nass-simscape/org/optimal_stiffness_disturbances.org

Determination of the optimal nano-hexapod's stiffness for reducing the effect of disturbances

• Introduction
• Disturbances
  • Introduction
  • Plots
• Effect of disturbances on the position error
  • Introduction
  • Initialization
  • Identification
  • Sensitivity to Stages vibration (Filtering)
  • Effect of Ground motion (Transmissibility).
  • Direct Forces (Compliance).
  • Conclusion
• Effect of granite stiffness
  • Introduction
  • Analytical Analysis
    • Simple mass-spring-damper model
    • General Case
  • Soft Granite
  • Effect of the Granite transfer function
  • Conclusion
• Open Loop Budget Error
  • Introduction
  • Noise Budgeting - Theory
  • Power Spectral Densities
  • Cumulative Amplitude Spectrum
  • Conclusion
• Closed Loop Budget Error
  • Introduction
  • Approximation of the effect of feedback on the motion error
  • Reduction thanks to feedback - Required bandwidth
• Conclusion

Introduction   ignore

In this document is studied how the stiffness of the nano-hexapod will impact the effect of disturbances on the position error of the sample.

It is divided in the following sections:

• Section sec:psd_disturbances: the disturbances are listed and their Power Spectral Densities (PSD) are shown
• Section sec:effect_disturbances: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses
• Section sec:granite_stiffness:
• Section sec:open_loop_budget_error: from both the PSD of the disturbances and the transfer function from disturbances to sample's position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)
• Section sec:closed_loop_budget_error: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values

Disturbances

<<sec:psd_disturbances>>

Introduction   ignore

The main disturbances considered here are:

• $D_w$: Ground displacement in the $x$, $y$ and $z$ directions
• $F_{ty}$: Forces applied by the Translation stage in the $x$ and $z$ directions
• $F_{rz}$: Forces applied by the Spindle in the $z$ direction
• $F_d$: Direct forces applied at the center of mass of the Payload

The level of these disturbances has been identified form experiments which are detailed in this document.

Plots   ignore

The measured Amplitude Spectral Densities (ASD) of these forces are shown in Figures fig:opt_stiff_dist_gm and fig:opt_stiff_dist_fty_frz.

In this study, the expected frequency content of the direct forces applied to the payload is not considered.

<<plt-matlab>>

Amplitude Spectral Density of the Ground Displacement (png, pdf)

<<plt-matlab>>

Amplitude Spectral Density of the "parasitic" forces comming from the Translation stage and the spindle (png, pdf)

Effect of disturbances on the position error

<<sec:effect_disturbances>>

Introduction   ignore

In this section, we use the Simscape model to identify the transfer function from disturbances to the position error of the sample. We do that for a wide range of nano-hexapod stiffnesses and we compare the obtained results.

Initialization

We initialize all the stages with the default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

We use a sample mass of 10kg.

  initializeSample('mass', 10);

We include gravity, and we use no controller.

  initializeSimscapeConfiguration('gravity', true);
  initializeController();
  initializeDisturbances('enable', false);
  initializeLoggingConfiguration('log', 'none');

Identification

The considered inputs are:

• Dwx: Ground displacement in the $x$ direction
• Dwy: Ground displacement in the $y$ direction
• Dwz: Ground displacement in the $z$ direction
• Fty_x: Forces applied by the Translation stage in the $x$ direction
• Fty_z: Forces applied by the Translation stage in the $z$ direction
• Frz_z: Forces applied by the Spindle in the $z$ direction
• Fd: Direct forces applied at the center of mass of the Payload

The outputs are Ex, Ey, Ez, Erx, Ery, Erz which are the 3 positions and 3 orientations errors of the sample.

We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.

  Ks = logspace(3,9,7); % [N/m]

Sensitivity to Stages vibration (Filtering)

The sensitivity the stage vibrations are displayed:

• Figure fig:opt_stiff_sensitivity_Frz: sensitivity to vertical spindle vibrations
• Figure fig:opt_stiff_sensitivity_Fty_z: sensitivity to vertical translation stage vibrations
• Figure fig:opt_stiff_sensitivity_Fty_x: sensitivity to horizontal (x) translation stage vibrations

<<plt-matlab>>

Sensitivity to Spindle vertical motion error ($F_{rz}$) to the vertical error position of the sample ($E_z$) (png, pdf)

<<plt-matlab>>

Sensitivity to Translation stage vertical motion error ($F_{ty,z}$) to the vertical error position of the sample ($E_z$) (png, pdf)

<<plt-matlab>>

Sensitivity to Translation stage $x$ motion error ($F_{ty,x}$) to the error position of the sample in the $x$ direction ($E_x$) (png, pdf)

Effect of Ground motion (Transmissibility).

The effect of Ground motion on the position error of the sample is shown in Figure fig:opt_stiff_sensitivity_Dw.

<<plt-matlab>>

Sensitivity to Ground motion ($D_{w}$) to the position error of the sample ($E_y$ and $E_z$) (png, pdf)

Direct Forces (Compliance).

The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure fig:opt_stiff_sensitivity_Fd.

<<plt-matlab>>

Sensitivity to Direct forces and torques applied to the sample ($F_d$, $M_d$) to the position error of the sample (png, pdf)

Conclusion

Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.

In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section sec:open_loop_budget_error.

Effect of granite stiffness

<<sec:granite_stiffness>>

Introduction   ignore

In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.

Analytical Analysis

Simple mass-spring-damper model

Let's consider the system shown in Figure fig:2dof_system_granite_stiffness consisting of two stacked mass-spring-damper systems. The bottom one represents the granite, and the top one all the positioning stages. We want the smallest stage "deformation" $d = x^\prime - x$ due to ground motion $w$.

  \begin{tikzpicture}
    % ====================
    % Parameters
    % ====================
    \def\massw{2.2}  % Width of the masses
    \def\massh{0.8}  % Height of the masses
    \def\spaceh{1.2} % Height of the springs/dampers
    \def\dispw{0.3}  % Width of the dashed line for the displacement
    \def\disph{0.5}  % Height of the arrow for the displacements
    \def\bracs{0.05} % Brace spacing vertically
    \def\brach{-10pt} % Brace shift horizontaly
    % ====================


    % ====================
    % Ground
    % ====================
    \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
    \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
    \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};

    % ====================
    % Micro Station
    % ====================
    \begin{scope}[shift={(0, 0)}]
      % Mass
      \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};

      % Spring, Damper, and Actuator
      \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
      \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};

      % Displacements
      \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
      \draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x$};

      % Legend
      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
    \end{scope}

    % ====================
    % Nano Station
    % ====================
    \begin{scope}[shift={(0, \spaceh+\massh)}]
      % Mass
      \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};

      % Spring, Damper, and Actuator
      \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
      \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c^\prime$};

      % Displacements
      \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
      \draw[->] (0.5*\massw+0.5*\dispw, \spaceh) -- ++(0, \disph) node[right]{$x^\prime$};

      % Legend
      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
    \end{scope}
  \end{tikzpicture}

Mass Spring Damper system consisting of a granite and a positioning stage

If we write the equation of motion of the system in Figure fig:2dof_system_granite_stiffness, we obtain:

\begin{align} m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\ ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x) \end{align}

If we note $d = x^\prime - x$, we obtain:

\begin{equation} \frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)} \end{equation}

General Case

Let's now considering a general positioning stage defined by:

• $G^\prime(s) = \frac{F}{x}$: its mechanical "impedance"
• $D^\prime(s) = \frac{d}{x}$: its "deformation" transfer function

  \begin{tikzpicture}
    \def\massw{2.2}  % Width of the masses
    \def\massh{0.8}  % Height of the masses
    \def\spaceh{1.2} % Height of the springs/dampers
    \def\dispw{0.3}  % Width of the dashed line for the displacement
    \def\disph{0.5}  % Height of the arrow for the displacements
    \def\bracs{0.05} % Brace spacing vertically
    \def\brach{-10pt} % Brace shift horizontaly

    % Mass
    \draw[fill=white] (-0.5*\massw, \spaceh) rectangle node[left=6pt]{$m$} (0.5*\massw, \spaceh+\massh);

    % Spring, Damper, and Actuator
    \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
    \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};

    % Ground
    \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
    % Groud Motion
    \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0);
    \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};

    % Displacements
    \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(dhigh);
    \draw[->] (0.5*\massw+1.5*\dispw, \spaceh+\massh) -- ++(0, \disph) node[right]{$x$};

    % Legend
    \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
      (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
      node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};

    \begin{scope}[shift={(0, \spaceh+\massh)}]
      \node[piezo={2.2}{1.5}{6}, anchor=south] (piezo) at (0, 0){};
      \draw[->] (0,0)node[branch]{} -- ++(0, -0.6)node[above right]{$F$}

      \draw[<->] (1.1+0.5*\dispw,0) -- node[midway, right]{$d$} ++(0,1.5);

      \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
        ($(piezo.south west) + (-10pt, 0)$) -- ($(piezo.north west) + (-10pt, 0)$) %
        node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
    \end{scope}
  \end{tikzpicture}

Mass Spring Damper representing the granite and a general representation of positioning stages

The equation of motion are:

\begin{align} ms^2 x &= (cs + k) (x - w) - F \\ F &= G^\prime(s) x \\ d &= D^\prime(s) x \end{align}

where:

• $F$ is the force applied by the position stages on the granite mass

We can express $d$ as a function of $w$:

\begin{equation} \frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)} \end{equation}

This is the transfer function that we would like to minimize.

Let's verify this formula for a simple mass/spring/damper positioning stage. In that case, we have:

\begin{align*} D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\ G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \end{align*}

And finally:

\begin{equation} \frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)} \end{equation}

which is the same as the previously derived equation.

Soft Granite

Let's initialize a soft granite and see how the sensitivity to disturbances will change.

  initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));

Effect of the Granite transfer function

From Figure fig:opt_stiff_soft_granite_Dw, we can see that having a "soft" granite suspension greatly lowers the sensitivity to ground motion. The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).

From Figures fig:opt_stiff_soft_granite_Frz and fig:opt_stiff_soft_granite_Fd, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.

<<plt-matlab>>

Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (png, pdf)

<<plt-matlab>>

Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (png, pdf)

<<plt-matlab>>

Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (png, pdf)

Conclusion

Having a soft granite suspension greatly decreases the sensitivity the ground motion. Also, it does not affect much the sensitivity to stage vibration and direct forces. Thus the level of sample vibration can be reduced by using a soft granite suspension if it is found that ground motion is the limiting factor.

Open Loop Budget Error

<<sec:open_loop_budget_error>>

Introduction   ignore

Now that the frequency content of disturbances have been estimated (Section sec:psd_disturbances) and the transfer functions from disturbances to the position error of the sample have been identified (Section sec:effect_disturbances), we can compute the level of sample vibration due to the disturbances.

We then can conclude and the nano-hexapod stiffness that will lower the sample position error.

Noise Budgeting - Theory

Let's consider Figure fig:psd_change_tf there $G_d(s)$ is the transfer function from a signal $d$ (the perturbation) to a signal $y$ (the sample's position error).

  \begin{tikzpicture}
    \node[block] (G) at (0, 0) {$G_d(s)$};

    \draw[<-] (G.west) -- ++(-1, 0) node[above right]{$d$};
    \draw[->] (G.east) -- ++( 1, 0) node[above left ]{$y$};
  \end{tikzpicture}

Signal $d$ going through and LTI transfer function $G_d(s)$ to give a signal $y$

We can compute the Power Spectral Density (PSD) of signal $y$ from the PSD of $d$ and the norm of $G_d(s)$:

\begin{equation} S_{y}(\omega) = \left|G_d(j\omega)\right|^2 S_{d}(\omega) \label{eq:psd_transfer_function} \end{equation}

If we now consider multiple disturbances $d_1, \dots, d_n$ as shown in Figure fig:psd_change_tf_multiple_pert, we have that:

\begin{equation} S_{y}(\omega) = \left|G_{d_1}(j\omega)\right|^2 S_{d_1}(\omega) + \dots + \left|G_{d_n}(j\omega)\right|^2 S_{d_n}(\omega) \label{eq:sum_psd} \end{equation}

Sometimes, we prefer to compute the Amplitude Spectral Density (ASD) which is related to the PSD by: \[ \Gamma_y(\omega) = \sqrt{S_y(\omega)} \]

  \begin{tikzpicture}
    \node[block] (Gm) at (0, 0) {$\dots$};
    \draw[<-] (Gm.west) -- ++(-1, 0);

    \node[block, above=0.5 of Gm] (G1) {$G_{d_1}(s)$};
    \draw[<-] (G1.west) -- ++(-1, 0) node[above right]{$d_1$};

    \node[block, below=0.5 of Gm] (Gn) {$G_{d_n}(s)$};
    \draw[<-] (Gn.west) -- ++(-1, 0) node[above right]{$d_n$};

    \node[addb, right= of Gm] (add) {};

    \draw[->] (G1.east) -| (add.north);
    \draw[->] (Gm.east) -- (add.west);
    \draw[->] (Gn.east) -| (add.south);

    \draw[->] (add) -- ++( 1, 0) node[above left]{$y$};
  \end{tikzpicture}

Block diagram showing and output $y$ resulting from the addition of multiple perturbations $d_i$

The Cumulative Power Spectrum (CPS) is here defined as:

\begin{equation} \Phi_y(\omega) = \int_\omega^\infty S_y(\nu) d\nu \end{equation}

And the Cumulative Amplitude Spectrum (CAS):

\begin{equation} \Psi(\omega) = \sqrt{\Phi(\omega)} = \sqrt{\int_\omega^\infty S_y(\nu) d\nu} \end{equation}

The CAS evaluation for all frequency corresponds to the rms value of the considered quantity: \[ y_{\text{rms}} = \Psi(\omega = 0) = \sqrt{\int_0^\infty S_y(\nu) d\nu} \]

Power Spectral Densities

We compute the effect of perturbations on the motion error thanks to Eq. eqref:eq:psd_transfer_function.

The result is shown in:

• Figure fig:opt_stiff_psd_dz_gm: PSD of the vertical sample's motion error due to vertical ground motion
• Figure fig:opt_stiff_psd_dz_rz: PSD of the vertical sample's motion error due to vertical vibrations of the Spindle

<<plt-matlab>>

Amplitude Spectral Density of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (png, pdf)

<<plt-matlab>>

Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (png, pdf)

We compute the effect of all perturbations on the vertical position error using Eq. eqref:eq:sum_psd and the resulting PSD is shown in Figure fig:opt_stiff_psd_dz_tot.

<<plt-matlab>>

Amplitude Spectral Density of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (png, pdf)

Cumulative Amplitude Spectrum

Similarly, the Cumulative Amplitude Spectrum of the sample vibrations are shown:

• Figure fig:opt_stiff_cas_dz_gm: due to vertical ground motion
• Figure fig:opt_stiff_cas_dz_rz: due to vertical vibrations of the Spindle
• Figure fig:opt_stiff_cas_dz_tot: due to all considered perturbations

The black dashed line corresponds to the performance objective of a sample vibration equal to $10\ nm [rms]$.

<<plt-matlab>>

Cumulative Amplitude Spectrum of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (png, pdf)

<<plt-matlab>>

Cumulative Amplitude Spectrum of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (png, pdf)

<<plt-matlab>>

Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (png, pdf)

Conclusion

From Figure fig:opt_stiff_cas_dz_tot, we can see that a soft nano-hexapod $k<10^6\ [N/m]$ significantly reduces the effect of perturbations from 20Hz to 300Hz.

Closed Loop Budget Error

<<sec:closed_loop_budget_error>>

Introduction   ignore

From the total open-loop power spectral density of the sample's motion error, we can estimate what is the required control bandwidth for the sample's motion error to be reduced down to $10nm$.

Approximation of the effect of feedback on the motion error

Let's consider Figure fig:effect_feedback_disturbance_diagram where a controller $K$ is used to reduce the effect of the disturbance $d$ on the position error $y$.

  \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[block, above=0.6 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[->] (adddy.east) -- ++(1, 0) node[above left]{$y$};
    \draw[->] ($(adddy.east)+(0.6, 0)$) node[branch]{} -- ++(0, -1) -| (addfb.south);
    \draw[<-] (Gd.north) -- ++(0, 0.6) node[below right]{$d$};
    \draw[->] (Gd.south) -- (adddy.north);
  \end{tikzpicture}

Feedback System

The reduction of the impact of $d$ on $y$ thanks to feedback is described by the following equation:

\begin{equation} \frac{y}{d} = \frac{G_d}{1 + KG} \end{equation}

The transfer functions corresponding to $G_d$ are those identified in Section sec:effect_disturbances.

As a first approximation, we can consider that the controller $K$ is designed in such a way that the loop gain $KG$ is a pure integrator: \[ L_1(s) = K_1(s) G(s) = \frac{\omega_c}{s} \] where $\omega_c$ is the crossover frequency.

We may then consider another controller in such a way that the loop gain corresponds to a double integrator with a lead centered with the crossover frequency $\omega_c$: \[ L_2(s) = K_2(s) G(s) = \left( \frac{\omega_c}{s} \right)^2 \cdot \frac{1 + \frac{s}{\omega_c/2}}{1 + \frac{s}{2\omega_c}} \]

In the next section, we see how the power spectral density of $y$ is reduced as a function of the control bandwidth $\omega_c$. This will help to determine what is the approximate control bandwidth required such that the rms value of $y$ is below $10nm$.

Reduction thanks to feedback - Required bandwidth

Let's first see how does the Cumulative Amplitude Spectrum of the sample's motion error is modified by the control.

In Figure fig:opt_stiff_cas_closed_loop is shown the Cumulative Amplitude Spectrum of the sample's motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses. The controller used in this simulation is $K_1$. The loop gain is then a pure integrator.

In Figure fig:opt_stiff_req_bandwidth_K1_K2 is shown the expected RMS value of the sample's position error as a function of the control bandwidth, both for $K_1$ (left plot) and $K_2$ (right plot). As expected, it is shown that $K_2$ performs better than $K_1$. This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.

The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample's motion error less than 10nm rms are gathered in Table tab:approx_required_wc_10nm.

<<plt-matlab>>

Cumulative Amplitude Spectrum of the sample's motion error in Open-Loop and in Closed Loop for several control bandwidth and 4 different nano-hexapod stiffnesses (png, pdf)

<<plt-matlab>>

Expected RMS value of the sample's motion error $E_z$ as a function of the control bandwidth when using $K_1$ and $K_2$ (png, pdf)

Nano-Hexapod stiffness [N/m]	10^3	10^4	10^5	10^6	10^7	10^8	10^9
Required wc with L1 [Hz]	152	305	1000	870	933	870	870
Required wc with L2 [Hz]	57	66	152	152	248	266	248

Approximate required control bandwidth such that the motion error is below $10nm$

Conclusion

From Figure fig:opt_stiff_req_bandwidth_K1_K2 and Table tab:approx_required_wc_10nm, we can clearly see three different results depending on the nano-hexapod stiffness:

• For a soft nano-hexapod ($k < 10^4\ [N/m]$), the required bandwidth is $\omega_c \approx 50-100\ Hz$
• For a nano-hexapods with $10^5 < k < 10^6\ [N/m]$, the required bandwidth is $\omega_c \approx 150-300\ Hz$
• For a stiff nano-hexapods ($k > 10^7\ [N/m]$), the required bandwidth is $\omega_c \approx 250-500\ Hz$