The overall objective is to design a nano-hexapod an the associated control architecture that allows the stabilization of samples down to \(\approx 10nm\) in presence of disturbances and system variability.
<li>the frequency content of the important source of disturbances in play such as vibration of stages and ground motion (Section <ahref="#org71cae32">3</a>)</li>
In Section <ahref="#orge13978b">1.2</a> is introduced the <b>dynamic error budgeting</b> which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources.
The use of feedback control as several advantages and pitfalls that are listed below (taken from <aclass='org-ref-reference'href="#schmidt14_desig_high_perfor_mechat_revis_edition">schmidt14_desig_high_perfor_mechat_revis_edition</a>):
<li><b>Reduction of the effect of disturbances</b>:
Disturbances affecting the sample vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them</li>
<li><b>Handling of uncertainties</b>:
Feedback controlled systems can also be designed for <i>robustness</i>, which means that the stability and performance requirements are guaranteed even for parameter variation of the controller mechatronics system</li>
</ul></li>
<li><b>Pitfalls</b>:
<ulclass="org-ul">
<li><b>Limited reaction speed</b>:
A feedback controller reacts on the difference between the reference signal (wanted motion) and the measurement (actual motion), which means that the error has to occur first <i>before</i> the controller can correct for it.
The limited reaction speed means that the controller will be able to compensate the positioning errors only in some frequency band, called the controller <i>bandwidth</i></li>
<li><b>Feedback of noise</b>:
By closing the loop, the sensor noise is also fed back and will induce positioning errors</li>
<li><b>Can introduce instability</b>:
Feedback control can destabilize a stable plant.
Thus the <i>robustness</i> properties of the feedback system must be carefully guaranteed</li>
If we write down the position error signal \(\epsilon = r - y\) as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\) (using the feedback diagram in Figure <ahref="#org2d835ae">1</a>), we obtain:
As shown in the previous section, the effect of disturbances is reduced <i>inside</i> the control bandwidth.
</p>
<p>
Moreover, the slope of \(|S(j\omega)|\) is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have large effects).
</p>
<p>
The next important question is <b>what effects do limit the attainable control bandwidth?</b>
</p>
<p>
The main issue it that for stability reasons, <b>the behavior of the mechanical system must be known with only small uncertainty in the vicinity of the crossover frequency</b>.
For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure <ahref="#org51dcadc">3</a>).
<p><spanclass="figure-number">Figure 3: </span>Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth. <aclass='org-ref-reference'href="#oomen18_advan_motion_contr_precis_mechat">oomen18_advan_motion_contr_precis_mechat</a></p>
This also means that <b>any possible change in the system should have a small impact on the system dynamics in the vicinity of the crossover</b>.
</p>
<p>
For the NASS, the possible changes in the system are:
</p>
<ulclass="org-ul">
<li>a modification of the payload mass and dynamics</li>
<li>a change of experimental condition: spindle’s rotation speed, position of each micro-station’s stage</li>
<li>a change in the micro-station dynamics (change of mechanical elements, aging effect, …)</li>
</ul>
<p>
The nano-hexapod and the control architecture have to be developed such that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system.
</p>
<p>
This problem of <b>robustness</b> represent one of the main challenge for the design of the NASS.
The <b>Power Spectral Density</b> (PSD) \(S_{xx}(f)\) of the time domain signal \(x(t)\) is defined as the Fourier transform of the autocorrelation function:
Thus, the total power in the signal can be obtained by integrating these infinitesimal contributions, the Root Mean Square (RMS) value of the signal \(x(t)\) is then:
One can also integrate the infinitesimal power \(S_{xx}(\omega)d\omega\) over a finite frequency band to obtain the power of the signal \(x\) in that frequency band:
The <b>Cumulative Power Spectrum</b> is the cumulative integral of the Power Spectral Density starting from \(0\ \text{Hz}\) with increasing frequency:
The Cumulative Power Spectrum will be used to determine in which frequency band the effect of disturbances should be reduced, and thus the approximate required control bandwidth.
Let’s consider a signal \(u\) with a PSD \(S_{uu}\) going through a LTI system \(G(s)\) that outputs a signal \(y\) with a PSD (Figure <ahref="#org50b219a">4</a>).
To do so, we need to identify the dynamics of the micro-station (Section <ahref="#orge98daae">2</a>), include this dynamics in a model (Section <ahref="#orgb3d8e70">4</a>) and add a model of the nano-hexapod to the model (Section <ahref="#org24cc2ee">5</a>)</li>
<li>The controller \(K\) that will be designed in Section <ahref="#org1a74334">6</a></li>
As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be coupled with the dynamics of the nano-hexapod and thus is very important for both the design of the nano-hexapod and controller.
All the measurements performed on the micro-station are detailed in <ahref="https://tdehaeze.github.io/meas-analysis/">this</a> document and summarized in the following sections.
To limit the number of degrees of freedom to be measured, we suppose that in the frequency range of interest (DC-300Hz), each of the positioning stage is behaving as a <b>solid body</b>.
Thus, to fully describe the dynamics of the station, we (only) need to measure 6 degrees of freedom on each of the positioning stage (that is 36 degrees of freedom for the 6 solid bodies).
</p>
<p>
In order to perform the <b>Modal Analysis</b>, the following devices were used:
From the measurements, we obtain all the transfer functions from forces applied at the location of the hammer impacts to the x-y-z acceleration of each solid body at the location of each accelerometer.
Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures <ahref="#orge621ac1">9</a><ahref="#orgd66a167">10</a>.
From the reduced transfer function matrix, we can re-synthesize the response at the 69 measured degrees of freedom and we find that we have an exact match.
<p><spanclass="figure-number">Figure 11: </span>Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction</p>
In Section <ahref="#orgb3d8e70">4</a>, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
Note that here we are not much interested by low frequency disturbances such as thermal effects and static guiding errors of each positioning stage.
This is because the frequency content of these errors will be located in the controller bandwidth and thus will be easily compensated by the nano-hexapod.
Open Loop Noise budget: <ahref="https://tdehaeze.github.io/nass-simscape/disturbances.html">https://tdehaeze.github.io/nass-simscape/disturbances.html</a>
The measured Power Spectral Density of the ground motion at the ID31 floor is compared with other measurements performed at ID09 and at CERN.
The low frequency differences between the ground motion at ID31 and ID09 is just due to the fact that for the later measurement, the low frequency sensitivity of the inertial sensor was not taken into account.
The goal is to see what noise is injected in the system due to the regulation loop of each stage.
</p>
<p>
Complete reports on these measurements are accessible <ahref="https://tdehaeze.github.io/meas-analysis/2018-10-15%20-%20Marc/index.html">here</a> and <ahref="https://tdehaeze.github.io/meas-analysis/disturbance-control-system/index.html">here</a>.
We consider here the vibrations induced by the scans of the translation stage and rotation of the spindle.
</p>
<p>
Details reports are accessible <ahref="https://tdehaeze.github.io/meas-analysis/disturbance-ty/index.html">here</a> for the translation stage and <ahref="https://tdehaeze.github.io/meas-analysis/disturbance-sr-rz/index.html">here</a> for the spindle/slip-ring.
We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the translation stage (Figure <ahref="#orgf64cc33">16</a>), and we measure the absolute velocity of both the sample and the granite.
<p><spanclass="figure-number">Figure 18: </span>Amplitude spectral density of the measure velocity corresponding to the geophone in the vertical direction located on the granite and at the sample location when the translation stage is scanning at 1Hz</p>
The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure <ahref="#orgb075a53">19</a>.
All the disturbance measurements were made with inertial sensors, and to obtain the relative motion sample/granite, two inertial sensors were used and the signals were subtracted.
As was shown during the modal analysis (Section <ahref="#orge98daae">2</a>), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies.
Thus, a <b>multi-body model</b> is perfect to represent such dynamics.
</p>
<p>
To do so, we use the Matlab’s <ahref="https://www.mathworks.com/products/simscape.html">Simscape</a> toolbox.
A small summary of the multi-body Simscape is available <ahref="https://tdehaeze.github.io/nass-simscape/simscape.html">here</a> and each of the modeled stage is described <ahref="https://tdehaeze.github.io/nass-simscape/simscape_subsystems.html">here</a>.
The (6dof) stiffness between two solid bodies is first guessed from either measurements of data-sheets.
Then, the values of the stiffness and damping of each joint is manually tuned until the obtained dynamics is sufficiently close to the measured dynamics.
It is very difficult the tune the dynamics of such model as there are more than 50 parameters and many curves to compare between the model and the measurements.
<p><spanclass="figure-number">Figure 22: </span>Frequency Response function from Hammer force in the X,Y and Z directions to the X,Y and Z displacements of the micro-hexapod’s top platform. The measurements are shown in blue and the Model in red.</p>
More detailed comparison between the model and the measured dynamics is performed <ahref="https://tdehaeze.github.io/nass-simscape/identification.html">here</a>.
</p>
<p>
Now that the multi-body model dynamics as been tuned, the following elements are included:
</p>
<ulclass="org-ul">
<li>Actuators to perform the motion of each stage (translation, tilt, spindle, hexapod)</li>
<li>Sensors to measure the motion of each stage and the relative motion of the sample with respect to the granite (metrology system)</li>
<li>Disturbances such as ground motion and stage’s vibrations</li>
</ul>
<p>
Then, using the model, we can
</p>
<ulclass="org-ul">
<li>perform simulation of experiments in presence of disturbances</li>
<li>measure the motion of the solid-bodies</li>
<li>identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod and control design</li>
<li>include a multi-body model of the nano-hexapod and closed-loop simulations</li>
<li>First, we need to determine the actual <b>wanted pose</b> (3 translations and 3 rotations) of the sample with respect to the granite.
This is determined from the wanted motion of each micro-station stage.
Each wanted stage motion is represented by an homogeneous transformation matrix (explain <ahref="http://planning.cs.uiuc.edu/node111.html">here</a>), then these matrices are combined to give to total wanted motion of the sample with respect to the granite.</li>
<li>Then, we need to determine the <b>actual pose</b> of the sample with respect to the granite.
This will be performed by several interferometers and several computation will be required to compute the pose of the sample from the interferometers measurements.
However we here directly measure the 3 translations and 3 rotations of the sample using a special simscape block.</li>
<li>Finally, we need to compare the wanted pose with the measured pose to compute the position error of the sample.
This position error can be expressed in the frame of the granite, or in the frame of the (rotating) nano-hexapod.
We first do a simulation where the nano-hexapod is considered to be a solid-body to estimate the sample’s motion that we have without an control.
<p><spanclass="figure-number">Figure 25: </span>Tomography Experiment using the Simscape Model - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))</p>
For the vertical rotation, this is due to the fact that we suppose perfect rotation of the Spindle, and anyway, no measurement of the sample with respect to the granite is made by the interferometers.
<p><spanclass="figure-number">Figure 26: </span>Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances</p>
<li>study many effects such as the change of dynamics due to the rotation, the sample mass, etc.</li>
<li>extract transfer function like \(G\) and \(G_d\)</li>
<li>simulate experiments to validate performance</li>
</ul>
<p>
This model will be used in the next sections to help the design of the nano-hexapod, to develop the robust control architecture and to perform simulation in order to validate.
The study presented here only consider changes in the nano-hexapod <b>stiffness</b>.
The nano-hexapod mass cannot be change too much, and will anyway be negligible compare to the metrology reflector and the payload masses.
The choice of the nano-hexapod architecture (e.g. orientations of the actuators and implementation of sensors) will be further studied in accord with the control architecture.
<h3id="orgeacf23e"><spanclass="section-number-3">5.1</span> Optimal Stiffness to reduce the effect of disturbances</h3>
<divclass="outline-text-3"id="text-5-1">
<p>
The nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of \(G_d\)).
</p>
<p>
For instance, it is quite obvious that a stiff nano-hexapod is better than a soft one when it comes to direct forces applied to the sample such as cable forces.
</p>
<p>
A complete study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility <ahref="https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html">here</a> and summarized below.
</p>
<p>
The sensibility to the spindle vibration as a function of the nano-hexapod stiffness is shown in Figure <ahref="#org9549b68">27</a> (similar curves are obtained for translation stage vibrations).
It is shown that a softer nano-hexapod it better to filter out stage vibrations.
More precisely, is start to filters the vibration at the first suspension mode of the payload on top of the nano-hexapod.
The sensibilities to ground motion in the Y and Z directions are shown in Figure <ahref="#orgfc71478">28</a>.
We can see that above the suspension mode of the nano-hexapod, the norm of the transmissibility is close to one until the suspension mode of the granite.
Then, we take the Power Spectral Density of all the sources of disturbances as identified in Section <ahref="#org71cae32">3</a>, and we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure <ahref="#org24f3472">29</a>).
We can see that the most important change is in the frequency range 30Hz to 300Hz where a stiffness smaller than \(10^5\,[N/m]\) greatly reduces the sensibility to disturbances.
<p><spanclass="figure-number">Figure 29: </span>Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses</p>
If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure <ahref="#org35084a7">30</a>, we can observe that a soft hexapod (\(k <10^5-10^6\,[N/m]\))helpsreducingthehighfrequencydisturbances,andthusasmallercontrolbandwidthwillsufficetoobtainthewantedperformance.
<p><spanclass="figure-number">Figure 30: </span>Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses</p>
<li>A change in the <b>Support’s compliance</b> (complete analysis <ahref="https://tdehaeze.github.io/nass-simscape/uncertainty_support.html">here</a>): if the micro-station dynamics is changing due to the change of parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change.</li>
<li>A change in the <b>Payload mass/dynamics</b> (complete analysis <ahref="https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html">here</a>).</li>
<li>A change of <b>experimental condition</b> such as the micro-station’s pose or the spindle rotation (complete analysis <ahref="https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html">here</a>)</li>
Fortunately, the nano-hexapod stiffness have an influence on the dynamical uncertainty induced by the above effects and we wish here to determine the optimal nano-hexapod stiffness.
<p><spanclass="figure-number">Figure 31: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod</p>
<p><spanclass="figure-number">Figure 32: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod</p>
<p><spanclass="figure-number">Figure 33: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance</p>
<p><spanclass="figure-number">Figure 34: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm</p>
<p><spanclass="figure-number">Figure 35: </span>Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness</p>
It is usually a good idea to maximize the mass, damping and stiffness of the isolation platform in order to be less sensible to the payload dynamics.
The best thing to do is to have a stiff isolation platform.
</p>
<p>
The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when the stiffness of the legs is less than \(10^6\,[N/m]\). When the nano-hexapod is stiff (\(k > 10^7\,[N/m]\)), the compliance of the micro-station appears in the primary plant.
<p><spanclass="figure-number">Figure 38: </span>Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture</p>
<p><spanclass="figure-number">Figure 39: </span>Tomography Experiment using the Simscape Model in Closed Loop with the HAC-LAC Control - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))</p>
<ulclass='org-ref-bib'><li><aid="schmidt14_desig_high_perfor_mechat_revis_edition">[schmidt14_desig_high_perfor_mechat_revis_edition]</a><aname="schmidt14_desig_high_perfor_mechat_revis_edition"></a>Schmidt, Schitter & Rankers, The Design of High Performance Mechatronics - 2nd Revised Edition, Ios Press (2014).</li>
<li><aid="oomen18_advan_motion_contr_precis_mechat">[oomen18_advan_motion_contr_precis_mechat]</a><aname="oomen18_advan_motion_contr_precis_mechat"></a>Tom Oomen, Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems, <i>IEEJ Journal of Industry Applications</i>, <b>7(2)</b>, 127-140 (2018). <ahref="https://doi.org/10.1541/ieejjia.7.127">link</a>. <ahref="http://dx.doi.org/10.1541/ieejjia.7.127">doi</a>.</li>
