178 lines
6.0 KiB
Mathematica
178 lines
6.0 KiB
Mathematica
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Path for functions, data and scripts
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addpath('./src/'); % Path for scripts
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addpath('./mat/'); % Path for data
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addpath('./STEPS/'); % Path for Simscape Model
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%% Linearization options
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opts = linearizeOptions;
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opts.SampleTime = 0;
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%% Open Simscape Model
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mdl = 'test_apa_simscape'; % Name of the Simulink File
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open(mdl); % Open Simscape Model
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%% Colors for the figures
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colors = colororder;
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%% Input/Output definition of the Model
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage
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io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
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io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
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% Tuning of the APA model
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% <<ssec:test_apa_2dof_model_tuning>>
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% 9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
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% #+name: fig:test_apa_2dof_model_simscape
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% #+caption: Schematic of the two degrees of freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$
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% [[file:figs/test_apa_2dof_model_simscape.png]]
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%% Stiffness values for the 2DoF APA model
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k1 = 0.38e6; % Estimated Shell Stiffness [N/m]
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w0 = 2*pi*95; % Resonance frequency [rad/s]
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m = 5.7; % Suspended mass [kg]
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ktot = m*(w0)^2; % Total Axial Stiffness to have to wanted resonance frequency [N/m]
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ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m]
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ke = 2*ka; % Stiffness of the Sensor stack [N/m]
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%% Damping values for the 2DoF APA model
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c1 = 20; % Damping for the Shell [N/(m/s)]
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ca = 100; % Damping of the actuators stacks [N/(m/s)]
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ce = 2*ca; % Damping of the sensor stack [N/(m/s)]
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%% Estimation ot the sensor and actuator gains
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% Initialize the structure with unitary sensor and actuator "gains"
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n_hexapod = struct();
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n_hexapod.actuator = initializeAPA(...
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'type', '2dof', ...
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'k', k1, ...
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'ka', ka, ...
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'ke', ke, ...
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'c', c1, ...
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'ca', ca, ...
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'ce', ce, ...
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'Ga', 1, ... % Actuator constant [N/V]
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'Gs', 1 ... % Sensor constant [V/m]
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);
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c_granite = 0; % Do not take into account damping added by the air bearing
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% Run the linearization
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G_norm = linearize(mdl, io, 0.0, opts);
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G_norm.InputName = {'u'};
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G_norm.OutputName = {'Vs', 'de'};
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% Load Identification Data to estimate the two gains
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load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
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% Estimation ot the Actuator Gain
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fa = 10; % Frequency where the two FRF should match [Hz]
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[~, i_f] = min(abs(f - fa));
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ga = -abs(enc_frf(i_f,1))./abs(evalfr(G_norm('de', 'u'), 1i*2*pi*fa));
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% Estimation ot the Sensor Gain
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fs = 600; % Frequency where the two FRF should match [Hz]
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[~, i_f] = min(abs(f - fs))
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gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga;
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% Obtained Dynamics
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% <<ssec:test_apa_2dof_model_result>>
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% The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model.
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% It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf.
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% A good match can be observed between the model and the experimental data, both for the encoder and for the force sensor.
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% This indicates that this model represents well the axial dynamics of the APA300ML.
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%% 2DoF APA300ML with optimized parameters
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n_hexapod = struct();
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n_hexapod.actuator = initializeAPA(...
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'type', '2dof', ...
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'k', k1, ...
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'ka', ka, ...
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'ke', ke, ...
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'c', c1, ...
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'ca', ca, ...
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'ce', ce, ...
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'Ga', ga, ...
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'Gs', gs ...
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);
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%% Identification of the APA300ML with optimized parameters
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G_2dof = exp(-s*Ts)*linearize(mdl, io, 0.0, opts);
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G_2dof.InputName = {'u'};
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G_2dof.OutputName = {'Vs', 'de'};
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%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML
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freqs = 5*logspace(0, 3, 1000);
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figure;
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tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
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for i = 1:length(apa_nums)
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plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-8, 1e-3]);
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legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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ax1b = nexttile([2,1]);
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hold on;
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plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
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for i = 2:length(apa_nums)
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plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-2, 1e2]);
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legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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ax2 = nexttile;
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hold on;
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for i = 1:length(apa_nums)
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plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
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end
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360); ylim([-180, 180]);
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ax2b = nexttile;
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hold on;
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for i = 1:length(apa_nums)
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plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]);
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end
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360); ylim([-180, 180]);
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linkaxes([ax1,ax2,ax1b,ax2b],'x');
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xlim([10, 2e3]);
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