Add conclusion

2025-04-20 18:15:02 +02:00 · 2025-04-20 18:15:02 +02:00 · 31a973a926
commit 31a973a926
parent 58b4d7fd33
10 changed files with 2356 additions and 125 deletions
--- a/figs/conclusion_maglev_dyck15.jpg
+++ b/figs/conclusion_maglev_dyck15.jpg
--- a/figs/conclusion_maglev_heyman23.jpg
+++ b/figs/conclusion_maglev_heyman23.jpg
--- a/figs/conclusion_nass_architecture.pdf
+++ b/figs/conclusion_nass_architecture.pdf
--- a/figs/conclusion_nass_architecture.png
+++ b/figs/conclusion_nass_architecture.png
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--- a/figs/inkscape/conclusion_nass_concept_schematic.svg
+++ b/figs/inkscape/conclusion_nass_concept_schematic.svg
--- a/phd-thesis.bib
+++ b/phd-thesis.bib
@ -457,50 +457,6 @@
@book{schmidt20_desig_high_perfor_mechat_third_revis_edition,
  author          = {Schmidt, R Munnig and Schitter, Georg and Rankers, Adrian},
  title           = {The Design of High Performance Mechatronics - Third Revised
                  Edition},
  year            = 2020,
  publisher       = {Ios Press},
  isbn            = {978-1-64368-050-7},
  keywords        = {favorite},
 }
@article{du14_piezo_actuat_high_precis_flexib,
  author          = {Zhijiang Du and Ruochong Shi and Wei Dong},
  title           = {A Piezo-Actuated High-Precision Flexible Parallel Pointing
                  Mechanism: Conceptual Design, Development, and Experiments},
  journal         = {IEEE Transactions on Robotics},
  volume          = 30,
  number          = 1,
  pages           = {131-137},
  year            = 2014,
  doi             = {10.1109/tro.2013.2288800},
  url             = {https://doi.org/10.1109/tro.2013.2288800},
  keywords        = {parallel robot},
 }
@article{hauge04_sensor_contr_space_based_six,
  author          = {G.S. Hauge and M.E. Campbell},
  title           = {Sensors and Control of a Space-Based Six-Axis Vibration
                  Isolation System},
  journal         = {Journal of Sound and Vibration},
  volume          = 269,
  number          = {3-5},
  pages           = {913-931},
  year            = 2004,
  doi             = {10.1016/s0022-460x(03)00206-2},
  url             = {https://doi.org/10.1016/s0022-460x(03)00206-2},
  keywords        = {parallel robot, favorite},
 }
@inproceedings{dehaeze18_sampl_stabil_for_tomog_exper,
  author          = {Dehaeze, T. and Magnin Mattenet, M. and Collette, C.},
  title           = {Sample Stabilization For Tomography Experiments In Presence
@ -1312,6 +1268,22 @@
@article{hauge04_sensor_contr_space_based_six,
  author          = {G.S. Hauge and M.E. Campbell},
  title           = {Sensors and Control of a Space-Based Six-Axis Vibration
                  Isolation System},
  journal         = {Journal of Sound and Vibration},
  volume          = 269,
  number          = {3-5},
  pages           = {913-931},
  year            = 2004,
  doi             = {10.1016/s0022-460x(03)00206-2},
  url             = {https://doi.org/10.1016/s0022-460x(03)00206-2},
  keywords        = {parallel robot, favorite},
 }
@article{geng95_intel_contr_system_multip_degree,
  author          = {Z. Jason Geng and George G. Pan and Leonard S. Haynes and
                  Ben K. Wada and John A. Garba},
@ -1489,6 +1461,22 @@
@article{du14_piezo_actuat_high_precis_flexib,
  author          = {Zhijiang Du and Ruochong Shi and Wei Dong},
  title           = {A Piezo-Actuated High-Precision Flexible Parallel Pointing
                  Mechanism: Conceptual Design, Development, and Experiments},
  journal         = {IEEE Transactions on Robotics},
  volume          = 30,
  number          = 1,
  pages           = {131-137},
  year            = 2014,
  doi             = {10.1109/tro.2013.2288800},
  url             = {https://doi.org/10.1109/tro.2013.2288800},
  keywords        = {parallel robot},
 }
@phdthesis{naves21_desig_optim_large_strok_flexur_mechan,
  author          = {Mark Naves},
  day             = 21,
@ -1664,6 +1652,18 @@
@book{schmidt20_desig_high_perfor_mechat_third_revis_edition,
  author          = {Schmidt, R Munnig and Schitter, Georg and Rankers, Adrian},
  title           = {The Design of High Performance Mechatronics - Third Revised
                  Edition},
  year            = 2020,
  publisher       = {Ios Press},
  isbn            = {978-1-64368-050-7},
  keywords        = {favorite},
 }
@inproceedings{li01_simul_vibrat_isolat_point_contr,
  author          = {Xiaochun Li and Jerry C. Hamann and John E. McInroy},
  title           = {Simultaneous Vibration Isolation and Pointing Control of
@ -2340,6 +2340,48 @@
@phdthesis{fahmy22_magnet_xy_theta_x,
  author          = {Fahmy, Abdel},
  school          = {The University of Texas at Austin},
  title           = {Magnetically levitated XY-THETA motion stage for X-ray
                  microscopy applications},
  year            = 2022,
 }
@article{heyman23_levcub,
  author          = {Ian L. Heyman and Jingjie Wu and Lei Zhou},
  title           = {Levcube: a Six-Degree-Of-Freedom Magnetically Levitated
                  Nanopositioning Stage With Centimeter-Range Xyz Motion},
  journal         = {Precision Engineering},
  volume          = 83,
  pages           = {102-111},
  year            = 2023,
  doi             = {10.1016/j.precisioneng.2023.04.008},
  url             = {http://dx.doi.org/10.1016/j.precisioneng.2023.04.008},
  keywords        = {maglev},
 }
@article{dyck15_magnet_levit_six_degree_freed_rotar_table,
  author          = {Mark Dyck and Xiaodong Lu and Yusuf Altintas},
  title           = {Magnetically Levitated Six Degree of Freedom Rotary Table},
  journal         = {CIRP Annals},
  volume          = 64,
  number          = 1,
  pages           = {353-356},
  year            = 2015,
  doi             = {10.1016/j.cirp.2015.04.107},
  url             = {http://dx.doi.org/10.1016/j.cirp.2015.04.107},
  keywords        = {maglev},
 }
@article{gustavsen99_ration_approx_frequen_domain_respon,
  author          = {Gustavsen, B.; Semlyen, A.},
  title           = {Rational Approximation of Frequency Domain Responses By
--- a/phd-thesis.org
+++ b/phd-thesis.org
@ -13642,8 +13642,146 @@ The system performed exceptionally well during vertical scans, though some limit
 With the implementation of an accurate online metrology system, the NASS will be ready for integration into the beamline environment, significantly enhancing the capabilities of high-precision X-ray experimentation on the ID31 beamline.
-* TODO Conclusion and Future Work
+* Conclusion and Future Work
 <<chap:conclusion>>
 #+LATEX: \begingroup
 #+LATEX: \def\clearpage{\par}
 ** Summary of Findings
 #+LATEX: \endgroup
 The primary objective of this research was to enhance the positioning accuracy of the ID31 micro-station by approximately two orders of magnitude, enabling full exploitation of the new $4^{\text{th}}$ generation light source, without compromising the system's mobility or its capacity to handle payloads up to $50\,\text{kg}$.
 To meet this demanding objective, the concept of a Nano Active Stabilization System (NASS) was proposed and developed.
 This system comprises an active stabilization platform positioned between the existing micro-station and the sample.
 Integrated with an external online metrology system and an custom control architecture, the NASS was designed to actively measure and compensate for positioning errors originating from various sources, including micro-station imperfections, thermal drift, and vibrations.
 The conceptual design phase rigorously evaluated the feasibility of the NASS concept.
 Through progressive modeling, from simplified uniaxial representations to complex multi-body dynamic simulations, key design insights were obtained.
 It was determined that an active platform with moderate stiffness offered an optimal compromise, decoupling the system from micro-station dynamics while mitigating gyroscopic effects from continuous rotation.
 The multi-body modeling approach, informed by experimental modal analysis of the micro-station, was essential for capturing the system's complex dynamics.
 The Stewart platform architecture was selected for the active stage, and its viability was confirmed through closed-loop simulations employing a High-Authority Control / Low-Authority Control (HAC-LAC) strategy.
 This strategy incorporated a modified form of Integral Force Feedback (IFF), adapted to provide robust active damping despite the platform rotation and varying payloads.
 These simulations demonstrated the NASS concept could meet the nanometer-level stability requirements under realistic operating conditions.
 Following the conceptual validation, the detailed design phase focused on translating the NASS concept into an optimized, physically realizable system.
 Geometric optimization studies refined the Stewart platform configuration.
 A hybrid modeling technique combining Finite Element Analysis (FEA) with multi-body dynamics simulation was applied and experimentally validated.
 This approach enabled detailed optimization of components, such as amplified piezoelectric actuators and flexible joints, while efficiently simulating the complete system dynamics.
 Work was also undertaken on the optimization of the control strategy for the active platform.
 Instrumentation selection (sensors, actuators, control hardware) was guided by dynamic error budgeting to ensure component noise levels met the overall nanometer-level performance target.
 The final phase of the project was dedicated to the experimental validation of the developed NASS.
 Component tests confirmed the performance of the selected actuators and flexible joints, validated their respective models.
 Dynamic testing of the assembled nano-hexapod on an isolated test bench provided essential experimental data that correlated well with the predictions of the multi-body model.
 The final validation was performed on the ID31 beamline, utilizing a short-stroke metrology system to assess performance under realistic experimental conditions.
 These tests demonstrated that the NASS, operating with the implemented HAC-LAC control architecture, successfully achieved the target positioning stability – maintaining residual errors below $30\,\text{nm RMS}$ laterally, $15\,\text{nm RMS}$ vertically, and $250\,\text{nrad RMS}$ in tilt – during various experiments, including tomography scans with significant payloads.
 Crucially, the system's robustness to variations in payload mass and operational modes was confirmed.
 ** Perspectives
 Although this research successfully validated the NASS concept, it concurrently highlighted specific areas where the system could be enhanced, alongside related topics that merit further investigation.
 ***** Automatic tuning of a multi-body model from an experimental modal analysis
 The manual tuning process employed to match the multi-body model dynamics with experimental measurements was found to be laborious.
 Systems like the micro-station can be conceptually modeled as interconnected solid bodies, springs, and dampers, with component inertia readily obtainable from CAD models.
 An interesting perspective is the development of methods for the automatic tuning of the multi-body model's stiffness matrix (representing the interconnecting spring stiffnesses) directly from experimental modal analysis data.
 Such a capability would enable the rapid generation of accurate dynamic models for existing end-stations, which could subsequently be utilized for detailed system analysis and simulation studies.
 ***** Better addressing plant uncertainty coming from a change of payload
 For most high-performance mechatronic systems like lithography machines or atomic force microscopes, payloads inertia are often known and fixed, allowing controllers to be precisely optimized.
 However, synchrotron end-stations frequently handle samples with widely varying masses and inertias – ID31 being an extreme example, but many require nanometer positioning for samples from very light masses up to 5kg.
 The conventional strategy involves implementing controllers with relatively small bandwidth to accommodate various payloads.
 When controllers are optimized for a specific payload, changing payloads may destabilize the feedback loops that needs to be re-tuned.
 In this thesis, the HAC-IFF robust control approach was employed to maintain stability despite payload variations, though this resulted in relatively modest bandwidth.
 Therefore, a key objective for future work is to enhance the management of payload-induced plant uncertainty, aiming for improved performance without sacrificing robustness.
 Potential strategies to be explored include adaptive control (involving automatic plant identification and controller tuning after a change of payload) and robust control techniques such as $\mu\text{-synthesis}$ (allowing the controller to be synthesized while explicitly considering a specified range of payload masses).
 ***** Control based on Complementary Filters
 The control architecture based on complementary filters (detailed in Section ref:sec:detail_control_cf) has been successfully implemented in several instruments at the ESRF.
 This approach has proven straightforward to implement and offers the valuable capability of modifying closed-loop behavior in real time, which proves advantageous for many applications.
 For instance, the controller can be optimized according to the scan type: constant velocity scans benefit from a $+2$ slope for the sensitivity transfer function, while ptychography may be better served by a $+1$ slope with slightly higher bandwidth to minimize point-to-point transition times.
 Nevertheless, a more rigorous analysis of this control architecture and its comparison with similar approaches documented in the literature is necessary to fully understand its capabilities and limitations.
 ***** Sensor Fusion
 While the HAC-LAC approach demonstrated a simple and comprehensive methodology for controlling the NASS, sensor fusion represents an interesting alternative that is worth investigating.
 While the synthesis method developed for complementary filters facilitates their design (Section ref:sec:detail_control_sensor), their application specifically for sensor fusion within the NASS context was not examined in detail.
 One potential approach involves fusing external metrology (utilized at low frequencies) with force sensors (employed at high frequencies).
 This configuration could enhance robustness through the collocation of force sensors with actuators.
 The integration of encoder feedback into the control architecture could also be explored.
 ***** Development of multi-DoF metrology systems
 Although experimental validation using the short-stroke metrology prototype was achieved, the NASS remains unsuitable for beamline applications due to the lack of a long stroke metrology system.
 Efforts were initiated during this project to develop such a metrology system, though these were not presented herein as the focus was directed toward the active platform, instrumentation, and controllers.
 The development process revealed that the metrology system constitutes a complex mechatronic system, which could benefit significantly from the design approach employed throughout this thesis.
 This challenge is particularly complex when continuous rotation is combined with long stroke movements.
 Yet, the development of such metrology systems is considered critical for future end-stations, especially for future tomography end stations where nano-meter accuracy is desired across larger strokes.
 Promising approaches have been presented in the literature.
 A ball lens retroreflector is used in [[cite:&schropp20_ptynam]], providing a $\approx 1\,\text{mm}^3$ measuring volume, but does not fully accommodate complete rotation.
 In [[cite:&geraldes23_sapot_carnaub_sirius_lnls]], an interesting metrology approach is presented, utilizing interferometers for long stroke/non-rotated movements and capacitive sensors for short stroke/rotated positioning.
 ***** Alternative Architecture for the NASS
 The original micro-station design was driven by optimizing positioning accuracy, utilizing dedicated actuators for different DoFs (leading to simple kinematics and a stacked configuration), and maximizing stiffness.
 This design philosophy ensured that the micro-station would remain functional for micro-focusing applications even if the NASS project did not meet expectations.
 Analyzing the NASS as an complete system reveals that the positioning accuracy is primarily determined by the metrology system and the feedback control.
 Consequently, the underlying micro-station's own positioning accuracy has minimal influence on the final performances (it does however impact the required mobility of the active platform).
 Nevertheless, it remains crucial that the micro-station itself does not generate detrimental high-frequency vibrations, particularly during movements, as evidenced by issues previously encountered with stepper motors.
 Designing a future end-station with the understanding that a functional NASS will ensure final positioning accuracy could allow for a significantly simplified long-stroke stage architecture, perhaps chosen primarily to facilitate the integration of the online metrology.
 One possible configuration, illustrated in Figure ref:fig:conclusion_nass_architecture, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that need not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle.
 #+name: fig:conclusion_nass_architecture
 #+caption: Proposed alternative configuration for an end-station including the Nano Active Stabilization System
 #+attr_latex: :options [h!tbp]
 [[file:figs/conclusion_nass_architecture.png]]
 With this architecture, the online metrology could be divided into two systems, as proposed by [[cite:&geraldes23_sapot_carnaub_sirius_lnls]]: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by [[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]].
 ***** Development of long stroke high performance stage
 As an alternative to the short-stroke/long-stroke architecture, the development of a high-performance long-stroke stage seems worth investigating.
 Stages based on voice coils, offering nano-positioning capabilities with $3\,mm$ stroke, have recently been reported in the literature [[cite:&schropp20_ptynam;&kelly22_delta_robot_long_travel_nano]].
 Magnetic levitation also emerges as a particularly interesting technology to be explored, especially for microscopy  [[cite:&fahmy22_magnet_xy_theta_x;&heyman23_levcub]] and tomography [[cite:&dyck15_magnet_levit_six_degree_freed_rotar_table;&fahmy22_magnet_xy_theta_x]] end-stations.
 Two notable designs illustrating these capabilities are shown in Figure ref:fig:conclusion_maglev.
 Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately $1\,\text{cm}^3$, is depicted in Figure ref:fig:conclusion_maglev_heyman23, while a 6DoF stage featuring infinite rotation, is shown in Figure ref:fig:conclusion_maglev_dyck15.
 However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature.
 #+name: fig:conclusion_maglev
 #+caption: Example of magnetic levitation stages. LevCube allowing for 6DoF control of the position with $\approx 1\,\text{cm}^3$ mobility (\subref{fig:conclusion_maglev_heyman23}). Magnetic levitation stage with infinite $R_z$ rotation mobility (\subref{fig:conclusion_maglev_dyck15})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:conclusion_maglev_heyman23}LevCube with $\approx 1\,\text{cm}^3$ mobility \cite{heyman23_levcub}}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
 #+attr_latex: :width 0.9\linewidth
 [[file:figs/conclusion_maglev_heyman23.jpg]]
 #+end_subfigure
 #+attr_latex: :caption \subcaption{\label{fig:conclusion_maglev_dyck15}Stage with infinite $R_z$ rotation \cite{dyck15_magnet_levit_six_degree_freed_rotar_table}}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
 #+attr_latex: :width 0.9\linewidth
 [[file:figs/conclusion_maglev_dyck15.jpg]]
 #+end_subfigure
 #+end_figure
 ***** Extending the design methodology to complete beamlines
 The application of dynamic error budgeting and the mechatronic design approach to an entire beamline represents an interesting direction for future work.
 During the early design phases of a beamline, performance metrics are typically expressed as integrated values (usually RMS values) rather than as functions of frequency.
 However, the frequency content of these performance metrics (such as beam stability, energy stability, and sample stability) is crucial, as factors like detector integration time can filter out high-frequency components.
 Therefore, adopting a design approach utilizing dynamic error budgets, cascading from overall beamline requirements down to individual component specifications, is considered a potentially valuable direction for future investigation.
 * Bibliography                                                        :ignore:
 #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
--- a/phd-thesis.pdf
+++ b/phd-thesis.pdf
--- a/phd-thesis.tex
+++ b/phd-thesis.tex
@ -1,4 +1,4 @@
-% Created 2025-04-18 Fri 16:53
+% Created 2025-04-20 Sun 18:06
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -14,7 +14,6 @@
 \newacronym{frf}{FRF}{Frequency Response Function}
 \newacronym{iff}{IFF}{Integral Force Feedback}
 \newacronym{rdc}{RDC}{Relative Damping Control}
 \newacronym{drga}{DRGA}{Dynamical Relative Gain Array}
 \newacronym{rga}{RGA}{Relative Gain Array}
 \newacronym{hpf}{HPF}{high-pass filter}
 \newacronym{lpf}{LPF}{low-pass filter}
@ -42,7 +41,7 @@
 \addbibresource{ref.bib}
 \addbibresource{phd-thesis.bib}
 \author{Dehaeze Thomas}
-\date{2025-04-18}
+\date{2025-04-20}
 \title{Nano Active Stabilization of samples for tomography experiments: A mechatronic design approach}
 \subtitle{PhD Thesis}
 \hypersetup{
@ -212,10 +211,10 @@ This global distribution of such facilities underscores the significant utility
 \caption{\label{fig:introduction_synchrotrons}Major synchrotron radiation facilities in the world. 3rd generation Synchrotrons are shown in blue. Planned upgrades to 4th generation are shown in green, and 4th generation Synchrotrons in operation are shown in red.}
 \end{figure}
-These facilities fundamentally comprise two main parts: the accelerator complex, where electron acceleration and light generation occur, and the beamlines, where the intense X-ray beams are conditioned and directed for experimental use.
+These facilities fundamentally comprise two main parts: the accelerator and storage ring, where electron acceleration and light generation occur, and the beamlines, where the intense X-ray beams are conditioned and directed for experimental use.
 The European Synchrotron Radiation Facility (ESRF), shown in Figure~\ref{fig:introduction_esrf_picture}, is a joint research institution supported by 19 member countries.
-The ESRF commenced user operations in 1994 as the world's first third-generation synchrotron.
+The ESRF started user operations in 1994 as the world's first third-generation synchrotron.
 Its accelerator complex, schematically depicted in Figure~\ref{fig:introduction_esrf_schematic}, includes a linear accelerator where electrons are initially generated and accelerated, a booster synchrotron to further accelerate the electrons, and an 844-meter circumference storage ring where electrons are maintained in a stable orbit.
 Synchrotron light are emitted in more than 40 beamlines surrounding the storage ring, each having specialized experimental stations.
@ -274,7 +273,7 @@ These components are housed in multiple Optical Hutches, as depicted in Figure~\
 \caption{\label{fig:introduction_id31_oh}Schematic of the two ID31 optical hutches: OH1 (\subref{fig:introduction_id31_oh1}) and OH2 (\subref{fig:introduction_id31_oh2}). Distance from the source (the insertion device) is indicated in meters.}
 \end{figure}
-Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure~\ref{fig:introduction_id31_cad}), where, for experiments pertinent to this work, focusing optics are utilized.
+Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure~\ref{fig:introduction_id31_cad}), where, for experiments pertinent to this work, focusing optics are used.
 The sample is mounted on a positioning stage, referred to as the ``end-station'', that enables precise alignment relative to the X-ray beam.
 Detectors are used to capture the X-rays transmitted through or scattered by the sample.
 Throughout this thesis, the standard ESRF coordinate system is adopted, wherein the X-axis aligns with the beam direction, Y is transverse horizontal, and Z is vertical upwards against gravity.
@ -282,7 +281,7 @@ Throughout this thesis, the standard ESRF coordinate system is adopted, wherein
 The specific end-station employed on the ID31 beamline is designated the ``micro-station''.
 As depicted in Figure~\ref{fig:introduction_micro_station_dof}, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a micro-hexapod (in purple).
 The sample itself (cyan), potentially housed within complex sample environments (e.g., for high pressure or extreme temperatures), is mounted on top of this assembly.
-Each stage serves distinct positioning functions; for example, the micro-hexapod enables fine static adjustments, while the \(T_y\) translation and \(R_z\) rotation stages are utilized for specific scanning applications.
+Each stage serves distinct positioning functions; for example, the micro-hexapod enables fine static adjustments, while the \(T_y\) translation and \(R_z\) rotation stages are used for specific scanning applications.
 \begin{figure}[h!tbp]
 \begin{subfigure}{0.52\textwidth}
@ -346,7 +345,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
 \caption{\label{fig:introduction_scanning}Exemple of a scanning experiment. The sample is scanned in the Y-Z plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of 20nm (\subref{fig:introduction_scanning_results}).}
 \end{figure}
 \subsubsection*{Need of Accurate Positioning End-Stations with High Dynamics}
-Continuous advancements in both synchrotron source technology and X-ray optics have led to the availability of smaller, more intense, and more stable X-ray beams.
+Continuous progress in both synchrotron source technology and X-ray optics have led to the availability of smaller, more intense, and more stable X-ray beams.
 The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source size, particularly in the horizontal dimension, coupled with increased brilliance, as illustrated in Figure~\ref{fig:introduction_beam_3rd_4th_gen}.
 \begin{figure}[h!tbp]
@ -410,7 +409,7 @@ With higher X-ray flux and reduced detector noise, integration times can now be
 This reduction in integration time has two major implications for positioning requirements.
 Firstly, for a given spatial sampling (``pixel size''), faster integration necessitates proportionally higher scanning velocities.
 Secondly, the shorter integration times make the measurements more susceptible to high-frequency vibrations.
-Therefore, not only must the sample position be stable against long-term drifts, but it must also be actively controlled to minimize vibrations, especially during dynamic fly-scan acquisitions.
+Therefore, not only the sample position must be stable against long-term drifts, but it must also be actively controlled to minimize vibrations, especially during dynamic fly-scan acquisitions.
 \subsubsection*{Existing Nano Positioning End-Stations}
 To contextualize the system developed within this thesis, a brief overview of existing strategies and technologies for high-accuracy, high-dynamics end-stations is provided.
 The aim is to identify the specific characteristics that distinguish the proposed system from current state-of-the-art implementations.
@ -465,11 +464,11 @@ The concept of using an external metrology to measure and potentially correct fo
 Ideally, the relative position between the sample's point of interest and the X-ray beam focus would be measured directly.
 In practice, direct measurement is often impossible; instead, the sample position is typically measured relative to a reference frame associated with the focusing optics, providing an indirect measurement.
-This measured position can be utilized in several ways: for post-processing correction of acquired data; for calibration routines to compensate for repeatable errors; or, most relevantly here, for real-time feedback control.
+This measured position can be used in several ways: for post-processing correction of acquired data; for calibration routines to compensate for repeatable errors; or, most relevantly here, for real-time feedback control.
 Various sensor technologies have been employed, with capacitive sensors~\cite{schroer17_ptynam,villar18_nanop_esrf_id16a_nano_imagin_beaml,schropp20_ptynam} and, increasingly, fiber-based interferometers~\cite{nazaretski15_pushin_limit,stankevic17_inter_charac_rotat_stages_x_ray_nanot,holler17_omny_pin_versat_sampl_holder,holler18_omny_tomog_nano_cryo_stage,engblom18_nanop_resul,schropp20_ptynam,nazaretski22_new_kirkp_baez_based_scann,kelly22_delta_robot_long_travel_nano,xu23_high_nsls_ii,geraldes23_sapot_carnaub_sirius_lnls} being prominent choices.
 Two examples illustrating the integration of online metrology are presented in Figure~\ref{fig:introduction_metrology_stations}.
-The system at NSLS X8C (Figure~\ref{fig:introduction_stages_wang}) utilized capacitive sensors for rotation stage calibration and image alignment during tomography post-processing~\cite{wang12_autom_marker_full_field_hard}.
+The system at NSLS X8C (Figure~\ref{fig:introduction_stages_wang}) used capacitive sensors for rotation stage calibration and image alignment during tomography post-processing~\cite{wang12_autom_marker_full_field_hard}.
 The PtiNAMi microscope at DESY P06 (Figure~\ref{fig:introduction_stages_schroer}) employs interferometers directed at a spherical target below the sample for position monitoring during tomography, with plans for future feedback loop implementation~\cite{schropp20_ptynam}.
 \begin{figure}[h!tbp]
@ -552,7 +551,7 @@ 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 kg.
-The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately \(10\,\mu m\) and \(10\,\mu\text{rad}\) due to inherent factors such as backlash, mechanical play, thermal expansion, imperfect guiding, and vibrations.
+The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately \(10\,\mu m\) and \(10\,\mu\text{rad}\) 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.
 \paragraph{The Nano Active Stabilization System Concept}
@ -632,26 +631,26 @@ This leads to strong dynamic coupling between the active platform and the micro-
 These variations in operating conditions and payload translate into significant uncertainty or changes in the plant dynamics that the controller must handle.
 Therefore, the feedback controller must be designed to be robust against this plant uncertainty while still delivering the required nanometer-level performance.
-\paragraph{Predictive Design / Mechatronics approach}
+\paragraph{Predictive Design}
 The overall performance achieved by the NASS is determined by numerous factors, such as external disturbances, the noise characteristics of the instrumentation, the dynamics resulting from the chosen mechanical architecture, and the achievable bandwidth dictated by the control architecture.
-Ensuring the final system met its stringent specifications requires the implementation of a predictive design methodology, also known as a mechatronics design approach.
+Ensuring the final system meets its stringent specifications requires the implementation of a predictive design methodology, also known as a mechatronic design approach.
 The goal is to rigorously evaluate different concepts, predict performance limitations, and guide the design process.
 Key challenges within this approach include developing appropriate design methodologies, creating accurate models capable of comparing different concepts quantitatively, and converging on a final design that achieves the target performance levels.
 \section{Original Contributions}
-This thesis presents several original contributions aimed at addressing the challenges inherent in the design, control, and implementation of the Nano Active Stabilization System, primarily within the fields of Control Theory, Mechatronics Design, and Experimental Validation.
+This thesis presents several original contributions aimed at addressing the challenges inherent in the design, control, and implementation of the Nano Active Stabilization System, primarily within the fields of Control Theory, Mechatronic Design, and Experimental Validation.
 \paragraph{6DoF vibration control of a rotating platform}
 Traditional long-stroke/short-stroke architectures typically operate in one or two degrees of freedom.
 This work extends the concept to six degrees of freedom, with the active platform designed not only to correct rotational errors but to simultaneously compensate for errors originating from all underlying micro-station stages.
 The application of a continuously rotating Stewart platform for active vibration control and error compensation in this manner is believed to be novel in the reviewed literature.
-\paragraph{Mechatronics design approach}
+\paragraph{Mechatronic design approach}
-A rigorous mechatronics design methodology was applied consistently throughout the NASS development lifecycle~\cite{dehaeze18_sampl_stabil_for_tomog_exper,dehaeze21_mechat_approac_devel_nano_activ_stabil_system}.
+A rigorous mechatronic design methodology was applied consistently throughout the NASS development life-cycle~\cite{dehaeze18_sampl_stabil_for_tomog_exper,dehaeze21_mechat_approac_devel_nano_activ_stabil_system}.
-Although the mechatronics approach itself is not new, its comprehensive application here, from initial concept evaluation using simplified models to detailed design optimization and experimental validation informed by increasingly sophisticated models, potentially offers useful insights to the existing literature.
+Although the mechatronic approach itself is not new, its comprehensive application here, from initial concept evaluation using simplified models to detailed design optimization and experimental validation informed by increasingly sophisticated models, potentially offers useful insights to the existing literature.
-This thesis documents this process chronologically, illustrating how models of varying complexity can be effectively utilized at different project phases and how design decisions were systematically based on quantitative model predictions and analyses.
+This thesis documents this process chronologically, illustrating how models of varying complexity can be effectively used at different project phases and how design decisions were systematically based on quantitative model predictions and analyses.
-While the resulting system is highly specific, the documented effectiveness of this design approach may contribute to the broader adoption of mechatronics methodologies in the design of future synchrotron instrumentation.
+While the resulting system is highly specific, the documented effectiveness of this design approach may contribute to the broader adoption of mechatronic methodologies in the design of future synchrotron instrumentation.
-\paragraph{Multi-body simulations with reduced order flexible bodies obtained by FEA}
+\paragraph{Experimental validation of multi-body simulations with reduced order flexible bodies obtained by FEA}
 A key tool employed extensively in this work was a combined multi-body simulation and Finite Element Analysis technique, specifically utilizing Component Mode Synthesis to represent flexible bodies within the multi-body framework~\cite{brumund21_multib_simul_reduc_order_flexib_bodies_fea}.
 This hybrid approach, while established, was experimentally validated in this work for components critical to the NASS, namely amplified piezoelectric actuators and flexible joints.
@ -686,8 +685,7 @@ Crucially, robustness to variations in sample mass and diverse experimental cond
 The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal utilization 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.
 \section{Outline}
-This thesis is structured chronologically, mirroring the phases of the mechatronics development approach employed for the NASS project.
+This is divided into three chapters, each corresponding to a distinct phase of this methodology: Conceptual Design, Detailed Design, and Experimental Validation.
 It is divided into three chapters, each corresponding to a distinct phase of this methodology: Conceptual Design, Detailed Design, and Experimental Validation.
 While the chapters follow this logical progression, care has been taken to structure each chapter such that its constitutive sections may also be consulted independently based on the reader's specific interests.
 \paragraph{Conceptual design development}
@ -1061,6 +1059,7 @@ From this analysis, it may be concluded that the stiffer the nano-hexapod the be
 However, what is more important is the \emph{closed-loop} residual vibration of \(d\) (i.e., while the feedback controller is used).
 The goal is to obtain a closed-loop residual vibration \(\epsilon_d \approx 20\,nm\,\text{RMS}\) (represented by an horizontal dashed black line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}).
 The bandwidth of the feedback controller leading to a closed-loop residual vibration of \(20\,nm\,\text{RMS}\) can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure~\ref{fig:uniaxial_cas_d_disturbances_payload_masses}.
 A closed loop bandwidth of \(\approx 10\,\text{Hz}\) is found for the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)), \(\approx 50\,\text{Hz}\) for the relatively stiff nano-hexapod (\(k_n = 1\,N/\mu m\)), and \(\approx 100\,\text{Hz}\) for the stiff nano-hexapod (\(k_n = 100\,N/\mu m\)).
 Therefore, while the \emph{open-loop} vibration is the lowest for the stiff nano-hexapod, it requires the largest feedback bandwidth to meet the specifications.
@ -1948,34 +1947,34 @@ The uniform rotation of the system induces two \emph{gyroscopic effects} as show
 \end{itemize}
 One can verify that without rotation (\(\Omega = 0\)), the system becomes equivalent to two \emph{uncoupled} one degree of freedom mass-spring-damper systems.
-To study the dynamics of the system, the two differential equations of motions~\eqref{eq:rotating_eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation~\eqref{eq:rotating_Gd_mimo_tf} is obtained.
+To study the dynamics of the system, the two differential equations of motions~\eqref{eq:rotating_eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\bm{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation~\eqref{eq:rotating_Gd_mimo_tf} is obtained.
-The four transfer functions in \(\mathbf{G}_d\) are shown in equation~\eqref{eq:rotating_Gd_indiv_el}.
+The four transfer functions in \(\bm{G}_d\) are shown in equation~\eqref{eq:rotating_Gd_indiv_el}.
 \begin{equation}\label{eq:rotating_Gd_mimo_tf}
-  \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
+  \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
 \end{equation}
 \begin{subequations}\label{eq:rotating_Gd_indiv_el}
 \begin{align}
-  \mathbf{G}_{d}(1,1) &=  \mathbf{G}_{d}(2,2) = \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \\
+  \bm{G}_{d}(1,1) &=  \bm{G}_{d}(2,2) = \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \\
-  \mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}
+  \bm{G}_{d}(1,2) &= -\bm{G}_{d}(2,1) = \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}
 \end{align}
 \end{subequations}
 To simplify the analysis, the undamped natural frequency \(\omega_0\) and the damping ratio \(\xi\) defined in~\eqref{eq:rotating_xi_and_omega} are used instead.
-The elements of the transfer function matrix \(\mathbf{G}_d\) are described by equation~\eqref{eq:rotating_Gd_w0_xi_k}.
+The elements of the transfer function matrix \(\bm{G}_d\) are described by equation~\eqref{eq:rotating_Gd_w0_xi_k}.
 \begin{equation} \label{eq:rotating_xi_and_omega}
  \omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}}
 \end{equation}
 \begin{subequations} \label{eq:rotating_Gd_w0_xi_k}
  \begin{align}
-    \mathbf{G}_{d}(1,1) &= \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \\
+    \bm{G}_{d}(1,1) &= \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \\
-    \mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
+    \bm{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
  \end{align}
 \end{subequations}
 \subsubsection{System Poles: Campbell Diagram}
-The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation~\eqref{eq:rotating_poles} (i.e. the roots of its denominator).
+The poles of \(\bm{G}_d\) are the complex solutions \(p\) of equation~\eqref{eq:rotating_poles} (i.e. the roots of its denominator).
 \begin{equation}\label{eq:rotating_poles}
  \left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0
@ -2012,7 +2011,7 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo
 \end{figure}
 \subsubsection{System Dynamics: Effect of rotation}
 The system dynamics from actuator forces \([F_u, F_v]\) to the relative motion \([d_u, d_v]\) is identified for several rotating velocities.
-Looking at the transfer function matrix \(\mathbf{G}_d\) in equation~\eqref{eq:rotating_Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
+Looking at the transfer function matrix \(\bm{G}_d\) in equation~\eqref{eq:rotating_Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
 The bode plots of these two terms are shown in Figure~\ref{fig:rotating_bode_plot} for several rotational speeds \(\Omega\).
 These plots confirm the expected behavior: the frequencies of the two pairs of complex conjugate poles are further separated as \(\Omega\) increases.
 For \(\Omega > \omega_0\), the low-frequency pair of complex conjugate poles \(p_{-}\) becomes unstable (shown be the 180 degrees phase lead instead of phase lag).
@ -2082,21 +2081,21 @@ The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force s
  \begin{bmatrix} d_u \\ d_v \end{bmatrix}
 \end{equation}
-The transfer function matrix \(\mathbf{G}_{f}\) from actuator forces to measured forces in equation~\eqref{eq:rotating_Gf_mimo_tf} can be obtained by inserting equation~\eqref{eq:rotating_Gd_w0_xi_k} into equation~\eqref{eq:rotating_measured_force}.
+The transfer function matrix \(\bm{G}_{f}\) from actuator forces to measured forces in equation~\eqref{eq:rotating_Gf_mimo_tf} can be obtained by inserting equation~\eqref{eq:rotating_Gd_w0_xi_k} into equation~\eqref{eq:rotating_measured_force}.
 Its elements are shown in equation~\eqref{eq:rotating_Gf}.
 \begin{equation}\label{eq:rotating_Gf_mimo_tf}
-  \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
+  \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \bm{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
 \end{equation}
 \begin{subequations}\label{eq:rotating_Gf}
  \begin{align}
-    \mathbf{G}_{f}(1,1) &= \mathbf{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \label{eq:rotating_Gf_diag_tf} \\
+    \bm{G}_{f}(1,1) &= \bm{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \label{eq:rotating_Gf_diag_tf} \\
-    \mathbf{G}_{f}(1,2) &= -\mathbf{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \label{eq:rotating_Gf_off_diag_tf}
+    \bm{G}_{f}(1,2) &= -\bm{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \label{eq:rotating_Gf_off_diag_tf}
  \end{align}
 \end{subequations}
-The zeros of the diagonal terms of \(\mathbf{G}_f\) in equation~\eqref{eq:rotating_Gf_diag_tf} are computed, and neglecting the damping for simplicity, two complex conjugated zeros \(z_{c}\) \eqref{eq:rotating_iff_zero_cc}, and two real zeros \(z_{r}\) \eqref{eq:rotating_iff_zero_real} are obtained.
+The zeros of the diagonal terms of \(\bm{G}_f\) in equation~\eqref{eq:rotating_Gf_diag_tf} are computed, and neglecting the damping for simplicity, two complex conjugated zeros \(z_{c}\) \eqref{eq:rotating_iff_zero_cc}, and two real zeros \(z_{r}\) \eqref{eq:rotating_iff_zero_real} are obtained.
 \begin{subequations}
  \begin{align}
@ -2111,12 +2110,12 @@ This is what usually gives the unconditional stability of IFF when collocated fo
 However, for non-null rotational speeds, the two real zeros \(z_r\) in equation~\eqref{eq:rotating_iff_zero_real} are inducing a \emph{non-minimum phase behavior}.
 This can be seen in the Bode plot of the diagonal terms (Figure~\ref{fig:rotating_iff_bode_plot_effect_rot}) where the low-frequency gain is no longer zero while the phase stays at \(\SI{180}{\degree}\).
-The low-frequency gain of \(\mathbf{G}_f\) increases with the rotational speed \(\Omega\) as shown in equation~\eqref{eq:rotating_low_freq_gain_iff_plan}.
+The low-frequency gain of \(\bm{G}_f\) increases with the rotational speed \(\Omega\) as shown in equation~\eqref{eq:rotating_low_freq_gain_iff_plan}.
 This can be explained as follows: a constant actuator force \(F_u\) induces a small displacement of the mass \(d_u = \frac{F_u}{k - m\Omega^2}\) (Hooke's law considering the negative stiffness induced by the rotation).
 This small displacement then increases the centrifugal force \(m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u\) which is then measured by the force sensors.
 \begin{equation}\label{eq:rotating_low_freq_gain_iff_plan}
-  \lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix}
+  \lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
  \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\
  0  & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2}
 \end{bmatrix}
@ -2152,7 +2151,7 @@ The decentralized \acrshort{iff} controller \(\bm{K}_F\) corresponds to a diagon
 \begin{equation} \label{eq:rotating_Kf_pure_int}
 \begin{aligned}
-  \mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
+  \bm{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
  K_{F}(s) &= g \cdot \frac{1}{s}
 \end{aligned}
 \end{equation}
@ -2278,23 +2277,23 @@ To keep the overall stiffness \(k = k_a + k_p\) constant, thus not modifying the
  k_p = \alpha k, \quad k_a = (1 - \alpha) k
 \end{equation}
-After the equations of motion are derived and transformed in the Laplace domain, the transfer function matrix \(\mathbf{G}_k\) in Eq.~\eqref{eq:rotating_Gk_mimo_tf} is computed.
+After the equations of motion are derived and transformed in the Laplace domain, the transfer function matrix \(\bm{G}_k\) in Eq.~\eqref{eq:rotating_Gk_mimo_tf} is computed.
 Its elements are shown in Eqs.~\eqref{eq:rotating_Gk_diag} and \eqref{eq:rotating_Gk_off_diag}.
 \begin{equation}\label{eq:rotating_Gk_mimo_tf}
  \begin{bmatrix} f_u \\ f_v \end{bmatrix} =
-  \mathbf{G}_k
+  \bm{G}_k
  \begin{bmatrix} F_u \\ F_v \end{bmatrix}
 \end{equation}
 \begin{subequations}\label{eq:rotating_Gk}
 \begin{align}
-\mathbf{G}_{k}(1,1) &= \mathbf{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}  \label{eq:rotating_Gk_diag} \\
+\bm{G}_{k}(1,1) &= \bm{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}  \label{eq:rotating_Gk_diag} \\
-\mathbf{G}_{k}(1,2) &= -\mathbf{G}_{k}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \label{eq:rotating_Gk_off_diag}
+\bm{G}_{k}(1,2) &= -\bm{G}_{k}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \label{eq:rotating_Gk_off_diag}
 \end{align}
 \end{subequations}
-Comparing \(\mathbf{G}_k\) in~\eqref{eq:rotating_Gk} with \(\mathbf{G}_f\) in~\eqref{eq:rotating_Gf} shows that while the poles of the system remain the same, the zeros of the diagonal terms change.
+Comparing \(\bm{G}_k\) in~\eqref{eq:rotating_Gk} with \(\bm{G}_f\) in~\eqref{eq:rotating_Gf} shows that while the poles of the system remain the same, the zeros of the diagonal terms change.
 The two real zeros \(z_r\) in~\eqref{eq:rotating_iff_zero_real} that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in~\eqref{eq:rotating_kp_cond_cc_zeros} holds.
 Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{negative stiffness} induced by centrifugal forces \(m \Omega^2\), the dynamics from the actuator to its collocated force sensor will show \emph{minimum phase behavior}.
@ -2404,13 +2403,13 @@ Let's note \(\bm{G}_d\) the transfer function between actuator forces and measur
 The elements of \(\bm{G}_d\) were derived in Section~\ref{sec:rotating_system_description} are shown in~\eqref{eq:rotating_rdc_plant_elements}.
 \begin{equation}\label{eq:rotating_rdc_plant_matrix}
-  \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
+  \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
 \end{equation}
 \begin{subequations}\label{eq:rotating_rdc_plant_elements}
  \begin{align}
-    \mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \\
+    \bm{G}_{d}(1,1) &= \bm{G}_{d}(2,2) = \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}  \\
-    \mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
+    \bm{G}_{d}(1,2) &= -\bm{G}_{d}(2,1) = \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
  \end{align}
 \end{subequations}
@ -2915,16 +2914,16 @@ Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta AD
 \label{ssec:modal_test_preparation}
 To obtain meaningful results, the modal analysis of the micro-station is performed \emph{in-situ}.
-To do so, all the micro-station stage controllers are turned ``ON''.
+To do so, all the micro-station stage controllers are turned on.
 This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
-If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
+If these local feedback controls were turned off, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
 The top part representing the active stabilization stage was disassembled as the active stabilization stage will be added in the multi-body model afterwards.
-To perform the modal analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\mathbf{H}\) needs to be measured, where \(n\) is the considered number of degrees of freedom.
+To perform the modal analysis from the measured responses, the \(n \times n\) frequency response function matrix \(\bm{H}\) needs to be measured, where \(n\) is the considered number of degrees of freedom.
 The \(H_{jk}\) element of this \acrfull{frf} matrix corresponds to the frequency response function from a force \(F_k\) applied at \acrfull{dof} \(k\) to the displacement of the structure \(X_j\) at \acrshort{dof} \(j\).
 Measuring this \acrshort{frf} matrix is time consuming as it requires to make \(n \times n\) measurements.
-However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
+However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\bm{H}\) \cite[chapt. 5.2]{ewins00_modal}.
 Therefore, a minimum set of \(n\) frequency response functions is required.
 This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
@ -3086,7 +3085,7 @@ After all measurements are conducted, a \(n \times p \times q\) \acrlongpl{frf}
 For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations~\eqref{eq:modal_frf_matrix_raw}.
 \begin{equation}\label{eq:modal_frf_matrix_raw}
-  \mathbf{H}(\omega_i) = \begin{bmatrix}
+  \bm{H}(\omega_i) = \begin{bmatrix}
  \frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\
  \frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\
  \frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\
@ -3176,10 +3175,10 @@ Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\
 \end{tabularx}
 \end{table}
-Using~\eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\mathbf{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the center of mass of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\mathbf{H}\).
+Using~\eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the center of mass of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\bm{H}\).
 \begin{equation}\label{eq:modal_frf_matrix_com}
-  \mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
+  \bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
  \frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
  \frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
  \frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
@ -3194,8 +3193,8 @@ Using~\eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\mathbf{H}_\
 \subsubsection{Verification of solid body assumption}
 \label{ssec:modal_solid_body_assumption}
-From the response of one solid body expressed by its 6 \acrshortpl{dof} (i.e. from \(\mathbf{H}_{\text{CoM}}\)), and using equation~\eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any considered location.
+From the response of one solid body expressed by its 6 \acrshortpl{dof} (i.e. from \(\bm{H}_{\text{CoM}}\)), and using equation~\eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any considered location.
-In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements \(\mathbf{H}\).
+In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements \(\bm{H}\).
 This is what is done here to check whether the solid body assumption is correct in the frequency band of interest.
 The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure~\ref{fig:modal_comp_acc_solid_body_frf}).
@ -3334,27 +3333,27 @@ The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation~\eqref{
 \subsubsection{Verification of the modal model validity}
 \label{ssec:modal_model_validity}
-To check the validity of the modal model, the complete \(n \times n\) \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) is first synthesized from the modal parameters.
+To check the validity of the modal model, the complete \(n \times n\) \acrshort{frf} matrix \(\bm{H}_{\text{syn}}\) is first synthesized from the modal parameters.
-Then, the elements of this \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) that were already measured can be compared to the measured \acrshort{frf} matrix \(\mathbf{H}\).
+Then, the elements of this \acrshort{frf} matrix \(\bm{H}_{\text{syn}}\) that were already measured can be compared to the measured \acrshort{frf} matrix \(\bm{H}\).
 In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r\) are first organized in matrix from as shown in equation~\eqref{eq:modal_eigvector_matrix}.
 \begin{equation}\label{eq:modal_eigvector_matrix}
-\Phi = \begin{bmatrix}
+\bm{\Phi} = \begin{bmatrix}
  & & & & &\\
  \phi_1 & \dots & \phi_N & \phi_1^* & \dots & \phi_N^* \\
  & & & & &
 \end{bmatrix}_{n \times 2m}
 \end{equation}
-The full \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) can be obtained using~\eqref{eq:modal_synthesized_frf}.
+The full \acrshort{frf} matrix \(\bm{H}_{\text{syn}}\) can be obtained using~\eqref{eq:modal_synthesized_frf}.
 \begin{equation}\label{eq:modal_synthesized_frf}
-  [\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^{\intercal}
+  \left[\bm{H}_{\text{syn}}(\omega)\right]_{n\times n} = \left[\bm{\Phi}\right]_{n\times2m} \left[\bm{H}_{\text{mod}}(\omega)\right]_{2m\times2m} \left[\bm{\Phi}\right]_{2m\times n}^{\intercal}
 \end{equation}
-With \(\mathbf{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the response of the different modes~\eqref{eq:modal_modal_resp}.
+With \(\bm{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the response of the different modes~\eqref{eq:modal_modal_resp}.
 \begin{equation}\label{eq:modal_modal_resp}
-\mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
+  \bm{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
 \end{equation}
 A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure~\ref{fig:modal_comp_acc_frf_modal}.
@ -5630,7 +5629,7 @@ As anticipated by the control analysis, some performance degradation was observe
 \minitoc
 \subsubsection*{Abstract}
 Following the validation of the Nano Active Stabilization System concept in the previous chapter through simulated tomography experiments, this chapter addresses the refinement of the preliminary conceptual model into an optimized implementation.
-The initial validation utilized a nano-hexapod with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise.
+The initial validation used a nano-hexapod with arbitrary geometry, where components such as flexible joints and actuators were modeled as ideal elements, employing simplified control strategies without consideration for instrumentation noise.
 This detailed design phase aims to optimize each component while ensuring none will limit the system's overall performance.
 This chapter begins by determining the optimal geometric configuration for the nano-hexapod (Section~\ref{sec:detail_kinematics}).
@ -5910,7 +5909,7 @@ However, theoretical frameworks for evaluating flexible joint contribution to th
  \bm{K} = \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J}
 \end{equation}
-It is assumed that the stiffness of all struts is the same: \(\bm{\mathcal{K}} = k \cdot \mathbf{I}_6\).
+It is assumed that the stiffness of all struts is the same: \(\bm{\mathcal{K}} = k \cdot \bm{I}_6\).
 In that case, the obtained stiffness matrix linearly depends on the strut stiffness \(k\), and is structured as shown in equation~\eqref{eq:detail_kinematics_stiffness_matrix_simplified}.
 \begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified}
@ -8753,7 +8752,7 @@ The measured noise characteristics are then incorporated into the multi-body mod
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.9\linewidth]{figs/detail_instrumentation_plant.png}
-\caption{\label{fig:detail_instrumentation_plant}Block diagram of the NASS with considered instrumentation. The RT controller is a Speedgoat machine.}
+\caption{\label{fig:detail_instrumentation_plant}Block diagram of the NASS with considered instrumentation. The real time controller is a Speedgoat machine.}
 \end{figure}
 \subsection{Dynamic Error Budgeting}
 \label{sec:detail_instrumentation_dynamic_error_budgeting}
@ -10074,7 +10073,7 @@ The rotation axes are represented by the dashed lines that intersect
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/test_joints_iso.png}
 \end{center}
-\subcaption{\label{fig:test_joints_iso}ISO view}
+\subcaption{\label{fig:test_joints_iso}Isometric view}
 \end{subfigure}
 \begin{subfigure}{0.29\textwidth}
 \begin{center}
@ -11280,13 +11279,13 @@ A modal analysis of the nano-hexapod is first performed in Section~\ref{ssec:tes
 The results of the modal analysis will be useful to better understand the measured dynamics from actuators to sensors.
 A block diagram of the (open-loop) system is shown in Figure~\ref{fig:test_nhexa_nano_hexapod_signals}.
-The frequency response functions from controlled signals \(\mathbf{u}\) to the force sensors voltages \(\mathbf{V}_s\) and to the encoders measured displacements \(\mathbf{d}_e\) are experimentally identified in Section~\ref{ssec:test_nhexa_identification}.
+The frequency response functions from controlled signals \(\bm{u}\) to the force sensors voltages \(\bm{V}_s\) and to the encoders measured displacements \(\bm{d}_e\) are experimentally identified in Section~\ref{ssec:test_nhexa_identification}.
 The effect of the payload mass on the dynamics is discussed in Section~\ref{ssec:test_nhexa_added_mass}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.9\linewidth]{figs/test_nhexa_nano_hexapod_signals.png}
-\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the studied system. The command signal generated by the speedgoat is \(\mathbf{u}\), and the measured dignals are \(\mathbf{d}_{e}\) and \(\mathbf{V}_s\). Units are indicated in square brackets.}
+\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the studied system. The command signal generated by the speedgoat is \(\bm{u}\), and the measured dignals are \(\bm{d}_{e}\) and \(\bm{V}_s\). Units are indicated in square brackets.}
 \end{figure}
 \subsubsection{Modal analysis}
 \label{ssec:test_nhexa_enc_struts_modal_analysis}
@ -11343,7 +11342,7 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}.
 The dynamics of the nano-hexapod from the six command signals (\(u_1\) to \(u_6\)) to the six measured displacement by the encoders (\(d_{e1}\) to \(d_{e6}\)) and to the six force sensors (\(V_{s1}\) to \(V_{s6}\)) were identified by generating low-pass filtered white noise for each command signal, one by one.
-The \(6 \times 6\) FRF matrix from \(\mathbf{u}\) ot \(\mathbf{d}_e\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_de}.
+The \(6 \times 6\) FRF matrix from \(\bm{u}\) ot \(\bm{d}_e\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_de}.
 The diagonal terms are displayed using colored lines, and all the 30 off-diagonal terms are displayed by gray lines.
 All six diagonal terms are well superimposed up to at least \(1\,kHz\), indicating good manufacturing and mounting uniformity.
@ -11362,10 +11361,10 @@ This would not have occurred if the encoders were fixed to the struts.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_de.png}
-\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{d}_e\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.}
+\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\bm{u}\) to \(\bm{d}_e\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.}
 \end{figure}
-Similarly, the \(6 \times 6\) FRF matrix from \(\mathbf{u}\) to \(\mathbf{V}_s\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs}.
+Similarly, the \(6 \times 6\) FRF matrix from \(\bm{u}\) to \(\bm{V}_s\) is shown in Figure~\ref{fig:test_nhexa_identified_frf_Vs}.
 Alternating poles and zeros can be observed up to at least 2kHz, which is a necessary characteristics for applying decentralized IFF.
 Similar to what was observed for the encoder outputs, all the ``diagonal'' terms are well superimposed, indicating that the same controller can be applied to all the struts.
 The first flexible mode of the struts as 235Hz has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF.
@ -11373,7 +11372,7 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_Vs.png}
-\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.}
+\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\bm{u}\) to \(\bm{V}_s\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.}
 \end{figure}
 \subsubsection{Effect of payload mass on the dynamics}
 \label{ssec:test_nhexa_added_mass}
@ -11442,7 +11441,7 @@ This is checked in Section~\ref{ssec:test_nhexa_comp_model_masses}.
 The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF APA, and rigid top and bottom plates.
 The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the FEM.
 The parameters of the APA model were determined from the test bench of the APA.
-The \(6 \times 6\) transfer function matrices from \(\mathbf{u}\) to \(\mathbf{d}_e\) and from \(\mathbf{u}\) to \(\mathbf{V}_s\) are then extracted from the multi-body model.
+The \(6 \times 6\) transfer function matrices from \(\bm{u}\) to \(\bm{d}_e\) and from \(\bm{u}\) to \(\bm{V}_s\) are then extracted from the multi-body model.
 First, is it evaluated how well the models matches the ``direct'' terms of the measured FRF matrix.
 To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure~\ref{fig:test_nhexa_comp_simscape_diag}.
@ -11469,9 +11468,9 @@ At higher frequencies, no resonances can be observed in the model, as the top pl
 \label{ssec:test_nhexa_comp_model_coupling}
 Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of MIMO systems.
-Instead of comparing the full 36 elements of the \(6 \times 6\) FFR matrix from \(\mathbf{u}\) to \(\mathbf{d}_e\), only the first ``column'' is compared (Figure~\ref{fig:test_nhexa_comp_simscape_de_all}), which corresponds to the transfer function from the command \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\).
+Instead of comparing the full 36 elements of the \(6 \times 6\) FRF matrix from \(\bm{u}\) to \(\bm{d}_e\), only the first ``column'' is compared (Figure~\ref{fig:test_nhexa_comp_simscape_de_all}), which corresponds to the transfer function from the command \(u_1\) to the six measured encoder displacements \(d_{e1}\) to \(d_{e6}\).
 It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at 237Hz.
-Similar results are observed for all other coupling terms and for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\).
+Similar results are observed for all other coupling terms and for the transfer function from \(\bm{u}\) to \(\bm{V}_s\).
 \begin{figure}[htbp]
 \centering
@ -11544,7 +11543,7 @@ The frequency response functions from the six DAC voltages \(\bm{u}\) to the six
 This indicates that it is possible to implement decentralized Integral Force Feedback in a robust manner.
 The developed multi-body model of the nano-hexapod was found to accurately represents the suspension modes of the Nano-Hexapod (Section~\ref{sec:test_nhexa_model}).
-Both FRF matrices from \(\mathbf{u}\) to \(\mathbf{V}_s\) and from \(\mathbf{u}\) to \(\mathbf{d}_e\) are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective.
+Both FRF matrices from \(\bm{u}\) to \(\bm{V}_s\) and from \(\bm{u}\) to \(\bm{d}_e\) are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective.
 At frequencies above the suspension modes, the Nano-Hexapod model became inaccurate because the flexible modes were not modeled.
 It was found that modeling the APA300ML using a \emph{super-element} allows to model the internal resonances of the struts.
 The same can be done with the top platform and the encoder supports; however, the model order would be higher and may become unpractical for simulation.
@ -11918,7 +11917,7 @@ It is interesting to note that the anti-resonances in the force sensor plant now
 \end{center}
 \subcaption{\label{fig:test_id31_picture_mass_m3}$m=39\,\text{kg}$}
 \end{subfigure}
-\caption{\label{fig:test_id31_picture_masses}The four tested payload conditions. (\subref{fig:test_id31_picture_mass_m0}) without payload. (\subref{fig:test_id31_picture_mass_m1}) with \(13\,\text{kg}\) payload. (\subref{fig:test_id31_picture_mass_m2}) with \(26\,\text{kg}\) payload. (\subref{fig:test_id31_picture_mass_m3}) with \(39\,\text{kg}\) payload.}
+\caption{\label{fig:test_id31_picture_masses}The four tested payload conditions: (\subref{fig:test_id31_picture_mass_m0}) no payload, (\subref{fig:test_id31_picture_mass_m1}) \(13\,\text{kg}\) payload, (\subref{fig:test_id31_picture_mass_m2}) \(26\,\text{kg}\) payload, (\subref{fig:test_id31_picture_mass_m3}) \(39\,\text{kg}\) payload.}
 \end{figure}
 \begin{figure}[htbp]
@ -12709,6 +12708,134 @@ The system performed exceptionally well during vertical scans, though some limit
 With the implementation of an accurate online metrology system, the NASS will be ready for integration into the beamline environment, significantly enhancing the capabilities of high-precision X-ray experimentation on the ID31 beamline.
 \chapter{Conclusion and Future Work}
 \label{chap:conclusion}
 \begingroup
 \def\clearpage{\par}
 \section{Summary of Findings}
 \endgroup
 The primary objective of this research was to enhance the positioning accuracy of the ID31 micro-station by approximately two orders of magnitude, enabling full exploitation of the new \(4^{\text{th}}\) generation light source, without compromising the system's mobility or its capacity to handle payloads up to \(50\,\text{kg}\).
 To meet this demanding objective, the concept of a Nano Active Stabilization System (NASS) was proposed and developed.
 This system comprises an active stabilization platform positioned between the existing micro-station and the sample.
 Integrated with an external online metrology system and an custom control architecture, the NASS was designed to actively measure and compensate for positioning errors originating from various sources, including micro-station imperfections, thermal drift, and vibrations.
 The conceptual design phase rigorously evaluated the feasibility of the NASS concept.
 Through progressive modeling, from simplified uniaxial representations to complex multi-body dynamic simulations, key design insights were obtained.
 It was determined that an active platform with moderate stiffness offered an optimal compromise, decoupling the system from micro-station dynamics while mitigating gyroscopic effects from continuous rotation.
 The multi-body modeling approach, informed by experimental modal analysis of the micro-station, was essential for capturing the system's complex dynamics.
 The Stewart platform architecture was selected for the active stage, and its viability was confirmed through closed-loop simulations employing a High-Authority Control / Low-Authority Control (HAC-LAC) strategy.
 This strategy incorporated a modified form of Integral Force Feedback (IFF), adapted to provide robust active damping despite the platform rotation and varying payloads.
 These simulations demonstrated the NASS concept could meet the nanometer-level stability requirements under realistic operating conditions.
 Following the conceptual validation, the detailed design phase focused on translating the NASS concept into an optimized, physically realizable system.
 Geometric optimization studies refined the Stewart platform configuration.
 A hybrid modeling technique combining Finite Element Analysis (FEA) with multi-body dynamics simulation was applied and experimentally validated.
 This approach enabled detailed optimization of components, such as amplified piezoelectric actuators and flexible joints, while efficiently simulating the complete system dynamics.
 Work was also undertaken on the optimization of the control strategy for the active platform.
 Instrumentation selection (sensors, actuators, control hardware) was guided by dynamic error budgeting to ensure component noise levels met the overall nanometer-level performance target.
 The final phase of the project was dedicated to the experimental validation of the developed NASS.
 Component tests confirmed the performance of the selected actuators and flexible joints, validated their respective models.
 Dynamic testing of the assembled nano-hexapod on an isolated test bench provided essential experimental data that correlated well with the predictions of the multi-body model.
 The final validation was performed on the ID31 beamline, utilizing a short-stroke metrology system to assess performance under realistic experimental conditions.
 These tests demonstrated that the NASS, operating with the implemented HAC-LAC control architecture, successfully achieved the target positioning stability – maintaining residual errors below \(30\,\text{nm RMS}\) laterally, \(15\,\text{nm RMS}\) vertically, and \(250\,\text{nrad RMS}\) in tilt – during various experiments, including tomography scans with significant payloads.
 Crucially, the system's robustness to variations in payload mass and operational modes was confirmed.
 \section{Perspectives}
 Although this research successfully validated the NASS concept, it concurrently highlighted specific areas where the system could be enhanced, alongside related topics that merit further investigation.
 \paragraph{Automatic tuning of a multi-body model from an experimental modal analysis}
 The manual tuning process employed to match the multi-body model dynamics with experimental measurements was found to be laborious.
 Systems like the micro-station can be conceptually modeled as interconnected solid bodies, springs, and dampers, with component inertia readily obtainable from CAD models.
 An interesting perspective is the development of methods for the automatic tuning of the multi-body model's stiffness matrix (representing the interconnecting spring stiffnesses) directly from experimental modal analysis data.
 Such a capability would enable the rapid generation of accurate dynamic models for existing end-stations, which could subsequently be utilized for detailed system analysis and simulation studies.
 \paragraph{Better addressing plant uncertainty coming from a change of payload}
 For most high-performance mechatronic systems like lithography machines or atomic force microscopes, payloads inertia are often known and fixed, allowing controllers to be precisely optimized.
 However, synchrotron end-stations frequently handle samples with widely varying masses and inertias – ID31 being an extreme example, but many require nanometer positioning for samples from very light masses up to 5kg.
 The conventional strategy involves implementing controllers with relatively small bandwidth to accommodate various payloads.
 When controllers are optimized for a specific payload, changing payloads may destabilize the feedback loops that needs to be re-tuned.
 In this thesis, the HAC-IFF robust control approach was employed to maintain stability despite payload variations, though this resulted in relatively modest bandwidth.
 Therefore, a key objective for future work is to enhance the management of payload-induced plant uncertainty, aiming for improved performance without sacrificing robustness.
 Potential strategies to be explored include adaptive control (involving automatic plant identification and controller tuning after a change of payload) and robust control techniques such as \(\mu\text{-synthesis}\) (allowing the controller to be synthesized while explicitly considering a specified range of payload masses).
 \paragraph{Control based on Complementary Filters}
 The control architecture based on complementary filters (detailed in Section \ref{sec:detail_control_cf}) has been successfully implemented in several instruments at the ESRF.
 This approach has proven straightforward to implement and offers the valuable capability of modifying closed-loop behavior in real time, which proves advantageous for many applications.
 For instance, the controller can be optimized according to the scan type: constant velocity scans benefit from a \(+2\) slope for the sensitivity transfer function, while ptychography may be better served by a \(+1\) slope with slightly higher bandwidth to minimize point-to-point transition times.
 Nevertheless, a more rigorous analysis of this control architecture and its comparison with similar approaches documented in the literature is necessary to fully understand its capabilities and limitations.
 \paragraph{Sensor Fusion}
 While the HAC-LAC approach demonstrated a simple and comprehensive methodology for controlling the NASS, sensor fusion represents an interesting alternative that is worth investigating.
 While the synthesis method developed for complementary filters facilitates their design (Section \ref{sec:detail_control_sensor}), their application specifically for sensor fusion within the NASS context was not examined in detail.
 One potential approach involves fusing external metrology (utilized at low frequencies) with force sensors (employed at high frequencies).
 This configuration could enhance robustness through the collocation of force sensors with actuators.
 The integration of encoder feedback into the control architecture could also be explored.
 \paragraph{Development of multi-DoF metrology systems}
 Although experimental validation using the short-stroke metrology prototype was achieved, the NASS remains unsuitable for beamline applications due to the lack of a long stroke metrology system.
 Efforts were initiated during this project to develop such a metrology system, though these were not presented herein as the focus was directed toward the active platform, instrumentation, and controllers.
 The development process revealed that the metrology system constitutes a complex mechatronic system, which could benefit significantly from the design approach employed throughout this thesis.
 This challenge is particularly complex when continuous rotation is combined with long stroke movements.
 Yet, the development of such metrology systems is considered critical for future end-stations, especially for future tomography end stations where nano-meter accuracy is desired across larger strokes.
 Promising approaches have been presented in the literature.
 A ball lens retroreflector is used in \cite{schropp20_ptynam}, providing a \(\approx 1\,\text{mm}^3\) measuring volume, but does not fully accommodate complete rotation.
 In \cite{geraldes23_sapot_carnaub_sirius_lnls}, an interesting metrology approach is presented, utilizing interferometers for long stroke/non-rotated movements and capacitive sensors for short stroke/rotated positioning.
 \paragraph{Alternative Architecture for the NASS}
 The original micro-station design was driven by optimizing positioning accuracy, utilizing dedicated actuators for different DoFs (leading to simple kinematics and a stacked configuration), and maximizing stiffness.
 This design philosophy ensured that the micro-station would remain functional for micro-focusing applications even if the NASS project did not meet expectations.
 Analyzing the NASS as an complete system reveals that the positioning accuracy is primarily determined by the metrology system and the feedback control.
 Consequently, the underlying micro-station's own positioning accuracy has minimal influence on the final performances (it does however impact the required mobility of the active platform).
 Nevertheless, it remains crucial that the micro-station itself does not generate detrimental high-frequency vibrations, particularly during movements, as evidenced by issues previously encountered with stepper motors.
 Designing a future end-station with the understanding that a functional NASS will ensure final positioning accuracy could allow for a significantly simplified long-stroke stage architecture, perhaps chosen primarily to facilitate the integration of the online metrology.
 One possible configuration, illustrated in Figure \ref{fig:conclusion_nass_architecture}, would comprise a long-stroke Stewart platform providing the required mobility without generating high-frequency vibrations; a spindle that need not deliver exceptional performance but should be stiff and avoid inducing high-frequency vibrations (an air-bearing spindle might not be essential); and a short-stroke Stewart platform for correcting errors from the long-stroke stage and spindle.
 \begin{figure}[htbp]
 \centering
 \includegraphics[h!tbp]{figs/conclusion_nass_architecture.png}
 \caption{\label{fig:conclusion_nass_architecture}Proposed alternative configuration for an end-station including the Nano Active Stabilization System}
 \end{figure}
 With this architecture, the online metrology could be divided into two systems, as proposed by \cite{geraldes23_sapot_carnaub_sirius_lnls}: a long-stroke metrology system potentially using interferometers, and a short-stroke metrology system using capacitive sensors, as successfully demonstrated by \cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}.
 \paragraph{Development of long stroke high performance stage}
 As an alternative to the short-stroke/long-stroke architecture, the development of a high-performance long-stroke stage seems worth investigating.
 Stages based on voice coils, offering nano-positioning capabilities with \(3\,mm\) stroke, have recently been reported in the literature \cite{schropp20_ptynam,kelly22_delta_robot_long_travel_nano}.
 Magnetic levitation also emerges as a particularly interesting technology to be explored, especially for microscopy  \cite{fahmy22_magnet_xy_theta_x,heyman23_levcub} and tomography \cite{dyck15_magnet_levit_six_degree_freed_rotar_table,fahmy22_magnet_xy_theta_x} end-stations.
 Two notable designs illustrating these capabilities are shown in Figure \ref{fig:conclusion_maglev}.
 Specifically, a compact 6DoF stage known as LevCube, providing a mobility of approximately \(1\,\text{cm}^3\), is depicted in Figure \ref{fig:conclusion_maglev_heyman23}, while a 6DoF stage featuring infinite rotation, is shown in Figure \ref{fig:conclusion_maglev_dyck15}.
 However, implementations of such magnetic levitation stages on synchrotron beamlines have yet to be documented in the literature.
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.9\linewidth]{figs/conclusion_maglev_heyman23.jpg}
 \end{center}
 \subcaption{\label{fig:conclusion_maglev_heyman23}LevCube with $\approx 1\,\text{cm}^3$ mobility \cite{heyman23_levcub}}
 \end{subfigure}
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.9\linewidth]{figs/conclusion_maglev_dyck15.jpg}
 \end{center}
 \subcaption{\label{fig:conclusion_maglev_dyck15}Stage with infinite $R_z$ rotation \cite{dyck15_magnet_levit_six_degree_freed_rotar_table}}
 \end{subfigure}
 \caption{\label{fig:conclusion_maglev}Example of magnetic levitation stages. LevCube allowing for 6DoF control of the position with \(\approx 1\,\text{cm}^3\) mobility (\subref{fig:conclusion_maglev_heyman23}). Magnetic levitation stage with infinite \(R_z\) rotation mobility (\subref{fig:conclusion_maglev_dyck15})}
 \end{figure}
 \paragraph{Extending the design methodology to complete beamlines}
 The application of dynamic error budgeting and the mechatronic design approach to an entire beamline represents an interesting direction for future work.
 During the early design phases of a beamline, performance metrics are typically expressed as integrated values (usually RMS values) rather than as functions of frequency.
 However, the frequency content of these performance metrics (such as beam stability, energy stability, and sample stability) is crucial, as factors like detector integration time can filter out high-frequency components.
 Therefore, adopting a design approach utilizing dynamic error budgets, cascading from overall beamline requirements down to individual component specifications, is considered a potentially valuable direction for future investigation.
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \chapter*{List of Publications}
 \begin{refsection}[ref.bib]