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\author{T. Dehaeze\textsuperscript{1,}\thanks{thomas.dehaeze@esrf.fr}, J. Bonnefoy, ESRF, Grenoble, France \\ C. Collette\textsuperscript{1}, Université Libre de Bruxelles, BEAMS department, Brussels, Belgium \\ \textsuperscript{1}also at Precision Mechatronics Laboratory, University of Liege, Belgium}
\date{2021-07-14}
\title{MECHATRONICS APPROACH FOR THE DEVELOPMENT OF A NANO-ACTIVE-STABILIZATION-SYSTEM}
\begin{document}

\maketitle

\begin{abstract}
With the growing number of fourth generation light sources, there is an increased need of fast positioning end-stations with nanometric precision.
Such systems are usually including dedicated control strategies, and many factors may limit their performances.
In order to design such complex systems in a predictive way, a mechatronic design approach also known as ``model based design'', may be utilized.
In this paper, we present how this mechatronic design approach was used for the development of a nano-hexapod for the ESRF ID31 beamline.
The chosen design approach consists of using models of the mechatronic system (including sensors, actuators and control strategies) to predict its behavior.
Based on this behavior and closed-loop simulations, the elements that are limiting the performances can be identified and re-designed accordingly.
This allows to make adequate choices concerning the design of the nano-hexapod and the overall mechatronic architecture early in the project and save precious time and resources.
Several test benches were used to validate the models and to gain confidence on the predictability of the final system's performances.
Measured nano-hexapod's dynamics was shown to be in very good agreement with the models.
Further tests should be done in order to confirm that the performances of the system match the predicted one.
The presented development approach is foreseen to be applied more frequently to future mechatronic system design at the ESRF.
\end{abstract}

\section{INTRODUCTION}
Such mechatronic approach is widely used in the dutch industry \cite{rankers98_machin} and much less in the Synchrotron's world.
In this paper, is presented how the mechatronic approach is used for the development of a nano active stabilization system.

\cite{dehaeze21_activ_dampin_rotat_platf_using}
\cite{souleille18_concep_activ_mount_space_applic}
\cite{brumund21_multib_simul_reduc_order_flexib_bodies_fea}
\cite{dehaeze18_sampl_stabil_for_tomog_exper}
\cite{schmidt20_desig_high_perfor_mechat_third_revis_edition}

\section{NASS - MECHATRONIC APPROACH}
\subsection{The ID31 Micro Station}
The ID31 Micro Station is used to position samples along complex trajectories \cite{dehaeze18_sampl_stabil_for_tomog_exper}.
It is composed of several stacked stages (represented in yellow in Fig.~\ref{fig:nass_concept_schematic}).
This allows this station to have high mobility, however, this limits the position accuracy to tens of \(\mu m\).

\subsection{The Nano Active Stabilization System}
The Nano Active Stabilization System (NASS) is a system whose goal is to improve the positioning accuracy of the ID31 Micro Station.

It is represented in Fig.~\ref{fig:nass_concept_schematic} and consists of three main elements:
\begin{itemize}
\item a nano-hexapod located between the sample to be positioned and the micro-station.
\item a interferometric metrology system measuring the sample's position with respect to the focusing optics
\item a control system (not represented), which base on the measured position, properly actuates the nano-hexapod in order to stabilize the sample's position
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/nass_concept_schematic.pdf}
\caption{\label{fig:nass_concept_schematic}Nano Active Stabilization System - Schematic representation. 1) micro-station, 2) nano-hexapod, 3) sample, 4) metrology system}
\end{figure}

\subsection{Mechatronic Approach - Overview}
In order to design the NASS in a predictive way, a mechatronic approach, schematically represented in Fig.~\ref{fig:nass_mechatronics_approach}, is used.

It consists of three main phases:
\begin{enumerate}
\item Conceptual phase: Simple models of both the micro-station and the nano-hexapod are used to first evaluate the performances of several concepts.
During this phase, the type of sensors to use and the approximate required dynamical characteristics of the nano-hexapod are determined.
\item Detail design phase: Once the concept is validated, the models are used to list specifications both for the mechanics and the instrumentation.
Each critical elements can then be properly designed.
The models are updated as the design progresses.
\item Experimental phase: Once the design is completed and the parts received, several test benches are used to verify the properties of the key elements.
Then the hexapod can be mounted and fully tested with the instrumentation and the control system.
\end{enumerate}

\begin{figure*}
\centering
\includegraphics[scale=1,width=\linewidth]{figs/nass_mechatronics_approach.pdf}
\caption{\label{fig:nass_mechatronics_approach}Overview of the mechatronic approach}
\end{figure*}

\subsection{Models}
As shown in Fig.~\ref{fig:nass_mechatronics_approach}, the models are at the core of the mechatronic approach.
Indeed, several models are used throughout the design with increasing level of complexity (Fig.~\ref{fig:nass_models}).

\begin{figure*}[htbp]
  \begin{subfigure}[t]{0.25\linewidth}
    \centering
    \includegraphics[width=0.68\linewidth]{figs/mass_spring_damper_hac_lac.pdf}
    \caption{\label{fig:mass_spring_damper_hac_lac} Mass Spring Damper Model}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.48\linewidth}
    \centering
    \includegraphics[width=0.89\linewidth]{figs/nass_simscape_3d.png}
    \caption{\label{fig:nass_simscape_3d} Multi Body Model}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.25\linewidth}
    \centering
    \includegraphics[width=0.93\linewidth]{figs/super_element_simscape_alt.pdf}
    \caption{\label{fig:super_element_simscape} Finite Element Model}
  \end{subfigure}
  \hfill
  \caption{\label{fig:nass_models}Schematic of several models used during all the mechatronic design process.}
  \centering
\end{figure*}

At the beginning of the conceptual phase, simple ``mass-spring-dampers'' models are used (Fig.~\ref{fig:mass_spring_damper_hac_lac}) in order to easily try different concepts.
Noise budgeting and closed-loop simulations were performed, and it was concluded that a nano-hexapod with low frequency ``suspension'' modes would help both for the reduction of the effects of disturbances and for the decoupling between the nano-hexapod dynamics and the complex micro-station dynamics.
Also, including a force sensor in series with the nano-hexapod's actuators can be used to actively damp the resonances using the ``Integral Force Feedback'' (IFF) strategy.
The goal is to obtain a ``plant'' dynamics which is easy to control in a robust way.

Rapidly, a more sophisticated multi-body model (Fig.~\ref{fig:nass_simscape_3d}) has been used.
This model is based on the 3D representation of the micro-station as well as on extensive dynamical measurements.
Time domain simulations can then be performed with each stage moving with the associated positioning errors and disturbances.
Such model is more realistic and permits to study effects which were not modeled with the previous model such as the coupling between directions and effect of the rotation of the spindle on the nano-hexapod's dynamics (gyroscopic effects \cite{dehaeze21_activ_dampin_rotat_platf_using}).
The multi-input multi-output control strategy can be developed and tested.

During the detail design phase, the nano-hexapod model is updated by importing the 3D parts exported from the CAD software.
The key elements of the nano-hexapod such as the flexible joints and the APA are optimized using a Finite Element Software.
As the flexible modes of the system are what generally limit the controller bandwidth, they are important to model in order to understand which ones are problematic and should be maximized.
In order to do so, a ``super-element'' can be exported using a finite element analysis software and imported in Simscape (Fig.~\ref{fig:super_element_simscape}).
Such process is described in \cite{brumund21_multib_simul_reduc_order_flexib_bodies_fea}.
The multi-body model with included flexible elements can be used to obtain very accurately the dynamics of the system.
However due to the large number of states included, it becomes non practical to perform time domain simulations.

Finally, during the experimental phase, the models are refined using experimental system identification.
These models can be used to understand the measurements, the associated performance limitations and to gain insight on which measures to take in order to overcome these limitations.

For instance, it has been found that when fixing the encoders to the struts (Fig.~\ref{fig:nano_hexapod_elements}), several flexible modes of the APA were appearing in the dynamics which render the control using the encoders very complex.
Therefore, an alternative configuration with the encoders fixed to the plates was used instead.

\section{NANO-HEXAPOD DESIGN}
\subsection{Nano-Hexapod Specifications}
The Nano-Hexapod must have a maximum height of \(95\,mm\), support samples up to \(50\,kg\) and have a stroke of \(\approx 100\,\mu m\).
it have few parts: two plates and 6 active struts in between.
Each strut is composed of one flexible joint at each end, and one actuator (Fig.~\ref{fig:picture_nano_hexapod_strut}).
A 3D view of the nano-hexapod is shown in Fig.~\ref{fig:nano_hexapod_elements}.

\begin{figure*}
\centering
\includegraphics[scale=1,width=0.9\linewidth]{figs/nano_hexapod_elements.pdf}
\caption{\label{fig:nano_hexapod_elements}CAD view of the nano-hexapod with key elements}
\end{figure*}

Based on the models used throughout the mechatronic approach, several specifications was obtained in order to maximize the performances of the system:
\begin{itemize}
\item Actuator: axial stiffness \(\approx \SI{2}{\um}\)
\item Flexible joints: bending stiffness \(< \SI{100}{Nm/rad}\) and axial stiffness \(> \SI{100}{N/\um}\)
\item Precise positioning of the \(b_i\) and \(\hat{s}_i\)
\item Flexible modes of the top-plate as high as possible
\item Integration of a force sensor in each strut
\end{itemize}

\subsection{Parts' Optimization}
The geometry of the flexible joint could be optimized using a finite element software.
The obtained stiffnesses are compliance with the requirements and the model was updated.

The top plate was manually optimized to maximize its flexible modes.
Flexible modes at around \(\SI{700}{Hz}\) could be obtained.

Amplified Piezoelectric Actuators (APA) were found to be the most suitable actuator for the nano-hexapod due to its compact size, large stroke and adequate stiffness.
The chosen model was the APA300ML from Cedrat Technologies (shown in Fig.~\ref{fig:picture_nano_hexapod_strut}).
It is composed of three piezoelectric stacks, a lever mechanism increasing the stroke up to \(\approx \SI{300}{\um}\) and decreasing the axial stiffness down to \(\approx \SI{1.8}{\um}\).
One of the three stacks can be used as a force sensor, at the price of loosing \(1/3\) of the stroke.
The main benefits is the good ``collocation'' of the sensor stack with the actuator stacks, meaning that the active damping controller will easily be made robust.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.9\linewidth]{figs/picture_nano_hexapod_strut.pdf}
\caption{\label{fig:picture_nano_hexapod_strut}Picture of a nano-hexapod's strut}
\end{figure}

\subsection{Nano-Hexapod Mounting}
After each element

The nano-hexapod mounted on top of the micro-station is shown in Fig.~\ref{fig:nano_hexapod_picture}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.9\linewidth]{figs/nano_hexapod_picture.jpg}
\caption{\label{fig:nano_hexapod_picture}Nano-Hexapod on top of the ID31 micro-station}
\end{figure}

\section{TEST-BENCHES}
\subsection{Flexible Joints and Instrumentation}
Several test benches were used to characterize the individual elements of the NASS.

The bending stiffness of the flexible joints was measured by applying a (measured) force to one end of the joint while measuring its deflection at the same time.
This helped exclude the ones not compliant with the requirement and pair the remaining ones.

The transfer function from input to output voltage of the voltage amplifier\footnote{PD200 from PiezoDrive} as well as its output noise was measured.
Similarly, the measurement noise of the encoders\footnote{Vionic from Renishaw} was also measured.

These simple measurements on individual elements are useful to refine their models, found any problem as early as possible, and will help analyzing the results once the nano-hexapod is mounted and all elements combined.

\subsection{APA and Struts Dynamics}

An other test bench schematically shown in Fig.~\ref{fig:test_bench_apa_schematic} was used to identify the dynamics of the APA.
It consist of a \(5\,\text{kg}\) granite vertical guided with an air bearing and fixed on top of the APA.
An excitation signal (low pass filtered white noise) is generated and applied to two of the piezoelectric stacks.
Both the voltage generated by the third piezoelectric stack and the displacement measured by the encoder are recorded.
The two obtained frequency response functions (FRF) can then be compared with the model (Fig.~\ref{fig:apa_test_bench_results}).

The piezoelectric constants describing the conversion from the mechanical domain (force, strain) easily accessible on the model to the electrical domain (voltages, charges) easily measured can be estimated.
With these constants, the match between the measured FRF and the model dynamics is very good (Fig.~\ref{fig:apa_test_bench_results})

The same bench was also used with the struts in order to study the added effects of the flexible joints.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/test_bench_apa_schematic.pdf}
\caption{\label{fig:test_bench_apa_schematic}Schematic of the bench used to identify the APA dynamics}
\end{figure}

\begin{figure}[htbp]
  \begin{subfigure}[t]{0.48\linewidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figs/apa_test_bench_results_de.pdf}
    \caption{\label{fig:apa_test_bench_results_de} Encoder}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.48\linewidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figs/apa_test_bench_results_Vs.pdf}
    \caption{\label{fig:apa_test_bench_results_Vs} Force Sensor}
  \end{subfigure}
  \caption{\label{fig:apa_test_bench_results}Measured Frequency Response functions compared with the Simscape model. From the actuator stacks voltage to the encoder (\subref{fig:apa_test_bench_results_de}) and to the force sensor stack (\subref{fig:apa_test_bench_results_Vs}).}
  \centering
\end{figure}

\subsection{Nano-Hexapod}

Once the nano-hexapod is mounted, its dynamics is identified.
To do so, each actuator is individually excited and the six force sensors and six encoders signals are recorded each time.
Two \(6\) by \(6\) FRF matrices are computed.
The diagonal elements of these two matrices are shown in Fig.~\ref{fig:nano_hexapod_identification_comp_simscape} and compared with the model.

From Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_de} one can observe the following modes:
\begin{itemize}
\item From \(\SI{100}{Hz}\) to \(\SI{200}{Hz}\): six suspension modes
\item At \(\SI{230}{Hz}\) and \(\SI{340}{Hz}\): flexible modes of the APA, also modeled thanks to the flexible model of the APA
\item At around \(\SI{700}{Hz}\): flexible modes of the top plate, not modeled (taken as a rigid body)
\end{itemize}

The transfer function from the actuator to the force sensors has alternating poles and zeros (Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_Vs}) which is confirming the good ``collocation'' between the stacks.

IFF is then applied individually on each pair of actuator/force sensor in order to actively damp the modes shown in Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_Vs}.
The optimal gain of the IFF controller is determined from the model.
After applying the active damping technique, the \(6\) by \(6\) FRF matrix from the actuator to the encoders is identified again and shown in Fig.~\ref{fig:nano_hexapod_identification_damp_comp_simscape}.

\begin{figure}[htbp]
  \begin{subfigure}[t]{0.48\linewidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figs/nano_hexapod_identification_comp_simscape_de.pdf}
    \caption{\label{fig:nano_hexapod_identification_comp_simscape_de} Encoder}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.48\linewidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figs/nano_hexapod_identification_comp_simscape_Vs.pdf}
    \caption{\label{fig:nano_hexapod_identification_comp_simscape_Vs} Force Sensor}
  \end{subfigure}
  \caption{\label{fig:nano_hexapod_identification_comp_simscape}Measured Frequency Response functions compared with the Simscape model. From the actuator stacks voltage to the encoder (\subref{fig:apa_test_bench_results_de}) and to the force sensor stack (\subref{fig:apa_test_bench_results_Vs}).}
  \centering
\end{figure}

\begin{figure}[htbp]
  \begin{subfigure}[t]{0.49\linewidth}
    \centering
    \includegraphics[height=5.5cm]{figs/nano_hexapod_identification_damp_comp_simscape_diag.pdf}
    \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape_diag} Diagonal}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.49\linewidth}
    \centering
    \includegraphics[height=5.5cm]{figs/nano_hexapod_identification_damp_comp_simscape_off_diag.pdf}
    \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape_off_diag} Off-Diagonal}
  \end{subfigure}
  \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape}}
  \centering
\end{figure}

\section{CONCLUSION}

Future work:
\begin{itemize}
\item actively damp the top plate flexible modes
\item make the controller robust to change of payload mass
\item integrate it on top of the micro-station
\end{itemize}

\section{ACKNOWLEDGMENTS}
This research was made possible by a grant from the FRIA.
Damien Coulomb
Youness Benya
Marc Lesourd
Philipp Brumund

\printbibliography{}
\end{document}
