#+TITLE: Design of the Nano-Hexapod and associated Control Architectures - Summary
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#+STARTUP: overview
#+DATE: 04-2020

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#+LATEX_HEADER: \newcommand{\authorFirstName}{Thomas}
#+LATEX_HEADER: \newcommand{\authorLastName}{Dehaeze}
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* Introduction                                                        :ignore:
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.


To understand the design challenges of such system, a short introduction to Feedback control is provided in Section [[sec:feedback_introduction]].
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]]).

* Introduction to Feedback Systems and Noise budgeting
<<sec:feedback_introduction>>

In this section, we first introduce some basics of *feedback systems* (Section [[sec:feedback]]).
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.

** Feedback System
<<sec:feedback>>

*** Introduction                                                    :ignore:
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
Let's consider the block diagram shown in Figure [[fig:classical_feedback_small]] where the signals are:
- $y$: the relative position of the sample with respect to the granite (the quantity we wish to control)
- $d$: the disturbances affecting $y$ (ground motion, vibration of stages)
- $n$: the noise of the sensor measuring $y$
- $r$: the reference signal, corresponding to the wanted $y$
- $\epsilon = r - y$: the position error

And the dynamical blocks are:
- $G$: representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite $y$
- $G_d$: representing how the disturbances (e.g. ground motion) are affecting the relative position sample/granite $y$
- $K$: representing the controller (to be designed)

#+begin_src latex :file classical_feedback_small.pdf
  \begin{tikzpicture}
    \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
    \node[block, right=0.6 of addfb] (K){$K$};
    \node[block, right=0.6 of K] (G){$G$};
    \node[addb={+}{}{}{}{}, right=0.6 of G] (adddy){};
    \node[addb={+}{}{}{}{}, below right=0.6 and 0.6 of adddy] (addn) {};
    \node[block, above=0.7 of adddy] (Gd){$G_d$};

    \draw[<-] (addfb.west) -- ++(-0.6, 0) node[above right]{$r$};
    \draw[->] (addfb.east) -- (K.west);
    \draw[->] (K.east) -- (G.west) node[above left]{$u$};
    \draw[->] (G.east) -- (adddy.west);
    \draw[<-] (addn.east) -- ++(0.6, 0) coordinate[](endpos) node[above left]{$n$};
    \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
    \draw[->] (adddy-|addn) node[branch]{} -- (addn.north);
    \draw[->] (addn.west) -| (addfb.south) node[below right]{$y_m$};
    \draw[<-] (adddy.north) -- (Gd.south);
    \draw[<-] (Gd.north) -- ++(0, 0.7) node[below right]{$d$};
  \end{tikzpicture}
#+end_src

#+name: fig:classical_feedback_small
#+caption: Block Diagram of a simple feedback system
#+RESULTS:
[[file:figs/classical_feedback_small.png]]

Without the use of feedback (i.e. nano-hexapod), the disturbances will induce a sample motion error equal to:
\begin{equation}
  y = G_d d \label{eq:open_loop_error}
\end{equation}
which is out of the specifications (micro-meter range compare to the required $\approx 10nm$).

In the next section, we see how the use of the feedback system permits to lower the effect of the disturbances $d$ on the sample motion error.

*** 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:
\[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]

We usually note:
\begin{align}
  S &= \frac{1}{1 + GK} \\
  T &= \frac{GK}{1 + GK}
\end{align}
where $S$ is called the sensibility transfer function and $T$ the transmissibility transfer function.

And the position error can be rewritten as:
\begin{equation}
  \epsilon = S r + T n - G_d S d \label{eq:closed_loop_error}
\end{equation}


From Eq. eqref:eq:closed_loop_error representing the closed-loop system behavior, we can see that:
- the effect of disturbances $d$ on $\epsilon$ is multiplied by a factor $S$ compared to the open-loop case
- the measurement noise $n$ is injected and multiplied by a factor $T$

Ideally, we would like to design the controller $K$ such that:
- $|S|$ is small to limit the effect of disturbances
- $|T|$ is small to limit the injection of sensor noise

As shown in the next section, there is a trade-off between the disturbance reduction and the noise injection.

*** Trade off: Disturbance Reduction / Noise Injection
We have from the definition of $S$ and $T$ that:
\begin{equation}
  S + T = \frac{1}{1 + GK} + \frac{GK}{1 + GK} = 1
\end{equation}
meaning that we cannot have $|S|$ and $|T|$ small at the same time.

There is therefore a *trade-off between the disturbance rejection and the measurement noise filtering*.


Typical shapes of $|S|$ and $|T|$ as a function of frequency are shown in Figure [[fig:h-infinity-2-blocs-constrains]].
We can observe that $|S|$ and $|T|$ exhibit different behaviors depending on the frequency band:
- *At low frequency* (inside the control bandwidth):
  - $|S|$ can be made small and thus the effect of disturbances is reduced
  - $|T| \approx 1$ and all the sensor noise is transmitted
- *At high frequency* (outside the control bandwidth):
  - $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances
  - $|T|$ is small and thus the sensor noise is filtered
- *Near the crossover frequency* (between the two frequency bands):
  - The effect of disturbances is increased

#+begin_src latex :file h-infinity-2-blocs-constrains.pdf
  \begin{tikzpicture}
    \begin{scope}[shift={(0, 0)}]
      \draw[dashed, fill=white] (-0.5, -3.4) rectangle (5.5, 1.4);
      \draw[] (2.5, 1.0) node[]{$\left| S(j\omega) \right|$};
      \draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
      \draw[] (0.6, -0.5) node[]{$\sim \left| GK \right|^{-1}$};
      \draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
      \draw[] (4.5, -0.5) node[]{$\sim 1$};
      \draw[fill=red!20] (2.5, 0.15) circle (0.15);
      \draw[dashed] (-0.4, 0) -- (5.4, 0);
      \draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3)
      to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
      \draw[<->] (-0.2, -2.8) -- node[midway, below, align=center]{\footnotesize Low Freq. } (1.8, -2.8);
      \draw[<->] (1.8, -2.8) --  node[midway, below, align=center]{\footnotesize Cross Over} (3.2, -2.8);
      \draw[<->] (3.2, -2.8) --  node[midway, below, align=center]{\footnotesize High Freq.} (5.2, -2.8);
    \end{scope}

    \begin{scope}[shift={(6.4, 0)}]
      \draw[dashed, fill=white] (-0.5, -3.4) rectangle (5.5, 1.4);
      \draw[] (2.5, 1.0) node[]{$\left| T(j\omega) \right|$};
      \draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
      \draw[] (0.6, -0.5) node[]{$\sim 1$};
      \draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
      \draw[] (4.5, -0.5) node[]{$\sim \left| GK \right|$};
      \draw[fill=red!20] (2.5, 0.15) circle (0.15);
      \draw[dashed] (-0.4, 0) -- (5.4, 0);
      \draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
      \draw[<->] (-0.2, -2.8) -- node[midway, below, align=center]{\footnotesize Low Freq. } (1.8, -2.8);
      \draw[<->] (1.8, -2.8) --  node[midway, below, align=center]{\footnotesize Cross Over} (3.2, -2.8);
      \draw[<->] (3.2, -2.8) --  node[midway, below, align=center]{\footnotesize High Freq.} (5.2, -2.8);
    \end{scope}
  \end{tikzpicture}
#+end_src

#+name: fig:h-infinity-2-blocs-constrains
#+caption: Typical shapes and constrain of the Sensibility and Transmibility closed-loop transfer functions
#+RESULTS:
[[file:figs/h-infinity-2-blocs-constrains.png]]

*** Trade off: Robustness / Performance
<<sec:perf_robust_tradeoff>>

As shown in the previous section, the effect of disturbances is reduced /inside/ the control bandwidth.

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 next important question is *what effects do limit the attainable control bandwidth?*


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]]).

#+name: 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
[[file:figs/oomen18_next_gen_loop_gain.png]]

This also means that *any possible change in the system should have a small impact on the system dynamics in the vicinity of the crossover*.

For the NASS, the possible changes in the system are:
- a modification of the payload mass and dynamics
- a change of experimental condition: spindle's rotation speed, position of each micro-station's stage
- a change in the micro-station dynamics (change of mechanical elements, aging effect, ...)

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.

This problem of *robustness* represent one of the main challenge for the design of the NASS.

# 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.

** Dynamic error budgeting
<<sec:noise_budget>>

*** Introduction                                                    :ignore:
The dynamic error budgeting is a powerful tool to study the effect of multiple error sources and to see how the feedback system does reduce the effect

To understand how to use and understand it, the Power Spectral Density and the Cumulative Power Spectrum are first introduced.
Then, is shown how does multiple error sources are combined and modified by dynamical systems.

Finally,

*** Power Spectral Density
The *Power Spectral Density* (PSD) $S_{xx}(f)$ of the time domain signal $x(t)$ is defined as the Fourier transform of the autocorrelation function:
\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \ \frac{[\text{unit of } x]^2}{\text{Hz}} \]

The PSD $S_{xx}(\omega)$ represents the *distribution of the (average) signal power over frequency*.

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:
\begin{equation}
  x_{\text{rms}} = \sqrt{\int_{0}^{\infty} S_{xx}(\omega) d\omega}
\end{equation}

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:
\begin{equation}
  P_{f_1,f_2} = \int_{f_1}^{f_2} S_{xx}(\omega) d\omega \quad [\text{unit of } x]^2
\end{equation}

*** Cumulative Power Spectrum
The *Cumulative Power Spectrum* is the cumulative integral of the Power Spectral Density starting from $0\ \text{Hz}$ with increasing frequency:
\begin{equation}
  CPS_x(f) = \int_0^f S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2
\end{equation}
The Cumulative Power Spectrum taken at frequency $f$ thus represent the power in the signal in the frequency band $0$ to $f$.


An alternative definition of the Cumulative Power Spectrum can be used where the PSD is integrated from $f$ to $\infty$:
\begin{equation}
  CPS_x(f) = \int_f^\infty S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2
\end{equation}
And thus $CPS_x(f)$ represents the power in the signal $x$ for frequencies above $f$.


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]]).

#+begin_src latex :file psd_lti_system.pdf
  \begin{tikzpicture}
    \node[block] (G) at (0, 0) {$G(s)$};

    \draw[<-] (G.west) -- node[midway, above]{$u$} ++(-1.4, 0);
    \draw[->] (G.east) -- node[midway, above]{$y$} ++(1.4, 0);
  \end{tikzpicture}
#+end_src

#+NAME: fig:psd_lti_system
#+CAPTION: LTI dynamical system $G(s)$ with input signal $u$ and output signal $y$
#+RESULTS:
[[file:figs/psd_lti_system.png]]

The Power Spectral Density of the output signal $y$ can be computed using:
\begin{equation}
  S_{yy}(\omega) = \left|G(j\omega)\right|^2 S_{uu}(\omega)
\end{equation}

*** PSD of combined signals
Let's consider a signal $y$ that is the sum of two *uncorrelated* signals $u$ and $v$ (Figure [[fig:psd_sum]]).

We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v$ (can be easily shown from the definition of the PSD):
\[ S_{yy} = S_{uu} + S_{vv} \]

#+begin_src latex :file psd_sum.pdf
  \begin{tikzpicture}
    \node[addb] (addb) at (0, 0) {};

    \draw[<-] (addb.north west) -- ++(-0.5,  0.5) -- node[midway, above]{$u$} ++(-1.4, 0);
    \draw[<-] (addb.south west) -- ++(-0.5, -0.5) -- node[midway, above]{$v$} ++(-1.4, 0);

    \draw[->] (addb.east) -- node[midway, above]{$y$} ++(1.4, 0);
  \end{tikzpicture}
#+end_src

#+name: fig:psd_sum
#+caption: $y$ as the sum of two signals $u$ and $v$
#+RESULTS:
[[file:figs/psd_sum.png]]

*** Dynamic Noise Budgeting
Let's consider the Feedback architecture in Figure [[fig:classical_feedback_small]] where the position error $\epsilon$ is equal to:
\[ \epsilon = S r + T n - G_d S d \]

If we suppose that the signals $r$, $n$ and $d$ are *uncorrelated* (which is a good approximation in our case), the PSD of $\epsilon$ is:
\[ S_{\epsilon \epsilon}(\omega) = |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \]

And we can compute the RMS value of the residual motion using:
\begin{align*}
  \epsilon_\text{rms} &= \sqrt{ \int_0^\infty S_{\epsilon\epsilon}(\omega) d\omega} \\
  &= \sqrt{ \int_0^\infty \Big( |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \Big) d\omega }
\end{align*}


To estimate the PSD of the position error $\epsilon$ and thus the RMS residual motion (in closed-loop), we need to determine:
- The Power Spectral Densities of the signals affecting the 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]]

* Identification of the Micro-Station Dynamics
<<sec:micro_station_dynamics>>

** Introduction                                                      :ignore:
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 the controller.
The estimated dynamics will also be used to tune the developed multi-body model of the micro-station with which the simulations will be performed.

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.


** Setup
<<sec:id_setup>>

To measure the dynamics of such complicated system, it as been chosen to perform a full modal analysis.

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:
- 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) which is way more that what was required.

We chose to have some redundancy in the measurement to be able to verify that the solid-body assumption was correct for each of the stage.

#+name: fig:hammer_z
#+caption: Example of one hammer impact
[[file:figs/hammer_z.gif]]

#+name: fig:accelerometers_ty_overview
#+caption: 3 tri axis accelerometers fixed to the translation stage
[[file:figs/accelerometers_ty_overview.jpg]]

** Results
<<sec:id_results>>

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.

Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures [[fig:mode1]] [[fig:mode6]].

#+name: fig:mode1
#+caption: First mode
[[file:figs/mode1.gif]]

#+name: fig:mode6
#+caption: Sixth mode
[[file:figs/mode6.gif]]


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.

This confirms the fact that the stages are indeed behaving as a *solid body* in the frequency band of interest.
This thus means 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:
In this section, we wish to list and identify all the disturbances affecting the system.

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 problem are on the high frequency disturbances.
In the following sections, we consider:
- the ground motion
- vibrations of each stage, due either to their control systems or their motion


https://tdehaeze.github.io/meas-analysis/
Open Loop Noise budget: https://tdehaeze.github.io/nass-simscape/disturbances.html

** Ground Motion
<<sec:ground_motion>>

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.

#+name: fig:ground_motion_compare
#+caption: Comparison of the PSD of the ground motion measured at different location
[[file:figs/ground_motion_compare.png]]

** Stage Vibration - Effect of Control systems
<<sec:stage_vibration_control>>

Control system of each stage has been tested

Each motor are turn off and then on.
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]].

25Hz vertical motion when the *Spindle* is turned on (even when not rotating).

** Stage Vibration - Effect of Motion
<<sec:stage_vibration_motion>>

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.
#+end_important

** Sum of all disturbances
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.

#+name: fig:dist_effect_relative_motion
#+caption: Amplitude Spectral Density fo the motion error due to disturbances
[[file:figs/dist_effect_relative_motion.png]]

The Cumulative Amplitude Spectrum is shown in Figure [[fig:dist_effect_relative_motion_cas]].
It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range.

#+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]]

We can also estimate the required bandwidth by seeing that $10\ nm [rms]$ motion is induced by the perturbations above 100Hz.

This means that if the controller compensate all the motion errors below 100Hz (ideal case), 10nm [rms] of motion will still remain.

From that, we can conclude that we will probably need a control bandwidth to around 100Hz.

** Better estimation of the disturbances
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.

This is not perfect as using only one geophone on the sample and one on the granite do not permit to separate the translations and the rotations.

An alternative could be to position a reference object at the sample location and to use the X-ray to measure its motion.

The detector requirement would be:
- Sample frequency above $400Hz$
- Resolution of $\approx 100nm$ (to be discussed)

** Conclusion
Main disturbance sources have been identified.
These disturbances will then be included in the multi-body model.


Other disturbance sources were not estimated such as cable forces and acoustic disturbances.
If heavy/stiff cables are to be fixed to the sample, this should be quantified and included in the model.


Having better estimation of the disturbances would allows to more precisely estimate the attainable performances.
This should however not change the conclusion of this study nor significantly change the nano-hexapod design.

* 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

** Simulation of Experiments
Now that the
- dynamics of the model is tuned
- disturbances are included in the model

We can perform simulation of experiments.

https://tdehaeze.github.io/nass-simscape/experiments.html

[[fig:exp_scans_rz_dist]]

#+name: fig:exp_scans_rz_dist
#+caption: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances
[[file:figs/exp_scans_rz_dist.png]]

** Conclusion
#+begin_important
Possible to study many effects.
Extraction of transfer function like $G$ and $G_d$.

Simulation of experiments to validate performance.
#+end_important

* Optimal Nano-Hexapod Design
<<sec:nano_hexapod_design>>

** Introduction                                                      :ignore:

As explain before, the nano-hexapod properties (mass, stiffness, architecture, ...) will influence:
- the plant dynamics $G$
- the effect of disturbances $G_d$

We which here to choose the nano-hexapod properties such that:
- has an easy
- minimize the
- minimize $|G_d|$

** Optimal Stiffness to reduce the effect of disturbances

** Optimal Stiffness
The goal is to design a system that is *robust*.

Thus, we have to identify the sources of uncertainty and try to minimize them.

Uncertainty in the system can be caused by:
- Effect of Support Compliance: https://tdehaeze.github.io/nass-simscape/uncertainty_support.html
- Effect of Payload Dynamics: https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html
- Effect of experimental condition (micro-station pose, spindle rotation): https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html

All these uncertainty will limit the maximum attainable bandwidth.

Fortunately, the nano-hexapod stiffness have an influence on the dynamical uncertainty induced by the above effects.

Determination of the optimal stiffness based on all the effects:
- https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html

#+begin_conclusion

#+end_conclusion


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


The nano-hexapod stiffness will also change the sensibility to disturbances.

Effect of Nano-hexapod stiffness on the Sensibility to disturbances: https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html

#+begin_conclusion

#+end_conclusion

** Sensors to be included

Ways to damp:
- Force Sensor
- Relative Velocity Sensors
- Inertial Sensor



https://tdehaeze.github.io/rotating-frame/index.html

Sensors to be included:


* Robust Control Architecture
<<sec:robust_control_architecture>>

** Introduction                                                      :ignore:
https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html


** Simulation of Tomography Experiments
<<sec:tomography_experiment>>

- Make several animations
  - [ ] One of a tomography experiment where we see all the station rotating
  - [ ] A zoom on at the nano-meter level to see how the wanted position is moving

** Conclusion
* Further notes
Soft granite

Sensible to detector motion?

Common metrology frame for the nano-focusing optics and the measurement of the sample position?

Cable forces?

Slip-Ring noise?