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@ -11,7 +11,6 @@
#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
#+STARTUP: overview
#+DATE: 05-2020
#+LATEX_CLASS: cleanreport
#+LATEX_CLASS_OPTIONS: [conf, hangsection, secbreak]
@ -42,9 +41,9 @@ To understand the design challenges of such system, a short introduction to Feed
The mathematical tools (Power Spectral Density, Noise Budgeting, ...) that will be used throughout this study are also introduced.
To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of:
- the micro-station dynamics (Section [[sec:micro_station_dynamics]])
- the frequency content of the sources of disturbances such as vibrations induced by the micro-station's stages and ground motion (Section [[sec:identification_disturbances]])
To develop both the nano-hexapod and the control architecture in an optimal way, precise estimation of the following is required:
- micro-station dynamics (Section [[sec:micro_station_dynamics]])
- frequency content of the sources of disturbances such as vibrations induced by the micro-station's stages and ground motion (Section [[sec:identification_disturbances]])
A model of the micro-station is then developed and tuned using the previous estimations (Section [[sec:multi_body_model]]).
@ -139,13 +138,13 @@ Without the use of feedback (i.e. without the nano-hexapod), the disturbances wi
\end{equation}
which is, in the case of the NASS out of the specifications (micro-meter range compare to the required $\approx 10nm$).
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.
In the next section, is explained how the use of the feedback lowers the effect of the disturbances $d$ on the sample motion error.
*** How does the feedback loop is modifying the system behavior?
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 [[fig:classical_feedback_small]]), we obtain:
From the feedback diagram in Figure [[fig:classical_feedback_small]], the position error signal $\epsilon = r - y$ can be written as a function of the reference signal $r$, the disturbances $d$ and the measurement noise $n$:
\[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
We usually note:
It is common to define the following two transfer functions:
\begin{align}
S &= \frac{1}{1 + GK} \\
T &= \frac{GK}{1 + GK}
@ -158,35 +157,35 @@ And the position error can be rewritten as:
\end{equation}
From Eq. eqref:eq:closed_loop_error representing the closed-loop system behavior, we can see that:
From Eq. eqref:eq:closed_loop_error representing the closed-loop system behavior, it is seen that:
- the effect of disturbances $d$ on $\epsilon$ is multiplied by a factor $S$ compared to the open-loop case
- the measurement noise $n$ is injected and multiplied by a factor $T$
Ideally, we would like to design the controller $K$ such that:
Ideally, it is desired to design the controller $K$ such that:
- $|S|$ is small to *reduce the effect of disturbances*
- $|T|$ is small to *limit the injection of sensor noise*
*** Trade off: Disturbance Reduction / Noise Injection
We have from the definition of $S$ and $T$ that:
From the definition of $S$ and $T$:
\begin{equation}
S + T = \frac{1}{1 + GK} + \frac{GK}{1 + GK} = 1
\end{equation}
meaning that we cannot have $|S|$ and $|T|$ small at the same time.
meaning that it is not possible to have $|S|$ and $|T|$ small at the same time.
There is therefore a *trade-off between the disturbance rejection and the measurement noise filtering*.
Typical shapes of $|S|$ and $|T|$ as a function of frequency are shown in Figure [[fig:h-infinity-2-blocs-constrains]].
We can observe that $|S|$ and $|T|$ exhibit different behaviors depending on the frequency band:
It is shown that $|S|$ and $|T|$ exhibit different behaviors depending on the frequency band:
*At low frequency* (inside the control bandwidth):
- $|S|$ can be made small and thus the effect of disturbances is reduced
- $|T| \approx 1$ and all the sensor noise is transmitted
*At high frequency* (outside the control bandwidth):
- $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances
- $|T|$ is small and thus the sensor noise is filtered
*Near the crossover frequency* (between the two frequency bands):
- The effect of disturbances is increased
- *At low frequency* (inside the control bandwidth):
- $|S|$ can be made small and thus the effect of disturbances is reduced
- $|T| \approx 1$ and all the sensor noise is transmitted
- *At high frequency* (outside the control bandwidth):
- $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances
- $|T|$ is small and thus the sensor noise is filtered
- *Near the crossover frequency* (between the two frequency bands):
- The effect of disturbances is increased
#+begin_src latex :file h-infinity-2-blocs-constrains.pdf
\begin{tikzpicture}
@ -342,7 +341,7 @@ The Power Spectral Density of the output signal $y$ can be computed using:
Let's consider a signal $y$ that is the sum of two *uncorrelated* signals $u$ and $v$ (Figure [[fig:psd_sum]]).
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):
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):
\[ S_{yy} = S_{uu} + S_{vv} \]
#+begin_src latex :file psd_sum.pdf
@ -367,24 +366,24 @@ We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v
Let's consider the Feedback architecture in Figure [[fig:classical_feedback_small]] where the position error $\epsilon$ is equal to:
\[ \epsilon = S r + T n - G_d S d \]
If we suppose that the signals $r$, $n$ and $d$ are *uncorrelated* (which is a good approximation in our case), the PSD of $\epsilon$ is:
Supposing that the signals $r$, $n$ and $d$ are *uncorrelated* (which is a good approximation in our case), the PSD of $\epsilon$ is equal to:
\[ S_{\epsilon \epsilon}(\omega) = |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \]
And we can compute the RMS value of the residual motion using:
And the RMS value of the residual motion can be computed using:
\begin{align*}
\epsilon_\text{rms} &= \sqrt{ \int_0^\infty S_{\epsilon\epsilon}(\omega) d\omega} \\
&= \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 }
\end{align*}
To estimate the PSD of the position error $\epsilon$ and thus the RMS residual motion (in closed-loop), we need to determine:
To estimate the PSD of the position error $\epsilon$ and thus the RMS residual motion (in closed-loop), the following needs to be determined:
- The Power Spectral Densities of the signals affecting the system:
- The disturbances $S_{dd}$: this will be done in Section [[sec:identification_disturbances]]
- The sensor noise $S_{nn}$: this can be estimated from the sensor data-sheet
- The wanted sample's motion $S_{rr}$: this is a deterministic signal that we choose.
For a simple tomography experiment, we can consider that it is equal to $0$ as we only want to compensate all the sample's vibrations
- The wanted sample's motion $S_{rr}$: this is a deterministic signal that is chosen by the "user".
For a simple tomography experiment, the wanted sample's motion can consider to be equal to $0$ (the point of interest should stay on the focus X-ray)
- The dynamics of the complete system comprising the micro-station and the nano-hexapod: $G$, $G_d$.
To do so, we need to identify the dynamics of the micro-station (Section [[sec:micro_station_dynamics]]), include this dynamics in a model (Section [[sec:multi_body_model]]) and add a model of the nano-hexapod to the model (Section [[sec:nano_hexapod_design]])
To do so, the dynamics of the micro-station (Section [[sec:micro_station_dynamics]]) should be identified and then included in a model (Section [[sec:multi_body_model]]). Then a model of the nano-hexapod is merged with the micro-station model (Section [[sec:nano_hexapod_design]])
- The controller $K$ that will be designed in Section [[sec:robust_control_architecture]]
* Identification of the Micro-Station Dynamics
@ -418,8 +417,8 @@ Instead, the model will be tuned using both the modal model and the response mod
To measure the dynamics of such complicated system, it as been chosen to perform a modal analysis.
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 *solid body*.
Thus, to fully describe the dynamics of the station, we (only) need to measure 6 degrees of freedom for each positioning stage (that is 36 degrees of freedom for the 6 considered solid bodies).
To limit the number of degrees of freedom to be measured, it is supposed that in the frequency range of interest (DC-300Hz), each of the positioning stage is behaving as a *solid body*.
Thus, to fully describe the dynamics of the station, only 6 degrees of freedom for each positioning stage (that is 36 degrees of freedom for the 6 considered solid bodies) should be measured.
In order to perform the modal analysis, the following devices were used:
@ -467,9 +466,10 @@ Example of the obtained micro-station's mode shapes are shown in Figures [[fig:m
[[file:figs/mode6.gif]]
We then reduce the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF).
The number of degrees of freedom is then reduced from 69 (23 accelerometers with each 3DOF) down to 36 (6 solid bodies with 6 DOF).
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.
From the reduced transfer function matrix, the responses at the 69 measured degrees of freedom are re-synthesized.
The original measurements and the synthesized one are compared and found to match.
#+begin_important
This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest.
@ -545,7 +545,7 @@ Complete reports on these measurements are accessible [[https://tdehaeze.github.
<<sec:stage_vibration_motion>>
*** Introduction :ignore:
We consider here the vibrations induced by *scans of the translation stage* and *rotation of the spindle*.
In this section, the vibrations induced by *scans of the translation stage* and *rotation of the spindle* and studied.
Details reports are accessible [[https://tdehaeze.github.io/meas-analysis/disturbance-ty/index.html][here]] for the translation stage and [[https://tdehaeze.github.io/meas-analysis/disturbance-sr-rz/index.html][here]] for the spindle/slip-ring.
@ -567,7 +567,7 @@ A geophone is fixed at the location of the sample and the motion is measured:
The obtained Power Spectral Densities of the sample's absolute velocity are shown in Figure [[fig:sr_sp_psd_sample_compare]].
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.
It can be seen that when using the Slip-ring motor to rotate the sample, only a little increase of the motion is observed above 100Hz.
However, when rotating with the Spindle (normal functioning mode):
- a very sharp peak at 23Hz is observed.
@ -611,7 +611,7 @@ The ASD contains any peaks starting from 1Hz showing the large spectral content
A smoother motion for the translation stage (such as a sinus motion, of a filtered triangular signal) could help reducing much of the vibrations.
The goal is to inject no motion outside the control bandwidth.
We should also note that away from the rapid change of velocity, the sample's vibrations are much reduced.
It should be noted that away from the rapid change of velocity, the sample's vibrations are much reduced.
Thus, if the detector is only used in between the triangular peaks, the vibrations are expected to be much lower than those estimated.
#+end_important
@ -622,11 +622,11 @@ The ASD contains any peaks starting from 1Hz showing the large spectral content
** Open Loop noise budgeting
<<sec:open_loop_noise_budget>>
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).
The effect of all the disturbance sources on the position error (relative motion of the sample with respect to the granite) are now compared.
The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure [[fig:dist_effect_relative_motion]].
We can see that the ground motion is quite small compare to the translation stage and spindle induced motions.
It can be seen that the ground motion is quite small compare to the translation stage and spindle induced motions.
#+name: fig:dist_effect_relative_motion
#+caption: Amplitude Spectral Density fo the motion error due to disturbances
@ -709,7 +709,7 @@ The comparison of three of the Frequency Response Functions are shown in Figure
Most of the other measured FRFs and identified transfer functions from the multi-body model have the same level of matching.
We believe that the model is representing the micro-station dynamics with sufficient precision for the current analysis.
We believe that the model is representing the micro-station dynamics sufficient well for the current analysis.
#+name: fig:identification_comp_top_stages
#+caption: 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.
@ -724,7 +724,7 @@ Now that the multi-body model dynamics as been tuned, the following elements are
- Disturbances such as ground motion and stage's vibrations
Then, using the model, we can
Then, using the model, it is possible to:
- perform simulation of experiments in presence of disturbances
- measure the motion of the solid-bodies
- identify the dynamics from inputs (forces, imposed displacement) to outputs (measured motion, force sensor, etc.) which will be useful for the nano-hexapod design and the control synthesis
@ -814,7 +814,7 @@ As explain before, the nano-hexapod properties (mass, stiffness, legs' orientati
- the effect of disturbances
- the plant dynamics
Thus, we here wish to find the optimal nano-hexapod properties such that:
The objective is here to find the optimal nano-hexapod properties such that:
- the effect of disturbances is minimized (Section [[sec:optimal_stiff_dist]])
- the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section [[sec:optimal_stiff_plant]])
- the plant has nice dynamical properties for control (Section [[sec:nano_hexapod_architecture]])
@ -912,9 +912,9 @@ What is more important than comparing the sensitivity to disturbances, is to com
This is the *dynamic noise budgeting*.
From the Power Spectral Density of all the sources of disturbances identified in Section [[sec:identification_disturbances]], we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure [[fig:opt_stiff_psd_dz_tot]]).
From the Power Spectral Density of all the sources of disturbances identified in Section [[sec:identification_disturbances]] is computed the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure [[fig:opt_stiff_psd_dz_tot]]).
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.
It can be seen 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.
#+name: fig:opt_stiff_psd_dz_tot
#+caption: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses
@ -926,7 +926,7 @@ We can see that the most important change is in the frequency range 30Hz to 300H
:END:
#+begin_important
If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure [[fig:opt_stiff_cas_dz_tot]], we can observe that a soft hexapod ($k < 10^5 - 10^6\,[N/m]$) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will be required to obtain the wanted performance.
It can be observe on the Cumulative amplitude spectrum of the vertical error motion in Figure [[fig:opt_stiff_cas_dz_tot]], that a soft hexapod ($k < 10^5 - 10^6\,[N/m]$) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will be required to obtain the wanted performance.
#+end_important
#+name: fig:opt_stiff_cas_dz_tot
@ -938,7 +938,7 @@ If we look at the Cumulative amplitude spectrum of the vertical error motion in
*** Introduction :ignore:
One of the most important design goal is to obtain a system that is *robust* 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.
Therefore, all changes that might occur in the system must be identified and the nano-hexapod stiffness that minimizes the uncertainties to these changes should be determined.
The uncertainty in the system can be caused by:
- A change in the *Support's compliance* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_support.html][here]]): if the micro-station dynamics is changing due to the change of mechanical parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change
@ -949,8 +949,8 @@ Because of the trade-off between robustness and performance, *the bigger the pla
In the next sections, the effect the considered changes on the *plant dynamics* is quantified and conclusions are drawn 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.
In the following study, when it is referred to /plant dynamics/, it means the dynamics from forces applied by the nano-hexapod to the measured sample's displacement by the metrology.
Only the plant dynamics will be compared 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.
*** Effect of Payload
@ -981,7 +981,7 @@ As the maximum payload's mass is $50\,kg$, this may however not be practical, an
In Figure [[fig:opt_stiffness_payload_freq_fz_dz]] is shown the effect of a change of payload dynamics.
The mass of the payload is fixed and its resonance frequency is changing from 50Hz to 500Hz.
We can see (more easily for the soft nano-hexapod), that resonance of the payload produces an anti-resonance for the considered dynamics.
It can be seen (more easily for the soft nano-hexapod), that resonance of the payload produces an anti-resonance for the considered dynamics.
#+name: fig:opt_stiffness_payload_freq_fz_dz
#+caption: 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
@ -996,7 +996,7 @@ For nano-hexapod stiffnesses below $10^6\,[N/m]$:
For nano-hexapod stiffnesses above $10^7\,[N/m]$:
- the dynamics is unchanged until the first resonance which is around 25Hz-35Hz
- 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) and it would be difficult to have a controller with high bandwidth which is robust to such change of dynamics
- above that frequency, the change of dynamics is quite chaotic (which this is due to the micro-station dynamics as shown in the next section) and it would be difficult to have a controller with high bandwidth which is robust to such change of dynamics
#+name: fig:opt_stiffness_payload_impedance_all_fz_dz
#+caption: Dynamics from $\mathcal{F}_z$ to $\mathcal{X}_z$ for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod
@ -1204,7 +1204,7 @@ with:
- $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$: vector of actuator forces applied in each strut
- $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$: external force/torque action on the mobile platform
And thus *the Jacobian matrix can be used to compute the forces that should be applied on each leg from forces and torques that we want to apply on the nano-hexapod's top platform*.
And thus *the Jacobian matrix can be used to compute the forces that should be applied by the Stewart platform's legs from the wanted total forces/torques that is to be applied to the sample*.
Linear transformations in Eq. eqref:eq:jacobian_L and eqref:eq:jacobian_F will be widely in the developed control architectures in Section [[sec:robust_control_architecture]].
@ -1224,7 +1224,7 @@ An example of the mobility considering only pure translations is shown in Figure
From a wanted mobility and a specific geometry, the required actuator stroke can be estimated.
Suppose we want the following mobility:
Suppose the following specifications on the mobility:
- x, y and z translations up to $50\,\mu m$
- x and y rotation up to $30\,\mu rad$
- no z rotation
@ -1246,7 +1246,7 @@ The stiffness of the actuator $k_i$ links the applied (constant) actuator force
\begin{equation*}
\tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6
\end{equation*}
If we combine these 6 relations:
Combining these 6 relations:
\begin{equation*}
\bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right]
\end{equation*}
@ -1397,13 +1397,17 @@ For instance, the flexible joint used for the ID16 nano-hexapod have the followi
In Section [[sec:optimal_stiff_dist]], 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.
In Section [[sec:optimal_stiff_plant]], 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.
In Section [[sec:optimal_stiff_plant]], it has been 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 by the control.
Thus, a leg's stiffness of $10^5\,[N/m]$ will be used in Section [[sec:robust_control_architecture]] 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 [[https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html][here]].
Finally, in section [[sec:nano_hexapod_architecture]] some insights on the wanted nano-hexapod geometry are given.
#+end_important
@ -1855,7 +1859,7 @@ Several observations can be made:
- The sample's vibrations are reduced within the control bandwidth as was expected
- The obtained performances for all the three considered masses are very similar.
This is an indication of the good system's robustness
- From the Cumulative Amplitude Spectrum (Figure [[fig:opt_stiff_hac_dvf_L_cas_disp_error]]), we see that Z motion is reduced down to $\approx 30\,nm\,[rms]$ and the Y motion down to $\approx 25\,nm\,[rms]$
- From the Cumulative Amplitude Spectrum (Figure [[fig:opt_stiff_hac_dvf_L_cas_disp_error]]), it can be seen that Z motion is reduced down to $\approx 30\,nm\,[rms]$ and the Y motion down to $\approx 25\,nm\,[rms]$
- An increase in the rotational vibrations is observed.
This is due to the fact that:
1. no perturbations inducing rotations were included in the simulation and thus the rotational vibrations are very small
@ -1880,7 +1884,7 @@ Several observations can be made:
The time domain sample's vibrations are shown in Figure [[fig:opt_stiff_hac_dvf_L_pos_error]].
The use of the nano-hexapod combined with the HAC-LAC architecture is shown to considerably reduce the sample's vibrations.
An animation of the experiment is shown in Figure [[fig:closed_loop_sim_zoom]] and we can see that the actual sample's position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure [[fig:open_loop_sim_zoom]] (same scale was used for both animations).
An animation of the experiment is shown in Figure [[fig:closed_loop_sim_zoom]] and it can be seen that the actual sample's position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure [[fig:open_loop_sim_zoom]] (same scale was used for both animations).
#+name: fig:opt_stiff_hac_dvf_L_pos_error
#+caption: Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture
@ -2036,7 +2040,7 @@ Table summarizing the nano-hexapod wanted characteristics:
- Static positioning errors of the stages
- Maximum tracking errors of the stages (mainly translation stage and tilt stage).
This is probably more difficult to obtain.
However, by limiting the acceleration of these stages, we may limit the dynamic tracking errors to acceptable levels
However, by limiting the acceleration of these stages, we may limit the dynamic tracking errors to acceptable levels
- Sensors to be included
#+name: fig:nano_hexapod_size