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Active Damping applied on the Simscape Model

Table of Contents

First, in section 1, we will looked at the undamped system.

Then, we will compare three active damping techniques:

For each of the active damping technique, we will:

The disturbances are:

1 Undamped System

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

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

1.1 Identification of the dynamics for Active Damping

1.1.1 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('Rz_type', 'rotating', 'Ry_period', 1);

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');

1.1.2 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, 1, 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');

1.1.3 Obtained Plants for Active Damping

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

nass_active_damping_iff_plant.png

Figure 1: G_iff: IFF Plant (png, pdf)

nass_active_damping_ine_plant.png

Figure 2: G_dvf: Plant for Direct Velocity Feedback (png, pdf)

nass_active_damping_inertial_plant.png

Figure 3: G_ine: Inertial Feedback Plant (png, pdf)

1.2 Tomography Experiment

1.2.1 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');

1.2.2 Results

We load the results of tomography experiments.

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

nass_act_damp_undamped_sim_tomo_trans.png

Figure 4: Position Error during tomography experiment - Translations (png, pdf)

nass_act_damp_undamped_sim_tomo_rot.png

Figure 5: Position Error during tomography experiment - Rotations (png, pdf)

2 Integral Force Feedback

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

Integral Force Feedback is applied on the simscape model.

2.1 Control Design

2.1.1 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 6).

iff_plant.png

Figure 6: Transfer function from forces applied in the legs to force sensor (png, pdf)

2.1.2 Control Design

The controller for each pair of actuator/sensor is:

K_iff = 1000/s;

The corresponding loop gains are shown in figure 7.

iff_open_loop.png

Figure 7: Loop Gain for the Integral Force Feedback (png, pdf)

2.1.3 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');

2.1.4 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 8.

iff_hpf_open_loop.png

Figure 8: 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');

2.2 Tomography Experiment

2.2.1 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');

2.2.2 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');

2.2.3 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)));

nass_act_damp_iff_sim_tomo_xy.png

Figure 9: Position Error during tomography experiment - XY Motion (png, pdf)

nass_act_damp_iff_sim_tomo_trans.png

Figure 10: Position Error during tomography experiment - Translations (png, pdf)

nass_act_damp_iff_sim_tomo_rot.png

Figure 11: Position Error during tomography experiment - Rotations (png, pdf)

2.3 Conclusion

Integral Force Feedback:

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

3 Direct Velocity Feedback

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

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.

3.1 Control Design

3.1.1 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 12).

dvf_plant.png

Figure 12: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

3.1.2 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 13.

dvf_open_loop.png

Figure 13: Loop Gain for the Integral Force Feedback (png, pdf)

3.1.3 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');

3.2 Tomography Experiment

3.2.1 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');

3.2.2 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');

3.2.3 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)));

nass_act_damp_dvf_sim_tomo_xy.png

Figure 14: Position Error during tomography experiment - XY Motion (png, pdf)

nass_act_damp_dvf_sim_tomo_trans.png

Figure 15: Position Error during tomography experiment - Translations (png, pdf)

nass_act_damp_dvf_sim_tomo_rot.png

Figure 16: Position Error during tomography experiment - Rotations (png, pdf)

3.3 Conclusion

Direct Velocity Feedback:

4 Inertial Control

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

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

4.1 Control Design

4.1.1 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 17).

ine_plant.png

Figure 17: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

4.1.2 Control Design

The controller is defined below and the obtained loop gain is shown in figure 18.

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

ine_open_loop_gain.png

Figure 18: Loop Gain for Inertial Control (png, pdf)

4.1.3 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');

4.2 Tomography Experiment

4.2.1 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');

4.2.2 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');

4.2.3 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)));

nass_act_damp_ine_sim_tomo_xy.png

Figure 19: Position Error during tomography experiment - XY Motion (png, pdf)

nass_act_damp_ine_sim_tomo_trans.png

Figure 20: Position Error during tomography experiment - Translations (png, pdf)

nass_act_damp_ine_sim_tomo_rot.png

Figure 21: Position Error during tomography experiment - Rotations (png, pdf)

4.3 Conclusion

Inertial Control:

5 Variability of the system dynamics for Active Damping

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

5.1 Variation of the Sample Mass

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

We identify the dynamics for the following sample mass.

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

act_damp_variability_iff_sample_mass.png

Figure 22: Variability of the IFF plant with the Sample Mass (png, pdf)

act_damp_variability_dvf_sample_mass.png

Figure 23: Variability of the DVF plant with the Sample Mass (png, pdf)

act_damp_variability_ine_sample_mass.png

Figure 24: Variability of the Inertial plant with the Sample Mass (png, pdf)

5.2 Variation of the Spindle Angle

We identify the dynamics for the following Spindle angles.

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

act_damp_variability_iff_spindle_angle.png

Figure 25: Variability of the IFF plant with the Spindle Angle (png, pdf)

act_damp_variability_dvf_spindle_angle.png

Figure 26: Variability of the DVF plant with the Spindle Angle (png, pdf)

act_damp_variability_ine_spindle_angle.png

Figure 27: Variability of the Inertial plant with the Spindle Angle (png, pdf)

5.3 Variation of the Spindle Rotation Speed

We identify the dynamics for the following Spindle rotation periods.

Rz_periods = [60, 10, 1]; % [s]

act_damp_variability_iff_spindle_speed.png

Figure 28: Variability of the IFF plant with the Spindle rotation speed (png, pdf)

act_damp_variability_iff_spindle_speed_zoom.png

Figure 29: Variability of the IFF plant with the Spindle rotation speed (png, pdf)

act_damp_variability_dvf_spindle_speed.png

Figure 30: Variability of the DVF plant with the Spindle rotation speed (png, pdf)

act_damp_variability_dvf_spindle_speed_zoom.png

Figure 31: Variability of the DVF plant with the Spindle rotation speed (png, pdf)

act_damp_variability_ine_spindle_speed.png

Figure 32: Variability of the Inertial plant with the Spindle rotation speed (png, pdf)

act_damp_variability_ine_spindle_speed_zoom.png

Figure 33: Variability of the Inertial plant with the Spindle rotation speed (png, pdf)

5.4 Variation of the Tilt Angle

We identify the dynamics for the following Tilt stage angles.

Ry_amplitudes = [0, 3*pi/180]; % [rad]

act_damp_variability_iff_tilt_angle.png

Figure 34: Variability of the IFF plant with the Spindle Angle (png, pdf)

act_damp_variability_dvf_tilt_angle.png

Figure 35: Variability of the DVF plant with the Spindle Angle (png, pdf)

act_damp_variability_ine_tilt_angle.png

Figure 36: Variability of the Inertial plant with the Spindle Angle (png, pdf)

5.5 Conclusion

6 Comparison

6.1 Load the plants

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

6.2 Sensitivity to Disturbance

sensitivity_comp_ground_motion_z.png

Figure 37: Sensitivity to ground motion in the Z direction on the Z motion error (png, pdf)

sensitivity_comp_direct_forces_z.png

Figure 38: Compliance in the Z direction: Sensitivity of direct forces applied on the sample in the Z direction on the Z motion error (png, pdf)

sensitivity_comp_spindle_z.png

Figure 39: Sensitivity to forces applied in the Z direction by the Spindle on the Z motion error (png, pdf)

sensitivity_comp_ty_z.png

Figure 40: Sensitivity to forces applied in the Z direction by the Y translation stage on the Z motion error (png, pdf)

sensitivity_comp_ty_x.png

Figure 41: Sensitivity to forces applied in the X direction by the Y translation stage on the X motion error (png, pdf)

6.3 Damped Plant

plant_comp_damping_z.png

Figure 42: Plant for the \(z\) direction for different active damping technique used (png, pdf)

plant_comp_damping_x.png

Figure 43: Plant for the \(x\) direction for different active damping technique used (png, pdf)

plant_comp_damping_coupling.png

Figure 44: Comparison of one off-diagonal plant for different damping technique applied (png, pdf)

6.4 Tomography Experiment

6.4.1 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)));

6.4.2 Frequency Domain Analysis

Window used for pwelch function.

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

act_damp_tomo_exp_comp_psd_trans.png

Figure 45: PSD of the translation errors in the X direction for applied Active Damping techniques (png, pdf)

act_damp_tomo_exp_comp_psd_rot.png

Figure 46: PSD of the rotation errors in the X direction for applied Active Damping techniques (png, pdf)

act_damp_tomo_exp_comp_cps_trans.png

Figure 47: CPS of the translation errors in the X direction for applied Active Damping techniques (png, pdf)

act_damp_tomo_exp_comp_cps_rot.png

Figure 48: CPS of the rotation errors in the X direction for applied Active Damping techniques (png, pdf)

7 Useful Functions

7.1 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');

Author: Dehaeze Thomas

Created: 2020-01-21 mar. 17:28