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="#org0b76375">3</a>)</li>
In Section <ahref="#org7f92e20">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="#org6d692eb">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="#orgd4cb426">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="#org9b5d986">4</a>).
To do so, we need to identify the dynamics of the micro-station (Section <ahref="#orgaa66df2">2</a>), include this dynamics in a model (Section <ahref="#org85dc720">4</a>) and add a model of the nano-hexapod to the model (Section <ahref="#org7dba516">5</a>)</li>
<li>The controller \(K\) that will be designed in Section <ahref="#org1c73efc">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="#orge757fad">9</a><ahref="#org286ad50">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="#org85dc720">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="#org98ae3bf">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="#org083d882">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="#orgaa66df2">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.
<li>the nano-hexapod mass cannot be change too much, and will anyway be negligible compare to the metrology reflector and the payload masses</li>
<li>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</li>
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.
The sensibility to the spindle’s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure <ahref="#orgb4ec7ff">27</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.
It will be suggested in Section <ahref="#orgc413fc2">7.1</a> that using soft mounts for the granite can greatly lower the sensibility to ground motion.
However, lowering the sensibility to some disturbance at a frequency where its effect is already small compare to the other disturbances sources is not really interesting.
What is more important than comparing the sensitivity to disturbances, is thus to compare the obtain power spectral density of the sample’s position error.
From the Power Spectral Density of all the sources of disturbances identified in Section <ahref="#org0b76375">3</a>, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure <ahref="#org53d2b5c">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 sample’s vibrations.
<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="#org0c62d13">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>
One of the most important design goal is to obtain a system that is <b>robust</b> to all changes in the system.
Therefore, we have to identify all changes that might occurs in the system and choose the nano-hexapod stiffness such that the uncertainty to these changes is minimized.
<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>): the sample’s mass is ranging from \(1\,kg\) to \(50\,kg\)</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>
In the next sections, the effect the considered changes on the <b>plant dynamics</b> is quantified and conclusions are made on the optimal stiffness for robustness properties.
In the following study, when we refer to plant dynamics, this means the dynamics from forces applied by the nano-hexapod to the measured sample’s position by the metrology.
We will only compare the plant dynamics as it is the most important dynamics for robustness and performance properties.
However, the dynamics from forces to sensors located in the nano-hexapod legs, such as force and relative motion sensors, have also been considered in a separate study.
In Figure <ahref="#orgb41dd83">31</a> the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg.
On the left side, the change of dynamics is computed for a very soft nano-hexapod, while on the right side, it is computed for a very stiff nano-hexapod.
<li>the first resonance (suspension mode of the nano-hexapod) is lowered with an increase of the sample’s mass.
This first resonance corresponds to \(\omega = \sqrt{\frac{k_n}{m_n + m_s}}\) where \(k_n\) is the vertical nano-hexapod stiffness, \(m_n\) the mass of the nano-hexapod’s top platform, and \(m_s\) the sample’s mass</li>
<li>the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample’s mass</li>
<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 (left) and a stiff nano-hexapod (right)</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>
The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure <ahref="#orgd2c1e28">33</a>.
</p>
<p>
For nano-hexapod stiffnesses below \(10^6\,[N/m]\):
</p>
<ulclass="org-ul">
<li>the phase stays between 0 and -180deg which is a very nice property for control</li>
<li>the dynamical change up until the resonance of the payload is mostly a change of gain</li>
</ul>
<p>
For nano-hexapod stiffnesses above \(10^7\,[N/m]\):
</p>
<ulclass="org-ul">
<li>the dynamics is unchanged until the first resonance which is around 25Hz-35Hz</li>
<li>above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics)</li>
<p><spanclass="figure-number">Figure 33: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod</p>
</div>
<divclass="important">
<p>
For soft nano-hexapods, the payload has an important impact on the dynamics.
This will have to be carefully taken into account for the controller design.
</p>
<p>
For stiff nano-hexapod, the dynamics doe not change with the payload until the first resonance frequency of the nano-hexapod or of the payload.
</p>
<p>
If possible, the first resonance frequency of the payload should be maximized (stiff fixation).
</p>
<p>
Heavy samples with low first resonance mode will be very problematic.
Thus, it would be much more robust if the plant dynamics were not depending on the micro-station dynamics.
This as several other advantages:
</p>
<ulclass="org-ul">
<li>the control could be develop on top on another support and then added to the micro-station without changing the controller</li>
<li>the nano-hexapod could be use on top of any other station much more easily</li>
</ul>
<p>
To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure <ahref="#org6505158">34</a>):
</p>
<ulclass="org-ul">
<li>without the micro-station (solid curves)</li>
<li>with the micro-station dynamics (dashed curves)</li>
</ul>
<p>
One can see that for nano-hexapod stiffnesses below \(10^6\,[N/m]\), the plant dynamics does not significantly changed due to the micro station dynamics (the solid and dashed curves are superimposed).
</p>
<p>
For nano-hexapod stiffnesses above \(10^7\,[N/m]\), the micro-station compliance appears in the plant dynamics starting at about 45Hz.
<p><spanclass="figure-number">Figure 34: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance</p>
If the resonance of the nano-hexapod is below the first resonance of the micro-station, then the micro-station dynamics if “filtered out” and does not appears in the dynamics to be controlled.
This renders the system robust to any possible change of the micro-station dynamics.
</p>
<p>
If a stiff nano-hexapod is used, the control bandwidth should probably be limited to around the first micro-station’s mode (\(\approx 45\,[Hz]\)) which will likely no give acceptable performance.
The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure <ahref="#orgd447ed7">35</a>.
</p>
<p>
One can see that for nano-hexapods with a stiffness above \(10^5\,[N/m]\), the dynamics is mostly not changing with the spindle’s rotating speed.
For very soft nano-hexapods, the main resonance is split into two resonances and one anti-resonance that are all moving at a function of the rotating speed.
<p><spanclass="figure-number">Figure 35: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm</p>
</div>
<divclass="important">
<p>
If the resonance of the nano-hexapod is (say a factor 5) above the maximum rotation speed, then the plant dynamics will be mostly not impacted by the rotation.
</p>
<p>
A very soft (\(k <10^4\,[N/m]\))nano-hexapodshouldnotbeusedduetotheeffectofthespindle’srotation.
Finally, let’s combined all the uncertainties and display the plant dynamics “spread” for all the nano-hexapod stiffnesses (Figure <ahref="#orgfad19e6">36</a>).
This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics.
<p><spanclass="figure-number">Figure 36: </span>Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness</p>
<li>the payload’s mass influence the plant dynamics above the first resonance of the nano-hexapod.
Thus a high nano-hexapod stiffness helps reducing the effect of a change of the payload’s mass</li>
<li>the payload’s first resonance is seen as an anti-resonance in the plant dynamics.
As this effect will largely be variable from one payload to the other, <b>the payload’s first resonance should be maximized</b> (above 300Hz if possible) for all used payloads</li>
<li>the dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when \(k <10^6\,[N/m]\)</li>
<li>the spindle’s rotating speed has no significant influence on the plant dynamics for nano-hexapods with a stiffness \(k > 10^5\,[N/m]\)</li>
Concerning the plant dynamic uncertainty, the resonance frequency of the nano-hexapod should be between 5Hz (way above the maximum rotating speed) and 50Hz (before the first micro-station resonance) for all the considered payloads.
This corresponds to an optimal nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\).
</p>
<p>
In such case, the main limitation will be heavy samples with small stiffnesses.
In Section <ahref="#org2885849">5.1</a>, it has been concluded that a nano-hexapod stiffness below \(10^5-10^6\,[N/m]\) helps reducing the high frequency vibrations induced by all sources of disturbances considered.
As the high frequency vibrations are the most difficult to compensate for when using feedback control, a soft hexapod will most certainly helps improving the performances.
</p>
<p>
In Section <ahref="#org80634c0">5.2</a>, we concluded that a nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\) is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed.
Provided that the samples used have a first mode that is sufficiently high in frequency, the total plant dynamic uncertainty should be manageable.
</p>
<p>
Thus, a stiffness of \(10^5\,[N/m]\) will be used in Section <ahref="#org1c73efc">6</a> to develop the robust control architecture and to perform simulations.
A more detailed study of the determination of the optimal stiffness based on all the effects is available <ahref="https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html">here</a>.
<h2id="orgdd1f59f"><spanclass="section-number-2">6</span> Robust Control Architecture</h2>
<divclass="outline-text-2"id="text-6">
<p>
<aid="org1c73efc"></a>
</p>
<p>
Before designing the control system, let’s summarize what has been done:
</p>
<ulclass="org-ul">
<li>The multi-body model of the micro-station has been tuned based on actual dynamical measurements</li>
<li>Ground motion and stage vibrations are included in the model with realistic measured values</li>
<li>The optimal nano-hexapod stiffness has been determined such that is minimized the effect of disturbances and at the same time reduces the plant dynamic uncertainty</li>
</ul>
<p>
The optimal nano-hexapod is now included in the model, and a robust control architecture that minimizes the vibrations of the sample is developed.
<li>the plant dynamics does depend quite a lot on the payload’s mass</li>
<li>as there is a trade-off robustness/performance, the bigger the plant dynamic spread, the lower the simultaneous attainable performance is for all the plants</li>
If it not possible to develop a robust controller that gives acceptable performance, an alternative would be to develop an <b>adaptive</b> controller that depends on the payload mass/inertia.
This would require to measure the mass/inertia of each used payload and manually choose the controller that was design for that particular mass/inertia.
</p>
<p>
For such system, the <b>High-Authority-Control/Low-Authority-Control</b> (HAC-LAC) architecture
</p>
<p>
from the following reasons explained in <aclass='org-ref-reference'href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</a>:
</p>
<blockquote>
<p>
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <ahref="#orgddcc138">37</a>.
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
A separate study (accessible <ahref="https://tdehaeze.github.io/rotating-frame/index.html">here</a>) for all three sensor type have been done, the conclusions are:
<li>the use of force sensors is to be avoided as it could induce instability in the system due to rotation of the nano-hexapod</li>
<li>the use of inertial sensor should not be used as it would tends to decouple the motion of the sample to the motion of the granite (which is not wanted) and it would be difficult to apply in a robust way</li>
<li>relative motion sensors can be used to damped the nano-hexapod’s modes in a robust way</li>
Relative motion sensors are then included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture as shown in figure <ahref="#org9525c66">38</a> is applied.
\(\bm{K}_{\text{DVF}}\) is a diagonal controller that consists of applying a force in each actuator proportional to the relative velocity of the associated leg.
This adds damping to the nano-hexapod’s modes.
The plant dynamics before (solid curves) and after (dashed curves) the Law-Authority-Control implementation are compared in Figure <ahref="#org735749d">39</a>.
It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz up too 100Hz.
This will make the primary controller more robust and easier to develop.
As shown in Figure <ahref="#org4696e73">40</a>, the use of the DVF control lowers the sensibility to disturbances in the vicinity of the nano-hexapod resonance but increases the sensibility at higher frequencies.
This is probably not the optimal gain that could be used, and further analysis and optimization will be performed.
<p><spanclass="figure-number">Figure 40: </span>Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied</p>
The complete HAC-LAC architecture is shown in Figure <ahref="#orgaa578f2">41</a> where an outer loop is added to the decentralized direct velocity feedback loop.
</p>
<p>
The block <code>Compute Position Error</code> is used to compute the position error of the sample with respect to the nano-hexapod’s base platform \(\bm{\epsilon}_{\mathcal{X}_n}\) from the actual measurement of the sample’s pose \(\bm{\mathcal{X}}\) and the wanted pose \(\bm{r}_\mathcal{X}\).
The computation done in such block was briefly explained in Section <ahref="#org1b03042">4.3</a>.
</p>
<p>
From the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) expressed in the frame of the nano-hexapod, the nano-hexapod’s Jacobian \(\bm{J}\) (which is a real matrix) is used to compute the corresponding length error of each of the nano hexapod’s leg \(\bm{\epsilon}_\mathcal{L}\).
</p>
<p>
Then, a diagonal controller \(\bm{K}_\mathcal{L}\) generates the required force in each leg to compensate the position error.
<p><spanclass="figure-number">Figure 41: </span>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</p>
<p><spanclass="figure-number">Figure 42: </span>Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses</p>
The diagonal controller \(\bm{K}_\mathcal{L}\) is then tuned in such a way that the control bandwidth is around 100Hz and such that enough stability margins are obtained for all the payload’s masses used.
<p><spanclass="figure-number">Figure 44: </span>Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)</p>
The same simulation of a tomography experiment performed in Section <ahref="#org03572e5">4.4</a> is now re-done with the used of the HAC-LAC architecture.
All the disturbances are included such as ground motion, spindle and translation stage vibrations.
</p>
<p>
After the simulation is performed, the Power Spectral Density of the sample’s position error is plotted in Figure <ahref="#org6fda8f6">45</a> and the Cumulative Amplitude Spectrum is shown in Figure <ahref="#org250a399">46</a>.
The top three plots corresponds to the X, Y and Z translations and the bottom three plots corresponds to the X,Y and Z rotations.
</p>
<p>
Several observations can be made:
</p>
<ulclass="org-ul">
<li>The sample’s vibrations are reduced within the control bandwidth</li>
<li><p>
An increase in the rotational vibrations is observed.
<li>no perturbations inducing rotations are included in the simulation: the vibrations in rotation are very small</li>
<li>the feedback control induces some coupling between the translations and rotations.
This means that it introduces some rotations due to translation vibrations</li>
</ol>
<p>
This increase in rotation is still very small and is not foreseen to be a problem
</p></li>
<li>The obtained performances for all the three considered masses are very similar.
That shows the robustness of the system</li>
<li>The vertical rotation plot is meaningless as the spindle rotation was considered to be perfect and no attempt is made to compensate vertical rotation by the nano-hexapod</li>
<li>From the Cumulative Amplitude Spectrum, we see that Z motion is reduced down to \(\approx 30\,nm\,[rms]\) and the Y motion down to \(\approx 25\,nm\,[rms]\)</li>
The time domain sample’s vibrations are shown in Figure <ahref="#orgab1cf67">47</a>.
The use of the nano-hexapod combined with the HAC-LAC architecture is shown to considerably reduce the sample’s vibrations.
</p>
<p>
An animation of the experiment is shown in Figure <ahref="#org3cf2484">48</a> and we can see that the actual sample’s position is more closely following the ideal position as was the case with the simulation of the micro-station alone in Figure <ahref="#org21907ec">25</a> (same scale was used for both simulations).
<p><spanclass="figure-number">Figure 47: </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 48: </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>
<p><spanclass="figure-number">Figure 49: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves)</p>
</div>
<p>
This means that above the suspension mode of the granite (here around 2Hz), the granite
<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>
<li><aid="preumont18_vibrat_contr_activ_struc_fourt_edition">[preumont18_vibrat_contr_activ_struc_fourt_edition]</a><aname="preumont18_vibrat_contr_activ_struc_fourt_edition"></a>Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).</li>
