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test-bench-apa.bib
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@ -11,6 +11,8 @@
} }
@book{fleming14_desig_model_contr_nanop_system, @book{fleming14_desig_model_contr_nanop_system,
author = {Andrew J. Fleming and Kam K. Leang}, author = {Andrew J. Fleming and Kam K. Leang},
title = {Design, Modeling and Control of Nanopositioning Systems}, title = {Design, Modeling and Control of Nanopositioning Systems},

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test-bench-apa.org
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@ -109,12 +109,15 @@ CLOSED: [2024-04-02 Tue 10:17] SCHEDULED: <2024-04-02 Tue>
** DONE [#C] Add sitffness of APA shell from FEM :@philipp: ** DONE [#C] Add sitffness of APA shell from FEM :@philipp:
CLOSED: [2024-04-02 Tue 10:01] CLOSED: [2024-04-02 Tue 10:01]
** TODO [#C] Check things about resistor in parallel with the force sensor ** DONE [#C] Check things about resistor in parallel with the force sensor
CLOSED: [2024-04-04 Thu 10:42]
Verify that everything interesting to say about that is either done before in the thesis or in this report. Verify that everything interesting to say about that is either done before in the thesis or in this report.
** TODO [#B] A lieu d'identifier le plant et de tracer sur le root locus, tracer le plant dampé depuis le modèle et comparer a la mesure ** CANC [#B] A lieu d'identifier le plant et de tracer sur le root locus, tracer le plant dampé depuis le modèle et comparer a la mesure
CLOSED: [2024-04-04 Thu 10:42]
- State "CANC" from "TODO" [2024-04-04 Thu 10:42]
* Introduction :ignore: * Introduction :ignore:
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 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.
@ -1861,16 +1864,25 @@ exportFig('figs/test_apa_super_element_comp_frf_force.pdf', 'width', 'half', 'he
* Conclusion * Conclusion
<<sec:test_apa_conclusion>> <<sec:test_apa_conclusion>>
The main characteristics of the APA300ML such as hysteresis and axial stiffness have been measured and were found to comply with the specifications. 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.
The dynamics of the received APA were measured and found to all be identical (Figure ref:fig:test_apa_frf_dynamics). Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section ref:sec:test_apa_basic_meas.
Even tough a non-minimum zero was observed on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_non_minimum_phase), it was not found to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure ref:fig:test_apa_iff). 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.
- Compare 2DoF and FEM models (usefulness of the two) In Section ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML.
- Good match between all the APA: will simplify the modeling and control of the nano-hexapod This time, the /super element/ represents the dynamics of the APA300ML in all directions.
- No advantage of the FEM model here (as only uniaxial behavior is checked), but may be useful later 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 /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.
* Bibliography :ignore: * Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}] #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
@ -2022,8 +2034,6 @@ if args.Ga == 0
switch args.type switch args.type
case '2dof' case '2dof'
actuator.Ga = -2.5796; actuator.Ga = -2.5796;
case 'flexible frame'
actuator.Ga = 1; % TODO
case 'flexible' case 'flexible'
actuator.Ga = 23.2; actuator.Ga = 23.2;
end end
@ -2037,8 +2047,6 @@ if args.Gs == 0
switch args.type switch args.type
case '2dof' case '2dof'
actuator.Gs = 466664; actuator.Gs = 466664;
case 'flexible frame'
actuator.Gs = 1; % TODO
case 'flexible' case 'flexible'
actuator.Gs = -4898341; actuator.Gs = -4898341;
end end

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test-bench-apa.tex
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@ -1,4 +1,4 @@
% Created 2024-04-03 Wed 18:15 % Created 2024-04-04 Thu 11:14
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -59,7 +59,7 @@ Section \ref{sec:test_apa_model_flexible} & \texttt{test\_apa\_4\_model\_flexibl
\end{table} \end{table}
\chapter{First Basic Measurements} \chapter{First Basic Measurements}
\label{sec:org399d3a7} \label{sec:org228596b}
\label{sec:test_apa_basic_meas} \label{sec:test_apa_basic_meas}
Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed. Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed.
@ -68,7 +68,7 @@ Then, the capacitance of the piezoelectric stacks is measured in Section \ref{ss
The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}. The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}.
Finally, in Section \ref{ssec:test_apa_spurious_resonances}, the flexible modes of the APA are measured and compared with a finite element model. Finally, in Section \ref{ssec:test_apa_spurious_resonances}, the flexible modes of the APA are measured and compared with a finite element model.
\section{Geometrical Measurements} \section{Geometrical Measurements}
\label{sec:org9db80da} \label{sec:org726c24d}
\label{ssec:test_apa_geometrical_measurements} \label{ssec:test_apa_geometrical_measurements}
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. 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.
@ -103,7 +103,7 @@ APA 7 & 18.7\\
\end{center} \end{center}
\end{minipage} \end{minipage}
\section{Electrical Measurements} \section{Electrical Measurements}
\label{sec:orgbec5e3a} \label{sec:orgd08d506}
\label{ssec:test_apa_electrical_measurements} \label{ssec:test_apa_electrical_measurements}
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 \(20\,\mu F\). 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 \(20\,\mu F\).
@ -144,7 +144,7 @@ APA 7 & 4.85 & 9.85\\
\end{center} \end{center}
\end{minipage} \end{minipage}
\section{Stroke and Hysteresis Measurement} \section{Stroke and Hysteresis Measurement}
\label{sec:org7997ce8} \label{sec:org1a772f2}
\label{ssec:test_apa_stroke_measurements} \label{ssec:test_apa_stroke_measurements}
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}. 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}.
@ -184,7 +184,7 @@ From now on, only the six APA that behave as expected will be used.
\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})} \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})}
\end{figure} \end{figure}
\section{Flexible Mode Measurement} \section{Flexible Mode Measurement}
\label{sec:orga2217ed} \label{sec:orgf072115}
\label{ssec:test_apa_spurious_resonances} \label{ssec:test_apa_spurious_resonances}
In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
@ -245,7 +245,7 @@ Another explanation is the shape difference between the manufactured APA300ML an
\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 \(412\,\text{Hz}\)} \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 \(412\,\text{Hz}\)}
\end{figure} \end{figure}
\chapter{Dynamical measurements} \chapter{Dynamical measurements}
\label{sec:orgdf7ee61} \label{sec:org2e49535}
\label{sec:test_apa_dynamics} \label{sec:test_apa_dynamics}
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. 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. 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.
@ -281,7 +281,7 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a
\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.} \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.}
\end{figure} \end{figure}
\section{Hysteresis} \section{Hysteresis}
\label{sec:org7aacb8e} \label{sec:org63e3610}
\label{ssec:test_apa_hysteresis} \label{ssec:test_apa_hysteresis}
As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload. As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
@ -295,7 +295,7 @@ This is the typical behavior expected from a PZT stack actuator where the hyster
\caption{\label{fig:test_apa_meas_hysteresis}Obtained hysteresis curves (displacement as a function of applied voltage) for multiple excitation amplitudes} \caption{\label{fig:test_apa_meas_hysteresis}Obtained hysteresis curves (displacement as a function of applied voltage) for multiple excitation amplitudes}
\end{figure} \end{figure}
\section{Axial stiffness} \section{Axial stiffness}
\label{sec:org5cd81d2} \label{sec:org68f94d3}
\label{ssec:test_apa_stiffness} \label{ssec:test_apa_stiffness}
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. 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.
@ -357,7 +357,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the `
It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\). It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
\section{Dynamics} \section{Dynamics}
\label{sec:org9be9924} \label{sec:org7856f12}
\label{ssec:test_apa_meas_dynamics} \label{ssec:test_apa_meas_dynamics}
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. 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.
@ -403,7 +403,7 @@ All the identified dynamics of the six APA300ML (both when looking at the encode
\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} \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}
\end{figure} \end{figure}
\section{Non Minimum Phase Zero?} \section{Non Minimum Phase Zero?}
\label{sec:org5d96397} \label{sec:org49578de}
\label{ssec:test_apa_non_minimum_phase} \label{ssec:test_apa_non_minimum_phase}
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 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}).
@ -433,7 +433,7 @@ However, this is not so important here as the zero is lightly damped (i.e. very
\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.} \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.}
\end{figure} \end{figure}
\section{Effect of the resistor on the IFF Plant} \section{Effect of the resistor on the IFF Plant}
\label{sec:orge8ed591} \label{sec:org8b60e04}
\label{ssec:test_apa_resistance_sensor_stack} \label{ssec:test_apa_resistance_sensor_stack}
A resistor \(R \approx 80.6\,k\Omega\) 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 \(\approx 5\,\mu F\)). A resistor \(R \approx 80.6\,k\Omega\) 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 \(\approx 5\,\mu F\)).
@ -449,7 +449,7 @@ It is confirmed that the added resistor as the effect of adding an high pass fil
\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} \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}
\end{figure} \end{figure}
\section{Integral Force Feedback} \section{Integral Force Feedback}
\label{sec:orge2d1f26} \label{sec:orga8f1ff3}
\label{ssec:test_apa_iff_locus} \label{ssec:test_apa_iff_locus}
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}. 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}.
@ -506,7 +506,7 @@ The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_loc
\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})} \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})}
\end{figure} \end{figure}
\chapter{APA300ML - 2 Degrees of Freedom Model} \chapter{APA300ML - 2 Degrees of Freedom Model}
\label{sec:org42297a7} \label{sec:org4cdbff1}
\label{sec:test_apa_model_2dof} \label{sec:test_apa_model_2dof}
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. 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.
@ -521,7 +521,7 @@ The obtained model dynamics is compared with the measurements in Section \ref{ss
\caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model} \caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model}
\end{figure} \end{figure}
\section{Two Degrees of Freedom APA Model} \section{Two Degrees of Freedom APA Model}
\label{sec:org97d98f4} \label{sec:org5d75a3a}
\label{ssec:test_apa_2dof_model} \label{ssec:test_apa_2dof_model}
The model of the amplified piezoelectric actuator is shown in Figure \ref{fig:test_apa_2dof_model}. The model of the amplified piezoelectric actuator is shown in Figure \ref{fig:test_apa_2dof_model}.
@ -548,7 +548,7 @@ Such simple model has some limitations:
\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}} \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}}
\end{figure} \end{figure}
\section{Tuning of the APA model} \section{Tuning of the APA model}
\label{sec:org29ff205} \label{sec:org10fafb5}
\label{ssec:test_apa_2dof_model_tuning} \label{ssec:test_apa_2dof_model_tuning}
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}. 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}.
@ -607,7 +607,7 @@ The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_mode
\end{table} \end{table}
\section{Obtained Dynamics} \section{Obtained Dynamics}
\label{sec:orgbff057f} \label{sec:org0831884}
\label{ssec:test_apa_2dof_model_result} \label{ssec:test_apa_2dof_model_result}
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. 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.
@ -631,7 +631,7 @@ This indicates that this model represents well the axial dynamics of the APA300M
\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})} \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})}
\end{figure} \end{figure}
\chapter{APA300ML - Super Element} \chapter{APA300ML - Super Element}
\label{sec:org703026b} \label{sec:org8292e07}
\label{sec:test_apa_model_flexible} \label{sec:test_apa_model_flexible}
In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}. In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
@ -651,7 +651,7 @@ It will be used to measure the strain experience by this stack, and model the se
\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.} \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.}
\end{figure} \end{figure}
\section{Identification of the Actuator and Sensor constants} \section{Identification of the Actuator and Sensor constants}
\label{sec:org1fc96a6} \label{sec:org28294e9}
\label{ssec:test_apa_flexible_ga_gs} \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 extracted. 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 extracted.
@ -694,7 +694,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\end{table} \end{table}
\section{Comparison of the obtained dynamics} \section{Comparison of the obtained dynamics}
\label{sec:orgd657b0f} \label{sec:org14d2335}
\label{ssec:test_apa_flexible_comp_frf} \label{ssec:test_apa_flexible_comp_frf}
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}. 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}.
@ -719,20 +719,27 @@ Using this simple test bench, it can be concluded that the \emph{super element}
\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})} \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})}
\end{figure} \end{figure}
\chapter{Conclusion} \chapter{Conclusion}
\label{sec:org0ba1df7} \label{sec:org10e068b}
\label{sec:test_apa_conclusion} \label{sec:test_apa_conclusion}
The main characteristics of the APA300ML such as hysteresis and axial stiffness have been measured and were found to comply with the specifications. 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.
The dynamics of the received APA were measured and found to all be identical (Figure \ref{fig:test_apa_frf_dynamics}). Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section \ref{sec:test_apa_basic_meas}.
Even tough a non-minimum zero was observed on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_non_minimum_phase}), it was not found to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure \ref{fig:test_apa_iff}). 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.
\begin{itemize} In Section \ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a finite element model was used to model the APA300ML.
\item Compare 2DoF and FEM models (usefulness of the two) This time, the \emph{super element} represents the dynamics of the APA300ML in all directions.
\item Good match between all the APA: will simplify the modeling and control of the nano-hexapod However, only the axial dynamics could be compared with the experimental results yielding a good match.
\item No advantage of the FEM model here (as only uniaxial behavior is checked), but may be useful later 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.
\end{itemize} 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.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]
\end{document} \end{document}