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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.
- a nano-hexapod located between the sample to be positioned and the micro-station.
- a interferometric metrology system measuring the sample's position with respect to the focusing optics
- a control system (not represented), which base 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 are used (Fig.\nbsp{}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.
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).
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.\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.
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.
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[fn:1] as well as its output noise was measured.
Similarly, the measurement noise of the encoders[fn:2] 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.
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.\nbsp{}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.
\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}).}
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.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape and compared with the model.
From Fig.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape_de one can observe the following modes:
- From $\SI{100}{Hz}$ to $\SI{200}{Hz}$: six suspension modes
- At $\SI{230}{Hz}$ and $\SI{340}{Hz}$: flexible modes of the APA, also modeled thanks to the flexible model of the APA
- At around $\SI{700}{Hz}$: flexible modes of the top plate, not modeled (taken as a rigid body)
The transfer function from the actuator to the force sensors has alternating poles and zeros (Fig.\nbsp{}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.\nbsp{}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.\nbsp{}ref:fig:nano_hexapod_identification_damp_comp_simscape.
# #+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_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}).}
