The overall objective is to design a nano-hexapod an the associated control architecture that allows the stabilization of samples down to $\approx 10nm$ in presence of disturbances and system variability.
The mathematical tools (Power Spectral Density, Noise Budgeting, ...) that will be used throughout this study are also introduced.
To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of:
- the micro-station dynamics (Section [[sec:micro_station_dynamics]])
- the frequency content of the important source of disturbances in play such as vibration of stages and ground motion (Section [[sec:identification_disturbances]])
We then develop a model of the system that must represent all the important physical effects in play.
Such model is presented in Section [[sec:multi_body_model]].
A modular model of the nano-hexapod is then included in the system.
The effects of the nano-hexapod characteristics on the dynamics are then studied.
Based on that, an optimal choice of the nano-hexapod stiffness is made (Section [[sec:nano_hexapod_design]]).
Finally, using the optimally designed nano-hexapod, a robust control architecture is developed.
Simulations are performed to show that this design gives acceptable performance and the required robustness (Section [[sec:robust_control_architecture]]).
This should highlight the challenges in terms of combined performance and robustness.
In Section [[sec:noise_budget]] is introduced the *dynamic error budgeting* which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources.
This tool will be widely used throughout this study to both predict the performances and identify the effects that do limit the performances.
The use of feedback control as several advantages and pitfalls that are listed below (taken from cite:schmidt14_desig_high_perfor_mechat_revis_edition):
- *Advantages*:
- *Reduction of the effect of disturbances*:
Disturbances affecting the sample vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them
- *Handling of uncertainties*:
Feedback controlled systems can also be designed for /robustness/, which means that the stability and performance requirements are guaranteed even for parameter variation of the controller mechatronics system
- *Pitfalls*:
- *Limited reaction speed*:
A feedback controller reacts on the difference between the reference signal (wanted motion) and the measurement (actual motion), which means that the error has to occur first /before/ the controller can correct for it.
The limited reaction speed means that the controller will be able to compensate the positioning errors only in some frequency band, called the controller /bandwidth/
- *Feedback of noise*:
By closing the loop, the sensor noise is also fed back and will induce positioning errors
- *Can introduce instability*:
Feedback control can destabilize a stable plant.
Thus the /robustness/ properties of the feedback system must be carefully guaranteed
*** Simplified Feedback Control Diagram for the NASS
*** How does the feedback loop is modifying the system behavior?
If we write down the position error signal $\epsilon = r - y$ as a function of the reference signal $r$, the disturbances $d$ and the measurement noise $n$ (using the feedback diagram in Figure [[fig:classical_feedback_small]]), we obtain:
Moreover, the slope of $|S(j\omega)|$ is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have large effects).
The main issue it that for stability reasons, *the behavior of the mechanical system must be known with only small uncertainty in the vicinity of the crossover frequency*.
For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure [[fig:oomen18_next_gen_loop_gain]]).
#+caption: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth. cite:oomen18_advan_motion_contr_precis_mechat
The nano-hexapod and the control architecture have to be developed such that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system.
# High performance mechatronics systems (e.g. Wafer stages, or Atomic Force Microscopes) are usually developed in such a way that their mechanical behavior is extremely well known up to high frequency and such that the experimental conditions are usually be carefully controlled.
Thus, the total power in the signal can be obtained by integrating these infinitesimal contributions, the Root Mean Square (RMS) value of the signal $x(t)$ is then:
One can also integrate the infinitesimal power $S_{xx}(\omega)d\omega$ over a finite frequency band to obtain the power of the signal $x$ in that frequency band:
The Cumulative Power Spectrum will be used to determine in which frequency band the effect of disturbances should be reduced, and thus the approximate required control bandwidth.
*** Modification of a signal's PSD when going through an LTI system
Let's consider a signal $u$ with a PSD $S_{uu}$ going through a LTI system $G(s)$ that outputs a signal $y$ with a PSD (Figure [[fig:psd_lti_system]]).
- $S_{dd}$: disturbances, this will be done in Section [[sec:identification_disturbances]]
- $S_{nn}$: sensor noise, this can be estimated from the sensor data-sheet
- $S_{rr}$: which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to $0$
- The dynamics of the complete system comprising the micro-station and the nano-hexapod: $G$, $G_d$.
To do so, we need to identify the dynamics of the micro-station (Section [[sec:micro_station_dynamics]]), include this dynamics in a model (Section [[sec:multi_body_model]]) and add a model of the nano-hexapod to the model (Section [[sec:nano_hexapod_design]])
- The controller $K$ that will be designed in Section [[sec:robust_control_architecture]]
The obtained dynamics will allows us to compare the dynamics of the model.
** Setup
In order to perform to *Modal Analysis* and to obtain first a response model, the following devices were used:
- An *acquisition system* (OROS) with 24bits ADCs
- 3 tri-axis *Accelerometers*
- An *Instrumented Hammer*
The measurement thus consists of:
- Exciting the structure at the same location with the Hammer (Figure [[fig:hammer_z]])
- Move the accelerometers to measure all the DOF of the structure.
The position of the accelerometers are:
- 4 on the first granite
- 4 on the second granite
- 4 on top of the translation stage (figure [[fig:accelerometers_ty_overview]])
- 4 on top of the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod
In total, 69 degrees of freedom are measured (23 tri axis accelerometers).
#+name: fig:accelerometers_ty_overview
#+caption: Figure caption
[[file:figs/accelerometers_ty_overview.jpg]]
#+name: fig:hammer_z
#+caption: Figure caption
[[file:figs/hammer_z.gif]]
** Results
From the measurements, we obtain
- Reduction of the
- solid body assumption
- verification of the assumption => ok
#+name: fig:mode1
#+caption: Figure caption
[[file:figs/mode1.gif]]
#+name: fig:mode6
#+caption: Figure caption
[[file:figs/mode6.gif]]
** Conclusion
The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well.
This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest. This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
* Identification of the Disturbances
<<sec:identification_disturbances>>
** Introduction :ignore:
https://tdehaeze.github.io/meas-analysis/
Open Loop Noise budget: https://tdehaeze.github.io/nass-simscape/disturbances.html
Static Guiding errors:
- measured at the PEL
- low frequency errors, will thus be compensated
The problem are on the high frequency disturbances
25Hz vertical motion when the *Spindle* is turned on (even when not rotating).
** Stage Vibration - Effect of Motion
<<sec:stage_vibration_motion>>
We consider:
- The rotation of the Spindle
- The translation of the Translation Stage
** Sum of all disturbances
#+name: fig:dist_effect_relative_motion
#+caption: Amplitude Spectral Density fo the motion error due to disturbances
[[file:figs/dist_effect_relative_motion.png]]
#+name: fig:dist_effect_relative_motion_cas
#+caption: Cumulative Amplitude Spectrum of the motion error due to disturbances
[[file:figs/dist_effect_relative_motion_cas.png]]
Expected required bandwidth
** Better measurement of the effect of disturbances
Here, the measurement were made with inertial sensors.
However, we are interested in the relative motion of the sample with respect to the granite and not the absolute motion.
The best measurement of the disturbances would be to have the metrology already functioning.
We could perform a measurement using the X-ray.
Detector Requirement:
- Sample frequency above $400Hz$
- Resolution of $\approx 20nm$
** Conclusion
* Multi Body Model
<<sec:multi_body_model>>
** Introduction :ignore:
https://tdehaeze.github.io/nass-simscape/
Multi-Body model
** Validity of the model
The mass/inertia of each stage is automatically computed from the geometry and the density of the materials.
The stiffness of each joint is first set to measured values or stiffness from data sheets.
Then, the values of the stiffness and damping of each joint is manually tuned until the obtained dynamics is sufficiently close to the measured dynamics.
We could, from the measurement, automatically extract the stiffness and damping values, we this would have required a lot of work and having a perfect match is not required here.
Comparison model - measurements : https://tdehaeze.github.io/nass-simscape/identification.html
#+name: fig:identification_comp_top_stages
#+caption: Figure caption
[[file:figs/identification_comp_top_stages.png]]
** Wanted position of the sample and position error
From the reference position of each stage, we can compute the wanted pose of the sample with respect to the granite.
This is done with multiple transformation matrices.
Then, from the measurement of the metrology corresponding to the position of the sample with respect to the granite, we can compute the position error of the sample expressed in a frame fixed to the nano-hexapod.
#+name: fig:control-schematic-nass
#+caption: Figure caption
[[file:figs/control-schematic-nass.png]]
Measurement of the sample's position - conversion of positioning error in the frame of the Nano-hexapod for control: https://tdehaeze.github.io/nass-simscape/positioning_error.html
