In this chapter, the goal is to make sure that the received APA300ML (shown in Figure \ref{fig:test_apa_received}) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics.
In section \ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. Flexible modes of the APA300ML are computed with a finite element model and compared with measurements.
Using a dedicated test bench, dynamical measurements are performed (Section \ref{sec:test_apa_dynamics}).
The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
Integral Force Feedback is experimentally applied and the damped plants are estimated for several feedback gains.
Two different models of the APA300ML are then presented.
First, in Section \ref{sec:test_apa_model_2dof}, a two degrees of freedom model is presented, tuned and compared with the measured dynamics.
This model is proven to accurately simulate the APA300ML's axial dynamics.
Then, in Section \ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a finite element model and imported in Simscape.
This more complex model is also shown to well capture the axial dynamics of the APA300ML.
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 excellent 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 \(\pm0.5\,\mu m\)} 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.
The measured flatness, summarized in Table \ref{tab:test_apa_flatness_meas}, are within the specifications.
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 piezoelectric stacks capacitance of 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}.
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 \cite{wehrsdorfer95_large_signal_measur_piezoel_stack}.
In order to verify that the stroke of the APA300ML is as specified in the datasheet, 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}.
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_voltage}).
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).
\subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
\end{subfigure}
\caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of the applied voltage (\subref{fig:test_apa_stroke_hysteresis})}
To experimentally estimate these modes, the APA is fixed on one end (see Figure \ref{fig:test_apa_meas_setup_modes}).
A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points 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_modes}Experimental setup to measured flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
The measured frequency response functions computed from the experimental setups of figures \ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure \ref{fig:test_apa_meas_freq_compare}.
The \(y\) bending mode is observed at \(280\,\text{Hz}\) and the \(x\) bending mode is at \(412\,\text{Hz}\).
These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure \ref{fig:test_apa_meas_setup_modes}).
This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
\caption{\label{fig:test_apa_meas_freq_compare}Obtained frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
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 dynamics of the APA300ML.
This test bench, depicted in Figure \ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing.
That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
The bench is schematically shown in Figure \ref{fig:test_apa_schematic} with all the associated signals.
It will be first used to estimate the hysteresis from the piezoelectric stack (Section \ref{ssec:test_apa_hysteresis}) as well as the axial stiffness of the APA300ML (Section \ref{ssec:test_apa_stiffness}).
Then, the frequency response functions from the DAC voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section \ref{ssec:test_apa_meas_dynamics}.
The presence of a non minimum phase zero found on the transfer function from \(u\) to \(V_s\) is investigated in Section \ref{ssec:test_apa_non_minimum_phase}.
In order to limit the low frequency gain of the transfer function from \(u\) to \(V_s\), a resistor is added across the force sensor stack (Section \ref{ssec:test_apa_resistance_sensor_stack}).
Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section \ref{ssec:test_apa_iff_locus}.
\caption{\label{fig:test_apa_schematic}Schematic of the Test Bench used to measured the dynamics of the APA300ML. \(u\) is the output DAC voltage, \(V_a\) the output amplifier voltage (i.e. voltage applied across the actuator stacks), \(d_e\) the measured displacement by the encoder and \(V_s\) the measured voltage across the sensor stack.}
Do to so, a quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)) and with an amplitude varying from \(4\,V\) up to \(80\,V\) is generated using the DAC.
For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage in Figure \ref{fig:test_apa_meas_hysteresis}.
This is the typical behavior expected from a PZT stack actuator where the hysteresis increases as a function of the applied voltage amplitude \cite[chap. 1.4]{fleming14_desig_model_contr_nanop_system}.
\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 APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}.
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.
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 specified nominal stiffness of the APA300ML \(k =1.8\,N/\mu m\).
\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding (at \(t \approx3\,s\)) and removing (at \(t \approx13\,s\)) the mass}
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 \(m_{\text{sus}}=5.7\,\text{kg}\).
It is important to note that changes to 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}.
It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}}\approx2.3\,N/\mu m\) while the the closed-circuit stiffness \(k_{\text{sc}}\approx1.7\,N/\mu m\).
In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified.
\item A ``mass line'' up to \(\approx800\,\text{Hz}\), above which additional 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.
Flexible modes studied in section \ref{ssec:test_apa_spurious_resonances} seems not to impact the measured axial motion of the actuator.
The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_force}.
A lightly damped resonance (pole) is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance (zero) at \(41\,\text{Hz}\).
No additional resonances is present up to at least \(2\,\text{kHz}\) indicating that Integral Force Feedback can be applied without stability issues from high frequency flexible modes.
The zero at \(41\,\text{Hz}\) seems to be non-minimum phase (the phase \emph{decreases} by 180 degrees whereas it should have \emph{increased} by 180 degrees for a minimum phase zero).
This is investigated in Section \ref{ssec:test_apa_non_minimum_phase}.
As illustrated by the Root Locus, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases.
The significance of this behavior varies on the type of sensor used as explained in \cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
Considering the transfer function from \(u\) to \(V_s\), if a controller with a very high gain is applied such that the sensor stack voltage \(V_s\) is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
Consequently, the closed-loop system would virtually corresponds to one where the piezoelectric stacks are absent, leaving only the mechanical shell.
From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}}= m \omega_0^2=5.7\cdot(2\pi\cdot41)^2=0.38\,N/\mu m\) (which is close to what is found using a finite element model).
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 shell.
\subcaption{\label{fig:test_apa_frf_force}FRF from $u$ to $V_s$}
\end{subfigure}
\caption{\label{fig:test_apa_frf_dynamics}Measured frequency response function from generated voltage \(u\) to the encoder displacement \(d_e\) (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage \(V_s\) (\subref{fig:test_apa_frf_force}) for the six APA300ML}
It was surprising to observe a non-minimum phase behavior for the zero on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}).
It was initially thought that this non-minimum phase behavior is an artifact coming from the measurement.
A longer measurement was performed with different excitation signals (noise, slow sine sweep, etc.) to see it the phase behavior of the zero changes.
Results of one long measurement is shown in Figure \ref{fig:test_apa_non_minimum_phase}.
The coherence (Figure \ref{fig:test_apa_non_minimum_phase_coherence}) is good even in the vicinity of the lightly damped zero, and the phase (Figure \ref{fig:test_apa_non_minimum_phase_zoom}) clearly indicates non-minimum phase behavior.
Such non-minimum phase zero when using load cells has also been observed on other mechanical systems \cite{spanos95_soft_activ_vibrat_isolat,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}.
It could be induced to small non-linearity in the system, but the reason of this non-minimum phase for the APA300ML is not yet clear.
However, this is not so important here as the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure \ref{fig:test_apa_iff_root_locus}) should not be unstable except for very large controller gains that will never be applied in practice.
\subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
\end{subfigure}
\caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found on the transfer function from \(u\) to \(V_s\). The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.}
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 piezoelectric stack (capacitance estimated at \(\approx5\,\mu F\)).
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}.
\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}
In order to implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation \eqref{eq:test_apa_iff_manual_fit}.
The parameters are manually tuned, and the obtained 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\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see \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.
The evolution of the pole in the complex plane as a function of the controller gain \(g\) (i.e. the ``root locus'') is computed both using the IFF plant model \eqref{eq:test_apa_iff_manual_fit} and the implemented controller \eqref{eq:test_apa_Kiff_formula} and from the fitted transfer functions of the damped plants experimentally identified for several controller 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.
\subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants to match the experimental data (dashed lines)}
\subcaption{\label{fig:test_apa_iff_root_locus}Root Locus plot using the plant model (black) and poles of the identified damped plants (color crosses)}
\end{subfigure}
\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus})}
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}.
\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees of freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}}
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 \(m\) supported by the APA300ML can 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 \(m =5.7\,\text{kg}\).
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}.
The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
A good match can be observed between the model and the experimental data, both for the encoder (Figure \ref{fig:test_apa_2dof_comp_frf_enc}) and for the force sensor (Figure \ref{fig:test_apa_2dof_comp_frf_force}).
\subcaption{\label{fig:test_apa_2dof_comp_frf_force}from $u$ to $V_s$}
\end{subfigure}
\caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in \ref{sec:test_apa_model_2dof}.
Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then make accessible in the Simscape model as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}.
For the APA300ML \emph{super element}, 5 \emph{remote points} are defined.
Two \emph{remote points} (\texttt{1} and \texttt{2}) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used for connecting the APA300ML with other mechanical elements.
Two \emph{remote points} (\texttt{3} and \texttt{4}) are located across two piezoelectric stacks and will be used to apply internal forces representing the actuator stacks.
Finally, two \emph{remote points} (\texttt{4} and \texttt{4}) are located across the third piezoelectric stack.
It will be used to measure the strain experience by this stack, and model the sensor stack.
\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.}
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}.
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 close to the identified constants using the experimentally identified transfer functions.
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.
It is however a bit surprising that the model is a bit ``softer'' than the measured system as finite element models are usually overestimating the stiffness.
Using this simple test bench, it can be concluded that the \emph{super element} model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
\subcaption{\label{fig:test_apa_super_element_comp_frf_force}from $u$ to $V_s$}
\end{subfigure}
\caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the ``flexible'' model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
In this study, the received amplified piezoelectric actuators ``APA300ML'' have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase.
Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section \ref{sec:test_apa_basic_meas}.
These simple measurements allowed for early detection of a manufacturing defect in one of the APA300ML.
Then in Section \ref{sec:test_apa_dynamics}, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure \ref{fig:test_apa_frf_dynamics}).
This consistency indicates good manufacturing tolerances, facilitating the modeling and control of the nano-hexapod.
Although a non-minimum zero was identified in the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_non_minimum_phase}), it was found not to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure \ref{fig:test_apa_iff}).
Then, two different models were used to represent the APA300ML dynamics.
In Section \ref{sec:test_apa_model_2dof}, a simple two degrees of freedom mass-spring-damper model was presented and tuned based on the measured dynamics.
After following a tuning procedure (see Section \ref{ssec:test_apa_2dof_model_tuning}), the model dynamics was shown to match very well with the experiment.
However, it is important to note that this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions.
In Section \ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a finite element model was used to model the APA300ML.
This time, the \emph{super element} represents the dynamics of the APA300ML in all directions.
However, only the axial dynamics could be compared with the experimental results yielding a good match.
The benefit of employing this model over the two degrees of freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved.
Nonetheless, the \emph{super element} model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important.
