\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}
In order to design such complex systems in a predictive way, a mechatronics design approach also known as ``model based design'', may be utilized.
In this paper, we present how this mechatronics 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 mechatronics system (including sensors, actuators and control strategies) to predict its behavior.
This allows to make adequate choices regarding the design of the nano-hexapod and the overall mechatronics architecture early in the project and therefore save precious time and resources.
With the new \(4^\text{th}\) generation machines, there is an increasing need of fast and accurate positioning systems \cite{dimper15_esrf_upgrad_progr_phase_ii}.
These systems are usually including feedback control loops and therefore their performances are not only depending on the quality of the mechanical design, but also on its correct integration with the actuators, sensors and control system.
In order to optimize the performances of such system, it is essential to consider a design approach in which the structural design and the control design are integrated.
This approach, also called the ``mechatronics approach'', was shown to be very effective for the design many complex systems \cite{rankers98_machin,schmidt20_desig_high_perfor_mechat_third_revis_edition}.
Such design methodology was recently used for the development of several systems used by the synchrotron community \cite{geraldes17_mechat_concep_new_high_dynam_dcm_sirius,holler18_omny_tomog_nano_cryo_stage,brendike19_esrf_doubl_cryst_monoc_protot}.
The present paper presents how the ``mechatronic approach'' was used for the design of a Nano Active Stabilization System (NASS) for the ESRF ID31 beamline.
\item A nano-hexapod located between the sample to be positioned and the micro-station
\item An interferometric metrology system measuring the sample's position with respect to the focusing optics
\item A control system (not represented), which based on the measured position, properly actuates the nano-hexapod in order to stabilize the sample's position.
This system should be able to actively stabilize the sample position down to tens of nanometers while the micro-station is performing complex trajectories.
In order to design the NASS in a predictive way, a mechatronics approach, schematically represented in Fig.~\ref{fig:nass_mechatronics_approach}, was used.
\item\emph{Conceptual phase}: Simple models of both the micro-station and the nano-hexapod are used to first evaluate the performances of several concepts.
\item\emph{Detail design phase}: Once the concept is validated, the models are used to list specifications both for the mechanics and the instrumentation.
\item\emph{Experimental phase}: Once the design is completed and the parts received, several test benches are used to verify the properties of the key elements.
At the beginning of the conceptual phase, simple ``mass-spring-damper'' models (Fig.~\ref{fig:mass_spring_damper_hac_lac}) were used in order to easily study multiple 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.
I was found that by including a force sensor in series with the nano-hexapod's actuators, ``Integral Force Feedback'' (IFF) strategy could be used to actively damp the nano hexapod's resonances without impacting the high frequency disturbance rejection.
Rapidly, a more sophisticated and more realistic multi-body model (Fig.~\ref{fig:nass_simscape_3d}) using Simscape \cite{matlab20} was used.
This model was based on the 3D representation of the micro-station as well as on extensive dynamical measurements.
Time domain simulations were performed with every stage of the micro-station moving and the nano hexapod actively stabilizing the sample against the many disturbances.
The multi-body model permitted to study effects such as the coupling between the actuators and the sensors as well as the effect of the spindle's rotational speed on the nano-hexapod's dynamics \cite{dehaeze21_activ_dampin_rotat_platf_using}.
The multi-input multi-output control strategy could be developed and tested.
During the detail design phase, the nano-hexapod model was updated using 3D parts exported from the CAD software as the mechanical design progressed.
The key elements of the nano-hexapod such as the flexible joints and the APA were optimized using a Finite Element Analysis (FEA) Software.
As the flexible modes of the mechanics are what generally limit the controller bandwidth, they are important to model in order to understand which modes are problematic and should be addressed.
To do so, a ``super-element'' can be exported using a FEA software and imported into the multi-body model (Fig.~\ref{fig:super_element_simscape}).
Finally, during the experimental phase, the models were refined using experimental system identification data.
At this phase of the development, models are still useful.
They can help with the controller optimization, 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, as in Fig.~\ref{fig:nano_hexapod_elements}, several flexible modes of the APA were appearing in the dynamics which would render the control using the encoders very complex.
The nano-hexapod is a ``Gough-Stewart platform'', which is a fully parallel manipulator composed of few parts as shown in Fig.~\ref{fig:nano_hexapod_elements}: only two plates linked by 6 active struts.
The main benefits of this architecture are its compact design, good dynamical properties, high load capability over weight ratio, and to possibility to control the motion in 6 degrees of freedom.
The nano-hexapod should have a maximum height of \(95\,mm\), support samples up to \(50\,kg\), have a stroke of \(\approx100\,\mu m\) and be fully compliant to avoid any wear, backlash, play and to have predictable dynamics.
Based on the models used throughout the mechatronics approach, several specifications were added in order to maximize the performances of the system:
\item Actuator axial stiffness \(\approx\SI{2}{N/\um}\) as it is a good trade-off between disturbance filtering, dynamic decoupling from the micro-station and insensibility to the spindle's rotational speed.
The flexible joint geometry was optimized using a finite element software while the top plate geometry was manually optimized to maximize the frequency of its flexible modes.
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 (Fig.~\ref{fig:nano_heaxpod_strut_picture}).
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 \(\approx\SI{1.8}{N/\um}\).
This has the benefits of providing good ``collocation'' between the sensor stack and the actuator stacks, meaning that the active damping controller will be robust \cite{souleille18_concep_activ_mount_space_applic}.
Using the multi-body model of the nano-hexapod with the APA modeled as a flexible element, it was found that a misalignment between the APA and the two flexible joints was adding several resonances to the dynamics that were difficult to control.
The bending stiffness of the flexible joints was measured by applying a controlled force to one end of the joint while measuring its deflection at the same time.
The transfer function from the input to the output voltage of the voltage amplifier\footnote{PD200 from PiezoDrive} as well as its output noise were measured.
These simple measurements on individual elements were useful to refine their models, to found any problem as early as possible, and to help analyzing the results obtained when the the nano-hexapod is mounted and all the elements combined.
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.
\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}).}
After the nano-hexapod has been mounted, its dynamics was identified by individually exciting each of the actuators and simultaneously recording the six force sensors and six encoders signals.
The transfer functions from the actuators to their ``collocated'' force sensors have alternating poles and zeros as expected (Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_Vs}).
IFF was then applied individually on each pair of actuator/force sensor in order to actively damp the suspension modes.
The optimal gain of the IFF controller was determined using the model.
After applying the active damping technique, the \(6\) by \(6\) FRF matrix from the actuator to the encoders was identified again and shown in Fig.~\ref{fig:nano_hexapod_identification_damp_comp_simscape}.
It is shown that all the suspension modes are well damped, and that the model is able to predict the closed-loop behavior of the system.
Even the off-diagonal elements (effect of one actuator on the encoder fixed in parallel to another strut) is very well modeled (Fig.~\ref{fig:nano_hexapod_identification_damp_comp_simscape_off_diag}).
\caption{\label{fig:nano_hexapod_identification_comp_simscape}Comparison of the measured Frequency Response functions (FRF) with the Simscape model. From the excitation voltage to the associated encoder (\subref{fig:apa_test_bench_results_de}) and to the associated force sensor stack (\subref{fig:apa_test_bench_results_Vs}).}
\caption{\label{fig:nano_hexapod_identification_damp_comp_simscape}Transfer functions from actuator to encoder with (input $u$) and without (input $u^\prime$) IFF applied.}
Measurements made on the nano-hexapod were found to match very well with the models indicating that the final performances should match the predicted one.
The current performance limitation is coming from the flexible modes of the top platform, so future work will focus on overcoming this limitation.
This design methodology can be easily transposed to other complex mechatronics systems and are foreseen to be applied for future mechatronics systems at the ESRF.
