Rework part of experimental results

test-bench-id31.tex

@@ -1,4 +1,4 @@
-% Created 2025-02-01 Sat 18:34
+% Created 2025-02-03 Mon 14:43
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -216,6 +216,7 @@ Results are summarized in Table \ref{tab:test_id31_metrology_acceptance}.
 The obtained lateral acceptance for pure displacements in any direction is estimated to be around \(+/-0.5\,mm\), 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
@@ -228,8 +229,6 @@ 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}
 
 
@@ -746,19 +745,39 @@ These results demonstrate both the effectiveness and limitations of implementing
 
 \chapter{Validation with Scientific experiments}
 \label{sec:test_id31_experiments}
-The online metrology prototype does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam.
+In this section, the goal is to evaluate the performances of the NASS and validate its use for real work scientific experiments.
-However, in order to fully validate the NASS, typical motion performed during scientific experiments can be mimicked, and the positioning performances can be evaluated.
 
-For tomography scans, performances were already evaluated in Section \ref{ssec:test_id31_iff_hac_perf}.
+However, the online metrology prototype (presented in Section \ref{sec:test_id31_metrology}) does not allow samples to be placed on top of the nano-hexapod while being illuminated by the x-ray beam.
-Here, other typical experiments are performed:
+Nevertheless, in order to fully validate the NASS, typical motion performed during scientific experiments can be mimicked, and the positioning performances can be evaluated.
+
+Several scientific experiments are mimicked, such as:
 \begin{itemize}
-\item \emph{Lateral scans}: the \(T_y\) translations stage performs \(D_y\) scans and the errors are corrected by the NASS in real time (Section \ref{ssec:test_id31_scans_dy})
+\item Tomography scans: continuous rotation of the Spindle along the vertical axis (Section \ref{ssec:test_id31_scans_tomography})
-\item \emph{Vertical layer scans}: the nano-hexapod is used to perform \(D_z\) step motion or ramp scans (Section \ref{ssec:test_id31_scans_dz})
+\item Reflectivity scans: \(R_y\) rotations using the tilt-stage (Section \ref{ssec:test_id31_scans_reflectivity})
-\item \emph{Reflectivity scans}: the tilt stage is doing \(R_y\) rotations and the errors are corrected by the NASS in real time (Section \ref{ssec:test_id31_scans_reflectivity})
+\item Vertical layer scans: the nano-hexapod is used to perform \(D_z\) step motion or ramp scans (Section \ref{ssec:test_id31_scans_dz})
-\item \emph{Diffraction Tomography}: the Spindle is performing continuous \(R_z\) rotation while the translation stage is performing lateral \(D_y\) scans at the same time.
+\item Lateral scans: \(D_y\) scans using the \(T_y\) translation stage (Section \ref{ssec:test_id31_scans_dy})
+\item Diffraction Tomography: the Spindle is performing continuous \(R_z\) rotation while the translation stage is performing lateral \(D_y\) scans at the same time.
 This is the experiment with the most stringent requirements (Section \ref{ssec:test_id31_scans_diffraction_tomo})
 \end{itemize}
-\section{\(R_z\) scans: Tomography}
+
+For each experiment, the obtained performances are compared to the specifications for the most depending case in which nano-focusing optics are used to focus the beam down to \(200\,nm\times 100\,nm\).
+In that case the goal is to keep the sample's point of interested in the beam, and therefore the \(D_y\) and \(D_z\) positioning errors should be less than \(200\,nm\) and \(100\,nm\) peak-to-peak respectively.
+The \(R_y\) error should be less than \(1.7\,\mu\text{rad}\) peak-to-peak.
+In terms of RMS errors, this corresponds to \(30\,nm\) in \(D_y\), \(15\,nm\) in \(D_z\) and \(250\,\text{nrad}\) in \(R_y\).
+
+\begin{table}[htbp]
+\caption{\label{tab:test_id31_experiments_specifications}Specifications for the Nano-Active-Stabilization-System}
+\centering
+\begin{tabularx}{0.5\linewidth}{Xccc}
+\toprule
+ & \(D_y\) & \(D_z\) & \(R_y\)\\
+\midrule
+peak 2 peak & 200nm & 100nm & \(1.7\,\mu\text{rad}\)\\
+RMS & 30nm & 15nm & \(250\,\text{nrad}\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
+\section{Tomography Scans}
 \label{ssec:test_id31_scans_tomography}
 \textbf{Issue with this control architecture (or controller?)}:
 \begin{itemize}
@@ -770,51 +789,6 @@ This is the experiment with the most stringent requirements (Section \ref{ssec:t
 \item 1rpm, 6rpm, 30rpm
 \item at 1rpm: m0, m1, m2, m3 (same robust controller!)
 \end{itemize}
-\paragraph{Previous results at 30rpm}
-
-Then the same tomography experiment (i.e. constant spindle rotation at 30rpm, and no payload) was performed experimentally.
-The measured position of the ``point of interest'' during the experiment are shown in Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}.
-
-\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}
-\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}
-\end{center}
-\subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz}YZ plane}
-\end{subfigure}
-\caption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}Experimental results of a tomography experiment at 30RPM without payload. 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}
-
-Even though the simulation (Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}) and the experimental results (Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}) are looking similar, the most important metric to compare is the RMS values of the positioning errors in closed-loop.
-These are computed for both the simulation and the experimental results and are compared in Table \ref{tab:test_id31_tomo_m0_30rpm_robust_hac_iff_rms}.
-The lateral and vertical errors are similar, however the tilt (\(R_y\)) errors are underestimated by the model, which is reasonable as disturbances in \(R_y\) were not modeled.
-
-Results obtained with this conservative HAC are already close to the specifications.
-
-\begin{table}[htbp]
-\centering
-\begin{tabularx}{0.7\linewidth}{Xccc}
-\toprule
- & \(D_y\) & \(D_z\) & \(R_y\)\\
-\midrule
-Experiment (OL) & \(1.8\,\mu\text{mRMS}\) & \(24\,\text{nmRMS}\) & \(10\,\mu\text{radRMS}\)\\
-\midrule
-Simulation (CL) & \(30\,\text{nmRMS}\) & \(8\,\text{nmRMS}\) & \(73\,\text{nradRMS}\)\\
-Experiment (CL) & \(39\,\text{nmRMS}\) & \(11\,\text{nmRMS}\) & \(130\,\text{nradRMS}\)\\
-\midrule
-Specifications & \(30\,\text{nmRMS}\) & \(15\,\text{nmRMS}\) & \(250\,\text{nradRMS}\)\\
-\bottomrule
-\end{tabularx}
-\caption{\label{tab:test_id31_tomo_m0_30rpm_robust_hac_iff_rms}RMS values of the errors for a tomography experiment at 30RPM and without payload. Experimental results and simulation are compared.}
-
-\end{table}
-
 \paragraph{Previous results at 1rpm}
 
 The tomography experiments that were simulated were then experimentally conducted.
@@ -850,6 +824,7 @@ The RMS values of the open-loop and closed-loop errors for all masses are summar
 The obtained closed-loop errors are fulfilling the requirements, except for the \(39\,\text{kg}\) payload in the lateral (\(D_y\)) direction.
 
 \begin{table}[htbp]
+\caption{\label{tab:test_id31_tomo_1rpm_robust_ol_cl_errors}RMS values of the measured errors during open-loop and closed-loop tomography scans (1rpm) for all considered payloads. Measured closed-Loop errors are indicated by ``bold'' font.}
 \centering
 \begin{tabularx}{0.9\linewidth}{Xccc}
 \toprule
@@ -863,11 +838,286 @@ The obtained closed-loop errors are fulfilling the requirements, except for the
 \textbf{Specifications} & \(30\,\text{nmRMS}\) & \(15\,\text{nmRMS}\) & \(250\,\text{nradRMS}\)\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:test_id31_tomo_1rpm_robust_ol_cl_errors}RMS values of the measured errors during open-loop and closed-loop tomography scans (1rpm) for all considered payloads. Measured closed-Loop errors are indicated by ``bold'' font.}
-
 \end{table}
 
-\section{\(D_y\) - Lateral Scans}
+\paragraph{Previous results at 30rpm}
+
+Then the same tomography experiment (i.e. constant spindle rotation at 30rpm, and no payload) was performed experimentally.
+The measured position of the ``point of interest'' during the experiment are shown in Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}.
+
+\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}
+\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}
+\end{center}
+\subcaption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp_yz}YZ plane}
+\end{subfigure}
+\caption{\label{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}Experimental results of a tomography experiment at 30RPM without payload. 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}
+
+Even though the simulation (Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_sim}) and the experimental results (Figure \ref{fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp}) are looking similar, the most important metric to compare is the RMS values of the positioning errors in closed-loop.
+These are computed for both the simulation and the experimental results and are compared in Table \ref{tab:test_id31_tomo_m0_30rpm_robust_hac_iff_rms}.
+The lateral and vertical errors are similar, however the tilt (\(R_y\)) errors are underestimated by the model, which is reasonable as disturbances in \(R_y\) were not modeled.
+
+Results obtained with this conservative HAC are already close to the specifications.
+
+\begin{table}[htbp]
+\caption{\label{tab:test_id31_tomo_m0_30rpm_robust_hac_iff_rms}RMS values of the errors for a tomography experiment at 30RPM and without payload. Experimental results and simulation are compared.}
+\centering
+\begin{tabularx}{0.7\linewidth}{Xccc}
+\toprule
+ & \(D_y\) & \(D_z\) & \(R_y\)\\
+\midrule
+Experiment (OL) & \(1.8\,\mu\text{mRMS}\) & \(24\,\text{nmRMS}\) & \(10\,\mu\text{radRMS}\)\\
+\midrule
+Simulation (CL) & \(30\,\text{nmRMS}\) & \(8\,\text{nmRMS}\) & \(73\,\text{nradRMS}\)\\
+Experiment (CL) & \(39\,\text{nmRMS}\) & \(11\,\text{nmRMS}\) & \(130\,\text{nradRMS}\)\\
+\midrule
+Specifications & \(30\,\text{nmRMS}\) & \(15\,\text{nmRMS}\) & \(250\,\text{nradRMS}\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
+
+\paragraph{Dynamic Error Budgeting}
+
+In this section, the noise budget is performed.
+The vibrations of the sample is measured in different conditions using the external metrology.
+
+\textbf{Tomography}:
+\begin{itemize}
+\item Beam size: 200nm x 100nm
+\item Keep the PoI in the beam: peak to peak errors of 200nm in Dy and 100nm in Dz
+\item RMS errors (/ by 6.6) gives 30nmRMS in Dy and 15nmRMS in Dz.
+\item Ry error <1.7urad, 250nrad RMS
+\end{itemize}
+
+\begin{center}
+\begin{tabular}{lllllll}
+ & Dx & Dy & Dz & Rx & Ry & Rz\\
+\hline
+peak 2 peak &  & 200nm & 100nm &  & 1.7 urad & \\
+RMS &  & 30nm & 15nm &  & 250 nrad & \\
+\end{tabular}
+\end{center}
+
+
+\begin{itemize}
+\item Effect of rotation.
+\item Comparison with measurement noise: should be higher
+\item Maybe say that we then focus on the high rotation velocity
+\item Also say that for the RMS errors, we don't take into account drifts (so we NASS we can correct drifts)
+\item Focus on 30rpm case
+\end{itemize}
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/test_id31_hac_cas_ol_dy.png}
+\end{center}
+\subcaption{\label{fig:test_id31_hac_cas_ol_dy} $D_y$}
+\end{subfigure}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/test_id31_hac_cas_ol_dz.png}
+\end{center}
+\subcaption{\label{fig:test_id31_hac_cas_ol_dz} $D_z$}
+\end{subfigure}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/test_id31_hac_cas_ol_ry.png}
+\end{center}
+\subcaption{\label{fig:test_id31_hac_cas_ol_ry} $R_y$}
+\end{subfigure}
+\caption{\label{fig:test_id31_hac_cas_ol}Cumulative Amplitude Spectrum of the measured positioning errors without any rotation, with \(\Omega_z = 36\,\text{deg}/s\) and with \(\Omega_z = 180\,\text{deg}/s\). Open-loop case. RMS values are indicated in the legend.}
+\end{figure}
+
+Effect of LAC:
+\begin{itemize}
+\item reduce amplitude around 80Hz
+\item Inject some noise between 200 and 700Hz?
+\end{itemize}
+
+Effect of HAC:
+\begin{itemize}
+\item Bandwidth is approximately 10Hz
+\end{itemize}
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{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}
+\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}
+\end{center}
+\subcaption{\label{fig:test_id31_hac_cas_cl_ry} $R_y$}
+\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. RSM values are indicated in the legend.}
+\end{figure}
+
+\section{Reflectivity Scans}
+\label{ssec:test_id31_scans_reflectivity}
+
+X-ray reflectivity consists of scanning the \(R_y\) angle of thin structures (typically solid/liquid interfaces) through the beam.
+Here, a \(R_y\) scan is performed with a rotational velocity of \(100\,\mu rad/s\) and the positioning errors in closed-loop are recorded (Figure \ref{fig:test_id31_reflectivity}).
+It is shown that the NASS is able to keep the point of interest in the beam within specifications.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{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}
+\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}
+\end{center}
+\subcaption{\label{fig:test_id31_reflectivity_ry}$R_y$}
+\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 with precise \(R_y\) positioning and that are then scanned vertically with precise \(D_z\) motion.
+The vertical scan can be performed continuously of using step-by-step motion.
+\paragraph{Step by Step \(D_z\) motion}
+
+Vertical steps are here performed using the nano-hexapod only.
+Step sizes from \(10\,nm\) to \(1\,\mu m\) are tested, and the results are shown in Figure \ref{fig:test_id31_dz_mim_steps}.
+10nm steps can be resolved if detectors are integrating over 50ms (see red curve in Figure \ref{fig:test_id31_dz_mim_10nm_steps}), which is reasonable for many experiments.
+
+When doing step-by-step scans, the time to reach the next value is quite critical as long settling time can render the total experiment excessively long.
+The response time to reach the wanted value (to within \(\pm 20\,nm\)) is around \(70\,ms\) as shown with the \(1\,\mu m\) step response in Figure \ref{fig:test_id31_dz_mim_1000nm_steps}.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{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}
+\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}
+\end{center}
+\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu$m step}
+\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}
+
+\paragraph{Continuous \(D_z\) motion: Dirty Layer Scans}
+
+\begin{itemize}
+\item[{$\square$}] In this section and the following experiments, the NASS performs ``ramp scans'' (i.e. constant velocity scans).
+In order to have no tracking errors, two integrators needs to be present in the feedback loop.
+As the plant present not integral action at low frequency, two integrators are included in the controller.
+\end{itemize}
+
+Instead of performing ``step-by-step'' scans, continuous scans can also be performed in the vertical direction.
+At \(10\,\mu m/s\), the errors are well within the specifications (see Figure \ref{fig:test_id31_dz_scan_10ums}).
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{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}
+\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}
+\end{center}
+\subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$}
+\end{subfigure}
+\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan with a velocity of \(10\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry})}
+\end{figure}
+
+The second tested velocity is \(100\,\mu m/s\), which is the fastest velocity for \(D_z\) scans when the ultimate performances is wanted (corresponding to a 1ms integration time and 100nm ``resolution'').
+At this velocity, the positioning errors are also within the specifications except for the very start and very end of the motion (i.e. during acceleration/deceleration phases, see Figure \ref{fig:test_id31_dz_scan_100ums}).
+However, the detectors are usually triggered only during the constant velocity phase, so this is not not an issue.
+The performances during acceleration phase may also be improved by using a feedforward controller.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{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}
+\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}
+\end{center}
+\subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$}
+\end{subfigure}
+\caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan with 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}
+
+\paragraph{Summary}
+
+\begin{center}
+\begin{tabular}{lrrr}
+ & \(D_y\) & \(D_z\) & \(R_y\)\\
+\hline
+Specs & 100.0 & 50.0 & 0.85\\
+10um/s & 82.35 & 17.94 & 0.41\\
+100um/s & 98.72 & 41.45 & 0.48\\
+\end{tabular}
+\end{center}
+
+\begin{center}
+\begin{tabular}{lrrr}
+ & \(D_y\) & \(D_z\) & \(R_y\)\\
+\hline
+Specs & 30.0 & 15.0 & 0.25\\
+10um/s & 25.11 & 5.04 & 0.11\\
+100um/s & 34.84 & 9.08 & 0.13\\
+\end{tabular}
+\end{center}
+
+\section{Lateral Scans}
 \label{ssec:test_id31_scans_dy}
 Lateral scans are performed with the \(T_y\) stage.
 The stepper motor controller\footnote{The ``IcePAP'' \cite{janvier13_icepap}  which is developed at the ESRF} outputs the setpoint which is received by the Speedgoat.
@@ -958,7 +1208,6 @@ Specs & 30.0 & 15.0 & 0.25\\
 100um/s (OL) & 1063.58 & 166.85 & 6.44\\
 100um/s (CL) & 731.63 & 19.91 & 0.36\\
 \end{tabular}
-
 \end{center}
 
 \begin{center}
@@ -971,153 +1220,9 @@ Specs & 100.0 & 50.0 & 0.85\\
 100um/s (OL) & 2687.67 & 328.45 & 11.26\\
 100um/s (CL) & 1339.31 & 69.5 & 0.91\\
 \end{tabular}
-
 \end{center}
 
-\section{\(D_z\) scans: Dirty Layer Scans}
+\section{Diffraction Tomography}
-\label{ssec:test_id31_scans_dz}
-In some cases, samples are composed of several atomic ``layers'' that are first aligned in the horizontal plane with precise \(R_y\) positioning and that are then scanned vertically with precise \(D_z\) motion.
-The vertical scan can be performed continuously of using step-by-step motion.
-\paragraph{Step by Step \(D_z\) motion}
-
-Vertical steps are here performed using the nano-hexapod only.
-Step sizes from \(10\,nm\) to \(1\,\mu m\) are tested, and the results are shown in Figure \ref{fig:test_id31_dz_mim_steps}.
-10nm steps can be resolved if detectors are integrating over 50ms (see red curve in Figure \ref{fig:test_id31_dz_mim_10nm_steps}), which is reasonable for many experiments.
-
-When doing step-by-step scans, the time to reach the next value is quite critical as long settling time can render the total experiment excessively long.
-The response time to reach the wanted value (to within \(\pm 20\,nm\)) is around \(70\,ms\) as shown with the \(1\,\mu m\) step response in Figure \ref{fig:test_id31_dz_mim_1000nm_steps}.
-
-\begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,scale=1]{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}
-\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}
-\end{center}
-\subcaption{\label{fig:test_id31_dz_mim_1000nm_steps}$1\,\mu$m step}
-\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}
-
-\paragraph{Continuous \(D_z\) motion: Dirty Layer Scans}
-
-Instead of performing ``step-by-step'' scans, continuous scans can also be performed in the vertical direction.
-At \(10\,\mu m/s\), the errors are well within the specifications (see Figure \ref{fig:test_id31_dz_scan_10ums}).
-
-\begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,scale=1]{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}
-\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}
-\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_10ums_ry}$R_y$}
-\end{subfigure}
-\caption{\label{fig:test_id31_dz_scan_10ums}\(D_z\) scan with a velocity of \(10\,\mu m/s\). \(D_z\) setpoint, measured position and error are shown in (\subref{fig:test_id31_dz_scan_10ums_dz}). Errors in \(D_y\) and \(R_y\) are respectively shown in (\subref{fig:test_id31_dz_scan_10ums_dy}) and (\subref{fig:test_id31_dz_scan_10ums_ry})}
-\end{figure}
-
-The second tested velocity is \(100\,\mu m/s\), which is the fastest velocity for \(D_z\) scans when the ultimate performances is wanted (corresponding to a 1ms integration time and 100nm ``resolution'').
-At this velocity, the positioning errors are also within the specifications except for the very start and very end of the motion (i.e. during acceleration/deceleration phases, see Figure \ref{fig:test_id31_dz_scan_100ums}).
-However, the detectors are usually triggered only during the constant velocity phase, so this is not not an issue.
-The performances during acceleration phase may also be improved by using a feedforward controller.
-
-\begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,scale=1]{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}
-\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}
-\end{center}
-\subcaption{\label{fig:test_id31_dz_scan_100ums_ry}$R_y$}
-\end{subfigure}
-\caption{\label{fig:test_id31_dz_scan_100ums}\(D_z\) scan with 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}
-
-\paragraph{Summary}
-
-\begin{center}
-\begin{tabular}{lrrr}
- & \(D_y\) & \(D_z\) & \(R_y\)\\
-\hline
-Specs & 100.0 & 50.0 & 0.85\\
-10um/s & 82.35 & 17.94 & 0.41\\
-100um/s & 98.72 & 41.45 & 0.48\\
-\end{tabular}
-
-\end{center}
-
-\begin{center}
-\begin{tabular}{lrrr}
- & \(D_y\) & \(D_z\) & \(R_y\)\\
-\hline
-Specs & 30.0 & 15.0 & 0.25\\
-10um/s & 25.11 & 5.04 & 0.11\\
-100um/s & 34.84 & 9.08 & 0.13\\
-\end{tabular}
-
-\end{center}
-
-\section{\(R_y\) scans: Reflectivity}
-\label{ssec:test_id31_scans_reflectivity}
-
-X-ray reflectivity consists of scanning the \(R_y\) angle of thin structures (typically solid/liquid interfaces) through the beam.
-Here, a \(R_y\) scan is performed with a rotational velocity of \(100\,\mu rad/s\) and the positioning errors in closed-loop are recorded (Figure \ref{fig:test_id31_reflectivity}).
-It is shown that the NASS is able to keep the point of interest in the beam within specifications.
-
-\begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,scale=1]{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}
-\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}
-\end{center}
-\subcaption{\label{fig:test_id31_reflectivity_ry}$R_y$}
-\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{Combined \(R_z\) and \(D_y\): Diffraction Tomography}
 \label{ssec:test_id31_scans_diffraction_tomo}
 
 In diffraction tomography, the micro-station performs combined \(R_z\) rotation and \(D_y\) lateral scans.
@@ -1169,7 +1274,6 @@ Specs & 100.0 & 50.0 & 0.85\\
 0.5 mm/s & 117.94 & 28.03 & 0.27\\
 1 mm/s & 186.88 & 33.02 & 0.53\\
 \end{tabular}
-
 \end{center}
 
 \begin{center}
@@ -1181,7 +1285,6 @@ Specs & 30.0 & 15.0 & 0.25\\
 0.5 mm/s & 28.58 & 7.52 & 0.08\\
 1 mm/s & 53.05 & 9.84 & 0.14\\
 \end{tabular}
-
 \end{center}
 
 \section*{Conclusion}
@@ -1190,6 +1293,7 @@ Specs & 30.0 & 15.0 & 0.25\\
 For each conducted experiments, the \(D_y\), \(D_z\) and \(R_y\) errors are computed and summarized in Table \ref{tab:id31_experiments_results_summary}.
 
 \begin{table}[htbp]
+\caption{\label{tab:id31_experiments_results_summary}Table caption}
 \centering
 \begin{tabularx}{\linewidth}{Xccc}
 \toprule
@@ -1212,8 +1316,6 @@ Diffraction Tomography (\(R_z\) 1rpm, \(D_y\) 0.1mm/s) & 75 & 9 & 118\\
 Diffraction Tomography (\(R_z\) 1rpm, \(D_y\) 1mm/s) & 428 & 11 & 169\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:id31_experiments_results_summary}Table caption}
-
 \end{table}
 
 \chapter*{Conclusion}