Nano-Hexapod Struts - Test Bench

1. Model of the Amplified Piezoelectric Actuator
2. First Basic Measurements
3. Dynamical measurements - APA
4. Test Bench APA300ML - Simscape Model
5. Dynamical measurements - Struts
- 5.1. Measurement on Strut 1
  - 5.1.1. Without Encoder
  - 5.1.2. With Encoder
- 5.2. Comparison of all the Struts
6. Test Bench Struts - Simscape Model
7. Function

This report is also available as a pdf.

In this document, a test-bench is used to characterize the struts of the nano-hexapod.

Each strut includes:

2 flexible joints at each ends. This flexible joints have been characterized in a separate test bench.
one Amplified Piezoelectric Actuator (APA300ML) (described in Section 1)
one encoder (Renishaw Vionci) that has been characterized in a separate test bench.

The goal here, is to first characterize the APA300ML in terms of:

The, geometric features, electrical capacitance, stroke, hysteresis, spurious resonances. This is performed in Section 2.
The dynamics from the voltage applied on the actuator stacks to the induced displacement, and to the measured voltage by the force sensor stack. Also the “actuator constant” and “sensor constant” are identified. This is done in Section 3.
Compare the measurements with the Simscape models (2DoF, Super-Element) in order to tuned/validate the models. This is explained in Section 4.

Then the struts are mounted, and are fixed to the same measurement bench. Similarly, the goals are to:

Identify the dynamics from the generated DAC voltage (going to the PD200 voltage amplitifer and then to the actuator stacks) to (Section 5):
- the sensors stack generated voltage
- the measured displacement by the encoder
- the measured displacement by the interferometer
Compare the measurements with the Simscape model of the struts and tune the models (Section 6)

The final goal of the work presented in this document is to have an accurate Simscape model of the struts that can then be included in the Simscape model of the nano-hexapod.

1 Model of the Amplified Piezoelectric Actuator

The Amplified Piezoelectric Actuator (APA) used is the APA300ML from Cedrat technologies (Figure 1).

Figure 1: Picture of the APA300ML

Simscape models of the APA300ML are developed:

Section 1.1: a simple 2 degrees of freedom (DoF) model
Section 1.2: a “flexible” model using a “super-element” extracted from a Finite Element Model of the APA

For both models, an “actuator constant” and a “sensor constant” are used. These constants are used to link the electrical domain and the mechanical domain. They are described in Section 1.3.

1.1 Two Degrees of Freedom Model

Figure 2: Schematic of the 2DoF model for the Amplified Piezoelectric Actuator

1.2 Flexible Model

Figure 3: Remote points for the APA300ML (Ansys)

Figure 4: From a finite Element Model (Ansys, bottom left) is extract the mass and stiffness matrices that are then used on Simscape (right)

1.3 Actuator and Sensor constants

Consider a schematic of the Amplified Piezoelectric Actuator in Figure 5.

Figure 5: 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 6.

Figure 6: Model of the APA with Simscape/Simulink

2 First Basic Measurements

Section 2.1:
Section 2.2:
Section 2.3:
Section 2.4:

2.1 Geometrical Measurements

The received APA are shown in Figure 7.

Figure 7: Received APA

2.1.1 Measurement Setup

The flatness corresponding to the two interface planes are measured as shown in Figure 8.

Figure 8: Measurement Setup

2.1.2 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 1.

Table 1: Estimated flatness
	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

2.2 Electrical Measurements

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

Figure 9: LCR Meter used for the measurements

The excitation frequency is set to be 1kHz.

Table 2: Capacitance measured with the LCR meter. The excitation signal is a sinus at 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.

2.3 Stroke measurement

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 10.

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

Figure 10: Bench to measured the APA stroke

2.3.1 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 11.

Figure 11: Applied voltage as a function of time

The obtained displacement is shown in Figure 12. The displacement is set to zero at initial time when the voltage applied is -20V.

Figure 12: Displacement as a function of time for all the APA300ML

Finally, the displacement is shown as a function of the applied voltage in Figure 13. We can clearly see that there is a problem with the APA 3. Also, there is a large hysteresis.

Figure 13: Displacement as a function of the applied voltage

We can clearly see from Figure 13 that there is a problem with the APA number 3.

2.3.2 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 14. The displacement is set to zero at initial time when the voltage applied is -20V.

Figure 14: Displacement as a function of time for all the APA300ML

Finally, the displacement is shown as a function of the applied voltage in Figure 15. We can clearly see that there is a problem with the APA 3. Also, there is a large hysteresis.

Figure 15: Displacement as a function of the applied voltage

2.3.3 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 16).

Figure 16: Displacement as a function of the applied voltage

The obtained maximum stroke for all the APA are summarized in Table 3.

Table 3: Measured maximum 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

2.4 Spurious resonances

2.4.1 Introduction

Three main resonances are foreseen to be problematic for the control of the APA300ML (Figure 17):

Mode in X-bending at 189Hz
Mode in Y-bending at 285Hz
Mode in Z-torsion at 400Hz

Figure 17: Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz

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.

2.4.2 Setup

The measurement setup is shown in Figure 18. 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

Figure 18: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer

2.4.3 Bending - X

The setup to measure the X-bending motion is shown in Figure 19. 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.

Figure 19: X-Bending measurement setup

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 20.

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).

Figure 20: Obtained FRF for the X-bending

2.4.4 Bending - Y

The setup to measure the Y-bending is shown in Figure 21.

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).

Figure 21: Y-Bending measurement setup

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 22. The main resonance is at 412Hz, and we also see the third “harmonic” at 1220Hz.

Figure 22: Obtained FRF for the Y-bending

2.4.5 Torsion - Z

Finally, we measure the Z-torsion resonance as shown in Figure 23.

The excitation is shown on the other side of the APA, on the side to excite the torsion motion.

Figure 23: Z-Torsion measurement setup

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 24. 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?).

Figure 24: 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 25. 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.

Figure 25: Obtained FRF for the Z-torsion

2.4.6 Compare

The three measurements are shown in Figure 26.

Figure 26: Obtained FRF - Comparison

2.4.7 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?).

Table 4: Measured frequency of the modes
Mode	Strut Mode	Measured Frequency
X-Bending	189Hz	280Hz
Y-Bending	285Hz	410Hz
Z-Torsion	400Hz	?

3 Dynamical measurements - APA

In this section, a measurement test bench is used 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

The bench is shown in Figure 27, and a zoom picture on the APA and encoder is shown in Figure 28.

Figure 27: Picture of the test bench

Figure 28: Zoom on the APA with the 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 29 and the signal used are summarized in Table 5.

Figure 29: Schematic of the Test Bench

Table 5: Variables used during the measurements
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 3.1: the Speedgoat setup is described (excitation signals, saved signals, etc.)
Section 3.2: the measurements are first performed on one APA.
Section 3.3: the same measurements are performed on all the APA and are compared.

3.1 Speedgoat Setup

3.1.1 `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));

Figure 30: 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));

Figure 31: Example of Sweep Sin excitation signal

In order to better estimate the high frequency dynamics, a band-limited noise can be used (Figure 32). 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);

Figure 32: 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);

Figure 33: 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');

3.1.2 `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');

3.2 Measurements on APA 1

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

3.2.1 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 34. 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.

Figure 34: Excitation voltage

3.2.2 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);

3.2.3 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 35. It is quite good from 10Hz up to 500Hz.

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

Figure 35: Coherence for the identification from \(V_a\) to \(d_e\)

The transfer functions are then estimated and shown in Figure 36.

%% 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);

Figure 36: Obtained transfer functions from \(V_a\) to both \(d_e\) and \(d_a\)

It seems that using the interferometer, we have a lot more time delay than when using the encoder.

3.2.4 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 37 and found very good from 10Hz up to 2kHz.

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

Figure 37: Coherence for the identification from \(V_a\) to \(V_s\)

The transfer function is estimated and shown in Figure 38.

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

Figure 38: Obtained transfer functions from \(V_a\) to \(V_s\)

3.2.5 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 39.

Figure 39: Expected Hysteresis poel10_explor_activ_hard_mount_vibrat

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 40.

Figure 40: 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 41.

Figure 41: 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.

3.2.6 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 42.

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));

Figure 42: 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]

3.2.7 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 43.

Figure 43: Measured displacement

And the stiffness is estimated in both case. The results are shown in Table 6.

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)));

Table 6: Measured stiffnesses on “open” and “closed” circuits
	\(k [N/\mu m]\)
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.

3.2.8 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 44.

Figure 44: Transfer function from \(V_a\) to \(V_s\) with and without the resistor \(k\)

The added resistor has indeed the expected effect.

3.3 Comparison of all the APA

The same measurements that was performed in Section 3.2 are now performed on all the APA and then compared.

3.3.1 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 45. 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?

Figure 45: 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 7.

Table 7: Measured stiffnesses
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.

3.3.2 FRF Identification - Setup

The identification is performed in three steps:

White noise excitation with small amplitude. This is used to determine the main resonance of the system.
Sweep sine excitation with the amplitude lowered around the resonance. The sweep sine is from 10Hz to 400Hz.
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);
i_lf = f <= 350;
i_hf = f >  350;

3.3.3 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_enc = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [coh_lf, ~] = mscohere(apa_sweep{i}.Va,    apa_sweep{i}.de,    win, [], [], 1/Ts);
    [coh_hf, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
    coh_enc(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
end

The coherence is shown in Figure 46. 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.

Figure 46: 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
enc_frf = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [frf_lf, ~] = tfestimate(apa_sweep{i}.Va,    apa_sweep{i}.de,    win, [], [], 1/Ts);
    [frf_hf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
    enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
end

The obtained transfer functions are shown in Figure 47. 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?

Figure 47: 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 48. 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?

Figure 48: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\)) - Zoom on the main resonance

3.3.4 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 49. 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_iff = zeros(length(f), length(apa_nums));
for i = 1:length(apa_nums)
    [coh_lf, ~] = mscohere(apa_sweep{i}.Va,    apa_sweep{i}.Vs,    win, [], [], 1/Ts);
    [coh_hf, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
    coh_iff(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
end

Figure 49: Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure 50

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

Figure 50: Identified IFF Plant

3.3.5 Save measured FRF

%% Remove the APA 7 (6 in the list) from measurements
apa_nums(6) = [];
enc_frf(:,6) = [];
iff_frf(:,6) = [];

save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');

4 Test Bench APA300ML - Simscape Model

4.1 Introduction

A simscape model (Figure 51) of the measurement bench is used.

Figure 51: Screenshot of the Simscape model

4.2 First Identification

The APA is first initialized with default parameters and the transfer function from excitation voltage \(V_a\) (before the amplification of 20 due to the PD200 amplifier) to the sensor stack voltage \(V_s\), encoder \(d_L\) and interferometer \(z\) is identified.

%% Initialize the structure containing the data used in the model
n_hexapod = struct();
n_hexapod.actuator = initializeAPA('type', '2dof');

%% 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'};

The obtain dynamics are shown in Figure 52 and 53.

Figure 52: Bode plot of the transfer function from \(V_a\) to \(V_s\)

Figure 53: Bode plot of the transfer function from \(V_a\) to \(d_L\) and to \(z\)

4.3 Identify Sensor/Actuator constants and compare with measured FRF

4.3.1 How to identify these constants?

4.3.1.1 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 \(d\).

\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}

4.3.1.2 Piezoelectric Sensor Constant

Similarly, it is easy to determine the gain from the excitation voltage \(V_a\) to the voltage generated by the sensor stack \(V_s\):

\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 gain can be computed from the dynamical identification and taking the gain at the wanted frequency (above the first resonance).

Using the simscape model, compute the gain at the same frequency 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 “sensor” constant 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}

4.3.2 Identification Data

load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');

4.3.3 2DoF APA

4.3.3.1 2DoF APA

%% Initialize a 2DoF APA with Ga=Gs=1
n_hexapod.actuator = initializeAPA('type', '2dof', 'ga', 1, 'gs', 1);

4.3.3.2 Identification without actuator or sensor constants

%% 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

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

4.3.3.3 Actuator Constant

%% Estimated Actuator Constant
ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]

ga = -32.2 [N/V]

n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V]

4.3.3.4 Sensor Constant

%% Estimated Sensor Constant
gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]

gs = 0.088 [V/m]

n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m]

4.3.3.5 Comparison

Identify the dynamics with included constants.

%% Identify again the dynamics with correct Ga,Gs
Gs = linearize(mdl, io, 0.0, options);
Gs = Gs*exp(-Ts*s);
Gs.InputName  = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};

Figure 54: Comparison of the experimental data and Simscape model (\(u\) to \(d\mathcal{L}_m\))

Figure 55: Comparison of the experimental data and Simscape model (\(V_a\) to \(V_s\))

4.3.4 Flexible APA

4.3.4.1 Flexible APA

%% Initialize the APA as a flexible body
n_hexapod.actuator = initializeAPA('type', 'flexible', 'ga', 1, 'gs', 1);

4.3.4.2 Identification without actuator or sensor constants

%% Identify the dynamics
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName  = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};

4.3.4.3 Actuator Constant

%% Actuator Constant
ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]

ga = 23.5 [N/V]

n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V]

4.3.4.4 Sensor Constant

%% Sensor Constant
gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]

gs = -4839841.756 [V/m]

n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m]

4.3.4.5 Comparison

%% Identify with updated constants
Gs = linearize(mdl, io, 0.0, options);
Gs = Gs*exp(-Ts*s);
Gs.InputName  = {'Va'};
Gs.OutputName = {'Vs', 'dL', 'z'};

Figure 56: Comparison of the experimental data and Simscape model (\(u\) to \(d\mathcal{L}_m\))

Figure 57: Comparison of the experimental data and Simscape model (\(u\) to \(\tau_m\))

The flexible model is a bit “soft” compared with the experimental results.

4.4 Optimize 2-DoF model to fit the experimental Data

%% Optimized parameters
n_hexapod.actuator = initializeAPA('type', '2dof', ...
                                   'Ga', -30.0, ...
                                   'Gs', 0.098, ...
                                   'k',  ones(6,1)*0.38e6, ...
                                   'ke', ones(6,1)*1.75e6, ...
                                   'ka', ones(6,1)*3e7, ...
                                   'c',  ones(6,1)*1.3e2, ...
                                   'ce', ones(6,1)*1e1, ...
                                   'ca', ones(6,1)*1e1 ...
                                   );

Figure 58: Comparison of the measured FRF and the optimized model

5 Dynamical measurements - Struts

The same bench used in Section 3 is here used with the strut instead of only the APA.

The bench is shown in Figure 59. Measurements are performed either when no encoder is fixed to the strut (Figure 60) or when one encoder is fixed to the strut (Figure 59).

Figure 59: Test Bench with Strut - Overview

Figure 60: Test Bench with Strut - Zoom on the strut

Figure 61: Test Bench with Strut - Zoom on the strut with the encoder

5.1 Measurement on Strut 1

Measurements are first performed on the strut 1 that contains:

APA 1
flex 1 and flex 2

5.1.1 Without Encoder

5.1.1.1 FRF Identification - Setup

The identification is performed in three steps:

White noise excitation with small amplitude. This is used to determine the main resonance of the system.
Sweep sine excitation with the amplitude lowered around the resonance. The sweep sine is from 10Hz to 400Hz.
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);

5.1.1.2 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);

Figure 62: 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 63.

[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);

Figure 63: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the interferometer \(d_a\))

5.1.1.3 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 64. 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);

Figure 64: Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure 65

[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);

Figure 65: Identified IFF Plant for the Strut 1

5.1.2 With Encoder

5.1.2.1 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');

5.1.2.2 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);

Figure 66: 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 67.

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?

Figure 67: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

5.1.2.3 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 68.

Figure 68: 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.

5.1.2.4 APA Resonances Frequency

As shown in Figure 69, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.

Figure 69: 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 (Figure 70):

Mode in X-bending at 189Hz
Mode in Y-bending at 285Hz
Mode in Z-torsion at 400Hz

Figure 70: Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz

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

5.1.2.5 Estimated Flexible Joint axial stiffness

load(sprintf('frf_data_leg_coder_%i_add_mass_closed_circuit.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
de = de - de(1);
da = da - da(1);

figure;
hold on;
plot(t, de, 'DisplayName', 'Encoder')
plot(t, da, 'DisplayName', 'Interferometer')
hold off;
xlabel('Time [s]'); ylabel('Displacement [m]');
legend('location', 'northeast');

de_steps = [mean(de(t > 13 & t < 14));
            mean(de(t > 19 & t < 20));
            mean(de(t > 24 & t < 25));
            mean(de(t > 28.5 & t < 29.5));
            mean(de(t > 49 & t < 50))] - mean(de(t > 13 & t < 14));

da_steps = [mean(da(t > 13 & t < 14));
            mean(da(t > 19 & t < 20));
            mean(da(t > 24 & t < 25));
            mean(da(t > 28.5 & t < 29.5));
            mean(da(t > 49 & t < 50))] - mean(da(t > 13 & t < 14));

added_mass = [0; 1; 2; 3; 4];

lin_fit = (added_mass\abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)));

figure;
hold on;
plot(added_mass, abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)), 'ok')
plot(added_mass, added_mass*lin_fit, '-r')
hold off;

What is strange is that the encoder is measuring a larger displacement than the interferometer. That should be the opposite. Maybe is is caused by the fact that the APA is experiencing some bending and therefore a larger motion is measured on the encoder.

5.1.2.6 FRF Identification - IFF

In this section, the dynamics from \(V_a\) to \(V_s\) is identified.

[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);

Figure 71: Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure 72

[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);

Figure 72: Identified IFF Plant

Let’s now compare the IFF plants whether the encoders are fixed to the APA or not (Figure 73).

Figure 73: 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.

5.2 Comparison of all the Struts

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

5.2.1 FRF Identification - Setup

The identification is performed in two steps:

White noise excitation with small amplitude. This is used to estimate the low frequency dynamics.
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);
i_lf = f <= 350;
i_hf = f >  350;

5.2.2 FRF Identification - Encoder

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_enc = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh_lf, ~] = mscohere(leg_noise{i}.Va,    leg_noise{i}.de,    win, [], [], 1/Ts);
    [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
    coh_enc(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
end

The coherence is shown in Figure 74. 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.

Figure 74: 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
enc_frf = zeros(length(f), length(leg_nums));

for i = 1:length(leg_nums)
    [frf_lf, ~] = tfestimate(leg_noise{i}.Va,    leg_noise{i}.de,    win, [], [], 1/Ts);
    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.de, win, [], [], 1/Ts);
    enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
end

The obtained transfer functions are shown in Figure 75.

Figure 75: 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 75. 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. We can also observe the presence or absence of complex conjugate zeros between the first two modes. This might also be dependent on the alignment.

5.2.3 FRF Identification - Interferometer

In this section, the dynamics from \(V_a\) to \(d_a\) is identified.

We compute the coherence.

%% Coherence computation
coh_int = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh_lf, ~] = mscohere(leg_noise{i}.Va,    leg_noise{i}.da,    win, [], [], 1/Ts);
    [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
    coh_int(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
end

The coherence is shown in Figure 76. 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.

Figure 76: 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
int_frf = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [frf_lf, ~] = tfestimate(leg_noise{i}.Va,    leg_noise{i}.da,    win, [], [], 1/Ts);
    [frf_hf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts);
    int_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
end

The obtained transfer functions are shown in Figure 77.

They are all superimposed except for the LEG7.

Figure 77: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

5.2.4 FRF Identification - Force Sensor

In this section, the dynamics from \(V_a\) to \(V_s\) is identified.

First the coherence is computed and shown in Figure 78. 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_iff = zeros(length(f), length(leg_nums));
for i = 1:length(leg_nums)
    [coh_lf, ~] = mscohere(leg_noise{i}.Va,    leg_noise{i}.Vs,    win, [], [], 1/Ts);
    [coh_hf, ~] = mscohere(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts);
    coh_iff(:, i) = [coh_lf(i_lf); coh_hf(i_hf)];
end

Figure 78: Obtained coherence for the IFF plant

Then the FRF are estimated and shown in Figure 79

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

Figure 79: Identified IFF Plant

5.2.5 Save measured FRF

save('mat/meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');

6 Test Bench Struts - Simscape Model

6.1 Introduction

Section 6.2:
Section 6.3:
Section 6.4:

6.2 Comparison with the 2-DoF Model

6.2.1 First Identification

The object containing all the data is created.

%% Initialize structure containing data for the Simscape model
n_hexapod = struct();
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '2dof');
n_hexapod.actuator = initializeAPA('type', '2dof');

%% 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'};

Figure 80: Identified transfer function from \(V_a\) to \(V_s\)

Figure 81: Identified transfer function from \(V_a\) to \(d_L\)

6.2.2 Effect of flexible joints

Perfect

%% Perfect Joints
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');

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

Top Flexible

%% Top joint with Axial stiffness
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');

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

Bottom Flexible

%% Bottom joint with Axial stiffness
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');

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

Both Flexible

%% Both joints with Axial stiffness
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');

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

Comparison in Figures 82, 83 and 84.

Figure 82: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(V_s\)

Figure 83: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(d_L\) (encoder)

Figure 84: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(z\) (interferometer)

Imperfection of the flexible joints has the largest impact on the transfer function from \(V_a\) to \(d_L\). However, it has relatively small impact on the transfer functions from \(V_a\) to \(V_s\) and to \(z\).

6.2.3 Comparison with the experimental Data

%% Load measured FRF
load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');

%% Initialize Simscape data
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
n_hexapod.actuator = initializeAPA('type', '2dof');

%% 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, '/z'],   1, 'openoutput'); io_i = io_i + 1; % Interferometer
io(io_i) = linio([mdl, '/dL'],   1, 'openoutput'); io_i = io_i + 1; % Encoder

%% Identification
Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
Gs.InputName  = {'Va'};
Gs.OutputName = {'Vs', 'z', 'dL'};

Figure 85: Comparison of the measured FRF and the optimized model

Figure 86: Comparison of the measured FRF and the optimized model

The 2-DoF model is quite effective in modelling the transfer function from actuator to force sensor and from actuator to interferometer (Figure 85). But it is not effective in modeling the transfer function from actuator to encoder (Figure 86). This is due to the fact that resonances greatly affecting the encoder reading are not modelled. In the next section, flexible model of the APA will be used to model such resonances.

6.3 Effect of a misalignment of the APA and flexible joints on the transfer function from actuator to encoder

As shown in Figure 75, the dynamics from actuator to encoder for all the struts is very different.

This could be explained by a large variability in the alignment of the flexible joints and the APA (at the time, the alignment pins were not used).

Depending on the alignment, the spurious resonances of the struts (Figure 17) can be excited differently.

Figure 87: Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz

For instance, consider Figure 88 where there is a misalignment in the \(y\) direction. In such case, the mode at 200Hz is foreseen to be more excited as the misalignment \(d_y\) increases and therefore the dynamics from the actuator to the encoder should also change around 200Hz.

Figure 88: Mis-alignement between the joints and the APA

If the misalignment is in the \(x\) direction, the mode at 300Hz should be more affected whereas a misalignment in the \(z\) direction should not affect these resonances.

Such statement is studied in this section.

6.3.1 Load Data

First, the measured FRF are loaded.

load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');

6.3.2 Perfectly aligned APA

Let’s first consider that the strut is perfectly mounted such that the two flexible joints and the APA are aligned.

%% Initialize Simscape data
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
n_hexapod.actuator = initializeAPA('type', 'flexible');

And define the inputs and outputs of the models:

Input: voltage generated by the DAC
Output: measured displacement by the encoder

%% 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, '/dL'],  1, 'openoutput'); io_i = io_i + 1; % Encoder

The transfer function is identified and shown in Figure 89.

%% Identification
Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
Gs.InputName  = {'Va'};
Gs.OutputName = {'dL'};

From Figure 89, it is clear that:

The model with perfect alignment is not matching the measured FRF
The measured FRF have different shapes

Figure 89: Comparison of the model with a perfectly aligned APA and flexible joints with the measured FRF from actuator to encoder

6.3.3 Effect of a misalignment in y

Let’s compute the transfer function from output DAC voltage to the measured displacement by the encoder for several misalignment in the \(y\) direction:

%% Considered misalignments
dy_aligns = [-0.5, -0.1, 0, 0.1, 0.5]*1e-3; % [m]

%% Transfer functions from u to dL for all the misalignment in y direction
Gs_align = {zeros(length(dy_aligns), 1)};

for i = 1:length(dy_aligns)
    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [0; dy_aligns(i); 0]);

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va'};
    G.OutputName = {'dL'};

    Gs_align(i) = {G};
end

The obtained dynamics are shown in Figure 90.

Figure 90: Effect of a misalignement in the \(y\) direction

The alignment of the APA with the flexible joints as a huge influence on the dynamics from actuator voltage to measured displacement by the encoder. The misalignment in the \(y\) direction mostly influences:

the presence of the flexible mode at 200Hz
the location of the complex conjugate zero between the first two resonances:
- if \(d_y < 0\): there is no zero between the two resonances and possibly not even between the second and third ones
- if \(d_y > 0\): there is a complex conjugate zero between the first two resonances
the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)

6.3.4 Effect of a misalignment in x

Let’s compute the transfer function from output DAC voltage to the measured displacement by the encoder for several misalignment in the \(x\) direction:

%% Considered misalignments
dx_aligns = [-0.1, -0.05, 0, 0.05, 0.1]*1e-3; % [m]

%% Transfer functions from u to dL for all the misalignment in x direction
Gs_align = {zeros(length(dx_aligns), 1)};

for i = 1:length(dx_aligns)
    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [dx_aligns(i); 0; 0]);

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va'};
    G.OutputName = {'dL'};

    Gs_align(i) = {G};
end

The obtained dynamics are shown in Figure 91.

Figure 91: Effect of a misalignement in the \(x\) direction

The misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz.

6.3.5 Find the misalignment of each strut

From the previous analysis on the effect of a \(x\) and \(y\) misalignment, it is possible to estimate the \(x,y\) misalignment of the measured struts.

The misalignment that gives the best match for the FRF are defined below.

%% Tuned misalignment [m]
d_aligns = [[-0.05,  -0.3,  0];
            [ 0,      0.5,  0];
            [-0.1,   -0.3,  0];
            [ 0,      0.3,  0];
            [-0.05,   0.05, 0]]'*1e-3;

For each misalignment, the dynamics from the DAC voltage to the encoder measurement is identified.

%% Idenfity the transfer function from actuator to encoder for all cases
Gs_align = {zeros(size(d_aligns,2), 1)};

for i = 1:size(d_aligns,2)
    n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', d_aligns(:,i));

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va'};
    G.OutputName = {'dL'};

    Gs_align(i) = {G};
end

The results are shown in Figure 92.

Figure 92: Comparison (model and measurements) of the FRF from DAC voltage u to measured displacement by the encoders for all the struts

By tuning the misalignment of the APA with respect to the flexible joints, it is relatively easy to obtain a good fit between the model and the measurements (Figure 92).

If encoders are to be used when fixed on the struts, it is therefore very important to properly align the APA and the flexible joints when mounting the struts.

This should be achieve by using alignment pins.

6.4 Effect of flexible joint’s stiffness

6.4.1 Effect of bending stiffness of the flexible joints

Let’s initialize an APA which is a little bit misaligned.

n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align', [0.1e-3; 0.5e-3; 0]);

The bending stiffnesses for which the dynamics is identified are defined below.

%% Tested bending stiffnesses [Nm/rad]
kRs = [3, 4, 5, 6, 7];

Then the identification is performed for all the values of the bending stiffnesses.

%% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
Gs = {zeros(length(kRs), 1)};

for i = 1:length(kRs)
    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
        'type', '4dof', ...
        'kRx', kRs(i), ...
        'kRy', kRs(i));
    n_hexapod.flex_top = initializeTopFlexibleJoint(...
        'type', '4dof', ...
        'kRx', kRs(i), ...
        'kRy', kRs(i));

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va'};
    G.OutputName = {'dL'};

    Gs(i) = {G};
end

The obtained dynamics from DAC voltage to encoder measurements are compared in Figure 93. It is clear that the bending stiffness has a negligible impact on the observed dynamics.

Figure 93: Dynamics from DAC output to encoder for several bending stiffnesses

6.4.2 Effect of axial stiffness of the flexible joints

The axial stiffnesses for which the dynamics is identified are defined below.

%% Tested axial stiffnesses [N/m]
kzs = [5e7 7.5e7 1e8 2.5e8];

Then the identification is performed for all the values of the bending stiffnesses.

%% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
Gs = {zeros(length(kzs), 1)};

for i = 1:length(kzs)
    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
        'type', '4dof', ...
        'kz', kzs(i));
    n_hexapod.flex_top = initializeTopFlexibleJoint(...
        'type', '4dof', ...
        'kz', kzs(i));

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va'};
    G.OutputName = {'dL'};

    Gs(i) = {G};
end

The obtained dynamics from DAC voltage to encoder measurements are compared in Figure 93.

Figure 94: Dynamics from DAC output to encoder for several axial stiffnesses

The axial stiffness of the flexible joint has a large impact on the frequency of the complex conjugate zero. Using the measured FRF on the test-bench, if is therefore possible to estimate the axial stiffness of the flexible joints from le location of the zero. This method gives nice match between the measured FRF and the one extracted from the simscape model, however it could give not so accurate values of the joint’s axial stiffness as other factors are also influencing the location of the zero.

Using this method, an axial stiffness of \(70 N/\mu m\) is found to give good results (and is reasonable based on the finite element models).

6.4.3 Effect of bending damping

7 Function

7.1 `initializeBotFlexibleJoint` - Initialize Flexible Joint

Function description

function [flex_bot] = initializeBotFlexibleJoint(args)
% initializeBotFlexibleJoint -
%
% Syntax: [flex_bot] = initializeBotFlexibleJoint(args)
%
% Inputs:
%    - args -
%
% Outputs:
%    - flex_bot -

Optional Parameters

arguments
    args.type      char   {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'

    args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
    args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
    args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260
    args.kz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*7e7

    args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
    args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
    args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
    args.cz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
end

Initialize the structure

flex_bot = struct();

Set the Joint’s type

switch args.type
  case '2dof'
    flex_bot.type = 1;
  case '3dof'
    flex_bot.type = 2;
  case '4dof'
    flex_bot.type = 3;
end

Set parameters

flex_bot.kRx = args.kRx;
flex_bot.kRy = args.kRy;
flex_bot.kRz = args.kRz;
flex_bot.kz  = args.kz;

flex_bot.cRx = args.cRx;
flex_bot.cRy = args.cRy;
flex_bot.cRz = args.cRz;
flex_bot.cz  = args.cz;

7.2 `initializeTopFlexibleJoint` - Initialize Flexible Joint

Function description

function [flex_top] = initializeTopFlexibleJoint(args)
% initializeTopFlexibleJoint -
%
% Syntax: [flex_top] = initializeTopFlexibleJoint(args)
%
% Inputs:
%    - args -
%
% Outputs:
%    - flex_top -

Optional Parameters

arguments
    args.type      char   {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'

    args.kRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
    args.kRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*5
    args.kRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*260
    args.kz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*7e7

    args.cRx (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
    args.cRy (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
    args.cRz (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
    args.cz  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.001
end

Initialize the structure

flex_top = struct();

Set the Joint’s type

switch args.type
  case '2dof'
    flex_top.type = 1;
  case '3dof'
    flex_top.type = 2;
  case '4dof'
    flex_top.type = 3;
end

Set parameters

flex_top.kRx = args.kRx;
flex_top.kRy = args.kRy;
flex_top.kRz = args.kRz;
flex_top.kz  = args.kz;

flex_top.cRx = args.cRx;
flex_top.cRy = args.cRy;
flex_top.cRz = args.cRz;
flex_top.cz  = args.cz;

7.3 `initializeAPA` - Initialize APA

Function description

function [actuator] = initializeAPA(args)
% initializeAPA -
%
% Syntax: [actuator] = initializeAPA(args)
%
% Inputs:
%    - args -
%
% Outputs:
%    - actuator -

Optional Parameters

arguments
    args.type      char   {mustBeMember(args.type,{'2dof', 'flexible frame', 'flexible'})} = '2dof'

    % Actuator and Sensor constants
    args.Ga  (1,1) double {mustBeNumeric} = 0
    args.Gs  (1,1) double {mustBeNumeric} = 0

    % For 2DoF
    args.k   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.38e6
    args.ke  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.75e6
    args.ka  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e7

    args.c   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e1
    args.ce  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1
    args.ca  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1

    args.Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056

    % Force Flexible APA
    args.xi (1,1) double {mustBeNumeric, mustBePositive} = 0.01
    args.d_align (3,1) double {mustBeNumeric} = zeros(3,1) % [m]

    % For Flexible Frame
    args.ks (1,1) double {mustBeNumeric, mustBePositive} = 235e6
    args.cs (1,1) double {mustBeNumeric, mustBePositive} = 1e1
end

Initialize Structure

actuator = struct();

Type

switch args.type
  case '2dof'
    actuator.type = 1;
  case 'flexible frame'
    actuator.type = 2;
  case 'flexible'
    actuator.type = 3;
end

Actuator/Sensor Constants

if args.Ga == 0
    switch args.type
      case '2dof'
        actuator.Ga = -30.0;
      case 'flexible frame'
        actuator.Ga = 1; % TODO
      case 'flexible'
        actuator.Ga = 23.4;
    end
else
    actuator.Ga = args.Ga; % Actuator gain [N/V]
end

if args.Gs == 0
    switch args.type
      case '2dof'
        actuator.Gs = 0.098;
      case 'flexible frame'
        actuator.Gs = 1; % TODO
      case 'flexible'
        actuator.Gs = -4674824;
    end
else
    actuator.Gs = args.Gs; % Sensor gain [V/m]
end

2DoF parameters

actuator.k  = args.k;  % [N/m]
actuator.ke = args.ke; % [N/m]
actuator.ka = args.ka; % [N/m]

actuator.c  = args.c;  % [N/(m/s)]
actuator.ce = args.ce; % [N/(m/s)]
actuator.ca = args.ca; % [N/(m/s)]

actuator.Leq = args.Leq; % [m]

Flexible frame and fully flexible

switch args.type
  case 'flexible frame'
    actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
    actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
    actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
  case 'flexible'
    actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix
    actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix
    actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m]
    actuator.d_align = args.d_align;
end

actuator.xi = args.xi; % Damping ratio

actuator.ks = args.ks; % Stiffness of one stack [N/m]
actuator.cs = args.cs; % Damping of one stack [N/m]

7.4 `generateSweepExc`: Generate sweep sinus excitation

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];

7.5 `generateShapedNoise`: Generate Shaped Noise excitation

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];

7.6 `generateSinIncreasingAmpl`: Generate Sinus with increasing amplitude

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];

Bibliography

Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” CEAS Space Journal 10 (2). Springer:157–65.

Nano-Hexapod Struts - Test Bench

Table of Contents

1 Model of the Amplified Piezoelectric Actuator

1.1 Two Degrees of Freedom Model

1.2 Flexible Model

1.3 Actuator and Sensor constants

2 First Basic Measurements

2.1 Geometrical Measurements

2.1.1 Measurement Setup

2.1.2 Measurement Results

2.2 Electrical Measurements

2.3 Stroke measurement

2.3.1 Voltage applied on one stack

2.3.2 Voltage applied on two stacks

2.3.3 Voltage applied on all three stacks

2.4 Spurious resonances

2.4.1 Introduction

2.4.2 Setup

2.4.3 Bending - X

2.4.4 Bending - Y

2.4.5 Torsion - Z

2.4.6 Compare

2.4.7 Conclusion

3 Dynamical measurements - APA

3.1 Speedgoat Setup

3.1.1 frf_setup.m - Measurement Setup

3.1.2 frf_save.m - Save Data

3.2 Measurements on APA 1

3.2.1 Excitation Signal

3.2.2 FRF Identification - Setup

3.2.3 FRF Identification - Displacement

3.2.4 FRF Identification - Force Sensor

3.2.5 Hysteresis

3.2.6 Estimation of the APA axial stiffness

3.2.7 Stiffness change due to electrical connections

3.2.8 Effect of the resistor on the IFF Plant

3.3 Comparison of all the APA

3.3.1 Axial Stiffnesses - Comparison

3.3.2 FRF Identification - Setup

3.3.3 FRF Identification - DVF

3.3.4 FRF Identification - IFF

3.3.5 Save measured FRF

4 Test Bench APA300ML - Simscape Model

4.1 Introduction

4.2 First Identification

4.3 Identify Sensor/Actuator constants and compare with measured FRF

4.3.1 How to identify these constants?

4.3.1.1 Piezoelectric Actuator Constant

4.3.1.2 Piezoelectric Sensor Constant

4.3.2 Identification Data

4.3.3 2DoF APA

4.3.3.1 2DoF APA

4.3.3.2 Identification without actuator or sensor constants

4.3.3.3 Actuator Constant

4.3.3.4 Sensor Constant

4.3.3.5 Comparison

4.3.4 Flexible APA

4.3.4.1 Flexible APA

4.3.4.2 Identification without actuator or sensor constants

4.3.4.3 Actuator Constant

4.3.4.4 Sensor Constant

4.3.4.5 Comparison

4.4 Optimize 2-DoF model to fit the experimental Data

5 Dynamical measurements - Struts

5.1 Measurement on Strut 1

5.1.1 Without Encoder

5.1.1.1 FRF Identification - Setup

5.1.1.2 FRF Identification - Displacement

5.1.1.3 FRF Identification - IFF

5.1.2 With Encoder

5.1.2.1 Measurement Data

5.1.2.2 FRF Identification - DVF

5.1.2.3 Comparison of the Encoder and Interferometer

5.1.2.4 APA Resonances Frequency

5.1.2.5 Estimated Flexible Joint axial stiffness

5.1.2.6 FRF Identification - IFF

5.2 Comparison of all the Struts

5.2.1 FRF Identification - Setup

5.2.2 FRF Identification - Encoder

5.2.3 FRF Identification - Interferometer

3.1.1 `frf_setup.m` - Measurement Setup

3.1.2 `frf_save.m` - Save Data

7.1 `initializeBotFlexibleJoint` - Initialize Flexible Joint

7.2 `initializeTopFlexibleJoint` - Initialize Flexible Joint

7.3 `initializeAPA` - Initialize APA

7.4 `generateSweepExc`: Generate sweep sinus excitation

7.5 `generateShapedNoise`: Generate Shaped Noise excitation

7.6 `generateSinIncreasingAmpl`: Generate Sinus with increasing amplitude