<li>Section <ahref="#org17d3d6a">1</a>: the disturbances are listed and their Power Spectral Densities (PSD) are shown</li>
<li>Section <ahref="#orgf9e4300">2</a>: the transfer functions from disturbances to the position error of the sample are computed for a wide range of nano-hexapod stiffnesses</li>
<li>Section <ahref="#orgd4105b6">3</a>:</li>
<li>Section <ahref="#org5d05990">4</a>: from both the PSD of the disturbances and the transfer function from disturbances to sample’s position errors, we compute the resulting PSD and Cumulative Amplitude Spectrum (CAS)</li>
<li>Section <ahref="#orgd3503fb">5</a>: from a simplistic model is computed the required control bandwidth to reduce the position error to acceptable values</li>
<p><spanclass="figure-number">Figure 1: </span>Amplitude Spectral Density of the Ground Displacement (<ahref="./figs/opt_stiff_dist_gm.png">png</a>, <ahref="./figs/opt_stiff_dist_gm.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 2: </span>Amplitude Spectral Density of the “parasitic” forces comming from the Translation stage and the spindle (<ahref="./figs/opt_stiff_dist_fty_frz.png">png</a>, <ahref="./figs/opt_stiff_dist_fty_frz.pdf">pdf</a>)</p>
The outputs are <code>Ex</code>, <code>Ey</code>, <code>Ez</code>, <code>Erx</code>, <code>Ery</code>, <code>Erz</code> which are the 3 positions and 3 orientations errors of the sample.
</p>
<p>
We initialize the set of the nano-hexapod stiffnesses, and for each of them, we identify the dynamics from defined inputs to defined outputs.
<p><spanclass="figure-number">Figure 3: </span>Sensitivity to Spindle vertical motion error (\(F_{rz}\)) to the vertical error position of the sample (\(E_z\)) (<ahref="./figs/opt_stiff_sensitivity_Frz.png">png</a>, <ahref="./figs/opt_stiff_sensitivity_Frz.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 4: </span>Sensitivity to Translation stage vertical motion error (\(F_{ty,z}\)) to the vertical error position of the sample (\(E_z\)) (<ahref="./figs/opt_stiff_sensitivity_Fty_z.png">png</a>, <ahref="./figs/opt_stiff_sensitivity_Fty_z.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 5: </span>Sensitivity to Translation stage \(x\) motion error (\(F_{ty,x}\)) to the error position of the sample in the \(x\) direction (\(E_x\)) (<ahref="./figs/opt_stiff_sensitivity_Fty_x.png">png</a>, <ahref="./figs/opt_stiff_sensitivity_Fty_x.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 6: </span>Sensitivity to Ground motion (\(D_{w}\)) to the position error of the sample (\(E_y\) and \(E_z\)) (<ahref="./figs/opt_stiff_sensitivity_Dw.png">png</a>, <ahref="./figs/opt_stiff_sensitivity_Dw.pdf">pdf</a>)</p>
<h3id="org0448746"><spanclass="section-number-3">2.5</span> Direct Forces (Compliance).</h3>
<divclass="outline-text-3"id="text-2-5">
<p>
The effect of direct forces/torques applied on the sample (cable forces for instance) on the position error of the sample is shown in Figure <ahref="#orga33395f">7</a>.
<p><spanclass="figure-number">Figure 7: </span>Sensitivity to Direct forces and torques applied to the sample (\(F_d\), \(M_d\)) to the position error of the sample (<ahref="./figs/opt_stiff_sensitivity_Fd.png">png</a>, <ahref="./figs/opt_stiff_sensitivity_Fd.pdf">pdf</a>)</p>
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section <ahref="#org5d05990">4</a>.
<h3id="org8878556"><spanclass="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
<divclass="outline-text-3"id="text-3-3">
<p>
From Figure <ahref="#org38146da">10</a>, we can see that having a “soft” granite suspension greatly lowers the sensitivity to ground motion.
The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
</p>
<p>
From Figures <ahref="#orgc4c14fb">11</a> and <ahref="#org533cc4b">12</a>, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
<p><spanclass="figure-number">Figure 10: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<ahref="./figs/opt_stiff_soft_granite_Dw.png">png</a>, <ahref="./figs/opt_stiff_soft_granite_Dw.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 11: </span>Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<ahref="./figs/opt_stiff_soft_granite_Frz.png">png</a>, <ahref="./figs/opt_stiff_soft_granite_Frz.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 12: </span>Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<ahref="./figs/opt_stiff_soft_granite_Fd.png">png</a>, <ahref="./figs/opt_stiff_soft_granite_Fd.pdf">pdf</a>)</p>
Now that the frequency content of disturbances have been estimated (Section <ahref="#org17d3d6a">1</a>) and the transfer functions from disturbances to the position error of the sample have been identified (Section <ahref="#orgf9e4300">2</a>), we can compute the level of sample vibration due to the disturbances.
</p>
<p>
We then can conclude and the nano-hexapod stiffness that will lower the sample position error.
Let’s consider Figure <ahref="#org7ff50a0">13</a> there \(G_d(s)\) is the transfer function from a signal \(d\) (the perturbation) to a signal \(y\) (the sample’s position error).
<p><spanclass="figure-number">Figure 14: </span>Block diagram showing and output \(y\) resulting from the addition of multiple perturbations \(d_i\)</p>
<p><spanclass="figure-number">Figure 15: </span>Amplitude Spectral Density of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_psd_dz_gm.png">png</a>, <ahref="./figs/opt_stiff_psd_dz_gm.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 16: </span>Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_psd_dz_rz.png">png</a>, <ahref="./figs/opt_stiff_psd_dz_rz.pdf">pdf</a>)</p>
We compute the effect of all perturbations on the vertical position error using Eq. \eqref{eq:sum_psd} and the resulting PSD is shown in Figure <ahref="#orgdbfb5e0">17</a>.
<p><spanclass="figure-number">Figure 17: </span>Amplitude Spectral Density of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_psd_dz_tot.png">png</a>, <ahref="./figs/opt_stiff_psd_dz_tot.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 18: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to Ground motion for multiple nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_cas_dz_gm.png">png</a>, <ahref="./figs/opt_stiff_cas_dz_gm.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 19: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_cas_dz_rz.png">png</a>, <ahref="./figs/opt_stiff_cas_dz_rz.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 20: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_cas_dz_tot.png">png</a>, <ahref="./figs/opt_stiff_cas_dz_tot.pdf">pdf</a>)</p>
From Figure <ahref="#orgf6888f0">20</a>, we can see that a soft nano-hexapod \(k<10^6\[N/m]\)significantlyreducestheeffectofperturbationsfrom20Hzto300Hz.
From the total open-loop power spectral density of the sample’s motion error, we can estimate what is the required control bandwidth for the sample’s motion error to be reduced down to \(10nm\).
Let’s consider Figure <ahref="#org6308d80">21</a> where a controller \(K\) is used to reduce the effect of the disturbance \(d\) on the position error \(y\).
We may then consider another controller in such a way that the loop gain corresponds to a double integrator with a lead centered with the crossover frequency \(\omega_c\):
In Figure <ahref="#orgcbef465">22</a> is shown the Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses.
The controller used in this simulation is \(K_1\). The loop gain is then a pure integrator.
In Figure <ahref="#orgd677910">23</a> is shown the expected RMS value of the sample’s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot).
As expected, it is shown that \(K_2\) performs better than \(K_1\).
This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.
The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample’s motion error less than 10nm rms are gathered in Table <ahref="#org5ab4860">1</a>.
<p><spanclass="figure-number">Figure 22: </span>Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth and 4 different nano-hexapod stiffnesses (<ahref="./figs/opt_stiff_cas_closed_loop.png">png</a>, <ahref="./figs/opt_stiff_cas_closed_loop.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 23: </span>Expected RMS value of the sample’s motion error \(E_z\) as a function of the control bandwidth when using \(K_1\) and \(K_2\) (<ahref="./figs/opt_stiff_req_bandwidth_K1_K2.png">png</a>, <ahref="./figs/opt_stiff_req_bandwidth_K1_K2.pdf">pdf</a>)</p>
<captionclass="t-above"><spanclass="table-number">Table 1:</span> Approximate required control bandwidth such that the motion error is below \(10nm\)</caption>
From Figure <ahref="#orgd677910">23</a> and Table <ahref="#org5ab4860">1</a>, we can clearly see three different results depending on the nano-hexapod stiffness:
</p>
<ulclass="org-ul">
<li>For a soft nano-hexapod (\(k <10^4\[N/m]\)),therequiredbandwidthis\(\omega_c\approx50-100\Hz\)</li>
