diff --git a/figs/detail_instrumentation_sensor_eddy_current.pdf b/figs/detail_instrumentation_sensor_eddy_current.pdf
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--- a/figs/inkscape/detail_instrumentation_sensor_eddy_current.svg
+++ b/figs/inkscape/detail_instrumentation_sensor_eddy_current.svg
@@ -3,12 +3,12 @@
+ d="m 98.85776,78.640785 c -2.957065,0.902899 -3.703562,2.531966 -3.703562,2.531966"
+ id="path3"
+ sodipodi:nodetypes="cc" />
diff --git a/figs/uniaxial_ustation_dynamical_id_setup.jpg b/figs/uniaxial_ustation_dynamical_id_setup.jpg
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diff --git a/phd-thesis.org b/phd-thesis.org
index e5f2ad2..f6ee0d6 100644
--- a/phd-thesis.org
+++ b/phd-thesis.org
@@ -892,7 +892,6 @@ The complete nano-hexapod assembly was tested on an isolated table, allowing acc
Finally, the integrated NASS was validated on the ID31 beamline using a purpose-built short-stroke metrology system (Section\nbsp{}ref:sec:test_id31).
The implemented control architecture was tested under realistic experimental scenarios, including tomography with heavy payloads, confirming the system's performance and robustness.
-
* Conceptual Design Development
<>
\minitoc
@@ -943,7 +942,6 @@ The confidence gained through this systematic investigation provides a solid fou
In this report, a uniaxial model of the acrfull:nass is developed and used to obtain a first idea of the challenges involved in this complex system.
Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions.
-The model is schematically shown in Figure\nbsp{}ref:fig:uniaxial_overview_model_sections where the colors represent the parts studied in different sections.
To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section\nbsp{}ref:sec:uniaxial_micro_station_model).
Then, a model of the nano-hexapod is added on top of the micro-station.
@@ -961,28 +959,16 @@ Once the system is well damped, a feedback position controller is applied and th
Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section\nbsp{}ref:sec:uniaxial_support_compliance) and the presence of dynamics between the nano-hexapod and the sample's point of interest (Section\nbsp{}ref:sec:uniaxial_payload_dynamics).
-#+name: fig:uniaxial_overview_model_sections
-#+caption: Uniaxial Micro-Station model in blue (Section\nbsp{}ref:sec:uniaxial_micro_station_model), Nano-Hexapod models in red (Section\nbsp{}ref:sec:uniaxial_nano_station_model), Disturbances in yellow (Section\nbsp{}ref:sec:uniaxial_disturbances), Active Damping in green (Section\nbsp{}ref:sec:uniaxial_active_damping), Position control in purple (Section\nbsp{}ref:sec:uniaxial_position_control) and Sample dynamics in cyan (Section\nbsp{}ref:sec:uniaxial_payload_dynamics)
-[[file:figs/uniaxial_overview_model_sections.png]]
-
*** Micro Station Model
<>
**** Introduction :ignore:
In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station.
-The measurement setup is shown in Figure\nbsp{}ref:fig:uniaxial_ustation_first_meas_dynamics where several geophones[fn:uniaxial_1] are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station.
-
-From the measured frequency response functions (FRF), the model can be tuned to approximate the uniaxial dynamics of the micro-station.
-
-#+name: fig:uniaxial_ustation_first_meas_dynamics
-#+caption: Experimental setup used for the first dynamical measurements on the Micro-Station. Geophones are fixed to different stages of the micro-station.
-#+attr_latex: :width \linewidth
-[[file:figs/uniaxial_ustation_first_meas_dynamics.jpg]]
**** Measured dynamics
The measurement setup is schematically shown in Figure\nbsp{}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 micro-hexapod's top platform (force $F_{h}$).
-The vertical inertial motion of the granite $x_{g}$ and the top platform of the micro-hexapod $x_{h}$ are measured using geophones.
+The vertical inertial motion of the granite $x_{g}$ and the top platform of the micro-hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1].
Three frequency response functions 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 frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model).
@@ -1018,7 +1004,7 @@ The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_
#+name: tab:uniaxial_ustation_parameters
#+caption: Physical parameters used for the micro-station uniaxial model
-#+attr_latex: :environment tabularx :width 0.9\linewidth :align lXXX
+#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| *Stage* | *Mass* | *Stiffness* | *Damping* |
|---------------------+-------------------------+----------------------+-----------------------------|
@@ -1039,6 +1025,7 @@ More accurate models will be used later on.
#+name: fig:uniaxial_comp_frf_meas_model
#+caption: Comparison of the measured FRF and identified ones from the uniaxial model
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_comp_frf_meas_model.png]]
*** Nano-Hexapod Model
@@ -1064,7 +1051,7 @@ The effect of resonances between the sample's point of interest and the nano-hex
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces $f$ to measured displacement $d$ by the metrology}
#+attr_latex: :options {0.59\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_plant_first_params.png]]
#+end_subfigure
#+end_figure
@@ -1089,19 +1076,19 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nb
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_fs}Direct forces}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_first_params_fs.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_ft}$\mu\text{-station}$ disturbances}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_first_params_ft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_xf}Floor motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_first_params_xf.png]]
#+end_subfigure
#+end_figure
@@ -1125,7 +1112,7 @@ The geophone located on the floor was used to measure the floor motion $x_f$ whi
#+attr_latex: :width 0.95\linewidth
[[file:figs/uniaxial_ustation_meas_disturbances.png]]
#+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:uniaxial_ustation_dynamical_id_setup}Two geophones are used to measure vibrations induced by $T_y$ and $R_z$ scans}
+#+attr_latex: :caption \subcaption{\label{fig:uniaxial_ustation_dynamical_id_setup}Geophones used to measure vibrations induced by $T_y$ and $R_z$ scans}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
@@ -1165,13 +1152,13 @@ The estimated amplitude spectral density $\Gamma_{x_f}$ of the floor motion $x_f
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_asd_floor_motion_id31}Estimated ASD of $x_f$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_asd_floor_motion_id31.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_asd_disturbance_force}Estimated ASD of $f_t$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_asd_disturbance_force.png]]
#+end_subfigure
#+end_figure
@@ -1187,6 +1174,7 @@ The sharp peak observed at $24\,\text{Hz}$ is believed to be induced by electrom
#+name: fig:uniaxial_asd_vibration_spindle_rotation
#+caption: Amplitude Spectral Density $\Gamma_{R_z}$ of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_asd_vibration_spindle_rotation.png]]
To compute the equivalent disturbance force $f_t$ (Figure\nbsp{}ref:fig:uniaxial_model_micro_station) that induces such motion, the transfer function $G_{f_t}(s)$ from $f_t$ to the relative motion between the micro-hexapod's top platform and the granite $(x_{h} - x_{g})$ is extracted from the model.
@@ -1227,19 +1215,19 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}Direct forces}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}$\mu\text{-station}$ disturbances}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf}Floor motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf.png]]
#+end_subfigure
#+end_figure
@@ -1260,13 +1248,13 @@ The conclusion is that the sample mass has little effect on the cumulative ampli
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_d_disturbances_stiffnesses}Effect of floor motion $x_f$ and stage disturbances $f_t$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_d_disturbances_stiffnesses.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Effect of nano-hexapod stiffness $k_n$ and payload mass $m_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_d_disturbances_payload_masses.png]]
#+end_subfigure
#+end_figure
@@ -1280,7 +1268,7 @@ From this analysis, it may be concluded that the stiffer the nano-hexapod the be
However, what is more important is the /closed-loop/ residual vibration of $d$ (i.e., while the feedback controller is used).
The goal is to obtain a closed-loop residual vibration $\epsilon_d \approx 20\,nm\,\text{RMS}$ (represented by an horizontal dashed black line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses).
The bandwidth of the feedback controller leading to a closed-loop residual vibration of $20\,nm\,\text{RMS}$ can be estimated as the frequency at which the cumulative amplitude spectrum crosses the black dashed line in Figure\nbsp{}ref:fig:uniaxial_cas_d_disturbances_payload_masses.
-# TODO - It would be important to link to a appendix where this is explained in more details, or add some references where this is explained
+
A closed loop bandwidth of $\approx 10\,\text{Hz}$ is found for the soft nano-hexapod ($k_n = 0.01\,N/\mu m$), $\approx 50\,\text{Hz}$ for the relatively stiff nano-hexapod ($k_n = 1\,N/\mu m$), and $\approx 100\,\text{Hz}$ for the stiff nano-hexapod ($k_n = 100\,N/\mu m$).
Therefore, while the /open-loop/ vibration is the lowest for the stiff nano-hexapod, it requires the largest feedback bandwidth to meet the specifications.
@@ -1421,25 +1409,23 @@ Therefore, it is expected that the micro-station dynamics might impact the achie
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_iff}IFF}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_plant_active_damping_techniques_iff.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_rdc}RDC}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_plant_active_damping_techniques_rdc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_plant_active_damping_techniques_dvf}DVF}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_plant_active_damping_techniques_dvf.png]]
#+end_subfigure
#+end_figure
-**** Active Damping Controller Optimization and Damped plants :noexport:
-
**** Achievable Damping and Damped Plants
<>
@@ -1471,25 +1457,26 @@ There is even some damping authority on micro-station modes in the following cas
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_root_locus_damping_techniques_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_root_locus_damping_techniques_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_root_locus_damping_techniques_stiff.png]]
#+end_subfigure
#+end_figure
#+name: fig:uniaxial_root_locus_damping_techniques_micro_station_mode
#+caption: Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png]]
The transfer functions from the plant input $f$ to the relative displacement $d$ while active damping is implemented are shown in Figure\nbsp{}ref:fig:uniaxial_damped_plant_three_active_damping_techniques.
@@ -1502,19 +1489,19 @@ All three active damping techniques yielded similar damped plants.
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_damped_plant_three_active_damping_techniques_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_damped_plant_three_active_damping_techniques_pz.png]]
#+end_subfigure
#+end_figure
@@ -1540,19 +1527,19 @@ Several conclusions can be drawn by comparing the obtained sensitivity transfer
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_fs}Direct forces}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_active_damping_fs.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_ft}$\mu\text{-station}$ disturbances}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_active_damping_ft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_xf}Floor motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_active_damping_xf.png]]
#+end_subfigure
#+end_figure
@@ -1568,19 +1555,19 @@ All three active damping methods give similar results.
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.37\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_active_damping_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_active_damping_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_active_damping_stiff.png]]
#+end_subfigure
#+end_figure
@@ -1603,7 +1590,7 @@ Which one will be used will be determined by the use of more accurate models and
#+name: tab:comp_active_damping
#+caption: Comparison of active damping strategies
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc
-#+attr_latex: :center t :booktabs t :font \scriptsize
+#+attr_latex: :center t :booktabs t
| | *IFF* | *RDC* | *DVF* |
|---------------------+-----------------------------+---------------------------+---------------------------------|
| *Sensor* | Force sensor | Relative motion sensor | Inertial sensor |
@@ -1637,7 +1624,7 @@ This control architecture applied to the uniaxial model is shown in Figure\nbsp{
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_lac_architecture}Typical HAC-LAC Architecture}
#+attr_latex: :options {0.54\textwidth}
#+begin_subfigure
-#+attr_latex: :width 1.0\linewidth
+#+attr_latex: :width \linewidth
[[file:figs/uniaxial_hac_lac_architecture.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_lac_model}Uniaxial model with HAC-IFF strategy}
@@ -1663,19 +1650,19 @@ This effect will be further explained in Section\nbsp{}ref:sec:uniaxial_support_
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.37\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_hac_iff_damped_plants_masses_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_hac_iff_damped_plants_masses_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_hac_iff_damped_plants_masses_stiff.png]]
#+end_subfigure
#+end_figure
@@ -1724,7 +1711,7 @@ K_{\text{stiff}}(s) &= g \cdot
#+name: tab:uniaxial_feedback_controller_parameters
#+caption: Parameters used for the position feedback controllers
-#+attr_latex: :environment tabularx :width \linewidth :align lXXX
+#+attr_latex: :environment tabularx :width 0.75\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| | *Soft* | *Moderately stiff* | *Stiff* |
|--------+-------------------------------------------+--------------------------------------------+------------------------------------------|
@@ -1749,19 +1736,19 @@ The goal is to have a first estimation of the attainable performance.
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_nyquist_hac_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_nyquist_hac_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_nyquist_hac_pz.png]]
#+end_subfigure
#+end_figure
@@ -1773,19 +1760,19 @@ The goal is to have a first estimation of the attainable performance.
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_loop_gain_hac_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_loop_gain_hac_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_loop_gain_hac_pz.png]]
#+end_subfigure
#+end_figure
@@ -1804,19 +1791,19 @@ As expected, the sensitivity to disturbances decreased in the controller bandwid
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_fs}Direct forces}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_hac_lac_fs.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_ft}$\mu\text{-station}$ disturbances}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_hac_lac_ft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_xf}Floor motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_sensitivity_dist_hac_lac_xf.png]]
#+end_subfigure
#+end_figure
@@ -1832,19 +1819,19 @@ Obtained root mean square values of the distance $d$ are better for the soft nan
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.37\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_hac_lac_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_hac_lac_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_cas_hac_lac_stiff.png]]
#+end_subfigure
#+end_figure
@@ -1904,19 +1891,19 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_soft}$\omega_{n} \ll \omega_{\mu}$}
#+attr_latex: :options {0.37\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_neglected_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_mid}$\omega_{n} = \omega_{\mu}$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_neglected_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_stiff}$\omega_{n} \gg \omega_{\mu}$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_neglected_stiff.png]]
#+end_subfigure
#+end_figure
@@ -1941,19 +1928,19 @@ If a soft nano-hexapod is used, the support dynamics appears in the dynamics bet
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_soft}$\omega_{n} \ll \omega_{\mu}$}
#+attr_latex: :options {0.37\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_dynamics_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_mid}$\omega_{n} = \omega_{\mu}$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_dynamics_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_stiff}$\omega_{n} \gg \omega_{\mu}$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_dynamics_stiff.png]]
#+end_subfigure
#+end_figure
@@ -1971,19 +1958,19 @@ Conversely, if a "stiff" nano-hexapod is used, the support dynamics appears in t
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_soft}$\omega_{n} \ll \omega_{\mu}$}
#+attr_latex: :options {0.37\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_dynamics_d_soft.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_mid}$\omega_{n} = \omega_{\mu}$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_dynamics_d_mid.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_stiff}$\omega_{n} \gg \omega_{\mu}$}
#+attr_latex: :options {0.31\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_effect_support_compliance_dynamics_d_stiff.png]]
#+end_subfigure
#+end_figure
@@ -2051,13 +2038,13 @@ The flexibility of the sample also changes the high frequency gain (the mass lin
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,N/\mu m$, $m_s = 1\,kg$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,N/\mu m$, $m_s = 50\,kg$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_payload_dynamics_soft_nano_hexapod_heavy.png]]
#+end_subfigure
#+end_figure
@@ -2074,13 +2061,13 @@ Even though the added sample's flexibility still shifts the high frequency mass
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,N/\mu m$, $m_s = 1\,kg$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,N/\mu m$, $m_s = 50\,kg$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/uniaxial_payload_dynamics_stiff_nano_hexapod_heavy.png]]
#+end_subfigure
#+end_figure
@@ -2116,13 +2103,13 @@ What happens is that above $\omega_s$, even though the motion $d$ can be control
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :width 0.8\linewidth
[[file:figs/uniaxial_sample_flexibility_noise_budget_d.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_y}Cumulative Amplitude Spectrum of $y$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :width 0.8\linewidth
[[file:figs/uniaxial_sample_flexibility_noise_budget_y.png]]
#+end_subfigure
#+end_figure
@@ -2186,11 +2173,6 @@ Up until this section, the study was performed on a very simplistic model that o
In the last section (Section\nbsp{}ref:sec:rotating_nass), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics.
The goal is to determine whether the rotation imposes performance limitation on the NASS.
-#+name: fig:rotating_overview
-#+caption: Overview of this chapter's organization. Sections are indicated by the red circles.
-#+attr_latex: :width \linewidth
-[[file:figs/rotating_overview.png]]
-
*** System Description and Analysis
<>
@@ -2294,13 +2276,13 @@ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forc
#+attr_latex: :caption \subcaption{\label{fig:rotating_campbell_diagram_real}Real part}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/rotating_campbell_diagram_real.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_campbell_diagram_imag}Imaginary part}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/rotating_campbell_diagram_imag.png]]
#+end_subfigure
#+end_figure
@@ -2321,13 +2303,13 @@ For $\Omega > \omega_0$, the low-frequency pair of complex conjugate poles $p_{-
#+attr_latex: :caption \subcaption{\label{fig:rotating_bode_plot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_bode_plot_direct.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_bode_plot_coupling}Coupling terms: $d_u/F_v$, $d_v/F_u$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_bode_plot_coupling.png]]
#+end_subfigure
#+end_figure
@@ -2441,13 +2423,13 @@ As expected from the derived equations of motion:
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_bode_plot_effect_rot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_bode_plot_effect_rot_direct.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/rotating_root_locus_iff_pure_int.png]]
#+end_subfigure
#+end_figure
@@ -2535,13 +2517,13 @@ For larger values of $\omega_i$, the attainable damping ratio decreases as a fun
#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_root_locus_iff_modified_effect_wi.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_hpf_optimal_gain.png]]
#+end_subfigure
#+end_figure
@@ -2559,13 +2541,13 @@ The same trade-off can be seen between achievable damping and loss of compliance
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}Obtained plants}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_hpf_damped_plant_effect_wi_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of $\omega_i$ on the compliance}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_hpf_effect_wi_compliance.png]]
#+end_subfigure
#+end_figure
@@ -2621,8 +2603,6 @@ Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiff
\boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2}
\end{equation}
-**** Identify plant with parallel stiffnesses :noexport:
-
**** Effect of parallel stiffness on the IFF plant
The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified without parallel stiffness $k_p = 0$, with a small parallel stiffness $k_p < m \Omega^2$ and with a large parallel stiffness $k_p > m \Omega^2$.
Bode plots of the obtained dynamics are shown in Figure\nbsp{}ref:fig:rotating_iff_effect_kp.
@@ -2639,13 +2619,13 @@ It is shown that if the added stiffness is higher than the maximum negative stif
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$}
#+attr_latex: :options {0.55\linewidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_effect_kp.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring}
#+attr_latex: :options {0.44\linewidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_kp_root_locus.png]]
#+end_subfigure
#+end_figure
@@ -2665,13 +2645,13 @@ This is confirmed by the Figure\nbsp{}ref:fig:rotating_iff_kp_optimal_gain where
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$}
#+attr_latex: :options {0.49\linewidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_kp_root_locus_effect_kp.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown because the system is unstable.}
#+attr_latex: :options {0.49\linewidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_kp_optimal_gain.png]]
#+end_subfigure
#+end_figure
@@ -2703,13 +2683,13 @@ The added high-pass filter gives almost the same damping properties to the suspe
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$}
#+attr_latex: :options {0.34\linewidth}
#+begin_subfigure
-#+attr_latex: :scale 0.95
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_kp_added_hpf_effect_damping.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF}
#+attr_latex: :options {0.65\linewidth}
#+begin_subfigure
-#+attr_latex: :scale 0.95
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_kp_added_hpf_damped_plant.png]]
#+end_subfigure
#+end_figure
@@ -2771,7 +2751,7 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
#+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control}
#+attr_latex: :options {0.49\linewidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/rotating_rdc_root_locus.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control}
@@ -2790,8 +2770,6 @@ These two proposed IFF modifications and relative damping control are compared i
For the following comparisons, the cut-off frequency for the added HPF is set to $\omega_i = 0.1 \omega_0$ and the stiffness of the parallel springs is set to $k_p = 5 m \Omega^2$ (corresponding to $\alpha = 0.05$).
These values are chosen one the basis of previous discussions about optimal parameters.
-**** Identify plants :noexport:
-
**** Root Locus
Figure\nbsp{}ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications and the relative damping control.
While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not.
@@ -2808,13 +2786,13 @@ It is interesting to note that the maximum added damping is very similar for bot
#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_comp_techniques_root_locus.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_dampled_plants}Damped plants}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_comp_techniques_dampled_plants.png]]
#+end_subfigure
#+end_figure
@@ -2843,13 +2821,13 @@ This is very well known characteristics of these common active damping technique
#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_comp_techniques_transmissibility.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_comp_techniques_compliance.png]]
#+end_subfigure
#+end_figure
@@ -2879,19 +2857,19 @@ The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually esti
#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nano_hexapod_dynamics_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nano_hexapod_dynamics_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nano_hexapod_dynamics_pz.png]]
#+end_subfigure
#+end_figure
@@ -2913,26 +2891,26 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_hpf_nass_optimal_gain_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_hpf_nass_optimal_gain_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_iff_hpf_nass_optimal_gain_pz.png]]
#+end_subfigure
#+end_figure
#+name: tab:rotating_iff_hpf_opt_iff_hpf_params_nass
#+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high-pass filter. The corresponding achievable simultaneous damping of the two modes $\xi$ is also shown.
-#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.3\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ |
|-----------------+------------+------+------------------|
@@ -2954,50 +2932,50 @@ This distance is larger for stiff nano-hexapod because the open-loop pole will b
Let's choose $k_p = 1\,N/mm$, $k_p = 0.01\,N/\mu m$ and $k_p = 1\,N/\mu m$ for the three considered nano-hexapods.
The corresponding optimal controller gains and achievable damping are summarized in Table\nbsp{}ref:tab:rotating_iff_kp_opt_iff_kp_params_nass.
-#+attr_latex: :options [t]{0.49\linewidth}
+#+attr_latex: :options [b]{0.49\linewidth}
#+begin_minipage
#+name: fig:rotating_iff_kp_nass_optimal_gain
-#+attr_latex: :width \linewidth :float nil
+#+attr_latex: :scale 0.8 :float nil
#+caption: Maximum damping $\xi$ as a function of the parallel stiffness $k_p$
[[file:figs/rotating_iff_kp_nass_optimal_gain.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.45\linewidth}
#+begin_minipage
-#+caption: Obtained optimal parameters for the IFF controller when using parallel stiffnesses
-#+name: tab:rotating_iff_kp_opt_iff_kp_params_nass
-#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
-#+attr_latex: :booktabs t :float nil
+#+latex: \centering
+#+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align cccc
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ |
|-----------------+-----------------+---------+--------------------|
| $0.01\,N/\mu m$ | $1\,N/mm$ | 47.9 | 0.44 |
| $1\,N/\mu m$ | $0.01\,N/\mu m$ | 465.57 | 0.97 |
| $100\,N/\mu m$ | $1\,N/\mu m$ | 4624.25 | 0.99 |
+#+latex: \captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses}
#+end_minipage
**** Optimal Relative Motion Control
For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure\nbsp{}ref:fig:rotating_rdc_optimal_gain).
The gain is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table\nbsp{}ref:tab:rotating_rdc_opt_params_nass).
-#+attr_latex: :options [t]{0.49\linewidth}
+#+attr_latex: :options [b]{0.49\linewidth}
#+begin_minipage
#+name: fig:rotating_rdc_optimal_gain
-#+attr_latex: :width \linewidth :float nil
+#+attr_latex: :scale 0.8 :float nil
#+caption: Maximum damping $\xi$ as a function of the RDC gain $g$
[[file:figs/rotating_rdc_optimal_gain.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.45\linewidth}
#+begin_minipage
-#+caption: Obtained optimal parameters for the RDC
-#+name: tab:rotating_rdc_opt_params_nass
-#+attr_latex: :environment tabularx :width 0.8\linewidth :placement [b] :align Xcc
-#+attr_latex: :booktabs t :float nil
+#+latex: \centering
+#+attr_latex: :environment tabularx :width 0.6\linewidth :placement [b] :align ccc
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| $k_n$ | $g$ | $\xi_{\text{opt}}$ |
|-----------------+-------+--------------------|
| $0.01\,N/\mu m$ | 1600 | 0.99 |
| $1\,N/\mu m$ | 8200 | 0.99 |
| $100\,N/\mu m$ | 80000 | 0.99 |
+#+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC}
#+end_minipage
**** Comparison of the obtained damped plants
@@ -3015,19 +2993,19 @@ Similar to what was concluded in the previous analysis:
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_damped_plant_comp_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_damped_plant_comp_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_damped_plant_comp_pz.png]]
#+end_subfigure
#+end_figure
@@ -3071,19 +3049,19 @@ It can be observed that:
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_plant_comp_stiffness_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_plant_comp_stiffness_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_plant_comp_stiffness_pz.png]]
#+end_subfigure
#+end_figure
@@ -3107,19 +3085,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_floor_motion_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_floor_motion_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_floor_motion_pz.png]]
#+end_subfigure
#+end_figure
@@ -3131,19 +3109,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_stage_vibration_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_stage_vibration_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_stage_vibration_pz.png]]
#+end_subfigure
#+end_figure
@@ -3155,19 +3133,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_direct_forces_vc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_direct_forces_md.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/rotating_nass_effect_direct_forces_pz.png]]
#+end_subfigure
#+end_figure
@@ -3177,27 +3155,27 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
:UNNUMBERED: t
:END:
-In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section\nbsp{}ref:sec:rotating_system_description).
-Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section\nbsp{}ref:sec:rotating_iff_pure_int).
+In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model.
+Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms.
Two modifications of the classical acrshort:iff control have been proposed to overcome this issue.
The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators.
This is equivalent to moving the controller pole to the left along the real axis.
-This allows the closed-loop system to be stable up to some value of the controller gain (Section\nbsp{}ref:sec:rotating_iff_pseudo_int).
+This allows the closed-loop system to be stable up to some value of the controller gain.
The second proposed modification concerns the mechanical system.
Additional springs are added in parallel with the actuators and force sensors.
-It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized acrshort:iff regains its unconditional stability property (Section\nbsp{}ref:sec:rotating_iff_parallel_stiffness).
+It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized acrshort:iff regains its unconditional stability property.
-These two modifications were compared with acrlong:rdc in Section\nbsp{}ref:sec:rotating_comp_act_damp.
+These two modifications were compared with acrlong:rdc.
While having very different implementations, both proposed modifications were found to be very similar with respect to the attainable damping and the obtained closed-loop system behavior.
-This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters (Section\nbsp{}ref:sec:rotating_nano_hexapod).
+This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters.
As for the uniaxial model, three nano-hexapod stiffnesses values were considered.
The dynamics of the soft nano-hexapod ($k_n = 0.01\,N/\mu m$) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects).
In addition, the attainable damping ratio of the soft nano-hexapod 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 (Section\nbsp{}ref:sec:rotating_nass).
+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 nano-hexapod due to the spindle's rotation.
For the moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that showed better results with the uniaxial model.
@@ -3318,10 +3296,9 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrsh
\hfill
#+attr_latex: :options [b]{0.36\linewidth}
#+begin_minipage
-#+begin_scriptsize
#+latex: \centering
-#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
-#+attr_latex: :booktabs t :float nil :center nil
+#+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align Xccc
+#+attr_latex: :booktabs t :float nil :center nil :font \scriptsize\sf
| | $x$ | $y$ | $z$ |
|-------------------+------+------+------|
| (17) Low. Granite | -730 | -526 | -951 |
@@ -3348,7 +3325,6 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrsh
| (3) Hexapod | 64 | 64 | -270 |
| (4) Hexapod | 64 | -64 | -270 |
#+latex: \captionof{table}{\label{tab:modal_position_accelerometers}Positions in mm}
-#+end_scriptsize
#+end_minipage
#+name: fig:modal_accelerometer_pictures
@@ -3420,13 +3396,13 @@ Similar results were obtained for all measured frequency response functions.
#+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_raw_meas.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_asd_acc_force.png]]
#+end_subfigure
#+end_figure
@@ -3442,13 +3418,13 @@ Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corres
#+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_frf_acc_force.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_coh_acc_force} Coherence}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_coh_acc_force.png]]
#+end_subfigure
#+end_figure
@@ -3539,7 +3515,7 @@ The position of each accelerometer with respect to the center of mass of the cor
#+name: tab:modal_com_solid_bodies
#+caption: Center of mass of considered solid bodies with respect to the "point of interest"
-#+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.45\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| | $X$ | $Y$ | $Z$ |
|-------------------+-----------------+------------------+--------------------|
@@ -3580,6 +3556,7 @@ This also validates the reduction in the number of degrees of freedom from 69 (2
#+name: fig:modal_comp_acc_solid_body_frf
#+caption: Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the $x$ direction
+#+attr_latex: :scale 0.8
[[file:figs/modal_comp_acc_solid_body_frf.png]]
*** Modal Analysis
@@ -3617,20 +3594,19 @@ The obtained acrshort:mif is shown on Figure\nbsp{}ref:fig:modal_indication_func
A total of 16 modes were found between 0 and $200\,\text{Hz}$.
The obtained natural frequencies and associated modal damping are summarized in Table\nbsp{}ref:tab:modal_obtained_modes_freqs_damps.
-#+attr_latex: :options [b]{0.70\linewidth}
+#+attr_latex: :options [b]{0.65\linewidth}
#+begin_minipage
#+name: fig:modal_indication_function
#+caption: Modal Indication Function
-#+attr_latex: :float nil :scale 1
+#+attr_latex: :float nil :scale 0.8
[[file:figs/modal_indication_function.png]]
#+end_minipage
\hfill
-#+attr_latex: :options [b]{0.28\linewidth}
+#+attr_latex: :options [b]{0.33\linewidth}
#+begin_minipage
-#+begin_scriptsize
#+latex: \centering
-#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align ccc
-#+attr_latex: :booktabs t :float nil :center nil
+#+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align ccc
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| Mode | Frequency | Damping |
|------+--------------------+------------|
| 1 | $11.9\,\text{Hz}$ | $12.2\,\%$ |
@@ -3650,7 +3626,6 @@ The obtained natural frequencies and associated modal damping are summarized in
| 15 | $150.5\,\text{Hz}$ | $2.4\,\%$ |
| 16 | $165.4\,\text{Hz}$ | $1.4\,\%$ |
#+latex: \captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes}
-#+end_scriptsize
#+end_minipage
**** Modal parameter extraction
@@ -3750,19 +3725,19 @@ This can be seen in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal_3 that shows t
#+attr_latex: :caption \subcaption{\label{fig:modal_comp_acc_frf_modal_1}From $F_{11,z}$ to $a_{11,z}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_comp_acc_frf_modal_1.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_comp_acc_frf_modal_2}From $F_{11,z}$ to $a_{15,z}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_comp_acc_frf_modal_2.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_comp_acc_frf_modal_3}From $F_{11,y}$ to $a_{2,x}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/modal_comp_acc_frf_modal_3.png]]
#+end_subfigure
#+end_figure
@@ -4163,6 +4138,7 @@ External forces can be used to model disturbances, and "sensors" can be used to
#+name: fig:ustation_simscape_stage_example
#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included
+#+attr_latex: :scale 0.8
[[file:figs/ustation_simscape_stage_example.png]]
Therefore, the micro-station is modeled by several solid bodies connected by joints.
@@ -4188,7 +4164,7 @@ The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_
#+name: tab:ustation_6dof_stiffness_values
#+caption: Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by "-". The frames in which the 6-DoF joints are defined are indicated in figures found in Section\nbsp{}ref:ssec:ustation_stages
-#+attr_latex: :environment tabularx :width \linewidth :align Xcccccc
+#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccccc
#+attr_latex: :center t :booktabs t
| *Stage* | $D_x$ | $D_y$ | $D_z$ | $R_x$ | $R_y$ | $R_z$ |
|-------------+-----------------+-----------------+-----------------+-------------------------+------------------------+-------------------------|
@@ -4216,19 +4192,19 @@ Tuning the numerous model parameters to better match the measurements is a highl
#+attr_latex: :caption \subcaption{\label{fig:ustation_comp_com_response_rz_x}Spindle, $x$ response}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_comp_com_response_rz_x.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_comp_com_response_hexa_y}Hexapod, $y$ response}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_comp_com_response_hexa_y.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_comp_com_response_ry_z}Tilt, $z$ response}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_comp_com_response_ry_z.png]]
#+end_subfigure
#+end_figure
@@ -4250,10 +4226,10 @@ For each impact position, 10 impacts were performed to average and improve the d
#+caption: Schematic of the measurement setup used to estimate the compliance of the micro-station. The top platform of the positioning hexapod is shown with four 3-axis accelerometers (shown in red) are on top. 10 hammer impacts are performed at different locations (shown in blue).
[[file:figs/ustation_compliance_meas.png]]
-To convert the 12 acceleration signals $a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]$ to the acceleration expressed in the frame $\{\mathcal{X}\}$ $a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix $\bm{J}_a$ is written based on the positions and orientations of the accelerometers\nbsp{}eqref:eq:ustation_compliance_acc_jacobian.
+To convert the 12 acceleration signals $a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]$ to the acceleration expressed in the $\{\mathcal{X}\}$ frame $a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix $\bm{J}_a$ is written based on the positions and orientations of the accelerometers\nbsp{}eqref:eq:ustation_compliance_acc_jacobian.
\begin{equation}\label{eq:ustation_compliance_acc_jacobian}
-\bm{J}_a = \begin{bmatrix}
+\bm{J}_a = \left[\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 &-d \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & d & 0 & 0 \\
@@ -4266,19 +4242,19 @@ To convert the 12 acceleration signals $a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & d \\
0 & 0 & 1 & 0 &-d & 0
-\end{bmatrix}
+\end{smallmatrix}\right]
\end{equation}
Then, the acceleration in the cartesian frame can be computed using\nbsp{}eqref:eq:ustation_compute_cart_acc.
\begin{equation}\label{eq:ustation_compute_cart_acc}
-a_{\mathcal{X}} = \bm{J}_a^\dagger \cdot a_{\mathcal{L}}
+ a_{\mathcal{X}} = \bm{J}_a^{-1} \cdot a_{\mathcal{L}}
\end{equation}
Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is computed\nbsp{}eqref:eq:ustation_compliance_force_jacobian and used to convert the individual hammer forces $F_{\mathcal{L}}$ to force and torques $F_{\mathcal{X}}$ applied at the center of the micro-hexapod top plate (defined by frame $\{\mathcal{X}\}$ in Figure\nbsp{}ref:fig:ustation_compliance_meas).
\begin{equation}\label{eq:ustation_compliance_force_jacobian}
-\bm{J}_F = \begin{bmatrix}
+\bm{J}_F = \left[\begin{smallmatrix}
0 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & -d & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0\\
@@ -4289,7 +4265,7 @@ Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is
0 & 0 & -1 & 0 & d & 0\\
-1 & 0 & 0 & 0 & 0 & -d\\
-1 & 0 & 0 & 0 & 0 & d
-\end{bmatrix}
+\end{smallmatrix}\right]
\end{equation}
The equivalent forces and torques applied at center of $\{\mathcal{X}\}$ are then computed using\nbsp{}eqref:eq:ustation_compute_cart_force.
@@ -4312,13 +4288,13 @@ Considering the complexity of the micro-station compliance dynamics, the model c
#+attr_latex: :caption \subcaption{\label{fig:ustation_frf_compliance_xyz_model}Compliance in translation}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_frf_compliance_xyz_model.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_frf_compliance_Rxyz_model}Compliance in rotation}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_frf_compliance_Rxyz_model.png]]
#+end_subfigure
#+end_figure
@@ -4358,7 +4334,7 @@ The obtained ground motion displacement is shown in Figure\nbsp{}ref:fig:ustatio
#+begin_minipage
#+name: fig:ustation_ground_disturbance
#+caption: Measured ground motion
-#+attr_latex: :scale 1 :float nil
+#+attr_latex: :scale 0.8 :float nil
[[file:figs/ustation_ground_disturbance.png]]
#+end_minipage
\hfill
@@ -4394,13 +4370,13 @@ Similar result is obtained for the $x$ lateral direction.
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical}Measured vertical error}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_dy_vertical.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing linear fit}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_dy_vertical_remove_mean.png]]
#+end_subfigure
#+end_figure
@@ -4445,19 +4421,19 @@ The vertical motion induced by scanning the spindle is in the order of $\pm 30\,
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_radial}Radial errors}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_spindle_radial.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_axial}Axial error}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_spindle_axial.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_spindle_tilt}Tilt errors}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_spindle_tilt.png]]
#+end_subfigure
#+end_figure
@@ -4476,19 +4452,19 @@ The obtained transfer functions are shown in Figure\nbsp{}ref:fig:ustation_model
#+attr_latex: :caption \subcaption{\label{fig:ustation_model_sensitivity_ground_motion}Ground motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_model_sensitivity_ground_motion.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_model_sensitivity_ty}Translation stage}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_model_sensitivity_ty.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_model_sensitivity_rz}Spindle}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_model_sensitivity_rz.png]]
#+end_subfigure
#+end_figure
@@ -4506,19 +4482,19 @@ The obtained power spectral density of the disturbances are shown in Figure\nbsp
#+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_ground_motion}Ground Motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_dist_source_ground_motion.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_translation_stage}Translation Stage}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_dist_source_translation_stage.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_spindle}Spindle}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_dist_source_spindle.png]]
#+end_subfigure
#+end_figure
@@ -4535,19 +4511,19 @@ Examples of the obtained time-domain disturbance signals are shown in Figure\nbs
#+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_ground_motion_time}Ground Motion}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_dist_source_ground_motion_time.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_translation_stage_time}Translation Stage}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_dist_source_translation_stage_time.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_dist_source_spindle_time}Spindle}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/ustation_dist_source_spindle_time.png]]
#+end_subfigure
#+end_figure
@@ -4578,13 +4554,13 @@ A good correlation with the measurements is observed both for radial errors (Fig
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_model_spindle_radial}Radial error}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_model_spindle_radial.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:ustation_errors_model_spindle_axial}Axial error}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_model_spindle_axial.png]]
#+end_subfigure
#+end_figure
@@ -4600,6 +4576,7 @@ A similar error amplitude was observed, thus indicating that the multi-body mode
#+name: fig:ustation_errors_model_dy_vertical
#+caption: Vertical errors during a constant-velocity scan of the translation stage. Comparison of the measurements and simulated errors.
+#+attr_latex: :scale 0.8
[[file:figs/ustation_errors_model_dy_vertical.png]]
*** Conclusion
@@ -4667,13 +4644,13 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac
#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope}
#+attr_latex: :options {0.36\textwidth}
#+begin_subfigure
-#+attr_latex: :height 6.5cm
+#+attr_latex: :height 6cm
[[file:figs/nhexa_stages_nazaretski.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_sapoti} SAPOTI sample stage}
#+attr_latex: :options {0.60\textwidth}
#+begin_subfigure
-#+attr_latex: :height 6.5cm
+#+attr_latex: :height 6cm
[[file:figs/nhexa_stages_sapoti.png]]
#+end_subfigure
#+end_figure
@@ -4691,7 +4668,7 @@ However, attempts to implement real-time feedback using YZ external metrology pr
#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station}
#+attr_latex: :options {0.54\textwidth}
#+begin_subfigure
-#+attr_latex: :height 6cm
+#+attr_latex: :height 5.5cm
[[file:figs/nhexa_stages_villar.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_stages_schroer} PtyNAMi microscope}
@@ -4708,7 +4685,7 @@ Although direct performance comparisons between these systems are challenging du
#+name: tab:nhexa_sample_stages
#+caption: End-Stations with integrated feedback loops based on online metrology. The stages used for feedback are indicated in bold font. Stages not used for scanning purposes are ommited or indicated between parentheses. The specifications for the NASS are indicated in the last row.
#+attr_latex: :environment tabularx :width 0.8\linewidth :align ccccc
-#+attr_latex: :placement [!ht] :center t :booktabs t :font \scriptsize
+#+attr_latex: :placement [!ht] :center t :booktabs t
| *Stacked Stages* | *Specifications* | *Measured DoFs* | *Bandwidth* | *Reference* |
|---------------------+------------------------------+---------------------+--------------------------+-------------------------------------------------------------------------------------------------------|
| Sample | light | Interferometers | 3 PID, n/a | APS |
@@ -4798,13 +4775,13 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha
#+attr_latex: :caption \subcaption{\label{fig:nhexa_serial_architecture_kenton} Serial positioning stage}
#+attr_latex: :options {0.41\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5cm
+#+attr_latex: :height 4.5cm
[[file:figs/nhexa_serial_architecture_kenton.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoF stage}
#+attr_latex: :options {0.55\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5cm
+#+attr_latex: :height 4.5cm
[[file:figs/nhexa_parallel_architecture_shen.png]]
#+end_subfigure
#+end_figure
@@ -4822,13 +4799,13 @@ Furthermore, the successful implementation of Integral Force Feedback (IFF) cont
#+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :width 0.9\linewidth
[[file:figs/nhexa_stewart_piezo_furutani.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_stewart_vc_preumont} Stewart platform for vibration isolation}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :width 0.9\linewidth
[[file:figs/nhexa_stewart_vc_preumont.png]]
#+end_subfigure
#+end_figure
@@ -4866,6 +4843,7 @@ The typical configuration consists of a universal joint at one end and a spheric
#+name: fig:nhexa_stewart_architecture
#+caption: Schematical representation of the Stewart platform architecture.
+#+attr_latex: :scale 0.9
[[file:figs/nhexa_stewart_architecture.png]]
To facilitate the rigorous analysis of the Stewart platform, four reference frames were defined:
@@ -4885,6 +4863,7 @@ This is summarized in Figure\nbsp{}ref:fig:nhexa_stewart_notations.
#+name: fig:nhexa_stewart_notations
#+caption: Frame and key notations for the Stewart platform
+#+attr_latex: :scale 0.9
[[file:figs/nhexa_stewart_notations.png]]
**** Kinematic Analysis
@@ -4902,6 +4881,7 @@ This equation links the pose[fn:nhexa_2] variables ${}^A\bm{P}$ and ${}^A\bm{R}_
#+name: fig:nhexa_stewart_loop_closure
#+caption: Notations to compute the kinematic loop closure
+#+attr_latex: :scale 0.9
[[file:figs/nhexa_stewart_loop_closure.png]]
***** Inverse Kinematics
@@ -5008,6 +4988,7 @@ It can be computed once at the rest position and used for both forward and inver
#+name: fig:nhexa_forward_kinematics_approximate_errors
#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,mm$ was used to perform this analysis. $\epsilon_D$ corresponds to the distance between the true positioin and the estimated position. $\epsilon_R$ corresponds to the angular motion between the true orientation and the estimated orientation.
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_forward_kinematics_approximate_errors.png]]
***** Static Forces
@@ -5165,16 +5146,15 @@ From these parameters, key kinematic properties can be derived: the strut orient
#+begin_minipage
#+name: fig:nhexa_stewart_model_def
#+caption: Geometry of the stewart platform
-#+attr_latex: :float nil :scale 1
+#+attr_latex: :float nil :scale 0.9
[[file:figs/nhexa_stewart_model_def.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.38\linewidth}
#+begin_minipage
-#+begin_scriptsize
#+latex: \centering
#+attr_latex: :environment tabularx :width 0.75\linewidth :placement [b] :align Xrrr
-#+attr_latex: :booktabs t :float nil :center nil
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| | $\bm{x}$ | $\bm{y}$ | $\bm{z}$ |
|-----------+----------+----------+----------|
| ${}^M\bm{O}_B$ | $0$ | $0$ | $150$ |
@@ -5192,7 +5172,6 @@ From these parameters, key kinematic properties can be derived: the strut orient
| ${}^M\bm{b}_5$ | $-78$ | $78$ | $-20$ |
| ${}^M\bm{b}_6$ | $-106$ | $28$ | $-20$ |
#+latex: \captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]}
-#+end_scriptsize
#+end_minipage
***** Inertia of Plates
@@ -5226,23 +5205,21 @@ This modular approach to actuator modeling allows for future refinements as the
#+begin_minipage
#+name: fig:nhexa_actuator_model
#+caption: Model of the nano-hexapod actuators
-#+attr_latex: :float nil :scale 1
+#+attr_latex: :float nil :scale 0.8
[[file:figs/nhexa_actuator_model.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.38\linewidth}
#+begin_minipage
-#+begin_scriptsize
#+latex: \centering
-#+attr_latex: :environment tabularx :width 0.4\linewidth :placement [b] :align Xl
-#+attr_latex: :booktabs t :float nil :center nil
+#+attr_latex: :environment tabularx :width 0.5\linewidth :placement [b] :align Xl
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| | Value |
|-------+-----------------|
| $k_a$ | $1\,N/\mu m$ |
| $c_a$ | $50\,N/(m/s)$ |
| $k_p$ | $0.05\,N/\mu m$ |
#+latex: \captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters}
-#+end_scriptsize
#+end_minipage
**** Validation of the multi-body model
@@ -5255,7 +5232,7 @@ A three-dimensional visualization of the model is presented in Figure\nbsp{}ref:
#+attr_latex: :options [b]{0.6\linewidth}
#+begin_minipage
#+name: fig:nhexa_stewart_model_input_outputs
-#+caption: Nano-Hexapod plant with inputs and outputs. Frames $\{F\}$ and $\{M\}$ can be connected to other elements in the multi-body models.
+#+caption: Nano-Hexapod plant with inputs and outputs. Frames $\{F\}$ and $\{M\}$ can be connected to other elements in the model.
#+attr_latex: :scale 1 :float nil
[[file:figs/nhexa_stewart_model_input_outputs.png]]
#+end_minipage
@@ -5264,7 +5241,7 @@ A three-dimensional visualization of the model is presented in Figure\nbsp{}ref:
#+begin_minipage
#+name: fig:nhexa_simscape_screenshot
#+caption: 3D representation of the multi-body model
-#+attr_latex: :width 0.90\linewidth :float nil
+#+attr_latex: :width 0.8\linewidth :float nil
[[file:figs/nhexa_simscape_screenshot.jpg]]
#+end_minipage
@@ -5293,6 +5270,7 @@ The close agreement between both approaches across the frequency spectrum valida
#+name: fig:nhexa_comp_multi_body_analytical
#+caption: Comparison of the analytical transfer functions and the multi-body model
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_comp_multi_body_analytical.png]]
**** Nano Hexapod Dynamics
@@ -5323,13 +5301,13 @@ The inclusion of parallel stiffness introduces an additional complex conjugate z
#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{\mathcal{L}}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_multi_body_plant_dL.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{n}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_multi_body_plant_fm.png]]
#+end_subfigure
#+end_figure
@@ -5378,6 +5356,7 @@ In the context of the nano-hexapod, two distinct control strategies were examine
#+name: fig:nhexa_stewart_decentralized_control
#+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.
+#+attr_latex: :scale 0.9
[[file:figs/nhexa_stewart_decentralized_control.png]]
**** Choice of the Control Space
@@ -5439,13 +5418,13 @@ More sophisticated control strategies will be explored during the detailed desig
#+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_plant_frame_struts.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_plant_frame_cartesian.png]]
#+end_subfigure
#+end_figure
@@ -5459,6 +5438,7 @@ The corresponding block diagram of the control loop is shown in Figure\nbsp{}ref
#+name: fig:nhexa_decentralized_iff_schematic
#+caption: Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs $\bm{f}^{\prime}$
+#+attr_latex: :scale 0.9
[[file:figs/nhexa_decentralized_iff_schematic.png]]
\begin{equation}\label{eq:nhexa_kiff}
@@ -5487,13 +5467,13 @@ This high gain, combined with the bounded phase, enables effective damping of th
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{n1}}{f_1}(s)$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.85
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_decentralized_iff_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.85
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_decentralized_iff_root_locus.png]]
#+end_subfigure
#+end_figure
@@ -5510,6 +5490,7 @@ A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these
#+name: fig:nhexa_hac_iff_schematic
#+caption: HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts
+#+attr_latex: :scale 0.9
[[file:figs/nhexa_hac_iff_schematic.png]]
The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions.
@@ -5524,13 +5505,13 @@ This damping of structural resonances serves two purposes: it reduces vibrations
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped plant in the frame of the struts}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_decentralized_hac_iff_plant_undamped.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped plant with Decentralized IFF}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_decentralized_hac_iff_plant_damped.png]]
#+end_subfigure
#+end_figure
@@ -5560,13 +5541,13 @@ Additionally, the distance of the loci from the $-1$ point provides information
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.85
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_decentralized_hac_iff_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Characteristic Loci}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.85
+#+attr_latex: :scale 0.8
[[file:figs/nhexa_decentralized_hac_iff_root_locus.png]]
#+end_subfigure
#+end_figure
@@ -5822,13 +5803,13 @@ However, their alternating pattern is preserved, which ensures the phase remains
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_effect_rotation}Effect of Spindle rotation}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_iff_plant_effect_rotation.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_effect_payload}Effect of payload mass}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_iff_plant_effect_payload.png]]
#+end_subfigure
#+end_figure
@@ -5858,6 +5839,7 @@ The overall gain was then increased to obtain a large loop gain around the reson
#+name: fig:nass_iff_loop_gain
#+caption: Loop gain for the decentralized IFF: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$
#+attr_latex: :options [h!tbp]
+#+attr_latex: :scale 0.8
[[file:figs/nass_iff_loop_gain.png]]
To verify stability, the root loci for the three payload configurations were computed, as shown in Figure\nbsp{}ref:fig:nass_iff_root_locus.
@@ -5870,19 +5852,19 @@ The results demonstrate that the closed-loop poles remain within the left-half p
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_iff_root_locus_1kg.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_25kg} $25\,\text{kg}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_iff_root_locus_25kg.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_iff_root_locus_50kg.png]]
#+end_subfigure
#+end_figure
@@ -5926,13 +5908,13 @@ This also validates the developed control strategy.
#+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_Wz}Effect of rotational velocity $\Omega_z$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_undamped_plant_effect_Wz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_mass}Effect of payload's mass}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_undamped_plant_effect_mass.png]]
#+end_subfigure
#+end_figure
@@ -5955,13 +5937,13 @@ For the undamped plants (shown in blue), achieving robust control with bandwidth
#+attr_latex: :caption \subcaption{\label{fig:nass_comp_undamped_damped_plant_m1}Effect of IFF - $m = 1\,\text{kg}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_comp_undamped_damped_plant_m1.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_hac_plants}Effect of IFF on the set of plants to control}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_hac_plants.png]]
#+end_subfigure
#+end_figure
@@ -5976,6 +5958,7 @@ This result confirms effective dynamic decoupling between the nano-hexapod and t
#+name: fig:nass_effect_ustation_compliance
#+caption: Effect of the micro-station limited compliance on the plant dynamics
#+attr_latex: :options [h!tbp]
+#+attr_latex: :scale 0.8
[[file:figs/nass_effect_ustation_compliance.png]]
**** Effect of Nano-Hexapod Stiffness on System Dynamics
@@ -6001,13 +5984,13 @@ The current approach of controlling the position in the strut frame is inadequat
#+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,N/\mu m$ - Coupling with the micro-station}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_stiff_nano_hexapod_coupling_ustation.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,N/\mu m$ - Effect of Spindle rotation}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_soft_nano_hexapod_effect_Wz.png]]
#+end_subfigure
#+end_figure
@@ -6033,13 +6016,13 @@ Second, the characteristic loci analysis presented in Figure\nbsp{}ref:fig:nass_
#+attr_latex: :caption \subcaption{\label{fig:nass_hac_loop_gain}Loop Gain}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_hac_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_hac_loci}Characteristic Loci}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/nass_hac_loci.png]]
#+end_subfigure
#+end_figure
@@ -6065,13 +6048,13 @@ The results demonstrate the system's capability to maintain the sample's positio
#+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_xy}XY plane}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/nass_tomo_1kg_60rpm_xy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_yz}YZ plane}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/nass_tomo_1kg_60rpm_yz.png]]
#+end_subfigure
#+end_figure
@@ -6090,19 +6073,19 @@ For higher mass configurations, rotational velocities are expected to be below 3
#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,kg$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/nass_tomography_hac_iff_m1.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,kg$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/nass_tomography_hac_iff_m25.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,kg$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/nass_tomography_hac_iff_m50.png]]
#+end_subfigure
#+end_figure
@@ -6126,8 +6109,6 @@ Simulations of tomography experiments have been performed, with positioning accu
The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples.
While some degradation in positioning accuracy has been observed with heavier payloads, as anticipated by the control analysis, the overall performance remains sufficient to validate the fundamental concept of the NASS.
-These results provide a solid foundation for advancing to the subsequent detailed design phase and experimental implementation.
-
** Conceptual Design - Conclusion
:PROPERTIES:
:UNNUMBERED: notoc
@@ -6542,8 +6523,8 @@ These results could have been easily deduced based on mechanical principles, but
These trade-offs provide important guidelines when choosing the Stewart platform geometry.
#+name: tab:detail_kinematics_geometry
-#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcc
-#+attr_latex: :center t :booktabs t :float t :font \small
+#+attr_latex: :environment tabularx :width 0.65\linewidth :align Xcc
+#+attr_latex: :center t :booktabs t :float t
#+caption: Effect of a change in geometry on the manipulator's stiffness and mobility
| *Struts* | *Vertically Oriented* | *Increased separation* |
|-------------------------------+-----------------------+------------------------|
@@ -6750,7 +6731,7 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m
#+name: fig:detail_kinematics_cubic_payload
#+caption: Cubic stewart platform with top cylindrical payload
-#+attr_latex: :width 0.6\linewidth
+#+attr_latex: :width 0.5\linewidth
[[file:figs/detail_kinematics_cubic_payload.png]]
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure\nbsp{}ref:fig:detail_kinematics_cubic_payload).
@@ -6765,13 +6746,13 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_cubic_cart_coupling_com.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_cubic_cart_coupling_cok.png]]
#+end_subfigure
#+end_figure
@@ -6798,7 +6779,7 @@ If a design similar to Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_pay
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_cubic_cart_coupling_com_cok.png]]
#+end_subfigure
#+end_figure
@@ -6829,7 +6810,7 @@ The second uses a non-cubic Stewart platform shown in Figure\nbsp{}ref:fig:detai
#+name: fig:detail_kinematics_non_cubic_payload
#+caption: Stewart platform with non-cubic architecture
-#+attr_latex: :width 0.6\linewidth
+#+attr_latex: :width 0.5\linewidth
[[file:figs/detail_kinematics_non_cubic_payload.png]]
***** Relative Displacement Sensors
@@ -6848,13 +6829,13 @@ The resonance frequencies differ between the two cases because the more vertical
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_non_cubic_decentralized_dL.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_cubic_decentralized_dL.png]]
#+end_subfigure
#+end_figure
@@ -6872,13 +6853,13 @@ The system demonstrates good decoupling at high frequency in both cases, with no
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_non_cubic_decentralized_fn.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_cubic_decentralized_fn.png]]
#+end_subfigure
#+end_figure
@@ -7084,13 +7065,13 @@ The positioning angles, as shown in Figure\nbsp{}ref:fig:detail_kinematics_nano_
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_nano_hexapod_iso}Isometric view}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.9
[[file:figs/detail_kinematics_nano_hexapod_iso.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_nano_hexapod_top}Top view}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.9
[[file:figs/detail_kinematics_nano_hexapod_top.png]]
#+end_subfigure
#+end_figure
@@ -7114,15 +7095,14 @@ The required mobility parameters include combined translations in the XYZ direct
Additionally, at any point within this workspace, combined $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$, with $R_z$ maintained at 0, should be possible.
Calculations based on the selected geometry indicate that an actuator stroke of $\pm 94\,\mu m$ is required to achieve the desired mobility.
-This specification will be used during the actuator selection process.
-# TODO - Add link to section
+This specification will be used during the actuator selection process in Section ref:sec:detail_fem_actuator.
Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of $\pm 94\,\mu m$.
The diagram confirms that the required workspace fits within the system's capabilities.
#+name: fig:detail_kinematics_nano_hexapod_mobility
#+caption: Specified translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/detail_kinematics_nano_hexapod_mobility.png]]
**** Required Joint angular stroke
@@ -7132,8 +7112,7 @@ With the nano-hexapod geometry and mobility requirements established, the flexib
This analysis focuses solely on bending stroke, as the torsional stroke of the flexible joints is expected to be minimal given the absence of vertical rotation requirements.
The required angular stroke for both fixed and mobile joints is estimated to be equal to $1\,\text{mrad}$.
-This specification will guide the design of the flexible joints.
-# TODO - Add link to section
+This specification will guide the design of the flexible joints in Section ref:sec:detail_fem_joint.
*** Conclusion
:PROPERTIES:
@@ -7226,8 +7205,8 @@ The specific design of the APA allows for the simultaneous modeling of a mechani
#+attr_latex: :options [b]{0.48\linewidth}
#+begin_minipage
#+latex: \centering
-#+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align Xc
-#+attr_latex: :booktabs t :float nil :center nil
+#+attr_latex: :environment tabularx :width 0.55\linewidth :placement [b] :align Xc
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| *Parameter* | *Value* |
|----------------+---------------|
| Nominal Stroke | $100\,\mu m$ |
@@ -7242,8 +7221,8 @@ The development of the finite element model for the APA95ML required the knowled
The finite element mesh, shown in Figure\nbsp{}ref:fig:detail_fem_apa95ml_mesh, was then generated.
#+name: tab:detail_fem_material_properties
-#+caption: Material properties used for FEA modal reduction model. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density
-#+attr_latex: :environment tabularx :width 0.7\linewidth :align lXXX
+#+caption: Material properties used for FEA. $E$ is the Young's modulus, $\nu$ the Poisson ratio and $\rho$ the material density
+#+attr_latex: :environment tabularx :width 0.55\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| | $E$ | $\nu$ | $\rho$ |
|------------------------------+-------------+--------+-----------------------|
@@ -7306,24 +7285,24 @@ Yet, based on the available properties of the stacks in the data-sheet (summariz
#+name: tab:detail_fem_stack_parameters
#+caption: Stack Parameters
-#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc
+#+attr_latex: :environment tabularx :width 0.3\linewidth :align Xc
#+attr_latex: :center t :booktabs t
-| Parameter | Unit | Value |
-|----------------+-----------+------------|
-| Nominal Stroke | $\mu m$ | 20 |
-| Blocked force | $N$ | 4700 |
-| Stiffness | $N/\mu m$ | 235 |
-| Voltage Range | $V$ | -20 to 150 |
-| Capacitance | $\mu F$ | 4.4 |
-| Length | $mm$ | 20 |
-| Stack Area | $mm^2$ | 10x10 |
+| *Parameter* | *Value* |
+|----------------+---------------------|
+| Nominal Stroke | $20\,\mu m$ |
+| Blocked force | $4700\,N$ |
+| Stiffness | $235\,N/\mu m$ |
+| Voltage Range | $-20/150\,V$ |
+| Capacitance | $4.4\,\mu F$ |
+| Length | $20\,mm$ |
+| Stack Area | $10\times 10\,mm^2$ |
The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table\nbsp{}ref:tab:detail_fem_piezo_properties.
From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained.
#+name: tab:detail_fem_piezo_properties
#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities
-#+attr_latex: :environment tabularx :width 1\linewidth :align ccX
+#+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX
#+attr_latex: :center t :booktabs t
| *Parameter* | *Value* | *Description* |
|----------------+----------------------------+--------------------------------------------------------------|
@@ -7347,6 +7326,7 @@ The multi-body model predicted a resonant frequency under block-free conditions
#+name: fig:detail_fem_apa95ml_compliance
#+caption: Estimated compliance of the APA95ML
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa95ml_compliance.png]]
In order to estimate the stroke of the APA95ML, the mechanical amplification factor, defined as the ratio between vertical displacement and horizontal stack displacement, was first determined.
@@ -7399,13 +7379,13 @@ Regarding the amplitude characteristics, the constants $g_a$ and $g_s$ could be
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa95ml_comp_plant_actuator.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa95ml_comp_plant_sensor.png]]
#+end_subfigure
#+end_figure
@@ -7434,13 +7414,13 @@ The close agreement between experimental measurements and theoretical prediction
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa95ml_iff_root_locus.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa95ml_damped_plants.png]]
#+end_subfigure
#+end_figure
@@ -7467,7 +7447,7 @@ Additional specifications arise from the control strategy and physical constrain
The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator.
The system's geometric constraints limit the actuator height to 50mm, given the nano-hexapod's maximum height of 95mm and the presence of flexible joints at each strut extremity.
Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility.
-An actuator stroke of $\approx 100\,\mu m$ is therefore required.
+An actuator stroke of $\approx 200\,\mu m$ is therefore required.
Three actuator technologies were evaluated (examples of such actuators are shown in Figure\nbsp{}ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity.
@@ -7479,13 +7459,13 @@ Variable reluctance actuators were not considered despite their superior efficie
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_voice_coil_picture}Voice Coil}
#+attr_latex: :options {0.25\textwidth}
#+begin_subfigure
-#+attr_latex: :height 4.5cm
+#+attr_latex: :height 4cm
[[file:figs/detail_fem_voice_coil_picture.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_piezo_picture}Piezoelectric stack}
#+attr_latex: :options {0.25\textwidth}
#+begin_subfigure
-#+attr_latex: :height 4.5cm
+#+attr_latex: :height 4cm
[[file:figs/detail_fem_piezo_picture.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_fpa_picture}Amplified Piezoelectric Actuator}
@@ -7496,7 +7476,7 @@ Variable reluctance actuators were not considered despite their superior efficie
#+end_subfigure
#+end_figure
-Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,N/\mu m$), would require forces in the order of $100\,N$ to achieve the specified $100\,\mu m$ displacement.
+Voice coil actuators (shown in Figure\nbsp{}ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness ($\approx 1\,N/\mu m$), would require forces in the order of $200\,N$ to achieve the specified $200\,\mu m$ displacement.
While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability.
Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions.
@@ -7517,10 +7497,10 @@ The demonstrated accuracy of the modeling approach for the APA95ML provides conf
#+name: tab:detail_fem_piezo_act_models
#+caption: List of some amplified piezoelectric actuators that could be used for the nano-hexapod
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccccc
-#+attr_latex: :center t :booktabs t :float t :font \scriptsize
+#+attr_latex: :center t :booktabs t :float t
| *Specification* | APA150M | *APA300ML* | APA400MML | FPA-0500E-P | FPA-0300E-S |
|------------------------------------+---------+------------+-----------+-------------+-------------|
-| Stroke $> 100\, [\mu m]$ | 187 | 304 | 368 | 432 | 240 |
+| Stroke $> 200\, [\mu m]$ | 187 | 304 | 368 | 432 | 240 |
| Stiffness $\approx 1\, [N/\mu m]$ | 0.7 | 1.8 | 0.55 | 0.87 | 0.58 |
| Resolution $< 2\, [nm]$ | 2 | 3 | 4 | | |
| Blocked Force $> 100\, [N]$ | 127 | 546 | 201 | 376 | 139 |
@@ -7592,7 +7572,7 @@ While higher-order modes and non-axial flexibility are not captured, the model a
#+name: tab:detail_fem_apa300ml_2dof_parameters
#+caption: Summary of the obtained parameters for the 2 DoF APA300ML model
-#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc
+#+attr_latex: :environment tabularx :width 0.25\linewidth :align cc
#+attr_latex: :center t :booktabs t
| *Parameter* | *Value* |
|-------------+-----------------|
@@ -7612,13 +7592,13 @@ While higher-order modes and non-axial flexibility are not captured, the model a
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}from $V_a$ to $d_i$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa300ml_comp_fem_2dof_actuator.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}from $V_a$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa300ml_comp_fem_2dof_force_sensor.png]]
#+end_subfigure
#+end_figure
@@ -7635,6 +7615,7 @@ The developed models of the APA do not represent such behavior, but as this effe
#+name: fig:detail_fem_apa95ml_effect_electrical_boundaries
#+caption: Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_apa95ml_effect_electrical_boundaries.png]]
However, the electrical characteristics of the APA remain crucial for instrumentation design.
@@ -7664,13 +7645,13 @@ These results validate both the selection of the APA300ML and the effectiveness
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_actuator_fem_vs_perfect_hac_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_actuator_fem_vs_perfect_iff_plant.png]]
#+end_subfigure
#+end_figure
@@ -7745,13 +7726,13 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_bending_stiffness_hac_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_plant}$\bm{f}$ to $\bm{f}_m$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_bending_stiffness_iff_plant.png]]
#+end_subfigure
#+end_figure
@@ -7763,13 +7744,13 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1DoF actuators}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_bending_stiffness_iff_locus_1dof.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}APA300ML actuators}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_bending_stiffness_iff_locus_apa300ml.png]]
#+end_subfigure
#+end_figure
@@ -7822,13 +7803,13 @@ Based on this analysis, an axial stiffness specification of $100\,N/\mu m$ was e
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root Locus}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_axial_stiffness_iff_locus.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}RGA number}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_axial_stiffness_rga_hac_plant.png]]
#+end_subfigure
#+end_figure
@@ -7842,7 +7823,7 @@ Based on the dynamic analysis presented in previous sections, quantitative speci
#+name: tab:detail_fem_joints_specs
#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model
-#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcc
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc
#+attr_latex: :center t :booktabs t :float t
| | *Specification* | *FEM* |
|-------------------------+------------------------+-------|
@@ -7905,13 +7886,13 @@ While additional degrees of freedom could potentially capture more dynamic featu
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_fem_vs_perfect_hac_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_fem_joints_fem_vs_perfect_iff_plant.png]]
#+end_subfigure
#+end_figure
@@ -8308,21 +8289,21 @@ To facilitate the expression of these specifications, formula\nbsp{}eqref:eq:det
The parameters in this formula are $G_0 = \lim_{\omega \to 0} |W(j\omega)|$ (the low-frequency gain), $G_\infty = \lim_{\omega \to \infty} |W(j\omega)|$ (the high-frequency gain), $G_c = |W(j\omega_c)|$ (the gain at a specific frequency $\omega_c$ in $\si{rad/s}$), and $n$ (the slope between high and low frequency, which also corresponds to the order of the weighting function).
The typical magnitude response of a weighting function generated using\nbsp{}eqref:eq:detail_control_sensor_weight_formula is illustrated in Figure\nbsp{}ref:fig:detail_control_sensor_weight_formula.
-#+attr_latex: :options []{0.49\linewidth}
+#+attr_latex: :options []{0.45\linewidth}
#+begin_minipage
#+name: fig:detail_control_sensor_weight_formula
#+caption: Magnitude of a weighting function generated using\nbsp{}eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$.
-#+attr_latex: :width 0.95\linewidth :float nil
+#+attr_latex: :scale 0.8 :float nil
[[file:figs/detail_control_sensor_weight_formula.png]]
#+end_minipage
\hfill
-#+attr_latex: :options []{0.49\linewidth}
+#+attr_latex: :options []{0.54\linewidth}
#+begin_minipage
\begin{equation}\label{eq:detail_control_sensor_weight_formula}
W(s) = \left( \frac{
\hfill{} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
}{
- \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
+ \frac{1}{G_\infty^{\frac{1}{n}}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \frac{1}{G_c^{\frac{1}{n}}}
}\right)^n
\end{equation}
#+end_minipage
@@ -8343,7 +8324,7 @@ The inverse magnitudes of the designed weighting functions, which represent the
#+attr_latex: :options [b]{0.44\linewidth}
#+begin_minipage
#+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align ccc
-#+attr_latex: :booktabs t :float nil
+#+attr_latex: :booktabs t :float nil :font \footnotesize\sf
| Parameter | $W_1(s)$ | $W_2(s)$ |
|-------------+------------------+------------------|
| $G_0$ | $0.1$ | $1000$ |
@@ -8357,7 +8338,7 @@ The inverse magnitudes of the designed weighting functions, which represent the
#+attr_latex: :options [b]{0.52\linewidth}
#+begin_minipage
#+name: fig:detail_control_sensor_hinf_filters_results
-#+attr_latex: :scale 1 :float nil
+#+attr_latex: :scale 0.8 :float nil
#+caption: Weights and obtained filters
[[file:figs/detail_control_sensor_hinf_filters_results.png]]
#+end_minipage
@@ -8461,7 +8442,7 @@ Consider the generalized plant $P_3(s)$ shown in Figure\nbsp{}ref:fig:detail_con
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_three_complementary_filters_results}Weights and obtained filters}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_sensor_three_complementary_filters_results.png]]
#+end_subfigure
#+end_figure
@@ -8542,10 +8523,9 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O
\hfill
#+attr_latex: :options [b]{0.36\linewidth}
#+begin_minipage
-#+begin_scriptsize
#+latex: \centering
#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align cXc
-#+attr_latex: :booktabs t :float nil :center nil
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| | *Description* | *Value* |
|-------+-----------------------+-------------------|
| $l_a$ | | $0.5\,m$ |
@@ -8555,7 +8535,6 @@ Two reference frames are defined within this model: frame $\{M\}$ with origin $O
| $m$ | Payload mass | $40\,\text{kg}$ |
| $I$ | Payload $R_z$ inertia | $5\,\text{kg}m^2$ |
#+latex: \captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
-#+end_scriptsize
#+end_minipage
The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass\nbsp{}eqref:eq:detail_control_decoupling_model_eom, where $\bm{\mathcal{X}}_{\{M\}}$ represents the two translations and one rotation with respect to the center of mass, and $\bm{\mathcal{F}}_{\{M\}}$ denotes the forces and torque applied at the center of mass.
@@ -8635,6 +8614,7 @@ Depending on the symmetry present in the system, certain diagonal elements may e
#+name: fig:detail_control_decoupling_coupled_plant_bode
#+caption: Model dynamics from actuator forces to relative displacement sensor of each strut.
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_decoupling_coupled_plant_bode.png]]
**** Jacobian Decoupling
@@ -8714,7 +8694,7 @@ This phenomenon is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_decoupling_jacobian_plant_CoM.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM}
@@ -8769,7 +8749,7 @@ When a high-frequency force is applied at a point not aligned with the center of
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+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}
@@ -8853,7 +8833,7 @@ Each of these diagonal elements corresponds to a specific mode, as shown in Figu
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_decoupling_modal_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes}
@@ -8940,6 +8920,7 @@ Additionally, the diagonal terms manifest as second-order dynamic systems, facil
#+name: fig:detail_control_decoupling_svd_plant
#+caption: Plant dynamics $\bm{G}_{\text{SVD}}(s)$ obtained after decoupling using Singular Value Decomposition
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_decoupling_svd_plant.png]]
As it was surprising to obtain such a good decoupling at all frequencies, a variant system with identical dynamics but different sensor configurations was examined.
@@ -8960,7 +8941,7 @@ Notably, the coupling demonstrates local minima near the decoupling frequency, c
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_decoupling_svd_alt_plant.png]]
#+end_subfigure
#+end_figure
@@ -9271,13 +9252,13 @@ These filters can also be implemented in the digital domain with analytical form
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_analytical_effect_alpha}Effect of $\alpha$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_analytical_effect_alpha.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_analytical_effect_w0}Effect of $\omega_0$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_analytical_effect_w0.png]]
#+end_subfigure
#+end_figure
@@ -9334,7 +9315,7 @@ Figure\nbsp{}ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the n
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set}
#+attr_latex: :options {0.66\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_bode_plot_mech_sys.png]]
#+end_subfigure
#+end_figure
@@ -9364,13 +9345,13 @@ There magnitudes are displayed in Figure\nbsp{}ref:fig:detail_control_cf_specs_S
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_specs_S_T.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_bode_Kfb.png]]
#+end_subfigure
#+end_figure
@@ -9474,6 +9455,7 @@ The measured noise characteristics are then incorporated into the multi-body mod
#+name: fig:detail_instrumentation_plant
#+caption: Block diagram of the NASS with considered instrumentation. The RT controller is a Speedgoat machine.
+#+attr_latex: :width 0.9\linewidth
[[file:figs/detail_instrumentation_plant.png]]
*** Dynamic Error Budgeting
@@ -9501,6 +9483,7 @@ The transfer functions from these three noise sources (for one strut) to the ver
#+name: fig:detail_instrumentation_noise_sensitivities
#+caption: Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy.
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_noise_sensitivities.png]]
**** Estimation of maximum instrumentation noise
@@ -9591,26 +9574,27 @@ 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 $10\,\text{kHz}$
The PD200 from PiezoDrive was ultimately selected because it meets all the requirements and is accompanied by clear documentation, particularly regarding noise characteristics and bandwidth specifications.
#+name: tab:detail_instrumentation_amp_choice
#+caption: Specifications for the Voltage amplifier and considered commercial products
-#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
+#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :float t
-| *Specification* | *PD200* | WMA-200 | LA75B | E-505 |
-| | PiezoDrive | Falco | Cedrat | PI |
-|--------------------------------------+-----------------------+-------------------------------+--------------+-----------|
-| Input Voltage Range: $\pm 10\,V$ | $\pm 10\,V$ | $\pm8.75\,V$ | $-1/7.5\,V$ | |
-| Output Voltage Range: $-20/150\,V$ | $-50/150\,V$ | $\pm 175\,V$ | $-20/150\,V$ | -30/130 |
-| Gain $>15$ | 20 | 20 | 20 | 10 |
-| Output Current $> 300\,mA$ | $900\,mA$ | $150\,mA$ | $360\,mA$ | $215\,mA$ |
-| Slew Rate $> 34\,V/ms$ | $150\,V/\mu s$ | $80\,V/\mu s$ | n/a | n/a |
-| Output noise $< 20\,mV\ \text{RMS}$ | $0.7\,mV\,\text{RMS}$ | $0.05\,mV$ | $3.4\,mV$ | $0.6\,mV$ |
-| (10uF load) | ($10\,\mu F$ load) | ($10\,\mu F$ load) | (n/a) | (n/a) |
-| Small Signal Bandwidth $> 5\,kHz$ | $6.4\,kHz$ | $300\,Hz$ | $30\,kHz$ | n/a |
-| ($10\,\mu F$ load) | ($10\,\mu F$ load) | [fn:detail_instrumentation_1] | (unloaded) | (n/a) |
-| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a |
+| *Specifications* | PD200 | WMA-200 | LA75B | E-505 |
+| | PiezoDrive | Falco | Cedrat | PI |
+|--------------------------------------+--------------------+--------------------+--------------+------------|
+| Input Voltage Range: $\pm 10\,V$ | $\pm 10\,V$ | $\pm8.75\,V$ | $-1/7.5\,V$ | $-2/12\,V$ |
+| Output Voltage Range: $-20/150\,V$ | $-50/150\,V$ | $\pm 175\,V$ | $-20/150\,V$ | -30/130 |
+| Gain $>15$ | 20 | 20 | 20 | 10 |
+| Output Current $> 300\,mA$ | $900\,mA$ | $150\,mA$ | $360\,mA$ | $215\,mA$ |
+| Slew Rate $> 34\,V/ms$ | $150\,V/\mu s$ | $80\,V/\mu s$ | n/a | n/a |
+| Output noise $< 20\,mV\ \text{RMS}$ | $0.7\,mV$ | $0.05\,mV$ | $3.4\,mV$ | $0.6\,mV$ |
+| (10uF load) | ($10\,\mu F$ load) | ($10\,\mu F$ load) | (n/a) | (n/a) |
+| Small Signal Bandwidth $> 5\,kHz$ | $6.4\,kHz$ | $300\,Hz$ | $30\,kHz$ | n/a |
+| ($10\,\mu F$ load) | ($10\,\mu F$ load) | ($10\,\mu F$ load) | (unloaded) | (n/a) |
+| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a |
**** ADC and DAC
***** Introduction :ignore:
@@ -9756,9 +9740,9 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic
#+name: tab:detail_instrumentation_sensor_specs
#+caption: Specifications for the relative displacement sensors and considered commercial products
-#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.65\linewidth :align Xccc
#+attr_latex: :center t :booktabs t :float t
-| *Specification* | *Renishaw Vionic* | LION CPL190 | Cedrat ECP500 |
+| *Specifications* | Renishaw Vionic | LION CPL190 | Cedrat ECP500 |
|-----------------------------+---------------------+-------------+---------------|
| Technology | Digital Encoder | Capacitive | Eddy Current |
| Bandwidth $> 5\,\text{kHz}$ | $> 500\,\text{kHz}$ | 10kHz | 20kHz |
@@ -9784,6 +9768,7 @@ This approach is effective because the noise approximates white noise and its am
#+name: fig:detail_instrumentation_adc_noise_measured
#+caption: Measured ADC noise (IO318)
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_adc_noise_measured.png]]
***** Reading of piezoelectric force sensor
@@ -9822,7 +9807,7 @@ With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu F$,
#+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_step_response_force_sensor}Measured Signals}
#+attr_latex: :options {0.35\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_step_response_force_sensor.png]]
#+end_subfigure
#+end_figure
@@ -9852,7 +9837,7 @@ These results validate both the model of the ADC and the effectiveness of the ad
#+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_step_response_force_sensor_R}Measured Signals}
#+attr_latex: :options {0.35\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_step_response_force_sensor_R.png]]
#+end_subfigure
#+end_figure
@@ -9862,7 +9847,7 @@ These results validate both the model of the ADC and the effectiveness of the ad
Because the ADC noise may be too low to measure the noise of other instruments (anything below $5.6\,\mu V/\sqrt{\text{Hz}}$ cannot be distinguished from the noise of the ADC itself), a low noise instrumentation amplifier was employed.
A Femto DLPVA-101-B-S amplifier with adjustable gains from 20dB up to 80dB was selected for this purpose.
-The first step was to characterize the input[fn:detail_instrumentation_2] noise of the amplifier.
+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 ADC (Figure\nbsp{}ref:fig:detail_instrumentation_femto_meas_setup).
The maximum amplifier gain of 80dB (equivalent to 10000) was used for this measurement.
@@ -9882,7 +9867,7 @@ The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the
#+begin_minipage
#+name: fig:detail_instrumentation_femto_input_noise
#+caption: Obtained ASD of the instrumentation amplifier input voltage noise
-#+attr_latex: :scale 1 :float nil
+#+attr_latex: :scale 0.8 :float nil
[[file:figs/detail_instrumentation_femto_input_noise.png]]
#+end_minipage
@@ -9916,13 +9901,13 @@ The observed frequency response function corresponds to exactly one sample delay
#+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_dac_output_noise}Output noise of the DAC}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_dac_output_noise.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_dac_adc_tf}Transfer function from DAC to ADC}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_dac_adc_tf.png]]
#+end_subfigure
#+end_figure
@@ -9954,6 +9939,7 @@ While the exact cause of these peaks is not fully understood, their amplitudes r
#+name: fig:detail_instrumentation_pd200_noise
#+caption: Measured output voltage noise of the PD200 amplifiers
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_pd200_noise.png]]
***** Small Signal Bandwidth
@@ -9970,6 +9956,7 @@ The identified dynamics shown in Figure\nbsp{}ref:fig:detail_instrumentation_pd2
#+name: fig:detail_instrumentation_pd200_tf
#+caption: Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_pd200_tf.png]]
**** Linear Encoders
@@ -9994,8 +9981,8 @@ The noise profile exhibits characteristics of white noise with an amplitude of a
#+attr_latex: :options [b]{0.48\linewidth}
#+begin_minipage
#+name: fig:detail_instrumentation_vionic_asd
-#+caption: Measured Amplitude Spectral Density of the encoder noise
-#+attr_latex: :width 0.95\linewidth :float nil
+#+caption: Measured encoder noise ASD
+#+attr_latex: :scale 0.8 :float nil
[[file:figs/detail_instrumentation_vionic_asd.png]]
#+end_minipage
@@ -10010,6 +9997,7 @@ This confirms that the selected instrumentation, with its measured noise charact
#+name: fig:detail_instrumentation_cl_noise_budget
#+caption: Closed-loop noise budgeting using measured noise of instrumentation
+#+attr_latex: :scale 0.8
[[file:figs/detail_instrumentation_cl_noise_budget.png]]
*** Conclusion
@@ -10160,17 +10148,16 @@ The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatnes
#+attr_latex: :options [b]{0.48\textwidth}
#+begin_minipage
#+name: fig:test_apa_flatness_setup
-#+attr_latex: :width 0.7\linewidth :float nil
+#+attr_latex: :width 0.6\linewidth :float nil
#+caption: Measurement setup for flatness estimation
[[file:figs/test_apa_flatness_setup.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.48\textwidth}
#+begin_minipage
-#+name: tab:test_apa_flatness_meas
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xc
-#+attr_latex: :booktabs t :float nil
-#+caption: Estimated flatness of the APA300ML interfaces
+#+latex: \centering
+#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xc
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| | *Flatness* $[\mu m]$ |
|-------+----------------------|
| APA 1 | 8.9 |
@@ -10180,6 +10167,7 @@ The measured flatness values, summarized in Table\nbsp{}ref:tab:test_apa_flatnes
| APA 5 | 1.9 |
| APA 6 | 7.1 |
| APA 7 | 18.7 |
+#+latex: \captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
#+end_minipage
**** Electrical Measurements
@@ -10199,17 +10187,16 @@ This may be because the manufacturer measures the capacitance with large signals
#+attr_latex: :options [b]{0.48\textwidth}
#+begin_minipage
#+name: fig:test_apa_lcr_meter
-#+attr_latex: :width 0.95\linewidth :float nil
+#+attr_latex: :width 0.8\linewidth :float nil
#+caption: Used LCR meter
[[file:figs/test_apa_lcr_meter.jpg]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.48\textwidth}
#+begin_minipage
-#+name: tab:test_apa_capacitance
-#+caption: Measured capacitance in $\mu F$
-#+attr_latex: :environment tabularx :width 0.95\linewidth :align lcc
-#+attr_latex: :center t :booktabs t :float nil
+#+latex: \centering
+#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcc
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| | *Sensor Stack* | *Actuator Stacks* |
|-------+----------------+-------------------|
| APA 1 | 5.10 | 10.03 |
@@ -10219,6 +10206,7 @@ This may be because the manufacturer measures the capacitance with large signals
| APA 5 | 4.90 | 9.66 |
| APA 6 | 4.99 | 9.91 |
| APA 7 | 4.85 | 9.85 |
+#+latex: \captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu F$}
#+end_minipage
**** Stroke and Hysteresis Measurement
@@ -10231,7 +10219,7 @@ Note that the voltage is slowly varied as the displacement probe has a very low
#+name: fig:test_apa_stroke_bench
#+caption: Bench to measure the APA stroke
-#+attr_latex: :width 0.7\linewidth
+#+attr_latex: :width 0.6\linewidth
[[file:figs/test_apa_stroke_bench.jpg]]
The measured APA displacement is shown as a function of the applied voltage in Figure\nbsp{}ref:fig:test_apa_stroke_hysteresis.
@@ -10246,19 +10234,19 @@ This unit was sent 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
-#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML \subref{fig:test_apa_stroke_voltage}. Measured displacement as a function of applied voltage \subref{fig:test_apa_stroke_hysteresis}
+#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of applied voltage (\subref{fig:test_apa_stroke_hysteresis})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_stroke_voltage.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_stroke_hysteresis.png]]
#+end_subfigure
#+end_figure
@@ -10298,7 +10286,7 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
#+end_figure
#+name: fig:test_apa_meas_setup_modes
-#+caption: Experimental setup to measure the flexible modes of the APA300ML. For the bending in the $X$ direction \subref{fig:test_apa_meas_setup_X_bending}, the impact point is at the back of the top measurement point. For the bending in the $Y$ direction \subref{fig:test_apa_meas_setup_Y_bending}, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
+#+caption: Experimental setup to measure the flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
@@ -10324,6 +10312,7 @@ Another explanation is the shape difference between the manufactured APA300ML an
#+name: fig:test_apa_meas_freq_compare
#+caption: Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_meas_freq_compare.png]]
*** Dynamical measurements
@@ -10374,6 +10363,7 @@ This is the typical behavior expected from a PZT stack actuator, where the hyste
#+name: fig:test_apa_meas_hysteresis
#+caption: Displacement as a function of applied voltage for multiple excitation amplitudes
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_meas_hysteresis.png]]
**** Axial stiffness
@@ -10397,16 +10387,15 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu
#+begin_minipage
#+name: fig:test_apa_meas_stiffness_time
#+caption: Measured displacement when adding (at $t \approx 3\,s$) and removing (at $t \approx 13\,s$) the mass
-#+attr_latex: :width 0.9\linewidth :float nil
+#+attr_latex: :scale 0.8 :float nil
[[file:figs/test_apa_meas_stiffness_time.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.37\textwidth}
#+begin_minipage
-#+name: tab:test_apa_measured_stiffnesses
-#+caption: Measured axial stiffnesses (in $N/\mu m$)
+#+latex: \centering
#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xcc
-#+attr_latex: :center t :booktabs t :float nil
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| APA | $k_1$ | $k_2$ |
|-----+-------+-------|
| 1 | 1.68 | 1.9 |
@@ -10415,6 +10404,7 @@ These estimated stiffnesses are summarized in Table\nbsp{}ref:tab:test_apa_measu
| 5 | 1.7 | 1.93 |
| 6 | 1.7 | 1.92 |
| 8 | 1.73 | 1.98 |
+#+latex: \captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $N/\mu m$}
#+end_minipage
The stiffness can also be computed using equation\nbsp{}eqref:eq:test_apa_res_freq by knowing the main vertical resonance frequency $\omega_z \approx 95\,\text{Hz}$ (estimated by the dynamical measurements shown in section\nbsp{}ref:ssec:test_apa_meas_dynamics) and the suspended mass $m_{\text{sus}} = 5.7\,\text{kg}$.
@@ -10462,19 +10452,19 @@ From this analysis, it can be inferred that the axial stiffness of the shell is
All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure\nbsp{}ref:fig:test_apa_frf_encoder and at the force sensor in Figure\nbsp{}ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
#+name: fig:test_apa_frf_dynamics
-#+caption: Measured frequency response function from generated voltage $u$ to the encoder displacement $d_e$ \subref{fig:test_apa_frf_encoder} and to the force sensor voltage $V_s$ \subref{fig:test_apa_frf_force} for the six APA300ML
+#+caption: Measured frequency response function from generated voltage $u$ to the encoder displacement $d_e$ (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage $V_s$ (\subref{fig:test_apa_frf_force}) for the six APA300ML
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_frf_encoder}FRF from $u$ to $d_e$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_frf_encoder.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_frf_force}FRF from $u$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_frf_force.png]]
#+end_subfigure
#+end_figure
@@ -10493,19 +10483,19 @@ It could be induced to small non-linearity in the system, but the reason for thi
However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure\nbsp{}ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice.
#+name: fig:test_apa_non_minimum_phase
-#+caption: Measurement of the anti-resonance found in the transfer function from $u$ to $V_s$. The coherence \subref{fig:test_apa_non_minimum_phase_coherence} is quite good around the anti-resonance frequency. The phase \subref{fig:test_apa_non_minimum_phase_zoom} shoes a non-minimum phase behavior.
+#+caption: Measurement of the anti-resonance found in the transfer function from $u$ to $V_s$. The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_non_minimum_phase_coherence.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_non_minimum_phase_zoom.png]]
#+end_subfigure
#+end_figure
@@ -10522,6 +10512,7 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil
#+name: fig:test_apa_effect_resistance
#+caption: Transfer function from $u$ to $V_s$ with and without the resistor $R$ in parallel with the piezoelectric stack used as the force sensor
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_effect_resistance.png]]
**** Integral Force Feedback
@@ -10538,6 +10529,7 @@ A comparison between the identified plant and the manually tuned transfer functi
#+name: fig:test_apa_iff_plant_comp_manual_fit
#+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_iff_plant_comp_manual_fit.png]]
The implemented Integral Force Feedback Controller transfer function is shown in equation\nbsp{}eqref:eq:test_apa_Kiff_formula.
@@ -10564,19 +10556,19 @@ Second using the fitted transfer functions of the damped plants experimentally i
The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were fitted using only a second-order transfer function.
#+name: fig:test_apa_iff
-#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant \subref{fig:test_apa_identified_damped_plants} and Root Locus \subref{fig:test_apa_iff_root_locus} corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}
+#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants that match the experimental data (dashed lines)}
#+attr_latex: :options {0.59\textwidth}
#+begin_subfigure
-#+attr_latex: :height 8cm
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_identified_damped_plants.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_iff_root_locus}Root Locus plot using the plant model (black) and poles of the identified damped plants (color crosses)}
#+attr_latex: :options {0.39\textwidth}
#+begin_subfigure
-#+attr_latex: :height 8cm
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_iff_root_locus.png]]
#+end_subfigure
#+end_figure
@@ -10592,7 +10584,7 @@ After the model is presented, the procedure for tuning the model is described, a
#+name: fig:test_apa_bench_model
#+caption: Screenshot of the multi-body model
-#+attr_latex: :width 0.8\linewidth
+#+attr_latex: :width 0.7\linewidth
[[file:figs/test_apa_bench_model.png]]
***** Two degrees-of-freedom APA Model
@@ -10652,7 +10644,7 @@ The obtained parameters of the model shown in Figure\nbsp{}ref:fig:test_apa_2dof
#+name: tab:test_apa_2dof_parameters
#+caption: Summary of the obtained parameters for the 2 DoF APA300ML model
-#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc
+#+attr_latex: :environment tabularx :width 0.25\linewidth :align cc
#+attr_latex: :center t :booktabs t
| *Parameter* | *Value* |
|-------------+------------------|
@@ -10674,19 +10666,19 @@ A good match can be observed between the model and the experimental data, both f
This indicates that this model represents well the axial dynamics of the APA300ML.
#+name: fig:test_apa_2dof_comp_frf
-#+caption: Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from $u$ to $d_e$ \subref{fig:test_apa_2dof_comp_frf_enc} \subref{fig:test_apa_2dof_comp_frf_force} and from $u$ to $V_s$ \subref{fig:test_apa_2dof_comp_frf_force}
+#+caption: Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_2dof_comp_frf_enc}from $u$ to $d_e$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_2dof_comp_frf_enc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_2dof_comp_frf_force}from $u$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_2dof_comp_frf_force.png]]
#+end_subfigure
#+end_figure
@@ -10735,7 +10727,7 @@ From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, w
#+name: tab:test_apa_piezo_properties
#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities
-#+attr_latex: :environment tabularx :width 1\linewidth :align ccX
+#+attr_latex: :environment tabularx :width 0.8\linewidth :align ccX
#+attr_latex: :center t :booktabs t
| *Parameter* | *Value* | *Description* |
|----------------+----------------------------+--------------------------------------------------------------|
@@ -10756,19 +10748,19 @@ It is however surprising that the model is "softer" than the measured system, as
Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
#+name: fig:test_apa_super_element_comp_frf
-#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from $u$ to $d_e$ \subref{fig:test_apa_super_element_comp_frf_enc} and from $u$ to $V_s$ \subref{fig:test_apa_super_element_comp_frf_force}
+#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_super_element_comp_frf_enc}) and from $u$ to $V_s$ (\subref{fig:test_apa_super_element_comp_frf_force})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_super_element_comp_frf_enc}from $u$ to $d_e$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_super_element_comp_frf_enc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_super_element_comp_frf_force}from $u$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_apa_super_element_comp_frf_force.png]]
#+end_subfigure
#+end_figure
@@ -10780,7 +10772,6 @@ Using this simple test bench, it can be concluded that the /super element/ model
<>
In this study, the amplified piezoelectric actuators "APA300ML" have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase.
-
Geometrical features such as the flatness of its interfaces, electrical capacitance, and achievable strokes were measured in Section\nbsp{}ref:sec:test_apa_basic_meas.
These simple measurements allowed for the early detection of a manufacturing defect in one of the APA300ML.
@@ -10811,7 +10802,7 @@ During the detailed design phase, specifications in terms of stiffness and strok
#+name: tab:test_joints_specs
#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model
-#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcc
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcc
#+attr_latex: :center t :booktabs t :float t
| | *Specification* | *FEM* |
|-------------------+------------------------+-------|
@@ -10894,7 +10885,7 @@ What is typically observed is shown in Figure\nbsp{}ref:fig:test_joints_profilom
It is then possible to estimate the dimension of the flexible beam with an accuracy of $\approx 5\,\mu m$,
#+name: fig:test_joints_profilometer
-#+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer \subref{fig:test_joints_profilometer_setup} and a image is obtained with which the gap can be estimated \subref{fig:test_joints_profilometer_image}
+#+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_profilometer_setup}Flexible joint fixed on the profilometer}
@@ -10923,6 +10914,7 @@ However, what is more important than the true value of the thickness is the cons
#+name: fig:test_joints_size_hist
#+caption: Histogram for the (16x2) measured beams' thicknesses
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_size_hist.png]]
**** Bad flexible joints
@@ -11105,7 +11097,7 @@ An overall accuracy of $\approx 5\,\%$ can be expected with this measurement ben
#+name: tab:test_joints_error_budget
#+caption: Summary of the error budget for estimating the bending stiffness
-#+attr_latex: :environment tabularx :width 0.4\linewidth :align lX
+#+attr_latex: :environment tabularx :width 0.35\linewidth :align Xc
#+attr_latex: :center t :booktabs t
| *Effect* | *Error* |
|----------------------+-------------------------|
@@ -11133,7 +11125,7 @@ The flexible joint can be rotated by $90^o$ in order to measure the bending stif
The obtained CAD design of the measurement bench is shown in Figure\nbsp{}ref:fig:test_joints_bench_overview while a zoom on the flexible joint with the associated important quantities is shown in Figure\nbsp{}ref:fig:test_joints_bench_side.
#+name: fig:test_joints_bench
-#+caption: CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in \subref{fig:test_joints_bench_overview} while a zoom on the flexible joint is shown in \subref{fig:test_joints_bench_side}
+#+caption: CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_bench_overview} Schematic of the test bench to measure the bending stiffness of the flexible joints}
@@ -11185,19 +11177,19 @@ The gain mismatch between the two load cells is approximately $4\,\%$ which is h
However, the estimated non-linearity is bellow $0.2\,\%$ for forces between $1\,N$ and $5\,N$.
#+name: fig:test_joints_force_sensor_calib
-#+caption: Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in \subref{fig:test_joints_force_sensor_calib_picture}. Comparison of the two measured forces and estimated non-linearity are shown in \subref{fig:test_joints_force_sensor_calib_fit}
-#+attr_latex: :options [htbp]
+#+caption: Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in (\subref{fig:test_joints_force_sensor_calib_picture}). Comparison of the two measured forces and estimated non-linearity are shown in (\subref{fig:test_joints_force_sensor_calib_fit})
+#+attr_latex: :options [h!tbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_force_sensor_calib_picture}Zoom on the two load cells in contact}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.5cm
+#+attr_latex: :height 5cm
[[file:figs/test_joints_force_sensor_calib_picture.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_force_sensor_calib_fit}Measured two forces}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.5cm
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_force_sensor_calib_fit.png]]
#+end_subfigure
#+end_figure
@@ -11210,19 +11202,19 @@ The measured displacement as a function of the measured force is shown in Figure
The load cell stiffness can then be estimated by computing a linear fit and is found to be $k_F \approx 0.68\,N/\mu m$.
#+name: fig:test_joints_meas_force_sensor_stiffness
-#+caption: Estimation of the load cell stiffness. The measurement setup is shown in \subref{fig:test_joints_meas_force_sensor_stiffness_picture}. The measurement results are shown in \subref{fig:test_joints_force_sensor_stiffness_fit}.
+#+caption: Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}), and results are shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the measurement bench}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.5cm
+#+attr_latex: :height 5cm
[[file:figs/test_joints_meas_force_sensor_stiffness_picture.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_force_sensor_stiffness_fit}Measured displacement as a function of the force}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.5cm
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_force_sensor_stiffness_fit.png]]
#+end_subfigure
#+end_figure
@@ -11238,19 +11230,19 @@ The bending stiffness of the flexible joint can be estimated by computing the sl
The bending stroke can also be estimated as shown in Figure\nbsp{}ref:fig:test_joints_meas_F_d_lin_fit and is found to be $\theta_{y,\text{max}} = 20.9\,\text{mrad}$.
#+name: fig:test_joints_meas_example
-#+caption: Results obtained on the first flexible joint. The measured force and displacement are shown in \subref{fig:test_joints_meas_bend_time}. The estimated angular displacement $\theta_x$ as a function of the estimated applied torque $T_{x}$ is shown in \subref{fig:test_joints_meas_F_d_lin_fit}. The bending stiffness $k_{R_x}$ of the flexible joint can be estimated by computing a best linear fit (red dashed line).
+#+caption: Results obtained on the first flexible joint. The measured force and displacement are shown in (\subref{fig:test_joints_meas_bend_time}). The estimated angular displacement $\theta_x$ as a function of the estimated applied torque $T_{x}$ is shown in (\subref{fig:test_joints_meas_F_d_lin_fit}). The bending stiffness $k_{R_x}$ of the flexible joint can be estimated by computing a best linear fit (red dashed line).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_bend_time}Force and displacement measured as a function of time}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.3cm
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_meas_bend_time.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_F_d_lin_fit}Angular displacement measured as a function of the applied torque}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.3cm
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_meas_F_d_lin_fit.png]]
#+end_subfigure
#+end_figure
@@ -11265,19 +11257,19 @@ A histogram of the measured bending stiffnesses is shown in Figure\nbsp{}ref:fig
Most of the bending stiffnesses are between $4.6\,Nm/rad$ and $5.0\,Nm/rad$.
#+name: fig:test_joints_meas_bending_results
-#+caption: Result of measured $k_{R_x}$ and $k_{R_y}$ stiffnesses for the 16 flexible joints. Raw data are shown in \subref{fig:test_joints_meas_bending_all_raw_data}. A histogram of the measured stiffnesses is shown in \subref{fig:test_joints_bend_stiff_hist}
+#+caption: Result of measured $k_{R_x}$ and $k_{R_y}$ stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_meas_bending_all_raw_data}Measured torque and angular motion for the flexible joints}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.3cm
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_meas_bending_all_raw_data.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness in the x and y directions}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :height 5.3cm
+#+attr_latex: :scale 0.8
[[file:figs/test_joints_bend_stiff_hist.png]]
#+end_subfigure
#+end_figure
@@ -11521,13 +11513,13 @@ The obtained frequency response functions for the three configurations (X-bendin
#+attr_latex: :caption \subcaption{\label{fig:test_struts_spur_res_frf_no_enc}without encoder}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_spur_res_frf_no_enc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_spur_res_frf_enc}with the encoder}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_spur_res_frf_enc.png]]
#+end_subfigure
#+end_figure
@@ -11585,7 +11577,7 @@ Finally, all measured struts are compared in terms of dynamics in Section\nbsp{}
System identification was performed without the encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_front) and with one encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_coder).
#+name: fig:test_struts_bench_leg_with_without_enc
-#+caption: Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts \subref{fig:test_struts_bench_leg_coder} or removed \subref{fig:test_struts_bench_leg_front}
+#+caption: Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder}
@@ -11609,25 +11601,25 @@ Similarly, it has little effect on the transfer function from $u$ to the sensor
This means that the encoder should have little effect on the effectiveness of the integral force feedback control strategy.
#+name: fig:test_struts_effect_encoder
-#+caption: Effect of having the encoder fixed to the struts on the measured dynamics from $u$ to $d_a$ \subref{fig:test_struts_effect_encoder_int} and from $u$ to $V_s$ \subref{fig:test_struts_effect_encoder_iff}. Comparison of the observed dynamics by the encoder and interferometers \subref{fig:test_struts_comp_enc_int}
+#+caption: Effect of having the encoder fixed to the struts on the measured dynamics from $u$ to $d_a$ (\subref{fig:test_struts_effect_encoder_int}) and from $u$ to $V_s$ (\subref{fig:test_struts_effect_encoder_iff}). Comparison of the observed dynamics by the encoder and interferometers (\subref{fig:test_struts_comp_enc_int})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_encoder_int}$u$ to $d_a$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_effect_encoder_int.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_encoder_iff}$u$ to $V_s$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_effect_encoder_iff.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_enc_int}$u$ to $d_e$, $d_a$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_enc_int.png]]
#+end_subfigure
#+end_figure
@@ -11657,19 +11649,19 @@ A very good match can be observed between the struts.
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_interf_plants.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_iff_plants.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_enc_plants.png]]
#+end_subfigure
#+end_figure
@@ -11718,19 +11710,19 @@ For the flexible model, it will be shown in the next section that by adding some
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_frf_flexible_model_int.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_frf_flexible_model_enc.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$}
#+attr_latex: :options {0.32\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_frf_flexible_model_iff.png]]
#+end_subfigure
#+end_figure
@@ -11765,19 +11757,19 @@ A comparison of the experimental frequency response functions in Figure\nbsp{}re
This similarity suggests that the identified differences in dynamics are caused by misalignment.
#+name: fig:test_struts_effect_misalignment
-#+caption: Effect of a misalignment between the flexible joints and the APA300ML in the $y$ direction \subref{fig:test_struts_effect_misalignment_y} and in the $x$ direction \subref{fig:test_struts_effect_misalignment_x}
+#+caption: Effect of a misalignment between the flexible joints and the APA300ML in the $y$ direction (\subref{fig:test_struts_effect_misalignment_y}) and in the $x$ direction (\subref{fig:test_struts_effect_misalignment_x})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_misalignment_y}Misalignment along $y$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_effect_misalignment_y.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_struts_effect_misalignment_x}Misalignment along $x$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_effect_misalignment_x.png]]
#+end_subfigure
#+end_figure
@@ -11800,7 +11792,7 @@ Thickness differences for all the struts were found to be between $0.94\,mm$ and
#+name: tab:test_struts_meas_y_misalignment
#+caption: Measured $y$ misalignment at the top and bottom of the APA. Measurements are in $mm$
-#+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc
+#+attr_latex: :environment tabularx :width 0.2\linewidth :align Xcc
#+attr_latex: :center t :booktabs t
| *Strut* | *Bot* | *Top* |
|---------+-------+-------|
@@ -11821,6 +11813,7 @@ With a better alignment, the amplitude of the spurious resonances is expected to
#+name: fig:test_struts_comp_dy_tuned_model_frf_enc
#+caption: Comparison of the frequency response functions from DAC voltage $u$ to measured displacement $d_e$ by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the $y$ misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the $x$ and $y$ misalignments are tuned
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_dy_tuned_model_frf_enc.png]]
**** Proper struts alignment
@@ -11855,6 +11848,7 @@ Therefore, fixing the encoders to the nano-hexapod plates instead may be an inte
#+name: fig:test_struts_comp_enc_frf_realign
#+caption: Comparison of the dynamics from $u$ to $d_e$ before and after proper alignment using the dowel pins
+#+attr_latex: :scale 0.8
[[file:figs/test_struts_comp_enc_frf_realign.png]]
*** Conclusion
@@ -11891,7 +11885,7 @@ The goal was to fix the six struts to the two nano-hexapod plates (shown in Figu
To do so, a precisely machined mounting tool (Figure\nbsp{}ref:fig:test_nhexa_center_part_hexapod_mounting) is used to position the two nano-hexapod plates during the assembly procedure.
#+name: fig:test_nhexa_received_parts
-#+caption: 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}
+#+caption: 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})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_nano_hexapod_plates}Top and bottom plates}
@@ -11916,7 +11910,7 @@ The two plates were then fixed to the mounting tool, as shown in Figure\nbsp{}re
The main goal of this "mounting tool" is to position the flexible joint interfaces (the "V" shapes) of both plates so that a cylinder can rest on the 4 flat interfaces at the same time.
#+name: fig:test_nhexa_dimensional_check
-#+caption: A FARO arm is used to dimensionally check the received parts \subref{fig:test_nhexa_plates_tolerances} and after mounting the two plates with the mounting part \subref{fig:test_nhexa_mounting_tool_hexapod_top_view}
+#+caption: A FARO arm is used to dimensionally check the received parts (\subref{fig:test_nhexa_plates_tolerances}) and after mounting the two plates with the mounting part (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_plates_tolerances}Dimensional check of the bottom plate}
@@ -11942,7 +11936,7 @@ The straightness was found to be better than $15\,\mu m$ for all struts[fn:test_
#+name: tab:measured_straightness
#+caption: Measured straightness between the two "V" shapes for the six struts. These measurements were performed twice for each strut.
-#+attr_latex: :environment tabularx :width 0.3\linewidth :align Xcc
+#+attr_latex: :environment tabularx :width 0.25\linewidth :align Xcc
#+attr_latex: :center t :booktabs t
| *Strut* | *Meas 1* | *Meas 2* |
|---------+--------------+--------------|
@@ -11956,7 +11950,7 @@ The straightness was found to be better than $15\,\mu m$ for all struts[fn:test_
The encoder rulers and heads were then fixed to the top and bottom plates, respectively (Figure\nbsp{}ref:fig:test_nhexa_mount_encoder), and the encoder heads were aligned to maximize the received contrast.
#+name: fig:test_nhexa_mount_encoder
-#+caption: Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate \subref{fig:test_nhexa_mount_encoder_rulers} while encoders heads are fixed to the bottom plate \subref{fig:test_nhexa_mount_encoder_heads}
+#+caption: Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate (\subref{fig:test_nhexa_mount_encoder_rulers}) while encoders heads are fixed to the bottom plate (\subref{fig:test_nhexa_mount_encoder_heads})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_mount_encoder_rulers}Encoder rulers}
@@ -12035,11 +12029,9 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{
\hfill
#+attr_latex: :options [b]{0.45\textwidth}
#+begin_minipage
-#+begin_scriptsize
-#+name: tab:test_nhexa_suspended_table_modes
-#+caption: Obtained modes of the suspended table
-#+attr_latex: :environment tabularx :width 0.8\linewidth :placement [b] :align clX
-#+attr_latex: :booktabs t :float nil :center t
+#+latex: \centering
+#+attr_latex: :environment tabularx :width 0.9\linewidth :placement [b] :align clX
+#+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf
| *Modes* | *Frequency* | *Description* |
|---------+-------------+------------------|
| 1,2 | 1.3 Hz | X-Y translations |
@@ -12050,7 +12042,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{
| 7 | 701 Hz | "Membrane" Mode |
| 8 | 989 Hz | Complex mode |
| 9 | 1025 Hz | Complex mode |
-#+end_scriptsize
+#+latex: \captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table}
#+end_minipage
#+name: fig:test_nhexa_table_flexible_modes
@@ -12120,7 +12112,7 @@ The effect of the payload mass on the dynamics is discussed in Section\nbsp{}ref
#+name: fig:test_nhexa_nano_hexapod_signals
#+caption: Block diagram of the studied system. The command signal generated by the speedgoat is $\mathbf{u}$, and the measured dignals are $\mathbf{d}_{e}$ and $\mathbf{V}_s$. Units are indicated in square brackets.
-#+attr_latex: :width \linewidth
+#+attr_latex: :width 0.9\linewidth
[[file:figs/test_nhexa_nano_hexapod_signals.png]]
**** Modal analysis
@@ -12194,7 +12186,7 @@ This would not have occurred if the encoders were fixed to the struts.
#+name: fig:test_nhexa_identified_frf_de
#+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{d}_e$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_identified_frf_de.png]]
Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs.
@@ -12204,7 +12196,7 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor
#+name: fig:test_nhexa_identified_frf_Vs
#+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.
-#+attr_latex: :width \linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_identified_frf_Vs.png]]
**** Effect of payload mass on the dynamics
@@ -12237,19 +12229,19 @@ The measured FRFs from $u_i$ to $V_{si}$ are shown in Figure\nbsp{}ref:fig:test_
For all tested payloads, the measured FRF always have alternating poles and zeros, indicating that IFF can be applied in a robust manner.
#+name: fig:test_nhexa_identified_frf_masses
-#+caption: Measured Frequency Response Functions from $u_i$ to $d_{ei}$ \subref{fig:test_nhexa_identified_frf_de_masses} and from $u_i$ to $V_{si}$ \subref{fig:test_nhexa_identified_frf_Vs_masses} for all 4 payload conditions. Only diagonal terms are shown.
+#+caption: Measured Frequency Response Functions from $u_i$ to $d_{ei}$ (\subref{fig:test_nhexa_identified_frf_de_masses}) and from $u_i$ to $V_{si}$ (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_identified_frf_de_masses.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_identified_frf_Vs_masses.png]]
#+end_subfigure
#+end_figure
@@ -12288,19 +12280,19 @@ The three resonances that were attributed to "internal" flexible modes of the st
At higher frequencies, no resonances can be observed in the model, as the top plate and the encoder supports are modeled as rigid bodies.
#+name: fig:test_nhexa_comp_simscape_diag
-#+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ \subref{fig:test_nhexa_comp_simscape_de_diag} and from $u$ to $V_s$ \subref{fig:test_nhexa_comp_simscape_Vs_diag}
+#+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_de_diag.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_Vs_diag.png]]
#+end_subfigure
#+end_figure
@@ -12315,6 +12307,7 @@ Similar results are observed for all other coupling terms and for the transfer f
#+name: fig:test_nhexa_comp_simscape_de_all
#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a 2-DoF mass-spring-damper system.
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_de_all.png]]
The APA300ML was then modeled with a /super-element/ extracted from a FE-software.
@@ -12325,6 +12318,7 @@ Therefore, if the modes of the struts are to be modeled, the /super-element/ of
#+name: fig:test_nhexa_comp_simscape_de_all_flex
#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a "super-element".
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_de_all_flex.png]]
**** Effect of payload mass
@@ -12340,19 +12334,19 @@ One option could be to tune the damping as a function of the mass (similar to wh
However, as decentralized IFF will be applied, the damping is actively brought, and the open-loop damping value should have very little impact on the obtained plant dynamics.
#+name: fig:test_nhexa_comp_simscape_diag_masses
-#+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ \subref{fig:test_nhexa_comp_simscape_de_diag} and from $u$ to $V_s$ \subref{fig:test_nhexa_comp_simscape_Vs_diag}
+#+caption: Comparison of the diagonal elements (i.e. "direct" terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from $u$ to $V_s$ (\subref{fig:test_nhexa_comp_simscape_Vs_diag})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_de_diag_masses.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_Vs_diag_masses.png]]
#+end_subfigure
#+end_figure
@@ -12363,6 +12357,7 @@ Therefore, the model effectively represents the system coupling for different pa
#+name: fig:test_nhexa_comp_simscape_de_all_high_mass
#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$
+#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_comp_simscape_de_all_high_mass.png]]
*** Conclusion
@@ -12580,13 +12575,13 @@ The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ i
#+attr_latex: :caption \subcaption{\label{fig:test_id31_metrology_align_rx_ry}Angular alignment}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_metrology_align_rx_ry.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_metrology_align_dx_dy}Lateral alignment}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_metrology_align_dx_dy.png]]
#+end_subfigure
#+end_figure
@@ -12602,7 +12597,7 @@ The obtained lateral acceptance for pure displacements in any direction is estim
#+name: tab:test_id31_metrology_acceptance
#+caption: Estimated measurement range for each interferometer, and for three different directions.
-#+attr_latex: :environment tabularx :width 0.45\linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| | $D_x$ | $D_y$ | $D_z$ |
|-----------+-------------+------------+-------|
@@ -12642,13 +12637,13 @@ The effect of noise on the translation and rotation measurements is estimated in
#+attr_latex: :caption \subcaption{\label{fig:test_id31_xy_map_sphere}Z measurement during an XY mapping}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_xy_map_sphere.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_interf_noise}Interferometer noise}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_interf_noise.png]]
#+end_subfigure
#+end_figure
@@ -12673,6 +12668,7 @@ Voltages generated by the force sensor piezoelectric stacks $\bm{V}_s = [V_{s1},
#+name: fig:test_id31_block_schematic_plant
#+caption: Schematic of the NASS plant
+#+attr_latex: :scale 0.9
[[file:figs/test_id31_block_schematic_plant.png]]
**** Open-Loop Plant Identification
@@ -12696,13 +12692,13 @@ This issue was later solved.
#+attr_latex: :caption \subcaption{\label{fig:test_id31_first_id_int}External Metrology}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_first_id_int.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_first_id_iff}Force Sensors}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_first_id_iff.png]]
#+end_subfigure
#+end_figure
@@ -12726,13 +12722,13 @@ Results shown in Figure\nbsp{}ref:fig:test_id31_Rz_align_correct are indeed indi
#+attr_latex: :caption \subcaption{\label{fig:test_id31_Rz_align_error}Before alignment}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_Rz_align_error.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_Rz_align_correct}After alignment}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_Rz_align_correct.png]]
#+end_subfigure
#+end_figure
@@ -12745,6 +12741,7 @@ The flexible modes of the top platform can be passively damped, whereas the mode
#+name: fig:test_id31_first_id_int_better_rz_align
#+caption: Decrease of the coupling with better Rz alignment
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_first_id_int_better_rz_align.png]]
**** Effect of Payload Mass
@@ -12764,25 +12761,25 @@ It is interesting to note that the anti-resonances in the force sensor plant now
#+attr_latex: :caption \subcaption{\label{fig:test_id31_picture_mass_m0}$m=0\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :width 0.95\linewidth
[[file:figs/test_id31_picture_mass_m0.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_picture_mass_m1}$m=13\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :width 0.95\linewidth
[[file:figs/test_id31_picture_mass_m1.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_picture_mass_m2}$m=26\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :width 0.95\linewidth
[[file:figs/test_id31_picture_mass_m2.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_picture_mass_m3}$m=39\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.99\linewidth
+#+attr_latex: :width 0.95\linewidth
[[file:figs/test_id31_picture_mass_m3.jpg]]
#+end_subfigure
#+end_figure
@@ -12823,13 +12820,13 @@ This also indicates that the metrology kinematics is correct and is working in r
#+attr_latex: :caption \subcaption{\label{fig:test_id31_effect_rotation_direct}Direct terms}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_effect_rotation_direct.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_effect_rotation_coupling}Coupling terms}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_effect_rotation_coupling.png]]
#+end_subfigure
#+end_figure
@@ -12877,6 +12874,7 @@ This confirms that the multi-body model can be used to tune the IFF controller.
#+name: fig:test_id31_comp_simscape_Vs
#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six force sensor voltages $V_{s1}$ to $V_{s6}$
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_comp_simscape_Vs.png]]
**** IFF Controller
@@ -12900,13 +12898,13 @@ It can be seen that the loop-gain is larger than $1$ around the suspension modes
#+attr_latex: :caption \subcaption{\label{fig:test_id31_Kiff_bode_plot}Bode plot of $K_{\text{IFF}}$}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_Kiff_bode_plot.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_Kiff_loop_gain}Decentralized Loop gains}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_Kiff_loop_gain.png]]
#+end_subfigure
#+end_figure
@@ -12925,25 +12923,25 @@ However, in this study, it was chosen to implement a "fixed" (i.e. non-adaptive)
#+attr_latex: :caption \subcaption{\label{fig:test_id31_iff_root_locus_m0}$m = 0\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_iff_root_locus_m0.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_iff_root_locus_m1}$m = 13\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_iff_root_locus_m1.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_iff_root_locus_m2}$m = 26\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_iff_root_locus_m2.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_iff_root_locus_m3}$m = 39\,\text{kg}$}
#+attr_latex: :options {0.24\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_iff_root_locus_m3.png]]
#+end_subfigure
#+end_figure
@@ -12965,13 +12963,13 @@ The obtained frequency response functions are compared with the model in Figure\
#+attr_latex: :caption \subcaption{\label{fig:test_id31_comp_ol_iff_plant_model}Effect of IFF on the plant}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_comp_ol_iff_plant_model.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_plant_effect_mass}Comparison of model and experimental results}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_plant_effect_mass.png]]
#+end_subfigure
#+end_figure
@@ -13016,6 +13014,7 @@ Considering the complexity of the system's dynamics, the model can be considered
#+name: fig:test_id31_comp_simscape_hac
#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal ($u_1^\prime$) of the damped plant to the estimated errors ($\epsilon_{\mathcal{L}_i}$) in the frame of the six struts by the external metrology
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_comp_simscape_hac.png]]
The challenge here is to tune a high authority controller such that it is robust to the change in dynamics due to different payloads being used.
@@ -13025,6 +13024,7 @@ This is one of the key benefits of using the HAC-LAC strategy.
#+name: fig:test_id31_comp_all_undamped_damped_plants
#+caption: Comparison of the (six) direct terms for all (four) payload conditions in the undamped case (in blue) and the damped case (i.e. with the decentralized IFF being implemented, in red).
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_comp_all_undamped_damped_plants.png]]
**** Interaction Analysis
@@ -13052,6 +13052,7 @@ This design choice, while beneficial for system simplicity, introduces inherent
#+name: fig:test_id31_hac_rga_number
#+caption: RGA-number for the damped plants - Comparison of all the payload conditions
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_rga_number.png]]
**** Robust Controller Design
@@ -13077,13 +13078,13 @@ However, small stability margins were observed for the highest mass, indicating
#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_loop_gain}Loop Gains}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_characteristic_loci}Characteristic Loci}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_characteristic_loci.png]]
#+end_subfigure
#+end_figure
@@ -13103,13 +13104,13 @@ The obtained closed-loop positioning accuracy was found to comply with the requi
#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy}XY plane}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}YZ plane}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz.png]]
#+end_subfigure
#+end_figure
@@ -13126,6 +13127,7 @@ However, it was decided that this controller should be tested experimentally and
#+name: fig:test_id31_hac_tomography_Wz36_simulation
#+caption: Positioning errors in the Y-Z plane during tomography experiments simulated using the multi-body model (in closed-loop)
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_tomography_Wz36_simulation.png]]
**** Conclusion
@@ -13172,7 +13174,7 @@ Results obtained for all experiments are summarized and compared to the specific
#+name: tab:test_id31_experiments_specifications
#+caption: Specifications for the Nano-Active-Stabilization-System
-#+attr_latex: :environment tabularx :width 0.45\linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| | $D_y$ | $D_z$ | $R_y$ |
|-------------+-------+-------+----------------------|
@@ -13199,13 +13201,13 @@ While this approach likely underestimates actual open-loop errors, as perfect al
#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in $(x,y)$ plane}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}Removed eccentricity}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed.png]]
#+end_subfigure
#+end_figure
@@ -13216,6 +13218,7 @@ These experimental findings are consistent with the predictions from the tomogra
#+name: fig:test_id31_tomo_Wz36_results
#+caption: Measured errors in the $Y-Z$ plane during tomography experiments at $6\,\text{deg/s}$ for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data.
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_Wz36_results.png]]
***** Fast Tomography scans
@@ -13232,13 +13235,13 @@ Nevertheless, even with this robust (i.e. conservative) HAC implementation, the
#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_xy}XY plane}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_m0_30rpm_robust_hac_iff_exp_xy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz}YZ plane}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 0.9
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz.png]]
#+end_subfigure
#+end_figure
@@ -13262,19 +13265,19 @@ This experiment also illustrates that when needed, performance can be enhanced b
#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_dy} $D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_cas_cl_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_dz} $D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_cas_cl_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_hac_cas_cl_ry} $R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_hac_cas_cl_ry.png]]
#+end_subfigure
#+end_figure
@@ -13293,19 +13296,19 @@ The results confirmed that the NASS successfully maintained the point of interes
#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dy}$D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_reflectivity_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_dz}$D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_reflectivity_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_reflectivity_ry}$R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_reflectivity_ry.png]]
#+end_subfigure
#+end_figure
@@ -13335,19 +13338,19 @@ The settling duration typically decreases for smaller step sizes.
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_10nm_steps}10nm steps}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_mim_10nm_steps.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_100nm_steps}100nm steps}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_mim_100nm_steps.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu$m step}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_mim_1000nm_steps.png]]
#+end_subfigure
#+end_figure
@@ -13366,19 +13369,19 @@ Initial testing at $10\,\mu m/s$ demonstrated positioning errors well within spe
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_scan_10ums_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_dz}$D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_scan_10ums_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_scan_10ums_ry.png]]
#+end_subfigure
#+end_figure
@@ -13394,19 +13397,19 @@ However, performance during acceleration phases could be enhanced through the im
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_scan_100ums_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_dz}$D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_scan_100ums_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dz_scan_100ums_ry.png]]
#+end_subfigure
#+end_figure
@@ -13438,19 +13441,19 @@ Under closed-loop control, positioning errors remain within specifications in al
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dy} $D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dy_10ums_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_dz} $D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dy_10ums_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_10ums_ry} $R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dy_10ums_ry.png]]
#+end_subfigure
#+end_figure
@@ -13474,19 +13477,19 @@ For applications requiring small $D_y$ scans, the nano-hexapod can be used exclu
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dy} $D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dy_100ums_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_dz} $D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dy_100ums_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_dy_100ums_ry} $R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_dy_100ums_ry.png]]
#+end_subfigure
#+end_figure
@@ -13501,6 +13504,7 @@ The system performance was evaluated at three lateral scanning velocities: $0.1\
#+name: fig:test_id31_diffraction_tomo_setpoint
#+caption: Dy motion for several configured velocities
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_diffraction_tomo_setpoint.png]]
The positioning errors measured along $D_y$, $D_z$, and $R_y$ directions are displayed in Figure\nbsp{}ref:fig:test_id31_diffraction_tomo.
@@ -13516,19 +13520,19 @@ Alternatively, a feedforward controller could improve the lateral positioning ac
#+attr_latex: :caption \subcaption{\label{fig:test_id31_diffraction_tomo_dy} $D_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_diffraction_tomo_dy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_diffraction_tomo_dz} $D_z$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_diffraction_tomo_dz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_diffraction_tomo_ry} $R_y$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
-#+attr_latex: :scale 1
+#+attr_latex: :scale 0.8
[[file:figs/test_id31_diffraction_tomo_ry.png]]
#+end_subfigure
#+end_figure
@@ -13558,7 +13562,7 @@ The identified limitations, primarily related to high-speed lateral scanning and
#+name: tab:test_id31_experiments_results_summary
#+caption: Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated by bold number.
-#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc
+#+attr_latex: :environment tabularx :width 0.85\linewidth :align Xccc
#+attr_latex: :center t :booktabs t
| *Experiments* | $\bm{D_y}$ *[nmRMS]* | $\bm{D_z}$ *[nmRMS]* | $\bm{R_y}$ *[nradRMS]* |
|---------------------------------------------------------+-----------------------------+---------------------------+-----------------------------|
@@ -13706,8 +13710,7 @@ With the implementation of an accurate online metrology system, the NASS will be
[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_2] 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:detail_instrumentation_1] The manufacturer proposed to remove the $50\,\Omega$ output resistor to improve to small signal bandwidth above $10\,kHz$
+[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 $20\,V/V$
[fn:test_apa_12]The DAC used is the one included in the IO131 card sold by Speedgoat. It has an output range of $\pm 10\,V$ and 16-bits resolution
diff --git a/phd-thesis.pdf b/phd-thesis.pdf
index 8a2d1d7..db62f95 100644
Binary files a/phd-thesis.pdf and b/phd-thesis.pdf differ
diff --git a/phd-thesis.tex b/phd-thesis.tex
index c006a54..95c696c 100644
--- a/phd-thesis.tex
+++ b/phd-thesis.tex
@@ -1,4 +1,4 @@
-% Created 2025-04-18 Fri 11:23
+% Created 2025-04-18 Fri 16:53
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -759,7 +759,6 @@ The confidence gained through this systematic investigation provides a solid fou
\label{sec:uniaxial}
In this report, a uniaxial model of the \acrfull{nass} is developed and used to obtain a first idea of the challenges involved in this complex system.
Note that in this study, only the vertical direction is considered (which is the most stiff), but other directions were considered as well, yielding to similar conclusions.
-The model is schematically shown in Figure~\ref{fig:uniaxial_overview_model_sections} where the colors represent the parts studied in different sections.
To have a relevant model, the micro-station dynamics is first identified and its model is tuned to match the measurements (Section~\ref{sec:uniaxial_micro_station_model}).
Then, a model of the nano-hexapod is added on top of the micro-station.
@@ -776,28 +775,13 @@ Three active damping techniques are studied (Section~\ref{sec:uniaxial_active_da
Once the system is well damped, a feedback position controller is applied and the obtained performance is analyzed (Section~\ref{sec:uniaxial_position_control}).
Two key effects that may limit that positioning performances are then considered: the limited micro-station compliance (Section~\ref{sec:uniaxial_support_compliance}) and the presence of dynamics between the nano-hexapod and the sample's point of interest (Section~\ref{sec:uniaxial_payload_dynamics}).
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1]{figs/uniaxial_overview_model_sections.png}
-\caption{\label{fig:uniaxial_overview_model_sections}Uniaxial Micro-Station model in blue (Section~\ref{sec:uniaxial_micro_station_model}), Nano-Hexapod models in red (Section~\ref{sec:uniaxial_nano_station_model}), Disturbances in yellow (Section~\ref{sec:uniaxial_disturbances}), Active Damping in green (Section~\ref{sec:uniaxial_active_damping}), Position control in purple (Section~\ref{sec:uniaxial_position_control}) and Sample dynamics in cyan (Section~\ref{sec:uniaxial_payload_dynamics})}
-\end{figure}
\subsection{Micro Station Model}
\label{sec:uniaxial_micro_station_model}
In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station.
-The measurement setup is shown in Figure~\ref{fig:uniaxial_ustation_first_meas_dynamics} where several geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{m/s}\) and a natural frequency of \(\approx 1\,\text{Hz}\)} are fixed to the micro-station and an instrumented hammer is used to inject forces on different stages of the micro-station.
-
-From the measured frequency response functions (FRF), the model can be tuned to approximate the uniaxial dynamics of the micro-station.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_ustation_first_meas_dynamics.jpg}
-\caption{\label{fig:uniaxial_ustation_first_meas_dynamics}Experimental setup used for the first dynamical measurements on the Micro-Station. Geophones are fixed to different stages of the micro-station.}
-\end{figure}
\subsubsection{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 micro-hexapod's top platform (force \(F_{h}\)).
-The vertical inertial motion of the granite \(x_{g}\) and the top platform of the micro-hexapod \(x_{h}\) are measured using geophones.
+The vertical inertial motion of the granite \(x_{g}\) and the top platform of the micro-hexapod \(x_{h}\) are measured using geophones\footnote{Mark Product L4-C geophones are used with a sensitivity of \(171\,\frac{V}{m/s}\) and a natural frequency of \(\approx 1\,\text{Hz}\)}.
Three frequency response functions 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 frequency response functions will only be shown between 20 and 200Hz (solid lines in Figure~\ref{fig:uniaxial_comp_frf_meas_model}).
@@ -831,7 +815,7 @@ The parameters obtained are summarized in Table~\ref{tab:uniaxial_ustation_param
\begin{table}[htbp]
\caption{\label{tab:uniaxial_ustation_parameters}Physical parameters used for the micro-station uniaxial model}
\centering
-\begin{tabularx}{0.9\linewidth}{lXXX}
+\begin{tabularx}{0.6\linewidth}{Xccc}
\toprule
\textbf{Stage} & \textbf{Mass} & \textbf{Stiffness} & \textbf{Damping}\\
\midrule
@@ -854,7 +838,7 @@ More accurate models will be used later on.
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/uniaxial_comp_frf_meas_model.png}
+\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 identified ones from the uniaxial model}
\end{figure}
\subsection{Nano-Hexapod Model}
@@ -874,7 +858,7 @@ The effect of resonances between the sample's point of interest and the nano-hex
\end{subfigure}
\begin{subfigure}{0.59\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_plant_first_params.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_plant_first_params.png}
\end{center}
\subcaption{\label{fig:uniaxial_plant_first_params}Bode Plot of the transfer function from actuator forces $f$ to measured displacement $d$ by the metrology}
\end{subfigure}
@@ -895,19 +879,19 @@ For further analysis, 9 ``configurations'' of the uniaxial NASS model of Figure~
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_first_params_fs.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_first_params_fs.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_fs}Direct forces}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_first_params_ft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_first_params_ft.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_ft}$\mu\text{-station}$ disturbances}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_first_params_xf.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_first_params_xf.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_first_params_xf}Floor motion}
\end{subfigure}
@@ -930,7 +914,7 @@ The geophone located on the floor was used to measure the floor motion \(x_f\) w
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_ustation_dynamical_id_setup.jpg}
\end{center}
-\subcaption{\label{fig:uniaxial_ustation_dynamical_id_setup}Two geophones are used to measure vibrations induced by $T_y$ and $R_z$ scans}
+\subcaption{\label{fig:uniaxial_ustation_dynamical_id_setup}Geophones used to measure vibrations induced by $T_y$ and $R_z$ scans}
\end{subfigure}
\caption{\label{fig:uniaxial_ustation_meas_disturbances_setup}Identification of the disturbances coming from the micro-station. The measurement schematic is shown in (\subref{fig:uniaxial_ustation_meas_disturbances}). A picture of the setup is shown in (\subref{fig:uniaxial_ustation_dynamical_id_setup})}
\end{figure}
@@ -964,13 +948,13 @@ The estimated amplitude spectral density \(\Gamma_{x_f}\) of the floor motion \(
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_asd_floor_motion_id31.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_asd_floor_motion_id31.png}
\end{center}
\subcaption{\label{fig:uniaxial_asd_floor_motion_id31}Estimated ASD of $x_f$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_asd_disturbance_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_asd_disturbance_force.png}
\end{center}
\subcaption{\label{fig:uniaxial_asd_disturbance_force}Estimated ASD of $f_t$}
\end{subfigure}
@@ -987,7 +971,7 @@ The sharp peak observed at \(24\,\text{Hz}\) is believed to be induced by electr
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/uniaxial_asd_vibration_spindle_rotation.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_asd_vibration_spindle_rotation.png}
\caption{\label{fig:uniaxial_asd_vibration_spindle_rotation}Amplitude Spectral Density \(\Gamma_{R_z}\) of the relative motion measured between the granite and the micro-hexapod's top platform during Spindle rotating}
\end{figure}
@@ -1026,19 +1010,19 @@ The obtained sensitivity to disturbances for the three nano-hexapod stiffnesses
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_fs}Direct forces}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_ft}$\mu\text{-station}$ disturbances}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_disturbances_nano_hexapod_stiffnesses_xf}Floor motion}
\end{subfigure}
@@ -1056,13 +1040,13 @@ The conclusion is that the sample mass has little effect on the cumulative ampli
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_cas_d_disturbances_stiffnesses.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_d_disturbances_stiffnesses.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_d_disturbances_stiffnesses}Effect of floor motion $x_f$ and stage disturbances $f_t$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_cas_d_disturbances_payload_masses.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_d_disturbances_payload_masses.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_d_disturbances_payload_masses}Effect of nano-hexapod stiffness $k_n$ and payload mass $m_s$}
\end{subfigure}
@@ -1199,19 +1183,19 @@ Therefore, it is expected that the micro-station dynamics might impact the achie
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/uniaxial_plant_active_damping_techniques_iff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_plant_active_damping_techniques_iff.png}
\end{center}
\subcaption{\label{fig:uniaxial_plant_active_damping_techniques_iff}IFF}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/uniaxial_plant_active_damping_techniques_rdc.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_plant_active_damping_techniques_rdc.png}
\end{center}
\subcaption{\label{fig:uniaxial_plant_active_damping_techniques_rdc}RDC}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/uniaxial_plant_active_damping_techniques_dvf.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_plant_active_damping_techniques_dvf.png}
\end{center}
\subcaption{\label{fig:uniaxial_plant_active_damping_techniques_dvf}DVF}
\end{subfigure}
@@ -1243,19 +1227,19 @@ The micro-station and the nano-hexapod masses are connected through a large damp
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_root_locus_damping_techniques_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_soft}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_root_locus_damping_techniques_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_mid}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_root_locus_damping_techniques_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_root_locus_damping_techniques_stiff}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1264,7 +1248,7 @@ The micro-station and the nano-hexapod masses are connected through a large damp
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_root_locus_damping_techniques_micro_station_mode.png}
\caption{\label{fig:uniaxial_root_locus_damping_techniques_micro_station_mode}Root Locus for the three damping techniques applied with the soft nano-hexapod. It is shown that the RDC active damping technique has some authority on one mode of the micro-station. This mode corresponds to the suspension mode of the micro-hexapod.}
\end{figure}
@@ -1274,19 +1258,19 @@ All three active damping techniques yielded similar damped plants.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_vc.png}
\end{center}
\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_damped_plant_three_active_damping_techniques_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_md.png}
\end{center}
\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_damped_plant_three_active_damping_techniques_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_damped_plant_three_active_damping_techniques_pz.png}
\end{center}
\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1311,19 +1295,19 @@ This is because the equivalent damper in parallel with the actuator (see Figure~
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_active_damping_fs.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_active_damping_fs.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_fs}Direct forces}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_active_damping_ft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_active_damping_ft.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_ft}$\mu\text{-station}$ disturbances}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_active_damping_xf.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_active_damping_xf.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_active_damping_xf}Floor motion}
\end{subfigure}
@@ -1337,19 +1321,19 @@ All three active damping methods give similar results.
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_cas_active_damping_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_active_damping_soft}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_cas_active_damping_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_active_damping_mid}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_cas_active_damping_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_active_damping_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_active_damping_stiff}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1373,7 +1357,6 @@ Which one will be used will be determined by the use of more accurate models and
\begin{table}[htbp]
\caption{\label{tab:comp_active_damping}Comparison of active damping strategies}
\centering
-\scriptsize
\begin{tabularx}{0.9\linewidth}{Xccc}
\toprule
& \textbf{IFF} & \textbf{RDC} & \textbf{DVF}\\
@@ -1408,7 +1391,7 @@ This control architecture applied to the uniaxial model is shown in Figure~\ref{
\begin{figure}[htbp]
\begin{subfigure}{0.54\textwidth}
\begin{center}
-\includegraphics[scale=1,width=1.0\linewidth]{figs/uniaxial_hac_lac_architecture.png}
+\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_hac_lac_architecture.png}
\end{center}
\subcaption{\label{fig:uniaxial_hac_lac_architecture}Typical HAC-LAC Architecture}
\end{subfigure}
@@ -1431,19 +1414,19 @@ This effect will be further explained in Section~\ref{sec:uniaxial_support_compl
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_hac_iff_damped_plants_masses_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_soft}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_hac_iff_damped_plants_masses_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_mid}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_hac_iff_damped_plants_masses_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_hac_iff_damped_plants_masses_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_hac_iff_damped_plants_masses_stiff}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1496,7 +1479,7 @@ K_{\text{stiff}}(s) &= g \cdot
\begin{table}[htbp]
\caption{\label{tab:uniaxial_feedback_controller_parameters}Parameters used for the position feedback controllers}
\centering
-\begin{tabularx}{\linewidth}{lXXX}
+\begin{tabularx}{0.75\linewidth}{Xccc}
\toprule
& \textbf{Soft} & \textbf{Moderately stiff} & \textbf{Stiff}\\
\midrule
@@ -1520,19 +1503,19 @@ The goal is to have a first estimation of the attainable performance.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_nyquist_hac_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_vc.png}
\end{center}
\subcaption{\label{fig:uniaxial_nyquist_hac_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_nyquist_hac_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_md.png}
\end{center}
\subcaption{\label{fig:uniaxial_nyquist_hac_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_nyquist_hac_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_nyquist_hac_pz.png}
\end{center}
\subcaption{\label{fig:uniaxial_nyquist_hac_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1542,19 +1525,19 @@ The goal is to have a first estimation of the attainable performance.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_loop_gain_hac_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_vc.png}
\end{center}
\subcaption{\label{fig:uniaxial_loop_gain_hac_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_loop_gain_hac_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_md.png}
\end{center}
\subcaption{\label{fig:uniaxial_loop_gain_hac_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_loop_gain_hac_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_loop_gain_hac_pz.png}
\end{center}
\subcaption{\label{fig:uniaxial_loop_gain_hac_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1570,19 +1553,19 @@ As expected, the sensitivity to disturbances decreased in the controller bandwid
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_hac_lac_fs.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_hac_lac_fs.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_fs}Direct forces}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_hac_lac_ft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_hac_lac_ft.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_ft}$\mu\text{-station}$ disturbances}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_sensitivity_dist_hac_lac_xf.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_sensitivity_dist_hac_lac_xf.png}
\end{center}
\subcaption{\label{fig:uniaxial_sensitivity_dist_hac_lac_xf}Floor motion}
\end{subfigure}
@@ -1596,19 +1579,19 @@ Obtained root mean square values of the distance \(d\) are better for the soft n
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_cas_hac_lac_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_hac_lac_soft}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_cas_hac_lac_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_hac_lac_mid}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_cas_hac_lac_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_cas_hac_lac_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_cas_hac_lac_stiff}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -1659,19 +1642,19 @@ When neglecting the support compliance, a large feedback bandwidth can be achiev
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_neglected_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_neglected_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_soft}$\omega_{n} \ll \omega_{\mu}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_neglected_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_neglected_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_mid}$\omega_{n} = \omega_{\mu}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_neglected_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_neglected_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_neglected_stiff}$\omega_{n} \gg \omega_{\mu}$}
\end{subfigure}
@@ -1691,19 +1674,19 @@ If a soft nano-hexapod is used, the support dynamics appears in the dynamics bet
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_dynamics_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_dynamics_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_soft}$\omega_{n} \ll \omega_{\mu}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_dynamics_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_dynamics_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_mid}$\omega_{n} = \omega_{\mu}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_dynamics_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_dynamics_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_stiff}$\omega_{n} \gg \omega_{\mu}$}
\end{subfigure}
@@ -1718,19 +1701,19 @@ Conversely, if a ``stiff'' nano-hexapod is used, the support dynamics appears in
\begin{figure}[htbp]
\begin{subfigure}{0.37\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_dynamics_d_soft.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_dynamics_d_soft.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_soft}$\omega_{n} \ll \omega_{\mu}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_dynamics_d_mid.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_dynamics_d_mid.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_mid}$\omega_{n} = \omega_{\mu}$}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/uniaxial_effect_support_compliance_dynamics_d_stiff.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_effect_support_compliance_dynamics_d_stiff.png}
\end{center}
\subcaption{\label{fig:uniaxial_effect_support_compliance_dynamics_d_stiff}$\omega_{n} \gg \omega_{\mu}$}
\end{subfigure}
@@ -1793,13 +1776,13 @@ The flexibility of the sample also changes the high frequency gain (the mass lin
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_light.png}
\end{center}
\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_light}$k_n = 0.01\,N/\mu m$, $m_s = 1\,kg$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_heavy.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_soft_nano_hexapod_heavy.png}
\end{center}
\subcaption{\label{fig:uniaxial_payload_dynamics_soft_nano_hexapod_heavy}$k_n = 0.01\,N/\mu m$, $m_s = 50\,kg$}
\end{subfigure}
@@ -1814,13 +1797,13 @@ Even though the added sample's flexibility still shifts the high frequency mass
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_light.png}
\end{center}
\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_light}$k_n = 100\,N/\mu m$, $m_s = 1\,kg$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_heavy.png}
+\includegraphics[scale=1,scale=0.8]{figs/uniaxial_payload_dynamics_stiff_nano_hexapod_heavy.png}
\end{center}
\subcaption{\label{fig:uniaxial_payload_dynamics_stiff_nano_hexapod_heavy}$k_n = 100\,N/\mu m$, $m_s = 50\,kg$}
\end{subfigure}
@@ -1855,13 +1838,13 @@ What happens is that above \(\omega_s\), even though the motion \(d\) can be con
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_sample_flexibility_noise_budget_d.png}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/uniaxial_sample_flexibility_noise_budget_d.png}
\end{center}
\subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_d}Cumulative Amplitude Spectrum of $d$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/uniaxial_sample_flexibility_noise_budget_y.png}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/uniaxial_sample_flexibility_noise_budget_y.png}
\end{center}
\subcaption{\label{fig:uniaxial_sample_flexibility_noise_budget_y}Cumulative Amplitude Spectrum of $y$}
\end{subfigure}
@@ -1916,12 +1899,6 @@ The previous analysis was applied to three considered nano-hexapod stiffnesses (
Up until this section, the study was performed on a very simplistic model that only captures the rotation aspect, and the model parameters were not tuned to correspond to the NASS.
In the last section (Section~\ref{sec:rotating_nass}), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics.
The goal is to determine whether the rotation imposes performance limitation on the NASS.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=\linewidth]{figs/rotating_overview.png}
-\caption{\label{fig:rotating_overview}Overview of this chapter's organization. Sections are indicated by the red circles.}
-\end{figure}
\subsection{System Description and Analysis}
\label{sec:rotating_system_description}
@@ -2021,13 +1998,13 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/rotating_campbell_diagram_real.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_campbell_diagram_real.png}
\end{center}
\subcaption{\label{fig:rotating_campbell_diagram_real}Real part}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/rotating_campbell_diagram_imag.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_campbell_diagram_imag.png}
\end{center}
\subcaption{\label{fig:rotating_campbell_diagram_imag}Imaginary part}
\end{subfigure}
@@ -2043,13 +2020,13 @@ For \(\Omega > \omega_0\), the low-frequency pair of complex conjugate poles \(p
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_bode_plot_direct.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_bode_plot_direct.png}
\end{center}
\subcaption{\label{fig:rotating_bode_plot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_bode_plot_coupling.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_bode_plot_coupling.png}
\end{center}
\subcaption{\label{fig:rotating_bode_plot_coupling}Coupling terms: $d_u/F_v$, $d_v/F_u$}
\end{subfigure}
@@ -2157,13 +2134,13 @@ A pair of (minimum phase) complex conjugate zeros appears between the two comple
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_bode_plot_effect_rot_direct.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_bode_plot_effect_rot_direct.png}
\end{center}
\subcaption{\label{fig:rotating_iff_bode_plot_effect_rot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/rotating_root_locus_iff_pure_int.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_root_locus_iff_pure_int.png}
\end{center}
\subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus}
\end{subfigure}
@@ -2242,13 +2219,13 @@ For larger values of \(\omega_i\), the attainable damping ratio decreases as a f
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_root_locus_iff_modified_effect_wi.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_root_locus_iff_modified_effect_wi.png}
\end{center}
\subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_optimal_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_optimal_gain.png}
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown}
\end{subfigure}
@@ -2263,13 +2240,13 @@ The same trade-off can be seen between achievable damping and loss of compliance
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_damped_plant_effect_wi_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_damped_plant_effect_wi_plant.png}
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}Obtained plants}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_effect_wi_compliance.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_effect_wi_compliance.png}
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of $\omega_i$ on the compliance}
\end{subfigure}
@@ -2336,13 +2313,13 @@ It is shown that if the added stiffness is higher than the maximum negative stif
\begin{figure}[htbp]
\begin{subfigure}{0.55\linewidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_effect_kp.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_effect_kp.png}
\end{center}
\subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$}
\end{subfigure}
\begin{subfigure}{0.44\linewidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_kp_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_root_locus.png}
\end{center}
\subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring}
\end{subfigure}
@@ -2359,13 +2336,13 @@ This is confirmed by the Figure~\ref{fig:rotating_iff_kp_optimal_gain} where the
\begin{figure}[htbp]
\begin{subfigure}{0.49\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/rotating_iff_kp_root_locus_effect_kp.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_root_locus_effect_kp.png}
\end{center}
\subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/rotating_iff_kp_optimal_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_optimal_gain.png}
\end{center}
\subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. The corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown because the system is unstable.}
\end{subfigure}
@@ -2394,13 +2371,13 @@ The added high-pass filter gives almost the same damping properties to the suspe
\begin{figure}[htbp]
\begin{subfigure}{0.34\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=0.95]{figs/rotating_iff_kp_added_hpf_effect_damping.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_added_hpf_effect_damping.png}
\end{center}
\subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$}
\end{subfigure}
\begin{subfigure}{0.65\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=0.95]{figs/rotating_iff_kp_added_hpf_damped_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_added_hpf_damped_plant.png}
\end{center}
\subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF}
\end{subfigure}
@@ -2457,7 +2434,7 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
\begin{figure}[htbp]
\begin{subfigure}{0.49\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/rotating_rdc_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_rdc_root_locus.png}
\end{center}
\subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control}
\end{subfigure}
@@ -2487,13 +2464,13 @@ It is interesting to note that the maximum added damping is very similar for bot
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_root_locus.png}
\end{center}
\subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_dampled_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_dampled_plants.png}
\end{center}
\subcaption{\label{fig:rotating_comp_techniques_dampled_plants}Damped plants}
\end{subfigure}
@@ -2518,13 +2495,13 @@ This is very well known characteristics of these common active damping technique
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_transmissibility.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_transmissibility.png}
\end{center}
\subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_compliance.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_comp_techniques_compliance.png}
\end{center}
\subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance}
\end{subfigure}
@@ -2547,19 +2524,19 @@ The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually es
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_vc.png}
\end{center}
\subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_md.png}
\end{center}
\subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nano_hexapod_dynamics_pz.png}
\end{center}
\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -2580,19 +2557,19 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_nass_optimal_gain_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_vc.png}
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_nass_optimal_gain_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_md.png}
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_nass_optimal_gain_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_hpf_nass_optimal_gain_pz.png}
\end{center}
\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -2602,7 +2579,7 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\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 of the two modes \(\xi\) is also shown.}
\centering
-\begin{tabularx}{0.4\linewidth}{Xccc}
+\begin{tabularx}{0.3\linewidth}{Xccc}
\toprule
\(k_n\) & \(\omega_i\) & \(g\) & \(\xi_\text{opt}\)\\
\midrule
@@ -2626,17 +2603,17 @@ This distance is larger for stiff nano-hexapod because the open-loop pole will b
Let's choose \(k_p = 1\,N/mm\), \(k_p = 0.01\,N/\mu m\) and \(k_p = 1\,N/\mu m\) for the three considered nano-hexapods.
The corresponding optimal controller gains and achievable damping are summarized in Table~\ref{tab:rotating_iff_kp_opt_iff_kp_params_nass}.
-\begin{minipage}[t]{0.49\linewidth}
+\begin{minipage}[b]{0.49\linewidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/rotating_iff_kp_nass_optimal_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_kp_nass_optimal_gain.png}
\captionof{figure}{\label{fig:rotating_iff_kp_nass_optimal_gain}Maximum damping \(\xi\) as a function of the parallel stiffness \(k_p\)}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
-\begin{center}
-\captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses}
-\begin{tabularx}{\linewidth}{Xccc}
+\centering
+{\footnotesize\sf
+\begin{tabularx}{0.9\linewidth}{cccc}
\toprule
\(k_n\) & \(k_p\) & \(g\) & \(\xi_{\text{opt}}\)\\
\midrule
@@ -2644,24 +2621,24 @@ The corresponding optimal controller gains and achievable damping are summarized
\(1\,N/\mu m\) & \(0.01\,N/\mu m\) & 465.57 & 0.97\\
\(100\,N/\mu m\) & \(1\,N/\mu m\) & 4624.25 & 0.99\\
\bottomrule
-\end{tabularx}
-\end{center}
+\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}
\subsubsection{Optimal Relative Motion Control}
For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure~\ref{fig:rotating_rdc_optimal_gain}).
The gain is chosen such that 99\% of modal damping is obtained (obtained gains are summarized in Table~\ref{tab:rotating_rdc_opt_params_nass}).
-\begin{minipage}[t]{0.49\linewidth}
+\begin{minipage}[b]{0.49\linewidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/rotating_rdc_optimal_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_rdc_optimal_gain.png}
\captionof{figure}{\label{fig:rotating_rdc_optimal_gain}Maximum damping \(\xi\) as a function of the RDC gain \(g\)}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\linewidth}
-\begin{center}
-\captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC}
-\begin{tabularx}{0.8\linewidth}{Xcc}
+\centering
+{\footnotesize\sf
+\begin{tabularx}{0.6\linewidth}{ccc}
\toprule
\(k_n\) & \(g\) & \(\xi_{\text{opt}}\)\\
\midrule
@@ -2669,8 +2646,8 @@ The gain is chosen such that 99\% of modal damping is obtained (obtained gains a
\(1\,N/\mu m\) & 8200 & 0.99\\
\(100\,N/\mu m\) & 80000 & 0.99\\
\bottomrule
-\end{tabularx}
-\end{center}
+\end{tabularx}}
+\captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC}
\end{minipage}
\subsubsection{Comparison of the obtained damped plants}
Now that the optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure~\ref{fig:rotating_nass_damped_plant_comp}.
@@ -2685,19 +2662,19 @@ Similar to what was concluded in the previous analysis:
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_damped_plant_comp_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_vc.png}
\end{center}
\subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_damped_plant_comp_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_md.png}
\end{center}
\subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_damped_plant_comp_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_damped_plant_comp_pz.png}
\end{center}
\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -2740,19 +2717,19 @@ It can be observed that:
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_comp_stiffness_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_vc.png}
\end{center}
\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_comp_stiffness_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_md.png}
\end{center}
\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_comp_stiffness_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_plant_comp_stiffness_pz.png}
\end{center}
\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -2779,19 +2756,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_floor_motion_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_vc.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_floor_motion_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_md.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_floor_motion_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_floor_motion_pz.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -2801,19 +2778,19 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_stage_vibration_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_vc.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_stage_vibration_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_md.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_stage_vibration_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_stage_vibration_pz.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
@@ -2823,46 +2800,46 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_direct_forces_vc.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_vc.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_direct_forces_md.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_md.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_direct_forces_pz.png}
+\includegraphics[scale=1,scale=0.8]{figs/rotating_nass_effect_direct_forces_pz.png}
\end{center}
\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$}
\end{subfigure}
\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low-frequency.}
\end{figure}
\subsection*{Conclusion}
-In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section~\ref{sec:rotating_system_description}).
-Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms (Section~\ref{sec:rotating_iff_pure_int}).
+In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model.
+Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms.
Two modifications of the classical \acrshort{iff} control have been proposed to overcome this issue.
The first modification concerns the controller and consists of adding a high-pass filter to the pure integrators.
This is equivalent to moving the controller pole to the left along the real axis.
-This allows the closed-loop system to be stable up to some value of the controller gain (Section~\ref{sec:rotating_iff_pseudo_int}).
+This allows the closed-loop system to be stable up to some value of the controller gain.
The second proposed modification concerns the mechanical system.
Additional springs are added in parallel with the actuators and force sensors.
-It was shown that if the stiffness \(k_p\) of the additional springs is larger than the negative stiffness \(m \Omega^2\) induced by centrifugal forces, the classical decentralized \acrshort{iff} regains its unconditional stability property (Section~\ref{sec:rotating_iff_parallel_stiffness}).
+It was shown that if the stiffness \(k_p\) of the additional springs is larger than the negative stiffness \(m \Omega^2\) induced by centrifugal forces, the classical decentralized \acrshort{iff} regains its unconditional stability property.
-These two modifications were compared with \acrlong{rdc} in Section~\ref{sec:rotating_comp_act_damp}.
+These two modifications were compared with \acrlong{rdc}.
While having very different implementations, both proposed modifications were found to be very similar with respect to the attainable damping and the obtained closed-loop system behavior.
-This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters (Section~\ref{sec:rotating_nano_hexapod}).
+This study has been applied to a rotating platform that corresponds to the nano-hexapod parameters.
As for the uniaxial model, three nano-hexapod stiffnesses values were considered.
The dynamics of the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)) was shown to be more depend more on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects).
In addition, the attainable damping ratio of the soft nano-hexapod 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 (Section~\ref{sec:rotating_nass}).
+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 nano-hexapod due to the spindle's rotation.
For the moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)), the gyroscopic effects only slightly affect the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that showed better results with the uniaxial model.
\section{Micro Station - Modal Analysis}
@@ -2972,9 +2949,9 @@ However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured \acrs
\end{minipage}
\hfill
\begin{minipage}[b]{0.36\linewidth}
-\begin{scriptsize}
\centering
-\begin{tabularx}{\linewidth}{Xccc}
+{\scriptsize\sf
+\begin{tabularx}{0.9\linewidth}{Xccc}
\toprule
& \(x\) & \(y\) & \(z\)\\
\midrule
@@ -3002,9 +2979,8 @@ 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{scriptsize}
\end{minipage}
\begin{figure}[htbp]
@@ -3066,13 +3042,13 @@ Similar results were obtained for all measured frequency response functions.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_raw_meas.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_raw_meas.png}
\end{center}
\subcaption{\label{fig:modal_raw_meas}Time domain signals}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_asd_acc_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_asd_acc_force.png}
\end{center}
\subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
\end{subfigure}
@@ -3086,13 +3062,13 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_frf_acc_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_frf_acc_force.png}
\end{center}
\subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/modal_coh_acc_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_coh_acc_force.png}
\end{center}
\subcaption{\label{fig:modal_coh_acc_force} Coherence}
\end{subfigure}
@@ -3186,7 +3162,7 @@ The position of each accelerometer with respect to the center of mass of the cor
\begin{table}[htbp]
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
\centering
-\begin{tabularx}{0.55\linewidth}{Xccc}
+\begin{tabularx}{0.45\linewidth}{Xccc}
\toprule
& \(X\) & \(Y\) & \(Z\)\\
\midrule
@@ -3229,7 +3205,7 @@ This also validates the reduction in the number of degrees of freedom from 69 (2
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/modal_comp_acc_solid_body_frf.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_comp_acc_solid_body_frf.png}
\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparison of the original accelerometer responses and the reconstructed responses from the solid body response. Accelerometers 1 to 4 corresponding to the micro-hexapod are shown. Input is a hammer force applied on the micro-hexapod in the \(x\) direction}
\end{figure}
\subsection{Modal Analysis}
@@ -3264,17 +3240,17 @@ The obtained \acrshort{mif} is shown on Figure~\ref{fig:modal_indication_functio
A total of 16 modes were found between 0 and \(200\,\text{Hz}\).
The obtained natural frequencies and associated modal damping are summarized in Table~\ref{tab:modal_obtained_modes_freqs_damps}.
-\begin{minipage}[b]{0.70\linewidth}
+\begin{minipage}[b]{0.65\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/modal_indication_function.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_indication_function.png}
\captionof{figure}{\label{fig:modal_indication_function}Modal Indication Function}
\end{center}
\end{minipage}
\hfill
-\begin{minipage}[b]{0.28\linewidth}
-\begin{scriptsize}
+\begin{minipage}[b]{0.33\linewidth}
\centering
-\begin{tabularx}{\linewidth}{ccc}
+{\footnotesize\sf
+\begin{tabularx}{0.9\linewidth}{ccc}
\toprule
Mode & Frequency & Damping\\
\midrule
@@ -3295,9 +3271,8 @@ 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{scriptsize}
\end{minipage}
\subsubsection{Modal parameter extraction}
\label{ssec:modal_parameter_extraction}
@@ -3390,19 +3365,19 @@ This can be seen in Figure~\ref{fig:modal_comp_acc_frf_modal_3} that shows the f
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_1.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_comp_acc_frf_modal_1.png}
\end{center}
\subcaption{\label{fig:modal_comp_acc_frf_modal_1}From $F_{11,z}$ to $a_{11,z}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_2.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_comp_acc_frf_modal_2.png}
\end{center}
\subcaption{\label{fig:modal_comp_acc_frf_modal_2}From $F_{11,z}$ to $a_{15,z}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/modal_comp_acc_frf_modal_3.png}
+\includegraphics[scale=1,scale=0.8]{figs/modal_comp_acc_frf_modal_3.png}
\end{center}
\subcaption{\label{fig:modal_comp_acc_frf_modal_3}From $F_{11,y}$ to $a_{2,x}$}
\end{subfigure}
@@ -3780,7 +3755,7 @@ External forces can be used to model disturbances, and ``sensors'' can be used t
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/ustation_simscape_stage_example.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_simscape_stage_example.png}
\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included}
\end{figure}
@@ -3809,7 +3784,7 @@ The spring values are summarized in Table~\ref{tab:ustation_6dof_stiffness_value
\begin{table}[htbp]
\caption{\label{tab:ustation_6dof_stiffness_values}Summary of the stage stiffnesses. The contrained degrees-of-freedom are indicated by ``-''. The frames in which the 6-DoF joints are defined are indicated in figures found in Section~\ref{ssec:ustation_stages}}
\centering
-\begin{tabularx}{\linewidth}{Xcccccc}
+\begin{tabularx}{0.9\linewidth}{Xcccccc}
\toprule
\textbf{Stage} & \(D_x\) & \(D_y\) & \(D_z\) & \(R_x\) & \(R_y\) & \(R_z\)\\
\midrule
@@ -3835,19 +3810,19 @@ Tuning the numerous model parameters to better match the measurements is a highl
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_comp_com_response_rz_x.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_comp_com_response_rz_x.png}
\end{center}
\subcaption{\label{fig:ustation_comp_com_response_rz_x}Spindle, $x$ response}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_comp_com_response_hexa_y.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_comp_com_response_hexa_y.png}
\end{center}
\subcaption{\label{fig:ustation_comp_com_response_hexa_y}Hexapod, $y$ response}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_comp_com_response_ry_z.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_comp_com_response_ry_z.png}
\end{center}
\subcaption{\label{fig:ustation_comp_com_response_ry_z}Tilt, $z$ response}
\end{subfigure}
@@ -3872,10 +3847,10 @@ For each impact position, 10 impacts were performed to average and improve the d
\caption{\label{fig:ustation_compliance_meas}Schematic of the measurement setup used to estimate the compliance of the micro-station. The top platform of the positioning hexapod is shown with four 3-axis accelerometers (shown in red) are on top. 10 hammer impacts are performed at different locations (shown in blue).}
\end{figure}
-To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]\) to the acceleration expressed in the frame \(\{\mathcal{X}\}\) \(a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]\), a Jacobian matrix \(\bm{J}_a\) is written based on the positions and orientations of the accelerometers~\eqref{eq:ustation_compliance_acc_jacobian}.
+To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]\) to the acceleration expressed in the \(\{\mathcal{X}\}\) frame \(a_{\mathcal{X}} = [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]\), a Jacobian matrix \(\bm{J}_a\) is written based on the positions and orientations of the accelerometers~\eqref{eq:ustation_compliance_acc_jacobian}.
\begin{equation}\label{eq:ustation_compliance_acc_jacobian}
-\bm{J}_a = \begin{bmatrix}
+\bm{J}_a = \left[\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 &-d \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & d & 0 & 0 \\
@@ -3888,19 +3863,19 @@ To convert the 12 acceleration signals \(a_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & d \\
0 & 0 & 1 & 0 &-d & 0
-\end{bmatrix}
+\end{smallmatrix}\right]
\end{equation}
Then, the acceleration in the cartesian frame can be computed using~\eqref{eq:ustation_compute_cart_acc}.
\begin{equation}\label{eq:ustation_compute_cart_acc}
-a_{\mathcal{X}} = \bm{J}_a^\dagger \cdot a_{\mathcal{L}}
+ a_{\mathcal{X}} = \bm{J}_a^{-1} \cdot a_{\mathcal{L}}
\end{equation}
Similar to what is done for the accelerometers, a Jacobian matrix \(\bm{J}_F\) is computed~\eqref{eq:ustation_compliance_force_jacobian} and used to convert the individual hammer forces \(F_{\mathcal{L}}\) to force and torques \(F_{\mathcal{X}}\) applied at the center of the micro-hexapod top plate (defined by frame \(\{\mathcal{X}\}\) in Figure~\ref{fig:ustation_compliance_meas}).
\begin{equation}\label{eq:ustation_compliance_force_jacobian}
-\bm{J}_F = \begin{bmatrix}
+\bm{J}_F = \left[\begin{smallmatrix}
0 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & -d & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0\\
@@ -3911,7 +3886,7 @@ Similar to what is done for the accelerometers, a Jacobian matrix \(\bm{J}_F\) i
0 & 0 & -1 & 0 & d & 0\\
-1 & 0 & 0 & 0 & 0 & -d\\
-1 & 0 & 0 & 0 & 0 & d
-\end{bmatrix}
+\end{smallmatrix}\right]
\end{equation}
The equivalent forces and torques applied at center of \(\{\mathcal{X}\}\) are then computed using~\eqref{eq:ustation_compute_cart_force}.
@@ -3930,13 +3905,13 @@ Considering the complexity of the micro-station compliance dynamics, the model c
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/ustation_frf_compliance_xyz_model.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_frf_compliance_xyz_model.png}
\end{center}
\subcaption{\label{fig:ustation_frf_compliance_xyz_model}Compliance in translation}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/ustation_frf_compliance_Rxyz_model.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_frf_compliance_Rxyz_model.png}
\end{center}
\subcaption{\label{fig:ustation_frf_compliance_Rxyz_model}Compliance in rotation}
\end{subfigure}
@@ -3970,7 +3945,7 @@ The obtained ground motion displacement is shown in Figure~\ref{fig:ustation_gro
\begin{minipage}[b]{0.54\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/ustation_ground_disturbance.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_ground_disturbance.png}
\captionof{figure}{\label{fig:ustation_ground_disturbance}Measured ground motion}
\end{center}
\end{minipage}
@@ -4003,13 +3978,13 @@ Similar result is obtained for the \(x\) lateral direction.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/ustation_errors_dy_vertical.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_dy_vertical.png}
\end{center}
\subcaption{\label{fig:ustation_errors_dy_vertical}Measured vertical error}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/ustation_errors_dy_vertical_remove_mean.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_dy_vertical_remove_mean.png}
\end{center}
\subcaption{\label{fig:ustation_errors_dy_vertical_remove_mean}Error after removing linear fit}
\end{subfigure}
@@ -4049,19 +4024,19 @@ The vertical motion induced by scanning the spindle is in the order of \(\pm 30\
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_errors_spindle_radial.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_spindle_radial.png}
\end{center}
\subcaption{\label{fig:ustation_errors_spindle_radial}Radial errors}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_errors_spindle_axial.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_spindle_axial.png}
\end{center}
\subcaption{\label{fig:ustation_errors_spindle_axial}Axial error}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_errors_spindle_tilt.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_spindle_tilt.png}
\end{center}
\subcaption{\label{fig:ustation_errors_spindle_tilt}Tilt errors}
\end{subfigure}
@@ -4077,19 +4052,19 @@ The obtained transfer functions are shown in Figure~\ref{fig:ustation_model_sens
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_model_sensitivity_ground_motion.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_model_sensitivity_ground_motion.png}
\end{center}
\subcaption{\label{fig:ustation_model_sensitivity_ground_motion}Ground motion}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_model_sensitivity_ty.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_model_sensitivity_ty.png}
\end{center}
\subcaption{\label{fig:ustation_model_sensitivity_ty}Translation stage}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_model_sensitivity_rz.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_model_sensitivity_rz.png}
\end{center}
\subcaption{\label{fig:ustation_model_sensitivity_rz}Spindle}
\end{subfigure}
@@ -4104,19 +4079,19 @@ The obtained power spectral density of the disturbances are shown in Figure~\ref
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_dist_source_ground_motion.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_dist_source_ground_motion.png}
\end{center}
\subcaption{\label{fig:ustation_dist_source_ground_motion}Ground Motion}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_dist_source_translation_stage.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_dist_source_translation_stage.png}
\end{center}
\subcaption{\label{fig:ustation_dist_source_translation_stage}Translation Stage}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_dist_source_spindle.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_dist_source_spindle.png}
\end{center}
\subcaption{\label{fig:ustation_dist_source_spindle}Spindle}
\end{subfigure}
@@ -4131,19 +4106,19 @@ Examples of the obtained time-domain disturbance signals are shown in Figure~\re
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_dist_source_ground_motion_time.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_dist_source_ground_motion_time.png}
\end{center}
\subcaption{\label{fig:ustation_dist_source_ground_motion_time}Ground Motion}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_dist_source_translation_stage_time.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_dist_source_translation_stage_time.png}
\end{center}
\subcaption{\label{fig:ustation_dist_source_translation_stage_time}Translation Stage}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/ustation_dist_source_spindle_time.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_dist_source_spindle_time.png}
\end{center}
\subcaption{\label{fig:ustation_dist_source_spindle_time}Spindle}
\end{subfigure}
@@ -4168,13 +4143,13 @@ A good correlation with the measurements is observed both for radial errors (Fig
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/ustation_errors_model_spindle_radial.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_model_spindle_radial.png}
\end{center}
\subcaption{\label{fig:ustation_errors_model_spindle_radial}Radial error}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/ustation_errors_model_spindle_axial.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_model_spindle_axial.png}
\end{center}
\subcaption{\label{fig:ustation_errors_model_spindle_axial}Axial error}
\end{subfigure}
@@ -4191,7 +4166,7 @@ A similar error amplitude was observed, thus indicating that the multi-body mode
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/ustation_errors_model_dy_vertical.png}
+\includegraphics[scale=1,scale=0.8]{figs/ustation_errors_model_dy_vertical.png}
\caption{\label{fig:ustation_errors_model_dy_vertical}Vertical errors during a constant-velocity scan of the translation stage. Comparison of the measurements and simulated errors.}
\end{figure}
\subsection*{Conclusion}
@@ -4245,13 +4220,13 @@ Similarly, at the Sirius facility, a tripod configuration based on voice coil ac
\begin{figure}[h!tbp]
\begin{subfigure}{0.36\textwidth}
\begin{center}
-\includegraphics[scale=1,height=6.5cm]{figs/nhexa_stages_nazaretski.png}
+\includegraphics[scale=1,height=6cm]{figs/nhexa_stages_nazaretski.png}
\end{center}
\subcaption{\label{fig:nhexa_stages_nazaretski} MLL microscope}
\end{subfigure}
\begin{subfigure}{0.60\textwidth}
\begin{center}
-\includegraphics[scale=1,height=6.5cm]{figs/nhexa_stages_sapoti.png}
+\includegraphics[scale=1,height=6cm]{figs/nhexa_stages_sapoti.png}
\end{center}
\subcaption{\label{fig:nhexa_stages_sapoti} SAPOTI sample stage}
\end{subfigure}
@@ -4267,7 +4242,7 @@ However, attempts to implement real-time feedback using YZ external metrology pr
\begin{figure}[h!tbp]
\begin{subfigure}{0.54\textwidth}
\begin{center}
-\includegraphics[scale=1,height=6cm]{figs/nhexa_stages_villar.png}
+\includegraphics[scale=1,height=5.5cm]{figs/nhexa_stages_villar.png}
\end{center}
\subcaption{\label{fig:nhexa_stages_villar} Simplified schematic of ID16a end-station}
\end{subfigure}
@@ -4286,7 +4261,6 @@ Although direct performance comparisons between these systems are challenging du
\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 ommited or indicated between parentheses. The specifications for the NASS are indicated in the last row.}
\centering
-\scriptsize
\begin{tabularx}{0.8\linewidth}{ccccc}
\toprule
\textbf{Stacked Stages} & \textbf{Specifications} & \textbf{Measured DoFs} & \textbf{Bandwidth} & \textbf{Reference}\\
@@ -4376,13 +4350,13 @@ Furthermore, hybrid architectures combining both serial and parallel elements ha
\begin{figure}[h!tbp]
\begin{subfigure}{0.41\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5cm]{figs/nhexa_serial_architecture_kenton.png}
+\includegraphics[scale=1,height=4.5cm]{figs/nhexa_serial_architecture_kenton.png}
\end{center}
\subcaption{\label{fig:nhexa_serial_architecture_kenton} Serial positioning stage}
\end{subfigure}
\begin{subfigure}{0.55\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5cm]{figs/nhexa_parallel_architecture_shen.png}
+\includegraphics[scale=1,height=4.5cm]{figs/nhexa_parallel_architecture_shen.png}
\end{center}
\subcaption{\label{fig:nhexa_parallel_architecture_shen} Hybrid 5-DoF stage}
\end{subfigure}
@@ -4398,13 +4372,13 @@ Furthermore, the successful implementation of Integral Force Feedback (IFF) cont
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/nhexa_stewart_piezo_furutani.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/nhexa_stewart_piezo_furutani.png}
\end{center}
\subcaption{\label{fig:nhexa_stewart_piezo_furutani} Stewart platform for Nano-positioning}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/nhexa_stewart_vc_preumont.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/nhexa_stewart_vc_preumont.png}
\end{center}
\subcaption{\label{fig:nhexa_stewart_vc_preumont} Stewart platform for vibration isolation}
\end{subfigure}
@@ -4440,7 +4414,7 @@ The typical configuration consists of a universal joint at one end and a spheric
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_stewart_architecture.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_architecture.png}
\caption{\label{fig:nhexa_stewart_architecture}Schematical representation of the Stewart platform architecture.}
\end{figure}
@@ -4463,7 +4437,7 @@ This is summarized in Figure~\ref{fig:nhexa_stewart_notations}.
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_stewart_notations.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_notations.png}
\caption{\label{fig:nhexa_stewart_notations}Frame and key notations for the Stewart platform}
\end{figure}
\subsubsection{Kinematic Analysis}
@@ -4481,7 +4455,7 @@ This equation links the pose\footnote{The \emph{pose} represents the position an
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_stewart_loop_closure.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_loop_closure.png}
\caption{\label{fig:nhexa_stewart_loop_closure}Notations to compute the kinematic loop closure}
\end{figure}
\paragraph{Inverse Kinematics}
@@ -4581,7 +4555,7 @@ It can be computed once at the rest position and used for both forward and inver
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_forward_kinematics_approximate_errors.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_forward_kinematics_approximate_errors.png}
\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,mm\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.}
\end{figure}
\paragraph{Static Forces}
@@ -4736,14 +4710,14 @@ From these parameters, key kinematic properties can be derived: the strut orient
\begin{minipage}[b]{0.6\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/nhexa_stewart_model_def.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_model_def.png}
\captionof{figure}{\label{fig:nhexa_stewart_model_def}Geometry of the stewart platform}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.38\linewidth}
-\begin{scriptsize}
\centering
+{\footnotesize\sf
\begin{tabularx}{0.75\linewidth}{Xrrr}
\toprule
& \(\bm{x}\) & \(\bm{y}\) & \(\bm{z}\)\\
@@ -4763,9 +4737,8 @@ 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{scriptsize}
\end{minipage}
\paragraph{Inertia of Plates}
@@ -4794,15 +4767,15 @@ This modular approach to actuator modeling allows for future refinements as the
\begin{minipage}[b]{0.6\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/nhexa_actuator_model.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_actuator_model.png}
\captionof{figure}{\label{fig:nhexa_actuator_model}Model of the nano-hexapod actuators}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.38\linewidth}
-\begin{scriptsize}
\centering
-\begin{tabularx}{0.4\linewidth}{Xl}
+{\footnotesize\sf
+\begin{tabularx}{0.5\linewidth}{Xl}
\toprule
& Value\\
\midrule
@@ -4810,9 +4783,8 @@ This modular approach to actuator modeling allows for future refinements as the
\(c_a\) & \(50\,N/(m/s)\)\\
\(k_p\) & \(0.05\,N/\mu m\)\\
\bottomrule
-\end{tabularx}
+\end{tabularx}}
\captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters}
-\end{scriptsize}
\end{minipage}
\subsubsection{Validation of the multi-body model}
\label{ssec:nhexa_model_validation}
@@ -4824,13 +4796,13 @@ A three-dimensional visualization of the model is presented in Figure~\ref{fig:n
\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_stewart_model_input_outputs.png}
-\captionof{figure}{\label{fig:nhexa_stewart_model_input_outputs}Nano-Hexapod plant with inputs and outputs. Frames \(\{F\}\) and \(\{M\}\) can be connected to other elements in the multi-body models.}
+\captionof{figure}{\label{fig:nhexa_stewart_model_input_outputs}Nano-Hexapod plant with inputs and outputs. Frames \(\{F\}\) and \(\{M\}\) can be connected to other elements in the model.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.35\linewidth}
\begin{center}
-\includegraphics[scale=1,width=0.90\linewidth]{figs/nhexa_simscape_screenshot.jpg}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/nhexa_simscape_screenshot.jpg}
\captionof{figure}{\label{fig:nhexa_simscape_screenshot}3D representation of the multi-body model}
\end{center}
\end{minipage}
@@ -4860,7 +4832,7 @@ The close agreement between both approaches across the frequency spectrum valida
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_comp_multi_body_analytical.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_comp_multi_body_analytical.png}
\caption{\label{fig:nhexa_comp_multi_body_analytical}Comparison of the analytical transfer functions and the multi-body model}
\end{figure}
\subsubsection{Nano Hexapod Dynamics}
@@ -4887,13 +4859,13 @@ The inclusion of parallel stiffness introduces an additional complex conjugate z
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/nhexa_multi_body_plant_dL.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_multi_body_plant_dL.png}
\end{center}
\subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/nhexa_multi_body_plant_fm.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_multi_body_plant_fm.png}
\end{center}
\subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{n}$}
\end{subfigure}
@@ -4941,7 +4913,7 @@ In the context of the nano-hexapod, two distinct control strategies were examine
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_stewart_decentralized_control.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_stewart_decentralized_control.png}
\caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.}
\end{figure}
\subsubsection{Choice of the Control Space}
@@ -4995,13 +4967,13 @@ More sophisticated control strategies will be explored during the detailed desig
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/nhexa_plant_frame_struts.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_plant_frame_struts.png}
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/nhexa_plant_frame_cartesian.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_plant_frame_cartesian.png}
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame}
\end{subfigure}
@@ -5016,7 +4988,7 @@ The corresponding block diagram of the control loop is shown in Figure~\ref{fig:
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_decentralized_iff_schematic.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_decentralized_iff_schematic.png}
\caption{\label{fig:nhexa_decentralized_iff_schematic}Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs \(\bm{f}^{\prime}\)}
\end{figure}
@@ -5042,13 +5014,13 @@ This high gain, combined with the bounded phase, enables effective damping of th
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_loop_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_iff_loop_gain.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{n1}}{f_1}(s)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus}
\end{subfigure}
@@ -5066,7 +5038,7 @@ A diagonal High Authority Controller \(\bm{K}_{\text{HAC}}\) then processes thes
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/nhexa_hac_iff_schematic.png}
+\includegraphics[scale=1,scale=0.9]{figs/nhexa_hac_iff_schematic.png}
\caption{\label{fig:nhexa_hac_iff_schematic}HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts}
\end{figure}
@@ -5078,13 +5050,13 @@ This damping of structural resonances serves two purposes: it reduces vibrations
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nhexa_decentralized_hac_iff_plant_undamped.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_hac_iff_plant_undamped.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped plant in the frame of the struts}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nhexa_decentralized_hac_iff_plant_damped.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_hac_iff_plant_damped.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped plant with Decentralized IFF}
\end{subfigure}
@@ -5112,13 +5084,13 @@ Additionally, the distance of the loci from the \(-1\) point provides informatio
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_hac_iff_loop_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_hac_iff_loop_gain.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_hac_iff_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/nhexa_decentralized_hac_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Characteristic Loci}
\end{subfigure}
@@ -5352,13 +5324,13 @@ However, their alternating pattern is preserved, which ensures the phase remains
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_iff_plant_effect_rotation.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_iff_plant_effect_rotation.png}
\end{center}
\subcaption{\label{fig:nass_iff_plant_effect_rotation}Effect of Spindle rotation}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_iff_plant_effect_payload.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_iff_plant_effect_payload.png}
\end{center}
\subcaption{\label{fig:nass_iff_plant_effect_payload}Effect of payload mass}
\end{subfigure}
@@ -5388,7 +5360,7 @@ The overall gain was then increased to obtain a large loop gain around the reson
\begin{figure}[htbp]
\centering
-\includegraphics[h!tbp]{figs/nass_iff_loop_gain.png}
+\includegraphics[h!tbp,scale=0.8]{figs/nass_iff_loop_gain.png}
\caption{\label{fig:nass_iff_loop_gain}Loop gain for the decentralized IFF: \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)}
\end{figure}
@@ -5398,19 +5370,19 @@ The results demonstrate that the closed-loop poles remain within the left-half p
\begin{figure}[h!tbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/nass_iff_root_locus_1kg.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_iff_root_locus_1kg.png}
\end{center}
\subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/nass_iff_root_locus_25kg.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_iff_root_locus_25kg.png}
\end{center}
\subcaption{\label{fig:nass_iff_root_locus_25kg} $25\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/nass_iff_root_locus_50kg.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_iff_root_locus_50kg.png}
\end{center}
\subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$}
\end{subfigure}
@@ -5448,13 +5420,13 @@ This also validates the developed control strategy.
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_undamped_plant_effect_Wz.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_undamped_plant_effect_Wz.png}
\end{center}
\subcaption{\label{fig:nass_undamped_plant_effect_Wz}Effect of rotational velocity $\Omega_z$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_undamped_plant_effect_mass.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_undamped_plant_effect_mass.png}
\end{center}
\subcaption{\label{fig:nass_undamped_plant_effect_mass}Effect of payload's mass}
\end{subfigure}
@@ -5475,13 +5447,13 @@ For the undamped plants (shown in blue), achieving robust control with bandwidth
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_comp_undamped_damped_plant_m1.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_comp_undamped_damped_plant_m1.png}
\end{center}
\subcaption{\label{fig:nass_comp_undamped_damped_plant_m1}Effect of IFF - $m = 1\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_hac_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_hac_plants.png}
\end{center}
\subcaption{\label{fig:nass_hac_plants}Effect of IFF on the set of plants to control}
\end{subfigure}
@@ -5497,7 +5469,7 @@ This result confirms effective dynamic decoupling between the nano-hexapod and t
\begin{figure}[htbp]
\centering
-\includegraphics[h!tbp]{figs/nass_effect_ustation_compliance.png}
+\includegraphics[h!tbp,scale=0.8]{figs/nass_effect_ustation_compliance.png}
\caption{\label{fig:nass_effect_ustation_compliance}Effect of the micro-station limited compliance on the plant dynamics}
\end{figure}
\subsubsection{Effect of Nano-Hexapod Stiffness on System Dynamics}
@@ -5519,13 +5491,13 @@ The current approach of controlling the position in the strut frame is inadequat
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_stiff_nano_hexapod_coupling_ustation.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_stiff_nano_hexapod_coupling_ustation.png}
\end{center}
\subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}$k_a = 100\,N/\mu m$ - Coupling with the micro-station}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_soft_nano_hexapod_effect_Wz.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_soft_nano_hexapod_effect_Wz.png}
\end{center}
\subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}$k_a = 0.01\,N/\mu m$ - Effect of Spindle rotation}
\end{subfigure}
@@ -5548,13 +5520,13 @@ Second, the characteristic loci analysis presented in Figure~\ref{fig:nass_hac_l
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_hac_loop_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_hac_loop_gain.png}
\end{center}
\subcaption{\label{fig:nass_hac_loop_gain}Loop Gain}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/nass_hac_loci.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_hac_loci.png}
\end{center}
\subcaption{\label{fig:nass_hac_loci}Characteristic Loci}
\end{subfigure}
@@ -5577,13 +5549,13 @@ The results demonstrate the system's capability to maintain the sample's positio
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/nass_tomo_1kg_60rpm_xy.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_tomo_1kg_60rpm_xy.png}
\end{center}
\subcaption{\label{fig:nass_tomo_1kg_60rpm_xy}XY plane}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/nass_tomo_1kg_60rpm_yz.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_tomo_1kg_60rpm_yz.png}
\end{center}
\subcaption{\label{fig:nass_tomo_1kg_60rpm_yz}YZ plane}
\end{subfigure}
@@ -5600,19 +5572,19 @@ For higher mass configurations, rotational velocities are expected to be below 3
\begin{figure}[h!tbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/nass_tomography_hac_iff_m1.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_tomography_hac_iff_m1.png}
\end{center}
\subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,kg$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/nass_tomography_hac_iff_m25.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_tomography_hac_iff_m25.png}
\end{center}
\subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,kg$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/nass_tomography_hac_iff_m50.png}
+\includegraphics[scale=1,scale=0.8]{figs/nass_tomography_hac_iff_m50.png}
\end{center}
\subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,kg$}
\end{subfigure}
@@ -5633,8 +5605,6 @@ This investigation has confirmed that the moderate actuator stiffness of \(1\,N/
Simulations of tomography experiments have been performed, with positioning accuracy requirements defined by the expected minimum beam dimensions of \(200\,\text{nm}\) by \(100\,\text{nm}\).
The system has demonstrated excellent performance at maximum rotational velocity with lightweight samples.
While some degradation in positioning accuracy has been observed with heavier payloads, as anticipated by the control analysis, the overall performance remains sufficient to validate the fundamental concept of the NASS.
-
-These results provide a solid foundation for advancing to the subsequent detailed design phase and experimental implementation.
\section*{Conceptual Design - Conclusion}
\label{sec:concept_conclusion}
@@ -6004,8 +5974,7 @@ These trade-offs provide important guidelines when choosing the Stewart platform
\begin{table}[htbp]
\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness and mobility}
\centering
-\small
-\begin{tabularx}{0.8\linewidth}{Xcc}
+\begin{tabularx}{0.65\linewidth}{Xcc}
\toprule
\textbf{Struts} & \textbf{Vertically Oriented} & \textbf{Increased separation}\\
\midrule
@@ -6201,7 +6170,7 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_cubic_payload.png}
+\includegraphics[scale=1,width=0.5\linewidth]{figs/detail_kinematics_cubic_payload.png}
\caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload}
\end{figure}
@@ -6213,13 +6182,13 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_cart_coupling_com.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_cok.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_cart_coupling_cok.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
\end{subfigure}
@@ -6243,7 +6212,7 @@ If a design similar to Figure~\ref{fig:detail_kinematics_cubic_centered_payload}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_cart_coupling_com_cok.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_cart_coupling_com_cok.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com_cok}Fully decoupled cartesian plant}
\end{subfigure}
@@ -6274,7 +6243,7 @@ The second uses a non-cubic Stewart platform shown in Figure~\ref{fig:detail_kin
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
+\includegraphics[scale=1,width=0.5\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture}
\end{figure}
\paragraph{Relative Displacement Sensors}
@@ -6289,13 +6258,13 @@ The resonance frequencies differ between the two cases because the more vertical
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_dL.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_non_cubic_decentralized_dL.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_dL}Non cubic architecture}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_dL.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_decentralized_dL.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_dL}Cubic architecture}
\end{subfigure}
@@ -6310,13 +6279,13 @@ The system demonstrates good decoupling at high frequency in both cases, with no
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_non_cubic_decentralized_fn.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_non_cubic_decentralized_fn.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_non_cubic_decentralized_fn}Non cubic architecture}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_cubic_decentralized_fn.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_cubic_decentralized_fn.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_decentralized_fn}Cubic architecture}
\end{subfigure}
@@ -6499,13 +6468,13 @@ The positioning angles, as shown in Figure~\ref{fig:detail_kinematics_nano_hexap
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/detail_kinematics_nano_hexapod_iso.png}
+\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_nano_hexapod_iso.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_nano_hexapod_iso}Isometric view}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/detail_kinematics_nano_hexapod_top.png}
+\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_nano_hexapod_top.png}
\end{center}
\subcaption{\label{fig:detail_kinematics_nano_hexapod_top}Top view}
\end{subfigure}
@@ -6528,13 +6497,14 @@ The required mobility parameters include combined translations in the XYZ direct
Additionally, at any point within this workspace, combined \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\), with \(R_z\) maintained at 0, should be possible.
Calculations based on the selected geometry indicate that an actuator stroke of \(\pm 94\,\mu m\) is required to achieve the desired mobility.
-This specification will be used during the actuator selection process.
+This specification will be used during the actuator selection process in Section \ref{sec:detail_fem_actuator}.
+
Figure~\ref{fig:detail_kinematics_nano_hexapod_mobility} illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of \(\pm 94\,\mu m\).
The diagram confirms that the required workspace fits within the system's capabilities.
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,scale=0.9]{figs/detail_kinematics_nano_hexapod_mobility.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_kinematics_nano_hexapod_mobility.png}
\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Specified translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).}
\end{figure}
\subsubsection{Required Joint angular stroke}
@@ -6544,7 +6514,7 @@ With the nano-hexapod geometry and mobility requirements established, the flexib
This analysis focuses solely on bending stroke, as the torsional stroke of the flexible joints is expected to be minimal given the absence of vertical rotation requirements.
The required angular stroke for both fixed and mobile joints is estimated to be equal to \(1\,\text{mrad}\).
-This specification will guide the design of the flexible joints.
+This specification will guide the design of the flexible joints in Section \ref{sec:detail_fem_joint}.
\subsection*{Conclusion}
\label{sec:detail_kinematics_conclusion}
@@ -6621,7 +6591,8 @@ The specific design of the APA allows for the simultaneous modeling of a mechani
\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering
-\begin{tabularx}{0.7\linewidth}{Xc}
+{\footnotesize\sf
+\begin{tabularx}{0.55\linewidth}{Xc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
@@ -6629,7 +6600,7 @@ Nominal Stroke & \(100\,\mu m\)\\
Blocked force & \(2100\,N\)\\
Stiffness & \(21\,N/\mu m\)\\
\bottomrule
-\end{tabularx}
+\end{tabularx}}
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
@@ -6638,9 +6609,9 @@ The development of the finite element model for the APA95ML required the knowled
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 modal reduction model. \(E\) is the Young's modulus, \(\nu\) the Poisson ratio and \(\rho\) the material density}
+\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}{0.7\linewidth}{lXXX}
+\begin{tabularx}{0.55\linewidth}{Xccc}
\toprule
& \(E\) & \(\nu\) & \(\rho\)\\
\midrule
@@ -6703,17 +6674,17 @@ Yet, based on the available properties of the stacks in the data-sheet (summariz
\begin{table}[htbp]
\caption{\label{tab:detail_fem_stack_parameters}Stack Parameters}
\centering
-\begin{tabularx}{0.4\linewidth}{Xcc}
+\begin{tabularx}{0.3\linewidth}{Xc}
\toprule
-Parameter & Unit & Value\\
+\textbf{Parameter} & \textbf{Value}\\
\midrule
-Nominal Stroke & \(\mu m\) & 20\\
-Blocked force & \(N\) & 4700\\
-Stiffness & \(N/\mu m\) & 235\\
-Voltage Range & \(V\) & -20 to 150\\
-Capacitance & \(\mu F\) & 4.4\\
-Length & \(mm\) & 20\\
-Stack Area & \(mm^2\) & 10x10\\
+Nominal Stroke & \(20\,\mu m\)\\
+Blocked force & \(4700\,N\)\\
+Stiffness & \(235\,N/\mu m\)\\
+Voltage Range & \(-20/150\,V\)\\
+Capacitance & \(4.4\,\mu F\)\\
+Length & \(20\,mm\)\\
+Stack Area & \(10\times 10\,mm^2\)\\
\bottomrule
\end{tabularx}
\end{table}
@@ -6724,7 +6695,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\begin{table}[htbp]
\caption{\label{tab:detail_fem_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities}
\centering
-\begin{tabularx}{1\linewidth}{ccX}
+\begin{tabularx}{0.8\linewidth}{ccX}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Description}\\
\midrule
@@ -6750,7 +6721,7 @@ The multi-body model predicted a resonant frequency under block-free conditions
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_fem_apa95ml_compliance.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_compliance.png}
\caption{\label{fig:detail_fem_apa95ml_compliance}Estimated compliance of the APA95ML}
\end{figure}
@@ -6797,13 +6768,13 @@ Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_comp_plant_actuator.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_comp_plant_actuator.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_comp_plant_sensor.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_comp_plant_sensor.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$}
\end{subfigure}
@@ -6829,13 +6800,13 @@ The close agreement between experimental measurements and theoretical prediction
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_iff_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_damped_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_damped_plants.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants}
\end{subfigure}
@@ -6862,7 +6833,7 @@ Additional specifications arise from the control strategy and physical constrain
The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator.
The system's geometric constraints limit the actuator height to 50mm, given the nano-hexapod's maximum height of 95mm and the presence of flexible joints at each strut extremity.
Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility.
-An actuator stroke of \(\approx 100\,\mu m\) is therefore required.
+An actuator stroke of \(\approx 200\,\mu m\) is therefore required.
Three actuator technologies were evaluated (examples of such actuators are shown in Figure~\ref{fig:detail_fem_actuator_pictures}): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity.
@@ -6870,13 +6841,13 @@ Variable reluctance actuators were not considered despite their superior efficie
\begin{figure}[htbp]
\begin{subfigure}{0.25\textwidth}
\begin{center}
-\includegraphics[scale=1,height=4.5cm]{figs/detail_fem_voice_coil_picture.jpg}
+\includegraphics[scale=1,height=4cm]{figs/detail_fem_voice_coil_picture.jpg}
\end{center}
\subcaption{\label{fig:detail_fem_voice_coil_picture}Voice Coil}
\end{subfigure}
\begin{subfigure}{0.25\textwidth}
\begin{center}
-\includegraphics[scale=1,height=4.5cm]{figs/detail_fem_piezo_picture.jpg}
+\includegraphics[scale=1,height=4cm]{figs/detail_fem_piezo_picture.jpg}
\end{center}
\subcaption{\label{fig:detail_fem_piezo_picture}Piezoelectric stack}
\end{subfigure}
@@ -6889,7 +6860,7 @@ Variable reluctance actuators were not considered despite their superior efficie
\caption{\label{fig:detail_fem_actuator_pictures}Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).}
\end{figure}
-Voice coil actuators (shown in Figure~\ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness (\(\approx 1\,N/\mu m\)), would require forces in the order of \(100\,N\) to achieve the specified \(100\,\mu m\) displacement.
+Voice coil actuators (shown in Figure~\ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness (\(\approx 1\,N/\mu m\)), would require forces in the order of \(200\,N\) to achieve the specified \(200\,\mu m\) displacement.
While these actuators offer excellent linearity and long strokes capabilities, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability.
Their advantages (linearity and long stroke) were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions.
@@ -6910,12 +6881,11 @@ The demonstrated accuracy of the modeling approach for the APA95ML provides conf
\begin{table}[htbp]
\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the nano-hexapod}
\centering
-\scriptsize
\begin{tabularx}{0.9\linewidth}{Xccccc}
\toprule
\textbf{Specification} & APA150M & \textbf{APA300ML} & APA400MML & FPA-0500E-P & FPA-0300E-S\\
\midrule
-Stroke \(> 100\, [\mu m]\) & 187 & 304 & 368 & 432 & 240\\
+Stroke \(> 200\, [\mu m]\) & 187 & 304 & 368 & 432 & 240\\
Stiffness \(\approx 1\, [N/\mu m]\) & 0.7 & 1.8 & 0.55 & 0.87 & 0.58\\
Resolution \(< 2\, [nm]\) & 2 & 3 & 4 & & \\
Blocked Force \(> 100\, [N]\) & 127 & 546 & 201 & 376 & 139\\
@@ -6989,7 +6959,7 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\begin{table}[htbp]
\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
\centering
-\begin{tabularx}{0.3\linewidth}{cc}
+\begin{tabularx}{0.25\linewidth}{cc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
@@ -7008,13 +6978,13 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa300ml_comp_fem_2dof_actuator.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa300ml_comp_fem_2dof_actuator.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}from $V_a$ to $d_i$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa300ml_comp_fem_2dof_force_sensor.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa300ml_comp_fem_2dof_force_sensor.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor}from $V_a$ to $V_s$}
\end{subfigure}
@@ -7032,7 +7002,7 @@ The developed models of the APA do not represent such behavior, but as this effe
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_fem_apa95ml_effect_electrical_boundaries.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_apa95ml_effect_electrical_boundaries.png}
\caption{\label{fig:detail_fem_apa95ml_effect_electrical_boundaries}Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency}
\end{figure}
@@ -7058,13 +7028,13 @@ These results validate both the selection of the APA300ML and the effectiveness
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_actuator_fem_vs_perfect_hac_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_actuator_fem_vs_perfect_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_actuator_fem_vs_perfect_iff_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_actuator_fem_vs_perfect_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
@@ -7131,13 +7101,13 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_hac_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_bending_stiffness_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_iff_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_bending_stiffness_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
@@ -7147,13 +7117,13 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_iff_locus_1dof.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_bending_stiffness_iff_locus_1dof.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}1DoF actuators}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_bending_stiffness_iff_locus_apa300ml.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_bending_stiffness_iff_locus_apa300ml.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml}APA300ML actuators}
\end{subfigure}
@@ -7201,13 +7171,13 @@ Based on this analysis, an axial stiffness specification of \(100\,N/\mu m\) was
\begin{figure}[h!tbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/detail_fem_joints_axial_stiffness_iff_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_axial_stiffness_iff_locus.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_iff_locus}Root Locus}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/detail_fem_joints_axial_stiffness_rga_hac_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_axial_stiffness_rga_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_axial_stiffness_rga_hac_plant}RGA number}
\end{subfigure}
@@ -7223,7 +7193,7 @@ Based on the dynamic analysis presented in previous sections, quantitative speci
\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}{0.5\linewidth}{Xcc}
+\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
& \textbf{Specification} & \textbf{FEM}\\
\midrule
@@ -7284,13 +7254,13 @@ While additional degrees of freedom could potentially capture more dynamic featu
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_fem_vs_perfect_hac_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_fem_vs_perfect_hac_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_hac_plant}$\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_fem_joints_fem_vs_perfect_iff_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_fem_joints_fem_vs_perfect_iff_plant.png}
\end{center}
\subcaption{\label{fig:detail_fem_joints_fem_vs_perfect_iff_plant}$\bm{f}$ to $\bm{f}_m$}
\end{subfigure}
@@ -7660,19 +7630,19 @@ To facilitate the expression of these specifications, formula~\eqref{eq:detail_c
The parameters in this formula are \(G_0 = \lim_{\omega \to 0} |W(j\omega)|\) (the low-frequency gain), \(G_\infty = \lim_{\omega \to \infty} |W(j\omega)|\) (the high-frequency gain), \(G_c = |W(j\omega_c)|\) (the gain at a specific frequency \(\omega_c\) in \(\si{rad/s}\)), and \(n\) (the slope between high and low frequency, which also corresponds to the order of the weighting function).
The typical magnitude response of a weighting function generated using~\eqref{eq:detail_control_sensor_weight_formula} is illustrated in Figure~\ref{fig:detail_control_sensor_weight_formula}.
-\begin{minipage}[]{0.49\linewidth}
+\begin{minipage}[]{0.45\linewidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_sensor_weight_formula.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_sensor_weight_formula.png}
\captionof{figure}{\label{fig:detail_control_sensor_weight_formula}Magnitude of a weighting function generated using~\eqref{eq:detail_control_sensor_weight_formula}, \(G_0 = 10^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
\end{center}
\end{minipage}
\hfill
-\begin{minipage}[]{0.49\linewidth}
+\begin{minipage}[]{0.54\linewidth}
\begin{equation}\label{eq:detail_control_sensor_weight_formula}
W(s) = \left( \frac{
\hfill{} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
}{
- \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
+ \frac{1}{G_\infty^{\frac{1}{n}}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \frac{1}{G_c^{\frac{1}{n}}}
}\right)^n
\end{equation}
\end{minipage}
@@ -7693,6 +7663,7 @@ The inverse magnitudes of the designed weighting functions, which represent the
\begin{minipage}[b]{0.44\linewidth}
\begin{center}
+\footnotesize\sf
\begin{tabularx}{0.7\linewidth}{ccc}
\toprule
Parameter & \(W_1(s)\) & \(W_2(s)\)\\
@@ -7710,7 +7681,7 @@ Parameter & \(W_1(s)\) & \(W_2(s)\)\\
\hfill
\begin{minipage}[b]{0.52\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/detail_control_sensor_hinf_filters_results.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_sensor_hinf_filters_results.png}
\captionof{figure}{\label{fig:detail_control_sensor_hinf_filters_results}Weights and obtained filters}
\end{center}
\end{minipage}
@@ -7807,7 +7778,7 @@ Consider the generalized plant \(P_3(s)\) shown in Figure~\ref{fig:detail_contro
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_sensor_three_complementary_filters_results.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_sensor_three_complementary_filters_results.png}
\end{center}
\subcaption{\label{fig:detail_control_sensor_three_complementary_filters_results}Weights and obtained filters}
\end{subfigure}
@@ -7883,8 +7854,8 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin
\end{minipage}
\hfill
\begin{minipage}[b]{0.36\linewidth}
-\begin{scriptsize}
\centering
+{\footnotesize\sf
\begin{tabularx}{\linewidth}{cXc}
\toprule
& \textbf{Description} & \textbf{Value}\\
@@ -7896,9 +7867,8 @@ Two reference frames are defined within this model: frame \(\{M\}\) with origin
\(m\) & Payload mass & \(40\,\text{kg}\)\\
\(I\) & Payload \(R_z\) inertia & \(5\,\text{kg}m^2\)\\
\bottomrule
-\end{tabularx}
+\end{tabularx}}
\captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
-\end{scriptsize}
\end{minipage}
The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass~\eqref{eq:detail_control_decoupling_model_eom}, where \(\bm{\mathcal{X}}_{\{M\}}\) represents the two translations and one rotation with respect to the center of mass, and \(\bm{\mathcal{F}}_{\{M\}}\) denotes the forces and torque applied at the center of mass.
@@ -7977,7 +7947,7 @@ Depending on the symmetry present in the system, certain diagonal elements may e
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_control_decoupling_coupled_plant_bode.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_coupled_plant_bode.png}
\caption{\label{fig:detail_control_decoupling_coupled_plant_bode}Model dynamics from actuator forces to relative displacement sensor of each strut.}
\end{figure}
\subsubsection{Jacobian Decoupling}
@@ -8056,7 +8026,7 @@ This phenomenon is illustrated in Figure~\ref{fig:detail_control_decoupling_mode
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_jacobian_plant_CoM.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_jacobian_plant_CoM.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM}
\end{subfigure}
@@ -8108,7 +8078,7 @@ When a high-frequency force is applied at a point not aligned with the center of
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_jacobian_plant_CoK.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_jacobian_plant_CoK.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK}
\end{subfigure}
@@ -8187,7 +8157,7 @@ Each of these diagonal elements corresponds to a specific mode, as shown in Figu
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_modal_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_modal_plant.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space}
\end{subfigure}
@@ -8275,7 +8245,7 @@ Additionally, the diagonal terms manifest as second-order dynamic systems, facil
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_control_decoupling_svd_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_svd_plant.png}
\caption{\label{fig:detail_control_decoupling_svd_plant}Plant dynamics \(\bm{G}_{\text{SVD}}(s)\) obtained after decoupling using Singular Value Decomposition}
\end{figure}
@@ -8293,7 +8263,7 @@ Notably, the coupling demonstrates local minima near the decoupling frequency, c
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_decoupling_svd_alt_plant.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_decoupling_svd_alt_plant.png}
\end{center}
\subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant}
\end{subfigure}
@@ -8594,13 +8564,13 @@ These filters can also be implemented in the digital domain with analytical form
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_analytical_effect_alpha.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_analytical_effect_alpha.png}
\end{center}
\subcaption{\label{fig:detail_control_cf_analytical_effect_alpha}Effect of $\alpha$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_analytical_effect_w0.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_analytical_effect_w0.png}
\end{center}
\subcaption{\label{fig:detail_control_cf_analytical_effect_w0}Effect of $\omega_0$}
\end{subfigure}
@@ -8652,7 +8622,7 @@ Figure~\ref{fig:detail_control_cf_bode_plot_mech_sys} illustrates both the nomin
\end{subfigure}
\begin{subfigure}{0.66\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/detail_control_cf_bode_plot_mech_sys.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_bode_plot_mech_sys.png}
\end{center}
\subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set}
\end{subfigure}
@@ -8681,13 +8651,13 @@ There magnitudes are displayed in Figure~\ref{fig:detail_control_cf_specs_S_T},
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_specs_S_T.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_specs_S_T.png}
\end{center}
\subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_bode_Kfb.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_bode_Kfb.png}
\end{center}
\subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$}
\end{subfigure}
@@ -8782,7 +8752,7 @@ The measured noise characteristics are then incorporated into the multi-body mod
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_instrumentation_plant.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_instrumentation_plant.png}
\caption{\label{fig:detail_instrumentation_plant}Block diagram of the NASS with considered instrumentation. The RT controller is a Speedgoat machine.}
\end{figure}
\subsection{Dynamic Error Budgeting}
@@ -8807,7 +8777,7 @@ The transfer functions from these three noise sources (for one strut) to the ver
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_instrumentation_noise_sensitivities.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_noise_sensitivities.png}
\caption{\label{fig:detail_instrumentation_noise_sensitivities}Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy.}
\end{figure}
\subsubsection{Estimation of maximum instrumentation noise}
@@ -8895,30 +8865,31 @@ 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 \(10\,\text{kHz}\)
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.
\begin{table}[htbp]
\caption{\label{tab:detail_instrumentation_amp_choice}Specifications for the Voltage amplifier and considered commercial products}
\centering
-\begin{tabularx}{0.9\linewidth}{Xcccc}
+\begin{tabularx}{0.8\linewidth}{Xcccc}
\toprule
-\textbf{Specification} & \textbf{PD200} & WMA-200 & LA75B & E-505\\
+\textbf{Specifications} & PD200 & WMA-200 & LA75B & E-505\\
& PiezoDrive & Falco & Cedrat & PI\\
\midrule
-Input Voltage Range: \(\pm 10\,V\) & \(\pm 10\,V\) & \(\pm8.75\,V\) & \(-1/7.5\,V\) & \\
+Input Voltage Range: \(\pm 10\,V\) & \(\pm 10\,V\) & \(\pm8.75\,V\) & \(-1/7.5\,V\) & \(-2/12\,V\)\\
Output Voltage Range: \(-20/150\,V\) & \(-50/150\,V\) & \(\pm 175\,V\) & \(-20/150\,V\) & -30/130\\
Gain \(>15\) & 20 & 20 & 20 & 10\\
Output Current \(> 300\,mA\) & \(900\,mA\) & \(150\,mA\) & \(360\,mA\) & \(215\,mA\)\\
Slew Rate \(> 34\,V/ms\) & \(150\,V/\mu s\) & \(80\,V/\mu s\) & n/a & n/a\\
-Output noise \(< 20\,mV\ \text{RMS}\) & \(0.7\,mV\,\text{RMS}\) & \(0.05\,mV\) & \(3.4\,mV\) & \(0.6\,mV\)\\
+Output noise \(< 20\,mV\ \text{RMS}\) & \(0.7\,mV\) & \(0.05\,mV\) & \(3.4\,mV\) & \(0.6\,mV\)\\
(10uF load) & (\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (n/a) & (n/a)\\
Small Signal Bandwidth \(> 5\,kHz\) & \(6.4\,kHz\) & \(300\,Hz\) & \(30\,kHz\) & n/a\\
-(\(10\,\mu F\) load) & (\(10\,\mu F\) load) & \footnotemark & (unloaded) & (n/a)\\
+(\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (unloaded) & (n/a)\\
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\
\bottomrule
\end{tabularx}
-\end{table}\footnotetext[27]{\label{org964a280}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)}
+\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.
The proper selection of these components is critical for system performance.
@@ -9055,9 +9026,9 @@ The specifications of the considered relative motion sensor, the Renishaw Vionic
\begin{table}[htbp]
\caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products}
\centering
-\begin{tabularx}{0.8\linewidth}{Xccc}
+\begin{tabularx}{0.65\linewidth}{Xccc}
\toprule
-\textbf{Specification} & \textbf{Renishaw Vionic} & LION CPL190 & Cedrat ECP500\\
+\textbf{Specifications} & Renishaw Vionic & LION CPL190 & Cedrat ECP500\\
\midrule
Technology & Digital Encoder & Capacitive & Eddy Current\\
Bandwidth \(> 5\,\text{kHz}\) & \(> 500\,\text{kHz}\) & 10kHz & 20kHz\\
@@ -9085,7 +9056,7 @@ This approach is effective because the noise approximates white noise and its am
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_instrumentation_adc_noise_measured.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_adc_noise_measured.png}
\caption{\label{fig:detail_instrumentation_adc_noise_measured}Measured ADC noise (IO318)}
\end{figure}
\paragraph{Reading of piezoelectric force sensor}
@@ -9122,7 +9093,7 @@ With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu F\
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_step_response_force_sensor.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_step_response_force_sensor.png}
\end{center}
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor}Measured Signals}
\end{subfigure}
@@ -9150,7 +9121,7 @@ These results validate both the model of the ADC and the effectiveness of the ad
\end{subfigure}
\begin{subfigure}{0.35\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_step_response_force_sensor_R.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_step_response_force_sensor_R.png}
\end{center}
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor_R}Measured Signals}
\end{subfigure}
@@ -9178,7 +9149,7 @@ The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and th
\hfill
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/detail_instrumentation_femto_input_noise.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_femto_input_noise.png}
\captionof{figure}{\label{fig:detail_instrumentation_femto_input_noise}Obtained ASD of the instrumentation amplifier input voltage noise}
\end{center}
\end{minipage}
@@ -9209,13 +9180,13 @@ The observed frequency response function corresponds to exactly one sample delay
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_dac_output_noise.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_dac_output_noise.png}
\end{center}
\subcaption{\label{fig:detail_instrumentation_dac_output_noise}Output noise of the DAC}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_dac_adc_tf.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_dac_adc_tf.png}
\end{center}
\subcaption{\label{fig:detail_instrumentation_dac_adc_tf}Transfer function from DAC to ADC}
\end{subfigure}
@@ -9250,7 +9221,7 @@ While the exact cause of these peaks is not fully understood, their amplitudes r
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_instrumentation_pd200_noise.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_pd200_noise.png}
\caption{\label{fig:detail_instrumentation_pd200_noise}Measured output voltage noise of the PD200 amplifiers}
\end{figure}
\paragraph{Small Signal Bandwidth}
@@ -9267,7 +9238,7 @@ The identified dynamics shown in Figure~\ref{fig:detail_instrumentation_pd200_tf
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_instrumentation_pd200_tf.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_pd200_tf.png}
\caption{\label{fig:detail_instrumentation_pd200_tf}Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier}
\end{figure}
\subsubsection{Linear Encoders}
@@ -9290,8 +9261,8 @@ The noise profile exhibits characteristics of white noise with an amplitude of a
\hfill
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_vionic_asd.png}
-\captionof{figure}{\label{fig:detail_instrumentation_vionic_asd}Measured Amplitude Spectral Density of the encoder noise}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_vionic_asd.png}
+\captionof{figure}{\label{fig:detail_instrumentation_vionic_asd}Measured encoder noise ASD}
\end{center}
\end{minipage}
\subsubsection{Noise budgeting from measured instrumentation noise}
@@ -9305,7 +9276,7 @@ This confirms that the selected instrumentation, with its measured noise charact
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/detail_instrumentation_cl_noise_budget.png}
+\includegraphics[scale=1,scale=0.8]{figs/detail_instrumentation_cl_noise_budget.png}
\caption{\label{fig:detail_instrumentation_cl_noise_budget}Closed-loop noise budgeting using measured noise of instrumentation}
\end{figure}
\subsection*{Conclusion}
@@ -9435,15 +9406,15 @@ The measured flatness values, summarized in Table~\ref{tab:test_apa_flatness_mea
\begin{minipage}[b]{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_flatness_setup.png}
+\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_flatness_setup.png}
\captionof{figure}{\label{fig:test_apa_flatness_setup}Measurement setup for flatness estimation}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\textwidth}
-\begin{center}
-\captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
-\begin{tabularx}{0.6\linewidth}{Xc}
+\centering
+{\footnotesize\sf
+\begin{tabularx}{0.5\linewidth}{Xc}
\toprule
& \textbf{Flatness} \([\mu m]\)\\
\midrule
@@ -9455,8 +9426,8 @@ APA 5 & 1.9\\
APA 6 & 7.1\\
APA 7 & 18.7\\
\bottomrule
-\end{tabularx}
-\end{center}
+\end{tabularx}}
+\captionof{table}{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
\end{minipage}
\subsubsection{Electrical Measurements}
\label{ssec:test_apa_electrical_measurements}
@@ -9474,15 +9445,15 @@ This may be because the manufacturer measures the capacitance with large signals
\begin{minipage}[b]{0.48\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_lcr_meter.jpg}
+\includegraphics[scale=1,width=0.8\linewidth]{figs/test_apa_lcr_meter.jpg}
\captionof{figure}{\label{fig:test_apa_lcr_meter}Used LCR meter}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\textwidth}
-\begin{center}
-\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in \(\mu F\)}
-\begin{tabularx}{0.95\linewidth}{lcc}
+\centering
+{\footnotesize\sf
+\begin{tabularx}{0.8\linewidth}{Xcc}
\toprule
& \textbf{Sensor Stack} & \textbf{Actuator Stacks}\\
\midrule
@@ -9494,8 +9465,8 @@ APA 5 & 4.90 & 9.66\\
APA 6 & 4.99 & 9.91\\
APA 7 & 4.85 & 9.85\\
\bottomrule
-\end{tabularx}
-\end{center}
+\end{tabularx}}
+\captionof{table}{\label{tab:test_apa_capacitance}Measured capacitance in $\mu F$}
\end{minipage}
\subsubsection{Stroke and Hysteresis Measurement}
\label{ssec:test_apa_stroke_measurements}
@@ -9507,7 +9478,7 @@ Note that the voltage is slowly varied as the displacement probe has a very low
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_stroke_bench.jpg}
+\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_stroke_bench.jpg}
\caption{\label{fig:test_apa_stroke_bench}Bench to measure the APA stroke}
\end{figure}
@@ -9525,17 +9496,17 @@ From now on, only the six remaining amplified piezoelectric actuators that behav
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_stroke_voltage.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_stroke_voltage.png}
\end{center}
\subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_stroke_hysteresis.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_stroke_hysteresis.png}
\end{center}
\subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
\end{subfigure}
-\caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML \subref{fig:test_apa_stroke_voltage}. Measured displacement as a function of applied voltage \subref{fig:test_apa_stroke_hysteresis}}
+\caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of applied voltage (\subref{fig:test_apa_stroke_hysteresis})}
\end{figure}
\subsubsection{Flexible Mode Measurement}
\label{ssec:test_apa_spurious_resonances}
@@ -9582,7 +9553,7 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
\end{center}
\subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
\end{subfigure}
-\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure the flexible modes of the APA300ML. For the bending in the \(X\) direction \subref{fig:test_apa_meas_setup_X_bending}, the impact point is at the back of the top measurement point. For the bending in the \(Y\) direction \subref{fig:test_apa_meas_setup_Y_bending}, the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
+\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure the flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
\end{figure}
The measured frequency response functions computed from the experimental setups of figures~\ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure~\ref{fig:test_apa_meas_freq_compare}.
@@ -9594,7 +9565,7 @@ Another explanation is the shape difference between the manufactured APA300ML an
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_apa_meas_freq_compare.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_freq_compare.png}
\caption{\label{fig:test_apa_meas_freq_compare}Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
\end{figure}
\subsection{Dynamical measurements}
@@ -9642,7 +9613,7 @@ This is the typical behavior expected from a PZT stack actuator, where the hyste
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_apa_meas_hysteresis.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_hysteresis.png}
\caption{\label{fig:test_apa_meas_hysteresis}Displacement as a function of applied voltage for multiple excitation amplitudes}
\end{figure}
\subsubsection{Axial stiffness}
@@ -9664,14 +9635,14 @@ These estimated stiffnesses are summarized in Table~\ref{tab:test_apa_measured_s
\begin{minipage}[b]{0.57\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_apa_meas_stiffness_time.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_meas_stiffness_time.png}
\captionof{figure}{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding (at \(t \approx 3\,s\)) and removing (at \(t \approx 13\,s\)) the mass}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{0.37\textwidth}
-\begin{center}
-\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses (in \(N/\mu m\))}
+\centering
+{\footnotesize\sf
\begin{tabularx}{0.6\linewidth}{Xcc}
\toprule
APA & \(k_1\) & \(k_2\)\\
@@ -9683,8 +9654,8 @@ APA & \(k_1\) & \(k_2\)\\
6 & 1.7 & 1.92\\
8 & 1.73 & 1.98\\
\bottomrule
-\end{tabularx}
-\end{center}
+\end{tabularx}}
+\captionof{table}{\label{tab:test_apa_measured_stiffnesses}Measured axial stiffnesses in $N/\mu m$}
\end{minipage}
The stiffness can also be computed using equation~\eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 95\,\text{Hz}\) (estimated by the dynamical measurements shown in section~\ref{ssec:test_apa_meas_dynamics}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).
@@ -9737,17 +9708,17 @@ All the identified dynamics of the six APA300ML (both when looking at the encode
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_frf_encoder.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_frf_encoder.png}
\end{center}
\subcaption{\label{fig:test_apa_frf_encoder}FRF from $u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_frf_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_frf_force.png}
\end{center}
\subcaption{\label{fig:test_apa_frf_force}FRF from $u$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:test_apa_frf_dynamics}Measured frequency response function from generated voltage \(u\) to the encoder displacement \(d_e\) \subref{fig:test_apa_frf_encoder} and to the force sensor voltage \(V_s\) \subref{fig:test_apa_frf_force} for the six APA300ML}
+\caption{\label{fig:test_apa_frf_dynamics}Measured frequency response function from generated voltage \(u\) to the encoder displacement \(d_e\) (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage \(V_s\) (\subref{fig:test_apa_frf_force}) for the six APA300ML}
\end{figure}
\subsubsection{Non Minimum Phase Zero?}
\label{ssec:test_apa_non_minimum_phase}
@@ -9765,17 +9736,17 @@ However, this is not so important here because the zero is lightly damped (i.e.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_non_minimum_phase_coherence.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_non_minimum_phase_coherence.png}
\end{center}
\subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_non_minimum_phase_zoom.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_non_minimum_phase_zoom.png}
\end{center}
\subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
\end{subfigure}
-\caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found in the transfer function from \(u\) to \(V_s\). The coherence \subref{fig:test_apa_non_minimum_phase_coherence} is quite good around the anti-resonance frequency. The phase \subref{fig:test_apa_non_minimum_phase_zoom} shoes a non-minimum phase behavior.}
+\caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found in the transfer function from \(u\) to \(V_s\). The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.}
\end{figure}
\subsubsection{Effect of the resistor on the IFF Plant}
\label{ssec:test_apa_resistance_sensor_stack}
@@ -9789,7 +9760,7 @@ It is confirmed that the added resistor has the effect of adding a high-pass fil
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_apa_effect_resistance.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_effect_resistance.png}
\caption{\label{fig:test_apa_effect_resistance}Transfer function from \(u\) to \(V_s\) with and without the resistor \(R\) in parallel with the piezoelectric stack used as the force sensor}
\end{figure}
\subsubsection{Integral Force Feedback}
@@ -9806,7 +9777,7 @@ A comparison between the identified plant and the manually tuned transfer functi
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_apa_iff_plant_comp_manual_fit.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_iff_plant_comp_manual_fit.png}
\caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\mu s\) is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.}
\end{figure}
@@ -9838,17 +9809,17 @@ The two obtained root loci are compared in Figure~\ref{fig:test_apa_iff_root_loc
\begin{figure}[htbp]
\begin{subfigure}{0.59\textwidth}
\begin{center}
-\includegraphics[scale=1,height=8cm]{figs/test_apa_identified_damped_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_identified_damped_plants.png}
\end{center}
\subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants that match the experimental data (dashed lines)}
\end{subfigure}
\begin{subfigure}{0.39\textwidth}
\begin{center}
-\includegraphics[scale=1,height=8cm]{figs/test_apa_iff_root_locus.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:test_apa_iff_root_locus}Root Locus plot using the plant model (black) and poles of the identified damped plants (color crosses)}
\end{subfigure}
-\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant \subref{fig:test_apa_identified_damped_plants} and Root Locus \subref{fig:test_apa_iff_root_locus} corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}}
+\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}}
\end{figure}
\subsection{APA300ML - 2 degrees-of-freedom Model}
\label{sec:test_apa_model_2dof}
@@ -9860,7 +9831,7 @@ After the model is presented, the procedure for tuning the model is described, a
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=0.8\linewidth]{figs/test_apa_bench_model.png}
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_bench_model.png}
\caption{\label{fig:test_apa_bench_model}Screenshot of the multi-body model}
\end{figure}
\paragraph{Two degrees-of-freedom APA Model}
@@ -9926,7 +9897,7 @@ The obtained parameters of the model shown in Figure~\ref{fig:test_apa_2dof_mode
\begin{table}[htbp]
\caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
\centering
-\begin{tabularx}{0.3\linewidth}{cc}
+\begin{tabularx}{0.25\linewidth}{cc}
\toprule
\textbf{Parameter} & \textbf{Value}\\
\midrule
@@ -9950,17 +9921,17 @@ This indicates that this model represents well the axial dynamics of the APA300M
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_2dof_comp_frf_enc.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_2dof_comp_frf_enc.png}
\end{center}
\subcaption{\label{fig:test_apa_2dof_comp_frf_enc}from $u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_2dof_comp_frf_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_2dof_comp_frf_force.png}
\end{center}
\subcaption{\label{fig:test_apa_2dof_comp_frf_force}from $u$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) \subref{fig:test_apa_2dof_comp_frf_enc} \subref{fig:test_apa_2dof_comp_frf_force} and from \(u\) to \(V_s\) \subref{fig:test_apa_2dof_comp_frf_force}}
+\caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
\end{figure}
\subsection{APA300ML - Super Element}
\label{sec:test_apa_model_flexible}
@@ -10006,7 +9977,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\begin{table}[htbp]
\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities}
\centering
-\begin{tabularx}{1\linewidth}{ccX}
+\begin{tabularx}{0.8\linewidth}{ccX}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Description}\\
\midrule
@@ -10031,23 +10002,22 @@ Using this simple test bench, it can be concluded that the \emph{super element}
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_super_element_comp_frf_enc.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_super_element_comp_frf_enc.png}
\end{center}
\subcaption{\label{fig:test_apa_super_element_comp_frf_enc}from $u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_apa_super_element_comp_frf_force.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_apa_super_element_comp_frf_force.png}
\end{center}
\subcaption{\label{fig:test_apa_super_element_comp_frf_force}from $u$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) \subref{fig:test_apa_super_element_comp_frf_enc} and from \(u\) to \(V_s\) \subref{fig:test_apa_super_element_comp_frf_force}}
+\caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_super_element_comp_frf_enc}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_super_element_comp_frf_force})}
\end{figure}
\subsection*{Conclusion}
\label{sec:test_apa_conclusion}
In this study, the amplified piezoelectric actuators ``APA300ML'' have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase.
-
Geometrical features such as the flatness of its interfaces, electrical capacitance, and achievable strokes were measured in Section~\ref{sec:test_apa_basic_meas}.
These simple measurements allowed for the early detection of a manufacturing defect in one of the APA300ML.
@@ -10076,7 +10046,7 @@ During the detailed design phase, specifications in terms of stiffness and strok
\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}{0.5\linewidth}{Xcc}
+\begin{tabularx}{0.4\linewidth}{Xcc}
\toprule
& \textbf{Specification} & \textbf{FEM}\\
\midrule
@@ -10171,7 +10141,7 @@ It is then possible to estimate the dimension of the flexible beam with an accur
\end{center}
\subcaption{\label{fig:test_joints_profilometer_image}Picture of the gap}
\end{subfigure}
-\caption{\label{fig:test_joints_profilometer}Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer \subref{fig:test_joints_profilometer_setup} and a image is obtained with which the gap can be estimated \subref{fig:test_joints_profilometer_image}}
+\caption{\label{fig:test_joints_profilometer}Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness. The flexible joint is fixed to the profilometer (\subref{fig:test_joints_profilometer_setup}) and a image is obtained with which the gap can be estimated (\subref{fig:test_joints_profilometer_image})}
\end{figure}
\subsubsection{Measurement Results}
The specified flexible beam thickness (gap) is \(250\,\mu m\).
@@ -10185,7 +10155,7 @@ However, what is more important than the true value of the thickness is the cons
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_joints_size_hist.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_size_hist.png}
\caption{\label{fig:test_joints_size_hist}Histogram for the (16x2) measured beams' thicknesses}
\end{figure}
\subsubsection{Bad flexible joints}
@@ -10358,7 +10328,7 @@ An overall accuracy of \(\approx 5\,\%\) can be expected with this measurement b
\begin{table}[htbp]
\caption{\label{tab:test_joints_error_budget}Summary of the error budget for estimating the bending stiffness}
\centering
-\begin{tabularx}{0.4\linewidth}{lX}
+\begin{tabularx}{0.35\linewidth}{Xc}
\toprule
\textbf{Effect} & \textbf{Error}\\
\midrule
@@ -10400,7 +10370,7 @@ The obtained CAD design of the measurement bench is shown in Figure~\ref{fig:tes
\end{center}
\subcaption{\label{fig:test_joints_bench_side} Zoom}
\end{subfigure}
-\caption{\label{fig:test_joints_bench}CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in \subref{fig:test_joints_bench_overview} while a zoom on the flexible joint is shown in \subref{fig:test_joints_bench_side}}
+\caption{\label{fig:test_joints_bench}CAD view of the test bench developed to measure the bending stiffness of the flexible joints. Different parts are shown in (\subref{fig:test_joints_bench_overview}) while a zoom on the flexible joint is shown in (\subref{fig:test_joints_bench_side})}
\end{figure}
\subsection{Bending Stiffness Measurement}
\label{sec:test_joints_bending_stiffness_meas}
@@ -10431,20 +10401,20 @@ The measured forces are compared in Figure~\ref{fig:test_joints_force_sensor_cal
The gain mismatch between the two load cells is approximately \(4\,\%\) which is higher than that specified in the data sheets.
However, the estimated non-linearity is bellow \(0.2\,\%\) for forces between \(1\,N\) and \(5\,N\).
-\begin{figure}[htbp]
+\begin{figure}[h!tbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.5cm]{figs/test_joints_force_sensor_calib_picture.png}
+\includegraphics[scale=1,height=5cm]{figs/test_joints_force_sensor_calib_picture.png}
\end{center}
\subcaption{\label{fig:test_joints_force_sensor_calib_picture}Zoom on the two load cells in contact}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.5cm]{figs/test_joints_force_sensor_calib_fit.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_force_sensor_calib_fit.png}
\end{center}
\subcaption{\label{fig:test_joints_force_sensor_calib_fit}Measured two forces}
\end{subfigure}
-\caption{\label{fig:test_joints_force_sensor_calib}Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in \subref{fig:test_joints_force_sensor_calib_picture}. Comparison of the two measured forces and estimated non-linearity are shown in \subref{fig:test_joints_force_sensor_calib_fit}}
+\caption{\label{fig:test_joints_force_sensor_calib}Estimation of the load cell accuracy by comparing the measured force of two load cells. A picture of the measurement bench is shown in (\subref{fig:test_joints_force_sensor_calib_picture}). Comparison of the two measured forces and estimated non-linearity are shown in (\subref{fig:test_joints_force_sensor_calib_fit})}
\end{figure}
\subsubsection{Load Cell Stiffness}
The objective of this measurement is to estimate the stiffness \(k_F\) of the force sensor.
@@ -10456,17 +10426,17 @@ The load cell stiffness can then be estimated by computing a linear fit and is f
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.5cm]{figs/test_joints_meas_force_sensor_stiffness_picture.jpg}
+\includegraphics[scale=1,height=5cm]{figs/test_joints_meas_force_sensor_stiffness_picture.jpg}
\end{center}
\subcaption{\label{fig:test_joints_meas_force_sensor_stiffness_picture}Picture of the measurement bench}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.5cm]{figs/test_joints_force_sensor_stiffness_fit.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_force_sensor_stiffness_fit.png}
\end{center}
\subcaption{\label{fig:test_joints_force_sensor_stiffness_fit}Measured displacement as a function of the force}
\end{subfigure}
-\caption{\label{fig:test_joints_meas_force_sensor_stiffness}Estimation of the load cell stiffness. The measurement setup is shown in \subref{fig:test_joints_meas_force_sensor_stiffness_picture}. The measurement results are shown in \subref{fig:test_joints_force_sensor_stiffness_fit}.}
+\caption{\label{fig:test_joints_meas_force_sensor_stiffness}Estimation of the load cell stiffness. Measurement setup is shown in (\subref{fig:test_joints_meas_force_sensor_stiffness_picture}), and results are shown in (\subref{fig:test_joints_force_sensor_stiffness_fit}).}
\end{figure}
\subsubsection{Bending Stiffness estimation}
The actual stiffness is now estimated by manually moving the translation stage from a start position where the force sensor is not yet in contact with the flexible joint to a position where the flexible joint is on its mechanical stop.
@@ -10481,17 +10451,17 @@ The bending stroke can also be estimated as shown in Figure~\ref{fig:test_joints
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.3cm]{figs/test_joints_meas_bend_time.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_meas_bend_time.png}
\end{center}
\subcaption{\label{fig:test_joints_meas_bend_time}Force and displacement measured as a function of time}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.3cm]{figs/test_joints_meas_F_d_lin_fit.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_meas_F_d_lin_fit.png}
\end{center}
\subcaption{\label{fig:test_joints_meas_F_d_lin_fit}Angular displacement measured as a function of the applied torque}
\end{subfigure}
-\caption{\label{fig:test_joints_meas_example}Results obtained on the first flexible joint. The measured force and displacement are shown in \subref{fig:test_joints_meas_bend_time}. The estimated angular displacement \(\theta_x\) as a function of the estimated applied torque \(T_{x}\) is shown in \subref{fig:test_joints_meas_F_d_lin_fit}. The bending stiffness \(k_{R_x}\) of the flexible joint can be estimated by computing a best linear fit (red dashed line).}
+\caption{\label{fig:test_joints_meas_example}Results obtained on the first flexible joint. The measured force and displacement are shown in (\subref{fig:test_joints_meas_bend_time}). The estimated angular displacement \(\theta_x\) as a function of the estimated applied torque \(T_{x}\) is shown in (\subref{fig:test_joints_meas_F_d_lin_fit}). The bending stiffness \(k_{R_x}\) of the flexible joint can be estimated by computing a best linear fit (red dashed line).}
\end{figure}
\subsubsection{Measured flexible joint stiffness}
@@ -10505,17 +10475,17 @@ Most of the bending stiffnesses are between \(4.6\,Nm/rad\) and \(5.0\,Nm/rad\).
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.3cm]{figs/test_joints_meas_bending_all_raw_data.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_meas_bending_all_raw_data.png}
\end{center}
\subcaption{\label{fig:test_joints_meas_bending_all_raw_data}Measured torque and angular motion for the flexible joints}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,height=5.3cm]{figs/test_joints_bend_stiff_hist.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_joints_bend_stiff_hist.png}
\end{center}
\subcaption{\label{fig:test_joints_bend_stiff_hist}Histogram of the measured bending stiffness in the x and y directions}
\end{subfigure}
-\caption{\label{fig:test_joints_meas_bending_results}Result of measured \(k_{R_x}\) and \(k_{R_y}\) stiffnesses for the 16 flexible joints. Raw data are shown in \subref{fig:test_joints_meas_bending_all_raw_data}. A histogram of the measured stiffnesses is shown in \subref{fig:test_joints_bend_stiff_hist}}
+\caption{\label{fig:test_joints_meas_bending_results}Result of measured \(k_{R_x}\) and \(k_{R_y}\) stiffnesses for the 16 flexible joints. Raw data are shown in (\subref{fig:test_joints_meas_bending_all_raw_data}). A histogram of the measured stiffnesses is shown in (\subref{fig:test_joints_bend_stiff_hist}).}
\end{figure}
\subsection*{Conclusion}
\label{sec:test_joints_conclusion}
@@ -10735,13 +10705,13 @@ The obtained frequency response functions for the three configurations (X-bendin
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_spur_res_frf_no_enc.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_spur_res_frf_no_enc.png}
\end{center}
\subcaption{\label{fig:test_struts_spur_res_frf_no_enc}without encoder}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_spur_res_frf_enc.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_spur_res_frf_enc.png}
\end{center}
\subcaption{\label{fig:test_struts_spur_res_frf_enc}with the encoder}
\end{subfigure}
@@ -10811,7 +10781,7 @@ System identification was performed without the encoder being fixed to the strut
\end{center}
\subcaption{\label{fig:test_struts_bench_leg_front}Strut without encoder}
\end{subfigure}
-\caption{\label{fig:test_struts_bench_leg_with_without_enc}Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts \subref{fig:test_struts_bench_leg_coder} or removed \subref{fig:test_struts_bench_leg_front}}
+\caption{\label{fig:test_struts_bench_leg_with_without_enc}Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})}
\end{figure}
The obtained frequency response functions are compared in Figure~\ref{fig:test_struts_effect_encoder}.
@@ -10823,23 +10793,23 @@ This means that the encoder should have little effect on the effectiveness of th
\begin{figure}[htbp]
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_encoder_int.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_effect_encoder_int.png}
\end{center}
\subcaption{\label{fig:test_struts_effect_encoder_int}$u$ to $d_a$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_encoder_iff.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_effect_encoder_iff.png}
\end{center}
\subcaption{\label{fig:test_struts_effect_encoder_iff}$u$ to $V_s$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_comp_enc_int.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_enc_int.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_enc_int}$u$ to $d_e$, $d_a$}
\end{subfigure}
-\caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) \subref{fig:test_struts_effect_encoder_int} and from \(u\) to \(V_s\) \subref{fig:test_struts_effect_encoder_iff}. Comparison of the observed dynamics by the encoder and interferometers \subref{fig:test_struts_comp_enc_int}}
+\caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) (\subref{fig:test_struts_effect_encoder_int}) and from \(u\) to \(V_s\) (\subref{fig:test_struts_effect_encoder_iff}). Comparison of the observed dynamics by the encoder and interferometers (\subref{fig:test_struts_comp_enc_int})}
\end{figure}
\subsubsection{Comparison of the encoder and interferometer}
\label{ssec:test_struts_comp_enc_int}
@@ -10861,19 +10831,19 @@ A very good match can be observed between the struts.
\begin{figure}[htbp]
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_comp_interf_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_interf_plants.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_comp_iff_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_iff_plants.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_comp_enc_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_enc_plants.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$}
\end{subfigure}
@@ -10917,19 +10887,19 @@ For the flexible model, it will be shown in the next section that by adding some
\begin{figure}[htbp]
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_comp_frf_flexible_model_int.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_frf_flexible_model_int.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_frf_flexible_model_int}$u$ to $d_a$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_comp_frf_flexible_model_enc.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_frf_flexible_model_enc.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_frf_flexible_model_enc}$u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_comp_frf_flexible_model_iff.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_frf_flexible_model_iff.png}
\end{center}
\subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$}
\end{subfigure}
@@ -10972,17 +10942,17 @@ This similarity suggests that the identified differences in dynamics are caused
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_misalignment_y.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_effect_misalignment_y.png}
\end{center}
\subcaption{\label{fig:test_struts_effect_misalignment_y}Misalignment along $y$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_misalignment_x.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_effect_misalignment_x.png}
\end{center}
\subcaption{\label{fig:test_struts_effect_misalignment_x}Misalignment along $x$}
\end{subfigure}
-\caption{\label{fig:test_struts_effect_misalignment}Effect of a misalignment between the flexible joints and the APA300ML in the \(y\) direction \subref{fig:test_struts_effect_misalignment_y} and in the \(x\) direction \subref{fig:test_struts_effect_misalignment_x}}
+\caption{\label{fig:test_struts_effect_misalignment}Effect of a misalignment between the flexible joints and the APA300ML in the \(y\) direction (\subref{fig:test_struts_effect_misalignment_y}) and in the \(x\) direction (\subref{fig:test_struts_effect_misalignment_x})}
\end{figure}
\subsubsection{Measured strut misalignment}
\label{ssec:test_struts_meas_misalignment}
@@ -11005,7 +10975,7 @@ Thickness differences for all the struts were found to be between \(0.94\,mm\) a
\begin{table}[htbp]
\caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment at the top and bottom of the APA. Measurements are in \(mm\)}
\centering
-\begin{tabularx}{0.25\linewidth}{Xcc}
+\begin{tabularx}{0.2\linewidth}{Xcc}
\toprule
\textbf{Strut} & \textbf{Bot} & \textbf{Top}\\
\midrule
@@ -11029,7 +10999,7 @@ With a better alignment, the amplitude of the spurious resonances is expected to
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_struts_comp_dy_tuned_model_frf_enc.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_dy_tuned_model_frf_enc.png}
\caption{\label{fig:test_struts_comp_dy_tuned_model_frf_enc}Comparison of the frequency response functions from DAC voltage \(u\) to measured displacement \(d_e\) by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the \(y\) misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the \(x\) and \(y\) misalignments are tuned}
\end{figure}
\subsubsection{Proper struts alignment}
@@ -11068,7 +11038,7 @@ Therefore, fixing the encoders to the nano-hexapod plates instead may be an inte
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_struts_comp_enc_frf_realign.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_struts_comp_enc_frf_realign.png}
\caption{\label{fig:test_struts_comp_enc_frf_realign}Comparison of the dynamics from \(u\) to \(d_e\) before and after proper alignment using the dowel pins}
\end{figure}
\subsection*{Conclusion}
@@ -11110,7 +11080,7 @@ To do so, a precisely machined mounting tool (Figure~\ref{fig:test_nhexa_center_
\end{center}
\subcaption{\label{fig:test_nhexa_center_part_hexapod_mounting}Mounting tool}
\end{subfigure}
-\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}}
+\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 \(\pm 13\mu m\)} (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.}.
@@ -11131,7 +11101,7 @@ The main goal of this ``mounting tool'' is to position the flexible joint interf
\end{center}
\subcaption{\label{fig:test_nhexa_mounting_tool_hexapod_top_view}Wanted coaxiality between interfaces}
\end{subfigure}
-\caption{\label{fig:test_nhexa_dimensional_check}A FARO arm is used to dimensionally check the received parts \subref{fig:test_nhexa_plates_tolerances} and after mounting the two plates with the mounting part \subref{fig:test_nhexa_mounting_tool_hexapod_top_view}}
+\caption{\label{fig:test_nhexa_dimensional_check}A FARO arm is used to dimensionally check the received parts (\subref{fig:test_nhexa_plates_tolerances}) and after mounting the two plates with the mounting part (\subref{fig:test_nhexa_mounting_tool_hexapod_top_view})}
\end{figure}
The quality of the positioning can be estimated by measuring the ``straightness'' of the top and bottom ``V'' interfaces.
@@ -11142,7 +11112,7 @@ The straightness was found to be better than \(15\,\mu m\) for all struts\footno
\begin{table}[htbp]
\caption{\label{tab:measured_straightness}Measured straightness between the two ``V'' shapes for the six struts. These measurements were performed twice for each strut.}
\centering
-\begin{tabularx}{0.3\linewidth}{Xcc}
+\begin{tabularx}{0.25\linewidth}{Xcc}
\toprule
\textbf{Strut} & \textbf{Meas 1} & \textbf{Meas 2}\\
\midrule
@@ -11171,7 +11141,7 @@ The encoder rulers and heads were then fixed to the top and bottom plates, respe
\end{center}
\subcaption{\label{fig:test_nhexa_mount_encoder_heads}Encoder heads}
\end{subfigure}
-\caption{\label{fig:test_nhexa_mount_encoder}Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate \subref{fig:test_nhexa_mount_encoder_rulers} while encoders heads are fixed to the bottom plate \subref{fig:test_nhexa_mount_encoder_heads}}
+\caption{\label{fig:test_nhexa_mount_encoder}Mounting of the encoders to the Nano-hexapod. The rulers are fixed to the top plate (\subref{fig:test_nhexa_mount_encoder_rulers}) while encoders heads are fixed to the bottom plate (\subref{fig:test_nhexa_mount_encoder_heads})}
\end{figure}
The six struts were then fixed to the bottom and top plates one by one.
@@ -11230,10 +11200,9 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\textwidth}
-\begin{scriptsize}
-\begin{center}
-\captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table}
-\begin{tabularx}{0.8\linewidth}{clX}
+\centering
+{\footnotesize\sf
+\begin{tabularx}{0.9\linewidth}{clX}
\toprule
\textbf{Modes} & \textbf{Frequency} & \textbf{Description}\\
\midrule
@@ -11246,9 +11215,8 @@ The next modes are the flexible modes of the breadboard as shown in Figure~\ref{
8 & 989 Hz & Complex mode\\
9 & 1025 Hz & Complex mode\\
\bottomrule
-\end{tabularx}
-\end{center}
-\end{scriptsize}
+\end{tabularx}}
+\captionof{table}{\label{tab:test_nhexa_suspended_table_modes}Obtained modes of the suspended table}
\end{minipage}
\begin{figure}[htbp]
@@ -11317,7 +11285,7 @@ The effect of the payload mass on the dynamics is discussed in Section~\ref{ssec
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_nano_hexapod_signals.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_nhexa_nano_hexapod_signals.png}
\caption{\label{fig:test_nhexa_nano_hexapod_signals}Block diagram of the studied system. The command signal generated by the speedgoat is \(\mathbf{u}\), and the measured dignals are \(\mathbf{d}_{e}\) and \(\mathbf{V}_s\). Units are indicated in square brackets.}
\end{figure}
\subsubsection{Modal analysis}
@@ -11393,7 +11361,7 @@ This would not have occurred if the encoders were fixed to the struts.
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_identified_frf_de.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_de.png}
\caption{\label{fig:test_nhexa_identified_frf_de}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{d}_e\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.}
\end{figure}
@@ -11404,7 +11372,7 @@ The first flexible mode of the struts as 235Hz has large amplitude, and therefor
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1,width=\linewidth]{figs/test_nhexa_identified_frf_Vs.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_Vs.png}
\caption{\label{fig:test_nhexa_identified_frf_Vs}Measured FRF for the transfer function from \(\mathbf{u}\) to \(\mathbf{V}_s\). The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.}
\end{figure}
\subsubsection{Effect of payload mass on the dynamics}
@@ -11439,17 +11407,17 @@ For all tested payloads, the measured FRF always have alternating poles and zero
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_identified_frf_de_masses.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_de_masses.png}
\end{center}
\subcaption{\label{fig:test_nhexa_identified_frf_de_masses}$u_i$ to $d_{ei}$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_identified_frf_Vs_masses.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_identified_frf_Vs_masses.png}
\end{center}
\subcaption{\label{fig:test_nhexa_identified_frf_Vs_masses}$u_i$ to $V_{si}$}
\end{subfigure}
-\caption{\label{fig:test_nhexa_identified_frf_masses}Measured Frequency Response Functions from \(u_i\) to \(d_{ei}\) \subref{fig:test_nhexa_identified_frf_de_masses} and from \(u_i\) to \(V_{si}\) \subref{fig:test_nhexa_identified_frf_Vs_masses} for all 4 payload conditions. Only diagonal terms are shown.}
+\caption{\label{fig:test_nhexa_identified_frf_masses}Measured Frequency Response Functions from \(u_i\) to \(d_{ei}\) (\subref{fig:test_nhexa_identified_frf_de_masses}) and from \(u_i\) to \(V_{si}\) (\subref{fig:test_nhexa_identified_frf_Vs_masses}) for all 4 payload conditions. Only diagonal terms are shown.}
\end{figure}
\subsection{Nano-Hexapod Model Dynamics}
\label{sec:test_nhexa_model}
@@ -11485,17 +11453,17 @@ At higher frequencies, no resonances can be observed in the model, as the top pl
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_de_diag.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_diag.png}
\end{center}
\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag}from $u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_Vs_diag.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_Vs_diag.png}
\end{center}
\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag}from $u$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:test_nhexa_comp_simscape_diag}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(d_e\) \subref{fig:test_nhexa_comp_simscape_de_diag} and from \(u\) to \(V_s\) \subref{fig:test_nhexa_comp_simscape_Vs_diag}}
+\caption{\label{fig:test_nhexa_comp_simscape_diag}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})}
\end{figure}
\subsubsection{Dynamical coupling}
\label{ssec:test_nhexa_comp_model_coupling}
@@ -11507,7 +11475,7 @@ Similar results are observed for all other coupling terms and for the transfer f
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_all.png}
\caption{\label{fig:test_nhexa_comp_simscape_de_all}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\). The APA are here modeled with a 2-DoF mass-spring-damper system.}
\end{figure}
@@ -11519,7 +11487,7 @@ Therefore, if the modes of the struts are to be modeled, the \emph{super-element
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all_flex.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_all_flex.png}
\caption{\label{fig:test_nhexa_comp_simscape_de_all_flex}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\). The APA are here modeled with a ``super-element''.}
\end{figure}
\subsubsection{Effect of payload mass}
@@ -11537,17 +11505,17 @@ However, as decentralized IFF will be applied, the damping is actively brought,
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_de_diag_masses.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_diag_masses.png}
\end{center}
\subcaption{\label{fig:test_nhexa_comp_simscape_de_diag_masses}from $u$ to $d_e$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_nhexa_comp_simscape_Vs_diag_masses.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_Vs_diag_masses.png}
\end{center}
\subcaption{\label{fig:test_nhexa_comp_simscape_Vs_diag_masses}from $u$ to $V_s$}
\end{subfigure}
-\caption{\label{fig:test_nhexa_comp_simscape_diag_masses}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(d_e\) \subref{fig:test_nhexa_comp_simscape_de_diag} and from \(u\) to \(V_s\) \subref{fig:test_nhexa_comp_simscape_Vs_diag}}
+\caption{\label{fig:test_nhexa_comp_simscape_diag_masses}Comparison of the diagonal elements (i.e. ``direct'' terms) of the measured FRF matrix and the dynamics identified from the multi-body model. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_nhexa_comp_simscape_de_diag}) and from \(u\) to \(V_s\) (\subref{fig:test_nhexa_comp_simscape_Vs_diag})}
\end{figure}
In order to also check if the model well represents the coupling when high payload masses are used, the transfer functions from \(u_1\) to \(d_{e1}\) to \(d_{e6}\) are compared in the case of the 39kg payload in Figure~\ref{fig:test_nhexa_comp_simscape_de_all_high_mass}.
@@ -11556,7 +11524,7 @@ Therefore, the model effectively represents the system coupling for different pa
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_nhexa_comp_simscape_de_all_high_mass.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_nhexa_comp_simscape_de_all_high_mass.png}
\caption{\label{fig:test_nhexa_comp_simscape_de_all_high_mass}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six encoders \(d_{e1}\) to \(d_{e6}\)}
\end{figure}
\subsection*{Conclusion}
@@ -11752,13 +11720,13 @@ The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\)
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_metrology_align_rx_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_metrology_align_rx_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_metrology_align_rx_ry}Angular alignment}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_metrology_align_dx_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_metrology_align_dx_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_metrology_align_dx_dy}Lateral alignment}
\end{subfigure}
@@ -11776,7 +11744,7 @@ The obtained lateral acceptance for pure displacements in any direction is estim
\begin{table}[htbp]
\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
\centering
-\begin{tabularx}{0.45\linewidth}{Xccc}
+\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
& \(D_x\) & \(D_y\) & \(D_z\)\\
\midrule
@@ -11814,13 +11782,13 @@ The effect of noise on the translation and rotation measurements is estimated in
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_xy_map_sphere.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_xy_map_sphere.png}
\end{center}
\subcaption{\label{fig:test_id31_xy_map_sphere}Z measurement during an XY mapping}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_interf_noise.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_interf_noise.png}
\end{center}
\subcaption{\label{fig:test_id31_interf_noise}Interferometer noise}
\end{subfigure}
@@ -11844,7 +11812,7 @@ Voltages generated by the force sensor piezoelectric stacks \(\bm{V}_s = [V_{s1}
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_block_schematic_plant.png}
+\includegraphics[scale=1,scale=0.9]{figs/test_id31_block_schematic_plant.png}
\caption{\label{fig:test_id31_block_schematic_plant}Schematic of the NASS plant}
\end{figure}
\subsubsection{Open-Loop Plant Identification}
@@ -11864,13 +11832,13 @@ This issue was later solved.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_first_id_int.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_first_id_int.png}
\end{center}
\subcaption{\label{fig:test_id31_first_id_int}External Metrology}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_first_id_iff.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_first_id_iff.png}
\end{center}
\subcaption{\label{fig:test_id31_first_id_iff}Force Sensors}
\end{subfigure}
@@ -11891,13 +11859,13 @@ Results shown in Figure~\ref{fig:test_id31_Rz_align_correct} are indeed indicati
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_Rz_align_error.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_Rz_align_error.png}
\end{center}
\subcaption{\label{fig:test_id31_Rz_align_error}Before alignment}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_Rz_align_correct.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_Rz_align_correct.png}
\end{center}
\subcaption{\label{fig:test_id31_Rz_align_correct}After alignment}
\end{subfigure}
@@ -11912,7 +11880,7 @@ The flexible modes of the top platform can be passively damped, whereas the mode
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_first_id_int_better_rz_align.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_first_id_int_better_rz_align.png}
\caption{\label{fig:test_id31_first_id_int_better_rz_align}Decrease of the coupling with better Rz alignment}
\end{figure}
\subsubsection{Effect of Payload Mass}
@@ -11928,25 +11896,25 @@ It is interesting to note that the anti-resonances in the force sensor plant now
\begin{figure}[htbp]
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/test_id31_picture_mass_m0.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_picture_mass_m0.jpg}
\end{center}
\subcaption{\label{fig:test_id31_picture_mass_m0}$m=0\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/test_id31_picture_mass_m1.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_picture_mass_m1.jpg}
\end{center}
\subcaption{\label{fig:test_id31_picture_mass_m1}$m=13\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/test_id31_picture_mass_m2.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_picture_mass_m2.jpg}
\end{center}
\subcaption{\label{fig:test_id31_picture_mass_m2}$m=26\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.99\linewidth]{figs/test_id31_picture_mass_m3.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_picture_mass_m3.jpg}
\end{center}
\subcaption{\label{fig:test_id31_picture_mass_m3}$m=39\,\text{kg}$}
\end{subfigure}
@@ -11982,13 +11950,13 @@ This also indicates that the metrology kinematics is correct and is working in r
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_effect_rotation_direct.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_effect_rotation_direct.png}
\end{center}
\subcaption{\label{fig:test_id31_effect_rotation_direct}Direct terms}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_effect_rotation_coupling.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_effect_rotation_coupling.png}
\end{center}
\subcaption{\label{fig:test_id31_effect_rotation_coupling}Coupling terms}
\end{subfigure}
@@ -12031,7 +11999,7 @@ This confirms that the multi-body model can be used to tune the IFF controller.
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_comp_simscape_Vs.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_comp_simscape_Vs.png}
\caption{\label{fig:test_id31_comp_simscape_Vs}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal \(u_1\) to the six force sensor voltages \(V_{s1}\) to \(V_{s6}\)}
\end{figure}
\subsubsection{IFF Controller}
@@ -12051,13 +12019,13 @@ It can be seen that the loop-gain is larger than \(1\) around the suspension mod
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_Kiff_bode_plot.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_Kiff_bode_plot.png}
\end{center}
\subcaption{\label{fig:test_id31_Kiff_bode_plot}Bode plot of $K_{\text{IFF}}$}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_Kiff_loop_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_Kiff_loop_gain.png}
\end{center}
\subcaption{\label{fig:test_id31_Kiff_loop_gain}Decentralized Loop gains}
\end{subfigure}
@@ -12074,25 +12042,25 @@ However, in this study, it was chosen to implement a ``fixed'' (i.e. non-adaptiv
\begin{figure}[htbp]
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_id31_iff_root_locus_m0.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_iff_root_locus_m0.png}
\end{center}
\subcaption{\label{fig:test_id31_iff_root_locus_m0}$m = 0\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_id31_iff_root_locus_m1.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_iff_root_locus_m1.png}
\end{center}
\subcaption{\label{fig:test_id31_iff_root_locus_m1}$m = 13\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_id31_iff_root_locus_m2.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_iff_root_locus_m2.png}
\end{center}
\subcaption{\label{fig:test_id31_iff_root_locus_m2}$m = 26\,\text{kg}$}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/test_id31_iff_root_locus_m3.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_iff_root_locus_m3.png}
\end{center}
\subcaption{\label{fig:test_id31_iff_root_locus_m3}$m = 39\,\text{kg}$}
\end{subfigure}
@@ -12111,13 +12079,13 @@ The obtained frequency response functions are compared with the model in Figure~
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_comp_ol_iff_plant_model.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_comp_ol_iff_plant_model.png}
\end{center}
\subcaption{\label{fig:test_id31_comp_ol_iff_plant_model}Effect of IFF on the plant}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_hac_plant_effect_mass.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_plant_effect_mass.png}
\end{center}
\subcaption{\label{fig:test_id31_hac_plant_effect_mass}Comparison of model and experimental results}
\end{subfigure}
@@ -12157,7 +12125,7 @@ Considering the complexity of the system's dynamics, the model can be considered
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_comp_simscape_hac.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_comp_simscape_hac.png}
\caption{\label{fig:test_id31_comp_simscape_hac}Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal (\(u_1^\prime\)) of the damped plant to the estimated errors (\(\epsilon_{\mathcal{L}_i}\)) in the frame of the six struts by the external metrology}
\end{figure}
@@ -12168,7 +12136,7 @@ This is one of the key benefits of using the HAC-LAC strategy.
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_comp_all_undamped_damped_plants.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_comp_all_undamped_damped_plants.png}
\caption{\label{fig:test_id31_comp_all_undamped_damped_plants}Comparison of the (six) direct terms for all (four) payload conditions in the undamped case (in blue) and the damped case (i.e. with the decentralized IFF being implemented, in red).}
\end{figure}
\subsubsection{Interaction Analysis}
@@ -12196,7 +12164,7 @@ This design choice, while beneficial for system simplicity, introduces inherent
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_hac_rga_number.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_rga_number.png}
\caption{\label{fig:test_id31_hac_rga_number}RGA-number for the damped plants - Comparison of all the payload conditions}
\end{figure}
\subsubsection{Robust Controller Design}
@@ -12218,13 +12186,13 @@ However, small stability margins were observed for the highest mass, indicating
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_hac_loop_gain.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_loop_gain.png}
\end{center}
\subcaption{\label{fig:test_id31_hac_loop_gain}Loop Gains}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_id31_hac_characteristic_loci.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_characteristic_loci.png}
\end{center}
\subcaption{\label{fig:test_id31_hac_characteristic_loci}Characteristic Loci}
\end{subfigure}
@@ -12241,13 +12209,13 @@ The obtained closed-loop positioning accuracy was found to comply with the requi
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy.png}
\end{center}
\subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy}XY plane}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz.png}
\end{center}
\subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}YZ plane}
\end{subfigure}
@@ -12265,7 +12233,7 @@ However, it was decided that this controller should be tested experimentally and
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_hac_tomography_Wz36_simulation.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_tomography_Wz36_simulation.png}
\caption{\label{fig:test_id31_hac_tomography_Wz36_simulation}Positioning errors in the Y-Z plane during tomography experiments simulated using the multi-body model (in closed-loop)}
\end{figure}
\subsubsection*{Conclusion}
@@ -12308,7 +12276,7 @@ Results obtained for all experiments are summarized and compared to the specific
\begin{table}[htbp]
\caption{\label{tab:test_id31_experiments_specifications}Specifications for the Nano-Active-Stabilization-System}
\centering
-\begin{tabularx}{0.45\linewidth}{Xccc}
+\begin{tabularx}{0.4\linewidth}{Xccc}
\toprule
& \(D_y\) & \(D_z\) & \(R_y\)\\
\midrule
@@ -12333,13 +12301,13 @@ While this approach likely underestimates actual open-loop errors, as perfect al
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit.png}
\end{center}
\subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit}Errors in $(x,y)$ plane}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed.png}
\end{center}
\subcaption{\label{fig:test_id31_tomo_m2_1rpm_robust_hac_iff_fit_removed}Removed eccentricity}
\end{subfigure}
@@ -12352,7 +12320,7 @@ These experimental findings are consistent with the predictions from the tomogra
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_tomo_Wz36_results.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_Wz36_results.png}
\caption{\label{fig:test_id31_tomo_Wz36_results}Measured errors in the \(Y-Z\) plane during tomography experiments at \(6\,\text{deg/s}\) for all considered payloads. In the open-loop case, the effect of eccentricity is removed from the data.}
\end{figure}
\paragraph{Fast Tomography scans}
@@ -12365,13 +12333,13 @@ Nevertheless, even with this robust (i.e. conservative) HAC implementation, the
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_exp_xy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_exp_xy.png}
\end{center}
\subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_xy}XY plane}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz.png}
\end{center}
\subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz}YZ plane}
\end{subfigure}
@@ -12392,19 +12360,19 @@ This experiment also illustrates that when needed, performance can be enhanced b
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_hac_cas_cl_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_cas_cl_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_hac_cas_cl_dy} $D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_hac_cas_cl_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_cas_cl_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_hac_cas_cl_dz} $D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_hac_cas_cl_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_hac_cas_cl_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_hac_cas_cl_ry} $R_y$}
\end{subfigure}
@@ -12420,19 +12388,19 @@ The results confirmed that the NASS successfully maintained the point of interes
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_reflectivity_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_reflectivity_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_reflectivity_dy}$D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_reflectivity_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_reflectivity_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_reflectivity_dz}$D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_reflectivity_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_reflectivity_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_reflectivity_ry}$R_y$}
\end{subfigure}
@@ -12456,19 +12424,19 @@ The settling duration typically decreases for smaller step sizes.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_mim_10nm_steps.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_10nm_steps.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_mim_10nm_steps}10nm steps}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_mim_100nm_steps.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_100nm_steps.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_mim_100nm_steps}100nm steps}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_mim_1000nm_steps.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_mim_1000nm_steps.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu$m step}
\end{subfigure}
@@ -12484,19 +12452,19 @@ Initial testing at \(10\,\mu m/s\) demonstrated positioning errors well within s
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_scan_10ums_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_10ums_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_scan_10ums_dy}$D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_scan_10ums_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_10ums_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_scan_10ums_dz}$D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_scan_10ums_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_10ums_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$}
\end{subfigure}
@@ -12510,19 +12478,19 @@ However, performance during acceleration phases could be enhanced through the im
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_scan_100ums_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_100ums_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_scan_100ums_dy}$D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_scan_100ums_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_100ums_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_scan_100ums_dz}$D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dz_scan_100ums_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dz_scan_100ums_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$}
\end{subfigure}
@@ -12548,19 +12516,19 @@ Under closed-loop control, positioning errors remain within specifications in al
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dy_10ums_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_10ums_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_dy_10ums_dy} $D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dy_10ums_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_10ums_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_dy_10ums_dz} $D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dy_10ums_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_10ums_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_dy_10ums_ry} $R_y$}
\end{subfigure}
@@ -12581,19 +12549,19 @@ For applications requiring small \(D_y\) scans, the nano-hexapod can be used exc
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dy_100ums_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_100ums_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_dy_100ums_dy} $D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dy_100ums_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_100ums_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_dy_100ums_dz} $D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_dy_100ums_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_dy_100ums_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_dy_100ums_ry} $R_y$}
\end{subfigure}
@@ -12609,7 +12577,7 @@ The system performance was evaluated at three lateral scanning velocities: \(0.1
\begin{figure}[htbp]
\centering
-\includegraphics[scale=1]{figs/test_id31_diffraction_tomo_setpoint.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_diffraction_tomo_setpoint.png}
\caption{\label{fig:test_id31_diffraction_tomo_setpoint}Dy motion for several configured velocities}
\end{figure}
@@ -12622,19 +12590,19 @@ Alternatively, a feedforward controller could improve the lateral positioning ac
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_diffraction_tomo_dy.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_diffraction_tomo_dy.png}
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_dy} $D_y$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_diffraction_tomo_dz.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_diffraction_tomo_dz.png}
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_dz} $D_z$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
-\includegraphics[scale=1,scale=1]{figs/test_id31_diffraction_tomo_ry.png}
+\includegraphics[scale=1,scale=0.8]{figs/test_id31_diffraction_tomo_ry.png}
\end{center}
\subcaption{\label{fig:test_id31_diffraction_tomo_ry} $R_y$}
\end{subfigure}
@@ -12663,7 +12631,7 @@ The identified limitations, primarily related to high-speed lateral scanning and
\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 by bold number.}
\centering
-\begin{tabularx}{0.9\linewidth}{Xccc}
+\begin{tabularx}{0.85\linewidth}{Xccc}
\toprule
\textbf{Experiments} & \(\bm{D_y}\) \textbf{{[}nmRMS]} & \(\bm{D_z}\) \textbf{{[}nmRMS]} & \(\bm{R_y}\) \textbf{{[}nradRMS]}\\
\midrule