\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.
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)}
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 \ref{fig:apa_mode_shapes_ter}):
\caption{\label{fig:apa300ml_meas_freq_bending_x}Obtained FRF for the X-bending}
\end{figure}
Then the APA is in the ``free-free'' condition, this bending mode is foreseen to be at 200Hz (Figure \ref{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.
\subsection{Y-Bending Mode}
The setup to measure the Y-bending is shown in Figure \ref{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).
\caption{\label{fig:apa300ml_meas_freq_bending_y}Obtained FRF for the Y-bending}
\end{figure}
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.
\subsection{Z-Torsion Mode}
Finally, we measure the Z-torsion resonance as shown in Figure \ref{fig:measurement_setup_torsion_bis}.
The excitation is shown on the other side of the APA, on the side to excite the torsion motion.
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?).
\begin{table}[htbp]
\caption{\label{tab:apa300ml_measured_modes_freq}Measured frequency of the modes}
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).
\item measure the dynamics of the APA (from \(V_a\) to \(d_e\) and \(d_a\) in Section \ref{ssec:test_apa_meas_frf_disp}, and from \(V_a\) to \(V_s\) in section \ref{ssec:test_apa_meas_frf_force})
\item estimate the added damping using Integral Force Feedback (Section \ref{ssec:test_apa_iff_locus})
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 \(\omega_z \approx94\,\text{Hz}\) (estimated by the dynamical measurements shown in section \ref{ssec:test_apa_meas_frf_disp}) 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
\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 identified dynamics are then fitted by second order transfer functions using the ``Vector Fitting'' toolbox \cite{gustavsen99_ration_approx_frequen_domain_respon}.
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: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 \ref{sec:simscape_bench_apa_first_id}), the comparison of the obtained dynamics with the measured FRF will permit to:
\begin{enumerate}
\item Estimate the ``actuator constant'' and ``sensor constant'' (Section \ref{sec:simscape_bench_apa_id_constants})
The obtain dynamics are shown in Figure \ref{fig:apa_model_bench_bode_vs} and \ref{fig:apa_model_bench_bode_dl_z}.
It can be seen that:
\begin{itemize}
\item the shape of these bode plots are very similar to the one measured in Section \ref{sec:dynamical_meas_apa} expect from a change in gain and exact location of poles and zeros
\item 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''.
\item 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.
\caption{\label{fig:apa_model_bench_bode_dl_z}Bode plot of the transfer function from \(V_a\) to \(d_L\) and to \(z\)}
\end{figure}
\section{Identify Sensor/Actuator constants and compare with measured FRF}
\label{sec:simscape_bench_apa_id_constants}
\subsection{How to identify these constants?}
\paragraph{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 \(g_a\):
Let's load the measured FRF from the DAC voltage to the measured encoder and to the sensor stack voltage.
\subsection{2DoF APA}
\paragraph{2DoF APA}
Let's initialize the APA as a 2DoF model with unity sensor and actuator gains.
\paragraph{Identification without actuator or sensor constants}
The transfer function from \(V_a\) to \(V_s\), \(d_e\) and \(d_a\) is identified.
\paragraph{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.
\begin{verbatim}
ga = -32.2 [N/V]
\end{verbatim}
\paragraph{Sensor Constant}
Similarly, the sensor constant can be estimated using Eq. \eqref{eq:sensor_constant_formula}.
\begin{verbatim}
gs = 0.088 [V/m]
\end{verbatim}
\paragraph{Comparison}
Let's now initialize the APA with identified sensor and actuator constant:
And identify the dynamics with included constants.
The transfer functions from \(V_a\) to \(d_e\) are compared in Figure \ref{fig:apa_act_constant_comp} and the one from \(V_a\) to \(V_s\) are compared in Figure \ref{fig:apa_sens_constant_comp}.
