nass-simscape/active_damping/index.org

Active Damping applied on the Simscape Model

• Introduction
• Undamped System
  • Introduction
  • Identification of the dynamics for Active Damping
    • Initialize the Simulation
    • Identification
    • Obtained Plants for Active Damping
  • Tomography Experiment
    • Simulation
    • Results
• Variability of the system dynamics for Active Damping
  • Introduction
  • Variation of the Sample Mass
    • Introduction
    • Identification
    • Plots
  • Variation of the Spindle Angle
    • Introduction
    • Identification
    • Plots
  • Variation of the Spindle Rotation Speed
    • Introduction
    • Identification
    • Dynamics of the Active Damping plants
    • Variation of the poles and zeros with the Spindle rotation frequency
  • Variation of the Tilt Angle
    • Introduction
    • Identification
    • Plots
  • Scans of the Translation Stage
    • Introduction
    • Identification
    • Plots
  • Conclusion
• Integral Force Feedback
  • ZIP file containing the data and matlab files
  • Introduction
  • Control Design
    • Plant
    • Control Design
    • Diagonal Controller
    • IFF with High Pass Filter
  • Tomography Experiment
    • Simulation with IFF Controller
    • Simulation with IFF Controller with added High Pass Filter
    • Compare with Undamped system
  • Conclusion
• Direct Velocity Feedback
  • ZIP file containing the data and matlab files
  • Introduction
  • Control Design
    • Plant
    • Control Design
    • Diagonal Controller
  • Tomography Experiment
    • Initialize the Simulation
    • Simulation
    • Compare with Undamped system
  • Conclusion
• Inertial Control
  • ZIP file containing the data and matlab files
  • Introduction
  • Control Design
    • Plant
    • Control Design
    • Diagonal Controller
  • Tomography Experiment
    • Initialize the Simulation
    • Simulation
    • Compare with Undamped system
  • Conclusion
• Comparison
  • Introduction
  • Load the plants
  • Sensitivity to Disturbance
  • Damped Plant
  • Tomography Experiment
    • Load the Simulation Data
    • Frequency Domain Analysis
• Useful Functions
  • prepareTomographyExperiment
    • Function Description
    • Optional Parameters
    • Initialize the Simulation

Introduction   ignore

First, in section sec:undamped_system, we will looked at the undamped system.

Then, we will compare three active damping techniques:

• In section sec:iff: the integral force feedback is used
• In section sec:dvf: the direct velocity feedback is used
• In section sec:ine: inertial control is used

For each of the active damping technique, we will:

• Look at the damped plant
• Simulate tomography experiments
• Compare the sensitivity from disturbances

The disturbances are:

• Ground motion
• Motion errors of all the stages

Undamped System

<<sec:undamped_system>>

Introduction   ignore

We first look at the undamped system. The performance of this undamped system will be compared with the damped system using various techniques.

Identification of the dynamics for Active Damping

Initialize the Simulation

We initialize all the stages with the default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

  initializeNanoHexapod('actuator', 'piezo');
  initializeSample('mass', 50);

We set the references to zero.

  initializeReferences();

  initializeDisturbances('enable', false);

And all the controllers are set to 0.

  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_ine = tf(zeros(6));
  save('./mat/controllers.mat', 'K_ine', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');

Identification

First, we identify the dynamics of the system using the linearize function.

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

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fnl'],           1, 'openinput');              io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Vlm');  io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, 0.5, options);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                  'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                  'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'};

We then create transfer functions corresponding to the active damping plants.

  G_iff = minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
  G_dvf = minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
  G_ine = minreal(G({'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));

And we save them for further analysis.

  save('./active_damping/mat/undamped_plants.mat', 'G_iff', 'G_dvf', 'G_ine');

Obtained Plants for Active Damping

  load('./active_damping/mat/undamped_plants.mat', 'G_iff', 'G_dvf', 'G_ine');

<<plt-matlab>>

G_iff: IFF Plant (png, pdf)

<<plt-matlab>>

G_dvf: Plant for Direct Velocity Feedback (png, pdf)

<<plt-matlab>>

G_ine: Inertial Feedback Plant (png, pdf)

Tomography Experiment

Simulation

We initialize elements for the tomography experiment.

  prepareTomographyExperiment();

We change the simulation stop time.

  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

  sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

  save('./active_damping/mat/tomo_exp.mat', 'En', 'Eg', '-append');

Results

We load the results of tomography experiments.

  load('./active_damping/mat/tomo_exp.mat', 'En');
  t = linspace(0, 3, length(En(:,1)));

<<plt-matlab>>

Position Error during tomography experiment - Translations (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Rotations (png, pdf)

Variability of the system dynamics for Active Damping

<<sec:act_damp_variability_plant>>

Introduction   ignore

The goal of this section is to study how the dynamics of the Active Damping plants are changing with the experimental conditions. These experimental conditions are:

• The mass of the sample (section sec:variability_sample_mass)
• The spindle angle with a null rotating speed (section sec:variability_spindle_angle)
• The spindle rotation speed (section sec:variability_rotation_speed)
• The tilt angle (section sec:variability_tilt_angle)
• The scans of the translation stage (section sec:variability_ty_scans)

For the identification of the dynamics, the system is simulation for $\approx 0.5s$ before the linearization is performed. This is done in order for the transient phase to be over.

Variation of the Sample Mass

<<sec:variability_sample_mass>>

Introduction   ignore

For all the identifications, the disturbances are disabled and no controller are used.

Identification   ignore

We identify the dynamics for the following sample mass.

  masses = [1, 10, 50]; % [kg]

Plots   ignore

<<plt-matlab>>

Variability of the IFF plant with the Sample Mass (png, pdf)

<<plt-matlab>>

Variability of the DVF plant with the Sample Mass (png, pdf)

<<plt-matlab>>

Variability of the Inertial plant with the Sample Mass (png, pdf)

Variation of the Spindle Angle

<<sec:variability_spindle_angle>>

Introduction   ignore

Identification   ignore

We identify the dynamics for the following Spindle angles.

  Rz_amplitudes = [0, pi/4, pi/2, pi]; % [rad]

Plots   ignore

<<plt-matlab>>

Variability of the IFF plant with the Spindle Angle (png, pdf)

<<plt-matlab>>

Variability of the DVF plant with the Spindle Angle (png, pdf)

<<plt-matlab>>

Variability of the Inertial plant with the Spindle Angle (png, pdf)

Variation of the Spindle Rotation Speed

<<sec:variability_rotation_speed>>

Introduction   ignore

Identification   ignore

We identify the dynamics for the following Spindle rotation periods.

  Rz_periods = [60, 6, 2, 1]; % [s]

The identification of the dynamics is done at the same Spindle angle position.

Dynamics of the Active Damping plants

<<plt-matlab>>

Variability of the IFF plant with the Spindle rotation speed (png, pdf)

<<plt-matlab>>

Variability of the IFF plant with the Spindle rotation speed (png, pdf)

<<plt-matlab>>

Variability of the DVF plant with the Spindle rotation speed (png, pdf)

<<plt-matlab>>

Variability of the DVF plant with the Spindle rotation speed (png, pdf)

<<plt-matlab>>

Variability of the Inertial plant with the Spindle rotation speed (png, pdf)

<<plt-matlab>>

Variability of the Inertial plant with the Spindle rotation speed (png, pdf)

Variation of the poles and zeros with the Spindle rotation frequency

<<plt-matlab>>

Evolution of the pole with respect to the spindle rotation speed (png, pdf)

<<plt-matlab>>

Evolution of the zero with respect to the spindle rotation speed (png, pdf)

Variation of the Tilt Angle

<<sec:variability_tilt_angle>>

Introduction   ignore

Identification   ignore

We identify the dynamics for the following Tilt stage angles.

  Ry_amplitudes = [-3*pi/180, 3*pi/180]; % [rad]

Plots   ignore

<<plt-matlab>>

Variability of the IFF plant with the Tilt stage Angle (png, pdf)

<<plt-matlab>>

Variability of the DVF plant with the Tilt Angle (png, pdf)

<<plt-matlab>>

Variability of the Inertial plant with the Tilt Angle (png, pdf)

Scans of the Translation Stage

<<sec:variability_ty_scans>>

Introduction   ignore

We want here to verify if the dynamics used for Active damping is varying when using the translation stage for scans.

Identification   ignore

We initialize the translation stage reference to be a sinus with an amplitude of 5mm and a period of 1s (Figure fig:ty_scanning_reference_sinus).

  initializeReferences('Dy_type', 'sinusoidal', ...
                       'Dy_amplitude', 5e-3, ... % [m]
                       'Dy_period', 1); % [s]

<<plt-matlab>>

Reference path for the translation stage (png, pdf)

We identify the dynamics at different positions (times) when scanning with the Translation stage.

  t_lin = [0.5, 0.75, 1, 1.25];

Plots   ignore

<<plt-matlab>>

Variability of the IFF plant at different Ty scan positions (png, pdf)

<<plt-matlab>>

Variability of the DVF plant at different Ty scan positions (png, pdf)

<<plt-matlab>>

Variability of the Inertial plant at different Ty scan positions (png, pdf)

Conclusion

	Change of Dynamics
Sample Mass	Large
Spindle Angle	Small
Spindle Rotation Speed	First resonance is split in two resonances
Tilt Angle	Negligible
Translation Stage scans	Negligible

Conclusion on the variability of the system dynamics for active damping

Also, for the Inertial Sensor, a RHP zero may appear when the spindle is rotating fast.

When using either a force sensor or a relative motion sensor for active damping, the only "parameter" that induce a large change for the dynamics to be controlled is the sample mass. Thus, the developed damping techniques should be robust to variations of the sample mass.

Integral Force Feedback

<<sec:iff>>

ZIP file containing the data and matlab files   ignore

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

Introduction   ignore

Integral Force Feedback is applied on the simscape model.

Control Design

Plant

Let's load the previously indentified undamped plant:

  load('./active_damping/mat/undamped_plants.mat', 'G_iff');

Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure fig:iff_plant).

  <<plt-matlab>>

Transfer function from forces applied in the legs to force sensor (png, pdf)

Control Design

The controller for each pair of actuator/sensor is:

  K_iff = 1000/s;

The corresponding loop gains are shown in figure fig:iff_open_loop.

  <<plt-matlab>>

Loop Gain for the Integral Force Feedback (png, pdf)

Diagonal Controller

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

  K_iff = -K_iff*eye(6);

We save the controller for further analysis.

  save('./active_damping/mat/K_iff.mat', 'K_iff');

IFF with High Pass Filter

  w_hpf = 2*pi*10; % Cut-off frequency for the high pass filter [rad/s]

  K_iff = 2*pi*200/s * (s/w_hpf)/(s/w_hpf + 1);

The corresponding loop gains are shown in figure fig:iff_hpf_open_loop.

  <<plt-matlab>>

Loop Gain for the Integral Force Feedback with an High pass filter (png, pdf)

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

  K_iff = -K_iff*eye(6);

We save the controller for further analysis.

  save('./active_damping/mat/K_iff_hpf.mat', 'K_iff');

Tomography Experiment

Simulation with IFF Controller

We initialize elements for the tomography experiment.

  prepareTomographyExperiment();

We set the IFF controller.

  load('./active_damping/mat/K_iff.mat', 'K_iff');
  save('./mat/controllers.mat', 'K_iff', '-append');

We change the simulation stop time.

  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

  sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

  En_iff = En;
  Eg_iff = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_iff', 'Eg_iff', '-append');

Simulation with IFF Controller with added High Pass Filter

We initialize elements for the tomography experiment.

  prepareTomographyExperiment();

We set the IFF controller with the High Pass Filter.

  load('./active_damping/mat/K_iff_hpf.mat', 'K_iff');
  save('./mat/controllers.mat', 'K_iff', '-append');

We change the simulation stop time.

  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

  sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

  En_iff_hpf = En;
  Eg_iff_hpf = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_iff_hpf', 'Eg_iff_hpf', '-append');

Compare with Undamped system

We load the results of tomography experiments.

  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_iff', 'En_iff_hpf');
  t = linspace(0, 3, length(En(:,1)));

<<plt-matlab>>

Position Error during tomography experiment - XY Motion (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Translations (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Rotations (png, pdf)

Conclusion

Integral Force Feedback:

• Robust (guaranteed stability)
• Acceptable Damping
• Increase the sensitivity to disturbances at low frequencies

Direct Velocity Feedback

<<sec:dvf>>

ZIP file containing the data and matlab files   ignore

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

Introduction   ignore

In the Direct Velocity Feedback (DVF), a derivative feedback is applied between the measured actuator displacement to the actuator force input. The actuator displacement can be measured with a capacitive sensor for instance.

Control Design

Plant

Let's load the undamped plant:

  load('./active_damping/mat/undamped_plants.mat', 'G_dvf');

Let's look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure fig:dvf_plant).

  <<plt-matlab>>

Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

Control Design

The Direct Velocity Feedback is defined below. A Low pass Filter is added to make the controller transfer function proper.

  K_dvf = s*20000/(1 + s/2/pi/10000);

The obtained loop gains are shown in figure fig:dvf_open_loop.

  <<plt-matlab>>

Loop Gain for the Integral Force Feedback (png, pdf)

Diagonal Controller

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

  K_dvf = -K_dvf*eye(6);

We save the controller for further analysis.

  save('./active_damping/mat/K_dvf.mat', 'K_dvf');

Tomography Experiment

Initialize the Simulation

We initialize elements for the tomography experiment.

  prepareTomographyExperiment();

We set the DVF controller.

  load('./active_damping/mat/K_dvf.mat', 'K_dvf');
  save('./mat/controllers.mat', 'K_dvf', '-append');

Simulation

We change the simulation stop time.

  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

  sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

  En_dvf = En;
  Eg_dvf = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_dvf', 'Eg_dvf', '-append');

Compare with Undamped system

We load the results of tomography experiments.

  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_dvf');
  t = linspace(0, 3, length(En(:,1)));

<<plt-matlab>>

Position Error during tomography experiment - XY Motion (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Translations (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Rotations (png, pdf)

Conclusion

Direct Velocity Feedback:

Inertial Control

<<sec:ine>>

ZIP file containing the data and matlab files   ignore

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

Introduction   ignore

In Inertial Control, a feedback is applied between the measured absolute motion (velocity or acceleration) of the platform to the actuator force input.

Control Design

Plant

Let's load the undamped plant:

  load('./active_damping/mat/undamped_plants.mat', 'G_ine');

Let's look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure fig:ine_plant).

  <<plt-matlab>>

Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

Control Design

The controller is defined below and the obtained loop gain is shown in figure fig:ine_open_loop_gain.

  K_ine = 1e4/(1+s/(2*pi*100));

  <<plt-matlab>>

Loop Gain for Inertial Control (png, pdf)

Diagonal Controller

We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.

  K_ine = -K_ine*eye(6);

We save the controller for further analysis.

  save('./active_damping/mat/K_ine.mat', 'K_ine');

Tomography Experiment

Initialize the Simulation

We initialize elements for the tomography experiment.

  prepareTomographyExperiment();

We set the Inertial controller.

  load('./active_damping/mat/K_ine.mat', 'K_ine');
  save('./mat/controllers.mat', 'K_ine', '-append');

Simulation

We change the simulation stop time.

  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');

And we simulate the system.

  sim('sim_nass_active_damping');

Finally, we save the simulation results for further analysis

  En_ine = En;
  Eg_ine = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_ine', 'Eg_ine', '-append');

Compare with Undamped system

We load the results of tomography experiments.

  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_ine');
  t = linspace(0, 3, length(En_ine(:,1)));

<<plt-matlab>>

Position Error during tomography experiment - XY Motion (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Translations (png, pdf)

<<plt-matlab>>

Position Error during tomography experiment - Rotations (png, pdf)

Conclusion

Inertial Control:

Comparison

<<sec:comparison>>

Introduction   ignore

Load the plants

  load('./active_damping/mat/plants.mat', 'G', 'G_iff', 'G_ine', 'G_dvf');

Sensitivity to Disturbance

  <<plt-matlab>>

Sensitivity to ground motion in the Z direction on the Z motion error (png, pdf)

  <<plt-matlab>>

Compliance in the Z direction: Sensitivity of direct forces applied on the sample in the Z direction on the Z motion error (png, pdf)

  <<plt-matlab>>

Sensitivity to forces applied in the Z direction by the Spindle on the Z motion error (png, pdf)

  <<plt-matlab>>

Sensitivity to forces applied in the Z direction by the Y translation stage on the Z motion error (png, pdf)

  <<plt-matlab>>

Sensitivity to forces applied in the X direction by the Y translation stage on the X motion error (png, pdf)

Damped Plant

  <<plt-matlab>>

Plant for the $z$ direction for different active damping technique used (png, pdf)

  <<plt-matlab>>

Plant for the $x$ direction for different active damping technique used (png, pdf)

  <<plt-matlab>>

Comparison of one off-diagonal plant for different damping technique applied (png, pdf)

Tomography Experiment

Load the Simulation Data

  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_iff_hpf', 'En_dvf', 'En_ine');
  En_iff = En_iff_hpf;
  t = linspace(0, 3, length(En(:,1)));

Frequency Domain Analysis

Window used for pwelch function.

  n_av = 8;
  han_win = hanning(ceil(length(En(:, 1))/n_av));

<<plt-matlab>>

PSD of the translation errors in the X direction for applied Active Damping techniques (png, pdf)

<<plt-matlab>>

PSD of the rotation errors in the X direction for applied Active Damping techniques (png, pdf)

<<plt-matlab>>

CPS of the translation errors in the X direction for applied Active Damping techniques (png, pdf)

<<plt-matlab>>

CPS of the rotation errors in the X direction for applied Active Damping techniques (png, pdf)

Useful Functions

prepareTomographyExperiment

<<sec:prepareTomographyExperiment>>

This Matlab function is accessible here.

Function Description

  function [] = prepareTomographyExperiment(args)

Optional Parameters

  arguments
      args.nass_actuator       char   {mustBeMember(args.nass_actuator,{'piezo', 'lorentz'})} = 'piezo'
      args.sample_mass   (1,1) double {mustBeNumeric, mustBePositive} = 50
      args.Ry_period     (1,1) double {mustBeNumeric, mustBePositive} = 1
  end

Initialize the Simulation

We initialize all the stages with the default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

  initializeNanoHexapod('actuator', args.nass_actuator);
  initializeSample('mass', args.sample_mass);

We set the references to zero.

  initializeReferences('Rz_type', 'rotating', 'Rz_period', args.Ry_period);

And all the controllers are set to 0.

  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_ine = tf(zeros(6));
  save('./mat/controllers.mat', 'K_ine', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');