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Identification

Introduction   ignore

The goal here is to make an identification of the micro-station in order to compare the model with the measurements on the real micro-station.

In order to do so:

  • Decide where to virtually excite the station and where to measure its motion
  • Extract transfer functions from the excitation forces to the measured motion
  • Compare those transfer functions with the modal analysis

For the excitation, we can choose the same excitation points as the one used for the modal test. For the measurement points, we can choose the Center of Mass of each solid body. The center of mass of each solid body is not easily defined using Simscape. Indeed, we can define the center of mass of any solid body but not of multiple solid bodies. However, one solid body is composed of multiple STEP files. One solution could be to use one STEP file for one solid body. However, the position of the center of mass can be exported using simulink and then defined on Simscape.

Identification of the Micro-Station

Introduction   ignore

Compute the transfer functions

We first define some parameters for the identification. The simulink file for the identification is sim_micro_station_id.slx.

  open 'simscape/sim_micro_station_id.slx'
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'sim_micro_station_id';
%% Micro-Hexapod
% Input/Output definition
io(1) = linio([mdl, '/Micro-Station/Fm_ext'],1,'openinput');
io(2) = linio([mdl, '/Micro-Station/Fg_ext'],1,'openinput');
io(3) = linio([mdl, '/Micro-Station/Dm_inertial'],1,'output');
io(4) = linio([mdl, '/Micro-Station/Ty_inertial'],1,'output');
io(5) = linio([mdl, '/Micro-Station/Ry_inertial'],1,'output');
io(6) = linio([mdl, '/Micro-Station/Dg_inertial'],1,'output');
% Run the linearization
G_ms = linearize(mdl, io, 0);

% Input/Output names
G_ms.InputName  = {'Fmx', 'Fmy', 'Fmz',...
                   'Fgx', 'Fgy', 'Fgz'};
G_ms.OutputName = {'Dmx', 'Dmy', 'Dmz', ...
                   'Tyx', 'Tyy', 'Tyz', ...
                   'Ryx', 'Ryy', 'Ryz', ...
                   'Dgx', 'Dgy', 'Dgz'};
%% Save the obtained transfer functions
save('./mat/id_micro_station.mat', 'G_ms');

Plots the transfer functions

Compare with the measurements

Modal Analysis of the Micro-Station

Simscape Model 

  open 'simscape/sim_micro_station_modal_analysis.slx'
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'sim_micro_station_modal_analysis';
%% Micro-Hexapod
% Input/Output definition
io(1) = linio([mdl, '/Micro-Station/F_hammer'],1,'openinput');
io(2) = linio([mdl, '/Micro-Station/acc9'],1,'output');
io(3) = linio([mdl, '/Micro-Station/acc10'],1,'output');
io(4) = linio([mdl, '/Micro-Station/acc11'],1,'output');
io(5) = linio([mdl, '/Micro-Station/acc12'],1,'output');
  % Run the linearization
  G_ms = linearize(mdl, io, 0);

  % Input/Output names
  G_ms.InputName  = {'Fx', 'Fy', 'Fz'};
  G_ms.OutputName = {'x9', 'y9', 'z9', ...
                     'x10', 'y10', 'z10', ...
                     'x11', 'y11', 'z11', ...
                     'x12', 'y12', 'z12'};

Plot Results 

figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_ms('x9', 'Fx'), freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');
hold off;

Compare with measurements 

  load('../meas/modal-analysis/mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
  dirs = {'x', 'y', 'z'};

  n_acc = 9;
  n_dir = 1; % x, y, z
  n_exc = 1; % x, y, z

  figure;
  hold on;
  plot(freqs, abs(squeeze(FRFs(3*(n_acc-1) + n_dir, n_exc, :)))./((2*pi*freqs).^2)');
  plot(freqs, abs(squeeze(freqresp(G_ms([dirs{n_dir}, num2str(n_acc)], ['F', dirs{n_dir}]), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]');
  hold off;

Compare with measurements at the CoM of each element

Init 

%% Initialize Ground
initializeGround();

%% Initialize Granite
initializeGranite();

%% Initialize Translation stage
initializeTy();

%% Initialize Tilt Stage
initializeRy();

%% Initialize Spindle
initializeRz();

%% Initialize Hexapod Symétrie
initializeMicroHexapod();

%% Initialize Center of gravity compensation
initializeAxisc();

Center of Mass of each solid body

granite bot granite top ty ry rz hexa
X [mm] 45 52 0 0 0 -4
Y [mm] 144 258 14 -5 0 6
Z [mm] -1251 -778 -600 -628 -580 -319 
  open 'simscape/sim_micro_station_modal_analysis_com.slx'

Simscape Model 

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_micro_station_modal_analysis_com';
  %% Micro-Hexapod
  % Input/Output definition
  io(1) = linio([mdl, '/Micro-Station/F_hammer'],1,'openinput');
  io(2) = linio([mdl, '/Micro-Station/acc_gtop'],1,'output');
  io(3) = linio([mdl, '/Micro-Station/acc_ty'],1,'output');
  io(4) = linio([mdl, '/Micro-Station/acc_ry'],1,'output');
  io(5) = linio([mdl, '/Micro-Station/acc_rz'],1,'output');
  io(6) = linio([mdl, '/Micro-Station/acc_hexa'],1,'output');
  % Run the linearization
  G_ms = linearize(mdl, io, 0);

  % Input/Output names
  G_ms.InputName  = {'Fx', 'Fy', 'Fz'};
  G_ms.OutputName = {'gtop_x', 'gtop_y', 'gtop_z', 'gtop_rx', 'gtop_ry', 'gtop_rz', ...
                     'ty_x', 'ty_y', 'ty_z', 'ty_rx', 'ty_ry', 'ty_rz', ...
                     'ry_x', 'ry_y', 'ry_z', 'ry_rx', 'ry_ry', 'ry_rz', ...
                     'rz_x', 'rz_y', 'rz_z', 'rz_rx', 'rz_ry', 'rz_rz', ...
                     'hexa_x', 'hexa_y', 'hexa_z', 'hexa_rx', 'hexa_ry', 'hexa_rz'};

Compare with measurements 

  load('../meas/modal-analysis/mat/frf_coh_matrices.mat', 'freqs');
  load('../meas/modal-analysis/mat/frf_com.mat', 'FRFs_CoM');
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'gbot', 'gtop', 'ty', 'ry', 'rz', 'hexa'}

  n_stg = 2;
  n_dir = 2; % x, y, z, Rx, Ry, Rz
  n_exc = 2; % x, y, z

  f = logspace(1, 3, 1000);

  figure;
  hold on;
  plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg-1) + n_dir, n_exc, :)))./((2*pi*freqs).^2)');
  plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_exc}]), f, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]');
  hold off;
  xlim([10, 1000]);
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'gtop', 'ty', 'ry', 'rz', 'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1:2
    for n_dir = 1:3
      subplot(3, 2, (n_dir-1)*2 + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'ry', 'rz', 'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1:2
    for n_dir = 1:3
      subplot(3, 2, (n_dir-1)*2 + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg+2) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1
    for n_dir = 1:3
      subplot(3, 2, (n_dir-1)*2 + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg+4) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end

Bis 

  %% Initialize Ground
  initializeGround();

  %% Initialize Granite
  initializeGranite(struct('rigid', false));

  %% Initialize Translation stage
  initializeTy(struct('rigid', false));

  %% Initialize Tilt Stage
  initializeRy(struct('rigid', false));

  %% Initialize Spindle
  initializeRz(struct('rigid', false));

  %% Initialize Hexapod Symétrie
  initializeMicroHexapod(struct('rigid', false));

  %% Initialize Center of gravity compensation
  initializeAxisc();

For Granite Fx to Tx/Ty/Tz/Rx/Ry/Rz

  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'gtop', 'ty', 'ry', 'rz', 'hexa'}

  n_stg = 5;
  n_exc = 3;

  f = logspace(0, 3, 1000);

  figure;
  for n_dir = 1:6
    subplot(2, 3, n_dir);
    title(['F', dirs{n_exc}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
    hold on;
    plot(freqs, abs(squeeze(FRFs_CoM(6*(n_stg) + n_dir, n_exc, :)))./((2*pi*freqs).^2)');
    plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_exc}]), f, 'Hz'))));
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    ylabel('Amplitude [m/N]');
    hold off;
    xlim([10, 1000]);
    ylim([1e-12, 1e-6]);
  end

ZIP file containing the data and matlab files   ignore

All the files (data and Matlab scripts) are accessible here.

Identification of the micro-station 

  simulinkproject('../');
  open sim_micro_station_id.slx

Plot the obtained transfer functions

Compare with the modal measurements

Modal Identification of the micro station