Design of the Nano-Hexapod and associated Control Architectures - Summary

Let’s consider the block diagram shown in Figure 1 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)

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.

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 1), 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.

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 2. 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

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 3).

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.

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}

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.

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 4).

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}

Let’s consider a signal \(y\) that is the sum of two uncorrelated signals \(u\) and \(v\) (Figure 5).

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): \[ S_{yy} = S_{uu} + S_{vv} \]

Let’s consider the Feedback architecture in Figure 1 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 3
  • \(S_{nn}\): sensor noise, this can be estimated from the sensor data-sheet
  • \(S_{rr}\):
• 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 2), include this dynamics in a model (Section 4) and add a model of the nano-hexapod to the model (Section 5)
• The controller \(K\) that will be designed in Section 6