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]]
As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be coupled with the dynamics of the nano-hexapod and thus is very important for both the design of the nano-hexapod and controller.
All the measurements performed on the micro-station are detailed in [[https://tdehaeze.github.io/meas-analysis/][this]] document and summarized in the following sections.
To limit the number of degrees of freedom to be measured, we suppose that in the frequency range of interest (DC-300Hz), each of the positioning stage is behaving as a *solid body*.
Thus, to fully describe the dynamics of the station, we (only) need to measure 6 degrees of freedom on each of the positioning stage (that is 36 degrees of freedom for the 6 solid bodies).
In order to perform the *Modal Analysis*, the following devices were used:
From the measurements, we obtain all the transfer functions from forces applied at the location of the hammer impacts to the x-y-z acceleration of each solid body at the location of each accelerometer.
We then reduce the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF).
From the reduced transfer function matrix, we can re-synthesize the response at the 69 measured degrees of freedom and we find that we have an exact match.
Many Frequency Response Functions (FRF) are obtained from the measurements.
Examples of FRF are shown in Figure [[fig:frf_all_bodies_one_direction]].
These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.
#+name: fig:frf_all_bodies_one_direction
#+caption: Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction
[[file:figs/frf_all_bodies_one_direction.png]]
** Conclusion
#+begin_important
The modal analysis of the micro-station confirmed the fact that a multi-body model should be able to correctly represents the micro-station dynamics.
In Section [[sec:multi_body_model]], the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
Note that here we are not much interested by low frequency disturbances such as thermal effects and static guiding errors of each positioning stage.
This is because the frequency content of these errors will be located in the controller bandwidth and thus will be easily compensated by the nano-hexapod.
The ground motion can easily be estimated using an inertial sensor with sufficient sensitivity.
To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure [[fig:geophones]]).
#+name: fig:geophones
#+caption: Huddle Test Setup
[[file:figs/geophones.jpg]]
The measured Power Spectral Density of the ground motion at the ID31 floor is compared with other measurements performed at ID09 and at CERN.
The low frequency differences between the ground motion at ID31 and ID09 is just due to the fact that for the later measurement, the low frequency sensitivity of the inertial sensor was not taken into account.
The goal is to see what noise is injected in the system due to the regulation loop of each stage.
Complete reports on these measurements are accessible [[https://tdehaeze.github.io/meas-analysis/2018-10-15%20-%20Marc/index.html][here]] and [[https://tdehaeze.github.io/meas-analysis/disturbance-control-system/index.html][here]].
We consider here the vibrations induced by the scans of the translation stage and rotation of the spindle.
Details reports are accessible [[https://tdehaeze.github.io/meas-analysis/disturbance-ty/index.html][here]] for the translation stage and [[https://tdehaeze.github.io/meas-analysis/disturbance-sr-rz/index.html][here]] for the spindle/slip-ring.
*** Spindle and Slip-Ring
:PROPERTIES:
:UNNUMBERED: t
:END:
The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure [[fig:rz_meas_errors]].
#+name: fig:rz_meas_errors
#+caption: Measurement of the sample's vertical motion when rotating at 6rpm
[[file:figs/rz_meas_errors.gif]]
A geophone is fixed at the location of the sample and we measure the motion:
- without rotation
- when rotating at 6rpm using the slip-ring motor
- when rotating at 6rpm using the spindle motor synchronized with the slip-ring motor
The obtained Power Spectral Density of the sample's absolute velocity are shown in Figure [[fig:sr_sp_psd_sample_compare]].
We can see that when using the Slip-ring motor to rotate the sample, only a little increase of the motion is observed above 100Hz.
However, when rotating with the Spindle (normal functioning mode):
- a very sharp peak at 23Hz is observed. Its cause has not been identified yet
- a general large increase in motion above 30Hz
#+name: fig:sr_sp_psd_sample_compare
#+caption: Comparison of the ASD of the measured voltage from the Geophone at the sample location
[[file:figs/sr_sp_psd_sample_compare.png]]
#+begin_important
Some investigation should be performed on the Spindle to determine where does this 23Hz motion comes from.
#+end_important
*** Translation Stage
:PROPERTIES:
:UNNUMBERED: t
:END:
The same setup is used (a geophone is located at the sample's location and another on the granite).
We impose a 1Hz triangle motion with an amplitude of $\pm 2.5mm$ on the translation stage (Figure [[fig:figure_name]]), and we measure the absolute velocity of both the sample and the granite.
#+name: fig:figure_name
#+caption: Y position of the translation stage measured by the encoders
[[file:figs/ty_position_time.png]]
The time domain absolute vertical velocity of the sample and granite are shown in Figure [[fig:ty_z_time]].
It is shown that quite large motion of the granite is induced by the translation stage scans.
This could be a problem if this is shown to excite the metrology frame of the nano-focusing lens position stage.
#+name: fig:ty_z_time
#+caption: Vertical velocity of the sample and marble when scanning with the translation stage
[[file:figs/ty_z_time.png]]
The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure [[fig:asd_z_direction]].
We can see many peaks starting from 1Hz showing the large spectral content probably due to the triangular reference of the translation stage.
#+name: fig:asd_z_direction
#+caption: Amplitude spectral density of the measure velocity corresponding to the geophone in the vertical direction located on the granite and at the sample location when the translation stage is scanning at 1Hz
[[file:figs/asd_z_direction.png]]
#+begin_important
A smoother motion for the translation stage (such as a sinus motion) could probably help reducing much of the vibrations produced.
We can now compare the effect of all the disturbance sources on the position error (relative motion of the sample with respect to the granite).
The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure [[fig:dist_effect_relative_motion]].
We can see that the ground motion is quite small compare to the translation stage and spindle induced motions.
All the disturbance measurements were made with inertial sensors, and to obtain the relative motion sample/granite, two inertial sensors were used and the signals were subtracted.
As was shown during the modal analysis (Section [[sec:micro_station_dynamics]]), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies.
To do so, we use the Matlab's [[https://www.mathworks.com/products/simscape.html][Simscape]] toolbox.
A small summary of the multi-body Simscape is available [[https://tdehaeze.github.io/nass-simscape/simscape.html][here]] and each of the modeled stage is described [[https://tdehaeze.github.io/nass-simscape/simscape_subsystems.html][here]].
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.
It is very difficult the tune the dynamics of such model as there are more than 50 parameters and many curves to compare between the model and the measurements.
The comparison of three of the Frequency Response Functions are shown in Figure [[fig:identification_comp_top_stages]].
Most of the other measured FRFs and identified transfer functions from the multi-body model have the same level of matching.
We believe that the model is representing the micro-station dynamics with sufficient precision for the current analysis.
#+caption: Frequency Response function from Hammer force in the X,Y and Z directions to the X,Y and Z displacements of the micro-hexapod's top platform. The measurements are shown in blue and the Model in red.
More detailed comparison between the model and the measured dynamics is performed [[https://tdehaeze.github.io/nass-simscape/identification.html][here]].
Now that the multi-body model dynamics as been tuned, the following elements are included:
- Actuators to perform the motion of each stage (translation, tilt, spindle, hexapod)
- Sensors to measure the motion of each stage and the relative motion of the sample with respect to the granite (metrology system)
- Disturbances such as ground motion and stage's vibrations
Then, using the model, we can
- perform simulation of experiments in presence of disturbances
- measure the motion of the solid-bodies
- identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod and control design
- include a multi-body model of the nano-hexapod and closed-loop simulations
** Wanted position of the sample and position error
For the control of the nano-hexapod, we need to now the sample position error (the motion to be compensated) in the frame of the nano-hexapod.
To do so, we need to perform several computations (summarized in Figure [[fig:control-schematic-nass]]):
- First, we need to determine the actual *wanted pose* (3 translations and 3 rotations) of the sample with respect to the granite.
This is determined from the wanted motion of each micro-station stage.
Each wanted stage motion is represented by an homogeneous transformation matrix (explain [[http://planning.cs.uiuc.edu/node111.html][here]]), then these matrices are combined to give to total wanted motion of the sample with respect to the granite.
- Then, we need to determine the *actual pose* of the sample with respect to the granite.
This will be performed by several interferometers and several computation will be required to compute the pose of the sample from the interferometers measurements.
However we here directly measure the 3 translations and 3 rotations of the sample using a special simscape block.
- Finally, we need to compare the wanted pose with the measured pose to compute the position error of the sample.
This position error can be expressed in the frame of the granite, or in the frame of the (rotating) nano-hexapod.
#+caption: Tomography Experiment using the Simscape Model
[[file:figs/open_loop_sim.gif]]
#+name: fig:open_loop_sim_zoom
#+caption: Tomography Experiment using the Simscape Model - Zoom on the sample's position (the full vertical scale is $\approx 10 \mu m$)
[[file:figs/open_loop_sim_zoom.gif]]
The position error of the sample with respect to the granite are shown in Figure [[fig:exp_scans_rz_dist]].
It is shown that the X-Y-Z position errors are in the micro-meter range.
For the rotation around X and Y, the errors are quite small.
This is explained by the fact that no torque disturbances is considered in the model.
For the vertical rotation, this is due to the fact that we suppose perfect rotation of the Spindle, and anyway, no measurement of the sample with respect to the granite is made by the interferometers.
- study many effects such as the change of dynamics due to the rotation, the sample mass, etc.
- extract transfer function like $G$ and $G_d$
- simulate experiments to validate performance
This model will be used in the next sections to help the design of the nano-hexapod, to develop the robust control architecture and to perform simulation in order to validate.
