Active Damping applied on the Simscape Model

1. Undamped System
- 1.1. Identification of the dynamics for Active Damping
- 1.2. Tomography Experiment
  - 1.2.1. Simulation
  - 1.2.2. Results
2. Variability of the system dynamics for Active Damping
3. Integral Force Feedback
4. Direct Velocity Feedback
5. Inertial Control
6. Comparison
7. Useful Functions
- 7.1. prepareTomographyExperiment

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

Then, we will compare three active damping techniques:

In section 3: the integral force feedback is used
In section 4: the direct velocity feedback is used
In section 5: 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

1 Undamped System

In this section, we identify the dynamic of the system from forces applied in the nano-hexapod legs to the various sensors included in the nano-hexapod that could be use for Active Damping, namely:

A relative motion sensor, measuring the relative displacement of each of the leg
A force sensor measuring the total force transmitted to the top part of the leg in the direction of the leg
A absolute velocity sensor measuring the absolute velocity of the top part of the leg in the direction of the leg

After that, a tomography experiment is simulation without any active damping 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();

No disturbance is included in the system.

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');

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; % Actuator Inputs
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm');  io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm');  io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Vlm');   io_i = io_i + 1; % Absolute Velocity Outputs
io(io_i) = linio([mdl, '/Compute Error in NASS base'], 2, 'openoutput'); io_i = io_i + 1; % Metrology Outputs

%% 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', ...
                'Dxn', 'Dyn', 'Dzn', 'Rxn', 'Ryn', 'Rzn'};

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'}));

load('mat/stages.mat', 'nano_hexapod');
G_cart = minreal(G({'Dxn', 'Dyn', 'Dzn', 'Rxn', 'Ryn', 'Rzn'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}))*inv(nano_hexapod.J');
G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};

And we save them for further analysis.

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

1.1.3 Obtained Plants for Active Damping

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

Figure 1: G_iff: Transfer functions from forces applied in the actuators to the force sensor in each actuator (png, pdf)

Figure 2: G_dvf: Transfer functions from forces applied in the actuators to the relative motion sensor in each actuator (png, pdf)

Figure 3: G_ine: Transfer functions from forces applied in the actuators to the geophone located in each leg measuring the absolute velocity of the top part of the leg in the direction of the leg (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)));

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

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

2 Variability of the system dynamics for Active Damping

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 2.1)
The spindle angle with a null rotating speed (section 2.2)
The spindle rotation speed (section 2.3)
The tilt angle (section 2.4)
The scans of the translation stage (section 2.5)

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.

2.1 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]

Figure 6: Variability of the IFF plant with the Sample Mass (png, pdf)

Figure 7: Variability of the DVF plant with the Sample Mass (png, pdf)

Figure 8: Variability of the Inertial plant with the Sample Mass (png, pdf)

2.2 Variation of the Spindle Angle

We identify the dynamics for the following Spindle angles.

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

Figure 9: Variability of the IFF plant with the Spindle Angle (png, pdf)

Figure 10: Variability of the DVF plant with the Spindle Angle (png, pdf)

Figure 11: Variability of the Inertial plant with the Spindle Angle (png, pdf)

2.3 Variation of the Spindle Rotation Speed

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.

2.3.1 Dynamics of the Active Damping plants

Figure 12: Variability of the IFF plant with the Spindle rotation speed (png, pdf)

Figure 13: Variability of the IFF plant with the Spindle rotation speed (png, pdf)

Figure 14: Variability of the DVF plant with the Spindle rotation speed (png, pdf)

Figure 15: Variability of the DVF plant with the Spindle rotation speed (png, pdf)

Figure 16: Variability of the Inertial plant with the Spindle rotation speed (png, pdf)

Figure 17: Variability of the Inertial plant with the Spindle rotation speed (png, pdf)

2.3.2 Variation of the poles and zeros with the Spindle rotation frequency

Figure 18: Evolution of the pole with respect to the spindle rotation speed (png, pdf)

Figure 19: Evolution of the zero with respect to the spindle rotation speed (png, pdf)

2.4 Variation of the Tilt Angle

We identify the dynamics for the following Tilt stage angles.

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

Figure 20: Variability of the IFF plant with the Tilt stage Angle (png, pdf)

Figure 21: Variability of the DVF plant with the Tilt Angle (png, pdf)

Figure 22: Variability of the Inertial plant with the Tilt Angle (png, pdf)

2.5 Scans of the Translation Stage

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

We initialize the translation stage reference to be a sinus with an amplitude of 5mm and a period of 1s (Figure 23).

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

Figure 23: 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];

Figure 24: Variability of the IFF plant at different Ty scan positions (png, pdf)

Figure 25: Variability of the DVF plant at different Ty scan positions (png, pdf)

Figure 26: Variability of the Inertial plant at different Ty scan positions (png, pdf)

2.6 Conclusion

Table 1: Conclusion on the variability of the system dynamics for active damping
	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

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.

3 Integral Force Feedback

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

Here, we study the use of Integral Force Feedback (IFF) to actively damp the resonances.

The IFF control is applied in a decentralized way: there is on controller for each leg.

The control architecture is represented in figure 27 where one of the 6 nano-hexapod legs is represented.

Figure 27: Integral Force Feedback applied to a 1dof system

3.1 Control Design

3.1.1 Plant

Let’s load the previously identified undamped plant:

load('./active_damping/mat/undamped_plants.mat', 'G_iff');
load('./active_damping/mat/plants_variable.mat', 'masses', 'Gm_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 28).

Figure 28: Transfer function from forces applied in the legs to force sensor (png, pdf)

3.1.2 Control Design

The controller for each pair of actuator/sensor is:

w0 = 2*pi*50;
K_iff = -5000/s * (s/w0)/(1 + s/w0);

The corresponding loop gains are shown in figure 29.

Figure 29: 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_iff = -K_iff*eye(6);

We save the controller for further analysis.

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

3.2 Tomography Experiment

3.2.1 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');

3.2.2 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');

3.2.3 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)));

Figure 30: Position Error during tomography experiment - XY Motion (png, pdf)

Figure 31: Position Error during tomography experiment - Translations (png, pdf)

Figure 32: Position Error during tomography experiment - Rotations (png, pdf)

3.3 Conclusion

Integral Force Feedback:

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

4 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.

4.1 Control Design

4.1.1 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 33).

Figure 33: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

4.1.2 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 34.

Figure 34: Loop Gain for the Integral Force Feedback (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_dvf = -K_dvf*eye(6);

We save the controller for further analysis.

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

4.2 Tomography Experiment

4.2.1 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');

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

4.2.3 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)));

Figure 35: Position Error during tomography experiment - XY Motion (png, pdf)

Figure 36: Position Error during tomography experiment - Translations (png, pdf)

Figure 37: Position Error during tomography experiment - Rotations (png, pdf)

4.3 Conclusion

Direct Velocity Feedback:

5 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.

5.1 Control Design

5.1.1 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 38).

Figure 38: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

5.1.2 Control Design

The controller is defined below and the obtained loop gain is shown in figure 39.

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

Figure 39: Loop Gain for Inertial Control (png, pdf)

5.1.3 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');

5.2 Tomography Experiment

5.2.1 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');

5.2.2 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');

5.2.3 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)));

Figure 40: Position Error during tomography experiment - XY Motion (png, pdf)

Figure 41: Position Error during tomography experiment - Translations (png, pdf)

Figure 42: Position Error during tomography experiment - Rotations (png, pdf)

5.3 Conclusion

Inertial Control:

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

Figure 43: Sensitivity to ground motion in the Z direction on the Z motion error (png, pdf)

Figure 44: Compliance in the Z direction: Sensitivity of direct forces applied on the sample in the Z direction on the Z motion error (png, pdf)

Figure 45: Sensitivity to forces applied in the Z direction by the Spindle on the Z motion error (png, pdf)

Figure 46: Sensitivity to forces applied in the Z direction by the Y translation stage on the Z motion error (png, pdf)

Figure 47: Sensitivity to forces applied in the X direction by the Y translation stage on the X motion error (png, pdf)

6.3 Damped Plant

Figure 48: Plant for the \(z\) direction for different active damping technique used (png, pdf)

Figure 49: Plant for the \(x\) direction for different active damping technique used (png, pdf)

Figure 50: 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));

Figure 51: PSD of the translation errors in the X direction for applied Active Damping techniques (png, pdf)

Figure 52: PSD of the rotation errors in the X direction for applied Active Damping techniques (png, pdf)

Figure 53: CPS of the translation errors in the X direction for applied Active Damping techniques (png, pdf)

Figure 54: 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 % [kg]
    args.Rz_period     (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
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.Rz_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');

Active Damping applied on the Simscape Model

Table of Contents

1 Undamped System

1.1 Identification of the dynamics for Active Damping

1.1.1 Initialize the Simulation

1.1.2 Identification

1.1.3 Obtained Plants for Active Damping

1.2 Tomography Experiment

1.2.1 Simulation

1.2.2 Results

2 Variability of the system dynamics for Active Damping

2.1 Variation of the Sample Mass

2.2 Variation of the Spindle Angle

2.3 Variation of the Spindle Rotation Speed

2.3.1 Dynamics of the Active Damping plants

2.3.2 Variation of the poles and zeros with the Spindle rotation frequency

2.4 Variation of the Tilt Angle

2.5 Scans of the Translation Stage

2.6 Conclusion

3 Integral Force Feedback

3.1 Control Design

3.1.1 Plant

3.1.2 Control Design

3.1.3 Diagonal Controller

3.2 Tomography Experiment

3.2.1 Simulation with IFF Controller

3.2.2 Simulation with IFF Controller with added High Pass Filter

3.2.3 Compare with Undamped system

3.3 Conclusion

4 Direct Velocity Feedback

4.1 Control Design

4.1.1 Plant

4.1.2 Control Design

4.1.3 Diagonal Controller

4.2 Tomography Experiment

4.2.1 Initialize the Simulation

4.2.2 Simulation

4.2.3 Compare with Undamped system

4.3 Conclusion

5 Inertial Control

5.1 Control Design

5.1.1 Plant

5.1.2 Control Design

5.1.3 Diagonal Controller

5.2 Tomography Experiment

5.2.1 Initialize the Simulation

5.2.2 Simulation

5.2.3 Compare with Undamped system

5.3 Conclusion

6 Comparison

6.1 Load the plants

6.2 Sensitivity to Disturbance

6.3 Damped Plant

6.4 Tomography Experiment

6.4.1 Load the Simulation Data

6.4.2 Frequency Domain Analysis

7 Useful Functions

7.1 prepareTomographyExperiment

Function Description

Optional Parameters

Initialize the Simulation