Control of the NASS with optimal stiffness

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

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

  initializeController('type', 'hac-dvf');

We set the stiffness of the payload fixation:

  Kp = 1e8; % [N/m]

  K = tf(zeros(6));
  Kdvf = tf(zeros(6));

We identify the system for the following payload masses:

  Ms = [1, 10, 50];

The nano-hexapod has the following leg’s stiffness and damping.

  initializeNanoHexapod('k', 1e5, 'c', 2e2);

The obtain dynamics from actuators forces \(\tau_i\) to the relative motion of the legs \(d\mathcal{L}_i\) is shown in Figure 2 for the three considered payload masses.

The Root Locus is shown in Figure 3 and wee see that we have unconditional stability.

In order to choose the gain such that we obtain good damping for all the three payload masses, we plot the damping ration of the modes as a function of the gain for all three payload masses in Figure 4.

Damping as function of the gain

Finally, we use the following controller for the Decentralized Direct Velocity Feedback:

  Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);

Let’s identify the dynamics from actuator forces \(\bm{\tau}\) to displacement as measured by the metrology \(\bm{\mathcal{X}}\): \[ \bm{G}(s) = \frac{\bm{\mathcal{X}}}{\bm{\tau}} \] We do so both when the DVF is applied and when it is not applied.

Then, we compute the transfer function from forces applied by the actuators \(\bm{\mathcal{F}}\) to the measured position error in the frame of the nano-hexapod \(\bm{\epsilon}_{\mathcal{X}_n}\): \[ \bm{G}_\mathcal{X}(s) = \frac{\bm{\epsilon}_{\mathcal{X}_n}}{\bm{\mathcal{F}}} = \bm{G}(s) \bm{J}^{-T} \] The obtained dynamics is shown in Figure 5.

And we compute the transfer function from actuator forces \(\bm{\tau}\) to position error of each leg \(\bm{\epsilon}_\mathcal{L}\): \[ \bm{G}_\mathcal{L} = \frac{\bm{\epsilon}_\mathcal{L}}{\bm{\tau}} = \bm{J} \bm{G}(s) \] The obtained dynamics is shown in Figure 6.

The coupling (off diagonal elements) of \(\bm{G}_\mathcal{X}\) are shown in Figure 7 both when DVF is applied and when it is not.

The coupling does not change a lot with DVF.

The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure 8.

We may now see how Decentralized Direct Velocity Feedback changes the sensibility to disturbances, namely:

• Ground motion
• Spindle and Translation stage vibrations
• Direct forces applied to the sample

To simplify the analysis, we here only consider the vertical direction, thus, we will look at the transfer functions:

• from vertical ground motion \(D_{w,z}\) to the vertical position error of the sample \(E_z\)
• from vertical vibration forces of the spindle \(F_{R_z,z}\) to \(E_z\)
• from vertical vibration forces of the translation stage \(F_{T_y,z}\) to \(E_z\)
• from vertical direct forces (such as cable forces) \(F_{d,z}\) to \(E_z\)

The norm of these transfer functions are shown in Figure 9.

We now look at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).

The diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses are shown in Figure 11.

The plant dynamics below \(100\ [Hz]\) is only slightly dependent on the payload mass.

We design a diagonal controller with all the same diagonal elements.

The requirements for the controller are:

• Crossover frequency of around 100Hz
• Stable for all the considered payload masses
• Sufficient phase and gain margin
• Integral action at low frequency

The design controller is as follows:

• Lead centered around the crossover
• An integrator below 10Hz
• A low pass filter at 250Hz

The loop gain is shown in Figure 12.

  h = 2.0;
  Kl = 2e7 * eye(6) * ...
       1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
       1/h*(s/(2*pi*200/h) + 1)/(s/(2*pi*200*h) + 1) * ...
       (s/2/pi/10 + 1)/(s/2/pi/10) * ...
       1/(1 + s/2/pi/300);

Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system.

  load('mat/stages.mat', 'nano_hexapod');
  K = Kl*nano_hexapod.kinematics.J*diag([1, 1, 1, 1, 1, 0]);

We identify the transfer function from disturbances to the position error of the sample when the HAC-LAC control is applied.

We compare the norm of these transfer function for the vertical direction when no control is applied and when HAC-LAC control is applied: Figure 13.

Then, we load the Power Spectral Density of the perturbations and we look at the obtained PSD of the displacement error in the vertical direction due to the disturbances:

• Figure 14: Amplitude Spectral Density of the vertical position error due to both the vertical ground motion and the vertical vibrations of the spindle
• Figure 15: Comparison of the Amplitude Spectral Density of the vertical position error in Open Loop and with the HAC-DVF Control
• Figure 16: Comparison of the Cumulative Amplitude Spectrum of the vertical position error in Open Loop and with the HAC-DVF Control

Let’s now simulate a tomography experiment. To do so, we include all disturbances except vibrations of the translation stage.

  initializeDisturbances();
  initializeSimscapeConfiguration('gravity', false);
  initializeLoggingConfiguration('log', 'all');

And we run the simulation for all three payload Masses.

Let’s now see how this controller performs.

First, we compute the Power Spectral Density of the sample’s position error and we compare it with the open loop case in Figure 17.

Similarly, the Cumulative Amplitude Spectrum is shown in Figure 18.

Finally, the time domain position error signals are shown in Figure 19.

Let’s look \(\bm{G}_\mathcal{X}(s)\).