From 5698fcd6f5b8078505e120deb6899a426d09fd28 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 22 Mar 2024 19:14:07 +0100 Subject: [PATCH] Update analysis --- .gitignore | 1 + figs/test_apa_2dof_comp_frf.pdf | 7485 ++++++++++++++++++ figs/test_apa_2dof_comp_frf.png | Bin 0 -> 182356 bytes figs/test_apa_2dof_model.pdf | Bin 0 -> 24817 bytes figs/test_apa_2dof_model.png | Bin 0 -> 103091 bytes figs/test_apa_2dof_model.svg | 1066 +++ figs/test_apa_2dof_model_simscape.pdf | Bin 0 -> 11656 bytes figs/test_apa_2dof_model_simscape.png | Bin 0 -> 69728 bytes figs/test_apa_2dof_model_simscape.svg | 1119 +++ figs/test_apa_ansys.jpg | Bin 46036 -> 0 bytes figs/test_apa_bench_model.pdf | Bin 13333 -> 10636 bytes figs/test_apa_bench_model.png | Bin 11808 -> 8865 bytes figs/test_apa_meas_freq_compare.pdf | Bin 87838 -> 89656 bytes figs/test_apa_meas_freq_compare.png | Bin 139310 -> 94228 bytes figs/test_apa_meas_setup_Y_bending.jpg | Bin 170774 -> 163292 bytes figs/test_apa_super_element_comp_frf.pdf | 7598 +++++++++++++++++++ figs/test_apa_super_element_comp_frf.png | Bin 0 -> 187291 bytes figs/test_apa_super_element_simscape.pdf | Bin 0 -> 1742190 bytes figs/test_apa_super_element_simscape.png | Bin 0 -> 1782444 bytes figs/test_apa_super_element_simscape.svg | 3818 ++++++++++ matlab/STEPS/full_APA300ML_K.CSV | 36 + matlab/STEPS/full_APA300ML_M.CSV | 36 + matlab/STEPS/full_APA300ML_out_nodes_3D.txt | 61 + matlab/test_apa_1_basic_meas.m | 204 +- matlab/test_apa_2_dynamics.m | 14 +- matlab/test_apa_3_model_2dof.m | 177 + matlab/test_apa_4_model_flexible.m | 202 + matlab/test_apa_simscape.slx | Bin 60187 -> 65996 bytes test-bench-apa.bib | 62 +- test-bench-apa.org | 2230 ++---- test-bench-apa.pdf | Bin 4396031 -> 8780074 bytes test-bench-apa.tex | 493 +- 32 files changed, 22409 insertions(+), 2193 deletions(-) create mode 100644 figs/test_apa_2dof_comp_frf.pdf create mode 100644 figs/test_apa_2dof_comp_frf.png create mode 100644 figs/test_apa_2dof_model.pdf create mode 100644 figs/test_apa_2dof_model.png create mode 100644 figs/test_apa_2dof_model.svg create mode 100644 figs/test_apa_2dof_model_simscape.pdf create mode 100644 figs/test_apa_2dof_model_simscape.png create mode 100644 figs/test_apa_2dof_model_simscape.svg delete mode 100644 figs/test_apa_ansys.jpg create mode 100644 figs/test_apa_super_element_comp_frf.pdf create mode 100644 figs/test_apa_super_element_comp_frf.png create mode 100644 figs/test_apa_super_element_simscape.pdf create mode 100644 figs/test_apa_super_element_simscape.png create mode 100644 figs/test_apa_super_element_simscape.svg create mode 100644 matlab/STEPS/full_APA300ML_K.CSV create mode 100644 matlab/STEPS/full_APA300ML_M.CSV create mode 100644 matlab/STEPS/full_APA300ML_out_nodes_3D.txt create mode 100644 matlab/test_apa_3_model_2dof.m create mode 100644 matlab/test_apa_4_model_flexible.m diff --git a/.gitignore b/.gitignore index 6b7e1a4..f834ecc 100644 --- a/.gitignore +++ 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/dev/null +++ b/matlab/STEPS/full_APA300ML_out_nodes_3D.txt @@ -0,0 +1,61 @@ + + LIST ALL SELECTED NODES. DSYS= 0 + + *** ANSYS - ENGINEERING ANALYSIS SYSTEM RELEASE 2020 R2 20.2 *** + DISTRIBUTED ANSYS Mechanical Enterprise + + 00208316 VERSION=WINDOWS x64 10:10:05 MAR 26, 2021 CP= 2.188 + + Unknown + + + + NODE X Y Z THXY THYZ THZX + 1 0.0000 0.0000 0.28000E-001 0.00 0.00 0.00 + 1228810 0.0000 0.0000 -0.28000E-001 0.00 0.00 0.00 + 1228811 -0.30000E-001 0.0000 0.0000 0.00 0.00 0.00 + 1228812 0.10000E-001 0.0000 0.0000 0.00 0.00 0.00 + 1228813 0.30000E-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 10:10:05 MAR 26, 2021 CP= 2.188 + + Unknown + + + NODE LABEL SUPPORT + 1 UX + 1 UY + 1 UZ + 1 ROTX + 1 ROTY + 1 ROTZ + 1228810 UX + 1228810 UY + 1228810 UZ + 1228810 ROTX + 1228810 ROTY + 1228810 ROTZ + 1228811 UX + 1228811 UY + 1228811 UZ + 1228811 ROTX + 1228811 ROTY + 1228811 ROTZ + 1228812 UX + 1228812 UY + 1228812 UZ + 1228812 ROTX + 1228812 ROTY + 1228812 ROTZ + 1228813 UX + 1228813 UY + 1228813 UZ + 1228813 ROTX + 1228813 ROTY + 1228813 ROTZ diff --git a/matlab/test_apa_1_basic_meas.m b/matlab/test_apa_1_basic_meas.m index 27720d0..fee1e01 100644 --- a/matlab/test_apa_1_basic_meas.m +++ b/matlab/test_apa_1_basic_meas.m @@ -61,7 +61,7 @@ for i = 1:7 apa_d(i) = min_d; end -% Stroke Measurement +% Stroke and Hysteresis Measurement % <> % The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet. @@ -111,177 +111,99 @@ xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]') legend('location', 'southwest', 'FontSize', 8) xlim([-20, 150]); ylim([-250, 0]); -% X-Bending Mode +% Flexible Mode Measurement +% SCHEDULED: <2024-03-27 Wed> +% <> -% The vibrometer is setup to measure the X-bending motion is shown in Figure ref:fig:test_apa_meas_setup_X_bending. -% The APA is excited with an instrumented hammer having a solid metallic tip. -% The impact point is on the back-side of the APA aligned with the top measurement point. +% In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. -% #+name: fig:test_apa_meas_setup_X_bending -% #+caption: X-Bending measurement setup -% #+attr_latex: :width 0.7\linewidth +% To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_torsion). +% A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer[fn:7] is used to excite the flexible modes. +% Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated. + +% The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure ref:fig:test_apa_mode_shapes. + +% #+name: fig:test_apa_mode_shapes +% #+caption: Spurious resonances - Change this with the updated FEM analysis of the APA300ML +% #+attr_latex: :width 0.9\linewidth +% [[file:figs/test_apa_mode_shapes.png]] + +% #+name: fig:test_apa_meas_setup_torsion +% #+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the $Z$ torsion is measured. +% #+attr_latex: :width 0.6\linewidth +% [[file:figs/test_apa_meas_setup_torsion.jpg]] + +% Two other similar measurements are performed to measured the bending of the APA around the $X$ direction and around the $Y$ direction (see Figure ref:fig:test_apa_meas_setup_modes). + +% #+name: fig:test_apa_meas_setup_modes +% #+caption: Experimental setup to measured flexible modes of the APA300ML. For the bending in the $X$ direction, the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points). +% #+begin_figure +% #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending} +% #+attr_latex: :options {0.49\textwidth} +% #+begin_subfigure +% #+attr_latex: :width 0.95\linewidth % [[file:figs/test_apa_meas_setup_X_bending.jpg]] +% #+end_subfigure +% #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending} +% #+attr_latex: :options {0.49\textwidth} +% #+begin_subfigure +% #+attr_latex: :width 0.95\linewidth +% [[file:figs/test_apa_meas_setup_Y_bending.jpg]] +% #+end_subfigure +% #+end_figure -% The data is loaded. -%% Load Data +%% X-Bending Identification +% Load Data bending_X = load('apa300ml_bending_X_top.mat'); - - -% The configuration (Sampling time and windows) for =tfestimate= is done: - -%% Spectral Analysis setup +% Spectral Analysis setup Ts = bending_X.Track1_X_Resolution; % Sampling Time [s] Nfft = floor(1/Ts); win = hanning(Nfft); Noverlap = floor(Nfft/2); - - -% The transfer function from the input force to the output "rotation" (difference between the two measured distances). - -%% Compute the transfer function from applied force to measured rotation +% Compute the transfer function from applied force to measured rotation [G_bending_X, f] = tfestimate(bending_X.Track1, bending_X.Track2, win, Noverlap, Nfft, 1/Ts); - - -% The result is shown in Figure ref:fig:test_apa_meas_freq_bending_x. - -% The can clearly observe a nice peak at 280Hz, and then peaks at the odd "harmonics" (third "harmonic" at 840Hz, and fifth "harmonic" at 1400Hz). - -%% Plot the transfer function -figure; -hold on; -plot(f, abs(G_bending_X), 'k-'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 2e-1]); -text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(840, 2.0e-3,{'840Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(1400, 7.0e-3,{'1400Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') - -% Y-Bending Mode - -% The setup to measure the Y-bending is shown in Figure ref:fig:test_apa_meas_setup_Y_bending. - -% The impact point of the instrumented hammer is located on the back surface of the top interface (on the back of the 2 measurements points). - -% #+name: fig:test_apa_meas_setup_Y_bending -% #+caption: Y-Bending measurement setup -% #+attr_latex: :width 0.7\linewidth -% [[file:figs/test_apa_meas_setup_Y_bending.jpg]] - -% The data is loaded, and the transfer function from the force to the measured rotation is computed. - -%% Load Data +%% Y-Bending identification +% Load Data bending_Y = load('apa300ml_bending_Y_top.mat'); -%% Compute the transfer function +% Compute the transfer function [G_bending_Y, ~] = tfestimate(bending_Y.Track1, bending_Y.Track2, win, Noverlap, Nfft, 1/Ts); +%% Z-Torsion identification +% Load data +torsion = load('apa300ml_torsion_top.mat'); +% Compute transfer function +[G_torsion_top, ~] = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts); -% The results are shown in Figure ref:fig:test_apa_meas_freq_bending_y. -% The main resonance is at 412Hz, and we also see the third "harmonic" at 1220Hz. - - -%% Plot the transfer function -figure; -hold on; -plot(f, abs(G_bending_Y), 'k-'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 3e-2]) -text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(1218, 1.5e-2,{'1220Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') - -% Z-Torsion Mode - -% Finally, we measure the Z-torsion resonance as shown in Figure ref:fig:test_apa_meas_setup_torsion_bis. - -% The excitation is shown on the other side of the APA, on the side to excite the torsion motion. - -% #+name: fig:test_apa_meas_setup_torsion_bis -% #+caption: Z-Torsion measurement setup -% #+attr_latex: :width 0.7\linewidth -% [[file:figs/test_apa_meas_setup_torsion_bis.jpg]] - -% The data is loaded, and the transfer function computed. - -%% Load Data +% Load Data torsion = load('apa300ml_torsion_left.mat'); -%% Compute transfer function +% Compute transfer function [G_torsion, ~] = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts); -% The results are shown in Figure ref:fig:test_apa_meas_freq_torsion_z. -% We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz. -% And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz. -% A third mode at 800Hz could correspond to this torsion mode. +% The three measured frequency response functions are shown in Figure ref:fig:test_apa_meas_freq_compare. +% - a clear $x$ bending mode at $280\,\text{Hz}$ +% - a clear $y$ bending mode at $412\,\text{Hz}$ +% - for the $z$ torsion test, the $y$ bending mode is also excited and observed, and we may see a mode at $800\,\text{Hz}$ -%% Plot the transfer function figure; hold on; -plot(f, abs(G_torsion), 'k-'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 2e-2]) -text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +plot(f, abs(G_bending_X), 'DisplayName', '$X$ bending'); +plot(f, abs(G_bending_Y), 'DisplayName', '$Y$ bending'); +plot(f, abs(G_torsion), 'DisplayName', '$Z$ torsion'); +text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') text(800, 6e-4,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') - - - -% #+name: fig:test_apa_meas_freq_torsion_z -% #+caption: Obtained FRF for the Z-torsion -% #+RESULTS: -% [[file:figs/test_apa_meas_freq_torsion_z.png]] - -% In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited. - -%% Load data -torsion = load('apa300ml_torsion_top.mat'); - -%% Compute transfer function -[G_torsion_top, ~] = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts); - - - -% The two FRF are compared in Figure ref:fig:test_apa_meas_freq_torsion_z_comp. -% It is clear that the first two modes does not correspond to the torsional mode. -% Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here. - -%% Plot the two transfer functions -figure; -hold on; -plot(f, abs(G_torsion), 'k-', 'DisplayName', 'Left excitation'); -plot(f, abs(G_torsion_top), '-', 'DisplayName', 'Top excitation'); hold off; set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 2e-2]) -text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') -text(800, 2e-3,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') -legend('location', 'northwest'); - -% Compare -% The three measurements are shown in Figure ref:fig:test_apa_meas_freq_compare. - -figure; -hold on; -plot(f, abs(G_torsion), 'DisplayName', 'Torsion'); -plot(f, abs(G_bending_X), 'DisplayName', 'Bending - X'); -plot(f, abs(G_bending_Y), 'DisplayName', 'Bending - Y'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 1e-1]); -legend('location', 'southeast'); +xlim([50, 2e3]); ylim([5e-5, 2e-1]); +legend('location', 'northeast', 'FontSize', 8) diff --git a/matlab/test_apa_2_dynamics.m b/matlab/test_apa_2_dynamics.m index 9a3054d..90974f3 100644 --- a/matlab/test_apa_2_dynamics.m +++ b/matlab/test_apa_2_dynamics.m @@ -119,7 +119,7 @@ xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]'); % | 6 | 1.7 | 1.92 | % | 8 | 1.73 | 1.98 | -% The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 94\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_frf_disp) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. +% The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. % \begin{equation} \label{eq:test_apa_res_freq} % \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} @@ -204,7 +204,7 @@ save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums'); % - A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$. % The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA % - A lightly damped resonance at $95\,\text{Hz}$ -% - A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear +% - A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support. %% Plot the FRF from u to de @@ -305,17 +305,17 @@ xlim([10, 2e3]); %% Load the data -wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va'); % With the resistor -wo_k = load('frf_data_1_sweep_lf.mat', 't', 'Vs', 'Va'); % Without the resistor +wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'u'); % With the resistor +wo_k = load('frf_data_1_sweep_lf.mat', 't', 'Vs', 'u'); % Without the resistor %% Large Hanning window for good low frequency estimate Nfft = floor(50/Ts); win = hanning(Nfft); Noverlap = floor(Nfft/2); -%% Compute the transfer functions from Va to Vs -[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, Noverlap, Nfft, 1/Ts); -[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, Noverlap, Nfft, 1/Ts); +%% Compute the transfer functions from u to Vs +[frf_wo_k, f] = tfestimate(wo_k.u, wo_k.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_wi_k, ~] = tfestimate(wi_k.u, wi_k.Vs, win, Noverlap, Nfft, 1/Ts); %% Model for the high pass filter C = 5.1e-6; % Sensor Stack capacitance [F] diff --git a/matlab/test_apa_3_model_2dof.m b/matlab/test_apa_3_model_2dof.m new file mode 100644 index 0000000..94efc21 --- /dev/null +++ b/matlab/test_apa_3_model_2dof.m @@ -0,0 +1,177 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +%% Path for functions, data and scripts +addpath('./src/'); % Path for scripts +addpath('./mat/'); % Path for data + +addpath('./STEPS/'); % Path for Simscape Model + +%% Linearization options +opts = linearizeOptions; +opts.SampleTime = 0; + +%% Open Simscape Model +mdl = 'test_apa_simscape'; % Name of the Simulink File +open(mdl); % Open Simscape Model + +%% Colors for the figures +colors = colororder; + +%% Input/Output definition of the Model +clear io; io_i = 1; +io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage +io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder + +% Tuning of the APA model +% <> + +% 9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics. + +% #+name: fig:test_apa_2dof_model_simscape +% #+caption: Schematic of the two degrees of freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$ +% [[file:figs/test_apa_2dof_model_simscape.png]] + + +%% Stiffness values for the 2DoF APA model +k1 = 0.38e6; % Estimated Shell Stiffness [N/m] + +w0 = 2*pi*95; % Resonance frequency [rad/s] +m = 5.7; % Suspended mass [kg] +ktot = m*(w0)^2; % Total Axial Stiffness to have to wanted resonance frequency [N/m] + +ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m] +ke = 2*ka; % Stiffness of the Sensor stack [N/m] + +%% Damping values for the 2DoF APA model +c1 = 20; % Damping for the Shell [N/(m/s)] +ca = 100; % Damping of the actuators stacks [N/(m/s)] +ce = 2*ca; % Damping of the sensor stack [N/(m/s)] + +%% Estimation ot the sensor and actuator gains +% Initialize the structure with unitary sensor and actuator "gains" +n_hexapod = struct(); +n_hexapod.actuator = initializeAPA(... + 'type', '2dof', ... + 'k', k1, ... + 'ka', ka, ... + 'ke', ke, ... + 'c', c1, ... + 'ca', ca, ... + 'ce', ce, ... + 'Ga', 1, ... % Actuator constant [N/V] + 'Gs', 1 ... % Sensor constant [V/m] +); + +c_granite = 0; % Do not take into account damping added by the air bearing + +% Run the linearization +G_norm = linearize(mdl, io, 0.0, opts); +G_norm.InputName = {'u'}; +G_norm.OutputName = {'Vs', 'de'}; + +% Load Identification Data to estimate the two gains +load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums'); + +% Estimation ot the Actuator Gain +fa = 10; % Frequency where the two FRF should match [Hz] +[~, i_f] = min(abs(f - fa)); +ga = -abs(enc_frf(i_f,1))./abs(evalfr(G_norm('de', 'u'), 1i*2*pi*fa)); + +% Estimation ot the Sensor Gain +fs = 600; % Frequency where the two FRF should match [Hz] +[~, i_f] = min(abs(f - fs)) +gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga; + +% Obtained Dynamics +% <> + +% The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model. +% It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf. + +% A good match can be observed between the model and the experimental data, both for the encoder and for the force sensor. +% This indicates that this model represents well the axial dynamics of the APA300ML. + + +%% 2DoF APA300ML with optimized parameters +n_hexapod = struct(); +n_hexapod.actuator = initializeAPA(... + 'type', '2dof', ... + 'k', k1, ... + 'ka', ka, ... + 'ke', ke, ... + 'c', c1, ... + 'ca', ca, ... + 'ce', ce, ... + 'Ga', ga, ... + 'Gs', gs ... +); + +%% Identification of the APA300ML with optimized parameters +G_2dof = exp(-s*Ts)*linearize(mdl, io, 0.0, opts); +G_2dof.InputName = {'u'}; +G_2dof.OutputName = {'Vs', 'de'}; + +%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML +freqs = 5*logspace(0, 3, 1000); +figure; +tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 1:length(apa_nums) + plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-8, 1e-3]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); + +ax1b = nexttile([2,1]); +hold on; +plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 2:length(apa_nums) + plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); + +ax2 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]); +end +plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 180]); + +ax2b = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]); +end +plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 180]); + +linkaxes([ax1,ax2,ax1b,ax2b],'x'); +xlim([10, 2e3]); diff --git a/matlab/test_apa_4_model_flexible.m b/matlab/test_apa_4_model_flexible.m new file mode 100644 index 0000000..b07cf48 --- /dev/null +++ b/matlab/test_apa_4_model_flexible.m @@ -0,0 +1,202 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +%% Path for functions, data and scripts +addpath('./src/'); % Path for scripts +addpath('./mat/'); % Path for data + +addpath('./STEPS/'); % Path for Simscape Model + +%% Linearization options +opts = linearizeOptions; +opts.SampleTime = 0; + +%% Open Simscape Model +mdl = 'test_apa_simscape'; % Name of the Simulink File +open(mdl); % Open Simscape Model + +%% Colors for the figures +colors = colororder; + +%% Input/Output definition of the Model +clear io; io_i = 1; +io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage +io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder + +% Identification of the Actuator and Sensor constants +% <> + +% Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be identified. +% The gains $g_a$ and $g_s$ can then be tuned such that the gain of the transfer functions are matching the identified ones. + +% By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained. + + +%% Identification of the actuator and sensor "constants" +% Initialize the APA as a flexible body with unity "constants" +n_hexapod.actuator = initializeAPA(... + 'type', 'flexible', ... + 'ga', 1, ... + 'gs', 1); + +c_granite = 100; % Rought estimation of the damping added by the air bearing + +% Identify the dynamics +G_norm = linearize(mdl, io, 0.0, opts); +G_norm.InputName = {'u'}; +G_norm.OutputName = {'Vs', 'de'}; + +% Load Identification Data to estimate the two gains +load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums'); + +% Actuator Constant in [N/V] +ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u')); + +% Sensor Constant in [V/m] +gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz')))); + + + + +% To make sure these "gains" are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material. + +% From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force. + +% \begin{subequations} +% \begin{align} +% V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\ +% F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force} +% \end{align} +% \end{subequations} + +% Parameters used in equations eqref:eq:test_apa_piezo_strain_to_voltage and eqref:eq:test_apa_piezo_voltage_to_force are described in Table ref:tab:test_apa_piezo_properties. + +% Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML. +% However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. +% The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties. + +% From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are very close to the identified constants using the experimentally identified transfer functions. + +% #+name: tab:test_apa_piezo_properties +% #+caption: Piezoelectric properties used for the estimation of the sensor and actuators "gains" +% #+attr_latex: :environment tabularx :width 1\linewidth :align ccX +% #+attr_latex: :center t :booktabs t +% | *Parameter* | *Value* | *Description* | +% |----------------+----------------------------+--------------------------------------------------------------| +% | $d_{33}$ | $680 \cdot 10^{-12}\,m/V$ | Piezoelectric constant | +% | $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,F/m$ | Permittivity under constant stress | +% | $s^{D}$ | $21 \cdot 10^{-12}\,m^2/N$ | Elastic compliance understand constant electric displacement | +% | $c^{E}$ | $48 \cdot 10^{9}\,N/m^2$ | Young's modulus of elasticity | +% | $L$ | $20\,mm$ per stack | Length of the stack | +% | $A$ | $10^{-4}\,m^2$ | Area of the piezoelectric stack | +% | $n$ | $160$ per stack | Number of layers in the piezoelectric stack | + + +%% Estimate "Sensor Constant" - (THP5H) +d33 = 680e-12; % Strain constant [m/V] +n = 160; % Number of layers per stack +eT = 4500*8.854e-12; % Permittivity under constant stress [F/m] +sD = 21e-12; % Compliance under constant electric displacement [m2/N] + +gs = d33/(eT*sD*n); % Sensor Constant [V/m] + +%% Estimate "Actuator Constant" - (THP5H) +d33 = 680e-12; % Strain constant [m/V] +n = 320; % Number of layers + +cE = 1/sD; % Youngs modulus [N/m^2] +A = (10e-3)^2; % Area of the stacks [m^2] +L = 40e-3; % Length of the two stacks [m] +ka = cE*A/L; % Stiffness of the two stacks [N/m] + +ga = d33*n*ka; % Actuator Constant [N/V] + +% Comparison of the obtained dynamics + +% The obtained dynamics using the /super element/ with the tuned "sensor gain" and "actuator gain" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf. + +% A good match between the model and the experimental results is observed. +% - the /super element/ + +% This model represents fairly + +% The flexible model is a bit "soft" as compared with the experimental results. + +% This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators. + + +%% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants" +% Initialize the APA +n_hexapod.actuator = initializeAPA(... + 'type', 'flexible', ... + 'ga', 23.2, ... % Actuator gain [N/V] + 'gs', -4.9e6); % Sensor gain [V/m] + +% Identify with updated constants +G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts); +G_flex.InputName = {'u'}; +G_flex.OutputName = {'Vs', 'de'}; + +%% Comparison of the measured FRF and the "Flexible" model of the APA300ML +freqs = 5*logspace(0, 3, 1000); +figure; +tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 1:length(apa_nums) + plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-8, 1e-3]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); + +ax1b = nexttile([2,1]); +hold on; +plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 2:length(apa_nums) + plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); + +ax2 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]); +end +plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 180]); + +ax2b = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]); +end +plot(freqs, 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APA 6 | 4.99 | 9.91 | | APA 7 | 4.85 | 9.85 | -** Stroke Measurement +** Stroke and Hysteresis Measurement <> The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet. @@ -343,281 +347,121 @@ exportFig('figs/test_apa_stroke_result.pdf', 'width', 'full', 'height', 'normal' #+RESULTS: [[file:figs/test_apa_stroke_result.png]] -** TODO Spurious resonances - APA :@philipp: +** Flexible Mode Measurement SCHEDULED: <2024-03-27 Wed> <> -*** Introduction -From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure ref:fig:test_apa_mode_shapes): -- Mode in X-bending at 189Hz -- Mode in Y-bending at 285Hz -- Mode in Z-torsion at 400Hz +In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. + +To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_torsion). +A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer[fn:7] is used to excite the flexible modes. +Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated. + +The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure ref:fig:test_apa_mode_shapes. #+name: fig:test_apa_mode_shapes -#+caption: Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz -#+attr_latex: :width \linewidth +#+caption: Spurious resonances - Change this with the updated FEM analysis of the APA300ML +#+attr_latex: :width 0.9\linewidth [[file:figs/test_apa_mode_shapes.png]] -These modes are present when flexible joints are fixed to the ends of the APA300ML. - -In this section, we try to find the resonance frequency of these modes when one end of the APA is fixed and the other is free. - -In the section ref:sec:spurious_resonances_struts, a similar measurement will be performed directly on the struts. - -*** Measurement Setup - -The measurement setup is shown in Figure ref:fig:test_apa_meas_setup_torsion. -A Laser vibrometer is measuring the difference of motion between two points. -The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed. - -#+begin_note -The instrumentation used are: -- Laser Doppler Vibrometer Polytec OFV512 -- Instrumented hammer -#+end_note - #+name: fig:test_apa_meas_setup_torsion -#+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer -#+attr_latex: :width 0.7\linewidth +#+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the $Z$ torsion is measured. +#+attr_latex: :width 0.6\linewidth [[file:figs/test_apa_meas_setup_torsion.jpg]] -*** X-Bending Mode +Two other similar measurements are performed to measured the bending of the APA around the $X$ direction and around the $Y$ direction (see Figure ref:fig:test_apa_meas_setup_modes). -The vibrometer is setup to measure the X-bending motion is shown in Figure ref:fig:test_apa_meas_setup_X_bending. -The APA is excited with an instrumented hammer having a solid metallic tip. -The impact point is on the back-side of the APA aligned with the top measurement point. - -#+name: fig:test_apa_meas_setup_X_bending -#+caption: X-Bending measurement setup -#+attr_latex: :width 0.7\linewidth +#+name: fig:test_apa_meas_setup_modes +#+caption: Experimental setup to measured flexible modes of the APA300ML. For the bending in the $X$ direction, the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points). +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth [[file:figs/test_apa_meas_setup_X_bending.jpg]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/test_apa_meas_setup_Y_bending.jpg]] +#+end_subfigure +#+end_figure -The data is loaded. #+begin_src matlab -%% Load Data +%% X-Bending Identification +% Load Data bending_X = load('apa300ml_bending_X_top.mat'); -#+end_src -The configuration (Sampling time and windows) for =tfestimate= is done: -#+begin_src matlab -%% Spectral Analysis setup +% Spectral Analysis setup Ts = bending_X.Track1_X_Resolution; % Sampling Time [s] Nfft = floor(1/Ts); win = hanning(Nfft); Noverlap = floor(Nfft/2); -#+end_src -The transfer function from the input force to the output "rotation" (difference between the two measured distances). -#+begin_src matlab -%% Compute the transfer function from applied force to measured rotation +% Compute the transfer function from applied force to measured rotation [G_bending_X, f] = tfestimate(bending_X.Track1, bending_X.Track2, win, Noverlap, Nfft, 1/Ts); -#+end_src -The result is shown in Figure ref:fig:test_apa_meas_freq_bending_x. - -The can clearly observe a nice peak at 280Hz, and then peaks at the odd "harmonics" (third "harmonic" at 840Hz, and fifth "harmonic" at 1400Hz). -#+begin_src matlab :exports none -%% Plot the transfer function -figure; -hold on; -plot(f, abs(G_bending_X), 'k-'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 2e-1]); -text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(840, 2.0e-3,{'840Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(1400, 7.0e-3,{'1400Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/test_apa_meas_freq_bending_x.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_apa_meas_freq_bending_x -#+caption: Obtained FRF for the X-bending -#+RESULTS: -[[file:figs/test_apa_meas_freq_bending_x.png]] - -Then the APA is in the "free-free" condition, this bending mode is foreseen to be at 200Hz (Figure ref:fig:test_apa_mode_shapes). -We are here in the "fixed-free" condition. -If we consider that we therefore double the stiffness associated with this mode, we should obtain a resonance a factor $\sqrt{2}$ higher than 200Hz which is indeed 280Hz. -Not sure this reasoning is correct though. - -*** Y-Bending Mode - -The setup to measure the Y-bending is shown in Figure ref:fig:test_apa_meas_setup_Y_bending. - -The impact point of the instrumented hammer is located on the back surface of the top interface (on the back of the 2 measurements points). - -#+name: fig:test_apa_meas_setup_Y_bending -#+caption: Y-Bending measurement setup -#+attr_latex: :width 0.7\linewidth -[[file:figs/test_apa_meas_setup_Y_bending.jpg]] - -The data is loaded, and the transfer function from the force to the measured rotation is computed. -#+begin_src matlab -%% Load Data +%% Y-Bending identification +% Load Data bending_Y = load('apa300ml_bending_Y_top.mat'); -%% Compute the transfer function +% Compute the transfer function [G_bending_Y, ~] = tfestimate(bending_Y.Track1, bending_Y.Track2, win, Noverlap, Nfft, 1/Ts); -#+end_src -The results are shown in Figure ref:fig:test_apa_meas_freq_bending_y. -The main resonance is at 412Hz, and we also see the third "harmonic" at 1220Hz. +%% Z-Torsion identification +% Load data +torsion = load('apa300ml_torsion_top.mat'); -#+begin_src matlab :exports none -%% Plot the transfer function -figure; -hold on; -plot(f, abs(G_bending_Y), 'k-'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 3e-2]) -text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(1218, 1.5e-2,{'1220Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -#+end_src +% Compute transfer function +[G_torsion_top, ~] = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts); -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/test_apa_meas_freq_bending_y.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_apa_meas_freq_bending_y -#+caption: Obtained FRF for the Y-bending -#+RESULTS: -[[file:figs/test_apa_meas_freq_bending_y.png]] - -We can apply the same reasoning as in the previous section and estimate the mode to be a factor $\sqrt{2}$ higher than the mode estimated in the "free-free" condition. -We would obtain a mode at 403Hz which is very close to the one estimated here. - -*** Z-Torsion Mode - -Finally, we measure the Z-torsion resonance as shown in Figure ref:fig:test_apa_meas_setup_torsion_bis. - -The excitation is shown on the other side of the APA, on the side to excite the torsion motion. - -#+name: fig:test_apa_meas_setup_torsion_bis -#+caption: Z-Torsion measurement setup -#+attr_latex: :width 0.7\linewidth -[[file:figs/test_apa_meas_setup_torsion_bis.jpg]] - -The data is loaded, and the transfer function computed. -#+begin_src matlab -%% Load Data +% Load Data torsion = load('apa300ml_torsion_left.mat'); -%% Compute transfer function +% Compute transfer function [G_torsion, ~] = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts); #+end_src -The results are shown in Figure ref:fig:test_apa_meas_freq_torsion_z. -We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz. -And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz. -A third mode at 800Hz could correspond to this torsion mode. +The three measured frequency response functions are shown in Figure ref:fig:test_apa_meas_freq_compare. +- a clear $x$ bending mode at $280\,\text{Hz}$ +- a clear $y$ bending mode at $412\,\text{Hz}$ +- for the $z$ torsion test, the $y$ bending mode is also excited and observed, and we may see a mode at $800\,\text{Hz}$ #+begin_src matlab :exports none -%% Plot the transfer function figure; hold on; -plot(f, abs(G_torsion), 'k-'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 2e-2]) -text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') +plot(f, abs(G_bending_X), 'DisplayName', '$X$ bending'); +plot(f, abs(G_bending_Y), 'DisplayName', '$Y$ bending'); +plot(f, abs(G_torsion), 'DisplayName', '$Z$ torsion'); +text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') text(800, 6e-4,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/test_apa_meas_freq_torsion_z.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_apa_meas_freq_torsion_z -#+caption: Obtained FRF for the Z-torsion -#+RESULTS: -[[file:figs/test_apa_meas_freq_torsion_z.png]] - -In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited. -#+begin_src matlab -%% Load data -torsion = load('apa300ml_torsion_top.mat'); - -%% Compute transfer function -[G_torsion_top, ~] = tfestimate(torsion.Track1, torsion.Track2, win, Noverlap, Nfft, 1/Ts); -#+end_src - -The two FRF are compared in Figure ref:fig:test_apa_meas_freq_torsion_z_comp. -It is clear that the first two modes does not correspond to the torsional mode. -Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here. -#+begin_src matlab :exports none -%% Plot the two transfer functions -figure; -hold on; -plot(f, abs(G_torsion), 'k-', 'DisplayName', 'Left excitation'); -plot(f, abs(G_torsion_top), '-', 'DisplayName', 'Top excitation'); hold off; set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 2e-2]) -text(415, 4.3e-3,{'415Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') -text(267, 8e-4,{'267Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') -text(800, 2e-3,{'800Hz'}, 'VerticalAlignment', 'bottom','HorizontalAlignment','center') -legend('location', 'northwest'); +xlim([50, 2e3]); ylim([5e-5, 2e-1]); +legend('location', 'northeast', 'FontSize', 8) #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/test_apa_meas_freq_torsion_z_comp.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_apa_meas_freq_torsion_z_comp -#+caption: Obtained FRF for the Z-torsion -#+RESULTS: -[[file:figs/test_apa_meas_freq_torsion_z_comp.png]] - -*** Compare -The three measurements are shown in Figure ref:fig:test_apa_meas_freq_compare. -#+begin_src matlab :exports none -figure; -hold on; -plot(f, abs(G_torsion), 'DisplayName', 'Torsion'); -plot(f, abs(G_bending_X), 'DisplayName', 'Bending - X'); -plot(f, abs(G_bending_Y), 'DisplayName', 'Bending - Y'); -hold off; -set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude'); -xlim([50, 2e3]); ylim([1e-5, 1e-1]); -legend('location', 'southeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:test_apa_meas_freq_compare -#+caption: Obtained FRF - Comparison +#+caption: Obtained frequency response functions for the 3 tests with the instrumented hammer #+RESULTS: [[file:figs/test_apa_meas_freq_compare.png]] -*** Conclusion - -When two flexible joints are fixed at each ends of the APA, the APA is mostly in a free/free condition in terms of bending/torsion (the bending/torsional stiffness of the joints being very small). - -In the current tests, the APA are in a fixed/free condition. -Therefore, it is quite obvious that we measured higher resonance frequencies than what is foreseen for the struts. -It is however quite interesting that there is a factor $\approx \sqrt{2}$ between the two (increased of the stiffness by a factor 2?). - -#+name: tab:apa300ml_measured_modes_freq +#+name: tab:test_apa_measured_modes_freq #+caption: Measured frequency of the modes -#+attr_latex: :environment tabularx :width 0.7\linewidth :align Xcc -#+attr_latex: :center t :booktabs t :float t -| *Mode* | *FEM - Strut mode* | *Measured Frequency* | -|-----------+--------------------+----------------------| -| X-Bending | 189Hz | 280Hz | -| Y-Bending | 285Hz | 410Hz | -| Z-Torsion | 400Hz | 800Hz? | +#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcc +#+attr_latex: :center t :booktabs t +| *Mode* | *FEM* | *Measured Frequency* | +|-------------+-------+----------------------| +| $X$ bending | | 280Hz | +| $Y$ bending | | 410Hz | +| $Z$ torsion | | 800Hz | ** Conclusion :ignore: @@ -826,7 +670,7 @@ These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_st | 6 | 1.7 | 1.92 | | 8 | 1.73 | 1.98 | -The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 94\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_frf_disp) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. +The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. \begin{equation} \label{eq:test_apa_res_freq} \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} @@ -918,7 +762,7 @@ The following can be observed: - A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$. The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA - A lightly damped resonance at $95\,\text{Hz}$ -- A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear +- A "mass line" up to $\approx 800\,\text{Hz}$, above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support. #+begin_src matlab :exports none %% Plot the FRF from u to de @@ -1034,17 +878,17 @@ It is confirmed that the added resistor as the effect of adding an high pass fil #+begin_src matlab %% Load the data -wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va'); % With the resistor -wo_k = load('frf_data_1_sweep_lf.mat', 't', 'Vs', 'Va'); % Without the resistor +wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'u'); % With the resistor +wo_k = load('frf_data_1_sweep_lf.mat', 't', 'Vs', 'u'); % Without the resistor %% Large Hanning window for good low frequency estimate Nfft = floor(50/Ts); win = hanning(Nfft); Noverlap = floor(Nfft/2); -%% Compute the transfer functions from Va to Vs -[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, Noverlap, Nfft, 1/Ts); -[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, Noverlap, Nfft, 1/Ts); +%% Compute the transfer functions from u to Vs +[frf_wo_k, f] = tfestimate(wo_k.u, wo_k.Vs, win, Noverlap, Nfft, 1/Ts); +[frf_wi_k, ~] = tfestimate(wi_k.u, wi_k.Vs, win, Noverlap, Nfft, 1/Ts); %% Model for the high pass filter C = 5.1e-6; % Sensor Stack capacitance [F] @@ -1331,20 +1175,21 @@ exportFig('figs/test_apa_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall') #+caption: Root Locus of the APA300ML with Integral Force Feedback - Comparison between the computed root locus from the plant model (black line) and the root locus estimated from the damped plant pole identification (colorful crosses) #+RESULTS: [[file:figs/test_apa_iff_root_locus.png]] -** Conclusion + +** Conclusion :ignore: #+begin_important So far, all the measured FRF are showing the dynamical behavior that was expected. #+end_important - -* Simscape Model +* 2 Degrees of Freedom Model :PROPERTIES: -:header-args:matlab+: :tangle matlab/test_apa_3_simscape.m +:header-args:matlab+: :tangle matlab/test_apa_3_model_2dof.m :END: -<> +<> + ** Introduction :ignore: -In this section, a simscape model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to compare the model of the APA with the measured FRF. +In this section, a simscape model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to compare the model of the APA with the measured frequency response functions. After the transfer functions are extracted from the model (Section ref:sec:simscape_bench_apa_first_id), the comparison of the obtained dynamics with the measured FRF will permit to: 1. Estimate the "actuator constant" and "sensor constant" (Section ref:sec:simscape_bench_apa_id_constants) @@ -1354,7 +1199,7 @@ After the transfer functions are extracted from the model (Section ref:sec:simsc #+name: fig:test_apa_bench_model #+caption: Screenshot of the Simscape model -#+attr_latex: :width 0.5\linewidth +#+attr_latex: :width 0.8\linewidth [[file:figs/test_apa_bench_model.png]] ** Matlab Init :noexport:ignore: @@ -1386,662 +1231,212 @@ After the transfer functions are extracted from the model (Section ref:sec:simsc <> #+end_src -** First Identification -<> +#+begin_src matlab :exports none +%% Input/Output definition of the Model +clear io; io_i = 1; +io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage +io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder +#+end_src + +** Two Degrees of Freedom APA Model +<> + +The APA model shown in Figure ref:fig:test_apa_2dof_model is adapted from cite:souleille18_concep_activ_mount_space_applic. + +It can be decomposed into three components: +- the shell whose axial properties are represented by $k_1$ and $c_1$ +- the actuator stacks whose contribution in the axial stiffness is represented by $k_a$ and $c_a$. + A force source $\tau$ represents the axial force induced by the force sensor stacks. + The gain $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $\tau$ +- the actuator stacks whose contribution in the axial stiffness is represented by $k_e$ and $c_e$. + A "strain sensor" $d_L$, and a gain $g_s$ (in $V/m$) that converts this strain into a generated voltage + +Such simple model has some limitations: +- it only represents the axial characteristics of the APA (infinitely rigid in other directions) +- some physical insights are lost such as the amplification factor, the real stress and strain on the piezoelectric stacks +- it is fully linear and therefore the creep and hysteresis of the piezoelectric stacks are not modelled + +#+name: fig:test_apa_2dof_model +#+caption: Schematic of the two degrees of freedom model of the APA300ML +[[file:figs/test_apa_2dof_model.png]] + +** Tuning of the APA model +<> + +9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics. + +#+name: fig:test_apa_2dof_model_simscape +#+caption: Schematic of the two degrees of freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$ +[[file:figs/test_apa_2dof_model_simscape.png]] -The APA is first initialized with default parameters: #+begin_src matlab -%% Initialize the structure with default values +%% Stiffness values for the 2DoF APA model +k1 = 0.38e6; % Estimated Shell Stiffness [N/m] + +w0 = 2*pi*95; % Resonance frequency [rad/s] +m = 5.7; % Suspended mass [kg] +ktot = m*(w0)^2; % Total Axial Stiffness to have to wanted resonance frequency [N/m] + +ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m] +ke = 2*ka; % Stiffness of the Sensor stack [N/m] + +%% Damping values for the 2DoF APA model +c1 = 20; % Damping for the Shell [N/(m/s)] +ca = 100; % Damping of the actuators stacks [N/(m/s)] +ce = 2*ca; % Damping of the sensor stack [N/(m/s)] +#+end_src + +#+begin_src matlab +%% Estimation ot the sensor and actuator gains +% Initialize the structure with unitary sensor and actuator "gains" n_hexapod = struct(); n_hexapod.actuator = initializeAPA(... 'type', '2dof', ... - 'Ga', 1, ... % Actuator constant [N/V] - 'Gs', 1); % Sensor constant [V/m] -#+end_src + 'k', k1, ... + 'ka', ka, ... + 'ke', ke, ... + 'c', c1, ... + 'ca', ca, ... + 'ce', ce, ... + 'Ga', 1, ... % Actuator constant [N/V] + 'Gs', 1 ... % Sensor constant [V/m] +); -The transfer function from excitation voltage $V_a$ (before the amplification of $20$ due to the PD200 amplifier) to: -1. the sensor stack voltage $V_s$ -2. the measured displacement by the encoder $d_e$ +c_granite = 0; % Do not take into account damping added by the air bearing -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage -io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage -io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder +% Run the linearization +G_norm = linearize(mdl, io, 0.0, opts); +G_norm.InputName = {'u'}; +G_norm.OutputName = {'Vs', 'de'}; -%% Linearization options -opts = linearizeOptions; -opts.SampleTime = 0; - -%% Run the linearization -Ga = linearize(mdl, io, 0.0, opts); -Ga.InputName = {'Va'}; -Ga.OutputName = {'Vs', 'de', 'da'}; -#+end_src - -The obtain dynamics are shown in Figure ref:fig:apa_model_bench_bode_vs and ref:fig:apa_model_bench_bode_dl_z. -It can be seen that: -- the shape of these bode plots are very similar to the one measured in Section ref:sec:dynamical_meas_apa expect from a change in gain and exact location of poles and zeros -- there is a sign error for the transfer function from $V_a$ to $V_s$. - This will be corrected by taking a negative "sensor gain". -- the low frequency zero of the transfer function from $V_a$ to $V_s$ is minimum phase as expected. - The measured FRF are showing non-minimum phase zero, but it is most likely due to measurements artifacts. - -#+begin_src matlab :exports none -%% Bode plot of the transfer function from u to taum -freqs = logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(freqs, abs(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'k-') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; - -ax2 = nexttile; -hold on; -plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'k-') -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:45:360); -ylim([-180, 0]) - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_model_bench_bode_vs.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:apa_model_bench_bode_vs -#+caption: Bode plot of the transfer function from $V_a$ to $V_s$ -#+RESULTS: -[[file:figs/apa_model_bench_bode_vs.png]] - -#+begin_src matlab :exports none -%% Bode plot of the transfer function from Va to de and da -freqs = logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(freqs, abs(squeeze(freqresp(Ga('de', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -hold off; -legend('location', 'southwest'); - -ax2 = nexttile; -hold on; -plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('de', 'Va'), freqs, 'Hz')))) -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:45:360); -ylim([-180, 0]) - -linkaxes([ax1,ax2],'x'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_model_bench_bode_dl_z.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:apa_model_bench_bode_dl_z -#+caption: Bode plot of the transfer function from $V_a$ to $d_L$ and to $z$ -#+RESULTS: -[[file:figs/apa_model_bench_bode_dl_z.png]] - -** Identify Sensor/Actuator constants and compare with measured FRF -<> -*** How to identify these constants? -**** Piezoelectric Actuator Constant - -Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage $V_a$ to the induced displacement $d$. -\begin{equation} - d = g_{d/V_a} \cdot V_a -\end{equation} - -Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force $F_a$ to the induced displacement $d$: -\begin{equation} - d = g_{d/F_a} \cdot F_a -\end{equation} - -From the two gains, it is then easy to determine $g_a$: -\begin{equation} \label{eq:actuator_constant_formula} - \boxed{g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}}} -\end{equation} - -**** Piezoelectric Sensor Constant - -Similarly, it is easy to determine the gain from the excitation voltage $V_a$ to the voltage generated by the sensor stack $V_s$: -\begin{equation} - V_s = g_{V_s/V_a} V_a -\end{equation} - -Note here that there is an high pass filter formed by the piezoelectric capacitor and parallel resistor. - -The gain can be computed from the dynamical identification and taking the gain at the wanted frequency (above the first resonance). - -Using the simscape model, compute the gain at the same frequency from the actuator force $F_a$ to the strain of the sensor stack $dl$: -\begin{equation} - dl = g_{dl/F_a} F_a -\end{equation} - -Then, the "sensor" constant is: -\begin{equation} \label{eq:sensor_constant_formula} - \boxed{g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}}} -\end{equation} - -*** Identification Data -Let's load the measured FRF from the DAC voltage to the measured encoder and to the sensor stack voltage. -#+begin_src matlab -%% Load Data +% Load Identification Data to estimate the two gains load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums'); + +% Estimation ot the Actuator Gain +fa = 10; % Frequency where the two FRF should match [Hz] +[~, i_f] = min(abs(f - fa)); +ga = -abs(enc_frf(i_f,1))./abs(evalfr(G_norm('de', 'u'), 1i*2*pi*fa)); + +% Estimation ot the Sensor Gain +fs = 600; % Frequency where the two FRF should match [Hz] +[~, i_f] = min(abs(f - fs)) +gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga; #+end_src -*** 2DoF APA -**** 2DoF APA -Let's initialize the APA as a 2DoF model with unity sensor and actuator gains. +First, the mass supported by the APA300ML can simply be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. +Both methods leads to an estimated mass of $5.7\,\text{kg}$. + +Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force. +Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $20\,Ns/m$. + +Then, it is reasonable to make the assumption that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:5]. +Therefore, we have $k_e = 2 k_a$ and $c_e = 2 c_a$ as the actuator stack is composed of two stacks in series. +In that case, the total stiffness of the APA model is described by eqref:eq:test_apa_2dof_stiffness. + +\begin{equation}\label{eq:test_apa_2dof_stiffness} +k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a +\end{equation} + +Knowing from eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,N/\mu m$, we get from eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,N/\mu m$ and $k_e = 5\,N/\mu m$. + +\begin{equation}\label{eq:test_apa_tot_stiffness} +\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,N/\mu m \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s +\end{equation} + +Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance. +$c_a = 100\,Ns/m$ and $c_e = 200\,Ns/m$ are obtained. + +Finally, the two gains $g_s$ and $g_a$ can be used to match the gain of the identified transfer functions. + +The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_simscape are summarized in Table ref:tab:test_apa_2dof_parameters. + +#+name: tab:test_apa_2dof_parameters +#+caption: Summary of the obtained parameters for the 2 DoF APA300ML model +#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc +#+attr_latex: :center t :booktabs t +| *Parameter* | *Value* | +|-------------+------------------| +| $m$ | $5.7\,\text{kg}$ | +| $k_1$ | $0.38\,N/\mu m$ | +| $k_e$ | $5.0\, N/\mu m$ | +| $k_a$ | $2.5\,N/\mu m$ | +| $c_1$ | $20\,Ns/m$ | +| $c_e$ | $200\,Ns/m$ | +| $c_a$ | $100\,Ns/m$ | +| $g_a$ | $-2.58\,N/V$ | +| $g_s$ | $4.6\,V/\mu m$ | + +** Obtained Dynamics +<> + +The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model. +It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf. + +A good match can be observed between the model and the experimental data, both for the encoder and for the force sensor. +This indicates that this model represents well the axial dynamics of the APA300ML. + #+begin_src matlab -%% Initialize a 2DoF APA with Ga=Gs=1 +%% 2DoF APA300ML with optimized parameters +n_hexapod = struct(); n_hexapod.actuator = initializeAPA(... 'type', '2dof', ... - 'ga', 1, ... - 'gs', 1); -#+end_src + 'k', k1, ... + 'ka', ka, ... + 'ke', ke, ... + 'c', c1, ... + 'ca', ca, ... + 'ce', ce, ... + 'Ga', ga, ... + 'Gs', gs ... +); -**** Identification without actuator or sensor constants -The transfer function from $V_a$ to $V_s$, $d_e$ and $d_a$ is identified. -#+begin_src matlab -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage -io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage -io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder -io(io_i) = linio([mdl, '/da'], 1, 'openoutput'); io_i = io_i + 1; % Attocube - -%% Identification -Gs = linearize(mdl, io, 0.0, options); -Gs.InputName = {'Va'}; -Gs.OutputName = {'Vs', 'de', 'da'}; -#+end_src - -**** Actuator Constant -Then, the actuator constant can be computed as shown in Eq. eqref:eq:actuator_constant_formula by dividing the measured DC gain of the transfer function from $V_a$ to $d_e$ by the estimated DC gain of the transfer function from $V_a$ (in truth the actuator force called $F_a$) to $d_e$ using the Simscape model. -#+begin_src matlab -%% Estimated Actuator Constant -ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('de', 'Va')); % [N/V] -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('ga = %.1f [N/V]', ga); -#+end_src - -#+RESULTS: -: ga = -32.2 [N/V] - -**** Sensor Constant -Similarly, the sensor constant can be estimated using Eq. eqref:eq:sensor_constant_formula. - -#+begin_src matlab -%% Estimated Sensor Constant -gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m] -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('gs = %.3f [V/m]', gs); -#+end_src - -#+RESULTS: -: gs = 0.088 [V/m] - -**** Comparison -Let's now initialize the APA with identified sensor and actuator constant: -#+begin_src matlab -%% Set the identified constants -n_hexapod.actuator = initializeAPA(... - 'type', '2dof', ... - 'ga', ga, ... % Actuator gain [N/V] - 'gs', gs); % Sensor gain [V/m] -#+end_src - -And identify the dynamics with included constants. -#+begin_src matlab -%% Identify again the dynamics with correct Ga,Gs -Gs = linearize(mdl, io, 0.0, options); -Gs = Gs*exp(-Ts*s); -Gs.InputName = {'Va'}; -Gs.OutputName = {'Vs', 'de', 'da'}; -#+end_src - -The transfer functions from $V_a$ to $d_e$ are compared in Figure ref:fig:apa_act_constant_comp and the one from $V_a$ to $V_s$ are compared in Figure ref:fig:apa_sens_constant_comp. - -#+begin_src matlab :exports none -%% Bode plot of the transfer function from u to de -freqs = logspace(1,4,1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:length(apa_nums) - plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-8, 1e-3]); - -ax2 = nexttile; -hold on; -for i = 1:length(apa_nums) - plot(f, 180/pi*angle(enc_frf(:,1)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_act_constant_comp.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa_act_constant_comp -#+caption: Comparison of the experimental data and Simscape model ($V_a$ to $d_e$) -#+RESULTS: -[[file:figs/apa_act_constant_comp.png]] - -#+begin_src matlab :exports none -%% Bode plot of the transfer function from Va to Vs (both Simscape and measured FRF) -freqs = logspace(1,4,1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:length(apa_nums) - plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-2, 1e2]); - -ax2 = nexttile; -hold on; -for i = 1:length(apa_nums) - plot(f, 180/pi*angle(iff_frf(:,1)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_sens_constant_comp.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa_sens_constant_comp -#+caption: Comparison of the experimental data and Simscape model ($V_a$ to $V_s$) -#+RESULTS: -[[file:figs/apa_sens_constant_comp.png]] - -#+begin_src matlab :exports none -%% Compare the FRF and identified dynamics from Va to Vs and da -colors = colororder; - -figure; -tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'FRF'); -for i = 2:length(apa_nums) - plot(f, abs(enc_frf(:, i)), 'color', [0,0,0, 0.2], ... - 'HandleVisibility', 'off'); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('da', 'Va'), freqs, 'Hz'))), '--', ... - 'DisplayName', 'Model') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_a/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-8, 1e-3]); -legend('location', 'southwest'); - -ax1b = nexttile([2,1]); -hold on; -plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'FRF'); -for i = 1:length(apa_nums) - plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), '--', ... - 'DisplayName', 'Model') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-2, 1e2]); -legend('location', 'southeast'); - -ax2 = nexttile; -hold on; -for i = 1:length(apa_nums) - plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0, 0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('da', 'Va'), freqs, 'Hz'))), '--') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); ylim([-180, 180]); - -ax2b = nexttile; -hold on; -for i = 1:length(apa_nums) - plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0, 0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), '--') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); ylim([-180, 180]); - -linkaxes([ax1,ax2,ax1b,ax2b],'x'); -xlim([10, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_comp_model_frf.pdf', 'width', 1500, 'height', 'tall'); -#+end_src - -#+name: fig:apa_comp_model_frf -#+caption: -#+RESULTS: -[[file:figs/apa_comp_model_frf.png]] - - -#+begin_important -The "actuator constant" and "sensor constant" can indeed be identified using this test bench. -After identifying these constants, the 2DoF model shows good agreement with the measured dynamics. -#+end_important - -*** Flexible APA -**** Introduction :ignore: -In this section, the sensor and actuator "constants" are also estimated for the flexible model of the APA. - -**** Flexible APA -The Simscape APA model is initialized as a flexible one with unity "constants". -#+begin_src matlab -%% Initialize the APA as a flexible body -n_hexapod.actuator = initializeAPA(... - 'type', 'flexible', ... - 'ga', 1, ... - 'gs', 1); -#+end_src - -**** Identification without actuator or sensor constants -The dynamics from $V_a$ to $V_s$, $d_e$ and $d_a$ is identified. -#+begin_src matlab -%% Identify the dynamics -Gs = linearize(mdl, io, 0.0, options); -Gs.InputName = {'Va'}; -Gs.OutputName = {'Vs', 'de', 'da'}; -#+end_src - -**** Actuator Constant -Then, the actuator constant can be computed as shown in Eq. eqref:eq:actuator_constant_formula: -#+begin_src matlab -%% Actuator Constant -ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('de', 'Va')); % [N/V] -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('ga = %.1f [N/V]', ga); -#+end_src - -#+RESULTS: -: ga = 23.5 [N/V] - -**** Sensor Constant -#+begin_src matlab -%% Sensor Constant -gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m] -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('gs = %.3f [V/m]', gs); -#+end_src - -#+RESULTS: -: gs = -4839841.756 [V/m] - -**** Comparison -Let's now initialize the flexible APA with identified sensor and actuator constant: -#+begin_src matlab -%% Set the identified constants -n_hexapod.actuator = initializeAPA(... - 'type', 'flexible', ... - 'ga', ga, ... % Actuator gain [N/V] - 'gs', gs); % Sensor gain [V/m] -#+end_src - -And identify the dynamics with included constants. -#+begin_src matlab -%% Identify with updated constants -Gs = linearize(mdl, io, 0.0, options); -Gs = Gs*exp(-Ts*s); -Gs.InputName = {'Va'}; -Gs.OutputName = {'Vs', 'de', 'da'}; -#+end_src - -The obtained dynamics is compared with the measured one in Figures ref:fig:apa_act_constant_comp_flex and ref:fig:apa_sens_constant_comp_flex. - -#+begin_src matlab :exports none -%% Bode plot of the transfer function from V_a to d_e (both Simscape and measured FRF) -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:length(apa_nums) - plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-9, 1e-3]); - -ax2 = nexttile; -hold on; -for i = 1:length(apa_nums) - plot(f, 180/pi*angle(enc_frf(:,1)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_act_constant_comp_flex.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa_act_constant_comp_flex -#+caption: Comparison of the experimental data and Simscape model ($u$ to $d\mathcal{L}_m$) -#+RESULTS: -[[file:figs/apa_act_constant_comp_flex.png]] - -#+begin_src matlab :exports none -%% Bode plot of the transfer function from Va to Vs (both Simscape and measured FRF) -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:length(apa_nums) - plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([1e-2, 1e2]); - -ax2 = nexttile; -hold on; -for i = 1:length(apa_nums) - plot(f, 180/pi*angle(iff_frf(:,1)), 'color', [0,0,0,0.2]); -end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/apa_sens_constant_comp_flex.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa_sens_constant_comp_flex -#+caption: Comparison of the experimental data and Simscape model ($u$ to $\tau_m$) -#+RESULTS: -[[file:figs/apa_sens_constant_comp_flex.png]] - -#+begin_important -The flexible model is a bit "soft" as compared with the experimental results. -#+end_important - -** Optimize 2-DoF model to fit the experimental Data -<> -The parameters of the 2DoF model presented in Section ref:sec:apa_2dof_model are now optimize such that the model best matches the measured FRF. - -After optimization, the following parameters are used: -#+begin_src matlab -%% Optimized parameters -n_hexapod.actuator = initializeAPA('type', '2dof', ... - 'Ga', -32.2, ... - 'Gs', 0.088, ... - 'k', ones(6,1)*0.38e6, ... - 'ke', ones(6,1)*1.75e6, ... - 'ka', ones(6,1)*3e7, ... - 'c', ones(6,1)*1.3e2, ... - 'ce', ones(6,1)*1e1, ... - 'ca', ones(6,1)*1e1 ... - ); +%% Identification of the APA300ML with optimized parameters +G_2dof = exp(-s*Ts)*linearize(mdl, io, 0.0, opts); +G_2dof.InputName = {'u'}; +G_2dof.OutputName = {'Vs', 'de'}; #+end_src #+begin_src matlab :exports none -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage -io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage -io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder - -%% Identification with optimized parameters -Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options); -Gs.InputName = {'Va'}; -Gs.OutputName = {'Vs', 'de'}; -#+end_src - -The dynamics is identified using the Simscape model and compared with the measured FRF in Figure ref:fig:comp_apa_plant_after_opt. -#+begin_src matlab :exports none -%% Comparison of the experimental data and Simscape Model +%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML freqs = 5*logspace(0, 3, 1000); figure; tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; +plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); for i = 1:length(apa_nums) - plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2]); + plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz')))) +plot(freqs, abs(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); ax1b = nexttile([2,1]); hold on; -for i = 1:length(apa_nums) - plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2]); +plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 2:length(apa_nums) + plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); end -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) +plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); ax2 = nexttile; hold on; for i = 1:length(apa_nums) plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]); end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('de', 'Va'), freqs, 'Hz')))) +plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); @@ -2053,8 +1448,7 @@ hold on; for i = 1:length(apa_nums) plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]); end -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) +plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); @@ -2066,942 +1460,272 @@ xlim([10, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_apa_plant_after_opt.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/test_apa_2dof_comp_frf.pdf', 'width', 'full', 'height', 'tall'); #+end_src -#+name: fig:comp_apa_plant_after_opt -#+caption: Comparison of the measured FRF and the optimized model +#+name: fig:test_apa_2dof_comp_frf +#+caption: Comparison of the measured FRF and the optimized 2DoF model of the APA300ML #+RESULTS: -[[file:figs/comp_apa_plant_after_opt.png]] +[[file:figs/test_apa_2dof_comp_frf.png]] -#+begin_important -The tuned 2DoF is very well representing the (axial) dynamics of the APA. -#+end_important +** Conclusion :ignore: + +* APA300ML - Super Element +:PROPERTIES: +:header-args:matlab+: :tangle matlab/test_apa_4_model_flexible.m +:END: +<> -* TODO Compare with the FEM/Simscape Model :noexport: ** Introduction :ignore: -In this section, the Amplified Piezoelectric Actuator APA300ML ([[file:doc/APA300ML.pdf][doc]]) is modeled using a Finite Element Software. -Then a /super element/ is exported and imported in Simscape where its dynamic is studied. +In this section, a /super element/ of the Amplified Piezoelectric Actuator "APA300ML" is extracted using a Finite Element Software. +It is then imported in Simscape (using the stiffness and mass matrices) and it is included in the same model that was used in ref:sec:test_apa_model_2dof. -A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure ref:fig:test_apa_ansys. -The remote point used are also shown in this figure. +This procedure is illustrated in Figure ref:fig:test_apa_super_element_simscape. -#+name: fig:test_apa_ansys -#+caption: Ansys FEM of the APA300ML -[[file:figs/test_apa_ansys.jpg]] +#+name: fig:test_apa_super_element_simscape +#+caption: Finite Element Model of the APA300ML with "remotes points" on the left. Simscape model with included "Reduced Order Flexible Solid" on the right. +#+attr_latex: :width 1.0\linewidth +[[file:figs/test_apa_super_element_simscape.png]] -** Matlab Init :noexport:ignore: +** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) - <> +<> #+end_src #+begin_src matlab :exports none :results silent :noweb yes - <> +<> #+end_src -#+begin_src matlab :tangle no - addpath('matlab/'); - addpath('matlab/APA300ML/'); +#+begin_src matlab :tangle no :noweb yes +<> #+end_src -#+begin_src matlab :eval no - addpath('APA300ML/'); +#+begin_src matlab :eval no :noweb yes +<> #+end_src -#+begin_src matlab - open('APA300ML.slx'); +#+begin_src matlab :tangle no :noweb yes +<> #+end_src -** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates -We first extract the stiffness and mass matrices. -#+begin_src matlab - K = readmatrix('APA300ML_mat_K.CSV'); - M = readmatrix('APA300ML_mat_M.CSV'); +#+begin_src matlab :eval no :noweb yes +<> #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no - data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g '); +#+begin_src matlab :noweb yes +<> #+end_src -#+caption: First 10x10 elements of the Stiffness matrix -#+RESULTS: -| 200000000.0 | 30000.0 | -20000.0 | -70.0 | 300000.0 | 40.0 | 10000000.0 | 10000.0 | -6000.0 | 30.0 | -| 30000.0 | 30000000.0 | 2000.0 | -200000.0 | 60.0 | -10.0 | 4000.0 | 2000000.0 | -500.0 | 9000.0 | -| -20000.0 | 2000.0 | 7000000.0 | -10.0 | -30.0 | 10.0 | 6000.0 | 900.0 | -500000.0 | 3 | -| -70.0 | -200000.0 | -10.0 | 1000.0 | -0.1 | 0.08 | -20.0 | -9000.0 | 3 | -30.0 | -| 300000.0 | 60.0 | -30.0 | -0.1 | 900.0 | 0.1 | 30000.0 | 20.0 | -10.0 | 0.06 | -| 40.0 | -10.0 | 10.0 | 0.08 | 0.1 | 10000.0 | 20.0 | 9 | -5 | 0.03 | -| 10000000.0 | 4000.0 | 6000.0 | -20.0 | 30000.0 | 20.0 | 200000000.0 | 10000.0 | 9000.0 | 50.0 | -| 10000.0 | 2000000.0 | 900.0 | -9000.0 | 20.0 | 9 | 10000.0 | 30000000.0 | -500.0 | 200000.0 | -| -6000.0 | -500.0 | -500000.0 | 3 | -10.0 | -5 | 9000.0 | -500.0 | 7000000.0 | -2 | -| 30.0 | 9000.0 | 3 | -30.0 | 0.06 | 0.03 | 50.0 | 200000.0 | -2 | 1000.0 | - - -#+begin_src matlab :exports results :results value table replace :tangle no - data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g '); -#+end_src - -#+caption: First 10x10 elements of the Mass matrix -#+RESULTS: -| 0.01 | -2e-06 | 1e-06 | 6e-09 | 5e-05 | -5e-09 | -0.0005 | -7e-07 | 6e-07 | -3e-09 | -| -2e-06 | 0.01 | 8e-07 | -2e-05 | -8e-09 | 2e-09 | -9e-07 | -0.0002 | 1e-08 | -9e-07 | -| 1e-06 | 8e-07 | 0.009 | 5e-10 | 1e-09 | -1e-09 | -5e-07 | 3e-08 | 6e-05 | 1e-10 | -| 6e-09 | -2e-05 | 5e-10 | 3e-07 | 2e-11 | -3e-12 | 3e-09 | 9e-07 | -4e-10 | 3e-09 | -| 5e-05 | -8e-09 | 1e-09 | 2e-11 | 6e-07 | -4e-11 | -1e-06 | -2e-09 | 1e-09 | -8e-12 | -| -5e-09 | 2e-09 | -1e-09 | -3e-12 | -4e-11 | 1e-07 | -2e-09 | -1e-09 | -4e-10 | -5e-12 | -| -0.0005 | -9e-07 | -5e-07 | 3e-09 | -1e-06 | -2e-09 | 0.01 | 1e-07 | -3e-07 | -2e-08 | -| -7e-07 | -0.0002 | 3e-08 | 9e-07 | -2e-09 | -1e-09 | 1e-07 | 0.01 | -4e-07 | 2e-05 | -| 6e-07 | 1e-08 | 6e-05 | -4e-10 | 1e-09 | -4e-10 | -3e-07 | -4e-07 | 0.009 | -2e-10 | -| -3e-09 | -9e-07 | 1e-10 | 3e-09 | -8e-12 | -5e-12 | -2e-08 | 2e-05 | -2e-10 | 3e-07 | - - -Then, we extract the coordinates of the interface nodes. -#+begin_src matlab - [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA300ML_out_nodes_3D.txt'); -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f '); -#+end_src - -#+caption: Coordinates of the interface nodes -#+RESULTS: -| Node i | Node Number | x [m] | y [m] | z [m] | -|--------+-------------+---------+-------+--------| -| 1.0 | 697783.0 | 0.0 | 0.0 | -0.015 | -| 2.0 | 697784.0 | 0.0 | 0.0 | 0.015 | -| 3.0 | 697785.0 | -0.0325 | 0.0 | 0.0 | -| 4.0 | 697786.0 | -0.0125 | 0.0 | 0.0 | -| 5.0 | 697787.0 | -0.0075 | 0.0 | 0.0 | -| 6.0 | 697788.0 | 0.0125 | 0.0 | 0.0 | -| 7.0 | 697789.0 | 0.0325 | 0.0 | 0.0 | - -#+begin_src matlab :exports results :results value table replace :tangle no - data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f '); -#+end_src - -#+caption: Some extracted parameters of the FEM -#+RESULTS: -| Total number of Nodes | 7 | -| Number of interface Nodes | 7 | -| Number of Modes | 120 | -| Size of M and K matrices | 162 | - -Using =K=, =M= and =int_xyz=, we can now use the =Reduced Order Flexible Solid= simscape block. - -** Piezoelectric parameters -#+begin_src matlab - Ga = 1; % [N/V] - Gs = 1; % [V/m] -#+end_src - -#+begin_src matlab - m = 0.1; % [kg] -#+end_src - -** Simscape Model -The flexible element is imported using the =Reduced Order Flexible Solid= simscape block. - -Let's say we use two stacks as a force sensor and one stack as an actuator: -- A =Relative Motion Sensor= block is added between the nodes A and C -- An =Internal Force= block is added between the remote points E and B - -The interface nodes are shown in Figure ref:fig:test_apa_ansys. - -One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G). - -** Identification of the APA Characteristics -*** Stiffness -#+begin_src matlab :exports none - m = 0.0001; -#+end_src - -The transfer function from vertical external force to the relative vertical displacement is identified. - #+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; - - G = linearize(mdl, io); +%% Input/Output definition of the Model +clear io; io_i = 1; +io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage +io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder #+end_src -The inverse of its DC gain is the axial stiffness of the APA: -#+begin_src matlab :results replace value - 1e-6/dcgain(G) % [N/um] +** Extraction of the super-element + +- Explain how the "remote points" are chosen +- Show some parts of the mass and stiffness matrices? +- Say which materials were used? +- Maybe this was already explain earlier in the manuscript + +** Identification of the Actuator and Sensor constants +<> + +Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be identified. +The gains $g_a$ and $g_s$ can then be tuned such that the gain of the transfer functions are matching the identified ones. + +By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained. + +#+begin_src matlab +%% Identification of the actuator and sensor "constants" +% Initialize the APA as a flexible body with unity "constants" +n_hexapod.actuator = initializeAPA(... + 'type', 'flexible', ... + 'ga', 1, ... + 'gs', 1); + +c_granite = 100; % Rought estimation of the damping added by the air bearing + +% Identify the dynamics +G_norm = linearize(mdl, io, 0.0, opts); +G_norm.InputName = {'u'}; +G_norm.OutputName = {'Vs', 'de'}; + +% Load Identification Data to estimate the two gains +load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums'); + +% Actuator Constant in [N/V] +ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u')); + +% Sensor Constant in [V/m] +gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz')))); #+end_src -#+RESULTS: -: 1.753 -The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$. +To make sure these "gains" are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material. -*** Resonance Frequency -The resonance frequency is specified to be between 650Hz and 840Hz. -This is also the case for the FEM model (Figure ref:fig:apa300ml_resonance). +From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force. + +\begin{subequations} +\begin{align} + V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\ + F_a &= \underbrace{d_{33} n k_a}_{g_a} \cdot V_a, \quad k_a = \frac{c^{E} A}{L} \label{eq:test_apa_piezo_voltage_to_force} +\end{align} +\end{subequations} + +Parameters used in equations eqref:eq:test_apa_piezo_strain_to_voltage and eqref:eq:test_apa_piezo_voltage_to_force are described in Table ref:tab:test_apa_piezo_properties. + +Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML. +However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. +The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties. + +From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are very close to the identified constants using the experimentally identified transfer functions. + +#+name: tab:test_apa_piezo_properties +#+caption: Piezoelectric properties used for the estimation of the sensor and actuators "gains" +#+attr_latex: :environment tabularx :width 1\linewidth :align ccX +#+attr_latex: :center t :booktabs t +| *Parameter* | *Value* | *Description* | +|----------------+----------------------------+--------------------------------------------------------------| +| $d_{33}$ | $680 \cdot 10^{-12}\,m/V$ | Piezoelectric constant | +| $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,F/m$ | Permittivity under constant stress | +| $s^{D}$ | $21 \cdot 10^{-12}\,m^2/N$ | Elastic compliance understand constant electric displacement | +| $c^{E}$ | $48 \cdot 10^{9}\,N/m^2$ | Young's modulus of elasticity | +| $L$ | $20\,mm$ per stack | Length of the stack | +| $A$ | $10^{-4}\,m^2$ | Area of the piezoelectric stack | +| $n$ | $160$ per stack | Number of layers in the piezoelectric stack | + +#+begin_src matlab +%% Estimate "Sensor Constant" - (THP5H) +d33 = 680e-12; % Strain constant [m/V] +n = 160; % Number of layers per stack +eT = 4500*8.854e-12; % Permittivity under constant stress [F/m] +sD = 21e-12; % Compliance under constant electric displacement [m2/N] + +gs = d33/(eT*sD*n); % Sensor Constant [V/m] + +%% Estimate "Actuator Constant" - (THP5H) +d33 = 680e-12; % Strain constant [m/V] +n = 320; % Number of layers + +cE = 1/sD; % Youngs modulus [N/m^2] +A = (10e-3)^2; % Area of the stacks [m^2] +L = 40e-3; % Length of the two stacks [m] +ka = cE*A/L; % Stiffness of the two stacks [N/m] + +ga = d33*n*ka; % Actuator Constant [N/V] +#+end_src + +** Comparison of the obtained dynamics + +The obtained dynamics using the /super element/ with the tuned "sensor gain" and "actuator gain" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf. + +A good match between the model and the experimental results is observed. +- the /super element/ + +This model represents fairly + +The flexible model is a bit "soft" as compared with the experimental results. + +This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators. + +#+begin_src matlab +%% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants" +% Initialize the APA +n_hexapod.actuator = initializeAPA(... + 'type', 'flexible', ... + 'ga', 23.2, ... % Actuator gain [N/V] + 'gs', -4.9e6); % Sensor gain [V/m] + +% Identify with updated constants +G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts); +G_flex.InputName = {'u'}; +G_flex.OutputName = {'Vs', 'de'}; +#+end_src #+begin_src matlab :exports none - freqs = logspace(2, 4, 5000); +%% Comparison of the measured FRF and the "Flexible" model of the APA300ML +freqs = 5*logspace(0, 3, 1000); +figure; +tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - figure; - hold on; - plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Amplitude'); - hold off; +ax1 = nexttile([2,1]); +hold on; +plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 1:length(apa_nums) + plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-8, 1e-3]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); + +ax1b = nexttile([2,1]); +hold on; +plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified'); +for i = 2:length(apa_nums) + plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off'); +end +plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); + +ax2 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]); +end +plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 180]); + +ax2b = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]); +end +plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:)) +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 180]); + +linkaxes([ax1,ax2,ax1b,ax2b],'x'); +xlim([10, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_resonance.pdf', 'width', 'wide', 'height', 'normal'); +exportFig('figs/test_apa_super_element_comp_frf.pdf', 'width', 'full', 'height', 'tall'); #+end_src -#+name: fig:apa300ml_resonance -#+caption: First resonance is around 800Hz +#+name: fig:test_apa_super_element_comp_frf +#+caption: Comparison of the measured FRF and the "Flexible" model of the APA300ML #+RESULTS: -[[file:figs/apa300ml_resonance.png]] +[[file:figs/test_apa_super_element_comp_frf.png]] -*** Amplification factor -The amplification factor is the ratio of the vertical displacement to the stack displacement. - -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% 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, '/z'], 1, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; - - G = linearize(mdl, io); -#+end_src - -The ratio of the two displacement is computed from the FEM model. -#+begin_src matlab :results replace value - abs(dcgain(G(1,1))./dcgain(G(2,1))) -#+end_src - -#+RESULTS: -: 5.0749 - -This is actually correct and approximately corresponds to the ratio of the piezo height and length: -#+begin_src matlab :results replace value - 75/15 -#+end_src - -#+RESULTS: -: 5 - -*** Stroke - -Estimation of the actuator stroke: -\[ \Delta H = A n \Delta L \] -with: -- $\Delta H$ Axial Stroke of the APA -- $A$ Amplification factor (5 for the APA300ML) -- $n$ Number of stack used -- $\Delta L$ Stroke of the stack (0.1% of its length) - -#+begin_src matlab :results replace value - 1e6 * 5 * 3 * 20e-3 * 0.1e-2 -#+end_src - -#+RESULTS: -: 300 - -This is exactly the specified stroke in the data-sheet. - -*** TODO Stroke BIS -- [ ] Identified the stroke form the transfer function from V to z - -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/V'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; - - G = linearize(mdl, io); - - 1e6*170*abs(dcgain(G)) -#+end_src - -** Identification of the Dynamics from actuator to replace displacement -We first set the mass to be approximately zero. -#+begin_src matlab :exports none - m = 0.01; -#+end_src - -The dynamics is identified from the applied force to the measured relative displacement. -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% 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, '/z'], 1, 'openoutput'); io_i = io_i + 1; - - Gh = -linearize(mdl, io); -#+end_src - -The same dynamics is identified for a payload mass of 10Kg. -#+begin_src matlab - m = 10; -#+end_src - -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% 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, '/z'], 1, 'openoutput'); io_i = io_i + 1; - - Ghm = -linearize(mdl, io); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(0, 4, 5000); - - figure; - tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile([2,1]); - hold on; - plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-'); - plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude'); set(gca, 'XTickLabel',[]); - hold off; - - ax2 = nexttile; - hold on; - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gh, freqs, 'Hz')))), '-', ... - 'DisplayName', '$m = 0kg$'); - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))), '-', ... - 'DisplayName', '$m = 10kg$'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - yticks(-360:90:360); - ylim([-360 0]); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - hold off; - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); - legend('location', 'southwest'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_plant_dynamics.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa300ml_plant_dynamics -#+caption: Transfer function from forces applied by the stack to the axial displacement of the APA -#+RESULTS: -[[file:figs/apa300ml_plant_dynamics.png]] - -The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure ref:fig:apa300ml_dvf_root_locus. -#+begin_src matlab :exports none - figure; - - gains = logspace(0, 5, 500); - - hold on; - plot(real(pole(Ghm)), imag(pole(G)), 'kx'); - plot(real(tzero(Ghm)), imag(tzero(G)), 'ko'); - for k = 1:length(gains) - cl_poles = pole(feedback(Ghm, gains(k)*s)); - plot(real(cl_poles), imag(cl_poles), 'k.'); - end - hold off; - axis square; - xlim([-500, 10]); ylim([0, 510]); - - xlabel('Real Part'); ylabel('Imaginary Part'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa300ml_dvf_root_locus -#+caption: Root Locus for Direct Velocity Feedback -#+RESULTS: -[[file:figs/apa300ml_dvf_root_locus.png]] - -** Identification of the Dynamics from actuator to force sensor -Let's use 2 stacks as a force sensor and 1 stack as force actuator. - -The transfer function from actuator voltage to sensor voltage is identified and shown in Figure ref:fig:apa300ml_iff_plant. -#+begin_src matlab :exports none - m = 10; -#+end_src - -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; - - Giff = -linearize(mdl, io); -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(0, 4, 5000); - - figure; - tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile([2,1]); - hold on; - plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude'); set(gca, 'XTickLabel',[]); - hold off; - - ax2 = nexttile; - hold on; - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - yticks(-360:90:360); - ylim([-180 180]); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - hold off; - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_iff_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa300ml_iff_plant -#+caption: Transfer function from actuator to force sensor -#+RESULTS: -[[file:figs/apa300ml_iff_plant.png]] - -For root locus corresponding to IFF is shown in Figure ref:fig:apa300ml_iff_root_locus. -#+begin_src matlab :exports none - figure; - - gains = logspace(0, 5, 500); - - hold on; - plot(real(pole(Giff)), imag(pole(Giff)), 'kx'); - plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko'); - for k = 1:length(gains) - cl_poles = pole(feedback(Giff, gains(k)/s)); - plot(real(cl_poles), imag(cl_poles), 'k.'); - end - hold off; - axis square; - xlim([-500, 10]); ylim([0, 510]); - - xlabel('Real Part'); ylabel('Imaginary Part'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:apa300ml_iff_root_locus -#+caption: Root Locus for IFF -#+RESULTS: -[[file:figs/apa300ml_iff_root_locus.png]] - -** Identification for a simpler model -The goal in this section is to identify the parameters of a simple APA model from the FEM. -This can be useful is a lower order model is to be used for simulations. - -The presented model is based on cite:souleille18_concep_activ_mount_space_applic. - -The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure ref:fig:souleille18_model_piezo). -The parameters are shown in the table below. - -#+name: fig:souleille18_model_piezo -#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator -[[file:./figs/souleille18_model_piezo.png]] - -#+caption:Parameters used for the model of the APA 100M -| | Meaning | -|-------+----------------------------------------------------------------| -| $k_e$ | Stiffness used to adjust the pole of the isolator | -| $k_1$ | Stiffness of the metallic suspension when the stack is removed | -| $k_a$ | Stiffness of the actuator | -| $c_1$ | Added viscous damping | - -The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model. - -\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] -\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] - -If we can fix $k_a$, we can determine $k_e$ and $k_1$ with: -\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \] -\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \] - -#+begin_src matlab :exports none - m = 10; -#+end_src - -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N] - io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m] - io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N] - io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m] - io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V] - io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m] - - G = linearize(mdl, io); - - G.InputName = {'Fd', 'w', 'Fa'}; - G.OutputName = {'y', 'Fs', 'd'}; -#+end_src - -From the identified dynamics, compute $\alpha$ and $\beta$ -#+begin_src matlab - alpha = abs(dcgain(G('y', 'Fa'))); - beta = abs(dcgain(G('y', 'Fd'))); -#+end_src - -$k_a$ is estimated using the following formula: -#+begin_src matlab - ka = 0.8/abs(dcgain(G('y', 'Fa'))); -#+end_src -The factor can be adjusted to better match the curves. - -Then $k_e$ and $k_1$ are computed. -#+begin_src matlab - ke = ka/(beta/alpha - 1); - k1 = 1/beta - ke*ka/(ke + ka); -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f '); -#+end_src - -#+RESULTS: -| | Value [N/um] | -|----+--------------| -| ka | 40.5 | -| ke | 1.5 | -| k1 | 0.4 | - -The damping in the system is adjusted to match the FEM model if necessary. -#+begin_src matlab - c1 = 1e2; -#+end_src - -The analytical model of the simpler system is defined below: -#+begin_src matlab - Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ... - [ 1 , k1 + c1*s + ke*ka/(ke + ka) , ke/(ke + ka) ; - -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke + ka)*(m*s^2 + c1*s + k1)]; - - Ga.InputName = {'Fd', 'w', 'Fa'}; - Ga.OutputName = {'y', 'Fs'}; -#+end_src - -And the DC gain is adjusted for the force sensor: -#+begin_src matlab - F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd')); -#+end_src - -The dynamics of the FEM model and the simpler model are compared in Figure ref:fig:apa300ml_comp_simpler_model. - -#+begin_src matlab :exports none - freqs = logspace(0, 5, 1000); - - figure; - tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('$x_1/w$ [m/m]'); - ylim([1e-6, 1e2]); - - ax2 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('$x_1/f$ [m/N]'); - ylim([1e-14, 1e-6]); - - ax3 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('$x_1/F$ [m/N]'); - ylim([1e-14, 1e-4]); - - ax4 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - ylabel('$F_s/w$ [m/m]'); - ylim([1e2, 1e8]); - - ax5 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - ylabel('$F_s/f$ [m/N]'); - ylim([1e-4, 1e1]); - - ax6 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - ylabel('$F_s/F$ [m/N]'); - ylim([1e-7, 1e2]); - - linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full'); -#+end_src - -#+name: fig:apa300ml_comp_simpler_model -#+caption: Comparison of the Dynamics between the FEM model and the simplified one -#+RESULTS: -[[file:figs/apa300ml_comp_simpler_model.png]] - -The simplified model has also been implemented in Simscape. - -The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure ref:fig:apa300ml_comp_simpler_simscape. -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML_simplified'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N] - io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m] - io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N] - io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m] - io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V] - - Gs = linearize(mdl, io); - - Gs.InputName = {'Fd', 'w', 'Fa'}; - Gs.OutputName = {'y', 'Fs'}; -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(0, 5, 1000); - - figure; - tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('$x_1/w$ [m/m]'); - ylim([1e-6, 1e2]); - - ax2 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('$x_1/f$ [m/N]'); - ylim([1e-14, 1e-6]); - - ax3 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); - ylabel('$x_1/F$ [m/N]'); - ylim([1e-14, 1e-4]); - - ax4 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'w'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - ylabel('$F_s/w$ [m/m]'); - ylim([1e2, 1e8]); - - ax5 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fa'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - ylabel('$F_s/f$ [m/N]'); - ylim([1e-4, 1e1]); - - ax6 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fd'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); - ylabel('$F_s/F$ [m/N]'); - ylim([1e-7, 1e2]); - - linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/apa300ml_comp_simpler_simscape.pdf', 'width', 'full', 'height', 'full'); -#+end_src - -#+name: fig:apa300ml_comp_simpler_simscape -#+caption: Comparison of the Dynamics between the FEM model and the simplified simscape model -#+RESULTS: -[[file:figs/apa300ml_comp_simpler_simscape.png]] - -** Integral Force Feedback -In this section, Integral Force Feedback control architecture is applied on the APA300ML. - -First, the plant (dynamics from voltage actuator to voltage sensor is identified). -#+begin_src matlab :exports none - Kiff = tf(0); -#+end_src - -The payload mass is set to 10kg. -#+begin_src matlab - m = 10; -#+end_src - -#+begin_src matlab :exports none - %% Name of the Simulink File - mdl = 'APA300ML_IFF'; - - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/APA300ML'], 1, 'openoutput'); io_i = io_i + 1; - - G_ol = linearize(mdl, io); - G_ol.InputName = {'w', 'f', 'F'}; - G_ol.OutputName = {'x1', 'Fs'}; - - G = G_ol({'Fs'}, {'f'}); -#+end_src - -The obtained dynamics is shown in Figure ref:fig:piezo_amplified_iff_plant. - -#+begin_src matlab :exports none - freqs = logspace(1, 5, 1000); - - figure; - tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile([2,1]); - hold on; - plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude'); set(gca, 'XTickLabel',[]); - hold off; - - ax2 = nexttile; - hold on; - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz'))))); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - yticks(-360:90:360); - ylim([-390 30]); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - hold off; - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/piezo_amplified_iff_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:piezo_amplified_iff_plant -#+caption: IFF Plant -#+RESULTS: -[[file:figs/piezo_amplified_iff_plant.png]] - -The controller is defined below and the loop gain is shown in Figure ref:fig:piezo_amplified_iff_loop_gain. -#+begin_src matlab - Kiff = -1e3/s; -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(1, 5, 1000); - - figure; - tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile([2,1]); - hold on; - plot(freqs, abs(squeeze(freqresp(G*Kiff, freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude'); set(gca, 'XTickLabel',[]); - hold off; - - ax2 = nexttile; - hold on; - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*Kiff, freqs, 'Hz'))))); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - yticks(-360:90:360); - ylim([-180 180]); - xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); - hold off; - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/piezo_amplified_iff_loop_gain.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:piezo_amplified_iff_loop_gain -#+caption: IFF Loop Gain -#+RESULTS: -[[file:figs/piezo_amplified_iff_loop_gain.png]] - -Now the closed-loop system is identified again and compare with the open loop system in Figure ref:fig:piezo_amplified_iff_comp. - -It is the expected behavior as shown in the Figure ref:fig:souleille18_results (from cite:souleille18_concep_activ_mount_space_applic). - -#+begin_src matlab :exports none - clear io; io_i = 1; - io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; - io(io_i) = linio([mdl, '/APA300ML'], 1, 'output'); io_i = io_i + 1; - - Giff = linearize(mdl, io); - Giff.InputName = {'w', 'f', 'F'}; - Giff.OutputName = {'x1', 'Fs'}; -#+end_src - -#+begin_src matlab :exports none - freqs = logspace(0, 3, 1000); - - figure; - tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - - ax1 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'w'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); ylabel('$x_1/w$ [m/m]') - - ax2 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'f'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); ylabel('$x_1/f$ [m/N]'); - - ax3 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G_ol('x1', 'F'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - set(gca, 'XTickLabel',[]); ylabel('$x_1/F$ [m/N]'); - - ax4 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'w'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('$F_s/w$ [N/m]'); - - ax5 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'f'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('$F_s/f$ [N/N]'); - - ax6 = nexttile; - hold on; - plot(freqs, abs(squeeze(freqresp(G_ol('Fs', 'F'), freqs, 'Hz')))); - plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('$F_s/F$ [N/N]'); - - linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/piezo_amplified_iff_comp.pdf', 'width', 'full', 'height', 'full'); -#+end_src - -#+name: fig:piezo_amplified_iff_comp -#+caption: OL and CL transfer functions -#+RESULTS: -[[file:figs/piezo_amplified_iff_comp.png]] - -#+name: fig:souleille18_results -#+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic -[[file:figs/souleille18_results.png]] +** Conclusion :ignore: * Conclusion <> +- Compare 2DoF and FEM models (usefulness of the two) +- Good match between all the APA: will simplify the modeling and control of the nano-hexapod +- No advantage of the FEM model here (as only uniaxial behavior is checked), but may be useful later + * Bibliography :ignore: #+latex: \printbibliography[heading=bibintoc,title={Bibliography}] @@ -3027,6 +1751,10 @@ addpath('./mat/'); % Path for data #+BEGIN_SRC matlab addpath('./matlab/STEPS/'); % Path for Simscape Model +%% Linearization options +opts = linearizeOptions; +opts.SampleTime = 0; + %% Open Simscape Model mdl = 'test_apa_simscape'; % Name of the Simulink File open(mdl); % Open Simscape Model @@ -3036,6 +1764,10 @@ open(mdl); % Open Simscape Model #+BEGIN_SRC matlab addpath('./STEPS/'); % Path for Simscape Model +%% Linearization options +opts = linearizeOptions; +opts.SampleTime = 0; + %% Open Simscape Model mdl = 'test_apa_simscape'; % Name of the Simulink File open(mdl); % Open Simscape Model @@ -3213,6 +1945,10 @@ actuator.cs = args.cs; % Damping of one stack [N/m] #+end_src * Footnotes + +[fn:7]Kistler 9722A +[fn:6]Polytec controller 3001 with sensor heads OFV512 +[fn:5]Note that this is not fully correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. 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