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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.
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 depending on the mechanical system alone, but also on its interaction with the actuators, sensors and control electronics.
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 is called the "mechatronic approach" and 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.
- A nano-hexapod located between the sample to be positioned and the micro-station
- An interferometric metrology system measuring the sample's position with respect to the focusing optics
- A control system (not represented), which based on the measured position, properly actuates the nano-hexapod in order to stabilize the sample's position.
In order to design the NASS in a predictive way, a mechatronic approach, schematically represented in Fig.\nbsp{}ref:fig:nass_mechatronics_approach, is used.
1. 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.
2. 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.
3. 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.
At the beginning of the conceptual phase, simple "mass-spring-dampers" models (Fig.\nbsp{}ref:fig:mass_spring_damper_hac_lac) are used in order to easily study 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.
Such model permits 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.
During the detail design phase, the nano-hexapod model is updated using 3D parts exported from the CAD software as the mechanical design progresses.
The key elements of the nano-hexapod such as the flexible joints and the APA are 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 ones are problematic and should be maximized.
To do so, a "super-element" can be exported using a FEA software and then imported in Simscape (Fig.\nbsp{}ref:fig:super_element_simscape).
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.\nbsp{}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.
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.\nbsp{}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.
This has the benefits providing good "collocation" between the sensor stack and the actuator stacks, meaning that the active damping controller will easily be made robust cite:souleille18_concep_activ_mount_space_applic.
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.
These simple measurements on individual elements are useful to refine their models, to found any problem as early as possible, and to help analyzing the results obtained with the nano-hexapod mounted and all 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}).}
Once the nano-hexapod is mounted, its dynamics is 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 (Fig.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape_Vs) as expected.
IFF is 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 is determined using 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.\nbsp{}ref:fig:nano_hexapod_identification_damp_comp_simscape.
It is shown that all the suspension modes are critically 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 to another strut) is very well modeled (Fig.\nbsp{}ref:fig:nano_hexapod_identification_damp_comp_simscape_off_diag).
# #+caption: HAC-LAC Strategy - Block Diagram. The signals are: $\bm{r}$ the wanted sample's position, $\bm{X}$ the measured sample's position, $\bm{\epsilon}_{\mathcal{X}}$ the sample's position error, $\bm{\epsilon}_{\mathcal{L}}$ the sample position error expressed in the "frame" of the nano-hexapod struts, $\bm{u}$ the generated DAC voltages applied to the voltage amplifiers and then to the piezoelectric actuator stacks, $\bm{u}^\prime$ the new inputs corresponding to the damped plant, $\bm{\tau}$ the measured sensor stack voltages. $\bm{T}$ is . $\bm{K}_{\tiny IFF}$ is the Low Authority Controller used for active damping. $\bm{K}_{L}$ is the High Authority Controller.
\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 and without the active damping technique applied.}
This design methodology can be easily transposed to other complex mechatronic systems and are foreseen to be applied for future mechatronic systems at the ESRF.
