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DSYS= 0 + + *** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 *** + DISTRIBUTED ANSYS Mechanical Enterprise + + 00208316 VERSION=WINDOWS x64 11:07:02 MAR 25, 2021 CP= 2.062 + + Unknown + + + + NODE X Y Z THXY THYZ THZX + 840914 0.0000 0.0000 0.28000E-001 0.00 0.00 0.00 + 840915 0.0000 0.0000 -0.28000E-001 0.00 0.00 0.00 + 840916 -0.34000E-001 0.0000 0.0000 0.00 0.00 0.00 + 840917 0.34000E-001 0.0000 0.0000 0.00 0.00 0.00 + + LIST MASTERS ON ALL SELECTED NODES. + CURRENT DOF SET= UX UY UZ ROTX ROTY ROTZ + + *** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 *** + DISTRIBUTED ANSYS Mechanical Enterprise + + 00208316 VERSION=WINDOWS x64 11:07:02 MAR 25, 2021 CP= 2.188 + + Unknown + + + NODE LABEL SUPPORT + 840914 UX + 840914 UY + 840914 UZ + 840914 ROTX + 840914 ROTY + 840914 ROTZ + 840915 UX + 840915 UY + 840915 UZ + 840915 ROTX + 840915 ROTY + 840915 ROTZ + 840916 UX + 840916 UY + 840916 UZ + 840916 ROTX + 840916 ROTY + 840916 ROTZ + 840917 UX + 840917 UY + 840917 UZ + 840917 ROTX + 840917 ROTY + 840917 ROTZ diff --git a/matlab/mat/full_APA300ML_K.CSV b/matlab/mat/full_APA300ML_K.CSV new file mode 100644 index 0000000..5a09907 --- /dev/null +++ b/matlab/mat/full_APA300ML_K.CSV @@ -0,0 +1,36 @@ 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b/matlab/mat/test_nhexa_identification_data_mass_3.mat new file mode 100644 index 0000000..2e5da3a Binary files /dev/null and b/matlab/mat/test_nhexa_identification_data_mass_3.mat differ diff --git a/matlab/mat/test_nhexa_identified_frf_masses.mat b/matlab/mat/test_nhexa_identified_frf_masses.mat new file mode 100644 index 0000000..9079309 Binary files /dev/null and b/matlab/mat/test_nhexa_identified_frf_masses.mat differ diff --git a/matlab/mat/test_nhexa_simscape_flexible_masses.mat b/matlab/mat/test_nhexa_simscape_flexible_masses.mat new file mode 100644 index 0000000..dd0ed32 Binary files /dev/null and b/matlab/mat/test_nhexa_simscape_flexible_masses.mat differ diff --git a/matlab/mat/test_nhexa_simscape_masses.mat b/matlab/mat/test_nhexa_simscape_masses.mat new file mode 100644 index 0000000..5b8ff68 Binary files /dev/null and b/matlab/mat/test_nhexa_simscape_masses.mat differ diff --git a/matlab/src/initializeNanoHexapodFinal.m b/matlab/src/initializeNanoHexapodFinal.m index 9c596f7..6904c2e 100644 --- a/matlab/src/initializeNanoHexapodFinal.m +++ b/matlab/src/initializeNanoHexapodFinal.m @@ -41,9 +41,9 @@ arguments args.actuator_k (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*380000 args.actuator_ke (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*4952605 args.actuator_ka (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2476302 - args.actuator_c (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*20 - args.actuator_ce (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*200 - args.actuator_ca (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100 + args.actuator_c (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5 + args.actuator_ce (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100 + args.actuator_ca (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*50 args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m] % For Flexible Frame args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m] diff --git a/matlab/subsystems/nano_hexapod_left_strut.slx b/matlab/subsystems/nano_hexapod_left_strut.slx index fc133e3..a2e4359 100644 Binary files a/matlab/subsystems/nano_hexapod_left_strut.slx and b/matlab/subsystems/nano_hexapod_left_strut.slx differ diff --git a/matlab/subsystems/nano_hexapod_right_strut.slx b/matlab/subsystems/nano_hexapod_right_strut.slx index 2ff3acb..5440489 100644 Binary files a/matlab/subsystems/nano_hexapod_right_strut.slx and b/matlab/subsystems/nano_hexapod_right_strut.slx differ diff --git a/matlab/subsystems/nano_hexapod_subsystem.slx b/matlab/subsystems/nano_hexapod_subsystem.slx index 17a7650..74edc8c 100644 Binary files a/matlab/subsystems/nano_hexapod_subsystem.slx and b/matlab/subsystems/nano_hexapod_subsystem.slx differ diff --git a/matlab/test_bench_nano_hexapod.slx b/matlab/test_bench_nano_hexapod.slx index 5df33d1..862248f 100644 Binary files a/matlab/test_bench_nano_hexapod.slx and b/matlab/test_bench_nano_hexapod.slx differ diff --git a/matlab/test_nhexa_1_suspended_table.m b/matlab/test_nhexa_1_suspended_table.m new file mode 100644 index 0000000..cf189e0 --- /dev/null +++ b/matlab/test_nhexa_1_suspended_table.m @@ -0,0 +1,67 @@ +% Matlab Init :noexport:ignore: + +%% test_nhexa_table.m + +%% 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 STEPS +addpath('./subsystems/'); % Path for Subsystems Simulink files + +%% Initialize Parameters for Simscape model +table_type = 'Rigid'; % On top of vibration table +device_type = 'None'; % On top of vibration table +payload_num = 0; % No Payload + +% Simulink Model name +mdl = 'test_bench_nano_hexapod'; + +%% Colors for the figures +colors = colororder; + +%% Frequency Vector +freqs = logspace(log10(10), log10(2e3), 1000); + +% Simscape Model of the suspended table +% <> + +% The Simscape model of the suspended table simply consists of two solid bodies connected by 4 springs. +% The 4 springs are here modelled with "bushing joints" that have stiffness and damping properties in x, y and z directions. +% The 3D representation of the model is displayed in Figure ref:fig:test_nhexa_suspended_table_simscape where the 4 "bushing joints" are represented by the blue cylinders. + +% #+name: fig:test_nhexa_suspended_table_simscape +% #+caption: 3D representation of the simscape model +% #+attr_latex: :width 0.8\linewidth +% [[file:figs/test_nhexa_suspended_table_simscape.png]] + +% The model order is 12, and it represents the 6 suspension modes. +% The inertia properties of the parts are set from the geometry and material densities. +% The stiffness of the springs was initially set from the datasheet nominal value of $17.8\,N/mm$ and then reduced down to $14\,N/mm$ to better match the measured suspension modes. +% The stiffness of the springs in the horizontal plane is set at $0.5\,N/mm$. +% The obtained suspension modes of the simscape model are compared with the measured ones in Table ref:tab:test_nhexa_suspended_table_simscape_modes. + + +%% Configure Simscape Model +table_type = 'Suspended'; % On top of vibration table +device_type = 'None'; % No device on the vibration table +payload_num = 0; % No Payload + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/F_v'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +G = linearize(mdl, io); +G.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +G.OutputName = {'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'}; + +%% Compute the resonance frequencies +ws = eig(G.A); +ws = ws(imag(ws) > 0); diff --git a/matlab/test_nhexa_2_dynamics.m b/matlab/test_nhexa_2_dynamics.m new file mode 100644 index 0000000..a011a8c --- /dev/null +++ b/matlab/test_nhexa_2_dynamics.m @@ -0,0 +1,332 @@ +% Matlab Init :noexport:ignore: + +%% test_nhexa_dynamics.m +% Identification of the nano-hexapod dynamics from u to dL and to Vs +% Encoders are fixed to the plates + +%% 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 STEPS +addpath('./subsystems/'); % Path for Subsystems Simulink files + +%% Colors for the figures +colors = colororder; + +%% Frequency Vector +freqs = logspace(log10(10), log10(2e3), 1000); + +% Identification of the dynamics +% <> + +% The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) are identified by generating a low pass filtered white noise for each of the command signals, one by one. + +% The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure ref:fig:test_nhexa_identified_frf_de. +% The diagonal terms are displayed using colorful lines, and all the 30 off-diagonal terms are displayed by grey lines. + +% All the six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity. +% Below the first suspension mode, good decoupling can be observed (the amplitude of the all of off-diagonal terms are $\approx 20$ times smaller than the diagonal terms). + +% From 10Hz up to 1kHz, around 10 resonance frequencies can be observed. +% The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section ref:ssec:test_nhexa_enc_struts_modal_analysis. +% Then, three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section ref:ssec:test_nhexa_comp_model_coupling). +% Except the mode at 237Hz, their amplitude is rather low. +% Two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform. +% Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. + +% Up to at least 1kHz, an alternating pole/zero pattern is observed, which renders the control easier to tune. +% This would not have been the case if the encoders were fixed to the struts. + + +%% Load identification data +load('test_nhexa_identification_data_mass_0.mat', 'data'); + +%% Setup useful variables +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); % Number of points for the FFT computation +win = hanning(Nfft); % Hanning window +Noverlap = floor(Nfft/2); % Overlap between frequency analysis + +% And we get the frequency vector +[~, f] = tfestimate(data{1}.u, data{1}.de, win, Noverlap, Nfft, 1/Ts); + +%% Transfer function from u to dLm +G_de = zeros(length(f), 6, 6); + +for i = 1:6 + G_de(:,:,i) = tfestimate(data{i}.u, data{i}.de, win, Noverlap, Nfft, 1/Ts); +end + +%% Transfer function from u to Vs +G_Vs = zeros(length(f), 6, 6); + +for i = 1:6 + G_Vs(:,:,i) = tfestimate(data{i}.u, data{i}.Vs, win, Noverlap, Nfft, 1/Ts); +end + +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_de(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_de(:,i, i)), ... + 'DisplayName', sprintf('$d_{e,%i}/u_%i$', i, i)); +end +plot(f, abs(G_de(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$d_{e,i}/u_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 5e-4]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_de(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); + + + +% #+name: fig:test_nhexa_identified_frf_de +% #+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{d}_e$. The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines. +% #+RESULTS: +% [[file:figs/test_nhexa_identified_frf_de.png]] + + +% Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure ref:fig:test_nhexa_identified_frf_Vs. +% Alternating poles and zeros is observed up to at least 2kHz, which is a necessary characteristics in order to apply decentralized IFF. +% Similar to what was observed for the encoder outputs, all the "diagonal" terms are well superimposed, indicating that the same controller can be applied for all the struts. +% The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF. + + +%% Bode plot of the IFF Plant (transfer function from u to Vs) +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f, abs(G_Vs(:, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, abs(G_Vs(:,i , i)), ... + 'DisplayName', sprintf('$V_{s%i}/u_%i$', i, i)); +end +plot(f, abs(G_Vs(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$V_{si}/u_j$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 4); +leg.ItemTokenSize(1) = 15; +ylim([1e-3, 6e1]); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f, 180/pi*angle(G_Vs(:,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); + +% Effect of payload mass on the dynamics +% <> + +% As one major challenge in the control of the NASS is the wanted robustness to change of payload mass, it is necessary to understand how the dynamics of the nano-hexapod changes with a change of payload mass. + +% In order to study this change of dynamics with the payload mass, up to three "cylindrical masses" of $13\,kg$ each can be added for a total of $\approx 40\,kg$. +% These three cylindrical masses on top of the nano-hexapod are shown in Figure ref:fig:test_nhexa_table_mass_3. + +% #+name: fig:test_nhexa_table_mass_3 +% #+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,kg$ +% #+attr_org: :width 800px +% #+attr_latex: :width 0.8\linewidth +% [[file:figs/test_nhexa_table_mass_3.jpg]] + + +%% Load identification Data +meas_added_mass = {... + load('test_nhexa_identification_data_mass_0.mat', 'data'), .... + load('test_nhexa_identification_data_mass_1.mat', 'data'), .... + load('test_nhexa_identification_data_mass_2.mat', 'data'), .... + load('test_nhexa_identification_data_mass_3.mat', 'data')}; + +%% Setup useful variables +Ts = 1e-4; % Sampling Time [s] +Nfft = floor(1/Ts); % Number of points for the FFT computation +win = hanning(Nfft); % Hanning window +Noverlap = floor(Nfft/2); % Overlap between frequency analysis + +% And we get the frequency vector +[~, f] = tfestimate(meas_added_mass{1}.data{1}.u, meas_added_mass{1}.data{1}.de, win, Noverlap, Nfft, 1/Ts); + +G_de = {}; + +for i_mass = [0:3] + G_de(i_mass+1) = {zeros(length(f), 6, 6)}; + for i_strut = 1:6 + G_de{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_mass+1}.data{i_strut}.u, meas_added_mass{i_mass+1}.data{i_strut}.de, win, Noverlap, Nfft, 1/Ts); + end +end + +%% IFF Plant (transfer function from u to Vs) +G_Vs = {}; + +for i_mass = [0:3] + G_Vs(i_mass+1) = {zeros(length(f), 6, 6)}; + for i_strut = 1:6 + G_Vs{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_mass+1}.data{i_strut}.u, meas_added_mass{i_mass+1}.data{i_strut}.Vs, win, Noverlap, Nfft, 1/Ts); + end +end + +save('./mat/test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de') + +%% Load the identified transfer functions +frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de'); + + + +% The obtained frequency response functions from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure ref:fig:test_nhexa_identified_frf_de_masses. +% As expected, the frequency of the suspension modes are decreasing with an increase of the payload mass. +% The low frequency gain does not change as it is linked to the stiffness property of the nano-hexapod, and not to its mass property. + +% The frequencies of the two flexible modes of the top plate are first decreased a lot when the first mass is added (from $\approx 700\,Hz$ to $\approx 400\,Hz$). +% This is due to the fact that the added mass is composed of two half cylinders which are not fixed together. +% It therefore adds a lot of mass to the top plate without adding stiffness in one direction. +% When more than one "mass layer" is added, the half cylinders are added with some angles such that rigidity are added in all directions (see how the three mass "layers" are positioned in Figure ref:fig:test_nhexa_table_mass_3). +% In that case, the frequency of these flexible modes are increased. +% In practice, the payload should be one solid body, and no decrease of the frequency of this flexible mode should be observed. +% The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass is increased. + +% The measured FRF from $u_i$ to $V_{si}$ are shown in Figure ref:fig:test_nhexa_identified_frf_Vs_masses. +% For all the tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust way. + + +%% Bode plot for the transfer function from u to dLm - Several payloads +masses = [0, 13, 26, 39]; +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = [0:3] + % Diagonal terms + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.5], ... + 'DisplayName', sprintf('$d_{ei}/u_i$ - %i kg', masses(i_mass+1))); + for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5], ... + 'HandleVisibility', 'off'); + end + % % Off-Diagonal terms + % for i = 1:5 + % for j = i+1:6 + % plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + % 'HandleVisibility', 'off'); + % end + % end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 5e-4]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i_mass = [0:3] + for i =1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-90, 180]) + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); + +%% Bode plot for the transfer function from u to dLm +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = [0:3] + % Diagonal terms + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.5], ... + 'DisplayName', sprintf('$V_{si}/u_i$ - %i kg', masses(i_mass+1))); + for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5], ... + 'HandleVisibility', 'off'); + end + % % Off-Diagonal terms + % for i = 1:5 + % for j = i+1:6 + % plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + % 'HandleVisibility', 'off'); + % end + % end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-2, 1e2]); +leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i_mass = [0:3] + for i =1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); diff --git a/matlab/test_nhexa_3_model.m b/matlab/test_nhexa_3_model.m new file mode 100644 index 0000000..098b086 --- /dev/null +++ b/matlab/test_nhexa_3_model.m @@ -0,0 +1,549 @@ +% Matlab Init :noexport:ignore: + +%% test_nhexa_3_model.m +% Compare the measured dynamics from u to de and to Vs with the Simscape model + +%% 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 STEPS +addpath('./subsystems/'); % Path for Subsystems Simulink files + +%% Initialize Parameters for Simscape model +table_type = 'Rigid'; % On top of vibration table +device_type = 'None'; % On top of vibration table +payload_num = 0; % No Payload + +% Simulink Model name +mdl = 'test_bench_nano_hexapod'; + +%% Colors for the figures +colors = colororder; + +%% Frequency Vector +freqs = logspace(log10(10), log10(2e3), 1000); + +% Extract transfer function matrices from the Simscape Model :noexport: + +%% Extract the transfer function matrix from the Simscape model +% Initialization of the Simscape model +table_type = 'Suspended'; % On top of vibration table +device_type = 'Hexapod'; % Nano-Hexapod +payload_num = 0; % No Payload + +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); + +% Identify the FRF matrix from u to [de,Vs] +clear io; io_i = 1; +io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoders +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Encoders + +G_de = {}; +G_Vs = {}; + +for i = [0:3] + payload_num = i; % Change the payload on the nano-hexapod + G = exp(-s*1e-4)*linearize(mdl, io, 0.0); + G.InputName = {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'}; + G.OutputName = {'de1', 'de2', 'de3', 'de4', 'de5', 'de6', ... + 'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'}; + G_de(i+1) = {G({'de1', 'de2', 'de3', 'de4', 'de5', 'de6'},:)}; + G_Vs(i+1) = {G({'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'},:)}; +end + +% Save the identified plants +save('./mat/test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de') + +%% The same identification is performed, but this time with +% "flexible" model of the APA +table_type = 'Suspended'; % On top of vibration table +device_type = 'Hexapod'; % Nano-Hexapod +payload_num = 0; % No Payload + +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible'); + +G_de = {}; +G_Vs = {}; + +for i = [0:3] + payload_num = i; % Change the payload on the nano-hexapod + G = exp(-s*1e-4)*linearize(mdl, io, 0.0); + G.InputName = {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'}; + G.OutputName = {'de1', 'de2', 'de3', 'de4', 'de5', 'de6', ... + 'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'}; + G_de(i+1) = {G({'de1', 'de2', 'de3', 'de4', 'de5', 'de6'},:)}; + G_Vs(i+1) = {G({'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'},:)}; +end + +% Save the identified plants +save('./mat/test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de') + +% Nano-Hexapod model dynamics +% <> + + +%% Load Simscape Model and measured FRF +sim_ol = load('test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de'); +frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de'); + + + +% The Simscape model of the nano-hexapod is first configured with 4-DoF flexible joints, 2-DoF APA and rigid top and bottom platforms. +% The stiffness of the flexible joints are chosen based on the values estimated using the test bench and based on FEM. +% The parameters of the APA model are the ones determined from the test bench of the APA. +% The $6 \times 6$ transfer function matrices from $\mathbf{u}$ to $\mathbf{d}_e$ and from $\mathbf{u}$ to $\mathbf{V}_s$ are extracted then from the Simscape model. + +% A first feature that should be checked is that the model well represents the "direct" terms of the measured FRF matrix. +% To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_diag. +% It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modelled. +% The three resonances that were attributed to "internal" flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable as the APA are here modelled as a simple uniaxial 2-DoF system. +% At higher frequencies, no resonances can be seen in the model, as the as the top plate and the encoder supports are modelled as rigid bodies. + + +%% Diagonal elements of the FRF matrix from u to de +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:,1, 1)), 'color', [colors(1,:),0.5], ... + 'DisplayName', '$d_{ei}/u_i$ - FRF') +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.5], ... + 'DisplayName', '$d_{ei}/u_i$ - Model') +for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_de{1}(:,i, i)), 'color', [colors(1,:),0.5], ... + 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5], ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 5e-4]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{1}(:,i, i)), 'color', [colors(1,:),0.5]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_de{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5]); +end +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([freqs(1), freqs(end)]); + +%% Diagonal elements of the FRF matrix from u to Vs +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_Vs{1}(:,1, 1)), 'color', [colors(1,:),0.5], ... + 'DisplayName', '$V_{si}/u_i$ - FRF') +plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{1}(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ... + 'DisplayName', '$V_{si}/u_i$ - Model') +for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_Vs{1}(:,i, i)), 'color', [colors(1,:),0.5], ... + 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{1}(:,i, i)), 'color', [colors(1,:),0.5]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_Vs{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]); +end +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([freqs(1), freqs(end)]); + +% Modelling dynamical coupling +% <> + +% Another wanted feature of the model is that it well represents the coupling in the system as this is often the limiting factor for the control of MIMO systems. +% Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\mathbf{u}$ to $\mathbf{d}_e$, only the first "column" is compared (Figure ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$. +% It can be seen that the coupling in the model is well matching the measurements up to the first un-modelled flexible mode at 237Hz. +% Similar results are observed for all the other coupling terms, as well as for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$. + + +%% Comparison of the plants (encoder output) when tuning the misalignment +i_input = 1; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight'); + +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(1, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(2, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 3, i_input)), ... + 'DisplayName', 'Measurements'); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model (2-DoF APA)'); +text(54, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(4, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xticks([50, 100, 200, 400]) + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(5, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xticks([50, 100, 200, 400]) + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(6, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xticks([50, 100, 200, 400]) + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([50, 5e2]); ylim([1e-8, 5e-4]); + + + +% #+name: fig:test_nhexa_comp_simscape_de_all +% #+caption: Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$ +% #+RESULTS: +% [[file:figs/test_nhexa_comp_simscape_de_all.png]] + +% The APA300ML are then modelled with a /super-element/ extracted from a FE-software. +% The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_de_all_flex. +% While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modelled. +% Even the mode 395Hz can be observed in the model. +% Therefore, if the modes of the struts are to be modelled, the /super-element/ of the APA300ML may be used, at the cost of obtaining a much higher order model. + + +%% Load the plant model with Flexible APA +flex_ol = load('test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de'); + +%% Comparison of the plants (encoder output) when tuning the misalignment +i_input = 1; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight'); + +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(1, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(2, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 3, i_input)), ... + 'DisplayName', 'Measurements'); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model (Flexible APA)'); +text(54, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(4, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xticks([50, 100, 200, 400]) + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(5, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xticks([50, 100, 200, 400]) + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(6, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xticks([50, 100, 200, 400]) + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([50, 5e2]); ylim([1e-8, 5e-4]); + +% Modelling the effect of payload mass +% <> + +% Another important characteristics of the model is that it should well represents the dynamics of the system for all considered payloads. +% The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure ref:fig:test_nhexa_comp_simscape_diag_masses. +% The observed shift to lower frequency of the suspension modes with an increased payload mass is well represented by the Simscape model. +% The complex conjugate zeros are also well matching with the experiments both for the encoder outputs (Figure ref:fig:test_nhexa_comp_simscape_de_diag_masses) and the force sensor outputs (Figure ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). + +% Note that the model displays smaller damping that what is observed experimentally for high values of the payload mass. +% One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping). +% However, as decentralized IFF will be applied, the damping will be brought actively, and the open-loop damping value should have very little impact on the obtained plant. + + +%% Bode plot for the transfer function from u to de +masses = [0, 13, 26, 39]; +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = [0:3] + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('Meas (%i kg)', masses(i_mass+1))); + for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + set(gca, 'ColorOrderIndex', i_mass+1) + plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--', ... + 'DisplayName', 'Simscape'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([5e-8, 1e-3]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i_mass = [0:3] + for i =1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + end + set(gca, 'ColorOrderIndex', i_mass+1) + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--'); +end +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([20, 2e2]); +xticks([20, 50, 100, 200]) + +%% Bode plot for the transfer function from u to Vs +masses = [0, 13, 26, 39]; +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = 0:3 + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('Meas (%i kg)', masses(i_mass+1))); + for i = 2:6 + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + 'HandleVisibility', 'off'); + end + plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(1,1), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:), ... + 'DisplayName', 'Simscape'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e2]); +leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i_mass = 0:3 + for i =1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + end + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(i,i), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e2]); +xticks([20, 50, 100, 200]) + + + +% #+name: fig:test_nhexa_comp_simscape_diag_masses +% #+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag}) +% #+attr_latex: :options [htbp] +% #+begin_figure +% #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$} +% #+attr_latex: :options {0.49\textwidth} +% #+begin_subfigure +% #+attr_latex: :width 0.95\linewidth +% [[file:figs/test_nhexa_comp_simscape_de_diag_masses.png]] +% #+end_subfigure +% #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$} +% #+attr_latex: :options {0.49\textwidth} +% #+begin_subfigure +% #+attr_latex: :width 0.95\linewidth +% [[file:figs/test_nhexa_comp_simscape_Vs_diag_masses.png]] +% #+end_subfigure +% #+end_figure + +% In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the 39kg payload in Figure ref:fig:test_nhexa_comp_simscape_de_all_high_mass. +% Excellent match between the experimental coupling and the model coupling is observed. +% The model therefore well represents the system dynamical coupling for different considered payloads. + + +%% Comparison of the plants (encoder output) when tuning the misalignment +i_input = 1; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight'); + +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(1, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(2, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 3, i_input)), ... + 'DisplayName', 'Measurements'); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model (2-DoF APA)'); +text(12, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(4, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xticks([10, 50, 100, 200, 400]) + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(5, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xticks([10, 50, 100, 200, 400]) + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(6, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xticks([10, 50, 100, 200, 400]) + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([10, 5e2]); ylim([1e-8, 5e-4]); diff --git a/test-bench-nano-hexapod.org b/test-bench-nano-hexapod.org index 8d5851a..d5790cb 100644 --- a/test-bench-nano-hexapod.org +++ b/test-bench-nano-hexapod.org @@ -134,11 +134,23 @@ Maybe the rest is not so interesting here as it will be presented again in the n - High Authority Controller HAC - Decoupling Strategy +** TODO [#C] See if the FEM in Simscape can model the struts modes ** TODO [#C] Add nice pictures [[file:~/Cloud/pictures/work/nano-hexapod/vibration-table]] -** TODO [#B] Proper analysis of the identified dynamics +** TODO [#B] If possible, correlate the modal analysis with FEM + +This could just be used to show that experimental measure of the flexible mode of the top plate has been done: +- [X] *This test was made using encoder fixed to the struts, is it relevant to put it here?* +- [ ] Also compare with the FEM + - [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/Assembly 20201020/Modal t=0.50mm]] + - [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/GitLab_nass-fem/dynamic-modal/assy-hexapod-20201022/t_0.25mm]] + - [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/GitLab_nass-fem/dynamic-modal/assy-hexapod-20201022/t_0.5mm]] + - [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/GitLab_nass-fem/plateau-superelement]] + +** DONE [#B] Proper analysis of the identified dynamics +CLOSED: [2024-10-28 Mon 11:13] - [ ] Top plate flexible modes (2 modes) - [ ] Modes of the encoder supports @@ -146,14 +158,14 @@ Maybe the rest is not so interesting here as it will be presented again in the n ** TODO [#C] Remove un-used matlab scripts and src files -** TODO [#B] Make nice subfigures for identified modes -SCHEDULED: <2024-10-26 Sat> +** DONE [#B] Make nice subfigures for identified modes +CLOSED: [2024-10-27 Sun 15:58] SCHEDULED: <2024-10-26 Sat> Maybe try to do similar thing as for the micro station: [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A3-micro-station-modal-analysis/mode_shapes-gif-to-jpg/gen_mode_1.sh]] -- [ ] Table: 6 rigid body modes + 3 flexible modes +- [X] Table: 6 rigid body modes + 3 flexible modes [[file:figs/modal-analysis-table]] -- [ ] Nano hexapod: 6 rigid body modes + 2 flexible modes +- [X] Nano hexapod: 6 rigid body modes + 2 flexible modes [[file:figs/modal-analysis-hexapod]] ** DONE [#A] Update the default APA parameters to have good match @@ -212,3778 +224,6 @@ CLOSED: [2024-10-26 Sat 15:26] - git submodule? - Maybe just copy paste the directory as it will not change a lot now -** Analysis backup of HAC - Decoupling analysis -<> - -*** Introduction :ignore: - -In this section is studied the HAC-IFF architecture for the Nano-Hexapod. -More precisely: -- The LAC control is a decentralized integral force feedback as studied in Section ref:sec:test_nhexa_enc_plates_iff -- The HAC control is a decentralized controller working in the frame of the struts - -The corresponding control architecture is shown in Figure ref:fig:test_nhexa_control_architecture_hac_iff_struts with: -- $\bm{r}_{\mathcal{X}_n}$: the $6 \times 1$ reference signal in the cartesian frame -- $\bm{r}_{d\mathcal{L}}$: the $6 \times 1$ reference signal transformed in the frame of the struts thanks to the inverse kinematic -- $\bm{\epsilon}_{d\mathcal{L}}$: the $6 \times 1$ length error of the 6 struts -- $\bm{u}^\prime$: input of the damped plant -- $\bm{u}$: generated DAC voltages -- $\bm{\tau}_m$: measured force sensors -- $d\bm{\mathcal{L}}_m$: measured displacement of the struts by the encoders - -#+begin_src latex :file control_architecture_hac_iff_struts.pdf -\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} -\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} -\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} - -\begin{tikzpicture} - % Blocs - \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; - \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); - \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); - \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); - - \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; - \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; - \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; - \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; - \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; - \node[block, align=center, left= of subr, fill=control!20!white] (J) {\tiny Inverse\\\tiny Kinematics}; - - % Connections and labels - \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; - \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); - \draw[->] (Kiff.west) -| (addF.south); - \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; - \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; - - \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; - \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); - \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; - \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; - - \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; - \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_control_architecture_hac_iff_struts -#+caption: HAC-LAC: IFF + Control in the frame of the legs -#+RESULTS: -[[file:figs/test_nhexa_control_architecture_hac_iff_struts.png]] - -This part is structured as follow: -- Section ref:sec:test_nhexa_hac_iff_struts_ref_track: some reference tracking tests are performed -- Section ref:sec:test_nhexa_hac_iff_struts_controller: the decentralized high authority controller is tuned using the Simscape model and is implemented and tested experimentally -- Section ref:sec:test_nhexa_interaction_analysis: an interaction analysis is performed, from which the best decoupling strategy can be determined -- Section ref:sec:test_nhexa_robust_hac_design: Robust High Authority Controller are designed - -*** Reference Tracking - Trajectories -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/reference_tracking_paths.m -:END: -<> -**** Introduction :ignore: -In this section, several trajectories representing the wanted pose (position and orientation) of the top platform with respect to the bottom platform are defined. - -These trajectories will be used to test the HAC-LAC architecture. - -In order to transform the wanted pose to the wanted displacement of the 6 struts, the inverse kinematic is required. -As a first approximation, the Jacobian matrix $\bm{J}$ can be used instead of using the full inverse kinematic equations. - -Therefore, the control architecture with the input trajectory $\bm{r}_{\mathcal{X}_n}$ is shown in Figure ref:fig:test_nhexa_control_architecture_hac_iff_L. - -#+begin_src latex :file control_architecture_hac_iff_struts_L.pdf -\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} -\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} -\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} - -\begin{tikzpicture} - % Blocs - \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; - \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); - \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); - \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); - - \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; - \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; - \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; - \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; - \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; - \node[block, align=center, left= of subr, fill=control!20!white] (J) {$\bm{J}$}; - - % Connections and labels - \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; - \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); - \draw[->] (Kiff.west) -| (addF.south); - \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; - \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; - - \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; - \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); - \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; - \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; - - \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; - \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_control_architecture_hac_iff_L -#+caption: HAC-LAC: IFF + Control in the frame of the legs -#+RESULTS: -[[file:figs/test_nhexa_control_architecture_hac_iff_struts_L.png]] - -In the following sections, several reference trajectories are defined: -- Section ref:sec:test_nhexa_yz_scans: simple scans in the Y-Z plane -- Section ref:sec:test_nhexa_tilt_scans: scans in tilt are performed -- Section ref:sec:test_nhexa_nass_scans: scans with X-Y-Z translations in order to draw the word "NASS" - -**** Matlab Init :noexport:ignore: -#+begin_src matlab -%% reference_tracking_paths.m -% Computation of several reference paths -#+end_src - -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -#+begin_src matlab :tangle no :noweb yes -<> -#+end_src - -#+begin_src matlab :eval no :noweb yes -<> -#+end_src - -#+begin_src matlab :noweb yes -<> -#+end_src - -**** Y-Z Scans -<> -A function =generateYZScanTrajectory= has been developed in order to easily generate scans in the Y-Z plane. - -For instance, the following generated trajectory is represented in Figure ref:fig:test_nhexa_yz_scan_example_trajectory_yz_plane. -#+begin_src matlab -%% Generate the Y-Z trajectory scan -Rx_yz = generateYZScanTrajectory(... - 'y_tot', 4e-6, ... % Length of Y scans [m] - 'z_tot', 4e-6, ... % Total Z distance [m] - 'n', 5, ... % Number of Y scans - 'Ts', 1e-3, ... % Sampling Time [s] - 'ti', 1, ... % Time to go to initial position [s] - 'tw', 0, ... % Waiting time between each points [s] - 'ty', 0.6, ... % Time for a scan in Y [s] - 'tz', 0.2); % Time for a scan in Z [s] -#+end_src - -#+begin_src matlab :exports none -%% Plot the trajectory in the Y-Z plane -figure; -plot(Rx_yz(:,3), Rx_yz(:,4)); -xlabel('y [m]'); ylabel('z [m]'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scan_example_trajectory_yz_plane.pdf', 'width', 'normal', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_yz_scan_example_trajectory_yz_plane -#+caption: Generated scan in the Y-Z plane -#+RESULTS: -[[file:figs/test_nhexa_yz_scan_example_trajectory_yz_plane.png]] - -The Y and Z positions as a function of time are shown in Figure ref:fig:test_nhexa_yz_scan_example_trajectory. - -#+begin_src matlab :exports none -%% Plot the Y-Z trajectory as a function of time -figure; -hold on; -plot(Rx_yz(:,1), Rx_yz(:,3), ... - 'DisplayName', 'Y motion') -plot(Rx_yz(:,1), Rx_yz(:,4), ... - 'DisplayName', 'Z motion') -hold off; -xlabel('Time [s]'); -ylabel('Displacement [m]'); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scan_example_trajectory.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_yz_scan_example_trajectory -#+caption: Y and Z trajectories as a function of time -#+RESULTS: -[[file:figs/test_nhexa_yz_scan_example_trajectory.png]] - -Using the Jacobian matrix, it is possible to compute the wanted struts lengths as a function of time: -\begin{equation} - \bm{r}_{d\mathcal{L}} = \bm{J} \bm{r}_{\mathcal{X}_n} -\end{equation} - -#+begin_src matlab :exports none -load('jacobian.mat', 'J'); -#+end_src - -#+begin_src matlab -%% Compute the reference in the frame of the legs -dL_ref = [J*Rx_yz(:, 2:7)']'; -#+end_src - -The reference signal for the strut length is shown in Figure ref:fig:test_nhexa_yz_scan_example_trajectory_struts. -#+begin_src matlab :exports none -%% Plot the reference in the frame of the legs -figure; -hold on; -for i=1:6 - plot(Rx_yz(:,1), dL_ref(:, i), ... - 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) -end -xlabel('Time [s]'); ylabel('Strut Motion [m]'); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); -yticks(1e-6*[-5:5]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_yz_scan_example_trajectory_struts -#+caption: Trajectories for the 6 individual struts -#+RESULTS: -[[file:figs/test_nhexa_yz_scan_example_trajectory_struts.png]] - -**** Tilt Scans -<> - -A function =generalSpiralAngleTrajectory= has been developed in order to easily generate $R_x,R_y$ tilt scans. - -For instance, the following generated trajectory is represented in Figure ref:fig:test_nhexa_tilt_scan_example_trajectory. -#+begin_src matlab -%% Generate the "tilt-spiral" trajectory scan -R_tilt = generateSpiralAngleTrajectory(... - 'R_tot', 20e-6, ... % Total Tilt [ad] - 'n_turn', 5, ... % Number of scans - 'Ts', 1e-3, ... % Sampling Time [s] - 't_turn', 1, ... % Turn time [s] - 't_end', 1); % End time to go back to zero [s] -#+end_src - -#+begin_src matlab :exports none -%% Plot the trajectory -figure; -plot(1e6*R_tilt(:,5), 1e6*R_tilt(:,6)); -xlabel('$R_x$ [$\mu$rad]'); ylabel('$R_y$ [$\mu$rad]'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/tilt_scan_example_trajectory.pdf', 'width', 'normal', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_tilt_scan_example_trajectory -#+caption: Generated "spiral" scan -#+RESULTS: -[[file:figs/test_nhexa_tilt_scan_example_trajectory.png]] - -#+begin_src matlab :exports none -%% Compute the reference in the frame of the legs -load('jacobian.mat', 'J'); -dL_ref = [J*R_tilt(:, 2:7)']'; -#+end_src - -The reference signal for the strut length is shown in Figure ref:fig:test_nhexa_tilt_scan_example_trajectory_struts. -#+begin_src matlab :exports none -%% Plot the reference in the frame of the legs -figure; -hold on; -for i=1:6 - plot(R_tilt(:,1), dL_ref(:, i), ... - 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) -end -xlabel('Time [s]'); ylabel('Strut Motion [m]'); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); -yticks(1e-6*[-5:5]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/tilt_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_tilt_scan_example_trajectory_struts -#+caption: Trajectories for the 6 individual struts - Tilt scan -#+RESULTS: -[[file:figs/test_nhexa_tilt_scan_example_trajectory_struts.png]] - -**** "NASS" reference path -<> -In this section, a reference path that "draws" the work "NASS" is developed. - -First, a series of points representing each letter are defined. -Between each letter, a negative Z motion is performed. -#+begin_src matlab -%% List of points that draws "NASS" -ref_path = [ ... - 0, 0,0; % Initial Position - 0,0,1; 0,4,1; 3,0,1; 3,4,1; % N - 3,4,0; 4,0,0; % Transition - 4,0,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,2,1; 4,2,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,0,1; % A - 7,0,0; 8,0,0; % Transition - 8,0,1; 11,0,1; 11,2,1; 8,2,1; 8,4,1; 11,4,1; % S - 11,4,0; 12,0,0; % Transition - 12,0,1; 15,0,1; 15,2,1; 12,2,1; 12,4,1; 15,4,1; % S - 15,4,0; - ]; - -%% Center the trajectory arround zero -ref_path = ref_path - (max(ref_path) - min(ref_path))/2; - -%% Define the X-Y-Z cuboid dimensions containing the trajectory -X_max = 10e-6; -Y_max = 4e-6; -Z_max = 2e-6; - -ref_path = ([X_max, Y_max, Z_max]./max(ref_path)).*ref_path; % [m] -#+end_src - -Then, using the =generateXYZTrajectory= function, the $6 \times 1$ trajectory signal is computed. -#+begin_src matlab -%% Generating the trajectory -Rx_nass = generateXYZTrajectory('points', ref_path); -#+end_src - -The trajectory in the X-Y plane is shown in Figure ref:fig:test_nhexa_ref_track_test_nass (the transitions between the letters are removed). -#+begin_src matlab :exports none -%% "NASS" trajectory in the X-Y plane -figure; -plot(1e6*Rx_nass(Rx_nass(:,4)>0, 2), 1e6*Rx_nass(Rx_nass(:,4)>0, 3), 'k.') -xlabel('X [$\mu m$]'); -ylabel('Y [$\mu m$]'); -axis equal; -xlim(1e6*[min(Rx_nass(:,2)), max(Rx_nass(:,2))]); -ylim(1e6*[min(Rx_nass(:,3)), max(Rx_nass(:,3))]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_test_nass.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_ref_track_test_nass -#+caption: Reference path corresponding to the "NASS" acronym -#+RESULTS: -[[file:figs/test_nhexa_ref_track_test_nass.png]] - -It can also be better viewed in a 3D representation as in Figure ref:fig:test_nhexa_ref_track_test_nass_3d. - -#+begin_src matlab :exports none -figure; -plot3(1e6*Rx_nass(:,2), 1e6*Rx_nass(:,3), 1e6*Rx_nass(:,4), 'k-'); -xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); -view(-13, 41) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_test_nass_3d.pdf', 'width', 'normal', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_ref_track_test_nass_3d -#+caption: Reference path that draws "NASS" - 3D view -#+RESULTS: -[[file:figs/test_nhexa_ref_track_test_nass_3d.png]] - -*** First Basic High Authority Controller -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/hac_lac_first_try.m -:END: -<> -**** Introduction :ignore: -In this section, a simple decentralized high authority controller $\bm{K}_{\mathcal{L}}$ is developed to work without any payload. - -The diagonal controller is tuned using classical Loop Shaping in Section ref:sec:test_nhexa_hac_iff_no_payload_tuning. -The stability is verified in Section ref:sec:test_nhexa_hac_iff_no_payload_stability using the Simscape model. - -**** Matlab Init :noexport:ignore: -#+begin_src matlab -%% hac_lac_first_try.m -% Development and analysis of a first basic High Authority Controller -#+end_src - -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -#+begin_src matlab :tangle no :noweb yes -<> -#+end_src - -#+begin_src matlab :eval no :noweb yes -<> -#+end_src - -#+begin_src matlab :noweb yes -<> -<> -#+end_src - -#+begin_src matlab -%% Load the identified FRF and Simscape model -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -**** HAC Controller -<> - -Let's first try to design a first decentralized controller with: -- a bandwidth of 100Hz -- sufficient phase margin -- simple and understandable components - -After some very basic and manual loop shaping, A diagonal controller is developed. -Each diagonal terms are identical and are composed of: -- A lead around 100Hz -- A first order low pass filter starting at 200Hz to add some robustness to high frequency modes -- A notch at 700Hz to cancel the flexible modes of the top plate -- A pure integrator - -#+begin_src matlab -%% Lead to increase phase margin -a = 2; % Amount of phase lead / width of the phase lead / high frequency gain -wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] - -H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); - -%% Low Pass filter to increase robustness -H_lpf = 1/(1 + s/2/pi/200); - -%% Notch at the top-plate resonance -gm = 0.02; -xi = 0.3; -wn = 2*pi*700; - -H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); - -%% Decentralized HAC -Khac_iff_struts = -(1/(2.87e-5)) * ... % Gain - H_lead * ... % Lead - H_notch * ... % Notch - (2*pi*100/s) * ... % Integrator - eye(6); % 6x6 Diagonal -#+end_src - -This controller is saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') -#+end_src - -The experimental loop gain is computed and shown in Figure ref:fig:test_nhexa_loop_gain_hac_iff_struts. -#+begin_src matlab -L_hac_iff_struts = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the Loop Gain -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - - -ax1 = nexttile([2,1]); -hold on; -% Diagonal Elements Model -plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,1,:))), 'color', colors(1,:), ... - 'DisplayName', 'Diagonal'); -for i = 2:6 - plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); -end -plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,2,:))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', 'Off-Diag'); -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,j,:))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e2]); -legend('location', 'northeast'); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:)); -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([2, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loop_gain_hac_iff_struts.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loop_gain_hac_iff_struts -#+caption: Diagonal and off-diagonal elements of the Loop gain for "HAC-IFF-Struts" -#+RESULTS: -[[file:figs/test_nhexa_loop_gain_hac_iff_struts.png]] - -**** Verification of the Stability using the Simscape model -<> - -The HAC-IFF control strategy is implemented using Simscape. -#+begin_src matlab -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', 'flexible', ... - 'controller_type', 'hac-iff-struts'); -#+end_src - -#+begin_src matlab :exports none -support.type = 1; % On top of vibration table -payload.type = 3; % Payload / 1 "mass layer" - -load('Kiff_opt.mat', 'Kiff'); -#+end_src - -#+begin_src matlab -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -We identify the closed-loop system. -#+begin_src matlab -%% Identification -Gd_iff_hac_opt = linearize(mdl, io, 0.0, options); -#+end_src - -And verify that it is indeed stable. -#+begin_src matlab :results value replace :exports both -%% Verify the stability -isstable(Gd_iff_hac_opt) -#+end_src - -#+RESULTS: -: 1 - -**** Experimental Validation -Both the Integral Force Feedback controller (developed in Section ref:sec:test_nhexa_enc_plates_iff) and the high authority controller working in the frame of the struts (developed in Section ref:sec:test_nhexa_hac_iff_struts_controller) are implemented experimentally. - -Two reference tracking experiments are performed to evaluate the stability and performances of the implemented control. - -#+begin_src matlab -%% Load the experimental data -load('hac_iff_struts_yz_scans.mat', 't', 'de') -#+end_src - -#+begin_src matlab :exports none -%% Reset initial time -t = t - t(1); -#+end_src - -The position of the top-platform is estimated using the Jacobian matrix: -#+begin_src matlab -%% Pose of the top platform from the encoder values -load('jacobian.mat', 'J'); -Xe = [inv(J)*de']'; -#+end_src - -#+begin_src matlab -%% Generate the Y-Z trajectory scan -Rx_yz = generateYZScanTrajectory(... - 'y_tot', 4e-6, ... % Length of Y scans [m] - 'z_tot', 8e-6, ... % Total Z distance [m] - 'n', 5, ... % Number of Y scans - 'Ts', 1e-3, ... % Sampling Time [s] - 'ti', 1, ... % Time to go to initial position [s] - 'tw', 0, ... % Waiting time between each points [s] - 'ty', 0.6, ... % Time for a scan in Y [s] - 'tz', 0.2); % Time for a scan in Z [s] -#+end_src - -The reference path as well as the measured position are partially shown in the Y-Z plane in Figure ref:fig:test_nhexa_yz_scans_exp_results_first_K. -#+begin_src matlab :exports none -%% Position and reference signal in the Y-Z plane -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile; -hold on; -plot(1e6*Xe(t>2,2), 1e6*Xe(t>2,3)); -plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--'); -hold off; -xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); -xlim([-2.05, 2.05]); ylim([-4.1, 4.1]); -axis equal; - -ax2 = nexttile([1,2]); -hold on; -plot(1e6*Xe(:,2), 1e6*Xe(:,3), ... - 'DisplayName', '$\mathcal{X}_n$'); -plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--', ... - 'DisplayName', '$r_{\mathcal{X}_n}$'); -hold off; -legend('location', 'northwest'); -xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); -axis equal; -xlim([1.6, 2.1]); ylim([-4.1, -3.6]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scans_exp_results_first_K.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_yz_scans_exp_results_first_K -#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ in the Y-Z plane - Zoom on a change of direction -#+RESULTS: -[[file:figs/test_nhexa_yz_scans_exp_results_first_K.png]] - -#+begin_important -It is clear from Figure ref:fig:test_nhexa_yz_scans_exp_results_first_K that the position of the nano-hexapod effectively tracks to reference signal. -However, oscillations with amplitudes as large as 50nm can be observe. - -It turns out that the frequency of these oscillations is 100Hz which is corresponding to the crossover frequency of the High Authority Control loop. -This clearly indicates poor stability margins. -In the next section, the controller is re-designed to improve the stability margins. -#+end_important - -**** Controller with increased stability margins -The High Authority Controller is re-designed in order to improve the stability margins. -#+begin_src matlab -%% Lead -a = 5; % Amount of phase lead / width of the phase lead / high frequency gain -wc = 2*pi*110; % Frequency with the maximum phase lead [rad/s] - -H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); - -%% Low Pass Filter -H_lpf = 1/(1 + s/2/pi/300); - -%% Notch -gm = 0.02; -xi = 0.5; -wn = 2*pi*700; - -H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); - -%% HAC Controller -Khac_iff_struts = -2.2e4 * ... % Gain - H_lead * ... % Lead - H_lpf * ... % Lead - H_notch * ... % Notch - (2*pi*100/s) * ... % Integrator - eye(6); % 6x6 Diagonal -#+end_src - -#+begin_src matlab :exports none -%% Load the FRF of the transfer function from u to dL with IFF -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -#+begin_src matlab :exports none -%% Compute the Loop Gain -L_frf = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); -#+end_src - -The bode plot of the new loop gain is shown in Figure ref:fig:test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K. -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -freqs = 2*logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -% Diagonal Elements FRF -plot(frf_iff.f, abs(squeeze(L_frf(1,1,:))), 'color', colors(1,:), ... - 'DisplayName', 'Diagonal'); -for i = 2:6 - plot(frf_iff.f, abs(squeeze(L_frf(i,i,:))), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); -end -plot(frf_iff.f, abs(squeeze(L_frf(1,2,:))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', 'Off-Diag'); -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(L_frf(i,j,:))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e2]); -legend('location', 'northeast'); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(squeeze(L_frf(i,i,:))), 'color', colors(1,:)); -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([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/hac_iff_plates_exp_loop_gain_redesigned_K.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K -#+caption: Loop Gain for the updated decentralized HAC controller -#+RESULTS: -[[file:figs/test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K.png]] - -This new controller is implemented experimentally and several tracking tests are performed. -#+begin_src matlab -%% Load Measurements -load('hac_iff_more_lead_nass_scan.mat', 't', 'de') -#+end_src - -#+begin_src matlab :exports none -%% Reset Time -t = t - t(1); -#+end_src - -The pose of the top platform is estimated from the encoder position using the Jacobian matrix. -#+begin_src matlab -%% Compute the pose of the top platform -load('jacobian.mat', 'J'); -Xe = [inv(J)*de']'; -#+end_src - -#+begin_src matlab :exports none -%% Load the reference path -load('reference_path.mat', 'Rx_nass') -#+end_src - -The measured motion as well as the trajectory are shown in Figure ref:fig:test_nhexa_nass_scans_first_test_exp. -#+begin_src matlab :exports none -%% Plot the X-Y-Z "NASS" trajectory -figure; -hold on; -plot3(Xe(1:100:end,1), Xe(1:100:end,2), Xe(1:100:end,3)) -plot3(Rx_nass(1:100:end,2), Rx_nass(1:100:end,3), Rx_nass(1:100:end,4)) -hold off; -xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); -view(-13, 41) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/nass_scans_first_test_exp.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_nass_scans_first_test_exp -#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ for the "NASS" trajectory -#+RESULTS: -[[file:figs/test_nhexa_nass_scans_first_test_exp.png]] - -The trajectory and measured motion are also shown in the X-Y plane in Figure ref:fig:test_nhexa_ref_track_nass_exp_hac_iff_struts. -#+begin_src matlab :exports none -%% Estimate when the hexpod is on top position and drawing the letters -i_top = Xe(:,3) > 1.9e-6; -i_rx = Rx_nass(:,4) > 0; -#+end_src - -#+begin_src matlab :exports none -%% Plot the reference as well as the measurement in the X-Y plane -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([1,2]); -hold on; -scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); -plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); -hold off; -xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); -axis equal; -xlim([-10.5, 10.5]); ylim([-4.5, 4.5]); - -ax2 = nexttile; -hold on; -scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); -plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); -hold off; -xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); -axis equal; -xlim([4.5, 4.7]); ylim([-0.15, 0.05]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_nass_exp_hac_iff_struts.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_ref_track_nass_exp_hac_iff_struts -#+caption: Reference path and measured motion in the X-Y plane -#+RESULTS: -[[file:figs/test_nhexa_ref_track_nass_exp_hac_iff_struts.png]] - -The orientation errors during all the scans are shown in Figure ref:fig:test_nhexa_nass_ref_rx_ry. -#+begin_src matlab :exports none -%% Orientation Errors -figure; -hold on; -plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,4), '-', 'DisplayName', '$\epsilon_{\theta_x}$'); -plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,5), '-', 'DisplayName', '$\epsilon_{\theta_y}$'); -plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,6), '-', 'DisplayName', '$\epsilon_{\theta_z}$'); -hold off; -xlabel('Time [s]'); ylabel('Orientation Error [$\mu$ rad]'); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :exports none -%% Orientation Errors -figure; -hold on; -plot(1e9*Xe(100000:100:end,4), 1e9*Xe(100000:100:end,5), '.'); -th = 0:pi/50:2*pi; -xunit = 90 * cos(th); -yunit = 90 * sin(th); -plot(xunit, yunit, '--'); -hold off; -xlabel('$R_x$ [nrad]'); ylabel('$R_y$ [nrad]'); -xlim([-100, 100]); -ylim([-100, 100]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/nass_ref_rx_ry.pdf', 'width', 500, 'height', 500); -#+end_src - -#+name: fig:test_nhexa_nass_ref_rx_ry -#+caption: Orientation errors during the scan -#+RESULTS: -[[file:figs/test_nhexa_nass_ref_rx_ry.png]] - -#+begin_important -Using the updated High Authority Controller, the nano-hexapod can follow trajectories with high accuracy (the position errors are in the order of 50nm peak to peak, and the orientation errors 300nrad peak to peak). -#+end_important - -*** Interaction Analysis and Decoupling -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/interaction_analysis_enc_plates.m -:END: -<> -**** Introduction :ignore: - -In this section, the interaction in the identified plant is estimated using the Relative Gain Array (RGA) [[cite:skogestad07_multiv_feedb_contr][Chap. 3.4]]. - -Then, several decoupling strategies are compared for the nano-hexapod. - -The RGA Matrix is defined as follow: -\begin{equation} - \text{RGA}(G(f)) = G(f) \times (G(f)^{-1})^T -\end{equation} - -Then, the RGA number is defined: -\begin{equation} -\text{RGA-num}(f) = \| \text{I - RGA(G(f))} \|_{\text{sum}} -\end{equation} - - -In this section, the plant with 2 added mass is studied. - -**** Matlab Init :noexport:ignore: -#+begin_src matlab -%% interaction_analysis_enc_plates.m -% Interaction analysis of several decoupling strategies -#+end_src - -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -#+begin_src matlab :tangle no :noweb yes -<> -#+end_src - -#+begin_src matlab :eval no :noweb yes -<> -#+end_src - -#+begin_src matlab :noweb yes -<> -#+end_src - -#+begin_src matlab -%% Load the identified FRF and Simscape model -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -**** Parameters -#+begin_src matlab -wc = 100; % Wanted crossover frequency [Hz] -[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc -#+end_src - -#+begin_src matlab -%% Plant to be decoupled -frf_coupled = frf_iff.G_dL{2}; -G_coupled = sim_iff.G_dL{2}; -#+end_src - -**** No Decoupling (Decentralized) -<> - -#+begin_src latex :file decoupling_arch_decentralized.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - - % Connections and labels - \draw[<-] (G.west) -- ++(-1.8, 0) node[above right]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(G.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gdec) {}; - \node[below right] at (Gdec.north west) {$\bm{G}_{\text{dec}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_decentralized -#+caption: Block diagram representing the plant. -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_decentralized.png]] - -#+begin_src matlab :exports none -%% Decentralized Plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(frf_coupled(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(frf_coupled(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(frf_coupled(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(frf_coupled(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_decentralized_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_decentralized_plant -#+caption: Bode Plot of the decentralized plant (diagonal and off-diagonal terms) -#+RESULTS: -[[file:figs/test_nhexa_interaction_decentralized_plant.png]] - -#+begin_src matlab :exports none -%% Decentralized RGA -RGA_dec = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_dec(i,:,:) = squeeze(frf_coupled(i,:,:)).*inv(squeeze(frf_coupled(i,:,:))).'; -end - -RGA_dec_sum = zeros(length(frf_iff), 1); -for i = 1:length(frf_iff.f) - RGA_dec_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% RGA for Decentralized plant -figure; -plot(frf_iff.f, RGA_dec_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_decentralized.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_decentralized -#+caption: RGA number for the decentralized plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_decentralized.png]] - -**** Static Decoupling -<> - -#+begin_src latex :file decoupling_arch_static.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j0)^{-1}$}; - - % Connections and labels - \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; - \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\text{static}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_static -#+caption: Decoupling using the inverse of the DC gain of the plant -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_static.png]] - -The DC gain is evaluated from the model as be have bad low frequency identification. - -#+begin_src matlab :exports none -%% Compute the inverse of the DC gain -G_model = G_coupled; -G_model.outputdelay = 0; % necessary for further inversion -dc_inv = inv(dcgain(G_model)); - -%% Compute the inversed plant -G_dL_sta = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_sta(i,:,:) = squeeze(frf_coupled(i,:,:))*dc_inv; -end -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(dc_inv, {}, {}, ' %.1f '); -#+end_src - -#+RESULTS: -| -62011.5 | 3910.6 | 4299.3 | 660.7 | -4016.5 | -4373.6 | -| 3914.4 | -61991.2 | -4356.8 | -4019.2 | 640.2 | 4281.6 | -| -4020.0 | -4370.5 | -62004.5 | 3914.6 | 4295.8 | 653.8 | -| 660.9 | 4292.4 | 3903.3 | -62012.2 | -4366.5 | -4008.9 | -| 4302.8 | 655.6 | -4025.8 | -4377.8 | -62006.0 | 3919.7 | -| -4377.9 | -4013.2 | 668.6 | 4303.7 | 3906.8 | -62019.3 | - -#+begin_src matlab :exports none -%% Bode plot of the static decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_sta(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(G_dL_sta(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_sta(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_sta(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_static_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_static_dec_plant -#+caption: Bode Plot of the static decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_static_dec_plant.png]] - -#+begin_src matlab :exports none -%% Compute RGA Matrix -RGA_sta = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_sta(i,:,:) = squeeze(G_dL_sta(i,:,:)).*inv(squeeze(G_dL_sta(i,:,:))).'; -end - -%% Compute RGA-number -RGA_sta_sum = zeros(length(frf_iff), 1); -for i = 1:size(RGA_sta, 1) - RGA_sta_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-number for statically decoupled plant -figure; -plot(frf_iff.f, RGA_sta_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_static_dec.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_static_dec -#+caption: RGA number for the statically decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_static_dec.png]] - -**** Decoupling at the Crossover -<> - -#+begin_src latex :file decoupling_arch_crossover.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j\omega_c)^{-1}$}; - - % Connections and labels - \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; - \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\omega_c}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_crossover -#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_crossover.png]] - -#+begin_src matlab :exports none -%% Take complex matrix corresponding to the plant at 100Hz -V = squeeze(frf_coupled(i_wc,:,:)); - -%% Real approximation of inv(G(100Hz)) -D = pinv(real(V'*V)); -H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); - -%% Compute the decoupled plant -G_dL_wc = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_wc(i,:,:) = squeeze(frf_coupled(i,:,:))*H1; -end -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(H1, {}, {}, ' %.1f '); -#+end_src - -#+RESULTS: -| 67229.8 | 3769.3 | -13704.6 | -23084.8 | -6318.2 | 23378.7 | -| 3486.2 | 67708.9 | 23220.0 | -6314.5 | -22699.8 | -14060.6 | -| -5731.7 | 22471.7 | 66701.4 | 3070.2 | -13205.6 | -21944.6 | -| -23305.5 | -14542.6 | 2743.2 | 70097.6 | 24846.8 | -5295.0 | -| -14882.9 | -22957.8 | -5344.4 | 25786.2 | 70484.6 | 2979.9 | -| 24353.3 | -5195.2 | -22449.0 | -14459.2 | 2203.6 | 69484.2 | - -#+begin_src matlab :exports none -%% Bode plot of the plant decoupled at the crossover -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_wc(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -for i = 1:6 - plot(frf_iff.f, abs(G_dL_wc(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_wc(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_wc(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_wc_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_wc_plant -#+caption: Bode Plot of the plant decoupled at the crossover -#+RESULTS: -[[file:figs/test_nhexa_interaction_wc_plant.png]] - -#+begin_src matlab -%% Compute RGA Matrix -RGA_wc = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_wc(i,:,:) = squeeze(G_dL_wc(i,:,:)).*inv(squeeze(G_dL_wc(i,:,:))).'; -end - -%% Compute RGA-number -RGA_wc_sum = zeros(size(RGA_wc, 1), 1); -for i = 1:size(RGA_wc, 1) - RGA_wc_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-Number for the plant decoupled at crossover -figure; -plot(frf_iff.f, RGA_wc_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_wc.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_wc -#+caption: RGA number for the plant decoupled at the crossover -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_wc.png]] - -**** SVD Decoupling -<> - -#+begin_src latex :file decoupling_arch_svd.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - - \node[block, left=0.8 of G.west] (V) {$V^{-T}$}; - \node[block, right=0.8 of G.east] (U) {$U^{-1}$}; - - % Connections and labels - \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$}; - \draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- (U.west) node[above left]{$d\bm{\mathcal{L}}$}; - \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$}; - - \begin{scope}[on background layer] - \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {}; - \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_svd -#+caption: Decoupling using the Singular Value Decomposition -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_svd.png]] - -#+begin_src matlab :exports none -%% Take complex matrix corresponding to the plant at 100Hz -V = squeeze(frf_coupled(i_wc,:,:)); - -%% Real approximation of G(100Hz) -D = pinv(real(V'*V)); -H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); - -%% Singular Value Decomposition -[U,S,V] = svd(H1); - -%% Compute the decoupled plant using SVD -G_dL_svd = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_svd(i,:,:) = inv(U)*squeeze(frf_coupled(i,:,:))*inv(V'); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_svd_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_svd_plant -#+caption: Bode Plot of the plant decoupled using the Singular Value Decomposition -#+RESULTS: -[[file:figs/test_nhexa_interaction_svd_plant.png]] - -#+begin_src matlab -%% Compute the RGA matrix for the SVD decoupled plant -RGA_svd = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_svd(i,:,:) = squeeze(G_dL_svd(i,:,:)).*inv(squeeze(G_dL_svd(i,:,:))).'; -end - -%% Compute the RGA-number -RGA_svd_sum = zeros(size(RGA_svd, 1), 1); -for i = 1:length(frf_iff.f) - RGA_svd_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd(i,:,:))))); -end -#+end_src - -#+begin_src matlab -%% RGA Number for the SVD decoupled plant -figure; -plot(frf_iff.f, RGA_svd_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_svd.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_svd -#+caption: RGA number for the plant decoupled using the SVD -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_svd.png]] - -**** Dynamic decoupling -<> - -#+begin_src latex :file decoupling_arch_dynamic.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}^{-1}$}; - - % Connections and labels - \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; - \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\text{inv}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_dynamic -#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_dynamic.png]] - -#+begin_src matlab :exports none -%% Compute the plant inverse from the model -G_model = G_coupled; -G_model.outputdelay = 0; % necessary for further inversion -G_inv = inv(G_model); - -%% Compute the decoupled plant -G_dL_inv = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_inv(i,:,:) = squeeze(frf_coupled(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the decoupled plant by full inversion -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_inv(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(G_dL_inv(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_inv(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-4, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_inv(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_dynamic_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_dynamic_dec_plant -#+caption: Bode Plot of the dynamically decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_dynamic_dec_plant.png]] - -#+begin_src matlab :exports none -%% Compute the RGA matrix for the inverse based decoupled plant -RGA_inv = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_inv(i,:,:) = squeeze(G_dL_inv(i,:,:)).*inv(squeeze(G_dL_inv(i,:,:))).'; -end - -%% Compute the RGA-number -RGA_inv_sum = zeros(size(RGA_inv, 1), 1); -for i = 1:size(RGA_inv, 1) - RGA_inv_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% RGA Number for the decoupled plant using full inversion -figure; -plot(frf_iff.f, RGA_inv_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_dynamic_dec.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_dynamic_dec -#+caption: RGA number for the dynamically decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_dynamic_dec.png]] - -**** Jacobian Decoupling - Center of Stiffness -<> - -#+begin_src latex :file decoupling_arch_jacobian_cok.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Jt) {$J_{s,\{K\}}^{-T}$}; - \node[block, right=0.8 of G] (Ja) {$J_{a,\{K\}}^{-1}$}; - - % Connections and labels - \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$}; - \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; - \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$}; - - \begin{scope}[on background layer] - \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_jacobian_cok -#+caption: Decoupling using Jacobian matrices evaluated at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_jacobian_cok.png]] - -#+begin_src matlab :exports none -%% Initialize the Nano-Hexapod -n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... - 'motion_sensor_type', 'plates'); - -%% Get the Jacobians -J_cok = n_hexapod.geometry.J; -Js_cok = n_hexapod.geometry.Js; - -%% Decouple plant using Jacobian (CoM) -G_dL_J_cok = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(frf_coupled(i,:,:))*inv(J_cok'); -end -#+end_src - -The obtained plant is shown in Figure ref:fig:test_nhexa_interaction_J_cok_plant_not_normalized. -We can see that the stiffness in the $x$, $y$ and $z$ directions are equal, which is due to the cubic architecture of the Stewart platform. - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 2e-2]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_J_cok_plant_not_normalized.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_J_cok_plant_not_normalized -#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" -#+RESULTS: -[[file:figs/test_nhexa_interaction_J_cok_plant_not_normalized.png]] - -Because the plant in translation and rotation has very different gains, we choose to normalize the plant inputs such that the gain of the diagonal term is equal to $1$ at 100Hz. - -The results is shown in Figure ref:fig:test_nhexa_interaction_J_cok_plant. -#+begin_src matlab :exports none -%% Normalize the plant input -[~, i_100] = min(abs(frf_iff.f - 100)); -input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); - -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-4, 1e1]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_J_cok_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_J_cok_plant -#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" -#+RESULTS: -[[file:figs/test_nhexa_interaction_J_cok_plant.png]] - -#+begin_src matlab :exports none -%% Compute RGA Matrix -RGA_cok = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:)).*inv(squeeze(G_dL_J_cok(i,:,:))).'; -end - -%% Compute RGA-number -RGA_cok_sum = zeros(length(frf_iff.f), 1); -for i = 1:length(frf_iff.f) - RGA_cok_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-Number for the Jacobian (CoK) decoupled plant -figure; -plot(frf_iff.f, RGA_cok_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_J_cok.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_J_cok -#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_J_cok.png]] - -**** Jacobian Decoupling - Center of Mass -<> - -#+begin_src latex :file decoupling_arch_jacobian_com.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Jt) {$J_{s,\{M\}}^{-T}$}; - \node[block, right=0.8 of G] (Ja) {$J_{a,\{M\}}^{-1}$}; - - % Connections and labels - \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; - \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; - \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; - - \begin{scope}[on background layer] - \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_jacobian_com -#+caption: Decoupling using Jacobian matrices evaluated at the Center of Mass -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_jacobian_com.png]] - -#+begin_src matlab :exports none -%% Initialize the Nano-Hexapod -n_hexapod = initializeNanoHexapodFinal('MO_B', 25e-3, ... - 'motion_sensor_type', 'plates'); - -%% Get the Jacobians -J_com = n_hexapod.geometry.J; -Js_com = n_hexapod.geometry.Js; - -%% Decouple plant using Jacobian (CoM) -G_dL_J_com = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_J_com(i,:,:) = inv(Js_com)*squeeze(frf_coupled(i,:,:))*inv(J_com'); -end - -%% Normalize the plant input -[~, i_100] = min(abs(frf_iff.f - 100)); -input_normalize = diag(1./diag(abs(squeeze(G_dL_J_com(i_100,:,:))))); - -for i = 1:length(frf_iff.f) - G_dL_J_com(i,:,:) = squeeze(G_dL_J_com(i,:,:))*input_normalize; -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_com(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_com(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_com(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_com(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_com(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_com(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_com(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_com(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_com(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_J_com_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_J_com_plant -#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the Center of Mass -#+RESULTS: -[[file:figs/test_nhexa_interaction_J_com_plant.png]] - -#+begin_src matlab :exports none -%% Compute RGA Matrix -RGA_com = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_com(i,:,:) = squeeze(G_dL_J_com(i,:,:)).*inv(squeeze(G_dL_J_com(i,:,:))).'; -end - -%% Compute RGA-number -RGA_com_sum = zeros(size(RGA_com, 1), 1); -for i = 1:size(RGA_com, 1) - RGA_com_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_com(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-Number for the Jacobian (CoM) decoupled plant -figure; -plot(frf_iff.f, RGA_com_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_J_com.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_J_com -#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Mass -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_J_com.png]] - -**** Decoupling Comparison -<> - -Let's now compare all of the decoupling methods (Figure ref:fig:test_nhexa_interaction_compare_rga_numbers). - -#+begin_important -From Figure ref:fig:test_nhexa_interaction_compare_rga_numbers, the following remarks are made: -- *Decentralized plant*: well decoupled below suspension modes -- *Static inversion*: similar to the decentralized plant as the decentralized plant has already a good decoupling at low frequency -- *Crossover inversion*: the decoupling is improved around the crossover frequency as compared to the decentralized plant. However, the decoupling is increased at lower frequency. -- *SVD decoupling*: Very good decoupling up to 235Hz. Especially between 100Hz and 200Hz. -- *Dynamic Inversion*: the plant is very well decoupled at frequencies where the model is accurate (below 235Hz where flexible modes are not modelled). -- *Jacobian - Stiffness*: good decoupling at low frequency. The decoupling increases at the frequency of the suspension modes, but is acceptable up to the strut flexible modes (235Hz). -- *Jacobian - Mass*: bad decoupling at low frequency. Better decoupling above the frequency of the suspension modes, and acceptable decoupling up to the strut flexible modes (235Hz). -#+end_important - -#+begin_src matlab :exports none -%% Comparison of the RGA-Numbers -figure; -hold on; -plot(frf_iff.f, RGA_dec_sum, 'DisplayName', 'Decentralized'); -plot(frf_iff.f, RGA_sta_sum, 'DisplayName', 'Static inv.'); -plot(frf_iff.f, RGA_wc_sum, 'DisplayName', 'Crossover inv.'); -plot(frf_iff.f, RGA_svd_sum, 'DisplayName', 'SVD'); -plot(frf_iff.f, RGA_inv_sum, 'DisplayName', 'Dynamic inv.'); -plot(frf_iff.f, RGA_cok_sum, 'DisplayName', 'Jacobian - CoK'); -plot(frf_iff.f, RGA_com_sum, 'DisplayName', 'Jacobian - CoM'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_compare_rga_numbers.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_compare_rga_numbers -#+caption: Comparison of the obtained RGA-numbers for all the decoupling methods -#+RESULTS: -[[file:figs/test_nhexa_interaction_compare_rga_numbers.png]] - -**** Decoupling Robustness -<> - -Let's now see how the decoupling is changing when changing the payload's mass. -#+begin_src matlab -frf_new = frf_iff.G_dL{3}; -#+end_src - -#+begin_src matlab :exports none -%% Decentralized RGA -RGA_dec_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_dec_b(i,:,:) = squeeze(frf_new(i,:,:)).*inv(squeeze(frf_new(i,:,:))).'; -end - -RGA_dec_sum_b = zeros(length(frf_iff), 1); -for i = 1:length(frf_iff.f) - RGA_dec_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Static Decoupling -G_dL_sta_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_sta_b(i,:,:) = squeeze(frf_new(i,:,:))*dc_inv; -end - -RGA_sta_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_sta_b(i,:,:) = squeeze(G_dL_sta_b(i,:,:)).*inv(squeeze(G_dL_sta_b(i,:,:))).'; -end - -RGA_sta_sum_b = zeros(size(RGA_sta_b, 1), 1); -for i = 1:size(RGA_sta_b, 1) - RGA_sta_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Crossover Decoupling -V = squeeze(frf_coupled(i_wc,:,:)); -D = pinv(real(V'*V)); -H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); - -G_dL_wc_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_wc_b(i,:,:) = squeeze(frf_new(i,:,:))*H1; -end - -RGA_wc_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_wc_b(i,:,:) = squeeze(G_dL_wc_b(i,:,:)).*inv(squeeze(G_dL_wc_b(i,:,:))).'; -end - -RGA_wc_sum_b = zeros(size(RGA_wc_b, 1), 1); -for i = 1:size(RGA_wc_b, 1) - RGA_wc_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% SVD -V = squeeze(frf_coupled(i_wc,:,:)); -D = pinv(real(V'*V)); -H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); -[U,S,V] = svd(H1); - -G_dL_svd_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_svd_b(i,:,:) = inv(U)*squeeze(frf_new(i,:,:))*inv(V'); -end - -RGA_svd_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_svd_b(i,:,:) = squeeze(G_dL_svd_b(i,:,:)).*inv(squeeze(G_dL_svd_b(i,:,:))).'; -end - -RGA_svd_sum_b = zeros(size(RGA_svd_b, 1), 1); -for i = 1:size(RGA_svd, 1) - RGA_svd_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Dynamic Decoupling -G_model = G_coupled; -G_model.outputdelay = 0; % necessary for further inversion -G_inv = inv(G_model); - -G_dL_inv_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_inv_b(i,:,:) = squeeze(frf_new(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); -end - -RGA_inv_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_inv_b(i,:,:) = squeeze(G_dL_inv_b(i,:,:)).*inv(squeeze(G_dL_inv_b(i,:,:))).'; -end - -RGA_inv_sum_b = zeros(size(RGA_inv_b, 1), 1); -for i = 1:size(RGA_inv_b, 1) - RGA_inv_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Jacobian (CoK) -G_dL_J_cok_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_J_cok_b(i,:,:) = inv(Js_cok)*squeeze(frf_new(i,:,:))*inv(J_cok'); -end - -RGA_cok_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_cok_b(i,:,:) = squeeze(G_dL_J_cok_b(i,:,:)).*inv(squeeze(G_dL_J_cok_b(i,:,:))).'; -end - -RGA_cok_sum_b = zeros(size(RGA_cok_b, 1), 1); -for i = 1:size(RGA_cok_b, 1) - RGA_cok_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Jacobian (CoM) -G_dL_J_com_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_J_com_b(i,:,:) = inv(Js_com)*squeeze(frf_new(i,:,:))*inv(J_com'); -end - -RGA_com_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_com_b(i,:,:) = squeeze(G_dL_J_com_b(i,:,:)).*inv(squeeze(G_dL_J_com_b(i,:,:))).'; -end - -RGA_com_sum_b = zeros(size(RGA_com_b, 1), 1); -for i = 1:size(RGA_com_b, 1) - RGA_com_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_com_b(i,:,:))))); -end -#+end_src - -The obtained RGA-numbers are shown in Figure ref:fig:test_nhexa_interaction_compare_rga_numbers_rob. - -#+begin_important -From Figure ref:fig:test_nhexa_interaction_compare_rga_numbers_rob: -- The decoupling using the Jacobian evaluated at the "center of stiffness" seems to give the most robust results. -#+end_important - -#+begin_src matlab :exports none -%% Robustness of the Decoupling method -figure; -hold on; -plot(frf_iff.f, RGA_dec_sum, '-', 'DisplayName', 'Decentralized'); -plot(frf_iff.f, RGA_sta_sum, '-', 'DisplayName', 'Static inv.'); -plot(frf_iff.f, RGA_wc_sum, '-', 'DisplayName', 'Crossover inv.'); -plot(frf_iff.f, RGA_svd_sum, '-', 'DisplayName', 'SVD'); -plot(frf_iff.f, RGA_inv_sum, '-', 'DisplayName', 'Dynamic inv.'); -plot(frf_iff.f, RGA_cok_sum, '-', 'DisplayName', 'Jacobian - CoK'); -plot(frf_iff.f, RGA_com_sum, '-', 'DisplayName', 'Jacobian - CoM'); -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, RGA_dec_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_sta_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_wc_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_svd_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_inv_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_cok_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_com_sum_b, '--', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_compare_rga_numbers_rob.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_compare_rga_numbers_rob -#+caption: Change of the RGA-number with a change of the payload. Indication of the robustness of the inversion method. -#+RESULTS: -[[file:figs/test_nhexa_interaction_compare_rga_numbers_rob.png]] - -**** Conclusion - -#+begin_important -Several decoupling methods can be used: -- SVD -- Inverse -- Jacobian (CoK) -#+end_important - -#+name: tab:interaction_analysis_conclusion -#+caption: Summary of the interaction analysis and different decoupling strategies -#+attr_latex: :environment tabularx :width \linewidth :align lccc -#+attr_latex: :center t :booktabs t -| *Method* | *RGA* | *Diag Plant* | *Robustness* | -|----------------+-------+--------------+--------------| -| Decentralized | -- | Equal | ++ | -| Static dec. | -- | Equal | ++ | -| Crossover dec. | - | Equal | 0 | -| SVD | ++ | Diff | + | -| Dynamic dec. | ++ | Unity, equal | - | -| Jacobian - CoK | + | Diff | ++ | -| Jacobian - CoM | 0 | Diff | + | - -*** Robust High Authority Controller -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/hac_lac_enc_plates_suspended_table.m -:END: -<> -**** Introduction :ignore: -In this section we wish to develop a robust High Authority Controller (HAC) that is working for all payloads. - -cite:indri20_mechat_robot - -**** Matlab Init :noexport:ignore: -#+begin_src matlab -%% hac_lac_enc_plates_suspended_table.m -% Development and analysis of a robust High Authority Controller -#+end_src - -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -#+begin_src matlab :tangle no :noweb yes -<> -#+end_src - -#+begin_src matlab :eval no :noweb yes -<> -#+end_src - -#+begin_src matlab :noweb yes -<> -#+end_src - -#+begin_src matlab -%% Load the identified FRF and Simscape model -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -**** Using Jacobian evaluated at the center of stiffness -***** Decoupled Plant -#+begin_src matlab -G_nom = frf_iff.G_dL{2}; % Nominal Plant -#+end_src - -#+begin_src matlab :exports none -%% Initialize the Nano-Hexapod -n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... - 'motion_sensor_type', 'plates'); - -%% Get the Jacobians -J_cok = n_hexapod.geometry.J; -Js_cok = n_hexapod.geometry.Js; - -%% Decouple plant using Jacobian (CoM) -G_dL_J_cok = zeros(size(G_nom)); -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(G_nom(i,:,:))*inv(J_cok'); -end - -%% Normalize the plant input -[~, i_100] = min(abs(frf_iff.f - 10)); -input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); - -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_plant_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_plant_jacobian_cok -#+caption: Bode plot of the decoupled plant using the Jacobian evaluated at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_plant_jacobian_cok.png]] - -***** SISO Controller Design -As the diagonal elements of the plant are not equal, several SISO controllers are designed and then combined to form a diagonal controller. -All the diagonal terms of the controller consists of: -- A double integrator to have high gain at low frequency -- A lead around the crossover frequency to increase stability margins -- Two second order low pass filters above the crossover frequency to increase the robustness to high frequency modes - -#+begin_src matlab :exports none -%% Controller Ry,Rz - -% Wanted crossover frequency -wc_Rxy = 2*pi*80; - -% Lead -a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = wc_Rxy; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -% Integrator -w0_int = wc_Rxy/2; % [rad/s] -xi_int = 0.3; - -Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); - -% Low Pass Filter (High frequency robustness) -w0_lpf = wc_Rxy*2; % Cut-off frequency [rad/s] -xi_lpf = 0.6; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc_Rxy*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -% Unity Gain frequency -[~, i_80] = min(abs(frf_iff.f - wc_Rxy/2/pi)); - -% Combination of all the elements -Kd_Rxy = ... - -1/abs(G_dL_J_cok(i_80,4,4)) * ... - Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Rxy)) * ... % Lead (gain of 1 at wc) - Kd_int /abs(evalfr(Kd_int, 1j*wc_Rxy)) * ... - Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Rxy)) * ... - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Rxy)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% Controller Dx,Dy,Rz - -% Wanted crossover frequency -wc_Dxy = 2*pi*100; - -% Lead -a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = wc_Dxy; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -% Integrator -w0_int = wc_Dxy/2; % [rad/s] -xi_int = 0.3; - -Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); - -% Low Pass Filter (High frequency robustness) -w0_lpf = wc_Dxy*2; % Cut-off frequency [rad/s] -xi_lpf = 0.6; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc_Dxy*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -% Unity Gain frequency -[~, i_100] = min(abs(frf_iff.f - wc_Dxy/2/pi)); - -% Combination of all the elements -Kd_Dyx_Rz = ... - -1/abs(G_dL_J_cok(i_100,1,1)) * ... - Kd_int /abs(evalfr(Kd_int, 1j*wc_Dxy)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) - Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dxy)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% Controller Dz - -% Wanted crossover frequency -wc_Dz = 2*pi*100; - -% Lead -a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = wc_Dz; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -% Integrator -w0_int = wc_Dz/2; % [rad/s] -xi_int = 0.3; - -Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); - -% Low Pass Filter (High frequency robustness) -w0_lpf = wc_Dz*2; % Cut-off frequency [rad/s] -xi_lpf = 0.6; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc_Dz*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -% Unity Gain frequency -[~, i_100] = min(abs(frf_iff.f - wc_Dz/2/pi)); - -% Combination of all the elements -Kd_Dz = ... - -1/abs(G_dL_J_cok(i_100,3,3)) * ... - Kd_int /abs(evalfr(Kd_int, 1j*wc_Dz)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) - Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dz)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% Diagonal Controller -Kd_diag = blkdiag(Kd_Dyx_Rz, Kd_Dyx_Rz, Kd_Dz, Kd_Rxy, Kd_Rxy, Kd_Dyx_Rz); -#+end_src - -***** Obtained Loop Gain -#+begin_src matlab :exports none -%% Experimental Loop Gain -Lmimo = permute(pagemtimes(permute(G_dL_J_cok, [2,3,1]), squeeze(freqresp(Kd_diag, frf_iff.f, 'Hz'))), [3,1,2]); -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the experimental Loop Gain -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e+3]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:45:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_jacobian_cok -#+caption: Bode plot of the Loop Gain when using the Jacobian evaluated at the Center of Stiffness to decouple the system -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_jacobian_cok.png]] - -#+begin_src matlab -%% Controller to be implemented -Kd = inv(J_cok')*input_normalize*ss(Kd_diag)*inv(Js_cok); -#+end_src - -***** Verification of the Stability -Now the stability of the feedback loop is verified using the generalized Nyquist criteria. - -#+begin_src matlab :exports none -%% Compute the Eigenvalues of the loop gain -Ldet = zeros(3, 6, length(frf_iff.f)); - -for i_mass = 1:3 - % Loop gain - Lmimo = pagemtimes(permute(frf_iff.G_dL{i_mass}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); - for i_f = 2:length(frf_iff.f) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Plot of the eigenvalues of L in the complex plane -figure; -hold on; -for i_mass = 2:3 - plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('%i masses', i_mass)); - plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end -plot(-1, 0, 'kx', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -legend('location', 'southeast'); -xlim([-3, 1]); ylim([-2, 2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loci_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loci_hac_iff_loop_gain_jacobian_cok -#+caption: Loci of $L(j\omega)$ in the complex plane. -#+RESULTS: -[[file:figs/test_nhexa_loci_hac_iff_loop_gain_jacobian_cok.png]] - -***** Save for further analysis -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') -#+end_src - -***** Sensitivity transfer function from the model -#+begin_src matlab :exports none -%% Open Simulink Model -mdl = 'nano_hexapod_simscape'; - -options = linearizeOptions; -options.SampleTime = 0; - -open(mdl) - -Rx = zeros(1, 7); - -colors = colororder; -#+end_src - -#+begin_src matlab :exports none -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'hac-iff-struts'); - -support.type = 1; % On top of vibration table -payload.type = 2; % Payload -#+end_src - -#+begin_src matlab :exports none -%% Load controllers -load('Kiff_opt.mat', 'Kiff'); -Kiff = c2d(Kiff, Ts, 'Tustin'); -load('Khac_iff_struts_jacobian_cok.mat', 'Kd') -Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); -#+end_src - -#+begin_src matlab :exports none -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -#+begin_src matlab :exports none -%% Identification of the dynamics -Gcl = linearize(mdl, io, 0.0, options); -#+end_src - -#+begin_src matlab :exports none -%% Computation of the sensitivity transfer function -S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; -#+end_src - -The results are shown in Figure ref:fig:test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -freqs = logspace(0, 3, 1000); - -figure; -hold on; -for i =1:6 - set(gca,'ColorOrderIndex',i); - plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... - 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); -ylim([1e-4, 1e1]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -xlim([1, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/sensitivity_hac_jacobian_cok_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model -#+caption: Estimated sensitivity transfer functions for the HAC controller using the Jacobian estimated at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model.png]] - -**** Using Singular Value Decomposition -***** Decoupled Plant -#+begin_src matlab -G_nom = frf_iff.G_dL{2}; % Nominal Plant -#+end_src - -#+begin_src matlab :exports none -%% Take complex matrix corresponding to the plant at 100Hz -wc = 100; % Wanted crossover frequency [Hz] -[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc - -V = squeeze(G_nom(i_wc,:,:)); - -%% Real approximation of G(100Hz) -D = pinv(real(V'*V)); -H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); - -%% Singular Value Decomposition -[U,S,V] = svd(H1); - -%% Compute the decoupled plant using SVD -G_dL_svd = zeros(size(G_nom)); -for i = 1:length(frf_iff.f) - G_dL_svd(i,:,:) = inv(U)*squeeze(G_nom(i,:,:))*inv(V'); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the decoupled plant using SVD -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,i,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, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_plant_svd.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_plant_svd -#+caption: Bode plot of the decoupled plant using the SVD -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_plant_svd.png]] - -***** Controller Design -#+begin_src matlab :exports none -%% Lead -a = 6.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -%% Integrator -Kd_int = ((2*pi*50 + s)/(2*pi*0.1 + s))^2; - -%% Low Pass Filter (High frequency robustness) -w0_lpf = 2*pi*200; % Cut-off frequency [rad/s] -xi_lpf = 0.3; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -%% Normalize Gain -Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); - -%% Diagonal Control -Kd_diag = ... - Kd_norm * ... % Normalize gain at 100Hz - Kd_int /abs(evalfr(Kd_int, 1j*2*pi*100)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*2*pi*100)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*2*pi*100)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% MIMO Controller -Kd = -inv(V') * ... % Output decoupling - ss(Kd_diag) * ... - inv(U); % Input decoupling -#+end_src - -***** Loop Gain -#+begin_src matlab :exports none -%% Experimental Loop Gain -Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); -#+end_src - -#+begin_src matlab :exports none -%% Loop gain when using SVD -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e+3]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:30:360); -ylim([-180, 0]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_svd -#+caption: Bode plot of Loop Gain when using the SVD -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_svd.png]] - -***** Stability Verification -#+begin_src matlab -%% Compute the Eigenvalues of the loop gain -Ldet = zeros(3, 6, length(frf_iff.f)); - -for i = 1:3 - Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); - for i_f = 2:length(frf_iff.f) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Plot of the eigenvalues of L in the complex plane -figure; -hold on; -for i_mass = 2:3 - plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('%i masses', i_mass)); - plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end -plot(-1, 0, 'kx', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -legend('location', 'southeast'); -xlim([-3, 1]); ylim([-2, 2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loci_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loci_hac_iff_loop_gain_svd -#+caption: Locis of $L(j\omega)$ in the complex plane. -#+RESULTS: -[[file:figs/test_nhexa_loci_hac_iff_loop_gain_svd.png]] - -***** Save for further analysis -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts_svd.mat', 'Kd') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts_svd.mat', 'Kd') -#+end_src - -***** Measured Sensitivity Transfer Function -The sensitivity transfer function is estimated by adding a reference signal $R_x$ consisting of a low pass filtered white noise, and measuring the position error $E_x$ at the same time. - -The transfer function from $R_x$ to $E_x$ is the sensitivity transfer function. - -In order to identify the sensitivity transfer function for all directions, six reference signals are used, one for each direction. - -#+begin_src matlab :exports none -%% Tested directions -labels = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; -#+end_src - -#+begin_src matlab :exports none -%% Load Identification Data -meas_hac_svd_3m = {}; - -for i = 1:6 - meas_hac_svd_3m(i) = {load(sprintf('T_S_meas_%s_3m_hac_svd_iff.mat', labels{i}), 't', 'Va', 'Vs', 'de', 'Rx')}; -end -#+end_src - -#+begin_src matlab :exports none -%% Setup useful variables -% Sampling Time [s] -Ts = (meas_hac_svd_3m{1}.t(end) - (meas_hac_svd_3m{1}.t(1)))/(length(meas_hac_svd_3m{1}.t)-1); - -% Sampling Frequency [Hz] -Fs = 1/Ts; - -% Hannning Windows -win = hanning(ceil(5*Fs)); - -% And we get the frequency vector -[~, f] = tfestimate(meas_hac_svd_3m{1}.Va, meas_hac_svd_3m{1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src - -#+begin_src matlab :exports none -%% Load Jacobian matrix -load('jacobian.mat', 'J'); - -%% Compute position error -for i = 1:6 - meas_hac_svd_3m{i}.Xm = [inv(J)*meas_hac_svd_3m{i}.de']'; - meas_hac_svd_3m{i}.Ex = meas_hac_svd_3m{i}.Rx - meas_hac_svd_3m{i}.Xm; -end -#+end_src - -An example is shown in Figure ref:fig:test_nhexa_ref_track_hac_svd_3m where both the reference signal and the measured position are shown for translations in the $x$ direction. - -#+begin_src matlab :exports none -figure; -hold on; -plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Xm(:,1), 'DisplayName', 'Pos.') -plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Rx(:,1), 'DisplayName', 'Ref.') -hold off; -xlabel('Time [s]'); ylabel('Dx motion [m]'); -xlim([20, 22]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_ref_track_hac_svd_3m -#+caption: Reference position and measured position -#+RESULTS: -[[file:figs/test_nhexa_ref_track_hac_svd_3m.png]] - -#+begin_src matlab :exports none -%% Transfer function estimate of S -S_hac_svd_3m = zeros(length(f), 6, 6); - -for i = 1:6 - S_hac_svd_3m(:,:,i) = tfestimate(meas_hac_svd_3m{i}.Rx, meas_hac_svd_3m{i}.Ex, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -The sensitivity transfer functions estimated for all directions are shown in Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -hold on; -for i =1:6 - plot(f, abs(S_hac_svd_3m(:,i,i)), ... - 'DisplayName', sprintf('$S_{%s}$', labels{i})); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); -ylim([1e-4, 1e1]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -xlim([0.5, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/sensitivity_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_sensitivity_hac_svd_3m -#+caption: Measured diagonal elements of the sensitivity transfer function matrix. -#+RESULTS: -[[file:figs/test_nhexa_sensitivity_hac_svd_3m.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m: -- The sensitivity transfer functions are similar for all directions -- The disturbance attenuation at 1Hz is almost a factor 1000 as wanted -- The sensitivity transfer functions for $R_x$ and $R_y$ have high peak values which indicate poor stability margins. -#+end_important - -***** Sensitivity transfer function from the model -The sensitivity transfer function is now estimated using the model and compared with the one measured. - -#+begin_src matlab :exports none -%% Open Simulink Model -mdl = 'nano_hexapod_simscape'; - -options = linearizeOptions; -options.SampleTime = 0; - -open(mdl) - -Rx = zeros(1, 7); - -colors = colororder; -#+end_src - -#+begin_src matlab :exports none -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'hac-iff-struts'); - -support.type = 1; % On top of vibration table -payload.type = 2; % Payload -#+end_src - -#+begin_src matlab :exports none -%% Load controllers -load('Kiff_opt.mat', 'Kiff'); -Kiff = c2d(Kiff, Ts, 'Tustin'); -load('Khac_iff_struts_svd.mat', 'Kd') -Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); -#+end_src - -#+begin_src matlab :exports none -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -#+begin_src matlab :exports none -%% Identification of the dynamics -Gcl = linearize(mdl, io, 0.0, options); -#+end_src - -#+begin_src matlab :exports none -%% Computation of the sensitivity transfer function -S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; -#+end_src - -The results are shown in Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m_comp_model. -The model is quite effective in estimating the sensitivity transfer functions except around 60Hz were there is a peak for the measurement. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -freqs = logspace(0,3,1000); - -figure; -hold on; -for i =1:6 - set(gca,'ColorOrderIndex',i); - plot(f, abs(S_hac_svd_3m(:,i,i)), ... - 'DisplayName', sprintf('$S_{%s}$', labels{i})); - set(gca,'ColorOrderIndex',i); - plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... - 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); -ylim([1e-4, 1e1]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -xlim([0.5, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/sensitivity_hac_svd_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_sensitivity_hac_svd_3m_comp_model -#+caption: Comparison of the measured sensitivity transfer functions with the model -#+RESULTS: -[[file:figs/test_nhexa_sensitivity_hac_svd_3m_comp_model.png]] - -**** Using (diagonal) Dynamical Inverse :noexport: -***** Decoupled Plant -#+begin_src matlab -G_nom = frf_iff.G_dL{2}; % Nominal Plant -G_model = sim_iff.G_dL{2}; % Model of the Plant -#+end_src - -#+begin_src matlab :exports none -%% Simplified model of the diagonal term -balred_opts = balredOptions('FreqIntervals', 2*pi*[0, 1000], 'StateElimMethod', 'Truncate'); - -G_red = balred(G_model(1,1), 8, balred_opts); -G_red.outputdelay = 0; % necessary for further inversion -#+end_src - -#+begin_src matlab -%% Inverse -G_inv = inv(G_red); -[G_z, G_p, G_g] = zpkdata(G_inv); -p_uns = real(G_p{1}) > 0; -G_p{1}(p_uns) = -G_p{1}(p_uns); -G_inv_stable = zpk(G_z, G_p, G_g); -#+end_src - -#+begin_src matlab :exports none -%% "Uncertainty" of inversed plant -freqs = logspace(0,3,1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = i_masses - for i = 1 - plot(freqs, abs(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-1, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); - -ax2 = nexttile; -hold on; -for i_mass = i_masses - for i = 1 - plot(freqs, 180/pi*angle(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:15:360); -ylim([-45, 45]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -***** Controller Design -#+begin_src matlab :exports none -% Wanted crossover frequency -wc = 2*pi*80; -[~, i_wc] = min(abs(frf_iff.f - wc/2/pi)); - -%% Lead -a = 20.0; % Amount of phase lead / width of the phase lead / high frequency gain -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -%% Integrator -Kd_int = ((wc)/(2*pi*0.2 + s))^2; - -%% Low Pass Filter (High frequency robustness) -w0_lpf = 2*wc; % Cut-off frequency [rad/s] -xi_lpf = 0.3; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -%% Normalize Gain -Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); - -%% Diagonal Control -Kd_diag = ... - G_inv_stable * ... % Normalize gain at 100Hz - Kd_int /abs(evalfr(Kd_int, 1j*wc)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*wc)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -Kd = ss(Kd_diag)*eye(6); -#+end_src - -***** Loop Gain -#+begin_src matlab :exports none -%% Experimental Loop Gain -Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); -#+end_src - -#+begin_src matlab :exports none -%% Loop gain when using SVD -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e+3]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:30:360); -ylim([-180, 0]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_diag_inverse -#+caption: Bode plot of Loop Gain when using the Diagonal inversion -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_diag_inverse.png]] - -***** Stability Verification -MIMO Nyquist with eigenvalues -#+begin_src matlab -%% Compute the Eigenvalues of the loop gain -Ldet = zeros(3, 6, length(frf_iff.f)); - -for i = 1:3 - Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); - for i_f = 2:length(frf_iff.f) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Plot of the eigenvalues of L in the complex plane -figure; -hold on; -for i_mass = 2:3 - plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('%i masses', i_mass)); - plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end -plot(-1, 0, 'kx', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -legend('location', 'southeast'); -xlim([-3, 1]); ylim([-2, 2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loci_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loci_hac_iff_loop_gain_diag_inverse -#+caption: Locis of $L(j\omega)$ in the complex plane. -#+RESULTS: -[[file:figs/test_nhexa_loci_hac_iff_loop_gain_diag_inverse.png]] - -#+begin_important -Even though the loop gain seems to be fine, the closed-loop system is unstable. -This might be due to the fact that there is large interaction in the plant. -We could look at the RGA-number to verify that. -#+end_important - -***** Save for further use -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') -#+end_src - -**** Closed Loop Stability (Model) :noexport: -Verify stability using Simscape model -#+begin_src matlab -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ... - 'flex_top_type', '3dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'hac-iff-struts'); -#+end_src - -#+begin_src matlab -%% IFF Controller -Kiff = -g_opt*Kiff_g1*eye(6); -Khac_iff_struts = Kd*eye(6); -#+end_src - -#+begin_src matlab -%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain -clear io; io_i = 1; -io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -#+begin_src matlab -GG_cl = {}; - -for i = i_masses - payload.type = i; - GG_cl(i+1) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)}; -end -#+end_src - -#+begin_src matlab -for i = i_masses - isstable(GG_cl{i+1}) -end -#+end_src - -MIMO Nyquist -#+begin_src matlab -Kdm = Kd*eye(6); - -Ldet = zeros(3, length(fb(i_lim))); - -for i = 1:3 - Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); - Ldet(i,:) = arrayfun(@(t) det(eye(6) + squeeze(Lmimo(:,:,t))), 1:size(Lmimo,3)); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -hold on; -for i_mass = 3 - for i = 1 - plot(real(Ldet(i_mass,:)), imag(Ldet(i_mass,:)), ... - '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -xlim([-10, 1]); ylim([-4, 4]); -#+end_src - -MIMO Nyquist with eigenvalues -#+begin_src matlab -Kdm = Kd*eye(6); - -Ldet = zeros(3, 6, length(fb(i_lim))); - -for i = 1:3 - Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); - for i_f = 1:length(fb(i_lim)) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -hold on; -for i_mass = 1 - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -xlim([-10, 1]); ylim([-4, 2]); -#+end_src -** Other Backups -*** Nano-Hexapod Compliance - Effect of IFF -<> - -In this section, we wish to estimate the effectiveness of the IFF strategy regarding the compliance. - -The top plate is excited vertically using the instrumented hammer two times: -1. no control loop is used -2. decentralized IFF is used - -The data are loaded. -#+begin_src matlab -frf_ol = load('Measurement_Z_axis.mat'); % Open-Loop -frf_iff = load('Measurement_Z_axis_damped.mat'); % IFF -#+end_src - -The mean vertical motion of the top platform is computed by averaging all 5 vertical accelerometers. -#+begin_src matlab -%% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers) -d_frf_ol = 10/5*(frf_ol.FFT1_H1_4_1_RMS_Y_Mod + frf_ol.FFT1_H1_7_1_RMS_Y_Mod + frf_ol.FFT1_H1_10_1_RMS_Y_Mod + frf_ol.FFT1_H1_13_1_RMS_Y_Mod + frf_ol.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_ol.FFT1_H1_16_1_RMS_X_Val).^2; -d_frf_iff = 10/5*(frf_iff.FFT1_H1_4_1_RMS_Y_Mod + frf_iff.FFT1_H1_7_1_RMS_Y_Mod + frf_iff.FFT1_H1_10_1_RMS_Y_Mod + frf_iff.FFT1_H1_13_1_RMS_Y_Mod + frf_iff.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_iff.FFT1_H1_16_1_RMS_X_Val).^2; -#+end_src - -The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure ref:fig:test_nhexa_compliance_vertical_comp_iff. -#+begin_src matlab :exports none -%% Comparison of the vertical compliances (OL and IFF) -figure; -hold on; -plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, 'DisplayName', 'OL'); -plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, 'DisplayName', 'IFF'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); -xlim([20, 2e3]); ylim([2e-9, 2e-5]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/compliance_vertical_comp_iff.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_compliance_vertical_comp_iff -#+caption: Measured vertical compliance with and without IFF -#+RESULTS: -[[file:figs/test_nhexa_compliance_vertical_comp_iff.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_compliance_vertical_comp_iff, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapod. -It also has the effect of (slightly) degrading the vertical compliance at low frequency. - -It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts. -#+end_important - -*** Comparison with the Simscape Model -<> - -Let's initialize the Simscape model such that it corresponds to the experiment. -#+begin_src matlab -%% Nano-Hexapod is fixed on a rigid granite -support.type = 0; - -%% No Payload on top of the Nano-Hexapod -payload.type = 0; - -%% Initialize Nano-Hexapod in Open Loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'struts', ... - 'actuator_type', '2dof'); - -#+end_src - -And let's compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model. - -The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model. -#+begin_src matlab :exports none -%% Identify the IFF Plant (transfer function from u to taum) -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Fz_ext'], 1, 'openinput'); io_i = io_i + 1; % External - Vertical force -io(io_i) = linio([mdl, '/Z_top_plat'], 1, 'openoutput'); io_i = io_i + 1; % Absolute vertical motion of top platform -#+end_src - -#+begin_src matlab :exports none -%% Perform the identifications -G_compl_z_ol = linearize(mdl, io, 0.0, options); -#+end_src - -#+begin_src matlab :exports none -%% Initialize Nano-Hexapod with IFF -Kiff = 400*(1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz) - (s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain - (1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances - eye(6); % Diagonal 6x6 controller - -%% Initialize the Nano-Hexapod with IFF -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'struts', ... - 'actuator_type', '2dof', ... - 'controller_type', 'iff'); - -%% Perform the identification -G_compl_z_iff = linearize(mdl, io, 0.0, options); -#+end_src - -The comparison is done in Figure ref:fig:test_nhexa_compliance_vertical_comp_model_iff. -Again, the model is quite accurate in predicting the (closed-loop) behavior of the system. - -#+begin_src matlab :exports none -%% Comparison of the measured compliance and the one obtained from the model -freqs = 2*logspace(1,3,1000); - -figure; -hold on; -plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, '-', 'DisplayName', 'OL - Meas.'); -plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, '-', 'DisplayName', 'IFF - Meas.'); -set(gca,'ColorOrderIndex',1) -plot(freqs, abs(squeeze(freqresp(G_compl_z_ol, freqs, 'Hz'))), '--', 'DisplayName', 'OL - Model') -plot(freqs, abs(squeeze(freqresp(G_compl_z_iff, freqs, 'Hz'))), '--', 'DisplayName', 'IFF - Model') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); -xlim([20, 2e3]); ylim([2e-9, 2e-5]); -legend('location', 'northeast', 'FontSize', 8); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/compliance_vertical_comp_model_iff.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_compliance_vertical_comp_model_iff -#+caption: Measured vertical compliance with and without IFF -#+RESULTS: -[[file:figs/test_nhexa_compliance_vertical_comp_model_iff.png]] - -*** Computation of the transmissibility from accelerometer data -**** Introduction :ignore: - -The goal is to compute the $6 \times 6$ transfer function matrix corresponding to the transmissibility of the Nano-Hexapod. - -To do so, several accelerometers are located both on the vibration table and on the top of the nano-hexapod. - -The vibration table is then excited using a Shaker and all the accelerometers signals are recorded. - -Using transformation (jacobian) matrices, it is then possible to compute both the motion of the top and bottom platform of the nano-hexapod. - -Finally, it is possible to compute the $6 \times 6$ transmissibility matrix. - -Such procedure is explained in cite:marneffe04_stewar_platf_activ_vibrat_isolat. - -**** Jacobian matrices - -How to compute the Jacobian matrices is explained in Section ref:sec:meas_transformation. - -#+begin_src matlab -%% Bottom Accelerometers -Opb = [-0.1875, -0.1875, -0.245; - -0.1875, -0.1875, -0.245; - 0.1875, -0.1875, -0.245; - 0.1875, -0.1875, -0.245; - 0.1875, 0.1875, -0.245; - 0.1875, 0.1875, -0.245]'; - -Osb = [0, 1, 0; - 0, 0, 1; - 1, 0, 0; - 0, 0, 1; - 1, 0, 0; - 0, 0, 1;]'; - -Jb = zeros(length(Opb), 6); - -for i = 1:length(Opb) - Ri = [0, Opb(3,i), -Opb(2,i); - -Opb(3,i), 0, Opb(1,i); - Opb(2,i), -Opb(1,i), 0]; - Jb(i, 1:3) = Osb(:,i)'; - Jb(i, 4:6) = Osb(:,i)'*Ri; -end - -Jbinv = inv(Jb); -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) -data2orgtable(Jbinv, {'$\dot{x}_x$', '$\dot{x}_y$', '$\dot{x}_z$', '$\dot{\omega}_x$', '$\dot{\omega}_y$', '$\dot{\omega}_z$'}, {'$a_1$', '$a_2$', '$a_3$', '$a_4$', '$a_5$', '$a_6$'}, ' %.1f '); -#+end_src - -#+RESULTS: -| | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | -|------------------+-------+-------+-------+-------+-------+-------| -| $\dot{x}_x$ | 0.0 | 0.7 | 0.5 | -0.7 | 0.5 | 0.0 | -| $\dot{x}_y$ | 1.0 | 0.0 | 0.5 | 0.7 | -0.5 | -0.7 | -| $\dot{x}_z$ | 0.0 | 0.5 | 0.0 | 0.0 | 0.0 | 0.5 | -| $\dot{\omega}_x$ | 0.0 | 0.0 | 0.0 | -2.7 | 0.0 | 2.7 | -| $\dot{\omega}_y$ | 0.0 | 2.7 | 0.0 | -2.7 | 0.0 | 0.0 | -| $\dot{\omega}_z$ | 0.0 | 0.0 | 2.7 | 0.0 | -2.7 | 0.0 | - -#+begin_src matlab -%% Top Accelerometers -Opt = [-0.1, 0, -0.150; - -0.1, 0, -0.150; - 0.05, 0.075, -0.150; - 0.05, 0.075, -0.150; - 0.05, -0.075, -0.150; - 0.05, -0.075, -0.150]'; - -Ost = [0, 1, 0; - 0, 0, 1; - 1, 0, 0; - 0, 0, 1; - 1, 0, 0; - 0, 0, 1;]'; - -Jt = zeros(length(Opt), 6); - -for i = 1:length(Opt) - Ri = [0, Opt(3,i), -Opt(2,i); - -Opt(3,i), 0, Opt(1,i); - Opt(2,i), -Opt(1,i), 0]; - Jt(i, 1:3) = Ost(:,i)'; - Jt(i, 4:6) = Ost(:,i)'*Ri; -end - -Jtinv = inv(Jt); -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) -data2orgtable(Jtinv, {'$\dot{x}_x$', '$\dot{x}_y$', '$\dot{x}_z$', '$\dot{\omega}_x$', '$\dot{\omega}_y$', '$\dot{\omega}_z$'}, {'$b_1$', '$b_2$', '$b_3$', '$b_4$', '$b_5$', '$b_6$'}, ' %.1f '); -#+end_src - -#+RESULTS: -| | $b_1$ | $b_2$ | $b_3$ | $b_4$ | $b_5$ | $b_6$ | -|------------------+-------+-------+-------+-------+-------+-------| -| $\dot{x}_x$ | 0.0 | 1.0 | 0.5 | -0.5 | 0.5 | -0.5 | -| $\dot{x}_y$ | 1.0 | 0.0 | -0.7 | -1.0 | 0.7 | 1.0 | -| $\dot{x}_z$ | 0.0 | 0.3 | 0.0 | 0.3 | 0.0 | 0.3 | -| $\dot{\omega}_x$ | 0.0 | 0.0 | 0.0 | 6.7 | 0.0 | -6.7 | -| $\dot{\omega}_y$ | 0.0 | 6.7 | 0.0 | -3.3 | 0.0 | -3.3 | -| $\dot{\omega}_z$ | 0.0 | 0.0 | -6.7 | 0.0 | 6.7 | 0.0 | - -**** Using =linearize= function - -#+begin_src matlab -acc_3d.type = 2; % 1: inertial mass, 2: perfect - -%% Name of the Simulink File -mdl = 'vibration_table'; - -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/F_shaker'], 1, 'openinput'); io_i = io_i + 1; -io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; -io(io_i) = linio([mdl, '/acc_top'], 1, 'openoutput'); io_i = io_i + 1; - -%% Run the linearization -G = linearize(mdl, io); -G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; -G.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6', ... - 'b1', 'b2', 'b3', 'b4', 'b5', 'b6'}; -#+end_src - -#+begin_src matlab -Gb = Jbinv*G({'a1', 'a2', 'a3', 'a4', 'a5', 'a6'}, :); -Gt = Jtinv*G({'b1', 'b2', 'b3', 'b4', 'b5', 'b6'}, :); -#+end_src - -#+begin_src matlab -T = inv(Gb)*Gt; -T = minreal(T); -T = prescale(T, {2*pi*0.1, 2*pi*1e3}); -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(0, 3, 1000); - -figure; -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(T(i, i), freqs, 'Hz')))); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(T(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Transmissibility'); -ylim([1e-4, 1e2]); -xlim([freqs(1), freqs(end)]); -#+end_src - -*** Comparison with "true" transmissibility - -#+begin_src matlab -%% Name of the Simulink File -mdl = 'test_transmissibility'; - -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/d'], 1, 'openinput'); io_i = io_i + 1; -io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; - -%% Run the linearization -G = linearize(mdl, io); -G.InputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; -G.OutputName = {'Ax', 'Ay', 'Az', 'Bx', 'By', 'Bz'}; -#+end_src - -#+begin_src matlab -Tp = G/s^2; -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(0, 3, 1000); - -figure; -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(Tp(i, i), freqs, 'Hz')))); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Tp(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Transmissibility'); -ylim([1e-4, 1e2]); -xlim([freqs(1), freqs(end)]); -#+end_src - -*** Rigidification of the added payloads -- [ ] figure - -#+begin_src matlab -%% Load Identification Data -meas_added_mass = {}; - -for i_strut = 1:6 - meas_added_mass(i_strut) = {load(sprintf('frf_data_exc_strut_%i_spindle_1m_solid.mat', i_strut), 't', 'Va', 'Vs', 'de')}; -end -#+end_src - -Finally the $6 \times 6$ transfer function matrices from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ and from $\bm{u}$ to $\bm{\tau}_m$ are identified: -#+begin_src matlab -%% DVF Plant (transfer function from u to dLm) - -G_dL = zeros(length(f), 6, 6); -for i_strut = 1:6 - G_dL(:,:,i_strut) = tfestimate(meas_added_mass{i_strut}.Va, meas_added_mass{i_strut}.de, win, Noverlap, Nfft, 1/Ts); -end - -%% IFF Plant (transfer function from u to taum) -G_tau = zeros(length(f), 6, 6); -for i_strut = 1:6 - G_tau(:,:,i_strut) = tfestimate(meas_added_mass{i_strut}.Va, meas_added_mass{i_strut}.Vs, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm - Several payloads -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -% Diagonal terms -for i = 1:6 - plot(frf_ol.f, abs(frf_ol.G_dL{1}(:,i, i)), 'color', colors(1,:)); - plot(f, abs(G_dL(:,i, i)), 'color', colors(2,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); -ylim([1e-8, 1e-3]); -xlim([20, 2e3]); -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_ol.f, abs(frf_ol.G_dL(:,i, i)), 'color', colors(1,:)); - plot(f, abs(G_dL(:,i, i)), 'color', colors(2,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); -ylim([1e-8, 1e-3]); -xlim([10, 1e3]); -#+end_src - - * Introduction :ignore: In the previous section, all the struts were mounted and individually characterized. @@ -4111,7 +351,7 @@ After all six struts are mounted, the mounting tool (Figure ref:fig:test_nhexa_c * Suspended Table :PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/test_nhexa_table.m +:header-args:matlab+: :tangle matlab/test_nhexa_1_suspended_table.m :END: <> ** Introduction @@ -4234,9 +474,6 @@ The next modes are flexible modes of the breadboard as shown in Figure ref:fig:t #+end_figure ** Simscape Model of the suspended table -:PROPERTIES: -:header-args:matlab+: :tangle matlab/simscape_model.m -:END: <> The Simscape model of the suspended table simply consists of two solid bodies connected by 4 springs. @@ -4284,15 +521,43 @@ ws = ws(imag(ws) > 0); | Experimental | 1.3 Hz | 2.0 Hz | 6.9 Hz | 9.5 Hz | | Simscape | 1.3 Hz | 1.8 Hz | 6.8 Hz | 9.5 Hz | -* Nano-Hexapod Dynamics +** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: + +In this section, a suspended table with low frequency suspension modes and high frequency flexible modes was presented. +This suspended table will be used in Section ref:sec:test_nhexa_dynamics for dynamical identification of the Nano-Hexapod. +The objective is to be able to accurately identify the dynamics of the nano-hexapod, isolated from complex support dynamics. +The key point of this strategy is to be able to accurately model the suspended table. + +To do so, a modal analysis of the suspended table was performed in Section ref:ssec:test_nhexa_table_identification, validating the low frequency suspension modes and high frequency flexible modes. +Then, a multi-body model of this suspended table was tuned to match with the measurements (Section ref:ssec:test_nhexa_table_model). + +* Nano-Hexapod Measured Dynamics +:PROPERTIES: +:header-args:matlab+: :tangle matlab/test_nhexa_2_dynamics.m +:END: <> ** Introduction :ignore: -In Figure ref:fig:test_nhexa_nano_hexapod_signals is shown a block diagram of the experimental setup. -When possible, the notations are consistent with this diagram and summarized in Table ref:tab:list_signals. +The Nano-Hexapod is then mounted on top of the suspended table as shown in Figure ref:fig:test_nhexa_hexa_suspended_table. +All the instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) are setup and connected to the nano-hexapod using many cables. -#+begin_src latex :file nano_hexapod_signals.pdf +#+name: fig:test_nhexa_hexa_suspended_table +#+caption: Mounted Nano-Hexapod on top of the suspended table +#+attr_latex: :width 0.7\linewidth +[[file:figs/test_nhexa_hexa_suspended_table.jpg]] + +A modal analysis of the nano-hexapod is first performed in Section ref:ssec:test_nhexa_enc_struts_modal_analysis. +It will be used to better understand the measured dynamics from actuators to sensors. + +A block diagram schematic of the (open-loop) system is shown in Figure ref:fig:test_nhexa_nano_hexapod_signals. +The transfer function from controlled signals $\mathbf{u}$ to the force sensors voltages $\mathbf{V}_s$ and to the encoders measured displacements $\mathbf{d}_e$ are identified in Section ref:ssec:test_nhexa_identification. +The effect of the payload mass on the dynamics is studied in Section ref:ssec:test_nhexa_added_mass. + +#+begin_src latex :file test_nhexa_nano_hexapod_signals.pdf \definecolor{instrumentation}{rgb}{0, 0.447, 0.741} \definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} @@ -4311,17 +576,17 @@ When possible, the notations are consistent with this diagram and summarized in \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; % Connections and labels - \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); - \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); - \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); - \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; + \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\mathbf{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); + \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\mathbf{u}}$}node[midway, below]{$[V]$} (PD200.west); + \draw[->] (PD200.east) -- node[midway, above]{$\mathbf{u}_a$}node[midway, below]{$[V]$} (F_stack.west); + \draw[->] (F_stack.east) -- (inputF) node[above left]{$\mathbf{\tau}$}node[below left]{$[N]$}; - \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; - \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); - \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; + \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\mathbf{\epsilon}$} node[below left]{$[m]$}; + \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\mathbf{V}}_s$}node[midway, below]{$[V]$} (Fm_ADC.west); + \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\mathbf{V}_s$}node[below left]{$[V]$}; - \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; - \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; + \draw[->] (outputL) -- (encoder.west) node[above left]{$\mathbf{d}_e$} node[below left]{$[m]$}; + \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$\mathbf{d}_{e}$}node[below left]{$[m]$}; % Nano-Hexapod \begin{scope}[on background layer] @@ -4332,121 +597,15 @@ When possible, the notations are consistent with this diagram and summarized in #+end_src #+name: fig:test_nhexa_nano_hexapod_signals -#+caption: Block diagram of the system with named signals +#+caption: Block diagram of the system. Command signal generated by the speedgoat is $\mathbf{u}$, the measured dignals are $\mathbf{d}_{e}$ and $\mathbf{V}_s$. Units are indicated in square brackets. #+attr_latex: :scale 1 +#+RESULTS: [[file:figs/test_nhexa_nano_hexapod_signals.png]] -#+name: tab:list_signals -#+caption: List of signals -#+attr_latex: :environment tabularx :width \linewidth :align Xllll -#+attr_latex: :center t :booktabs t -| | *Unit* | *Matlab* | *Vector* | *Elements* | -|------------------------------------+-----------+-----------+-----------------------+----------------------| -| Control Input (wanted DAC voltage) | =[V]= | =u= | $\bm{u}$ | $u_i$ | -| DAC Output Voltage | =[V]= | =u= | $\tilde{\bm{u}}$ | $\tilde{u}_i$ | -| PD200 Output Voltage | =[V]= | =ua= | $\bm{u}_a$ | $u_{a,i}$ | -| Actuator applied force | =[N]= | =tau= | $\bm{\tau}$ | $\tau_i$ | -|------------------------------------+-----------+-----------+-----------------------+----------------------| -| Strut motion | =[m]= | =dL= | $d\bm{\mathcal{L}}$ | $d\mathcal{L}_i$ | -| Encoder measured displacement | =[m]= | =dLm= | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ | -|------------------------------------+-----------+-----------+-----------------------+----------------------| -| Force Sensor strain | =[m]= | =epsilon= | $\bm{\epsilon}$ | $\epsilon_i$ | -| Force Sensor Generated Voltage | =[V]= | =taum= | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ | -| Measured Generated Voltage | =[V]= | =taum= | $\bm{\tau}_m$ | $\tau_{m,i}$ | -|------------------------------------+-----------+-----------+-----------------------+----------------------| -| Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | -| Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | - -#+name: fig:test_nhexa_enc_fixed_to_struts -#+caption: Nano-Hexapod with encoders fixed to the struts -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_IMG_20210625_083801.jpg]] - -It is structured as follow: -- Section ref:sec:test_nhexa_enc_plates_plant_id: The dynamics of the nano-hexapod is identified. -- Section ref:sec:test_nhexa_enc_plates_comp_simscape: The identified dynamics is compared with the Simscape model. - -** Modal Analysis :noexport: -<> - -This could just be used to show that experimental measure of the flexible mode of the top plate has been done: -- [ ] *This test was made using encoder fixed to the struts, is it relevant to put it here?* -- [ ] Also compare with the FEM - -*** Introduction :ignore: -Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure ref:fig:test_nhexa_compliance_vertical_comp_iff. - -#+name: fig:test_nhexa_accelerometers_nano_hexapod -#+caption: Location of the accelerometers on top of the nano-hexapod -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_accelerometers_nano_hexapod.jpg]] - -The top platform is then excited using an instrumented hammer as shown in Figure ref:fig:test_nhexa_hammer_excitation_compliance_meas. - -#+name: fig:test_nhexa_hammer_excitation_compliance_meas -#+caption: Example of an excitation using an instrumented hammer -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_hammer_excitation_compliance_meas.jpg]] - -From this experiment, the resonance frequencies and the associated mode shapes can be computed (Section ref:sec:test_nhexa_modal_analysis_mode_shapes). -Then, in Section ref:sec:test_nhexa_compliance_effect_iff, the vertical compliance of the nano-hexapod is experimentally estimated. -Finally, in Section ref:sec:test_nhexa_compliance_effect_iff_comp_model, the measured compliance is compare with the estimated one from the Simscape model. - -*** Obtained Mode Shapes -<> - -We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure ref:fig:test_nhexa_mode_shapes_annotated. - -#+name: fig:test_nhexa_mode_shapes_annotated -#+caption: Measured mode shapes for the first six modes -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_mode_shapes_annotated.gif]] - -Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure ref:fig:test_nhexa_mode_shapes_flexible_annotated). - -#+name: fig:test_nhexa_mode_shapes_flexible_annotated -#+caption: First flexible mode at 692Hz -#+attr_latex: :width 0.3\linewidth -[[file:figs/test_nhexa_ModeShapeFlex1_crop.gif]] - -The obtained modes are summarized in Table ref:tab:description_modes. - -#+name: tab:description_modes -#+caption: Description of the identified modes -#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX -#+attr_latex: :center t :booktabs t -| *Mode* | *Freq. [Hz]* | *Description* | -|--------+--------------+----------------------------------------------| -| 1 | 105 | Suspension Mode: Y-translation | -| 2 | 107 | Suspension Mode: X-translation | -| 3 | 131 | Suspension Mode: Z-translation | -| 4 | 161 | Suspension Mode: Y-tilt | -| 5 | 162 | Suspension Mode: X-tilt | -| 6 | 180 | Suspension Mode: Z-rotation | -| 7 | 692 | (flexible) Membrane mode of the top platform | - -*** FEM - -- [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/Assembly 20201020/Modal t=0.50mm]] -- [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/GitLab_nass-fem/dynamic-modal/assy-hexapod-20201022/t_0.25mm]] -- [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/GitLab_nass-fem/dynamic-modal/assy-hexapod-20201022/t_0.5mm]] -- [[file:/home/thomas/Cloud/work-projects/ID31-NASS/nass-fem/GitLab_nass-fem/plateau-superelement]] - -** Identification of the dynamics -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates.m -:END: -<> -*** Introduction :ignore: -In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is identified. - -First, the measurement data are loaded in Section ref:sec:test_nhexa_enc_plates_plant_id_setup, then the transfer function matrix from the actuators to the encoders are estimated in Section ref:sec:test_nhexa_enc_plates_plant_id_dvf. -Finally, the transfer function matrix from the actuators to the force sensors is estimated in Section ref:sec:test_nhexa_enc_plates_plant_id_iff. - -*** Matlab Init :noexport:ignore: +** Matlab Init :noexport:ignore: #+begin_src matlab -%% id_frf_enc_plates.m -% Identification of the nano-hexapod dynamics from u to dL and to taum +%% test_nhexa_dynamics.m +% Identification of the nano-hexapod dynamics from u to dL and to Vs % Encoders are fixed to the plates #+end_src @@ -4470,21 +629,79 @@ Finally, the transfer function matrix from the actuators to the force sensors is <> #+end_src -*** Data Loading and Spectral Analysis Setup -<> +** Modal analysis +<> + +In order to ease the future analysis of the measured plant dynamics, a basic modal analysis of the nano-hexapod is performed. +Five 3-axis accelerometers are fixed on the top platform of the nano-hexapod (Figure ref:fig:test_nhexa_modal_analysis) and the top platform is excited using an instrumented hammer. + +#+name: fig:test_nhexa_modal_analysis +#+caption: Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis +#+attr_latex: :width 0.7\linewidth +[[file:figs/test_nhexa_modal_analysis.jpg]] + +Between 100Hz and 200Hz, 6 suspension modes (i.e. rigid body modes of the top platform) are identified. +At around 700Hz, two flexible modes of the top plate are observed (see Figure ref:fig:test_nhexa_hexa_flexible_modes). +These modes are summarized in Table ref:tab:test_nhexa_hexa_modal_modes_list. + +#+name: tab:test_nhexa_hexa_modal_modes_list +#+caption: Description of the identified modes of the Nano-Hexapod +#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX +#+attr_latex: :center t :booktabs t +| *Mode* | *Frequency* | *Description* | +|--------+-------------+----------------------------------------------| +| 1 | 120 Hz | Suspension Mode: Y-translation | +| 2 | 120 Hz | Suspension Mode: X-translation | +| 3 | 145 Hz | Suspension Mode: Z-translation | +| 4 | 165 Hz | Suspension Mode: Y-rotation | +| 5 | 165 Hz | Suspension Mode: X-rotation | +| 6 | 190 Hz | Suspension Mode: Z-rotation | +| 7 | 692 Hz | (flexible) Membrane mode of the top platform | +| 8 | 709 Hz | Second flexible mode of the top platform | + +#+name: fig:test_nhexa_hexa_flexible_modes +#+caption: Two identified flexible modes of the top plate of the Nano-Hexapod +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at 692Hz} +#+attr_latex: :options {\textwidth} +#+begin_subfigure +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_hexa_flexible_mode_1.jpg]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at 709Hz} +#+attr_latex: :options {\textwidth} +#+begin_subfigure +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_hexa_flexible_mode_2.jpg]] +#+end_subfigure +#+end_figure + +** Identification of the dynamics +<> + +The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) are identified by generating a low pass filtered white noise for each of the command signals, one by one. + +The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure ref:fig:test_nhexa_identified_frf_de. +The diagonal terms are displayed using colorful lines, and all the 30 off-diagonal terms are displayed by grey lines. + +All the six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity. +Below the first suspension mode, good decoupling can be observed (the amplitude of the all of off-diagonal terms are $\approx 20$ times smaller than the diagonal terms). + +From 10Hz up to 1kHz, around 10 resonance frequencies can be observed. +The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section ref:ssec:test_nhexa_enc_struts_modal_analysis. +Then, three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section ref:ssec:test_nhexa_comp_model_coupling). +Except the mode at 237Hz, their amplitude is rather low. +Two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform. +Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. + +Up to at least 1kHz, an alternating pole/zero pattern is observed, which renders the control easier to tune. +This would not have been the case if the encoders were fixed to the struts. -The actuators are excited one by one using a low pass filtered white noise. -For each excitation, the 6 force sensors and 6 encoders are measured and saved. #+begin_src matlab -%% Load Identification Data -meas_data = {}; +%% Load identification data +load('test_nhexa_identification_data_mass_0.mat', 'data'); -for i = 1:6 - meas_data(i) = {load(sprintf('frf_data_exc_strut_%i_realigned_vib_table_0m.mat', i), 't', 'Va', 'Vs', 'de')}; -end -#+end_src - -#+begin_src matlab :exports none %% Setup useful variables Ts = 1e-4; % Sampling Time [s] Nfft = floor(1/Ts); % Number of points for the FFT computation @@ -4492,24 +709,23 @@ win = hanning(Nfft); % Hanning window Noverlap = floor(Nfft/2); % Overlap between frequency analysis % And we get the frequency vector -[~, f] = tfestimate(meas_data{1}.Va, meas_data{1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src +[~, f] = tfestimate(data{1}.u, data{1}.de, win, Noverlap, Nfft, 1/Ts); -*** Transfer function from Actuator to Encoder -<> - -The 6x6 transfer function matrix from the excitation voltage $\bm{u}$ and the displacement $d\bm{\mathcal{L}}_m$ as measured by the encoders is estimated. - -#+begin_src matlab %% Transfer function from u to dLm -G_dL = zeros(length(f), 6, 6); +G_de = zeros(length(f), 6, 6); for i = 1:6 - G_dL(:,:,i) = tfestimate(meas_data{i}.Va, meas_data{i}.de, win, Noverlap, Nfft, 1/Ts); + G_de(:,:,i) = tfestimate(data{i}.u, data{i}.de, win, Noverlap, Nfft, 1/Ts); +end + +%% Transfer function from u to Vs +G_Vs = zeros(length(f), 6, 6); + +for i = 1:6 + G_Vs(:,:,i) = tfestimate(data{i}.u, data{i}.Vs, win, Noverlap, Nfft, 1/Ts); end #+end_src -The diagonal and off-diagonal terms of this transfer function matrix are shown in Figure ref:fig:test_nhexa_enc_plates_dvf_frf. #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm figure; @@ -4519,28 +735,29 @@ ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 - plot(f, abs(G_dL(:, i, j)), 'color', [0, 0, 0, 0.2], ... + plot(f, abs(G_de(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) - plot(f, abs(G_dL(:,i, i)), ... - 'DisplayName', sprintf('$d\\mathcal{L}_%i/u_%i$', i, i)); + plot(f, abs(G_de(:,i, i)), ... + 'DisplayName', sprintf('$d_{e,%i}/u_%i$', i, i)); end -plot(f, abs(G_dL(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... - 'DisplayName', '$d\mathcal{L}_i/u_j$'); +plot(f, abs(G_de(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$d_{e,i}/u_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-3]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +ylim([1e-8, 5e-4]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); +leg.ItemTokenSize(1) = 15; ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) - plot(f, 180/pi*angle(G_dL(:,i, i))); + plot(f, 180/pi*angle(G_de(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -4549,41 +766,26 @@ hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); +xlim([10, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_dvf_frf.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/test_nhexa_identified_frf_de.pdf', 'width', 'wide', 'height', 600); #+end_src -#+name: fig:test_nhexa_enc_plates_dvf_frf -#+caption: Measured FRF for the transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ +#+name: fig:test_nhexa_identified_frf_de +#+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{d}_e$. The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines. #+RESULTS: -[[file:figs/enc_plates_dvf_frf.png]] +[[file:figs/test_nhexa_identified_frf_de.png]] -#+begin_important -From Figure ref:fig:test_nhexa_enc_plates_dvf_frf, we can draw few conclusions on the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ when the encoders are fixed to the plates: -- the decoupling is rather good at low frequency (below the first suspension mode). - The low frequency gain is constant for the off diagonal terms, whereas when the encoders where fixed to the struts, the low frequency gain of the off-diagonal terms were going to zero (Figure ref:fig:test_nhexa_enc_struts_dvf_frf). -- the flexible modes of the struts at 226Hz and 337Hz are indeed shown in the transfer functions, but their amplitudes are rather low. -- the diagonal terms have alternating poles and zeros up to at least 600Hz: the flexible modes of the struts are not affecting the alternating pole/zero pattern. This what not the case when the encoders were fixed to the struts (Figure ref:fig:test_nhexa_enc_struts_dvf_frf). -#+end_important -*** Transfer function from Actuator to Force Sensor -<> -Then the 6x6 transfer function matrix from the excitation voltage $\bm{u}$ and the voltage $\bm{\tau}_m$ generated by the Force senors is estimated. -#+begin_src matlab -%% IFF Plant -G_tau = zeros(length(f), 6, 6); +Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure ref:fig:test_nhexa_identified_frf_Vs. +Alternating poles and zeros is observed up to at least 2kHz, which is a necessary characteristics in order to apply decentralized IFF. +Similar to what was observed for the encoder outputs, all the "diagonal" terms are well superimposed, indicating that the same controller can be applied for all the struts. +The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF. -for i = 1:6 - G_tau(:,:,i) = tfestimate(meas_data{i}.Va, meas_data{i}.Vs, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -The bode plot of the diagonal and off-diagonal terms are shown in Figure ref:fig:test_nhexa_enc_plates_iff_frf. #+begin_src matlab :exports none -%% Bode plot of the IFF Plant (transfer function from u to taum) +%% Bode plot of the IFF Plant (transfer function from u to Vs) figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); @@ -4591,28 +793,29 @@ ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 - plot(f, abs(G_tau(:, i, j)), 'color', [0, 0, 0, 0.2], ... + plot(f, abs(G_Vs(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) - plot(f, abs(G_tau(:,i , i)), ... - 'DisplayName', sprintf('$\\tau_{m,%i}/u_%i$', i, i)); + plot(f, abs(G_Vs(:,i , i)), ... + 'DisplayName', sprintf('$V_{s%i}/u_%i$', i, i)); end -plot(f, abs(G_tau(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... - 'DisplayName', '$\tau_{m,i}/u_j$'); +plot(f, abs(G_Vs(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$V_{si}/u_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -ylim([1e-3, 1e2]); +leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 4); +leg.ItemTokenSize(1) = 15; +ylim([1e-3, 6e1]); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) - plot(f, 180/pi*angle(G_tau(:,i, i))); + plot(f, 180/pi*angle(G_Vs(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -4621,149 +824,40 @@ hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); +xlim([10, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/test_nhexa_identified_frf_Vs.pdf', 'width', 'wide', 'height', 600); #+end_src -#+name: fig:test_nhexa_enc_plates_iff_frf -#+caption: Measured FRF for the IFF plant +#+name: fig:test_nhexa_identified_frf_Vs +#+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$. The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines. #+RESULTS: -[[file:figs/enc_plates_iff_frf.png]] +[[file:figs/test_nhexa_identified_frf_Vs.png]] -#+begin_important -It is shown in Figure ref:fig:test_nhexa_enc_plates_iff_comp_simscape_all that: -- The IFF plant has alternating poles and zeros -- The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF -- The decoupling is quite good at low frequency (below the first model) as well as high frequency (above the last suspension mode, except near the flexible modes of the top plate) -#+end_important +** Effect of payload mass on the dynamics +<> -*** Save Identified Plants -The identified dynamics is saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/mat/data_frf/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL') -#+end_src +As one major challenge in the control of the NASS is the wanted robustness to change of payload mass, it is necessary to understand how the dynamics of the nano-hexapod changes with a change of payload mass. -#+begin_src matlab :eval no -save('data_frf/mat/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL') -#+end_src +In order to study this change of dynamics with the payload mass, up to three "cylindrical masses" of $13\,kg$ each can be added for a total of $\approx 40\,kg$. +These three cylindrical masses on top of the nano-hexapod are shown in Figure ref:fig:test_nhexa_table_mass_3. -** Effect of Payload mass on the Dynamics -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates_effect_payload.m -:END: -<> -*** Introduction :ignore: -In this section, the encoders are fixed to the plates, and we identify the dynamics for several payloads. -The added payload are half cylinders, and three layers can be added for a total of around 40kg (Figure ref:fig:test_nhexa_picture_added_3_masses). +#+name: fig:test_nhexa_table_mass_3 +#+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,kg$ +#+attr_org: :width 800px +#+attr_latex: :width 0.8\linewidth +[[file:figs/test_nhexa_table_mass_3.jpg]] -#+name: fig:test_nhexa_picture_added_3_masses -#+caption: Picture of the nano-hexapod with added mass -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_picture_added_3_masses.jpg]] - -First the dynamics from $\bm{u}$ to $d\mathcal{L}_m$ and $\bm{\tau}_m$ is identified. -Then, the Integral Force Feedback controller is developed and applied as shown in Figure ref:fig:test_nhexa_nano_hexapod_signals_iff. -Finally, the dynamics from $\bm{u}^\prime$ to $d\mathcal{L}_m$ is identified and the added damping can be estimated. - -#+begin_src latex :file nano_hexapod_signals_iff.pdf -\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} -\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} -\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} - -\begin{tikzpicture} - % Blocs - \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics}; - \coordinate[] (inputF) at (nano_hexapod.west); - \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$); - \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$); - - \node[block, left= 0.8 of inputF, fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks}; - \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200) {PD200}; - \node[DAC, left= 0.8 of PD200, fill=instrumentation!20!white] (F_DAC) {DAC}; - \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack}; - \node[ADC, right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC) {ADC}; - \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; - \node[addb, left= 0.8 of F_DAC, fill=control!20!white] (add_iff) {}; - \node[block, below=0.8 of add_iff, fill=control!20!white] (Kiff) {\tiny $K_{\text{IFF}}(s)$}; - - % Connections and labels - \draw[->] (add_iff.east) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); - \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); - \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); - \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; - - \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; - \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); - \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; - - \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; - \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; - - \draw[->] ($(Fm_ADC.east)+(0.14,0)$) node[branch]{} -- node[sloped]{$/$} ++(0, -1.8) -| (Kiff.south); - \draw[->] (Kiff.north) -- node[sloped]{$/$} (add_iff.south); - \draw[->] ($(add_iff.west)+(-0.8,0)$) node[above right]{$\bm{u}^\prime$} node[below right]{$[V]$} -- node[sloped]{$/$} (add_iff.west); - - % Nano-Hexapod - \begin{scope}[on background layer] - \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {}; - \node[above] at (system.north) {Nano-Hexapod}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_nano_hexapod_signals_iff -#+caption: Block Diagram of the experimental setup and model -#+RESULTS: -[[file:figs/test_nhexa_nano_hexapod_signals_iff.png]] - -*** Matlab Init :noexport:ignore: #+begin_src matlab -%% id_frf_enc_plates_effect_payload.m -% Identification of the nano-hexapod dynamics from u to dL and to taum for several payloads -% Encoders are fixed to the plates -#+end_src +%% Load identification Data +meas_added_mass = {... + load('test_nhexa_identification_data_mass_0.mat', 'data'), .... + load('test_nhexa_identification_data_mass_1.mat', 'data'), .... + load('test_nhexa_identification_data_mass_2.mat', 'data'), .... + load('test_nhexa_identification_data_mass_3.mat', 'data')}; -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -#+begin_src matlab :tangle no :noweb yes -<> -#+end_src - -#+begin_src matlab :eval no :noweb yes -<> -#+end_src - -#+begin_src matlab :noweb yes -<> -#+end_src - -*** Measured Frequency Response Functions -The following data are loaded: -- =Va=: the excitation voltage (corresponding to $u_i$) -- =Vs=: the generated voltage by the 6 force sensors (corresponding to $\bm{\tau}_m$) -- =de=: the measured motion by the 6 encoders (corresponding to $d\bm{\mathcal{L}}_m$) -#+begin_src matlab -%% Load Identification Data -meas_added_mass = {}; - -for i_mass = i_masses - for i_strut = 1:6 - meas_added_mass(i_strut, i_mass+1) = {load(sprintf('frf_data_exc_strut_%i_realigned_vib_table_%im.mat', i_strut, i_mass), 't', 'Va', 'Vs', 'de')}; - end -end -#+end_src - -The window =win= and the frequency vector =f= are defined. -#+begin_src matlab :exports none %% Setup useful variables Ts = 1e-4; % Sampling Time [s] Nfft = floor(1/Ts); % Number of points for the FFT computation @@ -4771,82 +865,93 @@ win = hanning(Nfft); % Hanning window Noverlap = floor(Nfft/2); % Overlap between frequency analysis % And we get the frequency vector -[~, f] = tfestimate(meas_added_mass{1,1}.Va, meas_added_mass{1,1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src +[~, f] = tfestimate(meas_added_mass{1}.data{1}.u, meas_added_mass{1}.data{1}.de, win, Noverlap, Nfft, 1/Ts); -Finally the $6 \times 6$ transfer function matrices from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ and from $\bm{u}$ to $\bm{\tau}_m$ are identified: -#+begin_src matlab -%% DVF Plant (transfer function from u to dLm) -G_dL = {}; +G_de = {}; -for i_mass = i_masses - G_dL(i_mass+1) = {zeros(length(f), 6, 6)}; +for i_mass = [0:3] + G_de(i_mass+1) = {zeros(length(f), 6, 6)}; for i_strut = 1:6 - G_dL{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_strut, i_mass+1}.Va, meas_added_mass{i_strut, i_mass+1}.de, win, Noverlap, Nfft, 1/Ts); + G_de{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_mass+1}.data{i_strut}.u, meas_added_mass{i_mass+1}.data{i_strut}.de, win, Noverlap, Nfft, 1/Ts); end end -%% IFF Plant (transfer function from u to taum) -G_tau = {}; +%% IFF Plant (transfer function from u to Vs) +G_Vs = {}; -for i_mass = i_masses - G_tau(i_mass+1) = {zeros(length(f), 6, 6)}; +for i_mass = [0:3] + G_Vs(i_mass+1) = {zeros(length(f), 6, 6)}; for i_strut = 1:6 - G_tau{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_strut, i_mass+1}.Va, meas_added_mass{i_strut, i_mass+1}.Vs, win, Noverlap, Nfft, 1/Ts); + G_Vs{i_mass+1}(:,:,i_strut) = tfestimate(meas_added_mass{i_mass+1}.data{i_strut}.u, meas_added_mass{i_mass+1}.data{i_strut}.Vs, win, Noverlap, Nfft, 1/Ts); end end #+end_src The identified dynamics are then saved for further use. #+begin_src matlab :exports none :tangle no -save('matlab/mat/data_frf/frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL') +save('matlab/mat/test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de') #+end_src #+begin_src matlab :eval no -save('data_frf/mat/frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL') +save('./mat/test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de') #+end_src #+begin_src matlab :exports none -frf_ol = load('frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL'); +%% Load the identified transfer functions +frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de'); #+end_src -*** Transfer function from Actuators to Encoders -The transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_{m}$ are shown in Figure ref:fig:test_nhexa_comp_plant_payloads_dvf. +The obtained frequency response functions from actuator signal $u_i$ to the associated encoder $d_{ei}$ for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure ref:fig:test_nhexa_identified_frf_de_masses. +As expected, the frequency of the suspension modes are decreasing with an increase of the payload mass. +The low frequency gain does not change as it is linked to the stiffness property of the nano-hexapod, and not to its mass property. + +The frequencies of the two flexible modes of the top plate are first decreased a lot when the first mass is added (from $\approx 700\,Hz$ to $\approx 400\,Hz$). +This is due to the fact that the added mass is composed of two half cylinders which are not fixed together. +It therefore adds a lot of mass to the top plate without adding stiffness in one direction. +When more than one "mass layer" is added, the half cylinders are added with some angles such that rigidity are added in all directions (see how the three mass "layers" are positioned in Figure ref:fig:test_nhexa_table_mass_3). +In that case, the frequency of these flexible modes are increased. +In practice, the payload should be one solid body, and no decrease of the frequency of this flexible mode should be observed. +The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass is increased. + +The measured FRF from $u_i$ to $V_{si}$ are shown in Figure ref:fig:test_nhexa_identified_frf_Vs_masses. +For all the tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust way. #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm - Several payloads +masses = [0, 13, 26, 39]; figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -for i_mass = i_masses +for i_mass = [0:3] % Diagonal terms - plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,1, 1)), 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u_i$ - %i', i_mass)); + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.5], ... + 'DisplayName', sprintf('$d_{ei}/u_i$ - %i kg', masses(i_mass+1))); for i = 2:6 - plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:), ... + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5], ... 'HandleVisibility', 'off'); end - % Off-Diagonal terms - for i = 1:5 - for j = i+1:6 - plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... - 'HandleVisibility', 'off'); - end - end + % % Off-Diagonal terms + % for i = 1:5 + % for j = i+1:6 + % plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + % 'HandleVisibility', 'off'); + % end + % end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 1e-3]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-8, 5e-4]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; ax2 = nexttile; hold on; -for i_mass = i_masses +for i_mass = [0:3] for i =1:6 - plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:)); + plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5]); end end hold off; @@ -4857,37 +962,13 @@ yticks(-360:90:360); ylim([-90, 180]) linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); +xlim([10, 2e3]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_plant_payloads_dvf.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/test_nhexa_identified_frf_de_masses.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:test_nhexa_comp_plant_payloads_dvf -#+caption: Measured Frequency Response Functions from $u_i$ to $d\mathcal{L}_{m,i}$ for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms. -#+RESULTS: -[[file:figs/test_nhexa_comp_plant_payloads_dvf.png]] - - -#+begin_important -From Figure ref:fig:test_nhexa_comp_plant_payloads_dvf, we can observe few things: -- The obtained dynamics is changing a lot between the case without mass and when there is at least one added mass. -- Between 1, 2 and 3 added masses, the dynamics is not much different, and it would be easier to design a controller only for these cases. -- The flexible modes of the top plate is first decreased a lot when the first mass is added (from 700Hz to 400Hz). - This is due to the fact that the added mass is composed of two half cylinders which are not fixed together. - Therefore is adds a lot of mass to the top plate without adding a lot of rigidity in one direction. - When more than 1 mass layer is added, the half cylinders are added with some angles such that rigidity are added in all directions (see Figure ref:fig:test_nhexa_picture_added_3_masses). - In that case, the frequency of these flexible modes are increased. - In practice, the payload should be one solid body, and we should not see a massive decrease of the frequency of this flexible mode. -- Flexible modes of the top plate are becoming less problematic as masses are added. -- First flexible mode of the strut at 230Hz is not much decreased when mass is added. - However, its apparent amplitude is much decreased. -#+end_important - -*** Transfer function from Actuators to Force Sensors -The transfer functions from $\bm{u}$ to $\bm{\tau}_{m}$ are shown in Figure ref:fig:test_nhexa_comp_plant_payloads_iff. - #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm figure; @@ -4895,33 +976,34 @@ tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -for i_mass = i_masses +for i_mass = [0:3] % Diagonal terms - plot(frf_ol.f, abs(frf_ol.G_tau{i_mass+1}(:,1, 1)), 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('$\\tau_{m,i}/u_i$ - %i', i_mass)); + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.5], ... + 'DisplayName', sprintf('$V_{si}/u_i$ - %i kg', masses(i_mass+1))); for i = 2:6 - plot(frf_ol.f, abs(frf_ol.G_tau{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:), ... + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5], ... 'HandleVisibility', 'off'); end - % Off-Diagonal terms - for i = 1:5 - for j = i+1:6 - plot(frf_ol.f, abs(frf_ol.G_tau{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... - 'HandleVisibility', 'off'); - end - end + % % Off-Diagonal terms + % for i = 1:5 + % for j = i+1:6 + % plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... + % 'HandleVisibility', 'off'); + % end + % end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); ylim([1e-2, 1e2]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); +leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; ax2 = nexttile; hold on; -for i_mass = i_masses +for i_mass = [0:3] for i =1:6 - plot(frf_ol.f, 180/pi*angle(frf_ol.G_tau{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:)); + plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.5]); end end hold off; @@ -4931,100 +1013,75 @@ hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); +xlim([10, 2e3]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_plant_payloads_iff.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/test_nhexa_identified_frf_Vs_masses.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:test_nhexa_comp_plant_payloads_iff -#+caption: Measured Frequency Response Functions from $u_i$ to $\tau_{m,i}$ for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms. -#+RESULTS: -[[file:figs/test_nhexa_comp_plant_payloads_iff.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_comp_plant_payloads_iff, we can see that for all added payloads, the transfer function from $\bm{u}$ to $\bm{\tau}_{m}$ always has alternating poles and zeros. -#+end_important - -*** Coupling of the transfer function from Actuator to Encoders -The RGA-number, which is a measure of the interaction in the system, is computed for the transfer function matrix from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ for all the payloads. -The obtained numbers are compared in Figure ref:fig:test_nhexa_rga_num_ol_masses. - -#+begin_src matlab :exports none -%% Decentralized RGA - Undamped Plant -RGA_num = zeros(length(frf_ol.f), length(i_masses)); -for i_mass = i_masses - for i = 1:length(frf_ol.f) - RGA_num(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_ol.G_dL{i_mass+1}(i,:,:)).*inv(squeeze(frf_ol.G_dL{i_mass+1}(i,:,:))).'))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% RGA for Decentralized plant -figure; -hold on; -for i_mass = i_masses - plot(frf_ol.f, RGA_num(:,i_mass+1), '-', 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('RGA-num - %i mass', i_mass)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rga_num_ol_masses.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_rga_num_ol_masses -#+caption: RGA-number for the open-loop transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ -#+RESULTS: -[[file:figs/test_nhexa_rga_num_ol_masses.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_rga_num_ol_masses, it is clear that the coupling is quite large starting from the first suspension mode of the nano-hexapod. -Therefore, is the payload's mass is increase, the coupling in the system start to become unacceptably large at lower frequencies. -#+end_important +#+name: fig:test_struts_mounting +#+caption: Measured Frequency Response Functions from $u_i$ to $d_{ei}$ (\subref{fig:test_nhexa_identified_frf_de_masses}) and from $u_i$ to $V_{si}$ (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown. +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_identified_frf_de_masses.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_identified_frf_Vs_masses.png]] +#+end_subfigure +#+end_figure ** Conclusion -#+begin_important -In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied. - -It has been found that: -- The measured dynamics is in agreement with the dynamics of the simscape model, up to the flexible modes of the top plate. - See figures ref:fig:test_nhexa_enc_plates_iff_comp_simscape and ref:fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape for the transfer function to the force sensors and Figures ref:fig:test_nhexa_enc_plates_dvf_comp_simscape and ref:fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape for the transfer functions to the encoders -- The Integral Force Feedback strategy is very effective in damping the suspension modes of the nano-hexapod (Figure ref:fig:test_nhexa_enc_plant_plates_effect_iff). -- The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ show nice dynamical properties and is a much better candidate for the high-authority-control than when the encoders were fixed to the struts. - At least up to the flexible modes of the top plate, the diagonal elements of the transfer function matrix have alternating poles and zeros, and the phase is moving smoothly. - Only the flexible modes of the top plates seems to be problematic for control. -#+end_important - -* Comparison with the Nano-Hexapod model? -<> -** Comparison with the Simscape Model :PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/frf_enc_plates_comp_simscape.m +:UNNUMBERED: t :END: -<> -*** Introduction :ignore: -In this section, the measured dynamics done in Section ref:sec:test_nhexa_enc_plates_plant_id is compared with the dynamics estimated from the Simscape model. -A configuration is added to be able to put the nano-hexapod on top of the vibration table as shown in Figure ref:fig:simscape_vibration_table. +After the Nano-Hexapod was fixed on top of the suspended table, its dynamics was identified. -#+name: fig:simscape_vibration_table -#+caption: 3D representation of the simscape model with the nano-hexapod +The frequency response functions from the six DAC voltages $\mathbf{u}$ to the six encoders measured displacements $\mathbf{d}_e$ displays alternating complex conjugate poles and complex conjugate zeros up to at least 1kHz. +At low frequency, the coupling is small, indicating correct assembly of all parts. +This should enables the design of a decentralized positioning controller based on the encoder for relative positioning purposes. +The suspension modes and flexible modes measured during the modal analysis (Section ref:ssec:test_nhexa_enc_struts_modal_analysis) are also observed in the dynamics. +Lot's of other modes are present above 700Hz, which will inevitably limit the achievable bandwidth. +The observed effect of the payload's mass on the dynamics is quite large, which also represent a complex control challenge. + +The frequency response functions from the six DAC voltages $\mathbf{u}$ to the six force sensors voltages $\mathbf{V}_s$ all have alternating complex conjugate poles and complex conjugate zeros. +This indicates that it should be possible to implement decentralized Integral Force Feedback in a robust way. +This alternating property holds for all the tested payloads. + +* Nano-Hexapod Model Dynamics +:PROPERTIES: +:header-args:matlab+: :tangle matlab/test_nhexa_3_model.m +:END: +<> + +** Introduction :ignore: + +In this section, the measured dynamics done in Section ref:sec:test_nhexa_dynamics is compared with the dynamics estimated from the Simscape model. +The nano-hexapod simscape model is therefore added on top of the vibration table Simscape model as shown in Figure ref:fig:test_nhexa_hexa_simscape. + +#+name: fig:test_nhexa_hexa_simscape +#+caption: 3D representation of the simscape model with the nano-hexapod on top of the suspended table. Three mass "layers" are here added #+attr_latex: :width 0.8\linewidth -[[file:figs/vibration_table_nano_hexapod_simscape.png]] +[[file:figs/test_nhexa_hexa_simscape.png]] -*** Matlab Init :noexport:ignore: +The model should exhibit certain characteristics that are verified in this section. +First, it should match the measured system dynamics from actuators to sensors that were presented in Section ref:sec:test_nhexa_dynamics. +Both the "direct" terms (Section ref:ssec:test_nhexa_comp_model) and "coupling" terms (Section ref:ssec:test_nhexa_comp_model_coupling) of the Simscape model are compared with the measured dynamics. +Second, it should also represents how the system dynamics changes when a payload is fixed to the top platform. +This is checked in Section ref:ssec:test_nhexa_comp_model_masses. + +** Matlab Init :noexport:ignore: #+begin_src matlab -%% frf_enc_plates_comp_simscape.m -% Compare the measured dynamics from u to dL and to taum with the Simscape model -% Encoders are fixed to the plates +%% test_nhexa_3_model.m +% Compare the measured dynamics from u to de and to Vs with the Simscape model #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -5051,180 +1108,164 @@ A configuration is added to be able to put the nano-hexapod on top of the vibrat <> #+end_src +** Extract transfer function matrices from the Simscape Model :noexport: #+begin_src matlab -%% Load identification data -frf_ol = load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL'); -#+end_src - -*** Identification with the Simscape Model -The nano-hexapod is initialized with the APA taken as 2dof models. -#+begin_src matlab -%% Initialize Simscape Model +%% Extract the transfer function matrix from the Simscape model +% Initialization of the Simscape model table_type = 'Suspended'; % On top of vibration table -device_type = 'Hexapod'; % On top of vibration table +device_type = 'Hexapod'; % Nano-Hexapod payload_num = 0; % No Payload n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'plates', ... 'actuator_type', '2dof'); -#+end_src -Now, the dynamics from the DAC voltage $\bm{u}$ to the encoders $d\bm{\mathcal{L}}_m$ is estimated using the Simscape model. -#+begin_src matlab -%% Identify the DVtransfer function from u to dLm +% Identify the FRF matrix from u to [de,Vs] clear io; io_i = 1; io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders +io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoders +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Encoders -G_dL = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); +G_de = {}; +G_Vs = {}; + +for i = [0:3] + payload_num = i; % Change the payload on the nano-hexapod + G = exp(-s*1e-4)*linearize(mdl, io, 0.0); + G.InputName = {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'}; + G.OutputName = {'de1', 'de2', 'de3', 'de4', 'de5', 'de6', ... + 'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'}; + G_de(i+1) = {G({'de1', 'de2', 'de3', 'de4', 'de5', 'de6'},:)}; + G_Vs(i+1) = {G({'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'},:)}; +end #+end_src -#+begin_src matlab :exports none -%% Comparison of the plants (encoder output) when tuning the misalignment -freqs = 2*logspace(1, 3, 1000); - -i_input = 1; - -figure; -hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, i_input))); -plot(freqs, abs(squeeze(freqresp(G_dL(1, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); -xlim([40, 4e2]); ylim([1e-8, 1e-2]); -#+end_src - -Then the transfer function from $\bm{u}$ to $\bm{\tau}_m$ is identified using the Simscape model. -#+begin_src matlab -%% Identify the transfer function from u to taum -clear io; io_i = 1; -io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors - -G_tau = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); -#+end_src - -The identified dynamics is saved for further use. #+begin_src matlab :exports none :tangle no -%% Save Identified Plants -save('matlab/mat/data_frf/simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +% Save the identified plants +save('matlab/mat/test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de') #+end_src #+begin_src matlab :eval no -save('mat/data_frf/simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +% Save the identified plants +save('./mat/test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de') #+end_src +#+begin_src matlab +%% The same identification is performed, but this time with +% "flexible" model of the APA +table_type = 'Suspended'; % On top of vibration table +device_type = 'Hexapod'; % Nano-Hexapod +payload_num = 0; % No Payload + +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible'); + +G_de = {}; +G_Vs = {}; + +for i = [0:3] + payload_num = i; % Change the payload on the nano-hexapod + G = exp(-s*1e-4)*linearize(mdl, io, 0.0); + G.InputName = {'u1', 'u2', 'u3', 'u4', 'u5', 'u6'}; + G.OutputName = {'de1', 'de2', 'de3', 'de4', 'de5', 'de6', ... + 'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'}; + G_de(i+1) = {G({'de1', 'de2', 'de3', 'de4', 'de5', 'de6'},:)}; + G_Vs(i+1) = {G({'Vs1', 'Vs2', 'Vs3', 'Vs4', 'Vs5', 'Vs6'},:)}; +end +#+end_src + +#+begin_src matlab :exports none :tangle no +% Save the identified plants +save('matlab/mat/test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de') +#+end_src + +#+begin_src matlab :eval no +% Save the identified plants +save('./mat/test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de') +#+end_src + +** Nano-Hexapod model dynamics +<> + #+begin_src matlab :exports none -%% Load the Simscape model -sim_ol = load('simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); +%% Load Simscape Model and measured FRF +sim_ol = load('test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de'); +frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de'); #+end_src -*** Dynamics from Actuator to Force Sensors -The identified dynamics is compared with the measured FRF: -- Figure ref:fig:test_nhexa_enc_plates_iff_comp_simscape_all: the individual transfer function from $u_1$ (the DAC voltage for the first actuator) to the force sensors of all 6 struts are compared -- Figure ref:fig:test_nhexa_enc_plates_iff_comp_simscape: all the diagonal elements are compared -- Figure ref:fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape: all the off-diagonal elements are compared +The Simscape model of the nano-hexapod is first configured with 4-DoF flexible joints, 2-DoF APA and rigid top and bottom platforms. +The stiffness of the flexible joints are chosen based on the values estimated using the test bench and based on FEM. +The parameters of the APA model are the ones determined from the test bench of the APA. +The $6 \times 6$ transfer function matrices from $\mathbf{u}$ to $\mathbf{d}_e$ and from $\mathbf{u}$ to $\mathbf{V}_s$ are extracted then from the Simscape model. + +A first feature that should be checked is that the model well represents the "direct" terms of the measured FRF matrix. +To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_diag. +It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modelled. +The three resonances that were attributed to "internal" flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable as the APA are here modelled as a simple uniaxial 2-DoF system. +At higher frequencies, no resonances can be seen in the model, as the as the top plate and the encoder supports are modelled as rigid bodies. #+begin_src matlab :exports none -%% Comparison of the plants (encoder output) when tuning the misalignment -freqs = 2*logspace(1, 3, 1000); - -i_input = 1; - -figure; -tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); -hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:, 1, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Amplitude [V/V]'); -title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input)); - -ax2 = nexttile(); -hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:, 2, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(2, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input)); - -ax3 = nexttile(); -hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:, 3, i_input)), ... - 'DisplayName', 'Meas.'); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(3, i_input), freqs, 'Hz'))), ... - 'DisplayName', 'Model'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -legend('location', 'southeast', 'FontSize', 8); -title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input)); - -ax4 = nexttile(); -hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:, 4, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(4, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); -title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input)); - -ax5 = nexttile(); -hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:, 5, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(5, i_input), freqs, 'Hz')))); -hold off; -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input)); - -ax6 = nexttile(); -hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:, 6, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(6, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input)); - -linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); -xlim([20, 2e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_iff_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_enc_plates_iff_comp_simscape_all -#+caption: IFF Plant for the first actuator input and all the force senosrs -#+RESULTS: -[[file:figs/test_nhexa_enc_plates_iff_comp_simscape_all.png]] - -#+begin_src matlab :exports none -%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data -freqs = 2*logspace(1, 3, 1000); - +%% Diagonal elements of the FRF matrix from u to de figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; -plot(frf_ol.f, abs(frf_ol.G_tau(:,1, 1)), 'color', [colors(1,:),0.2], ... - 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') +plot(frf_ol.f, abs(frf_ol.G_de{1}(:,1, 1)), 'color', [colors(1,:),0.5], ... + 'DisplayName', '$d_{ei}/u_i$ - FRF') +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.5], ... + 'DisplayName', '$d_{ei}/u_i$ - Model') for i = 2:6 - plot(frf_ol.f, abs(frf_ol.G_tau(:,i, i)), 'color', [colors(1,:),0.2], ... + plot(frf_ol.f, abs(frf_ol.G_de{1}(:,i, i)), 'color', [colors(1,:),0.5], ... + 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5], ... 'HandleVisibility', 'off'); end -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', '$\tau_{m,i}/u_i$ - Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 5e-4]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{1}(:,i, i)), 'color', [colors(1,:),0.5]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_de{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5]); +end +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([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/test_nhexa_comp_simscape_de_diag.pdf', 'width', 'half', 'height', 600); +#+end_src + +#+begin_src matlab :exports none +%% Diagonal elements of the FRF matrix from u to Vs +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(frf_ol.f, abs(frf_ol.G_Vs{1}(:,1, 1)), 'color', [colors(1,:),0.5], ... + 'DisplayName', '$V_{si}/u_i$ - FRF') +plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{1}(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ... + 'DisplayName', '$V_{si}/u_i$ - Model') for i = 2:6 - plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... + plot(frf_ol.f, abs(frf_ol.G_Vs{1}(:,i, i)), 'color', [colors(1,:),0.5], ... + 'HandleVisibility', 'off'); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ... 'HandleVisibility', 'off'); end hold off; @@ -5235,8 +1276,8 @@ legend('location', 'southeast'); ax2 = nexttile; hold on; for i = 1:6 - plot(frf_ol.f, 180/pi*angle(frf_ol.G_tau(:,i, i)), 'color', [colors(1,:),0.2]); - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_tau(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.2]); + plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{1}(:,i, i)), 'color', [colors(1,:),0.5]); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_Vs{1}(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -5249,378 +1290,254 @@ xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_iff_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/test_nhexa_comp_simscape_Vs_diag.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:test_nhexa_enc_plates_iff_comp_simscape -#+caption: Diagonal elements of the IFF Plant -#+RESULTS: -[[file:figs/test_nhexa_enc_plates_iff_comp_simscape.png]] +#+name: fig:test_nhexa_comp_simscape_diag +#+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag}) +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/test_nhexa_comp_simscape_de_diag.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/test_nhexa_comp_simscape_Vs_diag.png]] +#+end_subfigure +#+end_figure -#+begin_src matlab :exports none -%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements) -freqs = 2*logspace(1, 3, 1000); +** Modelling dynamical coupling +<> -figure; -hold on; -% Off diagonal terms -plot(frf_ol.f, abs(frf_ol.G_tau(:, 1, 2)), 'color', [colors(1,:),0.2], ... - 'DisplayName', '$\tau_{m,i}/u_j$ - FRF') -for i = 1:5 - for j = i+1:6 - plot(frf_ol.f, abs(frf_ol.G_tau(:, i, j)), 'color', [colors(1,:),0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(1, 2), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... - 'DisplayName', '$\tau_{m,i}/u_j$ - Model') -for i = 1:5 - for j = i+1:6 - set(gca,'ColorOrderIndex',2); - plot(freqs, abs(squeeze(freqresp(sim_ol.G_tau(i, j), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); -xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape -#+caption: Off diagonal elements of the IFF Plant -#+RESULTS: -[[file:figs/test_nhexa_enc_plates_iff_comp_offdiag_simscape.png]] - -*** Dynamics from Actuator to Encoder -The identified dynamics is compared with the measured FRF: -- Figure ref:fig:test_nhexa_enc_plates_dvf_comp_simscape_all: the individual transfer function from $u_3$ (the DAC voltage for the actuator number 3) to the six encoders -- Figure ref:fig:test_nhexa_enc_plates_dvf_comp_simscape: all the diagonal elements are compared -- Figure ref:fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape: all the off-diagonal elements are compared +Another wanted feature of the model is that it well represents the coupling in the system as this is often the limiting factor for the control of MIMO systems. +Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\mathbf{u}$ to $\mathbf{d}_e$, only the first "column" is compared (Figure ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$. +It can be seen that the coupling in the model is well matching the measurements up to the first un-modelled flexible mode at 237Hz. +Similar results are observed for all the other coupling terms, as well as for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$. #+begin_src matlab :exports none %% Comparison of the plants (encoder output) when tuning the misalignment -freqs = 2*logspace(1, 3, 1000); - -i_input = 3; +i_input = 1; figure; -tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); +tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight'); ax1 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1, i_input), freqs, 'Hz')))); +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(1, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); -title(sprintf('$d\\mathcal{L}_{m1}/u_{%i}$', i_input)); ax2 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 2, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(2, i_input), freqs, 'Hz')))); +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(2, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title(sprintf('$d\\mathcal{L}_{m2}/u_{%i}$', i_input)); ax3 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 3, i_input)), ... - 'DisplayName', 'Meas.'); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(3, i_input), freqs, 'Hz'))), ... - 'DisplayName', 'Model'); +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 3, i_input)), ... + 'DisplayName', 'Measurements'); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model (2-DoF APA)'); +text(54, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -legend('location', 'southeast', 'FontSize', 8); -title(sprintf('$d\\mathcal{L}_{m3}/u_{%i}$', i_input)); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; ax4 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 4, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(4, i_input), freqs, 'Hz')))); +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(4, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); -title(sprintf('$d\\mathcal{L}_{m4}/u_{%i}$', i_input)); +xticks([50, 100, 200, 400]) ax5 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 5, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(5, i_input), freqs, 'Hz')))); +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(5, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -title(sprintf('$d\\mathcal{L}_{m5}/u_{%i}$', i_input)); +xticks([50, 100, 200, 400]) ax6 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:, 6, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(6, i_input), freqs, 'Hz')))); +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{1}(6, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -title(sprintf('$d\\mathcal{L}_{m6}/u_{%i}$', i_input)); +xticks([50, 100, 200, 400]) linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); -xlim([40, 4e2]); ylim([1e-8, 1e-2]); +xlim([50, 5e2]); ylim([1e-8, 5e-4]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_dvf_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/test_nhexa_comp_simscape_de_all.pdf', 'width', 'full', 'height', 700); #+end_src -#+name: fig:test_nhexa_enc_plates_dvf_comp_simscape_all -#+caption: DVF Plant for the first actuator input and all the encoders +#+name: fig:test_nhexa_comp_simscape_de_all +#+caption: Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$ #+RESULTS: -[[file:figs/test_nhexa_enc_plates_dvf_comp_simscape_all.png]] +[[file:figs/test_nhexa_comp_simscape_de_all.png]] + +The APA300ML are then modelled with a /super-element/ extracted from a FE-software. +The obtained transfer functions from $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$ are compared with the measured FRF in Figure ref:fig:test_nhexa_comp_simscape_de_all_flex. +While the damping of the suspension modes for the /super-element/ is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modelled. +Even the mode 395Hz can be observed in the model. +Therefore, if the modes of the struts are to be modelled, the /super-element/ of the APA300ML may be used, at the cost of obtaining a much higher order model. + +#+begin_src matlab +%% Load the plant model with Flexible APA +flex_ol = load('test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de'); +#+end_src #+begin_src matlab :exports none -%% Diagonal elements of the DVF plant -freqs = 2*logspace(1, 3, 1000); +%% Comparison of the plants (encoder output) when tuning the misalignment +i_input = 1; figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); +tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight'); -ax1 = nexttile([2,1]); +ax1 = nexttile(); hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:,1, 1)), 'color', [colors(1,:),0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') -for i = 2:6 - plot(frf_ol.f, abs(frf_ol.G_dL(:,i, i)), 'color', [colors(1,:),0.2], ... - 'HandleVisibility', 'off'); -end -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') -for i = 2:6 - plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... - 'HandleVisibility', 'off'); -end +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(1, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 1e-3]); -legend('location', 'northeast'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); -ax2 = nexttile; +ax2 = nexttile(); hold on; -for i = 1:6 - plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL(:,i, i)), 'color', [colors(1,:),0.2]); - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); -end +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(2, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top') 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]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_enc_plates_dvf_comp_simscape -#+caption: Diagonal elements of the DVF Plant -#+RESULTS: -[[file:figs/test_nhexa_enc_plates_dvf_comp_simscape.png]] - -#+begin_src matlab :exports none -%% Off-diagonal elements of the DVF plant -freqs = 2*logspace(1, 3, 1000); - -figure; +ax3 = nexttile(); hold on; -% Off diagonal terms -plot(frf_ol.f, abs(frf_ol.G_dL(:, 1, 2)), 'color', [colors(1,:),0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') -for i = 1:5 - for j = i+1:6 - plot(frf_ol.f, abs(frf_ol.G_dL(:, i, j)), 'color', [colors(1,:),0.2], ... - 'HandleVisibility', 'off'); - end -end -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1, 2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i, j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 3, i_input)), ... + 'DisplayName', 'Measurements'); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model (Flexible APA)'); +text(54, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(4, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); -xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]); -legend('location', 'northeast'); +xticks([50, 100, 200, 400]) + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(5, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xticks([50, 100, 200, 400]) + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{1}(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(flex_ol.G_de{1}(6, i_input), freqs, 'Hz')))); +text(54, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xticks([50, 100, 200, 400]) + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([50, 5e2]); ylim([1e-8, 5e-4]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +exportFig('figs/test_nhexa_comp_simscape_de_all_flex.pdf', 'width', 'full', 'height', 700); #+end_src -#+name: fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape -#+caption: Off diagonal elements of the DVF Plant +#+name: fig:test_nhexa_comp_simscape_de_all_flex +#+caption: Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$ #+RESULTS: -[[file:figs/test_nhexa_enc_plates_dvf_comp_offdiag_simscape.png]] +[[file:figs/test_nhexa_comp_simscape_de_all_flex.png]] -*** Conclusion -#+begin_important -The Simscape model is quite accurate for the transfer function matrices from $\bm{u}$ to $\bm{\tau}_m$ and from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ except at frequencies of the flexible modes of the top-plate. -The Simscape model can therefore be used to develop the control strategies. -#+end_important +** Modelling the effect of payload mass +<> -** Comparison with the Simscape model -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates_effect_payload_comp_simscape.m -:END: -<> -*** Introduction :ignore: -Let's now compare the identified dynamics with the Simscape model. -We wish to verify if the Simscape model is still accurate for all the tested payloads. +Another important characteristics of the model is that it should well represents the dynamics of the system for all considered payloads. +The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure ref:fig:test_nhexa_comp_simscape_diag_masses. +The observed shift to lower frequency of the suspension modes with an increased payload mass is well represented by the Simscape model. +The complex conjugate zeros are also well matching with the experiments both for the encoder outputs (Figure ref:fig:test_nhexa_comp_simscape_de_diag_masses) and the force sensor outputs (Figure ref:fig:test_nhexa_comp_simscape_Vs_diag_masses). -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% id_frf_enc_plates_effect_payload_comp_simscape.m -% Comparison of the nano-hexapod dynamics from u to dL and to taum for several payloads - -% Measured FRF and extracted dynamics from the Simscape model -% Encoders are fixed to the plates -#+end_src - -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -#+begin_src matlab :tangle no :noweb yes -<> -#+end_src - -#+begin_src matlab :eval no :noweb yes -<> -#+end_src - -#+begin_src matlab :noweb yes -<> -<> -#+end_src - -#+begin_src matlab -%% Load the identified FRF -frf_ol_m = load('frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL'); -#+end_src - -*** System Identification -Let's initialize the simscape model with the nano-hexapod fixed on top of the vibration table. -#+begin_src matlab -%% Initialize Nano-Hexapod -table_type = 'Suspended'; % On top of vibration table -device_type = 'Hexapod'; % On top of vibration table -payload_num = 0; % No Payload - -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof'); -#+end_src - -First perform the identification for the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$: -#+begin_src matlab -%% Identify the DVF Plant (transfer function from u to dLm) -clear io; io_i = 1; -io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders - -%% Identification for all the added payloads -G_dL = {}; - -for i = i_masses - fprintf('i = %i\n', i) - payload_num = i; % Change the payload on the nano-hexapod - G_dL(i+1) = {exp(-s*frf_ol_m.Ts)*linearize(mdl, io, 0.0, options)}; -end -#+end_src - -#+begin_src matlab -%% Identify the IFF Plant (transfer function from u to taum) -clear io; io_i = 1; -io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors - -%% Identification for all the added payloads -G_tau = {}; - -for i = 0:3 - fprintf('i = %i\n', i) - payload_num = i; % Change the payload on the nano-hexapod - G_tau(i+1) = {exp(-s*frf_ol_m.Ts)*linearize(mdl, io, 0.0, options)}; -end -#+end_src - -The identified dynamics are then saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/mat/data_frf/sim_vib_table_m.mat', 'G_tau', 'G_dL') -#+end_src - -#+begin_src matlab :eval no -save('./mat/data_frf/sim_vib_table_m.mat', 'G_tau', 'G_dL') -#+end_src +Note that the model displays smaller damping that what is observed experimentally for high values of the payload mass. +One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping). +However, as decentralized IFF will be applied, the damping will be brought actively, and the open-loop damping value should have very little impact on the obtained plant. #+begin_src matlab :exports none -sim_ol_m = load('sim_vib_table_m.mat', 'G_tau', 'G_dL'); -#+end_src - -*** Transfer function from Actuators to Encoders -The measured FRF and the identified dynamics from $u_i$ to $d\mathcal{L}_{m,i}$ are compared in Figure ref:fig:test_nhexa_comp_masses_model_exp_dvf. -A zoom near the "suspension" modes is shown in Figure ref:fig:test_nhexa_comp_masses_model_exp_dvf_zoom. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm +%% Bode plot for the transfer function from u to de +masses = [0, 13, 26, 39]; figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); -freqs = 2*logspace(1,3,1000); - ax1 = nexttile([2,1]); hold on; -for i_mass = i_masses - plot(frf_ol_m.f, abs(frf_ol_m.G_dL{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u_i$ - FRF %i', i_mass)); +for i_mass = [0:3] + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('Meas (%i kg)', masses(i_mass+1))); for i = 2:6 - plot(frf_ol_m.f, abs(frf_ol_m.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + plot(frf_ol.f, abs(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... 'HandleVisibility', 'off'); end set(gca, 'ColorOrderIndex', i_mass+1) - plot(freqs, abs(squeeze(freqresp(sim_ol_m.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--', ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u_i$ - Sim %i', i_mass)); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--', ... + 'DisplayName', 'Simscape'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 1e-3]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([5e-8, 1e-3]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; ax2 = nexttile; hold on; -for i_mass = i_masses +for i_mass = [0:3] for i =1:6 - plot(frf_ol_m.f, 180/pi*angle(frf_ol_m.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + plot(frf_ol.f, 180/pi*angle(frf_ol.G_de{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); end set(gca, 'ColorOrderIndex', i_mass+1) - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol_m.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--'); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -5630,73 +1547,46 @@ yticks(-360:45:360); ylim([-45, 180]); linkaxes([ax1,ax2],'x'); -xlim([20, 1e3]); +xlim([20, 2e2]); +xticks([20, 50, 100, 200]) #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_masses_model_exp_dvf.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/test_nhexa_comp_simscape_de_diag_masses.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:test_nhexa_comp_masses_model_exp_dvf -#+caption: Comparison of the transfer functions from $u_i$ to $d\mathcal{L}_{m,i}$ - measured FRF and identification from the Simscape model -#+RESULTS: -[[file:figs/comp_masses_model_exp_dvf.png]] - -#+begin_src matlab :exports none :tangle no -ax1.YLim = [1e-6, 5e-4]; -xlim([40, 2e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_masses_model_exp_dvf_zoom.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_comp_masses_model_exp_dvf_zoom -#+caption: Comparison of the transfer functions from $u_i$ to $d\mathcal{L}_{m,i}$ - measured FRF and identification from the Simscape model (Zoom) -#+RESULTS: -[[file:figs/test_nhexa_comp_masses_model_exp_dvf_zoom.png]] - -#+begin_important -The Simscape model is very accurately representing the measured dynamics up. -Only the flexible modes of the struts and of the top plate are not represented here as these elements are modelled as rigid bodies. -#+end_important - -*** Transfer function from Actuators to Force Sensors -The measured FRF and the identified dynamics from $u_i$ to $\tau_{m,i}$ are compared in Figure ref:fig:test_nhexa_comp_masses_model_exp_iff. -A zoom near the "suspension" modes is shown in Figure ref:fig:test_nhexa_comp_masses_model_exp_iff_zoom. - #+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm +%% Bode plot for the transfer function from u to Vs +masses = [0, 13, 26, 39]; figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); -freqs = 2*logspace(1,3,1000); - ax1 = nexttile([2,1]); hold on; for i_mass = 0:3 - plot(frf_ol_m.f, abs(frf_ol_m.G_tau{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... - 'DisplayName', sprintf('$d\\tau_{m,i}/u_i$ - FRF %i', i_mass)); + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... + 'DisplayName', sprintf('Meas (%i kg)', masses(i_mass+1))); for i = 2:6 - plot(frf_ol_m.f, abs(frf_ol_m.G_tau{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... + plot(frf_ol.f, abs(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... 'HandleVisibility', 'off'); end - plot(freqs, abs(squeeze(freqresp(sim_ol_m.G_tau{i_mass+1}(1,1), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('$\\tau_{m,i}/u_i$ - Sim %i', i_mass)); + plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(1,1), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:), ... + 'DisplayName', 'Simscape'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-2, 1e2]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e2]); +leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +leg.ItemTokenSize(1) = 15; ax2 = nexttile; hold on; for i_mass = 0:3 for i =1:6 - plot(frf_ol_m.f, 180/pi*angle(frf_ol_m.G_tau{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); + plot(frf_ol.f, 180/pi*angle(frf_ol.G_Vs{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); end - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol_m.G_tau{i_mass+1}(i,i), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:)); + plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(i,i), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); @@ -5705,30 +1595,142 @@ hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); +xlim([20, 2e2]); +xticks([20, 50, 100, 200]) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/test_nhexa_comp_simscape_Vs_diag_masses.pdf', 'width', 'half', 'height', 600); +#+end_src + +#+name: fig:test_nhexa_comp_simscape_diag_masses +#+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag}) +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/test_nhexa_comp_simscape_de_diag_masses.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/test_nhexa_comp_simscape_Vs_diag_masses.png]] +#+end_subfigure +#+end_figure + +In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from $u_1$ to $d_{e1}$ to $d_{e6}$ are compared in the case of the 39kg payload in Figure ref:fig:test_nhexa_comp_simscape_de_all_high_mass. +Excellent match between the experimental coupling and the model coupling is observed. +The model therefore well represents the system dynamical coupling for different considered payloads. + +#+begin_src matlab :exports none +%% Comparison of the plants (encoder output) when tuning the misalignment +i_input = 1; + +figure; +tiledlayout(2, 3, 'TileSpacing', 'tight', 'Padding', 'tight'); + +ax1 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 1, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(1, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e1}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); + +ax2 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 2, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(2, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e2}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); + +ax3 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 3, i_input)), ... + 'DisplayName', 'Measurements'); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(3, i_input), freqs, 'Hz'))), ... + 'DisplayName', 'Model (2-DoF APA)'); +text(12, 4e-4, '$d_{e3}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +leg.ItemTokenSize(1) = 15; + +ax4 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 4, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(4, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e4}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xticks([10, 50, 100, 200, 400]) + +ax5 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 5, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(5, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e5}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xticks([10, 50, 100, 200, 400]) + +ax6 = nexttile(); +hold on; +plot(frf_ol.f, abs(frf_ol.G_de{4}(:, 6, i_input))); +plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{4}(6, i_input), freqs, 'Hz')))); +text(12, 4e-4, '$d_{e6}/u_{1}$', 'Horiz','left', 'Vert','top') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xticks([10, 50, 100, 200, 400]) + +linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); +xlim([10, 5e2]); ylim([1e-8, 5e-4]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_masses_model_exp_iff.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/test_nhexa_comp_simscape_de_all_high_mass.pdf', 'width', 'full', 'height', 700); #+end_src -#+name: fig:test_nhexa_comp_masses_model_exp_iff -#+caption: Comparison of the transfer functions from $u_i$ to $\tau_{m,i}$ - measured FRF and identification from the Simscape model +#+name: fig:test_nhexa_comp_simscape_de_all_high_mass +#+caption: Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$ #+RESULTS: -[[file:figs/test_nhexa_comp_masses_model_exp_iff.png]] +[[file:figs/test_nhexa_comp_simscape_de_all_high_mass.png]] -#+begin_src matlab :exports none :tangle no -xlim([40, 2e2]); -#+end_src +** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_masses_model_exp_iff_zoom.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src +As illustrated in this section, the developed Simscape model accurately represents the suspension modes of the Nano-Hexapod. +Both FRF matrices from $\mathbf{u}$ to $\mathbf{V}_s$ and from $\mathbf{u}$ to $\mathbf{d}_e$ are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. -#+name: fig:test_nhexa_comp_masses_model_exp_iff_zoom -#+caption: Comparison of the transfer functions from $u_i$ to $\tau_{m,i}$ - measured FRF and identification from the Simscape model (Zoom) -#+RESULTS: -[[file:figs/test_nhexa_comp_masses_model_exp_iff_zoom.png]] +At frequency above the suspension modes, the Nano-Hexapod model becomes inaccurate as the flexible modes are not modelled. +It was shown that modelling the APA300ML using a "super-element" allows to model the internal resonances of the struts. +The same could be done with the top platform and the encoder supports, but the model order would be higher and may become unpractical for simulation purposes. + +* Conclusion + +The goal of this test bench was to obtain an accurate model of the nano-hexapod that can then be included on top of the micro-station model. + +This strategy was to measure the nano-hexapod in conditions where all factors that could have impacted the nano-hexapod dynamics were taken into account. +This was done by developing a suspended table with low frequency suspension modes which can be accurately modelled. + +While the dynamics of the nano-hexapod was indeed impacted by the dynamics of the suspended platform, this impact was also taken into account in the Simscape model, and a good match was obtained. + +Obtaining a model accurately representing the complex dynamics of the Nano-Hexapod was made possible by the modelling approach used during this work. +This approach consisted of tuning and validating models of individual components (such as the APA and flexible joints) using dedicated test benches. +Only then, the different models could be combined to form the Nano-Hexapod dynamical model. +If a model of the nano-hexapod was developed in one time, it would be difficult to tune all model parameters to match the measured dynamics, or even to know if the model "structure" would be adequate to represents the system dynamics. * Bibliography :ignore: #+latex: \printbibliography[heading=bibintoc,title={Bibliography}] @@ -5747,9 +1749,7 @@ exportFig('figs/comp_masses_model_exp_iff_zoom.pdf', 'width', 'wide', 'height', addpath('./matlab/'); % Path for scripts %% Path for functions, data and scripts -addpath('./matlab/mat/data_frf/'); % Path for Computed FRF -addpath('./matlab/mat/data_sim/'); % Path for Simulation -addpath('./matlab/mat/data_meas/'); % Path for Measurements +addpath('./matlab/mat/'); % Path for Computed FRF addpath('./matlab/src/'); % Path for functions addpath('./matlab/STEPS/'); % Path for STEPS addpath('./matlab/subsystems/'); % Path for Subsystems Simulink files @@ -5758,9 +1758,7 @@ addpath('./matlab/subsystems/'); % Path for Subsystems Simulink files #+NAME: m-init-path-tangle #+BEGIN_SRC matlab %% Path for functions, data and scripts -addpath('./mat/data_frf/'); % Path for Computed FRF -addpath('./mat/data_sim/'); % Path for Simulation -addpath('./mat/data_meas/'); % Path for Measurements +addpath('./mat/'); % Path for Data addpath('./src/'); % Path for functions addpath('./STEPS/'); % Path for STEPS addpath('./subsystems/'); % Path for Subsystems Simulink files @@ -5774,13 +1772,8 @@ table_type = 'Rigid'; % On top of vibration table device_type = 'None'; % On top of vibration table payload_num = 0; % No Payload -%% Open Simulink Model +% Simulink Model name mdl = 'test_bench_nano_hexapod'; - -options = linearizeOptions; -options.SampleTime = 0; - -open(mdl) #+end_src ** Initialize other elements @@ -5789,11 +1782,8 @@ open(mdl) %% Colors for the figures colors = colororder; -%% Tested Masses -i_masses = 0:3; - %% Frequency Vector -freqs = 2*logspace(1, 3, 1000); +freqs = logspace(log10(10), log10(2e3), 1000); #+END_SRC * Footnotes diff --git a/test-bench-nano-hexapod.pdf b/test-bench-nano-hexapod.pdf index 498af5a..621773f 100644 Binary files a/test-bench-nano-hexapod.pdf and b/test-bench-nano-hexapod.pdf differ diff --git a/test-bench-nano-hexapod.tex b/test-bench-nano-hexapod.tex index 1dcc9f8..fb0e560 100644 --- a/test-bench-nano-hexapod.tex +++ b/test-bench-nano-hexapod.tex @@ -1,4 +1,4 @@ -% Created 2024-10-27 Sun 14:49 +% Created 2024-10-29 Tue 15:32 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -254,329 +254,322 @@ Simscape & 1.3 Hz & 1.8 Hz & 6.8 Hz & 9.5 Hz\\ \end{table} -\chapter{Nano-Hexapod Dynamics} +\section*{Conclusion} +In this section, a suspended table with low frequency suspension modes and high frequency flexible modes was presented. +This suspended table will be used in Section \ref{sec:test_nhexa_dynamics} for dynamical identification of the Nano-Hexapod. +The objective is to be able to accurately identify the dynamics of the nano-hexapod, isolated from complex support dynamics. +The key point of this strategy is to be able to accurately model the suspended table. + +To do so, a modal analysis of the suspended table was performed in Section \ref{ssec:test_nhexa_table_identification}, validating the low frequency suspension modes and high frequency flexible modes. +Then, a multi-body model of this suspended table was tuned to match with the measurements (Section \ref{ssec:test_nhexa_table_model}). + +\chapter{Nano-Hexapod Measured Dynamics} \label{sec:test_nhexa_dynamics} -In Figure \ref{fig:test_nhexa_nano_hexapod_signals} is shown a block diagram of the experimental setup. -When possible, the notations are consistent with this diagram and summarized in Table \ref{tab:list_signals}. +The Nano-Hexapod is then mounted on top of the suspended table as shown in Figure \ref{fig:test_nhexa_hexa_suspended_table}. +All the instrumentation (Speedgoat with ADC, DAC, piezoelectric voltage amplifiers and digital interfaces for the encoder) are setup and connected to the nano-hexapod using many cables. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_hexa_suspended_table.jpg} +\caption{\label{fig:test_nhexa_hexa_suspended_table}Mounted Nano-Hexapod on top of the suspended table} +\end{figure} + +A modal analysis of the nano-hexapod is first performed in Section \ref{ssec:test_nhexa_enc_struts_modal_analysis}. +It will be used to better understand the measured dynamics from actuators to sensors. + +A block diagram schematic of the (open-loop) system is shown in Figure \ref{fig:test_nhexa_nano_hexapod_signals}. +The transfer function from controlled signals \(\mathbf{u}\) to the force sensors voltages \(\mathbf{V}_s\) and to the encoders measured displacements \(\mathbf{d}_e\) are identified in Section \ref{ssec:test_nhexa_identification}. +The effect of the payload mass on the dynamics is studied in Section \ref{ssec:test_nhexa_added_mass}. \begin{figure}[htbp] \centering \includegraphics[scale=1,scale=1]{figs/test_nhexa_nano_hexapod_signals.png} -\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the system with named signals} +\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the system. Command signal generated by the speedgoat is \(\mathbf{u}\), the measured dignals are \(\mathbf{d}_{e}\) and \(\mathbf{V}_s\). Units are indicated in square brackets.} \end{figure} +\section{Modal analysis} +\label{ssec:test_nhexa_enc_struts_modal_analysis} + +In order to ease the future analysis of the measured plant dynamics, a basic modal analysis of the nano-hexapod is performed. +Five 3-axis accelerometers are fixed on the top platform of the nano-hexapod (Figure \ref{fig:test_nhexa_modal_analysis}) and the top platform is excited using an instrumented hammer. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,width=0.7\linewidth]{figs/test_nhexa_modal_analysis.jpg} +\caption{\label{fig:test_nhexa_modal_analysis}Five accelerometers fixed on top of the nano-hexapod to perform a modal analysis} +\end{figure} + +Between 100Hz and 200Hz, 6 suspension modes (i.e. rigid body modes of the top platform) are identified. +At around 700Hz, two flexible modes of the top plate are observed (see Figure \ref{fig:test_nhexa_hexa_flexible_modes}). +These modes are summarized in Table \ref{tab:test_nhexa_hexa_modal_modes_list}. + \begin{table}[htbp] \centering -\begin{tabularx}{\linewidth}{Xllll} +\begin{tabularx}{0.7\linewidth}{ccX} \toprule - & \textbf{Unit} & \textbf{Matlab} & \textbf{Vector} & \textbf{Elements}\\ +\textbf{Mode} & \textbf{Frequency} & \textbf{Description}\\ \midrule -Control Input (wanted DAC voltage) & \texttt{[V]} & \texttt{u} & \(\bm{u}\) & \(u_i\)\\ -DAC Output Voltage & \texttt{[V]} & \texttt{u} & \(\tilde{\bm{u}}\) & \(\tilde{u}_i\)\\ -PD200 Output Voltage & \texttt{[V]} & \texttt{ua} & \(\bm{u}_a\) & \(u_{a,i}\)\\ -Actuator applied force & \texttt{[N]} & \texttt{tau} & \(\bm{\tau}\) & \(\tau_i\)\\ -\midrule -Strut motion & \texttt{[m]} & \texttt{dL} & \(d\bm{\mathcal{L}}\) & \(d\mathcal{L}_i\)\\ -Encoder measured displacement & \texttt{[m]} & \texttt{dLm} & \(d\bm{\mathcal{L}}_m\) & \(d\mathcal{L}_{m,i}\)\\ -\midrule -Force Sensor strain & \texttt{[m]} & \texttt{epsilon} & \(\bm{\epsilon}\) & \(\epsilon_i\)\\ -Force Sensor Generated Voltage & \texttt{[V]} & \texttt{taum} & \(\tilde{\bm{\tau}}_m\) & \(\tilde{\tau}_{m,i}\)\\ -Measured Generated Voltage & \texttt{[V]} & \texttt{taum} & \(\bm{\tau}_m\) & \(\tau_{m,i}\)\\ -\midrule -Motion of the top platform & \texttt{[m,rad]} & \texttt{dX} & \(d\bm{\mathcal{X}}\) & \(d\mathcal{X}_i\)\\ -Metrology measured displacement & \texttt{[m,rad]} & \texttt{dXm} & \(d\bm{\mathcal{X}}_m\) & \(d\mathcal{X}_{m,i}\)\\ +1 & 120 Hz & Suspension Mode: Y-translation\\ +2 & 120 Hz & Suspension Mode: X-translation\\ +3 & 145 Hz & Suspension Mode: Z-translation\\ +4 & 165 Hz & Suspension Mode: Y-rotation\\ +5 & 165 Hz & Suspension Mode: X-rotation\\ +6 & 190 Hz & Suspension Mode: Z-rotation\\ +7 & 692 Hz & (flexible) Membrane mode of the top platform\\ +8 & 709 Hz & Second flexible mode of the top platform\\ \bottomrule \end{tabularx} -\caption{\label{tab:list_signals}List of signals} +\caption{\label{tab:test_nhexa_hexa_modal_modes_list}Description of the identified modes of the Nano-Hexapod} \end{table} \begin{figure}[htbp] -\centering -\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_IMG_20210625_083801.jpg} -\caption{\label{fig:test_nhexa_enc_fixed_to_struts}Nano-Hexapod with encoders fixed to the struts} +\begin{subfigure}{\textwidth} +\begin{center} +\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_hexa_flexible_mode_1.jpg} +\end{center} +\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_1}Flexible mode at 692Hz} +\end{subfigure} +\begin{subfigure}{\textwidth} +\begin{center} +\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_hexa_flexible_mode_2.jpg} +\end{center} +\subcaption{\label{fig:test_nhexa_hexa_flexible_mode_2}Flexible mode at 709Hz} +\end{subfigure} +\caption{\label{fig:test_nhexa_hexa_flexible_modes}Two identified flexible modes of the top plate of the Nano-Hexapod} \end{figure} -It is structured as follow: -\begin{itemize} -\item Section \ref{sec:test_nhexa_enc_plates_plant_id}: The dynamics of the nano-hexapod is identified. -\item Section \ref{sec:test_nhexa_enc_plates_comp_simscape}: The identified dynamics is compared with the Simscape model. -\end{itemize} - \section{Identification of the dynamics} -\label{sec:test_nhexa_enc_plates_plant_id} -In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is identified. +\label{ssec:test_nhexa_identification} -First, the measurement data are loaded in Section \ref{sec:test_nhexa_enc_plates_plant_id_setup}, then the transfer function matrix from the actuators to the encoders are estimated in Section \ref{sec:test_nhexa_enc_plates_plant_id_dvf}. -Finally, the transfer function matrix from the actuators to the force sensors is estimated in Section \ref{sec:test_nhexa_enc_plates_plant_id_iff}. -\subsection{Data Loading and Spectral Analysis Setup} -\label{sec:test_nhexa_enc_plates_plant_id_setup} +The dynamics of the nano-hexapod from the six command signals (\(u_1\) to \(u_6\)) the six measured displacement by the encoders (\(d_{e1}\) to \(d_{e6}\)) and to the six force sensors (\(V_{s1}\) to \(V_{s6}\)) are identified by generating a low pass filtered white noise for each of the command signals, one by one. -The actuators are excited one by one using a low pass filtered white noise. -For each excitation, the 6 force sensors and 6 encoders are measured and saved. -\subsection{Transfer function from Actuator to Encoder} -\label{sec:test_nhexa_enc_plates_plant_id_dvf} +The \(6 \times 6\) FRF matrix from \(\mathbf{u}\) ot \(\mathbf{d}_e\) is shown in Figure \ref{fig:test_nhexa_identified_frf_de}. +The diagonal terms are displayed using colorful lines, and all the 30 off-diagonal terms are displayed by grey lines. -The 6x6 transfer function matrix from the excitation voltage \(\bm{u}\) and the displacement \(d\bm{\mathcal{L}}_m\) as measured by the encoders is estimated. +All the six diagonal terms are well superimposed up to at least \(1\,kHz\), indicating good manufacturing and mounting uniformity. +Below the first suspension mode, good decoupling can be observed (the amplitude of the all of off-diagonal terms are \(\approx 20\) times smaller than the diagonal terms). -The diagonal and off-diagonal terms of this transfer function matrix are shown in Figure \ref{fig:test_nhexa_enc_plates_dvf_frf}. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/enc_plates_dvf_frf.png} -\caption{\label{fig:test_nhexa_enc_plates_dvf_frf}Measured FRF for the transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\)} -\end{figure} +From 10Hz up to 1kHz, around 10 resonance frequencies can be observed. +The first 4 are suspension modes (at 122Hz, 143Hz, 165Hz and 191Hz) which correlate the modes measured during the modal analysis in Section \ref{ssec:test_nhexa_enc_struts_modal_analysis}. +Then, three modes at 237Hz, 349Hz and 395Hz are attributed to the internal strut resonances (this will be checked in Section \ref{ssec:test_nhexa_comp_model_coupling}). +Except the mode at 237Hz, their amplitude is rather low. +Two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform. +Other modes can be observed above 1kHz, which can be attributed to flexible modes of the encoder supports or to flexible modes of the top platform. -\begin{important} -From Figure \ref{fig:test_nhexa_enc_plates_dvf_frf}, we can draw few conclusions on the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) when the encoders are fixed to the plates: -\begin{itemize} -\item the decoupling is rather good at low frequency (below the first suspension mode). -The low frequency gain is constant for the off diagonal terms, whereas when the encoders where fixed to the struts, the low frequency gain of the off-diagonal terms were going to zero (Figure \ref{fig:test_nhexa_enc_struts_dvf_frf}). -\item the flexible modes of the struts at 226Hz and 337Hz are indeed shown in the transfer functions, but their amplitudes are rather low. -\item the diagonal terms have alternating poles and zeros up to at least 600Hz: the flexible modes of the struts are not affecting the alternating pole/zero pattern. This what not the case when the encoders were fixed to the struts (Figure \ref{fig:test_nhexa_enc_struts_dvf_frf}). -\end{itemize} -\end{important} - -\subsection{Transfer function from Actuator to Force Sensor} -\label{sec:test_nhexa_enc_plates_plant_id_iff} -Then the 6x6 transfer function matrix from the excitation voltage \(\bm{u}\) and the voltage \(\bm{\tau}_m\) generated by the Force senors is estimated. -The bode plot of the diagonal and off-diagonal terms are shown in Figure \ref{fig:test_nhexa_enc_plates_iff_frf}. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/enc_plates_iff_frf.png} -\caption{\label{fig:test_nhexa_enc_plates_iff_frf}Measured FRF for the IFF plant} -\end{figure} - -\begin{important} -It is shown in Figure \ref{fig:test_nhexa_enc_plates_iff_comp_simscape_all} that: -\begin{itemize} -\item The IFF plant has alternating poles and zeros -\item The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF -\item The decoupling is quite good at low frequency (below the first model) as well as high frequency (above the last suspension mode, except near the flexible modes of the top plate) -\end{itemize} -\end{important} - -\subsection{Save Identified Plants} -The identified dynamics is saved for further use. -\section{Effect of Payload mass on the Dynamics} -\label{sec:test_nhexa_added_mass} -In this section, the encoders are fixed to the plates, and we identify the dynamics for several payloads. -The added payload are half cylinders, and three layers can be added for a total of around 40kg (Figure \ref{fig:test_nhexa_picture_added_3_masses}). +Up to at least 1kHz, an alternating pole/zero pattern is observed, which renders the control easier to tune. +This would not have been the case if the encoders were fixed to the struts. \begin{figure}[htbp] \centering -\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_picture_added_3_masses.jpg} -\caption{\label{fig:test_nhexa_picture_added_3_masses}Picture of the nano-hexapod with added mass} +\includegraphics[scale=1]{figs/test_nhexa_identified_frf_de.png} +\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{d}_e\). The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.} \end{figure} -First the dynamics from \(\bm{u}\) to \(d\mathcal{L}_m\) and \(\bm{\tau}_m\) is identified. -Then, the Integral Force Feedback controller is developed and applied as shown in Figure \ref{fig:test_nhexa_nano_hexapod_signals_iff}. -Finally, the dynamics from \(\bm{u}^\prime\) to \(d\mathcal{L}_m\) is identified and the added damping can be estimated. + +Similarly, the \(6 \times 6\) FRF matrix from \(\mathbf{u}\) to \(\mathbf{V}_s\) is shown in Figure \ref{fig:test_nhexa_identified_frf_Vs}. +Alternating poles and zeros is observed up to at least 2kHz, which is a necessary characteristics in order to apply decentralized IFF. +Similar to what was observed for the encoder outputs, all the ``diagonal'' terms are well superimposed, indicating that the same controller can be applied for all the struts. +The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF. \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/test_nhexa_nano_hexapod_signals_iff.png} -\caption{\label{fig:test_nhexa_nano_hexapod_signals_iff}Block Diagram of the experimental setup and model} +\includegraphics[scale=1]{figs/test_nhexa_identified_frf_Vs.png} +\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\). The 6 diagonal terms are the colorfull lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.} \end{figure} -\subsection{Measured Frequency Response Functions} -The following data are loaded: -\begin{itemize} -\item \texttt{Va}: the excitation voltage (corresponding to \(u_i\)) -\item \texttt{Vs}: the generated voltage by the 6 force sensors (corresponding to \(\bm{\tau}_m\)) -\item \texttt{de}: the measured motion by the 6 encoders (corresponding to \(d\bm{\mathcal{L}}_m\)) -\end{itemize} -The window \texttt{win} and the frequency vector \texttt{f} are defined. -Finally the \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) and from \(\bm{u}\) to \(\bm{\tau}_m\) are identified: + +\section{Effect of payload mass on the dynamics} +\label{ssec:test_nhexa_added_mass} + +As one major challenge in the control of the NASS is the wanted robustness to change of payload mass, it is necessary to understand how the dynamics of the nano-hexapod changes with a change of payload mass. + +In order to study this change of dynamics with the payload mass, up to three ``cylindrical masses'' of \(13\,kg\) each can be added for a total of \(\approx 40\,kg\). +These three cylindrical masses on top of the nano-hexapod are shown in Figure \ref{fig:test_nhexa_table_mass_3}. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_table_mass_3.jpg} +\caption{\label{fig:test_nhexa_table_mass_3}Picture of the nano-hexapod with the added three cylindrical masses for a total of \(\approx 40\,kg\)} +\end{figure} + The identified dynamics are then saved for further use. -\subsection{Transfer function from Actuators to Encoders} -The transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_{m}\) are shown in Figure \ref{fig:test_nhexa_comp_plant_payloads_dvf}. +The obtained frequency response functions from actuator signal \(u_i\) to the associated encoder \(d_{ei}\) for the four payload conditions (no mass, 13kg, 26kg and 39kg) are shown in Figure \ref{fig:test_nhexa_identified_frf_de_masses}. +As expected, the frequency of the suspension modes are decreasing with an increase of the payload mass. +The low frequency gain does not change as it is linked to the stiffness property of the nano-hexapod, and not to its mass property. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_comp_plant_payloads_dvf.png} -\caption{\label{fig:test_nhexa_comp_plant_payloads_dvf}Measured Frequency Response Functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms.} -\end{figure} - - -\begin{important} -From Figure \ref{fig:test_nhexa_comp_plant_payloads_dvf}, we can observe few things: -\begin{itemize} -\item The obtained dynamics is changing a lot between the case without mass and when there is at least one added mass. -\item Between 1, 2 and 3 added masses, the dynamics is not much different, and it would be easier to design a controller only for these cases. -\item The flexible modes of the top plate is first decreased a lot when the first mass is added (from 700Hz to 400Hz). +The frequencies of the two flexible modes of the top plate are first decreased a lot when the first mass is added (from \(\approx 700\,Hz\) to \(\approx 400\,Hz\)). This is due to the fact that the added mass is composed of two half cylinders which are not fixed together. -Therefore is adds a lot of mass to the top plate without adding a lot of rigidity in one direction. -When more than 1 mass layer is added, the half cylinders are added with some angles such that rigidity are added in all directions (see Figure \ref{fig:test_nhexa_picture_added_3_masses}). +It therefore adds a lot of mass to the top plate without adding stiffness in one direction. +When more than one ``mass layer'' is added, the half cylinders are added with some angles such that rigidity are added in all directions (see how the three mass ``layers'' are positioned in Figure \ref{fig:test_nhexa_table_mass_3}). In that case, the frequency of these flexible modes are increased. -In practice, the payload should be one solid body, and we should not see a massive decrease of the frequency of this flexible mode. -\item Flexible modes of the top plate are becoming less problematic as masses are added. -\item First flexible mode of the strut at 230Hz is not much decreased when mass is added. -However, its apparent amplitude is much decreased. -\end{itemize} -\end{important} +In practice, the payload should be one solid body, and no decrease of the frequency of this flexible mode should be observed. +The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass is increased. -\subsection{Transfer function from Actuators to Force Sensors} -The transfer functions from \(\bm{u}\) to \(\bm{\tau}_{m}\) are shown in Figure \ref{fig:test_nhexa_comp_plant_payloads_iff}. +The measured FRF from \(u_i\) to \(V_{si}\) are shown in Figure \ref{fig:test_nhexa_identified_frf_Vs_masses}. +For all the tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust way. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_comp_plant_payloads_iff.png} -\caption{\label{fig:test_nhexa_comp_plant_payloads_iff}Measured Frequency Response Functions from \(u_i\) to \(\tau_{m,i}\) for all 4 payload conditions. Diagonal terms are solid lines, and shaded lines are off-diagonal terms.} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_identified_frf_de_masses.png} +\end{center} +\subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_identified_frf_Vs_masses.png} +\end{center} +\subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$} +\end{subfigure} +\caption{\label{fig:test_struts_mounting}Measured Frequency Response Functions from \(u_i\) to \(d_{ei}\) (\subref{fig:test_nhexa_identified_frf_de_masses}) and from \(u_i\) to \(V_{si}\) (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown.} \end{figure} -\begin{important} -From Figure \ref{fig:test_nhexa_comp_plant_payloads_iff}, we can see that for all added payloads, the transfer function from \(\bm{u}\) to \(\bm{\tau}_{m}\) always has alternating poles and zeros. -\end{important} +\section*{Conclusion} +After the Nano-Hexapod was fixed on top of the suspended table, its dynamics was identified. -\subsection{Coupling of the transfer function from Actuator to Encoders} -The RGA-number, which is a measure of the interaction in the system, is computed for the transfer function matrix from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) for all the payloads. -The obtained numbers are compared in Figure \ref{fig:test_nhexa_rga_num_ol_masses}. +The frequency response functions from the six DAC voltages \(\mathbf{u}\) to the six encoders measured displacements \(\mathbf{d}_e\) displays alternating complex conjugate poles and complex conjugate zeros up to at least 1kHz. +At low frequency, the coupling is small, indicating correct assembly of all parts. +This should enables the design of a decentralized positioning controller based on the encoder for relative positioning purposes. +The suspension modes and flexible modes measured during the modal analysis (Section \ref{ssec:test_nhexa_enc_struts_modal_analysis}) are also observed in the dynamics. +Lot's of other modes are present above 700Hz, which will inevitably limit the achievable bandwidth. +The observed effect of the payload's mass on the dynamics is quite large, which also represent a complex control challenge. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_rga_num_ol_masses.png} -\caption{\label{fig:test_nhexa_rga_num_ol_masses}RGA-number for the open-loop transfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\)} -\end{figure} +The frequency response functions from the six DAC voltages \(\mathbf{u}\) to the six force sensors voltages \(\mathbf{V}_s\) all have alternating complex conjugate poles and complex conjugate zeros. +This indicates that it should be possible to implement decentralized Integral Force Feedback in a robust way. +This alternating property holds for all the tested payloads. -\begin{important} -From Figure \ref{fig:test_nhexa_rga_num_ol_masses}, it is clear that the coupling is quite large starting from the first suspension mode of the nano-hexapod. -Therefore, is the payload's mass is increase, the coupling in the system start to become unacceptably large at lower frequencies. -\end{important} - -\section{Conclusion} -\begin{important} -In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied. - -It has been found that: -\begin{itemize} -\item The measured dynamics is in agreement with the dynamics of the simscape model, up to the flexible modes of the top plate. -See figures \ref{fig:test_nhexa_enc_plates_iff_comp_simscape} and \ref{fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape} for the transfer function to the force sensors and Figures \ref{fig:test_nhexa_enc_plates_dvf_comp_simscape} and \ref{fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape} for the transfer functions to the encoders -\item The Integral Force Feedback strategy is very effective in damping the suspension modes of the nano-hexapod (Figure \ref{fig:test_nhexa_enc_plant_plates_effect_iff}). -\item The transfer function from \(\bm{u}^\prime\) to \(d\bm{\mathcal{L}}_m\) show nice dynamical properties and is a much better candidate for the high-authority-control than when the encoders were fixed to the struts. -At least up to the flexible modes of the top plate, the diagonal elements of the transfer function matrix have alternating poles and zeros, and the phase is moving smoothly. -Only the flexible modes of the top plates seems to be problematic for control. -\end{itemize} -\end{important} - -\chapter{Comparison with the Nano-Hexapod model?} +\chapter{Nano-Hexapod Model Dynamics} \label{sec:test_nhexa_model} -\section{Comparison with the Simscape Model} -\label{sec:test_nhexa_enc_plates_comp_simscape} -In this section, the measured dynamics done in Section \ref{sec:test_nhexa_enc_plates_plant_id} is compared with the dynamics estimated from the Simscape model. - -A configuration is added to be able to put the nano-hexapod on top of the vibration table as shown in Figure \ref{fig:simscape_vibration_table}. +In this section, the measured dynamics done in Section \ref{sec:test_nhexa_dynamics} is compared with the dynamics estimated from the Simscape model. +The nano-hexapod simscape model is therefore added on top of the vibration table Simscape model as shown in Figure \ref{fig:test_nhexa_hexa_simscape}. \begin{figure}[htbp] \centering -\includegraphics[scale=1,width=0.8\linewidth]{figs/vibration_table_nano_hexapod_simscape.png} -\caption{\label{fig:simscape_vibration_table}3D representation of the simscape model with the nano-hexapod} +\includegraphics[scale=1,width=0.8\linewidth]{figs/test_nhexa_hexa_simscape.png} +\caption{\label{fig:test_nhexa_hexa_simscape}3D representation of the simscape model with the nano-hexapod on top of the suspended table. Three mass ``layers'' are here added} \end{figure} -\subsection{Identification with the Simscape Model} -The nano-hexapod is initialized with the APA taken as 2dof models. -Now, the dynamics from the DAC voltage \(\bm{u}\) to the encoders \(d\bm{\mathcal{L}}_m\) is estimated using the Simscape model. -Then the transfer function from \(\bm{u}\) to \(\bm{\tau}_m\) is identified using the Simscape model. -The identified dynamics is saved for further use. -\subsection{Dynamics from Actuator to Force Sensors} -The identified dynamics is compared with the measured FRF: -\begin{itemize} -\item Figure \ref{fig:test_nhexa_enc_plates_iff_comp_simscape_all}: the individual transfer function from \(u_1\) (the DAC voltage for the first actuator) to the force sensors of all 6 struts are compared -\item Figure \ref{fig:test_nhexa_enc_plates_iff_comp_simscape}: all the diagonal elements are compared -\item Figure \ref{fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape}: all the off-diagonal elements are compared -\end{itemize} + +The model should exhibit certain characteristics that are verified in this section. +First, it should match the measured system dynamics from actuators to sensors that were presented in Section \ref{sec:test_nhexa_dynamics}. +Both the ``direct'' terms (Section \ref{ssec:test_nhexa_comp_model}) and ``coupling'' terms (Section \ref{ssec:test_nhexa_comp_model_coupling}) of the Simscape model are compared with the measured dynamics. +Second, it should also represents how the system dynamics changes when a payload is fixed to the top platform. +This is checked in Section \ref{ssec:test_nhexa_comp_model_masses}. + +\section{Nano-Hexapod model dynamics} +\label{ssec:test_nhexa_comp_model} + +The Simscape model of the nano-hexapod is first configured with 4-DoF flexible joints, 2-DoF APA and rigid top and bottom platforms. +The stiffness of the flexible joints are chosen based on the values estimated using the test bench and based on FEM. +The parameters of the APA model are the ones determined from the test bench of the APA. +The \(6 \times 6\) transfer function matrices from \(\mathbf{u}\) to \(\mathbf{d}_e\) and from \(\mathbf{u}\) to \(\mathbf{V}_s\) are extracted then from the Simscape model. + +A first feature that should be checked is that the model well represents the ``direct'' terms of the measured FRF matrix. +To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure \ref{fig:test_nhexa_comp_simscape_diag}. +It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modelled. +The three resonances that were attributed to ``internal'' flexible modes of the struts (at 237Hz, 349Hz and 395Hz) cannot be seen in the model, which is reasonable as the APA are here modelled as a simple uniaxial 2-DoF system. +At higher frequencies, no resonances can be seen in the model, as the as the top plate and the encoder supports are modelled as rigid bodies. + +\begin{figure}[htbp] +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_de_diag.png} +\end{center} +\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_Vs_diag.png} +\end{center} +\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$} +\end{subfigure} +\caption{\label{fig:test_nhexa_comp_simscape_diag}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})} +\end{figure} + +\section{Modelling dynamical coupling} +\label{ssec:test_nhexa_comp_model_coupling} + +Another wanted feature of the model is that it well represents the coupling in the system as this is often the limiting factor for the control of MIMO systems. +Instead of comparing the full 36 elements of the \(6 \times 6\) FFR matrix from \(\mathbf{u}\) to \(\mathbf{d}_e\), only the first ``column'' is compared (Figure \ref{fig:test_nhexa_comp_simscape_de_all}), which corresponds to the transfer function from the command \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\). +It can be seen that the coupling in the model is well matching the measurements up to the first un-modelled flexible mode at 237Hz. +Similar results are observed for all the other coupling terms, as well as for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\). \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/test_nhexa_enc_plates_iff_comp_simscape_all.png} -\caption{\label{fig:test_nhexa_enc_plates_iff_comp_simscape_all}IFF Plant for the first actuator input and all the force senosrs} +\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all.png} +\caption{\label{fig:test_nhexa_comp_simscape_de_all}Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)} \end{figure} +The APA300ML are then modelled with a \emph{super-element} extracted from a FE-software. +The obtained transfer functions from \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\) are compared with the measured FRF in Figure \ref{fig:test_nhexa_comp_simscape_de_all_flex}. +While the damping of the suspension modes for the \emph{super-element} is underestimated (which could be solved by properly tuning the proportional damping coefficients), the flexible modes of the struts at 237Hz and 349Hz are well modelled. +Even the mode 395Hz can be observed in the model. +Therefore, if the modes of the struts are to be modelled, the \emph{super-element} of the APA300ML may be used, at the cost of obtaining a much higher order model. + \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/test_nhexa_enc_plates_iff_comp_simscape.png} -\caption{\label{fig:test_nhexa_enc_plates_iff_comp_simscape}Diagonal elements of the IFF Plant} +\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all_flex.png} +\caption{\label{fig:test_nhexa_comp_simscape_de_all_flex}Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)} \end{figure} +\section{Modelling the effect of payload mass} +\label{ssec:test_nhexa_comp_model_masses} + +Another important characteristics of the model is that it should well represents the dynamics of the system for all considered payloads. +The model dynamics is therefore compared with the measured dynamics for 4 payloads (no payload, 13kg, 26kg and 39kg) in Figure \ref{fig:test_nhexa_comp_simscape_diag_masses}. +The observed shift to lower frequency of the suspension modes with an increased payload mass is well represented by the Simscape model. +The complex conjugate zeros are also well matching with the experiments both for the encoder outputs (Figure \ref{fig:test_nhexa_comp_simscape_de_diag_masses}) and the force sensor outputs (Figure \ref{fig:test_nhexa_comp_simscape_Vs_diag_masses}). + +Note that the model displays smaller damping that what is observed experimentally for high values of the payload mass. +One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping). +However, as decentralized IFF will be applied, the damping will be brought actively, and the open-loop damping value should have very little impact on the obtained plant. + +\begin{figure}[htbp] +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_de_diag_masses.png} +\end{center} +\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_Vs_diag_masses.png} +\end{center} +\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$} +\end{subfigure} +\caption{\label{fig:test_nhexa_comp_simscape_diag_masses}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the identified dynamics from the Simscape model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})} +\end{figure} + +In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from \(u_1\) to \(d_{e1}\) to \(d_{e6}\) are compared in the case of the 39kg payload in Figure \ref{fig:test_nhexa_comp_simscape_de_all_high_mass}. +Excellent match between the experimental coupling and the model coupling is observed. +The model therefore well represents the system dynamical coupling for different considered payloads. + \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/test_nhexa_enc_plates_iff_comp_offdiag_simscape.png} -\caption{\label{fig:test_nhexa_enc_plates_iff_comp_offdiag_simscape}Off diagonal elements of the IFF Plant} +\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all_high_mass.png} +\caption{\label{fig:test_nhexa_comp_simscape_de_all_high_mass}Comparison of the measured (in blue) and modelled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)} \end{figure} -\subsection{Dynamics from Actuator to Encoder} -The identified dynamics is compared with the measured FRF: -\begin{itemize} -\item Figure \ref{fig:test_nhexa_enc_plates_dvf_comp_simscape_all}: the individual transfer function from \(u_3\) (the DAC voltage for the actuator number 3) to the six encoders -\item Figure \ref{fig:test_nhexa_enc_plates_dvf_comp_simscape}: all the diagonal elements are compared -\item Figure \ref{fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape}: all the off-diagonal elements are compared -\end{itemize} +\section*{Conclusion} +As illustrated in this section, the developed Simscape model accurately represents the suspension modes of the Nano-Hexapod. +Both FRF matrices from \(\mathbf{u}\) to \(\mathbf{V}_s\) and from \(\mathbf{u}\) to \(\mathbf{d}_e\) are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_enc_plates_dvf_comp_simscape_all.png} -\caption{\label{fig:test_nhexa_enc_plates_dvf_comp_simscape_all}DVF Plant for the first actuator input and all the encoders} -\end{figure} +At frequency above the suspension modes, the Nano-Hexapod model becomes inaccurate as the flexible modes are not modelled. +It was shown that modelling the APA300ML using a ``super-element'' allows to model the internal resonances of the struts. +The same could be done with the top platform and the encoder supports, but the model order would be higher and may become unpractical for simulation purposes. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_enc_plates_dvf_comp_simscape.png} -\caption{\label{fig:test_nhexa_enc_plates_dvf_comp_simscape}Diagonal elements of the DVF Plant} -\end{figure} +\chapter{Conclusion} -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_enc_plates_dvf_comp_offdiag_simscape.png} -\caption{\label{fig:test_nhexa_enc_plates_dvf_comp_offdiag_simscape}Off diagonal elements of the DVF Plant} -\end{figure} +The goal of this test bench was to obtain an accurate model of the nano-hexapod that can then be included on top of the micro-station model. -\subsection{Conclusion} -\begin{important} -The Simscape model is quite accurate for the transfer function matrices from \(\bm{u}\) to \(\bm{\tau}_m\) and from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) except at frequencies of the flexible modes of the top-plate. -The Simscape model can therefore be used to develop the control strategies. -\end{important} +This strategy was to measure the nano-hexapod in conditions where all factors that could have impacted the nano-hexapod dynamics were taken into account. +This was done by developing a suspended table with low frequency suspension modes which can be accurately modelled. -\section{Comparison with the Simscape model} -\label{sec:test_nhexa_added_mass_simscape} -Let's now compare the identified dynamics with the Simscape model. -We wish to verify if the Simscape model is still accurate for all the tested payloads. -\subsection{System Identification} -Let's initialize the simscape model with the nano-hexapod fixed on top of the vibration table. -First perform the identification for the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\): -The identified dynamics are then saved for further use. -\subsection{Transfer function from Actuators to Encoders} -The measured FRF and the identified dynamics from \(u_i\) to \(d\mathcal{L}_{m,i}\) are compared in Figure \ref{fig:test_nhexa_comp_masses_model_exp_dvf}. -A zoom near the ``suspension'' modes is shown in Figure \ref{fig:test_nhexa_comp_masses_model_exp_dvf_zoom}. +While the dynamics of the nano-hexapod was indeed impacted by the dynamics of the suspended platform, this impact was also taken into account in the Simscape model, and a good match was obtained. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/comp_masses_model_exp_dvf.png} -\caption{\label{fig:test_nhexa_comp_masses_model_exp_dvf}Comparison of the transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) - measured FRF and identification from the Simscape model} -\end{figure} - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_comp_masses_model_exp_dvf_zoom.png} -\caption{\label{fig:test_nhexa_comp_masses_model_exp_dvf_zoom}Comparison of the transfer functions from \(u_i\) to \(d\mathcal{L}_{m,i}\) - measured FRF and identification from the Simscape model (Zoom)} -\end{figure} - -\begin{important} -The Simscape model is very accurately representing the measured dynamics up. -Only the flexible modes of the struts and of the top plate are not represented here as these elements are modelled as rigid bodies. -\end{important} - -\subsection{Transfer function from Actuators to Force Sensors} -The measured FRF and the identified dynamics from \(u_i\) to \(\tau_{m,i}\) are compared in Figure \ref{fig:test_nhexa_comp_masses_model_exp_iff}. -A zoom near the ``suspension'' modes is shown in Figure \ref{fig:test_nhexa_comp_masses_model_exp_iff_zoom}. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_comp_masses_model_exp_iff.png} -\caption{\label{fig:test_nhexa_comp_masses_model_exp_iff}Comparison of the transfer functions from \(u_i\) to \(\tau_{m,i}\) - measured FRF and identification from the Simscape model} -\end{figure} - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/test_nhexa_comp_masses_model_exp_iff_zoom.png} -\caption{\label{fig:test_nhexa_comp_masses_model_exp_iff_zoom}Comparison of the transfer functions from \(u_i\) to \(\tau_{m,i}\) - measured FRF and identification from the Simscape model (Zoom)} -\end{figure} +Obtaining a model accurately representing the complex dynamics of the Nano-Hexapod was made possible by the modelling approach used during this work. +This approach consisted of tuning and validating models of individual components (such as the APA and flexible joints) using dedicated test benches. +Only then, the different models could be combined to form the Nano-Hexapod dynamical model. +If a model of the nano-hexapod was developed in one time, it would be difficult to tune all model parameters to match the measured dynamics, or even to know if the model ``structure'' would be adequate to represents the system dynamics. \printbibliography[heading=bibintoc,title={Bibliography}]