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Thomas Dehaeze 2025-02-01 09:45:01 +01:00
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@ -181,6 +181,10 @@ CLOSED: [2025-01-30 Thu 11:16]
- RMS errors (/ by 6.6) gives 30nmRMS in Dy and 15nmRMS in Dz.
- Ry error <1.7urad, 250nrad RMS
** TODO [#A] Maybe should only put experimental results in last section
Some Tomography experiments are presented in ref:sec:test_id31_iff_hac
Maybe it should only be simulations, and put everything in the last "experimental" section.
** TODO [#A] Ask Veijo to give me a short summary (5 lines) for each experiment type with references
- Tomography
@ -423,18 +427,17 @@ CLOSED: [2024-11-12 Tue 16:03]
* Introduction :ignore:
Now that the nano-hexapod is mounted and that a good multi-body model of the nano-hexapod
The system is validated on the ID31 beamline.
Now that the nano-hexapod is mounted and that the the multi-body model of the nano-hexapod could be validated based on dynamics measurements, the complete NASS is mounted as shown in Figure ref:fig:test_id31_micro_station_nano_hexapod and the performances are evaluated on the ID31 beamline.
At the beginning of the project, it was planned to develop a long stroke 5-DoF metrology system to measure the pose of the sample with respect to the granite.
The development of such system was complex, and was not completed at the time of the experimental tests on ID31.
To still validate the developed nano active platform and the associated instrumentation and control architecture, a 5-DoF short stroke metrology system was developed (Section ref:sec:test_id31_metrology).
To still be able to validate the developed nano active platform and the associated instrumentation and control architecture, a 5-DoF short stroke metrology system is developed and presented in Section ref:sec:test_id31_metrology.
The identify dynamics of the nano-hexapod fixed on top of the micro-station was identified for different experimental conditions (payload masses, rotational velocities) and compared with the model (Section ref:sec:test_id31_open_loop_plant).
The identify dynamics of the nano-hexapod fixed on top of the micro-station is identified for different experimental conditions (payload masses, rotational velocities) and compared with the multi-body model in Section ref:sec:test_id31_open_loop_plant.
Decentralized Integral Force Feedback is then applied to actively damp the plant in a robust way (Section ref:sec:test_id31_iff).
In order to apply the developed HAC-LAC architecture, decentralized Integral Force Feedback is first applied to actively damp the plant in a robust way (Section ref:sec:test_id31_iff), and the high authority controller is then implemented (Section ref:sec:test_id31_hac).
High authority control is then applied (Section ref:sec:test_id31_iff_hac).
Finally, the positioning accuracy of the NASS is evaluated by performing scans corresponding to several scientific experiments (Section ref:sec:test_id31_experiments)
#+name: fig:test_id31_micro_station_nano_hexapod
#+caption: Picture of the micro-station without the nano-hexapod (\subref{fig:test_id31_micro_station_cables}) and with the nano-hexapod (\subref{fig:test_id31_fixed_nano_hexapod})
@ -466,7 +469,7 @@ As the long-stroke ($\approx 1 \,cm^3$) metrology system was not developed yet,
A first considered option was to use the "Spindle error analyzer" shown in Figure ref:fig:test_id31_lion.
This system comprises 5 capacitive sensors which are facing two reference spheres.
As the gap between the capacitive sensors and the spheres is very small[fn:1], the risk of damaging the spheres and the capacitive sensors is high.
But as the gap between the capacitive sensors and the spheres is very small[fn:1], the risk of damaging the spheres and the capacitive sensors is too high.
#+name: fig:test_id31_short_stroke_metrology
#+caption: Short stroke metrology system used to measure the sample position with respect to the granite in 5DoF. The system is based on a "Spindle error analyzer" (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). Interferometer heads are shown in (\subref{fig:test_id31_interf_head})
@ -494,10 +497,11 @@ As the gap between the capacitive sensors and the spheres is very small[fn:1], t
Instead of using capacitive sensors, 5 fibered interferometers were used in a similar way (Figure ref:fig:test_id31_interf).
At the end of each fiber, a sensor head[fn:2] (Figure ref:fig:test_id31_interf_head) is used, which consists of a lens precisely positioned with respect to the fiber's end.
The lens is focusing the light on the surface of the sphere, such that it comes back to the fiber and produces an interference.
This way, the gap between the sensor and the reference sphere is much larger (here around $40\,mm$), removing the risk of collision.
The lens is focusing the light on the surface of the sphere, such that the reflected light comes back into the fiber and produces an interference.
This way, the gap between the head and the reference sphere is much larger (here around $40\,mm$), removing the risk of collision.
Nevertheless, the metrology system still has limited measurement range, as when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and for some perpendicular displacement, the interference is too small to be detected.
Nevertheless, the metrology system still has limited measurement range due to limited angular acceptance of the fibered interferometers.
Indeed, when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and above some perpendicular displacement, the reflected light does not comes back into the fiber, and no interference is produced.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -526,7 +530,7 @@ Nevertheless, the metrology system still has limited measurement range, as when
The developed short-stroke metrology system is schematically shown in Figure ref:fig:test_id31_metrology_kinematics.
The point of interest is indicated by the blue frame $\{B\}$, which is located $H = 150\,mm$ above the nano-hexapod's top platform.
The spheres have a diameter $d = 25.4\,mm$, and indicated dimensions are $l_1 = 60\,mm$ and $l_2 = 16.2\,mm$.
In order to compute the pose of the $\{B\}$ frame with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured small displacements $[d_1,\ d_2,\ d_3,\ d_4,\ d_5]$ by the interferometers are first written as a function of the small linear and angular motion of the $\{B\}$ frame $[D_x,\ D_y,\ D_z,\ R_x,\ R_y]$ eqref:eq:test_id31_metrology_kinematics.
In order to compute the pose of the $\{B\}$ frame with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured (small) displacements $[d_1,\ d_2,\ d_3,\ d_4,\ d_5]$ by the interferometers are first written as a function of the (small) linear and angular motion of the $\{B\}$ frame $[D_x,\ D_y,\ D_z,\ R_x,\ R_y]$ eqref:eq:test_id31_metrology_kinematics.
\begin{equation}\label{eq:test_id31_metrology_kinematics}
d_1 = D_y - l_2 R_x, \quad d_2 = D_y + l_1 R_x, \quad d_3 = -D_x - l_2 R_y, \quad d_4 = -D_x + l_1 R_y, \quad d_5 = -D_z
@ -548,18 +552,18 @@ d_1 = D_y - l_2 R_x, \quad d_2 = D_y + l_1 R_x, \quad d_3 = -D_x - l_2 R_y, \qua
[[file:figs/test_id31_align_top_sphere_comparators.jpg]]
#+end_minipage
The five equations eqref:eq:test_id31_metrology_kinematics can be written in a matrix form, and then inverted to have the pose of $\{B\}$ frame as a linear combination of the measured five distances by the interferometers eqref:eq:test_id31_metrology_kinematics_inverse.
The five equations eqref:eq:test_id31_metrology_kinematics can be written in a matrix form, and then inverted to have the pose of the $\{B\}$ frame as a linear combination of the measured five distances by the interferometers eqref:eq:test_id31_metrology_kinematics_inverse.
\begin{equation}\label{eq:test_id31_metrology_kinematics_inverse}
\begin{bmatrix}
D_x \\ D_y \\ D_z \\ R_x \\ R_y
\end{bmatrix} = \begin{bmatrix}
\end{bmatrix} = {\underbrace{\begin{bmatrix}
0 & 1 & 0 & -l_2 & 0 \\
0 & 1 & 0 & l_1 & 0 \\
-1 & 0 & 0 & 0 & -l_2 \\
-1 & 0 & 0 & 0 & l_1 \\
0 & 0 & -1 & 0 & 0
\end{bmatrix}^{-1} \cdot \begin{bmatrix}
\end{bmatrix}}_{\bm{J_d}}}^{-1} \cdot \begin{bmatrix}
d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
\end{bmatrix}
\end{equation}
@ -585,43 +589,49 @@ The two reference spheres are aligned with the rotation axis of the spindle.
To do so, two measuring probes are used as shown in Figure ref:fig:align_top_sphere_comparators.
To not damage the sensitive sphere surface, the probes are instead positioned on the cylinder on which the sphere is mounted.
First, the probes are fixed to the bottom (fixed) cylinder to align its axis with the spindle axis.
First, the probes are fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis.
Then, the probes are fixed to the top (adjustable) cylinder, and the same alignment is performed.
With this setup, the precision of the alignment of both sphere better with the spindle axis is expected to limited to $\approx 10\,\mu m$.
This is probably limited due to the poor coaxiality between the cylinders and the spheres.
However, the alignment precision should be enough to stay in the acceptance of the interferometers.
With this setup, the alignment accuracy of both spheres with the spindle axis is expected to around $10\,\mu m$.
The accuracy is probably limited due to the poor coaxiality between the cylinders and the spheres.
However, this first alignment should permit to position the two sphere within the acceptance of the interferometers.
** Tip-Tilt adjustment of the interferometers
<<ssec:test_id31_metrology_alignment>>
The short stroke metrology system is placed on top of the main granite using a gantry made of granite blocs to have good vibration and thermal stability (Figure ref:fig:short_stroke_metrology_overview).
The short-stroke metrology system is placed on top of the main granite using a gantry made of granite blocs (Figure ref:fig:short_stroke_metrology_overview).
Granite is used to have good vibration and thermal stability.
#+name: fig:short_stroke_metrology_overview
#+caption: Granite gantry used to fix the short-stroke metrology system
#+attr_latex: :width 0.8\linewidth
[[file:figs/test_id31_short_stroke_metrology_overview.jpg]]
The interferometers need to be aligned with respect to the two reference spheres to approach as much as possible the ideal case shown in Figure ref:fig:test_id31_metrology_kinematics.
The vertical position of the spheres is adjusted using the micro-hexapod to match the height of the interferometers.
The interferometer beams need to be position with respect to the two reference spheres as close as possible to the ideal case shown in Figure ref:fig:test_id31_metrology_kinematics.
This means that their positions and angles needs to be well adjusted with respect to the two spheres.
First, the vertical position of the spheres is adjusted using the micro-hexapod to match the height of the interferometers.
Then, the horizontal position of the gantry is adjusted such that the coupling efficiency (i.e. the intensity of the light reflected back in the fiber) of the top interferometer is maximized.
This is equivalent as to optimize the perpendicularity between the interferometer beam and the sphere surface (i.e. the concentricity between the beam and the sphere center).
This is equivalent as to optimize the perpendicularity between the interferometer beam and the sphere surface (i.e. the concentricity between the top beam and the sphere center).
The lateral sensor heads (i.e. all except the top one), which are each fixed to a custom tip-tilt adjustment mechanism, are individually oriented such that the coupling efficient is maximized.
The lateral sensor heads (i.e. all except the top one) are each fixed to a custom tip-tilt adjustment mechanism.
This allow to individually orient them such that they all point to the spheres' center (i.e. perpendicular to the sphere surface).
This is done by maximizing the coupling efficiency of each interferometer.
After the alignment procedure, the top interferometer should coincide with with spindle axis, and the lateral interferometers should all be in the horizontal plane and intersect the spheres' center.
** Fine Alignment of reference spheres using interferometers
<<ssec:test_id31_metrology_sphere_fine_alignment>>
Thanks to the good alignment of the two reference spheres with the spindle axis and to the fine adjustment of the interferometers orientations, the interferometer measurement is made possible during complete spindle rotation.
This metrology and therefore be used to better align the axis defined by the two spheres' center with the spindle axis.
Thanks to the first alignment of the two reference spheres with the spindle axis (Section ref:ssec:test_id31_metrology_sphere_rought_alignment) and to the fine adjustment of the interferometers orientations (Section ref:ssec:test_id31_metrology_alignment), the spindle can perform complete rotations while still having interference for all five interferometers.
This metrology can therefore be used to better align the axis defined by the two spheres' center with the spindle axis.
The alignment process is made by few iterations.
First, the spindle is scanned and the alignment errors are recorded.
From the errors, the motion of the micro-hexapod to better align the spheres is determined and the micro-hexapod is moved.
From the errors, the motion of the micro-hexapod to better align the spheres with the spindle axis is computed and the micro-hexapod is positioned accordingly.
Then, the spindle is scanned again, and the new alignment errors are recorded.
This iterative process is first perform for angular errors (Figure ref:fig:test_id31_metrology_align_rx_ry) and then for lateral errors (Figure ref:fig:test_id31_metrology_align_dx_dy).
Remaining error after alignment is in the order of $\pm5\,\mu\text{rad}$ for angular errors, $\pm 1\,\mu m$ laterally and less than $0.1\,\mu m$ vertically.
This iterative process is first performed for angular errors (Figure ref:fig:test_id31_metrology_align_rx_ry) and then for lateral errors (Figure ref:fig:test_id31_metrology_align_dx_dy).
The remaining errors after alignment is in the order of $\pm5\,\mu\text{rad}$ in $R_x$ and $R_y$ orientations, $\pm 1\,\mu m$ in $D_x$ and $D_y$ directions and less than $0.1\,\mu m$ vertically.
#+begin_src matlab
%% Angular alignment
@ -725,9 +735,9 @@ exportFig('figs/test_id31_metrology_align_dx_dy.pdf', 'width', 'half', 'height',
Because the interferometers are pointing to spheres and not flat surfaces, the lateral acceptance is limited.
In order to estimate the metrology acceptance, the micro-hexapod is used to perform three accurate scans of $\pm 1\,mm$, respectively along the the $x$, $y$ and $z$ axes.
During these scans, the 5 interferometers are recorded, and the ranges in which each interferometer has enough coupling efficiency for measurement are estimated.
During these scans, the 5 interferometers are recorded individually, and the ranges in which each interferometer has enough coupling efficiency to be able to measure the displacement are estimated.
Results are summarized in Table ref:tab:test_id31_metrology_acceptance.
The obtained lateral acceptance for pure displacements in any direction is estimated to be around $+/-0.5\,mm$, which is enough for the current application as it is well above the micro-station errors to be actively corrected.
The obtained lateral acceptance for pure displacements in any direction is estimated to be around $+/-0.5\,mm$, which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS.
#+begin_src matlab
%% Estimated acceptance of the metrology
@ -803,24 +813,24 @@ end
** Estimated measurement errors
<<ssec:test_id31_metrology_errors>>
When using the NASS, the accuracy of the sample's positioning is linked to the accuracy of the external metrology.
However, to validate the nano-hexapod with the associated instrumentation and control architecture, the accuracy of the metrology is not an issue.
When using the NASS, the accuracy of the sample's positioning is determined by the accuracy of the external metrology.
However, the validation of the nano-hexapod, the associated instrumentation and the control architecture is independent of the accuracy of the metrology system.
Only the bandwidth and noise characteristics of the external metrology are important.
Yet, some elements effecting the accuracy of the metrology are discussed here.
First, the "metrology kinematics" (discussed in Section ref:ssec:test_id31_metrology_kinematics) is only approximate (i.e. valid for very small displacements).
This can be seen when performing lateral $[D_x,\,D_y]$ scans using the micro-hexapod while recording the vertical interferometer (Figure ref:fig:test_id31_xy_map_sphere).
This can be easily seen when performing lateral $[D_x,\,D_y]$ scans using the micro-hexapod while recording the vertical interferometer (Figure ref:fig:test_id31_xy_map_sphere).
As the interferometer is pointing to a sphere and not to a plane, lateral motion of the sphere is seen as a vertical motion by the top interferometer.
Then, the reference spheres have some deviations with respect to an ideal sphere.
They are meant to be used with capacitive sensors which are integrating the shape errors over large surfaces.
When using interferometers, the size of the "light spot" on the sphere surface is a circle with a diameter $\approx 50\,\mu m$, therefore the system is more sensitive to shape errors with small features.
Then, the reference spheres have some deviations with respect to an ideal sphere [fn:6].
They are initially meant to be used with capacitive sensors which are integrating the shape errors over large surfaces.
When using interferometers, the size of the "light spot" on the sphere surface is a circle with a diameter approximately equal to $50\,\mu m$, and therefore the measurement is more sensitive to shape errors with small features.
As the interferometer light is travelling in air, the measured distance is sensitive to any variation in the refractive index of the air.
As the light from the interferometer is travelling through air (as opposed to being in vacuum), the measured distance is sensitive to any variation in the refractive index of the air.
Therefore, any variation of air temperature, pressure or humidity will induce measurement errors.
For a measurement length of $40\,mm$, a temperature variation of $0.1\,{}^oC$ induces an errors in the distance measurement of $\approx 4\,nm$.
For instance, for a measurement length of $40\,mm$, a temperature variation of $0.1\,{}^oC$ (which is typical for the ID31 experimental hutch) induces an errors in the distance measurement of $\approx 4\,nm$.
Finally, even in vacuum and in the absence of target motion, the interferometers are affected by noise [[cite:&watchi18_review_compac_inter]].
Interferometers are also affected by noise [[cite:&watchi18_review_compac_inter]].
The effect of the noise on the translation and rotation measurements is estimated in Figure ref:fig:test_id31_interf_noise.
#+begin_src matlab
@ -859,7 +869,6 @@ leg.ItemTokenSize(1) = 15;
exportFig('figs/test_id31_interf_noise.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+begin_src matlab
%% X-Y scan with the micro-hexapod, and record of the vertical interferometer
data = h5scan(data_dir, 'metrology_acceptance', 'after_int_align_meshXY', 1);
@ -917,6 +926,8 @@ exportFig('figs/test_id31_xy_map_sphere.pdf', 'width', 'half', 'height', 'normal
<<sec:test_id31_open_loop_plant>>
** Introduction :ignore:
- Force sensors: $\bm{V}_s = [V_{s1},\ V_{s2},\ V_{s3},\ V_{s4},\ V_{s5},\ V_{s6}]$
- Encoders: $\bm{d}_e = [d_{e1},\ d_{e2},\ d_{e3},\ d_{e4},\ d_{e5},\ d_{e6}]$
- Interferometers: $\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]$
@ -2498,7 +2509,7 @@ exportFig('figs/test_id31_comp_ol_iff_plant_model.pdf', 'width', 'wide', 'height
:PROPERTIES:
:header-args:matlab+: :tangle matlab/test_id31_4_hac.m
:END:
<<sec:test_id31_iff_hac>>
<<sec:test_id31_hac>>
** Introduction :ignore:
The position of the sample is actively stabilized by implementing a High-Authority-Controller as shown in Figure ref:fig:test_id31_iff_hac_schematic.
@ -2733,7 +2744,7 @@ exportFig('figs/test_id31_comp_simscape_hac.pdf', 'width', 'full', 'height', 700
The six direct terms for all four payload conditions are compared with the model in Figure ref:fig:test_id31_hac_plant_effect_mass.
It is shown that the model accurately represents the dynamics for all payloads.
In Section ref:sec:test_id31_iff_hac, a High Authority Controller is tuned to be robust to the change of dynamics due to different payloads used.
In Section ref:sec:test_id31_iff, a High Authority Controller is tuned to be robust to the change of dynamics due to different payloads used.
Without decentralized IFF being applied, the controller would have had to be robust to all the undamped dynamics shown in Figure ref:fig:test_id31_comp_all_undamped_damped_plants, which is a very complex control problem.
With the applied decentralized IFF, the HAC instead had to be be robust to all the damped dynamics shown in Figure ref:fig:test_id31_comp_all_undamped_damped_plants, which is easier from a control perspective.
This is one of the key benefit of using the HAC-LAC strategy.
@ -3660,14 +3671,9 @@ data2orgtable([1e9*data_tomo_m0_Wz6.Dx_rms_cl, 1e9*data_tomo_m0_Wz6.Dy_rms_cl, 1
{'$m_0$', '$m_1$', '$m_2$', '$m_3$'}, {'$D_x$ [nm]', '$D_y$ [nm]', '$D_z$ [nm]', '$R_x$ [nrad]', '$R_y$ [nrad]'}, ' %.0f ');
#+end_src
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
* Dynamic Error Budgeting
<<sec:test_id31_error_budget>>
** Introduction :ignore:
** Dynamic Error Budgeting
<<sec:test_id31_iff_hac_error_budget>>
**** Introduction :ignore:
In this section, the noise budget is performed.
The vibrations of the sample is measured in different conditions using the external metrology.
@ -3685,28 +3691,7 @@ The vibrations of the sample is measured in different conditions using the exter
| peak 2 peak | | 200nm | 100nm | | 1.7 urad | |
| RMS | | 30nm | 15nm | | 250 nrad | |
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
** Open-Loop Noise Budget
**** Open-Loop Noise Budget
- Effect of rotation.
- Comparison with measurement noise: should be higher
@ -3827,7 +3812,7 @@ linkaxes([ax1,ax2,ax3],'xy');
xlim([0.1, 5e2]); ylim([1e-10, 3e-5]);
#+end_src
** Effect of LAC
**** Effect of LAC
- [ ] Maybe merge this with the HAC-LAC
#+begin_src matlab
@ -3896,7 +3881,7 @@ linkaxes([ax1,ax2,ax3],'xy');
xlim([0.1, 5e2]); ylim([1e-10, 3e-5]);
#+end_src
** Effect of HAC
**** Effect of HAC
#+begin_src matlab
%% Effect of HAC
@ -4006,12 +3991,24 @@ linkaxes([ax1,ax2,ax3],'xy');
xlim([0.1, 5e2]); ylim([1e-10, 3e-5]);
#+end_src
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
* Validation with Scientific experiments
:PROPERTIES:
:header-args:matlab+: :tangle matlab/test_id31_5_experiments.m
:END:
<<sec:test_id31_experiments>>
** Introduction :ignore:
# - [ ] Where are different experiment explained?
# - [ ] Explain which controller was used here: robust one that works for all payloads!
# - [ ] All results are not filtered (i.e. 10kHz)
# - [ ] For ramp scans, a higher performance controller was used with two integrators
The online metrology prototype does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam.
However, in order to fully validate the NASS, typical motion performed during scientific experiments can be mimicked, and the positioning performances can be evaluated.
@ -8481,6 +8478,8 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
#+end_src
* Footnotes
[fn:6]The roundness of the spheres is specified at $50\,nm$
[fn:5]The "IcePAP" [[cite:&janvier13_icepap]] which is developed at the ESRF
[fn:4]Note that the eccentricity of the "point of interest" with respect to the Spindle rotation axis has been tuned from the measurements.
[fn:3]The "PEPU" [[cite:&hino18_posit_encod_proces_unit]] was used for digital protocol conversion between the interferometers and the Speedgoat

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@ -1,4 +1,4 @@
% Created 2025-01-31 Fri 14:50
% Created 2025-01-31 Fri 18:54
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -24,18 +24,17 @@
\clearpage
Now that the nano-hexapod is mounted and that a good multi-body model of the nano-hexapod
The system is validated on the ID31 beamline.
Now that the nano-hexapod is mounted and that the the multi-body model of the nano-hexapod could be validated based on dynamics measurements, the complete NASS is mounted as shown in Figure \ref{fig:test_id31_micro_station_nano_hexapod} and the performances are evaluated on the ID31 beamline.
At the beginning of the project, it was planned to develop a long stroke 5-DoF metrology system to measure the pose of the sample with respect to the granite.
The development of such system was complex, and was not completed at the time of the experimental tests on ID31.
To still validate the developed nano active platform and the associated instrumentation and control architecture, a 5-DoF short stroke metrology system was developed (Section \ref{sec:test_id31_metrology}).
To still be able to validate the developed nano active platform and the associated instrumentation and control architecture, a 5-DoF short stroke metrology system is developed and presented in Section \ref{sec:test_id31_metrology}.
The identify dynamics of the nano-hexapod fixed on top of the micro-station was identified for different experimental conditions (payload masses, rotational velocities) and compared with the model (Section \ref{sec:test_id31_open_loop_plant}).
The identify dynamics of the nano-hexapod fixed on top of the micro-station is identified for different experimental conditions (payload masses, rotational velocities) and compared with the multi-body model in Section \ref{sec:test_id31_open_loop_plant}.
Decentralized Integral Force Feedback is then applied to actively damp the plant in a robust way (Section \ref{sec:test_id31_iff}).
In order to apply the developed HAC-LAC architecture, decentralized Integral Force Feedback is first applied to actively damp the plant in a robust way (Section \ref{sec:test_id31_iff}), and the high authority controller is then implemented (Section \ref{sec:test_id31_hac}).
High authority control is then applied (Section \ref{sec:test_id31_iff_hac}).
Finally, the positioning accuracy of the NASS is evaluated by performing scans corresponding to several scientific experiments (Section \ref{sec:test_id31_experiments})
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -60,7 +59,7 @@ As the long-stroke (\(\approx 1 \,cm^3\)) metrology system was not developed yet
A first considered option was to use the ``Spindle error analyzer'' shown in Figure \ref{fig:test_id31_lion}.
This system comprises 5 capacitive sensors which are facing two reference spheres.
As the gap between the capacitive sensors and the spheres is very small\footnote{Depending on the measuring range, gap can range from \(\approx 1\,\mu m\) to \(\approx 100\,\mu m\)}, the risk of damaging the spheres and the capacitive sensors is high.
But as the gap between the capacitive sensors and the spheres is very small\footnote{Depending on the measuring range, gap can range from \(\approx 1\,\mu m\) to \(\approx 100\,\mu m\)}, the risk of damaging the spheres and the capacitive sensors is too high.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -86,17 +85,18 @@ As the gap between the capacitive sensors and the spheres is very small\footnote
Instead of using capacitive sensors, 5 fibered interferometers were used in a similar way (Figure \ref{fig:test_id31_interf}).
At the end of each fiber, a sensor head\footnote{M12/F40 model from Attocube} (Figure \ref{fig:test_id31_interf_head}) is used, which consists of a lens precisely positioned with respect to the fiber's end.
The lens is focusing the light on the surface of the sphere, such that it comes back to the fiber and produces an interference.
This way, the gap between the sensor and the reference sphere is much larger (here around \(40\,mm\)), removing the risk of collision.
The lens is focusing the light on the surface of the sphere, such that the reflected light comes back into the fiber and produces an interference.
This way, the gap between the head and the reference sphere is much larger (here around \(40\,mm\)), removing the risk of collision.
Nevertheless, the metrology system still has limited measurement range, as when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and for some perpendicular displacement, the interference is too small to be detected.
Nevertheless, the metrology system still has limited measurement range due to limited angular acceptance of the fibered interferometers.
Indeed, when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and above some perpendicular displacement, the reflected light does not comes back into the fiber, and no interference is produced.
\section{Metrology Kinematics}
\label{ssec:test_id31_metrology_kinematics}
The developed short-stroke metrology system is schematically shown in Figure \ref{fig:test_id31_metrology_kinematics}.
The point of interest is indicated by the blue frame \(\{B\}\), which is located \(H = 150\,mm\) above the nano-hexapod's top platform.
The spheres have a diameter \(d = 25.4\,mm\), and indicated dimensions are \(l_1 = 60\,mm\) and \(l_2 = 16.2\,mm\).
In order to compute the pose of the \(\{B\}\) frame with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured small displacements \([d_1,\ d_2,\ d_3,\ d_4,\ d_5]\) by the interferometers are first written as a function of the small linear and angular motion of the \(\{B\}\) frame \([D_x,\ D_y,\ D_z,\ R_x,\ R_y]\) \eqref{eq:test_id31_metrology_kinematics}.
In order to compute the pose of the \(\{B\}\) frame with respect to the granite (i.e. with respect to the fixed interferometer heads), the measured (small) displacements \([d_1,\ d_2,\ d_3,\ d_4,\ d_5]\) by the interferometers are first written as a function of the (small) linear and angular motion of the \(\{B\}\) frame \([D_x,\ D_y,\ D_z,\ R_x,\ R_y]\) \eqref{eq:test_id31_metrology_kinematics}.
\begin{equation}\label{eq:test_id31_metrology_kinematics}
d_1 = D_y - l_2 R_x, \quad d_2 = D_y + l_1 R_x, \quad d_3 = -D_x - l_2 R_y, \quad d_4 = -D_x + l_1 R_y, \quad d_5 = -D_z
@ -116,18 +116,18 @@ d_1 = D_y - l_2 R_x, \quad d_2 = D_y + l_1 R_x, \quad d_3 = -D_x - l_2 R_y, \qua
\end{center}
\end{minipage}
The five equations \eqref{eq:test_id31_metrology_kinematics} can be written in a matrix form, and then inverted to have the pose of \(\{B\}\) frame as a linear combination of the measured five distances by the interferometers \eqref{eq:test_id31_metrology_kinematics_inverse}.
The five equations \eqref{eq:test_id31_metrology_kinematics} can be written in a matrix form, and then inverted to have the pose of the \(\{B\}\) frame as a linear combination of the measured five distances by the interferometers \eqref{eq:test_id31_metrology_kinematics_inverse}.
\begin{equation}\label{eq:test_id31_metrology_kinematics_inverse}
\begin{bmatrix}
D_x \\ D_y \\ D_z \\ R_x \\ R_y
\end{bmatrix} = \begin{bmatrix}
\end{bmatrix} = {\underbrace{\begin{bmatrix}
0 & 1 & 0 & -l_2 & 0 \\
0 & 1 & 0 & l_1 & 0 \\
-1 & 0 & 0 & 0 & -l_2 \\
-1 & 0 & 0 & 0 & l_1 \\
0 & 0 & -1 & 0 & 0
\end{bmatrix}^{-1} \cdot \begin{bmatrix}
\end{bmatrix}}_{\bm{J_d}}}^{-1} \cdot \begin{bmatrix}
d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
\end{bmatrix}
\end{equation}
@ -139,17 +139,18 @@ The two reference spheres are aligned with the rotation axis of the spindle.
To do so, two measuring probes are used as shown in Figure \ref{fig:align_top_sphere_comparators}.
To not damage the sensitive sphere surface, the probes are instead positioned on the cylinder on which the sphere is mounted.
First, the probes are fixed to the bottom (fixed) cylinder to align its axis with the spindle axis.
First, the probes are fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis.
Then, the probes are fixed to the top (adjustable) cylinder, and the same alignment is performed.
With this setup, the precision of the alignment of both sphere better with the spindle axis is expected to limited to \(\approx 10\,\mu m\).
This is probably limited due to the poor coaxiality between the cylinders and the spheres.
However, the alignment precision should be enough to stay in the acceptance of the interferometers.
With this setup, the alignment accuracy of both spheres with the spindle axis is expected to around \(10\,\mu m\).
The accuracy is probably limited due to the poor coaxiality between the cylinders and the spheres.
However, this first alignment should permit to position the two sphere within the acceptance of the interferometers.
\section{Tip-Tilt adjustment of the interferometers}
\label{ssec:test_id31_metrology_alignment}
The short stroke metrology system is placed on top of the main granite using a gantry made of granite blocs to have good vibration and thermal stability (Figure \ref{fig:short_stroke_metrology_overview}).
The short-stroke metrology system is placed on top of the main granite using a gantry made of granite blocs (Figure \ref{fig:short_stroke_metrology_overview}).
Granite is used to have good vibration and thermal stability.
\begin{figure}[htbp]
\centering
@ -157,26 +158,31 @@ The short stroke metrology system is placed on top of the main granite using a g
\caption{\label{fig:short_stroke_metrology_overview}Granite gantry used to fix the short-stroke metrology system}
\end{figure}
The interferometers need to be aligned with respect to the two reference spheres to approach as much as possible the ideal case shown in Figure \ref{fig:test_id31_metrology_kinematics}.
The vertical position of the spheres is adjusted using the micro-hexapod to match the height of the interferometers.
The interferometer beams need to be position with respect to the two reference spheres as close as possible to the ideal case shown in Figure \ref{fig:test_id31_metrology_kinematics}.
This means that their positions and angles needs to be well adjusted with respect to the two spheres.
First, the vertical position of the spheres is adjusted using the micro-hexapod to match the height of the interferometers.
Then, the horizontal position of the gantry is adjusted such that the coupling efficiency (i.e. the intensity of the light reflected back in the fiber) of the top interferometer is maximized.
This is equivalent as to optimize the perpendicularity between the interferometer beam and the sphere surface (i.e. the concentricity between the beam and the sphere center).
This is equivalent as to optimize the perpendicularity between the interferometer beam and the sphere surface (i.e. the concentricity between the top beam and the sphere center).
The lateral sensor heads (i.e. all except the top one), which are each fixed to a custom tip-tilt adjustment mechanism, are individually oriented such that the coupling efficient is maximized.
The lateral sensor heads (i.e. all except the top one) are each fixed to a custom tip-tilt adjustment mechanism.
This allow to individually orient them such that they all point to the spheres' center (i.e. perpendicular to the sphere surface).
This is done by maximizing the coupling efficiency of each interferometer.
After the alignment procedure, the top interferometer should coincide with with spindle axis, and the lateral interferometers should all be in the horizontal plane and intersect the spheres' center.
\section{Fine Alignment of reference spheres using interferometers}
\label{ssec:test_id31_metrology_sphere_fine_alignment}
Thanks to the good alignment of the two reference spheres with the spindle axis and to the fine adjustment of the interferometers orientations, the interferometer measurement is made possible during complete spindle rotation.
This metrology and therefore be used to better align the axis defined by the two spheres' center with the spindle axis.
Thanks to the first alignment of the two reference spheres with the spindle axis (Section \ref{ssec:test_id31_metrology_sphere_rought_alignment}) and to the fine adjustment of the interferometers orientations (Section \ref{ssec:test_id31_metrology_alignment}), the spindle can perform complete rotations while still having interference for all five interferometers.
This metrology can therefore be used to better align the axis defined by the two spheres' center with the spindle axis.
The alignment process is made by few iterations.
First, the spindle is scanned and the alignment errors are recorded.
From the errors, the motion of the micro-hexapod to better align the spheres is determined and the micro-hexapod is moved.
From the errors, the motion of the micro-hexapod to better align the spheres with the spindle axis is computed and the micro-hexapod is positioned accordingly.
Then, the spindle is scanned again, and the new alignment errors are recorded.
This iterative process is first perform for angular errors (Figure \ref{fig:test_id31_metrology_align_rx_ry}) and then for lateral errors (Figure \ref{fig:test_id31_metrology_align_dx_dy}).
Remaining error after alignment is in the order of \(\pm5\,\mu\text{rad}\) for angular errors, \(\pm 1\,\mu m\) laterally and less than \(0.1\,\mu m\) vertically.
This iterative process is first performed for angular errors (Figure \ref{fig:test_id31_metrology_align_rx_ry}) and then for lateral errors (Figure \ref{fig:test_id31_metrology_align_dx_dy}).
The remaining errors after alignment is in the order of \(\pm5\,\mu\text{rad}\) in \(R_x\) and \(R_y\) orientations, \(\pm 1\,\mu m\) in \(D_x\) and \(D_y\) directions and less than \(0.1\,\mu m\) vertically.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -200,9 +206,9 @@ Remaining error after alignment is in the order of \(\pm5\,\mu\text{rad}\) for a
Because the interferometers are pointing to spheres and not flat surfaces, the lateral acceptance is limited.
In order to estimate the metrology acceptance, the micro-hexapod is used to perform three accurate scans of \(\pm 1\,mm\), respectively along the the \(x\), \(y\) and \(z\) axes.
During these scans, the 5 interferometers are recorded, and the ranges in which each interferometer has enough coupling efficiency for measurement are estimated.
During these scans, the 5 interferometers are recorded individually, and the ranges in which each interferometer has enough coupling efficiency to be able to measure the displacement are estimated.
Results are summarized in Table \ref{tab:test_id31_metrology_acceptance}.
The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(+/-0.5\,mm\), which is enough for the current application as it is well above the micro-station errors to be actively corrected.
The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(+/-0.5\,mm\), which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS.
\begin{table}[htbp]
\centering
@ -225,24 +231,24 @@ The obtained lateral acceptance for pure displacements in any direction is estim
\section{Estimated measurement errors}
\label{ssec:test_id31_metrology_errors}
When using the NASS, the accuracy of the sample's positioning is linked to the accuracy of the external metrology.
However, to validate the nano-hexapod with the associated instrumentation and control architecture, the accuracy of the metrology is not an issue.
When using the NASS, the accuracy of the sample's positioning is determined by the accuracy of the external metrology.
However, the validation of the nano-hexapod, the associated instrumentation and the control architecture is independent of the accuracy of the metrology system.
Only the bandwidth and noise characteristics of the external metrology are important.
Yet, some elements effecting the accuracy of the metrology are discussed here.
First, the ``metrology kinematics'' (discussed in Section \ref{ssec:test_id31_metrology_kinematics}) is only approximate (i.e. valid for very small displacements).
This can be seen when performing lateral \([D_x,\,D_y]\) scans using the micro-hexapod while recording the vertical interferometer (Figure \ref{fig:test_id31_xy_map_sphere}).
This can be easily seen when performing lateral \([D_x,\,D_y]\) scans using the micro-hexapod while recording the vertical interferometer (Figure \ref{fig:test_id31_xy_map_sphere}).
As the interferometer is pointing to a sphere and not to a plane, lateral motion of the sphere is seen as a vertical motion by the top interferometer.
Then, the reference spheres have some deviations with respect to an ideal sphere.
They are meant to be used with capacitive sensors which are integrating the shape errors over large surfaces.
When using interferometers, the size of the ``light spot'' on the sphere surface is a circle with a diameter \(\approx 50\,\mu m\), therefore the system is more sensitive to shape errors with small features.
Then, the reference spheres have some deviations with respect to an ideal sphere \footnote{The roundness of the spheres is specified at \(50\,nm\)}.
They are initially meant to be used with capacitive sensors which are integrating the shape errors over large surfaces.
When using interferometers, the size of the ``light spot'' on the sphere surface is a circle with a diameter approximately equal to \(50\,\mu m\), and therefore the measurement is more sensitive to shape errors with small features.
As the interferometer light is travelling in air, the measured distance is sensitive to any variation in the refractive index of the air.
As the light from the interferometer is travelling through air (as opposed to being in vacuum), the measured distance is sensitive to any variation in the refractive index of the air.
Therefore, any variation of air temperature, pressure or humidity will induce measurement errors.
For a measurement length of \(40\,mm\), a temperature variation of \(0.1\,{}^oC\) induces an errors in the distance measurement of \(\approx 4\,nm\).
For instance, for a measurement length of \(40\,mm\), a temperature variation of \(0.1\,{}^oC\) (which is typical for the ID31 experimental hutch) induces an errors in the distance measurement of \(\approx 4\,nm\).
Finally, even in vacuum and in the absence of target motion, the interferometers are affected by noise \cite{watchi18_review_compac_inter}.
Interferometers are also affected by noise \cite{watchi18_review_compac_inter}.
The effect of the noise on the translation and rotation measurements is estimated in Figure \ref{fig:test_id31_interf_noise}.
\begin{figure}[htbp]
@ -553,7 +559,7 @@ The peak amplitudes corresponding to the suspension modes are approximately redu
\section*{Conclusion}
\chapter{High Authority Control in the frame of the struts}
\label{sec:test_id31_iff_hac}
\label{sec:test_id31_hac}
The position of the sample is actively stabilized by implementing a High-Authority-Controller as shown in Figure \ref{fig:test_id31_iff_hac_schematic}.
\begin{equation}\label{eq:eq:test_id31_hac_diagonal}
@ -584,7 +590,7 @@ To verify if the model accurately represents the damped plants, both direct term
The six direct terms for all four payload conditions are compared with the model in Figure \ref{fig:test_id31_hac_plant_effect_mass}.
It is shown that the model accurately represents the dynamics for all payloads.
In Section \ref{sec:test_id31_iff_hac}, a High Authority Controller is tuned to be robust to the change of dynamics due to different payloads used.
In Section \ref{sec:test_id31_iff}, a High Authority Controller is tuned to be robust to the change of dynamics due to different payloads used.
Without decentralized IFF being applied, the controller would have had to be robust to all the undamped dynamics shown in Figure \ref{fig:test_id31_comp_all_undamped_damped_plants}, which is a very complex control problem.
With the applied decentralized IFF, the HAC instead had to be be robust to all the damped dynamics shown in Figure \ref{fig:test_id31_comp_all_undamped_damped_plants}, which is easier from a control perspective.
This is one of the key benefit of using the HAC-LAC strategy.
@ -665,10 +671,6 @@ An open-loop simulation and a closed-loop simulation were performed and compared
Then the same tomography experiment (i.e. constant spindle rotation at 30rpm, and no payload) was performed experimentally.
The measured position of the ``point of interest'' during the experiment are shown in Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}.
\begin{itemize}
\item[{$\square$}] Add beam size (200x100nm)
\end{itemize}
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
@ -712,6 +714,10 @@ Specifications & \(30\,\text{nmRMS}\) & \(15\,\text{nmRMS}\) & \(250\,\text{nrad
\section{Robustness to change of payload}
\label{ssec:test_id31_iff_hac_robustness}
\begin{itemize}
\item[{$\square$}] Make simulations with all masses?
\end{itemize}
To verify the robustness to the change of payload mass, four simulations of tomography experiments were performed with payloads as shown Figure \ref{fig:test_id31_picture_masses} (i.e. \(0\,kg\), \(13\,kg\), \(26\,kg\) and \(39\,kg\)).
This time, the rotational velocity was set at 1rpm (i.e. 6deg/s), as it is the typical rotational velocity for heavy samples.
The closed-loop systems were stable for all payload conditions, indicating good control robustness.
@ -766,10 +772,8 @@ The obtained closed-loop errors are fulfilling the requirements, except for the
\end{table}
\section*{Conclusion}
\chapter{Dynamic Error Budgeting}
\label{sec:test_id31_error_budget}
\section{Dynamic Error Budgeting}
\label{sec:test_id31_iff_hac_error_budget}
In this section, the noise budget is performed.
The vibrations of the sample is measured in different conditions using the external metrology.
@ -790,7 +794,7 @@ RMS & & 30nm & 15nm & & 250 nrad & \\
\end{tabular}
\end{center}
\section{Open-Loop Noise Budget}
\paragraph{Open-Loop Noise Budget}
\begin{itemize}
\item Effect of rotation.
@ -799,7 +803,7 @@ RMS & & 30nm & 15nm & & 250 nrad & \\
\item Also say that for the RMS errors, we don't take into account drifts (so we NASS we can correct drifts)
\end{itemize}
\section{Effect of LAC}
\paragraph{Effect of LAC}
\begin{itemize}
\item[{$\square$}] Maybe merge this with the HAC-LAC
\end{itemize}
@ -810,41 +814,46 @@ Effect of LAC:
\item Inject some noise between 200 and 700Hz?
\end{itemize}
\section{Effect of HAC}
\paragraph{Effect of HAC}
Bandwidth is approximately 10Hz.
\section*{Conclusion}
\chapter{Validation with Scientific experiments}
The online metrology prototype does not allow samples to be placed on top of the nano-hexapod and to be illuminated by the x-ray beam.
\label{sec:test_id31_experiments}
The online metrology prototype does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam.
However, in order to fully validate the NASS, typical motion performed during scientific experiments can be mimicked, and the positioning performances can be evaluated.
Performances were already evaluated with tomography scans (Section \ref{ssec:test_id31_iff_hac_perf}).
For tomography scans, performances were already evaluated in Section \ref{ssec:test_id31_iff_hac_perf}.
Here, other typical experiments are performed:
\begin{itemize}
\item Lateral scans: the translations stage performs \(D_y\) scans, and the errors are corrected by the NASS in real time (Section \ref{ssec:test_id31_scans_dy})
\item Vertical layer scans: the nano-hexapod is used to perform \(D_z\) steps or ramp scans (Section \ref{ssec:test_id31_scans_dz})
\item Reflectivity scans: the tilt stage is doing \(R_y\) rotations and the errors are corrected by the NASS in real time (Section \ref{ssec:test_id31_scans_reflectivity})
\item Diffraction Tomography: the Spindle is performing continuous \(R_z\) rotation while the translation stage is performing lateral \(D_y\) scans at the same time. This is the experiment with the most stringent requirements (Section \ref{ssec:test_id31_scans_diffraction_tomo})
\item \emph{Lateral scans}: the \(T_y\) translations stage performs \(D_y\) scans and the errors are corrected by the NASS in real time (Section \ref{ssec:test_id31_scans_dy})
\item \emph{Vertical layer scans}: the nano-hexapod is used to perform \(D_z\) step motion or ramp scans (Section \ref{ssec:test_id31_scans_dz})
\item \emph{Reflectivity scans}: the tilt stage is doing \(R_y\) rotations and the errors are corrected by the NASS in real time (Section \ref{ssec:test_id31_scans_reflectivity})
\item \emph{Diffraction Tomography}: the Spindle is performing continuous \(R_z\) rotation while the translation stage is performing lateral \(D_y\) scans at the same time.
This is the experiment with the most stringent requirements (Section \ref{ssec:test_id31_scans_diffraction_tomo})
\end{itemize}
\section{\(D_y\) - Lateral Scans}
\label{ssec:test_id31_scans_dy}
Lateral scans are performed with the \(T_y\) stage.
The stepper motor controller\footnote{The ``IcePAP'' \cite{janvier13_icepap} which is developed at the ESRF} outputs the setpoint which is received by the Speedgoat.
Therefore, the Nano-Hexapod can be used to correct positioning errors induced by the scanning of the \(T_y\) stage.
In the Speedgoat, the setpoint is compared with the measured \(D_y\) position of the top-platform, and the Nano-Hexapod is used to correct positioning errors induced by the scanning of the \(T_y\) stage.
The stroke is here limited to \(\pm 100\,\mu m\) due to the limited acceptance of the metrology system.
\paragraph{Slow scan}
The \(T_y\) stage is first scanned at \(10\,\mu m/s\) which is typical for such experiments.
The \(T_y\) stage is first scanned with a velocity of \(10\,\mu m/s\) which is typical for such experiments.
The errors in open-loop (i.e. without using the NASS) and in closed-loop are compared in Figure \ref{fig:test_id31_dy_10ums}.
In the direction of motion, periodic errors can be observed in the open-loop case (Figure \ref{fig:test_id31_dy_10ums_dy}).
These are due to the stepper motor being used in the \(T_y\) stage.
Indeed, stepper motors inherently have ``micro-stepping'' errors which are periodic errors happening 200 times per motor rotation with an amplitude approximately equal to \(1\,\text{mrad}\).
These errors are induced by the stepper motor being used in the \(T_y\) stage.
Indeed, stepper motors inherently have ``micro-stepping errors'' which are periodic errors happening 200 times per motor rotation with an amplitude approximately equal to \(1\,\text{mrad}\).
As the lead screw for the \(T_y\) stage has a pitch of \(2\,mm\), this means that the micro-stepping errors have a period of \(10\,\mu m\) and an amplitude of \(\approx 300\,nm\) which can indeed be seen in open-loop.
In the vertical direction (Figure \ref{fig:test_id31_dy_10ums_dz}), open-loop errors are most likely due to measurement errors of the metrology itself (see Figure \ref{fig:test_id31_xy_map_sphere}).
In the vertical direction (Figure \ref{fig:test_id31_dy_10ums_dz}), open-loop errors are most likely due to measurement errors of the metrology itself as the top interferometer point at a sphere (see Figure \ref{fig:test_id31_xy_map_sphere}).
In closed-loop, the errors are within the specifications in all directions.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -871,13 +880,15 @@ In the vertical direction (Figure \ref{fig:test_id31_dy_10ums_dz}), open-loop er
\paragraph{Faster Scan}
The performance of the NASS is then tested for a scanning velocity of \(100\,\mu m/s\) and the results are shown in Figure \ref{fig:test_id31_dy_100ums}.
At this velocity, the micro-stepping errors have a frequency of \(10\,\text{Hz}\) and are inducing lot's of vibrations which are amplified by some resonances of the micro-station.
At this velocity, the micro-stepping errors have a frequency of \(10\,\text{Hz}\) and are inducing lot's of vibrations which are even amplified by some resonances of the micro-station.
These vibrations are outside the bandwidth of the NASS feedback controller and therefore not well reduced in closed-loop.
This is the main reason why stepper motors should be not be used for ``long-stroke / short-stroke'' systems when good scanning performances are wanted \cite{dehaeze22_fastj_uhv}.
In order to improve the scanning performances at high velocity, the stepper motor of the \(T_y\) stage could be replaced by a three-phase torque motor for instance.
As the closed-loop errors in \(D_z\) and \(R_y\) directions are within specifications (see Figures \ref{fig:test_id31_dy_100ums_dz} and \ref{fig:test_id31_dy_100ums_ry}), the detectors could be triggered based on the measured \(D_y\) position and therefore the experiment would be much less sensitive to \(D_y\) vibrations.
As the closed-loop errors in \(D_z\) and \(R_y\) directions are within specifications (see Figures \ref{fig:test_id31_dy_100ums_dz} and \ref{fig:test_id31_dy_100ums_ry}), another option would be to trigger the detectors based on the measured \(D_y\) position instead of based on time or on the \(T_y\) setpoint.
This would make the experiment less sensitive to \(D_y\) vibrations.
For small \(D_y\) scans, the nano-hexapod alone can be used for the scans, but with limited strokes.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -933,13 +944,13 @@ Specs & 100.0 & 50.0 & 0.85\\
\section{\(D_z\) scans: Dirty Layer Scans}
\label{ssec:test_id31_scans_dz}
In some cases, samples are composed of several atomic ``layers'' that are first aligned in the horizontal plane with precise \(R_y\) positioning and then scanned vertically with precise \(D_z\) motion.
The vertical scan can be performed step-by-step or continuously.
In some cases, samples are composed of several atomic ``layers'' that are first aligned in the horizontal plane with precise \(R_y\) positioning and that are then scanned vertically with precise \(D_z\) motion.
The vertical scan can be performed continuously of using step-by-step motion.
\paragraph{Step by Step \(D_z\) motion}
Vertical steps are here performed using the nano-hexapod.
Vertical steps are here performed using the nano-hexapod only.
Step sizes from \(10\,nm\) to \(1\,\mu m\) are tested, and the results are shown in Figure \ref{fig:test_id31_dz_mim_steps}.
10nm steps can be resolved if detectors are integrating over 50ms (see red curve in Figure \ref{fig:test_id31_dz_mim_10nm_steps}), which is very typical.
10nm steps can be resolved if detectors are integrating over 50ms (see red curve in Figure \ref{fig:test_id31_dz_mim_10nm_steps}), which is reasonable for many experiments.
When doing step-by-step scans, the time to reach the next value is quite critical as long settling time can render the total experiment excessively long.
The response time to reach the wanted value (to within \(\pm 20\,nm\)) is around \(70\,ms\) as shown with the \(1\,\mu m\) step response in Figure \ref{fig:test_id31_dz_mim_1000nm_steps}.
@ -969,14 +980,8 @@ The response time to reach the wanted value (to within \(\pm 20\,nm\)) is around
\paragraph{Continuous \(D_z\) motion: Dirty Layer Scans}
Instead of performing ``step-by-step'' scans, continuous scans can also be performed in the vertical direction.
At \(10\,\mu m/s\), the errors are well within the specifications (see Figure \ref{fig:test_id31_dz_scan_10ums}).
The second tested velocity is \(100\,\mu m/s\), which is typically the fastest velocity for \(D_z\) scans when the ultimate performances is wanted (1ms integration time and 100nm ``resolution'').
At this velocity, the positioning errors are also within the specifications except for the very start and very end of the motion (i.e. during acceleration/deceleration phases, see Figure \ref{fig:test_id31_dz_scan_100ums}).
However, the detectors are usually triggered only during the constant velocity phase, so this should not be an issue.
The performances during acceleration phase may also be improved by using a feedforward controller.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
@ -999,6 +1004,11 @@ The performances during acceleration phase may also be improved by using a feedf
\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan with a velocity of \(10\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry})}
\end{figure}
The second tested velocity is \(100\,\mu m/s\), which is the fastest velocity for \(D_z\) scans when the ultimate performances is wanted (corresponding to a 1ms integration time and 100nm ``resolution'').
At this velocity, the positioning errors are also within the specifications except for the very start and very end of the motion (i.e. during acceleration/deceleration phases, see Figure \ref{fig:test_id31_dz_scan_100ums}).
However, the detectors are usually triggered only during the constant velocity phase, so this is not not an issue.
The performances during acceleration phase may also be improved by using a feedforward controller.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
@ -1049,9 +1059,8 @@ Specs & 30.0 & 15.0 & 0.25\\
\label{ssec:test_id31_scans_reflectivity}
X-ray reflectivity consists of scanning the \(R_y\) angle of thin structures (typically solid/liquid interfaces) through the beam.
Here, a \(R_y\) scan is performed at \(100\,\mu rad/s\) velocity and the positioning errors are recorded (Figure \ref{fig:test_id31_reflectivity}).
It is shown that the NASS is able to keep the point of interest in the beam.
Here, a \(R_y\) scan is performed with a rotational velocity of \(100\,\mu rad/s\) and the positioning errors in closed-loop are recorded (Figure \ref{fig:test_id31_reflectivity}).
It is shown that the NASS is able to keep the point of interest in the beam within specifications.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -1079,80 +1088,69 @@ It is shown that the NASS is able to keep the point of interest in the beam.
\section{Combined \(R_z\) and \(D_y\): Diffraction Tomography}
\label{ssec:test_id31_scans_diffraction_tomo}
The goal of this experiment is to perform combined \(R_z\) rotation and \(D_z\) lateral scans.
Here the spindle is performing a continuous 1rpm rotation while the nano-hexapod is used to perform fast \(D_z\) scans.
In diffraction tomography, the micro-station performs combined \(R_z\) rotation and \(D_y\) lateral scans.
Here the spindle is performing a continuous 1rpm rotation while the nano-hexapod is used to perform fast \(D_y\) scans.
The \(T_y\) stage is here not used as the stepper motor would induce high frequency vibrations, therefore the stroke is here limited to \(\approx \pm 100\,\mu m/s\).
Several \(D_y\) velocities are tested: \(0.1\,mm/s\), \(0.5\,mm/s\) and \(1\,mm/s\).
The \(D_y\) setpoint and the measured positions are shown for all tested velocities in Figure \ref{fig:test_id31_diffraction_tomo_setpoint}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_id31_diffraction_tomo_setpoint.png}
\caption{\label{fig:test_id31_diffraction_tomo_setpoint}Dy motion for several configured velocities}
\end{figure}
The measured errors in \(D_y\), \(D_z\) and \(R_y\) directions are shown in Figure \ref{fig:test_id31_diffraction_tomo}.
While the \(D_z\) and \(R_y\) errors are within specifications (see Figures \ref{fig:test_id31_diffraction_tomo_dz} and \ref{fig:test_id31_diffraction_tomo_ry}), the lateral error goes outside of specifications during acceleration and deceleration phases (Figure \ref{fig:test_id31_diffraction_tomo_dy}).
However, it goes out of specifications during only during \(\approx 20\,ms\), and this could be optimized using feedforward and more appropriate setpoint signals.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/test_id31_diffraction_tomo_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_dy}$D_y$}
\subcaption{\label{fig:test_id31_diffraction_tomo_dy} $D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/test_id31_diffraction_tomo_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_dz}$D_z$}
\subcaption{\label{fig:test_id31_diffraction_tomo_dz} $D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/test_id31_diffraction_tomo_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_ry}$R_y$}
\subcaption{\label{fig:test_id31_diffraction_tomo_ry} $R_y$}
\end{subfigure}
\caption{\label{fig:test_id31_diffraction_tomo}Diffraction tomography scans (combined \(R_z\) and \(D_y\) motions) at several \(D_y\) velocities (\(R_z\) rotational velocity is 1rpm).}
\end{figure}
The corresponding ``repetition rate'' and \(D_y\) scan per spindle turn are shown in Table \ref{tab:diffraction_tomo_velocities}.
The main issue here is the ``waiting'' time between two scans that is in the order of 50ms.
By removing this waiting time (fairly easily), we can double the repetition rate at 10mm/s.
\begin{table}[htbp]
\centering
\begin{tabularx}{0.6\linewidth}{lXX}
\toprule
\(D_y\) Velocity & Repetition rate & Scans per turn (at 1RPM)\\
\midrule
0.1 mm/s & 4 s & 15\\
0.5 mm/s & 0.9 s & 65\\
1 mm/s & 0.5 s & 120\\
\bottomrule
\end{tabularx}
\caption{\label{tab:diffraction_tomo_velocities}\(D_y\) scaning repetition rate}
\end{table}
The scan results for a velocity of 1mm/s is shown in Figure \ref{fig:id31_diffraction_tomo_1mms}.
The \(D_z\) and \(R_y\) errors are quite small during the scan.
The \(D_y\) errors are quite large as the velocity is increased.
This type of scan can probably be massively improved by using feed-forward and optimizing the trajectory.
Also, if the detectors are triggered in position (the Speedgoat could generate an encoder signal for instance), we don't care about the \(D_y\) errors.
\begin{table}[htbp]
\centering
\begin{tabularx}{\linewidth}{lXX}
\toprule
\begin{center}
\begin{tabular}{lrrr}
Velocity & \(D_y\) [nmRMS] & \(D_z\) [nmRMS] & \(R_y\) [\(\mu\text{radRMS}\)]\\
\midrule
0.1 mm/s & 75.45 & 9.13 & 0.12\\
0.5 mm/s & 190.47 & 9.97 & 0.1\\
1 mm/s & 428.0 & 11.24 & 0.17\\
\bottomrule
\end{tabularx}
\caption{\label{tab:id31_diffraction_tomo_results}Obtained errors for several \(D_y\) velocities}
\hline
Specs & 100.0 & 50.0 & 0.85\\
0.1 mm/s & 208.25 & 35.33 & 0.73\\
0.5 mm/s & 117.94 & 28.03 & 0.27\\
1 mm/s & 186.88 & 33.02 & 0.53\\
\end{tabular}
\end{table}
\end{center}
\begin{center}
\begin{tabular}{lrrr}
Velocity & \(D_y\) [nmRMS] & \(D_z\) [nmRMS] & \(R_y\) [\(\mu\text{radRMS}\)]\\
\hline
Specs & 30.0 & 15.0 & 0.25\\
0.1 mm/s & 36.18 & 7.35 & 0.11\\
0.5 mm/s & 28.58 & 7.52 & 0.08\\
1 mm/s & 53.05 & 9.84 & 0.14\\
\end{tabular}
\end{center}
\section*{Conclusion}
\label{ssec:test_id31_scans_conclusion}
@ -1165,13 +1163,19 @@ For each conducted experiments, the \(D_y\), \(D_z\) and \(R_y\) errors are comp
\toprule
& \(D_y\) [nmRMS] & \(D_z\) [nmRMS] & \(R_y\) [nradRMS]\\
\midrule
Specifications & & & \\
\midrule
Tomography (\(R_z\) 1rpm) & 15 & 5 & 55\\
Tomography (\(R_z\) 6rpm) & 19 & 5 & 73\\
Tomography (\(R_z\) 30rpm) & 38 & 10 & 129\\
\midrule
Dirty Layer (\(D_z\) \(10\,\mu m/s\)) & 25 & 5 & 114\\
Dirty Layer (\(D_z\) \(100\,\mu m/s\)) & 34 & 15 & 130\\
\midrule
Reflectivity (\(R_y\) \(100\,\mu\text{rad}/s\)) & 28 & 6 & 118\\
\midrule
Lateral Scan (\(D_y\) \(10\,\mu m/s\)) & 21 & 10 & 37\\
\midrule
Diffraction Tomography (\(R_z\) 1rpm, \(D_y\) 0.1mm/s) & 75 & 9 & 118\\
Diffraction Tomography (\(R_z\) 1rpm, \(D_y\) 1mm/s) & 428 & 11 & 169\\
\bottomrule
@ -1180,5 +1184,8 @@ Diffraction Tomography (\(R_z\) 1rpm, \(D_y\) 1mm/s) & 428 & 11 & 169\\
\end{table}
\chapter*{Conclusion}
\label{ssec:test_id31_conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}