From 8cdcf87bcec05b58238a7ff35de21f5a57acab9b Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 23 Apr 2025 10:13:08 +0200 Subject: [PATCH] \mu => \upmu --- config.tex | 2 + phd-thesis.org | 594 ++++++++++++++++++++++++------------------------- setup.org | 2 + 3 files changed, 301 insertions(+), 297 deletions(-) diff --git a/config.tex b/config.tex index 8c0a9de..5470e0d 100644 --- a/config.tex +++ b/config.tex @@ -13,6 +13,8 @@ \DeclareSIUnit\rms{rms} \DeclareSIUnit\rad{rad} +\usepackage{upgreek} % useful for "mu" in units + \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} diff --git a/phd-thesis.org b/phd-thesis.org index d9f3794..8d62cb9 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -525,7 +525,7 @@ The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source s #+end_subfigure #+end_figure -Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of acrshort:esrf, where typical spot sizes were on the order of $10\,\mu\text{m}$ [[cite:&riekel89_microf_works_at_esrf]]. +Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of acrshort:esrf, where typical spot sizes were on the order of $10\,\upmu\text{m}$ [[cite:&riekel89_microf_works_at_esrf]]. Various technologies, including zone plates, Kirkpatrick-Baez mirrors, and compound refractive lenses, have been developed and refined, each presenting unique advantages and limitations\nbsp{}[[cite:&barrett16_reflec_optic_hard_x_ray]]. The historical reduction in achievable spot sizes is represented in Figure\nbsp{}ref:fig:introduction_moore_law_focus. Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines. @@ -695,7 +695,7 @@ In most reported cases, only translation errors are actively corrected. Payload capacities for these high-precision systems are usually limited, typically handling calibrated samples on the micron scale, although capacities up to 500g have been reported\nbsp{}[[cite:&nazaretski22_new_kirkp_baez_based_scann;&kelly22_delta_robot_long_travel_nano]]. The system developed in this thesis aims for payload capabilities approximately 100 times heavier (up to $50\,\text{kg}$) than previous stations with similar positioning requirements. -End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few acrshortpl:dof with strokes around $100\,\mu\text{m}$. +End-stations integrating online metrology for active nano-positioning often exhibit limited operational ranges, typically constrained to a few acrshortpl:dof with strokes around $100\,\upmu\text{m}$. Recently, acrfull:vc actuators were used to increase the stroke up to $3\,\text{mm}$ [[cite:&kelly22_delta_robot_long_travel_nano;&geraldes23_sapot_carnaub_sirius_lnls]] An alternative strategy involves a "long stroke-short stroke" architecture, illustrated conceptually in Figure\nbsp{}ref:fig:introduction_two_stage_schematic. In this configuration, a high-accuracy, high-bandwidth short-stroke stage is mounted on top of a less precise long-stroke stage. @@ -732,7 +732,7 @@ Given the high frame rates of modern detectors, these specified positioning erro These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to $50\,\text{kg}$. -The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately $\SI{10}{\mu\m}$ and $\SI{10}{\mu\rad}$ due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. +The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately $\SI{10}{\micro\m}$ and $\SI{10}{\micro\rad}$ due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. The primary objective of this project is therefore defined as enhancing the positioning accuracy and stability of the ID31 micro-station by roughly two orders of magnitude, to fully leverage the capabilities offered by the ESRF-EBS source and modern detectors, without compromising its existing mobility and payload capacity. @@ -875,7 +875,7 @@ The integration of such filters into feedback control architectures can also lea ***** Experimental validation of the Nano Active Stabilization System The conclusion of this work involved the experimental implementation and validation of the complete NASS on the ID31 beamline. -Experimental results, presented in Section\nbsp{}ref:sec:test_id31, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native $\approx 10\,\mu\text{m}$ level down to the target $\approx 100\,\text{nm}$ range during representative scientific experiments. +Experimental results, presented in Section\nbsp{}ref:sec:test_id31, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native $\approx 10\,\upmu\text{m}$ level down to the target $\approx 100\,\text{nm}$ range during representative scientific experiments. Crucially, robustness to variations in sample mass and diverse experimental conditions was verified. The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal use of the advanced capabilities of the ESRF-EBS beam and associated detectors. To the author's knowledge, this represents the first demonstration of such a 5-DoF active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. @@ -1033,9 +1033,9 @@ The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_ #+attr_latex: :center t :booktabs t | *Stage* | *Mass* | *Stiffness* | *Damping* | |---------------------+-------------------------+----------------------------+-------------------------------------------| -| Hexapod | $m_h = 15\,\text{kg}$ | $k_h = 61\,\text{N}/\mu\text{m}$ | $c_h = 3\,\frac{\text{kN}}{\text{m/s}}$ | -| $T_y$, $R_y$, $R_z$ | $m_t = 1200\,\text{kg}$ | $k_t = 520\,\text{N}/\mu\text{m}$ | $c_t = 80\,\frac{\text{kN}}{\text{m/s}}$ | -| Granite | $m_g = 2500\,\text{kg}$ | $k_g = 950\,\text{N}/\mu\text{m}$ | $c_g = 250\,\frac{\text{kN}}{\text{m/s}}$ | +| Hexapod | $m_h = 15\,\text{kg}$ | $k_h = 61\,\text{N}/\upmu\text{m}$ | $c_h = 3\,\frac{\text{kN}}{\text{m/s}}$ | +| $T_y$, $R_y$, $R_z$ | $m_t = 1200\,\text{kg}$ | $k_t = 520\,\text{N}/\upmu\text{m}$ | $c_t = 80\,\frac{\text{kN}}{\text{m/s}}$ | +| Granite | $m_g = 2500\,\text{kg}$ | $k_g = 950\,\text{N}/\upmu\text{m}$ | $c_g = 250\,\frac{\text{kN}}{\text{m/s}}$ | Two disturbances are considered which are shown in red: the floor motion $x_f$ and the stage vibrations represented by $f_t$. The hammer impacts $F_{h}, F_{g}$ are shown in blue, whereas the measured inertial motions $x_{h}, x_{g}$ are shown in black. @@ -1086,13 +1086,13 @@ The active platform is represented by a mass spring damper system (shown in blue Its mass gls:mn is set to $15\,\text{kg}$ while its stiffness $k_n$ can vary depending on the chosen architecture/technology. The sample is represented by a mass gls:ms that can vary from $1\,\text{kg}$ up to $50\,\text{kg}$. -As a first example, the active platform stiffness of is set at $k_n = 10\,\text{N}/\mu\text{m}$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. +As a first example, the active platform stiffness of is set at $k_n = 10\,\text{N}/\upmu\text{m}$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. ***** Obtained Dynamic Response The sensitivity to disturbances (i.e., the transfer functions from $x_f,f_t,f_s$ to $d$) can be extracted from the uniaxial model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass and are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_first_params. The /plant/ (i.e., the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure\nbsp{}ref:fig:uniaxial_plant_first_params. -For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three active platform stiffnesses ($k_n = 0.01\,\text{N}/\mu\text{m}$, $k_n = 1\,\text{N}/\mu\text{m}$ and $k_n = 100\,\text{N}/\mu\text{m}$) combined with three sample's masses ($m_s = 1\,\text{kg}$, $m_s = 25\,\text{kg}$ and $m_s = 50\,\text{kg}$). +For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$, $k_n = 1\,\text{N}/\upmu\text{m}$ and $k_n = 100\,\text{N}/\upmu\text{m}$) combined with three sample's masses ($m_s = 1\,\text{kg}$, $m_s = 25\,\text{kg}$ and $m_s = 50\,\text{kg}$). #+name: fig:uniaxial_sensitivity_dist_first_params #+caption: Sensitivity of the relative motion $d$ to disturbances: $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_first_params_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_first_params_fs}) @@ -1221,9 +1221,9 @@ This is very useful to identify what is limiting the performance of the system, From the uniaxial model of the acrshort:nass (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), the transfer function from the disturbances ($f_s$, $x_f$ and $f_t$) to the displacement $d$ are computed. This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and three active platform stiffnesses: -- $k_n = 0.01\,\text{N}/\mu\text{m}$ that represents a voice coil actuator with soft flexible guiding -- $k_n = 1\,\text{N}/\mu\text{m}$ that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator -- $k_n = 100\,\text{N}/\mu\text{m}$ that represents a stiff piezoelectric stack actuator +- $k_n = 0.01\,\text{N}/\upmu\text{m}$ that represents a voice coil actuator with soft flexible guiding +- $k_n = 1\,\text{N}/\upmu\text{m}$ that represents a voice coil actuator with a stiff flexible guiding or a mechanically amplified piezoelectric actuator +- $k_n = 100\,\text{N}/\upmu\text{m}$ that represents a stiff piezoelectric stack actuator The obtained sensitivity to disturbances for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the sample mass $m_s = 1\,\text{kg}$ (the same conclusions can be drawn with $m_s = 50\,\text{kg}$): - The soft active platform is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs) @@ -1283,14 +1283,14 @@ The conclusion is that the sample mass has little effect on the cumulative ampli ***** Conclusion The open-loop residual vibrations of $d$ can be estimated from the low-frequency value of the cumulative amplitude spectrum in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. -This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$) and $1\,\mu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$). +This residual vibration of $d$ is found to be in the order of $100\,\text{nm RMS}$ for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$), $200\,\text{nm RMS}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$) and $1\,\upmu\text{m}\,\text{RMS}$ for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$). From this analysis, it may be concluded that the stiffer the active platform the better. However, what is more important is the /closed-loop/ residual vibration of $d$ (i.e., while the feedback controller is used). The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,\text{nm RMS}$ (represented by an horizontal dashed black line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses). The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,\text{nm RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses. -A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$), $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$), and $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$). +A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$), $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$), and $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$). Therefore, while the /open-loop/ vibration is the lowest for the stiff active platform, it requires the largest feedback bandwidth to meet the specifications. The advantage of the soft active platform can be explained by its natural isolation from the micro-station vibration above its suspension mode, as shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft. @@ -1421,7 +1421,7 @@ For the stiff active platform (yellow curves), the micro-station dynamics can be Therefore, it is expected that the micro-station dynamics might impact the achievable damping if a stiff active platform is used. #+name: fig:uniaxial_plant_active_damping_techniques -#+caption: Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three active platform stiffnesses ($k_n = 0.01\,\text{N}/\mu\text{m}$ in blue, $k_n = 1\,\text{N}/\mu\text{m}$ in red and $k_n = 100\,\text{N}/\mu\text{m}$ in yellow) and three sample's masses ($m_s = 1\,\text{kg}$: solid curves, $m_s = 25\,\text{kg}$: dot-dashed curves, and $m_s = 50\,\text{kg}$: dashed curves). +#+caption: Plant dynamics for the three active damping techniques (IFF: \subref{fig:uniaxial_plant_active_damping_techniques_iff}, RDC: \subref{fig:uniaxial_plant_active_damping_techniques_rdc}, DVF: \subref{fig:uniaxial_plant_active_damping_techniques_dvf}), for three active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$ in blue, $k_n = 1\,\text{N}/\upmu\text{m}$ in red and $k_n = 100\,\text{N}/\upmu\text{m}$ in yellow) and three sample's masses ($m_s = 1\,\text{kg}$: solid curves, $m_s = 25\,\text{kg}$: dot-dashed curves, and $m_s = 50\,\text{kg}$: dashed curves). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_iff}IFF} @@ -1470,19 +1470,19 @@ There is even some damping authority on micro-station modes in the following cas #+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed in the suspension mode of the active platform. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_root_locus_damping_techniques_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_root_locus_damping_techniques_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1502,19 +1502,19 @@ All three active damping techniques yielded similar damped plants. #+caption: Obtained damped transfer function from $f$ to $d$ for the three damping techniques. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_damped_plant_three_active_damping_techniques_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1525,7 +1525,7 @@ All three active damping techniques yielded similar damped plants. ***** Sensitivity to Disturbances and Noise Budgeting Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped. The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping. -The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\mu\text{m}$. +The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\upmu\text{m}$. Several conclusions can be drawn by comparing the obtained sensitivity transfer functions: - IFF degrades the sensitivity to direct forces on the sample (i.e., the compliance) below the resonance of the active platform (Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping_fs). @@ -1566,19 +1566,19 @@ All three active damping methods give similar results. #+caption: Comparison of the cumulative amplitude spectrum (CAS) of the distance $d$ for all three active damping techniques (acrshort:ol in black, IFF in blue, RDC in red and DVF in yellow). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_active_damping_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_active_damping_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1650,8 +1650,8 @@ This control architecture applied to the uniaxial model is shown in Figure\nbsp{ ***** Damped Plant Dynamics The damped plants obtained for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses. -For $k_n = 0.01\,\text{N}/\mu\text{m}$ and $k_n = 1\,\text{N}/\mu\text{m}$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). -However, this is not the case for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). +For $k_n = 0.01\,\text{N}/\upmu\text{m}$ and $k_n = 1\,\text{N}/\upmu\text{m}$, the dynamics are quite simple and can be well approximated by a second-order plant (Figures\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft and ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). +However, this is not the case for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) where two modes can be seen (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). This is due to the interaction between the micro-station (modeled modes at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$) and the active platform. This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_compliance. @@ -1659,19 +1659,19 @@ This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_ #+caption: Obtained damped plant using Integral Force Feedback for three sample masses #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_hac_iff_damped_plants_masses_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_hac_iff_damped_plants_masses_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1685,14 +1685,14 @@ The objective is to design high-authority feedback controllers for the three act This controller must be robust to the change of sample's mass (from $1\,\text{kg}$ up to $50\,\text{kg}$). The required feedback bandwidths were estimated in Section\nbsp{}ref:sec:uniaxial_noise_budgeting: -- $f_b \approx 10\,\text{Hz}$ for the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$). +- $f_b \approx 10\,\text{Hz}$ for the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$). Near this frequency, the plants (shown in Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_soft) are equivalent to a mass line (i.e., slope of $-40\,\text{dB/dec}$ and a phase of -180 degrees). The gain of this mass line can vary up to a fact $\approx 5$ (suspended mass from $16\,\text{kg}$ up to $65\,\text{kg}$). This means that the designed controller will need to have /large gain margins/ to be robust to the change of sample's mass. -- $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$). +- $\approx 50\,\text{Hz}$ for the relatively stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$). Similar to the soft active platform, the plants near the crossover frequency are equivalent to a mass line (Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_mid). It will probably be easier to have a little bit more bandwidth in this configuration to be further away from the active platform suspension mode. -- $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$). +- $\approx 100\,\text{Hz}$ for the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$). Contrary to the two first active platform stiffnesses, here the plants have more complex dynamics near the desired crossover frequency (see Figure\nbsp{}ref:fig:uniaxial_hac_iff_damped_plants_masses_stiff). The micro-station is not stiff enough to have a clear stiffness line at this frequency. Therefore, there is both a change of phase and gain depending on the sample mass. @@ -1744,19 +1744,19 @@ The goal is to have a first estimation of the attainable performance. #+caption: Nyquist Plot for the high authority controller. The minimum modulus margin is illustrated by a black circle. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_nyquist_hac_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_nyquist_hac_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1768,19 +1768,19 @@ The goal is to have a first estimation of the attainable performance. #+caption: Loop gains for the High Authority Controllers #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_loop_gain_hac_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_loop_gain_hac_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1795,7 +1795,7 @@ These are compared with the open-loop and damped plants cases in Figure\nbsp{}re As expected, the sensitivity to disturbances decreased in the controller bandwidth and slightly increased outside this bandwidth. #+name: fig:uniaxial_sensitivity_dist_hac_lac -#+caption: Change of sensitivity to disturbances with acrshort:lac and with acrshort:haclac. An active platform with $k_n = 1\,\text{N}/\mu\text{m}$ and a sample mass of $25\,\text{kg}$ is used. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}) +#+caption: Change of sensitivity to disturbances with acrshort:lac and with acrshort:haclac. An active platform with $k_n = 1\,\text{N}/\upmu\text{m}$ and a sample mass of $25\,\text{kg}$ is used. $f_s$ the direct forces applied on the sample (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}), $f_t$ disturbances from the micro-station stages (\subref{fig:uniaxial_sensitivity_dist_hac_lac_ft}) and $x_f$ the floor motion (\subref{fig:uniaxial_sensitivity_dist_hac_lac_fs}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_fs}Direct forces} @@ -1826,19 +1826,19 @@ Obtained root mean square values of the distance $d$ are better for the soft act #+caption: Cumulative Amplitude Spectrum for all three active platform stiffnesses - Comparison of OL, IFF and acrshort:haclac cases #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.37\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_hac_lac_soft.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_cas_hac_lac_mid.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.31\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -1853,7 +1853,7 @@ To achieve such bandwidth, the acrshort:haclac strategy was followed, which cons In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass. The attainable vibration control performances were estimated for the three active platform stiffnesses and were found to be close to the required values. -However, the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. +However, the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. A slight advantage can be given to the soft active platform as it requires less feedback bandwidth while providing better stability results. *** Effect of Limited Support Compliance @@ -2035,7 +2035,7 @@ To study the impact of the flexibility between the active platform and the paylo Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered. The resonances of the payload are set at $\omega_s = 20\,\text{Hz}$ and at $\omega_s = 200\,\text{Hz}$ while its mass is either $m_s = 1\,\text{kg}$ or $m_s = 50\,\text{kg}$. -The transfer functions from the active platform force $f$ to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. +The transfer functions from the active platform force $f$ to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod. It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern). The frequency of the anti-resonance corresponds to the "free" resonance of the sample $\omega_s = \sqrt{k_s/m_s}$. The flexibility of the sample also changes the high frequency gain (the mass line is shifted from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$). @@ -2044,13 +2044,13 @@ The flexibility of the sample also changes the high frequency gain (the mass lin #+caption: Effect of the payload dynamics on the soft active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,\text{N}/\upmu\text{m}$, $m_s = 50\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2058,7 +2058,7 @@ The flexibility of the sample also changes the high frequency gain (the mass lin #+end_subfigure #+end_figure -The same transfer functions are now compared when using a stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod. +The same transfer functions are now compared when using a stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod. In this case, the sample's resonance $\omega_s$ is smaller than the active platform resonance $\omega_n$. This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to $\omega_s$). Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy). @@ -2067,13 +2067,13 @@ Even though the added sample's flexibility still shifts the high frequency mass #+caption: Effect of the payload dynamics on the stiff active platform. Light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 1\,\text{kg}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,\text{N}/\upmu\text{m}$, $m_s = 1\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,\text{N}/\mu\text{m}$, $m_s = 50\,\text{kg}$} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,\text{N}/\upmu\text{m}$, $m_s = 50\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2105,7 +2105,7 @@ However, the cumulative amplitude spectrum of the distance $y$ (Figure\nbsp{}ref What happens is that above $\omega_s$, even though the motion $d$ can be controlled perfectly, the sample's mass is "isolated" from the motion of the active platform and the control on $y$ is not effective. #+name: fig:uniaxial_sample_flexibility_noise_budget -#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affect much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ and a active platform stiffness of $100\,\text{N}/\mu\text{m}$ are used for the simulations. +#+caption: Cumulative Amplitude Spectrum of the distances $d$ and $y$. The effect of the sample's flexibility does not affect much $d$ but is detrimental to the stability of $y$. A sample mass $m_s = 1\,\text{kg}$ and a active platform stiffness of $100\,\text{N}/\upmu\text{m}$ are used for the simulations. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$} @@ -2151,7 +2151,7 @@ However, this model does not allow the determination of which one is most suited Position feedback controllers have been developed for three considered active platform stiffnesses (Section\nbsp{}ref:sec:uniaxial_position_control). These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances. Having a soft active platform makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section\nbsp{}ref:sec:uniaxial_support_compliance) and requires less position feedback bandwidth to fulfill the requirements. -The moderately stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$) is requiring a higher feedback bandwidth, but still gives acceptable results. +The moderately stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$) is requiring a higher feedback bandwidth, but still gives acceptable results. However, the stiff active platform is the most complex to control and gives the worst positioning performance. ** Effect of Rotation @@ -2176,7 +2176,7 @@ This study of adapting acrshort:iff for the damping of rotating platforms has be It is then shown that acrfull:rdc is less affected by gyroscopic effects (Section\nbsp{}ref:sec:rotating_relative_damp_control). Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, damped plant and closed-loop compliance and transmissibility (Section\nbsp{}ref:sec:rotating_comp_act_damp). -The previous analysis was applied to three considered active platform stiffnesses ($k_n = 0.01\,\text{N}/\mu\text{m}$, $k_n = 1\,\text{N}/\mu\text{m}$ and $k_n = 100\,\text{N}/\mu\text{m}$) and the optimal active damping controller was obtained in each case (Section\nbsp{}ref:sec:rotating_nano_hexapod). +The previous analysis was applied to three considered active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$, $k_n = 1\,\text{N}/\upmu\text{m}$ and $k_n = 100\,\text{N}/\upmu\text{m}$) and the optimal active damping controller was obtained in each case (Section\nbsp{}ref:sec:rotating_nano_hexapod). Up until this section, the study was performed on a very simplistic model that only captures the rotation aspect, and the model parameters were not tuned to correspond to the NASS. In the last section (Section\nbsp{}ref:sec:rotating_nass), a model of the micro-station is added below the active platform with a rotating spindle and parameters tuned to match the NASS dynamics. The goal is to determine whether the rotation imposes performance limitation on the NASS. @@ -2842,7 +2842,7 @@ This is very well known characteristics of these common active damping technique <> ***** Introduction :ignore: The previous analysis is now applied to a model representing a rotating active platform. -Three active platform stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. +Three active platform stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\micro\m}$, $k_n = \SI{1}{\N\per\micro\m}$ and $k_n = \SI{100}{\N\per\micro\m}$. Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) because this is the worst identified case scenario in terms of gyroscopic effects. ***** Nano-Active-Stabilization-System - Plant Dynamics @@ -2858,19 +2858,19 @@ The coupling (or interaction) in a acrshort:mimo $2 \times 2$ system can be visu #+caption: Effect of rotation on the active platform dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nano_hexapod_dynamics_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nano_hexapod_dynamics_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2882,9 +2882,9 @@ The coupling (or interaction) in a acrshort:mimo $2 \times 2$ system can be visu Integral Force Feedback with an added acrlong:hpf is applied to the three active platforms. First, the parameters ($\omega_i$ and $g$) of the IFF controller that yield the best simultaneous damping are determined from Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain. The IFF parameters are chosen as follows: -- for $k_n = \SI{0.01}{\N\per\mu\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain): $\omega_i$ is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. +- for $k_n = \SI{0.01}{\N\per\micro\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain): $\omega_i$ is chosen such that maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. This is done to have some control robustness. -- for $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain_md and ref:fig:rotating_iff_hpf_nass_optimal_gain_pz): the largest $\omega_i$ is chosen such that the obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. +- for $k_n = \SI{1}{\N\per\micro\m}$ and $k_n = \SI{100}{\N\per\micro\m}$ (Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain_md and ref:fig:rotating_iff_hpf_nass_optimal_gain_pz): the largest $\omega_i$ is chosen such that the obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. Large $\omega_i$ is chosen here to limit the loss of compliance and the increase of coupling at low-frequency as shown in Section\nbsp{}ref:sec:rotating_iff_pseudo_int. The obtained IFF parameters and the achievable damping are visually shown by large dots in Figure\nbsp{}ref:fig:rotating_iff_hpf_nass_optimal_gain and are summarized in Table\nbsp{}ref:tab:rotating_iff_hpf_opt_iff_hpf_params_nass. @@ -2892,19 +2892,19 @@ The obtained IFF parameters and the achievable damping are visually shown by lar #+caption: For each value of $\omega_i$, the maximum damping ratio $\xi$ is computed (blue), and the corresponding controller gain is shown (in red). The chosen controller parameters used for further analysis are indicated by the large dots. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_hpf_nass_optimal_gain_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_hpf_nass_optimal_gain_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2918,9 +2918,9 @@ The obtained IFF parameters and the achievable damping are visually shown by lar #+attr_latex: :center t :booktabs t | $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ | |-----------------------+------------+------+------------------| -| $0.01\,\text{N}/\mu\text{m}$ | 7.3 | 51 | 0.45 | -| $1\,\text{N}/\mu\text{m}$ | 39 | 427 | 0.93 | -| $100\,\text{N}/\mu\text{m}$ | 500 | 3775 | 0.94 | +| $0.01\,\text{N}/\upmu\text{m}$ | 7.3 | 51 | 0.45 | +| $1\,\text{N}/\upmu\text{m}$ | 39 | 427 | 0.93 | +| $100\,\text{N}/\upmu\text{m}$ | 500 | 3775 | 0.94 | ***** Optimal IFF with Parallel Stiffness For each considered active platform stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total active platform stiffness). @@ -2933,7 +2933,7 @@ For the two stiff options, the achievable damping decreases when the parallel st Such behavior can be explained by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chapt 7.2]]. This distance is larger for stiff active platform because the open-loop pole will be at higher frequencies while the open-loop zero, whereas depends on the value of the parallel stiffness, can only be made large for stiff active platforms. -Let's choose $k_p = 1\,\text{N/mm}$, $k_p = 0.01\,\text{N}/\mu\text{m}$ and $k_p = 1\,\text{N}/\mu\text{m}$ for the three considered active platforms. +Let's choose $k_p = 1\,\text{N/mm}$, $k_p = 0.01\,\text{N}/\upmu\text{m}$ and $k_p = 1\,\text{N}/\upmu\text{m}$ for the three considered active platforms. The corresponding optimal controller gains and achievable damping are summarized in Table\nbsp{}ref:tab:rotating_iff_kp_opt_iff_kp_params_nass. #+attr_latex: :options [b]{0.49\linewidth} @@ -2951,9 +2951,9 @@ The corresponding optimal controller gains and achievable damping are summarized #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf | $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ | |-----------------------+-----------------------+---------+--------------------| -| $0.01\,\text{N}/\mu\text{m}$ | $1\,\text{N/mm}$ | 47.9 | 0.44 | -| $1\,\text{N}/\mu\text{m}$ | $0.01\,\text{N}/\mu\text{m}$ | 465.57 | 0.97 | -| $100\,\text{N}/\mu\text{m}$ | $1\,\text{N}/\mu\text{m}$ | 4624.25 | 0.99 | +| $0.01\,\text{N}/\upmu\text{m}$ | $1\,\text{N/mm}$ | 47.9 | 0.44 | +| $1\,\text{N}/\upmu\text{m}$ | $0.01\,\text{N}/\upmu\text{m}$ | 465.57 | 0.97 | +| $100\,\text{N}/\upmu\text{m}$ | $1\,\text{N}/\upmu\text{m}$ | 4624.25 | 0.99 | #+latex: \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses} #+end_minipage @@ -2976,9 +2976,9 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf | $k_n$ | $g$ | $\xi_{\text{opt}}$ | |-----------------------+-------+--------------------| -| $0.01\,\text{N}/\mu\text{m}$ | 1600 | 0.99 | -| $1\,\text{N}/\mu\text{m}$ | 8200 | 0.99 | -| $100\,\text{N}/\mu\text{m}$ | 80000 | 0.99 | +| $0.01\,\text{N}/\upmu\text{m}$ | 1600 | 0.99 | +| $1\,\text{N}/\upmu\text{m}$ | 8200 | 0.99 | +| $100\,\text{N}/\upmu\text{m}$ | 80000 | 0.99 | #+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} #+end_minipage @@ -2994,19 +2994,19 @@ Similar to what was concluded in the previous analysis: #+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by solid lines, and the coupling terms are shown by the shaded lines. Three active platform stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_damped_plant_comp_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_damped_plant_comp_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3024,10 +3024,10 @@ In this section, the limited compliance of the micro-station is considered as we ***** Nano Active Stabilization System Model To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed -- a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$) -- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_t = \SI{600}{\kg}$) +- a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\micro\m}$, $m_g = \SI{2500}{\kg}$) +- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\micro\m}$, $m_t = \SI{600}{\kg}$) - a spindle (vertical rotation) stage whose rotation is imposed ($m_s = \SI{600}{\kg}$) -- a 2-DoF positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_h = \SI{15}{\kg}$) +- a 2-DoF positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\micro\m}$, $m_h = \SI{15}{\kg}$) A payload is rigidly fixed to the active platform and the $x,y$ motion of the payload is measured with respect to the granite. @@ -3050,19 +3050,19 @@ It can be observed that: #+caption: Bode plot of the transfer function from active platform actuator to measured motion by the external metrology #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_plant_comp_stiffness_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_plant_comp_stiffness_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3086,19 +3086,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: #+caption: Effect of floor motion $x_{f,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. IFF is shown to increase the sensitivity to floor motion at low-frequency. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_floor_motion_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_floor_motion_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3110,19 +3110,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: #+caption: Effect of micro-station vibrations $f_{t,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft active platform suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc}) #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_stage_vibration_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_stage_vibration_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3134,19 +3134,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model: #+caption: Effect of sample forces $f_{s,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three active platform stiffnesses. Integral Force Feedback degrades this compliance at low-frequency. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_direct_forces_vc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_nass_effect_direct_forces_md.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,\text{N}/\mu\text{m}$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,\text{N}/\upmu\text{m}$} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -3176,12 +3176,12 @@ While having very different implementations, both proposed modifications were fo This study has been applied to a rotating platform that corresponds to the active platform parameters. As for the uniaxial model, three active platform stiffnesses values were considered. -The dynamics of the soft active platform ($k_n = 0.01\,\text{N}/\mu\text{m}$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). +The dynamics of the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects). In addition, the attainable damping ratio of the soft active platform when using acrshort:iff is limited by gyroscopic effects. To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered. Results are similar to those of the uniaxial model except that come complexity is added for the soft active platform due to the spindle's rotation. -For the moderately stiff active platform ($k_n = 1\,\text{N}/\mu\text{m}$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model. +For the moderately stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model. ** Micro Station - Modal Analysis <> @@ -4170,13 +4170,13 @@ The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_ #+caption: Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section\nbsp{}ref:ssec:ustation_stages #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccccc #+attr_latex: :center t :booktabs t -| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | -|-------------+-----------------------------+-----------------------------+-----------------------------+--------------------------------+-------------------------------+--------------------------------| -| Granite | $5\,\text{kN}/\mu\text{m}$ | $5\,\text{kN}/\mu\text{m}$ | $5\,\text{kN}/\mu\text{m}$ | $25\,\text{Nm}/\mu\text{rad}$ | $25\,\text{Nm}/\mu\text{rad}$ | $10\,\text{Nm}/\mu\text{rad}$ | -| Translation | $200\,\text{N}/\mu\text{m}$ | - | $200\,\text{N}/\mu\text{m}$ | $60\,\text{Nm}/\mu\text{rad}$ | $90\,\text{Nm}/\mu\text{rad}$ | $60\,\text{Nm}/\mu\text{rad}$ | -| Tilt | $380\,\text{N}/\mu\text{m}$ | $400\,\text{N}/\mu\text{m}$ | $380\,\text{N}/\mu\text{m}$ | $120\,\text{Nm}/\mu\text{rad}$ | - | $120\,\text{Nm}/\mu\text{rad}$ | -| Spindle | $700\,\text{N}/\mu\text{m}$ | $700\,\text{N}/\mu\text{m}$ | $2\,\text{kN}/\mu\text{m}$ | $10\,\text{Nm}/\mu\text{rad}$ | $10\,\text{Nm}/\mu\text{rad}$ | - | -| Hexapod | $10\,\text{N}/\mu\text{m}$ | $10\,\text{N}/\mu\text{m}$ | $100\,\text{N}/\mu\text{m}$ | $1.5\,\text{Nm/rad}$ | $1.5\,\text{Nm/rad}$ | $0.27\,\text{Nm/rad}$ | +| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | +|-------------+-------------------------------+-------------------------------+-------------------------------+----------------------------------+---------------------------------+----------------------------------| +| Granite | $5\,\text{kN}/\upmu\text{m}$ | $5\,\text{kN}/\upmu\text{m}$ | $5\,\text{kN}/\upmu\text{m}$ | $25\,\text{Nm}/\upmu\text{rad}$ | $25\,\text{Nm}/\upmu\text{rad}$ | $10\,\text{Nm}/\upmu\text{rad}$ | +| Translation | $200\,\text{N}/\upmu\text{m}$ | - | $200\,\text{N}/\upmu\text{m}$ | $60\,\text{Nm}/\upmu\text{rad}$ | $90\,\text{Nm}/\upmu\text{rad}$ | $60\,\text{Nm}/\upmu\text{rad}$ | +| Tilt | $380\,\text{N}/\upmu\text{m}$ | $400\,\text{N}/\upmu\text{m}$ | $380\,\text{N}/\upmu\text{m}$ | $120\,\text{Nm}/\upmu\text{rad}$ | - | $120\,\text{Nm}/\upmu\text{rad}$ | +| Spindle | $700\,\text{N}/\upmu\text{m}$ | $700\,\text{N}/\upmu\text{m}$ | $2\,\text{kN}/\upmu\text{m}$ | $10\,\text{Nm}/\upmu\text{rad}$ | $10\,\text{Nm}/\upmu\text{rad}$ | - | +| Hexapod | $10\,\text{N}/\upmu\text{m}$ | $10\,\text{N}/\upmu\text{m}$ | $100\,\text{N}/\upmu\text{m}$ | $1.5\,\text{Nm/rad}$ | $1.5\,\text{Nm/rad}$ | $0.27\,\text{Nm/rad}$ | **** Comparison with the Measured Dynamics <> @@ -4546,7 +4546,7 @@ Second, a constant velocity scans with the translation stage was performed and a To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm. Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section\nbsp{}ref:sec:ustation_disturbances. -A radial offset of $\approx 1\,\mu\text{m}$ between the acrfull:poi and the spindle's rotation axis is introduced to represent what is experimentally observed. +A radial offset of $\approx 1\,\upmu\text{m}$ between the acrfull:poi and the spindle's rotation axis is introduced to represent what is experimentally observed. During the 10 second simulation (i.e. 10 spindle turns), the position of the acrshort:poi with respect to the granite was recorded. Results are shown in Figure\nbsp{}ref:fig:ustation_errors_model_spindle. A good correlation with the measurements is observed both for radial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_radial) and axial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_axial). @@ -4638,7 +4638,7 @@ To overcome this limitation, external metrology systems have been implemented to A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems. In many cases, sample position control is limited to translational acrshortpl:dof. -At NSLS-II, for instance, a system capable of $100\,\mu\text{m}$ stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure\nbsp{}ref:fig:nhexa_stages_nazaretski). +At NSLS-II, for instance, a system capable of $100\,\upmu\text{m}$ stroke has been developed for payloads up to 500g, using interferometric measurements for position feedback (Figure\nbsp{}ref:fig:nhexa_stages_nazaretski). Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately $100\,\text{Hz}$ (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). #+name: fig:nhexa_stages_translations @@ -4661,8 +4661,8 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac The integration of $R_z$ rotational capability, which is necessary for tomography experiments, introduces additional complexity. At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section\nbsp{}ref:sec:nhexa_stewart_platform) using piezoelectric actuators has been positioned below the spindle (Figure\nbsp{}ref:fig:nhexa_stages_villar). -While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\mu\text{m}$ due to the inherent constraints of piezoelectric actuators. -In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\mu\text{m}$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). +While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\upmu\text{m}$ due to the inherent constraints of piezoelectric actuators. +In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\upmu\text{m}$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration. #+name: fig:nhexa_stages_spindle @@ -4698,7 +4698,7 @@ Although direct performance comparisons between these systems are challenging du | Sample | light | Capacitive sensors | $\approx 10\,\text{Hz}$ | ESRF | | Spindle | $R_z: \pm 90\,\text{deg}$ | $D_{xyz},\ R_{xy}$ | | ID16a | | *Hexapod (piezo)* | $D_{xyz}: 0.05\,\text{mm}$ | | | \nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] | -| | $R_{xy}: 500\,\mu\text{rad}$ | | | | +| | $R_{xy}: 500\,\upmu\text{rad}$ | | | | |---------------------+------------------------------+---------------------+--------------------------+--------------------------------------------------------------------------------------------------------------| | Sample | light | Interferometers | n/a | PETRA III | | *XYZ stage (piezo)* | $D_{xyz}: 0.1\,\text{mm}$ | $D_{yz}$ | | P06 | @@ -4752,10 +4752,10 @@ The primary control requirements focus on $[D_y,\ D_z,\ R_y]$ motions; however, <> The choice of the active platform architecture for the NASS requires careful consideration of several critical specifications. -The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\mu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of $300\,\text{mm}$ diameter and $95\,\text{mm}$ height. +The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\upmu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of $300\,\text{mm}$ diameter and $95\,\text{mm}$ height. It must accommodate payloads up to $50\,\text{kg}$ while maintaining high dynamical performance. For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to $100\,\text{Hz}$. -However, achieving such resonance frequencies with a $50\,\text{kg}$ payload would require unrealistic stiffness values of approximately $2000\,\text{N}/\mu\text{m}$. +However, achieving such resonance frequencies with a $50\,\text{kg}$ payload would require unrealistic stiffness values of approximately $2000\,\text{N}/\upmu\text{m}$. This limitation necessitates alternative control approaches, and the High acrfull:haclac strategy is proposed to address this challenge. To this purpose, the design includes force sensors for active damping. Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities. @@ -4832,7 +4832,7 @@ Second, its compact design compared to serial manipulators makes it ideal for in Third, the good dynamical properties should enable high-bandwidth positioning control. While Stewart platforms excel in precision and stiffness, they typically exhibit a relatively limited workspace compared to serial manipulators. -However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of $10\,\mu\text{m}$. +However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station, which are in the order of $10\,\upmu\text{m}$. This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications. The analysis encompasses the platform's kinematic relationships (Section\nbsp{}ref:ssec:nhexa_stewart_platform_kinematics), the use of the Jacobian matrix (Section\nbsp{}ref:ssec:nhexa_stewart_platform_jacobian), static behavior (Section\nbsp{}ref:ssec:nhexa_stewart_platform_static), and dynamic characteristics (Section\nbsp{}ref:ssec:nhexa_stewart_platform_dynamics). @@ -4984,10 +4984,10 @@ The accuracy of the Jacobian-based forward kinematics solution was estimated by For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation\nbsp{}eqref:eq:nhexa_inverse_kinematics. These strut lengths are then used with the Jacobian to estimate the platform pose\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$. -For motion strokes from $1\,\mu\text{m}$ to $10\,\text{mm}$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure\nbsp{}ref:fig:nhexa_forward_kinematics_approximate_errors. -The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\mu\text{m}$ as the size of the Stewart platform is here $\approx 100\,\text{mm}$), the Jacobian approximation provides excellent accuracy. +For motion strokes from $1\,\upmu\text{m}$ to $10\,\text{mm}$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure\nbsp{}ref:fig:nhexa_forward_kinematics_approximate_errors. +The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\upmu\text{m}$ as the size of the Stewart platform is here $\approx 100\,\text{mm}$), the Jacobian approximation provides excellent accuracy. -Since the maximum required stroke of the active platform ($\approx 100\,\mu\text{m}$) is three orders of magnitude smaller than its overall size ($\approx 100\,\text{mm}$), the Jacobian matrix can be considered constant throughout the workspace. +Since the maximum required stroke of the active platform ($\approx 100\,\upmu\text{m}$) is three orders of magnitude smaller than its overall size ($\approx 100\,\text{mm}$), the Jacobian matrix can be considered constant throughout the workspace. It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy. #+name: fig:nhexa_forward_kinematics_approximate_errors @@ -5220,9 +5220,9 @@ This modular approach to actuator modeling allows for future refinements as the #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf | | Value | |-------+------------------------------| -| $k_a$ | $1\,\text{N}/\mu\text{m}$ | +| $k_a$ | $1\,\text{N}/\upmu\text{m}$ | | $c_a$ | $50\,\text{Ns}/\text{m}$ | -| $k_p$ | $0.05\,\text{N}/\mu\text{m}$ | +| $k_p$ | $0.05\,\text{N}/\upmu\text{m}$ | #+latex: \captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters} #+end_minipage @@ -5250,7 +5250,7 @@ A three-dimensional visualization of the model is presented in Figure\nbsp{}ref: #+end_minipage The validation of the multi-body model was performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section\nbsp{}ref:ssec:nhexa_stewart_platform_dynamics. -This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness $k_a = 1\,\text{N}/\mu\text{m}$ and damping $c_a = 10\,\text{N}/({\text{m}/\text{s}})$. +This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness $k_a = 1\,\text{N}/\upmu\text{m}$ and damping $c_a = 10\,\text{N}/({\text{m}/\text{s}})$. The geometric parameters remain as specified in Table\nbsp{}ref:tab:nhexa_actuator_parameters. While the moving platform itself is considered massless, a $10\,\text{kg}$ cylindrical payload is mounted on top with a radius of $r = 110\,\text{mm}$ and a height $h = 300\,\text{mm}$. @@ -5609,7 +5609,7 @@ Furthermore, a multi-body model of the active platform was created, that can the [[file:figs/nass_simscape_model.jpg]] Building upon these foundations, this chapter presents the validation of the NASS concept. -The investigation begins with the previously established active platform model with actuator stiffness $k_a = 1\,\text{N}/\mu\text{m}$. +The investigation begins with the previously established active platform model with actuator stiffness $k_a = 1\,\text{N}/\upmu\text{m}$. A thorough examination of the control kinematics is presented in Section\nbsp{}ref:sec:nass_kinematics, detailing how both external metrology and active platform internal sensors are used in the control architecture. The control strategy is then implemented in two steps: first, the decentralized IFF is used for active damping (Section\nbsp{}ref:sec:nass_active_damping), then a High Authority Control is develop to stabilize the sample's position in a large bandwidth (Section\nbsp{}ref:sec:nass_hac). @@ -5794,7 +5794,7 @@ Although both cases show significant coupling around the resonances, stability i #+end_subfigure #+end_figure -The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\mu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). +The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\upmu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). Figure\nbsp{}ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics. The poles and zeros shift in frequency as the payload mass varies. @@ -5969,29 +5969,29 @@ This result confirms effective dynamic decoupling between the active platform an <> The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models. -These models suggest that a moderate stiffness of approximately $1\,\text{N}/\mu\text{m}$ would provide better performance than either very stiff or very soft configurations. +These models suggest that a moderate stiffness of approximately $1\,\text{N}/\upmu\text{m}$ would provide better performance than either very stiff or very soft configurations. -For the stiff active platform analysis, a system with an actuator stiffness of $100\,\text{N}/\mu\text{m}$ was simulated with a $25\,\text{kg}$ payload. +For the stiff active platform analysis, a system with an actuator stiffness of $100\,\text{N}/\upmu\text{m}$ was simulated with a $25\,\text{kg}$ payload. The transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station. As shown in Figure\nbsp{}ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the active platform and micro-station dynamics. This coupling introduces complex behavior that is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model. -The soft active platform configuration was evaluated using a stiffness of $0.01\,\text{N}/\mu\text{m}$ with a $25\,\text{kg}$ payload. +The soft active platform configuration was evaluated using a stiffness of $0.01\,\text{N}/\upmu\text{m}$ with a $25\,\text{kg}$ payload. The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s. Figure\nbsp{}ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies. The current approach of controlling the position in the strut frame is inadequate for soft active platforms; but even shifting control to a frame matching the payload's acrlong:com would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. #+name: fig:nass_soft_stiff_hexapod -#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\mu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,\text{N}/\mu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}) +#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\upmu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,\text{N}/\upmu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}) #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\mu\text{m}$ - Coupling with the micro-station} +#+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\upmu\text{m}$ - Coupling with the micro-station} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/nass_stiff_nano_hexapod_coupling_ustation.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,\text{N}/\mu\text{m}$ - Effect of Spindle rotation} +#+attr_latex: :caption \subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,\text{N}/\upmu\text{m}$ - Effect of Spindle rotation} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -6107,7 +6107,7 @@ The control strategy implements a High Authority Control - Low Authority Control The decentralized Integral Force Feedback component has been demonstrated to provide robust active damping under various operating conditions. The addition of parallel springs to the force sensors has been shown to ensure stability during spindle rotation. The centralized High Authority Controller, operating in the frame of the struts for simplicity, has successfully achieved the desired performance objectives of maintaining a bandwidth of $10\,\text{Hz}$ while maintaining robustness against payload mass variations. -This investigation has confirmed that the moderate actuator stiffness of $1\,\text{N}/\mu\text{m}$ represents an adequate choice for the active platform, as both very stiff and very compliant configurations introduce significant performance limitations. +This investigation has confirmed that the moderate actuator stiffness of $1\,\text{N}/\upmu\text{m}$ represents an adequate choice for the active platform, as both very stiff and very compliant configurations introduce significant performance limitations. Simulations of tomography experiments have been performed, with positioning accuracy requirements defined by the expected minimum beam dimensions of $200\,\text{nm}$ by $100\,\text{nm}$. The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples. @@ -6124,7 +6124,7 @@ Through a systematic progression from simplified to increasingly complex models, Using the simple uniaxial model revealed that a very stiff stabilization stage was unsuitable due to its strong coupling with the complex micro-station dynamics. Conversely, the three-degree-of-freedom rotating model demonstrated that very soft stabilization stage designs are equally problematic due to the gyroscopic effects induced by spindle rotation. -A moderate stiffness of approximately $1\,\text{N}/\mu\text{m}$ was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection. +A moderate stiffness of approximately $1\,\text{N}/\upmu\text{m}$ was identified as the optimal configuration, providing an effective balance between decoupling from micro-station dynamics, insensitivity to spindle's rotation, and good disturbance rejection. The multi-body modeling approach proved essential for capturing the complex dynamics of both the micro-station and the active platform. This model was tuned based on extensive modal analysis and vibration measurements. @@ -6220,7 +6220,7 @@ Long stroke Stewart platforms are not addressed here as their design presents di In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators. Voice coil actuators, providing stroke ranges from $0.5\,\text{mm}$ to $10\,\text{mm}$, are commonly implemented in cubic architectures (as illustrated in Figures\nbsp{}ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) and are mainly used for vibration isolation\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax;&thayer98_stewar;&mcinroy99_dynam;&preumont07_six_axis_singl_stage_activ]]. -For applications requiring short stroke (typically smaller than $500\,\mu\text{m}$), piezoelectric actuators present an interesting alternative, as shown in\nbsp{}[[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]]. +For applications requiring short stroke (typically smaller than $500\,\upmu\text{m}$), piezoelectric actuators present an interesting alternative, as shown in\nbsp{}[[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]]. Examples of piezoelectric-actuated Stewart platforms are presented in Figures\nbsp{}ref:fig:detail_kinematics_ulb_pz, ref:fig:detail_kinematics_uqp and ref:fig:detail_kinematics_yang19. Although less frequently encountered, magnetostrictive actuators have been successfully implemented in\nbsp{}[[cite:&zhang11_six_dof]] (Figure\nbsp{}ref:fig:detail_kinematics_zhang11). @@ -7039,8 +7039,8 @@ This is the location where precise control of the sample's position is required, The design of the active platform must satisfy several constraints. The device should fit within a cylinder with radius of $120\,\text{mm}$ and height of $95\,\text{mm}$. -Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of $\pm 50\,\mu\text{m}$. -At any position, the system should be capable of performing $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$. +Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of $\pm 50\,\upmu\text{m}$. +At any position, the system should be capable of performing $R_x$ and $R_y$ rotations of $\pm 50\,\upmu \text{rad}$. Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of $2\pi\,\text{rad/s}$ to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. @@ -7095,13 +7095,13 @@ Given the impracticality of consistently aligning the acrlong:com with the cube' With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined. -The required mobility parameters include combined translations in the XYZ directions of $\pm 50\,\mu\text{m}$ (essentially a cubic workspace). -Additionally, at any point within this workspace, combined $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$, with $R_z$ maintained at 0, should be possible. +The required mobility parameters include combined translations in the XYZ directions of $\pm 50\,\upmu\text{m}$ (essentially a cubic workspace). +Additionally, at any point within this workspace, combined $R_x$ and $R_y$ rotations of $\pm 50\,\upmu\text{rad}$, with $R_z$ maintained at 0, should be possible. -Calculations based on the selected geometry indicate that an actuator stroke of $\pm 94\,\mu\text{m}$ is required to achieve the desired mobility. +Calculations based on the selected geometry indicate that an actuator stroke of $\pm 94\,\upmu\text{m}$ is required to achieve the desired mobility. This specification will be used during the actuator selection process in Section\nbsp{}ref:sec:detail_fem_actuator. -Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the active platform with an actuator stroke of $\pm 94\,\mu\text{m}$. +Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the active platform with an actuator stroke of $\pm 94\,\upmu\text{m}$. The diagram confirms that the required workspace fits within the system's capabilities. #+name: fig:detail_kinematics_nano_hexapod_mobility @@ -7213,9 +7213,9 @@ The specific design of the acrshort:apa allows for the simultaneous modeling of #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf | *Parameter* | *Value* | |----------------+----------------------------| -| Nominal Stroke | $100\,\mu\text{m}$ | +| Nominal Stroke | $100\,\upmu\text{m}$ | | Blocked force | $2100\,\text{N}$ | -| Stiffness | $21\,\text{N}/\mu\text{m}$ | +| Stiffness | $21\,\text{N}/\upmu\text{m}$ | #+latex: \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications} #+end_minipage @@ -7293,16 +7293,16 @@ Yet, based on the available properties of the stacks in the data-sheet (summariz #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | |----------------+-----------------------------| -| Nominal Stroke | $20\,\mu\text{m}$ | +| Nominal Stroke | $20\,\upmu\text{m}$ | | Blocked force | $4700\,\text{N}$ | -| Stiffness | $235\,\text{N}/\mu\text{m}$ | +| Stiffness | $235\,\text{N}/\upmu\text{m}$ | | Voltage Range | $-20/150\,\text{V}$ | -| Capacitance | $4.4\,\mu\text{F}$ | +| Capacitance | $4.4\,\upmu\text{F}$ | | Length | $20\,\text{mm}$ | | Stack Area | $10\times 10\,\text{mm}^2$ | The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:detail_fem_piezo_properties. -From these parameters, $g_s = 5.1\,\text{V}/\mu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained. +From these parameters, $g_s = 5.1\,\text{V}/\upmu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained. #+name: tab:detail_fem_piezo_properties #+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities @@ -7324,7 +7324,7 @@ Initial validation of the acrlong:fem and its integration as a reduced-order fle The stiffness of the APA95ML was estimated from the multi-body model by computing the axial compliance of the APA95ML (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which corresponds to the transfer function from a vertical force applied between the two interface frames to the relative vertical displacement between these two frames. The inverse of the DC gain this transfer function corresponds to the axial stiffness of the APA95ML. -A value of $23\,\text{N}/\mu\text{m}$ was found which is close to the specified stiffness in the datasheet of $k = 21\,\text{N}/\mu\text{m}$. +A value of $23\,\text{N}/\upmu\text{m}$ was found which is close to the specified stiffness in the datasheet of $k = 21\,\text{N}/\upmu\text{m}$. The multi-body model predicted a resonant frequency under block-free conditions of $\approx 2\,\text{kHz}$ (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which is in agreement with the nominal specification. @@ -7336,9 +7336,9 @@ The multi-body model predicted a resonant frequency under block-free conditions In order to estimate the stroke of the APA95ML, the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, was first determined. This characteristic was quantified through analysis of the transfer function relating horizontal stack motion to vertical actuator displacement, from which an amplification factor of $1.5$ was derived. -The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,\text{mm}$), produce an individual nominal stroke of $20\,\mu\text{m}$ (see data-sheet of the piezoelectric stacks on Table\nbsp{}ref:tab:detail_fem_stack_parameters, page\nbsp{}pageref:tab:detail_fem_stack_parameters). -As three stacks are used, the horizontal displacement is $60\,\mu\text{m}$. -Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of $90\,\mu\text{m}$ which falls within the manufacturer-specified range of $80\,\mu\text{m}$ and $120\,\mu\text{m}$. +The piezoelectric stacks, exhibiting a typical strain response of $0.1\,\%$ relative to their length (here equal to $20\,\text{mm}$), produce an individual nominal stroke of $20\,\upmu\text{m}$ (see data-sheet of the piezoelectric stacks on Table\nbsp{}ref:tab:detail_fem_stack_parameters, page\nbsp{}pageref:tab:detail_fem_stack_parameters). +As three stacks are used, the horizontal displacement is $60\,\upmu\text{m}$. +Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of $90\,\upmu\text{m}$ which falls within the manufacturer-specified range of $80\,\upmu\text{m}$ and $120\,\upmu\text{m}$. The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include acrshort:fem into multi-body model. @@ -7445,13 +7445,13 @@ The actuator selection process was driven by several critical requirements deriv A primary consideration is the actuator stiffness, which significantly impacts system dynamics through multiple mechanisms. The spindle rotation induces gyroscopic effects that modify plant dynamics and increase coupling, necessitating sufficient stiffness. Conversely, the actuator stiffness must be carefully limited to ensure the active platform's suspension modes remain below the problematic modes of the micro-station to limit the coupling between the two structures. -These competing requirements suggest an optimal stiffness of approximately $1\,\text{N}/\mu\text{m}$. +These competing requirements suggest an optimal stiffness of approximately $1\,\text{N}/\upmu\text{m}$. Additional specifications arise from the control strategy and physical constraints. The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator. The system's geometric constraints limit the actuator height to $50\,\text{mm}$, given the active platform's maximum height of $95\,\text{mm}$ and the presence of flexible joints at each strut extremity. Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility. -An actuator stroke of $\approx 200\,\mu\text{m}$ is therefore required. +An actuator stroke of $\approx 200\,\upmu\text{m}$ is therefore required. Three actuator technologies were evaluated (examples of such actuators are shown in Figure\nbsp{}ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity. @@ -7480,13 +7480,13 @@ Variable reluctance actuators were not considered despite their superior efficie #+end_subfigure #+end_figure -Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,\text{N}/\mu\text{m}$), would require forces in the order of $200\,\text{N}$ to achieve the specified $200\,\mu\text{m}$ displacement. +Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,\text{N}/\upmu\text{m}$), would require forces in the order of $200\,\text{N}$ to achieve the specified $200\,\upmu\text{m}$ displacement. While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability. Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions. Conventional piezoelectric stack actuators (shown in Figure\nbsp{}ref:fig:detail_fem_piezo_picture) present two significant limitations for the current application. -Their stroke is inherently limited to approximately $0.1\,\%$ of their length, meaning that even with the maximum allowable height of $50\,\text{mm}$, the achievable stroke would only be $50\,\mu\text{m}$, insufficient for the application. -Additionally, their extremely high stiffness, typically around $100\,\text{N}/\mu\text{m}$, exceeds the desired specifications by two orders of magnitude. +Their stroke is inherently limited to approximately $0.1\,\%$ of their length, meaning that even with the maximum allowable height of $50\,\text{mm}$, the achievable stroke would only be $50\,\upmu\text{m}$, insufficient for the application. +Additionally, their extremely high stiffness, typically around $100\,\text{N}/\upmu\text{m}$, exceeds the desired specifications by two orders of magnitude. Amplified Piezoelectric Actuators emerged as the optimal solution by addressing these limitations through a specific mechanical design. The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile. @@ -7504,8 +7504,8 @@ The demonstrated accuracy of the modeling approach for the APA95ML provides conf #+attr_latex: :center t :booktabs t :float t | *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S | |----------------------------------------------+---------+------------+-----------+-------------+-------------| -| Stroke $> 200\,\mu\text{m}$ | 187 | 304 | 368 | 432 | 240 | -| Stiffness $\approx 1\,\text{N}/\mu\text{m}$ | 0.7 | 1.8 | 0.55 | 0.87 | 0.58 | +| Stroke $> 200\,\upmu\text{m}$ | 187 | 304 | 368 | 432 | 240 | +| Stiffness $\approx 1\,\text{N}/\upmu\text{m}$ | 0.7 | 1.8 | 0.55 | 0.87 | 0.58 | | Resolution $< 2\,\text{nm}$ | 2 | 3 | 4 | | | | Blocked Force $> 100\,\text{N}$ | 127 | 546 | 201 | 376 | 139 | | Height $< 50\,\text{mm}$ | 22 | 30 | 24 | 27 | 16 | @@ -7580,14 +7580,14 @@ While higher-order modes and non-axial flexibility are not captured, the model a #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | |-------------+------------------------------| -| $k_1$ | $0.30\,\text{N}/\mu\text{m}$ | -| $k_e$ | $4.3\,\text{N}/\mu\text{m}$ | -| $k_a$ | $2.15\,\text{N}/\mu\text{m}$ | +| $k_1$ | $0.30\,\text{N}/\upmu\text{m}$ | +| $k_e$ | $4.3\,\text{N}/\upmu\text{m}$ | +| $k_a$ | $2.15\,\text{N}/\upmu\text{m}$ | | $c_1$ | $18\,\text{Ns/m}$ | | $c_e$ | $0.7\,\text{Ns/m}$ | | $c_a$ | $0.35\,\text{Ns/m}$ | | $g_a$ | $2.7\,\text{N}/V$ | -| $g_s$ | $0.53\,\text{V}/\mu\text{m}$ | +| $g_s$ | $0.53\,\text{V}/\upmu\text{m}$ | #+name: fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof #+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}) @@ -7705,7 +7705,7 @@ The validation process, detailed in Section\nbsp{}ref:ssec:detail_fem_joint_vali The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]] and can affect system dynamics. -To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of $1\,\text{N}/\mu\text{m}$) without parallel stiffness to the force sensors. +To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of $1\,\text{N}/\upmu\text{m}$) without parallel stiffness to the force sensors. Flexible joint bending stiffness was varied from 0 (ideal case) to $500\,\text{Nm}/\text{rad}$. Analysis of the plant dynamics reveals two significant effects. @@ -7767,7 +7767,7 @@ As explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, Therefore, determining the minimum acceptable axial stiffness that maintains active platform performance becomes crucial. The analysis incorporates the strut mass (112g per APA300ML) to accurately model internal resonance effects. -A parametric study was conducted by varying the axial stiffness from $1\,\text{N}/\mu\text{m}$ (matching actuator stiffness) to $1000\,\text{N}/\mu\text{m}$ (approximating rigid behavior). +A parametric study was conducted by varying the axial stiffness from $1\,\text{N}/\upmu\text{m}$ (matching actuator stiffness) to $1000\,\text{N}/\upmu\text{m}$ (approximating rigid behavior). The resulting dynamics (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_plants) reveal distinct effects on system dynamics. The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both acrshortpl:frf (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_plant) and root locus analysis (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_locus). @@ -7780,7 +7780,7 @@ First, the system exhibits strong coupling between control channels, making dece Second, control authority diminishes significantly near the resonant frequencies. These effects fundamentally limit achievable control bandwidth, making high axial stiffness essential for system performance. -Based on this analysis, an axial stiffness specification of $100\,\text{N}/\mu\text{m}$ was established for the active platform joints. +Based on this analysis, an axial stiffness specification of $100\,\text{N}/\upmu\text{m}$ was established for the active platform joints. #+name: fig:detail_fem_joints_axial_stiffness_plants #+caption: Effect of axial stiffness of the flexible joints on the plant dynamics. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_axial_stiffness_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_axial_stiffness_iff_plant}) @@ -7831,8 +7831,8 @@ Based on the dynamic analysis presented in previous sections, quantitative speci #+attr_latex: :center t :booktabs t :float t | | *Specification* | *FEM* | |-------------------------+-------------------------------+-------| -| Axial Stiffness $k_a$ | $> 100\,\text{N}/\mu\text{m}$ | 94 | -| Shear Stiffness $k_s$ | $> 1\,\text{N}/\mu\text{m}$ | 13 | +| Axial Stiffness $k_a$ | $> 100\,\text{N}/\upmu\text{m}$ | 94 | +| Shear Stiffness $k_s$ | $> 1\,\text{N}/\upmu\text{m}$ | 13 | | Bending Stiffness $k_f$ | $< 100\,\text{Nm}/\text{rad}$ | 5 | | Torsion Stiffness $k_t$ | $< 500\,\text{Nm}/\text{rad}$ | 260 | | Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | @@ -8534,7 +8534,7 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O |-------+-----------------------+----------------------------| | $l_a$ | | $0.5\,\text{m}$ | | $h_a$ | | $0.2\,\text{m}$ | -| $k$ | Actuator stiffness | $10\,\text{N}/\mu\text{m}$ | +| $k$ | Actuator stiffness | $10\,\text{N}/\upmu\text{m}$ | | $c$ | Actuator damping | $200\,\text{Ns/m}$ | | $m$ | Payload mass | $40\,\text{kg}$ | | $I$ | Payload $R_z$ inertia | $5\,\text{kgm}^2$ | @@ -9294,7 +9294,7 @@ The positioning stage itself is characterized by stiffness $k$, internal damping The model of the plant $G(s)$ from actuator force $F$ to displacement $y$ is described by Equation\nbsp{}eqref:eq:detail_control_cf_test_plant_tf. \begin{equation}\label{eq:detail_control_cf_test_plant_tf} - G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = 1\si{\N/\mu\m},\ c = 10^2\si{\N\per(\m\per\s)} + G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = \SI{1}{\N/\micro\m},\ c = \SI{100}{\N\per(\m\per\s)} \end{equation} The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics and payload dynamics. @@ -9507,7 +9507,7 @@ In order to derive specifications in terms of noise spectral density for each in The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met. While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications. -Based on this analysis, the obtained maximum noise levels are as follows: acrshort:dac maximum output noise acrshort:asd is established at $14\,\mu\text{V}/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise acrshort:asd at $280\,\mu\text{V}/\sqrt{\text{Hz}}$, and acrshort:adc maximum measurement noise acrshort:asd at $11\,\mu\text{V}/\sqrt{\text{Hz}}$. +Based on this analysis, the obtained maximum noise levels are as follows: acrshort:dac maximum output noise acrshort:asd is established at $14\,\upmu\text{V}/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise acrshort:asd at $280\,\upmu\text{V}/\sqrt{\text{Hz}}$, and acrshort:adc maximum measurement noise acrshort:asd at $11\,\upmu\text{V}/\sqrt{\text{Hz}}$. In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the acrshort:dac, less than $20\,\text{mV RMS}$ for the voltage amplifier, and less than $0.8\,\text{mV RMS}$ for the acrshort:adc. If the Amplitude Spectral Density of the noise of the acrshort:adc, acrshort:dac, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below $15\,\text{nm RMS}$. @@ -9539,7 +9539,7 @@ When combined with the piezoelectric load (represented as a capacitance $C_p$), [[file:figs/detail_instrumentation_amp_output_impedance.png]] Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases. -For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to $C_p = 8.8\,\mu\text{F}$. +For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to $C_p = 8.8\,\upmu\text{F}$. If a small signal bandwidth of $f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}$ is desired, the voltage amplifier output impedance should be less than $R_0 = 3.6\,\Omega$. ***** Large signal Bandwidth @@ -9565,7 +9565,7 @@ As established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_bud It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure\nbsp{}ref:fig:detail_instrumentation_amp_output_impedance. Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement\nbsp{}[[cite:&spengen20_high_voltag_amplif]]. -For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\mu\text{F}$ and a bandwidth exceeding $5\,\text{kHz}$. +For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\upmu\text{F}$ and a bandwidth exceeding $5\,\text{kHz}$. ***** Choice of voltage amplifier @@ -9586,19 +9586,19 @@ The PD200 from PiezoDrive was ultimately selected because it meets all the requi #+caption: Specifications for the Voltage amplifier and considered commercial products #+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcccc #+attr_latex: :center t :booktabs t :float t -| *Specifications* | PD200 | WMA-200 | LA75B | E-505 | -| | PiezoDrive | Falco | Cedrat | PI | -|--------------------------------------------+-----------------------------+----------------------------+---------------------+---------------------| -| Input Voltage Range: $\pm 10\,\text{V}$ | $\pm 10\,\text{V}$ | $\pm8.75\,\text{V}$ | $-1/7.5\,\text{V}$ | $-2/12\,\text{V}$ | -| Output Voltage Range: $-20/150\,\text{V}$ | $-50/150\,\text{V}$ | $\pm 175\,\text{V}$ | $-20/150\,\text{V}$ | $-30/130\,\text{V}$ | -| Gain $>15$ | 20 | 20 | 20 | 10 | -| Output Current $> 300\,\text{mA}$ | $900\,\text{mA}$ | $150\,\text{mA}$ | $360\,\text{mA}$ | $215\,\text{mA}$ | -| Slew Rate $> 34\,\text{V/ms}$ | $150\,\text{V}/\mu\text{s}$ | $80\,\text{V}/\mu\text{s}$ | n/a | n/a | -| Output noise $< 20\,\text{mV RMS}$ | $0.7\,\text{mV}$ | $0.05\,\text{mV}$ | $3.4\,\text{mV}$ | $0.6\,\text{mV}$ | -| (10uF load) | ($10\,\mu\text{F}$ load) | ($10\,\mu\text{F}$ load) | (n/a) | (n/a) | -| Small Signal Bandwidth $> 5\,\text{kHz}$ | $6.4\,\text{kHz}$ | $300\,\text{Hz}$ | $30\,\text{kHz}$ | n/a | -| ($10\,\mu\text{F}$ load) | ($10\,\mu\text{F}$ load) | ($10\,\mu\text{F}$ load) | (unloaded) | (n/a) | -| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a | +| *Specifications* | PD200 | WMA-200 | LA75B | E-505 | +| | PiezoDrive | Falco | Cedrat | PI | +|--------------------------------------------+-------------------------------+------------------------------+---------------------+---------------------| +| Input Voltage Range: $\pm 10\,\text{V}$ | $\pm 10\,\text{V}$ | $\pm8.75\,\text{V}$ | $-1/7.5\,\text{V}$ | $-2/12\,\text{V}$ | +| Output Voltage Range: $-20/150\,\text{V}$ | $-50/150\,\text{V}$ | $\pm 175\,\text{V}$ | $-20/150\,\text{V}$ | $-30/130\,\text{V}$ | +| Gain $>15$ | 20 | 20 | 20 | 10 | +| Output Current $> 300\,\text{mA}$ | $900\,\text{mA}$ | $150\,\text{mA}$ | $360\,\text{mA}$ | $215\,\text{mA}$ | +| Slew Rate $> 34\,\text{V/ms}$ | $150\,\text{V}/\upmu\text{s}$ | $80\,\text{V}/\upmu\text{s}$ | n/a | n/a | +| Output noise $< 20\,\text{mV RMS}$ | $0.7\,\text{mV}$ | $0.05\,\text{mV}$ | $3.4\,\text{mV}$ | $0.6\,\text{mV}$ | +| (10uF load) | ($10\,\upmu\text{F}$ load) | ($10\,\upmu\text{F}$ load) | (n/a) | (n/a) | +| Small Signal Bandwidth $> 5\,\text{kHz}$ | $6.4\,\text{kHz}$ | $300\,\text{Hz}$ | $30\,\text{kHz}$ | n/a | +| ($10\,\upmu\text{F}$ load) | ($10\,\upmu\text{F}$ load) | ($10\,\upmu\text{F}$ load) | (unloaded) | (n/a) | +| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a | **** ADC and DAC ***** Introduction :ignore: @@ -9628,7 +9628,7 @@ For real-time control applications, acrfull:sar remain the predominant choice du ***** ADC Noise -Based on the dynamic error budget established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise acrshort:asd should not exceed $11\,\mu V/\sqrt{\text{Hz}}$. +Based on the dynamic error budget established in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise acrshort:asd should not exceed $11\,\upmu V/\sqrt{\text{Hz}}$. acrshortpl:adc are subject to various noise sources. Quantization noise, which results from the discrete nature of digital representation, is one of these sources. @@ -9665,11 +9665,11 @@ From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the min n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right) \end{equation} -With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,\text{V}$ and a maximum allowed acrshort:asd $\Gamma_{\text{max}} = 11\,\mu\text{V}/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial acrshortpl:adc. +With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,\text{V}$ and a maximum allowed acrshort:asd $\Gamma_{\text{max}} = 11\,\upmu\text{V}/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial acrshortpl:adc. ***** DAC Output voltage noise -Similar to the acrshort:adc requirements, the acrshort:dac output voltage noise acrshort:asd should not exceed $14\,\mu\text{V}/\sqrt{\text{Hz}}$. +Similar to the acrshort:adc requirements, the acrshort:dac output voltage noise acrshort:asd should not exceed $14\,\upmu\text{V}/\sqrt{\text{Hz}}$. This specification corresponds to a $\pm 10\,\text{V}$ acrshort:dac with 13-bit resolution, which is easily attainable with current technology. ***** Choice of the ADC and DAC Board @@ -9678,14 +9678,14 @@ Based on the preceding analysis, the selection of suitable acrshort:adc and acrs For optimal synchronicity, a Speedgoat-integrated solution was chosen. The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, $\pm 10\,\text{V}$ range, maximum sampling rate of 200kSPS (acrlong:sps), simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity. -The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, $\pm 10\,\text{V}$ range, conversion time of $10\,\mu s$, and simultaneous update capability. +The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, $\pm 10\,\text{V}$ range, conversion time of $10\,\upmu s$, and simultaneous update capability. Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements. This will be experimentally verified in Section\nbsp{}ref:sec:detail_instrumentation_characterization. **** Relative Displacement Sensors -The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below $6\,\text{nm RMS}$ (derived from the $15\,\text{nm RMS}$ vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding $100\,\mu\text{m}$. +The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below $6\,\text{nm RMS}$ (derived from the $15\,\text{nm RMS}$ vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding $100\,\upmu\text{m}$. Several sensor technologies are capable of meeting these requirements\nbsp{}[[cite:&fleming13_review_nanom_resol_posit_sensor]]. These include optical encoders (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_encoder), capacitive sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_capacitive), and eddy current sensors (Figure\nbsp{}ref:fig:detail_instrumentation_sensor_eddy_current), each with their own advantages and implementation considerations. @@ -9746,14 +9746,14 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic #+caption: Specifications for the relative displacement sensors and considered commercial products #+attr_latex: :environment tabularx :width 0.65\linewidth :align Xccc #+attr_latex: :center t :booktabs t :float t -| *Specifications* | Renishaw Vionic | LION CPL190 | Cedrat ECP500 | -|-----------------------------+----------------------+---------------------+---------------------| -| Technology | Digital Encoder | Capacitive | Eddy Current | -| Bandwidth $> 5\,\text{kHz}$ | $> 500\,\text{kHz}$ | $10\,\text{kHz}$ | $20\,\text{kHz}$ | -| Noise $< 6\,\text{nm RMS}$ | $1.6\,\text{nm RMS}$ | $4\,\text{nm RMS}$ | $15\,\text{nm RMS}$ | -| Range $> 100\,\mu\text{m}$ | Ruler length | $250\,\mu \text{m}$ | $500\,\mu \text{m}$ | -| In line measurement | | $\times$ | $\times$ | -| Digital Output | $\times$ | | | +| *Specifications* | Renishaw Vionic | LION CPL190 | Cedrat ECP500 | +|-------------------------------+----------------------+-----------------------+-----------------------| +| Technology | Digital Encoder | Capacitive | Eddy Current | +| Bandwidth $> 5\,\text{kHz}$ | $> 500\,\text{kHz}$ | $10\,\text{kHz}$ | $20\,\text{kHz}$ | +| Noise $< 6\,\text{nm RMS}$ | $1.6\,\text{nm RMS}$ | $4\,\text{nm RMS}$ | $15\,\text{nm RMS}$ | +| Range $> 100\,\upmu\text{m}$ | Ruler length | $250\,\upmu \text{m}$ | $500\,\upmu \text{m}$ | +| In line measurement | | $\times$ | $\times$ | +| Digital Output | $\times$ | | | *** Characterization of Instrumentation <> @@ -9762,7 +9762,7 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic The measurement of acrshort:adc noise was performed by short-circuiting its input with a $50\,\Omega$ resistor and recording the digital values at a sampling rate of $10\,\text{kHz}$. The amplitude spectral density of the recorded values was computed and is presented in Figure\nbsp{}ref:fig:detail_instrumentation_adc_noise_measured. -The acrshort:adc noise exhibits characteristics of white noise with an amplitude spectral density of $5.6\,\mu\text{V}/\sqrt{\text{Hz}}$ (equivalent to $0.4\,\text{mV RMS}$), which satisfies the established specifications. +The acrshort:adc noise exhibits characteristics of white noise with an amplitude spectral density of $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (equivalent to $0.4\,\text{mV RMS}$), which satisfies the established specifications. All acrshort:adc channels demonstrated similar performance, so only one channel's noise profile is shown. If necessary, oversampling can be applied to further reduce the noise\nbsp{}[[cite:&lab13_improv_adc]]. @@ -9796,7 +9796,7 @@ The charge generated by the piezoelectric effect across the stack's capacitance Consequently, the transfer function from the generated voltage $V_p$ to the measured voltage $V_{\text{ADC}}$ is a first-order high-pass filter with the time constant $\tau$. An exponential curve was fitted to the experimental data, yielding a time constant $\tau = 6.5\,\text{s}$. -With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu\text{F}$, the internal impedance of the Speedgoat acrshort:adc was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet. +With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\upmu\text{F}$, the internal impedance of the Speedgoat acrshort:adc was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet. #+name: fig:detail_instrumentation_force_sensor #+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin]]. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}). @@ -9818,7 +9818,7 @@ With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu\tex The constant voltage offset can be explained by the input bias current $i_n$ of the acrshort:adc, represented in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc. At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the acrshort:adc, and therefore the input bias current $i_n$ passing through the internal resistance $R_i$ produces a constant voltage offset $V_{\text{off}} = R_i \cdot i_n$. -The input bias current $i_n$ is estimated from $i_n = V_{\text{off}}/R_i = 1.5\mu A$. +The input bias current $i_n$ is estimated from $i_n = V_{\text{off}}/R_i = 1.5\,\upmu\text{A}$. In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor $R_p$ can be added in parallel to the force sensor, as illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_R. This modification produces two beneficial effects: a reduction of input voltage offset through the relationship $V_{\text{off}} = (R_p R_i)/(R_p + R_i) i_n$, and an increase in the high pass corner frequency $f_c$ according to the equations $\tau = 1/(2\pi f_c) = (R_i R_p)/(R_i + R_p) C_p$. @@ -9848,7 +9848,7 @@ These results validate both the model of the acrshort:adc and the effectiveness **** Instrumentation Amplifier -Because the acrshort:adc noise may be too low to measure the noise of other instruments (anything below $5.6\,\mu\text{V}/\sqrt{\text{Hz}}$ cannot be distinguished from the noise of the acrshort:adc itself), a low noise instrumentation amplifier was employed. +Because the acrshort:adc noise may be too low to measure the noise of other instruments (anything below $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ cannot be distinguished from the noise of the acrshort:adc itself), a low noise instrumentation amplifier was employed. A Femto DLPVA-101-B-S amplifier with adjustable gains from $20\,text{dB}$ up to $80\,text{dB}$ was selected for this purpose. The first step was to characterize the input[fn:detail_instrumentation_1] noise of the amplifier. @@ -9883,7 +9883,7 @@ The acrshort:dac was configured to output a constant voltage (zero in this case) The Amplitude Spectral Density $\Gamma_{n_{da}}(\omega)$ of the measured signal was computed, and verification was performed to confirm that the contributions of acrshort:adc noise and amplifier noise were negligible in the measurement. The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure\nbsp{}ref:fig:detail_instrumentation_dac_output_noise. -The noise profile is predominantly white with an acrshort:asd of $0.6\,\mu\text{V}/\sqrt{\text{Hz}}$. +The noise profile is predominantly white with an acrshort:asd of $0.6\,\upmu\text{V}/\sqrt{\text{Hz}}$. Minor $50\,\text{Hz}$ noise is present, along with some low frequency $1/f$ noise, but these are not expected to pose issues as they are well within specifications. It should be noted that all acrshort:dac channels demonstrated similar performance, so only one channel measurement is presented. @@ -10019,7 +10019,7 @@ Despite these challenges, suitable components were identified that theoretically The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized. Initial measurements of the acrshort:adc system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit. -All components were found to meet or exceed their respective specifications. The acrshort:adc demonstrated noise levels of $5.6\,\mu\text{V}/\sqrt{\text{Hz}}$ (versus the $11\,\mu\text{V}/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\mu\text{V}/\sqrt{\text{Hz}}$ (versus $14\,\mu\text{V}/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\mu\text{V}/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification). +All components were found to meet or exceed their respective specifications. The acrshort:adc demonstrated noise levels of $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus the $11\,\upmu\text{V}/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus $14\,\upmu\text{V}/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\upmu\text{V}/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification). Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance. The combined effect of all noise sources was estimated to induce vertical sample vibrations of only $1.5\,\text{nm RMS}$, which is substantially below the $15\,\text{nm RMS}$ requirement. @@ -10271,7 +10271,7 @@ Therefore, a more sophisticated model of the optical encoder was necessary. The optical encoders operate based on the interaction between an encoder head and a graduated scale or ruler. The optical encoder head contains a light source that illuminates the ruler. A reference frame $\{E\}$ fixed to the scale, represents the the light position on the scale, as illustrated in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_model. -The ruler features a precise grating pattern (in this case, with a $20\,\mu\text{m}$ pitch), and its position is associated with the reference frame $\{R\}$. +The ruler features a precise grating pattern (in this case, with a $20\,\upmu\text{m}$ pitch), and its position is associated with the reference frame $\{R\}$. The displacement measured by the encoder corresponds to the relative position of the encoder frame $\{E\}$ (specifically, the point where the light interacts with the scale) with respect to the ruler frame $\{R\}$, projected along the measurement direction defined by the scale. An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_disp, can induce a measured displacement. @@ -10428,26 +10428,26 @@ The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatnes #+latex: \centering #+attr_latex: :environment tabularx :width 0.5\linewidth :align Xc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| | *Flatness* $[\mu\text{m}]$ | -|-------+----------------------| -| APA 1 | 8.9 | -| APA 2 | 3.1 | -| APA 3 | 9.1 | -| APA 4 | 3.0 | -| APA 5 | 1.9 | -| APA 6 | 7.1 | -| APA 7 | 18.7 | +| | *Flatness* $[\upmu\text{m}]$ | +|-------+------------------------------| +| APA 1 | 8.9 | +| APA 2 | 3.1 | +| APA 3 | 9.1 | +| APA 4 | 3.0 | +| APA 5 | 1.9 | +| APA 6 | 7.1 | +| APA 7 | 18.7 | #+latex: \captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces} #+end_minipage **** Electrical Measurements <> -From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu\text{F}$ and $26\,\mu\text{F}$ with a nominal capacitance of $20\,\mu\text{F}$. +From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\upmu\text{F}$ and $26\,\upmu\text{F}$ with a nominal capacitance of $20\,\upmu\text{F}$. The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:test_apa_1] shown in Figure\nbsp{}ref:fig:test_apa_lcr_meter. The two stacks used as the actuator and the stack used as the force sensor were measured separately. -The measured capacitance values are summarized in Table\nbsp{}ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu\text{F}$. +The measured capacitance values are summarized in Table\nbsp{}ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \upmu\text{F}$. However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance. This may indicate a manufacturing defect. @@ -10476,7 +10476,7 @@ This may be because the manufacturer measures the capacitance with large signals | APA 5 | 4.90 | 9.66 | | APA 6 | 4.99 | 9.91 | | APA 7 | 4.85 | 9.85 | -#+latex: \captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu\text{F}$} +#+latex: \captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\upmu\text{F}$} #+end_minipage **** Stroke and Hysteresis Measurement @@ -10494,9 +10494,9 @@ Note that the voltage is slowly varied as the displacement probe has a very low The measured acrshort:apa displacement is shown as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis. Typical hysteresis curves for piezoelectric stack actuators can be observed. -The measured stroke is approximately $250\,\mu\text{m}$ when using only two of the three stacks. -This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu\text{m}$, therefore $\approx 200\,\mu\text{m}$ if only two stacks are used). -For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu\text{m}$. +The measured stroke is approximately $250\,\upmu\text{m}$ when using only two of the three stacks. +This is even above what is specified as the nominal stroke in the data-sheet ($304\,\upmu\text{m}$, therefore $\approx 200\,\upmu\text{m}$ if only two stacks are used). +For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\upmu\text{m}$. It is clear from Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units. This confirms the abnormal electrical measurements made in Section\nbsp{}ref:ssec:test_apa_electrical_measurements. @@ -10651,7 +10651,7 @@ It can be seen that there are some drifts in the measured displacement (probably These two effects induce some uncertainties in the measured stiffness. The stiffnesses are computed for all acrshortpl:apa from the two displacements $d_1$ and $d_2$ (see Figure\nbsp{}ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$. -These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,\text{N}/\mu\text{m}$. +These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,\text{N}/\upmu\text{m}$. #+attr_latex: :options [b]{0.57\textwidth} #+begin_minipage @@ -10674,7 +10674,7 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu | 5 | 1.7 | 1.93 | | 6 | 1.7 | 1.92 | | 8 | 1.73 | 1.98 | -#+latex: \captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\mu\text{m}$} +#+latex: \captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\upmu\text{m}$} #+end_minipage The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section\nbsp{}ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. @@ -10683,7 +10683,7 @@ The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_fr \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} \end{equation} -The obtained stiffness is $k \approx 2\,\text{N}/\mu\text{m}$ which is close to the values found in the documentation and using the "static deflection" method. +The obtained stiffness is $k \approx 2\,\text{N}/\upmu\text{m}$ which is close to the values found in the documentation and using the "static deflection" method. It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]]. @@ -10691,7 +10691,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the " - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance acrshort:adc) - $k_{\text{sc}}$: piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance) -The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,\text{N}/\mu\text{m}$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,\text{N}/\mu\text{m}$. +The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,\text{N}/\upmu\text{m}$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,\text{N}/\upmu\text{m}$. **** Dynamics <> @@ -10700,7 +10700,7 @@ In this section, the dynamics from the excitation voltage $u$ to the encoder mea First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure\nbsp{}ref:fig:test_apa_frf_encoder. The obtained acrshortpl:frf are similar to those of a (second order) mass-spring-damper system with: -- A "stiffness line" indicating a static gain equal to $\approx -17\,\mu\text{m}/V$. +- A "stiffness line" indicating a static gain equal to $\approx -17\,\upmu\text{m/V}$. The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the acrshort:apa - A lightly damped resonance at $95\,\text{Hz}$ - A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the acrshort:apa support. @@ -10717,7 +10717,7 @@ As illustrated by the Root Locus plot, the poles of the /closed-loop/ system con The significance of this behavior varies with the type of sensor used, as explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]]. Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell. -From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,\text{N}/\mu\text{m}$ (which is close to what is found using a acrshort:fem). +From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,\text{N}/\upmu\text{m}$ (which is close to what is found using a acrshort:fem). All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure\nbsp{}ref:fig:test_apa_frf_encoder and at the force sensor in Figure\nbsp{}ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. @@ -10773,7 +10773,7 @@ However, this is not so important here because the zero is lightly damped (i.e. **** Effect of the resistor on the IFF Plant <> -A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu\text{F}$). +A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\upmu\text{F}$). As explained before, this is done to limit the voltage offset due to the input bias current of the acrshort:adc as well as to limit the low frequency gain. @@ -10798,7 +10798,7 @@ G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s A comparison between the identified plant and the manually tuned transfer function is shown in Figure\nbsp{}ref:fig:test_apa_iff_plant_comp_manual_fit. #+name: fig:test_apa_iff_plant_comp_manual_fit -#+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero. +#+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\upmu\text{s}$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero. #+attr_latex: :scale 0.8 [[file:figs/test_apa_iff_plant_comp_manual_fit.png]] @@ -10888,7 +10888,7 @@ Such a simple model has some limitations: First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$. -Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,\text{N}/\mu\text{m}$ in Section\nbsp{}ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. +Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,\text{N}/\upmu\text{m}$ in Section\nbsp{}ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,\text{Ns/m}$. Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:test_apa_5]. @@ -10899,10 +10899,10 @@ In this case, the total stiffness of the acrshort:apa model is described by\nbsp k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a \end{equation} -Knowing from\nbsp{}eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,\text{N}/\mu\text{m}$, we get from\nbsp{}eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,\text{N}/\mu\text{m}$ and $k_e = 5\,\text{N}/\mu\text{m}$. +Knowing from\nbsp{}eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\text{tot}} = 2\,\text{N}/\upmu\text{m}$, we get from\nbsp{}eqref:eq:test_apa_2dof_stiffness that $k_a = 2.5\,\text{N}/\upmu\text{m}$ and $k_e = 5\,\text{N}/\upmu\text{m}$. \begin{equation}\label{eq:test_apa_tot_stiffness} -\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,\text{N}/\mu\text{m} \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s +\omega_0 = \frac{k_{\text{tot}}}{m} \Longrightarrow k_{\text{tot}} = m \omega_0^2 = 2\,\text{N}/\upmu\text{m} \quad \text{with}\ m = 5.7\,\text{kg}\ \text{and}\ \omega_0 = 2\pi \cdot 95\, \text{rad}/s \end{equation} Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance. @@ -10916,17 +10916,17 @@ The obtained parameters of the model shown in Figure\nbsp{}ref:fig:test_apa_2dof #+caption: Summary of the obtained parameters for the 2 DoF APA300ML model #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t -| *Parameter* | *Value* | -|-------------+------------------------------| -| $m$ | $5.7\,\text{kg}$ | -| $k_1$ | $0.38\,\text{N}/\mu\text{m}$ | -| $k_e$ | $5.0\,\text{N}/\mu\text{m}$ | -| $k_a$ | $2.5\,\text{N}/\mu\text{m}$ | -| $c_1$ | $5\,\text{Ns/m}$ | -| $c_e$ | $100\,\text{Ns/m}$ | -| $c_a$ | $50\,\text{Ns/m}$ | -| $g_a$ | $-2.58\,\text{N/V}$ | -| $g_s$ | $0.46\,\text{V}/\mu\text{m}$ | +| *Parameter* | *Value* | +|-------------+--------------------------------| +| $m$ | $5.7\,\text{kg}$ | +| $k_1$ | $0.38\,\text{N}/\upmu\text{m}$ | +| $k_e$ | $5.0\,\text{N}/\upmu\text{m}$ | +| $k_a$ | $2.5\,\text{N}/\upmu\text{m}$ | +| $c_1$ | $5\,\text{Ns/m}$ | +| $c_e$ | $100\,\text{Ns/m}$ | +| $c_a$ | $50\,\text{Ns/m}$ | +| $g_a$ | $-2.58\,\text{N/V}$ | +| $g_s$ | $0.46\,\text{V}/\upmu\text{m}$ | ***** Obtained Dynamics :ignore: @@ -10976,7 +10976,7 @@ Finally, two /remote points/ (=4= and =5=) are located across the third piezoele Once the APA300ML /super element/ is included in the multi-body model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be extracted. The gains $g_a$ and $g_s$ are then tuned such that the gains of the transfer functions match the identified ones. -By doing so, $g_s = 4.9\,\text{V}/\mu\text{m}$ and $g_a = 23.2\,\text{N/V}$ are obtained. +By doing so, $g_s = 4.9\,\text{V}/\upmu\text{m}$ and $g_a = 23.2\,\text{N/V}$ are obtained. To ensure that the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material. @@ -10993,7 +10993,7 @@ Unfortunately, the manufacturer of the stack was not willing to share the piezoe However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:test_apa_piezo_properties. -From these parameters, $g_s = 5.1\,\text{V}/\mu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. +From these parameters, $g_s = 5.1\,\text{V}/\upmu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. #+name: tab:test_apa_piezo_properties #+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities @@ -11074,13 +11074,13 @@ During the detailed design phase, specifications in terms of stiffness and strok #+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t -| | *Specification* | *FEM* | -|-------------------+-------------------------------+-------| -| Axial Stiffness | $> 100\,\text{N}/\mu\text{m}$ | 94 | -| Shear Stiffness | $> 1\,\text{N}/\mu\text{m}$ | 13 | -| Bending Stiffness | $< 100\,\text{Nm}/\text{rad}$ | 5 | -| Torsion Stiffness | $< 500\,\text{Nm}/\text{rad}$ | 260 | -| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | +| | *Specification* | *FEM* | +|-------------------+---------------------------------+-------| +| Axial Stiffness | $> 100\,\text{N}/\upmu\text{m}$ | 94 | +| Shear Stiffness | $> 1\,\text{N}/\upmu\text{m}$ | 13 | +| Bending Stiffness | $< 100\,\text{Nm}/\text{rad}$ | 5 | +| Torsion Stiffness | $< 500\,\text{Nm}/\text{rad}$ | 260 | +| Bending Stroke | $> 1\,\text{mrad}$ | 24.5 | After optimization using a acrshort:fem, the geometry shown in Figure\nbsp{}ref:fig:test_joints_schematic has been obtained and the corresponding flexible joint characteristics are summarized in Table\nbsp{}ref:tab:test_joints_specs. This flexible joint is a monolithic piece of stainless steel[fn:test_joints_1] manufactured using wire electrical discharge machining. @@ -11152,7 +11152,7 @@ The dimensions of the flexible joint in the Y-Z plane will contribute to the X-b The setup used to measure the dimensions of the "X" flexible beam is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_setup. What is typically observed is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_image. -It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\mu\text{m}$, +It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\upmu\text{m}$, #+name: fig:test_joints_profilometer #+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image}) @@ -11173,12 +11173,12 @@ It is then possible to estimate the dimension of the flexible beam with an accur #+end_figure **** Measurement Results -The specified flexible beam thickness (gap) is $250\,\mu\text{m}$. +The specified flexible beam thickness (gap) is $250\,\upmu\text{m}$. Four gaps are measured for each flexible joint (2 in the $x$ direction and 2 in the $y$ direction). The "beam thickness" is then estimated as the mean between the gaps measured on opposite sides. A histogram of the measured beam thicknesses is shown in Figure\nbsp{}ref:fig:test_joints_size_hist. -The measured thickness is less than the specified value of $250\,\mu\text{m}$, but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. +The measured thickness is less than the specified value of $250\,\upmu\text{m}$, but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. However, what is more important than the true value of the thickness is the consistency between all flexible joints. @@ -11324,7 +11324,7 @@ The estimated bending stiffness $k_{\text{est}}$ then depends on the shear stiff k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_s h^2}}_{\epsilon_{s}} \Bigl) \end{equation} -With an estimated shear stiffness $k_s = 13\,\text{N}/\mu\text{m}$ from the acrshort:fem and an height $h=25\,\text{mm}$, the estimation errors of the bending stiffness due to shear is $\epsilon_s < 0.1\,\%$ +With an estimated shear stiffness $k_s = 13\,\text{N}/\upmu\text{m}$ from the acrshort:fem and an height $h=25\,\text{mm}$, the estimation errors of the bending stiffness due to shear is $\epsilon_s < 0.1\,\%$ ***** Effect of Load Cell Limited Stiffness As explained in the previous section, because the measurement of the flexible joint deflection is indirectly performed with the encoder, errors will be made if the load cell experiences some compression. @@ -11336,7 +11336,7 @@ The estimation error of the bending stiffness due to the limited stiffness of th k_{R_y,\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} \approx k_{R_y} \Bigl( 1 - \underbrace{\frac{k_{R_y}}{k_F h^2}}_{\epsilon_f} \Bigl) \end{equation} -With an estimated load cell stiffness of $k_f \approx 1\,\text{N}/\mu\text{m}$ (from the documentation), the errors due to the load cell limited stiffness is around $\epsilon_f = 1\,\%$. +With an estimated load cell stiffness of $k_f \approx 1\,\text{N}/\upmu\text{m}$ (from the documentation), the errors due to the load cell limited stiffness is around $\epsilon_f = 1\,\%$. ***** Height Estimation Error Now consider an error $\delta h$ in the estimation of the height $h$ as described by\nbsp{}eqref:eq:test_joints_est_h_error. @@ -11466,10 +11466,10 @@ However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\, **** Load Cell Stiffness The objective of this measurement is to estimate the stiffness $k_F$ of the force sensor. -To do so, a stiff element (much stiffer than the estimated $k_F \approx 1\,\text{N}/\mu\text{m}$) is mounted in front of the force sensor, as shown in Figure\nbsp{}ref:fig:test_joints_meas_force_sensor_stiffness_picture. +To do so, a stiff element (much stiffer than the estimated $k_F \approx 1\,\text{N}/\upmu\text{m}$) is mounted in front of the force sensor, as shown in Figure\nbsp{}ref:fig:test_joints_meas_force_sensor_stiffness_picture. Then, the force sensor is pushed against this stiff element while the force sensor and the encoder displacement are measured. The measured displacement as a function of the measured force is shown in Figure\nbsp{}ref:fig:test_joints_force_sensor_stiffness_fit. -The load cell stiffness can then be estimated by computing a linear fit and is found to be $k_F \approx 0.68\,\text{N}/\mu\text{m}$. +The load cell stiffness can then be estimated by computing a linear fit and is found to be $k_F \approx 0.68\,\text{N}/\upmu\text{m}$. #+name: fig:test_joints_meas_force_sensor_stiffness #+caption: Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}), and results are shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}). @@ -11598,7 +11598,7 @@ A mounting bench was developed to ensure: The mounting bench is shown in Figure\nbsp{}ref:fig:test_struts_mounting_bench_first_concept. It consists of a "main frame" (Figure\nbsp{}ref:fig:test_struts_mounting_step_0) precisely machined to ensure both correct strut length and strut coaxiality. -The coaxiality is ensured by good flatness (specified at $20\,\mu\text{m}$) between surfaces A and B and between surfaces C and D. +The coaxiality is ensured by good flatness (specified at $20\,\upmu\text{m}$) between surfaces A and B and between surfaces C and D. Such flatness was checked using a FARO arm[fn:test_struts_1] (see Figure\nbsp{}ref:fig:test_struts_check_dimensions_bench) and was found to comply with the requirements. The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes. @@ -12093,7 +12093,7 @@ After receiving the positioning pins, the struts were mounted again with the pos This should improve the alignment of the acrshort:apa with the two flexible joints. The alignment is then estimated using a length gauge, as described in the previous sections. -Measured $y$ alignments are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu\text{m}$ for all the struts, which is much better than before (see Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). +Measured $y$ alignments are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\upmu\text{m}$ for all the struts, which is much better than before (see Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). #+name: tab:test_struts_meas_y_misalignment_with_pin #+caption: Measured $y$ misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in $\text{mm}$. @@ -12202,20 +12202,20 @@ The main goal of this "mounting tool" is to position the flexible joint interfac The quality of the positioning can be estimated by measuring the "straightness" of the top and bottom "V" interfaces. This corresponds to the diameter of the smallest cylinder which contains all points along the measured axis. This was again done using the FARO arm, and the results for all six struts are summarized in Table\nbsp{}ref:tab:measured_straightness. -The straightness was found to be better than $15\,\mu\text{m}$ for all struts[fn:test_nhexa_4], which is sufficiently good to not induce significant stress of the flexible joint during assembly. +The straightness was found to be better than $15\,\upmu\text{m}$ for all struts[fn:test_nhexa_4], which is sufficiently good to not induce significant stress of the flexible joint during assembly. #+name: tab:measured_straightness #+caption: Measured straightness between the two "V" shapes for the six struts. These measurements were performed twice for each strut. #+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc #+attr_latex: :center t :booktabs t -| *Strut* | *Meas 1* | *Meas 2* | -|---------+--------------------+--------------------| -| 1 | $7\,\mu\text{m}$ | $3\, \mu\text{m}$ | -| 2 | $11\, \mu\text{m}$ | $11\, \mu\text{m}$ | -| 3 | $15\, \mu\text{m}$ | $14\, \mu\text{m}$ | -| 4 | $6\, \mu\text{m}$ | $6\, \mu\text{m}$ | -| 5 | $7\, \mu\text{m}$ | $5\, \mu\text{m}$ | -| 6 | $6\, \mu\text{m}$ | $7\, \mu\text{m}$ | +| *Strut* | *Meas 1* | *Meas 2* | +|---------+----------------------+----------------------| +| 1 | $7\,\upmu\text{m}$ | $3\, \upmu\text{m}$ | +| 2 | $11\, \upmu\text{m}$ | $11\, \upmu\text{m}$ | +| 3 | $15\, \upmu\text{m}$ | $14\, \upmu\text{m}$ | +| 4 | $6\, \upmu\text{m}$ | $6\, \upmu\text{m}$ | +| 5 | $7\, \upmu\text{m}$ | $5\, \upmu\text{m}$ | +| 6 | $6\, \upmu\text{m}$ | $7\, \upmu\text{m}$ | The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure\nbsp{}ref:fig:test_nhexa_mount_encoder), and the encoder heads were aligned to maximize the received contrast. @@ -12669,7 +12669,7 @@ This section presents a comprehensive experimental evaluation of the complete sy Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-DoF metrology system to measure the sample position relative to the granite base. However, the complexity of this development prevented its completion before the experimental testing phase on ID31. -To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\mu\text{m}^3$) metrology system was developed, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. +To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\upmu\text{m}^3$) metrology system was developed, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. Then, several key aspects of the system validation are examined. Section\nbsp{}ref:sec:test_id31_open_loop_plant analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities. @@ -12705,7 +12705,7 @@ These include tomography scans at various speeds and with different payload mass **** Introduction :ignore: The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite. -As a long-stroke ($\approx 1 \,\text{cm}^3$) metrology system was not yet developed, a stroke stroke ($\approx 100\,\mu\text{m}^3$) was used instead to validate the nano-hexapod control. +As a long-stroke ($\approx 1 \,\text{cm}^3$) metrology system was not yet developed, a stroke stroke ($\approx 100\,\upmu\text{m}^3$) was used instead to validate the nano-hexapod control. The first considered option was to use the "Spindle error analyzer" shown in Figure\nbsp{}ref:fig:test_id31_lion. This system comprises 5 capacitive sensors facing two reference spheres. @@ -12797,7 +12797,7 @@ To not damage the sensitive sphere surface, the probes are instead positioned on The probes are first fixed to the bottom (fixed) cylinder to align the first sphere with the spindle axis. The probes are then fixed to the top (adjustable) cylinder, and the same alignment is performed. -With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around $10\,\mu\text{m}$. +With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around $10\,\upmu\text{m}$. The accuracy was probably limited by the poor coaxiality between the cylinders and the spheres. However, this first alignment should be sufficient to position the two sphere within the acceptance range of the interferometers. @@ -12836,7 +12836,7 @@ From the errors, the motion of the positioning hexapod to better align the spher Then, the spindle is scanned again, and new alignment errors are recorded. This iterative process is first performed for angular errors (Figure\nbsp{}ref:fig:test_id31_metrology_align_rx_ry) and then for lateral errors (Figure\nbsp{}ref:fig:test_id31_metrology_align_dx_dy). -The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ in $R_x$ and $R_y$ orientations, $\pm 1\,\mu\text{m}$ in $D_x$ and $D_y$ directions, and less than $0.1\,\mu\text{m}$ vertically. +The remaining errors after alignment are in the order of $\pm5\,\upmu\text{rad}$ in $R_x$ and $R_y$ orientations, $\pm 1\,\upmu\text{m}$ in $D_x$ and $D_y$ directions, and less than $0.1\,\upmu\text{m}$ vertically. #+name: fig:test_id31_metrology_align #+caption: Measured angular (\subref{fig:test_id31_metrology_align_rx_ry}) and lateral (\subref{fig:test_id31_metrology_align_dx_dy}) errors during full spindle rotation. Between two rotations, the positioning hexapod is adjusted to better align the two spheres with the rotation axis. @@ -12891,7 +12891,7 @@ As the top interferometer points to a sphere and not to a plane, lateral motion Then, the reference spheres have some deviations relative to an ideal sphere [fn:test_id31_6]. These sphere are originally intended for use with capacitive sensors that integrate 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 $50\,\mu\text{m}$, and therefore the measurement is more sensitive to shape errors with small features. +When using interferometers, the size of the "light spot" on the sphere surface is a circle with a diameter approximately equal to $50\,\upmu\text{m}$, and therefore the measurement is more sensitive to shape errors with small features. As the light from the interferometer travels 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 in air temperature, pressure or humidity will induce measurement errors. @@ -13438,7 +13438,7 @@ Higher performance controllers using complementary filters are investigated in S For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to $200\,\text{nm}\times 100\,\text{nm}$. In this case, the goal is to keep the sample's acrshort:poi in the beam, and therefore the $D_y$ and $D_z$ positioning errors should be less than $200\,\text{nm}$ and $100\,\text{nm}$ peak-to-peak, respectively. -The $R_y$ error should be less than $1.7\,\mu\text{rad}$ peak-to-peak. +The $R_y$ error should be less than $1.7\,\upmu\text{rad}$ peak-to-peak. In terms of RMS errors, this corresponds to $30\,\text{nm}$ in $D_y$, $15\,\text{nm}$ in $D_z$ and $250\,\text{nrad}$ in $R_y$ (a summary of the specifications is given in Table\nbsp{}ref:tab:test_id31_experiments_specifications). Results obtained for all experiments are summarized and compared to the specifications in Section\nbsp{}ref:ssec:test_id31_scans_conclusion. @@ -13449,7 +13449,7 @@ Results obtained for all experiments are summarized and compared to the specific #+attr_latex: :center t :booktabs t | | $D_y$ | $D_z$ | $R_y$ | |-------------+------------------+------------------+----------------------| -| peak 2 peak | $200\,\text{nm}$ | $100\,\text{nm}$ | $1.7\,\mu\text{rad}$ | +| peak 2 peak | $200\,\text{nm}$ | $100\,\text{nm}$ | $1.7\,\upmu\text{rad}$ | | RMS | $30\,\text{nm}$ | $15\,\text{nm}$ | $250\,\text{nrad}$ | **** Tomography Scans @@ -13558,11 +13558,11 @@ This experiment also illustrates that when needed, performance can be enhanced b <> X-ray reflectivity measurements involve scanning thin structures, particularly solid/liquid interfaces, through the beam by varying the $R_y$ angle. -In this experiment, a $R_y$ scan was executed at a rotational velocity of $100\,\mu \text{rad/s}$, and the closed-loop positioning errors were monitored (Figure\nbsp{}ref:fig:test_id31_reflectivity). +In this experiment, a $R_y$ scan was executed at a rotational velocity of $100\,\upmu \text{rad/s}$, and the closed-loop positioning errors were monitored (Figure\nbsp{}ref:fig:test_id31_reflectivity). The results confirmed that the NASS successfully maintained the acrshort:poi within the specified beam parameters throughout the scanning process. #+name: fig:test_id31_reflectivity -#+caption: Reflectivity scan ($R_y$) with a rotational velocity of $100\,\mu \text{rad}/s$. +#+caption: Reflectivity scan ($R_y$) with a rotational velocity of $100\,\upmu \text{rad}/s$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dy}$D_y$} @@ -13595,16 +13595,16 @@ These vertical scans can be executed either continuously or in a step-by-step ma ***** Step by Step $D_z$ Motion The vertical step motion was performed exclusively with the nano-hexapod. -Testing was conducted across step sizes ranging from $10\,\text{nm}$ to $1\,\mu\text{m}$. +Testing was conducted across step sizes ranging from $10\,\text{nm}$ to $1\,\upmu\text{m}$. Results are presented in Figure\nbsp{}ref:fig:test_id31_dz_mim_steps. The system successfully resolved $10\,\text{nm}$ steps (red curve in Figure\nbsp{}ref:fig:test_id31_dz_mim_10nm_steps) if a 50ms integration time is considered for the detectors, which is compatible with many experimental requirements. In step-by-step scanning procedures, the settling time is a critical parameter as it significantly affects the total experiment duration. -The system achieved a response time of approximately $70\,\text{ms}$ to reach the target position (within $\pm 20\,\text{nm}$), as demonstrated by the $1\,\mu\text{m}$ step response in Figure\nbsp{}ref:fig:test_id31_dz_mim_1000nm_steps. +The system achieved a response time of approximately $70\,\text{ms}$ to reach the target position (within $\pm 20\,\text{nm}$), as demonstrated by the $1\,\upmu\text{m}$ step response in Figure\nbsp{}ref:fig:test_id31_dz_mim_1000nm_steps. The settling duration typically decreases for smaller step sizes. #+name: fig:test_id31_dz_mim_steps -#+caption: Vertical steps performed with the nano-hexapod. $10\,\text{nm}$ steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of $50\,\text{ms}$. $100\,\text{nm}$ steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of $\pm 20\,\text{nm}$ is $\approx 70\,\text{ms}$ as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a $1\,\mu\text{m}$ step. +#+caption: Vertical steps performed with the nano-hexapod. $10\,\text{nm}$ steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of $50\,\text{ms}$. $100\,\text{nm}$ steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of $\pm 20\,\text{nm}$ is $\approx 70\,\text{ms}$ as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a $1\,\upmu\text{m}$ step. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_10nm_steps}$10\,\text{nm}$ steps} @@ -13619,7 +13619,7 @@ The settling duration typically decreases for smaller step sizes. #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_mim_100nm_steps.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu\text{m}$ step} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\upmu\text{m}$ step} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13632,10 +13632,10 @@ The settling duration typically decreases for smaller step sizes. For these and subsequent experiments, the NASS performs "ramp scans" (constant velocity scans). To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies. -Initial testing at $10\,\mu\text{m/s}$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure\nbsp{}ref:fig:test_id31_dz_scan_10ums). +Initial testing at $10\,\upmu\text{m/s}$ demonstrated positioning errors well within specifications (indicated by dashed lines in Figure\nbsp{}ref:fig:test_id31_dz_scan_10ums). #+name: fig:test_id31_dz_scan_10ums -#+caption: $D_z$ scan at a velocity of $10\,\mu \text{m/s}$. $D_z$ 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}) +#+caption: $D_z$ scan at a velocity of $10\,\upmu \text{m/s}$. $D_z$ 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}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$} @@ -13658,12 +13658,12 @@ Initial testing at $10\,\mu\text{m/s}$ demonstrated positioning errors well with #+end_subfigure #+end_figure -A subsequent scan at $100\,\mu\text{m/s}$ - the maximum velocity for high-precision $D_z$ scans[fn:test_id31_8] - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_dz_scan_100ums). +A subsequent scan at $100\,\upmu\text{m/s}$ - the maximum velocity for high-precision $D_z$ scans[fn:test_id31_8] - maintains positioning errors within specifications during the constant velocity phase, with deviations occurring only during acceleration and deceleration phases (Figure\nbsp{}ref:fig:test_id31_dz_scan_100ums). Since detectors typically operate only during the constant velocity phase, these transient deviations do not compromise the measurement quality. However, performance during acceleration phases could be enhanced through the implementation of feedforward control. #+name: fig:test_id31_dz_scan_100ums -#+caption: $D_z$ scan at a velocity of $100\,\mu\text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}) +#+caption: $D_z$ scan at a velocity of $100\,\upmu\text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}) #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$} @@ -13693,21 +13693,21 @@ However, performance during acceleration phases could be enhanced through the im Lateral scans are executed using the $T_y$ stage. The stepper motor controller[fn:test_id31_5] generates a setpoint that is transmitted to the Speedgoat. Within the Speedgoat, the system computes the positioning error by comparing the measured $D_y$ sample position against the received setpoint, and the Nano-Hexapod compensates for positioning errors introduced during $T_y$ stage scanning. -The scanning range is constrained $\pm 100\,\mu\text{m}$ due to the limited acceptance of the metrology system. +The scanning range is constrained $\pm 100\,\upmu\text{m}$ due to the limited acceptance of the metrology system. ***** Slow Scan -Initial testing were made with a scanning velocity of $10\,\mu\text{m/s}$, which is typical for these experiments. +Initial testing were made with a scanning velocity of $10\,\upmu\text{m/s}$, which is typical for these experiments. Figure\nbsp{}ref:fig:test_id31_dy_10ums compares the positioning errors between open-loop (without NASS) and closed-loop operation. In the scanning direction, open-loop measurements reveal periodic errors (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy) attributable to the $T_y$ stage's stepper motor. These micro-stepping errors, which are inherent to stepper motor operation, occur 200 times per motor rotation with approximately $1\,\text{mrad}$ angular error amplitude. -Given the $T_y$ stage's lead screw pitch of $2\,\text{mm}$, these errors manifest as $10\,\mu\text{m}$ periodic oscillations with $\approx 300\,\text{nm}$ amplitude, which can indeed be seen in the open-loop measurements (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy). +Given the $T_y$ stage's lead screw pitch of $2\,\text{mm}$, these errors manifest as $10\,\upmu\text{m}$ periodic oscillations with $\approx 300\,\text{nm}$ amplitude, which can indeed be seen in the open-loop measurements (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dy). In the vertical direction (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dz), open-loop errors likely stem from metrology measurement error because the top interferometer points at a spherical target surface (see Figure\nbsp{}ref:fig:test_id31_xy_map_sphere). Under closed-loop control, positioning errors remain within specifications in all directions. #+name: fig:test_id31_dy_10ums -#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $10\,\mu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_10ums_dy}). +#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $10\,\upmu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_10ums_dy}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dy} $D_y$} @@ -13732,7 +13732,7 @@ Under closed-loop control, positioning errors remain within specifications in al ***** Fast Scan -The system performance was evaluated at an increased scanning velocity of $100\,\mu\text{m/s}$, and the results are presented in Figure\nbsp{}ref:fig:test_id31_dy_100ums. +The system performance was evaluated at an increased scanning velocity of $100\,\upmu\text{m/s}$, and the results are presented in Figure\nbsp{}ref:fig:test_id31_dy_100ums. At this velocity, the micro-stepping errors generate $10\,\text{Hz}$ vibrations, which are further amplified by micro-station resonances. These vibrations exceeded the NASS feedback controller bandwidth, resulting in limited attenuation under closed-loop control. This limitation exemplifies why stepper motors are suboptimal for "long-stroke/short-stroke" systems requiring precise scanning performance\nbsp{}[[cite:&dehaeze22_fastj_uhv]]. @@ -13743,7 +13743,7 @@ Alternatively, since closed-loop errors in $D_z$ and $R_y$ directions remain wit For applications requiring small $D_y$ scans, the nano-hexapod can be used exclusively, although with limited stroke capability. #+name: fig:test_id31_dy_100ums -#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $100\,\mu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_100ums_dy}). +#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $100\,\upmu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_100ums_dy}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dy} $D_y$} @@ -13771,7 +13771,7 @@ For applications requiring small $D_y$ scans, the nano-hexapod can be used exclu In diffraction tomography experiments, the micro-station performs combined motions: continuous rotation around the $R_z$ axis while performing lateral scans along $D_y$. For this validation, the spindle maintained a constant rotational velocity of $6\,\text{deg/s}$ while the nano-hexapod performs the lateral scanning motion. -To avoid high-frequency vibrations typically induced by the stepper motor, the $T_y$ stage was not used, which constrained the scanning range to approximately $\pm 100\,\mu\text{m/s}$. +To avoid high-frequency vibrations typically induced by the stepper motor, the $T_y$ stage was not used, which constrained the scanning range to approximately $\pm 100\,\upmu\text{m/s}$. The system performance was evaluated at three lateral scanning velocities: $0.1\,\text{mm/s}$, $0.5\,\text{mm/s}$, and $1\,\text{mm/s}$. Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. #+name: fig:test_id31_diffraction_tomo_setpoint @@ -13896,9 +13896,9 @@ For tomography experiments, the NASS successfully maintained good positioning ac The acrshort:haclac control architecture proved particularly effective, with the decentralized IFF providing damping of nano-hexapod suspension modes, while the high authority controller addressed low-frequency disturbances. The vertical scanning capabilities were validated in both step-by-step and continuous motion modes. -The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to $100\,\mu\text{m/s}$. +The system successfully resolved 10 nm steps with 50 ms detector integration time, while maintaining positioning accuracy during continuous scans at speeds up to $100\,\upmu\text{m/s}$. -For lateral scanning, the system performed well at moderate speeds ($10\,\mu\text{m/s}$) but showed limitations at higher velocities ($100\,\mu\text{m/s}$) due to stepper motor-induced vibrations in the $T_y$ stage. +For lateral scanning, the system performed well at moderate speeds ($10\,\upmu\text{m/s}$) but showed limitations at higher velocities ($100\,\upmu\text{m/s}$) due to stepper motor-induced vibrations in the $T_y$ stage. The most challenging test case - diffraction tomography combining rotation and lateral scanning - demonstrated the system's ability to maintain vertical and angular stability while highlighting some limitations in lateral positioning during rapid accelerations. These limitations could be addressed through feedforward control or alternative detector triggering strategies. @@ -13920,13 +13920,13 @@ The identified limitations, primarily related to high-speed lateral scanning and | Tomography ($180\,\text{deg/s}$) | $143 \Rightarrow \bm{38}$ | $24 \Rightarrow 11$ | $252 \Rightarrow 130$ | | Tomography ($180\,\text{deg/s}$, custom HAC) | $143 \Rightarrow 29$ | $24 \Rightarrow 5$ | $252 \Rightarrow 142$ | |----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Reflectivity ($100\,\mu\text{rad}/s$) | $28$ | $6$ | $118$ | +| Reflectivity ($100\,\upmu\text{rad}/s$) | $28$ | $6$ | $118$ | |----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| $D_z$ scan ($10\,\mu\text{m/s}$) | $25$ | $5$ | $108$ | -| $D_z$ scan ($100\,\mu\text{m/s}$) | $\bm{35}$ | $9$ | $132$ | +| $D_z$ scan ($10\,\upmu\text{m/s}$) | $25$ | $5$ | $108$ | +| $D_z$ scan ($100\,\upmu\text{m/s}$) | $\bm{35}$ | $9$ | $132$ | |----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| -| Lateral Scan ($10\,\mu\text{m/s}$) | $585 \Rightarrow 21$ | $155 \Rightarrow 10$ | $6300 \Rightarrow 60$ | -| Lateral Scan ($100\,\mu\text{m/s}$) | $1063 \Rightarrow \bm{732}$ | $167 \Rightarrow \bm{20}$ | $6445 \Rightarrow \bm{356}$ | +| Lateral Scan ($10\,\upmu\text{m/s}$) | $585 \Rightarrow 21$ | $155 \Rightarrow 10$ | $6300 \Rightarrow 60$ | +| Lateral Scan ($100\,\upmu\text{m/s}$) | $1063 \Rightarrow \bm{732}$ | $167 \Rightarrow \bm{20}$ | $6445 \Rightarrow \bm{356}$ | |----------------------------------------------------------------+-----------------------------+---------------------------+-----------------------------| | Diffraction tomography ($6\,\text{deg/s}$, $0.1\,\text{mm/s}$) | $\bm{36}$ | $7$ | $113$ | | Diffraction tomography ($6\,\text{deg/s}$, $0.5\,\text{mm/s}$) | $29$ | $8$ | $81$ | @@ -14207,8 +14207,8 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_apa_6]Polytec controller 3001 with sensor heads OFV512 [fn:test_apa_5]Note that this is not completely correct as it was shown in Section\nbsp{}ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. [fn:test_apa_4]The Matlab =fminsearch= command is used to fit the plane -[fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu\text{m}$ -[fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\mu\text{m}$ +[fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\upmu\text{m}$ +[fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\upmu\text{m}$ [fn:test_apa_1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$ [fn:test_joints_5]XFL212R-50N from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range @@ -14222,16 +14222,16 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_struts_5] APA300ML from Cedrat Technologies [fn:test_struts_4] Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools [fn:test_struts_3] Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called /17-4 PH/. Encoder and ruler support material is aluminium. -[fn:test_struts_2] Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu\text{m}$ -[fn:test_struts_1] FARO Arm Platinum 4ft, specified accuracy of $\pm 13\mu\text{m}$ +[fn:test_struts_2] Heidenhain MT25, specified accuracy of $\pm 0.5\,\upmu\text{m}$ +[fn:test_struts_1] FARO Arm Platinum 4ft, specified accuracy of $\pm 13\upmu\text{m}$ [fn:test_nhexa_7]PCB 356B18. Sensitivity is $1\,\text{V/g}$, measurement range is $\pm 5\,\text{g}$ and bandwidth is $0.5$ to $5\,\text{kHz}$. [fn:test_nhexa_6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,\text{N/mm}$ [fn:test_nhexa_5]The $450\,\text{mm} \times 450\,\text{mm} \times 60\,\text{mm}$ Nexus B4545A from Thorlabs. -[fn:test_nhexa_4]As the accuracy of the FARO arm is $\pm 13\,\mu\text{m}$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached. -[fn:test_nhexa_3]The height dimension is better than $40\,\mu\text{m}$. The diameter fitting of 182g6 and 24g6 with the two plates is verified. +[fn:test_nhexa_4]As the accuracy of the FARO arm is $\pm 13\,\upmu\text{m}$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached. +[fn:test_nhexa_3]The height dimension is better than $40\,\upmu\text{m}$. The diameter fitting of 182g6 and 24g6 with the two plates is verified. [fn:test_nhexa_2]Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked. -[fn:test_nhexa_1]FARO Arm Platinum 4ft, specified accuracy of $\pm 13\mu\text{m}$ +[fn:test_nhexa_1]FARO Arm Platinum 4ft, specified accuracy of $\pm 13\upmu\text{m}$ [fn:test_id31_8]Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and $100\,\text{nm}$ "resolution" (equal to the vertical beam size). [fn:test_id31_7]The highest rotational velocity of $360\,\text{deg/s}$ could not be tested due to an issue in the Spindle's controller. @@ -14240,4 +14240,4 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_id31_4]Note that the eccentricity of the "point of interest" with respect to the Spindle rotation axis has been tuned based on measurements. [fn:test_id31_3]The "PEPU"\nbsp{}[[cite:&hino18_posit_encod_proces_unit]] was used for digital protocol conversion between the interferometers and the Speedgoat. [fn:test_id31_2]M12/F40 model from Attocube. -[fn:test_id31_1]Depending on the measuring range, gap can range from $\approx 1\,\mu\text{m}$ to $\approx 100\,\mu\text{m}$. +[fn:test_id31_1]Depending on the measuring range, gap can range from $\approx 1\,\upmu\text{m}$ to $\approx 100\,\upmu\text{m}$. diff --git a/setup.org b/setup.org index fed9e38..a314184 100644 --- a/setup.org +++ b/setup.org @@ -21,6 +21,8 @@ \DeclareSIUnit\px{px} \DeclareSIUnit\rms{rms} \DeclareSIUnit\rad{rad} + +\usepackage{upgreek} % useful for "mu" in units #+end_src ** Mathematics