935 lines
28 KiB
Matlab
935 lines
28 KiB
Matlab
%% Clear Workspace and Close figures
|
|
clear; close all; clc;
|
|
|
|
%% Intialize Laplace variable
|
|
s = zpk('s');
|
|
|
|
%% Path for functions, data and scripts
|
|
addpath('./mat/'); % Path for data
|
|
addpath('./src/'); % Path for functions
|
|
|
|
addpath('./STEPS/'); % Path for Simscape Model
|
|
|
|
%% Linearization options
|
|
opts = linearizeOptions;
|
|
opts.SampleTime = 0;
|
|
|
|
%% Open Simscape Model
|
|
mdl = 'test_struts_simscape'; % Name of the Simulink File
|
|
open(mdl); % Open Simscape Model
|
|
|
|
%% Colors for the figures
|
|
colors = colororder;
|
|
|
|
%% Input/Output definition of the Model
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage
|
|
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
|
|
io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
|
|
io(io_i) = linio([mdl, '/da'], 1, 'openoutput'); io_i = io_i + 1; % Interferometer
|
|
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
% 2Dof model
|
|
% The strut is initialized with default parameters (optimized parameters identified from previous experiments).
|
|
|
|
%% Initialize structure containing data for the Simscape model
|
|
n_hexapod = struct();
|
|
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
|
|
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
|
|
n_hexapod.actuator = initializeAPA('type', '2dof');
|
|
|
|
c_granite = 0; % Do not take into account damping added by the air bearing
|
|
|
|
|
|
|
|
% The dynamics is identified and shown in Figure ref:fig:strut_bench_model_bode.
|
|
|
|
%% Run the linearization
|
|
Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
Gs.InputName = {'u'};
|
|
Gs.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
%% Bode plot of the transfer functions
|
|
figure;
|
|
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
|
|
plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d/u$ [V/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
legend('location', 'southwest');
|
|
|
|
ax1b = nexttile([2,1]);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), 'k-')
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))))
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))))
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:45:360);
|
|
ylim([-180, 180])
|
|
|
|
ax2b = nexttile;
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), 'k-')
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:45:360);
|
|
ylim([0, 180])
|
|
|
|
linkaxes([ax1,ax2,ax1b,ax2b],'x');
|
|
xlim([10, 2e3]);
|
|
|
|
|
|
|
|
% #+name: fig:strut_bench_model_bode
|
|
% #+caption: Identified transfer function from $u$ to $V_s$ and from $u$ to $d_e,d_a$ using the simple 2DoF model for the APA
|
|
% #+RESULTS:
|
|
% [[file:figs/strut_bench_model_bode.png]]
|
|
|
|
% The experimentally measured FRF are loaded.
|
|
|
|
%% Load measured FRF
|
|
load('meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align');
|
|
|
|
|
|
|
|
% The FRF from $u$ to $d_a$ as well as from $u$ to $V_s$ are shown in Figure ref:fig:comp_strut_plant_after_opt and compared with the model.
|
|
% They are both found to match quite well with the model.
|
|
|
|
%% Compare the FRF and identified dynamics from u to Vs and da
|
|
figure;
|
|
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(int_frf(:, 1)), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', 'Meas. FRF');
|
|
for i = 2:length(strut_nums)
|
|
plot(f, abs(int_frf(:, i)), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '-', ...
|
|
'DisplayName', 'Model')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax1b = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', 'Meas. FRF');
|
|
for i = 1:length(strut_nums)
|
|
plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '-', ...
|
|
'DisplayName', 'Model')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-2, 1e2]);
|
|
legend('location', 'southeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(strut_nums)
|
|
plot(f, 180/pi*angle(int_frf(:, i)), 'color', [0,0,0,0.2]);
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '-')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
ax2b = nexttile;
|
|
hold on;
|
|
for i = 1:length(strut_nums)
|
|
plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]);
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '-')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2,ax1b,ax2b],'x');
|
|
xlim([10, 2e3]);
|
|
|
|
|
|
|
|
% #+name: fig:comp_strut_plant_after_opt
|
|
% #+caption: Comparison of the measured FRF and the optimized model
|
|
% #+RESULTS:
|
|
% [[file:figs/comp_strut_plant_after_opt.png]]
|
|
|
|
% The measured FRF from $u$ to $d_e$ (encoder) is compared with the model in Figure ref:fig:comp_strut_plant_iff_after_opt.
|
|
|
|
%% Compare the FRF and identified dynamics from u to de
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', 'Meas. FRF');
|
|
for i = 2:length(strut_nums)
|
|
plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-', ...
|
|
'DisplayName', 'Model')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(strut_nums)
|
|
plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([20, 2e3]);
|
|
|
|
% Comparison with the Flexible Model
|
|
% The strut is initialized with default parameters (optimized parameters identified from previous experiments).
|
|
|
|
|
|
%% Initialize structure containing data for the Simscape model
|
|
n_hexapod = struct();
|
|
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
|
|
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible');
|
|
|
|
c_granite = 100; % Do not take into account damping added by the air bearing
|
|
|
|
|
|
|
|
% The dynamics is identified and shown in Figure ref:fig:strut_bench_model_bode.
|
|
|
|
%% Run the linearization
|
|
Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
Gs.InputName = {'u'};
|
|
Gs.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
|
|
|
|
% - [ ] Add encoder plot
|
|
|
|
% The FRF from $u$ to $d_a$ as well as from $u$ to $V_s$ are shown in Figure ref:fig:comp_strut_plant_after_opt and compared with the model.
|
|
% They are both found to match quite well with the model.
|
|
|
|
%% Compare the FRF and identified dynamics from u to Vs and da
|
|
figure;
|
|
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 1)), 'color', [colors(2,:), 0.5], ...
|
|
'DisplayName', 'FRF - Encoder');
|
|
plot(f, abs(int_frf(:, 1)), 'color', [0,0,0, 0.2], ...
|
|
'DisplayName', 'FRF - Interferometer');
|
|
for i = 2:length(strut_nums)
|
|
plot(f, abs(int_frf(:, i)), 'color', [0,0,0, 0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--', ...
|
|
'DisplayName', 'Model - Interferometer')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'southwest');
|
|
|
|
ax1b = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(iff_frf(:, i)), 'color', [colors(2,:), 0.2], ...
|
|
'DisplayName', 'Meas. FRF');
|
|
for i = 1:length(strut_nums)
|
|
plot(f, abs(iff_frf(:, i)), 'color', [0,0,0, 0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '--', ...
|
|
'DisplayName', 'Model')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-2, 1e2]);
|
|
legend('location', 'southeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plot(f, 180/pi*angle(enc_frf(:, 1)), 'color', [colors(2,:), 0.5], ...
|
|
'HandleVisibility', 'off');
|
|
for i = 1:length(strut_nums)
|
|
plot(f, 180/pi*(angle(int_frf(:, i))), 'color', [0,0,0, 0.2]);
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
ax2b = nexttile;
|
|
hold on;
|
|
for i = 1:length(strut_nums)
|
|
plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0, 0.2]);
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '--')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2,ax1b,ax2b],'x');
|
|
xlim([10, 2e3]);
|
|
|
|
% Perfectly aligned APA
|
|
% Let's first consider that the strut is perfectly mounted such that the two flexible joints and the APA are aligned.
|
|
|
|
%% Initialize Simscape data
|
|
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
|
|
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible');
|
|
|
|
|
|
|
|
% And define the inputs and outputs of the models:
|
|
% - Input: voltage generated by the DAC
|
|
% - Output: measured displacement by the encoder
|
|
|
|
% The transfer function is identified and shown in Figure ref:fig:comp_enc_frf_align_perfect.
|
|
|
|
%% Identification
|
|
Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
Gs.InputName = {'u'};
|
|
Gs.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
|
|
|
|
% From Figure ref:fig:comp_enc_frf_align_perfect, it is clear that:
|
|
% 1. The model with perfect alignment is not matching the measured FRF
|
|
% 2. The mode at 200Hz is not present in the identified dynamics of the Simscape model
|
|
% 3. The measured FRF have different shapes
|
|
|
|
|
|
%% Measured FRF from Vs to de and identified dynamics using the flexible APA
|
|
freqs = 2*logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', 'Meas. FRF');
|
|
for i = 2:length(strut_nums)
|
|
plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-', ...
|
|
'DisplayName', 'Model')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(strut_nums)
|
|
plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([10, 2e3]);
|
|
|
|
% Effect of a misalignment in y
|
|
% Let's compute the transfer function from output DAC voltage $V_s$ to the measured displacement by the encoder $d_e$ for several misalignment in the $y$ direction:
|
|
|
|
%% Considered misalignments
|
|
dy_aligns = [-0.5, -0.1, 0, 0.1, 0.5]*1e-3; % [m]
|
|
|
|
%% Transfer functions from u to de for all the misalignment in y direction
|
|
Gs_align = {zeros(length(dy_aligns), 1)};
|
|
|
|
for i = 1:length(dy_aligns)
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [0; dy_aligns(i); 0], 'd_align_top', [0; dy_aligns(i); 0]);
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs_align(i) = {G};
|
|
end
|
|
|
|
|
|
|
|
% The obtained dynamics are shown in Figure ref:fig:effect_misalignment_y.
|
|
|
|
|
|
%% Transfer function from Vs to de - effect of x-misalignment
|
|
freqs = 2*logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
for i = 1:length(dy_aligns)
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$d_y = %.1f$ [mm]', 1e3*dy_aligns(i)));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(dy_aligns)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([10, 2e3]);
|
|
|
|
% Effect of a misalignment in x
|
|
% Let's compute the transfer function from output DAC voltage to the measured displacement by the encoder for several misalignment in the $x$ direction:
|
|
|
|
%% Considered misalignments
|
|
dx_aligns = [-0.1, -0.05, 0, 0.05, 0.1]*1e-3; % [m]
|
|
|
|
%% Transfer functions from u to de for all the misalignment in x direction
|
|
Gs_align = {zeros(length(dx_aligns), 1)};
|
|
|
|
for i = 1:length(dx_aligns)
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [dx_aligns(i); 0; 0], 'd_align_top', [dx_aligns(i); 0; 0]);
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs_align(i) = {G};
|
|
end
|
|
|
|
|
|
|
|
% The obtained dynamics are shown in Figure ref:fig:effect_misalignment_x.
|
|
|
|
%% Transfer function from Vs to de - effect of x-misalignment
|
|
freqs = 2*logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
for i = 1:length(dx_aligns)
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$d_x = %.2f$ [mm]', 1e3*dx_aligns(i)));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(dx_aligns)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([10, 2e3]);
|
|
|
|
% Comparison with identified misalignment
|
|
|
|
strut_align = 1e-3*[[-0.60, -0.82, -0.40, -0.16]
|
|
[-0.30, -0.63, -0.67, -0.34]
|
|
[-0.88, -0.79, -0.07, -0.16]
|
|
[-0.48, 0.07, -0.46, -1.00]
|
|
[-0.33, -0.48, -0.64, -0.52]
|
|
[-0.34, -0.42, -0.63, -0.57]];
|
|
|
|
%% Idenfity the transfer function from actuator to encoder for all cases
|
|
Gs_align = {zeros(size(strut_align,1), 1)};
|
|
|
|
for i = 1:size(strut_align,1)
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible', ...
|
|
'd_align_bot', [0; strut_align(i, 2) - strut_align(i, 4); 0], ...
|
|
'd_align_top', [0; strut_align(i, 1) - strut_align(i, 3); 0]);
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs_align(i) = {G};
|
|
end
|
|
|
|
%% Comparison of the plants (encoder output) when tuning the misalignment
|
|
freqs = 2*logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
ax1 = nexttile();
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 1)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{1}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
|
|
|
|
ax2 = nexttile();
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 2)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{2}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
|
|
|
ax3 = nexttile(4);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 3)), 'DisplayName', 'Meas.');
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', 'Model');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
|
|
legend('location', 'southwest', 'FontSize', 8);
|
|
|
|
ax4 = nexttile(5);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 4)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{4}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax5 = nexttile(6);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 5)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{5}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5],'xy');
|
|
xlim([20, 2e3]); ylim([1e-8, 1e-3]);
|
|
|
|
% Find the misalignment of each strut
|
|
% From the previous analysis on the effect of a $x$ and $y$ misalignment, it is possible to estimate the $x,y$ misalignment of the measured struts.
|
|
|
|
% The misalignment that gives the best match for the FRF are defined below.
|
|
|
|
%% Tuned misalignment [m]
|
|
d_aligns = [[-0.05, -0.3, 0];
|
|
[ 0, 0.5, 0];
|
|
[-0.1, -0.3, 0];
|
|
[ 0, 0.3, 0];
|
|
[-0.05, 0.05, 0]]'*1e-3;
|
|
|
|
|
|
|
|
% For each misalignment, the dynamics from the DAC voltage to the encoder measurement is identified.
|
|
|
|
%% Idenfity the transfer function from actuator to encoder for all cases
|
|
Gs_align = {zeros(size(d_aligns,2), 1)};
|
|
|
|
for i = 1:5
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_top', d_aligns(:,i), 'd_align_bot', d_aligns(:,i));
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs_align(i) = {G};
|
|
end
|
|
|
|
|
|
|
|
% The results are shown in Figure ref:fig:comp_all_struts_corrected_misalign.
|
|
|
|
%% Comparison of the plants (encoder output) when tuning the misalignment
|
|
freqs = 2*logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
ax1 = nexttile();
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 1)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{1}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
|
|
|
|
ax2 = nexttile();
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 2)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{2}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
|
|
|
|
ax3 = nexttile();
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 3)), 'DisplayName', 'Meas.');
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', 'Model');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
|
|
legend('location', 'southwest', 'FontSize', 8);
|
|
|
|
ax4 = nexttile(5);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 4)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{4}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
|
|
ax5 = nexttile(6);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:, 5)));
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{5}('de', 'u'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5],'xy');
|
|
xlim([20, 2e3]); ylim([1e-8, 1e-3]);
|
|
|
|
% Paper :noexport:
|
|
|
|
%% Comparison of the plants (encoder output) when tuning the misalignment
|
|
freqs = 2*logspace(0, 3, 1000);
|
|
|
|
colors = colororder;
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(f, abs(enc_frf(:,1)), 'color', [colors(1,:),0.2], ...
|
|
'DisplayName', 'FRF - $d_{e,i}/V_{a,i}$')
|
|
for i = 2:5
|
|
plot(f, abs(enc_frf(:,i)), 'color', [colors(1,:),0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
|
|
plot(f, abs(int_frf(:,1)), 'color', [colors(2,:),0.2], ...
|
|
'DisplayName', 'FRF - $d_{a,i}/V_{a,i}$')
|
|
for i = 2:5
|
|
plot(f, abs(int_frf(:,i)), 'color', [colors(2,:),0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{1}('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(1,:), ...
|
|
'DisplayName', 'Model - $d_{e,i}/V_{a,i}$')
|
|
for i = 2:5
|
|
plot(freqs, abs(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(1,:), ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), ...
|
|
'DisplayName', 'Model - $d_{a,i}/V_{a,i}$')
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'southwest')
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:5
|
|
plot(f, 180/pi*angle(enc_frf(:,i)), 'color', [colors(1,:),0.2]);
|
|
plot(f, 180/pi*(angle(int_frf(:, i)) - angle(squeeze(freqresp(exp(-s*2*1e-4), f, 'Hz')))), 'color', [colors(2,:),0.2]);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(1,:));
|
|
end
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([20, 2e3]);
|
|
|
|
% Effect of bending stiffness of the flexible joints
|
|
% <<sec:struts_effect_bending_stiff_joints>>
|
|
|
|
% Let's initialize an APA which is a little bit misaligned.
|
|
|
|
%% APA Initialization
|
|
n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [0.1e-3; 0.5e-3; 0], 'd_align_top', [0.1e-3; 0.5e-3; 0]);
|
|
|
|
|
|
|
|
% The bending stiffnesses for which the dynamics is identified are defined below.
|
|
|
|
%% Tested bending stiffnesses [Nm/rad]
|
|
kRs = [3, 4, 5, 6, 7];
|
|
|
|
|
|
|
|
% Then the identification is performed for all the values of the bending stiffnesses.
|
|
|
|
%% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
|
|
Gs = {zeros(length(kRs), 1)};
|
|
|
|
for i = 1:length(kRs)
|
|
n_hexapod.flex_bot = initializeBotFlexibleJoint(...
|
|
'type', '4dof', ...
|
|
'kRx', kRs(i), ...
|
|
'kRy', kRs(i));
|
|
n_hexapod.flex_top = initializeTopFlexibleJoint(...
|
|
'type', '4dof', ...
|
|
'kRx', kRs(i), ...
|
|
'kRy', kRs(i));
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs(i) = {G};
|
|
end
|
|
|
|
|
|
|
|
% The obtained dynamics from DAC voltage to encoder measurements are compared in Figure ref:fig:effect_enc_bending_stiff.
|
|
|
|
%% Plot the obtained transfer functions for all the bending stiffnesses
|
|
freqs = 2*logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
for i = 1:length(kRs)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$k_R = %.0f$ [Nm/rad]', kRs(i)));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(kRs)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([20, 2e3]);
|
|
|
|
% Effect of axial stiffness of the flexible joints
|
|
% <<sec:struts_effect_axial_stiff_joints>>
|
|
|
|
% The axial stiffnesses for which the dynamics is identified are defined below.
|
|
|
|
%% Tested axial stiffnesses [N/m]
|
|
kzs = [5e7 7.5e7 1e8 2.5e8];
|
|
|
|
|
|
|
|
% Then the identification is performed for all the values of the bending stiffnesses.
|
|
|
|
%% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
|
|
Gs = {zeros(length(kzs), 1)};
|
|
|
|
for i = 1:length(kzs)
|
|
n_hexapod.flex_bot = initializeBotFlexibleJoint(...
|
|
'type', '4dof', ...
|
|
'kz', kzs(i));
|
|
n_hexapod.flex_top = initializeTopFlexibleJoint(...
|
|
'type', '4dof', ...
|
|
'kz', kzs(i));
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs(i) = {G};
|
|
end
|
|
|
|
|
|
|
|
% The obtained dynamics from DAC voltage to encoder measurements are compared in Figure ref:fig:effect_enc_axial_stiff.
|
|
|
|
%% Plot the obtained transfer functions for all the axial stiffnesses
|
|
freqs = 2*logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
for i = 1:length(kzs)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$k_z = %.1e$ [N/m]', kzs(i)));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(kzs)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([20, 2e3]);
|
|
|
|
% Effect of bending damping
|
|
% <<sec:struts_effect_bending_damping_joints>>
|
|
% Now let's study the effect of the bending damping of the flexible joints.
|
|
|
|
% The tested bending damping are defined below:
|
|
|
|
%% Tested bending dampings [Nm/(rad/s)]
|
|
cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
|
|
|
|
|
|
|
|
% Then the identification is performed for all the values of the bending damping.
|
|
|
|
%% Idenfity the transfer function from actuator to encoder for all bending dampins
|
|
Gs = {zeros(length(cRs), 1)};
|
|
|
|
for i = 1:length(cRs)
|
|
n_hexapod.flex_bot = initializeBotFlexibleJoint(...
|
|
'type', '4dof', ...
|
|
'cRx', cRs(i), ...
|
|
'cRy', cRs(i));
|
|
n_hexapod.flex_top = initializeTopFlexibleJoint(...
|
|
'type', '4dof', ...
|
|
'cRx', cRs(i), ...
|
|
'cRy', cRs(i));
|
|
|
|
G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
|
|
G.InputName = {'u'};
|
|
G.OutputName = {'Vs', 'de', 'da'};
|
|
|
|
Gs(i) = {G};
|
|
end
|
|
|
|
|
|
|
|
% The results are shown in Figure ref:fig:effect_enc_bending_damp.
|
|
|
|
%% Plot the obtained transfer functions for all the bending stiffnesses
|
|
freqs = 2*logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
for i = 1:length(cRs)
|
|
plot(freqs, abs(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))), ...
|
|
'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', cRs(i)));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
ylim([1e-8, 1e-3]);
|
|
legend('location', 'southwest');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i = 1:length(cRs)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
yticks(-360:90:360); ylim([-180, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([20, 2e3]);
|