\item 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.
\item Compare the measurements with the two Simscape models: 2DoF (Section \ref{sec:test_apa_model_2dof}) Super-Element (Section \ref{sec:test_apa_model_flexible})
To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with very good flatness.
As shown in Figure \ref{fig:test_apa_flatness_setup}, the APA is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(0.5\,\mu m\)} is used to measure the height of 4 points on each of the APA300ML interfaces.
From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.
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 \(20\,\mu F\).
The capacitance of the piezoelectric stacks found in the APA300ML have been measured with the LCR meter\footnote{LCR-819 from Gwinstek, specified accuracy of \(0.05\%\), measured frequency is set at \(1\,\text{kHz}\)} shown in Figure \ref{fig:test_apa_lcr_meter}.
The two stacks used as an actuator and the stack used as a force sensor are measured separately.
The measured capacitance is found to be lower than the specified one.
This may be due to the fact that the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)) while it was here measured with small signals.
The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
To do so, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}.
Then, the voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC and a voltage amplifier.
Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure \ref{fig:test_apa_stroke_bench}, left).
Typical hysteresis curves for piezoelectric stack actuators can be observed.
The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks, which is enough for the current application.
This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu m\), therefore \(\approx200\,\mu m\) if only two stacks are used).
\caption{\label{fig:test_apa_stroke_result}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (left). Measured displacement as a function of the applied voltage (right)}
To experimentally estimate these modes, the APA is fixed on one end (see Figure \ref{fig:test_apa_meas_setup_torsion}).
A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points (i.e. the torsion of the APA along the vertical direction) and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes.
Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated.
The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure \ref{fig:test_apa_mode_shapes}.
\caption{\label{fig:test_apa_meas_setup_torsion}Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer. Here the \(Z\) torsion is measured.}
Two other similar measurements are performed to measured the bending of the APA around the \(X\) direction and around the \(Y\) direction (see Figure \ref{fig:test_apa_meas_setup_modes}).
\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measured flexible modes of the APA300ML. For the bending in the \(X\) direction, the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
After the basic measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the APA.
This test bench is shown in Figure \ref{fig:test_bench_apa} and consists of the APA300ML fixed on one end to the fixed granite, and on the other end to the 5kg granite vertically guided with an air bearing.
An encoder is used to measure the relative motion between the two granites (i.e. the displacement of the APA).
These measurements will also be used to tune the developed models of the APA (in Section \ref{sec:test_apa_model_2dof} for the 2DoF model, and in Section \ref{sec:test_apa_model_flexible} for the flexible model).
A quasi static sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)), and an amplitude varying from \(4\,V\) up to \(80\,V\).
\caption{\label{fig:test_apa_meas_hysteresis}Obtained hysteresis curves (displacement as a function of applied voltage) for multiple excitation amplitudes}
In order to estimate the stiffness of the APA, a weight with known mass \(m_a =6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(d_e\) is measured using the encoder.
The measured displacement \(d_e\) as a function of time is shown in Figure \ref{fig:test_apa_meas_stiffness_time}.
It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
These two effects induce some uncertainties in the measured stiffness.
\caption{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding the mass (at \(t \approx3\,s\)) and removing the mass(at \(t \approx13\,s\))}
The stiffnesses are computed for all the APA from the two displacements \(d_1\) and \(d_2\) (see Figure \ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\).
These estimated stiffnesses are summarized in Table \ref{tab:test_apa_measured_stiffnesses} and are found to be close to the nominal stiffness \(k =1.8\,N/\mu m\) found in the APA300ML manual.
The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx95\,\text{Hz}\) (estimated by the dynamical measurements shown in section \ref{ssec:test_apa_meas_dynamics}) and the suspended mass \(m_{\text{sus}}=5.7\,\text{kg}\).
However, changes in the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack \cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}.
In this section, the dynamics of the system from the excitation voltage \(u\) to encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified.
The obtained transfer functions for the 6 APA between the excitation voltage \(u\) and the encoder displacement \(d_e\) are shown in Figure \ref{fig:test_apa_frf_encoder}.
The obtained transfer functions are close to a mass-spring-damper system.
The following can be observed:
\begin{itemize}
\item A ``stiffness line'' indicating a static gain equal to \(\approx-17\,\mu m/V\).
The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA
\item A lightly damped resonance at \(95\,\text{Hz}\)
\item A ``mass line'' up to \(\approx800\,\text{Hz}\), above which some resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
\caption{\label{fig:test_apa_frf_encoder}Estimated Frequency Response Function from generated voltage \(u\) to the encoder displacement \(d_e\) for the 6 APA300ML}
A lightly damped resonance is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance at \(41\,\text{Hz}\).
No additional resonances is present up to at least \(2\,\text{kHz}\) indicating at Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
As illustrated by the Root Locus, the poles of the closed-loop system converges to the zeros of the open-loop plant.
Suppose that a controller with a very high gain is implemented such that the voltage \(V_s\) across the sensor stack is zero.
In that case, because of the very high controller gain, no stress and strain is present on the sensor stack (and on the actuator stacks are well, as they are both in series).
Such closed-loop system would therefore virtually corresponds to a system for which the piezoelectric stacks have been removed and just the mechanical shell is kept.
From this analysis, the axial stiffness of the shell can be estimated to be \(k_{\text{shell}}=5.7\cdot(2\pi\cdot41)^2=0.38\,N/\mu m\).
Such reasoning can lead to very interesting insight into the system just from an open-loop identification.
\caption{\label{fig:test_apa_frf_force}Estimated Frequency Response Function from generated voltage \(u\) to the sensor stack voltage \(V_s\) for the 6 APA300ML}
All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure \ref{fig:test_apa_frf_encoder} and at the force sensor in Figure \ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical lever.
A resistor \(R \approx80.6\,k\Omega\) is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the stack.
The (low frequency) transfer function from \(u\) to \(V_s\) with and without this resistor have been measured and are compared in Figure \ref{fig:test_apa_effect_resistance}.
It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of \(\approx0.35\,\text{Hz}\).
\caption{\label{fig:test_apa_effect_resistance}Transfer function from u to \(V_s\) with and without the resistor \(R\) in parallel with the piezoelectric stack used as the force sensor}
First, the transfer function \eqref{eq:test_apa_iff_manual_fit} is manually tuned to match the identified dynamics from generated voltage \(u\) to the measured sensor stack voltage \(V_s\) in Section \ref{ssec:test_apa_meas_dynamics}.
The obtained parameter values are \(\omega_{\textsc{hpf}}=0.4\,\text{Hz}\), \(\omega_{z}=42.7\,\text{Hz}\), \(\xi_{z}=0.4\,\%\), \(\omega_{p}=95.2\,\text{Hz}\), \(\xi_{p}=2\,\%\) and \(g_0=0.64\).
\caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\mu s\) is added to the model of the plant to better match the identified phase)}
The implemented Integral Force Feedback Controller transfer function is shown in equation \eqref{eq:test_apa_Kiff_formula}.
It contains an high pass filter (cut-off frequency of \(2\,\text{Hz}\)) to limit the low frequency gain, a low pass filter to add integral action above \(20\,\text{Hz}\), a second low pass filter to add robustness to high frequency resonances and a tunable gain \(g\).
To estimate how the dynamics of the APA changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure \ref{fig:test_apa_iff_schematic} is used.
The transfer function from the ``damped'' plant input \(u\prime\) to the encoder displacement \(d_e\) is identified for several IFF controller gains \(g\).
The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure \ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
\caption{\label{fig:test_apa_identified_damped_plants}Identified dynamics (solid lines) and fitted transfer functions (dashed lines) from \(u\prime\) to \(d_e\) for different IFF gains}
The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_locus} and are in good agreement considering that the damped plants were only fitted using a second order transfer function.
\caption{\label{fig:test_apa_iff_root_locus}Root Locus of the APA300ML with Integral Force Feedback - Comparison between the computed root locus from the plant model (black line) and the root locus estimated from the damped plant pole identification (colorful crosses)}
In this section, a simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to compare the model of the APA with the measured frequency response functions.
A 2 degrees of freedom model is used to model the APA300ML.
This model is presented in Section \ref{ssec:test_apa_2dof_model} and the procedure to tuned the model is described in Section \ref{ssec:test_apa_2dof_model_tuning}.
The obtained model dynamics is compared with the measurements in Section \ref{ssec:test_apa_2dof_model_result}.
9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
\caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees of freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
First, the mass supported by the APA300ML can simply be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
Both methods leads to an estimated mass of \(5.7\,\text{kg}\).
Then, the axial stiffness of the shell was estimated at \(k_1=0.38\,N/\mu m\) in Section \ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure \ref{fig:test_apa_frf_force}.
Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(20\,Ns/m\).
Then, it is reasonable to make the assumption that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not fully correct as it was shown in Section \ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}.
Therefore, we have \(k_e =2 k_a\) and \(c_e =2 c_a\) as the actuator stack is composed of two stacks in series.
In that case, the total stiffness of the APA model is described by \eqref{eq:test_apa_2dof_stiffness}.
Knowing from \eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{\text{tot}}=2\,N/\mu m\), we get from \eqref{eq:test_apa_2dof_stiffness} that \(k_a =2.5\,N/\mu m\) and \(k_e =5\,N/\mu m\).
The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
\begin{table}[htbp]
\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
The dynamics of the 2DoF APA300ML model is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
It is compared with the experimental data in Figure \ref{fig:test_apa_2dof_comp_frf}.
A good match can be observed between the model and the experimental data, both for the encoder and for the force sensor.
This indicates that this model represents well the axial dynamics of the APA300ML.
In this section, a \emph{super element} of the Amplified Piezoelectric Actuator ``APA300ML'' is extracted using a Finite Element Software.
It is then imported in Simscape (using the stiffness and mass matrices) and it is included in the same model that was used in \ref{sec:test_apa_model_2dof}.
\caption{\label{fig:test_apa_super_element_simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
\item Explain how the ``remote points'' are chosen
\item Show some parts of the mass and stiffness matrices?
\item Say which materials were used?
\item Maybe this was already explain earlier in the manuscript
\end{itemize}
\section{Identification of the Actuator and Sensor constants}
\label{ssec:test_apa_flexible_ga_gs}
Once the APA300ML \emph{super element} is included in the Simscape model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be identified.
The gains \(g_a\) and \(g_s\) can then be tuned such that the gain of the transfer functions are matching the identified ones.
By doing so, \(g_s =4.9\,V/\mu m\) and \(g_a =23.2\,N/V\) are obtained.
To make sure these ``gains'' are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:test_apa_piezo_strain_to_voltage} and from \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:test_apa_piezo_voltage_to_force}.
Parameters used in equations \eqref{eq:test_apa_piezo_strain_to_voltage} and \eqref{eq:test_apa_piezo_voltage_to_force} are described in Table \ref{tab:test_apa_piezo_properties}.
Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}.
From these parameters, \(g_s =5.1\,V/\mu m\) and \(g_a =26\,N/V\) were obtained which are very close to the identified constants using the experimentally identified transfer functions.
\begin{table}[htbp]
\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators ``gains''}
The obtained dynamics using the \emph{super element} with the tuned ``sensor gain'' and ``actuator gain'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}.
A good match between the model and the experimental results is observed.
