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Thomas Dehaeze 2024-03-25 14:42:03 +01:00
parent b93a482278
commit 22049068c1
46 changed files with 60594 additions and 1035 deletions
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*** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 ***
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CURRENT DOF SET= UX UY UZ ROTX ROTY ROTZ
*** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 ***
DISTRIBUTED ANSYS Mechanical Enterprise
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Unknown
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matlab/STEPS/vionic_encoder.STEP Normal file
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103
matlab/src/initializeAPA.m Normal file
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function [actuator] = initializeAPA(args)
% initializeAPA -
%
% Syntax: [actuator] = initializeAPA(args)
%
% Inputs:
% - args -
%
% Outputs:
% - actuator -
arguments
args.type char {mustBeMember(args.type,{'2dof', 'flexible frame', 'flexible'})} = '2dof'
% Actuator and Sensor constants
args.Ga (1,1) double {mustBeNumeric} = 0
args.Gs (1,1) double {mustBeNumeric} = 0
% For 2DoF
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)*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_bot (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
args.d_align_top (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
% For Flexible Frame
args.ks (1,1) double {mustBeNumeric, mustBePositive} = 235e6
args.cs (1,1) double {mustBeNumeric, mustBePositive} = 1e1
end
actuator = struct();
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]

45
matlab/src/initializeBotFlexibleJoint.m Normal file
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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;

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matlab/src/initializeTopFlexibleJoint.m Normal file
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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;

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matlab/test_apa_simscape.slx Normal file
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matlab/test_struts_1_flexible_modes.m Normal file
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%% 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');

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matlab/test_struts_2_dynamical_meas.m Normal file
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%% 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])

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

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