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+ +switch args.type + case '2dof' + actuator.type = 1; + case 'flexible frame' + actuator.type = 2; + case 'flexible' + actuator.type = 3; +end + +if args.Ga == 0 + switch args.type + case '2dof' + actuator.Ga = -2.5796; + case 'flexible frame' + actuator.Ga = 1; % TODO + case 'flexible' + actuator.Ga = 23.2; + end +else + actuator.Ga = args.Ga; % Actuator gain [N/V] +end + +if args.Gs == 0 + switch args.type + case '2dof' + actuator.Gs = 466664; + case 'flexible frame' + actuator.Gs = 1; % TODO + case 'flexible' + actuator.Gs = -4898341; + end +else + actuator.Gs = args.Gs; % Sensor gain [V/m] +end + +actuator.k = args.k; % [N/m] +actuator.ke = args.ke; % [N/m] +actuator.ka = args.ka; % [N/m] + +actuator.c = args.c; % [N/(m/s)] +actuator.ce = args.ce; % [N/(m/s)] +actuator.ca = args.ca; % [N/(m/s)] + +actuator.Leq = args.Leq; % [m] + +switch args.type + case 'flexible frame' + actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix + actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix + actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m] + case 'flexible' + actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix + actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix + actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m] + actuator.d_align_bot = args.d_align_bot; + actuator.d_align_top = args.d_align_top; +end + +actuator.xi = args.xi; % Damping ratio + +actuator.ks = args.ks; % Stiffness of one stack [N/m] +actuator.cs = args.cs; % Damping of one stack [N/m] diff --git a/matlab/src/initializeBotFlexibleJoint.m b/matlab/src/initializeBotFlexibleJoint.m new file mode 100644 index 0000000..e5975b2 --- /dev/null +++ b/matlab/src/initializeBotFlexibleJoint.m @@ -0,0 +1,45 @@ +function [flex_bot] = initializeBotFlexibleJoint(args) +% initializeBotFlexibleJoint - +% +% Syntax: [flex_bot] = initializeBotFlexibleJoint(args) +% +% Inputs: +% - args - +% +% Outputs: +% - flex_bot - + +arguments + args.type char {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof' + + args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5 + args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5 + args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260 + args.kz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*7e7 + + args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 + args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 + args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 + args.cz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 +end + +flex_bot = struct(); + +switch args.type + case '2dof' + flex_bot.type = 1; + case '3dof' + flex_bot.type = 2; + case '4dof' + flex_bot.type = 3; +end + +flex_bot.kRx = args.kRx; +flex_bot.kRy = args.kRy; +flex_bot.kRz = args.kRz; +flex_bot.kz = args.kz; + +flex_bot.cRx = args.cRx; +flex_bot.cRy = args.cRy; +flex_bot.cRz = args.cRz; +flex_bot.cz = args.cz; diff --git a/matlab/src/initializeTopFlexibleJoint.m b/matlab/src/initializeTopFlexibleJoint.m new file mode 100644 index 0000000..f6a7b83 --- /dev/null +++ b/matlab/src/initializeTopFlexibleJoint.m @@ -0,0 +1,45 @@ +function [flex_top] = initializeTopFlexibleJoint(args) +% initializeTopFlexibleJoint - +% +% Syntax: [flex_top] = initializeTopFlexibleJoint(args) +% +% Inputs: +% - args - +% +% Outputs: +% - flex_top - + +arguments + args.type char {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof' + + args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5 + args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5 + args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260 + args.kz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*7e7 + + args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 + args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 + args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 + args.cz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001 +end + +flex_top = struct(); + +switch args.type + case '2dof' + flex_top.type = 1; + case '3dof' + flex_top.type = 2; + case '4dof' + flex_top.type = 3; +end + +flex_top.kRx = args.kRx; +flex_top.kRy = args.kRy; +flex_top.kRz = args.kRz; +flex_top.kz = args.kz; + +flex_top.cRx = args.cRx; +flex_top.cRy = args.cRy; +flex_top.cRz = args.cRz; +flex_top.cz = args.cz; diff --git a/matlab/test_apa_simscape.slx b/matlab/test_apa_simscape.slx new file mode 100644 index 0000000..bb48c53 Binary files /dev/null and b/matlab/test_apa_simscape.slx differ diff --git a/matlab/test_struts_1_flexible_modes.m b/matlab/test_struts_1_flexible_modes.m new file mode 100644 index 0000000..ed51ca8 --- /dev/null +++ b/matlab/test_struts_1_flexible_modes.m @@ -0,0 +1,65 @@ +%% 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 + +%% Colors for the figures +colors = colororder; + +% Without Encoder +% When the encoder is not fixed to the strut, the obtained FRF are shown in Figure ref:fig:test_struts_spur_res_frf. + + +%% Load Data (without the encoder) +bending_X = load('strut_spur_res_x_bending.mat'); +bending_Y = load('strut_spur_res_y_bending.mat'); +torsion_Z = load('strut_spur_res_z_torsion.mat'); + +%% Plot the responses (without the encoder) +figure; +hold on; +plot(bending_X.FFT1_AvSpc_1_RMS_X_Val, bending_X.FFT1_AvSpc_1_RMS_Y_Val, ... + 'DisplayName', 'X-Bending') +plot(bending_Y.FFT1_AvSpc_1_RMS_X_Val, bending_Y.FFT1_AvSpc_1_RMS_Y_Val, ... + 'DisplayName', 'Y-Bending') +plot(torsion_Z.FFT1_AvSpc_1_RMS_X_Val, torsion_Z.FFT1_AvSpc_1_RMS_Y_Val, ... + 'DisplayName', 'Z-torsion'); +text(226, 1.5e-4,{'226Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +text(337, 6e-5,{'337Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +text(398, 1.5e-4,{'398Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +hold off; +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude'); +xlim([50, 8e2]); ylim([5e-7, 3e-4]) +legend('location', 'northwest'); + +% With Encoder +% Then, one encoder is fixed to the strut and the FRF are measured again and shown in Figure ref:fig:test_struts_spur_res_frf_enc. + +%% Load Data (with the encoder) +bending_X_enc = load('strut_spur_res_x_bending_enc.mat'); +bending_Y_enc = load('strut_spur_res_y_bending_enc.mat'); +torsion_Z_enc = load('strut_spur_res_z_torsion_enc.mat'); + +%% Plot the responses (with the encoder) +figure; +hold on; +plot(bending_X_enc.FFT1_AvSpc_1_RMS_X_Val, bending_X_enc.FFT1_AvSpc_1_RMS_Y_Val, ... + 'DisplayName', 'X-Bending') +plot(bending_Y_enc.FFT1_AvSpc_1_RMS_X_Val, bending_Y_enc.FFT1_AvSpc_1_RMS_Y_Val, ... + 'DisplayName', 'Y-Bending') +plot(torsion_Z_enc.FFT1_AvSpc_1_RMS_X_Val, torsion_Z_enc.FFT1_AvSpc_1_RMS_Y_Val, ... + 'DisplayName', 'Z-torsion'); +text(198, 4e-5,{'198Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +text(293, 6e-5,{'293Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +text(381, 1e-4,{'381Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +hold off; +set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude'); +xlim([50, 8e2]); ylim([5e-7, 2e-4]) +legend('location', 'northwest'); diff --git a/matlab/test_struts_2_dynamical_meas.m b/matlab/test_struts_2_dynamical_meas.m new file mode 100644 index 0000000..5ced4d9 --- /dev/null +++ b/matlab/test_struts_2_dynamical_meas.m @@ -0,0 +1,687 @@ +%% 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 + +%% Colors for the figures +colors = colororder; + +% FRF Identification - Setup +% Similarly to what was done for the identification of the APA, the identification is performed in three steps: +% 1. White noise excitation with small amplitude. +% This is used to determine the main resonance of the system. +% 2. Sweep sine excitation with the amplitude lowered around the resonance. +% The sweep sine is from 10Hz to 400Hz. +% 3. High frequency noise. +% The noise is band-passed between 300Hz and 2kHz. + +% Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. + + +%% Sampling frequency/time +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); + +%% Load Data +leg_sweep = load('frf_data_leg_1_sweep.mat', 'u', 'Vs', 'de', 'da'); +leg_noise_hf = load('frf_data_leg_1_noise_hf.mat', 'u', 'Vs', 'de', 'da'); + +%% We get the frequency vector that will be the same for all the frequency domain analysis. +[~, f] = tfestimate(leg_sweep.u, leg_sweep.de, win, Noverlap, Nfft, 1/Ts); +i_lf = f <= 350; % Indices used for the low frequency +i_hf = f > 350; % Indices used for the low frequency + +% FRF Identification - Interferometer +% In this section, the dynamics from the excitation voltage $u$ to the interferometer $d_a$ is identified. +% The transfer function from $u$ to the interferometer measured displacement $d_a$ is estimated and shown in Figure ref:fig:strut_1_frf_dvf_plant_tf. + +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_sweep.u, leg_sweep.da, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_noise_hf.u, leg_noise_hf.da, win, Noverlap, Nfft, 1/Ts); + +%% Combine the FRF +int_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; + +%% Plot the measured FRF +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(int_frf), 'k-'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-9, 1e-3]); + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(int_frf), 'k-'); +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]); + +% FRF Identification - IFF +% In this section, the dynamics from $u$ to $V_s$ is identified. +% Then the FRF are estimated and shown in Figure ref:fig:strut_1_frf_iff_plant_tf + +%% Compute the FRF +[frf_sweep, ~] = tfestimate(leg_sweep.u, leg_sweep.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_noise_hf.u, leg_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); + +%% Combine the FRF +iff_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; + +%% Plot the measured FRF +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(iff_frf), 'k-'); +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]); + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(iff_frf), 'k-'); +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]); + +% Measurement Data +% The measurements are loaded. + +%% Load data +leg_enc_sweep = load('frf_data_leg_coder_1_noise.mat', 'u', 'Vs', 'de', 'da'); +leg_enc_noise_hf = load('frf_data_leg_coder_1_noise_hf.mat', 'u', 'Vs', 'de', 'da'); + +% FRF Identification - Interferometer +% In this section, the dynamics from $u$ to $d_a$ is identified. + +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_enc_sweep.u, leg_enc_sweep.da, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.da, win, Noverlap, Nfft, 1/Ts); + +%% Combine the FRF +int_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; + + + +% The obtained FRF is very close to the one that was obtained when no encoder was fixed to the struts as shown in Figure ref:fig:strut_leg_compare_int_frf. + +%% Plot the FRF from u to da with and without the encoder +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(int_with_enc_frf), '-', 'DisplayName', 'With encoder'); +plot(f, abs(int_frf), '-', 'DisplayName', 'Without encoder'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-7, 1e-3]); +legend('location', 'northeast') + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(int_with_enc_frf), '-'); +plot(f, 180/pi*angle(int_frf), '-'); +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]); + +% FRF Identification - Encoder +% In this section, the dynamics from $u$ to $d_e$ (encoder) is identified. + +% The FRF from $u$ to the encoder measured displacement $d_e$ is computed and shown in Figure ref:fig:strut_1_enc_frf_dvf_plant_tf. + +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_enc_sweep.u, leg_enc_sweep.de, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.de, win, Noverlap, Nfft, 1/Ts); + +%% Combine the FRF +enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; + +%% Plot the FRF from u to de +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(enc_frf), 'k-'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-7, 1e-3]); + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(enc_frf), 'k-'); +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]); + + + +% #+name: fig:strut_1_enc_frf_dvf_plant_tf +% #+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the encoder $d_e$) +% #+RESULTS: +% [[file:figs/strut_1_enc_frf_dvf_plant_tf.png]] + +% The transfer functions from $u$ to $d_e$ (encoder) and to $d_a$ (interferometer) are compared in Figure ref:fig:strut_1_comp_enc_int. + + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(enc_frf), 'DisplayName', 'Encoder'); +plot(f, abs(int_with_enc_frf), 'DisplayName', 'Interferometer'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-8, 1e-3]); + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(enc_frf)); +plot(f, 180/pi*angle(int_with_enc_frf)); +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]); + +% APA Resonances Frequency +% As shown in Figure ref:fig:strut_1_spurious_resonances, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz. + + +%% Transfer function from Vs to de with indicated resonances +figure; +hold on; +plot(f, abs(enc_frf), 'k-'); +text(93, 4e-4, {'93Hz'}, 'VerticalAlignment','bottom','HorizontalAlignment','center') +text(200, 1.3e-4,{'197Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +text(300, 4e-6, {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +text(400, 1.4e-6,{'376Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); xlabel('Frequency [Hz]'); +hold off; +ylim([1e-7, 1e-3]); xlim([10, 2e3]); + +% FRF Identification - Force Sensor +% In this section, the dynamics from $u$ to $V_s$ is identified. + + +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_enc_sweep.u, leg_enc_sweep.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); + +%% Combine the FRF +iff_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; + + + +% Let's now compare the IFF plants whether the encoders are fixed to the APA or not (Figure ref:fig:strut_1_frf_iff_comp_enc). + +%% Compare the IFF plant with and without the encoders +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(iff_with_enc_frf), 'DisplayName', 'With Encoder'); +plot(f, abs(iff_frf), 'DisplayName', 'Without Encoder'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8); +ylim([1e-2, 1e2]); + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(iff_with_enc_frf)); +plot(f, 180/pi*angle(iff_frf)); +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]); + +% Non-Minimum phase zero? + +% In order to determine if the complex conjugate zero of Figure ref:fig:strut_1_enc_frf_iff_plant_tf is minimum phase or non-minimum phase, longer measurements are performed. + + +long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'u', 'Vs'); + +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(10/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); + +%% Transfer function estimation +[frf_noise, f] = tfestimate(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts); + +%% Bode plot of the FRF from u to de +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(frf_noise), '.-'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(frf_noise), '.-'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([38, 45]); + +% FRF Identification - Setup +% The identification of the struts dynamics is performed in two steps: +% 1. The excitation signal is a white noise with small amplitude. +% This is used to estimate the low frequency dynamics. +% 2. Then a high frequency noise band-passed between 300Hz and 2kHz is used to estimate the high frequency dynamics. + +% Then, the result of the first identification is used between 10Hz and 350Hz and the result of the second identification if used between 350Hz and 2kHz. + +% Here are the leg numbers that have been measured. + +%% Numnbers of the measured legs +strut_nums = [1 2 3 4 5]; + + + +% The data are loaded for both the first and second identification: + +%% First identification (low frequency noise) +leg_noise = {}; +for i = 1:length(strut_nums) + leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', strut_nums(i)), 'u', 'Vs', 'de', 'da')}; +end + +%% Second identification (high frequency noise) +leg_noise_hf = {}; +for i = 1:length(strut_nums) + leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', strut_nums(i)), 'u', 'Vs', 'de', 'da')}; +end + +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); + + + +% We get the frequency vector that will be the same for all the frequency domain analysis. + +% Only used to have the frequency vector "f" +[~, f] = tfestimate(leg_noise{1}.u, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts); +i_lf = f <= 350; +i_hf = f > 350; + +% FRF Identification - Encoder +% In this section, the dynamics from $u$ to $d_e$ (encoder) is identified. + +% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the encoder $d_e$ is computed: + +%% Transfer function estimation +enc_frf = zeros(length(f), length(strut_nums)); + +for i = 1:length(strut_nums) + [frf_lf, ~] = tfestimate(leg_noise{i}.u, detrend(leg_noise{i}.de, 0), win, Noverlap, Nfft, 1/Ts); + [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, detrend(leg_noise_hf{i}.de, 0), win, Noverlap, Nfft, 1/Ts); + enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)]; +end + + + +% The obtained transfer functions are shown in Figure ref:fig:struts_frf_dvf_plant_tf. + +%% Bode plot of the FRF from u to de +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(strut_nums) + plot(f, abs(enc_frf(:, i)), ... + 'DisplayName', sprintf('Leg %i', strut_nums(i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-8, 1e-3]); + +ax2 = nexttile; +hold on; +for i = 1:length(strut_nums) + plot(f, 180/pi*angle(enc_frf(:, i))); +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]); + +% FRF Identification - Interferometer +% In this section, the dynamics from $u$ to $d_a$ (interferometer) is identified. + +% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the Attocube is computed for all the struts and shown in Figure ref:fig:struts_frf_int_plant_tf. +% All the struts are giving very similar FRF. + +%% Transfer function estimation +int_frf = zeros(length(f), length(strut_nums)); +for i = 1:length(strut_nums) + [frf_lf, ~] = tfestimate(leg_noise{i}.u, leg_noise{i}.da, win, Noverlap, Nfft, 1/Ts); + [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, leg_noise_hf{i}.da, win, Noverlap, Nfft, 1/Ts); + int_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)]; +end + +%% Plot the FRF from u to de (interferometer) +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(strut_nums) + plot(f, abs(int_frf(:, i)), ... + 'DisplayName', sprintf('Leg %i', strut_nums(i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-9, 1e-3]); + +ax2 = nexttile; +hold on; +for i = 1:length(strut_nums) + plot(f, 180/pi*angle(int_frf(:, i))); +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]); + +% FRF Identification - Force Sensor +% In this section, the dynamics from $u$ to $V_s$ is identified. +% Then the FRF are estimated and shown in Figure ref:fig:struts_frf_iff_plant_tf. +% They are also shown all to be very similar. + +%% FRF estimation of the transfer function from u to Vs +iff_frf = zeros(length(f), length(strut_nums)); +for i = 1:length(strut_nums) + [frf_lf, ~] = tfestimate(leg_noise{i}.u, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); + [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, leg_noise_hf{i}.Vs, win, Noverlap, Nfft, 1/Ts); + iff_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)]; +end + +%% Plot the FRF from u to Vs +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(strut_nums) + plot(f, abs(iff_frf(:, i)), ... + 'DisplayName', sprintf('Leg %i', strut_nums(i))); +end +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', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i = 1:length(strut_nums) + plot(f, 180/pi*angle(iff_frf(:, i))); +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]); + +% Misalignment of the APA and flexible joints + +% The misalignment between the two flexible joints and the APA has been measured for all the struts: +% - the strut is fixed to the mounting bench +% - using an indicator, the height difference from the flexible joints and the APA is measured both for the top and bottom joints and on both sides +% - then it is possible to obtain the misalignment for both flexible joints + +% The raw measurements are shown in Table ref:tab:meas_misalignment_struts_raw. + +% As the flexible joint's "thickness" is 1mm larger than the APA "thickness", ideally (i.e. if it were perfectly centered) we would measure =-0.50mm= each time. + + +strut_nums = [1, 2, 3, 4, 5]; + +% R Top B Top R Bot B Bot +strut_align = [[-0.40, -0.60, -0.16, -0.82] % Strut 1 + [-0.67, -0.30, -0.34, -0.63] % Strut 2 + [-0.07, -0.88, -0.16, -0.79] % Strut 3 + [-0.48, -0.46, 0.07, -1.00] % Strut 4 + [-0.33, -0.64, -0.48, -0.52]]; % Strut 5 + +%% Save the estimated FRF for further analysis +save('./mat/meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align'); + +% Measured misalignment of the APA and flexible joints +% The misalignment between the APA and the flexible joints are measured. + +% The results are defined below and summarized in Table ref:tab:meas_misalignment_struts_new_raw. + + +% R Top B Top R Bot B Bot +strut_align = [[-0.54, -0.50, -0.50, -0.52] % strut 1 + [-0.44, -0.55, -0.49, -0.49] % strut 2 + [-0.48, -0.50, -0.50, -0.46] % strut 3 + [-0.45, -0.51, -0.51, -0.45] % strut 4 + [-0.50, -0.50, -0.50, -0.50] % strut 5 + [-0.50, -0.49, -0.43, -0.54]]; % strut 6 + +% FRF Identification - Setup +% The excitation signal is a low pass filtered white noise. +% Both the encoder and the force sensor voltage are measured. + +% Here are the leg numbers that have been measured. + +%% Numnbers of the measured legs +strut_nums = [1 2 3 4 5 6]; + +%% First identification (low frequency noise) +leg_noise = {}; +for i = 1:length(strut_nums) + leg_noise(i) = {load(sprintf('frf_struts_align_%i_noise.mat', strut_nums(i)), 'u', 'Vs', 'de')}; +end + +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); + + + +% We get the frequency vector that will be the same for all the frequency domain analysis. + +% Only used to have the frequency vector "f" +[~, f] = tfestimate(leg_noise{1}.u, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts); + +% FRF Identification - Encoder +% In this section, the dynamics from $u$ to $d_e$ (encoder) is identified. + +% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the encoder $d_e$ is computed: + +%% Transfer function estimation +enc_frf = zeros(length(f), length(strut_nums)); + +for i = 1:length(strut_nums) + enc_frf(:, i) = tfestimate(leg_noise{i}.u, leg_noise{i}.de, win, Noverlap, Nfft, 1/Ts); +end + +%% Transfer function estimation +iff_frf = zeros(length(f), length(strut_nums)); + +for i = 1:length(strut_nums) + iff_frf(:, i) = tfestimate(leg_noise{i}.u, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); +end + + + +% The obtained transfer functions are shown in Figure ref:fig:struts_align_frf_dvf_plant_tf. + +%% Bode plot of the FRF from u to de +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(strut_nums) + plot(f, abs(enc_frf(:, i)), ... + 'DisplayName', sprintf('Leg %i', strut_nums(i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-8, 1e-3]); + +ax2 = nexttile; +hold on; +for i = 1:length(strut_nums) + plot(f, 180/pi*angle(enc_frf(:, i))); +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]); + +% TODO Noise measurement :noexport: + +%% Nothing connected to the actuator stacks +open_circuit = load('frf_struts_align_3_huddle_open_circuit.mat', 't', 'Vs', 'de'); + +%% PD200 connected but its input short-circuited +mid_voltage = load('frf_struts_align_3_huddle_mid_voltage_dac.mat', 't', 'Vs', 'de'); + +%% PD200 connected to the DAC that outputs 0V +zero_voltage = load('frf_struts_align_3_huddle_dac_zero.mat', 't', 'Vs', 'de'); + +%% PD200 connected to the DAC that outputs 3.25V +short_circuit = load('frf_struts_align_3_huddle_amp_short_circuit.mat', 't', 'Vs', 'de'); + +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(2/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); + +[pxx_oc, f] = pwelch(detrend(open_circuit.Vs, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_mv, ~] = pwelch(detrend(mid_voltage.Vs, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_zv, ~] = pwelch(detrend(zero_voltage.Vs, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_sc, ~] = pwelch(detrend(short_circuit.Vs, 0), win, Noverlap, Nfft, 1/Ts); + +figure; +hold on; +plot(f, sqrt(pxx_oc), 'DisplayName', 'Open Circuit') +plot(f, sqrt(pxx_sc), 'DisplayName', 'Amp Short-Circuited') +plot(f, sqrt(pxx_zv), 'DisplayName', 'Zero Voltage (DAC)') +plot(f, sqrt(pxx_mv), 'DisplayName', 'Mid Voltage (DAC)') +hold off; +xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]'); +set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); +legend('location', 'northeast'); +xlim([1, 5e3]); + +[pxx_oc, f] = pwelch(detrend(open_circuit.de, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_mv, ~] = pwelch(detrend(mid_voltage.de, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_zv, ~] = pwelch(detrend(zero_voltage.de, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_sc, ~] = pwelch(detrend(short_circuit.de, 0), win, Noverlap, Nfft, 1/Ts); + +figure; +hold on; +plot(f, sqrt(pxx_oc), 'DisplayName', 'Open Circuit') +plot(f, sqrt(pxx_sc), 'DisplayName', 'Amp Short-Circuited') +plot(f, sqrt(pxx_zv), 'DisplayName', 'Zero Voltage (DAC)') +plot(f, sqrt(pxx_mv), 'DisplayName', 'Mid Voltage (DAC)') +hold off; +xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); +set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); +legend('location', 'northeast'); +xlim([1, 5e3]) diff --git a/matlab/test_struts_3_simscape_model.m b/matlab/test_struts_3_simscape_model.m new file mode 100644 index 0000000..b2348f3 --- /dev/null +++ b/matlab/test_struts_3_simscape_model.m @@ -0,0 +1,934 @@ +%% 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 +% <> + +% 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 +% <> + +% 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 +% <> +% 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]); diff --git a/matlab/test_struts_simscape.slx b/matlab/test_struts_simscape.slx new file mode 100644 index 0000000..4b3e744 Binary files /dev/null and b/matlab/test_struts_simscape.slx differ diff --git a/test-bench-struts.org b/test-bench-struts.org index ce5adc0..0344bc6 100644 --- a/test-bench-struts.org +++ b/test-bench-struts.org @@ -480,72 +480,33 @@ Similarly to what was done for the identification of the APA, the identification Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. #+begin_src matlab -%% Load Data -leg_sweep = load('frf_data_leg_1_sweep.mat', 't', 'Va', 'Vs', 'de', 'da'); -leg_noise_hf = load('frf_data_leg_1_noise_hf.mat', 't', 'Va', 'Vs', 'de', 'da'); -#+end_src - -The time is the same for all measurements. -#+begin_src matlab -%% Time vector -t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s] - %% Sampling frequency/time Ts = 1e-4; % Sampling Time [s] -Fs = 1/Ts; % Sampling Frequency [Hz] Nfft = floor(1/Ts); win = hanning(Nfft); Noverlap = floor(Nfft/2); #+end_src -We get the frequency vector that will be the same for all the frequency domain analysis. #+begin_src matlab -% Only used to have the frequency vector "f" -[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, Noverlap, Nfft, 1/Ts); +%% Load Data +leg_sweep = load('frf_data_leg_1_sweep.mat', 'u', 'Vs', 'de', 'da'); +leg_noise_hf = load('frf_data_leg_1_noise_hf.mat', 'u', 'Vs', 'de', 'da'); +#+end_src + +#+begin_src matlab +%% We get the frequency vector that will be the same for all the frequency domain analysis. +[~, f] = tfestimate(leg_sweep.u, leg_sweep.de, win, Noverlap, Nfft, 1/Ts); i_lf = f <= 350; % Indices used for the low frequency i_hf = f > 350; % Indices used for the low frequency #+end_src **** FRF Identification - Interferometer -In this section, the dynamics from the excitation voltage $V_a$ to the interferometer $d_a$ is identified. - -We compute the coherence for 2nd and 3rd identification and combine them. +In this section, the dynamics from the excitation voltage $u$ to the interferometer $d_a$ is identified. +The transfer function from $u$ to the interferometer measured displacement $d_a$ is estimated and shown in Figure ref:fig:strut_1_frf_dvf_plant_tf. #+begin_src matlab -%% Compute the coherence for both excitation signals -[int_coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.da, win, Noverlap, Nfft, 1/Ts); -[int_coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, Noverlap, Nfft, 1/Ts); - -%% Combine the coherence -int_coh = [int_coh_sweep(i_lf); int_coh_noise_hf(i_hf)]; -#+end_src - -The combined coherence is shown in Figure ref:fig:strut_1_frf_dvf_plant_coh, and is found to be very good up to at least 1kHz. - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -plot(f, int_coh(:, 1), 'k-'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:strut_1_frf_dvf_plant_coh -#+caption: Obtained coherence for the plant from $V_a$ to $d_a$ -#+RESULTS: -[[file:figs/strut_1_frf_dvf_plant_coh.png]] - -The transfer function from $V_a$ to the interferometer measured displacement $d_a$ is estimated and shown in Figure ref:fig:strut_1_frf_dvf_plant_tf. -#+begin_src matlab -%% Compute FRF function from Va to da -[frf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.da, win, Noverlap, Nfft, 1/Ts); -[frf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, Noverlap, Nfft, 1/Ts); +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_sweep.u, leg_sweep.da, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_noise_hf.u, leg_noise_hf.da, win, Noverlap, Nfft, 1/Ts); %% Combine the FRF int_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; @@ -554,14 +515,14 @@ int_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; #+begin_src matlab :exports none %% Plot the measured FRF figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(int_frf), 'k-'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-9, 1e-3]); @@ -583,51 +544,17 @@ exportFig('figs/strut_1_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall' #+end_src #+name: fig:strut_1_frf_dvf_plant_tf -#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the interferometer $d_a$) +#+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the interferometer $d_a$) #+RESULTS: [[file:figs/strut_1_frf_dvf_plant_tf.png]] **** FRF Identification - IFF -In this section, the dynamics from $V_a$ to $V_s$ is identified. - -First the coherence is computed and shown in Figure ref:fig:strut_1_frf_iff_plant_coh. -The coherence is very nice from 10Hz to 2kHz. -It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). - -#+begin_src matlab -%% Compute the coherence for both excitation signals -[iff_coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.Vs, win, Noverlap, Nfft, 1/Ts); -[iff_coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); - -%% Combine the coherence -iff_coh = [iff_coh_sweep(i_lf); iff_coh_noise_hf(i_hf)]; -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -plot(f, iff_coh, 'k-'); -hold off; -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:strut_1_frf_iff_plant_coh -#+caption: Obtained coherence for the IFF plant -#+RESULTS: -[[file:figs/strut_1_frf_iff_plant_coh.png]] - +In this section, the dynamics from $u$ to $V_s$ is identified. Then the FRF are estimated and shown in Figure ref:fig:strut_1_frf_iff_plant_tf #+begin_src matlab %% Compute the FRF -[frf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, Noverlap, Nfft, 1/Ts); -[frf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_sweep, ~] = tfestimate(leg_sweep.u, leg_sweep.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_noise_hf.u, leg_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); %% Combine the FRF iff_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; @@ -636,14 +563,14 @@ iff_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; #+begin_src matlab :exports none %% Plot the measured FRF figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(iff_frf), 'k-'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-2, 1e2]); @@ -678,95 +605,26 @@ Now the encoder is fixed to the strut and the identification is performed. The measurements are loaded. #+begin_src matlab %% Load data -leg_enc_sweep = load(sprintf('frf_data_leg_coder_badly_align_%i_noise.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); -leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_badly_align_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +leg_enc_sweep = load('frf_data_leg_coder_1_noise.mat', 'u', 'Vs', 'de', 'da'); +leg_enc_noise_hf = load('frf_data_leg_coder_1_noise_hf.mat', 'u', 'Vs', 'de', 'da'); #+end_src **** FRF Identification - Interferometer -In this section, the dynamics from $V_a$ to $d_a$ is identified. - -First, the coherence is computed and shown in Figure ref:fig:strut_1_int_with_enc_frf_dvf_plant_coh. +In this section, the dynamics from $u$ to $d_a$ is identified. #+begin_src matlab -%% Compute the coherence for both excitation signals -[int_coh_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.da, win, Noverlap, Nfft, 1/Ts); -[int_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, Noverlap, Nfft, 1/Ts); - -%% Combine the coherinte -int_coh = [int_coh_sweep(i_lf); int_coh_noise_hf(i_hf)]; -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -plot(f, int_coh(:, 1), 'k-'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_int_with_enc_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:strut_1_int_with_enc_frf_dvf_plant_coh -#+caption: Obtained coherence for the plant from $V_a$ to $d_a$ -#+RESULTS: -[[file:figs/strut_1_int_with_enc_frf_dvf_plant_coh.png]] - -Then the FRF are computed and shown in Figure ref:fig:strut_1_int_with_enc_frf_dvf_plant_tf. -#+begin_src matlab -%% Compute FRF function from Va to da -[frf_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.da, win, Noverlap, Nfft, 1/Ts); -[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, Noverlap, Nfft, 1/Ts); +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_enc_sweep.u, leg_enc_sweep.da, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.da, win, Noverlap, Nfft, 1/Ts); %% Combine the FRF int_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; #+end_src -#+begin_src matlab :exports none -%% Plot the FRF from Va to de -figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(f, abs(int_with_enc_frf), 'k-'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-7, 1e-3]); - -ax2 = nexttile; -hold on; -plot(f, 180/pi*angle(int_with_enc_frf), 'k-'); -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]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_int_with_enc_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:strut_1_int_with_enc_frf_dvf_plant_tf -#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) -#+RESULTS: -[[file:figs/strut_1_int_with_enc_frf_dvf_plant_tf.png]] - The obtained FRF is very close to the one that was obtained when no encoder was fixed to the struts as shown in Figure ref:fig:strut_leg_compare_int_frf. - #+begin_src matlab :exports none -%% Plot the FRF from Va to da with and without the encoder +%% Plot the FRF from u to da with and without the encoder figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; @@ -774,7 +632,7 @@ plot(f, abs(int_with_enc_frf), '-', 'DisplayName', 'With encoder'); plot(f, abs(int_frf), '-', 'DisplayName', 'Without encoder'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-7, 1e-3]); legend('location', 'northeast') @@ -798,66 +656,34 @@ exportFig('figs/strut_leg_compare_int_frf.pdf', 'width', 'wide', 'height', 'tall #+end_src #+name: fig:strut_leg_compare_int_frf -#+caption: Comparison of the measured FRF from $V_a$ to $d_a$ with and without the encoders fixed to the struts +#+caption: Comparison of the measured FRF from $u$ to $d_a$ with and without the encoders fixed to the struts #+RESULTS: [[file:figs/strut_leg_compare_int_frf.png]] - - **** FRF Identification - Encoder -In this section, the dynamics from $V_a$ to $d_e$ (encoder) is identified. +In this section, the dynamics from $u$ to $d_e$ (encoder) is identified. -The coherence is computed and shown in Figure ref:fig:strut_1_enc_frf_dvf_plant_coh. +The FRF from $u$ to the encoder measured displacement $d_e$ is computed and shown in Figure ref:fig:strut_1_enc_frf_dvf_plant_tf. #+begin_src matlab -%% Compute the coherence for both excitation signals -[enc_coh_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.de, win, Noverlap, Nfft, 1/Ts); -[enc_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, Noverlap, Nfft, 1/Ts); - -%% Combine the coherence -enc_coh = [enc_coh_sweep(i_lf); enc_coh_noise_hf(i_hf)]; -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -plot(f, enc_coh(:, 1), 'k-'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_enc_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:strut_1_enc_frf_dvf_plant_coh -#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ and from $V_a$ to $d_a$ -#+RESULTS: -[[file:figs/strut_1_enc_frf_dvf_plant_coh.png]] - -The FRF from $V_a$ to the encoder measured displacement $d_e$ is computed and shown in Figure ref:fig:strut_1_enc_frf_dvf_plant_tf. -#+begin_src matlab -%% Compute FRF function from Va to da -[frf_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.de, win, Noverlap, Nfft, 1/Ts); -[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, Noverlap, Nfft, 1/Ts); +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_enc_sweep.u, leg_enc_sweep.de, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.de, win, Noverlap, Nfft, 1/Ts); %% Combine the FRF enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; #+end_src #+begin_src matlab :exports none -%% Plot the FRF from Va to de +%% Plot the FRF from u to de figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(enc_frf), 'k-'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-7, 1e-3]); @@ -879,15 +705,15 @@ exportFig('figs/strut_1_enc_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 't #+end_src #+name: fig:strut_1_enc_frf_dvf_plant_tf -#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the encoder $d_e$) #+RESULTS: [[file:figs/strut_1_enc_frf_dvf_plant_tf.png]] -The transfer functions from $V_a$ to $d_e$ (encoder) and to $d_a$ (interferometer) are compared in Figure ref:fig:strut_1_comp_enc_int. +The transfer functions from $u$ to $d_e$ (encoder) and to $d_a$ (interferometer) are compared in Figure ref:fig:strut_1_comp_enc_int. #+begin_src matlab :exports none figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; @@ -895,7 +721,7 @@ plot(f, abs(enc_frf), 'DisplayName', 'Encoder'); plot(f, abs(int_with_enc_frf), 'DisplayName', 'Interferometer'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ylim([1e-8, 1e-3]); @@ -919,12 +745,12 @@ exportFig('figs/strut_1_comp_enc_int.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:strut_1_comp_enc_int -#+caption: Comparison of the transfer functions from excitation voltage $V_a$ to either the encoder $d_e$ or the interferometer $d_a$ +#+caption: Comparison of the transfer functions from excitation voltage $u$ to either the encoder $d_e$ or the interferometer $d_a$ #+RESULTS: [[file:figs/strut_1_comp_enc_int.png]] #+begin_important -The dynamics from the excitation voltage $V_a$ to the measured displacement by the encoder $d_e$ presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure ref:fig:strut_1_comp_enc_int). +The dynamics from the excitation voltage $u$ to the measured displacement by the encoder $d_e$ presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure ref:fig:strut_1_comp_enc_int). It will be further investigated why the two dynamics as so different and what are causing all these resonances. #+end_important @@ -942,7 +768,7 @@ text(300, 4e-6, {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','c text(400, 1.4e-6,{'376Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); xlabel('Frequency [Hz]'); +ylabel('Amplitude $d_e/u$ [m/V]'); xlabel('Frequency [Hz]'); hold off; ylim([1e-7, 1e-3]); xlim([10, 2e3]); #+end_src @@ -952,7 +778,7 @@ exportFig('figs/strut_1_spurious_resonances.pdf', 'width', 'wide', 'height', 'ta #+end_src #+name: fig:strut_1_spurious_resonances -#+caption: Magnitude of the transfer function from excitation voltage $V_a$ to encoder measurement $d_e$. The frequency of the resonances are noted. +#+caption: Magnitude of the transfer function from excitation voltage $u$ to encoder measurement $d_e$. The frequency of the resonances are noted. #+RESULTS: [[file:figs/strut_1_spurious_resonances.png]] @@ -969,92 +795,22 @@ The resonances seen by the encoder in Figure ref:fig:strut_1_spurious_resonances #+end_important **** FRF Identification - Force Sensor -In this section, the dynamics from $V_a$ to $V_s$ is identified. - -First the coherence is computed and shown in Figure ref:fig:strut_1_frf_iff_with_enc_plant_coh. -The coherence is very nice from 10Hz to 2kHz. -It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). +In this section, the dynamics from $u$ to $V_s$ is identified. #+begin_src matlab -%% Compute the coherence for both excitation signals -[iff_coh_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, Noverlap, Nfft, 1/Ts); -[iff_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); - -%% Combine the coherence -iff_coh = [iff_coh_sweep(i_lf); iff_coh_noise_hf(i_hf)]; -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -plot(f, iff_coh, 'k-'); -hold off; -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_frf_iff_with_enc_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:strut_1_frf_iff_with_enc_plant_coh -#+caption: Obtained coherence for the IFF plant -#+RESULTS: -[[file:figs/strut_1_frf_iff_with_enc_plant_coh.png]] - -Then the FRF are estimated and shown in Figure ref:fig:strut_1_enc_frf_iff_plant_tf -#+begin_src matlab -%% Compute FRF function from Va to da -[frf_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, Noverlap, Nfft, 1/Ts); -[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); +%% Compute FRF function from u to da +[frf_sweep, ~] = tfestimate(leg_enc_sweep.u, leg_enc_sweep.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.u, leg_enc_noise_hf.Vs, win, Noverlap, Nfft, 1/Ts); %% Combine the FRF iff_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; #+end_src -#+begin_src matlab :exports none -%% Plot FRF of the transfer function from Va to Vs -figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(f, abs(iff_with_enc_frf), 'k-'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-2, 1e2]); - -ax2 = nexttile; -hold on; -plot(f, 180/pi*angle(iff_with_enc_frf), 'k'); -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]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/strut_1_enc_frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:strut_1_enc_frf_iff_plant_tf -#+caption:Identified IFF Plant -#+RESULTS: -[[file:figs/strut_1_enc_frf_iff_plant_tf.png]] - Let's now compare the IFF plants whether the encoders are fixed to the APA or not (Figure ref:fig:strut_1_frf_iff_comp_enc). #+begin_src matlab :exports none %% Compare the IFF plant with and without the encoders figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; @@ -1062,7 +818,7 @@ plot(f, abs(iff_with_enc_frf), 'DisplayName', 'With Encoder'); plot(f, abs(iff_frf), 'DisplayName', 'Without Encoder'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; legend('location', 'northeast', 'FontSize', 8); ylim([1e-2, 1e2]); @@ -1091,81 +847,46 @@ exportFig('figs/strut_1_frf_iff_effect_enc.pdf', 'width', 'wide', 'height', 'tal [[file:figs/strut_1_frf_iff_effect_enc.png]] #+begin_important -The transfer function from the excitation voltage $V_a$ to the generated voltage $V_s$ by the sensor stack is not influence by the fixation of the encoder. +The transfer function from the excitation voltage $u$ to the generated voltage $V_s$ by the sensor stack is not influence by the fixation of the encoder. This means that the IFF control strategy should be as effective whether or not the encoders are fixed to the struts. #+end_important -**** TODO Non-Minimum phase zero? +**** Non-Minimum phase zero? In order to determine if the complex conjugate zero of Figure ref:fig:strut_1_enc_frf_iff_plant_tf is minimum phase or non-minimum phase, longer measurements are performed. #+begin_src matlab -long_sweep = load('frf_struts_align_3_sweep_long.mat', 't', 'Va', 'Vs'); -long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'Va', 'Vs'); +long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'u', 'Vs'); #+end_src #+begin_src matlab -Ts = (long_sweep.t(end) - long_sweep.t(1))/(length(long_sweep.t)-1); % Sampling Time [s] -Fs = 1/Ts; -#+end_src - -#+begin_src matlab -win = hanning(ceil(50*Fs)); % Hannning Windows +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(10/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); #+end_src #+begin_src matlab %% Transfer function estimation -[coh_sweep, f] = mscohere(long_sweep.Va, long_sweep.Vs, win, Noverlap, Nfft, 1/Ts); -[coh_noise, ~] = mscohere(long_noise.Va, long_noise.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_noise, f] = tfestimate(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts); #+end_src #+begin_src matlab :exports none -%% Bode plot of the FRF from Va to de +%% Bode plot of the FRF from u to de figure; -hold on; -plot(f, coh_sweep, '.'); -plot(f, coh_noise, '.'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -hold off; -xlim([38, 45]); ylim([0, 1]); -#+end_src - -Remove time delay -#+begin_src matlab -long_sweep.Va(end) = []; -long_noise.Va(end) = []; - -long_sweep.Vs(1) = []; -long_noise.Vs(1) = []; -#+end_src - -#+begin_src matlab -%% Transfer function estimation -[frf_sweep, f] = tfestimate(long_sweep.Va, long_sweep.Vs, win, Noverlap, Nfft, 1/Ts); -[frf_noise, ~] = tfestimate(long_noise.Va, long_noise.Vs, win, Noverlap, Nfft, 1/Ts); -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the FRF from Va to de -figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -plot(f, abs(frf_sweep), '.'); -plot(f, abs(frf_noise), '.'); +plot(f, abs(frf_noise), '.-'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; -plot(f, 180/pi*angle(frf_sweep), '.'); -plot(f, 180/pi*angle(frf_noise), '.'); +plot(f, 180/pi*angle(frf_noise), '.-'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); @@ -1192,118 +913,76 @@ Then, the result of the first identification is used between 10Hz and 350Hz and Here are the leg numbers that have been measured. #+begin_src matlab %% Numnbers of the measured legs -leg_nums = [1 2 3 4 5]; +strut_nums = [1 2 3 4 5]; #+end_src The data are loaded for both the first and second identification: #+begin_src matlab %% First identification (low frequency noise) leg_noise = {}; -for i = 1:length(leg_nums) - leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')}; +for i = 1:length(strut_nums) + leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', strut_nums(i)), 'u', 'Vs', 'de', 'da')}; end %% Second identification (high frequency noise) leg_noise_hf = {}; -for i = 1:length(leg_nums) - leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')}; +for i = 1:length(strut_nums) + leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', strut_nums(i)), 'u', 'Vs', 'de', 'da')}; end #+end_src -The time is the same for all measurements. #+begin_src matlab -%% Time vector -t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s] - -%% Sampling -Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] -Fs = 1/Ts; % Sampling Frequency [Hz] -#+end_src - -Then we defined a "Hanning" windows that will be used for the spectral analysis: -#+begin_src matlab -win = hanning(ceil(0.5*Fs)); % Hannning Windows +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); #+end_src We get the frequency vector that will be the same for all the frequency domain analysis. #+begin_src matlab % Only used to have the frequency vector "f" -[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts); +[~, f] = tfestimate(leg_noise{1}.u, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts); i_lf = f <= 350; i_hf = f > 350; #+end_src *** FRF Identification - Encoder -In this section, the dynamics from $V_a$ to $d_e$ (encoder) is identified. +In this section, the dynamics from $u$ to $d_e$ (encoder) is identified. -The coherence is computed and shown in Figure ref:fig:struts_frf_dvf_plant_coh for all the measured struts. -#+begin_src matlab -%% Coherence computation -coh_enc = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - [coh_lf, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.de, win, Noverlap, Nfft, 1/Ts); - [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, Noverlap, Nfft, 1/Ts); - coh_enc(:, i) = [coh_lf(i_lf); coh_hf(i_hf)]; -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -for i = 1:length(leg_nums) - plot(f, coh_enc(:, i)); -end; -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/struts_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:struts_frf_dvf_plant_coh -#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ -#+RESULTS: -[[file:figs/struts_frf_dvf_plant_coh.png]] - - -Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoder $d_e$ is computed: +Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the encoder $d_e$ is computed: #+begin_src matlab %% Transfer function estimation -enc_frf = zeros(length(f), length(leg_nums)); +enc_frf = zeros(length(f), length(strut_nums)); -for i = 1:length(leg_nums) - [frf_lf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.de, win, Noverlap, Nfft, 1/Ts); - [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, Noverlap, Nfft, 1/Ts); +for i = 1:length(strut_nums) + [frf_lf, ~] = tfestimate(leg_noise{i}.u, detrend(leg_noise{i}.de, 0), win, Noverlap, Nfft, 1/Ts); + [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, detrend(leg_noise_hf{i}.de, 0), win, Noverlap, Nfft, 1/Ts); enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)]; end #+end_src The obtained transfer functions are shown in Figure ref:fig:struts_frf_dvf_plant_tf. #+begin_src matlab :exports none -%% Bode plot of the FRF from Va to de +%% Bode plot of the FRF from u to de figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, abs(enc_frf(:, i)), ... - 'DisplayName', sprintf('Leg %i', leg_nums(i))); + 'DisplayName', sprintf('Leg %i', strut_nums(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ylim([1e-8, 1e-3]); ax2 = nexttile; hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, 180/pi*angle(enc_frf(:, i))); end hold off; @@ -1321,7 +1000,7 @@ exportFig('figs/struts_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall') #+end_src #+name: fig:struts_frf_dvf_plant_tf -#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the encoder $d_e$) #+RESULTS: [[file:figs/struts_frf_dvf_plant_tf.png]] @@ -1334,74 +1013,41 @@ All of this will be explained in Section ref:sec:simscape_bench_struts thanks to #+end_important *** FRF Identification - Interferometer -In this section, the dynamics from $V_a$ to $d_a$ (interferometer) is identified. +In this section, the dynamics from $u$ to $d_a$ (interferometer) is identified. -The coherence is computed and shown in Figure ref:fig:struts_frf_int_plant_coh. -#+begin_src matlab -%% Coherence computation -coh_int = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - [coh_lf, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.da, win, Noverlap, Nfft, 1/Ts); - [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, Noverlap, Nfft, 1/Ts); - coh_int(:, i) = [coh_lf(i_lf); coh_hf(i_hf)]; -end -#+end_src - -#+begin_src matlab :exports none -%% Plot coherence -figure; -hold on; -for i = 1:length(leg_nums) - plot(f, coh_int(:, i)); -end; -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/struts_frf_int_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:struts_frf_int_plant_coh -#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ -#+RESULTS: -[[file:figs/struts_frf_int_plant_coh.png]] - -Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the Attocube is computed for all the struts and shown in Figure ref:fig:struts_frf_int_plant_tf. +Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the Attocube is computed for all the struts and shown in Figure ref:fig:struts_frf_int_plant_tf. All the struts are giving very similar FRF. #+begin_src matlab %% Transfer function estimation -int_frf = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - [frf_lf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.da, win, Noverlap, Nfft, 1/Ts); - [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, Noverlap, Nfft, 1/Ts); +int_frf = zeros(length(f), length(strut_nums)); +for i = 1:length(strut_nums) + [frf_lf, ~] = tfestimate(leg_noise{i}.u, leg_noise{i}.da, win, Noverlap, Nfft, 1/Ts); + [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, leg_noise_hf{i}.da, win, Noverlap, Nfft, 1/Ts); int_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)]; end #+end_src #+begin_src matlab :exports none -%% Plot the FRF from Va to de (interferometer) +%% Plot the FRF from u to de (interferometer) figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, abs(int_frf(:, i)), ... - 'DisplayName', sprintf('Leg %i', leg_nums(i))); + 'DisplayName', sprintf('Leg %i', strut_nums(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ylim([1e-9, 1e-3]); ax2 = nexttile; hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, 180/pi*angle(int_frf(:, i))); end hold off; @@ -1419,79 +1065,45 @@ exportFig('figs/struts_frf_int_plant_tf.pdf', 'width', 'wide', 'height', 'tall') #+end_src #+name: fig:struts_frf_int_plant_tf -#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the encoder $d_e$) #+RESULTS: [[file:figs/struts_frf_int_plant_tf.png]] *** FRF Identification - Force Sensor -In this section, the dynamics from $V_a$ to $V_s$ is identified. - -First the coherence is computed and shown in Figure ref:fig:struts_frf_iff_plant_coh. -#+begin_src matlab -%% Coherence -coh_iff = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - [coh_lf, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); - [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, Noverlap, Nfft, 1/Ts); - coh_iff(:, i) = [coh_lf(i_lf); coh_hf(i_hf)]; -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -for i = 1:length(leg_nums) - plot(f, coh_iff(:, i)); -end; -hold off; -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/struts_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:struts_frf_iff_plant_coh -#+caption: Obtained coherence for the IFF plant -#+RESULTS: -[[file:figs/struts_frf_iff_plant_coh.png]] - +In this section, the dynamics from $u$ to $V_s$ is identified. Then the FRF are estimated and shown in Figure ref:fig:struts_frf_iff_plant_tf. They are also shown all to be very similar. #+begin_src matlab -%% FRF estimation of the transfer function from Va to Vs -iff_frf = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - [frf_lf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); - [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, Noverlap, Nfft, 1/Ts); +%% FRF estimation of the transfer function from u to Vs +iff_frf = zeros(length(f), length(strut_nums)); +for i = 1:length(strut_nums) + [frf_lf, ~] = tfestimate(leg_noise{i}.u, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); + [frf_hf, ~] = tfestimate(leg_noise_hf{i}.u, leg_noise_hf{i}.Vs, win, Noverlap, Nfft, 1/Ts); iff_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)]; end #+end_src #+begin_src matlab :exports none -%% Plot the FRF from Va to Vs +%% Plot the FRF from u to Vs figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, abs(iff_frf(:, i)), ... - 'DisplayName', sprintf('Leg %i', leg_nums(i))); + 'DisplayName', sprintf('Leg %i', strut_nums(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-2, 1e2]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, 180/pi*angle(iff_frf(:, i))); end hold off; @@ -1598,22 +1210,22 @@ The misalignment of the APA and flexible joints is quite large and variable from *** Conclusion #+begin_important -All the struts are giving very consistent behavior from the excitation voltage $V_a$ to the force sensor generated voltage $V_s$ and to the interferometer measured displacement $d_a$. -However, the dynamics from $V_a$ to the encoder measurement $d_e$ is much more complex and variable from one strut to the other most likely due to poor alignment of the APA with respect to the flexible joints. +All the struts are giving very consistent behavior from the excitation voltage $u$ to the force sensor generated voltage $V_s$ and to the interferometer measured displacement $d_a$. +However, the dynamics from $u$ to the encoder measurement $d_e$ is much more complex and variable from one strut to the other most likely due to poor alignment of the APA with respect to the flexible joints. #+end_important The measured FRF are now saved for further use. #+begin_src matlab :tangle no :exports none -save('matlab/mat/meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums', 'apa_align'); +save('./matlab/mat/meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align'); #+end_src #+begin_src matlab :eval no %% Save the estimated FRF for further analysis -save('mat/meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums', 'apa_align'); +save('./mat/meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align'); #+end_src -** Comparison of all the (re-aligned) Struts +** TODO Comparison of all the (re-aligned) Struts <> *** Introduction :ignore: The struts are re-aligned and measured using the same test bench. @@ -1703,105 +1315,72 @@ Both the encoder and the force sensor voltage are measured. Here are the leg numbers that have been measured. #+begin_src matlab %% Numnbers of the measured legs -leg_nums = [1 2 3 4 5 6]; +strut_nums = [1 2 3 4 5 6]; #+end_src #+begin_src matlab %% First identification (low frequency noise) leg_noise = {}; -for i = 1:length(leg_nums) - leg_noise(i) = {load(sprintf('frf_struts_align_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de')}; +for i = 1:length(strut_nums) + leg_noise(i) = {load(sprintf('frf_struts_align_%i_noise.mat', strut_nums(i)), 'u', 'Vs', 'de')}; end #+end_src -The time is the same for all measurements. #+begin_src matlab -%% Time vector -t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s] - -%% Sampling -Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] -Fs = 1/Ts; % Sampling Frequency [Hz] -#+end_src - -Then we defined a "Hanning" windows that will be used for the spectral analysis: -#+begin_src matlab -win = hanning(ceil(0.5*Fs)); % Hannning Windows +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); #+end_src We get the frequency vector that will be the same for all the frequency domain analysis. #+begin_src matlab % Only used to have the frequency vector "f" -[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts); +[~, f] = tfestimate(leg_noise{1}.u, leg_noise{1}.de, win, Noverlap, Nfft, 1/Ts); #+end_src *** FRF Identification - Encoder -In this section, the dynamics from $V_a$ to $d_e$ (encoder) is identified. +In this section, the dynamics from $u$ to $d_e$ (encoder) is identified. -The coherence is computed and shown in Figure ref:fig:struts_align_frf_dvf_plant_coh for all the measured struts. -#+begin_src matlab -%% Coherence computation -coh_enc = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - coh_enc(:, i) = mscohere(leg_noise{i}.Va, leg_noise{i}.de, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -for i = 1:length(leg_nums) - plot(f, coh_enc(:, i)); -end; -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/struts_align_frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:struts_align_frf_dvf_plant_coh -#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ -#+RESULTS: -[[file:figs/struts_align_frf_dvf_plant_coh.png]] - - -Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoder $d_e$ is computed: +Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the encoder $d_e$ is computed: #+begin_src matlab %% Transfer function estimation -enc_frf = zeros(length(f), length(leg_nums)); +enc_frf = zeros(length(f), length(strut_nums)); -for i = 1:length(leg_nums) - enc_frf(:, i) = tfestimate(leg_noise{i}.Va, leg_noise{i}.de, win, Noverlap, Nfft, 1/Ts); +for i = 1:length(strut_nums) + enc_frf(:, i) = tfestimate(leg_noise{i}.u, leg_noise{i}.de, win, Noverlap, Nfft, 1/Ts); +end + +%% Transfer function estimation +iff_frf = zeros(length(f), length(strut_nums)); + +for i = 1:length(strut_nums) + iff_frf(:, i) = tfestimate(leg_noise{i}.u, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); end #+end_src The obtained transfer functions are shown in Figure ref:fig:struts_align_frf_dvf_plant_tf. #+begin_src matlab :exports none -%% Bode plot of the FRF from Va to de +%% Bode plot of the FRF from u to de figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, abs(enc_frf(:, i)), ... - 'DisplayName', sprintf('Leg %i', leg_nums(i))); + 'DisplayName', sprintf('Leg %i', strut_nums(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ylim([1e-8, 1e-3]); ax2 = nexttile; hold on; -for i = 1:length(leg_nums) +for i = 1:length(strut_nums) plot(f, 180/pi*angle(enc_frf(:, i))); end hold off; @@ -1819,100 +1398,14 @@ exportFig('figs/struts_align_frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', ' #+end_src #+name: fig:struts_align_frf_dvf_plant_tf -#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+caption: Estimated FRF for the DVF plant (transfer function from $u$ to the encoder $d_e$) #+RESULTS: [[file:figs/struts_align_frf_dvf_plant_tf.png]] #+begin_important -Even though the struts are much better aligned, we still observe high variability between the struts for the transfer function from $V_a$ to $d_e$. +Even though the struts are much better aligned, we still observe high variability between the struts for the transfer function from $u$ to $d_e$. #+end_important -*** FRF Identification - Force Sensor -In this section, the dynamics from $V_a$ to $V_s$ is identified. - -First the coherence is computed and shown in Figure ref:fig:struts_frf_iff_plant_coh. -#+begin_src matlab -%% Coherence -coh_iff = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - coh_iff(:, i) = mscohere(leg_noise{i}.Va, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the coherence -figure; -hold on; -for i = 1:length(leg_nums) - plot(f, coh_iff(:, i)); -end; -hold off; -xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlim([10, 2e3]); ylim([0, 1]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/struts_align_frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:struts_align_frf_iff_plant_coh -#+caption: Obtained coherence for the IFF plant -#+RESULTS: -[[file:figs/struts_align_frf_iff_plant_coh.png]] - -Then the FRF are estimated and shown in Figure ref:fig:struts_frf_iff_plant_tf. -They are also shown all to be very similar. -#+begin_src matlab -%% FRF estimation of the transfer function from Va to Vs -iff_frf = zeros(length(f), length(leg_nums)); -for i = 1:length(leg_nums) - iff_frf(:, i) = tfestimate(leg_noise{i}.Va, leg_noise{i}.Vs, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the FRF from Va to Vs -figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:length(leg_nums) - plot(f, abs(iff_frf(:, i)), ... - 'DisplayName', sprintf('Leg %i', leg_nums(i))); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-2, 1e2]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i = 1:length(leg_nums) - plot(f, 180/pi*angle(iff_frf(:, i))); -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]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/struts_align_frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:struts_align_frf_iff_plant_tf -#+caption:Identified IFF Plant -#+RESULTS: -[[file:figs/struts_align_frf_iff_plant_tf.png]] - *** Conclusion #+begin_important @@ -1921,15 +1414,6 @@ Having the struts well aligned does not change significantly the obtained dynami The measured FRF are now saved for further use. -#+begin_src matlab :tangle no :exports none -save('matlab/mat/meas_aligned_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'leg_nums', 'strut_align'); -#+end_src - -#+begin_src matlab :eval no -%% Save the estimated FRF for further analysis -save('mat/meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'leg_nums', 'strut_align'); -#+end_src - ** TODO Noise measurement :noexport: #+begin_src matlab %% Nothing connected to the actuator stacks @@ -1946,24 +1430,17 @@ short_circuit = load('frf_struts_align_3_huddle_amp_short_circuit.mat', 't', 'Vs #+end_src #+begin_src matlab -%% Time vector -t = open_circuit.t - open_circuit.t(1) ; % Time vector [s] - -%% Sampling -Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] -Fs = 1/Ts; % Sampling Frequency [Hz] -#+end_src - -Then we defined a "Hanning" windows that will be used for the spectral analysis: -#+begin_src matlab -win = hanning(ceil(1*Fs)); % Hannning Windows +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(2/Ts); +win = hanning(Nfft); +Noverlap = floor(Nfft/2); #+end_src #+begin_src matlab -[pxx_oc, f] = pwelch(open_circuit.Vs, win, 0, [], Fs); -[pxx_mv, ~] = pwelch(mid_voltage.Vs, win, 0, [], Fs); -[pxx_zv, ~] = pwelch(zero_voltage.Vs, win, 0, [], Fs); -[pxx_sc, ~] = pwelch(short_circuit.Vs, win, 0, [], Fs); +[pxx_oc, f] = pwelch(detrend(open_circuit.Vs, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_mv, ~] = pwelch(detrend(mid_voltage.Vs, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_zv, ~] = pwelch(detrend(zero_voltage.Vs, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_sc, ~] = pwelch(detrend(short_circuit.Vs, 0), win, Noverlap, Nfft, 1/Ts); #+end_src #+begin_src matlab @@ -1981,10 +1458,10 @@ xlim([1, 5e3]); #+end_src #+begin_src matlab -[pxx_oc, f] = pwelch(open_circuit.de, win, 0, [], Fs); -[pxx_mv, ~] = pwelch(mid_voltage.de, win, 0, [], Fs); -[pxx_zv, ~] = pwelch(zero_voltage.de, win, 0, [], Fs); -[pxx_sc, ~] = pwelch(short_circuit.de, win, 0, [], Fs); +[pxx_oc, f] = pwelch(detrend(open_circuit.de, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_mv, ~] = pwelch(detrend(mid_voltage.de, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_zv, ~] = pwelch(detrend(zero_voltage.de, 0), win, Noverlap, Nfft, 1/Ts); +[pxx_sc, ~] = pwelch(detrend(short_circuit.de, 0), win, Noverlap, Nfft, 1/Ts); #+end_src #+begin_src matlab @@ -1998,6 +1475,7 @@ hold off; xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); legend('location', 'northeast'); +xlim([1, 5e3]) #+end_src * Simscape Model @@ -2020,13 +1498,13 @@ This Simscape model is used to: - tune the model of the struts (APA, flexible joints, encoder) This study is structured as follow: -- Section ref:sec:struts_comp_2dof: the measured FRF are compared with the 2DoF APA model. -- Section ref:sec:struts_effect_misalignment: the flexible APA model is used, and the effect of a misalignment of the APA and flexible joints is studied. - It is found that the misalignment has a large impact on the dynamics from $V_a$ to $d_e$. -- Section ref:sec:struts_effect_joint_stiffness: the effect of the flexible joint's stiffness on the dynamics is studied. +- Section ref:ssec:test_struts_comp_model: the measured FRF are compared with the Simscape model. +- Section ref:ssec:test_struts_effect_misalignment: the flexible APA model is used, and the effect of a misalignment of the APA and flexible joints is studied. + It is found that the misalignment has a large impact on the dynamics from $u$ to $d_e$. +- Section ref:ssec:test_struts_effect_joint_stiffness: the effect of the flexible joint's stiffness on the dynamics is studied. It is found that the axial stiffness of the joints has a large impact on the location of the zeros on the transfer function from $V_s$ to $d_e$. -** Matlab Init :noexport:ignore: +** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src @@ -2035,68 +1513,58 @@ This study is structured as follow: <> #+end_src -#+begin_src matlab :tangle no -%% Add useful folders to the path -addpath('matlab/'); -addpath('matlab/test_struts/'); -addpath('matlab/mat/'); -addpath('matlab/src/'); -addpath('matlab/png/'); +#+begin_src matlab :tangle no :noweb yes +<> #+end_src -#+begin_src matlab :eval no -%% Add useful folders to the path -addpath('test_struts/'); -addpath('png/'); -addpath('mat/'); -addpath('src/'); +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no :noweb yes +<> +#+end_src + +#+begin_src matlab :eval no :noweb yes +<> +#+end_src + +#+begin_src matlab :noweb yes +<> +#+end_src + +#+begin_src matlab :exports none +%% 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 #+end_src #+begin_src matlab -%% Frequency vector used for many plots -freqs = 2*logspace(0, 3, 1000); +freqs = logspace(1, 3, 1000); #+end_src -#+begin_src matlab -%% Options for Linearized -options = linearizeOptions; -options.SampleTime = 0; - -%% Name of the Simulink File -mdl = 'test_struts'; - -%% Open the Simulink File -open(mdl) -#+end_src - -** Comparison with the 2-DoF Model -<> -*** First Identification +** Comparison with the Model +<> +*** 2Dof model The strut is initialized with default parameters (optimized parameters identified from previous experiments). - #+begin_src matlab %% 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'); -#+end_src -The inputs and outputs of the model are defined. -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator 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 +c_granite = 0; % Do not take into account damping added by the air bearing #+end_src The dynamics is identified and shown in Figure ref:fig:strut_bench_model_bode. #+begin_src matlab %% Run the linearization -Gs = linearize(mdl, io, 0.0, options); -Gs.InputName = {'Va'}; +Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); +Gs.InputName = {'u'}; Gs.OutputName = {'Vs', 'de', 'da'}; #+end_src @@ -2107,24 +1575,24 @@ tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder') -plot(freqs, abs(squeeze(freqresp(Gs('da', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer') +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 $d/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +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', 'Va'), freqs, 'Hz'))), 'k-') +plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), 'k-') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +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', 'Va'), freqs, 'Hz')))) -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'Va'), freqs, 'Hz')))) +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; @@ -2133,7 +1601,7 @@ ylim([-180, 180]) ax2b = nexttile; hold on; -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), 'k-') +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; @@ -2149,25 +1617,20 @@ exportFig('figs/strut_bench_model_bode.pdf', 'width', 'full', 'height', 'tall'); #+end_src #+name: fig:strut_bench_model_bode -#+caption: Identified transfer function from $V_a$ to $V_s$ and from $V_a$ to $d_e,d_a$ using the simple 2DoF model for the APA +#+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]] -*** Comparison with the experimental Data The experimentally measured FRF are loaded. #+begin_src matlab %% Load measured FRF +load('meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums', 'strut_align'); #+end_src -#+begin_src matlab -%% Add time delay to the Simscape model -Gs = exp(-s*Ts)*Gs; -#+end_src - -The FRF from $V_a$ to $d_a$ as well as from $V_a$ to $V_s$ are shown in Figure ref:fig:comp_strut_plant_after_opt and compared with the model. +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. #+begin_src matlab :exports none -%% Compare the FRF and identified dynamics from Va to Vs and da +%% Compare the FRF and identified dynamics from u to Vs and da figure; tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); @@ -2175,16 +1638,16 @@ 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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-', ... +plot(freqs, abs(squeeze(freqresp(Gs('da', 'u'), freqs, 'Hz'))), '-', ... 'DisplayName', 'Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'northeast'); @@ -2193,27 +1656,27 @@ 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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-', ... +plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '-', ... 'DisplayName', 'Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-') +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]'); @@ -2222,11 +1685,11 @@ yticks(-360:90:360); ylim([-180, 180]); ax2b = nexttile; hold on; -for i = 1:length(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-') +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]'); @@ -2246,37 +1709,37 @@ exportFig('figs/comp_strut_plant_after_opt.pdf', 'width', 'full', 'height', 'tal #+RESULTS: [[file:figs/comp_strut_plant_after_opt.png]] -The measured FRF from $V_a$ to $d_e$ (encoder) is compared with the model in Figure ref:fig:comp_strut_plant_iff_after_opt. +The measured FRF from $u$ to $d_e$ (encoder) is compared with the model in Figure ref:fig:comp_strut_plant_iff_after_opt. #+begin_src matlab :exports none -%% Compare the FRF and identified dynamics from Va to de +%% Compare the FRF and identified dynamics from u to de figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-', ... +plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-', ... 'DisplayName', 'Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-') +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]'); @@ -2303,9 +1766,7 @@ This is due to the fact that resonances greatly affecting the encoder reading ar In the next section, flexible model of the APA will be used to model such resonances. #+end_important -** Comparison with the Flexible Model -<> -*** First Identification +*** Comparison with the Flexible Model The strut is initialized with default parameters (optimized parameters identified from previous experiments). #+begin_src matlab @@ -2314,42 +1775,24 @@ n_hexapod = struct(); n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof'); n_hexapod.actuator = initializeAPA('type', 'flexible'); -#+end_src -The inputs and outputs of the model are defined. -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator 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 +c_granite = 100; % Do not take into account damping added by the air bearing #+end_src The dynamics is identified and shown in Figure ref:fig:strut_bench_model_bode. #+begin_src matlab %% Run the linearization -Gs = linearize(mdl, io, 0.0, options); -Gs.InputName = {'Va'}; +Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); +Gs.InputName = {'u'}; Gs.OutputName = {'Vs', 'de', 'da'}; #+end_src -*** Comparison with the experimental Data -The experimentally measured FRF are loaded. -#+begin_src matlab -%% Load measured FRF -load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums'); -#+end_src +- [ ] Add encoder plot -#+begin_src matlab -%% Add time delay to the Simscape model -Gs = exp(-s*Ts)*Gs; -#+end_src - -The FRF from $V_a$ to $d_a$ as well as from $V_a$ to $V_s$ are shown in Figure ref:fig:comp_strut_plant_after_opt and compared with the model. +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. #+begin_src matlab :exports none -%% Compare the FRF and identified dynamics from Va to Vs and da +%% Compare the FRF and identified dynamics from u to Vs and da figure; tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); @@ -2359,16 +1802,16 @@ 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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '--', ... +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 $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); @@ -2377,16 +1820,16 @@ ax1b = nexttile([2,1]); hold on; plot(f, abs(iff_frf(:, i)), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'Meas. FRF'); -for i = 1:length(leg_nums) +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', 'Va'), freqs, 'Hz'))), '--', ... +plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'u'), freqs, 'Hz'))), '--', ... 'DisplayName', 'Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-2, 1e2]); legend('location', 'southeast'); @@ -2395,11 +1838,11 @@ ax2 = nexttile; hold on; plot(f, 180/pi*angle(enc_frf(:, 1)), 'color', [colors(2,:), 0.5], ... 'HandleVisibility', 'off'); -for i = 1:length(leg_nums) - plot(f, 180/pi*(angle(int_frf(:, i)) - angle(squeeze(freqresp(exp(-s*2*Ts), f, 'Hz')))), 'color', [0,0,0, 0.2]); +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', 'Va'), freqs, 'Hz'))), '--') +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]'); @@ -2408,11 +1851,11 @@ yticks(-360:90:360); ylim([-180, 180]); ax2b = nexttile; hold on; -for i = 1:length(leg_nums) +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', 'Va'), freqs, 'Hz'))), '--') +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]'); @@ -2433,7 +1876,7 @@ exportFig('figs/strut_meas_frf_model_int_force.pdf', 'width', 1500, 'height', 't [[file:figs/strut_meas_frf_model_int_force.png]] ** Effect of a misalignment of the APA and flexible joints on the transfer function from actuator to encoder -<> +<> *** Introduction :ignore: As shown in Figure ref:fig:struts_frf_dvf_plant_tf, the dynamics from actuator to encoder for all the struts is very different. @@ -2442,24 +1885,18 @@ This could be explained by a large variability in the alignment of the flexible Depending on the alignment, the spurious resonances of the struts (Figure ref:fig:test_struts_mode_shapes) can be excited differently. -For instance, consider Figure ref:fig:strut_misalign_schematic where there is a misalignment in the $y$ direction. +For instance, consider Figure ref:fig:test_struts_misalign_schematic where there is a misalignment in the $y$ direction. In such case, the mode at 200Hz is foreseen to be more excited as the misalignment $d_y$ increases and therefore the dynamics from the actuator to the encoder should also change around 200Hz. -#+name: fig:strut_misalign_schematic +#+name: fig:test_struts_misalign_schematic #+caption: Mis-alignement between the joints and the APA #+attr_latex: :width 0.8\linewidth -[[file:figs/strut_misalign_schematic.png]] +[[file:figs/test_struts_misalign_schematic.png]] If the misalignment is in the $x$ direction, the mode at 285Hz should be more affected whereas a misalignment in the $z$ direction should not affect these resonances. Such statement is studied in this section. -But first, the measured FRF of the struts are loaded. -#+begin_src matlab -%% Load measured FRF of the struts -load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums'); -#+end_src - *** Perfectly aligned APA Let's first consider that the strut is perfectly mounted such that the two flexible joints and the APA are aligned. #+begin_src matlab @@ -2472,20 +1909,13 @@ 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 -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator 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; % Encoder -#+end_src The transfer function is identified and shown in Figure ref:fig:comp_enc_frf_align_perfect. #+begin_src matlab %% Identification -Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options); -Gs.InputName = {'Va'}; -Gs.OutputName = {'de', 'da'}; +Gs = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); +Gs.InputName = {'u'}; +Gs.OutputName = {'Vs', 'de', 'da'}; #+end_src From Figure ref:fig:comp_enc_frf_align_perfect, it is clear that: @@ -2498,33 +1928,33 @@ From Figure ref:fig:comp_enc_frf_align_perfect, it is clear that: freqs = 2*logspace(0, 3, 1000); figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-', ... +plot(freqs, abs(squeeze(freqresp(Gs('de', 'u'), freqs, 'Hz'))), '-', ... 'DisplayName', 'Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +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(leg_nums) +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', 'Va'), freqs, 'Hz'))), '-') +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]'); @@ -2563,11 +1993,11 @@ dy_aligns = [-0.5, -0.1, 0, 0.1, 0.5]*1e-3; % [m] Gs_align = {zeros(length(dy_aligns), 1)}; for i = 1:length(dy_aligns) - n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [0; dy_aligns(i); 0]); + 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*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de'}; + G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); + G.InputName = {'u'}; + G.OutputName = {'Vs', 'de', 'da'}; Gs_align(i) = {G}; end @@ -2580,17 +2010,17 @@ The obtained dynamics are shown in Figure ref:fig:effect_misalignment_y. freqs = 2*logspace(0, 3, 1000); figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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', 'Va'), freqs, 'Hz'))), ... + 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 $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'northeast'); @@ -2598,7 +2028,7 @@ legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:length(dy_aligns) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'Va'), freqs, 'Hz')))); + 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'); @@ -2619,7 +2049,6 @@ exportFig('figs/effect_misalignment_y.pdf', 'width', 'wide', 'height', 'tall'); #+RESULTS: [[file:figs/effect_misalignment_y.png]] - #+begin_important The alignment of the APA with the flexible joints as a *huge* influence on the dynamics from actuator voltage to measured displacement by the encoder. The misalignment in the $y$ direction mostly influences: @@ -2642,11 +2071,11 @@ dx_aligns = [-0.1, -0.05, 0, 0.05, 0.1]*1e-3; % [m] Gs_align = {zeros(length(dx_aligns), 1)}; for i = 1:length(dx_aligns) - n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [dx_aligns(i); 0; 0]); + 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*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de'}; + G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); + G.InputName = {'u'}; + G.OutputName = {'Vs', 'de', 'da'}; Gs_align(i) = {G}; end @@ -2658,17 +2087,17 @@ The obtained dynamics are shown in Figure ref:fig:effect_misalignment_x. freqs = 2*logspace(0, 3, 1000); figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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', 'Va'), freqs, 'Hz'))), ... + 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 $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'northeast'); @@ -2676,7 +2105,7 @@ legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:length(dx_aligns) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'Va'), freqs, 'Hz')))); + 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'); @@ -2701,6 +2130,85 @@ exportFig('figs/effect_misalignment_x.pdf', 'width', 'wide', 'height', 'tall'); The misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz. #+end_important +*** Comparison with identified misalignment +#+begin_src matlab +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]]; +#+end_src + +#+begin_src matlab +%% 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 +#+end_src + +#+begin_src matlab :exports none +%% 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]); +#+end_src + *** 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. @@ -2722,9 +2230,9 @@ 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*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de', 'da'}; + G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); + G.InputName = {'u'}; + G.OutputName = {'Vs', 'de', 'da'}; Gs_align(i) = {G}; end @@ -2740,7 +2248,7 @@ 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', 'Va'), freqs, 'Hz')))); +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]'); @@ -2748,15 +2256,15 @@ 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', 'Va'), freqs, 'Hz')))); +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); +ax3 = nexttile(); hold on; plot(f, abs(enc_frf(:, 3)), 'DisplayName', 'Meas.'); -plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'Va'), freqs, 'Hz'))), ... +plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'u'), freqs, 'Hz'))), ... 'DisplayName', 'Model'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); @@ -2766,7 +2274,7 @@ 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', 'Va'), freqs, 'Hz')))); +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'); @@ -2774,7 +2282,7 @@ 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', 'Va'), freqs, 'Hz')))); +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',[]); @@ -2809,7 +2317,7 @@ freqs = 2*logspace(0, 3, 1000); colors = colororder; figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(enc_frf(:,1)), 'color', [colors(1,:),0.2], ... @@ -2825,13 +2333,13 @@ 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', 'Va'), freqs, 'Hz'))), '--', 'color', colors(1,:), ... +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', 'Va'), freqs, 'Hz'))), '--', 'color', colors(1,:), ... + 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', 'Va'), freqs, 'Hz'))), '--', 'color', colors(2,:), ... +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'); @@ -2843,10 +2351,10 @@ 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*Ts), f, 'Hz')))), 'color', [colors(2,:),0.2]); - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_align{i}('de', 'Va'), freqs, 'Hz'))), '--', 'color', colors(1,:)); + 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', 'Va'), freqs, 'Hz'))), '--', 'color', colors(2,:)); +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]'); @@ -2867,51 +2375,8 @@ exportFig('figs/frf_struts_enc_int.pdf', 'width', 'normal', 'height', 'full'); [[file:figs/frf_struts_enc_int.png]] -#+begin_src matlab :exports none -%% Comparison of the plants (encoder output) when tuning the misalignment -freqs = 2*logspace(0, 3, 1000); - -colors = colororder; - -figure; -tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); -ax1 = nexttile(); -hold on; -plot(f, abs(enc_frf(:, 5))); -plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))), 'k-'); -plot(freqs, abs(squeeze(freqresp(Gs_align{5}('de', 'Va'), freqs, 'Hz'))), 'color', colors(2,:)); -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(:, 3)), 'DisplayName', 'Measured FRF'); -plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz'))), 'k-', ... - 'DisplayName', 'Perfectly Aligned'); -plot(freqs, abs(squeeze(freqresp(Gs_align{3}('de', 'Va'), freqs, 'Hz'))), 'color', colors(2,:), ... - 'DisplayName', 'Tuned Model'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); -legend('location', 'southwest', 'FontSize', 8); - -linkaxes([ax1,ax2],'xy'); -xlim([20, 2e3]); ylim([1e-8, 1e-3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/frf_model_encoder_strut.pdf', 'width', 'normal', 'height', 'full'); -#+end_src - -#+name: fig:frf_model_encoder_strut -#+caption: -#+RESULTS: -[[file:figs/frf_model_encoder_strut.png]] - - ** Effect of flexible joint's characteristics -<> +<> *** Introduction :ignore: As the struts are composed of one APA and two flexible joints, it is obvious that the flexible joint characteristics will change the dynamic behavior of the struts. @@ -2921,13 +2386,7 @@ Using the Simscape model, the effect of the flexible joint's characteristics on - Section ref:sec:struts_effect_axial_stiff_joints: the effects of a change of axial stiffness is studied - Section ref:sec:struts_effect_bending_damping_joints: the effects of a change of bending damping is studied -The studied dynamics is between $V_a$ and the encoder displacement $d_e$. -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage -io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder -#+end_src +The studied dynamics is between $u$ and the encoder displacement $d_e$. *** Effect of bending stiffness of the flexible joints <> @@ -2935,7 +2394,7 @@ io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder Let's initialize an APA which is a little bit misaligned. #+begin_src matlab %% APA Initialization -n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [0.1e-3; 0.5e-3; 0]); +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]); #+end_src The bending stiffnesses for which the dynamics is identified are defined below. @@ -2959,9 +2418,9 @@ for i = 1:length(kRs) 'kRx', kRs(i), ... 'kRy', kRs(i)); - G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de'}; + G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); + G.InputName = {'u'}; + G.OutputName = {'Vs', 'de', 'da'}; Gs(i) = {G}; end @@ -2973,17 +2432,17 @@ The obtained dynamics from DAC voltage to encoder measurements are compared in F freqs = 2*logspace(1, 3, 1000); figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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', 'Va'), freqs, 'Hz'))), ... + 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 $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'northeast'); @@ -2991,7 +2450,7 @@ legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:length(kRs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'Va'), freqs, 'Hz')))); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -3013,7 +2472,7 @@ exportFig('figs/effect_enc_bending_stiff.pdf', 'width', 'wide', 'height', 'tall' [[file:figs/effect_enc_bending_stiff.png]] #+begin_important -The bending stiffness of the joints has little impact on the transfer function from $V_a$ to $d_e$. +The bending stiffness of the joints has little impact on the transfer function from $u$ to $d_e$. #+end_important *** Effect of axial stiffness of the flexible joints @@ -3038,9 +2497,9 @@ for i = 1:length(kzs) 'type', '4dof', ... 'kz', kzs(i)); - G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de'}; + G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); + G.InputName = {'u'}; + G.OutputName = {'Vs', 'de', 'da'}; Gs(i) = {G}; end @@ -3052,17 +2511,17 @@ The obtained dynamics from DAC voltage to encoder measurements are compared in F freqs = 2*logspace(1, 3, 1000); figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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', 'Va'), freqs, 'Hz'))), ... + 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 $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'northeast'); @@ -3070,7 +2529,7 @@ legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:length(kzs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'Va'), freqs, 'Hz')))); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -3125,9 +2584,9 @@ for i = 1:length(cRs) 'cRx', cRs(i), ... 'cRy', cRs(i)); - G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de'}; + G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts); + G.InputName = {'u'}; + G.OutputName = {'Vs', 'de', 'da'}; Gs(i) = {G}; end @@ -3139,17 +2598,17 @@ The results are shown in Figure ref:fig:effect_enc_bending_damp. freqs = 2*logspace(1, 3, 1000); figure; -tiledlayout(3, 1, 'TileSpace', 'Compact', 'Padding', 'None'); +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', 'Va'), freqs, 'Hz'))), ... + 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 $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); @@ -3157,7 +2616,7 @@ legend('location', 'southwest'); ax2 = nexttile; hold on; for i = 1:length(cRs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'Va'), freqs, 'Hz')))); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('de', 'u'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -3189,92 +2648,6 @@ One idea would be to introduce a sheet of damping material inside the flexible j Not sure is would be effect though. #+end_question -** TODO Comparison with identified misalignment -#+begin_src matlab -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]]; -#+end_src - -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage -io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder -#+end_src - -#+begin_src matlab -%% 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*Ts)*linearize(mdl, io, 0.0, options); - G.InputName = {'Va'}; - G.OutputName = {'de'}; - - Gs_align(i) = {G}; -end -#+end_src - -#+begin_src matlab :exports none -%% 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', 'Va'), 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', 'Va'), 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', 'Va'), 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', 'Va'), 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', 'Va'), 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]); -#+end_src - * Conclusion <> @@ -3298,6 +2671,33 @@ addpath('./mat/'); % Path for data addpath('./src/'); % Path for functions #+END_SRC +** Initialize Simscape +#+NAME: m-init-path-simscape +#+BEGIN_SRC matlab +addpath('./matlab/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 +#+END_SRC + +#+NAME: m-init-path-simscape-tangle +#+BEGIN_SRC matlab +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 +#+END_SRC + ** Initialize other elements #+NAME: m-init-other #+BEGIN_SRC matlab @@ -3487,7 +2887,7 @@ flex_top.cz = args.cz; ** =initializeAPA= - Initialize APA :PROPERTIES: -:header-args:matlab: :tangle matlab/src/initializeAPA.m +:header-args:matlab+: :tangle matlab/src/initializeAPA.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: <> @@ -3524,19 +2924,18 @@ arguments args.Gs (1,1) double {mustBeNumeric} = 0 % For 2DoF - args.k (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.38e6 - args.ke (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.75e6 - args.ka (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e7 + args.k (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*380000 + args.ke (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*4952605 + args.ka (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2476302 - args.c (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e1 - args.ce (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1 - args.ca (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1 + args.c (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*20 + args.ce (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*200 + args.ca (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100 args.Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % Force Flexible APA args.xi (1,1) double {mustBeNumeric, mustBePositive} = 0.01 - args.d_align (3,1) double {mustBeNumeric} = zeros(3,1) % [m] args.d_align_bot (3,1) double {mustBeNumeric} = zeros(3,1) % [m] args.d_align_top (3,1) double {mustBeNumeric} = zeros(3,1) % [m] @@ -3580,11 +2979,11 @@ end if args.Ga == 0 switch args.type case '2dof' - actuator.Ga = -30.0; + actuator.Ga = -2.5796; case 'flexible frame' actuator.Ga = 1; % TODO case 'flexible' - actuator.Ga = 23.4; + actuator.Ga = 23.2; end else actuator.Ga = args.Ga; % Actuator gain [N/V] @@ -3595,11 +2994,11 @@ end if args.Gs == 0 switch args.type case '2dof' - actuator.Gs = 0.098; + actuator.Gs = 466664; case 'flexible frame' actuator.Gs = 1; % TODO case 'flexible' - actuator.Gs = -4674824; + actuator.Gs = -4898341; end else actuator.Gs = args.Gs; % Sensor gain [V/m] @@ -3638,7 +3037,6 @@ switch args.type actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m] - actuator.d_align = args.d_align; actuator.d_align_bot = args.d_align_bot; actuator.d_align_top = args.d_align_top; end @@ -3648,4 +3046,3 @@ actuator.xi = args.xi; % Damping ratio actuator.ks = args.ks; % Stiffness of one stack [N/m] actuator.cs = args.cs; % Damping of one stack [N/m] #+end_src -