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Thomas Dehaeze 2025-02-27 10:39:25 +01:00
parent b7f92acc79
commit 9b6dc8f9bf
16 changed files with 1339 additions and 1046 deletions
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figs/detail_fem_apa95ml_compliance.pdf
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matlab/detail_fem_1_flexible_body.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('./src/'); % Path for scripts
addpath('./mat/'); % Path for data
addpath('./STEPS/'); % Path for Simscape Model
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Linearization options
opts = linearizeOptions;
opts.SampleTime = 0;
%% Open Simscape Model
mdl = 'detail_fem_super_element'; % Name of the Simulink File
open(mdl); % Open Simscape Model
%% Colors for the figures
colors = colororder;
freqs = logspace(1,3,500); % Frequency vector [Hz]
%% Estimate "Sensor Constant" - (THP5H)
d33 = 680e-12; % Strain constant [m/V]
n = 160; % Number of layers per stack
eT = 4500*8.854e-12; % Permittivity under constant stress [F/m]
sD = 21e-12; % Compliance under constant electric displacement [m2/N]
gs = d33/(eT*sD*n); % Sensor Constant [V/m]
%% Estimate "Actuator Constant" - (THP5H)
d33 = 680e-12; % Strain constant [m/V]
n = 320; % Number of layers
cE = 1/sD; % Youngs modulus [N/m^2]
A = (10e-3)^2; % Area of the stacks [m^2]
L = 40e-3; % Length of the two stacks [m]
ka = cE*A/L; % Stiffness of the two stacks [N/m]
ga = d33*n*ka; % Actuator Constant [N/V]
%% Load reduced order model
K = readmatrix('APA95ML_K.CSV'); % order: 48
M = readmatrix('APA95ML_M.CSV');
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
%% Stiffness estimation
m = 0.0001; % block-free condition, no payload
k_support = 1e9;
c_support = 1e3;
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
% The inverse of the DC gain of the transfer function
% from vertical force to vertical displacement is the axial stiffness of the APA
k_est = 1/dcgain(G); % [N/m]
%% Estimated compliance of the APA95ML
freqs = logspace(2, log10(5000), 1000);
% Get first resonance indice
i_max = find(abs(squeeze(freqresp(G, freqs(2:end), 'Hz'))) - abs(squeeze(freqresp(G, freqs(1:end-1), 'Hz'))) < 0, 1);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'DisplayName', 'Compliance');
plot([freqs(1), freqs(end)], [1/k_est, 1/k_est], 'k--', 'DisplayName', sprintf('$1/k$ ($k = %.0f N/\\mu m$)', 1e-6*k_est))
xline(freqs(i_max), '--', 'linewidth', 1, 'color', [0,0,0], 'DisplayName', sprintf('$f_0 = %.0f$ Hz', freqs(i_max)))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
xlim([100, 5000]);
%% Estimation of the amplification factor and Stroke
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
% Estimated amplification factor
ampl_factor = abs(dcgain(G(1,1))./dcgain(G(2,1)));
% Estimated stroke
apa_stroke = ampl_factor * 3 * 20e-6; % [m]
%% Experimental plant identification
% with PD200 amplifier (gain of 20) - 2 stacks as an actuator, 1 as a sensor
load('apa95ml_5kg_2a_1s.mat')
Va = 20*u; % Voltage amplifier gain: 20
% Spectral Analysis parameters
Ts = t(end)/(length(t)-1);
Nfft = floor(1/Ts);
win = hanning(Nfft);
Noverlap = floor(Nfft/2);
% Identification of the transfer function from Va to di
[G_y, f] = tfestimate(detrend(Va, 0), detrend(y, 0), win, Noverlap, Nfft, 1/Ts);
[G_Vs, ~] = tfestimate(detrend(Va, 0), detrend(v, 0), win, Noverlap, Nfft, 1/Ts);
%% Plant Identification from Multi-Body model
% Load Reduced Order Matrices
K = readmatrix('APA95ML_K.CSV'); % order: 48
M = readmatrix('APA95ML_M.CSV');
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA95ML_out_nodes_3D.txt');
m = 5.5; % Mass of the suspended granite [kg]
k_support = 4e7;
c_support = 3e2;
% Compute transfer functions
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Voltage accros piezo stacks [V]
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor stack voltage [V]
Gm = linearize(mdl, io);
Gm.InputName = {'Va'};
Gm.OutputName = {'y', 'Vs'};
%% Comparison of the identified transfer function from Va to di to the multi-body model
freqs = logspace(1, 3, 500);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_y), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5, 'DisplayName', 'Measured FRF');
plot(freqs, abs(squeeze(freqresp(Gm('y', 'Va'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', 'Model')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $y/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-5]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(G_y), '-', 'color' , [colors(2,:), 0.5], 'linewidth', 2.5);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm('y', 'Va'), freqs, 'Hz'))), '--', 'color', colors(2,:))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360);
ylim([-45, 180]);
linkaxes([ax1,ax2],'x');
xlim([10, 1e3]);
%% Comparison of the identified transfer function from Va to Vs to the multi-body model
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_Vs), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5, 'DisplayName', 'Measured FRF');
plot(freqs, abs(squeeze(freqresp(Gm('Vs', 'Va'), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', 'Model')
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-3, 1e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(G_Vs), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gm('Vs', 'Va'), freqs, 'Hz'))), '--', 'color', colors(1,:))
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, 1e3]);
%% Integral Force Feedback Controller
K_iff = (1/(s + 2*2*pi))*(s/(s + 0.5*2*pi));
K_iff.inputname = {'Vs'};
K_iff.outputname = {'u_iff'};
% New damped plant input
S1 = sumblk("u = u_iff + u_damp");
% Voltage amplifier with gain of 20
voltage_amplifier = tf(20);
voltage_amplifier.inputname = {'u'};
voltage_amplifier.outputname = {'Va'};
%% Load experimental data with IFF implemented with different gains
load('apa95ml_iff_test.mat', 'results');
% Tested gains
g_iff = [0, 10, 50, 100, 500, 1000];
% Spectral Analysis parameters
Ts = t(end)/(length(t)-1);
Nfft = floor(1/Ts);
win = hanning(Nfft);
Noverlap = floor(Nfft/2);
%% Computed the identified FRF of the damped plants
tf_iff = {zeros(1, length(g_iff))};
for i=1:length(g_iff)
[tf_est, f] = tfestimate(results{i}.u, results{i}.y, win, Noverlap, Nfft, 1/Ts);
tf_iff(i) = {tf_est};
end
%% Estimate the damped plants from the multi-body model
Gm_iff = {zeros(1, length(g_iff))};
for i=1:length(g_iff)
K_iff_g = -K_iff*g_iff(i); K_iff_g.inputname = {'Vs'}; K_iff_g.outputname = {'u_iff'};
Gm_iff(i) = {connect(Gm, K_iff_g, S1, voltage_amplifier, {'u_damp'}, {'y'})};
end
%% Identify second order plants from the experimental data
% This is mandatory to estimate the experimental "poles"
% an place them in the root-locus plot
G_id = {zeros(1,length(results))};
f_start = 70; % [Hz]
f_end = 500; % [Hz]
for i = 1:length(results)
tf_id = tf_iff{i}(sum(f<f_start):length(f)-sum(f>f_end));
f_id = f(sum(f<f_start):length(f)-sum(f>f_end));
gfr = idfrd(tf_id, 2*pi*f_id, Ts);
G_id(i) = {procest(gfr,'P2UDZ')};
end
%% Comparison of the Root-Locus computed from the multi-body model and the identified closed-loop poles
gains = logspace(0, 5, 1000);
figure;
hold on;
plot(real( pole(Gm('Vs', 'Va'))), imag( pole(Gm('Vs', 'Va'))), 'kx', 'HandleVisibility', 'off');
plot(real(tzero(Gm('Vs', 'Va'))), imag(tzero(Gm('Vs', 'Va'))), 'ko', 'HandleVisibility', 'off');
for i = 1:length(gains)
cl_poles = pole(feedback(Gm('Vs', 'Va'), gains(i)*K_iff));
plot(real(cl_poles(imag(cl_poles)>100)), imag(cl_poles(imag(cl_poles)>100)), 'k.', 'HandleVisibility', 'off');
end
for i = 1:length(g_iff)
cl_poles = pole(Gm_iff{i});
plot(real(cl_poles(imag(cl_poles)>100)), imag(cl_poles(imag(cl_poles)>100)), '.', 'MarkerSize', 20, 'color', colors(i,:), 'HandleVisibility', 'off');
plot(real(pole(G_id{i})), imag(pole(G_id{i})), 'x', 'color', colors(i,:), 'DisplayName', sprintf('g = %0.f', g_iff(i)), 'DisplayName', sprintf('$g = %.0f$', g_iff(i)));
end
xlabel('Real Part');
ylabel('Imaginary Part');
axis equal;
ylim([-100, 2100]);
xlim([-2100,100]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
%% Experimental damped plant for several IFF gains and estimated damped plants from the model
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
plot(f, abs(tf_iff{1}), '-', 'DisplayName', '$g = 0$', 'color', [0,0,0, 0.5], 'linewidth', 2.5)
plot(f, abs(squeeze(freqresp(Gm_iff{1}, f, 'Hz'))), 'k--', 'HandleVisibility', 'off')
for i = 2:length(results)
plot(f, abs(tf_iff{i}), '-', 'DisplayName', sprintf('g = %0.f', g_iff(i)), 'color', [colors(i-1,:), 0.5], 'linewidth', 2.5)
end
for i = 2:length(results)
plot(f, abs(squeeze(freqresp(Gm_iff{i}, f, 'Hz'))), '--', 'color', colors(i-1,:), 'HandleVisibility', 'off')
end
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude $y/V_a$ [m/N]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-6, 2e-4]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(tf_iff{1}./squeeze(freqresp(exp(-s*2e-4), f, 'Hz'))), '-', 'color', [0,0,0, 0.5], 'linewidth', 2.5)
plot(f, 180/pi*angle(squeeze(freqresp(Gm_iff{1}, f, 'Hz'))), 'k--')
for i = 2:length(results)
plot(f, 180/pi*angle(tf_iff{i}./squeeze(freqresp(exp(-s*2e-4), f, 'Hz'))), '-', 'color', [colors(i-1,:), 0.5], 'linewidth', 2.5)
plot(f, 180/pi*angle(squeeze(freqresp(Gm_iff{i}, f, 'Hz'))), '--', 'color', colors(i-1,:))
end
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
hold off;
yticks(-360:45:360);
ylim([-10, 190]);
linkaxes([ax1,ax2], 'x');
xlim([150, 500]);

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matlab/detail_fem_2_actuators.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('./src/'); % Path for scripts
addpath('./mat/'); % Path for data
addpath('./STEPS/'); % Path for Simscape Model
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Linearization options
opts = linearizeOptions;
opts.SampleTime = 0;
%% Open Simscape Model
mdl = 'detail_fem_APA300ML'; % Name of the Simulink File
open(mdl); % Open Simscape Model
% Piezoelectric parameters
ga = -25.9; % [N/V]
gs = -5.08e6; % [V/m]
%% Colors for the figures
colors = colororder;
freqs = logspace(1,3,500); % Frequency vector [Hz]
%% Identify dynamics with "Reduced Order Flexible Body"
K = readmatrix('APA300ML_mat_K.CSV');
M = readmatrix('APA300ML_mat_M.CSV');
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA300ML_out_nodes_3D.txt');
m = 5; % [kg]
ga = 25.9; % [N/V]
gs = 5.08e6; % [V/m]
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
G_fem = linearize(mdl, io);
G_fem_z = G_fem('z','Va');
G_fem_Vs = G_fem('Vs', 'Va');
G_fem_comp = G_fem('z', 'Fd');
%% Determine c1 and k1 from the zero
G_zeros = zero(minreal(G_fem_Vs));
G_zeros = G_zeros(imag(G_zeros)>0);
[~, i_sort] = sort(imag(G_zeros));
G_zeros = G_zeros(i_sort);
G_zero = G_zeros(1);
% Solving 2nd order equations
c1 = -2*m*real(G_zero);
k1 = m*(imag(G_zero)^2 + real(G_zero)^2);
%% Determine ka, ke, ca, ce from the first pole
G_poles = pole(minreal(G_fem_z));
G_poles = G_poles(imag(G_poles)>0);
[~, i_sort] = sort(imag(G_poles));
G_poles = G_poles(i_sort);
G_pole = G_poles(1);
% Solving 2nd order equations
ce = 3*(-2*m*real(G_pole(1)) - c1);
ca = 1/2*ce;
ke = 3*(m*(imag(G_pole)^2 + real(G_pole)^2) - k1);
ka = 1/2*ke;
%% Matching sensor/actuator constants
% ga = dcgain(G_fem_z) / (1/(ka + k1*ke/(k1 + ke)));
clear io; io_i = 1;
io(io_i) = linio([mdl, '_2dof', '/Fa'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '_2dof', '/z'], 1, 'openoutput'); io_i = io_i + 1;
ga = dcgain(G_fem_z)/dcgain(linearize([mdl, '_2dof'], io));
clear io; io_i = 1;
io(io_i) = linio([mdl, '_2dof', '/Va'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '_2dof', '/dL'], 1, 'openoutput'); io_i = io_i + 1;
gs = dcgain(G_fem_Vs)/dcgain(linearize([mdl, '_2dof'], io));
%% Identify dynamics with tuned 2DoF model
clear io; io_i = 1;
io(io_i) = linio([mdl, '_2dof', '/Va'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '_2dof', '/Fd'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '_2dof', '/z'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '_2dof', '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
G_2dof = linearize([mdl, '_2dof'], io);
G_2dof_z = G_2dof('z','Va');
G_2dof_Vs = G_2dof('Vs', 'Va');
G_2dof_comp = G_2dof('z', 'Fd');
%% Comparison of the transfer functions from Va to vertical motion - FEM vs 2DoF
freqs = logspace(1, 3, 500);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_fem_z, freqs, 'Hz'))), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5, 'DisplayName', 'FEM')
plot(freqs, abs(squeeze(freqresp(G_2dof_z, freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $y/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 2e-4]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_fem_z, freqs, 'Hz')))), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_2dof_z, freqs, 'Hz')))), '--', 'color', colors(2,:))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360); ylim([-20, 200]);
linkaxes([ax1,ax2],'x');
xlim([10, 1e3]);
%% Comparison of the transfer functions from Va to Vs - FEM vs 2DoF
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_fem_Vs, freqs, 'Hz'))), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5, 'DisplayName', 'FEM');
plot(freqs, abs(squeeze(freqresp(G_2dof_Vs, freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '2DoF Model')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([6e-4, 3e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_fem_Vs, freqs, 'Hz')))), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_2dof_Vs, freqs, 'Hz')))), '--', 'color', colors(1,:))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:45:360); ylim([-20, 200]);
linkaxes([ax1,ax2],'x');
xlim([10, 1e3]);
%% Effect of electrical boundaries on the
oc = load('detail_fem_apa95ml_open_circuit.mat', 't', 'encoder', 'u');
sc = load('detail_fem_apa95ml_short_circuit.mat', 't', 'encoder', 'u');
% Spectral Analysis parameters
Ts = sc.t(end)/(length(sc.t)-1);
Nfft = floor(2/Ts);
win = hanning(Nfft);
Noverlap = floor(Nfft/2);
% Identification of the transfer function from Va to di
[G_oc, f] = tfestimate(detrend(oc.u, 0), detrend(oc.encoder, 0), win, Noverlap, Nfft, 1/Ts);
[G_sc, f] = tfestimate(detrend(sc.u, 0), detrend(sc.encoder, 0), win, Noverlap, Nfft, 1/Ts);
% Find resonance frequencies
[~, i_oc] = max(abs(G_oc(f<300)));
[~, i_sc] = max(abs(G_sc(f<300)));
%% Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_oc), '-', 'DisplayName', sprintf('Open-Circuit - $f_0 = %.1f Hz$', f(i_oc)))
plot(f, abs(G_sc), '-', 'DisplayName', sprintf('Short-Circuit - $f_0 = %.1f Hz$', f(i_sc)))
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-6, 1e-4]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(G_oc), '-')
plot(f, 180/pi*angle(G_sc), '-')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
ylabel('Phase'); xlabel('Frequency [Hz]');
hold off;
yticks(-360:45:360);
ylim([-20, 200]);
axis padded 'auto x'
linkaxes([ax1,ax2], 'x');
xlim([100, 300]);
%% Compare Dynamics between "Reduced Order" flexible joints and "2-dof and 3-dof" joints
% Let's initialize all the stages with default parameters.
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeSample('m', 50);
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
mdl = 'detail_fem_nass';
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Errors in the frame of the struts
io(io_i) = linio([mdl, '/NASS'], 3, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors
% Flexible actuators
initializeSimplifiedNanoHexapod('actuator_type', 'flexible', ...
'flex_type_F', '2dof', ...
'flex_type_M', '3dof');
G_flex = linearize(mdl, io);
G_flex.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_flex.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
% Actuators modeled as 2DoF system
initializeSimplifiedNanoHexapod('actuator_type', 'apa300ml', ...
'flex_type_F', '2dof', ...
'flex_type_M', '3dof');
G_ideal = linearize(mdl, io);
G_ideal.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_ideal.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
%% Comparison of the dynamics for actuators modeled using "reduced order flexible body" and using 2DoF system - HAC plant
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_flex("l"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_ideal("l"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_flex("l1","f1"), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'FEM');
plot(freqs, abs(squeeze(freqresp(G_ideal("l1","f1"), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '2-DoF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-10, 1e-4]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex("l1","f1"), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ideal("l1","f1"), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');
%% Comparison of the dynamics for actuators modeled using "reduced order flexible body" and using 2DoF system - IFF plant
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_flex("fm"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_ideal("fm"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_flex("fm1","f1"), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'FEM');
plot(freqs, abs(squeeze(freqresp(G_ideal("fm1","f1"), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '2-DoF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-5, 1e1]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex("fm1","f1"), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ideal("fm1","f1"), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');

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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('./src/'); % Path for scripts
addpath('./mat/'); % Path for data
addpath('./STEPS/'); % Path for Simscape Model
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Linearization options
opts = linearizeOptions;
opts.SampleTime = 0;
%% Open Simscape Model
mdl = 'detail_fem_nass'; % Name of the Simulink File
open(mdl); % Open Simscape Model
%% Colors for the figures
colors = colororder;
freqs = logspace(1,3,500); % Frequency vector [Hz]
%% Identify the dynamics for several considered bending stiffnesses
% Let's initialize all the stages with default parameters.
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeSample('m', 50);
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Errors in the frame of the struts
io(io_i) = linio([mdl, '/NASS'], 3, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors
% Effect of bending stiffness
Kf = [0, 50, 100, 500]; % [Nm/rad]
G_Kf = {zeros(length(Kf), 1)};
for i = 1:length(Kf)
% Limited joint axial compliance
initializeSimplifiedNanoHexapod('actuator_type', '1dof', ...
'flex_type_F', '2dof', ...
'flex_type_M', '3dof', ...
'actuator_k', 1e6, ...
'actuator_c', 1e1, ...
'actuator_kp', 0, ...
'actuator_cp', 0, ...
'Fsm', 56e-3, ... % APA300ML weight 112g
'Msm', 56e-3, ...
'Kf_F', Kf(i), ...
'Kf_M', Kf(i));
G_Kf(i) = {linearize(mdl, io)};
G_Kf{i}.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_Kf{i}.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
end
freqs = logspace(0, 3, 1000);
%% Effect of the flexible joint bending stiffness on the HAC-plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(Kf)
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_Kf{i}("l"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(i,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_Kf{i}("l1","f1"), freqs, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$k_f = %.0f $ [Nm/rad]', Kf(i)));
for j = 2:6
plot(freqs, abs(squeeze(freqresp(G_Kf{i}("l"+j,"f"+j), freqs, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-10, 1e-4]);
ax2 = nexttile;
hold on;
for i = 1:length(Kf)
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_Kf{i}(1, 1), freqs, 'Hz')))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');
%% Effect of the flexible joint bending stiffness on the IFF plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(Kf)
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_Kf{i}("fm"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(i,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_Kf{i}("fm1","f1"), freqs, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$k_f = %.0f $ [Nm/rad]', Kf(i)));
for j = 2:6
plot(freqs, abs(squeeze(freqresp(G_Kf{i}("fm"+j,"f"+j), freqs, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-4, 1e2]);
ax2 = nexttile();
hold on;
for i = 1:length(Kf)
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_Kf{i}("fm1", "f1"), freqs, 'Hz')))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');
%% Decentalized IFF
Kiff = -200 * ... % Gain
1/s * ... % LPF: provides integral action
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
%% Root Locus for decentralized IFF - 1dof actuator - Effect of joint bending stiffness
gains = logspace(-1, 2, 400);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
for i = 1:length(Kf)
plot(real(pole(G_Kf{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), imag(pole(G_Kf{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), 'x', 'color', colors(i,:), ...
'DisplayName', sprintf('$k_f = %.0f$ Nm/rad', Kf(i)));
plot(real(tzero(G_Kf{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), imag(tzero(G_Kf{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), 'o', 'color', colors(i,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_Kf{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}), g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(i,:), ...
'HandleVisibility', 'off');
end
end
xline(0, 'HandleVisibility', 'off'); yline(0, 'HandleVisibility', 'off');
hold off;
axis equal;
xlim(1.1*[-900, 100]); ylim(1.1*[-100, 900]);
xticks(1.1*[-900:100:0]);
yticks(1.1*[0:100:900]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
%% Identify the dynamics for several considered bending stiffnesses - APA300ML
G_Kf_apa300ml = {zeros(length(Kf), 1)};
for i = 1:length(Kf)
% Limited joint axial compliance
initializeSimplifiedNanoHexapod('actuator_type', 'apa300ml', ...
'flex_type_F', '2dof', ...
'flex_type_M', '3dof', ...
'Fsm', 56e-3, ... % APA300ML weight 112g
'Msm', 56e-3, ...
'Kf_F', Kf(i), ...
'Kf_M', Kf(i));
G_Kf_apa300ml(i) = {linearize(mdl, io)};
G_Kf_apa300ml{i}.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_Kf_apa300ml{i}.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
end
Kiff = -1000 * ... % Gain
1/(s) * ... % LPF: provides integral action
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
%% Root Locus for decentralized IFF - APA300ML actuator - Effect of joint bending stiffness
gains = logspace(-1, 2, 300);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
for i = 1:length(Kf)
plot(real(pole(G_Kf_apa300ml{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), imag(pole(G_Kf_apa300ml{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), 'x', 'color', colors(i,:), ...
'DisplayName', sprintf('$k_f = %.0f$ [Nm/rad]', Kf(i)));
plot(real(tzero(G_Kf_apa300ml{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), imag(tzero(G_Kf_apa300ml{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), 'o', 'color', colors(i,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_Kf_apa300ml{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}), g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(i,:), ...
'HandleVisibility', 'off');
end
end
xline(0, 'HandleVisibility', 'off'); yline(0, 'HandleVisibility', 'off');
hold off;
axis equal;
xlim(1.4*[-900, 100]); ylim(1.4*[-100, 900]);
xticks(1.4*[-900:100:0]);
yticks(1.4*[0:100:900]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
%% Identify the dynamics for several considered axial stiffnesses
% Let's initialize all the stages with default parameters.
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeSample('m', 50);
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Errors in the frame of the struts
io(io_i) = linio([mdl, '/NASS'], 3, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors
% Effect of bending stiffness
Ka = 1e6*[1000, 100, 10, 1]; % [Nm/rad]
G_Ka = {zeros(length(Ka), 1)};
for i = 1:length(Ka)
% Limited joint axial compliance
initializeSimplifiedNanoHexapod('actuator_type', '1dof', ...
'flex_type_F', '2dof_axial', ...
'flex_type_M', '4dof', ...
'actuator_k', 1e6, ...
'actuator_c', 1e1, ...
'actuator_kp', 0, ...
'actuator_cp', 0, ...
'Fsm', 56e-3, ... % APA300ML weight 112g
'Msm', 56e-3, ...
'Ca_F', 1, ...
'Ca_M', 1, ...
'Ka_F', Ka(i), ...
'Ka_M', Ka(i));
G_Ka(i) = {linearize(mdl, io)};
G_Ka{i}.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_Ka{i}.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
end
freqs = logspace(1, 4, 1000);
%% Effect of the flexible joint axial stiffness on the HAC-plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(Ka)
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_Ka{i}("l"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(i,:), 0.1], ...
'HandleVisibility', 'off');
end
end
end
for i = 1:length(Ka)
plot(freqs, abs(squeeze(freqresp(G_Ka{i}("l1","f1"), freqs, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$k_a = %.0f$ [N/$\\mu$m]', 1e-6*Ka(i)));
% for j = 2:6
% plot(freqs, abs(squeeze(freqresp(G_Ka{i}("l"+j,"f"+j), freqs, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
% end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-10, 1e-4]);
ax2 = nexttile;
hold on;
for i = 1:length(Ka)
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_Ka{i}(1, 1), freqs, 'Hz')))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');
%% Effect of the flexible joint axial stiffness on the IFF plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:length(Ka)
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_Ka{i}("fm"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(i,:), 0.1], ...
'HandleVisibility', 'off');
end
end
end
for i = 1:length(Ka)
plot(freqs, abs(squeeze(freqresp(G_Ka{i}("fm1","f1"), freqs, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$k_a = %.0f$ [N/$\\mu$m]', 1e-6*Ka(i)));
% for j = 2:6
% plot(freqs, abs(squeeze(freqresp(G_Ka{i}("fm"+j,"f"+j), freqs, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
% end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-4, 1e2]);
ax2 = nexttile();
hold on;
for i = 1:length(Ka)
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_Ka{i}("fm1", "f1"), freqs, 'Hz')))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');
%% Decentalized IFF
Kiff = -200 * ... % Gain
1/s * ... % LPF: provides integral action
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
%% Root Locus for decentralized IFF - 1dof actuator - Effect of joint bending stiffness
gains = logspace(-1, 2, 400);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
for i = 1:length(Ka)
plot(real(pole(G_Ka{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), imag(pole(G_Ka{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), 'x', 'color', colors(i,:), ...
'DisplayName', sprintf('$k_a = %.0f$ N/$\\mu$m', 1e-6*Ka(i)));
plot(real(tzero(G_Ka{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), imag(tzero(G_Ka{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}))), 'o', 'color', colors(i,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_Ka{i}({"fm1", "fm2", "fm3", "fm4", "fm5", "fm6"}, {"f1", "f2", "f3", "f4", "f5", "f6"}), g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(i,:), ...
'HandleVisibility', 'off');
end
end
xline(0, 'HandleVisibility', 'off'); yline(0, 'HandleVisibility', 'off');
hold off;
axis equal;
xlim(1.1*[-900, 100]); ylim(1.1*[-100, 900]);
xticks(1.1*[-900:100:0]);
yticks(1.1*[0:100:900]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
%% Compute the damped plants
Kiff = -500 * ... % Gain
1/(s + 2*pi*0.1) * ... % LPF: provides integral action
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'u1iff', 'u2iff', 'u3iff', 'u4iff', 'u5iff', 'u6iff'};
% New damped plant input
S1 = sumblk("f1 = u1iff + u1");
S2 = sumblk("f2 = u2iff + u2");
S3 = sumblk("f3 = u3iff + u3");
S4 = sumblk("f4 = u4iff + u4");
S5 = sumblk("f5 = u5iff + u5");
S6 = sumblk("f6 = u6iff + u6");
G_Ka_iff = {zeros(1,length(Ka))};
for i=1:length(Ka)
G_Ka_iff(i) = {connect(G_Ka{i}, Kiff, S1, S2, S3, S4, S5, S6, {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'}, {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'})};
end
%% Interaction Analysis - RGA Number
rga = zeros(length(Ka), length(freqs));
for i = 1:length(Ka)
for j = 1:length(freqs)
rga(i,j) = sum(sum(abs(inv(evalfr(G_Ka_iff{i}({"l1", "l2", "l3", "l4", "l5", "l6"}, {"u1", "u2", "u3", "u4", "u5", "u6"}), 1j*2*pi*freqs(j)).').*evalfr(G_Ka_iff{i}({"l1", "l2", "l3", "l4", "l5", "l6"}, {"u1", "u2", "u3", "u4", "u5", "u6"}), 1j*2*pi*freqs(j)) - eye(6))));
end
end
%% RGA number for the damped plants - Effect of the flexible joint axial stiffness
figure;
hold on;
for i = 1:length(Ka)
plot(freqs, rga(i,:), 'DisplayName', sprintf('$k_a = %.0f$ N/$\\mu$m', 1e-6*Ka(i)))
end
hold off;
xlabel('Frequency [Hz]'); ylabel('RGA number');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylim([0, 10]); xlim([10, 5e3]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
%% Extract stiffness of the joint from the reduced order model
% We first extract the stiffness and mass matrices.
K = readmatrix('flex025_mat_K.CSV');
M = readmatrix('flex025_mat_M.CSV');
% Then, we extract the coordinates of the interface nodes.
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('flex025_out_nodes_3D.txt');
m = 1;
%% Name of the Simulink File
mdl = 'detail_fem_joint';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/T'], 1, 'openinput'); io_i = io_i + 1; % Forces and Torques
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Translations and Rotations
G = linearize(mdl, io);
% Stiffness extracted from the Simscape model
k_a = 1/dcgain(G(3,3)); % Axial stiffness [N/m]
k_f = 1/dcgain(G(4,4)); % Bending stiffness [N/m]
k_t = 1/dcgain(G(6,6)); % Torsion stiffness [N/m]
% Stiffness extracted from the Stiffness matrix
k_s = K(1,1); % shear [N/m]
% k_s = K(2,2); % shear [N/m]
k_a = K(3,3); % axial [N/m]
k_f = K(4,4); % bending [Nm/rad]
% k_f = K(5,5); % bending [Nm/rad]
k_t = K(6,6); % torsion [Nm/rad]
%% Compare Dynamics between "Reduced Order" flexible joints and "2-dof and 3-dof" joints
% Let's initialize all the stages with default parameters.
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeSample('m', 50);
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
mdl = 'detail_fem_nass';
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Errors in the frame of the struts
io(io_i) = linio([mdl, '/NASS'], 3, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors
% Fully flexible joints
initializeSimplifiedNanoHexapod('actuator_type', 'apa300ml', ...
'flex_type_F', 'flexible', ...
'flex_type_M', 'flexible', ...
'Fsm', 56e-3, ... % APA300ML weight 112g
'Msm', 56e-3);
G_flex = linearize(mdl, io);
G_flex.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_flex.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
% Flexible joints modelled by 2DoF and 3DoF joints
initializeSimplifiedNanoHexapod('actuator_type', 'apa300ml', ...
'flex_type_F', '2dof_axial', ...
'flex_type_M', '4dof', ...
'Kf_F', k_f, ...
'Kt_F', k_t, ...
'Ka_F', k_a, ...
'Kf_M', k_f, ...
'Kt_M', k_t, ...
'Ka_M', k_a, ...
'Cf_F', 1e-2, ...
'Ct_F', 1e-2, ...
'Ca_F', 1e-2, ...
'Cf_M', 1e-2, ...
'Ct_M', 1e-2, ...
'Ca_M', 1e-2, ...
'Fsm', 56e-3, ... % APA300ML weight 112g
'Msm', 56e-3);
G_ideal = linearize(mdl, io);
G_ideal.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_ideal.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6', 'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
%% Comparison of the dynamics with joints modelled with FEM and modelled with "ideal joints" - HAC plant
freqs = logspace(1, 4, 1000);
%% Effect of the flexible joint axial stiffness on the HAC-plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_flex("l"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_ideal("l"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_flex("l1","f1"), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'Reduced Order Flexible Joints');
plot(freqs, abs(squeeze(freqresp(G_ideal("l1","f1"), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'Bot: $k_f$, $k_a$, Top: $k_f$, $k_t$, $k_a$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-10, 1e-4]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex("l1","f1"), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ideal("l1","f1"), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');
freqs = logspace(0, 3, 1000);
%% Effect of the flexible joint axial stiffness on the HAC-plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for j = 1:5
for k = j+1:6
plot(freqs, abs(squeeze(freqresp(G_flex("fm"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_ideal("fm"+k,"f"+j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_flex("fm1","f1"), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', 'Reduced Order Flexible Joints');
plot(freqs, abs(squeeze(freqresp(G_ideal("fm1","f1"), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', 'Bot: $k_f$, $k_a$, Top: $k_f$, $k_t$, $k_a$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ylim([1e-5, 1e1]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex("fm1","f1"), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ideal("fm1","f1"), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([-360:45:360]);
linkaxes([ax1,ax2],'x');

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nass-fem.bib
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@ -1,15 +1,15 @@
@article{souleille18_concep_activ_mount_space_applic,
author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and
Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues,
Gon{\c{c}}alo and Collette, Christophe},
title = {A Concept of Active Mount for Space Applications},
journal = {CEAS Space Journal},
volume = 10,
number = 2,
pages = {157--165},
year = 2018,
publisher = {Springer},
keywords = {parallel robot, iff},
@article{mcinroy02_model_desig_flexur_joint_stewar,
author = {J.E. McInroy},
title = {Modeling and Design of Flexure Jointed Stewart Platforms
for Control Purposes},
journal = {IEEE/ASME Transactions on Mechatronics},
volume = 7,
number = 1,
pages = {95-99},
year = 2002,
doi = {10.1109/3516.990892},
url = {https://doi.org/10.1109/3516.990892},
keywords = {parallel robot, flexure},
}
@ -109,6 +109,19 @@
@book{pintelon12_system_ident,
author = {Rik Pintelon and Johan Schoukens},
title = {System Identification : a Frequency Domain Approach},
year = 2012,
publisher = {Wiley IEEE Press},
url = {https://doi.org/10.1002/9781118287422},
address = {Hoboken, N.J. Piscataway, NJ},
doi = {10.1002/9781118287422},
isbn = 9780470640371,
}
@phdthesis{hanieh03_activ_stewar,
author = {Hanieh, Ahmed Abu},
keywords = {parallel robot},
@ -120,18 +133,29 @@
@article{mcinroy02_model_desig_flexur_joint_stewar,
author = {J.E. McInroy},
title = {Modeling and Design of Flexure Jointed Stewart Platforms
for Control Purposes},
journal = {IEEE/ASME Transactions on Mechatronics},
volume = 7,
number = 1,
pages = {95-99},
year = 2002,
doi = {10.1109/3516.990892},
url = {https://doi.org/10.1109/3516.990892},
keywords = {parallel robot, flexure},
@article{souleille18_concep_activ_mount_space_applic,
author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and
Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues,
Gon{\c{c}}alo and Collette, Christophe},
title = {A Concept of Active Mount for Space Applications},
journal = {CEAS Space Journal},
volume = 10,
number = 2,
pages = {157--165},
year = 2018,
publisher = {Springer},
keywords = {parallel robot, iff},
}
@book{schmidt20_desig_high_perfor_mechat_third_revis_edition,
author = {Schmidt, R Munnig and Schitter, Georg and Rankers, Adrian},
title = {The Design of High Performance Mechatronics - Third Revised
Edition},
year = 2020,
publisher = {Ios Press},
keywords = {favorite},
}
@ -187,3 +211,16 @@
keywords = {parallel robot},
}
@book{preumont18_vibrat_contr_activ_struc_fourt_edition,
author = {Andre Preumont},
title = {Vibration Control of Active Structures - Fourth Edition},
year = 2018,
publisher = {Springer International Publishing},
url = {https://doi.org/10.1007/978-3-319-72296-2},
doi = {10.1007/978-3-319-72296-2},
keywords = {favorite, parallel robot},
series = {Solid Mechanics and Its Applications},
}

1023
nass-fem.org
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29
nass-fem.tex
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@ -1,4 +1,4 @@
% Created 2025-02-27 Thu 01:24
% Created 2025-02-27 Thu 10:38
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -38,7 +38,6 @@ Section \ref{sec:detail_fem_joint} addresses the design of flexible joints, wher
In both cases, the hybrid modeling approach enables detailed component optimization while maintaining the ability to predict system-level dynamic behavior, particularly under closed-loop control conditions.
\chapter{Reduced order flexible bodies}
\label{sec:orgfd22661}
\label{sec:detail_fem_super_element}
Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models.
These components are traditionally analyzed using Finite Element Analysis (FEA) software.
@ -50,7 +49,6 @@ First, the fundamental principles and methodological approaches of this modeling
It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section \ref{ssec:detail_fem_super_element_example}).
Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section \ref{ssec:detail_fem_super_element_validation}).
\section{Procedure}
\label{sec:org93ab665}
\label{ssec:detail_fem_super_element_theory}
In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models.
@ -74,7 +72,6 @@ m = 6 \times n + p
\end{equation}
\section{Example with an Amplified Piezoelectric Actuator}
\label{sec:org1e66a5f}
\label{ssec:detail_fem_super_element_example}
The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons.
Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section \ref{sec:detail_fem_actuator}.
@ -105,7 +102,6 @@ Stiffness & \(21\,N/\mu m\)\\
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
\label{sec:orgce5afdb}
The development of the finite element model for the APA95ML necessitated the specification of appropriate material properties, as summarized in Table \ref{tab:detail_fem_material_properties}.
The finite element mesh, shown in Figure \ref{fig:detail_fem_apa95ml_mesh}, was then generated.
@ -142,11 +138,10 @@ The modal reduction procedure was then executed, yielding the reduced mass and s
\end{center}
\subcaption{\label{fig:detail_fem_apa_model_schematic}Inclusion in multi-body model}
\end{subfigure}
\caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_modal_schematic}).}
\caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}).}
\end{figure}
\paragraph{Super Element in the Multi-Body Model}
\label{sec:org809b5e7}
Previously computed reduced order mass and stiffness matrices were imported in a multi-body model block called ``Reduced Order Flexible Solid''.
This block has several interface frames corresponding to the ones defined in the FEA software.
@ -158,7 +153,6 @@ This is illustrated in Figure \ref{fig:detail_fem_apa_model_schematic}.
However, to have access to the physical voltage input of the actuators stacks \(V_a\) and to the generated voltage by the force sensor \(V_s\), conversion between the electrical and mechanical domains need to be determined.
\paragraph{Sensor and Actuator ``constants''}
\label{sec:orgb3a075f}
To link the electrical domain to the mechanical domain, an ``actuator constant'' \(g_a\) and a ``sensor constant'' \(g_s\) were introduced as shown in Figure \ref{fig:detail_fem_apa_model_schematic}.
@ -217,7 +211,6 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\end{table}
\paragraph{Identification of the APA Characteristics}
\label{sec:orge041867}
Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications.
@ -225,7 +218,7 @@ The stiffness of the APA95ML was estimated from the multi-body model by computin
The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML.
A value of \(23\,N/\mu m\) was found which is close to the specified stiffness in the datasheet of \(k = 21\,N/\mu m\).
The multi-body model predicted a resonant frequency under block-free conditions of \(2024\,\text{Hz}\) (Figure \ref{fig:detail_fem_apa95ml_compliance}), which is in agreement with the nominal specification of \(2000\,\text{Hz}\).
The multi-body model predicted a resonant frequency under block-free conditions of \(\approx 2\,\text{kHz}\) (Figure \ref{fig:detail_fem_apa95ml_compliance}), which is in agreement with the nominal specification of \(2\,\text{kHz}\).
\begin{figure}[htbp]
\centering
@ -243,7 +236,6 @@ Through the established amplification factor of 1.5, this translates to a predic
The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model.
\section{Experimental Validation}
\label{sec:org354cea4}
\label{ssec:detail_fem_super_element_validation}
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
The goal is to measure the dynamics of the APA95ML and compared it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
@ -269,7 +261,6 @@ Measurement of the sensor stack voltage \(V_s\) was performed using an analog-to
\caption{\label{fig:detail_fem_apa95ml_bench}Test bench used to validate ``reduced order solid bodies'' using an APA95ML. Picture of the bench is shown in (\subref{fig:detail_fem_apa95ml_bench_picture}). Schematic is shown in (\subref{fig:detail_fem_apa95ml_bench_schematic}).}
\end{figure}
\paragraph{Comparison of the dynamics}
\label{sec:orgc0ac08b}
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
The identification procedure necessitated careful choice of the excitation signal \cite[, chap. 5]{pintelon12_system_ident}.
@ -304,7 +295,6 @@ Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could
\end{figure}
\paragraph{Integral Force Feedback with APA}
\label{sec:org4b65d74}
To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally.
Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains.
@ -338,7 +328,6 @@ The close agreement between experimental measurements and theoretical prediction
\end{figure}
\section*{Conclusion}
\label{sec:org105aef7}
The modeling procedure presented in this section will demonstrate significant utility for the optimization of complex mechanical components within multi-body systems, particularly in the design of actuators (Section \ref{sec:detail_fem_actuator}) and flexible joints (Section \ref{sec:detail_fem_joint}).
Through experimental validation using an Amplified Piezoelectric Actuator, the methodology has been shown to accurately predict both open-loop and closed-loop dynamic behavior, thereby establishing its reliability for component design and system analysis.
@ -348,7 +337,6 @@ This is exemplified by the nano-hexapod configuration, where the implementation
However, the methodology remains valuable for system analysis, as the extraction of frequency domain characteristics can be efficiently performed even with such high-order models.
\chapter{Actuator Selection}
\label{sec:org3ec4809}
\label{sec:detail_fem_actuator}
The selection and modeling of actuators constitutes a critical step in the development of the nano-hexapod.
This chapter presents the approach to actuator selection and modeling.
@ -356,7 +344,6 @@ First, specifications for the nano-hexapod actuators are derived from previous a
Then, the chosen actuator is modeled using the reduced-order flexible body approach developed in the previous section, enabling validation of this selection through detailed dynamical analysis (Section \ref{ssec:detail_fem_actuator_apa300ml}).
Finally, a simplified two-degree-of-freedom model is developed to facilitate time-domain simulations while maintaining accurate representation of the actuator's essential characteristics (Section \ref{ssec:detail_fem_actuator_apa300ml_2dof}).
\section{Choice of the Actuator based on Specifications}
\label{sec:org929a34b}
\label{ssec:detail_fem_actuator_specifications}
The actuator selection process was driven by several critical requirements derived from previous dynamic analyses.
@ -431,7 +418,6 @@ Height \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
\end{table}
\section{APA300ML - Reduced Order Flexible Body}
\label{sec:orgeff9b1b}
\label{ssec:detail_fem_actuator_apa300ml}
The validation of the APA300ML started by incorporating a ``reduced order flexible body'' into the multi-body model as explained in Section \ref{sec:detail_fem_super_element}.
@ -459,7 +445,6 @@ While this high order provides excellent accuracy for validation purposes, it pr
The sensor and actuator ``constants'' (\(g_s\) and \(g_a\)) derived in Section \ref{ssec:detail_fem_super_element_example} for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks.
\section{Simpler 2DoF Model of the APA300ML}
\label{sec:org120c274}
\label{ssec:detail_fem_actuator_apa300ml_2dof}
To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model was developed, adapted from \cite{souleille18_concep_activ_mount_space_applic}.
@ -532,7 +517,6 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\end{figure}
\section{Electrical characteristics of the APA}
\label{sec:orga7af5a1}
\label{ssec:detail_fem_actuator_apa300ml_electrical}
The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain) \cite[, chapter 5.5]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
@ -553,7 +537,6 @@ Proper consideration must be given to voltage amplifier specifications and force
These aspects, being fundamental to system implementation, will be addressed in the instrumentation chapter.
\section{Validation with the Nano-Hexapod}
\label{sec:orgba951c9}
\label{ssec:detail_fem_actuator_apa300ml_validation}
The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation.
@ -586,7 +569,6 @@ These results validate both the selection of the APA300ML and the effectiveness
\end{figure}
\chapter{Flexible Joint Design}
\label{sec:org93c9b2c}
\label{sec:detail_fem_joint}
High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy.
This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation.
@ -622,7 +604,6 @@ The analysis of bending and axial stiffness effects enables the establishment of
These specifications guide the development and optimization of a flexible joint design through finite element analysis (Section \ref{ssec:detail_fem_joint_specs}).
The validation process, detailed in Section \ref{ssec:detail_fem_joint_validation}, begins with the integration of the joints as ``reduced order flexible bodies'' in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics.
\section{Bending and Torsional Stiffness}
\label{sec:org582c93a}
\label{ssec:detail_fem_joint_bending}
The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction.
@ -679,7 +660,6 @@ A parallel analysis of torsional stiffness revealed similar dynamic effects, tho
\end{figure}
\section{Axial Stiffness}
\label{sec:org05206a1}
\label{ssec:detail_fem_joint_axial}
The limited axial stiffness (\(k_a\)) of flexible joints introduces an additional compliance between the actuation point and the measurement point.
@ -735,7 +715,6 @@ Based on this analysis, an axial stiffness specification of \(100\,N/\mu m\) was
\end{figure}
\section{Specifications and Design flexible joints}
\label{sec:org2310ef0}
\label{ssec:detail_fem_joint_specs}
The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics.
@ -784,7 +763,6 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop
\end{figure}
\section{Validation with the Nano-Hexapod}
\label{sec:org4e972fc}
\label{ssec:detail_fem_joint_validation}
The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis.
@ -822,7 +800,6 @@ While additional degrees of freedom could potentially capture more dynamic featu
\end{figure}
\chapter*{Conclusion}
\label{sec:org5382fe7}
\label{sec:detail_fem_conclusion}
In this chapter, the methodology of combining finite element analysis with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design phase of the nano-hexapod.