204 lines
8.7 KiB
Matlab
204 lines
8.7 KiB
Matlab
%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Path for functions, data and scripts
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addpath('./src/'); % Path for scripts
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addpath('./mat/'); % Path for data
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addpath('./STEPS/'); % Path for Simscape Model
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%% Linearization options
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opts = linearizeOptions;
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opts.SampleTime = 0;
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%% Open Simscape Model
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mdl = 'test_apa_simscape'; % Name of the Simulink File
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open(mdl); % Open Simscape Model
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%% Colors for the figures
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colors = colororder;
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%% Input/Output definition of the Model
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage
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io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
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io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
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% Identification of the Actuator and Sensor constants
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% <<ssec:test_apa_flexible_ga_gs>>
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% 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.
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% The gains $g_a$ and $g_s$ can then be tuned such that the gain of the transfer functions are matching the identified ones.
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% By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained.
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%% Identification of the actuator and sensor "constants"
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% Initialize the APA as a flexible body with unity "constants"
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n_hexapod.actuator = initializeAPA(...
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'type', 'flexible', ...
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'ga', 1, ...
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'gs', 1);
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c_granite = 100; % Rought estimation of the damping added by the air bearing
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% Identify the dynamics
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G_norm = linearize(mdl, io, 0.0, opts);
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G_norm.InputName = {'u'};
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G_norm.OutputName = {'Vs', 'de'};
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% Load Identification Data to estimate the two gains
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load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
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% Actuator Constant in [N/V]
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ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u'));
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% Sensor Constant in [V/m]
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gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz'))));
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% To make sure these "gains" are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
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% 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.
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% \begin{subequations}
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% \begin{align}
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% V_s &= \underbrace{\frac{d_{33}}{\epsilon^T s^D n}}_{g_s} d_L \label{eq:test_apa_piezo_strain_to_voltage} \\
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% 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}
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% \end{align}
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% \end{subequations}
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% 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.
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% Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
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% 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.
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% The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties.
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% 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.
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% #+name: tab:test_apa_piezo_properties
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% #+caption: Piezoelectric properties used for the estimation of the sensor and actuators "gains"
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% #+attr_latex: :environment tabularx :width 1\linewidth :align ccX
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% #+attr_latex: :center t :booktabs t
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% | *Parameter* | *Value* | *Description* |
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% |----------------+----------------------------+--------------------------------------------------------------|
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% | $d_{33}$ | $680 \cdot 10^{-12}\,m/V$ | Piezoelectric constant |
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% | $\epsilon^{T}$ | $4.0 \cdot 10^{-8}\,F/m$ | Permittivity under constant stress |
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% | $s^{D}$ | $21 \cdot 10^{-12}\,m^2/N$ | Elastic compliance understand constant electric displacement |
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% | $c^{E}$ | $48 \cdot 10^{9}\,N/m^2$ | Young's modulus of elasticity |
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% | $L$ | $20\,mm$ per stack | Length of the stack |
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% | $A$ | $10^{-4}\,m^2$ | Area of the piezoelectric stack |
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% | $n$ | $160$ per stack | Number of layers in the piezoelectric stack |
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%% Estimate "Sensor Constant" - (THP5H)
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d33 = 680e-12; % Strain constant [m/V]
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n = 160; % Number of layers per stack
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eT = 4500*8.854e-12; % Permittivity under constant stress [F/m]
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sD = 21e-12; % Compliance under constant electric displacement [m2/N]
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gs = d33/(eT*sD*n); % Sensor Constant [V/m]
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%% Estimate "Actuator Constant" - (THP5H)
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d33 = 680e-12; % Strain constant [m/V]
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n = 320; % Number of layers
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cE = 1/sD; % Youngs modulus [N/m^2]
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A = (10e-3)^2; % Area of the stacks [m^2]
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L = 40e-3; % Length of the two stacks [m]
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ka = cE*A/L; % Stiffness of the two stacks [N/m]
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ga = d33*n*ka; % Actuator Constant [N/V]
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% Comparison of the obtained dynamics
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% <<ssec:test_apa_flexible_comp_frf>>
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% 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.
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% A good match between the model and the experimental results is observed.
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% - the /super element/
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% This model represents fairly
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% The flexible model is a bit "soft" as compared with the experimental results.
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% This method can be used to model piezoelectric stack actuators as well as amplified piezoelectric stack actuators.
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%% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants"
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% Initialize the APA
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n_hexapod.actuator = initializeAPA(...
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'type', 'flexible', ...
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'ga', 23.2, ... % Actuator gain [N/V]
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'gs', -4.9e6); % Sensor gain [V/m]
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% Identify with updated constants
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G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts);
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G_flex.InputName = {'u'};
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G_flex.OutputName = {'Vs', 'de'};
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%% Comparison of the measured FRF and the "Flexible" model of the APA300ML
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freqs = 5*logspace(0, 3, 1000);
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figure;
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tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
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for i = 1:length(apa_nums)
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plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-8, 1e-3]);
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legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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ax1b = nexttile([2,1]);
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hold on;
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plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
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for i = 2:length(apa_nums)
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plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-2, 1e2]);
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legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
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ax2 = nexttile;
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hold on;
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for i = 1:length(apa_nums)
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plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
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end
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360); ylim([-180, 180]);
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ax2b = nexttile;
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hold on;
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for i = 1:length(apa_nums)
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plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]);
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end
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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yticks(-360:90:360); ylim([-180, 180]);
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linkaxes([ax1,ax2,ax1b,ax2b],'x');
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xlim([10, 2e3]);
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