Analysis of the resonance frequencies of the APA

This commit is contained in:
Thomas Dehaeze 2021-05-06 16:17:50 +02:00
parent 69a11cfe94
commit c1cd73abe1
25 changed files with 1704 additions and 106 deletions

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@ -1612,5 +1612,872 @@ It is the expected behavior as shown in the Figure [[fig:souleille18_results]] (
#+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic
[[file:figs/souleille18_results.png]]
* Bibliography :ignore:
* Test Bench APA300ML - Simscape Model
** Introduction
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
addpath('matlab/test_bench_apa300ml/');
#+end_src
#+begin_src matlab :eval no
addpath('test_bench_apa300ml/');
#+end_src
#+begin_src matlab
open('test_bench_apa300ml.slx')
#+end_src
** Nano Hexapod object
#+begin_src matlab
n_hexapod = struct();
#+end_src
*** APA - 2 DoF
#+begin_src matlab
n_hexapod.actuator = struct();
n_hexapod.actuator.type = 1;
n_hexapod.actuator.k = ones(6,1)*0.35e6; % [N/m]
n_hexapod.actuator.ke = ones(6,1)*1.5e6; % [N/m]
n_hexapod.actuator.ka = ones(6,1)*43e6; % [N/m]
n_hexapod.actuator.c = ones(6,1)*3e1; % [N/(m/s)]
n_hexapod.actuator.ce = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.ca = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.Leq = ones(6,1)*0.056; % [m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
*** APA - Flexible Frame
#+begin_src matlab
n_hexapod.actuator.type = 2;
n_hexapod.actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
n_hexapod.actuator.ks = 235e6; % Stiffness of one stack [N/m]
n_hexapod.actuator.cs = 1e1; % Stiffness of one stack [N/m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
*** APA - Fully Flexible
#+begin_src matlab
n_hexapod.actuator.type = 3;
n_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
** Identification
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'test_bench_apa300ml';
%% 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, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Motion
%% Run the linearization
Ga = linearize(mdl, io, 0.0, options);
Ga.InputName = {'Va'};
Ga.OutputName = {'Vs', 'dL', 'z'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Ga('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', '')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('Vs', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Ga('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
plot(freqs, abs(squeeze(freqresp(Ga('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga('z', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
** Compare 2-DoF with flexible
*** APA - 2 DoF
#+begin_src matlab
n_hexapod = struct();
n_hexapod.actuator = struct();
n_hexapod.actuator.type = 1;
n_hexapod.actuator.k = ones(6,1)*0.35e6; % [N/m]
n_hexapod.actuator.ke = ones(6,1)*1.5e6; % [N/m]
n_hexapod.actuator.ka = ones(6,1)*43e6; % [N/m]
n_hexapod.actuator.c = ones(6,1)*3e1; % [N/(m/s)]
n_hexapod.actuator.ce = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.ca = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.Leq = ones(6,1)*0.056; % [m]
n_hexapod.actuator.Ga = ones(6,1)*-2.15; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*2.305e-08; % Sensor gain [V/m]
#+end_src
#+begin_src matlab
G_2dof = linearize(mdl, io, 0.0, options);
G_2dof.InputName = {'Va'};
G_2dof.OutputName = {'Vs', 'dL', 'z'};
#+end_src
*** APA - Fully Flexible
#+begin_src matlab
n_hexapod = struct();
n_hexapod.actuator.type = 3;
n_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
#+begin_src matlab
G_flex = linearize(mdl, io, 0.0, options);
G_flex.InputName = {'Va'};
G_flex.OutputName = {'Vs', 'dL', 'z'};
#+end_src
*** Comparison
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', '$G_a$')
plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', '$G_s$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('Vs', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_2dof('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', '$G_a$')
plot(freqs, abs(squeeze(freqresp(G_flex('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', '$G_s$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('dL', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_2dof('z', 'Va'), freqs, 'Hz'))), 'DisplayName', '$G_a$')
plot(freqs, abs(squeeze(freqresp(G_flex('z', 'Va'), freqs, 'Hz'))), 'DisplayName', '$G_s$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $z/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('z', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
* Test Bench Struts - Simscape Model
** Introduction
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
addpath('matlab/');
addpath('matlab/test_bench_struts/');
addpath('matlab/png/');
#+end_src
#+begin_src matlab :eval no
addpath('test_bench_struts/');
addpath('png/');
#+end_src
#+begin_src matlab
open('test_bench_struts.slx')
#+end_src
** Nano Hexapod object
#+begin_src matlab
n_hexapod = struct();
#+end_src
*** Flexible Joint - Bot
#+begin_src matlab
n_hexapod.flex_bot = struct();
n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_bot.kRx = ones(6,1)*5; % X bending stiffness [Nm/rad]
n_hexapod.flex_bot.kRy = ones(6,1)*5; % Y bending stiffness [Nm/rad]
n_hexapod.flex_bot.kRz = ones(6,1)*260; % Torsionnal stiffness [Nm/rad]
n_hexapod.flex_bot.kz = ones(6,1)*1e8; % Axial stiffness [N/m]
n_hexapod.flex_bot.cRx = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_bot.cRy = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_bot.cRz = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_bot.cz = ones(6,1)*1e2; %[N/(m/s)]
#+end_src
*** Flexible Joint - Top
#+begin_src matlab
n_hexapod.flex_top = struct();
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.kRx = ones(6,1)*5; % X bending stiffness [Nm/rad]
n_hexapod.flex_top.kRy = ones(6,1)*5; % Y bending stiffness [Nm/rad]
n_hexapod.flex_top.kRz = ones(6,1)*260; % Torsionnal stiffness [Nm/rad]
n_hexapod.flex_top.kz = ones(6,1)*1e8; % Axial stiffness [N/m]
n_hexapod.flex_top.cRx = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_top.cRy = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_top.cRz = ones(6,1)*0.1; % [Nm/(rad/s)]
n_hexapod.flex_top.cz = ones(6,1)*1e2; %[N/(m/s)]
#+end_src
*** APA - 2 DoF
#+begin_src matlab
n_hexapod.actuator = struct();
n_hexapod.actuator.type = 1;
n_hexapod.actuator.k = ones(6,1)*0.35e6; % [N/m]
n_hexapod.actuator.ke = ones(6,1)*1.5e6; % [N/m]
n_hexapod.actuator.ka = ones(6,1)*43e6; % [N/m]
n_hexapod.actuator.c = ones(6,1)*3e1; % [N/(m/s)]
n_hexapod.actuator.ce = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.ca = ones(6,1)*1e1; % [N/(m/s)]
n_hexapod.actuator.Leq = ones(6,1)*0.056; % [m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
*** APA - Flexible Frame
#+begin_src matlab
n_hexapod.actuator.type = 2;
n_hexapod.actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
n_hexapod.actuator.ks = 235e6; % Stiffness of one stack [N/m]
n_hexapod.actuator.cs = 1e1; % Stiffness of one stack [N/m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
*** APA - Fully Flexible
#+begin_src matlab
n_hexapod.actuator.type = 3;
n_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix
n_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix
n_hexapod.actuator.xi = 0.01; % Damping ratio
n_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m]
n_hexapod.actuator.Ga = ones(6,1)*1; % Actuator gain [N/V]
n_hexapod.actuator.Gs = ones(6,1)*1; % Sensor gain [V/m]
#+end_src
** Identification
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'test_bench_struts';
%% 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, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Motion
%% Run the linearization
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', '')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Encoder')
plot(freqs, abs(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Interferometer')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('z', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
** Compare flexible joints
*** Perfect
#+begin_src matlab
n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
#+end_src
#+begin_src matlab
Gp = linearize(mdl, io, 0.0, options);
Gp.InputName = {'Va'};
Gp.OutputName = {'Vs', 'dL', 'z'};
#+end_src
*** Top Flexible
#+begin_src matlab
n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
#+end_src
#+begin_src matlab
Gt = linearize(mdl, io, 0.0, options);
Gt.InputName = {'Va'};
Gt.OutputName = {'Vs', 'dL', 'z'};
#+end_src
*** Bottom Flexible
#+begin_src matlab
n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
#+end_src
#+begin_src matlab
Gb = linearize(mdl, io, 0.0, options);
Gb.InputName = {'Va'};
Gb.OutputName = {'Vs', 'dL', 'z'};
#+end_src
*** Both Flexible
#+begin_src matlab
n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
#+end_src
#+begin_src matlab
Gf = linearize(mdl, io, 0.0, options);
Gf.InputName = {'Va'};
Gf.OutputName = {'Vs', 'dL', 'z'};
#+end_src
*** Comparison
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gp('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Perfect')
plot(freqs, abs(squeeze(freqresp(Gt('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Top')
plot(freqs, abs(squeeze(freqresp(Gb('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Bot')
plot(freqs, abs(squeeze(freqresp(Gf('Vs', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flex')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('Vs', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vs', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gp('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Perfect')
plot(freqs, abs(squeeze(freqresp(Gt('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Top')
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Bot')
plot(freqs, abs(squeeze(freqresp(Gf('dL', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flex')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('dL', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('dL', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gp('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Perfect')
plot(freqs, abs(squeeze(freqresp(Gt('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Top')
plot(freqs, abs(squeeze(freqresp(Gb('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Bot')
plot(freqs, abs(squeeze(freqresp(Gf('z', 'Va'), freqs, 'Hz'))), 'DisplayName', 'Flex')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $z/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('z', 'Va'), freqs, 'Hz'))))
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('z', '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, 180])
linkaxes([ax1,ax2],'x');
#+end_src
* Resonance frequencies - APA300ML
** Introduction
Three main resonances are foreseen to be problematic for the control of the APA300ML:
- Mode in X-bending at 189Hz (Figure [[fig:mode_bending_x]])
- Mode in Y-bending at 285Hz (Figure [[fig:mode_bending_y]])
- Mode in Z-torsion at 400Hz (Figure [[fig:mode_torsion_z]])
#+name: fig:mode_bending_x
#+caption: X-bending mode (189Hz)
#+attr_latex: :width 0.9\linewidth
[[file:figs/mode_bending_x.gif]]
#+name: fig:mode_bending_y
#+caption: Y-bending mode (285Hz)
#+attr_latex: :width 0.9\linewidth
[[file:figs/mode_bending_y.gif]]
#+name: fig:mode_torsion_z
#+caption: Z-torsion mode (400Hz)
#+attr_latex: :width 0.9\linewidth
[[file:figs/mode_torsion_z.gif]]
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.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
addpath('matlab/');
addpath('matlab/mat/');
#+end_src
#+begin_src matlab :eval no
addpath('mat/');
#+end_src
** Setup
The measurement setup is shown in Figure [[fig:measurement_setup_torsion]].
A Laser vibrometer is measuring the difference of motion of 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
- Laser Doppler Vibrometer Polytec OFV512
- Instrumented hammer
#+end_note
#+name: fig:measurement_setup_torsion
#+caption: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_torsion.png]]
** Bending - X
The setup to measure the X-bending motion is shown in Figure [[fig:measurement_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:measurement_setup_X_bending
#+caption: X-Bending measurement setup
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_X_bending.png]]
The data is loaded.
#+begin_src matlab
bending_X = load('apa300ml_bending_X_top.mat')
#+end_src
#+begin_src matlab
Ts = bending_X.Track1_X_Resolution; % Sampling frequency [Hz]
#+end_src
The transfer function from the input force to the output "rotation" (difference between the two measured distances).
#+begin_src matlab
win = hann(ceil(1/Ts));
[G_bending_X, f] = tfestimate(bending_X.Track1, bending_X.Track2, win, [], [], 1/Ts);
#+end_src
The result is shown in Figure [[fig:apa300ml_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
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/apa300ml_meas_freq_bending_x.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_bending_x
#+caption: Obtained FRF for the X-bending
#+RESULTS:
[[file:figs/apa300ml_meas_freq_bending_x.png]]
** Bending - Y
The setup to measure the Y-bending is shown in Figure [[fig:measurement_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:measurement_setup_Y_bending
#+caption: Y-Bending measurement setup
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_Y_bending.png]]
The data is loaded, and the transfer function from the force to the measured rotation is computed.
#+begin_src matlab
bending_Y = load('apa300ml_bending_Y_top.mat')
[G_bending_Y, ~] = tfestimate(bending_Y.Track1, bending_Y.Track2, win, [], [], 1/Ts);
#+end_src
The results are shown in Figure [[fig:apa300ml_meas_freq_bending_y]].
The main resonance is at 412Hz, and we also see the third "harmonic" at 1220Hz.
#+begin_src matlab :exports none
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
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_bending_y.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_bending_y
#+caption: Obtained FRF for the Y-bending
#+RESULTS:
[[file:figs/apa300ml_meas_freq_bending_y.png]]
** Torsion - Z
Finally, we measure the Z-torsion resonance as shown in Figure [[fig:measurement_setup_torsion_bis]].
The excitation is shown on the other side of the APA, on the side to excite the torsion motion.
#+name: fig:measurement_setup_torsion_bis
#+caption: Z-Torsion measurement setup
#+attr_latex: :width 0.7\linewidth
[[file:figs/measurement_setup_torsion_bis.png]]
The data is loaded, and the transfer function computed.
#+begin_src matlab
torsion = load('apa300ml_torsion_left.mat')
[G_torsion, ~] = tfestimate(torsion_left.Track1, torsion_left.Track2, win, [], [], 1/Ts);
#+end_src
The results are shown in Figure [[fig:apa300ml_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.
The mode in pure torsion is probably at higher frequency (peak around 1kHz?).
#+begin_src matlab :exports none
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')
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/apa300ml_meas_freq_torsion_z.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:apa300ml_meas_freq_torsion_z
#+caption: Obtained FRF for the Z-torsion
#+RESULTS:
[[file:figs/apa300ml_meas_freq_torsion_z.png]]
** Compare
The three measurements are shown in Figure [[fig:apa300ml_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/apa300ml_meas_freq_compare.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:apa300ml_meas_freq_compare
#+caption: Obtained FRF - Comparison
#+RESULTS:
[[file:figs/apa300ml_meas_freq_compare.png]]
** Conclusion
#+name: tab:apa300ml_measured_modes_freq
#+caption: Measured frequency of the modes
#+attr_latex: :environment tabularx :width 0.3\linewidth :align lX
#+attr_latex: :center t :booktabs t :float t
| Mode | Measured Frequency |
|-----------+--------------------|
| X-Bending | 280Hz |
| Y-Bending | 410Hz |
| Z-Torsion | ? |
* Bibliography :ignore:
#+latex: \printbibliography