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@ -1,6 +1,4 @@
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% Matlab Init :noexport:ignore:
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%% test_nhexa_table.m
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%% test_nhexa_1_suspended_table.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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@ -28,25 +26,6 @@ colors = colororder;
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%% Frequency Vector
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freqs = logspace(log10(10), log10(2e3), 1000);
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% Simscape Model of the suspended table
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% <<ssec:test_nhexa_table_model>>
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% The Simscape model of the suspended table simply consists of two solid bodies connected by 4 springs.
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% The 4 springs are here modelled with "bushing joints" that have stiffness and damping properties in x, y and z directions.
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% 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.
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% #+name: fig:test_nhexa_suspended_table_simscape
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% #+caption: 3D representation of the simscape model
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% #+attr_latex: :width 0.8\linewidth
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% [[file:figs/test_nhexa_suspended_table_simscape.png]]
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% The model order is 12, and it represents the 6 suspension modes.
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% The inertia properties of the parts are set from the geometry and material densities.
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% 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.
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% The stiffness of the springs in the horizontal plane is set at $0.5\,N/mm$.
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% The obtained suspension modes of the simscape model are compared with the measured ones in Table ref:tab:test_nhexa_suspended_table_simscape_modes.
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%% Configure Simscape Model
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table_type = 'Suspended'; % On top of vibration table
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device_type = 'None'; % No device on the vibration table
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@ -1,7 +1,5 @@
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% Matlab Init :noexport:ignore:
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%% test_nhexa_dynamics.m
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% Identification of the nano-hexapod dynamics from u to dL and to Vs
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%% test_nhexa_2_dynamics.m
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% Identification of the nano-hexapod dynamics from u to de and to Vs
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% Encoders are fixed to the plates
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%% Clear Workspace and Close figures
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@ -22,32 +20,11 @@ colors = colororder;
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%% Frequency Vector
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freqs = logspace(log10(10), log10(2e3), 1000);
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% Identification of the dynamics
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% <<ssec:test_nhexa_identification>>
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% 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.
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% The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure ref:fig:test_nhexa_identified_frf_de.
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% The diagonal terms are displayed using colorful lines, and all the 30 off-diagonal terms are displayed by grey lines.
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% All the six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity.
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% 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).
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% From 10Hz up to 1kHz, around 10 resonance frequencies can be observed.
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% 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.
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% 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).
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% Except the mode at 237Hz, their amplitude is rather low.
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% Two modes at 665Hz and 695Hz are attributed to the flexible modes of the top platform.
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% 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.
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% Up to at least 1kHz, an alternating pole/zero pattern is observed, which renders the control easier to tune.
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% This would not have been the case if the encoders were fixed to the struts.
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%% Load identification data
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%% Identification of the transfer function from u to de and from u to Vs without payload
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% Load identification data
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load('test_nhexa_identification_data_mass_0.mat', 'data');
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%% Setup useful variables
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% Setup useful variables
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Ts = 1e-4; % Sampling Time [s]
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Nfft = floor(1/Ts); % Number of points for the FFT computation
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win = hanning(Nfft); % Hanning window
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@ -56,21 +33,21 @@ Noverlap = floor(Nfft/2); % Overlap between frequency analysis
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% And we get the frequency vector
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[~, f] = tfestimate(data{1}.u, data{1}.de, win, Noverlap, Nfft, 1/Ts);
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%% Transfer function from u to dLm
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% Transfer function from u to de
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G_de = zeros(length(f), 6, 6);
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for i = 1:6
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G_de(:,:,i) = tfestimate(data{i}.u, data{i}.de, win, Noverlap, Nfft, 1/Ts);
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end
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%% Transfer function from u to Vs
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% Transfer function from u to Vs
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G_Vs = zeros(length(f), 6, 6);
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for i = 1:6
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G_Vs(:,:,i) = tfestimate(data{i}.u, data{i}.Vs, win, Noverlap, Nfft, 1/Ts);
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end
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%% Bode plot for the transfer function from u to dLm
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%% Bode plot for the transfer function from u to de
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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@ -93,7 +70,7 @@ hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
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ylim([1e-8, 5e-4]);
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leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4);
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leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 4);
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leg.ItemTokenSize(1) = 15;
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ax2 = nexttile;
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@ -111,20 +88,6 @@ yticks(-360:90:360);
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linkaxes([ax1,ax2],'x');
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xlim([10, 2e3]);
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% #+name: fig:test_nhexa_identified_frf_de
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% #+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.
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% #+RESULTS:
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% [[file:figs/test_nhexa_identified_frf_de.png]]
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% 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.
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% Alternating poles and zeros is observed up to at least 2kHz, which is a necessary characteristics in order to apply decentralized IFF.
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% 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.
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% 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.
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%% Bode plot of the IFF Plant (transfer function from u to Vs)
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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@ -166,29 +129,20 @@ yticks(-360:90:360);
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linkaxes([ax1,ax2],'x');
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xlim([10, 2e3]);
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% Effect of payload mass on the dynamics
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% <<ssec:test_nhexa_added_mass>>
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%% Set to true only if all the FRF matrices should again computed
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% from the experimental data
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compute_frf = false;
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if compute_frf
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% 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.
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% 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 $\approx 40\,kg$.
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% These three cylindrical masses on top of the nano-hexapod are shown in Figure ref:fig:test_nhexa_table_mass_3.
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% #+name: fig:test_nhexa_table_mass_3
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% #+caption: Picture of the nano-hexapod with the added three cylindrical masses for a total of $\approx 40\,kg$
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% #+attr_org: :width 800px
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% #+attr_latex: :width 0.8\linewidth
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% [[file:figs/test_nhexa_table_mass_3.jpg]]
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%% Load identification Data
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%% Identification of the FRF matrices from u to de and to Vs
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% Load identification Data
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meas_added_mass = {...
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load('test_nhexa_identification_data_mass_0.mat', 'data'), ....
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load('test_nhexa_identification_data_mass_1.mat', 'data'), ....
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load('test_nhexa_identification_data_mass_2.mat', 'data'), ....
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load('test_nhexa_identification_data_mass_3.mat', 'data')};
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%% Setup useful variables
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% Setup useful variables
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Ts = 1e-4; % Sampling Time [s]
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Nfft = floor(1/Ts); % Number of points for the FFT computation
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win = hanning(Nfft); % Hanning window
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% And we get the frequency vector
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[~, f] = tfestimate(meas_added_mass{1}.data{1}.u, meas_added_mass{1}.data{1}.de, win, Noverlap, Nfft, 1/Ts);
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% FRF from u to de
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G_de = {};
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for i_mass = [0:3]
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@ -206,7 +161,7 @@ for i_mass = [0:3]
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end
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end
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%% IFF Plant (transfer function from u to Vs)
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% FRF from u to Vs
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G_Vs = {};
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for i_mass = [0:3]
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end
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end
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% The identified dynamics are then saved for further use.
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save('./mat/test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de')
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end
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%% Load the identified transfer functions
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frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de');
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% 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.
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% As expected, the frequency of the suspension modes are decreasing with an increase of the payload mass.
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% 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.
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% 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 $\approx 400\,Hz$).
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% This is due to the fact that the added mass is composed of two half cylinders which are not fixed together.
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% It therefore adds a lot of mass to the top plate without adding stiffness in one direction.
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% 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).
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% In that case, the frequency of these flexible modes are increased.
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% In practice, the payload should be one solid body, and no decrease of the frequency of this flexible mode should be observed.
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% The apparent amplitude of the flexible mode of the strut at 237Hz becomes smaller as the payload mass is increased.
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% The measured FRF from $u_i$ to $V_{si}$ are shown in Figure ref:fig:test_nhexa_identified_frf_Vs_masses.
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% For all the tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust way.
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%% Bode plot for the transfer function from u to dLm - Several payloads
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%% Bode plot for the transfer function from u to de - Several payloads
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masses = [0, 13, 26, 39];
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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@ -286,7 +226,7 @@ ylim([-90, 180])
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linkaxes([ax1,ax2],'x');
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xlim([10, 2e3]);
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%% Bode plot for the transfer function from u to dLm
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%% Bode plot for the transfer function from u to de
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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% Matlab Init :noexport:ignore:
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%% test_nhexa_3_model.m
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% Compare the measured dynamics from u to de and to Vs with the Simscape model
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%% Frequency Vector
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freqs = logspace(log10(10), log10(2e3), 1000);
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% Extract transfer function matrices from the Simscape Model :noexport:
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%% Set to true only if all the dynamics should again computed
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% from the simscape model
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extract_simscape_dynamics = false
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extract_simscape_dynamics = false;
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if extract_simscape_dynamics
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%% Extract the transfer function matrix from the Simscape model
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@ -98,28 +94,10 @@ if extract_simscape_dynamics
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end
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% Nano-Hexapod model dynamics
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% <<ssec:test_nhexa_comp_model>>
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%% Load Simscape Model and measured FRF
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sim_ol = load('test_nhexa_simscape_masses.mat', 'G_Vs', 'G_de');
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frf_ol = load('test_nhexa_identified_frf_masses.mat', 'f', 'G_Vs', 'G_de');
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% The Simscape model of the nano-hexapod is first configured with 4-DoF flexible joints, 2-DoF APA and rigid top and bottom platforms.
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% The stiffness of the flexible joints are chosen based on the values estimated using the test bench and based on FEM.
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% The parameters of the APA model are the ones determined from the test bench of the APA.
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% 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.
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% A first feature that should be checked is that the model well represents the "direct" terms of the measured FRF matrix.
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% 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.
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% It can be seen that the 4 suspension modes of the nano-hexapod (at 122Hz, 143Hz, 165Hz and 191Hz) are well modelled.
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% 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.
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% 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.
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%% Diagonal elements of the FRF matrix from u to de
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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@ -194,15 +172,6 @@ yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% Modelling dynamical coupling
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% <<ssec:test_nhexa_comp_model_coupling>>
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% 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.
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% 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 $d_{e6}$.
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% 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.
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% Similar results are observed for all the other coupling terms, as well as for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$.
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%% Comparison of the plants (encoder output) when tuning the misalignment
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i_input = 1;
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@ -273,20 +242,6 @@ xticks([50, 100, 200, 400])
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linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
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xlim([50, 5e2]); ylim([1e-8, 5e-4]);
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% #+name: fig:test_nhexa_comp_simscape_de_all
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% #+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 $d_{e6}$
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% #+RESULTS:
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% [[file:figs/test_nhexa_comp_simscape_de_all.png]]
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% The APA300ML are then modelled with a /super-element/ extracted from a FE-software.
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% 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.
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% 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.
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% Even the mode 395Hz can be observed in the model.
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% 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.
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%% Load the plant model with Flexible APA
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flex_ol = load('test_nhexa_simscape_flexible_masses.mat', 'G_Vs', 'G_de');
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@ -360,19 +315,6 @@ xticks([50, 100, 200, 400])
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linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
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xlim([50, 5e2]); ylim([1e-8, 5e-4]);
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% Modelling the effect of payload mass
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% <<ssec:test_nhexa_comp_model_masses>>
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% Another important characteristics of the model is that it should well represents the dynamics of the system for all considered payloads.
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% 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.
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% The observed shift to lower frequency of the suspension modes with an increased payload mass is well represented by the Simscape model.
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% 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).
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% Note that the model displays smaller damping that what is observed experimentally for high values of the payload mass.
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% One option could be to tune the damping as a function of the mass (similar to what is done with the Rayleigh damping).
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% 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.
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%% Bode plot for the transfer function from u to de
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masses = [0, 13, 26, 39];
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figure;
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@ -389,7 +331,7 @@ for i_mass = [0:3]
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end
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set(gca, 'ColorOrderIndex', i_mass+1)
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plot(freqs, abs(squeeze(freqresp(sim_ol.G_de{i_mass+1}(1,1), freqs, 'Hz'))), '--', ...
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'DisplayName', 'Simscape');
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'DisplayName', 'Model');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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@ -433,7 +375,7 @@ for i_mass = 0:3
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'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(sim_ol.G_Vs{i_mass+1}(1,1), freqs, 'Hz'))), '--', 'color', colors(i_mass+1,:), ...
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'DisplayName', 'Simscape');
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'DisplayName', 'Model');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
@ -460,31 +402,6 @@ 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;
|
||||
|
||||
|
||||
@ -22,7 +22,7 @@
|
||||
#+BIND: org-latex-bib-compiler "biber"
|
||||
|
||||
#+PROPERTY: header-args:matlab :session *MATLAB*
|
||||
#+PROPERTY: header-args:matlab+ :comments org
|
||||
#+PROPERTY: header-args:matlab+ :comments no
|
||||
#+PROPERTY: header-args:matlab+ :exports none
|
||||
#+PROPERTY: header-args:matlab+ :results none
|
||||
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||
|
||||
Reference in New Issue
Block a user