Unidirectional Model - Report

The measurement setup is schematically shown in Figure 2 where:

• Two hammer hits are performed, one on the Granite, and one on the micro-hexapod’s top platform
• The inertial motion of the granite and the micro-hexapod’s top platform 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 3).

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

The uni-axial model of the micro-station is shown in Figure 5, 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 comparison between the measurements and the model is done in Figure 5.

Ad the model is simplistic, the goal is not to match exactly the measurement but to have a first approximation. More accurate models will be developed and tuned 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 7. The plant (i.e. the transfer function from actuator force \(f\) to measured displacement \(d\)) is shown in Figure 8.

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

%% Load floor motion data
load('ground_motion_measurement.mat', 't', 'V');

The sensitivity of the geophone as well as the gain of the voltage amplifier are 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 10.

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
load('meas_spindle_on.mat', 't', 'vg', 'vh');

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]

It is then integrated to obtain the Amplitude Spectral Density of the relative motion (Figure 11).

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

%% Power Spectral Density of the equivalent force ft
psd_ft = (psd_vft./(2*pi*f).^2)./abs(squeeze(freqresp(G('Dh', 'ft') - G('Dg', 'ft'), f, 'Hz'))).^2;

The obtained amplitude spectral density of the disturbance force \(f_t\) is shown in Figure 12.

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 13 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 14 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 15. The conclusion is that the sample’s mass has little effect on the cumulative amplitude spectrum of the relative motion \(d\).

The plant dynamics for all three active damping techniques are shown in Figure 17.

All have alternating poles and zeros.

Controller used for the three active damping techniques are:

It is shown in the Root Locus in Figure 18 that all three active damping approach can lead to critical damping.

The sensitivity to disturbances (\(f_s\), \(f_t\) and \(x_f\)) for all three active damping techniques are compared in Figure 19.

Cumulative Amplitude Spectrum of the distance \(d\) with all three active damping techniques are compared in Figure 20. All three active damping methods are giving similar results.

Active Damping helps to lower the vibrations and less bandwidth is required for the position controller.