Imaginary Part');
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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/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:amplified_piezo_root_locus
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#+caption: Root Locus
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#+RESULTS:
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[[file:figs/amplified_piezo_root_locus.png]]
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** Analytical Model
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If we apply the Newton's second law of motion on the top mass, we obtain:
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\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]
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Then, we can write that the measured force $F_s$ is equal to:
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\[ F_s = k_a(w - x_e) + f = -k_e (x_1 - x_e) \]
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which gives:
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\[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \]
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Re-injecting that into the previous equations gives:
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\[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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\[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
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#+begin_src matlab
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Ga = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * ...
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[ 1 , k1 + ke*ka/(ke + ka) , ke/(ke + ka) ;
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-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke+ka)*(m*s^2 + k1)];
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Ga.InputName = {'F', 'w', 'f'};
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Ga.OutputName = {'x1', 'Fs'};
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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, 3, 1);
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title('$\displaystyle \frac{x_1}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('x1', 'w'), freqs, 'Hz'))), 'k--');
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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/m]');xlabel('Frequency [Hz]');
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ax2 = subplot(2, 3, 2);
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title('$\displaystyle \frac{x_1}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('x1', 'f'), freqs, 'Hz'))), 'k--');
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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]');xlabel('Frequency [Hz]');
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ax3 = subplot(2, 3, 3);
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title('$\displaystyle \frac{x_1}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('x1', 'F'), freqs, 'Hz'))), 'k--');
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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]');xlabel('Frequency [Hz]');
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ax4 = subplot(2, 3, 4);
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title('$\displaystyle \frac{F_s}{w}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'w'), freqs, 'Hz'))), 'k--');
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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/m]');xlabel('Frequency [Hz]');
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ax5 = subplot(2, 3, 5);
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title('$\displaystyle \frac{F_s}{f}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'f'), freqs, 'Hz'))), 'k--');
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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]');xlabel('Frequency [Hz]');
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ax6 = subplot(2, 3, 6);
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title('$\displaystyle \frac{F_s}{F}$')
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hold on;
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plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'F'), freqs, 'Hz'))), 'k--');
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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]'); xlabel('Frequency [Hz]');
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linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'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/comp_simscape_analytical.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:comp_simscape_analytical
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#+caption: Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)
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#+RESULTS:
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[[file:figs/comp_simscape_analytical.png]]
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** Analytical Analysis
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For Integral Force Feedback Control, the plant is:
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\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
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As high frequency, this converge to:
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\[ \frac{F_s}{f} \underset{\omega\to\infty}{\longrightarrow} \frac{k_e}{k_e + k_a} \]
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And at low frequency:
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\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
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It has two complex conjugate zeros at:
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\[ z = \pm j \sqrt{\frac{k_1}{m}} \]
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And two complex conjugate poles at:
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\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \]
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If maximal damping is to be attained with IFF, the distance between the zero and the pole is to be maximized.
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Thus, we wish to maximize $p/z$, which is equivalent as to minimize $k_1$ and have $k_e \approx k_a$ (supposing $k_e + k_a \approx \text{cst}$).
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#+begin_src matlab
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m = 1;
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k1 = 1e6;
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ka = 1e6;
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ke = 1e6;
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Giff.InputName = {'f'};
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Giff.OutputName = {'Fs'};
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#+end_src
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#+begin_src matlab :exports none
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m = 1;
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ka = 1e6;
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ke = 1e6;
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freqs = logspace(-1, 3, 1000);
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figure;
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hold on;
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for k1 = [1e5, 1e6, 1e7, 1e8]
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Giff = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * (-ke/(ke+ka))*(m*s^2 + k1);
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plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), ...
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'DisplayName', sprintf('$k_1 = %.1e$', k1));
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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 [N/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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#+end_src
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#+begin_src matlab :exports none
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m = 1;
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k1 = 1e6;
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ke = 1e6;
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freqs = logspace(-1, 3, 1000);
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figure;
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hold on;
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for ka = [1e5, 1e6, 1e7, 1e8]
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Giff = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * (-ke/(ke+ka))*(m*s^2 + k1);
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plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), ...
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'DisplayName', sprintf('$k_a = %.1e$', ka));
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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 [N/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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#+end_src
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#+begin_src matlab :exports none
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m = 1;
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k1 = 1e6;
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ka = 1e6;
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freqs = logspace(-1, 3, 1000);
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figure;
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hold on;
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for ke = [1e5, 1e6, 1e7, 1e8]
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Giff = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * (-ke/(ke+ka))*(m*s^2 + k1);
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plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), ...
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'DisplayName', sprintf('$k_e = %.1e$', ke));
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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 [N/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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#+end_src
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* Rotating X-Y platform
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** Introduction :ignore:
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This analysis gave rise to a paper cite:dehaeze20_activ_dampin_rotat_platf_integ_force_feedb.
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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
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+BEGIN_SRC matlab
|
|
simulinkproject('../');
|
|
#+END_SRC
|
|
|
|
#+begin_src matlab
|
|
open 'amplified_piezo_xy_rotating_stage.slx'
|
|
#+end_src
|
|
|
|
** Parameters
|
|
#+begin_src matlab
|
|
m = 1; % [kg]
|
|
|
|
ke = 4.8e6; % [N/m]
|
|
ce = 5; % [N/(m/s)]
|
|
me = 0.001; % [kg]
|
|
|
|
k1 = 0.96e6; % [N/m]
|
|
c1 = 10; % [N/(m/s)]
|
|
|
|
ka = 65e6; % [N/m]
|
|
ca = 5; % [N/(m/s)]
|
|
ma = 0.001; % [kg]
|
|
|
|
h = 0.2; % [m]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Kiff = tf(0);
|
|
#+end_src
|
|
|
|
** Identification
|
|
Rotating speed in rad/s:
|
|
#+begin_src matlab
|
|
Ws = 2*pi*[0, 1, 10, 100];
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Gs = {zeros(length(Ws), 1)};
|
|
#+end_src
|
|
|
|
Identification in open-loop.
|
|
#+begin_src matlab
|
|
%% Name of the Simulink File
|
|
mdl = 'amplified_piezo_xy_rotating_stage';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
|
|
|
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
|
|
|
for i = 1:length(Ws)
|
|
ws = Ws(i);
|
|
G = linearize(mdl, io, 0);
|
|
G.InputName = {'fx', 'fy'};
|
|
G.OutputName = {'Fsx', 'Fsy'};
|
|
Gs(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
title('$\displaystyle \frac{F_{s,x}}{f_x}$')
|
|
hold on;
|
|
for i = 1:length(Ws)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fx'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
title('$\displaystyle \frac{F_{s,y}}{f_x}$')
|
|
hold on;
|
|
for i = 1:length(Ws)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fx'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
title('$\displaystyle \frac{F_{s,x}}{f_y}$')
|
|
hold on;
|
|
for i = 1:length(Ws)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fy'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
title('$\displaystyle \frac{F_{s,y}}{f_y}$')
|
|
hold on;
|
|
for i = 1:length(Ws)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fy'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplitifed_piezo_xy_rotation_plant_iff.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplitifed_piezo_xy_rotation_plant_iff
|
|
#+caption: Transfer function matrix from forces to force sensors for multiple rotation speed
|
|
#+RESULTS:
|
|
[[file:figs/amplitifed_piezo_xy_rotation_plant_iff.png]]
|
|
|
|
** Root Locus
|
|
#+begin_src matlab :exports none :post
|
|
figure;
|
|
|
|
gains = logspace(1, 6, 500);
|
|
|
|
hold on;
|
|
for i = 1:length(Ws)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
|
|
for k = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',i);
|
|
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
|
end
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-2900, 100]); ylim([0, 3000]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_xy_rotation_root_locus.pdf', 'width', 'tall', 'height', 'wide');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_xy_rotation_root_locus
|
|
#+caption: Root locus for 3 rotating speed
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_xy_rotation_root_locus.png]]
|
|
|
|
** Analysis
|
|
The negative stiffness induced by the rotation is equal to $m \omega_0^2$.
|
|
Thus, the maximum rotation speed where IFF can be applied is:
|
|
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
|
|
|
|
Let's verify that.
|
|
#+begin_src matlab
|
|
Ws = 2*pi*[140, 160];
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gs = {zeros(length(Ws), 1)};
|
|
#+end_src
|
|
|
|
Identification
|
|
#+begin_src matlab
|
|
%% Name of the Simulink File
|
|
mdl = 'amplified_piezo_xy_rotating_stage';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
|
|
|
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
|
|
|
for i = 1:length(Ws)
|
|
ws = Ws(i);
|
|
G = linearize(mdl, io, 0);
|
|
G.InputName = {'fx', 'fy'};
|
|
G.OutputName = {'Fsx', 'Fsy'};
|
|
Gs(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
gains = logspace(1, 6, 500);
|
|
|
|
hold on;
|
|
for i = 1:length(Ws)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
|
|
for k = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',i);
|
|
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
|
end
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-100, 50]); ylim([0, 150]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_xy_rotating_unstable_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_xy_rotating_unstable_root_locus
|
|
#+caption: Root Locus for the two considered rotation speed. For the red curve, the system is unstable.
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_xy_rotating_unstable_root_locus.png]]
|
|
|
|
* Stewart Platform with Amplified Actuators
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('nass_model.slx')
|
|
#+end_src
|
|
|
|
** Initialization
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
|
|
initializeSimscapeConfiguration();
|
|
initializeDisturbances('enable', false);
|
|
initializeLoggingConfiguration('log', 'none');
|
|
|
|
initializeController('type', 'hac-iff');
|
|
#+end_src
|
|
|
|
We set the stiffness of the payload fixation:
|
|
#+begin_src matlab
|
|
Kp = 1e8; % [N/m]
|
|
#+end_src
|
|
|
|
** APA-100 Amplified Actuator
|
|
*** Identification
|
|
#+begin_src matlab
|
|
K = tf(zeros(6));
|
|
Kiff = tf(zeros(6));
|
|
#+end_src
|
|
|
|
We identify the system for the following payload masses:
|
|
#+begin_src matlab
|
|
Ms = [1, 10, 50];
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gm_iff = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
The nano-hexapod has the following leg's stiffness and damping.
|
|
#+begin_src matlab
|
|
initializeNanoHexapod('actuator', 'amplified');
|
|
#+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; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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_iff = linearize(mdl, io);
|
|
G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
|
|
Gm_iff(i) = {G_iff};
|
|
end
|
|
#+end_src
|
|
|
|
*** Controller Design
|
|
The loop gain for IFF is shown in Figure [[fig:amplified_piezo_iff_loop_gain]].
|
|
|
|
The corresponding root locus is shown in Figure [[fig:amplified_piezo_iff_root_locus]].
|
|
|
|
Finally, the damping as function of the gain is display in Figure [[fig:amplified_piezo_iff_damping_gain]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, abs(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i}(1, 1), 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/amplified_piezo_iff_loop_gain.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_loop_gain
|
|
#+caption: Dynamics for the Integral Force Feedback for three payload masses
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_loop_gain.png]]
|
|
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
gains = logspace(2, 4, 200);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(pole(Gm_iff{i})), imag(pole(Gm_iff{i})), 'x', ...
|
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(tzero(Gm_iff{i})), imag(tzero(Gm_iff{i})), 'o', ...
|
|
'HandleVisibility', 'off');
|
|
for k = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',i);
|
|
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
|
|
plot(real(cl_poles), imag(cl_poles), '.', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-400, 10]); ylim([0, 500]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
legend('location', 'northwest');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_root_locus
|
|
#+caption: Root Locus for the IFF control for three payload masses
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_root_locus.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
c1 = [ 0 0.4470 0.7410]; % Blue
|
|
c2 = [0.8500 0.3250 0.0980]; % Orange
|
|
c3 = [0.9290 0.6940 0.1250]; % Yellow
|
|
c4 = [0.4940 0.1840 0.5560]; % Purple
|
|
c5 = [0.4660 0.6740 0.1880]; % Green
|
|
c6 = [0.3010 0.7450 0.9330]; % Light Blue
|
|
c7 = [0.6350 0.0780 0.1840]; % Red
|
|
colors = [c1; c2; c3; c4; c5; c6; c7];
|
|
|
|
figure;
|
|
|
|
gains = logspace(2, 5, 100);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for k = 1:length(gains)
|
|
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(gains(k), sin(-pi/2 + angle(cl_poles)), '.', 'color', colors(i, :));
|
|
end
|
|
end
|
|
hold off;
|
|
xlabel('IFF Gain'); ylabel('Modal Damping');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylim([0, 1]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_iff_damping_gain.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_damping_gain
|
|
#+caption: Damping ratio of the poles as a function of the IFF gain
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_damping_gain.png]]
|
|
|
|
The following controller for the Decentralized Integral Force Feedback is used:
|
|
#+begin_src matlab
|
|
Kiff = -1e4/s*eye(6);
|
|
#+end_src
|
|
|
|
*** Effect of the Low Authority Control on the Primary Plant
|
|
**** Introduction :ignore:
|
|
#+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
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
#+end_src
|
|
|
|
**** Identification of the undamped plant :ignore:
|
|
#+begin_src matlab :exports none
|
|
Kiff_backup = Kiff;
|
|
Kiff = tf(zeros(6));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
G_x = {zeros(length(Ms), 1)};
|
|
G_l = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
G_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
G_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kiff = Kiff_backup;
|
|
#+end_src
|
|
|
|
**** Identification of the damped plant :ignore:
|
|
#+begin_src matlab :exports none
|
|
Gm_x = {zeros(length(Ms), 1)};
|
|
Gm_l = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
Gm_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
Gm_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
**** Effect of the Damping on the plant diagonal dynamics :ignore:
|
|
#+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(Gm_x{i}(1, 1), freqs, 'Hz'))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_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(Gm_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(Gm_x{i}(1, 1), freqs, 'Hz')))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_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(Gm_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/amplified_piezo_iff_plant_damped_X.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_plant_damped_X
|
|
#+caption: Primary plant in the task space with (dashed) and without (solid) IFF
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_plant_damped_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(Gm_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(Gm_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/amplified_piezo_iff_damped_plant_L.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_damped_plant_L
|
|
#+caption: Primary plant in the space of the legs with (dashed) and without (solid) IFF
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_damped_plant_L.png]]
|
|
|
|
|
|
**** Effect of the Damping on the coupling dynamics :ignore:
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:5
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(G_x{1}(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_x{1}(i, j), freqs, 'Hz'))), '--', 'color', [0, 0, 0, 0.2]);
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_x{1}(1, 1), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_x{1}(1, 1), freqs, 'Hz'))), '--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
ylim([1e-12, inf]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_iff_damped_coupling_X.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_damped_coupling_X
|
|
#+caption: Coupling in the primary plant in the task with (dashed) and without (solid) IFF
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_damped_coupling_X.png]]
|
|
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:5
|
|
for j = i+1:6
|
|
plot(freqs, abs(squeeze(freqresp(G_l{1}(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_l{1}(i, j), freqs, 'Hz'))), '--', 'color', [0, 0, 0, 0.2]);
|
|
end
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_l{1}(1, 1), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_l{1}(1, 1), freqs, 'Hz'))), '--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
ylim([1e-9, inf]);
|
|
#+end_src
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_iff_damped_coupling_L.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_damped_coupling_L
|
|
#+caption: Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_damped_coupling_L.png]]
|
|
|
|
*** Effect of the Low Authority Control on the Sensibility to Disturbances
|
|
**** Introduction :ignore:
|
|
|
|
**** Identification :ignore:
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Micro-Hexapod
|
|
clear io; io_i = 1;
|
|
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_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, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kiff_backup = Kiff;
|
|
Kiff = tf(zeros(6));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gd = {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 = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gd(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kiff = Kiff_backup;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gd_iff = {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 = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gd_iff(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
**** Results :ignore:
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 5000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 2, 1);
|
|
title('$D_{w,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Dwz'), freqs, 'Hz'))), '--', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'southeast');
|
|
|
|
subplot(2, 2, 2);
|
|
title('$F_{dz}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 3);
|
|
title('$F_{T_y,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 4);
|
|
title('$F_{R_z,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_iff_disturbances.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_iff_disturbances
|
|
#+caption: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_iff_disturbances.png]]
|
|
|
|
*** Conclusion :ignore:
|
|
#+begin_important
|
|
#+end_important
|
|
|
|
|
|
** Optimal Stiffnesses
|
|
*** Introduction :ignore:
|
|
Based on the analytical analysis, we can determine the parameters of the amplified piezoelectric actuator in order to be able to add a lots of damping using IFF:
|
|
- $k_1$ should be minimized.
|
|
- $k_e \approx k_a \approx 10^5 - 10^6\,[N/m]$
|
|
|
|
However, this might not be realizable.
|
|
|
|
*** Low Authority Controller
|
|
**** Identification
|
|
The nano-hexapod is initialized with the following parameters:
|
|
#+begin_src matlab
|
|
initializeNanoHexapod('actuator', 'amplified', ...
|
|
'k1', 1e4, ...
|
|
'ke', 1e6, ...
|
|
'ka', 1e6);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
initializeController('type', 'hac-iff');
|
|
|
|
K = tf(zeros(6));
|
|
Kiff = tf(zeros(6));
|
|
|
|
Ms = [1, 10, 50];
|
|
|
|
Gm_iff = {zeros(length(Ms), 1)};
|
|
|
|
%% 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; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
|
|
|
|
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_iff = linearize(mdl, io);
|
|
G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
|
|
Gm_iff(i) = {G_iff};
|
|
end
|
|
#+end_src
|
|
|
|
The obtain plan for the IFF control is shown in Figure [[fig:amplified_piezo_opt_stiff_iff_plant]].
|
|
The associated Root Locus is shown in Figure [[fig:amplified_piezo_opt_stiff_iff_root_locus]].
|
|
|
|
Based on that, the following IFF gain is chosen:
|
|
#+begin_src matlab
|
|
Kiff = -1e3/s*eye(6);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, abs(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_iff{i}(1, 1), 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/amplified_piezo_opt_stiff_iff_plant.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_opt_stiff_iff_plant
|
|
#+caption: Plant dynamics for IFF with the amplified piezoelectric stack actuator
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_opt_stiff_iff_plant.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
gains = logspace(1, 4, 400);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(pole(Gm_iff{i})), imag(pole(Gm_iff{i})), 'x', ...
|
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(tzero(Gm_iff{i})), imag(tzero(Gm_iff{i})), 'o', ...
|
|
'HandleVisibility', 'off');
|
|
for k = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',i);
|
|
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
|
|
plot(real(cl_poles), imag(cl_poles), '.', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-400, 10]); ylim([0, 410]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
legend('location', 'northwest');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_opt_stiff_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_opt_stiff_iff_root_locus
|
|
#+caption: Root Locus for IFF with the amplified piezoelectric stack actuator
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_opt_stiff_iff_root_locus.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
c1 = [ 0 0.4470 0.7410]; % Blue
|
|
c2 = [0.8500 0.3250 0.0980]; % Orange
|
|
c3 = [0.9290 0.6940 0.1250]; % Yellow
|
|
c4 = [0.4940 0.1840 0.5560]; % Purple
|
|
c5 = [0.4660 0.6740 0.1880]; % Green
|
|
c6 = [0.3010 0.7450 0.9330]; % Light Blue
|
|
c7 = [0.6350 0.0780 0.1840]; % Red
|
|
colors = [c1; c2; c3; c4; c5; c6; c7];
|
|
|
|
figure;
|
|
|
|
gains = logspace(2, 5, 100);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for k = 1:length(gains)
|
|
cl_poles = pole(feedback(Gm_iff{i}, -(gains(k)/s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(gains(k), sin(-pi/2 + angle(cl_poles)), '.', 'color', colors(i, :));
|
|
end
|
|
end
|
|
hold off;
|
|
xlabel('IFF Gain'); ylabel('Modal Damping');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylim([0, 1]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_opt_stiff_gain_damping.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_opt_stiff_gain_damping
|
|
#+caption: Damping of the modes as a function of the IFF gain
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_opt_stiff_gain_damping.png]]
|
|
|
|
**** Effect of the Low Authority Control on the Primary Plant
|
|
***** Introduction :ignore:
|
|
#+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
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
#+end_src
|
|
|
|
***** Identification of the undamped plant :ignore:
|
|
#+begin_src matlab :exports none
|
|
Kiff_backup = Kiff;
|
|
Kiff = tf(zeros(6));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
G_x = {zeros(length(Ms), 1)};
|
|
G_l = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
G_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
G_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kiff = Kiff_backup;
|
|
#+end_src
|
|
|
|
***** Identification of the damped plant :ignore:
|
|
#+begin_src matlab :exports none
|
|
Gm_x = {zeros(length(Ms), 1)};
|
|
Gm_l = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
Gm_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
Gm_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
***** Effect of the Damping on the plant diagonal dynamics :ignore:
|
|
#+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(Gm_x{i}(1, 1), freqs, 'Hz'))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_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(Gm_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(Gm_x{i}(1, 1), freqs, 'Hz')))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_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(Gm_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 :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(Gm_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(Gm_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
|
|
|
|
**** Effect of the Low Authority Control on the Sensibility to Disturbances
|
|
***** Introduction :ignore:
|
|
|
|
***** Identification :ignore:
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Micro-Hexapod
|
|
clear io; io_i = 1;
|
|
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_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, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kiff_backup = Kiff;
|
|
Kiff = tf(zeros(6));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gd = {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 = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gd(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kiff = Kiff_backup;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gd_iff = {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 = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gd_iff(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
***** Results :ignore:
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 5000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 2, 1);
|
|
title('$D_{w,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Dwz'), freqs, 'Hz'))), '--', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'southeast');
|
|
|
|
subplot(2, 2, 2);
|
|
title('$F_{dz}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 3);
|
|
title('$F_{T_y,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 4);
|
|
title('$F_{R_z,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_iff{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/amplified_piezo_opt_stiff_iff_dist.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:amplified_piezo_opt_stiff_iff_dist
|
|
#+caption: Effect of disturbance with and without IFF
|
|
#+RESULTS:
|
|
[[file:figs/amplified_piezo_opt_stiff_iff_dist.png]]
|
|
|
|
***** Conclusion :ignore:
|
|
#+begin_important
|
|
|
|
#+end_important
|
|
|
|
*** High Authority Controller
|
|
**** Introduction :ignore:
|
|
|
|
**** Controller Design
|
|
#+begin_src matlab
|
|
h = 2.5;
|
|
Kl = 5e6 * eye(6) * ...
|
|
1/h*(s/(2*pi*40/h) + 1)/(s/(2*pi*40*h) + 1) * ...
|
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
|
(s/2/pi/50 + 1)/(s/2/pi/50) * ...
|
|
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
|
|
1/(1 + s/2/pi/200);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Kl = 3e10 * eye(6) * ...
|
|
1/s * ...
|
|
(s+0.8)/s * ...
|
|
(s+50)/(s+0.01) * ...
|
|
(s+120)/(s+1000) * ...
|
|
(s+150)/(s+1000);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for j = 1:6
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
|
end
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for j = 1:6
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
|
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');
|
|
#+end_src
|
|
|
|
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.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
for i = 1:length(Ms)
|
|
isstable(feedback(nano_hexapod.kinematics.J\Gm_l{i}*K, eye(6), -1))
|
|
end
|
|
#+end_src
|
|
|
|
**** Sensibility to Disturbances and Noise Budget
|
|
***** Identification :ignore:
|
|
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Micro-Hexapod
|
|
clear io; io_i = 1;
|
|
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_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, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gd_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 = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gd_L(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
***** Obtained Sensibility to Disturbances :ignore:
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 5000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 2, 1);
|
|
title('$D_{w,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Dwz'), freqs, 'Hz'))), '--', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'southeast');
|
|
|
|
subplot(2, 2, 2);
|
|
title('$F_{dz}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 3);
|
|
title('$F_{T_y,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 4);
|
|
title('$F_{R_z,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
#+end_src
|
|
|
|
**** Simulations of Tomography Experiment
|
|
Let's now simulate a tomography experiment.
|
|
To do so, we include all disturbances except vibrations of the translation stage.
|
|
#+begin_src matlab
|
|
initializeDisturbances();
|
|
initializeSimscapeConfiguration('gravity', false);
|
|
initializeLoggingConfiguration('log', 'all');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
load('mat/conf_simulink.mat');
|
|
set_param(conf_simulink, 'StopTime', '2');
|
|
#+end_src
|
|
|
|
And we run the simulation for all three payload Masses.
|
|
#+begin_src matlab :exports none
|
|
hac_dvf_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', 'Rz_period', Ms(i));
|
|
|
|
sim('nass_model');
|
|
hac_iff_L(i) = {simout};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
save('./mat/tomo_exp_hac_iff_opt_stiff.mat', 'hac_dvf_L', 'Ms');
|
|
#+end_src
|
|
|
|
**** Results
|
|
#+begin_src matlab :exports none
|
|
load('./mat/experiment_tomography.mat', 'scans_rz_align_dist');
|
|
load('./mat/tomo_exp_hac_iff_opt_stiff.mat', 'hac_iff_L', 'Ms');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
n_av = 4;
|
|
han_win = hanning(ceil(length(hac_iff_L{1}.Em.En.Data(:,1))/n_av));
|
|
|
|
t = hac_iff_L{1}.Em.En.Time;
|
|
Ts = t(2)-t(1);
|
|
|
|
[pxx_ol, f] = pwelch(scans_rz_align_dist.Em.En.Data, han_win, [], [], 1/Ts);
|
|
|
|
pxx_iff_L = zeros(length(f), 6, length(Ms));
|
|
for i = 1:length(Ms)
|
|
[pxx, ~] = pwelch(hac_iff_L{i}.Em.En.Data(ceil(0.2/Ts):end,:), han_win, [], [], 1/Ts);
|
|
pxx_iff_L(:, :, i) = pxx;
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 1)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_iff_L(:, 1, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{D_x}$ [$m/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 2)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_iff_L(:, 2, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{D_y}$ [$m/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 3)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_iff_L(:, 3, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{D_z}$ [$m/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 4)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_iff_L(:, 4, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{R_x}$ [$rad/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 5)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_iff_L(:, 5, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{R_y}$ [$rad/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 6)), 'k-', 'DisplayName', '$\mu$-Station')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_iff_L(:, 6, i)), ...
|
|
'DisplayName', sprintf('HAC-IFF $m = %.0f kg$', Ms(i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{R_z}$ [$rad/\sqrt{Hz}$]');
|
|
legend('location', 'southwest');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
xlim([f(2), f(end)])
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 1))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_iff_L(:, 1, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $D_x$ [$m$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 2))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_iff_L(:, 2, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $D_y$ [$m$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 3))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_iff_L(:, 3, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $D_z$ [$m$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 4))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_iff_L(:, 4, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $R_x$ [$rad$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 5))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_iff_L(:, 5, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $R_y$ [$rad$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 6))))), 'k-', 'DisplayName', '$\mu$-Station')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_iff_L(:, 6, i))))), ...
|
|
'DisplayName', sprintf('HAC-IFF $m = %.0f kg$', Ms(i)));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $R_z$ [$rad$]');
|
|
legend('location', 'southwest');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
xlim([f(2), f(end)])
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 1), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_iff_L{i}.Em.En.Time, hac_iff_L{i}.Em.En.Data(:, 1));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dx [m]');
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 2), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_iff_L{i}.Em.En.Time, hac_iff_L{i}.Em.En.Data(:, 2));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dy [m]');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 3), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_iff_L{i}.Em.En.Time, hac_iff_L{i}.Em.En.Data(:, 3));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dz [m]');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 4), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_iff_L{i}.Em.En.Time, hac_iff_L{i}.Em.En.Data(:, 4));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Rx [rad]');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 5), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_iff_L{i}.Em.En.Time, hac_iff_L{i}.Em.En.Data(:, 5));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Ry [rad]');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 6), 'k-', ...
|
|
'DisplayName', '$\mu$-Station');
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_iff_L{i}.Em.En.Time, hac_iff_L{i}.Em.En.Data(:, 6), ...
|
|
'DisplayName', sprintf('HAC-IFF $m = %.0f kg$', Ms(i)));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Rz [rad]');
|
|
legend();
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
xlim([0.5, inf]);
|
|
#+end_src
|
|
|
|
** Direct Velocity Feedback with Amplified Actuators
|
|
Lack of collocation.
|
|
|
|
#+begin_src matlab
|
|
initializeController('type', 'hac-dvf');
|
|
K = tf(zeros(6));
|
|
Kdvf = tf(zeros(6));
|
|
#+end_src
|
|
|
|
We identify the system for the following payload masses:
|
|
#+begin_src matlab
|
|
Ms = [1, 10, 50];
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gm_dvf = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
initializeNanoHexapod('actuator', 'amplified', ...
|
|
'k1', 1e4, ...
|
|
'ke', 1e6, ...
|
|
'ka', 1e6);
|
|
#+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; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Force Sensors
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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_dvf = linearize(mdl, io);
|
|
G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
|
|
Gm_dvf(i) = {G_dvf};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 4, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, abs(squeeze(freqresp(Gm_dvf{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)
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_dvf{i}(1, 1), 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 :exports none
|
|
figure;
|
|
|
|
gains = logspace(2, 4, 100);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(pole(Gm_dvf{i})), imag(pole(Gm_dvf{i})), 'x', ...
|
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(tzero(Gm_dvf{i})), imag(tzero(Gm_dvf{i})), 'o', ...
|
|
'HandleVisibility', 'off');
|
|
for k = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',i);
|
|
cl_poles = pole(feedback(Gm_dvf{i}, (gains(k)*s)*eye(6)));
|
|
plot(real(cl_poles), imag(cl_poles), '.', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-400, 10]); ylim([0, 500]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
legend('location', 'northwest');
|
|
#+end_src
|
|
|
|
* APA300ML
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('nass_model.slx')
|
|
#+end_src
|
|
|
|
** Initialization
|
|
#+begin_src matlab
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
|
|
initializeSimscapeConfiguration();
|
|
initializeDisturbances('enable', false);
|
|
initializeLoggingConfiguration('log', 'none');
|
|
|
|
initializeController('type', 'hac-dvf');
|
|
#+end_src
|
|
|
|
We set the stiffness of the payload fixation:
|
|
#+begin_src matlab
|
|
Kp = 1e8; % [N/m]
|
|
#+end_src
|
|
|
|
** Identification
|
|
#+begin_src matlab
|
|
K = tf(zeros(6));
|
|
Kdvf = tf(zeros(6));
|
|
#+end_src
|
|
|
|
We identify the system for the following payload masses:
|
|
#+begin_src matlab
|
|
Ms = [1, 10, 50];
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gm_dvf = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
The nano-hexapod has the following leg's stiffness and damping.
|
|
#+begin_src matlab
|
|
initializeNanoHexapod('actuator', 'amplified', 'k1', 0.4e6, 'ka', 43e6, 'ke', 1.5e6);
|
|
#+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; % Actuator Inputs
|
|
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Force Sensors
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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_dvf = linearize(mdl, io);
|
|
G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
|
|
G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
|
|
Gm_dvf(i) = {G_dvf};
|
|
end
|
|
#+end_src
|
|
|
|
** Controller Design
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
plot(freqs, abs(squeeze(freqresp(Gm_dvf{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)
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_dvf{i}(1, 1), 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 :exports none
|
|
figure;
|
|
|
|
gains = logspace(2, 6, 100);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(pole(Gm_dvf{i})), imag(pole(Gm_dvf{i})), 'x', ...
|
|
'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(real(tzero(Gm_dvf{i})), imag(tzero(Gm_dvf{i})), 'o', ...
|
|
'HandleVisibility', 'off');
|
|
for k = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',i);
|
|
cl_poles = pole(feedback(Gm_dvf{i}, (gains(k)*s)*eye(6)));
|
|
plot(real(cl_poles), imag(cl_poles), '.', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-1e3, 10]); ylim([0, 1e3]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
legend('location', 'northwest');
|
|
#+end_src
|
|
|
|
Damping as function of the gain
|
|
#+begin_src matlab :exports none
|
|
c1 = [ 0 0.4470 0.7410]; % Blue
|
|
c2 = [0.8500 0.3250 0.0980]; % Orange
|
|
c3 = [0.9290 0.6940 0.1250]; % Yellow
|
|
c4 = [0.4940 0.1840 0.5560]; % Purple
|
|
c5 = [0.4660 0.6740 0.1880]; % Green
|
|
c6 = [0.3010 0.7450 0.9330]; % Light Blue
|
|
c7 = [0.6350 0.0780 0.1840]; % Red
|
|
colors = [c1; c2; c3; c4; c5; c6; c7];
|
|
|
|
figure;
|
|
|
|
gains = logspace(4, 7, 100);
|
|
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for k = 1:length(gains)
|
|
cl_poles = pole(feedback(Gm_dvf{i}, (gains(k)*s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(gains(k), sin(-pi/2 + angle(cl_poles)), '.', 'color', colors(i, :));
|
|
end
|
|
end
|
|
hold off;
|
|
xlabel('DVF Gain'); ylabel('Modal Damping');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylim([0, 1]);
|
|
#+end_src
|
|
|
|
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
|
|
#+begin_src matlab
|
|
Kdvf = 5e5*s/(1+s/2/pi/1e3)*eye(6);
|
|
#+end_src
|
|
|
|
** Effect of the Low Authority Control on the Primary Plant
|
|
*** Introduction :ignore:
|
|
#+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
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
#+end_src
|
|
|
|
*** Identification of the undamped plant :ignore:
|
|
#+begin_src matlab :exports none
|
|
Kdvf_backup = Kdvf;
|
|
Kdvf = tf(zeros(6));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
G_x = {zeros(length(Ms), 1)};
|
|
G_l = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
G_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
G_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Kdvf = Kdvf_backup;
|
|
#+end_src
|
|
|
|
*** Identification of the damped plant :ignore:
|
|
#+begin_src matlab :exports none
|
|
Gm_x = {zeros(length(Ms), 1)};
|
|
Gm_l = {zeros(length(Ms), 1)};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
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.kinematics.J');
|
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
|
Gm_x(i) = {Gx};
|
|
|
|
Gl = -nano_hexapod.kinematics.J*G;
|
|
Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'};
|
|
Gm_l(i) = {Gl};
|
|
end
|
|
#+end_src
|
|
|
|
*** Effect of the Damping on the plant diagonal dynamics :ignore:
|
|
#+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(Gm_x{i}(1, 1), freqs, 'Hz'))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_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(Gm_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(Gm_x{i}(1, 1), freqs, 'Hz')))), '--');
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm_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(Gm_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 :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(Gm_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(Gm_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
|
|
|
|
** Control in the leg space
|
|
We design a diagonal controller with all the same diagonal elements.
|
|
|
|
The requirements for the controller are:
|
|
- Crossover frequency of around 100Hz
|
|
- Stable for all the considered payload masses
|
|
- Sufficient phase and gain margin
|
|
- Integral action at low frequency
|
|
|
|
The design controller is as follows:
|
|
- Lead centered around the crossover
|
|
- An integrator below 10Hz
|
|
- A low pass filter at 250Hz
|
|
|
|
#+begin_src matlab
|
|
h = 2.0;
|
|
Kl = 1e9 * eye(6) * ...
|
|
1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
|
|
1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
|
|
(s/2/pi/10 + 1)/(s/2/pi/10) * ...
|
|
1/(1 + s/2/pi/300);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for j = 1:6
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
|
end
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
for j = 1:6
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_l{i}(j, j)*Kl(j,j), freqs, 'Hz'))));
|
|
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');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
load('mat/stages.mat', 'nano_hexapod');
|
|
K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
for i = 1:length(Ms)
|
|
isstable(feedback(nano_hexapod.kinematics.J\Gm_l{i}*K, eye(6), -1))
|
|
end
|
|
#+end_src
|
|
|
|
** Sensibility to Disturbances and Noise Budget
|
|
*** Identification :ignore:
|
|
We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'nass_model';
|
|
|
|
%% Micro-Hexapod
|
|
clear io; io_i = 1;
|
|
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_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, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gd_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 = {'Dwz', 'Fty_z', 'Frz_z', 'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
|
G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
|
|
|
|
Gd_L(i) = {G};
|
|
end
|
|
#+end_src
|
|
|
|
*** Obtained Sensibility to Disturbances :ignore:
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 5000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 2, 1);
|
|
title('$D_{w,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
% set(gca,'ColorOrderIndex',i);
|
|
% plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz'), freqs, 'Hz'))), ...
|
|
% 'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Dwz'), freqs, 'Hz'))), '--', ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'southeast');
|
|
|
|
subplot(2, 2, 2);
|
|
title('$F_{dz}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
% set(gca,'ColorOrderIndex',i);
|
|
% plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fdz'), freqs, 'Hz'))));
|
|
% set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 3);
|
|
title('$F_{T_y,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
% set(gca,'ColorOrderIndex',i);
|
|
% plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Fty_z'), freqs, 'Hz'))));
|
|
% set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Fty_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
|
|
subplot(2, 2, 4);
|
|
title('$F_{R_z,z}$ to $E_z$');
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
% set(gca,'ColorOrderIndex',i);
|
|
% plot(freqs, abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), freqs, 'Hz'))));
|
|
% set(gca,'ColorOrderIndex',i);
|
|
plot(freqs, abs(squeeze(freqresp(Gd_L{i}('Ez', 'Frz_z'), freqs, 'Hz'))), '--');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
#+end_src
|
|
|
|
*** Noise Budgeting :ignore:
|
|
|
|
#+begin_src matlab :exports none
|
|
load('./mat/dist_psd.mat', 'dist_f');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
hold on;
|
|
plot(dist_f.f, sqrt(dist_f.psd_gm).*abs(squeeze(freqresp(Gd_L{1}('Ez', 'Dwz' ), dist_f.f, 'Hz'))), 'DisplayName', '$D_w$')
|
|
plot(dist_f.f, sqrt(dist_f.psd_rz).*abs(squeeze(freqresp(Gd_L{1}('Ez', 'Frz_z'), dist_f.f, 'Hz'))), 'DisplayName', '$F_{R_z}$')
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('Amplitude Spectral Density [$m/\sqrt{Hz}$]');
|
|
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
|
legend('location', 'southwest');
|
|
xlim([1, 500]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/opt_stiff_primary_control_L_psd_dist.pdf', 'width', 'full', 'height', 'tall')
|
|
#+end_src
|
|
|
|
#+name: fig:opt_stiff_primary_control_L_psd_dist
|
|
#+caption: Amplitude Spectral Density of the vertical position error of the sample when the HAC-DVF control is applied due to both the ground motion and spindle vibrations
|
|
#+RESULTS:
|
|
[[file:figs/opt_stiff_primary_control_L_psd_dist.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
hold on;
|
|
plot(dist_f.f, sqrt(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{1}('Ez', 'Dwz' ), dist_f.f, 'Hz'))).^2 + ...
|
|
dist_f.psd_rz.*abs(squeeze(freqresp(Gd{1}('Ez', 'Frz_z'), dist_f.f, 'Hz'))).^2), 'DisplayName', 'Open-Loop')
|
|
plot(dist_f.f, sqrt(dist_f.psd_gm.*abs(squeeze(freqresp(Gd_L{1}('Ez', 'Dwz' ), dist_f.f, 'Hz'))).^2 + ...
|
|
dist_f.psd_rz.*abs(squeeze(freqresp(Gd_L{1}('Ez', 'Frz_z'), dist_f.f, 'Hz'))).^2), 'DisplayName', 'HAC-DVF')
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('Amplitude Spectral Density [$m/\sqrt{Hz}$]');
|
|
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
|
legend('location', 'northeast');
|
|
xlim([1, 500]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/opt_stiff_primary_control_L_psd_tot.pdf', 'width', 'full', 'height', 'tall')
|
|
#+end_src
|
|
|
|
#+name: fig:opt_stiff_primary_control_L_psd_tot
|
|
#+caption: Amplitude Spectral Density of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied
|
|
#+RESULTS:
|
|
[[file:figs/opt_stiff_primary_control_L_psd_tot.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
hold on;
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(dist_f.f, sqrt(flip(-cumtrapz(flip(dist_f.f), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd{i}('Ez', 'Dwz' ), dist_f.f, 'Hz'))).^2 + ...
|
|
dist_f.psd_rz.*abs(squeeze(freqresp(Gd{i}('Ez', 'Frz_z'), dist_f.f, 'Hz'))).^2)))), ...
|
|
'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i)));
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(dist_f.f, sqrt(flip(-cumtrapz(flip(dist_f.f), flip(dist_f.psd_gm.*abs(squeeze(freqresp(Gd_L{i}('Ez', 'Dwz' ), dist_f.f, 'Hz'))).^2 + ...
|
|
dist_f.psd_rz.*abs(squeeze(freqresp(Gd_L{i}('Ez', 'Frz_z'), dist_f.f, 'Hz'))).^2)))), '--', ...
|
|
'HandleVisibility', 'off')
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('Amplitude Spectral Density [$m/\sqrt{Hz}$]');
|
|
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
|
legend('location', 'southwest');
|
|
xlim([0.1, 500]); ylim([1e-12, 1e-6]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/opt_stiff_primary_control_L_cas_tot.pdf', 'width', 'full', 'height', 'tall')
|
|
#+end_src
|
|
|
|
#+name: fig:opt_stiff_primary_control_L_cas_tot
|
|
#+caption: Cumulative Amplitude Spectrum of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied
|
|
#+RESULTS:
|
|
[[file:figs/opt_stiff_primary_control_L_cas_tot.png]]
|
|
|
|
** Simulations of Tomography Experiment
|
|
Let's now simulate a tomography experiment.
|
|
To do so, we include all disturbances except vibrations of the translation stage.
|
|
#+begin_src matlab
|
|
initializeDisturbances();
|
|
initializeSimscapeConfiguration('gravity', false);
|
|
initializeLoggingConfiguration('log', 'all');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
load('mat/conf_simulink.mat');
|
|
set_param(conf_simulink, 'StopTime', '2');
|
|
#+end_src
|
|
|
|
And we run the simulation for all three payload Masses.
|
|
#+begin_src matlab :exports none
|
|
hac_dvf_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', 'Rz_period', Ms(i));
|
|
|
|
sim('nass_model');
|
|
hac_dvf_L(i) = {simout};
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
save('./mat/tomo_exp_hac_dvf_ampl.mat', 'hac_dvf_L', 'Ms');
|
|
#+end_src
|
|
|
|
** Results
|
|
#+begin_src matlab :exports none
|
|
load('./mat/experiment_tomography.mat', 'scans_rz_align_dist');
|
|
load('./mat/tomo_exp_hac_dvf_ampl.mat', 'hac_dvf_L', 'Ms');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
n_av = 4;
|
|
han_win = hanning(ceil(length(hac_dvf_L{1}.Em.En.Data(:,1))/n_av));
|
|
|
|
t = hac_dvf_L{1}.Em.En.Time;
|
|
Ts = t(2)-t(1);
|
|
|
|
[pxx_ol, f] = pwelch(scans_rz_align_dist.Em.En.Data, han_win, [], [], 1/Ts);
|
|
|
|
pxx_dvf_L = zeros(length(f), 6, length(Ms));
|
|
for i = 1:length(Ms)
|
|
[pxx, ~] = pwelch(hac_dvf_L{i}.Em.En.Data(ceil(0.2/Ts):end,:), han_win, [], [], 1/Ts);
|
|
pxx_dvf_L(:, :, i) = pxx;
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 1)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_dvf_L(:, 1, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{D_x}$ [$m/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 2)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_dvf_L(:, 2, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{D_y}$ [$m/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 3)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_dvf_L(:, 3, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{D_z}$ [$m/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 4)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_dvf_L(:, 4, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{R_x}$ [$rad/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 5)), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_dvf_L(:, 5, i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{R_y}$ [$rad/\sqrt{Hz}$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(f, sqrt(pxx_ol(:, 6)), 'k-', 'DisplayName', '$\mu$-Station')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(pxx_dvf_L(:, 6, i)), ...
|
|
'DisplayName', sprintf('HAC-DVF $m = %.0f kg$', Ms(i)))
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$\Gamma_{R_z}$ [$rad/\sqrt{Hz}$]');
|
|
legend('location', 'southwest');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
xlim([f(2), f(end)])
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 1))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_dvf_L(:, 1, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $D_x$ [$m$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 2))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_dvf_L(:, 2, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $D_y$ [$m$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 3))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_dvf_L(:, 3, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $D_z$ [$m$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 4))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_dvf_L(:, 4, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $R_x$ [$rad$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 5))))), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_dvf_L(:, 5, i))))));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $R_y$ [$rad$]');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_ol(:, 6))))), 'k-', 'DisplayName', '$\mu$-Station')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxx_dvf_L(:, 6, i))))), ...
|
|
'DisplayName', sprintf('HAC-DVF $m = %.0f kg$', Ms(i)));
|
|
end
|
|
hold off;
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('CAS $R_z$ [$rad$]');
|
|
legend('location', 'southwest');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-11, 1e-5]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
xlim([f(2), f(end)])
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
ax1 = subplot(2, 3, 1);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 1), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_dvf_L{i}.Em.En.Time, hac_dvf_L{i}.Em.En.Data(:, 1));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dx [m]');
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 2), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_dvf_L{i}.Em.En.Time, hac_dvf_L{i}.Em.En.Data(:, 2));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dy [m]');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 3), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_dvf_L{i}.Em.En.Time, hac_dvf_L{i}.Em.En.Data(:, 3));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Dz [m]');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 4), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_dvf_L{i}.Em.En.Time, hac_dvf_L{i}.Em.En.Data(:, 4));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Rx [rad]');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 5), 'k-')
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_dvf_L{i}.Em.En.Time, hac_dvf_L{i}.Em.En.Data(:, 5));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Ry [rad]');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(scans_rz_align_dist.Em.En.Time, scans_rz_align_dist.Em.En.Data(:, 6), 'k-', ...
|
|
'DisplayName', '$\mu$-Station');
|
|
for i = 1:length(Ms)
|
|
set(gca,'ColorOrderIndex',i);
|
|
plot(hac_dvf_L{i}.Em.En.Time, hac_dvf_L{i}.Em.En.Data(:, 6), ...
|
|
'DisplayName', sprintf('HAC-DVF $m = %.0f kg$', Ms(i)));
|
|
end
|
|
hold off;
|
|
xlabel('Time [s]');
|
|
ylabel('Rz [rad]');
|
|
legend();
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
xlim([0.5, inf]);
|
|
#+end_src
|