nass-simscape/org/optimal_stiffness_control.org

Control of the NASS with optimal stiffness

• Introduction
• Low Authority Control - Decentralized Direct Velocity Feedback
  • Introduction
  • Initialization
  • Identification
  • Controller Design
  • Effect of the Low Authority Control on the Primary Plant
    • Introduction
    • Identification of the undamped plant
    • Identification of the damped plant
    • Effect of the Damping on the plant diagonal dynamics
    • Effect of the Damping on the coupling dynamics
  • Effect of the Low Authority Control on the Sensibility to Disturbances
    • Introduction
    • Identification
    • Results
• Primary Control in the leg space
  • Introduction
  • Plant in the task space
  • Control in the leg space
  • Sensibility to Disturbances and Noise Budget
    • Identification
    • Obtained Sensibility to Disturbances
    • Noise Budgeting
  • Simulations
  • Results
• Primary Control in the task space
  • Introduction
  • Plant in the task space
  • Control in the task space
    • Stability
  • Simulation

Introduction   ignore

Low Authority Control - Decentralized Direct Velocity Feedback

<<sec:lac_dvf>>

Introduction   ignore

Low Authority Control: Decentralized Direct Velocity Feedback

Initialization

  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]

Identification

  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);

Controller Design

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

The Root Locus is shown in Figure fig:opt_stiff_dvf_root_locus 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 fig:opt_stiff_dvf_damping_gain.

Dynamics for the Direct Velocity Feedback active damping for three payload masses

Root Locus for the DVF controll for three payload masses

Damping as function of the gain

Damping ratio of the poles as a function of the DVF gain

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

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

Effect of the Low Authority Control on the Primary Plant

Introduction   ignore

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 fig:opt_stiff_primary_plant_damped_X.

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 fig:opt_stiff_primary_plant_damped_L.

Identification of the undamped plant   ignore

Identification of the damped plant   ignore

Effect of the Damping on the plant diagonal dynamics   ignore

Primary plant in the task space with (dashed) and without (solid) Direct Velocity Feedback

Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback

Effect of the Damping on the coupling dynamics   ignore

The coupling (off diagonal elements) of $\bm{G}_\mathcal{X}$ are shown in Figure fig:opt_stiff_primary_plant_damped_coupling_X 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 fig:opt_stiff_primary_plant_damped_coupling_L. The magnitude of the coupling around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.

Coupling in the primary plant in the task with (dashed) and without (solid) Direct Velocity Feedback

Coupling in the primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback

Effect of the Low Authority Control on the Sensibility to Disturbances

Introduction   ignore

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 fig:opt_stiff_sensibility_dist_dvf.

Identification   ignore

Results   ignore

Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied

Primary Control in the leg space

<<sec:primary_control_L>>

Introduction   ignore

In this section we implement the control architecture shown in Figure fig:control_architecture_hac_dvf_pos_L consisting of:

• an inner loop with a decentralized direct velocity feedback control
• an outer loop where the controller $\bm{K}_\mathcal{L}$ is designed in the frame of the legs

Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg's space

The controller for decentralized direct velocity feedback is the one designed in Section sec:lac_dvf.

Plant in the task space

We now loop 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 fig:opt_stiff_primary_plant_L.

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

Diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the three considered masses

Control in the leg space

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 fig:opt_stiff_primary_loop_gain_L.

  h = 2.5;
  Kl = 2e7 * eye(6) * ...
       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);

Loop gain for the primary plant

  load('mat/stages.mat', 'nano_hexapod');
  K = Kl*nano_hexapod.J;

Sensibility to Disturbances and Noise Budget

Identification   ignore

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

Obtained Sensibility to Disturbances   ignore

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 fig:opt_stiff_primary_control_L_senbility_dist.

Sensibility to disturbances when the HAC-LAC control is applied

Noise Budgeting   ignore

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

Amplitude Spectral Density of the vertical position error of the sample when the HAC-DVF control is applied due to both the ground motion and spindle vibrations

Amplitude Spectral Density of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied

Cumulative Amplitude Spectrum of the vertical position error of the sample in Open-Loop and when the HAC-DVF control is applied

Simulations

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

  load('mat/conf_simulink.mat');
  set_param(conf_simulink, 'StopTime', '2');

  hac_dvf_L = {zeros(length(Ms)), 1};

  for i = 1:length(Ms)
      initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1));
      initializeReferences('Rz_type', 'rotating', 'Rz_period', Ms(i));

      sim('nass_model');
      hac_dvf_L(i) = {simout};
  end

  save('./mat/tomo_exp_hac_dvf.mat', 'hac_dvf_L');

Results

  load('./mat/experiment_tomography.mat', 'tomo_align_dist');

  n_av = 4;
  han_win = hanning(ceil(length(simout.Em.En.Data(:,1))/n_av));

  t = simout.Em.En.Time;
  Ts = t(2)-t(1);

  [pxx_ol, f] = pwelch(tomo_align_dist.Em.En.Data, han_win, [], [], 1/Ts);

  pxx_dvf_L = zeros(length(f), 6, length(Ms));
  for i = 1:length(Ms)
      [pxx, ~] = pwelch(hac_dvf_L{i}.Em.En.Data(ceil(0.2/Ts):end,:), han_win, [], [], 1/Ts);
      pxx_dvf_L(:, :, i) = pxx;
  end

Primary Control in the task space

<<sec:primary_control_X>>

Introduction   ignore

Plant in the task space

Let's look $\bm{G}_\mathcal{X}(s)$.

Control in the task space

  Kx = tf(zeros(6));

  h = 2.5;
  Kx(1,1) = 3e7 * ...
            1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
            (s/2/pi/1 + 1)/(s/2/pi/1);

  Kx(2,2) = Kx(1,1);

  h = 2.5;
  Kx(3,3) = 3e7 * ...
            1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
            (s/2/pi/1 + 1)/(s/2/pi/1);

  h = 1.5;
  Kx(4,4) = 5e5 * ...
            1/h*(s/(2*pi*100/h) + 1)/(s/(2*pi*100*h) + 1) * ...
            (s/2/pi/1 + 1)/(s/2/pi/1);

  Kx(5,5) = Kx(4,4);

  h = 1.5;
  Kx(6,6) = 5e4 * ...
            1/h*(s/(2*pi*30/h) + 1)/(s/(2*pi*30*h) + 1) * ...
            (s/2/pi/1 + 1)/(s/2/pi/1);

Stability

  for i = 1:length(Ms)
      isstable(feedback(Gm_x{i}*Kx, eye(6), -1))
  end

Simulation