test-bench-apa300ml/test-bench-apa300ml.org

Amplifier Piezoelectric Actuator APA300ML - Test Bench

• Introduction
• Model of an Amplified Piezoelectric Actuator and Sensor
• First Basic Measurements
  • Introduction
  • Geometrical Measurements
    • Introduction
    • Measurement Setup
    • Measurement Results
  • Electrical Measurements
  • Stroke measurement
    • Introduction
    • Voltage applied on one stack
    • Voltage applied on two stacks
    • Voltage applied on all three stacks
  • Spurious resonances
    • Introduction
    • Setup
    • Bending - X
    • Bending - Y
    • Torsion - Z
    • Compare
    • Conclusion
• Dynamical measurements - APA
  • Introduction
  • Speedgoat Setup
    • Introduction
    • frf_setup.m - Measurement Setup
    • frf_save.m - Save Data
  • Measurements on APA 1
    • Introduction
    • Excitation Signal
    • FRF Identification - Setup
    • FRF Identification - Displacement
    • FRF Identification - Force Sensor
    • Extract Parameters (Actuator/Sensor constants)
      • Piezoelectric Actuator Constant
      • Piezoelectric Sensor Constant
      • Results
    • Hysteresis
    • Estimation of the APA axial stiffness
    • Stiffness change due to electrical connections
    • Effect of the resistor on the IFF Plant
  • Comparison of all the APA
    • Introduction
    • Axial Stiffnesses - Comparison
    • FRF Identification - Setup
    • FRF Identification - DVF
    • FRF Identification - IFF
• Dynamical measurements - Struts
  • Introduction
  • Measurement on Strut 1
    • Introduction
    • Without Encoder
      • FRF Identification - Setup
      • FRF Identification - Displacement
      • FRF Identification - IFF
    • With Encoder
      • Measurement Data
      • FRF Identification - DVF
      • Comparison of the Encoder and Interferometer
      • APA Resonances Frequency
      • Estimated Flexible Joint axial stiffness
      • FRF Identification - IFF
  • Comparison of all the Struts
    • Introduction
    • FRF Identification - Setup
    • FRF Identification - DVF
    • FRF Identification - DVF with interferometer
    • FRF Identification - IFF
• Test Bench APA300ML - Simscape Model
  • Introduction
  • Nano Hexapod object
    • APA - 2 DoF
    • APA - Flexible Frame
    • APA - Fully Flexible
  • Identification
  • Compare 2-DoF with flexible
    • APA - 2 DoF
    • APA - Fully Flexible
    • Comparison
• Test Bench Struts - Simscape Model
  • Introduction
  • Nano Hexapod object
    • Flexible Joint - Bot
    • Flexible Joint - Top
    • APA - 2 DoF
    • APA - Flexible Frame
    • APA - Fully Flexible
  • Identification
  • Compare flexible joints
    • Perfect
    • Top Flexible
    • Bottom Flexible
    • Both Flexible
    • Comparison
• Function
  • generateSweepExc: Generate sweep sinus excitation
    • Function description
    • Optional Parameters
    • Sweep Sine part
    • Smooth Ends
    • Combine Excitation signals
  • generateShapedNoise: Generate Shaped Noise excitation
    • Function description
    • Optional Parameters
    • Shaped Noise
    • Smooth Ends
    • Combine Excitation signals
  • generateSinIncreasingAmpl: Generate Sinus with increasing amplitude
    • Function description
    • Optional Parameters
    • Sinus excitation
    • Smooth Ends
    • Combine Excitation signals
• Bibliography

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This report is also available as a pdf.

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Introduction   ignore

The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML.

This include:

• Stroke
• Stiffness
• Hysteresis
• Gain from the applied voltage $V_a$ to the generated Force $F_a$
• Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$
• Dynamical behavior

Model of an Amplified Piezoelectric Actuator and Sensor

Consider a schematic of the Amplified Piezoelectric Actuator in Figure fig:apa_model_schematic.

Amplified Piezoelectric Actuator Schematic

A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$:

\begin{equation} F_a = g_a \cdot V_a \end{equation}

A change of length $dl$ of the sensor stack will induce a voltage $V_s$:

\begin{equation} V_s = g_s \cdot dl \end{equation}

We wish here to experimental measure $g_a$ and $g_s$.

The block-diagram model of the piezoelectric actuator is then as shown in Figure fig:apa-model-simscape-schematic.

  \begin{tikzpicture}
    \node[block={2.0cm}{2.0cm}, align=center] (model) at (0,0){Simscape\\Model};
    \node[block, left=1.0 of model] (ga){$g_a(s)$};
    \node[block, right=1.0 of model] (gs){$g_s(s)$};

    \draw[<-] (ga.west) -- node[midway, above]{$V_a$} node[midway, below]{$[V]$} ++(-1.0, 0);
    \draw[->] (ga.east) --node[midway, above]{$F_a$} node[midway, below]{$[N]$} (model.west);
    \draw[->] (model.east) --node[midway, above]{$dl$} node[midway, below]{$[m]$} (gs.west);
    \draw[->] (gs.east) -- node[midway, above]{$V_s$} node[midway, below]{$[V]$} ++(1.0, 0);
  \end{tikzpicture}

Model of the APA with Simscape/Simulink

First Basic Measurements

<<sec:first_measurements>>

Introduction   ignore

• Section sec:geometrical_measurements:
• Section sec:electrical_measurements:
• Section sec:stroke_measurements:
• Section sec:spurious_resonances:

Geometrical Measurements

<<sec:geometrical_measurements>>

Introduction   ignore

The received APA are shown in Figure fig:received_apa.

Measurement Setup

The flatness corresponding to the two interface planes are measured as shown in Figure fig:flatness_meas_setup.

Measurement Results

The height (Z) measurements at the 8 locations (4 points by plane) are defined below.

apa1  = 1e-6*[0, -0.5 , 3.5  , 3.5  , 42   , 45.5, 52.5 , 46];
apa2  = 1e-6*[0, -2.5 , -3   , 0    , -1.5 , 1   , -2   , -4];
apa3  = 1e-6*[0, -1.5 , 15   , 17.5 , 6.5  , 6.5 , 21   , 23];
apa4  = 1e-6*[0, 6.5  , 14.5 , 9    , 16   , 22  , 29.5 , 21];
apa5  = 1e-6*[0, -12.5, 16.5 , 28.5 , -43  , -52 , -22.5, -13.5];
apa6  = 1e-6*[0, -8   , -2   , 5    , -57.5, -62 , -55.5, -52.5];
apa7  = 1e-6*[0, 19.5 , -8   , -29.5, 75   , 97.5, 70   , 48];
apa7b = 1e-6*[0, 9    , -18.5, -30  , 31   , 46.5, 16.5 , 7.5];
apa = {apa1, apa2, apa3, apa4, apa5, apa6, apa7b};

The X/Y Positions of the 8 measurement points are defined below.

W = 20e-3;   % Width [m]
L = 61e-3;   % Length [m]
d = 1e-3;    % Distance from border [m]
l = 15.5e-3; % [m]

pos = [[-L/2 + d; W/2 - d], [-L/2 + l - d; W/2 - d], [-L/2 + l - d; -W/2 + d], [-L/2 + d; -W/2 + d], [L/2 - l + d; W/2 - d], [L/2 - d; W/2 - d], [L/2 - d; -W/2 + d], [L/2 - l + d; -W/2 + d]];

Finally, the flatness is estimated by fitting a plane through the 8 points using the fminsearch command.

apa_d = zeros(1, 7);
for i = 1:7
    fun = @(x)max(abs(([pos; apa{i}]-[0;0;x(1)])'*([x(2:3);1]/norm([x(2:3);1]))));
    x0 = [0;0;0];
    [x, min_d] = fminsearch(fun,x0);
    apa_d(i) = min_d;
end

The obtained flatness are shown in Table tab:flatness_meas.

	Flatness $[\mu m]$
APA 1	8.9
APA 2	3.1
APA 3	9.1
APA 4	3.0
APA 5	1.9
APA 6	7.1
APA 7	18.7

Electrical Measurements

<<sec:electrical_measurements>>

The capacitance of the stacks is measure with the LCR-800 Meter (doc)

The excitation frequency is set to be 1kHz.

	Sensor Stack	Actuator Stacks
APA 1	5.10	10.03
APA 2	4.99	9.85
APA 3	1.72	5.18
APA 4	4.94	9.82
APA 5	4.90	9.66
APA 6	4.99	9.91
APA 7	4.85	9.85

There is clearly a problem with APA300ML number 3

The APA number 3 has ben sent back to Cedrat, and a new APA300ML has been shipped back.

Stroke measurement

<<sec:stroke_measurements>>

Introduction   ignore

We here wish to estimate the stroke of the APA.

To do so, one side of the APA is fixed, and a displacement probe is located on the other side as shown in Figure fig:stroke_test_bench.

Then, a voltage is applied on either one or two stacks using a DAC and a voltage amplifier.

Here are the documentation of the equipment used for this test bench:

• Voltage Amplifier: PD200 with a gain of 20
• 16bits DAC: IO313 Speedgoat card
• Displacement Probe: Millimar C1216 electronics and Millimar 1318 probe

Voltage applied on one stack

Let's first look at the relation between the voltage applied to one stack to the displacement of the APA as measured by the displacement probe.

The applied voltage is shown in Figure fig:apa_stroke_voltage_time.

Applied voltage as a function of time

The obtained displacement is shown in Figure fig:apa_stroke_time_1s. The displacement is set to zero at initial time when the voltage applied is -20V.

Displacement as a function of time for all the APA300ML

Finally, the displacement is shown as a function of the applied voltage in Figure fig:apa_d_vs_V_1s. We can clearly see that there is a problem with the APA 3. Also, there is a large hysteresis.

Displacement as a function of the applied voltage

We can clearly see from Figure fig:apa_d_vs_V_1s that there is a problem with the APA number 3.

Voltage applied on two stacks

Now look at the relation between the voltage applied to the two other stacks to the displacement of the APA as measured by the displacement probe.

The obtained displacement is shown in Figure fig:apa_stroke_time_2s. The displacement is set to zero at initial time when the voltage applied is -20V.

Displacement as a function of time for all the APA300ML

Finally, the displacement is shown as a function of the applied voltage in Figure fig:apa_d_vs_V_2s. We can clearly see that there is a problem with the APA 3. Also, there is a large hysteresis.

Displacement as a function of the applied voltage

Voltage applied on all three stacks

Finally, we can combine the two measurements to estimate the relation between the displacement and the voltage applied to the three stacks (Figure fig:apa_d_vs_V_3s).

Displacement as a function of the applied voltage

The obtained maximum stroke for all the APA are summarized in Table tab:apa_measured_stroke.

	Stroke $[\mu m]$
APA 1	373.2
APA 2	365.5
APA 3	181.7
APA 4	359.7
APA 5	361.5
APA 6	363.9
APA 7	358.4

Spurious resonances

<<sec:spurious_resonances>>

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)

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.

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.

• Laser Doppler Vibrometer Polytec OFV512
• Instrumented hammer

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.

The data is loaded.

bending_X = load('apa300ml_bending_X_top.mat');

The config for tfestimate is performed:

Ts = bending_X.Track1_X_Resolution; % Sampling frequency [Hz]
win = hann(ceil(1/Ts));

The transfer function from the input force to the output "rotation" (difference between the two measured distances).

[G_bending_X, f]  = tfestimate(bending_X.Track1, bending_X.Track2, win, [], [], 1/Ts);

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).

Obtained FRF for the X-bending

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).

The data is loaded, and the transfer function from the force to the measured rotation is computed.

bending_Y = load('apa300ml_bending_Y_top.mat');
[G_bending_Y, ~]  = tfestimate(bending_Y.Track1, bending_Y.Track2, win, [], [], 1/Ts);

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.

Obtained FRF for the Y-bending

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.

The data is loaded, and the transfer function computed.

torsion = load('apa300ml_torsion_left.mat');
[G_torsion, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);

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?).

Obtained FRF for the Z-torsion

In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited.

torsion = load('apa300ml_torsion_top.mat');
[G_torsion_top, ~]  = tfestimate(torsion.Track1, torsion.Track2, win, [], [], 1/Ts);

The two FRF are compared in Figure fig:apa300ml_meas_freq_torsion_z_comp. It is clear that the first two modes does not correspond to the torsional mode. Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here.

Obtained FRF for the Z-torsion

Compare

The three measurements are shown in Figure fig:apa300ml_meas_freq_compare.

Obtained FRF - Comparison

Conclusion

When two flexible joints are fixed at each ends of the APA, the APA is mostly in a free/free condition in terms of bending/torsion (the bending/torsional stiffness of the joints being very small).

In the current tests, the APA are in a fixed/free condition. Therefore, it is quite obvious that we measured higher resonance frequencies than what is foreseen for the struts. It is however quite interesting that there is a factor $\approx \sqrt{2}$ between the two (increased of the stiffness by a factor 2?).

Mode	Strut Mode	Measured Frequency
X-Bending	189Hz	280Hz
Y-Bending	285Hz	410Hz
Z-Torsion	400Hz	?

Dynamical measurements - APA

<<sec:dynamical_meas_apa>>

Introduction   ignore

In this section, a measurement test bench is used to identify the dynamics of the APA.

The bench is shown in Figure fig:picture_apa_bench, and a zoom picture on the APA and encoder is shown in Figure fig:picture_apa_bench_encoder.

Here are the documentation of the equipment used for this test bench:

• Voltage Amplifier: PD200
• Amplified Piezoelectric Actuator: APA300ML
• DAC/ADC: Speedgoat IO313
• Encoder: Renishaw Vionic and used Ruler
• Interferometer: Attocube IDS3010

The bench is schematically shown in Figure fig:test_bench_apa_alone and the signal used are summarized in Table tab:test_bench_apa_variables.

Variable	Description	Unit	Hardware
Va	Output DAC voltage	[V]	DAC - Ch. 1 => PD200 => APA
Vs	Measured stack voltage (ADC)	[V]	APA => ADC - Ch. 1
de	Encoder Measurement	[m]	PEPU Ch. 1 - IO318(1) - Ch. 1
da	Attocube Measurement	[m]	PEPU Ch. 2 - IO318(1) - Ch. 2
t	Time	[s]

This section is structured as follows:

• Section sec:meas_apa_speedgoat_setup: the Speedgoat setup is described (excitation signals, saved signals, etc.)
• Section sec:meas_one_apa: the measurements are first performed on one APA.
• Section sec:meas_all_apa: the same measurements are performed on all the APA and are compared.

Speedgoat Setup

<<sec:meas_apa_speedgoat_setup>>

Introduction   ignore

frf_setup.m - Measurement Setup

First is defined the sampling frequency:

%% Simulation configuration
Fs   = 10e3; % Sampling Frequency [Hz]
Ts   = 1/Fs; % Sampling Time [s]

%% Data record configuration
Trec_start = 5;  % Start time for Recording [s]
Trec_dur   = 100; % Recording Duration [s]

Tsim = 2*Trec_start + Trec_dur;  % Simulation Time [s]

A white noise excitation signal can be very useful in order to obtain a first idea of the plant FRF. The gain can be gradually increased until satisfactory output is obtained.

%% Shaped Noise
V_noise = generateShapedNoise('Ts', 1/Fs, ...
                              'V_mean', 3.25, ...
                              't_start', Trec_start, ...
                              'exc_duration', Trec_dur, ...
                              'smooth_ends', true, ...
                              'V_exc', 0.05/(1 + s/2/pi/10));

Example of Shaped noise excitation signal

The maximum excitation voltage at resonance is 9Vrms, therefore corresponding to 0.6V of output DAC voltage.

%% Sweep Sine
gc = 0.1;
xi = 0.5;
wn = 2*pi*94.3;

% Notch filter at the resonance of the APA
G_sweep = 0.2*(s^2 + 2*gc*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);

V_sweep = generateSweepExc('Ts',      Ts, ...
                           'f_start', 10, ...
                           'f_end',   400, ...
                           'V_mean',  3.25, ...
                           't_start', Trec_start, ...
                           'exc_duration', Trec_dur, ...
                           'sweep_type',   'log', ...
                           'V_exc', G_sweep*1/(1 + s/2/pi/500));

Example of Sweep Sin excitation signal

In order to better estimate the high frequency dynamics, a band-limited noise can be used (Figure fig:frf_meas_noise_hf_exc). The frequency content of the noise can be precisely controlled.

%% High Frequency Shaped Noise
[b,a] = cheby1(10, 2, 2*pi*[300 2e3], 'bandpass', 's');
wL = 0.005*tf(b, a);

V_noise_hf = generateShapedNoise('Ts', 1/Fs, ...
                              'V_mean', 3.25, ...
                              't_start', Trec_start, ...
                              'exc_duration', Trec_dur, ...
                              'smooth_ends', true, ...
                              'V_exc', wL);

Example of band-limited noise excitation signal

Then a sinus excitation can be used to estimate the hysteresis.

%% Sinus excitation with increasing amplitude
V_sin = generateSinIncreasingAmpl('Ts', 1/Fs, ...
                                  'V_mean', 3.25, ...
                                  'sin_ampls', [0.1, 0.2, 0.4, 1, 2, 4], ...
                                  'sin_period', 1, ...
                                  'sin_num', 5, ...
                                  't_start', Trec_start, ...
                                  'smooth_ends', true);

Example of Shaped noise excitation signal

Then, we select the wanted excitation signal.

%% Select the excitation signal
V_exc = timeseries(V_noise(2,:), V_noise(1,:));

%% Save data that will be loaded in the Simulink file
save('./frf_data.mat', 'Fs', 'Ts', 'Tsim', 'Trec_start', 'Trec_dur', 'V_exc');

frf_save.m - Save Data

First, we get data from the Speedgoat:

tg = slrt;

f = SimulinkRealTime.openFTP(tg);
mget(f, 'data/data.dat');
close(f);

And we load the data on the Workspace:

data = SimulinkRealTime.utils.getFileScopeData('data/data.dat').data;

da = data(:, 1); % Excitation Voltage (input of PD200) [V]
de = data(:, 2); % Measured voltage (force sensor) [V]
Vs = data(:, 3); % Measurment displacement (encoder) [m]
Va = data(:, 4); % Measurement displacement (attocube) [m]
t  = data(:, end); % Time [s]

And we save this to a mat file:

apa_number = 1;

save(sprintf('mat/frf_data_%i_huddle.mat', apa_number), 't', 'Va', 'Vs', 'de', 'da');

Measurements on APA 1

<<sec:meas_one_apa>>

Introduction   ignore

Measurements are first performed on only one APA. Once the measurement procedure is validated, it is performed on all the other APA.

Excitation Signal

For this first measurement, a basic logarithmic sweep is used between 10Hz and 2kHz.

The data are loaded.

apa_sweep = load(sprintf('mat/frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'da', 'de');

The initial time is set to zero.

%% Time vector
t = apa_sweep.t - apa_sweep.t(1) ; % Time vector [s]

The excitation signal is shown in Figure fig:apa_bench_exc_sweep. It is a sweep sine from 10Hz up to 2kHz filtered with a notch centered with the main resonance of the system and a low pass filter.

Excitation voltage

FRF Identification - Setup

Let's define the sampling time/frequency.

%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]

Then we defined a "Hanning" windows that will be used for the spectral analysis:

win = hanning(ceil(1*Fs)); % Hannning Windows

We get the frequency vector that will be the same for all the frequency domain analysis.

% Only used to have the frequency vector "f"
[~, f] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);

FRF Identification - Displacement

In this section, the transfer function from the excitation voltage $V_a$ to the encoder measured displacement $d_e$ and interferometer measurement $d_a$.

The coherence from $V_a$ to $d_e$ is computed and shown in Figure fig:apa_1_coh_dvf. It is quite good from 10Hz up to 500Hz.

%% TF - Encoder
[coh_sweep, ~] = mscohere(apa_sweep.Va,    apa_sweep.de,    win, [], [], 1/Ts);

Coherence for the identification from $V_a$ to $d_e$

The transfer functions are then estimated and shown in Figure fig:apa_1_frf_dvf.

%% TF - Encoder
[dvf_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);

%% TF - Interferometer
[int_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.da, win, [], [], 1/Ts);

Obtained transfer functions from $V_a$ to both $d_e$ and $d_a$

FRF Identification - Force Sensor

Now the dynamics from excitation voltage $V_a$ to the force sensor stack voltage $V_s$ is identified.

The coherence is computed and shown in Figure fig:apa_1_coh_iff and found very good from 10Hz up to 2kHz.

%% TF - Encoder
[coh_sweep, ~] = mscohere(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);

Coherence for the identification from $V_a$ to $V_s$

The transfer function is estimated and shown in Figure fig:apa_1_frf_iff.

%% Transfer function estimation
[iff_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);

Obtained transfer functions from $V_a$ to $V_s$

TODO Extract Parameters (Actuator/Sensor constants)

Piezoelectric Actuator Constant

Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage $V_a$ to the induced displacement $d$. Use a quasi static (1Hz) excitation signal $V_a$ on the piezoelectric stack and measure the vertical displacement $d$. Perform a linear regression to obtain:

\begin{equation} d = g_{d/V_a} \cdot V_a \end{equation}

Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force $F_a$ to the induced displacement $d$:

\begin{equation} d = g_{d/F_a} \cdot F_a \end{equation}

From the two gains, it is then easy to determine $g_a$:

\begin{equation} g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}} \end{equation}

Piezoelectric Sensor Constant

From a quasi static excitation of the piezoelectric stack, measure the gain from $V_a$ to $V_s$:

\begin{equation} V_s = g_{V_s/V_a} V_a \end{equation}

Note here that there is an high pass filter formed by the piezo capacitor and parallel resistor. The excitation frequency should then be in between the cut-off frequency of this high pass filter and the first resonance.

Alternatively, the gain can be computed from the dynamical identification and taking the gain at the wanted frequency.

Using the simscape model, compute the static gain from the actuator force $F_a$ to the strain of the sensor stack $dl$:

\begin{equation} dl = g_{dl/F_a} F_a \end{equation}

Then, the static gain from the sensor stack strain $dl$ to the general voltage $V_s$ is:

\begin{equation} g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}} \end{equation}

Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain. This external force can be some weight added, or a piezo in parallel.

Results

Quasi static gain between $d$ and $V_a$:

g_d_Va = mean(abs(dvf_sweep(f > 10 & f < 15)));

g_d_Va = 1.7e-05 [m/V]

Quasi static gain between $V_s$ and $V_a$:

g_Vs_Va = mean(abs(iff_sweep(f > 10 & f < 15)));

g_Vs_Va = 5.7e-01 [V/V]

Hysteresis

We here wish to visually see the amount of hysteresis present in the APA.

To do so, a quasi static sinusoidal excitation $V_a$ at different voltages is used.

The offset is 65V, and the sin amplitude is ranging from 1V up to 80V.

For each excitation amplitude, the vertical displacement $d$ of the mass is measured.

Then, $d$ is plotted as a function of $V_a$ for all the amplitudes.

We expect to obtained something like the hysteresis shown in Figure fig:expected_hysteresis.

The data is loaded.

apa_hyst = load('frf_data_1_hysteresis.mat', 't', 'Va', 'de');
% Initial time set to zero
apa_hyst.t = apa_hyst.t - apa_hyst.t(1);

The excitation voltage amplitudes are:

ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes

The excitation voltage and the measured displacement are shown in Figure fig:hyst_exc_signal_time.

Excitation voltage and measured displacement

For each amplitude, we only take the last sinus in order to reduce possible transients. Also, it is centered on zero.

The measured displacement at a function of the output voltage are shown in Figure fig:hyst_results_multi_ampl.

Obtained hysteresis for multiple excitation amplitudes

It is quite clear that hysteresis is increasing with the excitation amplitude.

Also, no hysteresis is found on the sensor stack voltage.

Estimation of the APA axial stiffness

In order to estimate the stiffness of the APA, a weight with known mass $m_a$ is added on top of the suspended granite and the deflection $d_e$ is measured using the encoder. The APA stiffness is then:

\begin{equation} k_{\text{apa}} = \frac{m_a g}{d} \end{equation}

Here, a mass of 6.4 kg is used:

added_mass = 6.4; % Added mass [kg]

The data is loaded, and the measured displacement is shown in Figure fig:apa_1_meas_stiffness.

apa_mass = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');
apa_mass.de = apa_mass.de - mean(apa_mass.de(apa_mass.t<11));

Measured displacement when adding the mass and removing the mass

There is some imprecision in the measurement as there are some drifts that are probably due to some creep.

The stiffness is then computed as follows:

k = 9.8 * added_mass / (mean(apa_mass.de(apa_mass.t > 12 & apa_mass.t < 12.5)) - mean(apa_mass.de(apa_mass.t > 20 & apa_mass.t < 20.5)));

And the stiffness obtained is very close to the one specified in the documentation ($k = 1.794\,[N/\mu m]$).

k = 1.68 [N/um]

Stiffness change due to electrical connections

We wish here to see if the stiffness changes when the actuator stacks are not connected to the amplifier and the sensor stacks are not connected to the ADC.

Note here that the resistor in parallel to the sensor stack is present in both cases.

First, the data are loaded.

add_mass_oc = load(sprintf('frf_data_%i_add_mass_open_circuit.mat',   1), 't', 'de');
add_mass_cc = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');

And the initial displacement is set to zero.

add_mass_oc.de = add_mass_oc.de - mean(add_mass_oc.de(add_mass_oc.t<11));
add_mass_cc.de = add_mass_cc.de - mean(add_mass_cc.de(add_mass_cc.t<11));

The measured displacements are shown in Figure fig:apa_meas_k_time_oc_cc.

Measured displacement

And the stiffness is estimated in both case. The results are shown in Table tab:APA_measured_k_oc_cc.

apa_k_oc = 9.8 * added_mass / (mean(add_mass_oc.de(add_mass_oc.t > 12 & add_mass_oc.t < 12.5)) - mean(add_mass_oc.de(add_mass_oc.t > 20 & add_mass_oc.t < 20.5)));
apa_k_cc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass_cc.t < 12.5)) - mean(add_mass_cc.de(add_mass_cc.t > 20 & add_mass_cc.t < 20.5)));

	$k [N/\mu m]$
Not connected	2.3
Connected	1.7

Clearly, connecting the actuator stacks to the amplified (basically equivalent as to short circuiting them) lowers the stiffness.

Effect of the resistor on the IFF Plant

A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack. This has the effect to form a high pass filter with the capacitance of the stack.

We here measured the low frequency transfer function from $V_a$ to $V_s$ with and without this resistor.

% With the resistor
wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va');

% Without the resistor
wo_k = load('frf_data_1_sweep_lf.mat',        't', 'Vs', 'Va');

We use a very long "Hanning" window for the spectral analysis in order to estimate the low frequency behavior.

win = hanning(ceil(50*Fs)); % Hannning Windows

And we estimate the transfer function from $V_a$ to $V_s$ in both cases:

[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, [], [], 1/Ts);
[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, [], [], 1/Ts);

With the following values of the resistor and capacitance, we obtain a first order high pass filter with a crossover frequency equal to:

C = 5.1e-6; % Sensor Stack capacitance [F]
R = 80.6e3; % Parallel Resistor [Ohm]

f0 = 1/(2*pi*R*C); % Crossover frequency of RC HPF [Hz]

f0 = 0.39 [Hz]

The transfer function of the corresponding high pass filter is:

G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0);

Let's compare the transfer function from actuator stack to sensor stack with and without the added resistor in Figure fig:frf_iff_effect_R.

Transfer function from $V_a$ to $V_s$ with and without the resistor $k$

The added resistor has indeed the expected effect.

Comparison of all the APA

<<sec:meas_all_apa>>

Introduction   ignore

The same measurements that was performed in Section sec:meas_one_apa are now performed on all the APA and then compared.

Axial Stiffnesses - Comparison

Let's first compare the APA axial stiffnesses.

The added mass is:

added_mass = 6.4; % Added mass [kg]

Here are the number of the APA that have been measured:

apa_nums = [1 2 4 5 6 7 8];

The data are loaded.

apa_mass = {};
for i = 1:length(apa_nums)
    apa_mass(i) = {load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', apa_nums(i)), 't', 'de')};
    % The initial displacement is set to zero
    apa_mass{i}.de = apa_mass{i}.de - mean(apa_mass{i}.de(apa_mass{i}.t<11));
end

The raw measurements are shown in Figure fig:apa_meas_k_time. All the APA seems to have similar stiffness except the APA 7 which should have an higher stiffness.

It is however strange that the displacement $d_e$ when the mass is removed is higher for the APA 7 than for the other APA. What could cause that?

Raw measurements for all the APA. A mass of 6.4kg is added at arround 15s and removed at arround 22s

The stiffnesses are computed for all the APA and are summarized in Table tab:APA_measured_k.

APA Num	$k [N/\mu m]$
1	1.68
2	1.69
4	1.7
5	1.7
6	1.7
7	1.93
8	1.73

The APA300ML manual specifies the nominal stiffness to be $1.8\,[N/\mu m]$ which is very close to what have been measured. Only the APA number 7 is a little bit off.

FRF Identification - Setup

The identification is performed in three steps:

1. White noise excitation with small amplitude. This is used to determine the main resonance of the system.
2. Sweep sine excitation with the amplitude lowered around the resonance. The sweep sine is from 10Hz to 400Hz.
3. High frequency noise. The noise is band-passed between 300Hz and 2kHz.

Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.

Here are the APA numbers that have been measured.

apa_nums = [1 2 4 5 6 7 8];

The data are loaded for both the second and third identification:

%% Second identification
apa_sweep = {};
for i = 1:length(apa_nums)
    apa_sweep(i) = {load(sprintf('frf_data_%i_sweep.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end

%% Third identification
apa_noise_hf = {};
for i = 1:length(apa_nums)
    apa_noise_hf(i) = {load(sprintf('frf_data_%i_noise_hf.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end

The time is the same for all measurements.

%% Time vector
t = apa_sweep{1}.t - apa_sweep{1}.t(1) ; % Time vector [s]

%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]

Then we defined a "Hanning" windows that will be used for the spectral analysis:

win = hanning(ceil(0.5*Fs)); % Hannning Windows

We get the frequency vector that will be the same for all the frequency domain analysis.

% Only used to have the frequency vector "f"
[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts);

FRF Identification - DVF

In this section, the dynamics from excitation voltage $V_a$ to encoder measured displacement $d_e$ is identified.

We compute the coherence for 2nd and 3rd identification:

%% Coherence computation
coh_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
    coh_sweep(:, i) = coh;
end

coh_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
    coh_noise_hf(:, i) = coh;
end

The coherence is shown in Figure fig:apa_frf_dvf_plant_coh. It is clear that the Sweep sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency.

Obtained coherence for the plant from $V_a$ to $d_e$

Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoders is computed:

%% Transfer function estimation
dvf_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
    dvf_sweep(:, i) = frf;
end

dvf_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
    dvf_noise_hf(:, i) = frf;
end

The obtained transfer functions are shown in Figure fig:apa_frf_dvf_plant_tf. They are all superimposed except for the APA7.

Why is the APA7 off? We could think that the APA7 is stiffer, but also the mass line is off.

It seems that there is a "gain" problem. The encoder seems fine (it measured the same as the Interferometer). Maybe it could be due to the amplifier?

Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)

A zoom on the main resonance is shown in Figure fig:apa_frf_dvf_zoom_res_plant_tf. It is clear that expect for the APA 7, the response around the resonances are well matching for all the APA.

It is also clear that there is not a single resonance but two resonances, a first one at 95Hz and a second one at 105Hz.

Why is there a double resonance at around 94Hz?

Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) - Zoom on the main resonance

FRF Identification - IFF

In this section, the dynamics from $V_a$ to $V_s$ is identified.

First the coherence is computed and shown in Figure fig:apa_frf_iff_plant_coh. The coherence is very nice from 10Hz to 2kHz. It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).

%% Coherence
coh_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
    coh_sweep(:, i) = coh;
end

coh_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
    coh_noise_hf(:, i) = coh;
end

Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure fig:apa_frf_iff_plant_tf

%% FRF estimation of the transfer function from Va to Vs
iff_sweep = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
    iff_sweep(:, i) = frf;
end

iff_noise_hf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
    iff_noise_hf(:, i) = frf;
end

#+caption:Identified IFF Plant

Dynamical measurements - Struts

<<sec:dynamical_meas_struts>>

Introduction   ignore

The same bench used in Section sec:dynamical_meas_apa is here used with the strut instead of only the APA.

The bench is shown in Figure fig:test_bench_leg_overview. Measurements are performed either when no encoder is fixed to the strut (Figure fig:test_bench_leg_front) or when one encoder is fixed to the strut (Figure fig:test_bench_leg_overview).

Measurement on Strut 1

<<sec:meas_strut_1>>

Introduction   ignore

Measurements are first performed on the strut 1 that contains:

• APA 1
• flex 1 and flex 2

Without Encoder

<<sec:meas_strut_1_no_encoder>>

FRF Identification - Setup

The identification is performed in three steps:

1. White noise excitation with small amplitude. This is used to determine the main resonance of the system.
2. Sweep sine excitation with the amplitude lowered around the resonance. The sweep sine is from 10Hz to 400Hz.
3. High frequency noise. The noise is band-passed between 300Hz and 2kHz.

Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.

leg_sweep    = load(sprintf('frf_data_leg_%i_sweep.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
leg_noise_hf = load(sprintf('frf_data_leg_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');

The time is the same for all measurements.

%% Time vector
t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s]

%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]

Then we defined a "Hanning" windows that will be used for the spectral analysis:

win = hanning(ceil(0.5*Fs)); % Hannning Windows

We get the frequency vector that will be the same for all the frequency domain analysis.

% Only used to have the frequency vector "f"
[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts);

FRF Identification - Displacement

In this section, the dynamics from the excitation voltage $V_a$ to the interferometer $d_a$ is identified.

We compute the coherence for 2nd and 3rd identification:

[coh_sweep, ~]    = mscohere(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);

Obtained coherence for the plant from $V_a$ to $d_a$

The transfer function from $V_a$ to the interferometer measured displacement $d_a$ is estimated and shown in Figure fig:strut_1_frf_dvf_plant_tf.

[dvf_sweep, ~]    = tfestimate(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
[dvf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);

Estimated FRF for the DVF plant (transfer function from $V_a$ to the interferometer $d_a$)

FRF Identification - IFF

In this section, the dynamics from $V_a$ to $V_s$ is identified.

First the coherence is computed and shown in Figure fig:strut_1_frf_iff_plant_coh. The coherence is very nice from 10Hz to 2kHz. It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).

[coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);

Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure fig:strut_1_frf_iff_plant_tf

[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);

#+caption:Identified IFF Plant for the Strut 1

With Encoder

<<sec:meas_strut_1_encoder>>

Measurement Data

leg_enc_sweep    = load(sprintf('frf_data_leg_coder_badly_align_%i_noise.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_badly_align_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');

FRF Identification - DVF

In this section, the dynamics from $V_a$ to $d_e$ is identified.

We compute the coherence for 2nd and 3rd identification:

[coh_enc_sweep, ~]    = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.de,    win, [], [], 1/Ts);
[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);

Obtained coherence for the plant from $V_a$ to $d_e$

[dvf_enc_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.de,    win, [], [], 1/Ts);
[dvf_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);

[dvf_int_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.da,    win, [], [], 1/Ts);
[dvf_int_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);

The obtained transfer functions are shown in Figure fig:strut_1_enc_frf_dvf_plant_tf.

They are all superimposed except for the APA7.

Why is the APA7 off? We could think that the APA7 is stiffer, but also the mass line is off.

It seems that there is a "gain" problem. The encoder seems fine (it measured the same as the Interferometer). Maybe it could be due to the amplifier?

Why is there a double resonance at around 94Hz?

Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)

Comparison of the Encoder and Interferometer

The interferometer could here represent the case where the encoders are fixed to the plates and not the APA.

The dynamics from $V_a$ to $d_e$ and from $V_a$ to $d_a$ are compared in Figure fig:strut_1_comp_enc_int.

Comparison of the transfer functions from excitation voltage $V_a$ to either the encoder $d_e$ or the interferometer $d_a$

It will clearly be difficult to do something (except some low frequency positioning) with the encoders fixed to the APA.

APA Resonances Frequency

As shown in Figure fig:strut_1_spurious_resonances, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.

Magnitude of the transfer function from excitation voltage $V_a$ to encoder measurement $d_e$. The frequency of the resonances are noted.

These resonances correspond to parasitic resonances of the APA itself. They are very close to what was estimated using the FEM:

• X-bending mode at around 190Hz (Figure fig:mode_bending_x_bis)
• Y-bending mode at around 290Hz (Figure fig:mode_bending_y_bis)
• Z-torsion mode at around 400Hz (Figure fig:mode_torsion_z_bis)

The resonances are indeed due to limited stiffness of the APA.

TODO Estimated Flexible Joint axial stiffness

FRF Identification - IFF

In this section, the dynamics from $V_a$ to $V_s$ is identified.

First the coherence is computed and shown in Figure fig:strut_1_frf_iff_plant_coh. The coherence is very nice from 10Hz to 2kHz. It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).

[coh_enc_sweep,    ~] = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);

Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure fig:strut_1_enc_frf_iff_plant_tf

[iff_enc_sweep,    ~] = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
[iff_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);

#+caption:Identified IFF Plant

Let's now compare the IFF plants whether the encoders are fixed to the APA or not (Figure fig:strut_1_frf_iff_comp_enc).

Effect of the encoder on the IFF plant

We can see that the IFF does not change whether of not the encoder are fixed to the struts.

Comparison of all the Struts

<<sec:meas_all_struts>>

Introduction   ignore

Now all struts are measured using the same procedure and test bench.

FRF Identification - Setup

The identification is performed in two steps:

1. White noise excitation with small amplitude. This is used to estimate the low frequency dynamics.
2. High frequency noise. The noise is band-passed between 300Hz and 2kHz.

Then, the result of the first identification is used between 10Hz and 350Hz and the result of the second identification if used between 350Hz and 2kHz.

Here are the LEG numbers that have been measured.

leg_nums = [1 2 3 4 5];

The data are loaded for both the first and second identification:

%% Second identification
leg_noise = {};
for i = 1:length(leg_nums)
    leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end

%% Third identification
leg_noise_hf = {};
for i = 1:length(leg_nums)
    leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
end

The time is the same for all measurements.

%% Time vector
t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s]

%% Sampling
Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
Fs = 1/Ts; % Sampling Frequency [Hz]

Then we defined a "Hanning" windows that will be used for the spectral analysis:

win = hanning(ceil(0.5*Fs)); % Hannning Windows

We get the frequency vector that will be the same for all the frequency domain analysis.

% Only used to have the frequency vector "f"
[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, [], [], 1/Ts);

FRF Identification - DVF

In this section, the dynamics from $V_a$ to $d_e$ is identified.

We compute the coherence for 2nd and 3rd identification:

%% Coherence computation
coh_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.de, win, [], [], 1/Ts);
    coh_noise(:, i) = coh;
end

coh_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
    coh_noise_hf(:, i) = coh;
end

The coherence is shown in Figure fig:struts_frf_dvf_plant_coh. It is clear that the Noise sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency.

Obtained coherence for the plant from $V_a$ to $d_e$

Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoders is computed:

%% Transfer function estimation
dvf_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.de, win, [], [], 1/Ts);
    dvf_noise(:, i) = frf;
end

dvf_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
    dvf_noise_hf(:, i) = frf;
end

The obtained transfer functions are shown in Figure fig:struts_frf_dvf_plant_tf.

They are all superimposed except for the LEG7.

Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)

Depending on how the APA are mounted with the flexible joints, the dynamics can change a lot as shown in Figure fig:struts_frf_dvf_plant_tf. In the future, a "pin" will be used to better align the APA with the flexible joints. We can expect the amplitude of the spurious resonances to decrease.

FRF Identification - DVF with interferometer

In this section, the dynamics from $V_a$ to $d_a$ is identified.

We compute the coherence.

%% Coherence computation
coh_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.da, win, [], [], 1/Ts);
    coh_noise(:, i) = coh;
end

coh_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
    coh_noise_hf(:, i) = coh;
end

The coherence is shown in Figure fig:struts_frf_int_plant_coh. It is clear that the Noise sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency.

Obtained coherence for the plant from $V_a$ to $d_e$

Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the Attocube is computed:

%% Transfer function estimation
dvf_a_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.da, win, [], [], 1/Ts);
    dvf_a_noise(:, i) = frf;
end

dvf_a_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
    dvf_a_noise_hf(:, i) = frf;
end

The obtained transfer functions are shown in Figure fig:struts_frf_int_plant_tf.

They are all superimposed except for the LEG7.

Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$)

FRF Identification - IFF

In this section, the dynamics from $V_a$ to $V_s$ is identified.

First the coherence is computed and shown in Figure fig:struts_frf_iff_plant_coh. The coherence is very nice from 10Hz to 2kHz. It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered).

%% Coherence
coh_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh, ~] = mscohere(leg_noise{i}.Va, leg_noise{i}.Vs, win, [], [], 1/Ts);
    coh_noise(:, i) = coh;
end

coh_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
    coh_noise_hf(:, i) = coh;
end

Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure fig:struts_frf_iff_plant_tf

%% FRF estimation of the transfer function from Va to Vs
iff_noise = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.Vs, win, [], [], 1/Ts);
    iff_noise(:, i) = frf;
end

iff_noise_hf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
    iff_noise_hf(:, i) = frf;
end

#+caption:Identified IFF Plant

Test Bench APA300ML - Simscape Model

Introduction

Nano Hexapod object

n_hexapod = struct();

APA - 2 DoF

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]

APA - Flexible Frame

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]

APA - Fully Flexible

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]

Identification

%% 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'};

Compare 2-DoF with flexible

APA - 2 DoF

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]

G_2dof = linearize(mdl, io, 0.0, options);
G_2dof.InputName  = {'Va'};
G_2dof.OutputName = {'Vs', 'dL', 'z'};

APA - Fully Flexible

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]

G_flex = linearize(mdl, io, 0.0, options);
G_flex.InputName  = {'Va'};
G_flex.OutputName = {'Vs', 'dL', 'z'};

Comparison

Test Bench Struts - Simscape Model

Introduction

Nano Hexapod object

n_hexapod = struct();

Flexible Joint - Bot

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)]

Flexible Joint - Top

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)]

APA - 2 DoF

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]

APA - Flexible Frame

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]

APA - Fully Flexible

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]

Identification

%% 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'};

Compare flexible joints

Perfect

n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof

Gp = linearize(mdl, io, 0.0, options);
Gp.InputName  = {'Va'};
Gp.OutputName = {'Vs', 'dL', 'z'};

Top Flexible

n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof

Gt = linearize(mdl, io, 0.0, options);
Gt.InputName  = {'Va'};
Gt.OutputName = {'Vs', 'dL', 'z'};

Bottom Flexible

n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof

Gb = linearize(mdl, io, 0.0, options);
Gb.InputName  = {'Va'};
Gb.OutputName = {'Vs', 'dL', 'z'};

Both Flexible

n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof

Gf = linearize(mdl, io, 0.0, options);
Gf.InputName  = {'Va'};
Gf.OutputName = {'Vs', 'dL', 'z'};

Comparison

Function

generateSweepExc: Generate sweep sinus excitation

<<sec:generateSweepExc>>

Function description

function [U_exc] = generateSweepExc(args)
% generateSweepExc - Generate a Sweep Sine excitation signal
%
% Syntax: [U_exc] = generateSweepExc(args)
%
% Inputs:
%    - args - Optinal arguments:
%        - Ts              - Sampling Time                                              - [s]
%        - f_start         - Start frequency of the sweep                               - [Hz]
%        - f_end           - End frequency of the sweep                                 - [Hz]
%        - V_mean          - Mean value of the excitation voltage                       - [V]
%        - V_exc           - Excitation Amplitude for the Sweep, could be numeric or TF - [V]
%        - t_start         - Time at which the sweep begins                             - [s]
%        - exc_duration    - Duration of the sweep                                      - [s]
%        - sweep_type      - 'logarithmic' or 'linear'                                  - [-]
%        - smooth_ends     - 'true' or 'false': smooth transition between 0 and V_mean  - [-]

Optional Parameters

arguments
    args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
    args.f_start         (1,1) double  {mustBeNumeric, mustBePositive} = 1
    args.f_end           (1,1) double  {mustBeNumeric, mustBePositive} = 1e3
    args.V_mean          (1,1) double  {mustBeNumeric} = 0
    args.V_exc                                         = 1
    args.t_start         (1,1) double  {mustBeNumeric, mustBeNonnegative} = 5
    args.exc_duration    (1,1) double  {mustBeNumeric, mustBePositive} = 10
    args.sweep_type            char    {mustBeMember(args.sweep_type,{'log', 'lin'})} = 'lin'
    args.smooth_ends           logical {mustBeNumericOrLogical} = true
end

Sweep Sine part

t_sweep = 0:args.Ts:args.exc_duration;

if strcmp(args.sweep_type, 'log')
    V_exc = sin(2*pi*args.f_start * args.exc_duration/log(args.f_end/args.f_start) * (exp(log(args.f_end/args.f_start)*t_sweep/args.exc_duration) - 1));
elseif strcmp(args.sweep_type, 'lin')
    V_exc = sin(2*pi*(args.f_start + (args.f_end - args.f_start)/2/args.exc_duration*t_sweep).*t_sweep);
else
    error('sweep_type should either be equal to "log" or to "lin"');
end

if isnumeric(args.V_exc)
    V_sweep = args.V_mean + args.V_exc*V_exc;
elseif isct(args.V_exc)
    if strcmp(args.sweep_type, 'log')
        V_sweep = args.V_mean + abs(squeeze(freqresp(args.V_exc, args.f_start*(args.f_end/args.f_start).^(t_sweep/args.exc_duration), 'Hz')))'.*V_exc;
    elseif strcmp(args.sweep_type, 'lin')
        V_sweep = args.V_mean + abs(squeeze(freqresp(args.V_exc, args.f_start+(args.f_end-args.f_start)/args.exc_duration*t_sweep, 'Hz')))'.*V_exc;
    end
end

Smooth Ends

if args.t_start > 0
    t_smooth_start = args.Ts:args.Ts:args.t_start;

    V_smooth_start = zeros(size(t_smooth_start));
    V_smooth_end   = zeros(size(t_smooth_start));

    if args.smooth_ends
        Vd_max = args.V_mean/(0.7*args.t_start);

        V_d = zeros(size(t_smooth_start));
        V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
        V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
        V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);

        V_smooth_start = cumtrapz(V_d)*args.Ts;

        V_smooth_end = args.V_mean - V_smooth_start;
    end
else
    V_smooth_start = [];
    V_smooth_end = [];
end

Combine Excitation signals

V_exc = [V_smooth_start, V_sweep, V_smooth_end];
t_exc = args.Ts*[0:1:length(V_exc)-1];

U_exc = [t_exc; V_exc];

generateShapedNoise: Generate Shaped Noise excitation

<<sec:generateShapedNoise>>

Function description

function [U_exc] = generateShapedNoise(args)
% generateShapedNoise - Generate a Shaped Noise excitation signal
%
% Syntax: [U_exc] = generateShapedNoise(args)
%
% Inputs:
%    - args - Optinal arguments:
%        - Ts              - Sampling Time                                              - [s]
%        - V_mean          - Mean value of the excitation voltage                       - [V]
%        - V_exc           - Excitation Amplitude, could be numeric or TF               - [V rms]
%        - t_start         - Time at which the noise begins                             - [s]
%        - exc_duration    - Duration of the noise                                      - [s]
%        - smooth_ends     - 'true' or 'false': smooth transition between 0 and V_mean  - [-]

Optional Parameters

arguments
    args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
    args.V_mean          (1,1) double  {mustBeNumeric} = 0
    args.V_exc                                         = 1
    args.t_start         (1,1) double  {mustBeNumeric, mustBePositive} = 5
    args.exc_duration    (1,1) double  {mustBeNumeric, mustBePositive} = 10
    args.smooth_ends           logical {mustBeNumericOrLogical} = true
end

Shaped Noise

t_noise = 0:args.Ts:args.exc_duration;

if isnumeric(args.V_exc)
    V_noise = args.V_mean + args.V_exc*sqrt(1/args.Ts/2)*randn(length(t_noise), 1)';
elseif isct(args.V_exc)
    V_noise = args.V_mean + lsim(args.V_exc, sqrt(1/args.Ts/2)*randn(length(t_noise), 1), t_noise)';
end

Smooth Ends

t_smooth_start = args.Ts:args.Ts:args.t_start;

V_smooth_start = zeros(size(t_smooth_start));
V_smooth_end   = zeros(size(t_smooth_start));

if args.smooth_ends
    Vd_max = args.V_mean/(0.7*args.t_start);

    V_d = zeros(size(t_smooth_start));
    V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
    V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
    V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);

    V_smooth_start = cumtrapz(V_d)*args.Ts;

    V_smooth_end = args.V_mean - V_smooth_start;
end

Combine Excitation signals

V_exc = [V_smooth_start, V_noise, V_smooth_end];
t_exc = args.Ts*[0:1:length(V_exc)-1];

U_exc = [t_exc; V_exc];

generateSinIncreasingAmpl: Generate Sinus with increasing amplitude

<<sec:generateSinIncreasingAmpl>>

Function description

function [U_exc] = generateSinIncreasingAmpl(args)
% generateSinIncreasingAmpl - Generate Sinus with increasing amplitude
%
% Syntax: [U_exc] = generateSinIncreasingAmpl(args)
%
% Inputs:
%    - args - Optinal arguments:
%        - Ts              - Sampling Time                                              - [s]
%        - V_mean          - Mean value of the excitation voltage                       - [V]
%        - sin_ampls       - Excitation Amplitudes                                      - [V]
%        - sin_freq        - Excitation Frequency                                       - [Hz]
%        - sin_num         - Number of period for each amplitude                        - [-]
%        - t_start         - Time at which the excitation begins                        - [s]
%        - smooth_ends     - 'true' or 'false': smooth transition between 0 and V_mean  - [-]

Optional Parameters

arguments
    args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
    args.V_mean          (1,1) double  {mustBeNumeric} = 0
    args.sin_ampls             double  {mustBeNumeric, mustBePositive} = [0.1, 0.2, 0.3]
    args.sin_period      (1,1) double  {mustBeNumeric, mustBePositive} = 1
    args.sin_num         (1,1) double  {mustBeNumeric, mustBePositive, mustBeInteger} = 3
    args.t_start         (1,1) double  {mustBeNumeric, mustBePositive} = 5
    args.smooth_ends           logical {mustBeNumericOrLogical} = true
end

Sinus excitation

t_noise = 0:args.Ts:args.sin_period*args.sin_num;
sin_exc = [];

for sin_ampl = args.sin_ampls
    sin_exc = [sin_exc, args.V_mean + sin_ampl*sin(2*pi/args.sin_period*t_noise)];
end

Smooth Ends

t_smooth_start = args.Ts:args.Ts:args.t_start;

V_smooth_start = zeros(size(t_smooth_start));
V_smooth_end   = zeros(size(t_smooth_start));

if args.smooth_ends
    Vd_max = args.V_mean/(0.7*args.t_start);

    V_d = zeros(size(t_smooth_start));
    V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
    V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
    V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);

    V_smooth_start = cumtrapz(V_d)*args.Ts;

    V_smooth_end = args.V_mean - V_smooth_start;
end

Combine Excitation signals

V_exc = [V_smooth_start, sin_exc, V_smooth_end];
t_exc = args.Ts*[0:1:length(V_exc)-1];

U_exc = [t_exc; V_exc];

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