Control Requirements

Let’s consider the simple mechanical system in Figure 1.

The signals are described in table 1.

For the nano-hexapod alone, we have the following equations: \[ \begin{align*} x &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \\ F_m &= \frac{ms^2}{ms^2 + k} F - \frac{k}{ms^2 + k} F_d + \frac{k m s^2}{ms^2 + k} x_\mu \\ d &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d - \frac{ms^2}{ms^2 + k} x_\mu \end{align*} \]

We can write the equations function of \(\omega_\nu = \sqrt{\frac{k}{m}}\): \[ \begin{align*} x &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\ F_m &= \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} F - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{k \frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\ d &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d - \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \end{align*} \]

Assumptions:

• the forces applied by the nano-hexapod have no influence on the micro-station, specifically on the displacement of the top platform of the micro-hexapod.

This means that the nano-hexapod can be considered separately from the micro-station and that the motion \(x_\mu\) is imposed and considered as an external input.

The nano-hexapod can thus be represented as in Figure 2.

What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite.

Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station.

As explained, we consider \(x_\mu\) as an external input (\(F\) has no influence on \(x_\mu\)).

\(x_\mu\) is the motion of the micro-station’s top platform due to the motion of each stage of the micro-station.

We consider that \(x_\mu\) has the following form: \[ x_\mu = T_\mu r + d_\mu \] where:

• \(T_\mu r\) corresponds to the response of the stages due to the reference \(r\)
• \(d_\mu\) is the motion of the hexapod due to all the vibrations of the stages

\(T_\mu\) can be considered to be a low pass filter with a bandwidth corresponding approximatively to the bandwidth of the micro-station’s stages. To simplify, we can consider \(T_\mu\) to be a first order low pass filter: \[ T_\mu = \frac{1}{1 + s/\omega_\mu} \] where \(\omega_\mu\) corresponds to the tracking speed of the micro-station.

What is important to note is that while \(x_\mu\) is viewed as a perturbation from the nano-hexapod point of view, \(x_\mu\) does depend on the reference signal \(r\).

Also, here, we suppose that the granite is not moving.

If we now include the motion of the granite \(w\), we obtain the block diagram shown in Figure 3.

\(T_w\) is the mechanical transmissibility of the micro-station. We can approximate this transfer function by a second order low pass filter: \[ T_w = \frac{1}{1 + 2 \xi s/\omega_0 + s^2/\omega_0^2} \]

Let’s define the mass and stiffness of the nano-hexapod.

  m = 50; % [kg]
  k = 1e7; % [N/m]

Let’s define the Plant as shown in Figure 2:

  Gn = 1/(m*s^2 + k)*[-k, k*m*s^2, m*s^2; 1, -m*s^2, 1; 1, k, 1];
  Gn.InputName  = {'Fd', 'xmu', 'F'};
  Gn.OutputName = {'Fm', 'd', 'x'};

Now, define the transmissibility transfer function \(T_\mu\) corresponding to the micro-station motion.

  wmu = 2*pi*50; % [rad/s]

  Tmu = 1/(1 + s/wmu);
  Tmu.InputName = {'r1'};
  Tmu.OutputName = {'ymu'};

  w0 = 2*pi*40;
  xi = 0.5;
  Tw = 1/(1 + 2*xi*s/w0 + s^2/w0^2);
  Tw.InputName = {'w1'};
  Tw.OutputName = {'dw'};

We add the fact that \(x_\mu = d_\mu + T_\mu r + T_w w\):

  Wsplit = [tf(1); tf(1)];
  Wsplit.InputName = {'w'};
  Wsplit.OutputName = {'w1', 'w2'};

  S = sumblk('xmu = ymu + dmu + dw');

  Sw = sumblk('y = x - w2');

  Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {'Fd', 'dmu', 'r1', 'F', 'w'}, {'Fm', 'd', 'y'});

Wanted ASD of motion error

  y_wanted = 100e-9; % 10nm rms wanted
  y_bw = 2*pi*100; % bandwidth [rad/s]

  G_y = 2*y_wanted/sqrt(y_bw) * (1 + s/y_bw/10) / (1 + s/y_bw);

  sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))

sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
ans =
      9.47118350214793e-08

Let’s create a generalized weighted plant for controller synthesis.

Let’s start simple:

Add \(F\) as output.

  F = [tf(1); tf(1)];
  F.InputName = {'Fi'};
  F.OutputName = {'F', 'Fu'};

  P_pz = connect(F, Gpz_dvf, {'dmu', 'Fi'}, {'y', 'Fu', 'y'})
  P_vc = connect(F, Gvc_dvf, {'dmu', 'Fi'}, {'y', 'Fu', 'y'})

Normalization.

We multiply the plant input by \(G_{rz}\) and the plant output by \(G_y^{-1}\):

  P_pz_norm = blkdiag(inv(G_y), inv(G_F), 1)*P_pz*blkdiag(G_rz, 1);
  P_pz_norm.OutputName = {'z', 'F', 'y'};
  P_pz_norm.InputName  = {'w', 'u'};

  P_vc_norm = blkdiag(inv(G_y), inv(G_F), 1)*P_vc*blkdiag(G_rz, 1);
  P_vc_norm.OutputName = {'z', 'F', 'y'};
  P_vc_norm.InputName  = {'w', 'u'};