227 KiB
Nano-Hexapod Struts - Test Bench
- Introduction
- Model of the Amplified Piezoelectric Actuator
- First Basic Measurements
- Dynamical measurements - APA
- Test Bench APA300ML - Simscape Model
- Dynamical measurements - Struts
- Test Bench Struts - Simscape Model
- Function
initializeBotFlexibleJoint- Initialize Flexible JointinitializeTopFlexibleJoint- Initialize Flexible JointinitializeAPA- Initialize APAgenerateSweepExc: Generate sweep sinus excitationgenerateShapedNoise: Generate Shaped Noise excitationgenerateSinIncreasingAmpl: Generate Sinus with increasing amplitude
- Bibliography
This report is also available as a pdf.
Introduction ignore
In this document, a test-bench is used to characterize the struts of the nano-hexapod.
Each strut includes (Figure fig:picture_strut_top_view):
- 2 flexible joints at each ends. These flexible joints have been characterized in a separate test bench.
- 1 Amplified Piezoelectric Actuator (APA300ML) (described in Section sec:model_apa). Two stacks are used as an actuator and one stack as a (force) sensor.
- 1 encoder (Renishaw Vionic) that has been characterized in a separate test bench.
The first goal is to characterize the APA300ML in terms of:
- The, geometric features, electrical capacitance, stroke, hysteresis, spurious resonances. This is performed in Section sec:first_measurements.
- The dynamics from the generated DAC voltage (going to the voltage amplifiers and then 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 sec:dynamical_meas_apa.
- Compare the measurements with the Simscape models (2DoF, Super-Element) in order to tuned/validate the models. This is explained in Section sec:simscape_bench_apa.
Then the struts are mounted (procedure described here), and are fixed to the same measurement bench. Similarly, the goals are to:
-
Section sec:dynamical_meas_struts: Identify the dynamics from the generated DAC voltage to:
- the sensors stack generated voltage
- the measured displacement by the encoder
- the measured displacement by the interferometer (representing encoders that would be fixed to the nano-hexapod's plates instead of the struts)
- Section sec:simscape_bench_struts: Compare the measurements with the Simscape model of the struts and tune the models
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.
Model of the Amplified Piezoelectric Actuator
<<sec:model_apa>>
Introduction ignore
The Amplified Piezoelectric Actuator (APA) used is the APA300ML from Cedrat technologies (Figure fig:apa300ML).
Two simscape models of the APA300ML are developed:
- Section sec:apa_2dof_model: a simple 2 degrees of freedom (DoF) model
- Section sec:apa_flexible_model: 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 sec:apa_constants.
Two Degrees of Freedom Model
<<sec:apa_2dof_model>>
The presented model is based on cite:souleille18_concep_activ_mount_space_applic and represented in Figure fig:souleille18_model_piezo.
The parameters are described in Table tab:souleille18_model_params.
| Meaning | |
|---|---|
| $k_e$ | Stiffness used to adjust the pole of the isolator |
| $k_1$ | Stiffness of the metallic suspension when the stack is removed |
| $k_a$ | Stiffness of the actuator |
| $c_1$ | Added viscous damping |
The model is shown again in Figure fig:2dof_apa_model. As will be shown in the next section, such model can be quite accurate in modelling the axial behavior of the APA. However, it does not model the flexibility of the APA in the other directions.
Therefore this model can be useful for quick simulations as it contains a very limited number of states, but when more complex dynamics of the APA is to be modelled, a flexible model will be used.
Flexible Model
<<sec:apa_flexible_model>>
In order to model with high accuracy the behavior of the APA, a flexible model can be used.
The idea is to do a Finite element model of the structure, and to defined "remote points" as shown in Figure fig:apa300ml_ansys. Then, on the finite element software, a "super-element" can be extracted which consists of a mass matrix, a stiffness matrix, and the coordinates of the remote points.
This "super-element" can then be included in the Simscape model as shown in Figure fig:figure_name. The remotes points are defined as "frames" in Simscape, and the "super-element" can be connected with other Simscape elements (mechanical joints, masses, force actuators, etc..).
Actuator and Sensor constants
<<sec:apa_constants>>
On Simscape, we want to model both the actuator stacks and the sensors stack. We therefore need to link the electrical domain (voltages, charges) with the mechanical domain (forces, strain). To do so, we use the "actuator constant" and the "sensor constant".
Consider a schematic of the Amplified Piezoelectric Actuator in Figure fig:apa_model_schematic.
A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$:
\begin{equation} \boxed{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} \boxed{V_s = g_s \cdot dl} \end{equation}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}
The constants $g_a$ and $g_s$ will be experimentally estimated.
First Basic Measurements
<<sec:first_measurements>>
Introduction ignore
Before using the measurement bench to characterize the APA300ML, first simple measurements are performed:
- Section sec:geometrical_measurements: the geometric tolerances of the interface planes are checked
- Section sec:electrical_measurements: the capacitance of the stacks are measured
- Section sec:stroke_measurements: the stroke of the APA are measured
- Section sec:spurious_resonances: the "spurious" resonances of the APA are investigated
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.
%% Measured height for all the APA at the 8 locations
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.
%% X-Y positions of the measurements points
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.
%% Using fminsearch to find the best fitting plane
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;
endThe 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 |
The measured flatness of the APA300ML interface planes are within the specifications.
Electrical Measurements
<<sec:electrical_measurements>>
Measurement Setup
The capacitance of the stacks is measure with the LCR-800 Meter (doc) shown in Figure fig:LCR_meter. The excitation frequency is set to be 1kHz.
Measured Capacitance
From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\mu F$ and $26\mu F$ with a nominal capacitance of $20\mu F$. However, from the documentation of the stack themselves, it can be seen that the capacitance of a single stack should be $4.4\mu F$. Clearly, the total capacitance of the APA300ML if more than just three times the capacitance of one stack.
Could it be possible that the capacitance of the stacks increase that much when they are pre-stressed?
The measured capacitance of the stacks are summarized in Table tab:apa300ml_capacitance.
| 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 |
From the measurements (Table tab:apa300ml_capacitance), the capacitance of one stack is found to be $\approx 5 \mu F$.
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
From the documentation, the nominal stroke of the APA300ML is $304\,\mu m$.
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.
The obtained displacements for all the APA are shown in Figure fig:apa_stroke_time_1s. The displacement is set to zero at initial time when the voltage applied is -20V.
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.
We can clearly confirm 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.
Finally, the displacement is shown as a function of the applied voltage in Figure fig:apa_d_vs_V_2s.
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).
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 |
Conclusion
The except from APA 3 that has a problem, all the APA are similar when it comes to stroke and hysteresis. Also, the obtained stroke is more than specified in the documentation. Therefore, only two stacks can be used as an actuator.
Spurious resonances
<<sec:spurious_resonances>>
Introduction
From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure fig:apa_mode_shapes_ter):
- Mode in X-bending at 200Hz
- Mode in Y-bending at 285Hz
- Mode in Z-torsion 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.
Measurement Setup
The measurement setup is shown in Figure fig:measurement_setup_torsion. A Laser vibrometer is measuring the difference of motion between two points. The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.
The instrumentation used are:
- Laser Doppler Vibrometer Polytec OFV512
- Instrumented hammer
X-Bending Mode
The vibrometer is 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.
%% Load Data
bending_X = load('apa300ml_bending_X_top.mat');
The configuration (Sampling time and windows) for tfestimate is done:
%% Spectral Analysis setup
Ts = bending_X.Track1_X_Resolution; % Sampling Time [s]
win = hann(ceil(1/Ts));The transfer function from the input force to the output "rotation" (difference between the two measured distances).
%% Compute the transfer function from applied force to measured rotation
[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).
Then the APA is in the "free-free" condition, this bending mode is foreseen to be at 200Hz (Figure fig:apa_mode_shapes_ter). We are here in the "fixed-free" condition. If we consider that we therefore double the stiffness associated with this mode, we should obtain a resonance a factor $\sqrt{2}$ higher than 200Hz which is indeed 280Hz. Not sure this reasoning is correct though.
Y-Bending Mode
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.
%% Load Data
bending_Y = load('apa300ml_bending_Y_top.mat');
%% Compute the transfer function
[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.
We can apply the same reasoning as in the previous section and estimate the mode to be a factor $\sqrt{2}$ higher than the mode estimated in the "free-free" condition. We would obtain a mode at 403Hz which is very close to the one estimated here.
Z-Torsion Mode
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.
%% Load Data
torsion = load('apa300ml_torsion_left.mat');
%% Compute transfer function
[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. A third mode at 800Hz could correspond to this torsion mode.
In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited.
%% Load data
torsion = load('apa300ml_torsion_top.mat');
%% Compute transfer function
[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.
Compare
The three measurements are shown in Figure fig:apa300ml_meas_freq_compare.
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 | FEM - Strut mode | Measured Frequency |
|---|---|---|
| X-Bending | 189Hz | 280Hz |
| Y-Bending | 285Hz | 410Hz |
| Z-Torsion | 400Hz | 800Hz? |
Dynamical measurements - APA
<<sec:dynamical_meas_apa>>
Introduction ignore
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
- "Actuator constant": Gain from the applied voltage $V_a$ to the generated Force $F_a$
- "Sensor constant": Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$
- Dynamical behavior from the actuator to the force sensor and to the motion 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_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.
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 Signals
Different excitation signals are used to perform FRF estimations.
Typically, this is done in three steps:
- A low pass filtered white noise is used with rather small amplitudes (Figure fig:exc_signal_1_noise). This first excitation is used to estimate the main resonance of the system.
- A sweep-sine from 10Hz to 400Hz is used (Figure fig:exc_signal_2_sweep). The sweep-sine is is notched around the estimated resonance of the system.
- A band-limited white noise from 300Hz to 2kHz is used to estimate the high frequency behavior (Figure fig:exc_signal_3_hf_noise).
For all the excitation signals, before the excitation starts, the mean voltage is slowly increased halfway between the minimum voltage (-20V) and the maximum (150V).
The first measurement is only used to have a first estimation of the dynamics and verify that everything is setup correctly. The second excitation is done to estimate the dynamics from 10Hz to 350Hz and the third excitation from 350Hz to 2kHz. The second and third measurements are therefore combined in the frequency domain to form one good estimation of the dynamics from 10Hz up to 2kHz.
First Measurement
For this first measurement for the first APA, a basic logarithmic sweep is used between 10Hz and 2kHz.
The data are loaded.
%% Load data
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.
FRF - Setup
Let's define the sampling time/frequency.
%% Sampling Frequency / Time
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 WindowsWe 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 - Encoder and Interferometer
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 $d_e$ and from $V_a$ to $d_a$ are computed and shown in Figure fig:apa_1_coh_dvf. They are quite good from 10Hz up to 500Hz.
%% Compute the coherence
[enc_coh, ~] = mscohere(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
[int_coh, ~] = mscohere(apa_sweep.Va, apa_sweep.da, win, [], [], 1/Ts);
The transfer functions are then estimated and shown in Figure fig:apa_1_frf_dvf.
%% TF - Encoder and interferometer
[frf_enc, ~] = tfestimate(apa_sweep.Va, apa_sweep.de, win, [], [], 1/Ts);
[frf_int, ~] = tfestimate(apa_sweep.Va, apa_sweep.da, win, [], [], 1/Ts);It is shown than both the encoder and interferometers are measuring the same dynamics up to $\approx 700\,Hz$. Above that, it is possible that there is some flexible elements apart from the APA that is adding resonances into one or the other FRF.
The transfer functions obtained in Figure fig:apa_1_frf_dvf are very close to what was expected:
- constant gain at low frequency
- resonance at around 100Hz which corresponds to the APA axial mode
- no further resonance up until high frequency ($\approx 700\,Hz$) at which points several elements of the test bench can induces resonances in the measured FRF
However, it was not expected to observe a "double resonance" at around 95Hz (instead of only one resonance).
FRF - 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.
%% Compute the coherence from Va to Vs
[iff_coh, ~] = mscohere(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);
The transfer function is estimated and shown in Figure fig:apa_1_frf_iff.
%% Compute the TF from Va to Vs
[iff_sweep, ~] = tfestimate(apa_sweep.Va, apa_sweep.Vs, win, [], [], 1/Ts);
The obtained dynamics from the excitation voltage $V_a$ to the measured sensor stack voltage $V_s$ is corresponding to what was expected:
- constant gain at low frequency
- complex conjugate zero and then complex conjugate pole
- constant gain at high frequency
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 (halfway between -20V and 150V), and the sin amplitude is ranging from 1V up to 80V (full range).
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.
%% Load measured data - hysteresis
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 amplitudesThe excitation voltage and the measured displacement are shown in Figure fig:hyst_exc_signal_time.
For each amplitude, we only take the last sinus in order to reduce possible transients. Also, the motion is centered on zero.
The measured displacement at a function of the output voltage are shown in Figure fig:hyst_results_multi_ampl.
From Figure fig:hyst_results_multi_ampl, it is quite clear that hysteresis is increasing with the excitation amplitude. For small excitation amplitudes ($V_a < 0.4\,V$) the hysteresis stays reasonably small.
Also, it is quite interesting to see that no hysteresis is found on the sensor stack voltage when using the same excitation signal.
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 can then be estimated to be:
\begin{equation} k_{\text{apa}} = \frac{m_a g}{d} \end{equation}The data is loaded, and the measured displacement is shown in Figure fig:apa_1_meas_stiffness.
%% Load data for stiffness measurement
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));
From Figure fig:apa_1_meas_stiffness, it can be seen that there are some drifts that are probably due to some creep. This will induce some uncertainties in the measured stiffness.
Here, a mass of 6.4 kg was used:
added_mass = 6.4; % Added mass [kg]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]
The stiffness could also be estimated based on the main vertical resonance of the system at $\omega_z = 2\pi \cdot 94 \,[rad/s]$. The suspended mass is $m_{\text{sus}} = 5\,kg$. And therefore, the axial stiffness of the APA can be estimated to be:
\begin{equation} k_{\text{APA}} = m_{\text{sus}} \omega_z^2 \end{equation}wz = 2*pi*94; % [rad/s]
msus = 5.7; % [kg]
k = msus * wz^2;k = 1.99 [N/um]
The two values are found relatively close to each other. Anyway, the stiffness of the model will be tuned to match the measured FRF.
Stiffness change due to electrical connections
Changes in the electrical impedance connected to the piezoelectric actuator causes changes in the mechanical compliance (or stiffness) of the piezoelectric actuator.
In this section is measured the stiffness of the APA whether the piezoelectric actuator is connected to an open circuit or a short circuit (e.g. the output of a voltage amplifier).
Note here that the resistor in parallel to the sensor stack is present in both cases.
First, the data are loaded.
%% Load Data
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.
%% Zero displacement at initial time
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.
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 its 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.
This is done for two reasons (explained in details this document):
- Limit the voltage offset due to the input bias current of the ADC
- Limit the low frequency gain
The (low frequency) transfer function from $V_a$ to $V_s$ with and without this resistor have been measured.
%% Load the data
wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'Va'); % With the resistor
wo_k = load('frf_data_1_sweep_lf.mat', 't', 'Vs', 'Va'); % Without the resistorWe use a very long "Hanning" window for the spectral analysis in order to estimate the low frequency behavior.
win = hanning(ceil(50*Fs)); % Hannning WindowsAnd we estimate the transfer function from $V_a$ to $V_s$ in both cases:
%% Compute the transfer functions from Va to Vs
[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:
%% Model for the high pass filter
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.
The added resistor has indeed the expected effect of forming an high pass filter.
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 numbers of the APA that have been measured:
apa_nums = [1 2 4 5 6 7 8];The data are loaded.
%% Load Data
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));
endThe raw measurements are shown in Figure fig:apa_meas_k_time. All the APA seems to have similar stiffness except the APA 7 which show strange behavior.
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? It is probably due to the fact that the mechanical element holding the granite in place was not removed.
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, maybe there was a problem with the experimental setup.
FRF - 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.
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')};
endThe 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 WindowsWe 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;FRF - Encoder and Interferometer
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)];
endThe coherence is shown in Figure fig:apa_frf_dvf_plant_coh, and it is found that the coherence is good from low frequency up to 700Hz.
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)];
endThe 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 dynamical behavior is so different from the other? We could think that the APA7 is stiffer, but also the mass line is off, and it should indeed be identical.
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?
FRF - Force Sensor
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).
%% Compute the 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
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_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#+caption:Identified IFF Plant
Conclusion
Except the APA 7 which shows strange behavior, all the other APA are showing a very similar behavior.
So far, all the measured FRF are showing the dynamical behavior that was expected.
%% Remove the APA 7 (6 in the list) from measurements
apa_nums(6) = [];
enc_frf(:,6) = [];
iff_frf(:,6) = [];%% Save the measured FRF
save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');Test Bench APA300ML - Simscape Model
<<sec:simscape_bench_apa>>
Introduction ignore
In this section, a simscape model (Figure fig:model_bench_apa) of the measurement bench is used to compare the model of the APA with the measured FRF.
After the transfer functions are extracted from the model (Section sec:simscape_bench_apa_first_id), the comparison of the obtained dynamics with the measured FRF will permit to:
- Estimate the "actuator constant" and "sensor constant" (Section sec:simscape_bench_apa_id_constants)
- Tune the model of the APA to match the measured dynamics (Section sec:simscape_bench_apa_tune_2dof_model)
First Identification
<<sec:simscape_bench_apa_first_id>>
The APA is first initialized with default parameters:
%% Initialize the structure with default values
n_hexapod = struct();
n_hexapod.actuator = initializeAPA(...
'type', '2dof', ...
'Ga', 1, ... % Actuator constant [N/V]
'Gs', 1); % Sensor constant [V/m]The transfer function from excitation voltage $V_a$ (before the amplification of $20$ due to the PD200 amplifier) to:
- the sensor stack voltage $V_s$
- the measured displacement by the encoder $d_e$
- the measured displacement by the interferometer $d_a$
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % DAC Voltage
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
io(io_i) = linio([mdl, '/da'], 1, 'openoutput'); io_i = io_i + 1; % Interferometer
%% Run the linearization
Ga = linearize(mdl, io, 0.0, options);
Ga.InputName = {'Va'};
Ga.OutputName = {'Vs', 'de', 'da'};The obtain dynamics are shown in Figure fig:apa_model_bench_bode_vs and fig:apa_model_bench_bode_dl_z. It can be seen that:
- the shape of these bode plots are very similar to the one measured in Section sec:dynamical_meas_apa expect from a change in gain and exact location of poles and zeros
- there is a sign error for the transfer function from $V_a$ to $V_s$. This will be corrected by taking a negative "sensor gain".
- the low frequency zero of the transfer function from $V_a$ to $V_s$ is minimum phase as expected. The measured FRF are showing non-minimum phase zero, but it is most likely due to measurements artifacts.
Identify Sensor/Actuator constants and compare with measured FRF
<<sec:simscape_bench_apa_id_constants>>
How to identify these 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 $d$.
\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} \label{eq:actuator_constant_formula} \boxed{g_a = \frac{F_a}{V_a} = \frac{F_a}{d} ⋅ \frac{d}{V_a} = \frac{gd/V_a}{gd/F_a}}
\end{equation}
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 piezoelectric 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} \label{eq:sensor_constant_formula} \boxed{g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} ⋅ \frac{V_a}{F_a} ⋅ \frac{F_a}{dl} = \frac{gV_s/V_a}{g_a ⋅ gdl/F_a}}
\end{equation}
Identification Data
Let's load the measured FRF from the DAC voltage to the measured encoder and to the sensor stack voltage.
%% Load Data
load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');2DoF APA
2DoF APA
Let's initialize the APA as a 2DoF model with unity sensor and actuator gains.
%% Initialize a 2DoF APA with Ga=Gs=1
n_hexapod.actuator = initializeAPA(...
'type', '2dof', ...
'ga', 1, ...
'gs', 1);Identification without actuator or sensor constants
The transfer function from $V_a$ to $V_s$, $d_e$ and $d_a$ is identified.
%% 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, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
io(io_i) = linio([mdl, '/da'], 1, 'openoutput'); io_i = io_i + 1; % Attocube
%% Identification
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'de', 'da'};Actuator Constant
Then, the actuator constant can be computed as shown in Eq. eqref:eq:actuator_constant_formula by dividing the measured DC gain of the transfer function from $V_a$ to $d_e$ by the estimated DC gain of the transfer function from $V_a$ (in truth the actuator force called $F_a$) to $d_e$ using the Simscape model.
%% Estimated Actuator Constant
ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('de', 'Va')); % [N/V]ga = -32.2 [N/V]
Sensor Constant
Similarly, the sensor constant can be estimated using Eq. eqref:eq:sensor_constant_formula.
%% 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]
Comparison
Let's now initialize the APA with identified sensor and actuator constant:
%% Set the identified constants
n_hexapod.actuator = initializeAPA(...
'type', '2dof', ...
'ga', ga, ... % Actuator gain [N/V]
'gs', gs); % Sensor gain [V/m]And 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', 'de', 'da'};The transfer functions from $V_a$ to $d_e$ are compared in Figure fig:apa_act_constant_comp and the one from $V_a$ to $V_s$ are compared in Figure fig:apa_sens_constant_comp.
The "actuator constant" and "sensor constant" can indeed be identified using this test bench. After identifying these constants, the 2DoF model shows good agreement with the measured dynamics.
Flexible APA
Introduction ignore
In this section, the sensor and actuator "constants" are also estimated for the flexible model of the APA.
Flexible APA
The Simscape APA model is initialized as a flexible one with unity "constants".
%% Initialize the APA as a flexible body
n_hexapod.actuator = initializeAPA(...
'type', 'flexible', ...
'ga', 1, ...
'gs', 1);Identification without actuator or sensor constants
The dynamics from $V_a$ to $V_s$, $d_e$ and $d_a$ is identified.
%% Identify the dynamics
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'de', 'da'};Actuator Constant
Then, the actuator constant can be computed as shown in Eq. eqref:eq:actuator_constant_formula:
%% Actuator Constant
ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(Gs('de', 'Va')); % [N/V]ga = 23.5 [N/V]
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]
Comparison
Let's now initialize the flexible APA with identified sensor and actuator constant:
%% Set the identified constants
n_hexapod.actuator = initializeAPA(...
'type', 'flexible', ...
'ga', ga, ... % Actuator gain [N/V]
'gs', gs); % Sensor gain [V/m]And identify the dynamics with included constants.
%% Identify with updated constants
Gs = linearize(mdl, io, 0.0, options);
Gs = Gs*exp(-Ts*s);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'de', 'da'};The obtained dynamics is compared with the measured one in Figures fig:apa_act_constant_comp_flex and fig:apa_sens_constant_comp_flex.
The flexible model is a bit "soft" as compared with the experimental results.
Optimize 2-DoF model to fit the experimental Data
<<sec:simscape_bench_apa_tune_2dof_model>> The parameters of the 2DoF model presented in Section sec:apa_2dof_model are now optimize such that the model best matches the measured FRF.
After optimization, the following parameters are used:
%% Optimized parameters
n_hexapod.actuator = initializeAPA('type', '2dof', ...
'Ga', -32.2, ...
'Gs', 0.088, ...
'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 ...
);The dynamics is identified using the Simscape model and compared with the measured FRF in Figure fig:comp_apa_plant_after_opt.
The tuned 2DoF is very well representing the (axial) dynamics of the APA.
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_coder).
Variables are named the same as in Section sec:dynamical_meas_apa.
First, only one strut is measured in details (Section sec:meas_strut_1), and then all the struts are measured and compared (Section sec:meas_all_struts).
Measurement on Strut 1
<<sec:meas_strut_1>>
Introduction ignore
Measurements are first performed on one of the strut that contains:
- the Amplified Piezoelectric Actuator (APA) number 1
- flexible joints 1 and 2
In Section sec:meas_strut_1_no_encoder, the dynamics of the strut is measured without the encoder attached to it. Then in Section sec:meas_strut_1_encoder, the encoder is attached to the struts, and the dynamic is identified.
Without Encoder
<<sec:meas_strut_1_no_encoder>>
FRF Identification - Setup
Similarly to what was done for the identification of the APA, 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.
%% Load Data
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 frequency/time
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 WindowsWe 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);
i_lf = f <= 350; % Indices used for the low frequency
i_hf = f > 350; % Indices used for the low frequencyFRF Identification - Interferometer
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 and combine them.
%% Compute the coherence for both excitation signals
[int_coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts);
[int_coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
%% Combine the coherence
int_coh = [int_coh_sweep(i_lf); int_coh_noise_hf(i_hf)];The combined coherence is shown in Figure fig:strut_1_frf_dvf_plant_coh, and is found to be very good up to at least 1kHz.
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.
%% Compute FRF function from Va to da
[frf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts);
[frf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
%% Combine the FRF
int_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
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).
%% Compute the coherence for both excitation signals
[iff_coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[iff_coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
%% Combine the coherence
iff_coh = [iff_coh_sweep(i_lf); iff_coh_noise_hf(i_hf)];
Then the FRF are estimated and shown in Figure fig:strut_1_frf_iff_plant_tf
%% Compute the FRF
[frf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
[frf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
%% Combine the FRF
iff_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];#+caption:Identified IFF Plant for the Strut 1
With Encoder
<<sec:meas_strut_1_encoder>>
Introduction ignore
Now the encoder is fixed to the strut and the identification is performed.
Measurement Data
The measurements are loaded.
%% Load 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 - Interferometer
In this section, the dynamics from $V_a$ to $d_a$ is identified.
First, the coherence is computed and shown in Figure fig:strut_1_int_with_enc_frf_dvf_plant_coh.
%% Compute the coherence for both excitation signals
[int_coh_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.da, win, [], [], 1/Ts);
[int_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);
%% Combine the coherinte
int_coh = [int_coh_sweep(i_lf); int_coh_noise_hf(i_hf)];
Then the FRF are computed and shown in Figure fig:strut_1_int_with_enc_frf_dvf_plant_tf.
%% Compute FRF function from Va to da
[frf_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.da, win, [], [], 1/Ts);
[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);
%% Combine the FRF
int_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
The obtained FRF is very close to the one that was obtained when no encoder was fixed to the struts as shown in Figure fig:strut_leg_compare_int_frf.
FRF Identification - Encoder
In this section, the dynamics from $V_a$ to $d_e$ (encoder) is identified.
The coherence is computed and shown in Figure fig:strut_1_enc_frf_dvf_plant_coh.
%% Compute the coherence for both excitation signals
[enc_coh_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.de, win, [], [], 1/Ts);
[enc_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
%% Combine the coherence
enc_coh = [enc_coh_sweep(i_lf); enc_coh_noise_hf(i_hf)];
The FRF from $V_a$ to the encoder measured displacement $d_e$ is computed and shown in Figure fig:strut_1_enc_frf_dvf_plant_tf.
%% Compute FRF function from Va to da
[frf_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.de, win, [], [], 1/Ts);
[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
%% Combine the FRF
enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];
The transfer functions from $V_a$ to $d_e$ (encoder) and to $d_a$ (interferometer) are compared in Figure fig:strut_1_comp_enc_int.
The dynamics from the excitation voltage $V_a$ to the measured displacement by the encoder $d_e$ presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure fig:strut_1_comp_enc_int). It will be further investigated why the two dynamics as so different and what are causing all these resonances.
APA Resonances Frequency
As shown in Figure fig:strut_1_spurious_resonances, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
These resonances correspond to parasitic resonances of the strut itself.
They are very close to what was estimated using a finite element model of the strut (Figure fig:apa_mode_shapes_bis):
- Mode in X-bending at 189Hz
- Mode in Y-bending at 285Hz
- Mode in Z-torsion at 400Hz
The resonances seen by the encoder in Figure fig:strut_1_spurious_resonances are indeed corresponding to the modes of the strut as shown in Figure fig:apa_mode_shapes_bis.
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 fig:strut_1_frf_iff_with_enc_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).
%% Compute the coherence for both excitation signals
[iff_coh_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, [], [], 1/Ts);
[iff_coh_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
%% Combine the coherence
iff_coh = [iff_coh_sweep(i_lf); iff_coh_noise_hf(i_hf)];
Then the FRF are estimated and shown in Figure fig:strut_1_enc_frf_iff_plant_tf
%% Compute FRF function from Va to da
[frf_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, [], [], 1/Ts);
[frf_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
%% Combine the FRF
iff_with_enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)];#+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).
The transfer function from the excitation voltage $V_a$ to the generated voltage $V_s$ by the sensor stack is not influence by the fixation of the encoder. This means that the IFF control strategy should be as effective whether or not the encoders 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 as in Section sec:meas_strut_1.
FRF Identification - Setup
The identification of the struts dynamics is performed in two steps:
- The excitation signal is a white noise with small amplitude. This is used to estimate the low frequency dynamics.
- Then a high frequency noise band-passed between 300Hz and 2kHz is used to estimate the high frequency dynamics.
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.
%% Numnbers of the measured legs
leg_nums = [1 2 3 4 5];The data are loaded for both the first and second identification:
%% First identification (low frequency noise)
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
%% Second identification (high frequency noise)
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')};
endThe 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 WindowsWe 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;FRF Identification - Encoder
In this section, the dynamics from $V_a$ to $d_e$ (encoder) is identified.
The coherence is computed and shown in Figure fig:struts_frf_dvf_plant_coh for all the measured struts.
%% 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
Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoder $d_e$ 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)];
endThe obtained transfer functions are shown in Figure fig:struts_frf_dvf_plant_tf.
There is a very large variability of the dynamics as measured by the encoder as shown in Figure fig:struts_frf_dvf_plant_tf. Even-though the same peaks are seen for all of the struts (95Hz, 200Hz, 300Hz, 400Hz), the amplitude of the peaks are not the same. Moreover, the location or even the presence of complex conjugate zeros is changing from one strut to the other.
All of this will be explained in Section sec:simscape_bench_struts thanks to the Simscape model.
FRF Identification - Interferometer
In this section, the dynamics from $V_a$ to $d_a$ (interferometer) is identified.
The coherence is computed and shown in Figure fig:struts_frf_int_plant_coh.
%% 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
Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the Attocube is computed for all the struts and shown in Figure fig:struts_frf_int_plant_tf. All the struts are giving very similar FRF.
%% 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
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 fig:struts_frf_iff_plant_coh.
%% 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
Then the FRF are estimated and shown in Figure fig:struts_frf_iff_plant_tf. They are also shown all to be very similar.
%% 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#+caption:Identified IFF Plant
Conclusion
All the struts are giving very consistent behavior from the excitation voltage $V_a$ to the force sensor generated voltage $V_s$ and to the interferometer measured displacement $d_a$. However, the dynamics from $V_a$ to the encoder measurement $d_e$ is much more complex and variable from one strut to the other.
The measured FRF are now saved for further use.
%% Save the estimated FRF for further analysis
save('mat/meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');Test Bench Struts - Simscape Model
<<sec:simscape_bench_struts>>
Introduction ignore
The same simscape model that was presented in Section sec:simscape_bench_apa is here used. However, now the full strut is put instead of only the APA (see Figure fig:simscape_model_bench_struts).
This Simscape model is used to:
- compare the measured FRF with the modelled FRF
- help the correct understanding/interpretation of the results
- tune the model of the struts (APA, flexible joints, encoder)
This study is structured as follow:
- Section sec:struts_comp_2dof: the measured FRF are compared with the 2DoF APA model.
- Section sec:struts_effect_misalignment: the flexible APA model is used, and the effect of a misalignment of the APA and flexible joints is studied. It is found that the misalignment has a large impact on the dynamics from $V_a$ to $d_e$.
- Section sec:struts_effect_joint_stiffness: the effect of the flexible joint's stiffness on the dynamics is studied. It is found that the axial stiffness of the joints has a large impact on the location of the zeros on the transfer function from $V_s$ to $d_e$.
Comparison with the 2-DoF Model
<<sec:struts_comp_2dof>>
First Identification
The strut is initialized with default parameters (optimized parameters identified from previous experiments).
%% Initialize structure containing data for the Simscape model
n_hexapod = struct();
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
n_hexapod.actuator = initializeAPA('type', '2dof');The inputs and outputs of the model are defined.
%% 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, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
io(io_i) = linio([mdl, '/da'], 1, 'openoutput'); io_i = io_i + 1; % InterferometerThe dynamics is identified and shown in Figure fig:strut_bench_model_bode.
%% Run the linearization
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'Vs', 'de', 'da'};
Comparison with the experimental Data
The experimentally measured FRF are loaded.
%% Load measured FRF
load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');%% Add time delay to the Simscape model
Gs = exp(-s*Ts)*Gs;The FRF from $V_a$ to $d_a$ as well as from $V_a$ to $V_s$ are shown in Figure fig:comp_strut_plant_after_opt and compared with the model. They are both found to match quite well with the model.
The measured FRF from $V_a$ to $d_e$ (encoder) is compared with the model in Figure fig:comp_strut_plant_iff_after_opt.
The 2-DoF model is quite effective in modelling the transfer function from actuator to force sensor and from actuator to interferometer (Figure fig:comp_strut_plant_after_opt). But it is not effective in modeling the transfer function from actuator to encoder (Figure fig:comp_strut_plant_iff_after_opt). 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.
Effect of a misalignment of the APA and flexible joints on the transfer function from actuator to encoder
<<sec:struts_effect_misalignment>>
Introduction ignore
As shown in Figure fig:struts_frf_dvf_plant_tf, 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 fig:apa_mode_shapes) can be excited differently.
For instance, consider Figure fig:strut_misalign_schematic 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.
If the misalignment is in the $x$ direction, the mode at 285Hz should be more affected whereas a misalignment in the $z$ direction should not affect these resonances.
Such statement is studied in this section.
But first, the measured FRF of the struts are loaded.
%% Load measured FRF of the struts
load('meas_struts_frf.mat', 'f', 'Ts', 'enc_frf', 'int_frf', 'iff_frf', 'leg_nums');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, '/de'], 1, 'openoutput'); io_i = io_i + 1; % EncoderThe transfer function is identified and shown in Figure fig:comp_enc_frf_align_perfect.
%% Identification
Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
Gs.InputName = {'Va'};
Gs.OutputName = {'de'};From Figure fig:comp_enc_frf_align_perfect, it is clear that:
- The model with perfect alignment is not matching the measured FRF
- The mode at 200Hz is not present in the identified dynamics of the Simscape model
- The measured FRF have different shapes
Why is the flexible mode of the strut at 200Hz is not seen in the model in Figure fig:comp_enc_frf_align_perfect?
Probably because the presence of this mode is not due because of the "unbalanced" mass of the encoder, but rather because of the misalignment of the APA with respect to the two flexible joints. This will be verified in the next sections.
Effect of a misalignment in y
Let's compute the transfer function from output DAC voltage $V_s$ to the measured displacement by the encoder $d_e$ 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 de 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 = {'de'};
Gs_align(i) = {G};
endThe obtained dynamics are shown in Figure fig:effect_misalignment_y.
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)
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 de 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 = {'de'};
Gs_align(i) = {G};
endThe obtained dynamics are shown in Figure fig:effect_misalignment_x.
The misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz.
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 = {'de'};
Gs_align(i) = {G};
endThe results are shown in Figure fig:comp_all_struts_corrected_misalign.
By tuning the misalignment of the APA with respect to the flexible joints, it is possible to obtain a good fit between the model and the measurements (Figure fig:comp_all_struts_corrected_misalign).
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.
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.
Effect of flexible joint's stiffness
<<sec:struts_effect_joint_stiffness>>
Introduction ignore
As the struts are composed of one APA and two flexible joints, it is obvious that the flexible joint characteristics will change the dynamic behavior of the struts.
Using the Simscape model, the effect of the flexible joint's characteristics on the dynamics as measured on the test bench are studied:
- Section sec:struts_effect_bending_stiff_joints: the effects of a change of bending stiffness is studied
- Section sec:struts_effect_axial_stiff_joints: the effects of a change of axial stiffness is studied
- Section sec:struts_effect_bending_damping_joints: the effects of a change of bending damping is studied
The studied dynamics is between $V_a$ and the encoder displacement $d_e$.
%% 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, '/de'], 1, 'openoutput'); io_i = io_i + 1; % EncoderEffect of bending stiffness of the flexible joints
<<sec:struts_effect_bending_stiff_joints>>
Let's initialize an APA which is a little bit misaligned.
%% APA Initialization
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 = {'de'};
Gs(i) = {G};
endThe obtained dynamics from DAC voltage to encoder measurements are compared in Figure fig:effect_enc_bending_stiff.
The bending stiffness of the joints has little impact on the transfer function from $V_a$ to $d_e$.
Effect of axial stiffness of the flexible joints
<<sec:struts_effect_axial_stiff_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 = {'de'};
Gs(i) = {G};
endThe obtained dynamics from DAC voltage to encoder measurements are compared in Figure fig:effect_enc_axial_stiff.
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 the 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).
Effect of bending damping
<<sec:struts_effect_bending_damping_joints>> Now let's study the effect of the bending damping of the flexible joints.
The tested bending damping are defined below:
%% Tested bending dampings [Nm/(rad/s)]
cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];Then the identification is performed for all the values of the bending damping.
%% Idenfity the transfer function from actuator to encoder for all bending dampins
Gs = {zeros(length(kRs), 1)};
for i = 1:length(kRs)
n_hexapod.flex_bot = initializeBotFlexibleJoint(...
'type', '4dof', ...
'cRx', cRs(i), ...
'cRy', cRs(i));
n_hexapod.flex_top = initializeTopFlexibleJoint(...
'type', '4dof', ...
'cRx', cRs(i), ...
'cRy', cRs(i));
G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
G.InputName = {'Va'};
G.OutputName = {'de'};
Gs(i) = {G};
endThe results are shown in Figure fig:effect_enc_bending_damp.
Adding damping in bending for the flexible joints could be a nice way to reduce the effects of the spurious resonances of the struts.
How to effectively add damping to the flexible joints?
One idea would be to introduce a sheet of damping material inside the flexible joint. Not sure is would be effect though.
Function
<<sec:functions>>
initializeBotFlexibleJoint - Initialize Flexible Joint
<<sec:initializeBotFlexibleJoint>>
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
endInitialize 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;
endSet 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;
initializeTopFlexibleJoint - Initialize Flexible Joint
<<sec:initializeTopFlexibleJoint>>
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
endInitialize 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;
endSet 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;
initializeAPA - Initialize APA
<<sec:initializeAPA>>
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
endInitialize Structure
actuator = struct();Type
switch args.type
case '2dof'
actuator.type = 1;
case 'flexible frame'
actuator.type = 2;
case 'flexible'
actuator.type = 3;
endActuator/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]
endif 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]
end2DoF 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]
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
endSweep 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"');
endif 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
endSmooth 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 = [];
endCombine 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
endShaped 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)';
endSmooth 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;
endCombine 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
endSinus 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)];
endSmooth 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;
endCombine 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];