Add Instrumentation section

2025-04-06 18:29:30 +02:00 · 2025-04-06 18:29:30 +02:00 · 2f2bd4c3b9
commit 2f2bd4c3b9
parent 647d60a53d
4 changed files with 1491 additions and 56 deletions
--- a/phd-thesis.bib
+++ b/phd-thesis.bib
@ -1330,16 +1330,71 @@
-@article{wehrsdorfer95_large_signal_measur_piezoel_stack,
+@techreport{spengen20_high_voltag_amplif,
-  author          = {Wehrsdorfer, E and Borchhardt, G and Karthe, W and Helke,
+  author          = {W. Merlijn van Spengen},
-                  G},
+  institution     = {Falco Systems},
-  title           = {Large Signal Measurements on Piezoelectric Stacks},
+  title           = {High Voltage Amplifiers: So you think you have noise!},
-  journal         = {Ferroelectrics},
+  year            = 2020,
-  volume          = 174,
+}
-  number          = 1,
+
-  pages           = {259--275},
+
-  year            = 1995,
+
-  publisher       = {Taylor \& Francis},
+@book{abramovitch22_pract_method_real_world_contr_system,
  author          = {Daniel Y. Abramovitch},
  title           = {Practical Methods for Real World Control Systems},
  year            = 2022,
  publisher       = {self-publishing},
 }
@inproceedings{abramovitch23_tutor_real_time_comput_issues_contr_system,
  author          = {Daniel Y. Abramovitch and Sean Andersson and Kam K. Leang
                  and William Nagel and Shalom Ruben},
  title           = {A Tutorial on Real-Time Computing Issues for Control
                  Systems},
  booktitle       = {2023 American Control Conference (ACC)},
  year            = 2023,
  pages           = {3751-3768},
  doi             = {10.23919/acc55779.2023.10156102},
  url             = {http://dx.doi.org/10.23919/ACC55779.2023.10156102},
  DATE_ADDED      = {Tue Dec 17 10:56:35 2024},
  month           = 5,
 }
@article{fleming13_review_nanom_resol_posit_sensor,
  author          = {Andrew J. Fleming},
  title           = {A Review of Nanometer Resolution Position Sensors:
                  Operation and Performance},
  journal         = {Sensors and Actuators A: Physical},
  volume          = 190,
  pages           = {106-126},
  year            = 2013,
  doi             = {10.1016/j.sna.2012.10.016},
  url             = {https://doi.org/10.1016/j.sna.2012.10.016},
  keywords        = {favorite, metrology},
 }
@techreport{lab13_improv_adc,
  author          = {Silicon Lab},
  institution     = {Silicon Laboratories},
  title           = {Improving the {ADC} resolution by oversampling and
                  averaging},
  year            = 2013,
 }
@article{hauser91_princ_overs_conver,
  author          = {Max Hauser},
  title           = {Principles of Oversampling A/d Conversion},
  journal         = {Journal of Audio Engineering Society},
  year            = 1991,
 }
@ -1357,6 +1412,21 @@
@article{wehrsdorfer95_large_signal_measur_piezoel_stack,
  author          = {Wehrsdorfer, E and Borchhardt, G and Karthe, W and Helke,
                  G},
  title           = {Large Signal Measurements on Piezoelectric Stacks},
  journal         = {Ferroelectrics},
  volume          = 174,
  number          = 1,
  pages           = {259--275},
  year            = 1995,
  publisher       = {Taylor \& Francis},
 }
@article{watchi18_review_compac_inter,
  author          = {Watchi, Jennifer and Cooper, Sam and Ding, Binlei and
                  Mow-Lowry, Conor M. and Collette, Christophe},
--- a/phd-thesis.org
+++ b/phd-thesis.org
@ -9728,9 +9728,806 @@ Such model reduction, guided by detailed understanding of component behavior, pr
 <<sec:detail_control>>
 # [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/B3-control/nass-control.org][NASS - Control]]
-** TODO Choice of Instrumentation
+** Choice of Instrumentation
 <<sec:detail_instrumentation>>
-# [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/B4-nass-instrumentation/nass-instrumentation.org][NASS - Instrumentation]]
+*** Introduction                                                        :ignore:
 This chapter presents an approach to select and validate appropriate instrumentation for the Nano Active Stabilization System (NASS), ensuring each component meets specific performance requirements.
 Figure ref:fig:detail_instrumentation_plant illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis.
 The selection process follows a three-stage methodology.
 First, dynamic error budgeting is performed in Section ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier).
 This analysis utilizes the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements.
 From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived.
 Section ref:sec:detail_instrumentation_choice then presents the selection of appropriate components based on these noise specifications and additional requirements.
 Finally, Section ref:sec:detail_instrumentation_characterization validates the selected components through experimental testing.
 Each instrument is characterized individually, measuring actual noise levels and performance characteristics.
 The measured noise characteristics are then incorporated into the multi-body model to confirm that the combined effect of all instrumentation noise sources remains within acceptable limits.
 #+begin_src latex :file detail_instrumentation_plant.pdf
 \begin{tikzpicture}
  % Blocs
  \node[block={2.0cm}{2.0cm}, align=center] (plant) {NASS};
  \coordinate[] (inputVa)  at ($(plant.south west)!0.5!(plant.north west)$);
  \coordinate[] (outputVs) at ($(plant.south east)!0.7!(plant.north east)$);
  \coordinate[] (outputde) at ($(plant.south east)!0.3!(plant.north east)$);
  \node[addb={+}{}{}{}{}, left=0.8 of inputVa] (ampl_noise) {};
  \node[block={1.0cm}{1.0cm}, left=0.4 of ampl_noise] (ampl_tf) {$G_{\text{ampl}}$};
  \node[addb={+}{}{}{}{}, left=0.8 of ampl_tf] (dac_noise) {};
  \node[DAC, left=0.4 of dac_noise] (dac_tf) {};
  \node[addb={+}{}{}{}{}, left=1.0 of dac_tf] (iff_sum) {};
  \node[block={1.0cm}{1.0cm}, above=0.4 of iff_sum] (Kiff) {$\bm{K}_{\text{IFF}}$};
  \node[block={1.0cm}{1.0cm}, left=0.4 of iff_sum] (Khac) {$\bm{K}_{\text{HAC}}$};
  \node[addb={+}{}{}{}{}, right=0.8 of outputVs] (adc_noise) {};
  \node[ADC, right=0.4 of adc_noise] (adc_tf) {};
  \draw[->] (iff_sum.east)    --node[midway, above]{$\bm{u}$} node[near start, sloped]{$/$} (dac_tf.west);
  \draw[->] (dac_tf.east)     -- (dac_noise.west);
  \draw[->] (dac_noise.east)  -- (ampl_tf.west);
  \draw[->] (ampl_tf.east)    -- (ampl_noise.west);
  \draw[->] (ampl_noise.east)    -- (inputVa)node[above left]{$\bm{V}_a$};
  \draw[->] (outputVs)node[above right]{$\bm{V}_s$} -- (adc_noise.west);
  \draw[->] (adc_noise.east)     -- (adc_tf.west);
  \draw[->] (adc_tf.east)  -| ++(0.4, 1.8) -| node[near start, sloped]{$/$} (Kiff.north);
  \draw[->] (Kiff.south)       -- node[sloped]{$/$} (iff_sum.north);
  \draw[->] (outputde)node[above right]{$\bm{\epsilon}_{\mathcal{L}}$} -| ++(0.6, -1.0) -| node[near start, sloped]{$/$} ($(Khac.west)+(-0.6, 0)$) -- (Khac.west);
  \draw[->] (Khac.east)       -- node[sloped]{$/$} (iff_sum.west);
  \draw[<-] (dac_noise.north)  -- ++(0, 0.8)coordinate(dac_noise_input)  node[below left]{$n_{\text{da}}$};
  \draw[<-] (ampl_noise.north) -- ++(0, 0.8)coordinate(ampl_noise_input) node[below left]{$n_{\text{amp}}$};
  \draw[<-] (adc_noise.north)  -- ++(0, 0.8)coordinate(adc_noise_input)  node[below right]{$n_{\text{ad}}$};
  \begin{scope}[on background layer]
    \node[fit={(dac_tf.south west) (dac_noise.east|-dac_noise_input)}, fill=colorblue!20!white, draw, dashed, inner sep=4pt] (dac_system) {};
    \node[anchor={north}] at (dac_system.south){$\text{DAC}$};
  \end{scope}
  \begin{scope}[on background layer]
    \node[fit={(ampl_tf.south west) (ampl_noise.east|-ampl_noise_input)}, fill=colorred!20!white, draw, dashed, inner sep=4pt] (ampl_system) {};
    \node[anchor={north}] at (ampl_system.south){$\text{Amplifier}$};
  \end{scope}
  \begin{scope}[on background layer]
    \node[fit={(adc_noise.south -| adc_noise.west) (adc_tf.east|-adc_noise_input)}, fill=coloryellow!20!white, draw, dashed, inner sep=4pt] (adc_system) {};
    \node[anchor={north}] at (adc_system.south){$\text{ADC}$};
  \end{scope}
  \begin{scope}[on background layer]
    \node[fit={(Khac.south west) (Kiff.north east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (control_system) {};
    \node[anchor={north}] at (control_system.south){$\text{RT Controller}$};
  \end{scope}
 \end{tikzpicture}
 #+end_src
 #+name: fig:detail_instrumentation_plant
 #+caption: Block diagram of the NASS with considered instrumentation. The RT controller is a Speedgoat machine.
 #+RESULTS:
 [[file:figs/detail_instrumentation_plant.png]]
 *** Dynamic Error Budgeting
 <<sec:detail_instrumentation_dynamic_error_budgeting>>
 **** Introduction                                                      :ignore:
 The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system.
 The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section ref:ssec:detail_instrumentation_cl_sensitivity).
 This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs.
 Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints.
 From these transfer functions, the maximum acceptable Amplitude Spectral Density (ASD) of the noise sources is derived (Section ref:ssec:detail_instrumentation_max_noise_specs).
 Since the voltage amplifier gain affects the amplification of DAC noise, an assumption of an amplifier gain of 20 was made.
 **** Closed-Loop Sensitivity to Instrumentation Disturbances
 <<ssec:detail_instrumentation_cl_sensitivity>>
 Several key noise sources are considered in the analysis (Figure ref:fig:detail_instrumentation_plant).
 These include the output voltage noise of the DAC ($n_{da}$), the output voltage noise of the voltage amplifier ($n_{amp}$), and the voltage noise of the ADC measuring the force sensor stacks ($n_{ad}$).
 Encoder noise, which is only used to estimate $R_z$, has been found to have minimal impact on the vertical sample error and is therefore omitted from this analysis for clarity.
 The transfer functions from these three noise sources (for one strut) to the vertical error of the sample are estimated from the multi-body model, which includes the APA300ML and the designed flexible joints (Figure ref:fig:detail_instrumentation_noise_sensitivities).
 #+name: fig:detail_instrumentation_noise_sensitivities
 #+caption: Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy.
 #+RESULTS:
 [[file:figs/detail_instrumentation_noise_sensitivities.png]]
 **** Estimation of maximum instrumentation noise
 <<ssec:detail_instrumentation_max_noise_specs>>
 The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of $100\,\text{nm}$, which corresponds to a maximum allowed vibration of $15\,\text{nm RMS}$.
 Several assumptions regarding the noise characteristics have been made.
 The DAC, ADC, and amplifier noise are considered uncorrelated, which is a reasonable assumption.
 Similarly, the noise sources corresponding to each strut are also assumed to be uncorrelated.
 This means that the power spectral densities (PSD) of the different noise sources are summed.
 Since the effect of each strut on the vertical error is identical due to symmetry, only one strut is considered for this analysis, and the total effect of the six struts is calculated as six times the effect of one strut in terms of power, which translates to a factor of $\sqrt{6} \approx 2.5$ for RMS values.
 In order to derive specifications in terms of noise spectral density for each instrumentation component, a white noise profile is assumed, which is typical for these components.
 The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
 While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications.
 Based on this analysis, the obtained maximum noise levels are as follows: DAC maximum output noise ASD is established at $14\,\mu V/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise ASD at $280\,\mu V/\sqrt{\text{Hz}}$, and ADC maximum measurement noise ASD at $11\,\mu V/\sqrt{\text{Hz}}$.
 In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the DAC, less than $20\,\text{mV RMS}$ for the voltage amplifier, and less than $0.8\,\text{mV RMS}$ for the ADC.
 If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS.
 *** Choice of Instrumentation
 <<sec:detail_instrumentation_choice>>
 **** Piezoelectric Voltage Amplifier
 ***** Introduction                                                :ignore:
 Several characteristics of piezoelectric voltage amplifiers must be considered for this application.
 To take advantage of the full stroke of the piezoelectric actuator, the voltage output should range between $-20$ and $150\,V$.
 The amplifier should accept an analog input voltage, preferably in the range of $-10$ to $10\,V$, as this is standard for most DACs.
 ***** Small signal Bandwidth and Output Impedance
 Small signal bandwidth is particularly important for feedback applications as it can limit the overall bandwidth of the complete feedback system.
 A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure ref:fig:detail_instrumentation_amp_output_impedance.
 This model is valid for small signals and provides insight into the small signal bandwidth limitation [[cite:&fleming14_desig_model_contr_nanop_system, chap. 14]].
 In this model, $R_o$ represents the output impedance of the amplifier.
 When combined with the piezoelectric load (represented as a capacitance $C_p$), it forms a first order low pass filter described by eqref:eq:detail_instrumentation_amp_output_impedance.
 \begin{equation}\label{eq:detail_instrumentation_amp_output_impedance}
  \frac{V_a}{V_i}(s) = \frac{1}{1 + \frac{s}{\omega_0}}, \quad \omega_0 = \frac{1}{R_o C_p}
 \end{equation}
 #+name: fig:detail_instrumentation_amp_output_impedance
 #+caption: Electrical model of a voltage amplifier with output impedance $R_0$ connected to a piezoelectric stack with capacitance $C_p$
 [[file:figs/detail_instrumentation_amp_output_impedance.png]]
 Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases.
 For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to $C_p = 8.8\,\mu F$.
 If a small signal bandwidth of $f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}$ is desired, the voltage amplifier output impedance should be less than $R_0 = 3.6\,\Omega$.
 ***** Large signal Bandwidth
 Large signal bandwidth relates to the maximum output capabilities of the amplifier in terms of amplitude as a function of frequency.
 Since the primary function of the NASS is position stabilization rather than scanning, this specification is less critical than the small signal bandwidth.
 However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of $f_r = 100\,\text{Hz}$ using the full voltage range of the piezoelectric actuator ($V_{pp} = 170\,V$) is considered.
 There are two limiting factors for large signal bandwidth that should be evaluated:
 1. Slew rate, which should exceed $2 \cdot V_{pp} \cdot f_r = 34\,V/ms$.
   This requirement is typically easily met by commercial voltage amplifiers.
 2. Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies.
   To achieve high voltage at high frequency, the amplifier must therefore provide substantial current.
   The maximum required current can be calculated as $I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A$.
 Therefore, ideally, a voltage amplifier capable of providing $0.3\,A$ of current would be interesting for scanning applications.
 ***** Output voltage noise
 As established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the output noise of the voltage amplifier should be below $20\,\text{mV RMS}$.
 It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure ref:fig:detail_instrumentation_amp_output_impedance.
 Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement [[cite:&spengen20_high_voltag_amplif]].
 For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\mu F$ and a bandwidth exceeding $5\,\text{kHz}$.
 ***** Choice of voltage amplifier
 The specifications are summarized in Table ref:tab:detail_instrumentation_amp_choice.
 The most critical characteristics are the small signal bandwidth ($>5\,\text{kHz}$) and the output voltage noise ($<20\,\text{mV RMS}$).
 Several voltage amplifiers were considered, with their datasheet information presented in Table ref:tab:detail_instrumentation_amp_choice.
 One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the ASD of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies.
 This approach does not account for the frequency dependency of the noise, which is crucial for accurate error budgeting.
 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.
 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: :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       |
 **** ADC and DAC
 ***** Introduction                                                :ignore:
 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.
 ***** Synchronicity and Jitter
 For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements [[cite:&abramovitch22_pract_method_real_world_contr_system;&abramovitch23_tutor_real_time_comput_issues_contr_system]].
 Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals.
 Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration.
 ***** Sampling Frequency, Bandwidth and delays
 Several requirements that may initially appear similar are actually distinct in nature.
 First, the /sampling frequency/ defines the interval between two sampled points and determines the Nyquist frequency.
 Then, the /bandwidth/ specifies the maximum frequency of a measured signal (typically defined as the -3dB point) and is often limited by implemented anti-aliasing filters.
 Finally, /delay/ (or /latency/) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system.
 Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency.
 Typically, the latency can reach 20 times the sampling period [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 8.4]].
 Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor.
 For real-time control applications, SAR-ADCs (Successive Approximation ADCs) remain the predominant choice due to their single-sample latency characteristics.
 ***** ADC Noise
 Based on the dynamic error budget established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise ASD should not exceed $11\,\mu V/\sqrt{\text{Hz}}$.
 ADCs are subject to various noise sources.
 Quantization noise, which results from the discrete nature of digital representation, is one of these sources.
 To determine the minimum bit depth $n$ required to meet the noise specifications, an ideal ADC where quantization error is the only noise source is considered.
 The quantization step size, denoted as $q = \Delta V/2^n$, represents the voltage equivalent of the least significant bit, with $\Delta V$ the full range of the ADC in volts, and $F_s$ the sampling frequency in Hertz.
 The quantization noise ranges between $\pm q/2$, and its probability density function is constant across this range (uniform distribution).
 Since the integral of this probability density function $p(e)$ equals one, its value is $1/q$ for $-q/2 < e < q/2$, as illustrated in Figure ref:fig:detail_instrumentation_adc_quantization.
 #+begin_src latex :file detail_instrumentation_adc_quantization.pdf
 \begin{tikzpicture}
  \path[fill=black!20!white] (-1, 0) |- (1, 1) |- (-1, 0);
  \draw[->] (-2, 0) -- (2, 0) node[above left]{$e$};
  \draw[->] (0, -0.5) -- (0, 2) node[below right]{$p(e)$};
  \draw[dashed] (-2, 0) -- (-1, 0) |- (1, 1) |- (2, 0);
  \node[below] at (1, 0){$\frac{q}{2}$};
  \node[below] at (-1, 0){$-\frac{q}{2}$};
  \node[right] at (1, 1){$\frac{1}{q}$};
 \end{tikzpicture}
 #+end_src
 #+name: fig:detail_instrumentation_adc_quantization
 #+caption: Probability density function $p(e)$ of the ADC quantization error $e$
 #+RESULTS:
 [[file:figs/detail_instrumentation_adc_quantization.png]]
 The variance (or time-average power) of the quantization noise is expressed by eqref:eq:detail_instrumentation_quant_power.
 \begin{equation}\label{eq:detail_instrumentation_quant_power}
  P_q = \int_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12}
 \end{equation}
 To compute the power spectral density of the quantization noise, which is defined as the Fourier transform of the noise's autocorrelation function, it is assumed that noise samples are uncorrelated.
 Under this assumption, the autocorrelation function approximates a delta function in the time domain.
 Since the Fourier transform of a delta function equals one, the power spectral density becomes frequency-independent (white noise).
 By Parseval's theorem, the power spectral density of the quantization noise $\Phi_q$ can be linked to the ADC sampling frequency and quantization step size eqref:eq:detail_instrumentation_psd_quant_noise.
 \begin{equation}\label{eq:detail_instrumentation_psd_quant_noise}
  \int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_q = \frac{q^2}{12 F_s} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 F_s} \quad \text{in } \left[ \frac{V^2}{\text{Hz}} \right]
 \end{equation}
 From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the minimum number of bits required to keep quantization noise below $\Gamma_{\text{max}}$ is calculated using eqref:eq:detail_instrumentation_min_n.
 \begin{equation}\label{eq:detail_instrumentation_min_n}
  n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right)
 \end{equation}
 With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,V$ and a maximum allowed ASD $\Gamma_{\text{max}} = 11\,\mu V/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial ADCs.
 ***** DAC Output voltage noise
 Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed $14\,\mu V/\sqrt{\text{Hz}}$.
 This specification corresponds to a $\pm 10\,V$ DAC with 13-bit resolution, which is easily attainable with current technology.
 ***** Choice of the ADC and DAC Board
 Based on the preceding analysis, the selection of suitable ADC and DAC components is straightforward.
 For optimal synchronicity, a Speedgoat-integrated solution was chosen.
 The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, $\pm 10\,V$ range, maximum sampling rate of 200kSPS, simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity.
 The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, $\pm 10\,V$ range, conversion time of $10\,\mu s$, and simultaneous update capability.
 Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements.
 This will be experimentally verified in Section ref:sec:detail_instrumentation_characterization.
 **** Relative Displacement Sensors
 The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below $6\,\text{nm RMS}$ (derived from the $15\,\text{nm RMS}$ vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding $100\,\mu m$.
 Several sensor technologies are capable of meeting these requirements [[cite:&fleming13_review_nanom_resol_posit_sensor]].
 These include optical encoders (Figure ref:fig:detail_instrumentation_sensor_encoder), capacitive sensors (Figure ref:fig:detail_instrumentation_sensor_capacitive), and eddy current sensors (Figure ref:fig:detail_instrumentation_sensor_eddy_current), each with their own advantages and implementation considerations.
 #+name: fig:detail_instrumentation_sensor_examples
 #+caption: Relative motion sensors considered for measuring the nano-hexapod strut motion
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_sensor_encoder}Optical Linear Encoder}
 #+attr_latex: :options {0.33\textwidth}
 #+begin_subfigure
 #+attr_latex: :width 0.9\linewidth
 [[file:figs/detail_instrumentation_sensor_encoder.jpg]]
 #+end_subfigure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_sensor_eddy_current}Eddy Current Sensor}
 #+attr_latex: :options {0.33\textwidth}
 #+begin_subfigure
 #+attr_latex: :width 0.9\linewidth
 [[file:figs/detail_instrumentation_sensor_eddy_current.png]]
 #+end_subfigure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_sensor_capacitive}Capacitive Sensor}
 #+attr_latex: :options {0.33\textwidth}
 #+begin_subfigure
 #+attr_latex: :width 0.9\linewidth
 [[file:figs/detail_instrumentation_sensor_capacitive.jpg]]
 #+end_subfigure
 #+end_figure
 From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the APA, as illustrated in Figure ref:fig:detail_instrumentation_capacitive_implementation.
 In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure ref:fig:detail_instrumentation_encoder_implementation.
 #+name: fig:detail_instrumentation_sensor_implementation
 #+caption: Implementation of relative displacement sensor to measure the motion of the APA
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_encoder_implementation}Optical Encoder}
 #+attr_latex: :options {0.48\textwidth}
 #+begin_subfigure
 #+attr_latex: :scale 1
 [[file:figs/detail_instrumentation_encoder_implementation.png]]
 #+end_subfigure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_capacitive_implementation}Capacitive Sensor}
 #+attr_latex: :options {0.48\textwidth}
 #+begin_subfigure
 #+attr_latex: :scale 1
 [[file:figs/detail_instrumentation_capacitive_implementation.png]]
 #+end_subfigure
 #+end_figure
 A significant consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
 Measurements conducted on the slip-ring integrated in the micro-station revealed substantial cross-talk between different slip-ring channels.
 To mitigate this issue, preference was given to sensors that transmit displacement measurements digitally, as these are inherently less susceptible to noise and cross-talk.
 Based on this criterion, an optical encoder with digital output was selected, where signal interpolation is performed directly in the sensor head.
 The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table ref:tab:detail_instrumentation_sensor_specs, alongside alternative options that were considered.
 #+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: :center t :booktabs t :float t
 | *Specification*             | *Renishaw Vionic*   | LION CPL190 | Cedrat ECP500 |
 |-----------------------------+---------------------+-------------+---------------|
 | Technology                  | Digital Encoder     | Capacitive  | Eddy Current  |
 | Bandwidth $> 5\,\text{kHz}$ | $> 500\,\text{kHz}$ | 10kHz       | 20kHz         |
 | Noise $< 6\,nm\,\text{RMS}$ | 1.6 nm rms          | 4 nm rms    | 15 nm rms     |
 | Range  $> 100\,\mu m$       | Ruler length        | 250 um      | 500um         |
 | In line measurement         |                     | $\times$    | $\times$      |
 | Digital Output              | $\times$            |             |               |
 *** Characterization of Instrumentation
 <<sec:detail_instrumentation_characterization>>
 **** Analog to Digital Converters
 ***** Measured Noise
 The measurement of ADC noise was performed by short-circuiting its input with a $50\,\Omega$ resistor and recording the digital values at a sampling rate of $10\,\text{kHz}$.
 The amplitude spectral density of the recorded values was computed and is presented in Figure ref:fig:detail_instrumentation_adc_noise_measured.
 The ADC noise exhibits characteristics of white noise with an amplitude spectral density of $5.6\,\mu V/\sqrt{\text{Hz}}$ (equivalent to $0.4\,\text{mV RMS}$), which satisfies the established specifications.
 All ADC channels demonstrated similar performance, so only one channel's noise profile is shown.
 If necessary, oversampling can be applied to further reduce the noise cite:lab13_improv_adc.
 To gain $w$ additional bits of resolution, the oversampling frequency $f_{os}$ should be set to $f_{os} = 4^w \cdot F_s$.
 Given that the ADC can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4).
 This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV) [[cite:hauser91_princ_overs_conver]].
 #+name: fig:detail_instrumentation_adc_noise_measured
 #+caption: Measured ADC noise (IO318)
 #+RESULTS:
 [[file:figs/detail_instrumentation_adc_noise_measured.png]]
 ***** Reading of piezoelectric force sensor
 To further validate the ADC's capability to effectively measure voltage generated by a piezoelectric stack, a test was conducted using the APA95ML.
 The setup is illustrated in Figure ref:fig:detail_instrumentation_force_sensor_adc_setup, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor.
 The voltage amplifier employed in this setup has a gain of 20.
 #+name: fig:detail_instrumentation_force_sensor_adc_setup
 #+caption: Schematic of the setup to validate the use of the ADC for reading the force sensor volage
 [[file:figs/detail_instrumentation_force_sensor_adc_setup.png]]
 Step signals with an amplitude of $1\,V$ were generated using the DAC, and the ADC signal was recorded.
 The excitation signal (steps) and the measured voltage across the sensor stack are displayed in Figure ref:fig:detail_instrumentation_step_response_force_sensor.
 Two notable observations were made: an offset voltage of $2.26\,V$ was present, and the measured voltage exhibited an exponential decay response to the step input.
 These phenomena can be explained by examining the electrical schematic shown in Figure ref:fig:detail_instrumentation_force_sensor_adc, where the ADC has an input impedance $R_i$ and an input bias current $i_n$.
 The input impedance $R_i$ of the ADC, in combination with the capacitance $C_p$ of the piezoelectric stack sensor, forms an RC circuit with a time constant $\tau = R_i C_p$.
 The charge generated by the piezoelectric effect across the stack's capacitance gradually discharges into the input resistor of the ADC.
 Consequently, the transfer function from the generated voltage $V_p$ to the measured voltage $V_{\text{ADC}}$ is a first-order high-pass filter with the time constant $\tau$.
 An exponential curve was fitted to the experimental data, yielding a time constant $\tau = 6.5\,s$.
 With the capacitance of the piezoelectric sensor stack being $C_p = 4.4\,\mu F$, the internal impedance of the Speedgoat ADC was calculated as $R_i = \tau/C_p = 1.5\,M\Omega$, which closely aligns with the specified value of $1\,M\Omega$ found in the datasheet.
 #+name: fig:detail_instrumentation_force_sensor
 #+caption: Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from cite:reza06_piezoel_trans_vibrat_contr_dampin. Measured voltage $V_s$ while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}).
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_force_sensor_adc}Electrical Schematic}
 #+attr_latex: :options {0.61\textwidth}
 #+begin_subfigure
 #+attr_latex: :scale 1
 [[file:figs/detail_instrumentation_force_sensor_adc.png]]
 #+end_subfigure
 #+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
 [[file:figs/detail_instrumentation_step_response_force_sensor.png]]
 #+end_subfigure
 #+end_figure
 The constant voltage offset can be explained by the input bias current $i_n$ of the ADC, represented in Figure ref:fig:detail_instrumentation_force_sensor_adc.
 At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the ADC, and therefore the input bias current $i_n$ passing through the internal resistance $R_i$ produces a constant voltage offset $V_{\text{off}} = R_i \cdot i_n$.
 The input bias current $i_n$ is estimated from $i_n = V_{\text{off}}/R_i = 1.5\mu A$.
 In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor $R_p$ can be added in parallel to the force sensor, as illustrated in Figure ref:fig:detail_instrumentation_force_sensor_adc_R.
 This modification produces two beneficial effects: a reduction of input voltage offset through the relationship $V_{\text{off}} = (R_p R_i)/(R_p + R_i) i_n$, and an increase in the high pass corner frequency $f_c$ according to the equations $\tau = 1/(2\pi f_c) = (R_i R_p)/(R_i + R_p) C_p$.
 To validate this approach, a resistor $R_p \approx 82\,k\Omega$ was added in parallel with the force sensor as shown in Figure ref:fig:detail_instrumentation_force_sensor_adc_R.
 After incorporating this resistor, the same step response tests were performed, with results displayed in Figure ref:fig:detail_instrumentation_step_response_force_sensor_R.
 The measurements confirmed the expected improvements, with a substantially reduced offset voltage ($V_{\text{off}} = 0.15\,V$) and a much faster time constant ($\tau = 0.45\,s$).
 These results validate both the model of the ADC and the effectiveness of the added parallel resistor as a solution.
 #+name: fig:detail_instrumentation_force_sensor_R
 #+caption: Effect of an added resistor $R_p$ in parallel to the force sensor. The electrical schematic is shown in (\subref{fig:detail_instrumentation_force_sensor_adc_R}) and the measured signals in (\subref{fig:detail_instrumentation_step_response_force_sensor_R}).
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:detail_instrumentation_force_sensor_adc_R}Electrical Schematic}
 #+attr_latex: :options {0.61\textwidth}
 #+begin_subfigure
 #+attr_latex: :scale 1
 [[file:figs/detail_instrumentation_force_sensor_adc_R.png]]
 #+end_subfigure
 #+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
 [[file:figs/detail_instrumentation_step_response_force_sensor_R.png]]
 #+end_subfigure
 #+end_figure
 **** Instrumentation Amplifier
 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.
 This was accomplished by short-circuiting its input with a $50\,\Omega$ resistor and measuring the output voltage with the ADC (Figure ref:fig:detail_instrumentation_femto_meas_setup).
 The maximum amplifier gain of 80dB (equivalent to 10000) was used for this measurement.
 The measured voltage $n$ was then divided by 10000 to determine the equivalent noise at the input of the voltage amplifier $n_a$.
 In this configuration, the noise contribution from the ADC $q_{ad}$ is rendered negligible due to the high gain employed.
 The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the (negligible) contribution of the ADC noise are presented in Figure ref:fig:detail_instrumentation_femto_input_noise.
 #+begin_src latex :file detail_instrumentation_femto_meas_setup.pdf
 \begin{tikzpicture}
  \node[block={0.6cm}{0.6cm}] (const) {$0$};
  % Pre Amp
  \node[addb, right=0.4 of const] (addna) {};
  \node[block, right=0.3 of addna] (Ga) {$G_a(s)$};
  % ADC
  \node[addb, right=0.8 of Ga] (addqad){};
  \node[ADC, right=0.3 of addqad] (ADC) {ADC};
  \draw[->] (const.east) -- (addna.west);
  \draw[->] (addna.east) -- (Ga.west);
  \draw[->] (Ga.east)    -- (addqad.west);
  \draw[->] (addqad.east) -- (ADC.west);
  \draw[->] (ADC.east)    -- node[sloped]{$/$} ++(0.8, 0) node[above left]{$n$};
  \draw[<-] (addna.north)  -- ++(0, 0.6) node[below right](na){$n_{a}$};
  \draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
  \coordinate[] (top) at (na.north);
  \coordinate[] (bot) at (Ga.south);
  % 5113
  \begin{scope}[on background layer]
    \node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=4pt, draw, dashed, fill=colorgreen!20!white] (P) {};
    \node[above] at (P.north) {Pre Amp};
  \end{scope}
  % ADC
  \begin{scope}[on background layer]
    \node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=4pt, draw, dashed, fill=coloryellow!20!white] (P) {};
    \node[above] at (P.north) {ADC};
  \end{scope}
 \end{tikzpicture}
 #+end_src
 #+attr_latex: :options [b]{0.48\linewidth}
 #+begin_minipage
 #+name: fig:detail_instrumentation_femto_meas_setup
 #+caption: Measurement of the instrumentation amplifier input voltage noise
 #+attr_latex: :scale 1 :float nil
 [[file:figs/detail_instrumentation_femto_meas_setup.png]]
 #+end_minipage
 \hfill
 #+attr_latex: :options [b]{0.48\linewidth}
 #+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
 [[file:figs/detail_instrumentation_femto_input_noise.png]]
 #+end_minipage
 **** Digital to Analog Converters
 ***** Output Voltage Noise
 To measure the output noise of the DAC, the setup schematically represented in Figure ref:fig:detail_instrumentation_dac_setup was utilized.
 The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC.
 The Amplitude Spectral Density $\Gamma_{n_{da}}(\omega)$ of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement.
 The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure ref:fig:detail_instrumentation_dac_output_noise.
 The noise profile is predominantly white with an ASD of $0.6\,\mu V/\sqrt{\text{Hz}}$.
 Minor $50\,\text{Hz}$ noise is present, along with some low frequency $1/f$ noise, but these are not expected to pose issues as they are well within specifications.
 It should be noted that all DAC channels demonstrated similar performance, so only one channel measurement is presented.
 #+begin_src latex :file detail_instrumentation_dac_setup.pdf
 \begin{tikzpicture}
  \node[block={0.6cm}{0.6cm}] (const) {$0$};
  % DAC
  \node[DAC, right=0.4 of const] (DAC) {DAC};
  \node[addb, right=0.3 of DAC] (addnda){};
  % Pre Amp
  \node[addb, right=0.8 of addnda] (addna) {};
  \node[block, right=0.3 of addna] (Ga) {$G_a(s)$};
  % ADC
  \node[addb, right=0.8 of Ga] (addqad){};
  \node[ADC, right=0.3 of addqad] (ADC) {ADC};
  \draw[->] (const.east) -- node[sloped]{$/$} (DAC.west);
  \draw[->] (DAC.east) -- (addnda.west);
  \draw[->] (addnda.east) -- (addna.west);
  \draw[->] (addna.east) -- (Ga.west);
  \draw[->] (Ga.east) -- (addqad.west);
  \draw[->] (addqad.east) -- (ADC.west);
  \draw[->] (ADC.east) -- node[sloped]{$/$} ++(0.8, 0);
  \draw[<-] (addnda.north) -- ++(0, 0.6) node[below left](nda){$n_{da}$};
  \draw[<-] (addna.north)  -- ++(0, 0.6) node[below right](na){$n_{a}$};
  \draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
  \coordinate[] (top) at (na.north);
  \coordinate[] (bot) at (Ga.south);
  % DAC
  \begin{scope}[on background layer]
    \node[fit={(DAC.west|-bot) (addnda.east|-top)}, inner sep=4pt, draw, dashed, fill=colorblue!20!white] (P) {};
    \node[above] at (P.north) {DAC};
  \end{scope}
  % 5113
  \begin{scope}[on background layer]
    \node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=4pt, draw, dashed, fill=colorgreen!20!white] (P) {};
    \node[above] at (P.north) {Pre Amp};
  \end{scope}
  % ADC
  \begin{scope}[on background layer]
    \node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=4pt, draw, dashed, fill=coloryellow!20!white] (P) {};
    \node[above] at (P.north) {ADC};
  \end{scope}
 \end{tikzpicture}
 #+end_src
 #+name: fig:detail_instrumentation_dac_setup
 #+caption: Measurement of the DAC output voltage noise. A pre-amplifier with a gain of 1000 is used before measuring the signal with the ADC.
 #+RESULTS:
 [[file:figs/detail_instrumentation_dac_setup.png]]
 ***** Delay from ADC to DAC
 To measure the transfer function from DAC to ADC and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the DAC output and the ADC input.
 A white noise signal was generated by the DAC, and the ADC response was recorded.
 The resulting frequency response function from the digital DAC signal to the digital ADC signal is presented in Figure ref:fig:detail_instrumentation_dac_adc_tf.
 The observed frequency response function corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer.
 #+name: fig:detail_instrumentation_dac
 #+caption: Measurement of the output voltage noise of the ADC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a "1-sample" delay.
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+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
 [[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
 [[file:figs/detail_instrumentation_dac_adc_tf.png]]
 #+end_subfigure
 #+end_figure
 **** Piezoelectric Voltage Amplifier
 ***** Output Voltage Noise
 The measurement setup for evaluating the PD200 amplifier noise is illustrated in Figure ref:fig:detail_instrumentation_pd200_setup.
 The input of the PD200 amplifier was shunted with a $50\,\Ohm$ resistor to ensure that only the inherent noise of the amplifier itself was measured.
 The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the ADC.
 Two piezoelectric stacks from the APA95ML were connected to the PD200 output to provide an appropriate load for the amplifier.
 #+begin_src latex :file detail_instrumentation_pd200_setup.pdf
 \begin{tikzpicture}
  \node[block={0.6cm}{0.6cm}] (const) {$0$};
  % PD200
  \node[block, right=0.4 of const] (Gp){$G_p(s)$};
  \node[addb, right=0.3 of Gp] (addnp){};
  % Pre Amp
  \node[addb, right=0.8 of addnp] (addna) {};
  \node[block, right=0.3 of addna] (Ga) {$G_a(s)$};
  % ADC
  \node[addb, right=0.8 of Ga] (addqad){};
  \node[ADC, right=0.3 of addqad] (ADC) {ADC};
  \draw[->] (const.east) -- (Gp.west);
  \draw[->] (Gp.east) -- (addnp.west);
  \draw[->] (addnp.east) -- (addna.west);
  \draw[->] (addna.east) -- (Ga.west);
  \draw[->] (Ga.east) -- (addqad.west);
  \draw[->] (addqad.east) -- (ADC.west);
  \draw[->] (ADC.east) -- node[sloped]{$/$} ++(0.8, 0) node[above left]{$n$};
  \draw[<-] (addnp.north) -- ++(0, 0.6) node[below left](np){$n_{p}$};
  \draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
  \draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
  \coordinate[] (top) at (na.north);
  \coordinate[] (bot) at (Ga.south);
  % PD200
  \begin{scope}[on background layer]
    \node[fit={(addnp.east|-bot) (Gp.west|-top)}, inner sep=4pt, draw, dashed, fill=colorred!20!white] (P) {};
    \node[above] at (P.north) {PD200};
  \end{scope}
  % 5113
  \begin{scope}[on background layer]
    \node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=4pt, draw, dashed, fill=colorgreen!20!white] (P) {};
    \node[above] at (P.north) {Pre Amp};
  \end{scope}
  % ADC
  \begin{scope}[on background layer]
    \node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=4pt, draw, dashed, fill=coloryellow!20!white] (P) {};
    \node[above] at (P.north) {ADC};
  \end{scope}
 \end{tikzpicture}
 #+end_src
 #+name: fig:detail_instrumentation_pd200_setup
 #+caption: Setup used to measured the output voltage noise of the PD200 voltage amplifier. A gain $G_a = 1000$ was used for the instrumentation amplifier.
 #+RESULTS:
 [[file:figs/detail_instrumentation_pd200_setup.png]]
 The Amplitude Spectral Density $\Gamma_{n}(\omega)$ of the signal measured by the ADC was computed.
 From this, the Amplitude Spectral Density of the output voltage noise of the PD200 amplifier $n_p$ was derived, accounting for the gain of the pre-amplifier according to eqref:eq:detail_instrumentation_amp_asd.
 \begin{equation}\label{eq:detail_instrumentation_amp_asd}
  \Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_p(j\omega) G_a(j\omega)|}
 \end{equation}
 The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure ref:fig:detail_instrumentation_pd200_noise.
 Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise.
 The measurements from all six amplifiers are displayed in this figure.
 The noise spectrum of the PD200 amplifiers exhibits several sharp peaks.
 While the exact cause of these peaks is not fully understood, their amplitudes remain below the specified limits and should not adversely affect system performance.
 #+name: fig:detail_instrumentation_pd200_noise
 #+caption: Measured output voltage noise of the PD200 amplifiers
 #+RESULTS:
 [[file:figs/detail_instrumentation_pd200_noise.png]]
 ***** Small Signal Bandwidth
 The small signal dynamics of all six PD200 amplifiers were characterized through frequency response measurements.
 A logarithmic sweep sine excitation voltage was generated using the Speedgoat DAC with an amplitude of $0.1\,V$, spanning frequencies from $1\,\text{Hz}$ to $5\,\text{kHz}$.
 The output voltage of the PD200 amplifier was measured via the monitor voltage output of the amplifier, while the input voltage (generated by the DAC) was measured with a separate ADC channel of the Speedgoat system.
 This measurement approach eliminates the influence of ADC-DAC-related time delays in the results.
 All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to 500Hz, well within the specified requirements.
 The identified dynamics shown in Figure ref:fig:detail_instrumentation_pd200_tf can be accurately modeled as either a first-order low-pass filter or as a simple constant gain.
 #+name: fig:detail_instrumentation_pd200_tf
 #+caption: Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier
 #+RESULTS:
 [[file:figs/detail_instrumentation_pd200_tf.png]]
 **** Linear Encoders
 To measure the noise of the encoder, the head and ruler were rigidly fixed together to ensure that no relative motion would be detected.
 Under these conditions, any measured signal would correspond solely to the encoder noise.
 The measurement setup is shown in Figure ref:fig:detail_instrumentation_vionic_bench.
 To minimize environmental disturbances, the entire bench was covered with a plastic bubble sheet during measurements.
 The amplitude spectral density of the measured displacement (which represents the measurement noise) is presented in Figure ref:fig:detail_instrumentation_vionic_asd.
 The noise profile exhibits characteristics of white noise with an amplitude of approximately $1\,\text{nm RMS}$, which complies with the system requirements.
 #+attr_latex: :options [b]{0.48\linewidth}
 #+begin_minipage
 #+name: fig:detail_instrumentation_vionic_bench
 #+caption: Test bench used to measured the encoder noise
 #+attr_latex: :width 0.95\linewidth :float nil
 [[file:figs/detail_instrumentation_vionic_bench.jpg]]
 #+end_minipage
 \hfill
 #+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
 [[file:figs/detail_instrumentation_vionic_asd.png]]
 #+end_minipage
 **** Noise budgeting from measured instrumentation noise
 After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier.
 The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure ref:fig:detail_instrumentation_cl_noise_budget.
 The total motion induced by all noise sources combined is approximately $1.5\,\text{nm RMS}$, which remains well within the specified limit of $15\,\text{nm RMS}$.
 This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application.
 #+name: fig:detail_instrumentation_cl_noise_budget
 #+caption: Closed-loop noise budgeting using measured noise of instrumentation
 #+RESULTS:
 [[file:figs/detail_instrumentation_cl_noise_budget.png]]
 *** Conclusion
 <<sec:detail_instrumentation_conclusion>>
 This section has presented a comprehensive approach to the selection and characterization of instrumentation for the nano active stabilization system.
 The multi-body model created earlier served as a key tool for embedding instrumentation components and their associated noise sources within the system analysis.
 From the most stringent requirement (i.e. the specification on vertical sample motion limited to 15 nm RMS), detailed specifications for each noise source were methodically derived through dynamic error budgeting.
 Based on these specifications, appropriate instrumentation components were selected for the system.
 The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions.
 Despite these challenges, suitable components were identified that theoretically met all requirements.
 The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized.
 Initial measurements of the ADC system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit.
 All components were found to meet or exceed their respective specifications. The ADC demonstrated noise levels of $5.6\,\mu V/\sqrt{\text{Hz}}$ (versus the $11\,\mu V/\sqrt{\text{Hz}}$ specification), the DAC showed $0.6\,\mu V/\sqrt{\text{Hz}}$ (versus $14\,\mu V/\sqrt{\text{Hz}}$ required), the voltage amplifiers exhibited noise well below the $280\,\mu V/\sqrt{\text{Hz}}$ limit, and the encoders achieved $1\,\text{nm RMS}$ noise (versus the $6\,\text{nm RMS}$ specification).
 Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance.
 The combined effect of all noise sources was estimated to induce vertical sample vibrations of only $1.5\,\text{nm RMS}$, which is substantially below the $15\,\text{nm RMS}$ requirement.
 This rigorous methodology spanning requirement formulation, component selection, and experimental characterization validates the instrumentation's ability to fulfill the nano active stabilization system's demanding performance specifications.
 ** TODO Obtained Design
 <<sec:detail_design>>
@ -12061,11 +12858,8 @@ This approach involved tuning and validating models of individual components (su
 The different models could then be combined to form the Nano-Hexapod dynamical model.
 If a model of the nano-hexapod was developed in one time, it would be difficult to tune all the model parameters to match the measured dynamics, or even to know if the model "structure" would be adequate to represent the system dynamics.
-** Experimental Validation - Conclusion
+** Nano Active Stabilization System
-:PROPERTIES:
+<<sec:test_id31>>
 :UNNUMBERED: notoc
 :END:
 <<sec:test_conclusion>>
 *** Introduction                                                    :ignore:
 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.
@ -13405,6 +14199,11 @@ The successful validation of the NASS demonstrates that once an accurate online
 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.
 ** Experimental Validation - Conclusion
 :PROPERTIES:
 :UNNUMBERED: notoc
 :END:
 <<sec:concept_conclusion>>
 * Conclusion and Future Work
 <<chap:conclusion>>
@ -13477,6 +14276,9 @@ Moreover, the systematic approach to system development and validation, along wi
 [fn:detail_fem_2]Cedrat technologies
 [fn:detail_fem_1]The manufacturer of the APA95ML was not willing to share the piezoelectric material properties of the stack.
 [fn:detail_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: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
 [fn:test_apa_11]Ansys\textsuperscript{\textregistered} was used
--- a/phd-thesis.pdf
+++ b/phd-thesis.pdf
--- a/phd-thesis.tex
+++ b/phd-thesis.tex
@ -1,4 +1,4 @@
-% Created 2025-04-04 Fri 17:55
+% Created 2025-04-06 Sun 18:19
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -42,7 +42,7 @@
 \addbibresource{ref.bib}
 \addbibresource{phd-thesis.bib}
 \author{Dehaeze Thomas}
-\date{2025-04-04}
+\date{2025-04-06}
 \title{Mechatronic approach for the design of a Nano Active Stabilization System}
 \subtitle{PhD Thesis}
 \hypersetup{
@ -6651,6 +6651,567 @@ Such model reduction, guided by detailed understanding of component behavior, pr
 \label{sec:detail_control}
 \section{Choice of Instrumentation}
 \label{sec:detail_instrumentation}
 This chapter presents an approach to select and validate appropriate instrumentation for the Nano Active Stabilization System (NASS), ensuring each component meets specific performance requirements.
 Figure \ref{fig:detail_instrumentation_plant} illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis.
 The selection process follows a three-stage methodology.
 First, dynamic error budgeting is performed in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting} to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier).
 This analysis utilizes the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements.
 From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived.
 Section \ref{sec:detail_instrumentation_choice} then presents the selection of appropriate components based on these noise specifications and additional requirements.
 Finally, Section \ref{sec:detail_instrumentation_characterization} validates the selected components through experimental testing.
 Each instrument is characterized individually, measuring actual noise levels and performance characteristics.
 The measured noise characteristics are then incorporated into the multi-body model to confirm that the combined effect of all instrumentation noise sources remains within acceptable limits.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{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}
 \label{sec:detail_instrumentation_dynamic_error_budgeting}
 The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system.
 The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section \ref{ssec:detail_instrumentation_cl_sensitivity}).
 This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs.
 Only the vertical direction is considered in this analysis as it presents the most stringent requirements; the horizontal directions are subject to less demanding constraints.
 From these transfer functions, the maximum acceptable Amplitude Spectral Density (ASD) of the noise sources is derived (Section \ref{ssec:detail_instrumentation_max_noise_specs}).
 Since the voltage amplifier gain affects the amplification of DAC noise, an assumption of an amplifier gain of 20 was made.
 \subsubsection{Closed-Loop Sensitivity to Instrumentation Disturbances}
 \label{ssec:detail_instrumentation_cl_sensitivity}
 Several key noise sources are considered in the analysis (Figure \ref{fig:detail_instrumentation_plant}).
 These include the output voltage noise of the DAC (\(n_{da}\)), the output voltage noise of the voltage amplifier (\(n_{amp}\)), and the voltage noise of the ADC measuring the force sensor stacks (\(n_{ad}\)).
 Encoder noise, which is only used to estimate \(R_z\), has been found to have minimal impact on the vertical sample error and is therefore omitted from this analysis for clarity.
 The transfer functions from these three noise sources (for one strut) to the vertical error of the sample are estimated from the multi-body model, which includes the APA300ML and the designed flexible joints (Figure \ref{fig:detail_instrumentation_noise_sensitivities}).
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{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}
 \label{ssec:detail_instrumentation_max_noise_specs}
 The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of \(100\,\text{nm}\), which corresponds to a maximum allowed vibration of \(15\,\text{nm RMS}\).
 Several assumptions regarding the noise characteristics have been made.
 The DAC, ADC, and amplifier noise are considered uncorrelated, which is a reasonable assumption.
 Similarly, the noise sources corresponding to each strut are also assumed to be uncorrelated.
 This means that the power spectral densities (PSD) of the different noise sources are summed.
 Since the effect of each strut on the vertical error is identical due to symmetry, only one strut is considered for this analysis, and the total effect of the six struts is calculated as six times the effect of one strut in terms of power, which translates to a factor of \(\sqrt{6} \approx 2.5\) for RMS values.
 In order to derive specifications in terms of noise spectral density for each instrumentation component, a white noise profile is assumed, which is typical for these components.
 The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
 While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications.
 Based on this analysis, the obtained maximum noise levels are as follows: DAC maximum output noise ASD is established at \(14\,\mu V/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise ASD at \(280\,\mu V/\sqrt{\text{Hz}}\), and ADC maximum measurement noise ASD at \(11\,\mu V/\sqrt{\text{Hz}}\).
 In terms of RMS noise, these translate to less than \(1\,\text{mV RMS}\) for the DAC, less than \(20\,\text{mV RMS}\) for the voltage amplifier, and less than \(0.8\,\text{mV RMS}\) for the ADC.
 If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS.
 \subsection{Choice of Instrumentation}
 \label{sec:detail_instrumentation_choice}
 \subsubsection{Piezoelectric Voltage Amplifier}
 Several characteristics of piezoelectric voltage amplifiers must be considered for this application.
 To take advantage of the full stroke of the piezoelectric actuator, the voltage output should range between \(-20\) and \(150\,V\).
 The amplifier should accept an analog input voltage, preferably in the range of \(-10\) to \(10\,V\), as this is standard for most DACs.
 \paragraph{Small signal Bandwidth and Output Impedance}
 Small signal bandwidth is particularly important for feedback applications as it can limit the overall bandwidth of the complete feedback system.
 A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure \ref{fig:detail_instrumentation_amp_output_impedance}.
 This model is valid for small signals and provides insight into the small signal bandwidth limitation \cite[, chap. 14]{fleming14_desig_model_contr_nanop_system}.
 In this model, \(R_o\) represents the output impedance of the amplifier.
 When combined with the piezoelectric load (represented as a capacitance \(C_p\)), it forms a first order low pass filter described by \eqref{eq:detail_instrumentation_amp_output_impedance}.
 \begin{equation}\label{eq:detail_instrumentation_amp_output_impedance}
  \frac{V_a}{V_i}(s) = \frac{1}{1 + \frac{s}{\omega_0}}, \quad \omega_0 = \frac{1}{R_o C_p}
 \end{equation}
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_instrumentation_amp_output_impedance.png}
 \caption{\label{fig:detail_instrumentation_amp_output_impedance}Electrical model of a voltage amplifier with output impedance \(R_0\) connected to a piezoelectric stack with capacitance \(C_p\)}
 \end{figure}
 Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases.
 For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to \(C_p = 8.8\,\mu F\).
 If a small signal bandwidth of \(f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}\) is desired, the voltage amplifier output impedance should be less than \(R_0 = 3.6\,\Omega\).
 \paragraph{Large signal Bandwidth}
 Large signal bandwidth relates to the maximum output capabilities of the amplifier in terms of amplitude as a function of frequency.
 Since the primary function of the NASS is position stabilization rather than scanning, this specification is less critical than the small signal bandwidth.
 However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of \(f_r = 100\,\text{Hz}\) using the full voltage range of the piezoelectric actuator (\(V_{pp} = 170\,V\)) is considered.
 There are two limiting factors for large signal bandwidth that should be evaluated:
 \begin{enumerate}
 \item Slew rate, which should exceed \(2 \cdot V_{pp} \cdot f_r = 34\,V/ms\).
 This requirement is typically easily met by commercial voltage amplifiers.
 \item Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies.
 To achieve high voltage at high frequency, the amplifier must therefore provide substantial current.
 The maximum required current can be calculated as \(I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A\).
 \end{enumerate}
 Therefore, ideally, a voltage amplifier capable of providing \(0.3\,A\) of current would be interesting for scanning applications.
 \paragraph{Output voltage noise}
 As established in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}, the output noise of the voltage amplifier should be below \(20\,\text{mV RMS}\).
 It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure \ref{fig:detail_instrumentation_amp_output_impedance}.
 Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement \cite{spengen20_high_voltag_amplif}.
 For this application, the output noise must remain below \(20\,\text{mV RMS}\) with a load of \(8.8\,\mu F\) and a bandwidth exceeding \(5\,\text{kHz}\).
 \paragraph{Choice of voltage amplifier}
 The specifications are summarized in Table \ref{tab:detail_instrumentation_amp_choice}.
 The most critical characteristics are the small signal bandwidth (\(>5\,\text{kHz}\)) and the output voltage noise (\(<20\,\text{mV RMS}\)).
 Several voltage amplifiers were considered, with their datasheet information presented in Table \ref{tab:detail_instrumentation_amp_choice}.
 One challenge encountered during the selection process was that manufacturers typically do not specify output noise as a function of frequency (i.e., the ASD of the noise), but instead provide only the RMS value, which represents the integrated value across all frequencies.
 This approach does not account for the frequency dependency of the noise, which is crucial for accurate error budgeting.
 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.
 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}
 \toprule
 \textbf{Specification} & \textbf{PD200} & WMA-200 & LA75B & E-505\\
 & PiezoDrive & Falco & Cedrat & PI\\
 \midrule
 Input Voltage Range:  \(\pm 10\,V\) & \(\pm 10\,V\) & \(\pm8.75\,V\) & \(-1/7.5\,V\) & \\
 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) & \footnotemark & (unloaded) & (n/a)\\
 Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\
 \bottomrule
 \end{tabularx}
 \end{table}\footnotetext[26]{\label{org61bd042}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)}
 \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.
 \paragraph{Synchronicity and Jitter}
 For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements \cite{abramovitch22_pract_method_real_world_contr_system,abramovitch23_tutor_real_time_comput_issues_contr_system}.
 Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals.
 Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration.
 \paragraph{Sampling Frequency, Bandwidth and delays}
 Several requirements that may initially appear similar are actually distinct in nature.
 First, the \emph{sampling frequency} defines the interval between two sampled points and determines the Nyquist frequency.
 Then, the \emph{bandwidth} specifies the maximum frequency of a measured signal (typically defined as the -3dB point) and is often limited by implemented anti-aliasing filters.
 Finally, \emph{delay} (or \emph{latency}) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system.
 Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency.
 Typically, the latency can reach 20 times the sampling period \cite[, chapt. 8.4]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
 Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor.
 For real-time control applications, SAR-ADCs (Successive Approximation ADCs) remain the predominant choice due to their single-sample latency characteristics.
 \paragraph{ADC Noise}
 Based on the dynamic error budget established in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise ASD should not exceed \(11\,\mu V/\sqrt{\text{Hz}}\).
 ADCs are subject to various noise sources.
 Quantization noise, which results from the discrete nature of digital representation, is one of these sources.
 To determine the minimum bit depth \(n\) required to meet the noise specifications, an ideal ADC where quantization error is the only noise source is considered.
 The quantization step size, denoted as \(q = \Delta V/2^n\), represents the voltage equivalent of the least significant bit, with \(\Delta V\) the full range of the ADC in volts, and \(F_s\) the sampling frequency in Hertz.
 The quantization noise ranges between \(\pm q/2\), and its probability density function is constant across this range (uniform distribution).
 Since the integral of this probability density function \(p(e)\) equals one, its value is \(1/q\) for \(-q/2 < e < q/2\), as illustrated in Figure \ref{fig:detail_instrumentation_adc_quantization}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_instrumentation_adc_quantization.png}
 \caption{\label{fig:detail_instrumentation_adc_quantization}Probability density function \(p(e)\) of the ADC quantization error \(e\)}
 \end{figure}
 The variance (or time-average power) of the quantization noise is expressed by \eqref{eq:detail_instrumentation_quant_power}.
 \begin{equation}\label{eq:detail_instrumentation_quant_power}
  P_q = \int_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12}
 \end{equation}
 To compute the power spectral density of the quantization noise, which is defined as the Fourier transform of the noise's autocorrelation function, it is assumed that noise samples are uncorrelated.
 Under this assumption, the autocorrelation function approximates a delta function in the time domain.
 Since the Fourier transform of a delta function equals one, the power spectral density becomes frequency-independent (white noise).
 By Parseval's theorem, the power spectral density of the quantization noise \(\Phi_q\) can be linked to the ADC sampling frequency and quantization step size \eqref{eq:detail_instrumentation_psd_quant_noise}.
 \begin{equation}\label{eq:detail_instrumentation_psd_quant_noise}
  \int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_q = \frac{q^2}{12 F_s} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 F_s} \quad \text{in } \left[ \frac{V^2}{\text{Hz}} \right]
 \end{equation}
 From a specified noise amplitude spectral density \(\Gamma_{\text{max}}\), the minimum number of bits required to keep quantization noise below \(\Gamma_{\text{max}}\) is calculated using \eqref{eq:detail_instrumentation_min_n}.
 \begin{equation}\label{eq:detail_instrumentation_min_n}
  n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right)
 \end{equation}
 With a sampling frequency \(F_s = 10\,\text{kHz}\), an input range \(\Delta V = 20\,V\) and a maximum allowed ASD \(\Gamma_{\text{max}} = 11\,\mu V/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is readily achievable with commercial ADCs.
 \paragraph{DAC Output voltage noise}
 Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed \(14\,\mu V/\sqrt{\text{Hz}}\).
 This specification corresponds to a \(\pm 10\,V\) DAC with 13-bit resolution, which is easily attainable with current technology.
 \paragraph{Choice of the ADC and DAC Board}
 Based on the preceding analysis, the selection of suitable ADC and DAC components is straightforward.
 For optimal synchronicity, a Speedgoat-integrated solution was chosen.
 The selected model is the IO131, which features 16 analog inputs based on the AD7609 with 16-bit resolution, \(\pm 10\,V\) range, maximum sampling rate of 200kSPS, simultaneous sampling, and differential inputs allowing the use of shielded twisted pairs for enhanced noise immunity.
 The board also includes 8 analog outputs based on the AD5754R with 16-bit resolution, \(\pm 10\,V\) range, conversion time of \(10\,\mu s\), and simultaneous update capability.
 Although noise specifications are not explicitly provided in the datasheet, the 16-bit resolution should ensure performance well below the established requirements.
 This will be experimentally verified in Section \ref{sec:detail_instrumentation_characterization}.
 \subsubsection{Relative Displacement Sensors}
 The specifications for the relative displacement sensors include sufficient compactness for integration within each strut, noise levels below \(6\,\text{nm RMS}\) (derived from the \(15\,\text{nm RMS}\) vertical error requirement for the system divided by the contributions of six struts), and a measurement range exceeding \(100\,\mu m\).
 Several sensor technologies are capable of meeting these requirements \cite{fleming13_review_nanom_resol_posit_sensor}.
 These include optical encoders (Figure \ref{fig:detail_instrumentation_sensor_encoder}), capacitive sensors (Figure \ref{fig:detail_instrumentation_sensor_capacitive}), and eddy current sensors (Figure \ref{fig:detail_instrumentation_sensor_eddy_current}), each with their own advantages and implementation considerations.
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.9\linewidth]{figs/detail_instrumentation_sensor_encoder.jpg}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_sensor_encoder}Optical Linear Encoder}
 \end{subfigure}
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.9\linewidth]{figs/detail_instrumentation_sensor_eddy_current.png}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_sensor_eddy_current}Eddy Current Sensor}
 \end{subfigure}
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.9\linewidth]{figs/detail_instrumentation_sensor_capacitive.jpg}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_sensor_capacitive}Capacitive Sensor}
 \end{subfigure}
 \caption{\label{fig:detail_instrumentation_sensor_examples}Relative motion sensors considered for measuring the nano-hexapod strut motion}
 \end{figure}
 From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the APA, as illustrated in Figure \ref{fig:detail_instrumentation_capacitive_implementation}.
 In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure \ref{fig:detail_instrumentation_encoder_implementation}.
 \begin{figure}[htbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/detail_instrumentation_encoder_implementation.png}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_encoder_implementation}Optical Encoder}
 \end{subfigure}
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/detail_instrumentation_capacitive_implementation.png}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_capacitive_implementation}Capacitive Sensor}
 \end{subfigure}
 \caption{\label{fig:detail_instrumentation_sensor_implementation}Implementation of relative displacement sensor to measure the motion of the APA}
 \end{figure}
 A significant consideration in the sensor selection process was the fact that sensor signals must pass through an electrical slip-ring due to the continuous spindle rotation.
 Measurements conducted on the slip-ring integrated in the micro-station revealed substantial cross-talk between different slip-ring channels.
 To mitigate this issue, preference was given to sensors that transmit displacement measurements digitally, as these are inherently less susceptible to noise and cross-talk.
 Based on this criterion, an optical encoder with digital output was selected, where signal interpolation is performed directly in the sensor head.
 The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table \ref{tab:detail_instrumentation_sensor_specs}, alongside alternative options that were considered.
 \begin{table}[htbp]
 \caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products}
 \centering
 \begin{tabularx}{0.8\linewidth}{Xccc}
 \toprule
 \textbf{Specification} & \textbf{Renishaw Vionic} & LION CPL190 & Cedrat ECP500\\
 \midrule
 Technology & Digital Encoder & Capacitive & Eddy Current\\
 Bandwidth \(> 5\,\text{kHz}\) & \(> 500\,\text{kHz}\) & 10kHz & 20kHz\\
 Noise \(< 6\,nm\,\text{RMS}\) & 1.6 nm rms & 4 nm rms & 15 nm rms\\
 Range  \(> 100\,\mu m\) & Ruler length & 250 um & 500um\\
 In line measurement &  & \(\times\) & \(\times\)\\
 Digital Output & \(\times\) &  & \\
 \bottomrule
 \end{tabularx}
 \end{table}
 \subsection{Characterization of Instrumentation}
 \label{sec:detail_instrumentation_characterization}
 \subsubsection{Analog to Digital Converters}
 \paragraph{Measured Noise}
 The measurement of ADC noise was performed by short-circuiting its input with a \(50\,\Omega\) resistor and recording the digital values at a sampling rate of \(10\,\text{kHz}\).
 The amplitude spectral density of the recorded values was computed and is presented in Figure \ref{fig:detail_instrumentation_adc_noise_measured}.
 The ADC noise exhibits characteristics of white noise with an amplitude spectral density of \(5.6\,\mu V/\sqrt{\text{Hz}}\) (equivalent to \(0.4\,\text{mV RMS}\)), which satisfies the established specifications.
 All ADC channels demonstrated similar performance, so only one channel's noise profile is shown.
 If necessary, oversampling can be applied to further reduce the noise \cite{lab13_improv_adc}.
 To gain \(w\) additional bits of resolution, the oversampling frequency \(f_{os}\) should be set to \(f_{os} = 4^w \cdot F_s\).
 Given that the ADC can operate at 200kSPS while the real-time controller runs at 10kSPS, an oversampling factor of 16 can be employed to gain approximately two additional bits of resolution (reducing noise by a factor of 4).
 This approach is effective because the noise approximates white noise and its amplitude exceeds 1 LSB (0.3 mV) \cite{hauser91_princ_overs_conver}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{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}
 To further validate the ADC's capability to effectively measure voltage generated by a piezoelectric stack, a test was conducted using the APA95ML.
 The setup is illustrated in Figure \ref{fig:detail_instrumentation_force_sensor_adc_setup}, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor.
 The voltage amplifier employed in this setup has a gain of 20.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_instrumentation_force_sensor_adc_setup.png}
 \caption{\label{fig:detail_instrumentation_force_sensor_adc_setup}Schematic of the setup to validate the use of the ADC for reading the force sensor volage}
 \end{figure}
 Step signals with an amplitude of \(1\,V\) were generated using the DAC, and the ADC signal was recorded.
 The excitation signal (steps) and the measured voltage across the sensor stack are displayed in Figure \ref{fig:detail_instrumentation_step_response_force_sensor}.
 Two notable observations were made: an offset voltage of \(2.26\,V\) was present, and the measured voltage exhibited an exponential decay response to the step input.
 These phenomena can be explained by examining the electrical schematic shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc}, where the ADC has an input impedance \(R_i\) and an input bias current \(i_n\).
 The input impedance \(R_i\) of the ADC, in combination with the capacitance \(C_p\) of the piezoelectric stack sensor, forms an RC circuit with a time constant \(\tau = R_i C_p\).
 The charge generated by the piezoelectric effect across the stack's capacitance gradually discharges into the input resistor of the ADC.
 Consequently, the transfer function from the generated voltage \(V_p\) to the measured voltage \(V_{\text{ADC}}\) is a first-order high-pass filter with the time constant \(\tau\).
 An exponential curve was fitted to the experimental data, yielding a time constant \(\tau = 6.5\,s\).
 With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4\,\mu F\), the internal impedance of the Speedgoat ADC was calculated as \(R_i = \tau/C_p = 1.5\,M\Omega\), which closely aligns with the specified value of \(1\,M\Omega\) found in the datasheet.
 \begin{figure}[htbp]
 \begin{subfigure}{0.61\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/detail_instrumentation_force_sensor_adc.png}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_force_sensor_adc}Electrical Schematic}
 \end{subfigure}
 \begin{subfigure}{0.35\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_step_response_force_sensor.png}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_step_response_force_sensor}Measured Signals}
 \end{subfigure}
 \caption{\label{fig:detail_instrumentation_force_sensor}Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from \cite{reza06_piezoel_trans_vibrat_contr_dampin}. Measured voltage \(V_s\) while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}).}
 \end{figure}
 The constant voltage offset can be explained by the input bias current \(i_n\) of the ADC, represented in Figure \ref{fig:detail_instrumentation_force_sensor_adc}.
 At DC, the impedance of the piezoelectric stack is much larger than the input impedance of the ADC, and therefore the input bias current \(i_n\) passing through the internal resistance \(R_i\) produces a constant voltage offset \(V_{\text{off}} = R_i \cdot i_n\).
 The input bias current \(i_n\) is estimated from \(i_n = V_{\text{off}}/R_i = 1.5\mu A\).
 In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor \(R_p\) can be added in parallel to the force sensor, as illustrated in Figure \ref{fig:detail_instrumentation_force_sensor_adc_R}.
 This modification produces two beneficial effects: a reduction of input voltage offset through the relationship \(V_{\text{off}} = (R_p R_i)/(R_p + R_i) i_n\), and an increase in the high pass corner frequency \(f_c\) according to the equations \(\tau = 1/(2\pi f_c) = (R_i R_p)/(R_i + R_p) C_p\).
 To validate this approach, a resistor \(R_p \approx 82\,k\Omega\) was added in parallel with the force sensor as shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc_R}.
 After incorporating this resistor, the same step response tests were performed, with results displayed in Figure \ref{fig:detail_instrumentation_step_response_force_sensor_R}.
 The measurements confirmed the expected improvements, with a substantially reduced offset voltage (\(V_{\text{off}} = 0.15\,V\)) and a much faster time constant (\(\tau = 0.45\,s\)).
 These results validate both the model of the ADC and the effectiveness of the added parallel resistor as a solution.
 \begin{figure}[htbp]
 \begin{subfigure}{0.61\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/detail_instrumentation_force_sensor_adc_R.png}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_force_sensor_adc_R}Electrical Schematic}
 \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}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_step_response_force_sensor_R}Measured Signals}
 \end{subfigure}
 \caption{\label{fig:detail_instrumentation_force_sensor_R}Effect of an added resistor \(R_p\) in parallel to the force sensor. The electrical schematic is shown in (\subref{fig:detail_instrumentation_force_sensor_adc_R}) and the measured signals in (\subref{fig:detail_instrumentation_step_response_force_sensor_R}).}
 \end{figure}
 \subsubsection{Instrumentation Amplifier}
 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\footnote{For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain.} noise of the amplifier.
 This was accomplished by short-circuiting its input with a \(50\,\Omega\) resistor and measuring the output voltage with the ADC (Figure \ref{fig:detail_instrumentation_femto_meas_setup}).
 The maximum amplifier gain of 80dB (equivalent to 10000) was used for this measurement.
 The measured voltage \(n\) was then divided by 10000 to determine the equivalent noise at the input of the voltage amplifier \(n_a\).
 In this configuration, the noise contribution from the ADC \(q_{ad}\) is rendered negligible due to the high gain employed.
 The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and the (negligible) contribution of the ADC noise are presented in Figure \ref{fig:detail_instrumentation_femto_input_noise}.
 \begin{minipage}[b]{0.48\linewidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{figs/detail_instrumentation_femto_meas_setup.png}
 \captionof{figure}{\label{fig:detail_instrumentation_femto_meas_setup}Measurement of the instrumentation amplifier input voltage noise}
 \end{center}
 \end{minipage}
 \hfill
 \begin{minipage}[b]{0.48\linewidth}
 \begin{center}
 \includegraphics[scale=1,scale=1]{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}
 \subsubsection{Digital to Analog Converters}
 \paragraph{Output Voltage Noise}
 To measure the output noise of the DAC, the setup schematically represented in Figure \ref{fig:detail_instrumentation_dac_setup} was utilized.
 The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC.
 The Amplitude Spectral Density \(\Gamma_{n_{da}}(\omega)\) of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement.
 The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure \ref{fig:detail_instrumentation_dac_output_noise}.
 The noise profile is predominantly white with an ASD of \(0.6\,\mu V/\sqrt{\text{Hz}}\).
 Minor \(50\,\text{Hz}\) noise is present, along with some low frequency \(1/f\) noise, but these are not expected to pose issues as they are well within specifications.
 It should be noted that all DAC channels demonstrated similar performance, so only one channel measurement is presented.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_instrumentation_dac_setup.png}
 \caption{\label{fig:detail_instrumentation_dac_setup}Measurement of the DAC output voltage noise. A pre-amplifier with a gain of 1000 is used before measuring the signal with the ADC.}
 \end{figure}
 \paragraph{Delay from ADC to DAC}
 To measure the transfer function from DAC to ADC and verify that the bandwidth and latency of both instruments is sufficient, a direct connection was established between the DAC output and the ADC input.
 A white noise signal was generated by the DAC, and the ADC response was recorded.
 The resulting frequency response function from the digital DAC signal to the digital ADC signal is presented in Figure \ref{fig:detail_instrumentation_dac_adc_tf}.
 The observed frequency response function corresponds to exactly one sample delay, which aligns with the specifications provided by the manufacturer.
 \begin{figure}[htbp]
 \begin{subfigure}{0.48\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{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}
 \end{center}
 \subcaption{\label{fig:detail_instrumentation_dac_adc_tf}Transfer function from DAC to ADC}
 \end{subfigure}
 \caption{\label{fig:detail_instrumentation_dac}Measurement of the output voltage noise of the ADC (\subref{fig:detail_instrumentation_dac_output_noise}) and measured transfer function from DAC to ADC (\subref{fig:detail_instrumentation_dac_adc_tf}) which corresponds to a ``1-sample'' delay.}
 \end{figure}
 \subsubsection{Piezoelectric Voltage Amplifier}
 \paragraph{Output Voltage Noise}
 The measurement setup for evaluating the PD200 amplifier noise is illustrated in Figure \ref{fig:detail_instrumentation_pd200_setup}.
 The input of the PD200 amplifier was shunted with a \(50\,\Ohm\) resistor to ensure that only the inherent noise of the amplifier itself was measured.
 The pre-amplifier gain was increased to produce a signal substantially larger than the noise floor of the ADC.
 Two piezoelectric stacks from the APA95ML were connected to the PD200 output to provide an appropriate load for the amplifier.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_instrumentation_pd200_setup.png}
 \caption{\label{fig:detail_instrumentation_pd200_setup}Setup used to measured the output voltage noise of the PD200 voltage amplifier. A gain \(G_a = 1000\) was used for the instrumentation amplifier.}
 \end{figure}
 The Amplitude Spectral Density \(\Gamma_{n}(\omega)\) of the signal measured by the ADC was computed.
 From this, the Amplitude Spectral Density of the output voltage noise of the PD200 amplifier \(n_p\) was derived, accounting for the gain of the pre-amplifier according to \eqref{eq:detail_instrumentation_amp_asd}.
 \begin{equation}\label{eq:detail_instrumentation_amp_asd}
  \Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_p(j\omega) G_a(j\omega)|}
 \end{equation}
 The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure \ref{fig:detail_instrumentation_pd200_noise}.
 Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise.
 The measurements from all six amplifiers are displayed in this figure.
 The noise spectrum of the PD200 amplifiers exhibits several sharp peaks.
 While the exact cause of these peaks is not fully understood, their amplitudes remain below the specified limits and should not adversely affect system performance.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{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}
 The small signal dynamics of all six PD200 amplifiers were characterized through frequency response measurements.
 A logarithmic sweep sine excitation voltage was generated using the Speedgoat DAC with an amplitude of \(0.1\,V\), spanning frequencies from \(1\,\text{Hz}\) to \(5\,\text{kHz}\).
 The output voltage of the PD200 amplifier was measured via the monitor voltage output of the amplifier, while the input voltage (generated by the DAC) was measured with a separate ADC channel of the Speedgoat system.
 This measurement approach eliminates the influence of ADC-DAC-related time delays in the results.
 All six amplifiers demonstrated consistent transfer function characteristics. The amplitude response remains constant across a wide frequency range, and the phase shift is limited to less than 1 degree up to 500Hz, well within the specified requirements.
 The identified dynamics shown in Figure \ref{fig:detail_instrumentation_pd200_tf} can be accurately modeled as either a first-order low-pass filter or as a simple constant gain.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{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}
 To measure the noise of the encoder, the head and ruler were rigidly fixed together to ensure that no relative motion would be detected.
 Under these conditions, any measured signal would correspond solely to the encoder noise.
 The measurement setup is shown in Figure \ref{fig:detail_instrumentation_vionic_bench}.
 To minimize environmental disturbances, the entire bench was covered with a plastic bubble sheet during measurements.
 The amplitude spectral density of the measured displacement (which represents the measurement noise) is presented in Figure \ref{fig:detail_instrumentation_vionic_asd}.
 The noise profile exhibits characteristics of white noise with an amplitude of approximately \(1\,\text{nm RMS}\), which complies with the system requirements.
 \begin{minipage}[b]{0.48\linewidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_vionic_bench.jpg}
 \captionof{figure}{\label{fig:detail_instrumentation_vionic_bench}Test bench used to measured the encoder noise}
 \end{center}
 \end{minipage}
 \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}
 \end{center}
 \end{minipage}
 \subsubsection{Noise budgeting from measured instrumentation noise}
 After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier.
 The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure \ref{fig:detail_instrumentation_cl_noise_budget}.
 The total motion induced by all noise sources combined is approximately \(1.5\,\text{nm RMS}\), which remains well within the specified limit of \(15\,\text{nm RMS}\).
 This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{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}
 \label{sec:detail_instrumentation_conclusion}
 This section has presented a comprehensive approach to the selection and characterization of instrumentation for the nano active stabilization system.
 The multi-body model created earlier served as a key tool for embedding instrumentation components and their associated noise sources within the system analysis.
 From the most stringent requirement (i.e. the specification on vertical sample motion limited to 15 nm RMS), detailed specifications for each noise source were methodically derived through dynamic error budgeting.
 Based on these specifications, appropriate instrumentation components were selected for the system.
 The selection process revealed certain challenges, particularly with voltage amplifiers, where manufacturer datasheets often lacked crucial information needed for accurate noise budgeting, such as amplitude spectral densities under specific load conditions.
 Despite these challenges, suitable components were identified that theoretically met all requirements.
 The selected instrumentation (including the IO131 ADC/DAC from Speedgoat, PD200 piezoelectric voltage amplifiers from PiezoDrive, and Vionic linear encoders from Renishaw) was procured and thoroughly characterized.
 Initial measurements of the ADC system revealed an issue with force sensor readout related to input bias current, which was successfully addressed by adding a parallel resistor to optimize the measurement circuit.
 All components were found to meet or exceed their respective specifications. The ADC demonstrated noise levels of \(5.6\,\mu V/\sqrt{\text{Hz}}\) (versus the \(11\,\mu V/\sqrt{\text{Hz}}\) specification), the DAC showed \(0.6\,\mu V/\sqrt{\text{Hz}}\) (versus \(14\,\mu V/\sqrt{\text{Hz}}\) required), the voltage amplifiers exhibited noise well below the \(280\,\mu V/\sqrt{\text{Hz}}\) limit, and the encoders achieved \(1\,\text{nm RMS}\) noise (versus the \(6\,\text{nm RMS}\) specification).
 Finally, the measured noise characteristics of all instrumentation components were included into the multi-body model to predict the actual system performance.
 The combined effect of all noise sources was estimated to induce vertical sample vibrations of only \(1.5\,\text{nm RMS}\), which is substantially below the \(15\,\text{nm RMS}\) requirement.
 This rigorous methodology spanning requirement formulation, component selection, and experimental characterization validates the instrumentation's ability to fulfill the nano active stabilization system's demanding performance specifications.
 \section{Obtained Design}
 \label{sec:detail_design}
 \begin{itemize}
@ -8861,8 +9422,8 @@ Obtaining a model that accurately represents the complex dynamics of the Nano-He
 This approach involved tuning and validating models of individual components (such as the APA and flexible joints) using dedicated test benches.
 The different models could then be combined to form the Nano-Hexapod dynamical model.
 If a model of the nano-hexapod was developed in one time, it would be difficult to tune all the model parameters to match the measured dynamics, or even to know if the model ``structure'' would be adequate to represent the system dynamics.
-\section*{Experimental Validation - Conclusion}
+\section{Nano Active Stabilization System}
-\label{sec:test_conclusion}
+\label{sec:test_id31}
 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.
@ -8896,7 +9457,7 @@ 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}
-\subsection*{Short Stroke Metrology System}
+\subsection{Short Stroke Metrology System}
 \label{sec:test_id31_metrology}
 The control of the nano-hexapod requires an external metrology system that measures the relative position of the nano-hexapod top platform with respect to the granite.
 As a long-stroke (\(\approx 1 \,cm^3\)) metrology system was not yet developed, a stroke stroke (\(\approx 100\,\mu m^3\)) was used instead to validate the nano-hexapod control.
@ -8934,7 +9495,7 @@ In this way, the gap between the head and the reference sphere is much larger (h
 Nevertheless, the metrology system still has a limited measurement range because of the limited angular acceptance of the fibered interferometers.
 Indeed, when the spheres are moving perpendicularly to the beam axis, the reflected light does not coincide with the incident light, and above some perpendicular displacement, the reflected light does not come back into the fiber, and no interference is produced.
-\subsubsection*{Metrology Kinematics}
+\subsubsection{Metrology Kinematics}
 \label{ssec:test_id31_metrology_kinematics}
 The proposed short-stroke metrology system is schematized in Figure \ref{fig:test_id31_metrology_kinematics}.
@ -8975,7 +9536,7 @@ 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}
-\subsubsection*{Rough alignment of the reference spheres}
+\subsubsection{Rough alignment of the reference spheres}
 \label{ssec:test_id31_metrology_sphere_rought_alignment}
 The two reference spheres must be well aligned with the rotation axis of the spindle.
@ -8988,7 +9549,7 @@ 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.
-\subsubsection*{Tip-Tilt adjustment of the interferometers}
+\subsubsection{Tip-Tilt adjustment of the interferometers}
 \label{ssec:test_id31_metrology_alignment}
 The short-stroke metrology system was placed on top of the main granite using granite blocs (Figure \ref{fig:test_id31_short_stroke_metrology_overview}).
@ -9011,7 +9572,7 @@ 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.
-\subsubsection*{Fine Alignment of reference spheres using interferometers}
+\subsubsection{Fine Alignment of reference spheres using interferometers}
 \label{ssec:test_id31_metrology_sphere_fine_alignment}
 Thanks to the first alignment of the two reference spheres with the spindle axis (Section \ref{ssec:test_id31_metrology_sphere_rought_alignment}) and to the fine adjustment of the interferometer orientations (Section \ref{ssec:test_id31_metrology_alignment}), the spindle can perform complete rotations while still having interference for all five interferometers.
@ -9040,7 +9601,7 @@ 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}
-\subsubsection*{Estimated measurement volume}
+\subsubsection{Estimated measurement volume}
 \label{ssec:test_id31_metrology_acceptance}
 Because the interferometers point to spheres and not flat surfaces, the lateral acceptance is limited.
@ -9064,7 +9625,7 @@ The obtained lateral acceptance for pure displacements in any direction is estim
 \bottomrule
 \end{tabularx}
 \end{table}
-\subsubsection*{Estimated measurement errors}
+\subsubsection{Estimated measurement errors}
 \label{ssec:test_id31_metrology_errors}
 When using the NASS, the accuracy of the sample positioning is determined by the accuracy of the external metrology.
@ -9102,7 +9663,7 @@ 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}
-\subsection*{Open Loop Plant}
+\subsection{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}.
 The input \(\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]\) is the command signal, which corresponds to the voltages generated for each piezoelectric actuator.
@ -9123,7 +9684,7 @@ Voltages generated by the force sensor piezoelectric stacks \(\bm{V}_s = [V_{s1}
 \includegraphics[scale=1]{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}
+\subsubsection{Open-Loop Plant Identification}
 \label{ssec:test_id31_open_loop_plant_first_id}
 The dynamics of the plant is first identified for a fixed spindle angle (at \(0\,\text{deg}\)) and without any payload.
@ -9152,7 +9713,7 @@ 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}
-\subsubsection*{Better Angular Alignment}
+\subsubsection{Better Angular Alignment}
 \label{ssec:test_id31_open_loop_plant_rz_alignment}
 One possible explanation of the increased coupling observed in Figure \ref{fig:test_id31_first_id_int} is the poor alignment between the external metrology axes (i.e. the interferometer supports) and the nano-hexapod axes.
@ -9191,7 +9752,7 @@ 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}
-\subsubsection*{Effect of Payload Mass}
+\subsubsection{Effect of Payload Mass}
 \label{ssec:test_id31_open_loop_plant_mass}
 To determine how the system dynamics changes with the payload, open-loop identification was performed for four payload conditions shown in Figure \ref{fig:test_id31_picture_masses}.
@ -9244,7 +9805,7 @@ 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}
-\subsubsection*{Effect of Spindle Rotation}
+\subsubsection{Effect of Spindle Rotation}
 \label{ssec:test_id31_open_loop_plant_rotation}
 To verify that all the kinematics in Figure \ref{fig:test_id31_block_schematic_plant} are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at \(36\,\text{deg}/s\) and at \(180\,\text{deg}/s\).
@ -9274,7 +9835,7 @@ This also indicates that the metrology kinematics is correct and is working in r
 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.
-\subsection*{Decentralized Integral Force Feedback}
+\subsection{Decentralized Integral Force Feedback}
 \label{sec:test_id31_iff}
 In this section, the low authority control part is first validated.
 It consists of a decentralized Integral Force Feedback controller \(\bm{K}_{\text{IFF}}\), with all the diagonal terms being equal \eqref{eq:test_id31_Kiff}.
@ -9294,7 +9855,7 @@ The decentralized Integral Force Feedback is implemented as shown in the block d
 \includegraphics[scale=1]{figs/test_id31_iff_schematic.png}
 \caption{\label{fig:test_id31_iff_block_diagram}Block diagram of the implemented decentralized IFF controller. The controller \(\bm{K}_{\text{IFF}}\) is a diagonal controller with the same elements for every diagonal term \(K_{\text{IFF}}\).}
 \end{figure}
-\subsubsection*{IFF Plant}
+\subsubsection{IFF Plant}
 \label{ssec:test_id31_iff_plant}
 As the multi-body model is used to evaluate the stability of the IFF controller and to optimize the achievable damping, it is first checked whether this model accurately represents the system dynamics.
@ -9310,7 +9871,7 @@ 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}
-\subsubsection*{IFF Controller}
+\subsubsection{IFF Controller}
 \label{ssec:test_id31_iff_controller}
 A decentralized IFF controller was designed to add damping to the suspension modes of the nano-hexapod for all considered payloads.
@ -9374,7 +9935,7 @@ 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}
-\subsubsection*{Damped Plant}
+\subsubsection{Damped Plant}
 \label{ssec:test_id31_iff_perf}
 As the model accurately represents the system dynamics, it can be used to estimate the damped plant, i.e. the transfer functions from \(\bm{u}^\prime\) to \(\bm{\mathcal{L}}\).
@ -9405,7 +9966,7 @@ Using the multi-body model, the controller was designed and optimized to ensure
 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.
-\subsection*{High Authority Control in the frame of the struts}
+\subsection{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.
 The corresponding control architecture is shown in Figure \ref{fig:test_id31_iff_hac_schematic}.
@ -9425,7 +9986,7 @@ K_{\text{HAC}} & & 0 \\
 \includegraphics[scale=1]{figs/test_id31_iff_hac_schematic.png}
 \caption{\label{fig:test_id31_iff_hac_schematic}Block diagram of the implemented HAC-IFF controllers. The controller \(\bm{K}_{\text{HAC}}\) is a diagonal controller with the same elements on every diagonal term \(K_{\text{HAC}}\).}
 \end{figure}
-\subsubsection*{Damped Plant}
+\subsubsection{Damped Plant}
 \label{ssec:test_id31_iff_hac_plant}
 To verify whether the multi-body model accurately represents the measured damped dynamics, both the direct terms and coupling terms corresponding to the first actuator are compared in Figure \ref{fig:test_id31_comp_simscape_hac}.
@ -9447,7 +10008,7 @@ 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}
-\subsubsection*{Interaction Analysis}
+\subsubsection{Interaction Analysis}
 \label{sec:test_id31_hac_interaction_analysis}
 The control strategy here is to apply a diagonal control in the frame of the struts; thus, it is important to determine the frequency at which the multivariable effects become significant, as this represents a critical limitation of the control approach.
@ -9475,7 +10036,7 @@ 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}
-\subsubsection*{Robust Controller Design}
+\subsubsection{Robust Controller Design}
 \label{ssec:test_id31_iff_hac_controller}
 A diagonal controller was designed to be robust against changes in payload mass, which means that every damped plant shown in Figure \ref{fig:test_id31_comp_all_undamped_damped_plants} must be considered during the controller design.
@ -9506,7 +10067,7 @@ 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}
-\subsubsection*{Performance estimation with simulation of Tomography scans}
+\subsubsection{Performance estimation with simulation of Tomography scans}
 \label{ssec:test_id31_iff_hac_perf}
 To estimate the performances that can be expected with this HAC-LAC architecture and the designed controller, simulations of tomography experiments were performed\footnote{Note that the eccentricity of the ``point of interest'' with respect to the Spindle rotation axis has been tuned based on measurements.}.
@ -9529,7 +10090,7 @@ 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}
-\subsubsection*{Robustness estimation with simulation of Tomography scans}
+\subsubsection{Robustness estimation with simulation of Tomography scans}
 \label{ssec:test_id31_iff_hac_robustness}
 To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure \ref{fig:test_id31_picture_masses} (i.e. \(0\,kg\), \(13\,kg\), \(26\,kg\) and \(39\,kg\)).
@ -9557,7 +10118,7 @@ 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.
-\subsection*{Validation with Scientific experiments}
+\subsection{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.
 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.
@ -9593,9 +10154,9 @@ RMS & 30nm & 15nm & \(250\,\text{nrad}\)\\
 \bottomrule
 \end{tabularx}
 \end{table}
-\subsubsection*{Tomography Scans}
+\subsubsection{Tomography Scans}
 \label{ssec:test_id31_scans_tomography}
-\paragraph*{Slow Tomography scans}
+\paragraph{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.
@ -9631,7 +10192,7 @@ 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}
-\paragraph*{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}.
@ -9653,7 +10214,7 @@ 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}
-\paragraph*{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.
@ -9686,7 +10247,7 @@ 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}
-\subsubsection*{Reflectivity Scans}
+\subsubsection{Reflectivity Scans}
 \label{ssec:test_id31_scans_reflectivity}
 X-ray reflectivity measurements involve scanning thin structures, particularly solid/liquid interfaces, through the beam by varying the \(R_y\) angle.
@ -9714,11 +10275,11 @@ 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}
-\subsubsection*{Dirty Layer Scans}
+\subsubsection{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}
+\paragraph{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 \(1\,\mu m\).
@ -9750,7 +10311,7 @@ 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}
-\paragraph*{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.
@ -9804,13 +10365,13 @@ 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}
-\subsubsection*{Lateral Scans}
+\subsubsection{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}
+\paragraph{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.
@ -9842,7 +10403,7 @@ 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}
-\paragraph*{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.
@ -9875,7 +10436,7 @@ 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}
-\subsubsection*{Diffraction Tomography}
+\subsubsection{Diffraction Tomography}
 \label{ssec:test_id31_scans_diffraction_tomo}
 In diffraction tomography experiments, the micro-station performs combined motions: continuous rotation around the \(R_z\) axis while performing lateral scans along \(D_y\).
@ -9987,6 +10548,8 @@ 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.
 \section*{Experimental Validation - Conclusion}
 \label{sec:concept_conclusion}
 \chapter{Conclusion and Future Work}
 \label{chap:conclusion}
 \section{Alternative Architecture}