NASS - Uniaxial Model

The measurement setup is schematically shown in Figure fig:micro_station_meas_dynamics_schematic where:

• Two hammer hits are performed, one on the Granite (force \(F_g\)), and one on the micro-hexapod’s top platform (force \(F_h\))
• The inertial motion of the granite \(x_g\) and the micro-hexapod’s top platform \(x_h\) are measured using geophones.

From the forces applied by the instrumented hammer and the responses of the geophones, the following frequency response functions can be computed:

• from \(F_h\) to \(d_h\) (i.e. the compliance of the micro-station)
• from \(F_g\) to \(d_h\) (or from \(F_h\) to \(d_g\))
• from \(F_g\) to \(d_g\)

Due to the bad coherence at low frequency, the frequency response functions are only shown between 20 and 200Hz (Figure fig:uniaxial_measured_frf_vertical).

%% Load measured FRF
load('meas_microstation_frf.mat');

The uni-axial model of the micro-station is shown in Figure fig:uniaxial_comp_frf_meas_model, with:

• Disturbances:
  • \(x_f\): Floor motion
  • \(f_t\): Stage vibrations
• Hammer impacts: \(F_h\) and \(F_g\).
• Geophones: \(x_h\) and \(x_g\)

Masses are estimated from the CAD.

%% Parameters - Mass
mh = 15;   % Micro Hexapod [kg]
mt = 1200; % Ty + Ry + Rz [kg]
mg = 2500; % Granite [kg]

And stiffnesses from the data-sheet of stage manufacturers.

%% Parameters - Stiffnesses
kh = 6.11e+07; % [N/m]
kt = 5.19e+08; % [N/m]
kg = 9.50e+08; % [N/m]

The damping coefficients are tuned to match the identified damping from the measurements.

%% Parameters - damping
ch = 2*0.05*sqrt(kh*mh); % [N/(m/s)]
ct = 2*0.05*sqrt(kt*mt); % [N/(m/s)]
cg = 2*0.08*sqrt(kg*mg); % [N/(m/s)]

The comparison between the measurements and the model is done in Figure fig:uniaxial_comp_frf_meas_model.

As the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation. More accurate models will be used later on.

The parameters for the nano-hexapod and sample are:

• \(m_s\) the sample mass that can vary from 1kg up to 50kg
• \(m_n\) the nano-hexapod mass which is set to 15kg
• \(k_n\) the nano-hexapod stiffness, which can vary depending on the chosen architecture/technology

As a first example, let’s choose a nano-hexapod stiffness of \(10\,N/\mu m\) and a sample mass of 10kg.

%% Nano-Hexapod Parameters
mn = 15; % [kg]
kn = 1e7; % [N/m]
cn = 2*0.01*sqrt(mn*kn); % [N/(m/s)]

%% Sample Mass
ms = 10; % [kg]

The sensitivity to disturbances (i.e. \(x_f\), \(f_t\) and \(f_s\)) are shown in Figure fig:uniaxial_sensitivity_dist_first_params. The plant (i.e. the transfer function from actuator force \(f\) to measured displacement \(d\)) is shown in Figure fig:uniaxial_plant_first_params.

For further analysis, 9 configurations are considered: three nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) combined with three sample’s masses (\(m_s = 1\,kg\), \(m_s = 25\,kg\) and \(m_s = 50\,kg\)).

The geophone fixed to the floor to measure the floor motion.

%% Load floor motion data
% t: time in [s]
% V: measured voltage genrated by the geophone and amplified by a 60dB gain voltage amplifier [V]
load('ground_motion_measurement.mat', 't', 'V');

The voltage generated by each geophone is amplified using a voltage amplifier (gain of 60dB) before going to the ADC. The sensitivity of the geophone as well as the gain of the voltage amplifier are then taken into account to reconstruct the floor displacement.

%% Sensitivity of the geophone
S0 = 88; % Sensitivity [V/(m/s)]
f0 = 2; % Cut-off frequency [Hz]

S = S0*(s/2/pi/f0)/(1+s/2/pi/f0); % Geophone's transfer function [V/(m/s)]

%% Gain of the voltage amplifier
G0_db = 60; % [dB]
G0 = 10^(G0_db/20); % [abs]

%% Transfer function from measured voltage to displacement
G_geo = 1/S/G0/s; % [m/V]

The PSD \(S_{V_f}\) of the measured voltage \(V_f\) is computed.

%% Compute measured voltage PSD
Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
win = hanning(ceil(2*Fs)); % Hanning window

[psd_V, f] = pwelch(V, win, [], [], Fs); % [V^2/Hz]

The PSD of the corresponding displacement can be computed as follows:

with:

• \(S_{\text{geo}}\) the sensitivity of the Geophone in \([Vs/m]\)
• \(G_{\text{amp}}\) the gain of the voltage amplifier
• \(\omega\) is here to integrate and have the displacement instead of the velocity

%% Ground Motion ASD
psd_xf = psd_V.*abs(squeeze(freqresp(G_geo, f, 'Hz'))).^2; % [m^2/Hz]

The amplitude spectral density \(\Gamma_{x_f}\) of the measured displacement \(x_f\) is shown in Figure fig:asd_floor_motion_id31.

During Spindle rotation (here at 6rpm), the granite velocity and micro-hexapod’s top platform velocity are measured with the geophones.

%% Measured velocity of granite and hexapod during spindle rotation
% t: time in [s]
% vg: measured granite velocity [m/s]
% vg: measured micro-hexapod's top platform velocity [m/s]
load('meas_spindle_on.mat', 't', 'vg', 'vh');
spindle_off = load('meas_spindle_off.mat', 't', 'vg', 'vh'); % No Rotation

The Power Spectral Density of the relative velocity between the hexapod and the granite is computed.

%% Compute Power Spectral Density of the relative velocity between granite and hexapod during spindle rotation
Fs = 1/(t(2)-t(1)); % Sampling Frequency [Hz]
win = hanning(ceil(2*Fs)); % Hanning window

[psd_vft, f] = pwelch(vh-vg, win, [], [], Fs); % [(m/s)^2/Hz]
[psd_off, ~] = pwelch(spindle_off.vh-spindle_off.vg, win, [], [], Fs); % [(m/s)^2/Hz]

It is then integrated to obtain the Amplitude Spectral Density of the relative motion which is compared with a non-rotating case (Figure fig:asd_vibration_spindle_rotation). It is shown that the spindle rotation induces vibrations in a wide frequency spectrum.

In order to compute the equivalent disturbance force \(f_t\) that induces such motion, the transfer function from \(f_t\) to the relative motion of the hexapod’s top platform and the granite is extracted from the model. The power spectral density \(\Gamma_{f_{t}}\) of the disturbance force can be computed as follows:

with:

• \(\Gamma_{v_{t}}\) the measured power spectral density of the relative motion between the micro-hexapod’s top platform and the granite during the spindle’s rotation
• \(G_{\text{model}}\) the transfer function (extracted from the uniaxial model) from \(f_t\) to the relative motion between the micro-hexapod’s top platform and the granite

The obtained amplitude spectral density of the disturbance force \(f_t\) is shown in Figure fig:asd_disturbance_force.

The vibrations induced by the \(T_y\) stage are not considered here as:

• the induced vibrations have less amplitude than the vibrations induced by the \(R_z\) stage
• it can be scanned at lower velocities if the induced vibrations are an issue

From the Uni-axial model, the transfer function from the disturbances (\(f_s\), \(x_f\) and \(f_t\)) to the displacement \(d\) are computed.

This is done for two extreme sample masses \(m_s = 1\,\text{kg}\) and \(m_s = 50\,\text{kg}\) and three nano-hexapod stiffnesses:

• \(k_n = 0.01\,N/\mu m\) that could represent a voice coil actuator with soft flexible guiding
• \(k_n = 1\,N/\mu m\) that could represent a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator
• \(k_n = 100\,N/\mu m\) that could represent a stiff piezoelectric stack actuator

The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses are shown in Figure fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the light sample (same conclusions can be drawn with the heavy one).

Now, the power spectral density of the disturbances is taken into account to estimate the residual motion \(d\) in each case.

The Cumulative Amplitude Spectrum of the relative motion \(d\) due to both the floor motion \(x_f\) and the stage vibrations \(f_t\) are shown in Figure fig:uniaxial_cas_d_disturbances_stiffnesses for the three nano-hexapod stiffnesses.

It is shown that the effect of the floor motion is much less than the stage vibrations, except for the soft nano-hexapod below 5Hz.

The total cumulative amplitude spectrum for the three nano-hexapod stiffnesses and for the two sample’s masses are shown in Figure fig:uniaxial_cas_d_disturbances_payload_masses. The conclusion is that the sample’s mass has little effect on the cumulative amplitude spectrum of the relative motion \(d\).

The Integral Force Feedback strategy consists of using a force sensor in series with the actuator (see Figure fig:uniaxial_active_damping_iff_equiv, left).

The control strategy consists of integrating the measured force and feeding it back to the actuator:

The mechanical equivalent of this strategy is to add a dashpot in series with the actuator stiffness with a damping coefficient equal to the stiffness of the actuator divided by the controller gain \(k/g\) (see Figure fig:uniaxial_active_damping_iff_equiv, right).

For the Relative Damping Control strategy, a relative motion sensor that measures the motion of the actuator is used (see Figure fig:uniaxial_active_damping_rdc_equiv, left).

The derivative of this relative motion is used for the feedback signal:

The mechanical equivalent is to add a dashpot in parallel with the actuator with a damping coefficient equal to the controller gain \(g\) (see Figure fig:uniaxial_active_damping_rdc_equiv, right).

Finally, the Direct Velocity Feedback strategy consists of using an inertial sensor (usually a geophone), that measured the “absolute” velocity of the body fixed on top of the actuator (se Figure fig:uniaxial_active_damping_dvf_equiv, left).

The measured velocity is then fed back to the actuator:

This is equivalent as to fix a dashpot (with a damping coefficient equal to the controller gain \(g\)) between the body (one which the inertial sensor is fixed) and an inertial reference frame (see Figure fig:uniaxial_active_damping_dvf_equiv, right). This is usually refers to as “sky hook damper”.

The plant dynamics for all three active damping techniques are shown in Figure fig:uniaxial_plant_active_damping_techniques. All have alternating poles and zeros meaning that the phase is bounded to \(\pm 90\,\text{deg}\) which makes the controller very robust.

When the nano-hexapod’s suspension modes are at lower frequencies than the resonances of the micro-station (blue and red curves in Figure fig:uniaxial_plant_active_damping_techniques), the resonances of the micro-stations have little impact on the transfer functions from IFF and DVF.

For the stiff nano-hexapod, the micro-station dynamics can be seen on the transfer functions in Figure fig:uniaxial_plant_active_damping_techniques. Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff nano-hexapod is used.

The Root Locus are computed for the three nano-hexapod stiffnesses and for the three active damping techniques. They are shown in Figure fig:uniaxial_root_locus_damping_techniques.

All three active damping approach can lead to critical damping of the nano-hexapod suspension mode.

There is even a little bit of authority on micro-station modes with IFF and DVF applied on the stiff nano-hexapod (Figure fig:uniaxial_root_locus_damping_techniques, right) and with RDC for a soft nano-hexapod (Figure fig:uniaxial_root_locus_damping_techniques_micro_station_mode). This can be explained by the fact that above the suspension mode of the soft nano-hexapod, the relative motion sensor acts as an inertial sensor for the micro-station top platform. Therefore, it is like DVF was applied (the nano-hexapod acts as a geophone!).

The sensitivity to disturbances (direct forces \(f_s\), stage vibrations \(f_t\) and floor motion \(x_f\)) for all three active damping techniques are compared in Figure fig:uniaxial_sensitivity_dist_active_damping. The comparison is done with the nano-hexapod having a stiffness \(k_n = 1\,N/\mu m\).

Cumulative Amplitude Spectrum of the distance \(d\) with all three active damping techniques are compared in Figure fig:uniaxial_cas_active_damping. All three active damping methods are giving similar results (except the RDC which is a little bit worse for the stiff nano-hexapod).

Compare to the open-loop case, the active damping helps to lower the vibrations induced by the nano-hexapod resonance.

The transfer functions from the plant input \(f\) to the relative displacement \(d\) while the active damping is implemented are shown in Figure fig:uniaxial_damped_plant_three_active_damping_techniques. All three active damping techniques yield similar damped plants.

The damped plants are shown in Figure fig:uniaxial_damped_plant_change_sample_mass for all three techniques, with the three considered nano-hexapod stiffnesses and sample’s masses.

The Root Locus for the three damping techniques are shown in Figure fig:uniaxial_active_damping_robustness_mass_root_locus for three sample’s mass (1kg, 25kg and 50kg). The closed-loop poles are shown by the squares for a specific gain.

We can see that having heavier samples yields larger damping for IFF and smaller damping for RDC and DVF.

As was shown in Section sec:uniaxial_active_damping, all three proposed active damping techniques yield similar damping plants. Therefore, Integral Force Feedback will be used in this section to study the HAC-LAC performances.

The obtained damped plants for the three nano-hexapod stiffnesses are shown in Figure fig:uniaxial_hac_iff_damped_plants_masses.

The objective now is to design a position feedback controller for each of the three nano-hexapods that are robust to the change of sample’s mass.

The required feedback bandwidth was approximately determined un Section sec:uniaxial_noise_budgeting:

• \(\approx 10\,\text{Hz}\) for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)). Near this frequency, the plants are equivalent to a mass line. The gain of the mass line can vary up to a fact \(\approx 5\) (suspended mass from \(16\,kg\) up to \(65\,kg\)). This mean that the designed controller will need to have large gain margins to be robust to the change of sample’s mass.
• \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)). Similarly to the soft nano-hexapod, the plants near the crossover frequency are equivalent to a mass line. It will be probably easier to have a little bit more bandwidth in this configuration to be further away from the nano-hexapod suspension mode.
• \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)). Contrary to the two first nano-hexapod stiffnesses, here the plants have more complex dynamics near the wanted crossover frequency. The micro-station is not stiff enough to have a clear stiffness line at this frequency. Therefore, there are both a change of phase and gain depending on the sample’s mass. This makes the robust design of the controller a little bit more complicated.

Position feedback controllers are designed for each nano-hexapod such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure fig:uniaxial_nyquist_hac). These high authority controllers are generally composed of a two integrators at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency and a low pass filter to increase the robustness to high frequency dynamics. The loop gains for the three nano-hexapod are shown in Figure fig:uniaxial_loop_gain_hac. We can see that:

• for the soft and moderately stiff nano-hexapod, the crossover frequency varies a lot with the sample mass. This is due to the fact that the crossover frequency corresponds to the mass line of the plant.
• for the stiff nano-hexapod, the obtained crossover frequency is not at high as what was estimated necessary. The crossover frequency in that case is close to the stiffness line of the plant, which makes the robust design of the controller easier.

Note that these controller were quickly tuned by hand and not designed using any optimization methods. The goal is just to have a first estimation of the attainable performances.

The high authority position feedback controllers are then implemented and the closed-loop sensitivity to disturbances are computed. These are compared with the open-loop and damped plants cases in Figure fig:uniaxial_sensitivity_dist_hac_lac for just one configuration (moderately stiff nano-hexapod with 25kg sample’s mass). As expected, the sensitivity to disturbances is decreased in the controller bandwidth and slightly increase outside this bandwidth.

The cumulative amplitude spectrum of the motion \(d\) is computed for all nano-hexapod configurations, all sample masses and in the open-loop (OL), damped (IFF) and position controlled (HAC-IFF) cases. The results are shown in Figure fig:uniaxial_cas_hac_lac. Obtained root mean square values of the distance \(d\) are better for the soft nano-hexapod (\(\approx 25\,nm\) to \(\approx 35\,nm\) depending on the sample’s mass) than for the stiffer nano-hexapod (from \(\approx 30\,nm\) to \(\approx 70\,nm\)).