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Add text about optimal stiff w.r.t. uncertainty

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Thomas Dehaeze 2020-04-28 19:50:42 +02:00
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@ -11,7 +11,7 @@
#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
#+STARTUP: overview
#+DATE: 04-2020
#+DATE: 05-2020
#+LATEX_CLASS: cleanreport
#+LATEX_CLASS_OPTIONS: [conf, hangsection, secbreak]
@ -764,51 +764,67 @@ As explain before, the nano-hexapod properties (mass, stiffness, architecture, .
- the plant dynamics $G$ (important for the control robustness properties)
Thus, we here wish to find the optimal nano-hexapod properties such that:
- the effect of disturbances is minimized
- the plant uncertainty due to a change of payload mass and experimental conditions is minimized
- the effect of disturbances is minimized (Section [[sec:optimal_stiff_dist]])
- the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section [[sec:optimal_stiff_plant]])
The study presented here only consider changes in the nano-hexapod *stiffness*.
The nano-hexapod mass cannot be change too much, and will anyway be negligible compare to the metrology reflector and the payload masses.
The choice of the nano-hexapod architecture (e.g. orientations of the actuators and implementation of sensors) will be further studied in accord with the control architecture.
The study presented here only consider changes in the nano-hexapod *stiffness* for two reasons:
- the nano-hexapod mass cannot be change too much, and will anyway be negligible compare to the metrology reflector and the payload masses
- the choice of the nano-hexapod architecture (e.g. orientations of the actuators and implementation of sensors) will be further studied in accord with the control architecture
** Optimal Stiffness to reduce the effect of disturbances
The nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of $G_d$).
<<sec:optimal_stiff_dist>>
*** Introduction :ignore:
As will be seen, the nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of $G_d$).
For instance, it is quite obvious that a stiff nano-hexapod is better than a soft one when it comes to direct forces applied to the sample such as cable forces.
A complete study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility [[https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html][here]] and summarized below.
*** Sensibility to stage vibrations
:PROPERTIES:
:UNNUMBERED: t
:END:
The sensibility to the spindle vibration as a function of the nano-hexapod stiffness is shown in Figure [[fig:opt_stiff_sensitivity_Frz]] (similar curves are obtained for translation stage vibrations).
It is shown that a softer nano-hexapod it better to filter out stage vibrations.
The sensibility to the spindle's vibration for all the considered nano-hexapod stiffnesses (from $10^3\,[N/m]$ to $10^9\,[N/m]$) is shown in Figure [[fig:opt_stiff_sensitivity_Frz]].
It is shown that a softer nano-hexapod it better to filter out vertical vibrations of the spindle.
More precisely, is start to filters the vibration at the first suspension mode of the payload on top of the nano-hexapod.
The same conclusion is made for vibrations of the translation stage.
#+name: fig:opt_stiff_sensitivity_Frz
#+caption: Sensitivity to Spindle vertical motion error to the vertical error position of the sample
[[file:figs/opt_stiff_sensitivity_Frz.png]]
*** Sensibility to ground motion
:PROPERTIES:
:UNNUMBERED: t
:END:
The sensibilities to ground motion in the Y and Z directions are shown in Figure [[fig:opt_stiff_sensitivity_Dw]].
We can see that above the suspension mode of the nano-hexapod, the norm of the transmissibility is close to one until the suspension mode of the granite.
Thus, a stiff nano-hexapod is better for reducing the effect of ground motion at low frequency.
It will be further suggested that using soft mounts for the granite can greatly improve the sensibility to ground motion.
It will be further suggested that using soft mounts for the granite can greatly lower the sensibility to ground motion.
#+name: fig:opt_stiff_sensitivity_Dw
#+caption: Sensitivity to Ground motion to the position error of the sample
[[file:figs/opt_stiff_sensitivity_Dw.png]]
*** Dynamic Noise Budgeting considering all the disturbances
:PROPERTIES:
:UNNUMBERED: t
:END:
However, lowering the sensibility to some disturbance at a frequency where its effect is already small compare to the other disturbances sources is not really interesting.
What is more important than comparing the sensitivity to disturbances, is thus to compare the obtain power spectral density of the sample's position error.
From the Power Spectral Density of all the sources of disturbances identified in Section [[sec:identification_disturbances]], we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure [[fig:opt_stiff_psd_dz_tot]]).
Then, we take the Power Spectral Density of all the sources of disturbances as identified in Section [[sec:identification_disturbances]], and we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure [[fig:opt_stiff_psd_dz_tot]]).
We can see that the most important change is in the frequency range 30Hz to 300Hz where a stiffness smaller than $10^5\,[N/m]$ greatly reduces the sensibility to disturbances.
We can see that the most important change is in the frequency range 30Hz to 300Hz where a stiffness smaller than $10^5\,[N/m]$ greatly reduces the sample's vibrations.
#+name: fig:opt_stiff_psd_dz_tot
#+caption: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses
[[file:figs/opt_stiff_psd_dz_tot.png]]
If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure [[fig:opt_stiff_cas_dz_tot]], we can observe that a soft hexapod ($k < 10^5 - 10^6\,[N/m]$) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance.
#+name: fig:opt_stiff_cas_dz_tot
@ -816,102 +832,185 @@ If we look at the Cumulative amplitude spectrum of the vertical error motion in
[[file:figs/opt_stiff_cas_dz_tot.png]]
** Optimal Stiffness to reduce the plant uncertainty
<<sec:optimal_stiff_plant>>
*** Introduction :ignore:
One of the primary design goal is to obtain a system that is *robust* to all changes in the system.
To design a robust system, we have to identify the sources of uncertainty and try to minimize them.
One of the most important design goal is to obtain a system that is *robust* to all changes in the system.
Therefore, we have to identify all changes that might occurs in the system and choose the nano-hexapod stiffness such that the uncertainty to these changes is minimized.
The uncertainty in the system can be caused by:
- A change in the *Support's compliance* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_support.html][here]]): if the micro-station dynamics is changing due to the change of parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change.
- A change in the *Payload mass/dynamics* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html][here]]).
- A change in the *Support's compliance* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_support.html][here]]): if the micro-station dynamics is changing due to the change of parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change
- A change in the *Payload mass/dynamics* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html][here]]): the sample's mass is ranging from $1\,kg$ to $50\,kg$
- A change of *experimental condition* such as the micro-station's pose or the spindle rotation (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html][here]])
All these uncertainties will limit the attainable bandwidth and hence the obtained performance.
All these uncertainties will limit the attainable bandwidth and hence the performances.
In the next sections, the effect the considered changes on the *plant dynamics* is quantified and conclusions are made on the optimal stiffness for robustness properties.
Fortunately, the nano-hexapod stiffness have an influence on the dynamical uncertainty induced by the above effects and we wish here to determine the optimal nano-hexapod stiffness.
Separate studies has been conducted to see how the support's compliance appears in
In the following study, when we refer to plant dynamics, this means the dynamics from forces applied by the nano-hexapod to the measured sample's position by the metrology.
We will only compare the plant dynamics as it is the most important dynamics for robustness and performance properties.
However, the dynamics from forces to sensors located in the nano-hexapod legs, such as force and relative motion sensors, have also been considered in a separate study.
*** Effect of Payload
:PROPERTIES:
:UNNUMBERED: t
:END:
The most obvious change in the system is the change of payload.
In Figure [[fig:opt_stiffness_payload_mass_fz_dz]] the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg.
On the left side, the change of dynamics is computed for a very soft nano-hexapod, while on the right side, it is computed for a very stiff nano-hexapod.
One can see that for the soft nano-hexapod:
- the first resonance (suspension mode of the nano-hexapod) is lowered with an increase of the sample's mass.
This first resonance corresponds to $\omega = \sqrt{\frac{k_n}{m_n + m_s}}$ where $k_n$ is the vertical nano-hexapod stiffness, $m_n$ the mass of the nano-hexapod's top platform, and $m_s$ the sample's mass
- the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample's mass
For the stiff-nano-hexapod, the change of payload mass has very little effect (the vertical scale for the amplitude is quite small).
#+name: fig:opt_stiffness_payload_mass_fz_dz
#+caption: Dynamics from $\mathcal{F}_z$ to $\mathcal{X}_z$ for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod
#+caption: Dynamics from $\mathcal{F}_z$ to $\mathcal{X}_z$ for varying payload mass, both for a soft nano-hexapod (left) and a stiff nano-hexapod (right)
[[file:figs/opt_stiffness_payload_mass_fz_dz.png]]
In Figure [[fig:opt_stiffness_payload_freq_fz_dz]] is shown the effect of a change of payload dynamics.
The mass of the payload is fixed and its resonance frequency is changing from 50Hz to 500Hz.
We can see (more easily for the soft nano-hexapod), that resonance of the payload produces an anti-resonance for the considered dynamics.
#+name: fig:opt_stiffness_payload_freq_fz_dz
#+caption: Dynamics from $\mathcal{F}_z$ to $\mathcal{X}_z$ for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod
[[file:figs/opt_stiffness_payload_freq_fz_dz.png]]
The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure [[fig:opt_stiffness_payload_impedance_all_fz_dz]].
For nano-hexapod stiffnesses below $10^6\,[N/m]$:
- the phase stays between 0 and -180deg which is a very nice property for control
- the dynamical change up until the resonance of the payload is mostly a change of gain
For nano-hexapod stiffnesses above $10^7\,[N/m]$:
- the dynamics is unchanged until the first resonance which is around 25Hz-35Hz
- above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics)
#+name: fig:opt_stiffness_payload_impedance_all_fz_dz
#+caption: Dynamics from $\mathcal{F}_z$ to $\mathcal{X}_z$ for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod
[[file:figs/opt_stiffness_payload_impedance_all_fz_dz.png]]
#+begin_important
For soft nano-hexapods, the payload has an important impact on the dynamics.
This will have to be carefully taken into account for the controller design.
For stiff nano-hexapod, the dynamics doe not change with the payload until the first resonance frequency of the nano-hexapod or of the payload.
If possible, the first resonance frequency of the payload should be maximized (stiff fixation).
Heavy samples with low first resonance mode will be very problematic.
#+end_important
*** Effect of Micro-Station Compliance
:PROPERTIES:
:UNNUMBERED: t
:END:
The micro-station dynamics is quite complex as was shown in Section [[sec:micro_station_dynamics]], moreover, its dynamics can change due to:
- a change in some mechanical elements
- a change in the position of one stage.
For instance, a large displacement of the micro-hexapod can change the micro-station compliance
- a change in a control loop
Thus, it would be much more robust if the plant dynamics were not depending on the micro-station dynamics.
This as several other advantages:
- the control could be develop on top on another support and then added to the micro-station without changing the controller
- the nano-hexapod could be use on top of any other station much more easily
To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure [[fig:opt_stiffness_micro_station_fx_dx]]):
- without the micro-station (solid curves)
- with the micro-station dynamics (dashed curves)
One can see that for nano-hexapod stiffnesses below $10^6\,[N/m]$, the plant dynamics does not significantly changed due to the micro station dynamics (the solid and dashed curves are superimposed).
For nano-hexapod stiffnesses above $10^7\,[N/m]$, the micro-station compliance appears in the plant dynamics starting at about 45Hz.
#+name: fig:opt_stiffness_micro_station_fx_dx
#+caption: Change of dynamics from force $\mathcal{F}_x$ to displacement $\mathcal{X}_x$ due to the micro-station compliance
[[file:figs/opt_stiffness_micro_station_fx_dx.png]]
#+begin_important
If the resonance of the nano-hexapod is below the first resonance of the micro-station, then the micro-station dynamics if "filtered out" and does not appears in the dynamics to be controlled.
This renders the system robust to any possible change of the micro-station dynamics.
If a stiff nano-hexapod is used, the control bandwidth should probably be limited to around the first micro-station's mode ($\approx 45\,[Hz]$) which will likely no give acceptable performance.
#+end_important
*** Effect of Spindle Rotating Speed
:PROPERTIES:
:UNNUMBERED: t
:END:
Let's now consider the rotation of the Spindle.
The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure [[fig:opt_stiffness_wz_fx_dx]].
One can see that for nano-hexapods with a stiffness above $10^5\,[N/m]$, the dynamics is mostly not changing with the spindle's rotating speed.
For very soft nano-hexapods, the main resonance is split into two resonances and one anti-resonance that are all moving at a function of the rotating speed.
#+name: fig:opt_stiffness_wz_fx_dx
#+caption: Change of dynamics from force $\mathcal{F}_x$ to displacement $\mathcal{X}_x$ for a spindle rotation speed from 0rpm to 60rpm
[[file:figs/opt_stiffness_wz_fx_dx.png]]
#+begin_important
If the resonance of the nano-hexapod is (say a factor 5) above the maximum rotation speed, then the plant dynamics will be mostly not impacted by the rotation.
*** Total Uncertainty
A very soft ($k < 10^4\,[N/m]$) nano-hexapod should not be used due to the effect of the spindle's rotation.
#+end_important
*** Total Plant Uncertainty
:PROPERTIES:
:UNNUMBERED: t
:END:
Finally, let's combined all the uncertainties and display the plant dynamics "spread" for all the nano-hexapod stiffnesses (Figure [[fig:opt_stiffness_plant_dynamics_task_space]]).
This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics.
#+name: fig:opt_stiffness_plant_dynamics_task_space
#+caption: Variability of the dynamics from $\bm{\mathcal{F}}_x$ to $\bm{\mathcal{X}}_x$ with varying nano-hexapod stiffness
[[file:figs/opt_stiffness_plant_dynamics_task_space.gif]]
#+begin_important
The leg stiffness should be at higher than $k = 10^4\,[N/m]$ such that the main resonance frequency does not shift too much when rotating.
Let's summarize the findings:
- the payload's mass influence the plant dynamics above the first resonance of the nano-hexapod.
Thus a high nano-hexapod stiffness helps reducing the effect of a change of the payload's mass
- the payload's first resonance is seen as an anti-resonance in the plant dynamics.
As this effect will largely be variable from one payload to the other, *the payload's first resonance should be maximized* (above 300Hz if possible) for all used payloads
- the dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when $k < 10^6\,[N/m]$
- the spindle's rotating speed has no significant influence on the plant dynamics for nano-hexapods with a stiffness $k > 10^5\,[N/m]$
Concerning the plant dynamic uncertainty, the resonance frequency of the nano-hexapod should be between 5Hz (way above the maximum rotating speed) and 50Hz (before the first micro-station resonance) for all the considered payloads.
This corresponds to an optimal nano-hexapod leg stiffness in the range $10^5 - 10^6\,[N/m]$.
In such case, the main limitation will be heavy samples with small stiffnesses.
#+end_important
#+begin_important
It is usually a good idea to maximize the mass, damping and stiffness of the isolation platform in order to be less sensible to the payload dynamics.
The best thing to do is to have a stiff isolation platform.
The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when the stiffness of the legs is less than $10^6\,[N/m]$. When the nano-hexapod is stiff ($k > 10^7\,[N/m]$), the compliance of the micro-station appears in the primary plant.
#+end_important
Determination of the optimal stiffness based on all the effects:
- https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html
The main performance limitation are payload variability
#+begin_question
Main problem: heavy samples with small stiffness.
The first resonance frequency of the sample will limit the performance.
#+end_question
#+begin_conclusion
#+end_conclusion
It is preferred that *one* controller is working for all the payloads.
If not possible, the alternative would be to develop an adaptive controller that depends on the payload mass/inertia.
** Conclusion
#+begin_important
In Section [[sec:optimal_stiff_dist]], it has been concluded that a nano-hexapod stiffness
Section [[sec:optimal_stiff_plant]]
A stiffness of $10^5\,[N/m]$ will be used.
#+end_important
#+begin_important
It is preferred that *one* controller is working for all the payloads.
If not possible, the alternative would be to develop an adaptive controller that depends on the payload mass/inertia.
#+end_important
A more detailed study of the determination of the optimal stiffness based on all the effects is available [[https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html][here]].
* Robust Control Architecture
<<sec:robust_control_architecture>>
@ -919,6 +1018,7 @@ If not possible, the alternative would be to develop an adaptive controller that
** Introduction :ignore:
https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html
stiffness 10^5
** Active Damping and Sensors to be included
Ways to damp:
@ -963,6 +1063,7 @@ Common metrology frame for the nano-focusing optics and the measurement of the s
Cable forces?
Slip-Ring noise?
* Bibliography :ignore:
bibliographystyle:unsrt
bibliography:ref.bib