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

index.org

@@ -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 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
+- 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 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 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 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'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]].
-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 vertical vibrations of the spindle.
-It is shown that a softer nano-hexapod it better to filter out stage vibrations.
 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 sample's vibrations.
-
-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.
 
 #+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.
+One of the most important 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.
+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 *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 *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.
 
-
+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.
-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.
+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.
-Separate studies has been conducted to see how the support's compliance appears in
 
 *** 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