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diff --git a/phd-thesis.org b/phd-thesis.org
index 9efe35d..227dd97 100644
--- a/phd-thesis.org
+++ b/phd-thesis.org
@@ -307,8 +307,8 @@ The research presented in this manuscript has been possible thanks to the Fonds
**** Synchrotron Radiation Facilities :ignore:
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.
+As these electrons interact with magnetic fields, typically generated by insertion devices or bending magnets, they produce exceptionally bright light known as synchrotron light.
+This intense electromagnetic radiation, centered mainly in the X-ray spectrum domain, 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.
This global distribution of such facilities underscores the significant utility of synchrotron light for the scientific community.
@@ -320,7 +320,7 @@ This global distribution of such facilities underscores the significant utility
These facilities fundamentally comprise two main parts: the accelerator and storage ring, where electron acceleration and light generation occur, and the beamlines, where the intense X-ray beams are conditioned and directed for experimental use.
-The acrfull:esrf, shown in Figure\nbsp{}ref:fig:introduction_esrf_picture, is a joint research institution supported by 19 member countries.
+The acrfull:esrf, shown in Figure\nbsp{}ref:fig:introduction_esrf_picture, is a joint research institution supported by 19 partner nations.
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.
@@ -349,7 +349,7 @@ In August 2020, following an extensive 20-month upgrade period, the acrshort:esr
This upgrade implemented a novel storage ring concept that substantially increases the brilliance and coherence of the X-ray beams.
Brilliance, a measure of the photon flux, is a key figure of merit for synchrotron facilities.
-It experienced an approximate 100-fold increase with the implementation of acrshort:ebs, as shown in the historical evolution depicted in Figure\nbsp{}ref:fig:introduction_moore_law_brillance.
+It experienced an approximate 30-fold increase with the implementation of acrshort:ebs, as shown in the historical evolution depicted in Figure\nbsp{}ref:fig:introduction_moore_law_brillance.
While this enhanced beam quality presents unprecedented scientific opportunities, it concurrently introduces considerable engineering challenges, particularly regarding experimental instrumentation and sample positioning systems.
#+name: fig:introduction_moore_law_brillance
@@ -389,11 +389,12 @@ These components are housed in multiple Optical Hutches, as depicted in Figure\n
#+end_figure
Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure\nbsp{}ref:fig:introduction_id31_cad), where, for experiments pertinent to this work, focusing optics are used.
-The sample is mounted on a positioning stage, referred to as the "end-station", that enables precise alignment relative to the X-ray beam.
-Detectors are used to capture the X-rays transmitted through or scattered by the sample.
+The sample is mounted on a positioning stage, referred to as the "end-station", which enables precise alignment relative to the X-ray beam.
+Detectors are used to capture the X-rays beam after interaction with the sample.
+
Throughout this thesis, the standard acrshort:esrf coordinate system is adopted, wherein the X-axis aligns with the beam direction, Y is transverse horizontal, and Z is vertical upwards against gravity.
-The specific end-station employed on the ID31 beamline is designated the "micro-station".
+The specific end-station employed on the ID31 beamline is referred to as the "micro-station".
As depicted in Figure\nbsp{}ref:fig:introduction_micro_station_dof, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a positioning hexapod (in purple).
The sample itself (cyan), potentially housed within complex sample environments (e.g., for high pressure or extreme temperatures), is mounted on top of this assembly.
Each stage serves distinct positioning functions; for example, the positioning hexapod enables fine static adjustments, while the $T_y$ translation and $R_z$ rotation stages are used for specific scanning applications.
@@ -421,10 +422,10 @@ Two illustrative examples are provided.
Tomography experiments, schematically represented in Figure\nbsp{}ref:fig:introduction_tomography_schematic, involve placing a sample in the X-ray beam path while controlling its vertical rotation angle using a dedicated stage.
Detector images are captured at numerous rotation angles, allowing the reconstruction of three-dimensional sample structure (Figure\nbsp{}ref:fig:introduction_tomography_results)\nbsp{}[[cite:&schoeppler17_shapin_highl_regul_glass_archit]].
-This reconstruction depends critically on maintaining the sample's acrfull:poi within the beam throughout the rotation process.
+This reconstruction depends critically on maintaining the sample's acrfull:poi within the beam during the rotation process.
Mapping or scanning experiments, depicted in Figure\nbsp{}ref:fig:introduction_scanning_schematic, typically use focusing optics to have a small beam size at the sample's location.
-The sample is then translated perpendicular to the beam (along Y and Z axes), while data is collected at each position.
+The sample is then translated perpendicular to the beam (along Y and Z axes), while data are collected at each position.
An example\nbsp{}[[cite:&sanchez-cano17_synch_x_ray_fluor_nanop]] of a resulting two-dimensional map, acquired with $20\,\text{nm}$ step increments, is shown in Figure\nbsp{}ref:fig:introduction_scanning_results.
The fidelity and resolution of such images are intrinsically linked to the focused beam size and the positioning precision of the sample relative to the focused beam.
Positional instabilities, such as vibrations and thermal drifts, inevitably lead to blurring and distortion in the obtained image.
@@ -472,7 +473,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
:END:
Continuous progress in both synchrotron source technology and X-ray optics have led to the availability of smaller, more intense, and more stable X-ray beams.
-The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source size, particularly in the horizontal dimension, coupled with increased brilliance, as illustrated in Figure\nbsp{}ref:fig:introduction_beam_3rd_4th_gen.
+The ESRF-EBS upgrade, for instance, resulted in a significantly reduction of the horizontal source size, coupled with a decrease of the beam horizontal divergence, leading to an increased brilliance, as illustrated in Figure\nbsp{}ref:fig:introduction_beam_3rd_4th_gen.
#+name: fig:introduction_beam_3rd_4th_gen
#+caption: View of the ESRF X-ray beam before the EBS upgrade (\subref{fig:introduction_beam_3rd_gen}) and after the EBS upgrade (\subref{fig:introduction_beam_4th_gen}). The brilliance is increased, whereas the horizontal size and emittance are reduced.
@@ -493,9 +494,9 @@ The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source s
#+end_figure
Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of acrshort:esrf, where typical spot sizes were on the order of $10\,\upmu\text{m}$ [[cite:&riekel89_microf_works_at_esrf]].
-Various technologies, including zone plates, Kirkpatrick-Baez mirrors, and compound refractive lenses, have been developed and refined, each presenting unique advantages and limitations\nbsp{}[[cite:&barrett16_reflec_optic_hard_x_ray]].
+Various technologies, including Fresnel Zone Plates (FZP), Kirkpatrick-Baez (KB) mirrors, Multilayer Laue Lenses (MLL), and Compound Refractive Lenses (CRL), have been developed and refined, each presenting unique advantages and limitations\nbsp{}[[cite:&barrett16_reflec_optic_hard_x_ray]].
The historical reduction in achievable spot sizes is represented in Figure\nbsp{}ref:fig:introduction_moore_law_focus.
-Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines.
+Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) may be 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. Adapted from [[cite:&barrett24_x_optic_accel_based_light_sourc]].
@@ -504,7 +505,7 @@ Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Ha
[[file:figs/introduction_moore_law_focus.png]]
The increased brilliance introduces challenges related to radiation damage, particularly at high-energy beamlines like ID31.
-Consequently, prolonged exposure of a single sample area to the focused beam must be avoided.
+Consequently, long exposure of a single sample area to the focused beam must be avoided.
Traditionally, experiments were conducted in a "step-scan" mode, illustrated in Figure\nbsp{}ref:fig:introduction_scan_step.
In this mode, the sample is moved to the desired position, the detector acquisition is initiated, and a beam shutter is opened for a brief, controlled duration to limit radiation damage before closing; this cycle is repeated for each measurement point.
While effective for mitigating radiation damage, this sequential process can be time-consuming, especially for high-resolution maps requiring numerous points.
@@ -531,14 +532,14 @@ An alternative, more efficient approach is the "fly-scan" or "continuous-scan" m
Here, the sample is moved continuously while the detector is triggered to acquire data "on the fly" at predefined positions or time intervals.
This technique significantly accelerates data acquisition, enabling better use of valuable beamtime while potentially enabling finer spatial resolution\nbsp{}[[cite:&huang15_fly_scan_ptych]].
-Recent developments in detector technology have yielded sensors with improved spatial resolution, lower noise characteristics, and substantially higher frame rates\nbsp{}[[cite:&hatsui15_x_ray_imagin_detec_synch_xfel_sourc]].
+Recent developments in detector technology have yielded sensors with improved spatial resolution, lower noise characteristics, better efficiency, and substantially higher frame rates\nbsp{}[[cite:&hatsui15_x_ray_imagin_detec_synch_xfel_sourc]].
Historically, detector integration times for scanning and tomography experiments were in the range of 0.1 to 1 second.
This extended integration effectively filtered high-frequency vibrations in beam or sample position, resulting in apparently stable but larger beam.
-With higher X-ray flux and reduced detector noise, integration times can now be shortened to approximately 1 millisecond, with frame rates exceeding $100\,\text{Hz}$.
+With higher X-ray flux and reduced detector noise, integration times can now be shortened down to approximately 1 millisecond, with frame rates exceeding $100\,\text{Hz}$.
This reduction in integration time has two major implications for positioning requirements.
Firstly, for a given spatial sampling ("pixel size"), faster integration necessitates proportionally higher scanning velocities.
-Secondly, the shorter integration times make the measurements more susceptible to high-frequency vibrations.
+Secondly, the shorter integration times make the measurements more sensitive to high-frequency vibrations.
Therefore, not only the sample position must be stable against long-term drifts, but it must also be actively controlled to minimize vibrations, especially during dynamic fly-scan acquisitions.
**** Existing Nano Positioning End-Stations
@@ -599,7 +600,7 @@ However, when a large number of DoFs are required, the cumulative errors and lim
#+end_subfigure
#+end_figure
-The concept of using an external metrology to measure and potentially correct for positioning errors is increasing used for nano-positioning end-stations.
+The concept of using an external metrology to measure and potentially correct for positioning errors is increasingly used for nano-positioning end-stations.
Ideally, the relative position between the sample's acrfull:poi and the X-ray beam focus would be measured directly.
In practice, direct measurement is often impossible; instead, the sample position is typically measured relative to a reference frame associated with the focusing optics, providing an indirect measurement.
@@ -695,18 +696,18 @@ The advent of fourth-generation light sources, coupled with advancements in focu
With ID31's anticipated minimum beam dimensions of approximately $200\,\text{nm}\times 100\,\text{nm}$, the primary experimental objective is maintaining the sample's acrshort:poi within this beam.
This necessitates peak-to-peak positioning errors below $200\,\text{nm}$ in $D_y$ and $200\,\text{nm}$ in $D_z$, corresponding to acrfull:rms errors of $30\,\text{nm}$ and $15\,\text{nm}$, respectively.
Additionally, the $R_y$ tilt angle error must remain below $0.1\,\text{mdeg}$ ($250\,\text{nrad RMS}$).
-Given the high frame rates of modern detectors, these specified positioning errors must be maintained even when considering high-frequency vibrations.
+Given the high frame rates of modern detectors, these specified positioning errors must be maintained even when considering high-frequency vibrations (typically up to $1\,\text{kHz}$).
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, 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 existing micro-station, despite being composed of high-performance stages, has a positioning accuracy limited to approximately $10\,\upmu m$ and $10\,\upmu\text{rad}$ (peak to peak) 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-DoFs 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 three 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.
@@ -963,7 +964,7 @@ The measurement setup is schematically shown in Figure\nbsp{}ref:fig:uniaxial_us
The vertical inertial motion of the granite $x_{g}$ and the top platform of the positioning hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1].
Three acrfullpl:frf were computed: one from $F_{h}$ to $x_{h}$ (i.e., the compliance of the micro-station), one from $F_{g}$ to $x_{h}$ (or from $F_{h}$ to $x_{g}$) and one from $F_{g}$ to $x_{g}$.
-Due to the poor coherence at low frequencies, these acrlongpl:frf will only be shown between 20 and $200\,\text{Hz}$ (solid lines in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model).
+Due to the poor coherence[fn:uniaxial_2] at low frequencies, these acrlongpl:frf will only be shown between 20 and $200\,\text{Hz}$ (solid lines in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model).
#+name: fig:micro_station_uniaxial_model
#+caption: Schematic of the Micro-Station measurement setup and uniaxial model.
@@ -1016,7 +1017,7 @@ However, the goal is not to have a perfect match with the measurement (this woul
More accurate models will be used later on.
#+name: fig:uniaxial_comp_frf_meas_model
-#+caption: Comparison of the measured FRF and the uniaxial model dynamics.
+#+caption: Comparison of the measured Frequency Response Functions (FRF) and the uniaxial model dynamics.
#+attr_latex: :scale 0.8
[[file:figs/uniaxial_comp_frf_meas_model.png]]
@@ -1088,7 +1089,7 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nb
*** Identification of Disturbances
<>
***** Introduction :ignore:
-To quantify disturbances (red signals in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_2] are used.
+To quantify disturbances (red signals in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_3] are used.
One is located on the floor, another one on the granite, and the last one on the positioning hexapod's top platform (see Figure\nbsp{}ref:fig:uniaxial_ustation_meas_disturbances).
The geophone located on the floor was used to measure the floor motion $x_f$ while the other two geophones were used to measure vibrations introduced by scanning of the $T_y$ stage and $R_z$ stage (see Figure\nbsp{}ref:fig:uniaxial_ustation_dynamical_id_setup).
@@ -1112,7 +1113,7 @@ The geophone located on the floor was used to measure the floor motion $x_f$ whi
***** Ground Motion
To acquire the geophone signals, the measurement setup shown in Figure\nbsp{}ref:fig:uniaxial_geophone_meas_chain is used.
-The voltage generated by the geophone is amplified using a low noise voltage amplifier[fn:uniaxial_3] with a gain of $60\,\text{dB}$ before going to the acrfull:adc.
+The voltage generated by the geophone is amplified using a low noise voltage amplifier[fn:uniaxial_4] with a gain of $60\,\text{dB}$ before going to the acrfull:adc.
This is done to improve the signal-to-noise ratio.
To reconstruct the displacement $x_f$ from the measured voltage $\hat{V}_{x_f}$, the transfer function of the measurement chain from $x_f$ to $\hat{V}_{x_f}$ needs to be estimated.
@@ -1194,7 +1195,7 @@ This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\
The obtained sensitivity to disturbances for the three active platform stiffnesses are shown in Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses for the sample mass $m_s = 1\,\text{kg}$ (the same conclusions can be drawn with $m_s = 50\,\text{kg}$):
- The soft active platform is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs)
-- Between the suspension mode of the active platform (here at $5\,\text{Hz}$) and the first mode of the micro-station (here at $70\,\text{Hz}$), the disturbances induced by the stage vibrations are filtered out (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft)
+- Between the suspension mode[fn:uniaxial_5] of the active platform (here at $5\,\text{Hz}$) and the first mode of the micro-station (here at $70\,\text{Hz}$), the disturbances induced by the stage vibrations are filtered out (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft)
- Above the suspension mode of the active platform, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to $1$ (Figure\nbsp{}ref:fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf)
#+name: fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses
@@ -1426,7 +1427,7 @@ All three active damping approaches can lead to /critical damping/ of the active
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) ::
This can be understood from the mechanical equivalent of IFF shown in Figure\nbsp{}ref:fig:uniaxial_active_damping_iff_equiv considering an high stiffness $k$.
- The micro-station top platform is connected to an inertial mass (the active platform) through a damper, which dampens the micro-station suspension suspension mode.
+ The micro-station top platform is connected to an inertial mass (the active platform) through a damper, which dampens the micro-station suspension mode.
- DVF with a stiff active platform (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_stiff) ::
In that case, the "sky hook damper" (see mechanical equivalent of acrshort:dvf in Figure\nbsp{}ref:fig:uniaxial_active_damping_dvf_equiv) is connected to the micro-station top platform through the stiff active platform.
- RDC with a soft active platform (Figure\nbsp{}ref:fig:uniaxial_root_locus_damping_techniques_micro_station_mode) ::
@@ -1667,7 +1668,7 @@ The required feedback bandwidths were estimated in Section\nbsp{}ref:sec:uniaxia
Position feedback controllers are designed for each active platform such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure\nbsp{}ref:fig:uniaxial_nyquist_hac).
An arbitrary minimum modulus margin of $0.25$ was chosen when designing the controllers.
-These acrfullpl:hac are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a acrfull:lpf to increase the robustness to high frequency dynamics.
+These acrfullpl:hac are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a acrfull:lpf to increase the robustness to high-frequency dynamics.
The controllers used for the three active platform are shown in Equation\nbsp{}eqref:eq:uniaxial_hac_formulas, and the parameters used are summarized in Table\nbsp{}ref:tab:uniaxial_feedback_controller_parameters.
\begin{subequations} \label{eq:uniaxial_hac_formulas}
@@ -1885,7 +1886,7 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev
#+end_subfigure
#+end_figure
-***** Effect of support compliance on $L/F$
+***** Effect of support compliance on $L/f$
Some support compliance is now added and the model shown in Figure\nbsp{}ref:fig:uniaxial_support_compliance_test_system is used.
The parameters of the support (i.e., $m_{\mu}$, $c_{\mu}$ and $k_{\mu}$) are chosen to match the vertical mode at $70\,\text{Hz}$ seen on the micro-station (Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model).
@@ -1973,38 +1974,38 @@ 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).
+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_payload_dynamics_rigid_schematic).
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.
+To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_schematic are used.
#+name: fig:uniaxial_payload_dynamics_models
#+caption: Models used to study the effect of payload dynamics.
#+attr_latex: :options [htbp]
#+begin_figure
-#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_rigid_schematic}Rigid payload}
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_rigid_schematic}Rigid payload}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
-[[file:figs/uniaxial_paylaod_dynamics_rigid_schematic.png]]
+[[file:figs/uniaxial_payload_dynamics_rigid_schematic.png]]
#+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Flexible payload}
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_schematic}Flexible payload}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
-[[file:figs/uniaxial_paylaod_dynamics_schematic.png]]
+[[file:figs/uniaxial_payload_dynamics_schematic.png]]
#+end_subfigure
#+end_figure
***** Impact on Plant Dynamics
-To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used.
-Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered.
+To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_rigid_schematic is used.
+Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_schematic are considered.
The resonances of the payload are set at $\omega_s = 20\,\text{Hz}$ and at $\omega_s = 200\,\text{Hz}$ while its mass is either $m_s = 1\,\text{kg}$ or $m_s = 50\,\text{kg}$.
-The transfer functions from the active platform force $f$ to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod.
+The transfer functions from the active platform force $F$ to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_soft_nano_hexapod.
It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern).
The frequency of the anti-resonance corresponds to the "free" resonance of the sample $\omega_s = \sqrt{k_s/m_s}$.
-The flexibility of the sample also changes the high frequency gain (the mass line is shifted from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$).
+The flexibility of the sample also changes the high-frequency gain (the mass line is shifted from $\frac{1}{(m_n + m_s)s^2}$ to $\frac{1}{m_ns^2}$).
#+name: fig:uniaxial_payload_dynamics_soft_nano_hexapod
#+caption: Effect of the payload dynamics on the soft active platform with light sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}).
@@ -2027,7 +2028,7 @@ The flexibility of the sample also changes the high frequency gain (the mass lin
The same transfer functions are now compared when using a stiff active platform ($k_n = 100\,\text{N}/\upmu\text{m}$) in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod.
In this case, the sample's resonance $\omega_s$ is smaller than the active platform resonance $\omega_n$.
This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to $\omega_s$).
-Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy).
+Even though the added sample's flexibility still shifts the high-frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure\nbsp{}ref:fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy).
#+name: fig:uniaxial_payload_dynamics_stiff_nano_hexapod
#+caption: Effect of the payload dynamics on the stiff active platform with light sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}), and heavy sample (\subref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}).
@@ -2093,7 +2094,7 @@ What happens is that above $\omega_s$, even though the motion $d$ can be control
Payload dynamics is usually a major concern when designing a positioning system.
In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample $\omega_s$ and of the active platform $\omega_n$.
-The larger the sample mass, the larger the effect (i.e., change of high frequency gain, appearance of additional resonances and anti-resonances).
+The larger the sample mass, the larger the effect (i.e., change of high-frequency gain, appearance of additional resonances and anti-resonances).
A zero/pole pattern is observed if $\omega_s > \omega_n$ and a pole/zero pattern if $\omega_s > \omega_n$.
Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in\nbsp{}[[cite:&rankers98_machin Section 4.2]].
The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload.
@@ -2167,7 +2168,7 @@ After the dynamics of this system is studied, the objective will be to dampen th
To obtain the equations of motion for the system represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic, the Lagrangian equation\nbsp{}eqref:eq:rotating_lagrangian_equations is used.
$L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$.
These terms are derived in\nbsp{}eqref:eq:rotating_energy_functions_lagrange.
-Note that the equation of motion corresponding to constant rotation along $\vec{i}_w$ is disregarded because this motion is imposed by the rotation stage.
+Note that the equation of motion corresponding to constant rotation around $\vec{i}_w$ is disregarded because this motion is imposed by the rotation stage.
\begin{equation}\label{eq:rotating_lagrangian_equations}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
@@ -2175,7 +2176,7 @@ Note that the equation of motion corresponding to constant rotation along $\vec{
\begin{equation} \label{eq:rotating_energy_functions_lagrange}
\begin{aligned}
- T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \quad Q_1 = F_u, \quad Q_2 = F_v, \\
+ T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \quad Q_u = F_u, \quad Q_v = F_v, \\
V &= \frac{1}{2} k \big( {d_u}^2 + {d_v}^2 \big), \quad D = \frac{1}{2} c \big( \dot{d}_u{}^2 + \dot{d}_v{}^2 \big)
\end{aligned}
\end{equation}
@@ -2422,7 +2423,7 @@ As explained in\nbsp{}[[cite:&preumont08_trans_zeros_struc_contr_with;&skogestad
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.
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.
+The control system is thus canceling the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability.
*** Integral Force Feedback with a High-Pass Filter
<>
@@ -2782,7 +2783,7 @@ This is a useful metric when disturbances are directly applied to the payload.
Here, it is defined as the transfer function from external forces applied on the payload along $\vec{i}_x$ to the displacement of the payload along the same direction.
Very similar results were obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure\nbsp{}ref:fig:rotating_comp_techniques_trans_compliance).
-Using IFF degrades the compliance at low frequencies, whereas using relative damping control degrades the transmissibility at high frequencies.
+Using IFF degrades the compliance at low frequencies, whereas using relative damping control degrades the transmissibility at high-frequencies.
This is very well known characteristics of these common active damping techniques that hold when applied to rotating platforms.
#+name: fig:rotating_comp_techniques_trans_compliance
@@ -3145,7 +3146,7 @@ The dynamics of the soft active platform ($k_n = 0.01\,\text{N}/\upmu\text{m}$)
In addition, the attainable damping ratio of the soft active platform when using acrshort:iff is limited by gyroscopic effects.
To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been considered.
-Results are similar to those of the uniaxial model except that come complexity is added for the soft active platform due to the spindle's rotation.
+Results are similar to those of the uniaxial model except that some complexity is added for the soft active platform due to the spindle's rotation.
For the moderately stiff active platform ($k_n = 1\,\text{N}/\upmu\text{m}$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model.
** Micro Station - Modal Analysis
@@ -3159,7 +3160,7 @@ Although the inertia of each solid body can easily be estimated from its geometr
Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
The tuning approach for the multi-body model based on measurements is illustrated in Figure\nbsp{}ref:fig:modal_vibration_analysis_procedure.
-First, a /response model/ is obtained, which corresponds to a set of acrshortpl:frf computed from experimental measurements.
+First, a /response model/ is obtained, which corresponds to a set of acrfullpl:frf computed from experimental measurements.
From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies.
@@ -3600,7 +3601,7 @@ The obtained natural frequencies and associated modal damping are summarized in
**** Modal Parameter Extraction
<>
-Generally, modal identification is using curve-fitting a theoretical expression to the actual measured acrshort:frf data.
+Generally, modal identification involves curve-fitting a theoretical model to the measured acrshort:frf data.
However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many acrshort:frf plots all obtained from the same structure.
Here, the last method is used because it provides a unique and consistent model.
@@ -3748,7 +3749,7 @@ The kinematics of the micro-station (i.e. how the motion of the stages are combi
Then, the multi-body model is presented and tuned to match the measured dynamics of the micro-station (Section\nbsp{}ref:sec:ustation_modeling).
Disturbances affecting the positioning accuracy also need to be modeled properly.
-To do so, the effects of these disturbances were first measured experimental and then injected into the multi-body model (Section\nbsp{}ref:sec:ustation_disturbances).
+To do so, the effects of these disturbances were first measured experimentally and then injected into the multi-body model (Section\nbsp{}ref:sec:ustation_disturbances).
To validate the accuracy of the micro-station model, "real world" experiments are simulated and compared with measurements in Section\nbsp{}ref:sec:ustation_experiments.
@@ -3907,7 +3908,7 @@ The rotation matrix can be used to express the coordinates of a point $P$ in a f
{}^AP = {}^A\bm{R}_B {}^BP
\end{equation}
-For rotations along $x$, $y$ or $z$ axis, the formulas of the corresponding rotation matrices are given in Equation\nbsp{}eqref:eq:ustation_rotation_matrices_xyz.
+For rotations around $x$, $y$ or $z$ axis, the formulas of the corresponding rotation matrices are given in Equation\nbsp{}eqref:eq:ustation_rotation_matrices_xyz.
\begin{subequations}\label{eq:ustation_rotation_matrices_xyz}
\begin{align}
@@ -4733,7 +4734,7 @@ These limitations generally make serial architectures unsuitable for nano-positi
In contrast, parallel mechanisms, which connect the mobile platform to the fixed base through multiple parallel struts, offer several advantages for precision positioning.
Their closed-loop kinematic structure provides inherently higher structural stiffness, as the platform is simultaneously supported by multiple struts\nbsp{}[[cite:&taghirad13_paral]].
Although parallel mechanisms typically exhibit limited workspace compared to serial architectures, this limitation is not critical for NASS given its modest stroke requirements.
-Numerous parallel kinematic architectures have been developed\nbsp{}[[cite:&dong07_desig_precis_compl_paral_posit]] to address various positioning requirements, with designs varying based on the desired acrshortpl:dof and specific application constraints.
+Numerous parallel kinematic architectures have been developed\nbsp{}[[cite:&dong07_desig_precis_compl_paral_posit]] to address various positioning requirements, with designs varying based on the intended acrshortpl:dof and specific application constraints.
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
@@ -4755,7 +4756,7 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha
#+end_figure
After evaluating the different options, the Stewart platform architecture was selected for several reasons.
-In addition to providing control over all required acrshortpl:dof, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints.
+In addition to allow control over all required acrshortpl:dof, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints.
Stewart platforms have been implemented in a wide variety of configurations, as illustrated in Figure\nbsp{}ref:fig:nhexa_stewart_examples, which shows two distinct implementations: one implementing piezoelectric actuators for nano-positioning applications, and another based on voice coil actuators for vibration isolation.
These examples demonstrate the architecture's versatility in terms of geometry, actuator selection, and scale, all of which can be optimized for specific applications.
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.
@@ -4947,7 +4948,7 @@ For a series of platform positions, the exact strut lengths are computed using t
These strut lengths are then used with the Jacobian to estimate the platform pose\nbsp{}eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$.
For motion strokes from $1\,\upmu\text{m}$ to $10\,\text{mm}$, the errors are estimated for all direction of motion, and the worst case errors are shown in Figure\nbsp{}ref:fig:nhexa_forward_kinematics_approximate_errors.
-The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\upmu\text{m}$ as the size of the Stewart platform is here $\approx 100\,\text{mm}$), the Jacobian approximation provides excellent accuracy.
+The results demonstrate that for displacements up to approximately $0.1\,\%$ of the hexapod's size (which corresponds to $100\,\upmu\text{m}$ as the size of the Stewart platform is here $\approx 100\,\text{mm}$), the Jacobian approximation provides excellent accuracy.
Since the maximum required stroke of the active platform ($\approx 100\,\upmu\text{m}$) is three orders of magnitude smaller than its overall size ($\approx 100\,\text{mm}$), the Jacobian matrix can be considered constant throughout the workspace.
It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy.
@@ -5254,7 +5255,7 @@ This reduction from six to four observable modes is attributed to the system's s
The system's behavior can be characterized in three frequency regions.
At low frequencies, well below the first resonance, the plant demonstrates good decoupling between actuators, with the response dominated by the strut stiffness: $\bm{G}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}}^{-1}$.
In the mid-frequency range, the system exhibits coupled dynamics through its resonant modes, reflecting the complex interactions between the platform's degrees of freedom.
-At high frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: $\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-\intercal} \bm{J}^{\intercal} \frac{-1}{\omega^2}$
+At high-frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: $\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-\intercal} \bm{J}^{\intercal} \frac{-1}{\omega^2}$
The force sensor transfer functions, shown in Figure\nbsp{}ref:fig:nhexa_multi_body_plant_fm, display characteristics typical of collocated actuator-sensor pairs.
Each actuator's transfer function to its associated force sensor exhibits alternating complex conjugate poles and zeros.
@@ -5653,7 +5654,9 @@ The external metrology system measures the sample position relative to the fixed
Due to the system's symmetry, this metrology provides measurements for five acrshortpl:dof: three translations ($D_x$, $D_y$, $D_z$) and two rotations ($R_x$, $R_y$).
The sixth acrshort:dof ($R_z$) is still required to compute the errors in the frame of the active platform struts (i.e. to compute the active platform inverse kinematics).
-This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology, which consists of relative motion sensors in each strut (note that the positioning hexapod is not used for $R_z$ rotation, and is therefore ignored for $R_z$ estimation).
+This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology.
+The active platform's metrology consists of relative motion sensors in each strut, such that the $R_z$ rotation of the active platform can be estimated by solving the forward kinematics eqref:eq:nhexa_forward_kinematics_approximate.
+Note that the positioning hexapod is not used for $R_z$ rotation, and is therefore ignored for $R_z$ estimation.
The measured sample pose is represented by the homogeneous transformation matrix $\bm{T}_{\text{sample}}$, as shown in equation\nbsp{}eqref:eq:nass_sample_pose.
@@ -5965,7 +5968,7 @@ The current approach of controlling the position in the strut frame is inadequat
<>
A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure\nbsp{}ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at $10\,\text{Hz}$) for high performance operation.
-The controller structure is defined in Equation\nbsp{}eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes.
+The controller structure is defined in Equation\nbsp{}eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high-frequency modes.
\begin{equation}\label{eq:nass_robust_hac}
K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right)
@@ -6434,8 +6437,8 @@ In that case, the obtained stiffness matrix linearly depends on the strut stiffn
As shown by equation\nbsp{}eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal}$.
In the extreme case where all struts are vertical ($s_i = [0\ 0\ 1]$), a vertical stiffness of $6k$ is achieved, but with null stiffness in the horizontal directions.
-If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions.
-This configuration corresponds to the cubic architecture presented in Section\nbsp{}ref:sec:detail_kinematics_cubic.
+If two struts are oriented along the X axis, two struts along the Y axis, and two struts along the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3$ and the stiffness is well distributed along all directions.
+This configuration corresponds to the cubic architecture, that is presented in Section\nbsp{}ref:sec:detail_kinematics_cubic.
When the struts are oriented more vertically, as shown in Figure\nbsp{}ref:fig:detail_kinematics_stewart_mobility_vert_struts, the vertical stiffness increases while the horizontal stiffness decreases.
Additionally, $R_x$ and $R_y$ stiffness increases while $R_z$ stiffness decreases.
@@ -6672,7 +6675,7 @@ When relative motion sensors are integrated in each strut (measuring $\bm{\mathc
#+caption: Typical control architecture in the cartesian frame.
[[file:figs/detail_kinematics_centralized_control.png]]
-***** Low Frequency and High Frequency Coupling
+***** Low Frequency and High-Frequency Coupling
As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart.
At low frequency, the behavior of the platform depends on the stiffness matrix\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart_low_freq.
@@ -6683,7 +6686,7 @@ At low frequency, the behavior of the platform depends on the stiffness matrix\n
In Section\nbsp{}ref:ssec:detail_kinematics_cubic_static, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is positioned at the cube's center.
In this case, the "Cartesian" plant is decoupled at low frequency.
-At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$)\nbsp{}eqref:eq:detail_kinematics_transfer_function_high_freq.
+At high-frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$)\nbsp{}eqref:eq:detail_kinematics_transfer_function_high_freq.
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
@@ -6699,7 +6702,7 @@ To achieve a diagonal mass matrix, the acrlong:com of the mobile components must
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure\nbsp{}ref:fig:detail_kinematics_cubic_payload).
Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames.
When the $\{B\}$ frame was positioned at the acrlong:com, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure\nbsp{}ref:fig:detail_kinematics_cubic_cart_coupling_com).
-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).
+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}).
@@ -6779,7 +6782,7 @@ The second uses a non-cubic Stewart platform shown in Figure\nbsp{}ref:fig:detai
The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure\nbsp{}ref:fig:detail_kinematics_decentralized_dL.
As anticipated from the equations of motion from $\bm{f}$ to $\bm{\mathcal{L}}$ eqref:eq:detail_kinematics_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency.
-At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
+At high-frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
No significant advantage is evident for the cubic architecture (Figure\nbsp{}ref:fig:detail_kinematics_cubic_decentralized_dL) compared to the non-cubic architecture (Figure\nbsp{}ref:fig:detail_kinematics_non_cubic_decentralized_dL).
The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes.
@@ -6806,7 +6809,7 @@ The resonance frequencies differ between the two cases because the more vertical
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
The results are presented in Figure\nbsp{}ref:fig:detail_kinematics_decentralized_fn.
-The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
+The system demonstrates good decoupling at high-frequency in both cases, with no clear advantage for the cubic architecture.
#+name: fig:detail_kinematics_decentralized_fn
#+caption: Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn}).
@@ -7133,7 +7136,7 @@ Initially, the component is modeled in a finite element software with appropriat
Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component.
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.
+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 still presenting the main dynamical characteristics.
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.
@@ -7728,7 +7731,7 @@ The resulting dynamics (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_p
The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both acrshortpl:frf (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_plant) and root locus analysis (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_iff_locus).
-However, the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes.
+However, the transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ demonstrates significant effects: internal strut modes appear at high-frequencies, introducing substantial cross-coupling between axes.
This coupling is quantified through acrfull:rga analysis of the damped system (Figure\nbsp{}ref:fig:detail_fem_joints_axial_stiffness_rga_hac_plant), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance.
Above this resonance frequency, two critical limitations emerge.
@@ -7940,7 +7943,7 @@ Similarly, in\nbsp{}[[cite:&wang16_inves_activ_vibrat_isolat_stewar]], piezoelec
In\nbsp{}[[cite:&xie17_model_contr_hybrid_passiv_activ]], force sensors are integrated in the struts for decentralized force feedback while accelerometers fixed to the top platform are employed for centralized control.
The second approach, sensor fusion (illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_arch_sensor_fusion), involves filtering signals from two sensors using complementary filters[fn:detail_control_1] and summing them to create an improved sensor signal.
-In\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six]], geophones (used at low frequency) are merged with force sensors (used at high frequency).
+In\nbsp{}[[cite:&hauge04_sensor_contr_space_based_six]], geophones (used at low frequency) are merged with force sensors (used at high-frequency).
It is demonstrated that combining both sensors using sensor fusion can improve performance compared to using only one of the two sensors.
In\nbsp{}[[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]], sensor fusion architecture is implemented with an accelerometer and a force sensor.
This implementation is shown to simultaneously achieve high damping of structural modes (through the force sensors) while maintaining very low vibration transmissibility (through the accelerometers).
@@ -8562,7 +8565,7 @@ The obtained transfer function from $\bm{\mathcal{\tau}}$ to $\bm{\mathcal{L}}$
\end{equation}
At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are equal to the actuator stiffnesses\nbsp{}eqref:eq:detail_control_decoupling_plant_decentralized_low_freq.
-At high frequencies, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal.
+At high-frequency, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal.
\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
\bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
@@ -8629,7 +8632,7 @@ Analytical formula of the plant $\bm{G}_{\{M\}}(s)$ is derived\nbsp{}eqref:eq:de
\frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
\end{equation}
-At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix\nbsp{}eqref:eq:detail_control_decoupling_plant_CoM_high_freq.
+At high-frequency, the plant converges to the inverse of the mass matrix, which is a diagonal matrix\nbsp{}eqref:eq:detail_control_decoupling_plant_CoM_high_freq.
\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
\bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
@@ -8703,7 +8706,7 @@ The physical reason for high-frequency coupling is illustrated in Figure\nbsp{}r
When a high-frequency force is applied at a point not aligned with the acrlong:com, it induces rotation around the acrlong:com.
#+name: fig:detail_control_decoupling_jacobian_plant_CoK_results
-#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).
+#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high-frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK}
@@ -8712,7 +8715,7 @@ When a high-frequency force is applied at a point not aligned with the acrlong:c
#+attr_latex: :scale 0.8
[[file:figs/detail_control_decoupling_jacobian_plant_CoK.png]]
#+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK}
+#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High-frequency force applied at the CoK}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
@@ -8746,7 +8749,7 @@ The inherent mathematical structure of the mass, damping, and stiffness matrices
This diagonalization transforms equation\nbsp{}eqref:eq:detail_control_decoupling_equation_modal_coordinates into a set of $n$ decoupled equations, enabling independent control of each mode without cross-interaction.
To implement this approach from a decentralized plant, the architecture shown in Figure\nbsp{}ref:fig:detail_control_decoupling_modal is employed.
-Inputs of the decoupling plant are the modal modal inputs $\bm{\tau}_m$ and the outputs are the modal amplitudes $\bm{\mathcal{X}}_m$.
+Inputs of the decoupling plant are the modal inputs $\bm{\tau}_m$ and the outputs are the modal amplitudes $\bm{\mathcal{X}}_m$.
This implementation requires knowledge of the system's equations of motion, from which the mode shapes matrix $\bm{\Phi}$ is derived.
The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually.
@@ -8929,7 +8932,7 @@ Modal decoupling provides a natural framework when specific vibrational modes re
SVD decoupling generally results in a loss of physical meaning for the "control space", potentially complicating the process of relating controller design to practical system requirements.
The quality of decoupling achieved through these methods also exhibits distinct characteristics.
-Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the acrlong:cok, or at high frequencies when aligned with the acrlong:com.
+Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low-frequency when aligned at the acrlong:cok, or at high-frequency when aligned with the acrlong:com.
Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach.
Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the model accuracy, with discrepancies potentially resulting in significant off-diagonal elements.
SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point.
@@ -8994,7 +8997,7 @@ Finally, in Section\nbsp{}ref:ssec:detail_control_cf_simulations, a numerical ex
The idea of using complementary filters in the control architecture originates from sensor fusion techniques\nbsp{}[[cite:&collette15_sensor_fusion_method_high_perfor]], where two sensors are combined using complementary filters.
Building upon this concept, "virtual sensor fusion"\nbsp{}[[cite:&verma20_virtual_sensor_fusion_high_precis_contr]] replaces one physical sensor with a model $G$ of the plant.
The corresponding control architecture is illustrated in Figure\nbsp{}ref:fig:detail_control_cf_arch, where $G^\prime$ represents the physical plant to be controlled, $G$ is a model of the plant, $k$ is the controller, and $H_L$ and $H_H$ are complementary filters satisfying $H_L(s) + H_H(s) = 1$.
-In this arrangement, the physical plant is controlled at low frequencies, while the plant model is used at high frequencies to enhance robustness.
+In this arrangement, the physical plant is controlled at low frequencies, while the plant model is used at high-frequency to enhance robustness.
#+name: fig:detail_control_cf_arch_and_eq
#+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 $u$.
@@ -9092,7 +9095,7 @@ For the nominal system, $S = H_H$ and $T = H_L$, hence the performance specifica
\end{equation}
For disturbance rejection, the magnitude of the sensitivity function $|S(j\omega)| = |H_H(j\omega)|$ should be minimized, particularly at low frequencies where disturbances are usually most prominent.
-Similarly, for noise attenuation, the magnitude of the complementary sensitivity function $|T(j\omega)| = |H_L(j\omega)|$ should be minimized, especially at high frequencies where measurement noise typically dominates.
+Similarly, for noise attenuation, the magnitude of the complementary sensitivity function $|T(j\omega)| = |H_L(j\omega)|$ should be minimized, especially at high-frequency where measurement noise typically dominates.
Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function.
Typically, maintaining $|S|_{\infty} \le 2$ ensures a gain margin of at least 2 and a phase margin of at least $\SI{29}{\degree}$.
@@ -9499,8 +9502,8 @@ However, considering potential scanning capabilities, a worst-case scenario of a
There are two limiting factors for large signal bandwidth that should be evaluated:
1. Slew rate, which should exceed $2 \cdot V_{pp} \cdot f_r = 34\,\text{V/ms}$.
This requirement is typically easily met by commercial voltage amplifiers.
-2. Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies.
- To achieve high voltage at high frequency, the amplifier must therefore provide substantial current.
+2. Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high-frequency.
+ To achieve high voltage at high-frequency, the amplifier must therefore provide substantial current.
The maximum required current can be calculated as $I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,\text{A}$.
Therefore, ideally, a voltage amplifier capable of providing $0.3\,\text{A}$ of current would be interesting for scanning applications.
@@ -9525,9 +9528,9 @@ This approach does not account for the frequency dependency of the noise, which
Additionally, the load conditions used to estimate bandwidth and noise specifications are often not explicitly stated.
In many cases, bandwidth is reported with minimal load while noise is measured with substantial load, making direct comparisons between different models more complex.
-Note that for the WMA-200, the manufacturer proposed to remove the $50\,\Omega$ output resistor to improve to small signal bandwidth above $10\,\text{kHz}$
+Note that for the WMA-200 amplifier, the manufacturer proposed to remove the $50\,\Omega$ output resistor to improve to small signal bandwidth above $10\,\text{kHz}$
-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.
+The PD200 amplifier 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.
@@ -9682,7 +9685,7 @@ In contrast, optical encoders are bigger and they must be offset from the strut'
#+end_subfigure
#+end_figure
-A significant consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
+A practical consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
Measurements conducted on the slip-ring integrated in the micro-station revealed substantial cross-talk between different slip-ring channels.
To mitigate this issue, preference was given to sensors that transmit displacement measurements digitally, as these are inherently less susceptible to noise and cross-talk.
Based on this criterion, an optical encoder with digital output was selected, where signal interpolation is performed directly in the sensor head.
@@ -9796,7 +9799,7 @@ These results validate both the model of the acrshort:adc and the effectiveness
**** Instrumentation Amplifier
Because the acrshort:adc noise may be too low to measure the noise of other instruments (anything below $5.6\,\upmu\text{V}/\sqrt{\text{Hz}}$ cannot be distinguished from the noise of the acrshort:adc itself), a low noise instrumentation amplifier was employed.
-A Femto DLPVA-101-B-S amplifier with adjustable gains from $20\,text{dB}$ up to $80\,text{dB}$ was selected for this purpose.
+A Femto DLPVA-101-B-S amplifier with adjustable gains from $20\,\text{dB}$ up to $80\,\text{dB}$ was selected for this purpose.
The first step was to characterize the input[fn:detail_instrumentation_1] noise of the amplifier.
This was accomplished by short-circuiting its input with a $50\,\Omega$ resistor and measuring the output voltage with the acrshort:adc (Figure\nbsp{}ref:fig:detail_instrumentation_femto_meas_setup).
@@ -10447,7 +10450,7 @@ For the NASS, this stroke is sufficient because the positioning errors to be cor
It is clear from Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units.
This confirms the abnormal electrical measurements made in Section\nbsp{}ref:ssec:test_apa_electrical_measurements.
-This unit was sent sent back to Cedrat, and a new one was shipped back.
+This unit was sent back to Cedrat, and a new one was shipped back.
From now on, only the six remaining amplified piezoelectric actuators that behave as expected will be used.
#+name: fig:test_apa_stroke
@@ -10471,7 +10474,7 @@ From now on, only the six remaining amplified piezoelectric actuators that behav
**** Flexible Mode Measurement
<>
-In this section, the flexible modes of the APA300ML are investigated both experimentally and using a acrshort:fem.
+In this section, the flexible modes of the APA300ML are investigated both experimentally and through finite element modeling.
To experimentally estimate these modes, the acrshort:apa is fixed at one end (see Figure\nbsp{}ref:fig:test_apa_meas_setup_modes).
A Laser Doppler Vibrometer[fn:test_apa_6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:test_apa_7] is used to excite the flexible modes.
Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated.
@@ -10631,7 +10634,7 @@ 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 further investigated.
+This is 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.
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]].
@@ -11934,7 +11937,7 @@ The misalignment in the $y$ direction mostly influences:
- the location of the complex conjugate zero between the first two resonances:
- if $d_{y} < 0$: there is no zero between the two resonances and possibly not even between the second and third resonances
- if $d_{y} > 0$: there is a complex conjugate zero between the first two resonances
-- the location of the high frequency complex conjugate zeros at $500\,\text{Hz}$ (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
+- the location of the high-frequency complex conjugate zeros at $500\,\text{Hz}$ (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
The same can be done for misalignments in the $x$ direction.
The obtained dynamics (Figure\nbsp{}ref:fig:test_struts_effect_misalignment_x) are showing that misalignment in the $x$ direction mostly influences the presence of the flexible mode at $300\,\text{Hz}$ (see mode shape in Figure\nbsp{}ref:fig:test_struts_mode_shapes_2).
@@ -12585,7 +12588,7 @@ This section presents a comprehensive experimental evaluation of the complete sy
Initially, the project planned to develop a long-stroke ($\approx 1 \, \text{cm}^3$) 5-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.
+To validate the nano-hexapod and its associated control architecture, an alternative short-stroke ($\approx 100\,\upmu\text{m}^3$) metrology system was developed instead, which is presented in Section\nbsp{}ref:sec:test_id31_metrology.
Then, several key aspects of the system validation are examined.
Section\nbsp{}ref:sec:test_id31_open_loop_plant analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities.
@@ -12621,7 +12624,7 @@ These include tomography scans at various speeds and with different payload mass
**** Introduction :ignore:
The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite.
-As a long-stroke ($\approx 1 \,\text{cm}^3$) metrology system was not yet developed, a stroke stroke ($\approx 100\,\upmu\text{m}^3$) was used instead to validate the nano-hexapod control.
+As a long-stroke ($\approx 1 \,\text{cm}^3$) metrology system was not yet developed, a short stroke ($\approx 100\,\upmu\text{m}^3$) was used instead to validate the nano-hexapod control.
The first considered option was to use the "Spindle error analyzer" shown in Figure\nbsp{}ref:fig:test_id31_lion.
This system comprises 5 capacitive sensors facing two reference spheres.
@@ -12864,7 +12867,7 @@ The dynamics of the plant is first identified for a fixed spindle angle (at $0\,
The model dynamics is also identified under the same conditions.
A comparison between the model and the measured dynamics is presented in Figure\nbsp{}ref:fig:test_id31_first_id.
-A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled).
+A good match can be observed for the diagonal dynamics (except the high-frequency modes which are not modeled).
However, the coupling of the transfer function from command signals $\bm{u}$ to the estimated strut motion from the external metrology $\bm{\epsilon\mathcal{L}}$ is larger than expected (Figure\nbsp{}ref:fig:test_id31_first_id_int).
The experimental time delay estimated from the acrshort:frf (Figure\nbsp{}ref:fig:test_id31_first_id_int) is larger than expected.
@@ -12896,7 +12899,7 @@ One possible explanation of the increased coupling observed in Figure\nbsp{}ref:
To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders was implemented.
This allowed to perform two straight motions of the nano-hexapod along its $x$ and $y$ axes.
During these two motions, external metrology measurements were recorded and the results are shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_error_and_correct.
-It was found that there was a misalignment of 2.7 degrees (rotation along the vertical axis) between the interferometer axes and nano-hexapod axes.
+It was found that there was a misalignment of 2.7 degrees (rotation around the vertical axis) between the interferometer axes and nano-hexapod axes.
This was corrected by adding an offset to the spindle angle.
After alignment, the same motion was performed using the nano-hexapod while recording the signal of the external metrology.
Results shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_correct are indeed indicating much better alignment.
@@ -12991,7 +12994,7 @@ It is interesting to note that the anti-resonances in the force sensor plant now
**** Effect of Spindle Rotation
<>
-To verify that all the kinematics in Figure\nbsp{}ref:fig:test_id31_block_schematic_plant are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at $36\,\text{deg}/s$ and at $180\,\text{deg}/s$.
+To verify that all the kinematics in Figure\nbsp{}ref:fig:test_id31_block_schematic_plant are correct and to check whether the system dynamics is affected by Spindle rotation or not, three identification experiments were performed: no spindle rotation, spindle rotation at $36\,\text{deg}/s$ and at $180\,\text{deg}/s$.
The obtained dynamics from command signal $u$ to estimated strut error $\epsilon\mathcal{L}$ are displayed in Figure\nbsp{}ref:fig:test_id31_effect_rotation.
Both direct terms (Figure\nbsp{}ref:fig:test_id31_effect_rotation_direct) and coupling terms (Figure\nbsp{}ref:fig:test_id31_effect_rotation_coupling) are unaffected by the rotation.
@@ -13024,7 +13027,7 @@ This also indicates that the metrology kinematics is correct and is working in r
The identified acrshortpl:frf from command signals $\bm{u}$ to the force sensors $\bm{V}_s$ and to the estimated strut errors $\bm{\epsilon\mathcal{L}}$ are well matching the dynamics of the developed multi-body model.
The effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used.
-The spindle rotation had no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation.
+The spindle rotation has no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation.
*** Decentralized Integral Force Feedback
<>
@@ -13234,7 +13237,7 @@ The results indicate that higher payload masses increase the coupling when imple
This indicates that achieving high bandwidth feedback control is increasingly challenging as the payload mass increases.
This behavior can be attributed to the fundamental approach of implementing control in the frame of the struts.
Above the suspension modes of the nano-hexapod, the motion induced by the piezoelectric actuators is no longer dictated by kinematics but rather by the inertia of the different parts.
-This design choice, while beneficial for system simplicity, introduces inherent limitations in the system's ability to handle larger masses at high frequency.
+This design choice, while beneficial for system simplicity, introduces inherent limitations in the system's ability to handle larger masses at high-frequency.
#+name: fig:test_id31_hac_rga_number
#+caption: RGA-number for the damped plants - Comparison of all the payload conditions.
@@ -13246,7 +13249,7 @@ This design choice, while beneficial for system simplicity, introduces inherent
A diagonal controller was designed to be robust against changes in payload mass, which means that every damped plant shown in Figure\nbsp{}ref:fig:test_id31_comp_all_undamped_damped_plants must be considered during the controller design.
For this controller design, a crossover frequency of $5\,\text{Hz}$ was chosen to limit the multivariable effects, as explain in Section\nbsp{}ref:sec:test_id31_hac_interaction_analysis.
-One integrator is added to increase the low-frequency gain, a lead is added around $5\,\text{Hz}$ to increase the stability margins and a first-order low-pass filter with a cut-off frequency of $30\,\text{Hz}$ is added to improve the robustness to dynamical uncertainty at high frequency.
+One integrator is added to increase the low-frequency gain, a lead is added around $5\,\text{Hz}$ to increase the stability margins and a first-order low-pass filter with a cut-off frequency of $30\,\text{Hz}$ is added to improve the robustness to dynamical uncertainty at high-frequency.
The controller transfer function is shown in\nbsp{}eqref:eq:test_id31_robust_hac.
\begin{equation}\label{eq:test_id31_robust_hac}
@@ -14073,14 +14076,16 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro
* Footnotes
-[fn:uniaxial_3]DLPVA-100-B from Femto with a voltage input noise is $2.4\,\text{nV}/\sqrt{\text{Hz}}$
-[fn:uniaxial_2]Mark Product L-22D geophones are used with a sensitivity of $88\,\frac{V}{\text{m/s}}$ and a natural frequency of $\approx 2\,\text{Hz}$
-[fn:uniaxial_1]Mark Product L4-C geophones are used with a sensitivity of $171\,\frac{V}{\text{m/s}}$ and a natural frequency of $\approx 1\,\text{Hz}$
+[fn:uniaxial_5]In this work, the "suspension mode" of a platform refers to a low-frequency vibration mode in which the supported payload behaves as a rigid body, while the platform acts as a compliant support.
+[fn:uniaxial_4]DLPVA-100-B from Femto with a voltage input noise is $2.4\,\text{nV}/\sqrt{\text{Hz}}$.
+[fn:uniaxial_3]Mark Product L-22D geophones are used with a sensitivity of $88\,\frac{V}{\text{m/s}}$ and a natural frequency of $\approx 2\,\text{Hz}$.
+[fn:uniaxial_2]Coherence is a statistical measure (ranging from $0$ to $1$) used in system identification to assess how well the output of a linear system can be predicted from its input. Values near $1$ indicate strong linear correlation, while noise or non-linearities reduce coherence and indicate poor data quality.
+[fn:uniaxial_1]Mark Product L4-C geophones are used with a sensitivity of $171\,\frac{V}{\text{m/s}}$ and a natural frequency of $\approx 1\,\text{Hz}$.
[fn:modal_5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.
[fn:modal_4]NVGate software from OROS company.
[fn:modal_3]OROS OR36. 24bits signal-delta ADC.
-[fn:modal_2]Kistler 9722A2000. Sensitivity of $2.3\,\text{mV/N}$ and measurement range of $2\,\text{kN}$
+[fn:modal_2]Kistler 9722A2000. Sensitivity of $2.3\,\text{mV/N}$ and measurement range of $2\,\text{kN}$.
[fn:modal_1]PCB 356B18. Sensitivity is $1\,\text{V/g}$, measurement range is $\pm 5\,\text{g}$ and bandwidth is $0.5$ to $5\,\text{kHz}$.
[fn:ustation_11]It was probably caused by rust of the linear guides along its stroke.
@@ -14095,53 +14100,53 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro
[fn:ustation_2]Made by LAB Motion Systems.
[fn:ustation_1]HCR 35 A C1, from THK.
-[fn:nhexa_3]Such equation is called the /velocity loop closure/
-[fn:nhexa_2]The /pose/ represents the position and orientation of an object
-[fn:nhexa_1]Different architecture exists, typically referred as "6-SPS" (Spherical, Prismatic, Spherical) or "6-UPS" (Universal, Prismatic, Spherical)
+[fn:nhexa_3]Such equation is called the /velocity loop closure/.
+[fn:nhexa_2]The /pose/ represents the position and orientation of an object.
+[fn:nhexa_1]Different architecture exists, typically referred as "6-SPS" (Spherical, Prismatic, Spherical) or "6-UPS" (Universal, Prismatic, Spherical).
-[fn:detail_fem_2]Cedrat technologies
+[fn:detail_fem_2]Cedrat technologies.
[fn:detail_fem_1]The manufacturer of the APA95ML was not willing to share the piezoelectric material properties of the stack.
-[fn:detail_control_2]$n$ corresponds to the number of degrees of freedom, here $n = 3$
+[fn:detail_control_2]$n$ corresponds to the number of degrees of freedom, here $n = 3$.
[fn:detail_control_1]A set of two complementary filters are two transfer functions that sum to one.
[fn:detail_instrumentation_1] For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain.
-[fn:test_apa_13]PD200 from PiezoDrive. The gain is $20\,\text{V/V}$
-[fn:test_apa_12]The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of $\pm 10\,\text{V}$ and 16-bits resolution
-[fn:test_apa_11]Ansys\textsuperscript{\textregistered} was used
-[fn:test_apa_10]The transfer function fitting was computed using the =vectfit3= routine, see\nbsp{}[[cite:&gustavsen99_ration_approx_frequen_domain_respon]]
-[fn:test_apa_9]Frequency of the sinusoidal wave is $1\,\text{Hz}$
-[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_13]PD200 from PiezoDrive. The gain is $20\,\text{V/V}$.
+[fn:test_apa_12]The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of $\pm 10\,\text{V}$ and 16-bits resolution.
+[fn:test_apa_11]Ansys\textsuperscript{\textregistered} was used.
+[fn:test_apa_10]The transfer function fitting was computed using the =vectfit3= routine, see\nbsp{}[[cite:&gustavsen99_ration_approx_frequen_domain_respon]].
+[fn:test_apa_9]Frequency of the sinusoidal wave is $1\,\text{Hz}$.
+[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 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 $1\,\upmu\text{m}$
-[fn:test_apa_1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$
+[fn:test_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 $1\,\upmu\text{m}$.
+[fn:test_apa_1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$.
-[fn:test_joints_5]XFL212R-50N from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range
-[fn:test_joints_4]Resolute\texttrademark{} encoder with $1\,\text{nm}$ resolution and $\pm 40\,\text{nm}$ maximum non-linearity
+[fn:test_joints_5]XFL212R-50N from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range.
+[fn:test_joints_4]Resolute\texttrademark{} encoder with $1\,\text{nm}$ resolution and $\pm 40\,\text{nm}$ maximum non-linearity.
[fn:test_joints_3]V-408 PIMag\textsuperscript{\textregistered} linear stage is used. Crossed rollers are used to guide the motion.
-[fn:test_joints_2]The load cell is FC22 from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range
+[fn:test_joints_2]The load cell is FC22 from TE Connectivity. The measurement range is $50\,\text{N}$. The specified accuracy is $1\,\%$ of the full range.
[fn:test_joints_1]The alloy used is called /F16PH/, also refereed as "1.4542"
-[fn:test_struts_7] OFV-3001 controller and OFV512 sensor head from Polytec
-[fn:test_struts_6] Vionic from Renishaw
-[fn:test_struts_5] APA300ML from Cedrat Technologies
-[fn:test_struts_4] Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools
+[fn:test_struts_7] OFV-3001 controller and OFV512 sensor head from Polytec.
+[fn:test_struts_6] Vionic from Renishaw.
+[fn:test_struts_5] APA300ML from Cedrat Technologies.
+[fn:test_struts_4] Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools.
[fn:test_struts_3] Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called /17-4 PH/. Encoder and ruler support material is aluminium.
-[fn:test_struts_2] Heidenhain MT25, specified accuracy of $\pm 0.5\,\upmu\text{m}$
-[fn:test_struts_1] FARO Arm Platinum 4ft, specified accuracy of $\pm 13\upmu\text{m}$
+[fn:test_struts_2] Heidenhain MT25, specified accuracy of $\pm 0.5\,\upmu\text{m}$.
+[fn:test_struts_1] FARO Arm Platinum 4ft, specified accuracy of $\pm 13\upmu\text{m}$.
[fn:test_nhexa_7]PCB 356B18. Sensitivity is $1\,\text{V/g}$, measurement range is $\pm 5\,\text{g}$ and bandwidth is $0.5$ to $5\,\text{kHz}$.
-[fn:test_nhexa_6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,\text{N/mm}$
+[fn:test_nhexa_6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,\text{N/mm}$.
[fn:test_nhexa_5]The $450\,\text{mm} \times 450\,\text{mm} \times 60\,\text{mm}$ Nexus B4545A from Thorlabs.
[fn:test_nhexa_4]As the accuracy of the FARO arm is $\pm 13\,\upmu\text{m}$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.
[fn:test_nhexa_3]The height dimension is better than $40\,\upmu\text{m}$. The diameter fitting of 182g6 and 24g6 with the two plates is verified.
[fn:test_nhexa_2]Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.
-[fn:test_nhexa_1]FARO Arm Platinum 4ft, specified accuracy of $\pm 13\upmu\text{m}$
+[fn:test_nhexa_1]FARO Arm Platinum 4ft, specified accuracy of $\pm 13\upmu\text{m}$.
[fn:test_id31_8]Such scan could corresponding to a 1ms integration time (which is typically the smallest integration time) and $100\,\text{nm}$ "resolution" (equal to the vertical beam size).
[fn:test_id31_7]The highest rotational velocity of $360\,\text{deg/s}$ could not be tested due to an issue in the Spindle's controller.
diff --git a/phd-thesis.pdf b/phd-thesis.pdf
index 54ece65..18ff11f 100644
Binary files a/phd-thesis.pdf and b/phd-thesis.pdf differ
diff --git a/phd-thesis.tex b/phd-thesis.tex
index 77c850d..d5e5ece 100644
--- a/phd-thesis.tex
+++ b/phd-thesis.tex
@@ -1,4 +1,4 @@
-% Created 2025-04-23 Wed 23:54
+% Created 2025-06-13 Fri 14:33
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -55,7 +55,7 @@
\addbibresource{ref.bib}
\addbibresource{phd-thesis.bib}
\author{Dehaeze Thomas}
-\date{2025-04-23}
+\date{2025-06-13}
\title{Nano Active Stabilization of samples for tomography experiments: A mechatronic design approach}
\subtitle{PhD Thesis}
\hypersetup{
@@ -63,7 +63,7 @@
pdftitle={Nano Active Stabilization of samples for tomography experiments: A mechatronic design approach},
pdfkeywords={},
pdfsubject={},
- pdfcreator={Emacs 30.1 (Org mode 9.7.26)},
+ pdfcreator={Emacs 30.1 (Org mode 9.7.29)},
pdflang={English}}
\usepackage{biblatex}
@@ -214,8 +214,8 @@ The research presented in this manuscript has been possible thanks to the Fonds
\section{Context of this thesis}
\endgroup
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.
+As these electrons interact with magnetic fields, typically generated by insertion devices or bending magnets, they produce exceptionally bright light known as synchrotron light.
+This intense electromagnetic radiation, centered mainly in the X-ray spectrum domain, is subsequently used for the detailed study of matter.
Approximately 70 synchrotron light sources are operational worldwide, some of which are indicated in Figure~\ref{fig:introduction_synchrotrons}.
This global distribution of such facilities underscores the significant utility of synchrotron light for the scientific community.
@@ -227,7 +227,7 @@ This global distribution of such facilities underscores the significant utility
These facilities fundamentally comprise two main parts: the accelerator and storage ring, where electron acceleration and light generation occur, and the beamlines, where the intense X-ray beams are conditioned and directed for experimental use.
-The \acrfull{esrf}, shown in Figure~\ref{fig:introduction_esrf_picture}, is a joint research institution supported by 19 member countries.
+The \acrfull{esrf}, shown in Figure~\ref{fig:introduction_esrf_picture}, is a joint research institution supported by 19 partner nations.
The \acrshort{esrf} started user operations in 1994 as the world's first third-generation synchrotron.
Its accelerator complex, schematically depicted in Figure~\ref{fig:introduction_esrf_schematic}, includes a linear accelerator where electrons are initially generated and accelerated, a booster synchrotron to further accelerate the electrons, and an 844-meter circumference storage ring where electrons are maintained in a stable orbit.
@@ -254,7 +254,7 @@ In August 2020, following an extensive 20-month upgrade period, the \acrshort{es
This upgrade implemented a novel storage ring concept that substantially increases the brilliance and coherence of the X-ray beams.
Brilliance, a measure of the photon flux, is a key figure of merit for synchrotron facilities.
-It experienced an approximate 100-fold increase with the implementation of \acrshort{ebs}, as shown in the historical evolution depicted in Figure~\ref{fig:introduction_moore_law_brillance}.
+It experienced an approximate 30-fold increase with the implementation of \acrshort{ebs}, as shown in the historical evolution depicted in Figure~\ref{fig:introduction_moore_law_brillance}.
While this enhanced beam quality presents unprecedented scientific opportunities, it concurrently introduces considerable engineering challenges, particularly regarding experimental instrumentation and sample positioning systems.
\begin{figure}[htbp]
@@ -288,11 +288,12 @@ These components are housed in multiple Optical Hutches, as depicted in Figure~\
\end{figure}
Following the optical hutches, the conditioned beam enters the Experimental Hutch (Figure~\ref{fig:introduction_id31_cad}), where, for experiments pertinent to this work, focusing optics are used.
-The sample is mounted on a positioning stage, referred to as the ``end-station'', that enables precise alignment relative to the X-ray beam.
-Detectors are used to capture the X-rays transmitted through or scattered by the sample.
+The sample is mounted on a positioning stage, referred to as the ``end-station'', which enables precise alignment relative to the X-ray beam.
+Detectors are used to capture the X-rays beam after interaction with the sample.
+
Throughout this thesis, the standard \acrshort{esrf} coordinate system is adopted, wherein the X-axis aligns with the beam direction, Y is transverse horizontal, and Z is vertical upwards against gravity.
-The specific end-station employed on the ID31 beamline is designated the ``micro-station''.
+The specific end-station employed on the ID31 beamline is referred to as the ``micro-station''.
As depicted in Figure~\ref{fig:introduction_micro_station_dof}, it comprises a stack of positioning stages: a translation stage (in blue), a tilt stage (in red), a spindle for continuous rotation (in yellow), and a positioning hexapod (in purple).
The sample itself (cyan), potentially housed within complex sample environments (e.g., for high pressure or extreme temperatures), is mounted on top of this assembly.
Each stage serves distinct positioning functions; for example, the positioning hexapod enables fine static adjustments, while the \(T_y\) translation and \(R_z\) rotation stages are used for specific scanning applications.
@@ -318,10 +319,10 @@ Two illustrative examples are provided.
Tomography experiments, schematically represented in Figure~\ref{fig:introduction_tomography_schematic}, involve placing a sample in the X-ray beam path while controlling its vertical rotation angle using a dedicated stage.
Detector images are captured at numerous rotation angles, allowing the reconstruction of three-dimensional sample structure (Figure~\ref{fig:introduction_tomography_results})~\cite{schoeppler17_shapin_highl_regul_glass_archit}.
-This reconstruction depends critically on maintaining the sample's \acrfull{poi} within the beam throughout the rotation process.
+This reconstruction depends critically on maintaining the sample's \acrfull{poi} within the beam during the rotation process.
Mapping or scanning experiments, depicted in Figure~\ref{fig:introduction_scanning_schematic}, typically use focusing optics to have a small beam size at the sample's location.
-The sample is then translated perpendicular to the beam (along Y and Z axes), while data is collected at each position.
+The sample is then translated perpendicular to the beam (along Y and Z axes), while data are collected at each position.
An example~\cite{sanchez-cano17_synch_x_ray_fluor_nanop} of a resulting two-dimensional map, acquired with \(20\,\text{nm}\) step increments, is shown in Figure~\ref{fig:introduction_scanning_results}.
The fidelity and resolution of such images are intrinsically linked to the focused beam size and the positioning precision of the sample relative to the focused beam.
Positional instabilities, such as vibrations and thermal drifts, inevitably lead to blurring and distortion in the obtained image.
@@ -360,7 +361,7 @@ Other advanced imaging modalities practiced on ID31 include reflectivity, diffra
\end{figure}
\subsubsection*{Need of Accurate Positioning End-Stations with High Dynamics}
Continuous progress in both synchrotron source technology and X-ray optics have led to the availability of smaller, more intense, and more stable X-ray beams.
-The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source size, particularly in the horizontal dimension, coupled with increased brilliance, as illustrated in Figure~\ref{fig:introduction_beam_3rd_4th_gen}.
+The ESRF-EBS upgrade, for instance, resulted in a significantly reduction of the horizontal source size, coupled with a decrease of the beam horizontal divergence, leading to an increased brilliance, as illustrated in Figure~\ref{fig:introduction_beam_3rd_4th_gen}.
\begin{figure}[h!tbp]
\begin{subfigure}{0.69\textwidth}
@@ -379,9 +380,9 @@ The ESRF-EBS upgrade, for instance, resulted in a significantly reduced source s
\end{figure}
Concurrently, substantial progress has been made in micro- and nano-focusing optics since the early days of \acrshort{esrf}, where typical spot sizes were on the order of \(10\,\upmu\text{m}\) \cite{riekel89_microf_works_at_esrf}.
-Various technologies, including zone plates, Kirkpatrick-Baez mirrors, and compound refractive lenses, have been developed and refined, each presenting unique advantages and limitations~\cite{barrett16_reflec_optic_hard_x_ray}.
+Various technologies, including Fresnel Zone Plates (FZP), Kirkpatrick-Baez (KB) mirrors, Multilayer Laue Lenses (MLL), and Compound Refractive Lenses (CRL), have been developed and refined, each presenting unique advantages and limitations~\cite{barrett16_reflec_optic_hard_x_ray}.
The historical reduction in achievable spot sizes is represented in Figure~\ref{fig:introduction_moore_law_focus}.
-Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) are routinely achieved on specialized nano-focusing beamlines.
+Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Half Maximum, FWHM) may be achieved on specialized nano-focusing beamlines.
\begin{figure}[htbp]
\centering
@@ -390,7 +391,7 @@ Presently, focused beam dimensions in the range of 10 to 20 nm (Full Width at Ha
\end{figure}
The increased brilliance introduces challenges related to radiation damage, particularly at high-energy beamlines like ID31.
-Consequently, prolonged exposure of a single sample area to the focused beam must be avoided.
+Consequently, long exposure of a single sample area to the focused beam must be avoided.
Traditionally, experiments were conducted in a ``step-scan'' mode, illustrated in Figure~\ref{fig:introduction_scan_step}.
In this mode, the sample is moved to the desired position, the detector acquisition is initiated, and a beam shutter is opened for a brief, controlled duration to limit radiation damage before closing; this cycle is repeated for each measurement point.
While effective for mitigating radiation damage, this sequential process can be time-consuming, especially for high-resolution maps requiring numerous points.
@@ -415,14 +416,14 @@ An alternative, more efficient approach is the ``fly-scan'' or ``continuous-scan
Here, the sample is moved continuously while the detector is triggered to acquire data ``on the fly'' at predefined positions or time intervals.
This technique significantly accelerates data acquisition, enabling better use of valuable beamtime while potentially enabling finer spatial resolution~\cite{huang15_fly_scan_ptych}.
-Recent developments in detector technology have yielded sensors with improved spatial resolution, lower noise characteristics, and substantially higher frame rates~\cite{hatsui15_x_ray_imagin_detec_synch_xfel_sourc}.
+Recent developments in detector technology have yielded sensors with improved spatial resolution, lower noise characteristics, better efficiency, and substantially higher frame rates~\cite{hatsui15_x_ray_imagin_detec_synch_xfel_sourc}.
Historically, detector integration times for scanning and tomography experiments were in the range of 0.1 to 1 second.
This extended integration effectively filtered high-frequency vibrations in beam or sample position, resulting in apparently stable but larger beam.
-With higher X-ray flux and reduced detector noise, integration times can now be shortened to approximately 1 millisecond, with frame rates exceeding \(100\,\text{Hz}\).
+With higher X-ray flux and reduced detector noise, integration times can now be shortened down to approximately 1 millisecond, with frame rates exceeding \(100\,\text{Hz}\).
This reduction in integration time has two major implications for positioning requirements.
Firstly, for a given spatial sampling (``pixel size''), faster integration necessitates proportionally higher scanning velocities.
-Secondly, the shorter integration times make the measurements more susceptible to high-frequency vibrations.
+Secondly, the shorter integration times make the measurements more sensitive to high-frequency vibrations.
Therefore, not only the sample position must be stable against long-term drifts, but it must also be actively controlled to minimize vibrations, especially during dynamic fly-scan acquisitions.
\subsubsection*{Existing Nano Positioning End-Stations}
To contextualize the system developed within this thesis, a brief overview of existing strategies and technologies for high-accuracy, high-dynamics end-stations is provided.
@@ -474,7 +475,7 @@ However, when a large number of DoFs are required, the cumulative errors and lim
\caption{\label{fig:introduction_passive_stations}Example of two nano end-stations lacking online metrology for measuring the sample's position.}
\end{figure}
-The concept of using an external metrology to measure and potentially correct for positioning errors is increasing used for nano-positioning end-stations.
+The concept of using an external metrology to measure and potentially correct for positioning errors is increasingly used for nano-positioning end-stations.
Ideally, the relative position between the sample's \acrfull{poi} and the X-ray beam focus would be measured directly.
In practice, direct measurement is often impossible; instead, the sample position is typically measured relative to a reference frame associated with the focusing optics, providing an indirect measurement.
@@ -561,17 +562,17 @@ The advent of fourth-generation light sources, coupled with advancements in focu
With ID31's anticipated minimum beam dimensions of approximately \(200\,\text{nm}\times 100\,\text{nm}\), the primary experimental objective is maintaining the sample's \acrshort{poi} within this beam.
This necessitates peak-to-peak positioning errors below \(200\,\text{nm}\) in \(D_y\) and \(200\,\text{nm}\) in \(D_z\), corresponding to \acrfull{rms} errors of \(30\,\text{nm}\) and \(15\,\text{nm}\), respectively.
Additionally, the \(R_y\) tilt angle error must remain below \(0.1\,\text{mdeg}\) (\(250\,\text{nrad RMS}\)).
-Given the high frame rates of modern detectors, these specified positioning errors must be maintained even when considering high-frequency vibrations.
+Given the high frame rates of modern detectors, these specified positioning errors must be maintained even when considering high-frequency vibrations (typically up to \(1\,\text{kHz}\)).
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, 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 existing micro-station, despite being composed of high-performance stages, has a positioning accuracy limited to approximately \(10\,\upmu m\) and \(10\,\upmu\text{rad}\) (peak to peak) due to inherent factors such as backlash, thermal expansion, imperfect guiding, and vibrations.
The primary objective of this project is therefore defined as enhancing the positioning accuracy and stability of the ID31 micro-station by roughly two orders of magnitude, to fully leverage the capabilities offered by the ESRF-EBS source and modern detectors, without compromising its existing mobility and payload capacity.
\paragraph{The Nano Active Stabilization System Concept}
To address these challenges, the concept of a \acrfull{nass} is proposed.
-As schematically illustrated in Figure~\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).
+As schematically illustrated in Figure~\ref{fig:introduction_nass_concept_schematic}, the \acrshort{nass} comprises three 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.
@@ -793,10 +794,10 @@ In this section, a uniaxial model of the micro-station is tuned to match measure
\paragraph{Measured Dynamics}
The measurement setup is schematically shown in Figure~\ref{fig:uniaxial_ustation_meas_dynamics_schematic} where two vertical hammer hits are performed, one on the Granite (force \(F_{g}\)) and the other on the positioning hexapod's top platform (force \(F_{h}\)).
-The vertical inertial motion of the granite \(x_{g}\) and the top platform of the positioning hexapod \(x_{h}\) are measured using geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 1\,\text{Hz}\)}.
+The vertical inertial motion of the granite \(x_{g}\) and the top platform of the positioning hexapod \(x_{h}\) are measured using geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 1\,\text{Hz}\).}.
Three \acrfullpl{frf} were computed: one from \(F_{h}\) to \(x_{h}\) (i.e., the compliance of the micro-station), one from \(F_{g}\) to \(x_{h}\) (or from \(F_{h}\) to \(x_{g}\)) and one from \(F_{g}\) to \(x_{g}\).
-Due to the poor coherence at low frequencies, these \acrlongpl{frf} will only be shown between 20 and \(200\,\text{Hz}\) (solid lines in Figure~\ref{fig:uniaxial_comp_frf_meas_model}).
+Due to the poor coherence\footnote{Coherence is a statistical measure (ranging from \(0\) to \(1\)) used in system identification to assess how well the output of a linear system can be predicted from its input. Values near \(1\) indicate strong linear correlation, while noise or non-linearities reduce coherence and indicate poor data quality.} at low frequencies, these \acrlongpl{frf} will only be shown between 20 and \(200\,\text{Hz}\) (solid lines in Figure~\ref{fig:uniaxial_comp_frf_meas_model}).
\begin{figure}[htbp]
\begin{subfigure}{0.69\textwidth}
@@ -825,6 +826,7 @@ The damping coefficients were tuned to match the damping identified from the mea
The parameters obtained are summarized in Table~\ref{tab:uniaxial_ustation_parameters}.
\begin{table}[htbp]
+\caption{\label{tab:uniaxial_ustation_parameters}Physical parameters used for the micro-station uniaxial model.}
\centering
\begin{tabularx}{0.6\linewidth}{Xccc}
\toprule
@@ -835,8 +837,6 @@ Hexapod & \(m_h = 15\,\text{kg}\) & \(k_h = 61\,\text{N}/\upmu\text{m}\) & \(c_h
Granite & \(m_g = 2500\,\text{kg}\) & \(k_g = 950\,\text{N}/\upmu\text{m}\) & \(c_g = 250\,\frac{\text{kN}}{\text{m/s}}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:uniaxial_ustation_parameters}Physical parameters used for the micro-station uniaxial model.}
-
\end{table}
Two disturbances are considered which are shown in red: the floor motion \(x_f\) and the stage vibrations represented by \(f_t\).
@@ -852,7 +852,7 @@ More accurate models will be used later on.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=0.8]{figs/uniaxial_comp_frf_meas_model.png}
-\caption{\label{fig:uniaxial_comp_frf_meas_model}Comparison of the measured FRF and the uniaxial model dynamics.}
+\caption{\label{fig:uniaxial_comp_frf_meas_model}Comparison of the measured Frequency Response Functions (FRF) and the uniaxial model dynamics.}
\end{figure}
\subsection{Active Platform Model}
\label{sec:uniaxial_nano_station_model}
@@ -912,7 +912,7 @@ For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~
\end{figure}
\subsection{Identification of Disturbances}
\label{sec:uniaxial_disturbances}
-To quantify disturbances (red signals in Figure~\ref{fig:uniaxial_model_micro_station_nass}), three geophones\footnote{Mark Product L-22D geophones are used with a sensitivity of \(88\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 2\,\text{Hz}\)} are used.
+To quantify disturbances (red signals in Figure~\ref{fig:uniaxial_model_micro_station_nass}), three geophones\footnote{Mark Product L-22D geophones are used with a sensitivity of \(88\,\frac{V}{\text{m/s}}\) and a natural frequency of \(\approx 2\,\text{Hz}\).} are used.
One is located on the floor, another one on the granite, and the last one on the positioning hexapod's top platform (see Figure~\ref{fig:uniaxial_ustation_meas_disturbances}).
The geophone located on the floor was used to measure the floor motion \(x_f\) while the other two geophones were used to measure vibrations introduced by scanning of the \(T_y\) stage and \(R_z\) stage (see Figure~\ref{fig:uniaxial_ustation_dynamical_id_setup}).
@@ -933,7 +933,7 @@ The geophone located on the floor was used to measure the floor motion \(x_f\) w
\end{figure}
\paragraph{Ground Motion}
To acquire the geophone signals, the measurement setup shown in Figure~\ref{fig:uniaxial_geophone_meas_chain} is used.
-The voltage generated by the geophone is amplified using a low noise voltage amplifier\footnote{DLPVA-100-B from Femto with a voltage input noise is \(2.4\,\text{nV}/\sqrt{\text{Hz}}\)} with a gain of \(60\,\text{dB}\) before going to the \acrfull{adc}.
+The voltage generated by the geophone is amplified using a low noise voltage amplifier\footnote{DLPVA-100-B from Femto with a voltage input noise is \(2.4\,\text{nV}/\sqrt{\text{Hz}}\).} with a gain of \(60\,\text{dB}\) before going to the \acrfull{adc}.
This is done to improve the signal-to-noise ratio.
To reconstruct the displacement \(x_f\) from the measured voltage \(\hat{V}_{x_f}\), the transfer function of the measurement chain from \(x_f\) to \(\hat{V}_{x_f}\) needs to be estimated.
@@ -1015,7 +1015,7 @@ This is done for two extreme sample masses \(m_s = 1\,\text{kg}\) and \(m_s = 50
The obtained sensitivity to disturbances for the three active platform stiffnesses are shown in Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses} for the sample mass \(m_s = 1\,\text{kg}\) (the same conclusions can be drawn with \(m_s = 50\,\text{kg}\)):
\begin{itemize}
\item The soft active platform is more sensitive to forces applied on the sample (cable forces for instance), which is expected due to its lower stiffness (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs})
-\item Between the suspension mode of the active platform (here at \(5\,\text{Hz}\)) and the first mode of the micro-station (here at \(70\,\text{Hz}\)), the disturbances induced by the stage vibrations are filtered out (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft})
+\item Between the suspension mode\footnote{In this work, the ``suspension mode'' of a platform refers to a low-frequency vibration mode in which the supported payload behaves as a rigid body, while the platform acts as a compliant support.} of the active platform (here at \(5\,\text{Hz}\)) and the first mode of the micro-station (here at \(70\,\text{Hz}\)), the disturbances induced by the stage vibrations are filtered out (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft})
\item Above the suspension mode of the active platform, the sample's inertial motion is unaffected by the floor motion; therefore, the sensitivity to floor motion is close to \(1\) (Figure~\ref{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf})
\end{itemize}
@@ -1225,7 +1225,7 @@ All three active damping approaches can lead to \emph{critical damping} of the a
There is even some damping authority on micro-station modes in the following cases:
\begin{description}
\item[{IFF with a stiff active platform (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_stiff})}] This can be understood from the mechanical equivalent of IFF shown in Figure~\ref{fig:uniaxial_active_damping_iff_equiv} considering an high stiffness \(k\).
-The micro-station top platform is connected to an inertial mass (the active platform) through a damper, which dampens the micro-station suspension suspension mode.
+The micro-station top platform is connected to an inertial mass (the active platform) through a damper, which dampens the micro-station suspension mode.
\item[{DVF with a stiff active platform (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_stiff})}] In that case, the ``sky hook damper'' (see mechanical equivalent of \acrshort{dvf} in Figure~\ref{fig:uniaxial_active_damping_dvf_equiv}) is connected to the micro-station top platform through the stiff active platform.
\item[{RDC with a soft active platform (Figure~\ref{fig:uniaxial_root_locus_damping_techniques_micro_station_mode})}] At the frequency of the micro-station mode, the active platform top mass behaves as an inertial reference because the suspension mode of the soft active platform is at much lower frequency.
The micro-station and the active platform masses are connected through a large damper induced by \acrshort{rdc} (see mechanical equivalent in Figure~\ref{fig:uniaxial_active_damping_rdc_equiv}) which allows some damping of the micro-station.
@@ -1359,6 +1359,7 @@ It is difficult to conclude on the best active damping strategy for the \acrfull
The one used will be determined by the use of more accurate models and will depend on which is easiest to implement in practice
\begin{table}[htbp]
+\caption{\label{tab:comp_active_damping}Comparison of active damping strategies for the NASS.}
\centering
\begin{tabularx}{0.9\linewidth}{Xccc}
\toprule
@@ -1375,8 +1376,6 @@ The one used will be determined by the use of more accurate models and will depe
\(x_f\) \textbf{Disturbance} & \(\nearrow\) at low frequency & \(\searrow\) near resonance & \(\nearrow\) at low frequency\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:comp_active_damping}Comparison of active damping strategies for the NASS.}
-
\end{table}
\subsection{Position Feedback Controller}
\label{sec:uniaxial_position_control}
@@ -1459,7 +1458,7 @@ This makes the robust design of the controller more complicated.
Position feedback controllers are designed for each active platform such that it is stable for all considered sample masses with similar stability margins (see Nyquist plots in Figure~\ref{fig:uniaxial_nyquist_hac}).
An arbitrary minimum modulus margin of \(0.25\) was chosen when designing the controllers.
-These \acrfullpl{hac} are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a \acrfull{lpf} to increase the robustness to high frequency dynamics.
+These \acrfullpl{hac} are generally composed of a lag at low frequency for disturbance rejection, a lead to increase the phase margin near the crossover frequency, and a \acrfull{lpf} to increase the robustness to high-frequency dynamics.
The controllers used for the three active platform are shown in Equation~\eqref{eq:uniaxial_hac_formulas}, and the parameters used are summarized in Table~\ref{tab:uniaxial_feedback_controller_parameters}.
\begin{subequations} \label{eq:uniaxial_hac_formulas}
@@ -1480,6 +1479,7 @@ K_{\text{stiff}}(s) &= g \cdot
\end{subequations}
\begin{table}[htbp]
+\caption{\label{tab:uniaxial_feedback_controller_parameters}Parameters used for the position feedback controllers.}
\centering
\begin{tabularx}{0.75\linewidth}{Xccc}
\toprule
@@ -1491,8 +1491,6 @@ K_{\text{stiff}}(s) &= g \cdot
\textbf{LPF} & \(\omega_l = 200\,\text{Hz}\) & \(\omega_l = 300\,\text{Hz}\) & \(\omega_l = 500\,\text{Hz}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:uniaxial_feedback_controller_parameters}Parameters used for the position feedback controllers.}
-
\end{table}
The loop gains corresponding to the designed \acrlongpl{hac} for the three active platform are shown in Figure~\ref{fig:uniaxial_loop_gain_hac}.
@@ -1663,7 +1661,7 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev
\end{subfigure}
\caption{\label{fig:uniaxial_effect_support_compliance_neglected}Obtained transfer functions from \(F\) to \(L^{\prime}\) when neglecting support compliance.}
\end{figure}
-\paragraph{Effect of support compliance on \(L/F\)}
+\paragraph{Effect of support compliance on \(L/f\)}
Some support compliance is now added and the model shown in Figure~\ref{fig:uniaxial_support_compliance_test_system} is used.
The parameters of the support (i.e., \(m_{\mu}\), \(c_{\mu}\) and \(k_{\mu}\)) are chosen to match the vertical mode at \(70\,\text{Hz}\) seen on the micro-station (Figure~\ref{fig:uniaxial_comp_frf_meas_model}).
@@ -1732,6 +1730,7 @@ For the \acrfull{nass}, having the suspension mode of the active platform at low
Note that the observations made in this section are also affected by the ratio between the support mass \(m_{\mu}\) and the active platform mass \(m_n\) (the effect is more pronounced when the ratio \(m_n/m_{\mu}\) increases).
\begin{table}[htbp]
+\caption{\label{tab:uniaxial_effect_compliance}Impact of the support dynamics on the plant dynamics.}
\centering
\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
@@ -1741,41 +1740,39 @@ Note that the observations made in this section are also affected by the ratio b
\(L/F\) & large & large & small\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:uniaxial_effect_compliance}Impact of the support dynamics on the plant dynamics.}
-
\end{table}
\subsection{Effect of Payload Dynamics}
\label{sec:uniaxial_payload_dynamics}
-Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic}).
+Up to this section, the sample was modeled as a mass rigidly fixed to the active platform (as shown in Figure~\ref{fig:uniaxial_payload_dynamics_rigid_schematic}).
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~\ref{fig:uniaxial_paylaod_dynamics_schematic} are used.
+To study the effect of the sample dynamics, the models shown in Figure~\ref{fig:uniaxial_payload_dynamics_schematic} are used.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_paylaod_dynamics_rigid_schematic.png}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_payload_dynamics_rigid_schematic.png}
\end{center}
-\subcaption{\label{fig:uniaxial_paylaod_dynamics_rigid_schematic}Rigid payload}
+\subcaption{\label{fig:uniaxial_payload_dynamics_rigid_schematic}Rigid payload}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_paylaod_dynamics_schematic.png}
+\includegraphics[scale=1,scale=1]{figs/uniaxial_payload_dynamics_schematic.png}
\end{center}
-\subcaption{\label{fig:uniaxial_paylaod_dynamics_schematic}Flexible payload}
+\subcaption{\label{fig:uniaxial_payload_dynamics_schematic}Flexible payload}
\end{subfigure}
\caption{\label{fig:uniaxial_payload_dynamics_models}Models used to study the effect of payload dynamics.}
\end{figure}
\paragraph{Impact on Plant Dynamics}
-To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_rigid_schematic} is used.
-Then ``flexible'' payload whose model is shown in Figure~\ref{fig:uniaxial_paylaod_dynamics_schematic} are considered.
+To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure~\ref{fig:uniaxial_payload_dynamics_rigid_schematic} is used.
+Then ``flexible'' payload whose model is shown in Figure~\ref{fig:uniaxial_payload_dynamics_schematic} are considered.
The resonances of the payload are set at \(\omega_s = 20\,\text{Hz}\) and at \(\omega_s = 200\,\text{Hz}\) while its mass is either \(m_s = 1\,\text{kg}\) or \(m_s = 50\,\text{kg}\).
-The transfer functions from the active platform force \(f\) to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_soft_nano_hexapod}.
+The transfer functions from the active platform force \(F\) to the motion of the active platform top platform are computed for all the above configurations and are compared for a soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_soft_nano_hexapod}.
It can be seen that the mode of the sample adds an anti-resonance followed by a resonance (zero/pole pattern).
The frequency of the anti-resonance corresponds to the ``free'' resonance of the sample \(\omega_s = \sqrt{k_s/m_s}\).
-The flexibility of the sample also changes the high frequency gain (the mass line is shifted from \(\frac{1}{(m_n + m_s)s^2}\) to \(\frac{1}{m_ns^2}\)).
+The flexibility of the sample also changes the high-frequency gain (the mass line is shifted from \(\frac{1}{(m_n + m_s)s^2}\) to \(\frac{1}{m_ns^2}\)).
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -1796,7 +1793,7 @@ The flexibility of the sample also changes the high frequency gain (the mass lin
The same transfer functions are now compared when using a stiff active platform (\(k_n = 100\,\text{N}/\upmu\text{m}\)) in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod}.
In this case, the sample's resonance \(\omega_s\) is smaller than the active platform resonance \(\omega_n\).
This changes the zero/pole pattern to a pole/zero pattern (the frequency of the zero still being equal to \(\omega_s\)).
-Even though the added sample's flexibility still shifts the high frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}).
+Even though the added sample's flexibility still shifts the high-frequency mass line as for the soft active platform, the dynamics below the active platform resonance is much less impacted, even when the sample mass is high and when the sample resonance is at low frequency (see yellow curve in Figure~\ref{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}).
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@@ -1857,7 +1854,7 @@ What happens is that above \(\omega_s\), even though the motion \(d\) can be con
Payload dynamics is usually a major concern when designing a positioning system.
In this section, the impact of the sample dynamics on the plant was found to vary with the sample mass and the relative resonance frequency of the sample \(\omega_s\) and of the active platform \(\omega_n\).
-The larger the sample mass, the larger the effect (i.e., change of high frequency gain, appearance of additional resonances and anti-resonances).
+The larger the sample mass, the larger the effect (i.e., change of high-frequency gain, appearance of additional resonances and anti-resonances).
A zero/pole pattern is observed if \(\omega_s > \omega_n\) and a pole/zero pattern if \(\omega_s > \omega_n\).
Such additional dynamics can induce stability issues depending on their position relative to the desired feedback bandwidth, as explained in~\cite[Section 4.2]{rankers98_machin}.
The general conclusion is that the stiffer the active platform, the less it is impacted by the payload's dynamics, which would make the feedback controller more robust to a change of payload.
@@ -1922,7 +1919,7 @@ After the dynamics of this system is studied, the objective will be to dampen th
To obtain the equations of motion for the system represented in Figure~\ref{fig:rotating_3dof_model_schematic}, the Lagrangian equation~\eqref{eq:rotating_lagrangian_equations} is used.
\(L = T - V\) is the Lagrangian, \(T\) the kinetic coenergy, \(V\) the potential energy, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\).
These terms are derived in~\eqref{eq:rotating_energy_functions_lagrange}.
-Note that the equation of motion corresponding to constant rotation along \(\vec{i}_w\) is disregarded because this motion is imposed by the rotation stage.
+Note that the equation of motion corresponding to constant rotation around \(\vec{i}_w\) is disregarded because this motion is imposed by the rotation stage.
\begin{equation}\label{eq:rotating_lagrangian_equations}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
@@ -1930,7 +1927,7 @@ Note that the equation of motion corresponding to constant rotation along \(\vec
\begin{equation} \label{eq:rotating_energy_functions_lagrange}
\begin{aligned}
- T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \quad Q_1 = F_u, \quad Q_2 = F_v, \\
+ T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \quad Q_u = F_u, \quad Q_v = F_v, \\
V &= \frac{1}{2} k \big( {d_u}^2 + {d_v}^2 \big), \quad D = \frac{1}{2} c \big( \dot{d}_u{}^2 + \dot{d}_v{}^2 \big)
\end{aligned}
\end{equation}
@@ -2166,7 +2163,7 @@ As explained in~\cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv
Whereas collocated IFF is usually associated with unconditional stability~\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~\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~\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.
+The control system is thus canceling the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability.
\subsection{Integral Force Feedback with a High-Pass Filter}
\label{sec:rotating_iff_pseudo_int}
@@ -2492,7 +2489,7 @@ This is a useful metric when disturbances are directly applied to the payload.
Here, it is defined as the transfer function from external forces applied on the payload along \(\vec{i}_x\) to the displacement of the payload along the same direction.
Very similar results were obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure~\ref{fig:rotating_comp_techniques_trans_compliance}).
-Using IFF degrades the compliance at low frequencies, whereas using relative damping control degrades the transmissibility at high frequencies.
+Using IFF degrades the compliance at low frequencies, whereas using relative damping control degrades the transmissibility at high-frequencies.
This is very well known characteristics of these common active damping techniques that hold when applied to rotating platforms.
\begin{figure}[htbp]
@@ -2580,6 +2577,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\end{figure}
\begin{table}[htbp]
+\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping \(\xi_{\text{opt}}\) of the two modes is also shown.}
\centering
\begin{tabularx}{0.3\linewidth}{Xccc}
\toprule
@@ -2590,8 +2588,6 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\(100\,\text{N}/\upmu\text{m}\) & 500 & 3775 & 0.94\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping \(\xi_{\text{opt}}\) of the two modes is also shown.}
-
\end{table}
\paragraph{Optimal IFF with Parallel Stiffness}
For each considered active platform stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total active platform stiffness).
@@ -2625,8 +2621,7 @@ The corresponding optimal controller gains and achievable damping are summarized
\(1\,\text{N}/\upmu\text{m}\) & \(0.01\,\text{N}/\upmu\text{m}\) & 465 & 0.97\\
\(100\,\text{N}/\upmu\text{m}\) & \(1\,\text{N}/\upmu\text{m}\) & 4624 & 0.99\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses}
\end{minipage}
\paragraph{Optimal Relative Motion Control}
@@ -2651,8 +2646,7 @@ The gain is chosen such that 99\% of modal damping is obtained (obtained gains a
\(1\,\text{N}/\upmu\text{m}\) & 8200 & 0.99\\
\(100\,\text{N}/\upmu\text{m}\) & 80000 & 0.99\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the acrlong:rdc}
\end{minipage}
\paragraph{Comparison of the Obtained Damped Plants}
@@ -2846,7 +2840,7 @@ The dynamics of the soft active platform (\(k_n = 0.01\,\text{N}/\upmu\text{m}\)
In addition, the attainable damping ratio of the soft active platform when using \acrshort{iff} is limited by gyroscopic effects.
To be closer to the \acrlong{nass} dynamics, the limited compliance of the micro-station has been considered.
-Results are similar to those of the uniaxial model except that come complexity is added for the soft active platform due to the spindle's rotation.
+Results are similar to those of the uniaxial model except that some complexity is added for the soft active platform due to the spindle's rotation.
For the moderately stiff active platform (\(k_n = 1\,\text{N}/\upmu\text{m}\)), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft active platform that showed better results with the uniaxial model.
\section{Micro Station - Modal Analysis}
\label{sec:modal}
@@ -2857,7 +2851,7 @@ Although the inertia of each solid body can easily be estimated from its geometr
Experimental modal analysis will be used to tune the model, and to verify that a multi-body model can accurately represent the dynamics of the micro-station.
The tuning approach for the multi-body model based on measurements is illustrated in Figure~\ref{fig:modal_vibration_analysis_procedure}.
-First, a \emph{response model} is obtained, which corresponds to a set of \acrshortpl{frf} computed from experimental measurements.
+First, a \emph{response model} is obtained, which corresponds to a set of \acrfullpl{frf} computed from experimental measurements.
From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considered solid bodies and the springs and dampers connecting the solid bodies.
@@ -2912,7 +2906,7 @@ These accelerometers were glued to the micro-station using a thin layer of wax f
\caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis.}
\end{figure}
-Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,\text{mV/N}\) and measurement range of \(2\,\text{kN}\)} (figure~\ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled manner.
+Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,\text{mV/N}\) and measurement range of \(2\,\text{kN}\).} (figure~\ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled manner.
Tests were conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved.
@@ -2985,8 +2979,7 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs
(3) Hexapod & 64 & 64 & -270\\
(4) Hexapod & 64 & -64 & -270\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
\end{minipage}
@@ -3167,6 +3160,7 @@ From the 3D model, the position of the \acrlong{com} of each solid body is compu
The position of each accelerometer with respect to the \acrlong{com} of the corresponding solid body can easily be determined.
\begin{table}[htbp]
+\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the \acrlong{poi}.}
\centering
\begin{tabularx}{0.45\linewidth}{Xccc}
\toprule
@@ -3180,8 +3174,6 @@ Spindle & \(0\) & \(0\) & \(-580\,\text{mm}\)\\
Positioning Hexapod & \(-4\,\text{mm}\) & \(6\,\text{mm}\) & \(-319\,\text{mm}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the \acrlong{poi}.}
-
\end{table}
Using~\eqref{eq:modal_cart_to_acc}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response at the \acrlong{com} of each solid body \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) can be computed from the initial \acrshort{frf} matrix \(\bm{H}\).
@@ -3279,14 +3271,13 @@ Mode & Frequency & Damping\\
15 & \(150.5\,\text{Hz}\) & \(2.4\,\%\)\\
16 & \(165.4\,\text{Hz}\) & \(1.4\,\%\)\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
\end{minipage}
\subsubsection{Modal Parameter Extraction}
\label{ssec:modal_parameter_extraction}
-Generally, modal identification is using curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
+Generally, modal identification involves curve-fitting a theoretical model to the measured \acrshort{frf} data.
However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many \acrshort{frf} plots all obtained from the same structure.
Here, the last method is used because it provides a unique and consistent model.
@@ -3423,7 +3414,7 @@ The kinematics of the micro-station (i.e. how the motion of the stages are combi
Then, the multi-body model is presented and tuned to match the measured dynamics of the micro-station (Section~\ref{sec:ustation_modeling}).
Disturbances affecting the positioning accuracy also need to be modeled properly.
-To do so, the effects of these disturbances were first measured experimental and then injected into the multi-body model (Section~\ref{sec:ustation_disturbances}).
+To do so, the effects of these disturbances were first measured experimentally and then injected into the multi-body model (Section~\ref{sec:ustation_disturbances}).
To validate the accuracy of the micro-station model, ``real world'' experiments are simulated and compared with measurements in Section~\ref{sec:ustation_experiments}.
\subsection{Micro-Station Kinematics}
@@ -3565,7 +3556,7 @@ The rotation matrix can be used to express the coordinates of a point \(P\) in a
{}^AP = {}^A\bm{R}_B {}^BP
\end{equation}
-For rotations along \(x\), \(y\) or \(z\) axis, the formulas of the corresponding rotation matrices are given in Equation~\eqref{eq:ustation_rotation_matrices_xyz}.
+For rotations around \(x\), \(y\) or \(z\) axis, the formulas of the corresponding rotation matrices are given in Equation~\eqref{eq:ustation_rotation_matrices_xyz}.
\begin{subequations}\label{eq:ustation_rotation_matrices_xyz}
\begin{align}
@@ -3791,6 +3782,7 @@ The springs and dampers values were first estimated from the joint/stage specifi
The spring values are summarized in Table~\ref{tab:ustation_6dof_stiffness_values}.
\begin{table}[htbp]
+\caption{\label{tab:ustation_6dof_stiffness_values}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}.}
\centering
\begin{tabularx}{0.9\linewidth}{Xcccccc}
\toprule
@@ -3803,8 +3795,6 @@ Spindle & \(700\,\text{N}/\upmu\text{m}\) & \(700\,\text{N}/\upmu\text{m}\) & \(
Hexapod & \(10\,\text{N}/\upmu\text{m}\) & \(10\,\text{N}/\upmu\text{m}\) & \(100\,\text{N}/\upmu\text{m}\) & \(1.5\,\text{Nm/rad}\) & \(1.5\,\text{Nm/rad}\) & \(0.27\,\text{Nm/rad}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:ustation_6dof_stiffness_values}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}.}
-
\end{table}
\subsubsection{Comparison with the Measured Dynamics}
\label{ssec:ustation_model_comp_dynamics}
@@ -4269,6 +4259,7 @@ Table~\ref{tab:nhexa_sample_stages} provides an overview of existing end-station
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.
\begin{table}[!ht]
+\caption{\label{tab:nhexa_sample_stages}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.}
\centering
\begin{tabularx}{0.8\linewidth}{ccccc}
\toprule
@@ -4314,8 +4305,6 @@ Tilt-Stage & \(R_y: \pm 3\,\text{deg}\) & & & \\
Translation Stage & \(D_y: \pm 10\,\text{mm}\) & & & \\
\bottomrule
\end{tabularx}
-\caption{\label{tab:nhexa_sample_stages}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.}
-
\end{table}
The first key distinction of the NASS is in the continuous rotation of the active vibration platform.
@@ -4355,7 +4344,7 @@ These limitations generally make serial architectures unsuitable for nano-positi
In contrast, parallel mechanisms, which connect the mobile platform to the fixed base through multiple parallel struts, offer several advantages for precision positioning.
Their closed-loop kinematic structure provides inherently higher structural stiffness, as the platform is simultaneously supported by multiple struts~\cite{taghirad13_paral}.
Although parallel mechanisms typically exhibit limited workspace compared to serial architectures, this limitation is not critical for NASS given its modest stroke requirements.
-Numerous parallel kinematic architectures have been developed~\cite{dong07_desig_precis_compl_paral_posit} to address various positioning requirements, with designs varying based on the desired \acrshortpl{dof} and specific application constraints.
+Numerous parallel kinematic architectures have been developed~\cite{dong07_desig_precis_compl_paral_posit} to address various positioning requirements, with designs varying based on the intended \acrshortpl{dof} and specific application constraints.
Furthermore, hybrid architectures combining both serial and parallel elements have been proposed~\cite{shen19_dynam_analy_flexur_nanop_stage}, as illustrated in Figure~\ref{fig:nhexa_serial_parallel_examples}, offering potential compromises between the advantages of both approaches.
\begin{figure}[h!tbp]
@@ -4375,7 +4364,7 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha
\end{figure}
After evaluating the different options, the Stewart platform architecture was selected for several reasons.
-In addition to providing control over all required \acrshortpl{dof}, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints.
+In addition to allow control over all required \acrshortpl{dof}, its compact design and predictable dynamic characteristics make it particularly suitable for nano-positioning when combined with flexible joints.
Stewart platforms have been implemented in a wide variety of configurations, as illustrated in Figure~\ref{fig:nhexa_stewart_examples}, which shows two distinct implementations: one implementing piezoelectric actuators for nano-positioning applications, and another based on voice coil actuators for vibration isolation.
These examples demonstrate the architecture's versatility in terms of geometry, actuator selection, and scale, all of which can be optimized for specific applications.
Furthermore, the successful implementation of Integral Force Feedback (IFF) control on Stewart platforms has been well documented~\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.
@@ -4421,7 +4410,7 @@ 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~\ref{fig:nhexa_stewart_architecture}).
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\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical)}.
+The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical).}.
\begin{figure}[htbp]
\centering
@@ -4462,7 +4451,7 @@ For each strut \(i\) (illustrated in Figure~\ref{fig:nhexa_stewart_loop_closure}
{}^A\bm{P}_B = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - \underbrace{{}^B\bm{b}_i}_{{}^A\bm{R}_B {}^B\bm{b}_i} \quad \text{for } i=1 \text{ to } 6
\end{equation}
-This equation links the pose\footnote{The \emph{pose} represents the position and orientation of an object} 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 \(l_i {}^A\hat{\bm{s}}_i\):
+This equation links the pose\footnote{The \emph{pose} represents the position and orientation of an object.} 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 \(l_i {}^A\hat{\bm{s}}_i\):
\begin{figure}[htbp]
\centering
@@ -4497,7 +4486,7 @@ While the previously derived kinematic relationships are essential for position
As discussed in Section~\ref{ssec:nhexa_stewart_platform_kinematics}, the strut lengths \(\bm{\mathcal{L}}\) and the platform pose \(\bm{\mathcal{X}}\) are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts.
-By taking the time derivative of the position loop close~\eqref{eq:nhexa_loop_closure}, equation~\eqref{eq:nhexa_loop_closure_velocity} is obtained\footnote{Such equation is called the \emph{velocity loop closure}}.
+By taking the time derivative of the position loop close~\eqref{eq:nhexa_loop_closure}, equation~\eqref{eq:nhexa_loop_closure_velocity} is obtained\footnote{Such equation is called the \emph{velocity loop closure}.}.
\begin{equation}\label{eq:nhexa_loop_closure_velocity}
{}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0}
@@ -4559,7 +4548,7 @@ For a series of platform positions, the exact strut lengths are computed using t
These strut lengths are then used with the Jacobian to estimate the platform pose~\eqref{eq:nhexa_forward_kinematics_approximate}, from which the error between the estimated and true poses can be calculated, both in terms of position \(\epsilon_D\) and orientation \(\epsilon_R\).
For motion strokes from \(1\,\upmu\text{m}\) to \(10\,\text{mm}\), the errors are estimated for all direction of motion, and the worst case errors are shown in Figure~\ref{fig:nhexa_forward_kinematics_approximate_errors}.
-The results demonstrate that for displacements up to approximately \(1\,\%\) of the hexapod's size (which corresponds to \(100\,\upmu\text{m}\) as the size of the Stewart platform is here \(\approx 100\,\text{mm}\)), the Jacobian approximation provides excellent accuracy.
+The results demonstrate that for displacements up to approximately \(0.1\,\%\) of the hexapod's size (which corresponds to \(100\,\upmu\text{m}\) as the size of the Stewart platform is here \(\approx 100\,\text{mm}\)), the Jacobian approximation provides excellent accuracy.
Since the maximum required stroke of the active platform (\(\approx 100\,\upmu\text{m}\)) is three orders of magnitude smaller than its overall size (\(\approx 100\,\text{mm}\)), the Jacobian matrix can be considered constant throughout the workspace.
It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy.
@@ -4748,8 +4737,7 @@ From these parameters, key kinematic properties can be derived: the strut orient
\({}^M\bm{b}_5\) & \(-78\) & \(78\) & \(-20\)\\
\({}^M\bm{b}_6\) & \(-106\) & \(28\) & \(-20\)\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]}
\end{minipage}
\paragraph{Inertia of Plates}
@@ -4795,8 +4783,7 @@ This modular approach to actuator modeling allows for future refinements as the
\(c_a\) & \(50\,\text{Ns}/\text{m}\)\\
\(k_p\) & \(0.05\,\text{N}/\upmu\text{m}\)\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters}
\end{minipage}
\subsubsection{Validation of the Multi-body Model}
@@ -4863,7 +4850,7 @@ This reduction from six to four observable modes is attributed to the system's s
The system's behavior can be characterized in three frequency regions.
At low frequencies, well below the first resonance, the plant demonstrates good decoupling between actuators, with the response dominated by the strut stiffness: \(\bm{G}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}}^{-1}\).
In the mid-frequency range, the system exhibits coupled dynamics through its resonant modes, reflecting the complex interactions between the platform's degrees of freedom.
-At high frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: \(\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-\intercal} \bm{J}^{\intercal} \frac{-1}{\omega^2}\)
+At high-frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: \(\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-\intercal} \bm{J}^{\intercal} \frac{-1}{\omega^2}\)
The force sensor transfer functions, shown in Figure~\ref{fig:nhexa_multi_body_plant_fm}, display characteristics typical of collocated actuator-sensor pairs.
Each actuator's transfer function to its associated force sensor exhibits alternating complex conjugate poles and zeros.
@@ -5233,7 +5220,9 @@ The external metrology system measures the sample position relative to the fixed
Due to the system's symmetry, this metrology provides measurements for five \acrshortpl{dof}: three translations (\(D_x\), \(D_y\), \(D_z\)) and two rotations (\(R_x\), \(R_y\)).
The sixth \acrshort{dof} (\(R_z\)) is still required to compute the errors in the frame of the active platform struts (i.e. to compute the active platform inverse kinematics).
-This \(R_z\) rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology, which consists of relative motion sensors in each strut (note that the positioning hexapod is not used for \(R_z\) rotation, and is therefore ignored for \(R_z\) estimation).
+This \(R_z\) rotation is estimated by combining measurements from the spindle encoder and the active platform's internal metrology.
+The active platform's metrology consists of relative motion sensors in each strut, such that the \(R_z\) rotation of the active platform can be estimated by solving the forward kinematics \eqref{eq:nhexa_forward_kinematics_approximate}.
+Note that the positioning hexapod is not used for \(R_z\) rotation, and is therefore ignored for \(R_z\) estimation.
The measured sample pose is represented by the homogeneous transformation matrix \(\bm{T}_{\text{sample}}\), as shown in equation~\eqref{eq:nass_sample_pose}.
@@ -5520,7 +5509,7 @@ The current approach of controlling the position in the strut frame is inadequat
\label{ssec:nass_hac_controller}
A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure~\ref{fig:nass_hac_plants}), and achievement of sufficient bandwidth (targeted at \(10\,\text{Hz}\)) for high performance operation.
-The controller structure is defined in Equation~\eqref{eq:nass_robust_hac}, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes.
+The controller structure is defined in Equation~\eqref{eq:nass_robust_hac}, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high-frequency modes.
\begin{equation}\label{eq:nass_robust_hac}
K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right)
@@ -5940,8 +5929,8 @@ In that case, the obtained stiffness matrix linearly depends on the strut stiffn
As shown by equation~\eqref{eq:detail_kinematics_stiffness_matrix_simplified}, the translation stiffnesses (the \(3 \times 3\) top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal}\).
In the extreme case where all struts are vertical (\(s_i = [0\ 0\ 1]\)), a vertical stiffness of \(6k\) is achieved, but with null stiffness in the horizontal directions.
-If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3\), resulting in well-distributed stiffness along all directions.
-This configuration corresponds to the cubic architecture presented in Section~\ref{sec:detail_kinematics_cubic}.
+If two struts are oriented along the X axis, two struts along the Y axis, and two struts along the Z axis, then \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3\) and the stiffness is well distributed along all directions.
+This configuration corresponds to the cubic architecture, that is presented in Section~\ref{sec:detail_kinematics_cubic}.
When the struts are oriented more vertically, as shown in Figure~\ref{fig:detail_kinematics_stewart_mobility_vert_struts}, the vertical stiffness increases while the horizontal stiffness decreases.
Additionally, \(R_x\) and \(R_y\) stiffness increases while \(R_z\) stiffness decreases.
@@ -5985,6 +5974,7 @@ These results could have been easily deduced based on mechanical principles, but
These trade-offs provide important guidelines when choosing the Stewart platform geometry.
\begin{table}[htbp]
+\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness and mobility.}
\centering
\begin{tabularx}{0.65\linewidth}{Xcc}
\toprule
@@ -6001,8 +5991,6 @@ Vertical rotation mobility & \(\nearrow\) & \(\searrow\)\\
Horizontal rotation mobility & \(\searrow\) & \(\searrow\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness and mobility.}
-
\end{table}
\subsection{The Cubic Architecture}
\label{sec:detail_kinematics_cubic}
@@ -6163,7 +6151,7 @@ When relative motion sensors are integrated in each strut (measuring \(\bm{\math
\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame.}
\end{figure}
-\paragraph{Low Frequency and High Frequency Coupling}
+\paragraph{Low Frequency and High-Frequency Coupling}
As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation~\eqref{eq:detail_kinematics_transfer_function_cart}.
At low frequency, the behavior of the platform depends on the stiffness matrix~\eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
@@ -6174,7 +6162,7 @@ At low frequency, the behavior of the platform depends on the stiffness matrix~\
In Section~\ref{ssec:detail_kinematics_cubic_static}, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame \(\{B\}\) is positioned at the cube's center.
In this case, the ``Cartesian'' plant is decoupled at low frequency.
-At high frequency, the behavior is governed by the mass matrix (evaluated at frame \(\{B\}\))~\eqref{eq:detail_kinematics_transfer_function_high_freq}.
+At high-frequency, the behavior is governed by the mass matrix (evaluated at frame \(\{B\}\))~\eqref{eq:detail_kinematics_transfer_function_high_freq}.
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1}
@@ -6191,7 +6179,7 @@ To achieve a diagonal mass matrix, the \acrlong{com} of the mobile components mu
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure~\ref{fig:detail_kinematics_cubic_payload}).
Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames.
When the \(\{B\}\) frame was positioned at the \acrlong{com}, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_com}).
-Conversely, when positioned at the \acrlong{cok}, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_cok}).
+Conversely, when positioned at the \acrlong{cok}, coupling occurred at high-frequency due to the non-diagonal mass matrix (Figure~\ref{fig:detail_kinematics_cubic_cart_coupling_cok}).
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@@ -6264,7 +6252,7 @@ The second uses a non-cubic Stewart platform shown in Figure~\ref{fig:detail_kin
The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure~\ref{fig:detail_kinematics_decentralized_dL}.
As anticipated from the equations of motion from \(\bm{f}\) to \(\bm{\mathcal{L}}\) \eqref{eq:detail_kinematics_transfer_function_struts}, the \(6 \times 6\) plant is decoupled at low frequency.
-At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
+At high-frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
No significant advantage is evident for the cubic architecture (Figure~\ref{fig:detail_kinematics_cubic_decentralized_dL}) compared to the non-cubic architecture (Figure~\ref{fig:detail_kinematics_non_cubic_decentralized_dL}).
The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes.
@@ -6288,7 +6276,7 @@ The resonance frequencies differ between the two cases because the more vertical
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
The results are presented in Figure~\ref{fig:detail_kinematics_decentralized_fn}.
-The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
+The system demonstrates good decoupling at high-frequency in both cases, with no clear advantage for the cubic architecture.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@@ -6579,7 +6567,7 @@ Initially, the component is modeled in a finite element software with appropriat
Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component.
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~\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.
+Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method~\cite{craig68_coupl_subst_dynam_analy} (also known as the ``fixed-interface method''), a technique that significantly reduces the number of DoF while still presenting the main dynamical characteristics.
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~\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.
@@ -6615,8 +6603,7 @@ Nominal Stroke & \(100\,\upmu\text{m}\)\\
Blocked force & \(2100\,\text{N}\)\\
Stiffness & \(21\,\text{N}/\upmu\text{m}\)\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
@@ -6625,6 +6612,7 @@ The development of the \acrfull{fem} for the APA95ML required the knowledge of t
The finite element mesh, shown in Figure~\ref{fig:detail_fem_apa95ml_mesh}, was then generated.
\begin{table}[htbp]
+\caption{\label{tab:detail_fem_material_properties}Material properties used for FEA. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density.}
\centering
\begin{tabularx}{0.55\linewidth}{Xccc}
\toprule
@@ -6634,8 +6622,6 @@ Stainless Steel & \(190\,\text{GPa}\) & \(0.31\) & \(7800\,\text{kg}/\text{m}^3\
Piezoelectric Ceramics (PZT) & \(49.5\,\text{GPa}\) & \(0.31\) & \(7800\,\text{kg}/\text{m}^3\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_fem_material_properties}Material properties used for FEA. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density.}
-
\end{table}
The definition of interface frames constitutes a critical aspect of the model preparation.
@@ -6689,6 +6675,7 @@ Unfortunately, it is difficult to know exactly which material is used for the pi
Yet, based on the available properties of the stacks in the data-sheet (summarized in Table~\ref{tab:detail_fem_stack_parameters}), the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
\begin{table}[htbp]
+\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters.}
\centering
\begin{tabularx}{0.3\linewidth}{Xc}
\toprule
@@ -6703,14 +6690,13 @@ Length & \(20\,\text{mm}\)\\
Stack Area & \(10\times 10\,\text{mm}^2\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters.}
-
\end{table}
The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table~\ref{tab:detail_fem_piezo_properties}.
From these parameters, \(g_s = 5.1\,\text{V}/\upmu\text{m}\) and \(g_a = 26\,\text{N/V}\) were obtained.
\begin{table}[htbp]
+\caption{\label{tab:detail_fem_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuator sensitivities.}
\centering
\begin{tabularx}{0.8\linewidth}{ccX}
\toprule
@@ -6725,8 +6711,6 @@ From these parameters, \(g_s = 5.1\,\text{V}/\upmu\text{m}\) and \(g_a = 26\,\te
\(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_fem_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuator sensitivities.}
-
\end{table}
\paragraph{Identification of the APA Characteristics}
@@ -6894,10 +6878,11 @@ Moreover, using \acrshort{apa} for active damping has been successfully demonstr
Several specific \acrshort{apa} models were evaluated against the established specifications (Table~\ref{tab:detail_fem_piezo_act_models}).
The APA300ML emerged as the optimal choice.
-This selection was further reinforced by previous experience with \acrshortpl{apa} from the same manufacturer\footnote{Cedrat technologies}, and particularly by the successful validation of the modeling methodology with a similar actuator (Section~\ref{ssec:detail_fem_super_element_example}).
+This selection was further reinforced by previous experience with \acrshortpl{apa} from the same manufacturer\footnote{Cedrat technologies.}, and particularly by the successful validation of the modeling methodology with a similar actuator (Section~\ref{ssec:detail_fem_super_element_example}).
The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses.
\begin{table}[htbp]
+\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the active platform.}
\centering
\begin{tabularx}{0.9\linewidth}{Xccccc}
\toprule
@@ -6910,8 +6895,6 @@ Blocked Force \(> 100\,\text{N}\) & 127 & 546 & 201 & 376 & 139\\
Height \(< 50\,\text{mm}\) & 22 & 30 & 24 & 27 & 16\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the active platform.}
-
\end{table}
\subsubsection{APA300ML - Reduced Order Flexible Body}
\label{ssec:detail_fem_actuator_apa300ml}
@@ -6977,6 +6960,7 @@ The resulting parameters, listed in Table~\ref{tab:detail_fem_apa300ml_2dof_para
While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range.
\begin{table}[htbp]
+\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2-DoFs APA300ML model.}
\centering
\begin{tabularx}{0.25\linewidth}{cc}
\toprule
@@ -6992,8 +6976,6 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\(g_s\) & \(0.53\,\text{V}/\upmu\text{m}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2-DoFs APA300ML model.}
-
\end{table}
\begin{figure}[htbp]
@@ -7163,7 +7145,7 @@ The resulting dynamics (Figure~\ref{fig:detail_fem_joints_axial_stiffness_plants
The force-sensor (IFF) plant exhibits minimal sensitivity to axial compliance, as evidenced by both \acrshortpl{frf} (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_plant}) and root locus analysis (Figure~\ref{fig:detail_fem_joints_axial_stiffness_iff_locus}).
-However, the transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) demonstrates significant effects: internal strut modes appear at high frequencies, introducing substantial cross-coupling between axes.
+However, the transfer function from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) demonstrates significant effects: internal strut modes appear at high-frequencies, introducing substantial cross-coupling between axes.
This coupling is quantified through \acrfull{rga} analysis of the damped system (Figure~\ref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}), which confirms increasing interaction between control channels at frequencies above the joint-induced resonance.
Above this resonance frequency, two critical limitations emerge.
@@ -7212,6 +7194,7 @@ Critical specifications include sufficient bending stroke to ensure long-term op
Based on the dynamic analysis presented in previous sections, quantitative specifications were established and are summarized in Table~\ref{tab:detail_fem_joints_specs}.
\begin{table}[htbp]
+\caption{\label{tab:detail_fem_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model.}
\centering
\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
@@ -7224,8 +7207,6 @@ Torsion Stiffness \(k_t\) & \(< 500\,\text{Nm}/\text{rad}\) & 260\\
Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_fem_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model.}
-
\end{table}
Among various possible flexible joint architectures, the design shown in Figure~\ref{fig:detail_fem_joints_design} was selected for three key advantages.
@@ -7360,7 +7341,7 @@ Similarly, in~\cite{wang16_inves_activ_vibrat_isolat_stewar}, piezoelectric actu
In~\cite{xie17_model_contr_hybrid_passiv_activ}, force sensors are integrated in the struts for decentralized force feedback while accelerometers fixed to the top platform are employed for centralized control.
The second approach, sensor fusion (illustrated in Figure~\ref{fig:detail_control_sensor_arch_sensor_fusion}), involves filtering signals from two sensors using complementary filters\footnote{A set of two complementary filters are two transfer functions that sum to one.} and summing them to create an improved sensor signal.
-In~\cite{hauge04_sensor_contr_space_based_six}, geophones (used at low frequency) are merged with force sensors (used at high frequency).
+In~\cite{hauge04_sensor_contr_space_based_six}, geophones (used at low frequency) are merged with force sensors (used at high-frequency).
It is demonstrated that combining both sensors using sensor fusion can improve performance compared to using only one of the two sensors.
In~\cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip}, sensor fusion architecture is implemented with an accelerometer and a force sensor.
This implementation is shown to simultaneously achieve high damping of structural modes (through the force sensors) while maintaining very low vibration transmissibility (through the accelerometers).
@@ -7697,7 +7678,6 @@ The inverse magnitudes of the designed weighting functions, which represent the
\(n\) & \(2\) & \(3\)\\
\bottomrule
\end{tabularx}
-
\end{center}
\captionof{table}{\label{tab:detail_control_sensor_weights_params}Parameters for \(W_1(s)\) and \(W_2(s)\)}
\end{minipage}
@@ -7890,8 +7870,7 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin
\(m\) & Payload mass & \(40\,\text{kg}\)\\
\(I\) & Payload \(R_z\) inertia & \(5\,\text{kgm}^2\)\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
\end{minipage}
@@ -7959,7 +7938,7 @@ The obtained transfer function from \(\bm{\mathcal{\tau}}\) to \(\bm{\mathcal{L}
\end{equation}
At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are equal to the actuator stiffnesses~\eqref{eq:detail_control_decoupling_plant_decentralized_low_freq}.
-At high frequencies, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal.
+At high-frequency, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal.
\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
\bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
@@ -8029,7 +8008,7 @@ Analytical formula of the plant \(\bm{G}_{\{M\}}(s)\) is derived~\eqref{eq:detai
\frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
\end{equation}
-At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix~\eqref{eq:detail_control_decoupling_plant_CoM_high_freq}.
+At high-frequency, the plant converges to the inverse of the mass matrix, which is a diagonal matrix~\eqref{eq:detail_control_decoupling_plant_CoM_high_freq}.
\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
\bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
@@ -8110,9 +8089,9 @@ When a high-frequency force is applied at a point not aligned with the \acrlong{
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_decoupling_model_test_CoK.png}
\end{center}
-\subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK}
+\subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High-frequency force applied at the CoK}
\end{subfigure}
-\caption{\label{fig:detail_control_decoupling_jacobian_plant_CoK_results}Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).}
+\caption{\label{fig:detail_control_decoupling_jacobian_plant_CoK_results}Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high-frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).}
\end{figure}
\subsubsection{Modal Decoupling}
\label{ssec:detail_control_decoupling_modal}
@@ -8123,7 +8102,7 @@ To convert the dynamics in the modal space, the equation of motion are first wri
\bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
\end{equation}
-For modal decoupling, a change of variables is introduced~\eqref{eq:detail_control_decoupling_modal_coordinates} where \(\bm{\mathcal{X}}_{m}\) represents the modal amplitudes and \(\bm{\Phi}\) is a \(n \times n\)\footnote{\(n\) corresponds to the number of degrees of freedom, here \(n = 3\)} matrix whose columns correspond to the mode shapes of the system, computed from \(\bm{M}_{\{M\}}\) and \(\bm{K}_{\{M\}}\).
+For modal decoupling, a change of variables is introduced~\eqref{eq:detail_control_decoupling_modal_coordinates} where \(\bm{\mathcal{X}}_{m}\) represents the modal amplitudes and \(\bm{\Phi}\) is a \(n \times n\)\footnote{\(n\) corresponds to the number of degrees of freedom, here \(n = 3\).} matrix whose columns correspond to the mode shapes of the system, computed from \(\bm{M}_{\{M\}}\) and \(\bm{K}_{\{M\}}\).
\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates}
\bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m}
@@ -8138,7 +8117,7 @@ The inherent mathematical structure of the mass, damping, and stiffness matrices
This diagonalization transforms equation~\eqref{eq:detail_control_decoupling_equation_modal_coordinates} into a set of \(n\) decoupled equations, enabling independent control of each mode without cross-interaction.
To implement this approach from a decentralized plant, the architecture shown in Figure~\ref{fig:detail_control_decoupling_modal} is employed.
-Inputs of the decoupling plant are the modal modal inputs \(\bm{\tau}_m\) and the outputs are the modal amplitudes \(\bm{\mathcal{X}}_m\).
+Inputs of the decoupling plant are the modal inputs \(\bm{\tau}_m\) and the outputs are the modal amplitudes \(\bm{\mathcal{X}}_m\).
This implementation requires knowledge of the system's equations of motion, from which the mode shapes matrix \(\bm{\Phi}\) is derived.
The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually.
@@ -8316,12 +8295,13 @@ Modal decoupling provides a natural framework when specific vibrational modes re
SVD decoupling generally results in a loss of physical meaning for the ``control space'', potentially complicating the process of relating controller design to practical system requirements.
The quality of decoupling achieved through these methods also exhibits distinct characteristics.
-Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the \acrlong{cok}, or at high frequencies when aligned with the \acrlong{com}.
+Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low-frequency when aligned at the \acrlong{cok}, or at high-frequency when aligned with the \acrlong{com}.
Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach.
Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the model accuracy, with discrepancies potentially resulting in significant off-diagonal elements.
SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point.
\begin{table}[htbp]
+\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies.}
\centering
\scriptsize
\begin{tabularx}{\linewidth}{lXXX}
@@ -8347,8 +8327,6 @@ SVD decoupling can be implemented using measured data without requiring a model,
\textbf{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\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_control_decoupling_strategies_comp}Comparison of decoupling strategies.}
-
\end{table}
\subsection{Closed-Loop Shaping using Complementary Filters}
\label{sec:detail_control_cf}
@@ -8383,7 +8361,7 @@ Finally, in Section~\ref{ssec:detail_control_cf_simulations}, a numerical exampl
The idea of using complementary filters in the control architecture originates from sensor fusion techniques~\cite{collette15_sensor_fusion_method_high_perfor}, where two sensors are combined using complementary filters.
Building upon this concept, ``virtual sensor fusion''~\cite{verma20_virtual_sensor_fusion_high_precis_contr} replaces one physical sensor with a model \(G\) of the plant.
The corresponding control architecture is illustrated in Figure~\ref{fig:detail_control_cf_arch}, where \(G^\prime\) represents the physical plant to be controlled, \(G\) is a model of the plant, \(k\) is the controller, and \(H_L\) and \(H_H\) are complementary filters satisfying \(H_L(s) + H_H(s) = 1\).
-In this arrangement, the physical plant is controlled at low frequencies, while the plant model is used at high frequencies to enhance robustness.
+In this arrangement, the physical plant is controlled at low frequencies, while the plant model is used at high-frequency to enhance robustness.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@@ -8476,7 +8454,7 @@ For the nominal system, \(S = H_H\) and \(T = H_L\), hence the performance speci
\end{equation}
For disturbance rejection, the magnitude of the sensitivity function \(|S(j\omega)| = |H_H(j\omega)|\) should be minimized, particularly at low frequencies where disturbances are usually most prominent.
-Similarly, for noise attenuation, the magnitude of the complementary sensitivity function \(|T(j\omega)| = |H_L(j\omega)|\) should be minimized, especially at high frequencies where measurement noise typically dominates.
+Similarly, for noise attenuation, the magnitude of the complementary sensitivity function \(|T(j\omega)| = |H_L(j\omega)|\) should be minimized, especially at high-frequency where measurement noise typically dominates.
Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function.
Typically, maintaining \(|S|_{\infty} \le 2\) ensures a gain margin of at least 2 and a phase margin of at least \(\SI{29}{\degree}\).
@@ -8856,8 +8834,8 @@ There are two limiting factors for large signal bandwidth that should be evaluat
\begin{enumerate}
\item Slew rate, which should exceed \(2 \cdot V_{pp} \cdot f_r = 34\,\text{V/ms}\).
This requirement is typically easily met by commercial voltage amplifiers.
-\item Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies.
-To achieve high voltage at high frequency, the amplifier must therefore provide substantial current.
+\item Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high-frequency.
+To achieve high voltage at high-frequency, the amplifier must therefore provide substantial current.
The maximum required current can be calculated as \(I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,\text{A}\).
\end{enumerate}
@@ -8881,11 +8859,12 @@ This approach does not account for the frequency dependency of the noise, which
Additionally, the load conditions used to estimate bandwidth and noise specifications are often not explicitly stated.
In many cases, bandwidth is reported with minimal load while noise is measured with substantial load, making direct comparisons between different models more complex.
-Note that for the WMA-200, the manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,\text{kHz}\)
+Note that for the WMA-200 amplifier, the manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,\text{kHz}\)
-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.
+The PD200 amplifier from PiezoDrive was ultimately selected because it meets all the requirements and is accompanied by clear documentation, particularly regarding noise characteristics and bandwidth specifications.
\begin{table}[htbp]
+\caption{\label{tab:detail_instrumentation_amp_choice}Specifications for the voltage amplifier and considered commercial products.}
\centering
\begin{tabularx}{0.8\linewidth}{Xcccc}
\toprule
@@ -8904,8 +8883,6 @@ Small Signal Bandwidth \(> 5\,\text{kHz}\) & \(6.4\,\text{kHz}\) & \(300\,\tex
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_instrumentation_amp_choice}Specifications for the voltage amplifier and considered commercial products.}
-
\end{table}
\subsubsection{ADC and DAC}
Analog-to-digital converters and digital-to-analog converters play key roles in the system, serving as the interface between the digital RT controller and the analog physical plant.
@@ -9033,7 +9010,7 @@ In contrast, optical encoders are bigger and they must be offset from the strut'
\caption{\label{fig:detail_instrumentation_sensor_implementation}Implementation of relative displacement sensors to measure the motion of the APA.}
\end{figure}
-A significant consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
+A practical consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
Measurements conducted on the slip-ring integrated in the micro-station revealed substantial cross-talk between different slip-ring channels.
To mitigate this issue, preference was given to sensors that transmit displacement measurements digitally, as these are inherently less susceptible to noise and cross-talk.
Based on this criterion, an optical encoder with digital output was selected, where signal interpolation is performed directly in the sensor head.
@@ -9041,6 +9018,7 @@ Based on this criterion, an optical encoder with digital output was selected, wh
The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table~\ref{tab:detail_instrumentation_sensor_specs}, alongside alternative options that were considered.
\begin{table}[htbp]
+\caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products.}
\centering
\begin{tabularx}{0.65\linewidth}{Xccc}
\toprule
@@ -9054,8 +9032,6 @@ In line measurement & & \(\times\) & \(\times\)\\
Digital Output & \(\times\) & & \\
\bottomrule
\end{tabularx}
-\caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products.}
-
\end{table}
\subsection{Characterization of Instrumentation}
\label{sec:detail_instrumentation_characterization}
@@ -9148,7 +9124,7 @@ These results validate both the model of the \acrshort{adc} and the effectivenes
\subsubsection{Instrumentation Amplifier}
Because the \acrshort{adc} noise may be too low to measure the noise of other instruments (anything below \(5.6\,\upmu\text{V}/\sqrt{\text{Hz}}\) cannot be distinguished from the noise of the \acrshort{adc} itself), a low noise instrumentation amplifier was employed.
-A Femto DLPVA-101-B-S amplifier with adjustable gains from \(20\,text{dB}\) up to \(80\,text{dB}\) was selected for this purpose.
+A Femto DLPVA-101-B-S amplifier with adjustable gains from \(20\,\text{dB}\) up to \(80\,\text{dB}\) was selected for this purpose.
The first step was to characterize the input\footnote{For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain.} noise of the amplifier.
This was accomplished by short-circuiting its input with a \(50\,\Omega\) resistor and measuring the output voltage with the \acrshort{adc} (Figure~\ref{fig:detail_instrumentation_femto_meas_setup}).
@@ -9660,8 +9636,8 @@ Finally, in Section~\ref{ssec:test_apa_spurious_resonances}, the flexible modes
\label{ssec:test_apa_geometrical_measurements}
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~\ref{fig:test_apa_flatness_setup}, the \acrshort{apa} is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\upmu\text{m}\)} is used to measure the height of four points on each of the APA300ML interfaces.
-From the XYZ coordinates of the measured eight points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.
+As shown in Figure~\ref{fig:test_apa_flatness_setup}, the \acrshort{apa} is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\upmu\text{m}\).} is used to measure the height of four points on each of the APA300ML interfaces.
+From the XYZ coordinates of the measured eight points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane.} a plane through all the points.
The measured flatness values, summarized in Table~\ref{tab:test_apa_flatness_meas}, are within the specifications.
\begin{minipage}[b]{0.48\textwidth}
@@ -9686,8 +9662,7 @@ APA 5 & 1.9\\
APA 6 & 7.1\\
APA 7 & 18.7\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
\end{minipage}
\subsubsection{Electrical Measurements}
@@ -9695,7 +9670,7 @@ APA 7 & 18.7\\
From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\upmu\text{F}\) and \(26\,\upmu\text{F}\) with a nominal capacitance of \(20\,\upmu\text{F}\).
-The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter\footnote{LCR-819 from Gwinstek, with a specified accuracy of \(0.05\%\). The measured frequency is set at \(1\,\text{kHz}\)} shown in Figure~\ref{fig:test_apa_lcr_meter}.
+The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter\footnote{LCR-819 from Gwinstek, with a specified accuracy of \(0.05\%\). The measured frequency is set at \(1\,\text{kHz}\).} shown in Figure~\ref{fig:test_apa_lcr_meter}.
The two stacks used as the actuator and the stack used as the force sensor were measured separately.
The measured capacitance values are summarized in Table~\ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \upmu\text{F}\).
However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance.
@@ -9726,16 +9701,15 @@ APA 5 & 4.90 & 9.66\\
APA 6 & 4.99 & 9.91\\
APA 7 & 4.85 & 9.85\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\upmu\text{F}$}
\end{minipage}
\subsubsection{Stroke and Hysteresis Measurement}
\label{ssec:test_apa_stroke_measurements}
-To compare the stroke of the APA300ML with the datasheet specifications, one side of the \acrshort{apa} is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\upmu\text{m}\)} is located on the other side as shown in Figure~\ref{fig:test_apa_stroke_bench}.
+To compare the stroke of the APA300ML with the datasheet specifications, one side of the \acrshort{apa} is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\upmu\text{m}\).} is located on the other side as shown in Figure~\ref{fig:test_apa_stroke_bench}.
-The voltage across the two actuator stacks is varied from \(-20\,\text{V}\) to \(150\,\text{V}\) using a DAC\footnote{The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of \(\pm 10\,\text{V}\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,\text{V/V}\)}.
+The voltage across the two actuator stacks is varied from \(-20\,\text{V}\) to \(150\,\text{V}\) using a DAC\footnote{The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of \(\pm 10\,\text{V}\) and 16-bits resolution.} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,\text{V/V}\).}.
Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure~\ref{fig:test_apa_stroke_voltage}).
\begin{figure}[htbp]
@@ -9752,7 +9726,7 @@ For the NASS, this stroke is sufficient because the positioning errors to be cor
It is clear from Figure~\ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared with the other units.
This confirms the abnormal electrical measurements made in Section~\ref{ssec:test_apa_electrical_measurements}.
-This unit was sent sent back to Cedrat, and a new one was shipped back.
+This unit was sent back to Cedrat, and a new one was shipped back.
From now on, only the six remaining amplified piezoelectric actuators that behave as expected will be used.
\begin{figure}[htbp]
@@ -9773,9 +9747,9 @@ From now on, only the six remaining amplified piezoelectric actuators that behav
\subsubsection{Flexible Mode Measurement}
\label{ssec:test_apa_spurious_resonances}
-In this section, the flexible modes of the APA300ML are investigated both experimentally and using a \acrshort{fem}.
+In this section, the flexible modes of the APA300ML are investigated both experimentally and through finite element modeling.
To experimentally estimate these modes, the \acrshort{apa} is fixed at one end (see Figure~\ref{fig:test_apa_meas_setup_modes}).
-A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes.
+A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512.} is used to measure the difference of motion between two ``red'' points and an instrumented hammer\footnote{Kistler 9722A.} is used to excite the flexible modes.
Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated.
The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software, and the results are shown in Figure~\ref{fig:test_apa_mode_shapes}.
@@ -9835,7 +9809,7 @@ Another explanation is the shape difference between the manufactured APA300ML an
After the measurements on the \acrshort{apa} were performed in Section~\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~\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\footnote{Renishaw Vionic, resolution of \(2.5\,\text{nm}\)} is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the \acrshort{apa}.
+An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,\text{nm}\).} is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the \acrshort{apa}.
\begin{figure}[htbp]
\begin{subfigure}{0.3\textwidth}
@@ -9889,8 +9863,7 @@ APA & \(k_1\) & \(k_2\)\\
6 & 1.7 & 1.92\\
8 & 1.73 & 1.98\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $\text{N}/\upmu\text{m}$}
\end{minipage}
@@ -9930,7 +9903,7 @@ The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\)
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 \emph{decreases} by 180 degrees whereas it should have \emph{increased} by 180 degrees for a minimum phase zero).
-This is investigated further investigated.
+This is further investigated.
As illustrated by the root locus plot, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases.
The significance of this behavior varies with the type of sensor used, as explained in~\cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
@@ -10029,7 +10002,7 @@ The transfer function from the ``damped'' plant input \(u\prime\) to the encoder
\caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\).}
\end{figure}
-The identified dynamics were then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see~\cite{gustavsen99_ration_approx_frequen_domain_respon}}.
+The identified dynamics were then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see~\cite{gustavsen99_ration_approx_frequen_domain_respon}.}.
A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure~\ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
@@ -10127,6 +10100,7 @@ In the last step, \(g_s\) and \(g_a\) can be tuned to match the gain of the iden
The obtained parameters of the model shown in Figure~\ref{fig:test_apa_2dof_model_simscape} are summarized in Table~\ref{tab:test_apa_2dof_parameters}.
\begin{table}[htbp]
+\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2-DoFs APA300ML model.}
\centering
\begin{tabularx}{0.25\linewidth}{cc}
\toprule
@@ -10143,8 +10117,6 @@ The obtained parameters of the model shown in Figure~\ref{fig:test_apa_2dof_mode
\(g_s\) & \(0.46\,\text{V}/\upmu\text{m}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2-DoFs APA300ML model.}
-
\end{table}
The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table~\ref{tab:test_apa_2dof_parameters}) from the multi-body model.
This is compared with the experimental data in Figure~\ref{fig:test_apa_2dof_comp_frf}.
@@ -10169,7 +10141,7 @@ This indicates that this model represents well the axial dynamics of the APA300M
\subsection{Reduced Order Flexible Model}
\label{sec:test_apa_model_flexible}
-In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
+In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used.}.
It is then imported into multi-body (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in~\ref{sec:test_apa_model_2dof}.
This procedure is illustrated in Figure~\ref{fig:test_apa_super_element_simscape}.
Several \emph{remote points} are defined in the \acrshort{fem} (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then made accessible in the multi-body software as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}.
@@ -10208,6 +10180,7 @@ The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) ar
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.
\begin{table}[htbp]
+\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuator sensitivities.}
\centering
\begin{tabularx}{0.8\linewidth}{ccX}
\toprule
@@ -10222,8 +10195,6 @@ From these parameters, \(g_s = 5.1\,\text{V}/\upmu\text{m}\) and \(g_a = 26\,\te
\(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuator sensitivities.}
-
\end{table}
\paragraph{Comparison of the Obtained Dynamics}
@@ -10278,6 +10249,7 @@ Deviations from these ideal properties will impact the dynamics of the Nano-Hexa
During the detailed design phase, specifications in terms of stiffness and stroke were determined and are summarized in Table~\ref{tab:test_joints_specs}.
\begin{table}[htbp]
+\caption{\label{tab:test_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model.}
\centering
\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
@@ -10290,8 +10262,6 @@ Torsion Stiffness & \(< 500\,\text{Nm}/\text{rad}\) & 260\\
Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_joints_specs}Specifications for the flexible joints and estimated characteristics from the Finite Element Model.}
-
\end{table}
After optimization using a \acrshort{fem}, the geometry shown in Figure~\ref{fig:test_joints_schematic} has been obtained and the corresponding flexible joint characteristics are summarized in Table~\ref{tab:test_joints_specs}.
@@ -10560,6 +10530,7 @@ 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.
\begin{table}[htbp]
+\caption{\label{tab:test_joints_error_budget}Summary of the error budget for the estimation of the bending stiffness.}
\centering
\begin{tabularx}{0.35\linewidth}{Xc}
\toprule
@@ -10572,21 +10543,19 @@ Displacement sensor & \(\epsilon_d < 0.01\,\%\)\\
Force sensor & \(\epsilon_F < 1\,\%\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_joints_error_budget}Summary of the error budget for the estimation of the bending stiffness.}
-
\end{table}
\subsubsection{Mechanical Design}
\label{ssec:test_joints_test_bench}
As explained in Section~\ref{ssec:test_joints_meas_principle}, the flexible joint's bending stiffness is estimated by applying a known force to the flexible joint's tip and by measuring its deflection at the same point.
-The force is applied using a load cell\footnote{The load cell is FC22 from TE Connectivity. The measurement range is \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range} such that the applied force to the flexible joint's tip is directly measured.
+The force is applied using a load cell\footnote{The load cell is FC22 from TE Connectivity. The measurement range is \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range.} such that the applied force to the flexible joint's tip is directly measured.
To control the height and direction of the applied force, a cylinder cut in half is fixed at the tip of the force sensor (pink element in Figure~\ref{fig:test_joints_bench_side}) that initially had a flat surface.
Doing so, the contact between the flexible joint cylindrical tip and the force sensor is a point (intersection of two cylinders) at a precise height, and the force is applied in a known direction.
To translate the load cell at a constant height, it is fixed to a translation stage\footnote{V-408 PIMag\textsuperscript{\textregistered} linear stage is used. Crossed rollers are used to guide the motion.} which is moved by hand.
Instead of measuring the displacement directly at the tip of the flexible joint (with a probe or an interferometer for instance), the displacement of the load cell itself is measured.
-To do so, an encoder\footnote{Resolute\texttrademark{} encoder with \(1\,\text{nm}\) resolution and \(\pm 40\,\text{nm}\) maximum non-linearity} is used, which measures the motion of a ruler.
+To do so, an encoder\footnote{Resolute\texttrademark{} encoder with \(1\,\text{nm}\) resolution and \(\pm 40\,\text{nm}\) maximum non-linearity.} 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~\ref{fig:test_joints_bench_overview}).
The flexible joint can be rotated by \(\SI{90}{\degree}\) in order to measure the bending stiffness in the two directions.
@@ -10628,7 +10597,7 @@ A closer view of the force sensor tip is shown in Figure~\ref{fig:test_joints_pi
\caption{\label{fig:test_joints_picture_bench}Manufactured test bench for compliance measurement of the flexible joints.}
\end{figure}
\subsubsection{Load Cell Calibration}
-In order to estimate the measured errors of the load cell ``FC2231'', it is compared against another load cell\footnote{XFL212R-50N from TE Connectivity. The measurement range is \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range}.
+In order to estimate the measured errors of the load cell ``FC2231'', it is compared against another load cell\footnote{XFL212R-50N from TE Connectivity. The measurement range is \(50\,\text{N}\). The specified accuracy is \(1\,\%\) of the full range.}.
The two load cells are measured simultaneously while they are pushed against each other (see Figure~\ref{fig:test_joints_force_sensor_calib_picture}).
The contact between the two load cells is well defined as one has a spherical interface and the other has a flat surface.
@@ -10736,7 +10705,7 @@ These measurements are helpful for refining the model of the flexible joints, th
Furthermore, the data obtained from these measurements have provided the necessary information to select the most suitable flexible joints for the nano-hexapod, ensuring optimal performance.
\section{Struts}
\label{sec:test_struts}
-The Nano-Hexapod struts (shown in Figure~\ref{fig:test_struts_picture_strut}) are composed of two flexible joints that are fixed at the two ends of the strut, one Amplified Piezoelectric Actuator\footnote{APA300ML from Cedrat Technologies} and one optical encoder\footnote{Vionic from Renishaw}.
+The Nano-Hexapod struts (shown in Figure~\ref{fig:test_struts_picture_strut}) are composed of two flexible joints that are fixed at the two ends of the strut, one Amplified Piezoelectric Actuator\footnote{APA300ML from Cedrat Technologies.} and one optical encoder\footnote{Vionic from Renishaw.}.
\begin{figure}[htbp]
\centering
@@ -10773,7 +10742,7 @@ This is important not to loose to much angular stroke during their mounting into
The mounting bench is shown in Figure~\ref{fig:test_struts_mounting_bench_first_concept}.
It consists of a ``main frame'' (Figure~\ref{fig:test_struts_mounting_step_0}) precisely machined to ensure both correct strut length and strut coaxiality.
The coaxiality is ensured by good flatness (specified at \(20\,\upmu\text{m}\)) between surfaces A and B and between surfaces C and D.
-Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\upmu\text{m}\)} (see Figure~\ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements.
+Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\upmu\text{m}\).} (see Figure~\ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements.
The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes.
\begin{figure}[htbp]
@@ -10905,7 +10874,7 @@ The mode shapes are displayed in Figure~\ref{fig:test_struts_mode_shapes}: an ``
\caption{\label{fig:test_struts_mode_shapes}Flexible modes of the struts estimated from a Finite Element Model.}
\end{figure}
-To experimentally measure these mode shapes, a Laser vibrometer\footnote{OFV-3001 controller and OFV512 sensor head from Polytec} was used.
+To experimentally measure these mode shapes, a Laser vibrometer\footnote{OFV-3001 controller and OFV512 sensor head from Polytec.} was used.
It measures the difference of motion between two beam path (red points in Figure~\ref{fig:test_struts_meas_modes}).
The strut is then excited by an instrumented hammer, and the transfer function from the hammer to the measured rotation is computed.
@@ -10959,6 +10928,7 @@ In addition, the computed resonance frequencies from the \acrshort{fem} are very
This validates the quality of the \acrshort{fem}.
\begin{table}[htbp]
+\caption{\label{tab:test_struts_spur_mode_freqs}Estimated and measured frequencies of the flexible modes of the struts.}
\centering
\begin{tabularx}{0.7\linewidth}{Xccc}
\toprule
@@ -10969,8 +10939,6 @@ Y-Bending & \(285\,\text{Hz}\) & \(293\,\text{Hz}\) & \(337\,\text{Hz}\)\\
Z-Torsion & \(400\,\text{Hz}\) & \(381\,\text{Hz}\) & \(398\,\text{Hz}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_struts_spur_mode_freqs}Estimated and measured frequencies of the flexible modes of the struts.}
-
\end{table}
\subsection{Dynamical Measurements}
\label{sec:test_struts_dynamical_meas}
@@ -10978,7 +10946,7 @@ In order to measure the dynamics of the strut, the test bench used to measure th
The strut mounted on the bench is shown in Figure~\ref{fig:test_struts_bench_leg_overview}
A schematic of the bench and the associated signals are shown in Figure~\ref{fig:test_struts_bench_schematic}.
-A fiber interferometer\footnote{Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools} is used to measure the motion of the granite (i.e. the axial motion of the strut).
+A fiber interferometer\footnote{Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools.} is used to measure the motion of the granite (i.e. the axial motion of the strut).
\begin{figure}[htbp]
\begin{subfigure}{0.3\textwidth}
@@ -11166,7 +11134,7 @@ The misalignment in the \(y\) direction mostly influences:
\item if \(d_{y} < 0\): there is no zero between the two resonances and possibly not even between the second and third resonances
\item if \(d_{y} > 0\): there is a complex conjugate zero between the first two resonances
\end{itemize}
-\item the location of the high frequency complex conjugate zeros at \(500\,\text{Hz}\) (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
+\item the location of the high-frequency complex conjugate zeros at \(500\,\text{Hz}\) (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
\end{itemize}
The same can be done for misalignments in the \(x\) direction.
@@ -11199,7 +11167,7 @@ Therefore, large \(y\) misalignments are expected.
To estimate the misalignments between the two flexible joints and the \acrshort{apa}:
\begin{itemize}
\item the struts were fixed horizontally on the mounting bench, as shown in Figure~\ref{fig:test_struts_mounting_step_3} but without the encoder
-\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\upmu\text{m}\)}, the height difference between the flexible joints surface and the \acrshort{apa} shell surface was measured for both the top and bottom joints and for both sides
+\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\upmu\text{m}\).}, the height difference between the flexible joints surface and the \acrshort{apa} shell surface was measured for both the top and bottom joints and for both sides
\item as the thickness of the flexible joint is \(21\,\text{mm}\) and the thickness of the \acrshort{apa} shell is \(20\,\text{mm}\), \(0.5\,\text{mm}\) of height difference should be measured if the two are perfectly aligned
\end{itemize}
@@ -11209,6 +11177,7 @@ 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~\ref{tab:test_struts_meas_y_misalignment}.
\begin{table}[htbp]
+\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment for each strut. Measurements are in \(\text{mm}\).}
\centering
\begin{tabularx}{0.2\linewidth}{Xcc}
\toprule
@@ -11221,8 +11190,6 @@ Thickness differences for all the struts were found to be between \(0.94\,\text{
5 & 0.15 & 0.02\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment for each strut. Measurements are in \(\text{mm}\).}
-
\end{table}
By using the measured \(y\) misalignment in the model with the flexible \acrshort{apa} model, the model dynamics from \(u\) to \(d_e\) is closer to the measured dynamics, as shown in Figure~\ref{fig:test_struts_comp_dy_tuned_model_frf_enc}.
@@ -11249,6 +11216,7 @@ The alignment is then estimated using a length gauge, as described in the previo
Measured \(y\) alignments are summarized in Table~\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~\ref{tab:test_struts_meas_y_misalignment}).
\begin{table}[htbp]
+\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment after realigning the struts using dowel pins. Measurements are in \(\text{mm}\).}
\centering
\begin{tabularx}{0.25\linewidth}{Xcc}
\toprule
@@ -11262,8 +11230,6 @@ Measured \(y\) alignments are summarized in Table~\ref{tab:test_struts_meas_y_mi
6 & -0.005 & 0.055\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment after realigning the struts using dowel pins. Measurements are in \(\text{mm}\).}
-
\end{table}
The dynamics of the re-aligned struts were then measured on the same test bench (Figure~\ref{fig:test_struts_bench_leg}).
@@ -11321,7 +11287,7 @@ To do so, a precisely machined mounting tool (Figure~\ref{fig:test_nhexa_center_
\caption{\label{fig:test_nhexa_received_parts}Nano-Hexapod plates (\subref{fig:test_nhexa_nano_hexapod_plates}) and mounting tool used to position the two plates during assembly (\subref{fig:test_nhexa_center_part_hexapod_mounting}).}
\end{figure}
-The mechanical tolerances of the received plates were checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\upmu\text{m}\)} (Figure~\ref{fig:test_nhexa_plates_tolerances}) and were found to comply with the requirements\footnote{Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.}.
+The mechanical tolerances of the received plates were checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm 13\upmu\text{m}\).} (Figure~\ref{fig:test_nhexa_plates_tolerances}) and were found to comply with the requirements\footnote{Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.}.
The same was done for the mounting tool\footnote{The height dimension is better than \(40\,\upmu\text{m}\). The diameter fitting of 182g6 and 24g6 with the two plates is verified.}.
The two plates were then fixed to the mounting tool, as shown in Figure~\ref{fig:test_nhexa_mounting_tool_hexapod_top_view}.
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.
@@ -11348,6 +11314,7 @@ 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\footnote{As the accuracy of the FARO arm is \(\pm 13\,\upmu\text{m}\), the true straightness is probably better than the values indicated. The limitation of the instrument is here reached.}, which is sufficiently good to not induce significant stress of the flexible joint during assembly.
\begin{table}[htbp]
+\caption{\label{tab:measured_straightness}Measured straightness between the V grooves for the six struts. Measurements were performed twice for each strut.}
\centering
\begin{tabularx}{0.25\linewidth}{Xcc}
\toprule
@@ -11361,8 +11328,6 @@ The straightness was found to be better than \(15\,\upmu\text{m}\) for all strut
6 & \(6\, \upmu\text{m}\) & \(7\, \upmu\text{m}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:measured_straightness}Measured straightness between the V grooves for the six struts. Measurements were performed twice for each strut.}
-
\end{table}
The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure~\ref{fig:test_nhexa_mount_encoder}), and the encoder heads were aligned to maximize the received contrast.
@@ -11415,7 +11380,7 @@ Finally, the multi-body model representing the suspended table was tuned to matc
The design of the suspended table is quite straightforward.
First, an optical table with high frequency flexible mode was selected\footnote{The \(450\,\text{mm} \times 450\,\text{mm} \times 60\,\text{mm}\) Nexus B4545A from Thorlabs.}.
-Then, four springs\footnote{``SZ8005 20 x 044'' from Steinel. The spring rate is specified at \(17.8\,\text{N/mm}\)} were selected with low spring rate such that the suspension modes are below \(10\,\text{Hz}\).
+Then, four springs\footnote{``SZ8005 20 x 044'' from Steinel. The spring rate is specified at \(17.8\,\text{N/mm}\).} were selected with low spring rate such that the suspension modes are below \(10\,\text{Hz}\).
Finally, some interface elements were designed, and mechanical lateral mechanical stops were added (Figure~\ref{fig:test_nhexa_suspended_table_cad}).
\begin{figure}[htbp]
@@ -11454,8 +11419,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{
8 & \(989\,\text{Hz}\) & Complex mode\\
9 & \(1025\,\text{Hz}\) & Complex mode\\
\bottomrule
-\end{tabularx}
-}
+\end{tabularx}}
\captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table}
\end{minipage}
@@ -11493,6 +11457,7 @@ The stiffness of the springs in the horizontal plane is set at \(0.5\,\text{N/mm
The obtained suspension modes of the multi-body model are compared with the measured modes in Table~\ref{tab:test_nhexa_suspended_table_simscape_modes}.
\begin{table}[htbp]
+\caption{\label{tab:test_nhexa_suspended_table_simscape_modes}Comparison of suspension modes of the multi-body model and the measured ones.}
\centering
\begin{tabularx}{0.5\linewidth}{Xcccc}
\toprule
@@ -11502,8 +11467,6 @@ Multi-body & \(1.3\,\text{Hz}\) & \(1.8\,\text{Hz}\) & \(6.8\,\text{Hz}\) & \(9.
Experimental & \(1.3\,\text{Hz}\) & \(2.0\,\text{Hz}\) & \(6.9\,\text{Hz}\) & \(9.5\,\text{Hz}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_nhexa_suspended_table_simscape_modes}Comparison of suspension modes of the multi-body model and the measured ones.}
-
\end{table}
\subsection{Measured Active Platform Dynamics}
\label{sec:test_nhexa_dynamics}
@@ -11546,6 +11509,7 @@ At around \(700\,\text{Hz}\), two flexible modes of the top plate were observed
These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}.
\begin{table}[htbp]
+\caption{\label{tab:test_nhexa_hexa_modal_modes_list}Description of the identified modes of the Nano-Hexapod.}
\centering
\begin{tabularx}{0.6\linewidth}{ccX}
\toprule
@@ -11561,8 +11525,6 @@ These modes are summarized in Table~\ref{tab:test_nhexa_hexa_modal_modes_list}.
8 & \(709\,\text{Hz}\) & Second flexible mode of the top platform\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_nhexa_hexa_modal_modes_list}Description of the identified modes of the Nano-Hexapod.}
-
\end{table}
\begin{figure}[htbp]
@@ -11802,7 +11764,7 @@ This section presents a comprehensive experimental evaluation of the complete sy
Initially, the project planned to develop a long-stroke (\(\approx 1 \, \text{cm}^3\)) 5-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~\ref{sec:test_id31_metrology}.
+To validate the nano-hexapod and its associated control architecture, an alternative short-stroke (\(\approx 100\,\upmu\text{m}^3\)) metrology system was developed instead, which is presented in Section~\ref{sec:test_id31_metrology}.
Then, several key aspects of the system validation are examined.
Section~\ref{sec:test_id31_open_loop_plant} analyzes the identified dynamics of the nano-hexapod mounted on the micro-station under various experimental conditions, including different payload masses and rotational velocities.
@@ -11833,7 +11795,7 @@ These include tomography scans at various speeds and with different payload mass
\subsection{Short Stroke Metrology System}
\label{sec:test_id31_metrology}
The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite.
-As a long-stroke (\(\approx 1 \,\text{cm}^3\)) metrology system was not yet developed, a stroke stroke (\(\approx 100\,\upmu\text{m}^3\)) was used instead to validate the nano-hexapod control.
+As a long-stroke (\(\approx 1 \,\text{cm}^3\)) metrology system was not yet developed, a short stroke (\(\approx 100\,\upmu\text{m}^3\)) was used instead to validate the nano-hexapod control.
The first considered option was to use the ``Spindle error analyzer'' shown in Figure~\ref{fig:test_id31_lion}.
This system comprises 5 capacitive sensors facing two reference spheres.
@@ -11984,6 +11946,7 @@ Results are summarized in Table~\ref{tab:test_id31_metrology_acceptance}.
The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(\pm0.5\,\text{mm}\), which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS.
\begin{table}[htbp]
+\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
\centering
\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
@@ -11996,8 +11959,6 @@ The obtained lateral acceptance for pure displacements in any direction is estim
\(d_5\) (z) & \(1.33\,\text{mm}\) & \(1.06\,\text{mm}\) & \(>2\,\text{mm}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
-
\end{table}
\subsubsection{Estimated Measurement Errors}
\label{ssec:test_id31_metrology_errors}
@@ -12065,7 +12026,7 @@ The dynamics of the plant is first identified for a fixed spindle angle (at \(0\
The model dynamics is also identified under the same conditions.
A comparison between the model and the measured dynamics is presented in Figure~\ref{fig:test_id31_first_id}.
-A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled).
+A good match can be observed for the diagonal dynamics (except the high-frequency modes which are not modeled).
However, the coupling of the transfer function from command signals \(\bm{u}\) to the estimated strut motion from the external metrology \(\bm{\epsilon\mathcal{L}}\) is larger than expected (Figure~\ref{fig:test_id31_first_id_int}).
The experimental time delay estimated from the \acrshort{frf} (Figure~\ref{fig:test_id31_first_id_int}) is larger than expected.
@@ -12094,7 +12055,7 @@ One possible explanation of the increased coupling observed in Figure~\ref{fig:t
To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders was implemented.
This allowed to perform two straight motions of the nano-hexapod along its \(x\) and \(y\) axes.
During these two motions, external metrology measurements were recorded and the results are shown in Figure~\ref{fig:test_id31_Rz_align_error_and_correct}.
-It was found that there was a misalignment of 2.7 degrees (rotation along the vertical axis) between the interferometer axes and nano-hexapod axes.
+It was found that there was a misalignment of 2.7 degrees (rotation around the vertical axis) between the interferometer axes and nano-hexapod axes.
This was corrected by adding an offset to the spindle angle.
After alignment, the same motion was performed using the nano-hexapod while recording the signal of the external metrology.
Results shown in Figure~\ref{fig:test_id31_Rz_align_correct} are indeed indicating much better alignment.
@@ -12182,7 +12143,7 @@ It is interesting to note that the anti-resonances in the force sensor plant now
\subsubsection{Effect of Spindle Rotation}
\label{ssec:test_id31_open_loop_plant_rotation}
-To verify that all the kinematics in Figure~\ref{fig:test_id31_block_schematic_plant} are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at \(36\,\text{deg}/s\) and at \(180\,\text{deg}/s\).
+To verify that all the kinematics in Figure~\ref{fig:test_id31_block_schematic_plant} are correct and to check whether the system dynamics is affected by Spindle rotation or not, three identification experiments were performed: no spindle rotation, spindle rotation at \(36\,\text{deg}/s\) and at \(180\,\text{deg}/s\).
The obtained dynamics from command signal \(u\) to estimated strut error \(\epsilon\mathcal{L}\) are displayed in Figure~\ref{fig:test_id31_effect_rotation}.
Both direct terms (Figure~\ref{fig:test_id31_effect_rotation_direct}) and coupling terms (Figure~\ref{fig:test_id31_effect_rotation_coupling}) are unaffected by the rotation.
@@ -12208,7 +12169,7 @@ This also indicates that the metrology kinematics is correct and is working in r
\subsubsection*{Conclusion}
The identified \acrshortpl{frf} from command signals \(\bm{u}\) to the force sensors \(\bm{V}_s\) and to the estimated strut errors \(\bm{\epsilon\mathcal{L}}\) are well matching the dynamics of the developed multi-body model.
The effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used.
-The spindle rotation had no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation.
+The spindle rotation has no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation.
\subsection{Decentralized Integral Force Feedback}
\label{sec:test_id31_iff}
In this section, the low authority control part is first validated.
@@ -12403,7 +12364,7 @@ The results indicate that higher payload masses increase the coupling when imple
This indicates that achieving high bandwidth feedback control is increasingly challenging as the payload mass increases.
This behavior can be attributed to the fundamental approach of implementing control in the frame of the struts.
Above the suspension modes of the nano-hexapod, the motion induced by the piezoelectric actuators is no longer dictated by kinematics but rather by the inertia of the different parts.
-This design choice, while beneficial for system simplicity, introduces inherent limitations in the system's ability to handle larger masses at high frequency.
+This design choice, while beneficial for system simplicity, introduces inherent limitations in the system's ability to handle larger masses at high-frequency.
\begin{figure}[htbp]
\centering
@@ -12415,7 +12376,7 @@ This design choice, while beneficial for system simplicity, introduces inherent
A diagonal controller was designed to be robust against changes in payload mass, which means that every damped plant shown in Figure~\ref{fig:test_id31_comp_all_undamped_damped_plants} must be considered during the controller design.
For this controller design, a crossover frequency of \(5\,\text{Hz}\) was chosen to limit the multivariable effects, as explain in Section~\ref{sec:test_id31_hac_interaction_analysis}.
-One integrator is added to increase the low-frequency gain, a lead is added around \(5\,\text{Hz}\) to increase the stability margins and a first-order low-pass filter with a cut-off frequency of \(30\,\text{Hz}\) is added to improve the robustness to dynamical uncertainty at high frequency.
+One integrator is added to increase the low-frequency gain, a lead is added around \(5\,\text{Hz}\) to increase the stability margins and a first-order low-pass filter with a cut-off frequency of \(30\,\text{Hz}\) is added to improve the robustness to dynamical uncertainty at high-frequency.
The controller transfer function is shown in~\eqref{eq:test_id31_robust_hac}.
\begin{equation}\label{eq:test_id31_robust_hac}
@@ -12518,6 +12479,7 @@ In terms of RMS errors, this corresponds to \(30\,\text{nm}\) in \(D_y\), \(15\,
Results obtained for all experiments are summarized and compared to the specifications in Section~\ref{ssec:test_id31_scans_conclusion}.
\begin{table}[htbp]
+\caption{\label{tab:test_id31_experiments_specifications}Positioning specifications for the Nano-Active-Stabilization-System.}
\centering
\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
@@ -12527,8 +12489,6 @@ peak 2 peak & \(200\,\text{nm}\) & \(100\,\text{nm}\) & \(1.7\,\upmu\text{rad}\)
RMS & \(30\,\text{nm}\) & \(15\,\text{nm}\) & \(250\,\text{nrad}\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_id31_experiments_specifications}Positioning specifications for the Nano-Active-Stabilization-System.}
-
\end{table}
\subsubsection{Tomography Scans}
\label{ssec:test_id31_scans_tomography}
@@ -12942,6 +12902,7 @@ 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.
\begin{table}[htbp]
+\caption{\label{tab:test_id31_experiments_results_summary}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.}
\centering
\begin{tabularx}{0.85\linewidth}{Xccc}
\toprule
@@ -12970,8 +12931,6 @@ Diffraction tomography (\(6\,\text{deg/s}\), \(1\,\text{mm/s}\)) & \(\bm{53}\) &
\textbf{Specifications} & \(30\) & \(15\) & \(250\)\\
\bottomrule
\end{tabularx}
-\caption{\label{tab:test_id31_experiments_results_summary}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.}
-
\end{table}
\subsection*{Conclusion}
\label{ssec:test_id31_conclusion}