From ae41f80b2477b73be2381af153a454d34dd70b03 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 23 Apr 2025 15:02:52 +0200 Subject: [PATCH] Review of all captions --- phd-thesis.org | 836 +++++++++++++++++++++++-------------------------- 1 file changed, 387 insertions(+), 449 deletions(-) diff --git a/phd-thesis.org b/phd-thesis.org index 26beb39..4b6ae51 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -28,18 +28,6 @@ #+LATEX_HEADER_EXTRA: \input{config_extra.tex} #+LATEX_HEADER_EXTRA: \addbibresource{ref.bib} #+LATEX_HEADER_EXTRA: \addbibresource{phd-thesis.bib} - -#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") -#+PROPERTY: header-args:latex+ :imagemagick t :fit yes -#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 -#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 -#+PROPERTY: header-args:latex+ :results file raw replace -#+PROPERTY: header-args:latex+ :buffer no -#+PROPERTY: header-args:latex+ :eval no-export -#+PROPERTY: header-args:latex+ :exports results -#+PROPERTY: header-args:latex+ :mkdirp yes -#+PROPERTY: header-args:latex+ :output-dir figs -#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") :END: * Build :noexport: @@ -87,24 +75,6 @@ * Glossary and Acronyms - Tables :ignore: -#+name: glossary -| label | name | description | -|-------+-------------------------+----------------------------------------------------------| -| ms | \ensuremath{m_s} | Mass of the sample | -| mn | \ensuremath{m_n} | Mass of the nano-hexapod | -| mh | \ensuremath{m_h} | Mass of the positioning hexapod | -| mt | \ensuremath{m_t} | Mass of the micro-station stages | -| mg | \ensuremath{m_g} | Mass of the granite | -| xf | \ensuremath{x_f} | Floor motion | -| ft | \ensuremath{f_t} | Disturbance force of the micro-station | -| fs | \ensuremath{f_s} | Direct forces applied on the sample | -| d | \ensuremath{d} | Measured motion between the nano-hexapod and the granite | -| fn | \ensuremath{f_n} | Force sensor on the nano-hexapod | -| psdx | \ensuremath{\Phi_{x}} | Power spectral density of signal $x$ | -| asdx | \ensuremath{\Gamma_{x}} | Amplitude spectral density of signal $x$ | -| cpsx | \ensuremath{\Phi_{x}} | Cumulative Power Spectrum of signal $x$ | -| casx | \ensuremath{\Gamma_{x}} | Cumulative Amplitude Spectrum of signal $x$ | - #+name: acronyms | key | abbreviation | full form | |--------+--------------+------------------------------------------------| @@ -149,6 +119,7 @@ | rdc | RDC | Relative Damping Control | | rga | RGA | Relative Gain Array | | rms | RMS | Root Mean Square | +| rpm | RPM | Rotations Per Minute | | rp | RP | Robust Performance | | rs | RS | Robust Stability | | siso | SISO | Single Input Single Output | @@ -216,7 +187,7 @@ European Synchrotron Radiation Facility (Grenoble, France) :UNNUMBERED: notoc :END: -The $4^{\text{th}}$ generation synchrotron light sources has yielded X-ray beams with a 100-fold increase in brightness and sub-micron focusing capabilities, offering unprecedented scientific opportunities while requiring end-stations with enhanced sample positioning accuracy. +The $4^{\text{th}}$ generation synchrotron light sources have yielded X-ray beams with a 100-fold increase in brightness and sub-micron focusing capabilities, offering unprecedented scientific opportunities while requiring end-stations with enhanced sample positioning accuracy. At the European Synchrotron (ESRF), the ID31 beamline features an end-station for positioning samples along complex trajectories. However, its micrometer-range accuracy, limited by thermal drifts and mechanical vibrations, prevents maintaining the point of interest on the focused beam during experiments. @@ -322,11 +293,11 @@ The research presented in this manuscript has been possible thanks to the Fonds \dominitoc \tableofcontents -\clearpage -\listoftables +% \clearpage +% \listoftables -\clearpage -\listoffigures +% \clearpage +% \listoffigures #+end_export * Introduction @@ -337,7 +308,7 @@ The research presented in this manuscript has been possible thanks to the Fonds #+LATEX: \endgroup **** Synchrotron Radiation Facilities :ignore: -Synchrotron radiation facilities, are particle accelerators where electrons are accelerated to near the speed of light. +Synchrotron radiation facilities are particle accelerators where electrons are accelerated to near the speed of light. As these electrons traverse magnetic fields, typically generated by insertion devices or bending magnets, they produce exceptionally bright light known as synchrotron light. This intense electromagnetic radiation, particularly in the X-ray spectrum, is subsequently used for the detailed study of matter. Approximately 70 synchrotron light sources are operational worldwide, some of which are indicated in Figure\nbsp{}ref:fig:introduction_synchrotrons. @@ -355,14 +326,14 @@ The acrfull:esrf, shown in Figure\nbsp{}ref:fig:introduction_esrf_picture, is a The acrshort:esrf started user operations in 1994 as the world's first third-generation synchrotron. Its accelerator complex, schematically depicted in Figure\nbsp{}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. +Synchrotron light is emitted in more than 40 beamlines surrounding the storage ring, each having specialized experimental stations. These beamlines host diverse instrumentation that enables a wide spectrum of scientific investigations, including structural biology, materials science, and study of matter under extreme conditions. #+name: fig:instroduction_esrf #+caption: Schematic (\subref{fig:introduction_esrf_schematic}) and picture (\subref{fig:introduction_esrf_picture}) of the European Synchrotron Radiation Facility, situated in Grenoble, France. #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:introduction_esrf_schematic} Schematic of the ESRF. The linear accelerator is shown in blue, the booster synchrotron in purple and the storage ring in green. There are over 40 beamlines, the ID31 beamline is highlighted in red} +#+attr_latex: :caption \subcaption{\label{fig:introduction_esrf_schematic} Schematic of the ESRF. The linear accelerator is shown in blue, the booster synchrotron in purple and the storage ring in green. There are over 40 beamlines. The ID31 beamline is highlighted in red} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth @@ -430,7 +401,7 @@ The sample itself (cyan), potentially housed within complex sample environments Each stage serves distinct positioning functions; for example, the positioning hexapod enables fine static adjustments, while the $T_y$ translation and $R_z$ rotation stages are used for specific scanning applications. #+name: fig:introduction_micro_station -#+caption: 3D view of the ID31 Experimal Hutch (\subref{fig:introduction_id31_cad}). There are typically four main elements: the focusing optics in yellow, the sample stage in green, the sample itself in purple and the detector in blue. All these elements are fixed to the same granite. 3D view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}). +#+caption: 3D view of the ID31 Experimental Hutch (\subref{fig:introduction_id31_cad}). There are typically four main elements: the focusing optics in yellow, the sample stage in green, the sample itself in purple and the detector in blue. All these elements are fixed to the same granite. 3D view of the micro-station with associated degrees of freedom (\subref{fig:introduction_micro_station_dof}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_id31_cad} Experimental Hutch} @@ -462,10 +433,10 @@ Positional instabilities, such as vibrations and thermal drifts, inevitably lead Other advanced imaging modalities practiced on ID31 include reflectivity, diffraction tomography, and small/wide-angle X-ray scattering (SAXS/WAXS). #+name: fig:introduction_tomography -#+caption: Exemple of a tomography experiment. The sample is rotated and images are taken at several angles (\subref{fig:introduction_tomography_schematic}). Example of one 3D image obtained using tomography (\subref{fig:introduction_tomography_results}). +#+caption: Example of a tomography experiment. The sample is rotated and images are taken at several angles (\subref{fig:introduction_tomography_schematic}). Example of one 3D image obtained using tomography (\subref{fig:introduction_tomography_results}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:introduction_tomography_schematic} Typical tomography experimental setup} +#+attr_latex: :caption \subcaption{\label{fig:introduction_tomography_schematic} Typical experimental setup for tomography experiment} #+attr_latex: :options {0.65\textwidth} #+begin_subfigure #+attr_latex: :scale 0.9 @@ -480,10 +451,10 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra #+end_figure #+name: fig:introduction_scanning -#+caption: 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 $20\,\text{nm}$ (\subref{fig:introduction_scanning_results}). +#+caption: Example of a scanning experiment. The sample is scanned in the YZ plane (\subref{fig:introduction_scanning_schematic}). Example of one 2D image obtained after scanning with a step size of $20\,\text{nm}$ (\subref{fig:introduction_scanning_results}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:introduction_scanning_schematic} Typical experimental setup for scanning experiment} +#+attr_latex: :caption \subcaption{\label{fig:introduction_scanning_schematic} Typical experimental setup for a scanning experiment} #+attr_latex: :options {0.65\textwidth} #+begin_subfigure #+attr_latex: :scale 0.9 @@ -529,7 +500,7 @@ The historical reduction in achievable spot sizes is represented in Figure\nbsp{ Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines. #+name: fig:introduction_moore_law_focus -#+caption: Evolution of the measured spot size for different hard X-ray focusing elements. Adapated from\nbsp{}[[cite:&barrett24_x_optic_accel_based_light_sourc]]. +#+caption: Evolution of the measured spot size for different hard X-ray focusing elements. Adapted from [[cite:&barrett24_x_optic_accel_based_light_sourc]]. #+attr_latex: :scale 0.9 #+attr_latex: :options [h!tbp] [[file:figs/introduction_moore_law_focus.png]] @@ -541,7 +512,7 @@ In this mode, the sample is moved to the desired position, the detector acquisit While effective for mitigating radiation damage, this sequential process can be time-consuming, especially for high-resolution maps requiring numerous points. #+name: fig:introduction_scan_mode -#+caption: Two acquisition modes. In step-by-step mode (\subref{fig:introduction_scan_step}), the motor moves at the wanted imaged position, the detector acquisition is started, the shutter is openned briefly to have the wanted exposure, the detector acquisition is stopped, and the motor can move to a new position. In /fly-scan/ mode (\subref{fig:introduction_scan_fly}), the shutter is openned while the sample is in motion, and the detector is acquired only at the wanted positions, on the /fly/. +#+caption: Two acquisition modes. In step-by-step mode (\subref{fig:introduction_scan_step}), the motor moves to the desired imaged position, the detector acquisition is started, the shutter is opened briefly to have the wanted exposure, the detector acquisition is stopped, and the motor can move to a new position. In /fly-scan/ mode (\subref{fig:introduction_scan_fly}), the shutter is opened while the sample is in motion, and the detector acquires data only at the desired positions while in motion ("on the fly"). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_scan_step} Step by step scan} @@ -613,7 +584,7 @@ Examples of such end-stations, including those at beamlines ID16B\nbsp{}[[cite:& However, when a large number of DoFs are required, the cumulative errors and limited dynamic stiffness of stacked configurations can make experiments with nano-focused beams extremely challenging or infeasible. #+name: fig:introduction_passive_stations -#+caption: Example of two nano end-stations without online metrology measuring the sample's position. +#+caption: Example of two nano end-stations lacking online metrology for measuring the sample's position. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_endstation_id16b}ID16b end-station \cite{martinez-criado16_id16b}} @@ -698,13 +669,13 @@ Recently, acrfull:vc actuators were used to increase the stroke up to $3\,\text{ An alternative strategy involves a "long stroke-short stroke" architecture, illustrated conceptually in Figure\nbsp{}ref:fig:introduction_two_stage_schematic. In this configuration, a high-accuracy, high-bandwidth short-stroke stage is mounted on top of a less precise long-stroke stage. The short-stroke stage actively compensates for errors based on metrology feedback, while the long-stroke stage performs the larger movements. -This approach allows combining extended travel with high precision and good dynamical response, but is often implemented for only one or a few DoFs, as seen in Figures\nbsp{}ref:fig:introduction_two_stage_schematic and\nbsp{}ref:fig:introduction_two_stage_control_h_bridge. +This approach allows the combination of extended travel with high precision and good dynamical response, but is often implemented for only one or a few DoFs, as seen in Figures\nbsp{}ref:fig:introduction_two_stage_schematic and\nbsp{}ref:fig:introduction_two_stage_control_h_bridge. #+name: fig:introduction_two_stage_example -#+caption: Schematic of a typical Long stroke - Short stroke control architecture (\subref{fig:introduction_two_stage_schematic}). A 3DoF long stroke - short stroke is shown in (\subref{fig:introduction_two_stage_control_h_bridge}) where $y_1$, $y_2$ and $x$ are 3-phase linear motors and short stroke actuators are voice coils. +#+caption: Schematic of a typical Long stroke-Short stroke control architecture (\subref{fig:introduction_two_stage_schematic}). A 3-DoFs long stroke-short stroke is shown in (\subref{fig:introduction_two_stage_control_h_bridge}) where $y_1$, $y_2$ and $x$ are 3-phase linear motors and short stroke actuators are voice coils. #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:introduction_two_stage_schematic} Typical Long Stroke - Short Stroke control architecture} +#+attr_latex: :caption \subcaption{\label{fig:introduction_two_stage_schematic} Typical Long Stroke-Short Stroke control architecture} #+attr_latex: :options {0.64\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth @@ -730,14 +701,14 @@ Given the high frame rates of modern detectors, these specified positioning erro These demanding stability requirements must be achieved within the specific context of the ID31 beamline, which necessitates the integration with the existing micro-station, accommodating a wide range of experimental configurations requiring high mobility, and handling substantial payloads up to $50\,\text{kg}$. -The existing micro-station, despite being composed of high-performance stages, exhibits positioning accuracy limited to approximately $\SI{10}{\micro\m}$ and $\SI{10}{\micro\rad}$ due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. +The existing micro-station, despite being composed of high-performance stages, has a positioning accuracy limited to approximately $\SI{10}{\micro\m}$ and $\SI{10}{\micro\rad}$ due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations. The primary objective of this project is therefore defined as enhancing the positioning accuracy and stability of the ID31 micro-station by roughly two orders of magnitude, to fully leverage the capabilities offered by the ESRF-EBS source and modern detectors, without compromising its existing mobility and payload capacity. ***** The Nano Active Stabilization System Concept To address these challenges, the concept of a acrfull:nass is proposed. -As schematically illustrated in Figure\nbsp{}ref:fig:introduction_nass_concept_schematic, the acrshort:nass comprises four principal components integrated with the existing micro-station (yellow): a 5-DoF online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). +As schematically illustrated in Figure\nbsp{}ref:fig:introduction_nass_concept_schematic, the acrshort:nass comprises four principal components integrated with the existing micro-station (yellow): a 5-DoFs online metrology system (red), an active stabilization platform (blue), and the associated control system and instrumentation (purple). This system essentially functions as a high-performance, multi-axis vibration isolation and error correction platform situated between the micro-station and the sample. It actively compensates for positioning errors measured by the external metrology system. @@ -749,7 +720,7 @@ It actively compensates for positioning errors measured by the external metrolog ***** Online Metrology system The performance of the acrshort:nass is fundamentally reliant on the accuracy and bandwidth of its online metrology system, as the active control is based directly on these measurements. -This metrology system must fulfill several criteria: measure the sample position in 5 DoF (excluding rotation about the vertical Z-axis); possess a measurement range compatible with the micro-station's extensive mobility and continuous spindle rotation; achieve an accuracy compatible with the sub-100 nm positioning target; and offer high bandwidth for real-time control. +This metrology system must fulfill several criteria: measure the sample position in 5-DoFs (excluding rotation about the vertical Z-axis); possess a measurement range compatible with the micro-station's extensive mobility and continuous spindle rotation; achieve an accuracy compatible with the sub-100 nm positioning target; and offer high bandwidth for real-time control. #+name: fig:introduction_nass_metrology #+caption: 2D representation of the NASS metrology system. @@ -760,7 +731,7 @@ Fiber interferometers target both surfaces. A tracking system maintains perpendicularity between the interferometer beams and the mirror, such that Abbe errors are eliminated. Interferometers pointing at the spherical surface provides translation measurement, while the ones pointing at the flat bottom surface yield tilt angles. The development of this complex metrology system constitutes a significant mechatronic project in itself and is currently ongoing; it is not further detailed within this thesis. -For the work presented herein, the metrology system is assumed to provide accurate, high-bandwidth 5-DoF position measurements. +For the work presented herein, the metrology system is assumed to provide accurate, high-bandwidth 5-DoFs position measurements. ***** Active Stabilization Platform Design @@ -770,14 +741,14 @@ It must possess excellent dynamic properties to enable high-bandwidth control ca Consequently, it must be free from backlash and play, and its active components (e.g., actuators) should introduce minimal vibrations. Critically, it must accommodate payloads up to $50\,\text{kg}$. -A suitable candidate architecture for this platform is the Stewart platform (also known as "hexapod"), a parallel kinematic mechanism capable of 6-DoF motion. +A suitable candidate architecture for this platform is the Stewart platform (also known as "hexapod"), a parallel kinematic mechanism capable of 6-DoFs motion. Stewart platforms are widely employed in positioning and vibration isolation applications due to their inherent stiffness and potential for high precision. Various designs exist, differing in geometry, actuation technology, sensing methods, and control strategies, as exemplified in Figure\nbsp{}ref:fig:introduction_stewart_platform_piezo. A central challenge addressed in this thesis is the optimal mechatronic design of such an active platform tailored to the specific requirements of the NASS. A more detailed review of Stewart platform and its main components will be given in Section\nbsp{}ref:sec:detail_kinematics_stewart_review. #+name: fig:introduction_stewart_platform_piezo -#+caption: Two examples of very different Stewart platforms geometries and struts' configuration. +#+caption: Two examples of very different Stewart platforms geometries and strut configurations. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:introduction_stewart_du14}Piezo based, for positioning purposes \cite{du14_piezo_actuat_high_precis_flexib}} @@ -826,7 +797,7 @@ Key challenges within this approach include developing appropriate design method 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. -***** 6DoF vibration control of a rotating platform +***** 6-DoFs 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. @@ -876,7 +847,7 @@ The conclusion of this work involved the experimental implementation and validat Experimental results, presented in Section\nbsp{}ref:sec:test_id31, demonstrate that the system successfully improves the effective positioning accuracy of the micro-station from its native $\approx 10\,\upmu\text{m}$ level down to the target $\approx 100\,\text{nm}$ range during representative scientific experiments. Crucially, robustness to variations in sample mass and diverse experimental conditions was verified. The NASS thus provides a versatile end-station solution, uniquely combining high payload capacity with nanometer-level accuracy, enabling optimal use of the advanced capabilities of the ESRF-EBS beam and associated detectors. -To the author's knowledge, this represents the first demonstration of such a 5-DoF active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. +To the author's knowledge, this represents the first demonstration of such a 5-DoFs active stabilization platform being used to enhance the accuracy of a complex positioning system to this level. ** Outline ***** Introduction :ignore: @@ -888,7 +859,7 @@ While the chapters follow this logical progression, care has been taken to struc The conceptual design phase, detailed in Chapter\nbsp{}ref:chap:concept, followed a methodical progression from simplified uniaxial models to more complex multi-body representations. Initial uniaxial analysis (Section\nbsp{}ref:sec:uniaxial) provided fundamental insights, particularly regarding the influence of active platform stiffness on performance. -The introduction of rotation in a 3-DoF model (Section\nbsp{}ref:sec:rotating) allowed investigation of gyroscopic effects, revealing challenges for softer platform designs. +The introduction of rotation in a 3-DoFs model (Section\nbsp{}ref:sec:rotating) allowed investigation of gyroscopic effects, revealing challenges for softer platform designs. Experimental modal analysis of the existing micro-station (Section\nbsp{}ref:sec:modal) confirmed its complex dynamics but supported a rigid-body assumption for the different stages, justifying the development of a detailed multi-body model. This model, tuned against experimental data and incorporating measured disturbances, was validated through simulation (Section\nbsp{}ref:sec:ustation). The Stewart platform architecture was selected for the active stage, and its kinematics, dynamics, and control were analyzed (Section\nbsp{}ref:sec:nhexa). @@ -926,7 +897,7 @@ The implemented control architecture was tested under realistic experimental sce The conceptual design of the Nano Active Stabilization System (NASS) follows a methodical progression from simple to more accurate modeling approaches, as illustrated in Figure\nbsp{}ref:fig:chapter1_overview. #+name: fig:chapter1_overview -#+caption: Overview of the conceptual design development. The approach evolves from simplified analytical models to a multi-body model tuned from experimental modal analysis. It is concluded by closed-loop simulations of tomography experiments, validating the conceptual design. +#+caption: Overview of the conceptual design development. The approach evolves from simplified analytical models to a multi-body model tuned from experimental modal analysis. Closed-loop simulations of tomography experiments are used to validate the concept. #+attr_org: :width 800px #+attr_latex: :options [h!tbp] #+attr_latex: :width \linewidth @@ -964,7 +935,7 @@ The confidence gained through this systematic investigation provides a solid fou *** Introduction :ignore: In this report, a uniaxial model of the acrfull:nass is developed and used to obtain a first idea of the challenges involved in this complex system. -Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions. +Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, leading to similar conclusions. To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model). Then, a model of the active platform is added on top of the micro-station. @@ -1056,7 +1027,7 @@ More accurate models will be used later on. ***** Introduction :ignore: A model of the active platform and sample is now added on top of the uniaxial model of the micro-station (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). -Disturbances (shown in red) are gls:fs the direct forces applied to the sample (for example cable forces), gls:ft representing the vibrations induced when scanning the different stages and gls:xf the floor motion. +Disturbances (shown in red) are $f_s$ the direct forces applied to the sample (for example cable forces), $f_t$ representing the vibrations induced when scanning the different stages and $x_f$ the floor motion. The control signal is the force applied by the active platform $f$ and the measurement is the relative motion between the sample and the granite $d$. The sample is here considered as a rigid body and rigidly fixed to the active platform. The effect of resonances between the sample's acrshort:poi and the active platform actuator will be considered in Section\nbsp{}ref:sec:uniaxial_payload_dynamics. @@ -1081,14 +1052,14 @@ The effect of resonances between the sample's acrshort:poi and the active platfo ***** Active Platform Parameters The active platform is represented by a mass spring damper system (shown in blue in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass). -Its mass gls:mn is set to $15\,\text{kg}$ while its stiffness $k_n$ can vary depending on the chosen architecture/technology. -The sample is represented by a mass gls:ms that can vary from $1\,\text{kg}$ up to $50\,\text{kg}$. +Its mass $m_n$ is set to $15\,\text{kg}$ while its stiffness $k_n$ can vary depending on the chosen architecture/technology. +The sample is represented by a mass $m_s$ that can vary from $1\,\text{kg}$ up to $50\,\text{kg}$. As a first example, the active platform stiffness of is set at $k_n = 10\,\text{N}/\upmu\text{m}$ and the sample mass is chosen at $m_s = 10\,\text{kg}$. ***** Obtained Dynamic Response The sensitivity to disturbances (i.e., the transfer functions from $x_f,f_t,f_s$ to $d$) can be extracted from the uniaxial model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass and are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_first_params. -The /plant/ (i.e., the transfer function from actuator force $f$ to measured displacement $d$) is shown in Figure\nbsp{}ref:fig:uniaxial_plant_first_params. +The /plant/ (i.e., the transfer function from actuator force $f$ to displacement $d$) is shown in Figure\nbsp{}ref:fig:uniaxial_plant_first_params. For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass will be considered: three active platform stiffnesses ($k_n = 0.01\,\text{N}/\upmu\text{m}$, $k_n = 1\,\text{N}/\upmu\text{m}$ and $k_n = 100\,\text{N}/\upmu\text{m}$) combined with three sample's masses ($m_s = 1\,\text{kg}$, $m_s = 25\,\text{kg}$ and $m_s = 50\,\text{kg}$). @@ -1186,7 +1157,7 @@ The estimated acrshort:asd $\Gamma_{x_f}$ of the floor motion $x_f$ is shown in ***** Stage Vibration To estimate the vibrations induced by scanning the micro-station stages, two geophones are used, as shown in Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup. -The vertical relative velocity between the top platform of the positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6rpm. +The vertical relative velocity between the top platform of the positioning hexapod and the granite is estimated in two cases: without moving the micro-station stages, and then during a Spindle rotation at 6 acrfull:rpm. The vibrations induced by the $T_y$ stage are not considered here because they have less amplitude than the vibrations induced by the $R_z$ stage and because the $T_y$ stage can be scanned at lower velocities if the induced vibrations are found to be an issue. The amplitude spectral density of the relative motion with and without the Spindle rotation are compared in Figure\nbsp{}ref:fig:uniaxial_asd_vibration_spindle_rotation. @@ -1452,7 +1423,7 @@ This is illustrated in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniqu \xi = \sin(\phi) \end{equation} -The Root Locus for the three active platform stiffnesses and the three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques. +The root locus for the three active platform stiffnesses and the three active damping techniques are shown in Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques. All three active damping approaches can lead to /critical damping/ of the active platform suspension mode (angle $\phi$ can be increased up to 90 degrees). There is even some damping authority on micro-station modes in the following cases: - IFF with a stiff active platform (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) :: @@ -1465,7 +1436,7 @@ There is even some damping authority on micro-station modes in the following cas The micro-station and the active platform masses are connected through a large damper induced by acrshort:rdc (see mechanical equivalent in Figure\nbsp{}ref:fig:uniaxial_active_damping_rdc_equiv) which allows some damping of the micro-station. #+name: fig:uniaxial_root_locus_damping_techniques -#+caption: Root Loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The Root Loci are zoomed on the suspension mode of the active platform. +#+caption: Root loci for the three active damping techniques (IFF in blue, RDC in red and DVF in yellow). This is shown for the three active platform stiffnesses. The root loci are zoomed on the suspension mode of the active platform. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,\text{N}/\upmu\text{m}$} @@ -1489,7 +1460,7 @@ There is even some damping authority on micro-station modes in the following cas #+end_figure #+name: fig:uniaxial_root_locus_damping_techniques_micro_station_mode -#+caption: Root Locus for the three damping techniques applied with the soft active platform. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the positioning hexapod. +#+caption: Root locus for the three damping techniques applied with the soft active platform. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the positioning hexapod. #+attr_latex: :scale 0.8 [[file:figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png]] @@ -1596,7 +1567,7 @@ The damping strategies were then compared in terms of disturbance reduction. The comparison between the three active damping strategies is summarized in Table\nbsp{}ref:tab:comp_active_damping. It is difficult to conclude on the best active damping strategy for the acrfull:nass yet. -Which one will be used will be determined by the use of more accurate models and will depend on which is the easiest to implement in practice +The one used will be determined by the use of more accurate models and will depend on which is easiest to implement in practice #+name: tab:comp_active_damping #+caption: Comparison of active damping strategies for the NASS. @@ -1617,7 +1588,7 @@ Which one will be used will be determined by the use of more accurate models and *** Position Feedback Controller <> ***** Introduction :ignore: -The gls:haclac architecture is shown in Figure\nbsp{}ref:fig:uniaxial_hac_lac_architecture. +The acrfull:haclac architecture is shown in Figure\nbsp{}ref:fig:uniaxial_hac_lac_architecture. This corresponds to a /two step/ control strategy: - First, an active damping controller $\bm{K}_{\textsc{LAC}}$ is implemented (see Section\nbsp{}ref:sec:uniaxial_active_damping). It allows the vibration level to be reduced, and it also makes the damped plant (transfer function from $u^{\prime}$ to $y$) easier to control than the undamped plant (transfer function from $u$ to $y$). @@ -1851,7 +1822,7 @@ To achieve such bandwidth, the acrshort:haclac strategy was followed, which cons In this section, feedback controllers were designed in such a way that the required closed-loop bandwidth was reached while being robust to changes in the payload mass. The attainable vibration control performances were estimated for the three active platform stiffnesses and were found to be close to the required values. -However, the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. +However, the stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) requires the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. A slight advantage can be given to the soft active platform as it requires less feedback bandwidth while providing better stability results. *** Effect of Limited Support Compliance @@ -2005,7 +1976,7 @@ Note that the observations made in this section are also affected by the ratio b ***** Introduction :ignore: Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic). -However, such a sample may present internal dynamics, and its fixation to the active platform may have limited stiffness. +However, such a sample may present internal dynamics, and its mounting on the active platform may have limited stiffness. To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are used. #+name: fig:uniaxial_payload_dynamics_models @@ -2018,7 +1989,7 @@ To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref #+attr_latex: :scale 1 [[file:figs/uniaxial_paylaod_dynamics_rigid_schematic.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Payload having some flexibility} +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Flexible payload} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -2149,7 +2120,7 @@ However, this model does not allow the determination of which one is most suited Position feedback controllers have been developed for three considered active platform stiffnesses (Section\nbsp{}ref:sec:uniaxial_position_control). These controllers were shown to be robust to the change of sample's masses, and to provide good rejection of disturbances. Having a soft active platform makes the plant dynamics easier to control (because its dynamics is decoupled from the micro-station dynamics, see Section\nbsp{}ref:sec:uniaxial_support_compliance) and requires less position feedback bandwidth to fulfill the requirements. -The moderately stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$) is requiring a higher feedback bandwidth, but still gives acceptable results. +The moderately stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$) requires a higher feedback bandwidth, but still gives acceptable results. However, the stiff active platform is the most complex to control and gives the worst positioning performance. ** Effect of Rotation @@ -2159,7 +2130,7 @@ However, the stiff active platform is the most complex to control and gives the An important aspect of the acrfull:nass is that the active platform continuously rotates around a vertical axis, whereas the external metrology is not. Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance. To study these effects, a model of a rotating suspended platform is first presented (Section\nbsp{}ref:sec:rotating_system_description) -This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects in play. +This model is simple enough to be able to derive its dynamics analytically and to understand its behavior, while still allowing the capture of important physical effects at play. acrfull:iff is then applied to the rotating platform, and it is shown that the unconditional stability of acrshort:iff is lost due to the gyroscopic effects induced by the rotation (Section\nbsp{}ref:sec:rotating_iff_pure_int). Two modifications of the Integral Force Feedback are then proposed. @@ -2192,7 +2163,7 @@ The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the r After the dynamics of this system is studied, the objective will be to dampen the two suspension modes of the payload while the rotating stage performs a constant rotation. #+name: fig:rotating_3dof_model_schematic -#+caption: Schematic of the studied 2-DoF translation stage on top of a rotation stage. +#+caption: Schematic of the studied 2-DoFs translation stage on top of a rotation stage. #+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic.png]] @@ -2301,7 +2272,7 @@ These plots confirm the expected behavior: the frequencies of the two pairs of c 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). #+name: fig:rotating_bode_plot -#+caption: Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities. +#+caption: Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_coupling}) terms for several rotating velocities. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_bode_plot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} @@ -2323,7 +2294,7 @@ For $\Omega > \omega_0$, the low-frequency pair of complex conjugate poles $p_{- ***** Introduction :ignore: The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. -As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances\nbsp{}[[cite:&collette11_review_activ_vibrat_isolat_strat]] and to make the plant easier to control for the high authority controller. +As was explained with the uniaxial model, such an active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances\nbsp{}[[cite:&collette11_review_activ_vibrat_isolat_strat]] and for making the plant easier to control for the high authority controller. Many active damping techniques have been developed over the years, such as Positive Position Feedback (PPF)\nbsp{}[[cite:&lin06_distur_atten_precis_hexap_point;&fanson90_posit_posit_feedb_contr_large_space_struc]], Integral Force Feedback (IFF)\nbsp{}[[cite:&preumont91_activ]] and Direct Velocity Feedback (DVF)\nbsp{}[[cite:&karnopp74_vibrat_contr_using_semi_activ_force_gener;&serrand00_multic_feedb_contr_isolat_base_excit_vibrat;&preumont02_force_feedb_versus_accel_feedb]]. In\nbsp{}[[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator, and an integral controller are used to increase the damping of a mechanical system. @@ -2430,7 +2401,7 @@ As expected from the derived equations of motion: #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_bode_plot_effect_rot_direct.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus} +#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root locus} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2449,11 +2420,11 @@ The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal c \end{aligned} \end{equation} -To determine how the acrshort:iff controller affects the poles of the closed-loop system, a Root Locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously. +To determine how the acrshort:iff controller affects the poles of the closed-loop system, a Root locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously. As explained in\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with;&skogestad07_multiv_feedb_contr]], the closed-loop poles start at the open-loop poles (shown by crosses) for $g = 0$ and coincide with the transmission zeros (shown by circles) as $g \to \infty$. Whereas collocated IFF is usually associated with unconditional stability\nbsp{}[[cite:&preumont91_activ]], this property is lost due to gyroscopic effects as soon as the rotation velocity becomes non-null. -This can be seen in the Root Locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. +This can be seen in the Root locus plot (Figure\nbsp{}ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot). The control system is thus cancels the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. @@ -2476,7 +2447,7 @@ The Integral Force Feedback Controller is modified such that instead of using pu The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure\nbsp{}ref:fig:rotating_iff_modified_loop_gain. The effect of the added HPF limits the low-frequency gain to finite values as expected. -The Root Locus plots for the decentralized acrshort:iff with and without the acrshort:hpf are displayed in Figure\nbsp{}ref:fig:rotating_iff_root_locus_hpf_large. +The Root locus plots for the decentralized acrshort:iff with and without the acrshort:hpf are displayed in Figure\nbsp{}ref:fig:rotating_iff_root_locus_hpf_large. With the added acrshort:hpf, the poles of the closed-loop system are shown to be stable up to some value of the gain $g_\text{max}$ given by equation\nbsp{}eqref:eq:rotating_gmax_iff_hpf. It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low-frequency loop gain reaches one (for instance the gain $g$ can be increased by a factor $5$ in Figure\nbsp{}ref:fig:rotating_iff_modified_loop_gain before the system becomes unstable). @@ -2494,7 +2465,7 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the controll #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_modified_loop_gain.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root locus} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2506,7 +2477,7 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the controll Two parameters can be tuned for the modified controller in equation\nbsp{}eqref:eq:rotating_iff_lhf: the gain $g$ and the pole's location $\omega_i$. The optimal values of $\omega_i$ and $g$ are considered here as the values for which the damping of all the closed-loop poles is simultaneously maximized. -To visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure\nbsp{}ref:fig:rotating_root_locus_iff_modified_effect_wi. +To visualize how $\omega_i$ does affect the attainable damping, the Root locus plots for several $\omega_i$ are displayed in Figure\nbsp{}ref:fig:rotating_root_locus_iff_modified_effect_wi. It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes (see Root locus in Figure\nbsp{}ref:fig:rotating_root_locus_iff_modified_effect_wi), the control gain $g$ may be limited to small values due to equation\nbsp{}eqref:eq:rotating_gmax_iff_hpf. To study this trade-off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$. The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure\nbsp{}ref:fig:rotating_iff_hpf_optimal_gain). @@ -2518,13 +2489,13 @@ For larger values of $\omega_i$, the attainable damping ratio decreases as a fun #+caption: Root loci for several high-pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases (\subref{fig:rotating_iff_hpf_optimal_gain}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus} +#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root locus} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_root_locus_iff_modified_effect_wi.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio as a function of $\omega_i/\omega_0$. Maximum and optical control gains are also shown} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2613,7 +2584,7 @@ Bode plots of the obtained dynamics are shown in Figure\nbsp{}ref:fig:rotating_i The two real zeros for $k_p < m \Omega^2$ are transformed into two complex conjugate zeros for $k_p > m \Omega^2$. In that case, the system shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case. -Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator, as shown in Eq.\nbsp{}eqref:eq:rotating_Kf_pure_int. +Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus shows the Root locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator, as shown in Eq.\nbsp{}eqref:eq:rotating_Kf_pure_int. It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the unconditional stability of acrshort:iff is recovered. #+name: fig:rotating_iff_plant_effect_kp @@ -2626,7 +2597,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif #+attr_latex: :scale 0.8 [[file:figs/rotating_iff_effect_kp.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with soft parallel spring and with stiff parallel spring} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus}Root locus for IFF without parallel spring, with soft parallel spring and with stiff parallel spring} #+attr_latex: :options {0.44\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2637,7 +2608,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif ***** Effect of $k_p$ on the Attainable Damping Even though the parallel stiffness $k_p$ has no impact on the open-loop poles (as the overall stiffness $k$ is kept constant), it has a large impact on the transmission zeros. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition]], the parallel stiffness $k_p$ is expected to have some impact on the attainable damping. -To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus_effect_kp. +To study this effect, Root locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus_effect_kp. The frequencies of the transmission zeros of the system increase with an increase in the parallel stiffness $k_p$ (thus getting closer to the poles), and the associated attainable damping is reduced. Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Omega^2$ for stability reasons, it should not be taken too large as this would limit the attainable damping. This is confirmed by the Figure\nbsp{}ref:fig:rotating_iff_kp_optimal_gain where the attainable closed-loop damping ratio $\xi_{\text{cl}}$ and the associated optimal control gain $g_\text{opt}$ are computed as a function of the parallel stiffness. @@ -2646,7 +2617,7 @@ This is confirmed by the Figure\nbsp{}ref:fig:rotating_iff_kp_optimal_gain where #+caption: Effect of the parallel stiffness on the achievable damping with IFF. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness, $\Omega = 0.1 \omega_0$} +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root locus: Effect of parallel stiffness, $\Omega = 0.1 \omega_0$} #+attr_latex: :options {0.49\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2681,7 +2652,7 @@ Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped pla The added acrshort:hpf gives almost the same damping properties to the suspension while exhibiting good low-frequency behavior. #+name: fig:rotating_iff_optimal_hpf -#+caption:Effect of high-pass filter cut-off frequency on the obtained damping (\subref{fig:rotating_iff_kp_added_hpf_effect_damping}) and on the dampled plant (\subref{fig:rotating_iff_kp_added_hpf_damped_plant}). +#+caption:Effect of high-pass filter cut-off frequency on the obtained damping (\subref{fig:rotating_iff_kp_added_hpf_effect_damping}) and on the damped plant (\subref{fig:rotating_iff_kp_added_hpf_damped_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$} @@ -2749,10 +2720,10 @@ The rotating aspect does not add any complexity to the use of Relative Damping C It does not increase the low-frequency coupling as compared to the Integral Force Feedback. #+name: fig:rotating_rdc_result -#+caption: Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant}). +#+caption: Relative Damping Control. Root locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control} +#+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_root_locus}Root locus for Relative Damping Control} #+attr_latex: :options {0.49\linewidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -2775,7 +2746,7 @@ For the following comparisons, the cut-off frequency for the added HPF is set to These values are chosen one the basis of previous discussions about optimal parameters. ***** Root Locus -Figure\nbsp{}ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications and the relative damping control. +Figure\nbsp{}ref:fig:rotating_comp_techniques_root_locus shows the root locus plots for the two proposed IFF modifications and the relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. This means that the closed-loop behavior of both systems will differ when large control gains are used. @@ -2787,22 +2758,22 @@ It is interesting to note that the maximum added damping is very similar for bot #+caption: Comparison of active damping techniques for rotating platform. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus} +#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_root_locus}Root locus} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/rotating_comp_techniques_root_locus.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_dampled_plants}Damped plants} +#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_damped_plants}Damped plants} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 -[[file:figs/rotating_comp_techniques_dampled_plants.png]] +[[file:figs/rotating_comp_techniques_damped_plants.png]] #+end_subfigure #+end_figure ***** Obtained Damped Plant -The actively damped plants are computed for the three techniques and compared in Figure\nbsp{}ref:fig:rotating_comp_techniques_dampled_plants. +The actively damped plants are computed for the three techniques and compared in Figure\nbsp{}ref:fig:rotating_comp_techniques_damped_plants. It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, the off-diagonal (coupling) terms are not. The acrshort:iff strategy is adding some coupling at low-frequency, which may negatively impact the positioning performance. @@ -3020,12 +2991,12 @@ While quite simplistic, this allowed us to study the effects of rotation and the In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. ***** Nano Active Stabilization System Model -To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): +To have a more realistic dynamics model of the NASS, the 2-DoFs active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed -- a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\micro\m}$, $m_g = \SI{2500}{\kg}$) -- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\micro\m}$, $m_t = \SI{600}{\kg}$) +- a 2-DoFs granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\micro\m}$, $m_g = \SI{2500}{\kg}$) +- a 2-DoFs $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\micro\m}$, $m_t = \SI{600}{\kg}$) - a spindle (vertical rotation) stage whose rotation is imposed ($m_s = \SI{600}{\kg}$) -- a 2-DoF positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\micro\m}$, $m_h = \SI{15}{\kg}$) +- a 2-DoFs positioning hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\micro\m}$, $m_h = \SI{15}{\kg}$) A payload is rigidly fixed to the active platform and the $x,y$ motion of the payload is measured with respect to the granite. @@ -3201,7 +3172,7 @@ This modal model can then be used to tune the spatial model (i.e. the multi-body [[file:figs/modal_vibration_analysis_procedure.png]] The measurement setup used to obtain the response model is described in Section\nbsp{}ref:sec:modal_meas_setup. -This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals. +This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planning, and a first analysis of the obtained signals. In Section\nbsp{}ref:sec:modal_frf_processing, the obtained acrshortpl:frf between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed. These measurements are projected at the acrfull:com of each considered solid body to facilitate the further use of the results. @@ -3257,7 +3228,7 @@ The softer tip was found to give best results as it injects more energy in the l Finally, an /acquisition system/[fn:modal_3] (figure\nbsp{}ref:fig:modal_oros) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise. -**** Structure Preparation and Test Planing +**** Structure Preparation and Test Planning <> To obtain meaningful results, the modal analysis of the micro-station is performed /in-situ/. @@ -3269,7 +3240,7 @@ The top part representing the active stabilization stage was disassembled as the To perform the modal analysis from the measured responses, the $n \times n$ acrshort:frf matrix $\bm{H}$ needs to be measured, where $n$ is the considered number of acrshortpl:dof. The $H_{jk}$ element of this acrfull:frf matrix corresponds to the acrshort:frf 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. +Measuring this acrshort:frf matrix is time consuming as it requires making $n \times n$ measurements. However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /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:&ewins00_modal chapt. 5.2]]. Therefore, a minimum set of $n$ acrshortpl:frf 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. @@ -3469,7 +3440,7 @@ Let us consider the schematic shown in Figure\nbsp{}ref:fig:modal_local_to_globa The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort:dof of the solid body expressed in the frame $\{O\}$. #+name: fig:modal_local_to_global_coordinates -#+caption: Schematic of the measured motion of a solid body at 4 distinc locations. +#+caption: Schematic of the measured motion of a solid body at 4 distinct locations. [[file:figs/modal_local_to_global_coordinates.png]] The motion of the rigid body of figure\nbsp{}ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$. @@ -3557,7 +3528,7 @@ Similar results were obtained for the other solid bodies, indicating that the so This also validates the reduction in the number of acrshortpl:dof from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof). #+name: fig:modal_comp_acc_solid_body_frf -#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4, which are corresponding to the positioning hexapod, are shown. Input is a hammer force applied on the positioning hexapod in the $x$ direction. +#+caption: Comparison of the original accelerometer responses with responses reconstructed from the solid body response. Accelerometers 1 to 4, corresponding to the positioning hexapod, are shown. Input is a hammer force applied on the positioning hexapod in the $x$ direction. #+attr_latex: :scale 0.8 [[file:figs/modal_comp_acc_solid_body_frf.png]] @@ -3768,9 +3739,9 @@ However, the measurements are useful for tuning the parameters of the micro-stat From the start of this work, it became increasingly clear that an accurate micro-station model was necessary. First, during the uniaxial study, it became clear that the micro-station dynamics affects the active platform dynamics. -Then, using the 3-DoF rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled. +Then, using the 3-DoFs rotating model, it was discovered that the rotation of the active platform induces gyroscopic effects that affect the system dynamics and should therefore be modeled. Finally, a modal analysis of the micro-station showed how complex the dynamics of the station is. -The modal analysis also confirm that each stage behaves as a rigid body in the frequency range of interest. +The modal analysis also confirms that each stage behaves as a rigid body in the frequency range of interest. Therefore, a multi-body model is a good candidate to accurately represent the micro-station dynamics. In this report, the development of such a multi-body model is presented. @@ -3799,7 +3770,7 @@ Such a stacked architecture allows high mobility, but the overall stiffness is r [[file:figs/ustation_cad_view.png]] There are different ways of modeling the stage dynamics in a multi-body model. -The one chosen in this work consists of modeling each stage by two solid bodies connected by one 6-DoF joint. +The one chosen in this work consists of modeling each stage by two solid bodies connected by one 6-DoFs joint. The stiffness and damping properties of the joint s can be tuned separately for each DoF. @@ -4057,7 +4028,7 @@ Similarly, the mobile frame of the tilt stage is equal to the fixed frame of the [[file:figs/ustation_stage_motion.png]] The motion induced by a positioning stage can be described by a homogeneous transformation matrix from frame $\{A\}$ to frame $\{B\}$ as explain in Section\nbsp{}ref:ssec:ustation_kinematics. -As any motion stage induces parasitic motion in all 6 DoF, the transformation matrix representing its induced motion can be written as in\nbsp{}eqref:eq:ustation_translation_stage_errors. +As any motion stage induces parasitic motion in all 6-DoFs, the transformation matrix representing its induced motion can be written as in\nbsp{}eqref:eq:ustation_translation_stage_errors. \begin{equation}\label{eq:ustation_translation_stage_errors} {}^A\bm{T}_B(D_x, D_y, D_z, \theta_x, \theta_y, \theta_z) = @@ -4124,7 +4095,7 @@ The inertia of the solid bodies and the stiffness properties of the guiding mech The obtained dynamics is then compared with the modal analysis performed on the micro-station (Section\nbsp{}ref:ssec:ustation_model_comp_dynamics). As the dynamics of the active platform is impacted by the micro-station compliance (see Section ref:sec:uniaxial_support_compliance), the most important dynamical characteristic that should be well modeled is the overall compliance of the micro-station. -To do so, the 6-DoF compliance of the micro-station is measured and then compared with the 6-DoF compliance extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_model_compliance). +To do so, the 6-DoFs compliance of the micro-station is measured and then compared with the 6-DoFs compliance extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_model_compliance). **** Multi-Body Model <> @@ -4138,13 +4109,13 @@ Joints are used to impose kinematic constraints between solid bodies and to spec External forces can be used to model disturbances, and "sensors" can be used to measure the relative pose between two defined frames. #+name: fig:ustation_simscape_stage_example -#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simulink - Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be "controlled", the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included. +#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simulink - Simscape). It is composed of two solid bodies connected by a 6-DoFs joint. One joint DoF (here the tilt angle) can be "controlled", the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included. #+attr_latex: :scale 0.8 [[file:figs/ustation_simscape_stage_example.png]] Therefore, the micro-station is modeled by several solid bodies connected by joints. -A typical stage (here the tilt-stage) is modeled as shown in Figure\nbsp{}ref:fig:ustation_simscape_stage_example where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoF joint. -One DoF of the 6-DoF joint is "imposed" by a setpoint (i.e. modeled as infinitely stiff), while the other 5 are each modeled by a spring and damper. +A typical stage (here the tilt-stage) is modeled as shown in Figure\nbsp{}ref:fig:ustation_simscape_stage_example where two solid bodies (the fixed part and the mobile part) are connected by a 6-DoFs joint. +One DoF of the 6-DoFs joint is "imposed" by a setpoint (i.e. modeled as infinitely stiff), while the other 5 are each modeled by a spring and damper. Additional forces can be used to model disturbances induced by the stage motion. The obtained 3D representation of the multi-body model is shown in Figure\nbsp{}ref:fig:ustation_simscape_model. @@ -4154,17 +4125,17 @@ The obtained 3D representation of the multi-body model is shown in Figure\nbsp{} [[file:figs/ustation_simscape_model.jpg]] The ground is modeled by a solid body connected to the "world frame" through a joint only allowing 3 translations. -The granite was then connected to the ground using a 6-DoF joint. -The translation stage is connected to the granite by a 6-DoF joint, but the $D_y$ motion is imposed. -Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoF joint, with 1 imposed DoF each time. -Finally, the positioning hexapod has 6-DoF. +The granite was then connected to the ground using a 6-DoFs joint. +The translation stage is connected to the granite by a 6-DoFs joint, but the $D_y$ motion is imposed. +Similarly, the tilt-stage and the spindle are connected to the stage below using a 6-DoFs joint, with 1 imposed DoF each time. +Finally, the positioning hexapod has 6-DoFs. The total number of "free" acrshortpl:dof is 27, so the model has 54 states. The springs and dampers values were first estimated from the joint/stage specifications and were later fined-tuned based on the measurements. The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_values. #+name: tab:ustation_6dof_stiffness_values -#+caption: Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section\nbsp{}ref:ssec:ustation_stages. +#+caption: Summary of the stage stiffnesses. The constrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoFs joints are defined are indicated in figures found in Section ref:ssec:ustation_stages. #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccccc #+attr_latex: :center t :booktabs t | *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ | @@ -4283,7 +4254,7 @@ These results are compared with the measurements in Figure\nbsp{}ref:fig:ustatio Considering the complexity of the micro-station compliance dynamics, the model compliance matches sufficiently well for the current application. #+name: fig:ustation_frf_compliance_model -#+caption: Compliance of the micro-station expressed in frame $\{\mathcal{X}\}$. The measured FRFs are display by translucent lines, while the FRFs extracted from the multi-body models are shown by opaque lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed. +#+caption: Compliance of the micro-station expressed in frame $\{\mathcal{X}\}$. The measured FRFs are displayed by translucent lines, while the FRFs extracted from the multi-body models are shown by opaque lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_frf_compliance_xyz_model}Compliance in translation} @@ -4365,7 +4336,7 @@ A vertical error of $\pm300\,\text{nm}$ induced by the translation stage is expe Similar result is obtained for the $x$ lateral direction. #+name: fig:ustation_errors_dy -#+caption: Measurement of the vertical error of the translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}). +#+caption: Measurement of the straightness (vertical error) of the translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical}Measured vertical error} @@ -4374,7 +4345,7 @@ Similar result is obtained for the $x$ lateral direction. #+attr_latex: :scale 0.8 [[file:figs/ustation_errors_dy_vertical.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing linear fit} +#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing a linear fit} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -4393,7 +4364,7 @@ From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translati #+caption: Experimental setup used to estimate the errors induced by the Spindle rotation (\subref{fig:ustation_rz_meas_lion}). The motion of the two reference spheres is measured using 5 capacitive sensors (\subref{fig:ustation_rz_meas_lion_zoom}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoF metrology} +#+attr_latex: :caption \subcaption{\label{fig:ustation_rz_meas_lion}Micro-station and 5-DoFs metrology} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width 0.9\linewidth @@ -4407,7 +4378,7 @@ From the 5 measured displacements $[d_1,\,d_2,\,d_3,\,d_4,\,d_5]$, the translati #+end_subfigure #+end_figure -A measurement was performed during a constant rotational velocity of the spindle of 60rpm and during 10 turns. +A measurement was performed during a constant rotational velocity of the spindle of $60\,\text{rpm}$ and during 10 turns. The obtained results are shown in Figure\nbsp{}ref:fig:ustation_errors_spindle. A large fraction of the radial (Figure\nbsp{}ref:fig:ustation_errors_spindle_radial) and tilt (Figure\nbsp{}ref:fig:ustation_errors_spindle_tilt) errors is linked to the fact that the two spheres are not perfectly aligned with the rotation axis of the Spindle. This is displayed by the dashed circle. @@ -4416,7 +4387,7 @@ However, some misalignment between the acrshort:poi of the sample and the rotati The vertical motion induced by scanning the spindle is in the order of $\pm 30\,\text{nm}$ (Figure\nbsp{}ref:fig:ustation_errors_spindle_axial). #+name: fig:ustation_errors_spindle -#+caption: Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis. +#+caption: Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) errors during a Spindle rotation at $60\,\text{rpm}$. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_radial}Radial errors} @@ -4541,7 +4512,7 @@ Second, a constant velocity scans with the translation stage was performed and a **** Tomography Experiment <> -To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of 60rpm. +To simulate a tomography experiment, the setpoint of the Spindle is configured to perform a constant rotation with a rotational velocity of $60\,\text{rpm}$. Both ground motion and spindle vibration disturbances were simulated based on what was computed in Section\nbsp{}ref:sec:ustation_disturbances. A radial offset of $\approx 1\,\upmu\text{m}$ between the acrfull:poi and the spindle's rotation axis is introduced to represent what is experimentally observed. During the 10 second simulation (i.e. 10 spindle turns), the position of the acrshort:poi with respect to the granite was recorded. @@ -4549,7 +4520,7 @@ Results are shown in Figure\nbsp{}ref:fig:ustation_errors_model_spindle. A good correlation with the measurements is observed both for radial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_radial) and axial errors (Figure\nbsp{}ref:fig:ustation_errors_model_spindle_axial). #+name: fig:ustation_errors_model_spindle -#+caption: Simulation results for a tomography experiment at constant velocity of 60rpm. The comparison is made with measurements for both radial (\subref{fig:ustation_errors_model_spindle_radial}) and axial errors (\subref{fig:ustation_errors_model_spindle_axial}). +#+caption: Simulation results for a tomography experiment at constant velocity of $60\,\text{rpm}$. The comparison is made with measurements for both radial (\subref{fig:ustation_errors_model_spindle_radial}) and axial errors (\subref{fig:ustation_errors_model_spindle_axial}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:ustation_errors_model_spindle_radial}Radial error} @@ -4639,16 +4610,16 @@ At NSLS-II, for instance, a system capable of $100\,\upmu\text{m}$ stroke has be Similarly, at the Sirius facility, a tripod configuration based on voice coil actuators has been implemented for XYZ position control, achieving feedback bandwidths of approximately $100\,\text{Hz}$ (Figure\nbsp{}ref:fig:nhexa_stages_sapoti). #+name: fig:nhexa_stages_translations -#+caption: Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope\nbsp{}[[cite:&nazaretski15_pushin_limit]] at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline\nbsp{}[[cite:&geraldes23_sapot_carnaub_sirius_lnls]] at Sirius facility (\subref{fig:nhexa_stages_sapoti}). +#+caption: Example of sample stage with active XYZ corrections based on external metrology. The MLL microscope at NSLS-II (\subref{fig:nhexa_stages_nazaretski}). Sample stage on SAPOTI beamline at Sirius facility (\subref{fig:nhexa_stages_sapoti}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope \cite{nazaretski15_pushin_limit}} #+attr_latex: :options {0.36\textwidth} #+begin_subfigure #+attr_latex: :height 6cm [[file:figs/nhexa_stages_nazaretski.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_sapoti} SAPOTI sample stage} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_sapoti} SAPOTI sample stage \cite{geraldes23_sapot_carnaub_sirius_lnls}} #+attr_latex: :options {0.60\textwidth} #+begin_subfigure #+attr_latex: :height 6cm @@ -4658,21 +4629,21 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac The integration of $R_z$ rotational capability, which is necessary for tomography experiments, introduces additional complexity. At ESRF's ID16A beamline, a Stewart platform (whose architecture will be presented in Section\nbsp{}ref:sec:nhexa_stewart_platform) using piezoelectric actuators has been positioned below the spindle (Figure\nbsp{}ref:fig:nhexa_stages_villar). -While this configuration enables the correction of spindle motion errors through 5-DoF control based on capacitive sensor measurements, the stroke is limited to $50\,\upmu\text{m}$ due to the inherent constraints of piezoelectric actuators. +While this configuration enables the correction of spindle motion errors through 5-DoFs control based on capacitive sensor measurements, the stroke is limited to $50\,\upmu\text{m}$ due to the inherent constraints of piezoelectric actuators. In contrast, at PETRA III, an alternative approach places a XYZ-stacked stage above the spindle, offering $100\,\upmu\text{m}$ stroke (Figure\nbsp{}ref:fig:nhexa_stages_schroer). However, attempts to implement real-time feedback using YZ external metrology proved challenging, possibly due to the poor dynamical response of the serial stage configuration. #+name: fig:nhexa_stages_spindle -#+caption: Example of two sample stages for tomography experiments. ID16a endstation\nbsp{}[[cite:&villar18_nanop_esrf_id16a_nano_imagin_beaml]] at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope\nbsp{}[[cite:&schropp20_ptynam;&schroer17_ptynam]] at PETRA III (\subref{fig:nhexa_stages_schroer}). +#+caption: Example of two sample stages for tomography experiments. ID16a endstation at the ESRF (\subref{fig:nhexa_stages_villar}). PtyNAMi microscope at PETRA III (\subref{fig:nhexa_stages_schroer}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station \cite{villar18_nanop_esrf_id16a_nano_imagin_beaml}} #+attr_latex: :options {0.54\textwidth} #+begin_subfigure #+attr_latex: :height 5.5cm [[file:figs/nhexa_stages_villar.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_schroer} PtyNAMi microscope} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_schroer} PtyNAMi microscope \cite{schropp20_ptynam,schroer17_ptynam}} #+attr_latex: :options {0.40\textwidth} #+begin_subfigure #+attr_latex: :height 6cm @@ -4684,7 +4655,7 @@ Table\nbsp{}ref:tab:nhexa_sample_stages provides an overview of existing end-sta Although direct performance comparisons between these systems are challenging due to their varying experimental requirements, scanning velocities, and specific use cases, several distinctive characteristics of the NASS can be identified. #+name: tab:nhexa_sample_stages -#+caption: End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are ommited or indicated between parentheses. The specifications for the NASS are indicated in the last row. +#+caption: End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are omitted or indicated between parentheses. The specifications for the NASS are indicated in the last row. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccccc #+attr_latex: :placement [!ht] :center t :booktabs t | *Stacked Stages* | *Specifications* | *Measured DoFs* | *Bandwidth* | *Reference* | @@ -4770,16 +4741,16 @@ Numerous parallel kinematic architectures have been developed\nbsp{}[[cite:&dong Furthermore, hybrid architectures combining both serial and parallel elements have been proposed\nbsp{}[[cite:&shen19_dynam_analy_flexur_nanop_stage]], as illustrated in Figure\nbsp{}ref:fig:nhexa_serial_parallel_examples, offering potential compromises between the advantages of both approaches. #+name: fig:nhexa_serial_parallel_examples -#+caption: Examples of an XYZ serial positioning stage\nbsp{}[[cite:&kenton12_desig_contr_three_axis_serial]] (\subref{fig:nhexa_serial_architecture_kenton}) and of a 5-DoF hybrid (parallel/serial) positioning platform\nbsp{}[[cite:&shen19_dynam_analy_flexur_nanop_stage]] (\subref{fig:nhexa_parallel_architecture_shen}). +#+caption: Examples of a serial positioning stage (\subref{fig:nhexa_serial_architecture_kenton}) and of a hybrid (parallel/serial) positioning platform (\subref{fig:nhexa_parallel_architecture_shen}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_serial_architecture_kenton} Serial positioning stage} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_serial_architecture_kenton} XYZ Serial positioning stage \cite{kenton12_desig_contr_three_axis_serial}} #+attr_latex: :options {0.41\textwidth} #+begin_subfigure #+attr_latex: :height 4.5cm [[file:figs/nhexa_serial_architecture_kenton.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoF stage} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoFs stage \cite{shen19_dynam_analy_flexur_nanop_stage}} #+attr_latex: :options {0.55\textwidth} #+begin_subfigure #+attr_latex: :height 4.5cm @@ -4794,16 +4765,16 @@ These examples demonstrate the architecture's versatility in terms of geometry, Furthermore, the successful implementation of Integral Force Feedback (IFF) control on Stewart platforms has been well documented\nbsp{}[[cite:&abu02_stiff_soft_stewar_platf_activ;&hanieh03_activ_stewar;&preumont07_six_axis_singl_stage_activ]], and the extensive body of research on this architecture enables thorough optimization specifically for the NASS. #+name: fig:nhexa_stewart_examples -#+caption: Two examples of Stewart platform. A Stewart platform based on piezoelectric stack actuators and used for nano-positioning is shown in (\subref{fig:nhexa_stewart_piezo_furutani})\nbsp{}[[cite:&furutani04_nanom_cuttin_machin_using_stewar]]. A Stewart platform based on voice coil actuators and used for vibration isolation is shown in (\subref{fig:nhexa_stewart_vc_preumont})\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ;&preumont18_vibrat_contr_activ_struc_fourt_edition]]. +#+caption: Two examples of Stewart platforms. (\subref{fig:nhexa_stewart_piezo_furutani}) Stewart platform based on piezoelectric actuators and used for nano-positioning. (\subref{fig:nhexa_stewart_vc_preumont}) Stewart platform based on voice coil actuators and used for vibration isolation. #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning \cite{furutani04_nanom_cuttin_machin_using_stewar}} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.9\linewidth [[file:figs/nhexa_stewart_piezo_furutani.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_vc_preumont} Stewart platform for vibration isolation} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_vc_preumont} Stewart platform for vibration isolation \cite{preumont07_six_axis_singl_stage_activ,preumont18_vibrat_contr_activ_struc_fourt_edition}} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.9\linewidth @@ -4839,11 +4810,11 @@ These theoretical foundations form the basis for subsequent design decisions and <> The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure\nbsp{}ref:fig:nhexa_stewart_architecture). -Each strut is modelled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. +Each strut is modeled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom[fn:nhexa_1]. #+name: fig:nhexa_stewart_architecture -#+caption: Schematical representation of the Stewart platform architecture. +#+caption: Schematic representation of the Stewart platform architecture. #+attr_latex: :scale 0.9 [[file:figs/nhexa_stewart_architecture.png]] @@ -4863,7 +4834,7 @@ The struts' orientations are represented by the unit vectors $\hat{\bm{s}}_i$ an This is summarized in Figure\nbsp{}ref:fig:nhexa_stewart_notations. #+name: fig:nhexa_stewart_notations -#+caption: Frame and key notations for the Stewart platform. +#+caption: Typical defined frames for the Stewart platform and key notations. #+attr_latex: :scale 0.9 [[file:figs/nhexa_stewart_notations.png]] @@ -4881,7 +4852,7 @@ For each strut $i$ (illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_loop_closu This equation links the pose[fn:nhexa_2] variables ${}^A\bm{P}$ and ${}^A\bm{R}_B$, the position vectors describing the known geometry of the base and the moving platform, $\bm{a}_i$ and $\bm{b}_i$, and the strut vector $l_i {}^A\hat{\bm{s}}_i$: #+name: fig:nhexa_stewart_loop_closure -#+caption: Notations to compute the kinematic loop closure. +#+caption: Geometrical representation of the loop closure. #+attr_latex: :scale 0.9 [[file:figs/nhexa_stewart_loop_closure.png]] @@ -4986,7 +4957,7 @@ Since the maximum required stroke of the active platform ($\approx 100\,\upmu\te It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy. #+name: fig:nhexa_forward_kinematics_approximate_errors -#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,\text{mm}$ was used to perform this analysis. $\epsilon_D$ corresponds to the distance between the true positioin and the estimated position. $\epsilon_R$ corresponds to the angular motion between the true orientation and the estimated orientation. +#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,\text{mm}$ was used to perform this analysis. $\epsilon_D$ corresponds to the distance between the true position and the estimated position. $\epsilon_R$ corresponds to the angular motion between the true orientation and the estimated orientation. #+attr_latex: :scale 0.8 [[file:figs/nhexa_forward_kinematics_approximate_errors.png]] @@ -5144,7 +5115,7 @@ From these parameters, key kinematic properties can be derived: the strut orient #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage #+name: fig:nhexa_stewart_model_def -#+caption: Geometry of the stewart platform. +#+caption: Geometrical parameters of the Stewart platform. #+attr_latex: :float nil :scale 0.9 [[file:figs/nhexa_stewart_model_def.png]] #+end_minipage @@ -5193,7 +5164,7 @@ These joints are considered massless and exhibit no stiffness along their degree The actuator model comprises several key elements (Figure\nbsp{}ref:fig:nhexa_actuator_model). At its core, each actuator is modeled as a prismatic joint with internal stiffness $k_a$ and damping $c_a$, driven by a force source $f$. -Similarly to what was found using the rotating 3-DoF model, a parallel stiffness $k_p$ is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. +Similarly to what was found using the rotating 3-DoFs model, a parallel stiffness $k_p$ is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. Each actuator is equipped with two sensors: a force sensor providing measurements $f_n$ and a relative motion sensor that measures the strut length $l_i$. The actuator parameters used in the conceptual phase are listed in Table\nbsp{}ref:tab:nhexa_actuator_parameters. @@ -5294,7 +5265,7 @@ Each actuator's transfer function to its associated force sensor exhibits altern The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, which was previously observed in the three-degree-of-freedom rotating model. #+name: fig:nhexa_multi_body_plant -#+caption: Bode plot of the transfer functions computed using the active platform multi-body model. +#+caption: Bode plot of the transfer functions computed using the multi-body model. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{\mathcal{L}}$} @@ -5354,7 +5325,7 @@ In the context of the active platform, two distinct control strategies were exam - acrfull:hac, which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section\nbsp{}ref:ssec:nhexa_control_hac_lac) #+name: fig:nhexa_stewart_decentralized_control -#+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity. +#+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown. #+attr_latex: :scale 0.9 [[file:figs/nhexa_stewart_decentralized_control.png]] @@ -5376,7 +5347,7 @@ This simplifies the control design because only one controller needs to be tuned Furthermore, at low frequencies, the plant exhibits good decoupling between the struts, allowing for effective independent control of each axis. #+name: fig:nhexa_control_frame -#+caption: Two control strategies. +#+caption: Two control strategies using the Jacobian matrix. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_control_strut}Control in the frame of the struts. $\bm{J}$ is used to project errors in the frame of the struts} @@ -5387,7 +5358,7 @@ Furthermore, at low frequencies, the plant exhibits good decoupling between the #+end_subfigure \bigskip -#+attr_latex: :caption \subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-\intercal}$ is used to project force and torques on each strut} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-\intercal}$ is used to project forces and torques on each strut} #+attr_latex: :options {0.98\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -5411,7 +5382,7 @@ For the conceptual validation of the acrshort:nass, control in the strut space w More sophisticated control strategies will be explored during the detailed design phase. #+name: fig:nhexa_plant_frame -#+caption: Bode plot of the transfer functions computed using the active platform multi-body model. +#+caption: Bode plots of plants corresponding to the two control strategies shown in Figure ref:fig:nhexa_control_frame. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts} @@ -5436,7 +5407,7 @@ The decentralized Integral Force Feedback (IFF) control strategy is implemented The corresponding block diagram of the control loop is shown in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_schematic, in which the controller $\bm{K}_{\text{IFF}}(s)$ is a diagonal matrix, where each diagonal element is a pure integrator\nbsp{}eqref:eq:nhexa_kiff. #+name: fig:nhexa_decentralized_iff_schematic -#+caption: Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs $\bm{f}^{\prime}$. +#+caption: Schematic of the implemented decentralized IFF controller. The damped plant has input $\bm{f}^{\prime}$. #+attr_latex: :scale 0.9 [[file:figs/nhexa_decentralized_iff_schematic.png]] @@ -5451,16 +5422,16 @@ The corresponding block diagram of the control loop is shown in Figure\nbsp{}ref In this section, the stiffness in parallel with the force sensor was omitted since the Stewart platform is not subjected to rotation. The effect of this parallel stiffness is examined in the next section when the platform is integrated into the complete NASS. -Root Locus analysis, shown in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$. +Root locus analysis, shown in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$. A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. This property is particularly valuable because the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone. -The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)$), presented in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$. +The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)$), presented in Figure\nbsp{}ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $\SI{-90}{\degree}$ and $\SI{+90}{\degree}$. The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes. This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability. #+name: fig:nhexa_decentralized_iff_results -#+caption: Decentralized IFF. +#+caption: Decentralized IFF. Loop Gain for an individual controller (\subref{fig:nhexa_decentralized_iff_loop_gain}) and root locus (\subref{fig:nhexa_decentralized_iff_root_locus}). Black crosses are indicating the closed-loop poles for the chosen controller gain. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{n1}}{f_1}(s)$} @@ -5469,7 +5440,7 @@ This high gain, combined with the bounded phase, enables effective damping of th #+attr_latex: :scale 0.8 [[file:figs/nhexa_decentralized_iff_loop_gain.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root locus} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -5534,7 +5505,7 @@ As shown in Figure\nbsp{}ref:fig:nhexa_decentralized_hac_iff_root_locus, all loc Additionally, the distance of the loci from the $-1$ point provides information about stability margins of the coupled system. #+name: fig:nhexa_decentralized_hac_iff_results -#+caption: Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection performances. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop. +#+caption: Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection level. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain} @@ -5756,7 +5727,7 @@ Then, the high authority controller uses the computed errors in the frame of the **** Introduction :ignore: Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the acrshort:haclac strategy. -The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3DoF rotating model. +The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3-DoFs rotating model. The objective here is to design a decentralized IFF controller that provides good damping of the active platform modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$. The payloads used for validation have a cylindrical shape with $250\,\text{mm}$ height and with masses of $1\,\text{kg}$, $25\,\text{kg}$, and $50\,\text{kg}$. @@ -5772,7 +5743,7 @@ Adding parallel stiffness (Figure\nbsp{}ref:fig:nass_iff_plant_kp) transforms th Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the actuators and sensors\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with]]. #+name: fig:nass_iff_plant_effect_kp -#+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of $25\,\text{kg}$. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}). +#+caption: Effect of stiffness in parallel with the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of $25\,\text{kg}$. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_no_kp}without parallel stiffness} @@ -5789,7 +5760,7 @@ Although both cases show significant coupling around the resonances, stability i #+end_subfigure #+end_figure -The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\upmu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model). +The effect of rotation, as shown in Figure\nbsp{}ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,\text{N}/\upmu\text{m}$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3-DoFs rotating model). Figure\nbsp{}ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics. The poles and zeros shift in frequency as the payload mass varies. @@ -5816,7 +5787,7 @@ However, their alternating pattern is preserved, which ensures the phase remains **** Controller Design <> -The previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation. +The previous analysis using the 3-DoFs rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation. This finding was also confirmed with the multi-body model of the NASS: the system was unstable when using pure integrators and without parallel stiffness. This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors. @@ -5845,7 +5816,7 @@ To verify stability, the root loci for the three payload configurations were com The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robustness of the applied decentralized IFF. #+name: fig:nass_iff_root_locus -#+caption: Root Loci for Decentralized IFF for three payload masses. The closed-loop poles are shown by the black crosses. +#+caption: Root loci for decentralized IFF for three payload masses. The closed-loop poles are shown by the black crosses. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$} @@ -5930,7 +5901,7 @@ More importantly, in the vicinity of the desired high authority control bandwidt For the undamped plants (shown in blue), achieving robust control with bandwidth above $10\,\text{Hz}$ while maintaining stability across different payload masses would be practically impossible. #+name: fig:nass_hac_plant -#+caption: Effect of Decentralized Integral Force Feedback on the positioning plant for a $1\,\text{kg}$ sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}). +#+caption: Effect of decentralized Integral Force Feedback on the positioning plant for a $1\,\text{kg}$ sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). Direct terms are shown by solid lines while coupling terms are shown by shaded lines. The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_comp_undamped_damped_plant_m1}Effect of IFF - $m = 1\,\text{kg}$} @@ -5963,7 +5934,7 @@ This result confirms effective dynamic decoupling between the active platform an **** Effect of Active Platform Stiffness on System Dynamics <> -The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models. +The influence of active platform stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3-DoFs) models. These models suggest that a moderate stiffness of approximately $1\,\text{N}/\upmu\text{m}$ would provide better performance than either very stiff or very soft configurations. For the stiff active platform analysis, a system with an actuator stiffness of $100\,\text{N}/\upmu\text{m}$ was simulated with a $25\,\text{kg}$ payload. @@ -5977,7 +5948,7 @@ Figure\nbsp{}ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation The current approach of controlling the position in the strut frame is inadequate for soft active platforms; but even shifting control to a frame matching the payload's acrlong:com would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects. #+name: fig:nass_soft_stiff_hexapod -#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\upmu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a compliance ($k_a = 0.01\,\text{N}/\upmu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}). +#+caption: Coupling between a stiff active platform ($k_a = 100\,\text{N}/\upmu\text{m}$) and the micro-station (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}). Large effect of the spindle rotational velocity for a soft ($k_a = 0.01\,\text{N}/\upmu\text{m}$) active platform (\subref{fig:nass_soft_nano_hexapod_effect_Wz}). #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,\text{N}/\upmu\text{m}$ - Coupling with the micro-station} @@ -6062,11 +6033,11 @@ The robustness of the NASS to payload mass variation was evaluated through addit As illustrated in Figure\nbsp{}ref:fig:nass_tomography_hac_iff, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis. While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions. -It should be noted that the maximum rotational velocity of 360deg/s is primarily intended for lightweight payload applications. -For higher mass configurations, rotational velocities are expected to be below 36deg/s. +It should be noted that the maximum rotational velocity of $360\,\text{deg/s}$ is primarily intended for lightweight payload applications. +For higher mass configurations, rotational velocities are expected to be below $36\,\text{deg/s}$. #+name: fig:nass_tomography_hac_iff -#+caption: Simulation of tomography experiments - 360deg/s. Beam size is indicated by the dashed black ellipse. +#+caption: Simulation of tomography experiments at $360\,\text{deg/s}$. Beam size is indicated by the dashed black ellipse. #+attr_latex: :options [h!tbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,\text{kg}$} @@ -6217,7 +6188,7 @@ Examples of piezoelectric-actuated Stewart platforms are presented in Figures\nb Although less frequently encountered, magnetostrictive actuators have been successfully implemented in\nbsp{}[[cite:&zhang11_six_dof]] (Figure\nbsp{}ref:fig:detail_kinematics_zhang11). #+name: fig:detail_kinematics_stewart_examples_cubic -#+caption: Some examples of developped Stewart platform with Cubic geometry. +#+caption: Some examples of developed Stewart platform with Cubic geometry. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_jpl}California Institute of Technology - USA \cite{spanos95_soft_activ_vibrat_isolat}} @@ -6267,7 +6238,7 @@ The second category comprises non-cubic architectures (Figure\nbsp{}ref:fig:deta The influence of strut orientation and joint positioning on Stewart platform properties is analyzed in Section\nbsp{}ref:sec:detail_kinematics_geometry. #+name: fig:detail_kinematics_stewart_examples_non_cubic -#+caption: Some examples of developped Stewart platform with non-cubic geometry. +#+caption: Some examples of developed Stewart platform with non-cubic geometry. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_pph}Naval Postgraduate School - USA \cite{chen03_payload_point_activ_vibrat_isolat}} @@ -6368,11 +6339,11 @@ The vertically oriented struts configuration leads to greater stroke in the hori Conversely, horizontal oriented struts configuration provides more stroke in the vertical direction. It may seem counterintuitive that less stroke is available in the direction of the struts. -This phenomenon occurs because the struts form a lever mechanism that amplifies the motion. +This phenomenon occurs because the struts form a lever arm mechanism that amplifies the motion. The amplification factor increases when the struts have a high angle with the direction of motion and equals one (i.e. is minimal) when aligned with the direction of motion. #+name: fig:detail_kinematics_stewart_mobility_translation_examples -#+caption: Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}). +#+caption: Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometries are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented horizontally (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_stewart_mobility_vert_struts}Vertical struts} @@ -6407,7 +6378,7 @@ The mobility for pure rotations is compared in Figure\nbsp{}ref:fig:detail_kinem Having struts further apart decreases the "lever arm" and therefore reduces the rotational mobility. #+name: fig:detail_kinematics_stewart_mobility_rotation_examples -#+caption: Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}). +#+caption: Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometries are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further apart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_stewart_mobility_close_struts}Struts close together} @@ -6545,7 +6516,7 @@ Practical implementations of such configurations can be observed in Figures\nbsp It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure\nbsp{}ref:fig:detail_kinematics_cubic_architecture_example_small), as exemplified in Figure\nbsp{}ref:fig:detail_kinematics_ulb_pz. #+name: fig:detail_kinematics_cubic_architecture_examples -#+caption: Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}). +#+caption: Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges' length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture} @@ -6588,7 +6559,7 @@ The unit vectors corresponding to the edges of the cube are described by equatio \end{equation} #+name: fig:detail_kinematics_cubic_schematic_cases -#+caption: Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic}). +#+caption: Cubic architecture. Struts are represented in blue. The cube's center is indicated by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube} @@ -6725,7 +6696,7 @@ At high frequency, the behavior is governed by the mass matrix (evaluated at fra To achieve a diagonal mass matrix, the acrlong:com of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame. #+name: fig:detail_kinematics_cubic_payload -#+caption: Cubic stewart platform with top cylindrical payload. +#+caption: Cubic Stewart platform with cylindrical payload located on the top platform. #+attr_latex: :width 0.5\linewidth [[file:figs/detail_kinematics_cubic_payload.png]] @@ -6735,7 +6706,7 @@ When the $\{B\}$ frame was positioned at the acrlong:com, coupling at low freque Conversely, when positioned at the acrlong:cok, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_cok). #+name: fig:detail_kinematics_cubic_cart_coupling -#+caption: Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}). +#+caption: Transfer functions for a cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass} @@ -7144,7 +7115,7 @@ Through this approach, system-level dynamic behavior under closed-loop control c Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models. These components are traditionally analyzed using acrshort:fea software. -However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be successfully integrated into multi-body models\nbsp{}[[cite:&hatch00_vibrat_matlab_ansys]]. +However, a methodological bridge between these two analytical approaches has been established, whereby components whose dynamical properties have been determined through FEA can be integrated into multi-body models\nbsp{}[[cite:&hatch00_vibrat_matlab_ansys]]. This combined multibody-FEA modeling approach presents significant advantages, as it enables the accurate FE modeling to specific elements while maintaining the computational efficiency of multi-body analysis for the broader system\nbsp{}[[cite:&rankers98_machin]]. The investigation of this hybrid modeling approach is structured in three sections. @@ -7167,7 +7138,7 @@ Subsequently, interface frames are defined at locations where the multi-body mod These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames. Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method\nbsp{}[[cite:&craig68_coupl_subst_dynam_analy]] (also known as the "fixed-interface method"), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics. -This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF. +This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100-DoFs. The number of acrshortpl:dof in the reduced model is determined by\nbsp{}eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled. The outcome of this procedure is an $m \times m$ set of reduced mass and stiffness matrices, $m$ being the total retained number of acrshortpl:dof, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. @@ -7229,7 +7200,7 @@ Six additional modes were considered, resulting in total model order of $48$. The modal reduction procedure was then executed, yielding the reduced mass and stiffness matrices that form the foundation of the component's representation in the multi-body simulation environment. #+name: fig:detail_fem_apa95ml_model -#+caption: Obtained mesh and defined interface frames (or "remote points") in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}). +#+caption: Obtained mesh and defined interface frames (or "remote points") in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interfaces with the multi-body model are shown in (\subref{fig:detail_fem_apa_model_schematic}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_mesh} } @@ -7294,7 +7265,7 @@ The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are list From these parameters, $g_s = 5.1\,\text{V}/\upmu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained. #+name: tab:detail_fem_piezo_properties -#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities. +#+caption: Piezoelectric properties used for the estimation of the sensor and actuator sensitivities. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | *Description* | @@ -7318,7 +7289,7 @@ A value of $23\,\text{N}/\upmu\text{m}$ was found which is close to the specifie The multi-body model predicted a resonant frequency under block-free conditions of $\approx 2\,\text{kHz}$ (Figure\nbsp{}ref:fig:detail_fem_apa95ml_compliance), which is in agreement with the nominal specification. #+name: fig:detail_fem_apa95ml_compliance -#+caption: Estimated compliance of the APA95ML. +#+caption: Estimated axial compliance of the APA95ML. #+attr_latex: :scale 0.8 [[file:figs/detail_fem_apa95ml_compliance.png]] @@ -7339,12 +7310,12 @@ Further validation of the reduced-order flexible body methodology was undertaken The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element. The test bench illustrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_bench_schematic was used, which consists of a $5.7\,\text{kg}$ granite suspended on top of the APA95ML. -The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$. +The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measure its vertical displacement $y$. A acrfull:dac was used to generate the control signal $u$, which was subsequently conditioned through a voltage amplifier with a gain of $20$, ultimately yielding the effective voltage $V_a$ across the two piezoelectric stacks. Measurement of the sensor stack voltage $V_s$ was performed using an acrshort:adc. #+name: fig:detail_fem_apa95ml_bench_schematic -#+caption: Test bench used to validate "reduced order solid bodies" using an APA95ML. +#+caption: Test bench used to validate the presented modeling strategy. #+attr_latex: :width \linewidth [[file:figs/detail_fem_apa95ml_bench_schematic.png]] @@ -7401,10 +7372,10 @@ The measured acrshortpl:frf for each gain configuration were compared with model The close agreement between experimental measurements and theoretical predictions across all gain configurations demonstrates the model's capability to accurately predict both open-loop and closed-loop system dynamics. #+name: fig:detail_fem_apa95ml_iff_results -#+caption: Results using Integral Force Feedback with the APA95ML. Closed-loop poles as a function of the controller gain $g$ are predicted by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains. +#+caption: Results using Integral Force Feedback with the APA95ML. Closed-loop poles as a function of the controller gain $g$ are predicted by the root locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot} +#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root locus plot} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -7491,13 +7462,13 @@ The demonstrated accuracy of the modeling approach for the APA95ML provides conf #+caption: List of some amplified piezoelectric actuators that could be used for the active platform. #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccccc #+attr_latex: :center t :booktabs t :float t -| *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S | -|----------------------------------------------+---------+------------+-----------+-------------+-------------| -| Stroke $> 200\,\upmu\text{m}$ | 187 | 304 | 368 | 432 | 240 | +| *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S | +|------------------------------------------------+---------+------------+-----------+-------------+-------------| +| Stroke $> 200\,\upmu\text{m}$ | 187 | 304 | 368 | 432 | 240 | | Stiffness $\approx 1\,\text{N}/\upmu\text{m}$ | 0.7 | 1.8 | 0.55 | 0.87 | 0.58 | -| Resolution $< 2\,\text{nm}$ | 2 | 3 | 4 | | | -| Blocked Force $> 100\,\text{N}$ | 127 | 546 | 201 | 376 | 139 | -| Height $< 50\,\text{mm}$ | 22 | 30 | 24 | 27 | 16 | +| Resolution $< 2\,\text{nm}$ | 2 | 3 | 4 | n/a | n/a | +| Blocked Force $> 100\,\text{N}$ | 127 | 546 | 201 | 376 | 139 | +| Height $< 50\,\text{mm}$ | 22 | 30 | 24 | 27 | 16 | **** APA300ML - Reduced Order Flexible Body <> @@ -7528,19 +7499,19 @@ While this high order provides excellent accuracy for validation purposes, it pr The sensor and actuator "constants" ($g_s$ and $g_a$) derived in Section\nbsp{}ref:ssec:detail_fem_super_element_example for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks. -**** Simpler 2DoF Model of the APA300ML +**** Simpler 2-DoFs Model of the APA300ML <> To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]], was developed. This model, illustrated in Figure\nbsp{}ref:fig:detail_fem_apa_2dof_model, comprises three components. The mechanical shell is characterized by its axial stiffness $k_1$ and damping $c_1$. -The actuator is modelled with stiffness $k_a$ and damping $c_a$, incorporating a force source $f$. +The actuator is modeled with stiffness $k_a$ and damping $c_a$, incorporating a force source $f$. This force is related to the applied voltage $V_a$ through the actuator constant $g_a$. The sensor stack is modeled with stiffness $k_e$ and damping $c_e$, with its deformation $d_L$ being converted to the output voltage $V_s$ through the sensor sensitivity $g_s$. #+name: fig:detail_fem_apa_2dof_model -#+caption: Schematic of the 2DoF model of the Amplified Piezoelectric Actuator. +#+caption: Schematic of the 2-DoFs model of the Amplified Piezoelectric Actuator. [[file:figs/detail_fem_apa_2dof_model.png]] While providing computational efficiency, this simplified model has inherent limitations. @@ -7551,7 +7522,7 @@ Nevertheless, the model's primary advantage lies in its simplicity, adding only The model requires tuning of 8 parameters ($k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$, and $g_a$) to match the dynamics extracted from the acrshort:fea. The shell parameters $k_1$ and $c_1$ were determined first through analysis of the zero in the $V_a$ to $V_s$ transfer function. -The physical interpretation of this zero can be understood through Root Locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros. +The physical interpretation of this zero can be understood through root locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros. The open-loop zero therefore corresponds to the poles of the system with a theoretical infinite-gain controller that ensures zero force in the sensor stack. This condition effectively represents the dynamics of an acrshort:apa without the force sensor stack (i.e. an acrshort:apa with only the shell). This physical interpretation enables straightforward parameter tuning: $k_1$ determines the frequency of the zero, while $c_1$ defines its damping characteristic. @@ -7564,7 +7535,7 @@ The resulting parameters, listed in Table\nbsp{}ref:tab:detail_fem_apa300ml_2dof While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range. #+name: tab:detail_fem_apa300ml_2dof_parameters -#+caption: Summary of the obtained parameters for the 2 DoF APA300ML model. +#+caption: Summary of the obtained parameters for the 2-DoFs APA300ML model. #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | @@ -7579,7 +7550,7 @@ While higher-order modes and non-axial flexibility are not captured, the model a | $g_s$ | $0.53\,\text{V}/\upmu\text{m}$ | #+name: fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof -#+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}). +#+caption: Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2-DoFs model. Both for the dynamics from $V_a$ to $d_i$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}from $V_a$ to $d_i$} @@ -7607,7 +7578,7 @@ As demonstrated in Figure\nbsp{}ref:fig:detail_fem_apa95ml_effect_electrical_bou The developed models of the acrshort:apa do not represent such behavior, but as this effect is quite small, this validates the simplifying assumption made in the models. #+name: fig:detail_fem_apa95ml_effect_electrical_boundaries -#+caption: Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency. +#+caption: Effect of the electrical boundaries of the force sensor stack on the APA95ML resonance frequency. #+attr_latex: :scale 0.8 [[file:figs/detail_fem_apa95ml_effect_electrical_boundaries.png]] @@ -7618,21 +7589,21 @@ These aspects will be addressed in the instrumentation chapter. **** Validation with the Active Platform <> -The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with acrshort:apa modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full acrshort:fem implementation. +The integration of the APA300ML model within the active platform simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with acrshort:apa modeled as flexible bodies, and to validate the simplified 2-DoFs model through comparative analysis with the full acrshort:fem implementation. The dynamics predicted using the flexible body model align well with the design requirements established during the conceptual phase. The dynamics from $\bm{u}$ to $\bm{V}_s$ exhibits the desired alternating pole-zero pattern (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_hac_plant), a critical characteristic for implementing robust decentralized Integral Force Feedback. Additionally, the model predicts no problematic high-frequency modes in the dynamics from $\bm{u}$ to $\bm{\epsilon}_{\mathcal{L}}$ (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant), maintaining consistency with earlier conceptual simulations. These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator. -Comparative analysis between the high-order acrshort:fem implementation and the simplified 2DoF model (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest. +Comparative analysis between the high-order acrshort:fem implementation and the simplified 2-DoFs model (Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest. This validates the use of the simplified model for time-domain simulations. -The reduction in model order is substantial: while the acrshort:fem implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete active platform. +The reduction in model order is substantial: while the acrshort:fem implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2-DoFs model requires only 24 states for the complete active platform. These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the active platform. #+name: fig:detail_fem_actuator_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between an active platform having the actuators modeled with FEM and an active platform having actuators modelled a 2DoF system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between an active platform having the actuators modeled with FEM and an active platform having actuators modeled as 2-DoFs system. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7659,7 +7630,7 @@ For Stewart platforms requiring nanometric precision, numerous flexible joint de For design simplicity and component standardization, identical joints are employed at both ends of the active platform struts. #+name: fig:detail_fem_joints_examples -#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having "notch curves" are also used\nbsp{}[[cite:&du14_piezo_actuat_high_precis_flexib]]. +#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in [[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitly specified as done in [[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having "notch curves" [[cite:&du14_piezo_actuat_high_precis_flexib]]. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_preumont}} @@ -7694,7 +7665,7 @@ The validation process, detailed in Section\nbsp{}ref:ssec:detail_fem_joint_vali The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction\nbsp{}[[cite:&mcinroy02_model_desig_flexur_joint_stewar]] and can affect system dynamics. -To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1DoF actuators (stiffness of $1\,\text{N}/\upmu\text{m}$) without parallel stiffness to the force sensors. +To quantify these effects, simulations were conducted with the micro-station considered rigid and using simplified 1-DoF actuators (stiffness of $1\,\text{N}/\upmu\text{m}$) without parallel stiffness to the force sensors. Flexible joint bending stiffness was varied from 0 (ideal case) to $500\,\text{Nm}/\text{rad}$. Analysis of the plant dynamics reveals two significant effects. @@ -7707,7 +7678,7 @@ For the force sensor plant, bending stiffness introduces complex conjugate zeros This behavior resembles having parallel stiffness to the force sensor as was the case with the APA300ML (see Figure\nbsp{}ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant). However, this time the parallel stiffness does not comes from the considered strut, but from the bending stiffness of the flexible joints of the other five struts. This characteristic impacts the achievable damping using decentralized Integral Force Feedback\nbsp{}[[cite:&preumont07_six_axis_singl_stage_activ]]. -This is confirmed by the Root Locus plot in Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof. +This is confirmed by the root locus plot in Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_1dof. This effect becomes less significant when using the selected APA300ML actuators (Figure\nbsp{}ref:fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml), which already incorporate parallel stiffness by design which is higher than the one induced by flexible joint stiffness. A parallel analysis of torsional stiffness revealed similar effects, though these proved less critical for system performance. @@ -7731,10 +7702,10 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes #+end_figure #+name: fig:detail_fem_joints_bending_stiffness_iff_locus -#+caption: Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. When having an actuator modelled as 1DoF without parallel stiffness to the force sensor (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2DoF model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}). +#+caption: Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. For 1-DoF actuators (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2-DoFs model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1DoF actuators} +#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1-DoF actuators} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -7793,7 +7764,7 @@ Based on this analysis, an axial stiffness specification of $100\,\text{N}/\upmu #+caption: Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}). #+attr_latex: :options [h!tbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root Locus} +#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root locus} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -7861,7 +7832,7 @@ This high-fidelity representation was created by defining two interface frames ( The computed transfer functions from actuator forces to both force sensor measurements ($\bm{f}$ to $\bm{f}_m$) and external metrology ($\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$) demonstrate dynamics consistent with predictions from earlier analyses (Figure\nbsp{}ref:fig:detail_fem_joints_fem_vs_perfect_plants), thereby validating the joint design. #+name: fig:detail_fem_joints_frames -#+caption: Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames $\{F\}$ and $\{M\}$ both located at the center of the rotation. +#+caption: Defined frames for the reduced order flexible body. The two flat interfaces are considered rigid, and are linked to the two frames $\{F\}$ and $\{M\}$ both located at the center of rotation. [[file:figs/detail_fem_joints_frames.png]] While this detailed modeling approach provides high accuracy, it results in a significant increase in system model order. @@ -7873,7 +7844,7 @@ This simplification reduces the total model order to 48 states: 12 for the paylo While additional acrshortpl:dof could potentially capture more dynamic features, the selected configuration preserves essential system characteristics while minimizing computational complexity. #+name: fig:detail_fem_joints_fem_vs_perfect_plants -#+caption: Comparison of the dynamics obtained between an active platform including joints modelled with FEM and an active platform having bottom joint modelled by bending stiffness $k_f$ and axial stiffness $k_a$ and top joints modelled by bending stiffness $k_f$, torsion stiffness $k_t$ and axial stiffness $k_a$. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}). +#+caption: Comparison of the dynamics obtained between an active platform including joints modeled with FEM and an active platform having 2-DoFs bottom joints and 3-DoFs top joints. Both from actuator force $\bm{f}$ to strut motion measured by external metrology $\bm{\epsilon}_{\mathcal{L}}$ (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}) and to the force sensors $\bm{f}_m$ (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$} @@ -7941,7 +7912,7 @@ In cases where multiple control objectives must be achieved simultaneously, as i From the literature, three principal approaches for combining sensors have been identified: acrlong:haclac, sensor fusion, and two-sensor control architectures. #+name: fig:detail_control_control_multiple_sensors -#+caption: Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}). +#+caption: Different control architectures combining multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}), Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}) and Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC-LAC} @@ -8078,7 +8049,7 @@ The sensor dynamics estimate $\hat{G}_i(s)$ may be a simple gain or a more compl #+caption: Sensor models with and without normalization. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model}Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model}Model with noise $n_i$ and acrshort:lti transfer function $G_i(s)$} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 1 @@ -8177,7 +8148,7 @@ The dynamical uncertainty of the super sensor can be graphically represented in \end{equation} #+name: fig:detail_control_sensor_uncertainty -#+caption: Sensor fusion architecture with sensor dynamics uncertainty (\subref{fig:detail_control_sensor_fusion_dynamic_uncertainty}). Uncertainty region (\subref{fig:detail_control_sensor_uncertainty_set_super_sensor}) of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted. +#+caption: Sensor fusion architecture with sensor dynamics uncertainty (\subref{fig:detail_control_sensor_fusion_dynamic_uncertainty}). Uncertainty region (\subref{fig:detail_control_sensor_uncertainty_set_super_sensor}) of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The uncertainty region is function of frequency, which is omitted. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_dynamic_uncertainty}Sensor Fusion Architecture} @@ -8186,7 +8157,7 @@ The dynamical uncertainty of the super sensor can be graphically represented in #+attr_latex: :width 0.95\linewidth [[file:figs/detail_control_sensor_fusion_dynamic_uncertainty.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_uncertainty_set_super_sensor}Uncertainty regions} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_uncertainty_set_super_sensor}Uncertainty region} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth @@ -8194,7 +8165,7 @@ The dynamical uncertainty of the super sensor can be graphically represented in #+end_subfigure #+end_figure -The super sensor dynamical uncertainty, and consequently the robustness of the fusion, clearly depends on the complementary filters' norm. +The super sensor dynamical uncertainty, and consequently the robustness of the fusion clearly depends on the complementary filters' norm. As it is generally desired to limit the dynamical uncertainty of the super sensor, the norm of the complementary filter $|H_i(j\omega)|$ should be made small when $|w_i(j\omega)|$ is large, i.e., at frequencies where the sensor dynamics is uncertain. **** Complementary Filters Shaping @@ -8285,7 +8256,7 @@ The typical magnitude response of a weighting function generated using\nbsp{}eqr #+attr_latex: :options []{0.45\linewidth} #+begin_minipage #+name: fig:detail_control_sensor_weight_formula -#+caption: Magnitude of a weighting function generated using\nbsp{}eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$. +#+caption: Magnitude of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$. #+attr_latex: :scale 0.8 :float nil [[file:figs/detail_control_sensor_weight_formula.png]] #+end_minipage @@ -8318,7 +8289,7 @@ The inverse magnitudes of the designed weighting functions, which represent the #+begin_minipage #+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align ccc #+attr_latex: :booktabs t :float nil :font \footnotesize\sf -| Parameter | $W_1(s)$ | $W_2(s)$ | +| | $W_1(s)$ | $W_2(s)$ | |--------------+------------------+------------------| | $G_0$ | $0.1$ | $1000$ | | $G_{\infty}$ | $1000$ | $0.1$ | @@ -8356,7 +8327,7 @@ Previous literature has offered only simple analytical formulas for this purpose This section presents a generalization of the proposed complementary filter synthesis method to address this gap. #+name: fig:detail_control_sensor_fusion_three -#+caption: Possible sensor fusion architecture when more than two sensors are to be merged. +#+caption: Sensor fusion architectures when more than two sensors are to be merged. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_three_sequential}Sequential fusion} @@ -8496,7 +8467,7 @@ The decentralized plant (transfer functions from actuators to sensors integrated Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_jacobian), modal decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_modal), and Singular Value Decomposition (SVD) decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_svd). Finally, a comparative analysis with concluding observations is provided in Section\nbsp{}ref:ssec:detail_control_decoupling_comp. -**** 3-DoF Test Model +**** 3-DoFs Test Model <> Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate the analysis. @@ -8628,7 +8599,7 @@ The resulting plant (Figure\nbsp{}ref:fig:detail_control_jacobian_decoupling_arc - $\bm{\mathcal{X}}_{\{O\}}$ represents translations/rotation of the payload expressed in frame $\{O\}$ #+name: fig:detail_control_jacobian_decoupling_arch -#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{O\}}$ to $\bm{\mathcal{X}}_{\{O\}}$. +#+caption: Block diagram of the decoupling the plant in a frame $\{O\}$ using Jacobian matrix $\bm{J}_{\{O\}}$ [[file:figs/detail_control_decoupling_control_jacobian.png]] The transfer function from $\bm{\mathcal{F}}_{\{O\}$ to $\bm{\mathcal{X}}_{\{O\}}$, denoted $\bm{G}_{\{O\}}(s)$ can be computed using\nbsp{}eqref:eq:detail_control_decoupling_plant_jacobian. @@ -8820,7 +8791,7 @@ The two computed matrices were implemented in the control architecture of Figure Each of these diagonal elements corresponds to a specific mode, as shown in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test_modal, resulting in a perfectly decoupled system. #+name: fig:detail_control_decoupling_modal_plant_modes -#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal}). +#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}). Decoupled elements can be used to invidiually address the modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space} @@ -8877,7 +8848,7 @@ This information can be obtained either experimentally or derived from a model. While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency. Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping. -***** Test on the 3-DoF model +***** Test on the 3-DoFs model Plant decoupling using the Singular Value Decomposition was then applied on the test model. A decoupling frequency of $\SI{100}{Hz}$ was used. @@ -8922,7 +8893,7 @@ Although Jacobian matrices could theoretically be used to map these sensors to t Notably, the coupling demonstrates local minima near the decoupling frequency, consistent with the fact that the decoupling matrices were derived specifically for that frequency point. #+name: fig:detail_control_svd_decoupling_not_symmetrical -#+caption: Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}). +#+caption: SVD decoupling applied on the system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_alt}Alternative location of sensors} @@ -8989,7 +8960,7 @@ SVD decoupling can be implemented using measured data without requiring a model, |-----------------------+-----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------| | *Pros* | Retain physical meaning of inputs / outputs. Controller acts on a meaningfully "frame" | Ability to target specific modes. Simple $2^{nd}$ order diagonal plants | Good Decoupling near the crossover. Very General and requires no model | |-----------------------+-----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------| -| *Cons* | Good decoupling at all frequency can only be obtained for specific mechanical architecture | Relies on the accuracy of equation of motions. Robustness to unmodelled dynamics may be poor | Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor | +| *Cons* | Good decoupling at all frequency can only be obtained for specific mechanical architecture | Relies on the accuracy of equation of motions. Robustness to unmodeled dynamics may be poor | Loss of physical meaning of inputs /outputs. Decoupling away from the chosen frequency may be poor | *** Closed-Loop Shaping using Complementary Filters <> @@ -9030,7 +9001,7 @@ The corresponding control architecture is illustrated in Figure\nbsp{}ref:fig:de In this arrangement, the physical plant is controlled at low frequencies, while the plant model is used at high frequencies to enhance robustness. #+name: fig:detail_control_cf_arch_and_eq -#+caption: Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}). An equivalent architecture is shown in (\subref{fig:detail_control_cf_arch_eq}). The signals are the reference signal $r$, the output perturbation $d_y$, the measurement noise $n$ and the control input $u$. +#+caption: Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}) and equivalent architecture (\subref{fig:detail_control_cf_arch_eq}). Signals are the reference input $r$, the output perturbation $d_y$, the measurement noise $n$ and the control input $u$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_arch}Virtual Sensor Fusion} @@ -9144,7 +9115,7 @@ The set of possible plants $\Pi_i$ is described by\nbsp{}eqref:eq:detail_control \end{equation} #+name: fig:detail_control_cf_input_uncertainty_nyquist -#+caption: Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty}). +#+caption: Input multiplicative uncertainty used to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is illustrated on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_input_uncertainty}Input multiplicative uncertainty} @@ -9287,7 +9258,7 @@ These uncertainties are represented using a multiplicative input uncertainty wei Figure\nbsp{}ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the nominal plant dynamics and the complete set of possible plants $\Pi_i$ encompassed by the uncertainty model. #+name: fig:detail_control_cf_test_model_plant -#+caption: Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function $G(s)$ from $F$ to $y$ and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}). +#+caption: Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of $G(s) = y/F$ and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_test_model}Test model} @@ -9323,7 +9294,7 @@ The second-order complementary filters from Equation\nbsp{}eqref:eq:detail_contr There magnitudes are displayed in Figure\nbsp{}ref:fig:detail_control_cf_specs_S_T, confirming that these complementary filters are fulfilling the specifications. #+name: fig:detail_control_cf_specs_S_T_obtained_filters -#+caption: Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}). +#+caption: Performance requirements are compared with the complementary filters in (\subref{fig:detail_control_cf_specs_S_T}). The bode plot of the obtained controller is shown in (\subref{fig:detail_control_cf_bode_Kfb}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters} @@ -9364,16 +9335,16 @@ Performance is evaluated by examining the closed-loop sensitivity and complement It is shown that the sensitivity transfer function achieves the desired $+2$ slope at low frequencies and that the complementary sensitivity transfer function nominally provides the wanted noise filtering. #+name: fig:detail_control_cf_simulation_results -#+caption: Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf}). +#+caption: Validation of robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust stability} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/detail_control_cf_nyquist_robustness.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and Robust performance} +#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and robust performance} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -9428,7 +9399,7 @@ Figure\nbsp{}ref:fig:detail_instrumentation_plant illustrates the control diagra The selection process follows a three-stage methodology. First, dynamic error budgeting is performed in Section\nbsp{}ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (acrshort:adc, acrshort:dac, and voltage amplifier). -This analysis is based on the multi-body model with a 2DoF acrshort:apa model, focusing particularly on the vertical direction due to its more stringent requirements. +This analysis is based on the multi-body model with a 2-DoFs acrshort:apa model, focusing particularly on the vertical direction due to its more stringent requirements. From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived. Section\nbsp{}ref:sec:detail_instrumentation_choice then presents the selection of appropriate components based on these noise specifications and additional requirements. @@ -9449,7 +9420,7 @@ The measured noise characteristics are then incorporated into the multi-body mod The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (acrshort:adc, acrshort:dac, and voltage amplifier) that would result in acceptable vibration levels in the system. The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section\nbsp{}ref:ssec:detail_instrumentation_cl_sensitivity). -This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs. +This analysis is conducted using the multi-body model with a 2-DoFs Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs. Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints. From these transfer functions, the maximum acceptable acrfull:asd of the noise sources is derived (Section\nbsp{}ref:ssec:detail_instrumentation_max_noise_specs). @@ -9466,7 +9437,7 @@ Encoder noise, which is only used to estimate $R_z$, has been found to have mini The transfer functions from these three noise sources (for one strut) to the vertical error of the sample are estimated from the multi-body model, which includes the APA300ML and the designed flexible joints (Figure\nbsp{}ref:fig:detail_instrumentation_noise_sensitivities). #+name: fig:detail_instrumentation_noise_sensitivities -#+caption: Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy. +#+caption: Transfer function from noise sources to vertical error, in closed-loop with the implemented HAC-LAC strategy. #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_noise_sensitivities.png]] @@ -9515,7 +9486,7 @@ When combined with the piezoelectric load (represented as a capacitance $C_p$), \end{equation} #+name: fig:detail_instrumentation_amp_output_impedance -#+caption: Electrical model of a voltage amplifier with output impedance $R_0$ connected to a piezoelectric stack with capacitance $C_p$. +#+caption: Electrical model of an amplifier with output impedance $R_0$ connected to a piezoelectric stack with capacitance $C_p$. [[file:figs/detail_instrumentation_amp_output_impedance.png]] Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases. @@ -9563,7 +9534,7 @@ Note that for the WMA-200, the manufacturer proposed to remove the $50\,\Omega$ The PD200 from PiezoDrive was ultimately selected because it meets all the requirements and is accompanied by clear documentation, particularly regarding noise characteristics and bandwidth specifications. #+name: tab:detail_instrumentation_amp_choice -#+caption: Specifications for the Voltage amplifier and considered commercial products. +#+caption: Specifications for the voltage amplifier and considered commercial products. #+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcccc #+attr_latex: :center t :booktabs t :float t | *Specifications* | PD200 | WMA-200 | LA75B | E-505 | @@ -9604,7 +9575,7 @@ Sigma-Delta acrshortpl:adc can provide excellent noise characteristics, high ban Typically, the latency can reach 20 times the sampling period\nbsp{}[[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 8.4]]. Consequently, while Sigma-Delta acrshortpl:adc are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor. -For real-time control applications, acrfull:sar remain the predominant choice due to their single-sample latency characteristics. +For real-time control applications, successive-approximation ADC remain the predominant choice due to their single-sample latency characteristics. ***** ADC Noise @@ -9698,7 +9669,7 @@ From an implementation perspective, capacitive and eddy current sensors offer a In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the acrshort:apa, as shown in Figure\nbsp{}ref:fig:detail_instrumentation_encoder_implementation. #+name: fig:detail_instrumentation_sensor_implementation -#+caption: Implementation of relative displacement sensor to measure the motion of the APA. +#+caption: Implementation of relative displacement sensors to measure the motion of the APA. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_encoder_implementation}Optical Encoder} @@ -9762,7 +9733,7 @@ The setup is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_se The voltage amplifier employed in this setup has a gain of 20. #+name: fig:detail_instrumentation_force_sensor_adc_setup -#+caption: Schematic of the setup to validate the use of the ADC for reading the force sensor volage. +#+caption: Schematic of the setup to validate the use of the ADC for reading the force sensor voltage. [[file:figs/detail_instrumentation_force_sensor_adc_setup.png]] Step signals with an amplitude of $1\,\text{V}$ were generated using the acrshort:dac, and the acrshort:adc signal was recorded. @@ -9779,7 +9750,7 @@ An exponential curve was fitted to the experimental data, yielding a time consta With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\upmu\text{F}$, the internal impedance of the Speedgoat acrshort:adc was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet. #+name: fig:detail_instrumentation_force_sensor -#+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from\nbsp{}[[cite:&reza06_piezoel_trans_vibrat_contr_dampin]]. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}). +#+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from [[cite:&reza06_piezoel_trans_vibrat_contr_dampin]]. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_force_sensor_adc}Electrical Schematic} @@ -9879,7 +9850,7 @@ The resulting acrshort:frf from the digital acrshort:dac signal to the digital a The observed acrshort:frf corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer. #+name: fig:detail_instrumentation_dac -#+caption: Measurement of the output voltage noise of the ADC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a "1-sample" delay. +#+caption: Measurement of the output voltage noise of the DAC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a "1-sample" delay. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_dac_output_noise}Output noise of the DAC} @@ -9904,7 +9875,7 @@ The pre-amplifier gain was increased to produce a signal substantially larger th Two piezoelectric stacks from the APA95ML were connected to the PD200 output to provide an appropriate load for the amplifier. #+name: fig:detail_instrumentation_pd200_setup -#+caption: Setup used to measured the output voltage noise of the PD200 voltage amplifier. A gain $G_a = 1000$ was used for the instrumentation amplifier. +#+caption: Setup used to measure the output voltage noise of the PD200 voltage amplifier. A gain $G_a = 1000$ was used for the instrumentation amplifier. [[file:figs/detail_instrumentation_pd200_setup.png]] The Amplitude Spectral Density $\Gamma_{n}(\omega)$ of the signal measured by the acrshort:adc was computed. @@ -9957,7 +9928,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a #+attr_latex: :options [b]{0.48\linewidth} #+begin_minipage #+name: fig:detail_instrumentation_vionic_bench -#+caption: Test bench used to measured the encoder noise. +#+caption: Test bench used to measure the encoder noise. #+attr_latex: :width 0.95\linewidth :float nil [[file:figs/detail_instrumentation_vionic_bench.jpg]] #+end_minipage @@ -9972,7 +9943,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a **** Noise Budgeting from Measured Instrumentation Noise -After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier. +After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model. The vertical motion induced by the noise sources, specifically the acrshort:adc noise, acrshort:dac noise, and voltage amplifier noise, is presented in Figure\nbsp{}ref:fig:detail_instrumentation_cl_noise_budget. The total motion induced by all noise sources combined is approximately $1.5\,\text{nm RMS}$, which remains well within the specified limit of $15\,\text{nm RMS}$. This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application. @@ -9996,7 +9967,7 @@ Based on these specifications, appropriate instrumentation components were selec The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions. Despite these challenges, suitable components were identified that theoretically met all requirements. -The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized. +The selected instrumentation was procured and thoroughly characterized. Initial measurements of the acrshort:adc system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit. All components were found to meet or exceed their respective specifications. The acrshort:adc demonstrated noise levels of $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus the $11\,\upmu\text{V}/\sqrt{\text{Hz}}$ specification), the acrshort:dac showed $0.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ (versus $14\,\upmu\text{V}/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\upmu\text{V}/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification). @@ -10014,11 +9985,11 @@ Several primary objectives guided the mechanical design. First, to ensure a well-defined Jacobian matrix used in the control architecture, accurate positioning of the top flexible joint rotation points and correct orientation of the struts were required. Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of $120\,\text{mm}$ and a height of $95\,\text{mm}$. Thirdly, because performance predicted by the multi-body model was fulfilling the requirements, the final design was intended to approximate the behavior of this "idealized" active platform as closely as possible. -This objective implies that the frequencies of (un-modelled) flexible modes potentially detrimental to control performance needed to be maximized. +This objective implies that the frequencies of (un-modeled) flexible modes potentially detrimental to control performance needed to be maximized. Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure. #+name: fig:detail_design_nano_hexapod_elements -#+caption: Obtained mechanical design of the Active platform, called the "nano-hexapod". +#+caption: Obtained mechanical design of the active platform, called the "nano-hexapod". #+attr_latex: :width 0.95\linewidth [[file:figs/detail_design_nano_hexapod_elements.png]] @@ -10032,10 +10003,10 @@ Due to the limited angular stroke of the flexible joints, it was critical that t To facilitate this alignment, cylindrical washers (Figure\nbsp{}ref:fig:detail_design_strut_without_enc) were integrated into the design to compensate for potential deviations from perfect flatness between the two acrshort:apa interface planes (Figure\nbsp{}ref:fig:detail_design_apa). Furthermore, a dedicated mounting bench was developed to enable precise alignment of each strut, even when accounting for typical machining inaccuracies. The mounting procedure is described in Section\nbsp{}ref:sec:test_struts_mounting. -Lastly, the design needed to permit the fixation of an encoder parallel to the strut axis, as shown in Figure\nbsp{}ref:fig:detail_design_strut_with_enc. +Lastly, the design needed to permit the mounting of an encoder parallel to the strut axis, as shown in Figure\nbsp{}ref:fig:detail_design_strut_with_enc. #+name: fig:detail_design_strut -#+caption: Design of the Nano-Hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration. +#+caption: Design of the nano-hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_design_strut_without_enc}Before encoder integration} @@ -10095,7 +10066,7 @@ To maximize the natural frequencies associated with plate flexibility, a network Although topology optimization methods were considered, the implemented ribbed design was found to provide sufficiently high natural frequencies for the flexible modes. #+name: fig:detail_design_top_plate -#+caption: The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is detailed later. +#+caption: The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is later detailed. #+attr_latex: :scale 1 [[file:figs/detail_design_top_plate.png]] @@ -10105,7 +10076,7 @@ These grooves consequently serve to define the nominal orientation of the struts High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the nano-hexapod is assembled. #+name: fig:detail_design_fixation_flexible_joints_platform -#+caption: Fixation of the flexible points to the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joint is in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joint is just oriented (\subref{fig:detail_design_location_bot_flex}). +#+caption: Clamping of the flexible points on the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joints are in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joints are just oriented (\subref{fig:detail_design_location_bot_flex}). #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_design_fixation_flexible_joints}Flexible Joint Clamping} #+attr_latex: :options {0.33\textwidth} @@ -10147,7 +10118,7 @@ The extent to which these modes might be detrimental is difficult to establish a Finally, the FEA indicated that flexible modes of the top plate itself begin to appear at frequencies above $650\,\text{Hz}$, with the first such mode shown in Figure\nbsp{}ref:fig:detail_design_fem_plate_mode. #+name: fig:detail_design_fem_nano_hexapod -#+caption: Measurement of strut flexible modes. First six modes are "suspension" modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from $205\,\text{Hz}$ to $420\,\text{Hz}$ (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at $650\,\text{Hz}$ (\subref{fig:detail_design_fem_plate_mode}). +#+caption: Finite Element Model of the nano-hexapod. The first six modes are "suspension" modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from $205\,\text{Hz}$ to $420\,\text{Hz}$ (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at $650\,\text{Hz}$ (\subref{fig:detail_design_fem_plate_mode}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension mode} @@ -10172,11 +10143,11 @@ Finally, the FEA indicated that flexible modes of the top plate itself begin to ***** Alternative Encoder Placement -In anticipation of potential issues arising from the local modes of the struts affecting encoder measurements, an alternative fixation strategy for the encoders was designed. +In anticipation of potential issues arising from the local modes of the struts affecting encoder measurements, an alternative mounting strategy for the encoders was designed. In this configuration, the encoders are fixed directly to the top and bottom plates instead of the struts, as illustrated in Figure\nbsp{}ref:fig:detail_design_enc_plates_design. #+name: fig:detail_design_enc_plates_design -#+caption: Alternative way of using the encoders: they are fixed directly to the plates. +#+caption: Alternative location of the encoders: fixed to the plates. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_design_enc_plates}Nano-Hexapod with encoders fixed to the plates} @@ -10185,7 +10156,7 @@ In this configuration, the encoders are fixed directly to the top and bottom pla #+attr_latex: :height 5cm [[file:figs/detail_design_enc_plates.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:detail_design_encoders_plates}Zoom on encoder fixation} +#+attr_latex: :caption \subcaption{\label{fig:detail_design_encoders_plates}Zoom on encoder mounting} #+attr_latex: :options {0.39\textwidth} #+begin_subfigure #+attr_latex: :height 5cm @@ -10229,18 +10200,18 @@ In these models, the top and bottom plates were represented as rigid bodies, wit Several levels of detail were considered for modeling the flexible joints within the multi-body model. Models with two acrshortpl:dof incorporating only bending stiffnesses, models with three acrshortpl:dof adding torsional stiffness, and models with four acrshortpl:dof further adding axial stiffness were evaluated. -The multi-body representation corresponding to the 4DoF configuration is shown in Figure\nbsp{}ref:fig:detail_design_simscape_model_flexible_joint. +The multi-body representation corresponding to the 4-DoFs configuration is shown in Figure\nbsp{}ref:fig:detail_design_simscape_model_flexible_joint. This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from acrshort:fea of the joint component. #+name: fig:detail_design_simscape_model_flexible_joint -#+caption: 4DoF multi-body model of the flexible joints. +#+caption: 4-DoFs multi-body model of the flexible joints. Axial, bending and torsional stiffnesses are modeled. #+attr_latex: :scale 1 [[file:figs/detail_design_simscape_model_flexible_joint.png]] ***** Amplified Piezoelectric Actuators The acrlongpl:apa were incorporated into the multi-body model following the methodology detailed in Section\nbsp{}ref:sec:detail_fem_actuator. -Two distinct representations of the acrshort:apa can be used within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. +Two distinct representations of the acrshort:apa can be used within the simulation: a simplified 2-DoFs model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a acrshort:fem. ***** Encoders @@ -10257,7 +10228,7 @@ The displacement measured by the encoder corresponds to the relative position of An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure\nbsp{}ref:fig:detail_design_simscape_encoder_disp, can induce a measured displacement. #+name: fig:detail_design_simscape_encoder_model -#+caption: Representation of the encoder model in the multi-body model. Measurement $d_i$ corresponds to the $x$ position of the encoder frame $\{E\}$ expresssed in the ruller frame $\{R\}$ (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}). +#+caption: Representation of the encoder multi-body model. Measurement $d_i$ corresponds to the $x$ position of the encoder frame $\{E\}$ expresssed in the ruller frame $\{R\}$ (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_design_simscape_encoder}Aligned encoder and ruler} @@ -10345,7 +10316,7 @@ A short-stroke metrology system is developed to measure the sample position rela The acrshort:haclac control architecture is implemented and tested under various experimental conditions, including payload masses up to $39\,\text{kg}$ and for typical experiments, including tomography scans, reflectivity measurements, and diffraction tomography. #+name: fig:chapter3_overview -#+caption: Overview of the Experimental validation phase. The actuators and flexible joints and individual tested and then integrated into the struts. The Nano-hexapod is then mounted and the complete system is validated on the ID31 beamline. +#+caption: Overview of the experimental validation phase. The actuators and flexible joints and individual tested and then integrated into the struts. The Nano-hexapod is then mounted and the complete system is validated on the ID31 beamline. #+attr_org: :width 800px #+attr_latex: :options [h!tbp] #+attr_latex: :width \linewidth @@ -10373,7 +10344,7 @@ This more complex model also captures well capture the axial dynamics of the APA #+name: fig:test_apa_received #+attr_latex: :width 0.7\linewidth -#+caption: Picture of 5 out of the 7 received APA300ML. +#+caption: 5 of the 7 received APA300ML. [[file:figs/test_apa_received.jpg]] *** Static Measurements @@ -10392,7 +10363,7 @@ Finally, in Section\nbsp{}ref:ssec:test_apa_spurious_resonances, the flexible mo To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness. As shown in Figure\nbsp{}ref:fig:test_apa_flatness_setup, the acrshort:apa is fixed to a clamp while a measuring probe[fn:test_apa_3] is used to measure the height of four points on each of the APA300ML interfaces. -From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting[fn:test_apa_4] a plane through all the points. +From the XYZ coordinates of the measured eight points, the flatness is estimated by best fitting[fn:test_apa_4] a plane through all the points. The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatness_meas, are within the specifications. #+attr_latex: :options [b]{0.48\textwidth} @@ -10468,8 +10439,8 @@ The voltage across the two actuator stacks is varied from $-20\,\text{V}$ to $15 Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure\nbsp{}ref:fig:test_apa_stroke_voltage). #+name: fig:test_apa_stroke_bench -#+caption: Bench to measure the APA stroke. -#+attr_latex: :width 0.6\linewidth +#+caption: Test bench to measure the APA stroke. +#+attr_latex: :width 0.5\linewidth [[file:figs/test_apa_stroke_bench.jpg]] The measured acrshort:apa displacement is shown as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis. @@ -10518,25 +10489,25 @@ The flexible modes for the same condition (i.e. one mechanical interface of the #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_1}Y-bending mode ($268\,\text{Hz}$)} #+attr_latex: :options {0.35\textwidth} #+begin_subfigure -#+attr_latex: :height 4.3cm +#+attr_latex: :height 4cm [[file:figs/test_apa_mode_shapes_1.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_2}X-bending mode ($399\,\text{Hz}$)} #+attr_latex: :options {0.27\textwidth} #+begin_subfigure -#+attr_latex: :height 4.3cm +#+attr_latex: :height 4cm [[file:figs/test_apa_mode_shapes_2.png]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:test_apa_mode_shapes_3}Z-axial mode ($706\,\text{Hz}$)} #+attr_latex: :options {0.35\textwidth} #+begin_subfigure -#+attr_latex: :height 4.3cm +#+attr_latex: :height 4cm [[file:figs/test_apa_mode_shapes_3.png]] #+end_subfigure #+end_figure #+name: fig:test_apa_meas_setup_modes -#+caption: Experimental setup to measure the flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points). +#+caption: Experimental setup to measure the flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the hammer impact point is at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the hammer impact point is located at the back of the top measurement point. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending} @@ -10567,57 +10538,31 @@ Another explanation is the shape difference between the manufactured APA300ML an *** Dynamical Measurements <> -**** Introduction :ignore: +***** Introduction :ignore: After the measurements on the acrshort:apa were performed in Section\nbsp{}ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML. This test bench, depicted in Figure\nbsp{}ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a $5\,\text{kg}$ granite block that is vertically guided by an air bearing. Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors. An encoder[fn:test_apa_8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the acrshort:apa. #+name: fig:test_bench_apa -#+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML. +#+caption: Test bench used to measure the dynamics of the APA300ML. $u$ is the output DAC voltage, $V_a$ the output amplifier voltage (i.e. voltage applied across the actuator stacks), $d_e$ the measured displacement by the encoder and $V_s$ the measured voltage across the sensor stack. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_bench_picture}Picture of the test bench} #+attr_latex: :options {0.3\textwidth} #+begin_subfigure -#+attr_latex: :height 8cm +#+attr_latex: :height 7cm [[file:figs/test_apa_bench_picture.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_apa_bench_picture_encoder}Zoom on the APA with the encoder} +#+attr_latex: :caption \subcaption{\label{fig:test_apa_schematic}Schematic of the test bench} #+attr_latex: :options {0.69\textwidth} #+begin_subfigure -#+attr_latex: :height 8cm -[[file:figs/test_apa_bench_picture_encoder.jpg]] +#+attr_latex: :height 7cm +[[file:figs/test_apa_schematic.png]] #+end_subfigure #+end_figure -The bench is schematically shown in Figure\nbsp{}ref:fig:test_apa_schematic with the associated signals. -It will be first used to estimate the hysteresis from the piezoelectric stack (Section\nbsp{}ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section\nbsp{}ref:ssec:test_apa_stiffness). -The acrshortpl:frf from the acrshort:dac voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section\nbsp{}ref:ssec:test_apa_meas_dynamics. -The presence of a non-minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section\nbsp{}ref:ssec:test_apa_non_minimum_phase. -To limit the low-frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section\nbsp{}ref:ssec:test_apa_resistance_sensor_stack). -Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section\nbsp{}ref:ssec:test_apa_iff_locus. - -#+name: fig:test_apa_schematic -#+caption: Schematic of the Test Bench used to measure the dynamics of the APA300ML. $u$ is the output DAC voltage, $V_a$ the output amplifier voltage (i.e. voltage applied across the actuator stacks), $d_e$ the measured displacement by the encoder and $V_s$ the measured voltage across the sensor stack. -#+attr_latex: :scale 1 -[[file:figs/test_apa_schematic.png]] - -**** Hysteresis -<> - -Because the payload is vertically guided without friction, the hysteresis of the acrshort:apa can be estimated from the motion of the payload. -A quasi static[fn:test_apa_9] sinusoidal excitation $V_a$ with an offset of $65\,\text{V}$ (halfway between $-20\,\text{V}$ and $150\,\text{V}$) and with an amplitude varying from $4\,\text{V}$ up to $80\,\text{V}$ is generated using the acrshort:dac. -For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_meas_hysteresis. -This is the typical behavior expected from a acrfull:pzt stack actuator, where the hysteresis increases as a function of the applied voltage amplitude\nbsp{}[[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]]. - -#+name: fig:test_apa_meas_hysteresis -#+caption: Displacement as a function of applied voltage for multiple excitation amplitudes. -#+attr_latex: :scale 0.8 -[[file:figs/test_apa_meas_hysteresis.png]] - -**** Axial stiffness -<> +***** Axial stiffness To estimate the stiffness of the acrshort:apa, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $\Delta d_e$ is measured using the encoder. The acrshort:apa stiffness can then be estimated from equation\nbsp{}eqref:eq:test_apa_stiffness, with $g \approx 9.8\,\text{m}/\text{s}^2$ the acceleration of gravity. @@ -10636,7 +10581,7 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu #+attr_latex: :options [b]{0.57\textwidth} #+begin_minipage #+name: fig:test_apa_meas_stiffness_time -#+caption: Measured displacement when adding (at $t \approx 3\,\text{s}$) and removing (at $t \approx 13\,\text{s}$) the mass. +#+caption: Displacement when adding and removing the payload. #+attr_latex: :scale 0.8 :float nil [[file:figs/test_apa_meas_stiffness_time.png]] #+end_minipage @@ -10657,7 +10602,7 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu #+latex: \captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\upmu\text{m}$} #+end_minipage -The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section\nbsp{}ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. +The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated from the dynamical measurements shown in Figure\nbsp{}ref:fig:test_apa_frf_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$. \begin{equation} \label{eq:test_apa_res_freq} \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} @@ -10673,8 +10618,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the " The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,\text{N}/\upmu\text{m}$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,\text{N}/\upmu\text{m}$. -**** Dynamics -<> +***** Dynamics In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified. @@ -10691,9 +10635,9 @@ The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for A lightly damped resonance (pole) is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance (zero) at $41\,\text{Hz}$. No additional resonances are present up to at least $2\,\text{kHz}$ indicating that Integral Force Feedback can be applied without stability issues from high-frequency flexible modes. The zero at $41\,\text{Hz}$ seems to be non-minimum phase (the phase /decreases/ by 180 degrees whereas it should have /increased/ by 180 degrees for a minimum phase zero). -This is investigated in Section\nbsp{}ref:ssec:test_apa_non_minimum_phase. +This is investigated further investigated. -As illustrated by the Root Locus plot, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases. +As illustrated by the root locus plot, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases. The significance of this behavior varies with the type of sensor used, as explained in\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]]. Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell. @@ -10719,8 +10663,7 @@ All the identified dynamics of the six APA300ML (both when looking at the encode #+end_subfigure #+end_figure -**** Non Minimum Phase Zero? -<> +***** Non Minimum Phase Zero? It was surprising to observe a non-minimum phase zero on the transfer function from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_frf_force). It was initially thought that this non-minimum phase behavior was an artifact arising from the measurement. @@ -10730,10 +10673,10 @@ The coherence (Figure\nbsp{}ref:fig:test_apa_non_minimum_phase_coherence) is goo Such non-minimum phase zero when using load cells has also been observed on other mechanical systems\nbsp{}[[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear. -However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure\nbsp{}ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice. +However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the root locus plot in Figure\nbsp{}ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice. #+name: fig:test_apa_non_minimum_phase -#+caption: Measurement of the anti-resonance found in the transfer function from $u$ to $V_s$. The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior. +#+caption: Measurement of the anti-resonance found in the transfer function from $u$ to $V_s$. The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shows a non-minimum phase behavior. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence} @@ -10750,8 +10693,7 @@ However, this is not so important here because the zero is lightly damped (i.e. #+end_subfigure #+end_figure -**** Effect of the resistor on the IFF Plant -<> +***** Effect of the resistor on the IFF Plant A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\upmu\text{F}$). @@ -10765,8 +10707,7 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil #+attr_latex: :scale 0.8 [[file:figs/test_apa_effect_resistance.png]] -**** Integral Force Feedback -<> +***** Integral Force Feedback To implement the Integral Force Feedback strategy, the measured acrshort:frf from $u$ to $V_s$ (Figure\nbsp{}ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation\nbsp{}eqref:eq:test_apa_iff_manual_fit. The parameters were manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$. @@ -10806,7 +10747,7 @@ Second using the fitted transfer functions of the damped plants experimentally i The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were fitted using only a second-order transfer function. #+name: fig:test_apa_iff -#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}. +#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and root locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants that match the experimental data (dashed lines)} @@ -10815,7 +10756,7 @@ The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_roo #+attr_latex: :scale 0.8 [[file:figs/test_apa_identified_damped_plants.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_apa_iff_root_locus}Root Locus plot using the plant model (black) and poles of the identified damped plants (color crosses)} +#+attr_latex: :caption \subcaption{\label{fig:test_apa_iff_root_locus}Root locus plot using the plant model (black) and poles of the identified damped plants (color crosses)} #+attr_latex: :options {0.39\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -10854,7 +10795,7 @@ Such a simple model has some limitations: - the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear #+name: fig:test_apa_2dof_model -#+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from\nbsp{}[[cite:&souleille18_concep_activ_mount_space_applic]]. +#+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from [[cite:&souleille18_concep_activ_mount_space_applic]]. [[file:figs/test_apa_2dof_model.png]] ***** Tuning of the APA model :ignore: @@ -10868,7 +10809,7 @@ Such a simple model has some limitations: First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$. -Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,\text{N}/\upmu\text{m}$ in Section\nbsp{}ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. +Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,\text{N}/\upmu\text{m}$ in Section\nbsp{}ref:sec:test_apa_dynamics from the frequency of the anti-resonance seen on Figure\nbsp{}ref:fig:test_apa_frf_force. Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $5\,\text{Ns/m}$. Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:test_apa_5]. @@ -10893,7 +10834,7 @@ In the last step, $g_s$ and $g_a$ can be tuned to match the gain of the identifi The obtained parameters of the model shown in Figure\nbsp{}ref:fig:test_apa_2dof_model_simscape are summarized in Table\nbsp{}ref:tab:test_apa_2dof_parameters. #+name: tab:test_apa_2dof_parameters -#+caption: Summary of the obtained parameters for the 2 DoF APA300ML model. +#+caption: Summary of the obtained parameters for the 2-DoFs APA300ML model. #+attr_latex: :environment tabularx :width 0.25\linewidth :align cc #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | @@ -10916,7 +10857,7 @@ A good match can be observed between the model and the experimental data, both f This indicates that this model represents well the axial dynamics of the APA300ML. #+name: fig:test_apa_2dof_comp_frf -#+caption: Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force}). +#+caption: Comparison of the measured frequency response functions and the identified dynamics from the 2-DoFs model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_apa_2dof_comp_frf_enc}from $u$ to $d_e$} @@ -10945,7 +10886,7 @@ Several /remote points/ are defined in the acrshort:fem (here illustrated by col For the APA300ML /super element/, 5 /remote points/ are defined. Two /remote points/ (=1= and =2=) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used to connect the APA300ML with other mechanical elements. Two /remote points/ (=3= and =4=) are located across two piezoelectric stacks and are used to apply internal forces representing the actuator stacks. -Finally, two /remote points/ (=4= and =5=) are located across the third piezoelectric stack, and will be used to measured the strain of the sensor stack. +Finally, two /remote points/ (=4= and =5=) are located across the third piezoelectric stack, and will be used to measure the strain of the sensor stack. #+name: fig:test_apa_super_element_simscape #+attr_latex: :width 1.0\linewidth @@ -10976,7 +10917,7 @@ The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are list From these parameters, $g_s = 5.1\,\text{V}/\upmu\text{m}$ and $g_a = 26\,\text{N/V}$ were obtained, which are close to the constants identified using the experimentally identified transfer functions. #+name: tab:test_apa_piezo_properties -#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities. +#+caption: Piezoelectric properties used for the estimation of the sensor and actuator sensitivities. #+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX #+attr_latex: :center t :booktabs t | *Parameter* | *Value* | *Description* | @@ -11128,14 +11069,14 @@ Results are shown in Section\nbsp{}ref:sec:test_joints_bending_stiffness_meas Two dimensions are critical for the bending stiffness of the flexible joints. These dimensions can be measured using a profilometer. -The dimensions of the flexible joint in the Y-Z plane will contribute to the X-bending stiffness, whereas the dimensions in the X-Z plane will contribute to the Y-bending stiffness. +The dimensions of the flexible joint in the YZ plane will contribute to the X-bending stiffness, whereas the dimensions in the X-Z plane will contribute to the Y-bending stiffness. The setup used to measure the dimensions of the "X" flexible beam is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_setup. What is typically observed is shown in Figure\nbsp{}ref:fig:test_joints_profilometer_image. It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\upmu\text{m}$, #+name: fig:test_joints_profilometer -#+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image}). +#+caption: Setup to measure the dimensions of the flexible "neck" corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and an image is obtained with which the "neck" size can be estimated (\subref{fig:test_joints_profilometer_image}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_profilometer_setup}Flexible joint fixed on the profilometer} @@ -11159,11 +11100,10 @@ The "beam thickness" is then estimated as the mean between the gaps measured on A histogram of the measured beam thicknesses is shown in Figure\nbsp{}ref:fig:test_joints_size_hist. The measured thickness is less than the specified value of $250\,\upmu\text{m}$, but this optical method may not be very accurate because the estimated gap can depend on the lighting of the part and of its proper alignment. - However, what is more important than the true value of the thickness is the consistency between all flexible joints. #+name: fig:test_joints_size_hist -#+caption: Histogram for the (16x2) measured beams' thicknesses. +#+caption: Histogram for the measured beams' thicknesses. #+attr_latex: :scale 0.8 [[file:figs/test_joints_size_hist.png]] @@ -11178,13 +11118,13 @@ Using this profilometer allowed to detect flexible joints with manufacturing def #+attr_latex: :caption \subcaption{\label{fig:test_joints_bad_shape}Non-Symmetrical shape} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :height 6cm +#+attr_latex: :height 5cm [[file:figs/test_joints_bad_shape.jpg]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:test_joints_bad_chips}"Chips" stuck in the air gap} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure -#+attr_latex: :height 6cm +#+attr_latex: :height 5cm [[file:figs/test_joints_bad_chips.jpg]] #+end_subfigure #+end_figure @@ -11234,7 +11174,7 @@ One part of the flexible joint is fixed to a rigid frame while a (known) force $ The deflection of the joint $d_x$ is measured using a displacement sensor. #+name: fig:test_joints_bench_working_principle -#+caption: Working principle of the test bench used to estimate the bending stiffness $k_{R_y}$ of the flexible joints by measuring $F_x$, $d_x$ and $h$. +#+caption: Test bench used to estimate the bending stiffness $k_{R_y}$ of the flexible joints by measuring $F_x$, $d_x$ and $h$. [[file:figs/test_joints_bench_working_principle.png]] ***** Required External Applied Force @@ -11346,7 +11286,7 @@ The most important source of error is the estimation error of the distance betwe An overall accuracy of $\approx 5\,\%$ can be expected with this measurement bench, which should be sufficient for an estimation of the bending stiffness of the flexible joints. #+name: tab:test_joints_error_budget -#+caption: Summary of the error budget for estimating the bending stiffness. +#+caption: Summary of the error budget for the estimation of the bending stiffness. #+attr_latex: :environment tabularx :width 0.35\linewidth :align Xc #+attr_latex: :center t :booktabs t | *Effect* | *Error* | @@ -11371,7 +11311,7 @@ Instead of measuring the displacement directly at the tip of the flexible joint To do so, an encoder[fn:test_joints_4] is used, which measures the motion of a ruler. This ruler is fixed to the translation stage in line (i.e. at the same height) with the application point to reduce Abbe errors (see Figure\nbsp{}ref:fig:test_joints_bench_overview). -The flexible joint can be rotated by $90^o$ in order to measure the bending stiffness in the two directions. +The flexible joint can be rotated by $\SI{90}{\degree}$ in order to measure the bending stiffness in the two directions. The obtained design of the measurement bench is shown in Figure\nbsp{}ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure\nbsp{}ref:fig:test_joints_bench_side. #+name: fig:test_joints_bench @@ -11507,7 +11447,7 @@ A histogram of the measured bending stiffnesses is shown in Figure\nbsp{}ref:fig Most of the bending stiffnesses are between $4.6\,\text{Nm/rad}$ and $5.0\,\text{Nm/rad}$. #+name: fig:test_joints_meas_bending_results -#+caption: Result of measured $k_{R_x}$ and $k_{R_y}$ stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}). +#+caption: Measured $k_{R_x}$ and $k_{R_y}$ stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_bending_all_raw_data}Measured torque and angular motion for the flexible joints} @@ -11601,7 +11541,7 @@ The strut length (defined by the distance between the rotation points of the two #+end_figure #+name: fig:test_struts_mounting_base_part -#+caption: Main element of the mounting bench for the struts that ensure good coaxiality of the two flexible joints and correct struts length. +#+caption: Part that ensures good coaxiality of the two flexible joints and correct struts length. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_0}Useful features of the main mounting element} @@ -11623,16 +11563,16 @@ The goal of these "sleeves" is to avoid mechanical stress that could damage the These "sleeves" have one dowel groove (that are fitted to the dowel holes shown in Figure\nbsp{}ref:fig:test_struts_mounting_step_0) that will determine the length of the mounted strut. #+name: fig:test_struts_cylindrical_mounting -#+caption: Preparation of the flexible joints by fixing them in their cylindrical "sleeve". +#+caption: Preparation of the flexible joints by fixing them in their cylindrical "sleeves". #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_cylindrical_mounting_part_top}Cylindral Interface (Top)} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_cylindrical_mounting_part_top}Cylindrical Interface (Top)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :height 4.5cm [[file:figs/test_struts_cylindrical_mounting_part_top.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_cylindrical_mounting_part_bot}Cylindrlcal Interface (Bottom)} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_cylindrical_mounting_part_bot}Cylindrical Interface (Bottom)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :height 4.5cm @@ -11662,13 +11602,13 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle #+caption: Steps for mounting the struts. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_1}Step 1} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_1}Fix the flexible joints} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth [[file:figs/test_struts_mounting_step_1.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_2}Step 2} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_2}Mount the APA with the cylindrical washers} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth @@ -11676,13 +11616,13 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle #+end_subfigure \bigskip -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_3}Step 3} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_3}Mount and align the encoder} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth [[file:figs/test_struts_mounting_step_3.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_4}Step 4} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_mounting_step_4}Obtained mounted strut} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth @@ -11699,7 +11639,7 @@ The two cylindrical interfaces were fixed (boundary conditions), and the first t The mode shapes are displayed in Figure\nbsp{}ref:fig:test_struts_mode_shapes: an "X-bending" mode at $189\,\text{Hz}$, a "Y-bending" mode at $285\,\text{Hz}$ and a "Z-torsion" mode at $400\,\text{Hz}$. #+name: fig:test_struts_mode_shapes -#+caption: Spurious resonances of the struts estimated from a Finite Element Model. +#+caption: Flexible modes of the struts estimated from a Finite Element Model. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode ($189\,\text{Hz}$)} @@ -11731,7 +11671,7 @@ The "Y-bending" mode is measured as shown in Figure\nbsp{}ref:fig:test_struts_me These tests were performed with and without the encoder being fixed to the strut. #+name: fig:test_struts_meas_modes -#+caption: Measurement of strut flexible modes. +#+caption: Measurement of the flexible modes of the struts using a laser vibrometer. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_meas_x_bending}X-bending mode} @@ -11757,7 +11697,7 @@ These tests were performed with and without the encoder being fixed to the strut The obtained acrshortpl:frf for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure\nbsp{}ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut. #+name: fig:test_struts_spur_res_frf -#+caption: Measured frequency response functions without the encoder\nbsp{}ref:fig:test_struts_spur_res_frf and with the encoder\nbsp{}ref:fig:test_struts_spur_res_frf_enc. +#+caption: Measured frequency response functions without the encoder (\subref{fig:test_struts_spur_res_frf_no_enc}) and with the encoder (\subref{fig:test_struts_spur_res_frf_enc}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_spur_res_frf_no_enc}without encoder} @@ -11780,7 +11720,7 @@ In addition, the computed resonance frequencies from the acrshort:fem are very c This validates the quality of the acrshort:fem. #+name: tab:test_struts_spur_mode_freqs -#+caption: Measured frequency of the flexible modes of the strut. +#+caption: Estimated and measured frequencies of the flexible modes of the struts. #+attr_latex: :environment tabularx :width 0.7\linewidth :align Xccc #+attr_latex: :center t :booktabs t :float t | *Mode* | *FEM with Encoder* | *Exp. with Encoder* | *Exp. without Encoder* | @@ -11806,13 +11746,13 @@ A fiber interferometer[fn:test_struts_4] is used to measure the motion of the gr #+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_leg_overview}Overview Picture} #+attr_latex: :options {0.3\textwidth} #+begin_subfigure -#+attr_latex: :height 210px +#+attr_latex: :height 6cm [[file:figs/test_struts_bench_leg_overview.jpg]] #+end_subfigure #+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_schematic}Schematic} #+attr_latex: :options {0.66\textwidth} #+begin_subfigure -#+attr_latex: :height 210px +#+attr_latex: :height 6cm [[file:figs/test_struts_bench_schematic.png]] #+end_subfigure #+end_figure @@ -11827,7 +11767,7 @@ Finally, all measured struts are compared in terms of dynamics in Section\nbsp{} System identification was performed without the encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_front) and with one encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_coder). #+name: fig:test_struts_bench_leg_with_without_enc -#+caption: Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front}). +#+caption: Strut fixed to the test bench with clamped flexible joints. The encoder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder} @@ -11893,22 +11833,22 @@ The obtained dynamics from $u$ to $d_a$ are compared in Figure\nbsp{}ref:fig:tes A very good match can be observed between the struts. #+name: fig:test_struts_comp_plants -#+caption: Comparison of the measured plants. +#+caption: Comparison of the measured dynamics for five of the struts.. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$ (interferometer)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_interf_plants.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$ (force sensor)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_iff_plants.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$ (encoder)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -11935,7 +11875,7 @@ This misalignment is estimated and measured in Section\nbsp{}ref:ssec:test_strut The struts were then disassembled and reassemble a second time to optimize alignment (Section\nbsp{}ref:sec:test_struts_meas_all_aligned_struts). #+name: fig:test_struts_simscape_model -#+caption: Screenshot of the multi-body model of the strut fixed to the bench. +#+caption: Multi-body model of the strut fixed to the bench. #+attr_latex: :width 0.65\linewidth [[file:figs/test_struts_simscape_model.png]] @@ -11944,32 +11884,32 @@ The struts were then disassembled and reassemble a second time to optimize align Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a acrlong:fem. These two models of the APA300ML were tuned to best match the measured acrshortpl:frf of the acrshort:apa alone. -The flexible joints were modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness). +The flexible joints were modeled with the 4-DoFs model (axial stiffness, two bending stiffnesses and one torsion stiffness). These two models are compared using the measured acrshortpl:frf in Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model. The model dynamics from DAC voltage $u$ to the axial motion of the strut $d_a$ (Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model_int) and from DAC voltage $u$ to the force sensor voltage $V_s$ (Figure\nbsp{}ref:fig:test_struts_comp_frf_flexible_model_iff) are well matching the experimental identification. However, the transfer function from $u$ to encoder displacement $d_e$ are not well matching for both models. -For the 2DoF model, this is normal because the resonances affecting the dynamics are not modelled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis). +For the 2-DoFs model, this is normal because the resonances affecting the dynamics are not modeled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis). For the flexible model, it will be shown in the next section that by adding some misalignment between the flexible joints and the APA300ML, this model can better represent the observed dynamics. #+name: fig:test_struts_comp_frf_flexible_model -#+caption: Comparison of the measured frequency response functions, the multi-body model using the 2 DoF APA model, and using the "flexible" APA300ML model (Super-Element extracted from a Finite Element Model). +#+caption: Comparison of the measured dynamics of the struts (black) with dynamics extracted from the multi-body model using the 2-DoFs APA model (blue), and using the reduced order flexible model of the APA300ML model (red). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$ (interferometer)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_frf_flexible_model_int.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$ (encoder)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_frf_flexible_model_enc.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$} +#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$ (force sensor)} #+attr_latex: :options {0.32\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -11986,7 +11926,7 @@ For instance, consider Figure\nbsp{}ref:fig:test_struts_misalign_schematic where In this case, the "x-bending" mode at $200\,\text{Hz}$ (see Figure\nbsp{}ref:fig:test_struts_meas_x_bending) can be expected to have greater impact on the dynamics from the actuator to the encoder. #+name: fig:test_struts_misalign_schematic -#+caption: Mis-alignement between the joints and the APA. +#+caption: Misalignment between the joints and the APA. #+attr_latex: :width 0.8\linewidth [[file:figs/test_struts_misalign_schematic.png]] @@ -12007,7 +11947,7 @@ A comparison of the experimental acrshortpl:frf in Figure\nbsp{}ref:fig:test_str This similarity suggests that the identified differences in dynamics are caused by misalignment. #+name: fig:test_struts_effect_misalignment -#+caption: Effect of a misalignment between the flexible joints and the APA300ML in the $y$ direction (\subref{fig:test_struts_effect_misalignment_y}) and in the $x$ direction (\subref{fig:test_struts_effect_misalignment_x}). +#+caption: Effect of a misalignment between the flexible joints and the APA300ML in the $y$ (\subref{fig:test_struts_effect_misalignment_y}) and in the $x$ direction (\subref{fig:test_struts_effect_misalignment_x}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_misalignment_y}Misalignment along $y$} @@ -12041,7 +11981,7 @@ To check the validity of the measurement, it can be verified that the sum of the Thickness differences for all the struts were found to be between $0.94\,\text{mm}$ and $1.00\,\text{mm}$ which indicate low errors compared to the misalignments found in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment. #+name: tab:test_struts_meas_y_misalignment -#+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $\text{mm}$. +#+caption: Measured $y$ misalignment for each strut. Measurements are in $\text{mm}$. #+attr_latex: :environment tabularx :width 0.2\linewidth :align Xcc #+attr_latex: :center t :booktabs t | *Strut* | *Bot* | *Top* | @@ -12062,7 +12002,7 @@ In the next section, the struts are re-assembled with a "positioning pin" to bet With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure\nbsp{}ref:fig:test_struts_effect_misalignment_y. #+name: fig:test_struts_comp_dy_tuned_model_frf_enc -#+caption: Comparison of the frequency response functions from DAC voltage $u$ to measured displacement $d_e$ by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the $y$ misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the $x$ and $y$ misalignments are tuned. +#+caption: Comparison of the frequency response functions from DAC voltage $u$ to measure displacement $d_e$ by the encoders for three struts. The measured dynamics is shown in blue, the dynamics extracted from the model with the $y$ misalignment estimated from measurements is shown in red, and the dynamics extracted from the model when both the $x$ and $y$ misalignments are tuned is shown in yellow. #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_dy_tuned_model_frf_enc.png]] @@ -12076,7 +12016,7 @@ The alignment is then estimated using a length gauge, as described in the previo Measured $y$ alignments are summarized in Table\nbsp{}ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\upmu\text{m}$ for all the struts, which is much better than before (see Table\nbsp{}ref:tab:test_struts_meas_y_misalignment). #+name: tab:test_struts_meas_y_misalignment_with_pin -#+caption: Measured $y$ misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in $\text{mm}$. +#+caption: Measured $y$ misalignment after realigning the struts using dowel pins. Measurements are in $\text{mm}$. #+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc #+attr_latex: :center t :booktabs t | *Strut* | *Bot* | *Top* | @@ -12160,7 +12100,7 @@ The two plates were then fixed to the mounting tool, as shown in Figure\nbsp{}re The main goal of this "mounting tool" is to position the flexible joint interfaces (the "V" shapes) of both plates so that a cylinder can rest on the 4 flat interfaces at the same time. #+name: fig:test_nhexa_dimensional_check -#+caption: A FARO arm is used to dimensionally check the received parts (\subref{fig:test_nhexa_plates_tolerances}) and after mounting the two plates with the mounting part (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view}). +#+caption: A FARO arm is used to dimensionally check the plates (\subref{fig:test_nhexa_plates_tolerances}) and to verify coaxiality of the strut interfaces (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_nhexa_plates_tolerances}Dimensional check of the bottom plate} @@ -12170,7 +12110,7 @@ The main goal of this "mounting tool" is to position the flexible joint interfac #+attr_latex: :width 0.95\linewidth [[file:figs/test_nhexa_plates_tolerances.jpg]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_mounting_tool_hexapod_top_view}Wanted coaxiality between interfaces} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_mounting_tool_hexapod_top_view}Wanted coaxiality between strut interfaces} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_org: :width 800px @@ -12185,7 +12125,7 @@ This was again done using the FARO arm, and the results for all six struts are s The straightness was found to be better than $15\,\upmu\text{m}$ for all struts[fn:test_nhexa_4], which is sufficiently good to not induce significant stress of the flexible joint during assembly. #+name: tab:measured_straightness -#+caption: Measured straightness between the two "V" shapes for the six struts. These measurements were performed twice for each strut. +#+caption: Measured straightness between the V grooves for the six struts. Measurements were performed twice for each strut. #+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc #+attr_latex: :center t :booktabs t | *Strut* | *Meas 1* | *Meas 2* | @@ -12269,7 +12209,7 @@ Using an instrumented hammer, the first 9 modes could be identified and are summ The first 6 modes are suspension modes (i.e. rigid body mode of the breadboard) and are located below $10\,\text{Hz}$. The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{}ref:fig:test_nhexa_table_flexible_modes, and are located above $700\,\text{Hz}$. -#+attr_latex: :options [t]{0.45\textwidth} +#+attr_latex: :options [b]{0.45\textwidth} #+begin_minipage #+name: fig:test_nhexa_suspended_table #+caption: Mounted suspended table. Only 1 or the 15 accelerometer is mounted on top. @@ -12361,7 +12301,7 @@ The acrshortpl:frf from controlled signals $\bm{u}$ to the force sensors voltage The effect of the payload mass on the dynamics is discussed in Section\nbsp{}ref:ssec:test_nhexa_added_mass. #+name: fig:test_nhexa_nano_hexapod_signals -#+caption: 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. +#+caption: Block diagram of the studied system. The command signal is $\bm{u}$, and the measured signals are $\bm{d}_{e}$ and $\bm{V}_s$. #+attr_latex: :width 0.9\linewidth [[file:figs/test_nhexa_nano_hexapod_signals.png]] @@ -12435,7 +12375,7 @@ Up to at least $1\,\text{kHz}$, an alternating pole/zero pattern is observed, wh This would not have occurred if the encoders were fixed to the struts. #+name: fig:test_nhexa_identified_frf_de -#+caption: 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. +#+caption: Measured FRFs from $\bm{u}$ to $\bm{d}_e$. The 6 direct terms are the colored lines, and the 30 coupling terms are the gray lines. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_identified_frf_de.png]] @@ -12445,7 +12385,7 @@ Similar to what was observed for the encoder outputs, all the "diagonal" terms a The first flexible mode of the struts as $235\,\text{Hz}$ has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF. #+name: fig:test_nhexa_identified_frf_Vs -#+caption: 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. +#+caption: Measured FRF from $\bm{u}$ to $\bm{V}_s$. The 6 direct terms are the colored lines, and the 30 coupling terms are the gray lines. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_identified_frf_Vs.png]] @@ -12482,13 +12422,13 @@ For all tested payloads, the measured acrshort:frf always have alternating poles #+caption: Measured Frequency Response Functions from $u_i$ to $d_{ei}$ (\subref{fig:test_nhexa_identified_frf_de_masses}) and from $u_i$ to $V_{si}$ (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$ (encoder)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_identified_frf_de_masses.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$ (force sensor)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -12505,7 +12445,7 @@ In this section, the dynamics measured in Section\nbsp{}ref:sec:test_nhexa_dynam The nano-hexapod multi-body model was therefore added on top of the vibration table multi-body model, as shown in Figure\nbsp{}ref:fig:test_nhexa_hexa_simscape. #+name: fig:test_nhexa_hexa_simscape -#+caption: 3D representation of the multi-body model with the nano-hexapod on top of the suspended table. Three mass "layers" are here added. +#+caption: Multi-body model of the nano-hexapod on top of the suspended table. Three mass "layers" are here added. #+attr_latex: :width 0.8\linewidth [[file:figs/test_nhexa_hexa_simscape.png]] @@ -12518,7 +12458,7 @@ This is checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_masses. **** Nano-Hexapod Model Dynamics <> -The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF acrshort:apa, and rigid top and bottom plates. +The multi-body model of the nano-hexapod was first configured with 4-DoFs flexible joints, 2-DoFs acrshort: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 acrshort:fem. The parameters of the acrshort:apa model were determined from the test bench of the acrshort:apa. 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. @@ -12526,20 +12466,20 @@ The $6 \times 6$ transfer function matrices from $\bm{u}$ to $\bm{d}_e$ and from First, is it evaluated how well the models matches the "direct" terms of the measured acrshort:frf matrix. To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured acrshort:frf in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag. It can be seen that the 4 suspension modes of the nano-hexapod (at $122\,\text{Hz}$, $143\,\text{Hz}$, $165\,\text{Hz}$ and $191\,\text{Hz}$) are well modeled. -The three resonances that were attributed to "internal" flexible modes of the struts (at $237\,\text{Hz}$, $349\,\text{Hz}$ and $395\,\text{Hz}$) cannot be seen in the model, which is reasonable because the acrshortpl:apa are here modeled as a simple uniaxial 2-DoF system. +The three resonances that were attributed to "internal" flexible modes of the struts (at $237\,\text{Hz}$, $349\,\text{Hz}$ and $395\,\text{Hz}$) cannot be seen in the model, which is reasonable because the acrshortpl:apa are here modeled as a simple uniaxial 2-DoFs system. At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies. #+name: fig:test_nhexa_comp_simscape_diag #+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$ (encoder)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_diag.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$ (force sensor)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -12556,7 +12496,7 @@ It can be seen that the coupling in the model matches the measurements well up t Similar results are observed for all other coupling terms and for the transfer function from $\bm{u}$ to $\bm{V}_s$. #+name: fig:test_nhexa_comp_simscape_de_all -#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a 2-DoF mass-spring-damper system. +#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a 2-DoFs mass-spring-damper system. No payload us used. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all.png]] @@ -12567,7 +12507,7 @@ Even the mode $395\,\text{Hz}$ can be observed in the model. Therefore, if the modes of the struts are to be modeled, the /super-element/ of the APA300ML can be used at the cost of obtaining a much higher order model. #+name: fig:test_nhexa_comp_simscape_de_all_flex -#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a "super-element". +#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a "super-element". No payload us used. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all_flex.png]] @@ -12587,13 +12527,13 @@ However, as decentralized IFF will be applied, the damping is actively brought, #+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$ (encoder)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_diag_masses.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$} +#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$ (force sensor)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -12606,7 +12546,7 @@ Excellent match between experimental and model coupling is observed. Therefore, the model effectively represents the system coupling for different payloads. #+name: fig:test_nhexa_comp_simscape_de_all_high_mass -#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. +#+caption: Comparison of the measured (in blue) and modeled (in red) FRF from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. $39\,\text{kg}$ payload is used. #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all_high_mass.png]] @@ -12647,7 +12587,7 @@ If a model of the nano-hexapod was developed in one time, it would be difficult To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure\nbsp{}ref:fig:test_id31_micro_station_nano_hexapod. This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions. -Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-DoF metrology system to measure the sample position relative to the granite base. +Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-DoFs metrology system to measure the sample position relative to the granite base. However, the complexity of this development prevented its completion before the experimental testing phase on ID31. To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\upmu\text{m}^3$) metrology system was developed, which is presented in Section\nbsp{}ref:sec:test_id31_metrology. @@ -12692,7 +12632,7 @@ This system comprises 5 capacitive sensors facing two reference spheres. However, as the gap between the capacitive sensors and the spheres is very small[fn:test_id31_1], the risk of damaging the spheres and the capacitive sensors is too high. #+name: fig:test_id31_short_stroke_metrology -#+caption: Short stroke metrology system used to measure the sample position with respect to the granite in 5DoF. The system is based on a "Spindle error analyzer" (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). The interferometer heads are shown in (\subref{fig:test_id31_interf_head}). +#+caption: Short stroke metrology system used to measure the sample position with respect to the granite in 5-DoFs. The system is based on a "Spindle error analyzer" (\subref{fig:test_id31_lion}), but the capacitive sensors are replaced with fibered interferometers (\subref{fig:test_id31_interf}). One interferometer head is shown in (\subref{fig:test_id31_interf_head}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_lion}Capacitive Sensors} @@ -12875,7 +12815,7 @@ When using interferometers, the size of the "light spot" on the sphere surface i As the light from the interferometer travels through air (as opposed to being in vacuum), the measured distance is sensitive to any variation in the refractive index of the air. Therefore, any variation in air temperature, pressure or humidity will induce measurement errors. -For instance, for a measurement length of $40\,\text{mm}$, a temperature variation of $0.1\,{}^oC$ (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of $\approx 4\,\text{nm}$. +For instance, for a measurement length of $40\,\text{mm}$, a temperature variation of $\SI{0.1}{\degree}$ (which is typical for the ID31 experimental hutch) induces errors in the distance measurement of $\approx 4\,\text{nm}$. Interferometers are also affected by noise\nbsp{}[[cite:&watchi18_review_compac_inter]]. The effect of noise on the translation and rotation measurements is estimated in Figure\nbsp{}ref:fig:test_id31_interf_noise. @@ -12936,7 +12876,7 @@ After investigation, it was found that the additional delay was due to a digital This issue was later solved. #+name: fig:test_id31_first_id -#+caption: Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and force sensors (\subref{fig:test_id31_first_id_iff}). Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are displayed with shaded lines. +#+caption: Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and for force sensors (\subref{fig:test_id31_first_id_iff}). Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are displayed with shaded lines. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_first_id_int}External Metrology} @@ -12966,7 +12906,7 @@ After alignment, the same motion was performed using the nano-hexapod while reco Results shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_correct are indeed indicating much better alignment. #+name: fig:test_id31_Rz_align_error_and_correct -#+caption: Measurement of the Nano-Hexapod axes in the frame of the external metrology. Before alignment (\subref{fig:test_id31_Rz_align_error}) and after alignment (\subref{fig:test_id31_Rz_align_correct}). +#+caption: Measurement of nano-hexapod's axes in the frame of the external metrology. Before (\subref{fig:test_id31_Rz_align_error}) and after alignment (\subref{fig:test_id31_Rz_align_correct}). #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_Rz_align_error}Before alignment} @@ -12990,7 +12930,7 @@ Between $650\,\text{Hz}$ and $1000\,\text{Hz}$, several modes can be observed, w The flexible modes of the top platform can be passively damped, whereas the modes of the two reference spheres should not be present in the final application. #+name: fig:test_id31_first_id_int_better_rz_align -#+caption: Decrease of the coupling with better Rz alignment. +#+caption: Decrease of the coupling after better $R_z$ alignment. #+attr_latex: :scale 0.8 [[file:figs/test_id31_first_id_int_better_rz_align.png]] @@ -13005,7 +12945,7 @@ Therefore, the model can be used for model-based control if necessary. It is interesting to note that the anti-resonances in the force sensor plant now appear as minimum-phase, as the model predicts (Figure\nbsp{}ref:fig:test_id31_comp_simscape_iff_diag_masses). #+name: fig:test_id31_picture_masses -#+caption: 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. +#+caption: 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. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_picture_mass_m0}$m=0\,\text{kg}$} @@ -13038,13 +12978,13 @@ It is interesting to note that the anti-resonances in the force sensor plant now #+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $\epsilon\mathcal{L}$ (\subref{fig:test_id31_comp_simscape_int_diag_masses}) and from $u$ to $V_s$ (\subref{fig:test_id31_comp_simscape_iff_diag_masses}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_simscape_int_diag_masses}from $u$ to $\epsilon\mathcal{L}$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_simscape_int_diag_masses}from $u$ to $\epsilon\mathcal{L}$ (strut error)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth [[file:figs/test_id31_comp_simscape_int_diag_masses.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_simscape_iff_diag_masses}from $u$ to $V_s$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_simscape_iff_diag_masses}from $u$ to $V_s$ (force sensor)} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :width 0.95\linewidth @@ -13108,7 +13048,7 @@ K_{\text{IFF}} & & 0 \\ The decentralized Integral Force Feedback is implemented as shown in the block diagram of Figure\nbsp{}ref:fig:test_id31_iff_block_diagram. #+name: fig:test_id31_iff_block_diagram -#+caption: Block diagram of the implemented decentralized IFF controller. The controller $\bm{K}_{\text{IFF}}$ is a diagonal controller with the same elements for every diagonal term $K_{\text{IFF}}$. +#+caption: Block diagram of the implemented decentralized IFF controller. The controller $\bm{K}_{\text{IFF}}$ is a diagonal controller. [[file:figs/test_id31_iff_schematic.png]] **** IFF Plant @@ -13123,7 +13063,7 @@ Similar results were obtained for all other 30 elements and for the different pa This confirms that the multi-body model can be used to tune the IFF controller. #+name: fig:test_id31_comp_simscape_Vs -#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six force sensor voltages $V_{s1}$ to $V_{s6}$. +#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal $u_1$ to the six force sensor voltages $V_{s1}$ to $V_{s6}$. #+attr_latex: :scale 0.8 [[file:figs/test_id31_comp_simscape_Vs.png]] @@ -13151,7 +13091,7 @@ It can be seen that the loop-gain is larger than $1$ around the suspension modes #+attr_latex: :scale 0.8 [[file:figs/test_id31_Kiff_bode_plot.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_Kiff_loop_gain}Decentralized Loop gains} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_Kiff_loop_gain}Decentralized loop gains} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13167,7 +13107,7 @@ For low payload masses, a higher IFF controller gain can lead to better damping. However, in this study, it was chosen to implement a "fixed" (i.e. non-adaptive) decentralized IFF controller. #+name: fig:test_id31_iff_root_locus -#+caption: Root Locus plots for the designed decentralized IFF controller, computed using the multy-body model. Black crosses indicate the closed-loop poles for the choosen value of the gain. +#+caption: Root loci for the decentralized IFF controller, computed using the multi-body model. Black crosses indicate the closed-loop poles for the chosen value of the gain. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_iff_root_locus_m0}$m = 0\,\text{kg}$} @@ -13207,7 +13147,7 @@ To experimentally validate the Decentralized IFF controller, it was implemented The obtained acrshortpl:frf are compared with the model in Figure\nbsp{}ref:fig:test_id31_hac_plant_effect_mass verifying the good correlation between the predicted damped plant using the multi-body model and the experimental results. #+name: fig:test_id31_hac_plant_effect_mass_comp_model -#+caption: Comparison of the open-loop plants and the damped plant with Decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped and modeled plants for all considered payloads (\subref{fig:test_id31_hac_plant_effect_mass}). Only "direct" terms ($\epsilon\mathcal{L}_i/u_i^\prime$) are displayed for simplificty. +#+caption: Comparison of the open-loop plant and the damped plant with decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped and modeled plants for all considered payloads (\subref{fig:test_id31_hac_plant_effect_mass}). Only "direct" terms ($\epsilon\mathcal{L}_i/u_i^\prime$) are displayed for simplicity. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_ol_iff_plant_model}Effect of IFF on the plant} @@ -13253,7 +13193,7 @@ K_{\text{HAC}} & & 0 \\ \end{equation} #+name: fig:test_id31_iff_hac_schematic -#+caption: Block diagram of the implemented HAC-IFF controllers. The controller $\bm{K}_{\text{HAC}}$ is a diagonal controller with the same elements on every diagonal term $K_{\text{HAC}}$. +#+caption: Block diagram of the implemented HAC-IFF controllers. The controller $\bm{K}_{\text{HAC}}$ is a diagonal controller. [[file:figs/test_id31_iff_hac_schematic.png]] **** Damped Plant @@ -13263,7 +13203,7 @@ To verify whether the multi-body model accurately represents the measured damped Considering the complexity of the system's dynamics, the model can be considered to represent the system's dynamics with good accuracy, and can therefore be used to tune the feedback controller and evaluate its performance. #+name: fig:test_id31_comp_simscape_hac -#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal ($u_1^\prime$) of the damped plant to the estimated errors ($\epsilon_{\mathcal{L}_i}$) in the frame of the six struts by the external metrology. +#+caption: Comparison of the measured (in blue) and modeled (in red) FRFs from the first control signal ($u_1^\prime$) of the damped plant to the estimated errors ($\epsilon_{\mathcal{L}_i}$) in the frame of the six struts by the external metrology. #+attr_latex: :scale 0.8 [[file:figs/test_id31_comp_simscape_hac.png]] @@ -13319,10 +13259,10 @@ K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot The obtained "decentralized" loop-gains (i.e. the diagonal element of the controller times the diagonal terms of the plant) are shown in Figure\nbsp{}ref:fig:test_id31_hac_loop_gain. The closed-loop stability was verified by computing the characteristic Loci (Figure\nbsp{}ref:fig:test_id31_hac_characteristic_loci). -However, small stability margins were observed for the highest mass, indicating that some multivariable effects are in play. +However, small stability margins were observed for the highest mass, indicating that some multivariable effects are at play. #+name: fig:test_id31_hac_loop_gain_loci -#+caption: Robust High Authority Controller. "Decentralized loop-gains" are shown in (\subref{fig:test_id31_hac_loop_gain}) and characteristic loci are shown in (\subref{fig:test_id31_hac_characteristic_loci}). +#+caption: "Decentralized loop-gains" (\subref{fig:test_id31_hac_loop_gain}) and characteristic loci (\subref{fig:test_id31_hac_characteristic_loci}) for the robust high authority controller. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_loop_gain}Loop Gains} @@ -13376,7 +13316,7 @@ However, the positioning errors worsen as the payload mass increases, especially However, it was decided that this controller should be tested experimentally and improved only if necessary. #+name: fig:test_id31_hac_tomography_Wz36_simulation -#+caption: Positioning errors in the Y-Z plane during tomography experiments simulated using the multi-body model (in closed-loop). +#+caption: Positioning errors in the YZ plane during closed-loop simulations of tomography experiments. #+attr_latex: :scale 0.8 [[file:figs/test_id31_hac_tomography_Wz36_simulation.png]] @@ -13424,7 +13364,7 @@ In terms of RMS errors, this corresponds to $30\,\text{nm}$ in $D_y$, $15\,\text Results obtained for all experiments are summarized and compared to the specifications in Section\nbsp{}ref:ssec:test_id31_scans_conclusion. #+name: tab:test_id31_experiments_specifications -#+caption: Specifications for the Nano-Active-Stabilization-System. +#+caption: Positioning specifications for the Nano-Active-Stabilization-System. #+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t | | $D_y$ | $D_z$ | $R_y$ | @@ -13446,10 +13386,10 @@ This idealized case was simulated by first calculating the eccentricity through While this approach likely underestimates actual open-loop errors, as perfect alignment is practically unattainable, it enables a more balanced comparison with closed-loop performance. #+name: fig:test_id31_tomo_m2_1rpm_robust_hac_iff -#+caption: Tomography experiment with a rotation velocity of $6\,\text{deg/s}$, and payload mass of $26\,\text{kg}$. Errors in the $(x,y)$ plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}). +#+caption: Tomography experiment with a rotation velocity of $6\,\text{deg/s}$, and payload mass of $26\,\text{kg}$. Errors in the XY plane are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}). The estimated eccentricity is represented by the black dashed circle. The errors with subtracted eccentricity are shown in (\subref{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in $(x,y)$ plane} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in XY plane} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13463,12 +13403,12 @@ While this approach likely underestimates actual open-loop errors, as perfect al #+end_subfigure #+end_figure -The residual motion (i.e. after compensating for eccentricity) in the $Y-Z$ is compared against the minimum beam size, as illustrated in Figure\nbsp{}ref:fig:test_id31_tomo_Wz36_results. +The residual motion (i.e. after compensating for eccentricity) in the YZ is compared against the minimum beam size, as illustrated in Figure\nbsp{}ref:fig:test_id31_tomo_Wz36_results. Results are indicating the NASS succeeds in keeping the sample's acrshort:poi on the beam, except for the highest mass of $39\,\text{kg}$ for which the lateral motion is a bit too high. These experimental findings are consistent with the predictions from the tomography simulations presented in Section\nbsp{}ref:ssec:test_id31_iff_hac_robustness. #+name: fig:test_id31_tomo_Wz36_results -#+caption: Measured errors in the $Y-Z$ plane during tomography experiments at $6\,\text{deg/s}$ for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data. +#+caption: Measured errors in the YZ plane during tomography experiments at $6\,\text{deg/s}$ for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data. #+attr_latex: :scale 0.8 [[file:figs/test_id31_tomo_Wz36_results.png]] @@ -13507,25 +13447,25 @@ Figure\nbsp{}ref:fig:test_id31_hac_cas_cl presents the cumulative amplitude spec The results reveal two distinct control contributions: the decentralized IFF effectively attenuates vibrations near the nano-hexapod suspension modes (an achievement not possible with HAC alone), while the high authority controller suppresses low-frequency vibrations primarily arising from Spindle guiding errors. Notably, the spectral patterns in Figure\nbsp{}ref:fig:test_id31_hac_cas_cl closely resemble the cumulative amplitude spectra computed in the project's early stages (Figure\nbsp{}ref:fig:uniaxial_cas_hac_lac_mid in page\nbps{}pageref:fig:uniaxial_cas_hac_lac_mid). -This experiment also illustrates that when needed, performance can be enhanced by designing controllers for specific experimental conditions rather than relying solely on robust controllers that can accommodate all payload ranges. +This experiment also illustrates that when needed, performance can be enhanced by designing controllers for specific experimental conditions rather than relying solely on robust controllers able to accommodate all payloads. #+name: fig:test_id31_hac_cas_cl #+caption: Cumulative Amplitude Spectrum for tomography experiments at $180\,\text{deg}/s$. Open-Loop case, IFF, and HAC-LAC are compared. Specifications are indicated by black dashed lines. The RMS values are indicated in the legend. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_dy} $D_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_dy}Lateral ($D_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_hac_cas_cl_dy.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_dz} $D_z$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_dz}Vertical ($D_z$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_hac_cas_cl_dz.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_ry} $R_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_ry}Tilt ($R_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13544,19 +13484,19 @@ The results confirmed that the NASS successfully maintained the acrshort:poi wit #+caption: Reflectivity scan ($R_y$) with a rotational velocity of $100\,\upmu \text{rad}/s$. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dy}$D_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dy}Lateral ($D_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_reflectivity_dy.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dz}$D_z$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dz}Vertical ($D_z$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_reflectivity_dz.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_ry}$R_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_ry}Tilt ($R_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13617,19 +13557,19 @@ Initial testing at $10\,\upmu\text{m/s}$ demonstrated positioning errors well wi #+caption: $D_z$ scan at a velocity of $10\,\upmu \text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dy}Vertical ($D_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_scan_10ums_dy.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dz}$D_z$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dz}Horizontal ($D_z$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_scan_10ums_dz.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_ry}Tilt ($R_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13645,19 +13585,19 @@ However, performance during acceleration phases could be enhanced through the im #+caption: $D_z$ scan at a velocity of $100\,\upmu\text{m/s}$. $D_z$ setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in $D_y$ and $R_y$ are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dy}Lateral ($D_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_scan_100ums_dy.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dz}$D_z$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dz}Vertical ($D_z$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dz_scan_100ums_dz.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_ry}Tilt ($R_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13686,22 +13626,22 @@ In the vertical direction (Figure\nbsp{}ref:fig:test_id31_dy_10ums_dz), open-loo Under closed-loop control, positioning errors remain within specifications in all directions. #+name: fig:test_id31_dy_10ums -#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $10\,\upmu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_10ums_dy}). +#+caption: Open-loop (in blue) and closed-loop (i.e. using the NASS, in red) during a $10\,\upmu\text{m/s}$ scan with the $T_y$ stage. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dy} $D_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dy}Lateral ($D_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dy_10ums_dy.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dz} $D_z$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dz}Vertical ($D_z$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dy_10ums_dz.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_ry} $R_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_ry}Tilt ($R_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13722,22 +13662,22 @@ Alternatively, since closed-loop errors in $D_z$ and $R_y$ directions remain wit For applications requiring small $D_y$ scans, the nano-hexapod can be used exclusively, although with limited stroke capability. #+name: fig:test_id31_dy_100ums -#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $100\,\upmu\text{m/s}$ scan with the $T_y$ stage. Errors in $D_y$ is shown in (\subref{fig:test_id31_dy_100ums_dy}). +#+caption: Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a $100\,\upmu\text{m/s}$ scan with the $T_y$ stage. #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dy} $D_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dy}Lateral ($D_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dy_100ums_dy.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dz} $D_z$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dz}Vertical ($D_z$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 [[file:figs/test_id31_dy_100ums_dz.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_ry} $R_y$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_ry}Tilt ($R_y$)} #+attr_latex: :options {0.33\textwidth} #+begin_subfigure #+attr_latex: :scale 0.8 @@ -13754,7 +13694,7 @@ To avoid high-frequency vibrations typically induced by the stepper motor, the $ The system performance was evaluated at three lateral scanning velocities: $0.1\,\text{mm/s}$, $0.5\,\text{mm/s}$, and $1\,\text{mm/s}$. Figure\nbsp{}ref:fig:test_id31_diffraction_tomo_setpoint presents both the $D_y$ position setpoints and the corresponding measured $D_y$ positions for all tested velocities. #+name: fig:test_id31_diffraction_tomo_setpoint -#+caption: Dy motion for several configured velocities. +#+caption: Lateral ($D_y$) motion for several configured velocities. #+attr_latex: :scale 0.8 [[file:figs/test_id31_diffraction_tomo_setpoint.png]] @@ -13765,7 +13705,7 @@ These large errors occurred only during $\approx 20\,\text{ms}$ intervals; thus, Alternatively, a feedforward controller could improve the lateral positioning accuracy during these transient phases. #+name: fig:test_id31_diffraction_tomo -#+caption: Diffraction tomography scans (combined $R_z$ and $D_y$ motions) at several $D_y$ velocities ($R_z$ rotational velocity is $6\,\text{deg/s}$). +#+caption: Diffraction tomography scans (combined $R_z$ and $D_y$ motions) at several $D_y$ velocities, $\Omega_z = 6\,\text{deg/s}$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_diffraction_tomo_dy} $D_y$} @@ -13800,11 +13740,11 @@ Control performance in each of these directions can be tuned independently. A schematic of the proposed control architecture is illustrated in Figure\nbsp{}ref:fig:test_id31_cf_control. #+name: fig:test_id31_cf_control -#+caption: Control architecture in the Cartesian frame. Only the controller corresponding to the $D_z$ direction is shown. $H_L$ and $H_H$ are complementary filters. +#+caption: Control architecture using complementary filters proposed in Section\nbsp{}ref:sec:detail_control_cf, here adapted for the NASS. Jacobian matrices are used to have the control in the Cartesian frame. Only the $D_z$ controller is shown. $H_L$ and $H_H$ are complementary filters. [[file:figs/test_id31_cf_control.png]] Implementation of this control architecture necessitates a plant model, which must subsequently be inverted. -This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF acrshort:apa (Section\nbsp{}ref:sec:test_apa_model_2dof) model and 4-DoF flexible joints, such that the model order stays relatively low. +This plant model was derived from the multi-body model incorporating the previously detailed 2-DoFs acrshort:apa (Section\nbsp{}ref:sec:test_apa_model_2dof) model and 4-DoFs flexible joints, such that the model order stays relatively low. Analytical formulas for complementary filters having $40\,\text{dB/dec}$ slopes, proposed in Section\nbsp{}ref:ssec:detail_control_cf_analytical_complementary_filters, were used during this experimental validation. An initial experimental validation was conducted under no-payload conditions, with control applied solely to the $D_y$, $D_z$, and $R_y$ directions. @@ -13821,7 +13761,7 @@ It also shows that the parameter $\alpha$ provides a mechanism for managing the #+caption: Measured closed-loop transfer functions. Different bandwidth can be specified for different directions using $\omega_0$ (\subref{fig:test_id31_cf_control_dy_dz_diff}). The shape can be adjusted using parameter $\alpha$ (\subref{fig:test_id31_cf_control_alpha}). #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:test_id31_cf_control_dy_dz_diff}Chose of bandwidth using $\omega_0$, $m = 0\,\text{kg}$} +#+attr_latex: :caption \subcaption{\label{fig:test_id31_cf_control_dy_dz_diff}Choice of bandwidth using $\omega_0$, $m = 0\,\text{kg}$} #+attr_latex: :options {0.49\textwidth} #+begin_subfigure #+attr_latex: :scale 0.9 @@ -13842,7 +13782,7 @@ Measured closed-loop transfer functions are shown in Figure\nbsp{}ref:fig:test_i For higher values of $\omega_0$, the system became unstable in the vertical direction, probably because of the resonance at $250\,\text{Hz}$ that is not well captured with the multi-body model (Figure\nbsp{}ref:fig:test_id31_hac_plant_effect_mass). #+name: fig:test_id31_high_bandwidth -#+caption: Measured Closed-Loop Sensitivity (\subref{fig:test_id31_high_bandwidth_S}) and Complementary Sensitivity (\subref{fig:test_id31_high_bandwidth_T}) transfer functions for the highest test bandwidth $\omega_0 = 2\pi\cdot 60\,\text{rad/s}$. +#+caption: Measured Closed-Loop Sensitivity (\subref{fig:test_id31_high_bandwidth_S}) and Complementary Sensitivity (\subref{fig:test_id31_high_bandwidth_T}) transfer functions for the highest tested parameter $\omega_0 = 2\pi\cdot 60\,\text{rad/s}$. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_high_bandwidth_S}Sensitivity} @@ -13883,7 +13823,7 @@ Overall, the experimental results validate the effectiveness of the developed co The identified limitations, primarily related to high-speed lateral scanning and heavy payload handling, provide clear directions for future improvements. #+name: tab:test_id31_experiments_results_summary -#+caption: Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated by bold number. +#+caption: Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated in bold. #+attr_latex: :environment tabularx :width 0.85\linewidth :align Xccc #+attr_latex: :center t :booktabs t | *Experiments* | $\bm{D_y}$ *[nmRMS]* | $\bm{D_z}$ *[nmRMS]* | $\bm{R_y}$ *[nradRMS]* | @@ -13919,7 +13859,7 @@ The identified limitations, primarily related to high-speed lateral scanning and This chapter presented a comprehensive experimental validation of the Nano Active Stabilization System (NASS) on the ID31 beamline, demonstrating its capability to maintain precise sample positioning during various experimental scenarios. The implementation and testing followed a systematic approach, beginning with the development of a short-stroke metrology system to measure the sample position, followed by the successful implementation of a acrshort:haclac control architecture, and concluding in extensive performance validation across diverse experimental conditions. -The short-stroke metrology system, while designed as a temporary solution, proved effective in providing high-bandwidth and low-noise 5-DoF position measurements. +The short-stroke metrology system, while designed as a temporary solution, proved effective in providing high-bandwidth and low-noise 5-DoFs position measurements. The careful alignment of the fibered interferometers targeting the two reference spheres ensured reliable measurements throughout the testing campaign. The implementation of the control architecture validated the theoretical framework developed earlier in this project. @@ -14080,11 +14020,11 @@ Stages based on voice coils, offering nano-positioning capabilities with $3\,\te Magnetic levitation also emerges as a particularly interesting technology to be explored, especially for microscopy\nbsp{}[[cite:&fahmy22_magnet_xy_theta_x;&heyman23_levcub]] and tomography\nbsp{}[[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\nbsp{}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\nbsp{}ref:fig:conclusion_maglev_heyman23, while a 6DoF stage featuring infinite rotation, is shown in Figure\nbsp{}ref:fig:conclusion_maglev_dyck15. +Specifically, a compact 6-DoFs stage known as LevCube, providing a mobility of approximately $1\,\text{cm}^3$, is depicted in Figure\nbsp{}ref:fig:conclusion_maglev_heyman23, while a 6-DoFs stage featuring infinite rotation, is shown in Figure\nbsp{}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}). +#+caption: Example of magnetic levitation stages. LevCube allowing for 6-DoFs 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}} @@ -14113,10 +14053,11 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro * List of Publications :PROPERTIES: -:UNNUMBERED: notoc +:UNNUMBERED: t :END: #+begin_export latex +\addcontentsline{toc}{chapter}{List of Publications} % Put the list of publications in the ToC \begin{refsection}[ref.bib] \renewcommand{\clearpage}{} % Désactive \clearpage temporairement % List all papers even if not cited @@ -14130,12 +14071,9 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro \end{refsection} #+end_export -* Glossary :ignore: -[[printglossaries:]] +* Acronyms :ignore: -# #+latex: \printglossary[type=\acronymtype] -# #+latex: \printglossary[type=\glossarytype] -# #+latex: \printglossary +\printglossary[type=\acronymtype] * Footnotes @@ -14181,7 +14119,7 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_apa_8]Renishaw Vionic, resolution of $2.5\,\text{nm}$ [fn:test_apa_7]Kistler 9722A [fn:test_apa_6]Polytec controller 3001 with sensor heads OFV512 -[fn:test_apa_5]Note that this is not completely correct as it was shown in Section\nbsp{}ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. +[fn:test_apa_5]Note that this is not completely correct as electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. [fn:test_apa_4]The Matlab =fminsearch= command is used to fit the plane [fn:test_apa_3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\upmu\text{m}$ [fn:test_apa_2]Millimar 1318 probe, specified linearity better than $1\,\upmu\text{m}$