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< h1 class = "title" > Design of the Nano-Hexapod and associated Control Architectures - Summary< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orge0ac2c3" > 1. Introduction to Feedback Systems and Noise budgeting< / a >
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< ul >
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< li > < a href = "#org3c0d370" > 1.1. Feedback System< / a >
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< ul >
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< li > < a href = "#org89fd894" > 1.1.1. Simplified Feedback Control Diagram for the NASS< / a > < / li >
< li > < a href = "#org749d418" > 1.1.2. How does the feedback loop is modifying the system behavior?< / a > < / li >
< li > < a href = "#org28611b5" > 1.1.3. Trade off: Disturbance Reduction / Noise Injection< / a > < / li >
< li > < a href = "#org668a4e8" > 1.1.4. Trade off: Robustness / Performance< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc811e11" > 1.2. Dynamic error budgeting< / a >
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< ul >
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< li > < a href = "#org898c6aa" > 1.2.1. Power Spectral Density< / a > < / li >
< li > < a href = "#org3993d5c" > 1.2.2. Cumulative Power Spectrum< / a > < / li >
< li > < a href = "#orga56f04c" > 1.2.3. Modification of a signal’ s PSD when going through an LTI system< / a > < / li >
< li > < a href = "#org37d8eed" > 1.2.4. PSD of combined signals< / a > < / li >
< li > < a href = "#org933f999" > 1.2.5. Dynamic Noise Budgeting< / a > < / li >
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< / ul >
< / li >
< / ul >
< / li >
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< li > < a href = "#org470f127" > 2. Identification of the Micro-Station Dynamics< / a >
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< ul >
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< li > < a href = "#org0d4e438" > 2.1. Setup< / a > < / li >
< li > < a href = "#org3ea6a8c" > 2.2. Results< / a > < / li >
< li > < a href = "#org3648889" > 2.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga49cd71" > 3. Identification of the Disturbances< / a >
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< ul >
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< li > < a href = "#orga79046a" > 3.1. Ground Motion< / a > < / li >
< li > < a href = "#org7605758" > 3.2. Stage Vibration - Effect of Control systems< / a > < / li >
< li > < a href = "#org491343d" > 3.3. Stage Vibration - Effect of Motion< / a >
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< ul >
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< li > < a href = "#orgdbdbeaf" > Spindle and Slip-Ring< / a > < / li >
< li > < a href = "#orgd96580d" > Translation Stage< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org2348309" > 3.4. Sum of all disturbances< / a > < / li >
< li > < a href = "#org25a7b3f" > 3.5. Better estimation of the disturbances< / a > < / li >
< li > < a href = "#org2e7835b" > 3.6. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org12a2fe0" > 4. Multi Body Model< / a >
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< ul >
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< li > < a href = "#org0e7965a" > 4.1. Multi-Body model< / a > < / li >
< li > < a href = "#orgec68dd4" > 4.2. Validity of the model’ s dynamics< / a > < / li >
< li > < a href = "#orgd0b55c5" > 4.3. Wanted position of the sample and position error< / a > < / li >
< li > < a href = "#org82db829" > 4.4. Simulation of Experiments< / a > < / li >
< li > < a href = "#orga05f525" > 4.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org463efb0" > 5. Optimal Nano-Hexapod Design< / a >
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< ul >
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< li > < a href = "#org39a6d2e" > 5.1. Optimal Stiffness to reduce the effect of disturbances< / a > < / li >
< li > < a href = "#orgbaf53ab" > 5.2. Optimal Stiffness< / a > < / li >
< li > < a href = "#orgd572056" > 5.3. Sensors to be included< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgdaa0983" > 6. Robust Control Architecture< / a >
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< ul >
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< li > < a href = "#orgd9f9771" > 6.1. Simulation of Tomography Experiments< / a > < / li >
< li > < a href = "#org790f59d" > 6.2. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd091d56" > 7. Further notes< / a > < / li >
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< / ul >
< / div >
< / div >
< p >
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.
< / p >
< p >
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To understand the design challenges of such system, a short introduction to Feedback control is provided in Section < a href = "#orgf42c82b" > 1< / a > .
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The mathematical tools (Power Spectral Density, Noise Budgeting, … ) that will be used throughout this study are also introduced.
< / p >
< p >
To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of:
< / p >
< ul class = "org-ul" >
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< li > the micro-station dynamics (Section < a href = "#org16c9690" > 2< / a > )< / li >
< li > the frequency content of the important source of disturbances in play such as vibration of stages and ground motion (Section < a href = "#org8dba979" > 3< / a > )< / li >
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< / ul >
< p >
We then develop a model of the system that must represent all the important physical effects in play.
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Such model is presented in Section < a href = "#org7c0f438" > 4< / a > .
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< / p >
< p >
A modular model of the nano-hexapod is then included in the system.
The effects of the nano-hexapod characteristics on the dynamics are then studied.
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Based on that, an optimal choice of the nano-hexapod stiffness is made (Section < a href = "#orgd2bab4c" > 5< / a > ).
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< / p >
< p >
Finally, using the optimally designed nano-hexapod, a robust control architecture is developed.
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Simulations are performed to show that this design gives acceptable performance and the required robustness (Section < a href = "#orgba39228" > 6< / a > ).
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< / p >
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< div id = "outline-container-orge0ac2c3" class = "outline-2" >
< h2 id = "orge0ac2c3" > < span class = "section-number-2" > 1< / span > Introduction to Feedback Systems and Noise budgeting< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
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< a id = "orgf42c82b" > < / a >
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< / p >
< p >
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In this section, we first introduce some basics of < b > feedback systems< / b > (Section < a href = "#org6a45622" > 1.1< / a > ).
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This should highlight the challenges in terms of combined performance and robustness.
< / p >
< p >
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In Section < a href = "#org60ead65" > 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.
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This tool will be widely used throughout this study to both predict the performances and identify the effects that do limit the performances.
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< / p >
< / div >
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< div id = "outline-container-org3c0d370" class = "outline-3" >
< h3 id = "org3c0d370" > < span class = "section-number-3" > 1.1< / span > Feedback System< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< p >
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< a id = "org6a45622" > < / a >
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< / p >
< p >
The use of feedback control as several advantages and pitfalls that are listed below (taken from < a class = 'org-ref-reference' href = "#schmidt14_desig_high_perfor_mechat_revis_edition" > schmidt14_desig_high_perfor_mechat_revis_edition< / a > ):
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< / p >
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< ul class = "org-ul" >
< li > < b > Advantages< / b > :
< ul class = "org-ul" >
< 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 > :
< ul class = "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 >
< / ul > < / li >
< / ul >
< / div >
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< div id = "outline-container-org89fd894" class = "outline-4" >
< h4 id = "org89fd894" > < span class = "section-number-4" > 1.1.1< / span > Simplified Feedback Control Diagram for the NASS< / h4 >
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< div class = "outline-text-4" id = "text-1-1-1" >
< p >
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Let’ s consider the block diagram shown in Figure < a href = "#orgb906f00" > 1< / a > where the signals are:
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< / p >
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< ul class = "org-ul" >
< li > \(y\): the relative position of the sample with respect to the granite (the quantity we wish to control)< / li >
< li > \(d\): the disturbances affecting \(y\) (ground motion, vibration of stages)< / li >
< li > \(n\): the noise of the sensor measuring \(y\)< / li >
< li > \(r\): the reference signal, corresponding to the wanted \(y\)< / li >
< li > \(\epsilon = r - y\): the position error< / li >
< / ul >
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< p >
And the dynamical blocks are:
< / p >
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< ul class = "org-ul" >
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< li > \(G\): representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite \(y\)< / li >
< li > \(G_d\): representing how the disturbances (e.g. ground motion) are affecting the relative position sample/granite \(y\)< / li >
< li > \(K\): representing the controller (to be designed)< / li >
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< / ul >
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< div id = "orgb906f00" class = "figure" >
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< p > < img src = "figs/classical_feedback_small.png" alt = "classical_feedback_small.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Block Diagram of a simple feedback system< / p >
< / div >
< p >
Without the use of feedback (i.e. nano-hexapod), the disturbances will induce a sample motion error equal to:
< / p >
\begin{equation}
y = G_d d \label{eq:open_loop_error}
\end{equation}
< p >
which is out of the specifications (micro-meter range compare to the required \(\approx 10nm\)).
< / p >
< p >
In the next section, we see how the use of the feedback system permits to lower the effect of the disturbances \(d\) on the sample motion error.
< / p >
< / div >
< / div >
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< div id = "outline-container-org749d418" class = "outline-4" >
< h4 id = "org749d418" > < span class = "section-number-4" > 1.1.2< / span > How does the feedback loop is modifying the system behavior?< / h4 >
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< div class = "outline-text-4" id = "text-1-1-2" >
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< p >
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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 < a href = "#orgb906f00" > 1< / a > ), we obtain:
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\[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
< / p >
< p >
We usually note:
< / p >
\begin{align}
S & = \frac{1}{1 + GK} \\
T & = \frac{GK}{1 + GK}
\end{align}
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< p >
where \(S\) is called the sensibility transfer function and \(T\) the transmissibility transfer function.
< / p >
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< p >
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And the position error can be rewritten as:
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< / p >
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\begin{equation}
\epsilon = S r + T n - G_d S d \label{eq:closed_loop_error}
\end{equation}
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< p >
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From Eq. \eqref{eq:closed_loop_error} representing the closed-loop system behavior, we can see that:
< / p >
< ul class = "org-ul" >
< li > the effect of disturbances \(d\) on \(\epsilon\) is multiplied by a factor \(S\) compared to the open-loop case< / li >
< li > the measurement noise \(n\) is injected and multiplied by a factor \(T\)< / li >
< / ul >
< p >
Ideally, we would like to design the controller \(K\) such that:
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< / p >
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< ul class = "org-ul" >
< li > \(|S|\) is small to limit the effect of disturbances< / li >
< li > \(|T|\) is small to limit the injection of sensor noise< / li >
< / ul >
< p >
As shown in the next section, there is a trade-off between the disturbance reduction and the noise injection.
< / p >
< / div >
< / div >
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< div id = "outline-container-org28611b5" class = "outline-4" >
< h4 id = "org28611b5" > < span class = "section-number-4" > 1.1.3< / span > Trade off: Disturbance Reduction / Noise Injection< / h4 >
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< div class = "outline-text-4" id = "text-1-1-3" >
< p >
We have from the definition of \(S\) and \(T\) that:
< / p >
\begin{equation}
S + T = \frac{1}{1 + GK} + \frac{GK}{1 + GK} = 1
\end{equation}
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< p >
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meaning that we cannot have \(|S|\) and \(|T|\) small at the same time.
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< / p >
< p >
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There is therefore a < b > trade-off between the disturbance rejection and the measurement noise filtering< / b > .
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< / p >
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< p >
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Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure < a href = "#org4eb9aca" > 2< / a > .
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We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on the frequency band:
< / p >
< ul class = "org-ul" >
< li > < b > At low frequency< / b > (inside the control bandwidth):
< ul class = "org-ul" >
< li > \(|S|\) can be made small and thus the effect of disturbances is reduced< / li >
< li > \(|T| \approx 1\) and all the sensor noise is transmitted< / li >
< / ul > < / li >
< li > < b > At high frequency< / b > (outside the control bandwidth):
< ul class = "org-ul" >
< li > \(|S| \approx 1\) and the feedback system does not reduce the effect of disturbances< / li >
< li > \(|T|\) is small and thus the sensor noise is filtered< / li >
< / ul > < / li >
< li > < b > Near the crossover frequency< / b > (between the two frequency bands):
< ul class = "org-ul" >
< li > The effect of disturbances is increased< / li >
< / ul > < / li >
< / ul >
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< div id = "org4eb9aca" class = "figure" >
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< p > < img src = "figs/h-infinity-2-blocs-constrains.png" alt = "h-infinity-2-blocs-constrains.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 2: < / span > Typical shapes and constrain of the Sensibility and Transmibility closed-loop transfer functions< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org668a4e8" class = "outline-4" >
< h4 id = "org668a4e8" > < span class = "section-number-4" > 1.1.4< / span > Trade off: Robustness / Performance< / h4 >
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< div class = "outline-text-4" id = "text-1-1-4" >
< p >
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< a id = "org59587ec" > < / a >
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< / p >
< p >
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 > .
< / p >
< p >
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For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure < a href = "#org3deed0a" > 3< / a > ).
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< / p >
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< div id = "org3deed0a" class = "figure" >
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< p > < img src = "figs/oomen18_next_gen_loop_gain.png" alt = "oomen18_next_gen_loop_gain.png" / >
< / p >
< p > < span class = "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. < a class = 'org-ref-reference' href = "#oomen18_advan_motion_contr_precis_mechat" > oomen18_advan_motion_contr_precis_mechat< / a > < / p >
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< / div >
< p >
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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 >
< ul class = "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.
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< / p >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgc811e11" class = "outline-3" >
< h3 id = "orgc811e11" > < span class = "section-number-3" > 1.2< / span > Dynamic error budgeting< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< p >
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< a id = "org60ead65" > < / a >
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< / p >
< p >
The dynamic error budgeting is a powerful tool to study the effect of multiple error sources and to see how the feedback system does reduce the effect
< / p >
< p >
To understand how to use and understand it, the Power Spectral Density and the Cumulative Power Spectrum are first introduced.
Then, is shown how does multiple error sources are combined and modified by dynamical systems.
< / p >
< p >
Finally,
< / p >
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< / div >
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< div id = "outline-container-org898c6aa" class = "outline-4" >
< h4 id = "org898c6aa" > < span class = "section-number-4" > 1.2.1< / span > Power Spectral Density< / h4 >
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< div class = "outline-text-4" id = "text-1-2-1" >
< p >
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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:
\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \ \frac{[\text{unit of } x]^2}{\text{Hz}} \]
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< / p >
< p >
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The PSD \(S_{xx}(\omega)\) represents the < b > distribution of the (average) signal power over frequency< / b > .
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< / p >
< p >
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:
< / p >
\begin{equation}
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x_{\text{rms}} = \sqrt{\int_{0}^{\infty} S_{xx}(\omega) d\omega}
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\end{equation}
< p >
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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:
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< / p >
\begin{equation}
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P_{f_1,f_2} = \int_{f_1}^{f_2} S_{xx}(\omega) d\omega \quad [\text{unit of } x]^2
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\end{equation}
< / div >
< / div >
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< div id = "outline-container-org3993d5c" class = "outline-4" >
< h4 id = "org3993d5c" > < span class = "section-number-4" > 1.2.2< / span > Cumulative Power Spectrum< / h4 >
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< div class = "outline-text-4" id = "text-1-2-2" >
< p >
The < b > Cumulative Power Spectrum< / b > is the cumulative integral of the Power Spectral Density starting from \(0\ \text{Hz}\) with increasing frequency:
< / p >
\begin{equation}
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CPS_x(f) = \int_0^f S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2
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\end{equation}
< p >
The Cumulative Power Spectrum taken at frequency \(f\) thus represent the power in the signal in the frequency band \(0\) to \(f\).
< / p >
< p >
An alternative definition of the Cumulative Power Spectrum can be used where the PSD is integrated from \(f\) to \(\infty\):
< / p >
\begin{equation}
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CPS_x(f) = \int_f^\infty S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2
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\end{equation}
< p >
And thus \(CPS_x(f)\) represents the power in the signal \(x\) for frequencies above \(f\).
< / p >
< p >
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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.
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< / p >
< / div >
< / div >
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< div id = "outline-container-orga56f04c" class = "outline-4" >
< h4 id = "orga56f04c" > < span class = "section-number-4" > 1.2.3< / span > Modification of a signal’ s PSD when going through an LTI system< / h4 >
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< div class = "outline-text-4" id = "text-1-2-3" >
< p >
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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 < a href = "#org22dc248" > 4< / a > ).
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< / p >
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< div id = "org22dc248" class = "figure" >
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< p > < img src = "figs/psd_lti_system.png" alt = "psd_lti_system.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > LTI dynamical system \(G(s)\) with input signal \(u\) and output signal \(y\)< / p >
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< / div >
< p >
The Power Spectral Density of the output signal \(y\) can be computed using:
< / p >
\begin{equation}
S_{yy}(\omega) = \left|G(j\omega)\right|^2 S_{uu}(\omega)
\end{equation}
< / div >
< / div >
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< div id = "outline-container-org37d8eed" class = "outline-4" >
< h4 id = "org37d8eed" > < span class = "section-number-4" > 1.2.4< / span > PSD of combined signals< / h4 >
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< div class = "outline-text-4" id = "text-1-2-4" >
< p >
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Let’ s consider a signal \(y\) that is the sum of two < b > uncorrelated< / b > signals \(u\) and \(v\) (Figure < a href = "#orge9f3044" > 5< / a > ).
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< / p >
< p >
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We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD of \(v\) (can be easily shown from the definition of the PSD):
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\[ S_{yy} = S_{uu} + S_{vv} \]
< / p >
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< div id = "orge9f3044" class = "figure" >
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< p > < img src = "figs/psd_sum.png" alt = "psd_sum.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > \(y\) as the sum of two signals \(u\) and \(v\)< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org933f999" class = "outline-4" >
< h4 id = "org933f999" > < span class = "section-number-4" > 1.2.5< / span > Dynamic Noise Budgeting< / h4 >
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< div class = "outline-text-4" id = "text-1-2-5" >
< p >
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Let’ s consider the Feedback architecture in Figure < a href = "#orgb906f00" > 1< / a > where the position error \(\epsilon\) is equal to:
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\[ \epsilon = S r + T n - G_d S d \]
< / p >
< p >
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If we suppose that the signals \(r\), \(n\) and \(d\) are < b > uncorrelated< / b > (which is a good approximation in our case), the PSD of \(\epsilon\) is:
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\[ S_{\epsilon \epsilon}(\omega) = |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \]
< / p >
< p >
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And we can compute the RMS value of the residual motion using:
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< / p >
\begin{align*}
\epsilon_\text{rms} & = \sqrt{ \int_0^\infty S_{\epsilon\epsilon}(\omega) d\omega} \\
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& = \sqrt{ \int_0^\infty \Big( |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \Big) d\omega }
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\end{align*}
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< p >
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To estimate the PSD of the position error \(\epsilon\) and thus the RMS residual motion (in closed-loop), we need to determine:
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< / p >
< ul class = "org-ul" >
< li > The Power Spectral Densities of the signals affecting the system:
< ul class = "org-ul" >
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< li > \(S_{dd}\): disturbances, this will be done in Section < a href = "#org8dba979" > 3< / a > < / li >
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< li > \(S_{nn}\): sensor noise, this can be estimated from the sensor data-sheet< / li >
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< li > \(S_{rr}\): which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to \(0\)< / li >
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< / ul > < / li >
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< li > The dynamics of the complete system comprising the micro-station and the nano-hexapod: \(G\), \(G_d\).
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To do so, we need to identify the dynamics of the micro-station (Section < a href = "#org16c9690" > 2< / a > ), include this dynamics in a model (Section < a href = "#org7c0f438" > 4< / a > ) and add a model of the nano-hexapod to the model (Section < a href = "#orgd2bab4c" > 5< / a > )< / li >
< li > The controller \(K\) that will be designed in Section < a href = "#orgba39228" > 6< / a > < / li >
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< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org470f127" class = "outline-2" >
< h2 id = "org470f127" > < span class = "section-number-2" > 2< / span > Identification of the Micro-Station Dynamics< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
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< a id = "org16c9690" > < / a >
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< / p >
< p >
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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.
< / p >
< p >
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The estimated dynamics will also be used to tune the developed multi-body model of the micro-station with which the simulations will be performed.
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< / p >
< p >
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All the measurements performed on the micro-station are detailed in < a href = "https://tdehaeze.github.io/meas-analysis/" > this< / a > document and summarized in the following sections.
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< / p >
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< p >
The general procedure to identify the dynamics of the micro-station is shown in Figure < a href = "#org58e7ef7" > 6< / a > .
< / p >
< p >
The steps are:
< / p >
< ol class = "org-ol" >
< li > extract a Response Model (Frequency Response Functions) from measurements< / li >
< li > convert the Response Model into a Modal Model (Natural Frequencies and Mode Shapes)< / li >
< li > extract a Spatial Model from the Modal Model (Mass/Damping/Stiffness matrices)< / li >
< / ol >
< div id = "org58e7ef7" class = "figure" >
< p > < img src = "figs/vibration_analysis_procedure.png" alt = "vibration_analysis_procedure.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Vibration Analysis Procedure< / p >
< / div >
< p >
The extraction of the Spatial Model (3rd step) was not performed as it requires a lot of time and was not judge necessary.
< / p >
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< / div >
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< div id = "outline-container-org0d4e438" class = "outline-3" >
< h3 id = "org0d4e438" > < span class = "section-number-3" > 2.1< / span > Setup< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
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< p >
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< a id = "orgcb9db1c" > < / a >
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< / p >
< p >
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To measure the dynamics of such complicated system, it as been chosen to perform a modal analysis.
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< / p >
< p >
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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 > .
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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:
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< / p >
< ul class = "org-ul" >
< li > An < b > acquisition system< / b > (OROS) with 24bits ADCs< / li >
< li > 3 tri-axis < b > Accelerometers< / b > < / li >
< li > An < b > Instrumented Hammer< / b > < / li >
< / ul >
< p >
The measurement thus consists of:
< / p >
< ul class = "org-ul" >
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< li > Exciting the structure at the same location with the Hammer (Figure < a href = "#org1bfe19e" > 7< / a > )< / li >
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< li > Move the accelerometers to measure all the DOF of the structure.
The position of the accelerometers are:
< ul class = "org-ul" >
< li > 4 on the first granite< / li >
< li > 4 on the second granite< / li >
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< li > 4 on top of the translation stage (figure < a href = "#org8c09e31" > 8< / a > )< / li >
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< li > 4 on top of the tilt stage< / li >
< li > 3 on top of the spindle< / li >
< li > 4 on top of the hexapod< / li >
< / ul > < / li >
< / ul >
< p >
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In total, 69 degrees of freedom are measured (23 tri axis accelerometers) which is more that what was required.
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< / p >
< p >
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We chose to have some redundancy in the measurement to be able to verify that the solid-body assumption is correct for each of the stage.
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< / p >
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< div id = "org1bfe19e" class = "figure" >
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< p > < img src = "figs/hammer_z.gif" alt = "hammer_z.gif" / >
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< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Example of one hammer impact< / p >
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< / div >
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< div id = "org8c09e31" class = "figure" >
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< p > < img src = "figs/accelerometers_ty_overview.jpg" alt = "accelerometers_ty_overview.jpg" / >
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< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > 3 tri axis accelerometers fixed to the translation stage< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org3ea6a8c" class = "outline-3" >
< h3 id = "org3ea6a8c" > < span class = "section-number-3" > 2.2< / span > Results< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< p >
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< a id = "orgee943d8" > < / a >
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< / p >
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< p >
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.
< / p >
< p >
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Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures < a href = "#orge554e13" > 9< / a > < a href = "#orgb87ae23" > 10< / a > .
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< / p >
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< div id = "orge554e13" class = "figure" >
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< p > < img src = "figs/mode1.gif" alt = "mode1.gif" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > First mode that shows a suspension mode, probably due to bad leveling of one Airloc< / p >
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< / div >
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< div id = "orgb87ae23" class = "figure" >
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< p > < img src = "figs/mode6.gif" alt = "mode6.gif" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Sixth mode< / p >
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< / div >
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< p >
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We then reduce the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF).
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< / p >
< p >
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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 >
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< div class = "important" >
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< p >
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This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest.
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This thus means that a multi-body model can be used to represent the dynamics of the micro-station.
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< / p >
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< / div >
< p >
Many Frequency Response Functions (FRF) are obtained from the measurements.
Examples of FRF are shown in Figure < a href = "#org7586abc" > 11< / a > .
These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.
< / p >
< div id = "org7586abc" class = "figure" >
< p > < img src = "figs/frf_all_bodies_one_direction.png" alt = "frf_all_bodies_one_direction.png" / >
< / p >
< p > < span class = "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 >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org3648889" class = "outline-3" >
< h3 id = "org3648889" > < span class = "section-number-3" > 2.3< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
< div class = "important" >
< p >
The modal analysis of the micro-station confirmed the fact that a multi-body model should be able to correctly represents the micro-station dynamics.
In Section < a href = "#org7c0f438" > 4< / a > , the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orga49cd71" class = "outline-2" >
< h2 id = "orga49cd71" > < span class = "section-number-2" > 3< / span > Identification of the Disturbances< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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< a id = "org8dba979" > < / a >
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< / p >
< p >
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In this section, we wish to list and identify all the disturbances affecting the system.
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< / p >
< p >
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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.
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< / p >
< p >
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The problem are on the high frequency disturbances.
In the following sections, we consider:
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< / p >
< ul class = "org-ul" >
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< li > the ground motion< / li >
< li > vibrations of each stage, due either to their control systems or their motion< / li >
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< / ul >
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< p >
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< a href = "https://tdehaeze.github.io/meas-analysis/" > https://tdehaeze.github.io/meas-analysis/< / a >
Open Loop Noise budget: < a href = "https://tdehaeze.github.io/nass-simscape/disturbances.html" > https://tdehaeze.github.io/nass-simscape/disturbances.html< / a >
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< / p >
< / div >
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< div id = "outline-container-orga79046a" class = "outline-3" >
< h3 id = "orga79046a" > < span class = "section-number-3" > 3.1< / span > Ground Motion< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< p >
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< a id = "org7be3491" > < / a >
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< / p >
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< p >
The ground motion can easily be estimated using an inertial sensor with sufficient sensitivity.
< / p >
< p >
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To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure < a href = "#org8219925" > 12< / a > ).
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< / p >
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< div id = "org8219925" class = "figure" >
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< p > < img src = "figs/geophones.jpg" alt = "geophones.jpg" / >
< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Huddle Test Setup< / p >
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< / div >
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< p >
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.
< / p >
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< div id = "org6a73c9d" class = "figure" >
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< p > < img src = "figs/ground_motion_compare.png" alt = "ground_motion_compare.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Comparison of the PSD of the ground motion measured at different location< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org7605758" class = "outline-3" >
< h3 id = "org7605758" > < span class = "section-number-3" > 3.2< / span > Stage Vibration - Effect of Control systems< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
< p >
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< a id = "org0b6e3e2" > < / a >
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< / p >
< p >
Control system of each stage has been tested
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< / p >
< p >
Each motor are turn off and then on.
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 < a href = "https://tdehaeze.github.io/meas-analysis/2018-10-15%20-%20Marc/index.html" > here< / a > and < a href = "https://tdehaeze.github.io/meas-analysis/disturbance-control-system/index.html" > here< / a > .
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< / p >
< p >
25Hz vertical motion when the < b > Spindle< / b > is turned on (even when not rotating).
< / p >
< / div >
< / div >
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< div id = "outline-container-org491343d" class = "outline-3" >
< h3 id = "org491343d" > < span class = "section-number-3" > 3.3< / span > Stage Vibration - Effect of Motion< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
< p >
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< a id = "org756755b" > < / a >
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< / p >
< p >
We consider here the vibrations induced by the scans of the translation stage and rotation of the spindle.
< / p >
< p >
Details reports are accessible < a href = "https://tdehaeze.github.io/meas-analysis/disturbance-ty/index.html" > here< / a > for the translation stage and < a href = "https://tdehaeze.github.io/meas-analysis/disturbance-sr-rz/index.html" > here< / a > for the spindle/slip-ring.
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< / p >
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< / div >
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< div id = "outline-container-orgdbdbeaf" class = "outline-4" >
< h4 id = "orgdbdbeaf" > Spindle and Slip-Ring< / h4 >
< div class = "outline-text-4" id = "text-orgdbdbeaf" >
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< p >
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The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure < a href = "#org3d732d4" > 14< / a > .
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< / p >
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< div id = "org3d732d4" class = "figure" >
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< p > < img src = "figs/rz_meas_errors.gif" alt = "rz_meas_errors.gif" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Measurement of the sample’ s vertical motion when rotating at 6rpm< / p >
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< / div >
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< p >
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A geophone is fixed at the location of the sample and we measure the motion:
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< / p >
< ul class = "org-ul" >
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< li > without rotation< / li >
< li > when rotating at 6rpm using the slip-ring motor< / li >
< li > when rotating at 6rpm using the spindle motor synchronized with the slip-ring motor< / li >
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< / ul >
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< p >
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The obtained Power Spectral Density of the sample’ s absolute velocity are shown in Figure < a href = "#orgbc9d6c2" > 15< / a > .
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< / p >
< p >
We can see that when using the Slip-ring motor to rotate the sample, only a little increase of the motion is observed above 100Hz.
< / p >
< p >
However, when rotating with the Spindle (normal functioning mode):
< / p >
< ul class = "org-ul" >
< li > a very sharp peak at 23Hz is observed. Its cause has not been identified yet< / li >
< li > a general large increase in motion above 30Hz< / li >
< / ul >
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< div id = "orgbc9d6c2" class = "figure" >
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< p > < img src = "figs/sr_sp_psd_sample_compare.png" alt = "sr_sp_psd_sample_compare.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Comparison of the ASD of the measured voltage from the Geophone at the sample location< / p >
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< / div >
< div class = "important" >
< p >
Some investigation should be performed on the Spindle to determine where does this 23Hz motion comes from.
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd96580d" class = "outline-4" >
< h4 id = "orgd96580d" > Translation Stage< / h4 >
< div class = "outline-text-4" id = "text-orgd96580d" >
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< p >
The same setup is used (a geophone is located at the sample’ s location and another on the granite).
< / p >
< p >
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We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the translation stage (Figure < a href = "#orgd1dcde2" > 16< / a > ), and we measure the absolute velocity of both the sample and the granite.
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< / p >
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< div id = "orgd1dcde2" class = "figure" >
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< p > < img src = "figs/ty_position_time.png" alt = "ty_position_time.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Y position of the translation stage measured by the encoders< / p >
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< / div >
< p >
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The time domain absolute vertical velocity of the sample and granite are shown in Figure < a href = "#org4560f79" > 17< / a > .
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It is shown that quite large motion of the granite is induced by the translation stage scans.
This could be a problem if this is shown to excite the metrology frame of the nano-focusing lens position stage.
< / p >
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< div id = "org4560f79" class = "figure" >
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< p > < img src = "figs/ty_z_time.png" alt = "ty_z_time.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 17: < / span > Vertical velocity of the sample and marble when scanning with the translation stage< / p >
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< / div >
< p >
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The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure < a href = "#org4709c12" > 18< / a > .
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We can see many peaks starting from 1Hz showing the large spectral content probably due to the triangular reference of the translation stage.
< / p >
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< div id = "org4709c12" class = "figure" >
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< p > < img src = "figs/asd_z_direction.png" alt = "asd_z_direction.png" / >
< / p >
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< p > < span class = "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 >
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< / div >
< div class = "important" >
< p >
A smoother motion for the translation stage (such as a sinus motion) could probably help reducing much of the vibrations produced.
< / p >
< / div >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org2348309" class = "outline-3" >
< h3 id = "org2348309" > < span class = "section-number-3" > 3.4< / span > Sum of all disturbances< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
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< p >
We can now compare the effect of all the disturbance sources on the position error (relative motion of the sample with respect to the granite).
< / p >
< p >
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The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure < a href = "#org08e075f" > 19< / a > .
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< / p >
< p >
We can see that the ground motion is quite small compare to the translation stage and spindle induced motions.
< / p >
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< div id = "org08e075f" class = "figure" >
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< p > < img src = "figs/dist_effect_relative_motion.png" alt = "dist_effect_relative_motion.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 19: < / span > Amplitude Spectral Density fo the motion error due to disturbances< / p >
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< / div >
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< p >
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The Cumulative Amplitude Spectrum is shown in Figure < a href = "#orgaa075dd" > 20< / a > .
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It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range.
< / p >
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< div id = "orgaa075dd" class = "figure" >
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< p > < img src = "figs/dist_effect_relative_motion_cas.png" alt = "dist_effect_relative_motion_cas.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 20: < / span > Cumulative Amplitude Spectrum of the motion error due to disturbances< / p >
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< / div >
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< p >
We can also estimate the required bandwidth by seeing that \(10\ nm [rms]\) motion is induced by the perturbations above 100Hz.
< / p >
< p >
This means that if the controller compensate all the motion errors below 100Hz (ideal case), 10nm [rms] of motion will still remain.
< / p >
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< p >
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From that, we can conclude that we will probably need a control bandwidth to around 100Hz.
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< / p >
< / div >
< / div >
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< div id = "outline-container-org25a7b3f" class = "outline-3" >
< h3 id = "org25a7b3f" > < span class = "section-number-3" > 3.5< / span > Better estimation of the disturbances< / h3 >
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< div class = "outline-text-3" id = "text-3-5" >
< p >
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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.
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< / p >
< p >
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This is not perfect as using only one geophone on the sample and one on the granite do not permit to separate the translations and the rotations.
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< / p >
< p >
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An alternative could be to position a reference object at the sample location and to use the X-ray to measure its motion.
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< / p >
< p >
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The detector requirement would be:
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< / p >
< ul class = "org-ul" >
< li > Sample frequency above \(400Hz\)< / li >
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< li > Resolution of \(\approx 100nm\) (to be discussed)< / li >
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< / ul >
< / div >
< / div >
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< div id = "outline-container-org2e7835b" class = "outline-3" >
< h3 id = "org2e7835b" > < span class = "section-number-3" > 3.6< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-3-6" >
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< div class = "important" >
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< p >
Main disturbance sources have been identified.
These disturbances will then be included in the multi-body model.
< / p >
< p >
Other disturbance sources were not estimated such as cable forces and acoustic disturbances.
If heavy/stiff cables are to be fixed to the sample, this should be quantified and included in the model.
< / p >
< p >
Having better estimation of the disturbances would allows to more precisely estimate the attainable performances.
This should however not change the conclusion of this study nor significantly change the nano-hexapod design.
< / p >
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< / div >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org12a2fe0" class = "outline-2" >
< h2 id = "org12a2fe0" > < span class = "section-number-2" > 4< / span > Multi Body Model< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< p >
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< a id = "org7c0f438" > < / a >
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< / p >
< p >
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As was shown during the modal analysis (Section < a href = "#org16c9690" > 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.
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< / p >
< p >
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Thus, a < b > multi-body model< / b > is perfect to represent such dynamics.
< / p >
< p >
To do so, we use the Matlab’ s < a href = "https://www.mathworks.com/products/simscape.html" > Simscape< / a > toolbox.
A small summary of the multi-body Simscape is available < a href = "https://tdehaeze.github.io/nass-simscape/simscape.html" > here< / a > and each of the modeled stage is described < a href = "https://tdehaeze.github.io/nass-simscape/simscape_subsystems.html" > here< / a > .
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< / p >
< / div >
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< div id = "outline-container-org0e7965a" class = "outline-3" >
< h3 id = "org0e7965a" > < span class = "section-number-3" > 4.1< / span > Multi-Body model< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
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The mass/inertia of each stage is automatically computed from the imported geometry and the material’ s density.
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< / p >
< p >
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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.
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< / p >
< p >
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The 3D representation of the simscape model is shown in Figure < a href = "#org507761c" > 21< / a > .
< / p >
< div id = "org507761c" class = "figure" >
< p > < img src = "figs/simscape_picture.png" alt = "simscape_picture.png" / >
< / p >
< p > < span class = "figure-number" > Figure 21: < / span > 3D representation of the simscape model< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgec68dd4" class = "outline-3" >
< h3 id = "orgec68dd4" > < span class = "section-number-3" > 4.2< / span > Validity of the model’ s dynamics< / h3 >
< div class = "outline-text-3" id = "text-4-2" >
< p >
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.
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< / p >
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< p >
The comparison of three of the Frequency Response Functions are shown in Figure < a href = "#org5afd70e" > 22< / a > .
< / p >
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< p >
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Most of the other measured FRFs and identified transfer functions from the multi-body model have the same level of matching.
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< / p >
< p >
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We believe that the model is representing the micro-station dynamics with sufficient precision for the current analysis.
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< / p >
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< div id = "org5afd70e" class = "figure" >
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< p > < img src = "figs/identification_comp_top_stages.png" alt = "identification_comp_top_stages.png" / >
< / p >
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< p > < span class = "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 >
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< / div >
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< p >
More detailed comparison between the model and the measured dynamics is performed < a href = "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 >
< ul class = "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 >
< ul class = "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 >
< / ul >
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< / div >
< / div >
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< div id = "outline-container-orgd0b55c5" class = "outline-3" >
< h3 id = "orgd0b55c5" > < span class = "section-number-3" > 4.3< / span > Wanted position of the sample and position error< / h3 >
< div class = "outline-text-3" id = "text-4-3" >
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< p >
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For the control of the nano-hexapod, we need to now the sample position error (the motion to be compensated) in the frame of the nano-hexapod.
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< / p >
< p >
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To do so, we need to perform several computations (summarized in Figure < a href = "#orgf18ac4e" > 23< / a > ):
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< / p >
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< ul class = "org-ul" >
< 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 < a href = "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.
Both computation are performed.< / li >
< / ul >
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< div id = "orgf18ac4e" class = "figure" >
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< p > < img src = "figs/control-schematic-nass.png" alt = "control-schematic-nass.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 23: < / span > Figure caption< / p >
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< / div >
< p >
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More details about these computations are accessible < a href = "https://tdehaeze.github.io/nass-simscape/positioning_error.html" > here< / a > .
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< / p >
< / div >
< / div >
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< div id = "outline-container-org82db829" class = "outline-3" >
< h3 id = "org82db829" > < span class = "section-number-3" > 4.4< / span > Simulation of Experiments< / h3 >
< div class = "outline-text-3" id = "text-4-4" >
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< p >
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Now that the dynamics of the model is tuned and the disturbances included in the model, we can perform simulation of experiments.
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< / p >
< p >
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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.
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< / p >
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< p >
An animation of the obtained motion is shown in Figure .
< / p >
< div id = "orgf4e5f2b" class = "figure" >
< p > < img src = "figs/open_loop_sim.gif" alt = "open_loop_sim.gif" / >
< / p >
< p > < span class = "figure-number" > Figure 24: < / span > Figure caption< / p >
< / div >
< div id = "org27781e7" class = "figure" >
< p > < img src = "figs/open_loop_sim_zoom.gif" alt = "open_loop_sim_zoom.gif" / >
< / p >
< p > < span class = "figure-number" > Figure 25: < / span > Figure caption< / p >
< / div >
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< p >
< a href = "https://tdehaeze.github.io/nass-simscape/experiments.html" > https://tdehaeze.github.io/nass-simscape/experiments.html< / a >
< / p >
< p >
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< a href = "#orge429fe9" > 26< / a >
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< / p >
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< div id = "orge429fe9" class = "figure" >
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< p > < img src = "figs/exp_scans_rz_dist.png" alt = "exp_scans_rz_dist.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 26: < / span > Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orga05f525" class = "outline-3" >
< h3 id = "orga05f525" > < span class = "section-number-3" > 4.5< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-4-5" >
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< div class = "important" >
< p >
Possible to study many effects.
Extraction of transfer function like \(G\) and \(G_d\).
< / p >
< p >
Simulation of experiments to validate performance.
< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org463efb0" class = "outline-2" >
< h2 id = "org463efb0" > < span class = "section-number-2" > 5< / span > Optimal Nano-Hexapod Design< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< p >
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< a id = "orgd2bab4c" > < / a >
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< / p >
< p >
As explain before, the nano-hexapod properties (mass, stiffness, architecture, … ) will influence:
< / p >
< ul class = "org-ul" >
< li > the plant dynamics \(G\)< / li >
< li > the effect of disturbances \(G_d\)< / li >
< / ul >
< p >
We which here to choose the nano-hexapod properties such that:
< / p >
< ul class = "org-ul" >
< li > has an easy< / li >
< li > minimize the< / li >
< li > minimize \(|G_d|\)< / li >
< / ul >
< / div >
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< div id = "outline-container-org39a6d2e" class = "outline-3" >
< h3 id = "org39a6d2e" > < span class = "section-number-3" > 5.1< / span > Optimal Stiffness to reduce the effect of disturbances< / h3 >
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< / div >
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< div id = "outline-container-orgbaf53ab" class = "outline-3" >
< h3 id = "orgbaf53ab" > < span class = "section-number-3" > 5.2< / span > Optimal Stiffness< / h3 >
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< div class = "outline-text-3" id = "text-5-2" >
< p >
The goal is to design a system that is < b > robust< / b > .
< / p >
< p >
Thus, we have to identify the sources of uncertainty and try to minimize them.
< / p >
< p >
Uncertainty in the system can be caused by:
< / p >
< ul class = "org-ul" >
< li > Effect of Support Compliance: < a href = "https://tdehaeze.github.io/nass-simscape/uncertainty_support.html" > https://tdehaeze.github.io/nass-simscape/uncertainty_support.html< / a > < / li >
< li > Effect of Payload Dynamics: < a href = "https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html" > https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html< / a > < / li >
< li > Effect of experimental condition (micro-station pose, spindle rotation): < a href = "https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html" > https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html< / a > < / li >
< / ul >
< p >
All these uncertainty will limit the maximum attainable bandwidth.
< / p >
< p >
Fortunately, the nano-hexapod stiffness have an influence on the dynamical uncertainty induced by the above effects.
< / p >
< p >
Determination of the optimal stiffness based on all the effects:
< / p >
< ul class = "org-ul" >
< li > < a href = "https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html" > https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html< / a > < / li >
< / ul >
< div class = "conclusion" >
< p >
< / p >
< / div >
< p >
The main performance limitation are payload variability
< / p >
< div class = "question" >
< p >
Main problem: heavy samples with small stiffness.
The first resonance frequency of the sample will limit the performance.
< / p >
< / div >
< p >
The nano-hexapod stiffness will also change the sensibility to disturbances.
< / p >
< p >
Effect of Nano-hexapod stiffness on the Sensibility to disturbances: < a href = "https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html" > https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html< / a >
< / p >
< div class = "conclusion" >
< p >
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd572056" class = "outline-3" >
< h3 id = "orgd572056" > < span class = "section-number-3" > 5.3< / span > Sensors to be included< / h3 >
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< div class = "outline-text-3" id = "text-5-3" >
< p >
Ways to damp:
< / p >
< ul class = "org-ul" >
< li > Force Sensor< / li >
< li > Relative Velocity Sensors< / li >
< li > Inertial Sensor< / li >
< / ul >
< p >
< a href = "https://tdehaeze.github.io/rotating-frame/index.html" > https://tdehaeze.github.io/rotating-frame/index.html< / a >
< / p >
< p >
Sensors to be included:
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgdaa0983" class = "outline-2" >
< h2 id = "orgdaa0983" > < span class = "section-number-2" > 6< / span > Robust Control Architecture< / h2 >
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< div class = "outline-text-2" id = "text-6" >
< p >
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< a id = "orgba39228" > < / a >
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< / p >
< p >
< a href = "https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html" > https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html< / a >
< / p >
< / div >
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< div id = "outline-container-orgd9f9771" class = "outline-3" >
< h3 id = "orgd9f9771" > < span class = "section-number-3" > 6.1< / span > Simulation of Tomography Experiments< / h3 >
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< div class = "outline-text-3" id = "text-6-1" >
< p >
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< a id = "org00bfc04" > < / a >
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< / p >
< ul class = "org-ul" >
< li > Make several animations
< ul class = "org-ul" >
< li class = "off" > < code > [  ]< / code > One of a tomography experiment where we see all the station rotating< / li >
< li class = "off" > < code > [  ]< / code > A zoom on at the nano-meter level to see how the wanted position is moving< / li >
< / ul > < / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org790f59d" class = "outline-3" >
< h3 id = "org790f59d" > < span class = "section-number-3" > 6.2< / span > Conclusion< / h3 >
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< / div >
< / div >
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< div id = "outline-container-orgd091d56" class = "outline-2" >
< h2 id = "orgd091d56" > < span class = "section-number-2" > 7< / span > Further notes< / h2 >
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< div class = "outline-text-2" id = "text-7" >
< p >
Soft granite
< / p >
< p >
Sensible to detector motion?
< / p >
< p >
Common metrology frame for the nano-focusing optics and the measurement of the sample position?
< / p >
< p >
Cable forces?
< / p >
< p >
Slip-Ring noise?
< / p >
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< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "date" > Date: 04-2020< / p >
< p class = "author" > Author: Thomas Dehaeze< / p >
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< p class = "date" > Created: 2020-04-27 lun. 16:22< / p >
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< / div >
< / body >
< / html >
