Change transpose notation

test-bench-id31.org

@@ -2981,7 +2981,7 @@ The control strategy here is to apply a diagonal control in the frame of the str
 To conduct this interaction analysis, the acrfull:rga $\bm{\Lambda_G}$ is first computed using eqref:eq:test_id31_rga for the plant dynamics identified with the multiple payload masses.
 
 \begin{equation}\label{eq:test_id31_rga}
-\bm{\Lambda_G}(\omega) = \bm{G}(j\omega) \star \left(\bm{G}(j\omega)^{-1}\right)^{T}, \quad (\star \text{ means element wise multiplication})
+\bm{\Lambda_G}(\omega) = \bm{G}(j\omega) \star \left(\bm{G}(j\omega)^{-1}\right)^{\intercal}, \quad (\star \text{ means element wise multiplication})
 \end{equation}
 
 Then, acrshort:rga numbers are computed using eqref:eq:test_id31_rga_number and are use as a metric for interaction [[cite:&skogestad07_multiv_feedb_contr chapt. 3.4]].

test-bench-id31.tex

@@ -1,4 +1,4 @@
-% Created 2025-02-20 Thu 10:55
+% Created 2025-04-07 Mon 14:38
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -18,7 +18,7 @@
  pdftitle={Experimental Validation on the ID31 Beamline},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.4 (Org mode 9.6)}, 
+ pdfcreator={Emacs 30.1 (Org mode 9.7.26)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -28,7 +28,6 @@
 \tableofcontents
 
 \clearpage
-
 To proceed with the full validation of the Nano Active Stabilization System (NASS), the nano-hexapod was mounted on top of the micro-station on ID31, as illustrated in figure \ref{fig:test_id31_micro_station_nano_hexapod}.
 This section presents a comprehensive experimental evaluation of the complete system's performance on the ID31 beamline, focusing on its ability to maintain precise sample positioning under various experimental conditions.
 
@@ -62,7 +61,6 @@ These include tomography scans at various speeds and with different payload mass
 \end{subfigure}
 \caption{\label{fig:test_id31_micro_station_nano_hexapod}Picture of the micro-station without the nano-hexapod (\subref{fig:test_id31_micro_station_cables}) and with the nano-hexapod (\subref{fig:test_id31_fixed_nano_hexapod})}
 \end{figure}
-
 \chapter{Short Stroke Metrology System}
 \label{sec:test_id31_metrology}
 The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite.
@@ -142,7 +140,6 @@ The five equations \eqref{eq:test_id31_metrology_kinematics} can be written in m
   d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5
 \end{bmatrix}
 \end{equation}
-
 \section{Rough alignment of the reference spheres}
 \label{ssec:test_id31_metrology_sphere_rought_alignment}
 
@@ -156,7 +153,6 @@ The probes are then fixed to the top (adjustable) cylinder, and the same alignme
 With this setup, the alignment accuracy of both spheres with the spindle axis was expected to around \(10\,\mu m\).
 The accuracy was probably limited by the poor coaxiality between the cylinders and the spheres.
 However, this first alignment should be sufficient to position the two sphere within the acceptance range of the interferometers.
-
 \section{Tip-Tilt adjustment of the interferometers}
 \label{ssec:test_id31_metrology_alignment}
 
@@ -180,7 +176,6 @@ This allows them to be individually oriented so that they all point to the spher
 This is achieved by maximizing the intensity of the reflected light of each interferometer.
 
 After the alignment procedure, the top interferometer should coincide with the spindle axis, and the lateral interferometers should all be in the horizontal plane and intersect the centers of the spheres.
-
 \section{Fine Alignment of reference spheres using interferometers}
 \label{ssec:test_id31_metrology_sphere_fine_alignment}
 
@@ -210,8 +205,6 @@ The remaining errors after alignment are in the order of \(\pm5\,\mu\text{rad}\)
 \end{subfigure}
 \caption{\label{fig:test_id31_metrology_align}Measured angular (\subref{fig:test_id31_metrology_align_rx_ry}) and lateral (\subref{fig:test_id31_metrology_align_dx_dy}) errors during full spindle rotation. Between two rotations, the micro-hexapod is adjusted to better align the two spheres with the rotation axis.}
 \end{figure}
-
-
 \section{Estimated measurement volume}
 \label{ssec:test_id31_metrology_acceptance}
 
@@ -222,7 +215,6 @@ Results are summarized in Table \ref{tab:test_id31_metrology_acceptance}.
 The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(+/-0.5\,mm\), which is enough for the current application as it is well above the micro-station errors to be actively corrected by the NASS.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
 \centering
 \begin{tabularx}{0.45\linewidth}{Xccc}
 \toprule
@@ -235,9 +227,9 @@ The obtained lateral acceptance for pure displacements in any direction is estim
 \(d_5\) (z) & \(1.33\, mm\) & \(1.06\,mm\) & \(>2\,mm\)\\
 \bottomrule
 \end{tabularx}
+\caption{\label{tab:test_id31_metrology_acceptance}Estimated measurement range for each interferometer, and for three different directions.}
+
 \end{table}
-
-
 \section{Estimated measurement errors}
 \label{ssec:test_id31_metrology_errors}
 
@@ -276,7 +268,6 @@ The effect of noise on the translation and rotation measurements is estimated in
 \end{subfigure}
 \caption{\label{fig:test_id31_metrology_errors}Estimated measurement errors of the metrology. Cross-coupling between lateral motion and vertical measurement is shown in (\subref{fig:test_id31_xy_map_sphere}). The effect of interferometer noise on the measured translations and rotations is shown in (\subref{fig:test_id31_interf_noise}).}
 \end{figure}
-
 \chapter{Open Loop Plant}
 \label{sec:test_id31_open_loop_plant}
 The NASS plant is schematically illustrated in Figure \ref{fig:test_id31_block_schematic_plant}.
@@ -327,7 +318,6 @@ This issue was later solved.
 \end{subfigure}
 \caption{\label{fig:test_id31_first_id}Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and force sensors (\subref{fig:test_id31_first_id_iff}). Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are displayed with shaded lines.}
 \end{figure}
-
 \section{Better Angular Alignment}
 \label{ssec:test_id31_open_loop_plant_rz_alignment}
 
@@ -367,7 +357,6 @@ The flexible modes of the top platform can be passively damped, whereas the mode
 \includegraphics[scale=1]{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}
-
 \section{Effect of Payload Mass}
 \label{ssec:test_id31_open_loop_plant_mass}
 
@@ -421,7 +410,6 @@ It is interesting to note that the anti-resonances in the force sensor plant now
 \end{subfigure}
 \caption{\label{fig:test_id31_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 \(\epsilon\mathcal{L}\) (\subref{fig:test_id31_comp_simscape_int_diag_masses}) and from \(u\) to \(V_s\) (\subref{fig:test_id31_comp_simscape_iff_diag_masses})}
 \end{figure}
-
 \section{Effect of Spindle Rotation}
 \label{ssec:test_id31_open_loop_plant_rotation}
 
@@ -448,12 +436,10 @@ This also indicates that the metrology kinematics is correct and is working in r
 \end{subfigure}
 \caption{\label{fig:test_id31_effect_rotation}Effect of the spindle rotation on the plant dynamics from \(u\) to \(\epsilon\mathcal{L}\). Three rotational velocities are tested (\(0\,\text{deg}/s\), \(36\,\text{deg}/s\) and \(180\,\text{deg}/s\)). Both direct terms (\subref{fig:test_id31_effect_rotation_direct}) and coupling terms (\subref{fig:test_id31_effect_rotation_coupling}) are displayed.}
 \end{figure}
-
 \section*{Conclusion}
 The identified frequency response functions from command signals \(\bm{u}\) to the force sensors \(\bm{V}_s\) and to the estimated strut errors \(\bm{\epsilon\mathcal{L}}\) are well matching the dynamics of the developed multi-body model.
 The effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used.
 The spindle rotation had no visible effect on the measured dynamics, indicating that controllers should be robust against spindle rotation.
-
 \chapter{Decentralized Integral Force Feedback}
 \label{sec:test_id31_iff}
 In this section, the low authority control part is first validated.
@@ -490,7 +476,6 @@ This confirms that the multi-body model can be used to tune the IFF controller.
 \includegraphics[scale=1]{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}
-
 \section{IFF Controller}
 \label{ssec:test_id31_iff_controller}
 
@@ -555,7 +540,6 @@ However, in this study, it was chosen to implement a ``fixed'' (i.e. non-adaptiv
 \end{subfigure}
 \caption{\label{fig:test_id31_iff_root_locus}Root Locus plots for the designed decentralized IFF controller, computed using the multy-body model. Black crosses indicate the closed-loop poles for the choosen value of the gain.}
 \end{figure}
-
 \section{Damped Plant}
 \label{ssec:test_id31_iff_perf}
 
@@ -581,14 +565,12 @@ The obtained frequency response functions are compared with the model in Figure
 \end{subfigure}
 \caption{\label{fig:test_id31_hac_plant_effect_mass_comp_model}Comparison of the open-loop plants and the damped plant with Decentralized IFF, estimated from the multi-body model (\subref{fig:test_id31_comp_ol_iff_plant_model}). Comparison of measured damped and modeled plants for all considered payloads (\subref{fig:test_id31_hac_plant_effect_mass}). Only ``direct'' terms (\(\epsilon\mathcal{L}_i/u_i^\prime\)) are displayed for simplificty}
 \end{figure}
-
 \section*{Conclusion}
 The implementation of a decentralized Integral Force Feedback controller was successfully demonstrated.
 Using the multi-body model, the controller was designed and optimized to ensure stability across all payload conditions while providing significant damping of suspension modes.
 The experimental results validated the model predictions, showing a reduction in peak amplitudes by approximately a factor of 10 across the full payload range (0-39 kg).
 Although higher gains could achieve better damping performance for lighter payloads, the chosen fixed-gain configuration represents a robust compromise that maintains stability and performance under all operating conditions.
 The good correlation between the modeled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction.
-
 \chapter{High Authority Control in the frame of the struts}
 \label{sec:test_id31_hac}
 In this section, a High-Authority-Controller is developed to actively stabilize the sample position.
@@ -631,7 +613,6 @@ This is one of the key benefits of using the HAC-LAC strategy.
 \includegraphics[scale=1]{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}
-
 \section{Interaction Analysis}
 \label{sec:test_id31_hac_interaction_analysis}
 
@@ -639,7 +620,7 @@ The control strategy here is to apply a diagonal control in the frame of the str
 To conduct this interaction analysis, the \acrfull{rga} \(\bm{\Lambda_G}\) is first computed using \eqref{eq:test_id31_rga} for the plant dynamics identified with the multiple payload masses.
 
 \begin{equation}\label{eq:test_id31_rga}
-\bm{\Lambda_G}(\omega) = \bm{G}(j\omega) \star \left(\bm{G}(j\omega)^{-1}\right)^{T}, \quad (\star \text{ means element wise multiplication})
+\bm{\Lambda_G}(\omega) = \bm{G}(j\omega) \star \left(\bm{G}(j\omega)^{-1}\right)^{\intercal}, \quad (\star \text{ means element wise multiplication})
 \end{equation}
 
 Then, \acrshort{rga} numbers are computed using \eqref{eq:test_id31_rga_number} and are use as a metric for interaction \cite[chapt. 3.4]{skogestad07_multiv_feedb_contr}.
@@ -660,7 +641,6 @@ This design choice, while beneficial for system simplicity, introduces inherent
 \includegraphics[scale=1]{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}
-
 \section{Robust Controller Design}
 \label{ssec:test_id31_iff_hac_controller}
 
@@ -692,7 +672,6 @@ However, small stability margins were observed for the highest mass, indicating
 \end{subfigure}
 \caption{\label{fig:test_id31_hac_loop_gain_loci}Robust High Authority Controller. ``Decentralized loop-gains'' are shown in (\subref{fig:test_id31_hac_loop_gain}) and characteristic loci are shown in (\subref{fig:test_id31_hac_characteristic_loci})}
 \end{figure}
-
 \section{Performance estimation with simulation of Tomography scans}
 \label{ssec:test_id31_iff_hac_perf}
 
@@ -716,7 +695,6 @@ The obtained closed-loop positioning accuracy was found to comply with the requi
 \end{subfigure}
 \caption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}Position error of the sample in the XY (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_xy}) and YZ (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim_yz}) planes during a simulation of a tomography experiment at \(180\,\text{deg/s}\). No payload is placed on top of the nano-hexapod.}
 \end{figure}
-
 \section{Robustness estimation with simulation of Tomography scans}
 \label{ssec:test_id31_iff_hac_robustness}
 
@@ -732,7 +710,6 @@ However, it was decided that this controller should be tested experimentally and
 \includegraphics[scale=1]{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}
-
 \section*{Conclusion}
 In this section, a High-Authority-Controller was developed to actively stabilize the sample position.
 The multi-body model was first validated by comparing it with the measured frequency responses of the damped plant, which showed good agreement for both direct terms and coupling terms.
@@ -746,7 +723,6 @@ The closed-loop system remained stable under all tested payload conditions (0 to
 With no payload at \(180\,\text{deg/s}\), the NASS successfully maintained the sample point of interest in the beam, which fulfilled the specifications.
 At \(6\,\text{deg/s}\), although the positioning errors increased with the payload mass (particularly in the lateral direction), the system remained stable.
 These results demonstrate both the effectiveness and limitations of implementing control in the frame of the struts.
-
 \chapter{Validation with Scientific experiments}
 \label{sec:test_id31_experiments}
 In this section, the goal is to evaluate the performance of the NASS and validate its use to perform typical scientific experiments.
@@ -772,7 +748,6 @@ In terms of RMS errors, this corresponds to \(30\,nm\) in \(D_y\), \(15\,nm\) in
 Results obtained for all experiments are summarized and compared to the specifications in Section \ref{ssec:test_id31_scans_conclusion}.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_id31_experiments_specifications}Specifications for the Nano-Active-Stabilization-System}
 \centering
 \begin{tabularx}{0.45\linewidth}{Xccc}
 \toprule
@@ -782,10 +757,12 @@ peak 2 peak & 200nm & 100nm & \(1.7\,\mu\text{rad}\)\\
 RMS & 30nm & 15nm & \(250\,\text{nrad}\)\\
 \bottomrule
 \end{tabularx}
+\caption{\label{tab:test_id31_experiments_specifications}Specifications for the Nano-Active-Stabilization-System}
+
 \end{table}
 \section{Tomography Scans}
 \label{ssec:test_id31_scans_tomography}
-\paragraph{Slow Tomography scans}
+\subsubsection{Slow Tomography scans}
 
 First, tomography scans were performed with a rotational velocity of \(6\,\text{deg/s}\) for all considered payload masses (shown in Figure \ref{fig:test_id31_picture_masses}).
 Each experimental sequence consisted of two complete spindle rotations: an initial open-loop rotation followed by a closed-loop rotation.
@@ -821,8 +798,7 @@ These experimental findings are consistent with the predictions from the tomogra
 \includegraphics[scale=1]{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}
-
+\subsubsection{Fast Tomography scans}
-\paragraph{Fast Tomography scans}
 
 A tomography experiment was then performed with the highest rotational velocity of the Spindle: \(180\,\text{deg/s}\)\footnote{The highest rotational velocity of \(360\,\text{deg/s}\) could not be tested due to an issue in the Spindle's controller.}.
 The trajectory of the point of interest during the fast tomography scan is shown in Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}.
@@ -844,8 +820,7 @@ Nevertheless, even with this robust (i.e. conservative) HAC implementation, the
 \end{subfigure}
 \caption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}Experimental results of tomography experiment at 180 deg/s without payload. The position error of the sample is shown in the XY (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_xy}) and YZ (\subref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz}) planes.}
 \end{figure}
-
+\subsubsection{Cumulative Amplitude Spectra}
-\paragraph{Cumulative Amplitude Spectra}
 
 A comparative analysis was conducted using three tomography scans at \(180\,\text{deg/s}\) to evaluate the effectiveness of the HAC-LAC strategy in reducing positioning errors.
 The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete HAC-LAC strategy.
@@ -878,7 +853,6 @@ This experiment also illustrates that when needed, performance can be enhanced b
 \end{subfigure}
 \caption{\label{fig:test_id31_hac_cas_cl}Cumulative Amplitude Spectrum for tomography experiments at \(180\,\text{deg}/s\). Open-Loop case, IFF, and HAC-LAC are compared. Specifications are indicated by black dashed lines. The RMS values are indicated in the legend.}
 \end{figure}
-
 \section{Reflectivity Scans}
 \label{ssec:test_id31_scans_reflectivity}
 
@@ -907,12 +881,11 @@ The results confirmed that the NASS successfully maintained the point of interes
 \end{subfigure}
 \caption{\label{fig:test_id31_reflectivity}Reflectivity scan (\(R_y\)) with a rotational velocity of \(100\,\mu \text{rad}/s\).}
 \end{figure}
-
 \section{Dirty Layer Scans}
 \label{ssec:test_id31_scans_dz}
 In some cases, samples are composed of several atomic ``layers'' that are first aligned in the horizontal plane through \(R_x\) and \(R_y\) positioning, followed by vertical scanning with precise \(D_z\) motion.
 These vertical scans can be executed either continuously or in a step-by-step manner.
-\paragraph{Step by Step \(D_z\) motion}
+\subsubsection{Step by Step \(D_z\) motion}
 
 The vertical step motion was performed exclusively with the nano-hexapod.
 Testing was conducted across step sizes ranging from \(10\,nm\) to \(1\,\mu m\).
@@ -944,8 +917,7 @@ The settling duration typically decreases for smaller step sizes.
 \end{subfigure}
 \caption{\label{fig:test_id31_dz_mim_steps}Vertical steps performed with the nano-hexapod. 10nm steps are shown in (\subref{fig:test_id31_dz_mim_10nm_steps}) with the low-pass filtered data corresponding to an integration time of \(50\,ms\). 100nm steps are shown in (\subref{fig:test_id31_dz_mim_100nm_steps}). The response time to reach a peak-to-peak error of \(\pm 20\,nm\) is \(\approx 70\,ms\) as shown in (\subref{fig:test_id31_dz_mim_1000nm_steps}) for a \(1\,\mu m\) step.}
 \end{figure}
-
+\subsubsection{Continuous \(D_z\) motion: Dirty Layer Scans}
-\paragraph{Continuous \(D_z\) motion: Dirty Layer Scans}
 
 For these and subsequent experiments, the NASS performs ``ramp scans'' (constant velocity scans).
 To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies.
@@ -999,14 +971,13 @@ However, performance during acceleration phases could be enhanced through the im
 \end{subfigure}
 \caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan at a velocity of \(100\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_100ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_100ums_dy}) and (\subref{fig:test_id31_dz_scan_100ums_ry})}
 \end{figure}
-
 \section{Lateral Scans}
 \label{ssec:test_id31_scans_dy}
 Lateral scans are executed using the \(T_y\) stage.
 The stepper motor controller\footnote{The ``IcePAP'' \cite{janvier13_icepap}  which is developed at the ESRF.} generates a setpoint that is transmitted to the Speedgoat.
 Within the Speedgoat, the system computes the positioning error by comparing the measured \(D_y\) sample position against the received setpoint, and the Nano-Hexapod compensates for positioning errors introduced during \(T_y\) stage scanning.
 The scanning range is constrained \(\pm 100\,\mu m\) due to the limited acceptance of the metrology system.
-\paragraph{Slow scan}
+\subsubsection{Slow scan}
 
 Initial testing utilized a scanning velocity of \(10\,\mu m/s\), which is typical for these experiments.
 Figure \ref{fig:test_id31_dy_10ums} compares the positioning errors between open-loop (without NASS) and closed-loop operation.
@@ -1038,8 +1009,7 @@ Under closed-loop control, positioning errors remain within specifications in al
 \end{subfigure}
 \caption{\label{fig:test_id31_dy_10ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(10\,\mu m/s\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_10ums_dy}).}
 \end{figure}
-
+\subsubsection{Fast Scan}
-\paragraph{Fast Scan}
 
 The system performance was evaluated at an increased scanning velocity of \(100\,\mu m/s\), and the results are presented in Figure \ref{fig:test_id31_dy_100ums}.
 At this velocity, the micro-stepping errors generate \(10\,\text{Hz}\) vibrations, which are further amplified by micro-station resonances.
@@ -1072,7 +1042,6 @@ For applications requiring small \(D_y\) scans, the nano-hexapod can be used exc
 \end{subfigure}
 \caption{\label{fig:test_id31_dy_100ums}Open-Loop (in blue) and Closed-loop (i.e. using the NASS, in red) during a \(100\,\mu m/s\) scan with the \(T_y\) stage. Errors in \(D_y\) is shown in (\subref{fig:test_id31_dy_100ums_dy}).}
 \end{figure}
-
 \section{Diffraction Tomography}
 \label{ssec:test_id31_scans_diffraction_tomo}
 
@@ -1114,7 +1083,6 @@ Alternatively, a feedforward controller could improve the lateral positioning ac
 \end{subfigure}
 \caption{\label{fig:test_id31_diffraction_tomo}Diffraction tomography scans (combined \(R_z\) and \(D_y\) motions) at several \(D_y\) velocities (\(R_z\) rotational velocity is \(6\,\text{deg/s}\)).}
 \end{figure}
-
 \section*{Conclusion}
 \label{ssec:test_id31_scans_conclusion}
 
@@ -1136,11 +1104,10 @@ Overall, the experimental results validate the effectiveness of the developed co
 The identified limitations, primarily related to high-speed lateral scanning and heavy payload handling, provide clear directions for future improvements.
 
 \begin{table}[htbp]
-\caption{\label{tab:test_id31_experiments_results_summary}Summary of the experimental results performed using the NASS on ID31. Open-loop errors are indicated on the left of the arrows. Closed-loop errors that are outside the specifications are indicated by bold number.}
 \centering
 \begin{tabularx}{\linewidth}{Xccc}
 \toprule
-\textbf{Experiments} & \(\bm{D_y}\) \textbf{[nmRMS]} & \(\bm{D_z}\) \textbf{[nmRMS]} & \(\bm{R_y}\) \textbf{[nradRMS]}\\
+\textbf{Experiments} & \(\bm{D_y}\) \textbf{{[}nmRMS]} & \(\bm{D_z}\) \textbf{{[}nmRMS]} & \(\bm{R_y}\) \textbf{{[}nradRMS]}\\
 \midrule
 Tomography (\(6\,\text{deg/s}\)) & \(142 \Rightarrow 15\) & \(32 \Rightarrow 5\) & \(464 \Rightarrow 56\)\\
 Tomography (\(6\,\text{deg/s}\), 13kg) & \(149 \Rightarrow 25\) & \(26 \Rightarrow 6\) & \(471 \Rightarrow 55\)\\
@@ -1165,8 +1132,9 @@ Diffraction tomography (\(6\,\text{deg/s}\), \(1\,mm/s\)) & \(\bm{53}\) & \(10\)
 \textbf{Specifications} & \(30\) & \(15\) & \(250\)\\
 \bottomrule
 \end{tabularx}
-\end{table}
+\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.}
 
+\end{table}
 \chapter*{Conclusion}
 \label{ssec:test_id31_conclusion}
 
@@ -1187,6 +1155,5 @@ Some limitations were identified, particularly in handling heavy payloads during
 The successful validation of the NASS demonstrates that once an accurate online metrology system is developed, it will be ready for integration into actual beamline operations.
 The system's ability to maintain precise sample positioning across a wide range of experimental conditions, combined with its robust performance and adaptive capabilities, suggests that it will significantly enhance the quality and efficiency of X-ray experiments at ID31.
 Moreover, the systematic approach to system development and validation, along with a detailed understanding of performance limitations, provides valuable insights for future improvements and potential applications in similar high-precision positioning systems.
-
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}