Work on test bench section + rework figures

paper/dehaeze21_mechatronics_approach_nass.org

@@ -21,6 +21,7 @@
 #+LATEX_HEADER: \usepackage{graphicx,tabularx,booktabs}
 #+LATEX_HEADER: \usepackage{blindtext,bm}
 #+LATEX_HEADER: \usepackage{subcaption}
+#+LATEX_HEADER: \usepackage{siunitx}
 #+LATEX_HEADER: \usepackage[USenglish]{babel}
 #+LATEX_HEADER: \setcounter{footnote}{1}
 #+LATEX_HEADER_EXTRA: \usepackage[colorlinks=true, allcolors=blue]{hyperref}
@@ -138,7 +139,7 @@ It consists of three main phases:
 
 ** Models
 As shown in Fig.\nbsp{}ref:fig:nass_mechatronics_approach, the models are at the core of the mechatronic approach.
-Not only one, but several models are used throughout the design with increasing level of complexity (Fig.\nbsp{fig:nass_models}).
+Indeed, several models are used throughout the design with increasing level of complexity (Fig.\nbsp{}ref:fig:nass_models).
 
 #+begin_export latex
 \begin{figure*}[htbp]
@@ -165,77 +166,76 @@ Not only one, but several models are used throughout the design with increasing
 \end{figure*}
 #+end_export
 
-At the beginning of the conceptual phase, simple "mass-spring-dampers" models (Fig.\nbsp{}ref:fig:mass_spring_damper_hac_lac) are used in order to evaluate the performances of different concepts.
+At the beginning of the conceptual phase, simple "mass-spring-dampers" models are used (Fig.\nbsp{}ref:fig:mass_spring_damper_hac_lac) in order to easily try different concepts.
-Based on this model, it has been concluded that a nano-hexapod with low frequency "suspension" modes would help both for the reduction of the effects of several disturbances and for the decoupling between the nano-hexapod dynamics and the complex micro-station dynamics.
+Noise budgeting and closed-loop simulations were performed, and it was concluded that a nano-hexapod with low frequency "suspension" modes would help both for the reduction of the effects of disturbances and for the decoupling between the nano-hexapod dynamics and the complex micro-station dynamics.
-This will greatly help simplifying the control.
+Also, including a force sensor in series with the nano-hexapod's actuators can be used to actively damp the resonances using the "Integral Force Feedback" (IFF) strategy.
-# Say that HAC-LAC is tested with the model => should include force sensor
+The goal is to obtain a "plant" dynamics which is easy to control in a robust way.
 
 Rapidly, a more sophisticated multi-body model (Fig.\nbsp{}ref:fig:nass_simscape_3d) has been used.
 This model is based on the 3D representation of the micro-station as well as on extensive dynamical measurements.
-Time domain simulations can then be performed where each stage is moving with the associated positioning errors and disturbances.
+Time domain simulations can then be performed with each stage moving with the associated positioning errors and disturbances.
+Such model is more realistic and permits to study effects which were not modeled with the previous model such as the coupling between directions and effect of the rotation of the spindle on the nano-hexapod's dynamics (gyroscopic effects cite:dehaeze21_activ_dampin_rotat_platf_using).
 The multi-input multi-output control strategy can be developed and tested.
 
 During the detail design phase, the nano-hexapod model is updated by importing the 3D parts exported from the CAD software.
 The key elements of the nano-hexapod such as the flexible joints and the APA are optimized using a Finite Element Software.
-As the flexible modes of the system are what generally limit the controller bandwidth, they are important to model in order to understand which are problematic and which are to be maximized.
+As the flexible modes of the system are what generally limit the controller bandwidth, they are important to model in order to understand which ones are problematic and should be maximized.
-In order to do so, a "super-element" can be exported and imported in Simscape (Fig.\nbsp{}ref:fig:super_element_simscape).
+In order to do so, a "super-element" can be exported using a finite element analysis software and imported in Simscape (Fig.\nbsp{}ref:fig:super_element_simscape).
 Such process is described in cite:brumund21_multib_simul_reduc_order_flexib_bodies_fea.
-
+The multi-body model with included flexible elements can be used to obtain very accurately the dynamics of the system.
-# - [ ] Table that compares the three models in terms of:
+However due to the large number of states included, it becomes non practical to perform time domain simulations.
-#   - time simulation
-#   - FRF
-#   - accuracy
-#   - easy to use
 
 Finally, during the experimental phase, the models are refined using experimental system identification.
-The models are still very useful to understand the measurements and the associated performance limitations.
+These models can be used to understand the measurements, the associated performance limitations and to gain insight on which measures to take in order to overcome these limitations.
-They are used to have a better insight on which measures to take in order to overcome the current limitations.
-
-For instance, it has been found that when fixing encoders to the struts (Fig.\nbsp{}ref:fig:nano_hexapod_elements), several flexible modes of the APA were appearing the dynamics which render the control using the encoders very complex.
-Therefore, an alternative configuration with the encoders fixed to the plates instead was used.
 
+For instance, it has been found that when fixing the encoders to the struts (Fig.\nbsp{}ref:fig:nano_hexapod_elements), several flexible modes of the APA were appearing in the dynamics which render the control using the encoders very complex.
+Therefore, an alternative configuration with the encoders fixed to the plates was used instead.
 
 * NANO-HEXAPOD DESIGN
 ** Nano-Hexapod Specifications
-A CAD view of the nano-hexapod is shown in Fig.\nbsp{}ref:fig:nano_hexapod_elements.
+The Nano-Hexapod must have a maximum height of $95\,mm$, support samples up to $50\,kg$ and have a stroke of $\approx 100\,\mu m$.
-It is composed of 6 struts fixed in between two plates.
+it have few parts: two plates and 6 active struts in between.
-Each strut is composed of one flexible joints at each end, and one actuator (Fig.\nbsp{}ref:fig:picture_nano_hexapod_strut).
+Each strut is composed of one flexible joint at each end, and one actuator (Fig.\nbsp{}ref:fig:picture_nano_hexapod_strut).
-).
+A 3D view of the nano-hexapod is shown in Fig.\nbsp{}ref:fig:nano_hexapod_elements.
-And encoder can be fixed to the struts as shown, but can also be directly fixed to the plates (not represented here).
-
-Basic specifications:
-- Limited height (95mm)
-- Stroke $\approx 100\,\mu m$
-- Load up to $50\,kg$
-
-Based on the models used throughout the mechatronic approach, several specifications was obtained in order to maximize the performances of the system:
-- Axial stiffness of the struts $\approx 2\,\mu m/N$ such that the nano-hexapod dynamics is insensible to the rotation as well as decoupled from the micro-station dynamics
-- Small bending stiffness and high axial stiffness of the flexible joints
-- Precise positioning of the $b_i$ and $\hat{s}_i$
-- Flexible modes of the top-plate as high as possible
-- Integration of a force sensor for active damping purposes (more in the next section)
-
-** Parts' Optimization
-- APA / Flexible Joints / Plates
-
-The flexible joints and the top plates have been optimize using a Finite Element Model combine with the multi-body model of the nano-hexapod.
-
-The actuators are APA300ML from Cedrat Technologies.
-Three stacks: two as actuator one as sensor
 
 #+name: fig:nano_hexapod_elements
-#+attr_latex: :float multicolumn :width \linewidth
+#+attr_latex: :float multicolumn :width 0.9\linewidth
 #+caption: CAD view of the nano-hexapod with key elements
 [[file:figs/nano_hexapod_elements.pdf]]
 
-** Mounted Nano-Hexapod
+Based on the models used throughout the mechatronic approach, several specifications was obtained in order to maximize the performances of the system:
-- Mounting benches
+- Actuator: axial stiffness $\approx \SI{2}{\um}$
+- Flexible joints: bending stiffness $< \SI{100}{Nm/rad}$ and axial stiffness $> \SI{100}{N/\um}$
+- Precise positioning of the $b_i$ and $\hat{s}_i$
+- Flexible modes of the top-plate as high as possible
+- Integration of a force sensor in each strut
+
+** Parts' Optimization
+The geometry of the flexible joint could be optimized using a finite element software.
+The obtained stiffnesses are compliance with the requirements and the model was updated.
+
+The top plate was manually optimized to maximize its flexible modes.
+Flexible modes at around $\SI{700}{Hz}$ could be obtained.
+
+Amplified Piezoelectric Actuators (APA) were found to be the most suitable actuator for the nano-hexapod due to its compact size, large stroke and adequate stiffness.
+The chosen model was the APA300ML from Cedrat Technologies (shown in Fig.\nbsp{}ref:fig:picture_nano_hexapod_strut).
+It is composed of three piezoelectric stacks, a lever mechanism increasing the stroke up to $\approx \SI{300}{\um}$ and decreasing the axial stiffness down to $\approx \SI{1.8}{\um}$.
+One of the three stacks can be used as a force sensor, at the price of loosing $1/3$ of the stroke.
+The main benefits is the good "collocation" of the sensor stack with the actuator stacks, meaning that the active damping controller will easily be made robust.
 
 #+name: fig:picture_nano_hexapod_strut
 #+attr_latex: :width 0.9\linewidth
 #+caption: Picture of a nano-hexapod's strut
 [[file:figs/picture_nano_hexapod_strut.pdf]]
 
+** Nano-Hexapod Mounting
+After each element
+
+# Mounting bench for the struts
+# Mounting tool for the nano-hexapod
+
+The nano-hexapod mounted on top of the micro-station is shown in Fig.\nbsp{}ref:fig:nano_hexapod_picture.
+
 #+name: fig:nano_hexapod_picture
 #+attr_latex: :width 0.9\linewidth
 #+caption: Nano-Hexapod on top of the ID31 micro-station
@@ -243,20 +243,28 @@ Three stacks: two as actuator one as sensor
 
 * TEST-BENCHES
 ** Flexible Joints and Instrumentation
-** APA/Struts Dynamics
+Several test benches were used to characterize the individual elements of the NASS.
-Several test benches were used for all the critical elements of the nano-hexapod.
-For instant, the bending stiffness of the flexible joints are measured, and the model is refined.
-The measurement noise of the encoders are also measured, and the input/output relationship and the output voltage noise of the voltage amplifiers are measured.
 
-Perhaps the most important test bench was the one used to identify the dynamics of the amplified piezoelectric actuator (shown in Fig.\nbsp{}ref:fig:test_bench_apa_schematic).
+The bending stiffness of the flexible joints was measured by applying a (measured) force to one end of the joint while measuring its deflection at the same time.
+This helped exclude the ones not compliant with the requirement and pair the remaining ones.
+
+The transfer function from input to output voltage of the voltage amplifier[fn:1] as well as its output noise was measured.
+Similarly, the measurement noise of the encoders[fn:2] was also measured.
+
+These simple measurements on individual elements are useful to refine their models, found any problem as early as possible, and will help analyzing the results once the nano-hexapod is mounted and all elements combined.
+
+** APA and Struts Dynamics
+
+An other test bench schematically shown in Fig.\nbsp{}ref:fig:test_bench_apa_schematic was used to identify the dynamics of the APA.
 It consist of a $5\,\text{kg}$ granite vertical guided with an air bearing and fixed on top of the APA.
 An excitation signal (low pass filtered white noise) is generated and applied to two of the piezoelectric stacks.
 Both the voltage generated by the third piezoelectric stack and the displacement measured by the encoder are recorded.
-The two obtained FRF can then be compared with the model and the piezoelectric constant are identified.
+The two obtained frequency response functions (FRF) can then be compared with the model (Fig.\nbsp{}ref:fig:apa_test_bench_results).
-These constants are used to do the conversion from the mechanical domain (force, strain) easily accessible on the model to the electrical domain (voltages, charges) easily measured.
-After identification of these constant, the match between the measured FRF and the model dynamics is quite good (Fig.\nbsp{}ref:fig:apa_test_bench_results)
 
-The same bench was also used with the struts in order to study the effects of the flexible joints.
+The piezoelectric constants describing the conversion from the mechanical domain (force, strain) easily accessible on the model to the electrical domain (voltages, charges) easily measured can be estimated.
+With these constants, the match between the measured FRF and the model dynamics is very good (Fig.\nbsp{}ref:fig:apa_test_bench_results)
+
+The same bench was also used with the struts in order to study the added effects of the flexible joints.
 
 #+name: fig:test_bench_apa_schematic
 #+attr_latex: :scale 1
@@ -283,28 +291,81 @@ The same bench was also used with the struts in order to study the effects of th
 
 ** Nano-Hexapod
 
-#+name: fig:nass_hac_lac_schematic_test
+Once the nano-hexapod is mounted, its dynamics is identified.
-#+attr_latex: :width \linewidth
+To do so, each actuator is individually excited and the six force sensors and six encoders signals are recorded each time.
-#+caption: HAC-LAC Strategy - Block Diagram. The signals are: $\bm{r}$ the wanted sample's position, $\bm{X}$ the measured sample's position, $\bm{\epsilon}_{\mathcal{X}}$ the sample's position error, $\bm{\epsilon}_{\mathcal{L}}$ the sample position error expressed in the "frame" of the nano-hexapod struts, $\bm{u}$ the generated DAC voltages applied to the voltage amplifiers and then to the piezoelectric actuator stacks, $\bm{u}^\prime$ the new inputs corresponding to the damped plant, $\bm{\tau}$ the measured sensor stack voltages. $\bm{T}$ is . $\bm{K}_{\tiny IFF}$ is the Low Authority Controller used for active damping. $\bm{K}_{L}$ is the High Authority Controller.
+Two $6$ by $6$ FRF matrices are computed.
-[[file:figs/nass_hac_lac_block_diagram_without_elec.pdf]]
+The diagonal elements of these two matrices are shown in Fig.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape and compared with the model.
 
+From Fig.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape_de one can observe the following modes:
+- From $\SI{100}{Hz}$ to $\SI{200}{Hz}$: six suspension modes
+- At $\SI{230}{Hz}$ and $\SI{340}{Hz}$: flexible modes of the APA, also modeled thanks to the flexible model of the APA
+- At around $\SI{700}{Hz}$: flexible modes of the top plate, not modeled (taken as a rigid body)
 
-#+name: fig:nano_hexapod_identification_comp_simscape
+The transfer function from the actuator to the force sensors has alternating poles and zeros (Fig.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape_Vs) which is confirming the good "collocation" between the stacks.
-#+attr_latex: :width \linewidth
-#+caption: Measured FRF and Simscape dynamics.
-[[file:figs/nano_hexapod_identification_comp_simscape.pdf]]
 
+IFF is then applied individually on each pair of actuator/force sensor in order to actively damp the modes shown in Fig.\nbsp{}ref:fig:nano_hexapod_identification_comp_simscape_Vs.
+The optimal gain of the IFF controller is determined from the model.
+After applying the active damping technique, the $6$ by $6$ FRF matrix from the actuator to the encoders is identified again and shown in Fig.\nbsp{}ref:fig:nano_hexapod_identification_damp_comp_simscape.
 
-#+name: fig:nano_hexapod_identification_damp_comp_simscape
+# #+name: fig:nass_hac_lac_schematic_test
-#+attr_latex: :width \linewidth
+# #+attr_latex: :width \linewidth
-#+caption: Undamped and Damped plant using IFF (measured FRF and Simscape model).
+# #+caption: HAC-LAC Strategy - Block Diagram. The signals are: $\bm{r}$ the wanted sample's position, $\bm{X}$ the measured sample's position, $\bm{\epsilon}_{\mathcal{X}}$ the sample's position error, $\bm{\epsilon}_{\mathcal{L}}$ the sample position error expressed in the "frame" of the nano-hexapod struts, $\bm{u}$ the generated DAC voltages applied to the voltage amplifiers and then to the piezoelectric actuator stacks, $\bm{u}^\prime$ the new inputs corresponding to the damped plant, $\bm{\tau}$ the measured sensor stack voltages. $\bm{T}$ is . $\bm{K}_{\tiny IFF}$ is the Low Authority Controller used for active damping. $\bm{K}_{L}$ is the High Authority Controller.
-[[file:figs/nano_hexapod_identification_damp_comp_simscape.pdf]]
+# [[file:figs/nass_hac_lac_block_diagram_without_elec.pdf]]
+
+#+begin_export latex
+\begin{figure}[htbp]
+  \begin{subfigure}[t]{0.48\linewidth}
+    \centering
+    \includegraphics[width=0.95\linewidth]{figs/nano_hexapod_identification_comp_simscape_de.pdf}
+    \caption{\label{fig:nano_hexapod_identification_comp_simscape_de} Encoder}
+  \end{subfigure}
+  \hfill
+  \begin{subfigure}[t]{0.48\linewidth}
+    \centering
+    \includegraphics[width=0.95\linewidth]{figs/nano_hexapod_identification_comp_simscape_Vs.pdf}
+    \caption{\label{fig:nano_hexapod_identification_comp_simscape_Vs} Force Sensor}
+  \end{subfigure}
+  \caption{\label{fig:nano_hexapod_identification_comp_simscape}Measured Frequency Response functions compared with the Simscape model. From the actuator stacks voltage to the encoder (\subref{fig:apa_test_bench_results_de}) and to the force sensor stack (\subref{fig:apa_test_bench_results_Vs}).}
+  \centering
+\end{figure}
+#+end_export
+
+#+begin_export latex
+\begin{figure}[htbp]
+  \begin{subfigure}[t]{0.49\linewidth}
+    \centering
+    \includegraphics[height=5.5cm]{figs/nano_hexapod_identification_damp_comp_simscape_diag.pdf}
+    \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape_diag} Diagonal}
+  \end{subfigure}
+  \hfill
+  \begin{subfigure}[t]{0.49\linewidth}
+    \centering
+    \includegraphics[height=5.5cm]{figs/nano_hexapod_identification_damp_comp_simscape_off_diag.pdf}
+    \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape_off_diag} Off-Diagonal}
+  \end{subfigure}
+  \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape}}
+  \centering
+\end{figure}
+#+end_export
 
 * CONCLUSION
 
+Future work:
+- actively damp the top plate flexible modes
+- make the controller robust to change of payload mass
+- integrate it on top of the micro-station
+
 * ACKNOWLEDGMENTS
 This research was made possible by a grant from the FRIA.
-We thank the following people for their support, without whose help this work would never have been possible: V. Honkimaki, L. Ducotte and M. Lessourd and the whole team of the Precision Mechatronic Laboratory.
+Damien Coulomb
+Youness Benya
+Marc Lesourd
+Philipp Brumund
 
 * REFERENCES                                                         :ignore:
 \printbibliography{}
+
+* Footnotes                                                           :ignore:
+
+[fn:1]PD200 from PiezoDrive
+[fn:2]Vionic from Renishaw

paper/dehaeze21_mechatronics_approach_nass.tex

@@ -1,4 +1,4 @@
-% Created 2021-07-14 mer. 12:41
+% Created 2021-07-14 mer. 18:47
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, keeplastbox, biblatex, boxit]{jacow}
 
@@ -6,6 +6,7 @@
 \usepackage{graphicx,tabularx,booktabs}
 \usepackage{blindtext,bm}
 \usepackage{subcaption}
+\usepackage{siunitx}
 \usepackage[USenglish, english]{babel}
 \setcounter{footnote}{1}
 \usepackage[colorlinks=true, allcolors=blue]{hyperref}
@@ -85,7 +86,7 @@ Then the hexapod can be mounted and fully tested with the instrumentation and th
 
 \subsection{Models}
 As shown in Fig.~\ref{fig:nass_mechatronics_approach}, the models are at the core of the mechatronic approach.
-Not only one, but several models are used throughout the design with increasing level of complexity (Fig.~\{fig:nass\_models\}).
+Indeed, several models are used throughout the design with increasing level of complexity (Fig.~\ref{fig:nass_models}).
 
 \begin{figure*}[htbp]
   \begin{subfigure}[t]{0.25\linewidth}
@@ -110,80 +111,77 @@ Not only one, but several models are used throughout the design with increasing
   \centering
 \end{figure*}
 
-At the beginning of the conceptual phase, simple ``mass-spring-dampers'' models (Fig.~\ref{fig:mass_spring_damper_hac_lac}) are used in order to evaluate the performances of different concepts.
+At the beginning of the conceptual phase, simple ``mass-spring-dampers'' models are used (Fig.~\ref{fig:mass_spring_damper_hac_lac}) in order to easily try different concepts.
-Based on this model, it has been concluded that a nano-hexapod with low frequency ``suspension'' modes would help both for the reduction of the effects of several disturbances and for the decoupling between the nano-hexapod dynamics and the complex micro-station dynamics.
+Noise budgeting and closed-loop simulations were performed, and it was concluded that a nano-hexapod with low frequency ``suspension'' modes would help both for the reduction of the effects of disturbances and for the decoupling between the nano-hexapod dynamics and the complex micro-station dynamics.
-This will greatly help simplifying the control.
+Also, including a force sensor in series with the nano-hexapod's actuators can be used to actively damp the resonances using the ``Integral Force Feedback'' (IFF) strategy.
+The goal is to obtain a ``plant'' dynamics which is easy to control in a robust way.
 
 Rapidly, a more sophisticated multi-body model (Fig.~\ref{fig:nass_simscape_3d}) has been used.
 This model is based on the 3D representation of the micro-station as well as on extensive dynamical measurements.
-Time domain simulations can then be performed where each stage is moving with the associated positioning errors and disturbances.
+Time domain simulations can then be performed with each stage moving with the associated positioning errors and disturbances.
+Such model is more realistic and permits to study effects which were not modeled with the previous model such as the coupling between directions and effect of the rotation of the spindle on the nano-hexapod's dynamics (gyroscopic effects \cite{dehaeze21_activ_dampin_rotat_platf_using}).
 The multi-input multi-output control strategy can be developed and tested.
 
 During the detail design phase, the nano-hexapod model is updated by importing the 3D parts exported from the CAD software.
 The key elements of the nano-hexapod such as the flexible joints and the APA are optimized using a Finite Element Software.
-As the flexible modes of the system are what generally limit the controller bandwidth, they are important to model in order to understand which are problematic and which are to be maximized.
+As the flexible modes of the system are what generally limit the controller bandwidth, they are important to model in order to understand which ones are problematic and should be maximized.
-In order to do so, a ``super-element'' can be exported and imported in Simscape (Fig.~\ref{fig:super_element_simscape}).
+In order to do so, a ``super-element'' can be exported using a finite element analysis software and imported in Simscape (Fig.~\ref{fig:super_element_simscape}).
 Such process is described in \cite{brumund21_multib_simul_reduc_order_flexib_bodies_fea}.
+The multi-body model with included flexible elements can be used to obtain very accurately the dynamics of the system.
+However due to the large number of states included, it becomes non practical to perform time domain simulations.
 
 Finally, during the experimental phase, the models are refined using experimental system identification.
-The models are still very useful to understand the measurements and the associated performance limitations.
+These models can be used to understand the measurements, the associated performance limitations and to gain insight on which measures to take in order to overcome these limitations.
-They are used to have a better insight on which measures to take in order to overcome the current limitations.
-
-For instance, it has been found that when fixing encoders to the struts (Fig.~\ref{fig:nano_hexapod_elements}), several flexible modes of the APA were appearing the dynamics which render the control using the encoders very complex.
-Therefore, an alternative configuration with the encoders fixed to the plates instead was used.
 
+For instance, it has been found that when fixing the encoders to the struts (Fig.~\ref{fig:nano_hexapod_elements}), several flexible modes of the APA were appearing in the dynamics which render the control using the encoders very complex.
+Therefore, an alternative configuration with the encoders fixed to the plates was used instead.
 
 \section{NANO-HEXAPOD DESIGN}
 \subsection{Nano-Hexapod Specifications}
-A CAD view of the nano-hexapod is shown in Fig.~\ref{fig:nano_hexapod_elements}.
+The Nano-Hexapod must have a maximum height of \(95\,mm\), support samples up to \(50\,kg\) and have a stroke of \(\approx 100\,\mu m\).
-It is composed of 6 struts fixed in between two plates.
+it have few parts: two plates and 6 active struts in between.
-Each strut is composed of one flexible joints at each end, and one actuator (Fig.~\ref{fig:picture_nano_hexapod_strut}).
+Each strut is composed of one flexible joint at each end, and one actuator (Fig.~\ref{fig:picture_nano_hexapod_strut}).
-).
+A 3D view of the nano-hexapod is shown in Fig.~\ref{fig:nano_hexapod_elements}.
-And encoder can be fixed to the struts as shown, but can also be directly fixed to the plates (not represented here).
-
-Basic specifications:
-\begin{itemize}
-\item Limited height (95mm)
-\item Stroke \(\approx 100\,\mu m\)
-\item Load up to \(50\,kg\)
-\end{itemize}
-
-Based on the models used throughout the mechatronic approach, several specifications was obtained in order to maximize the performances of the system:
-\begin{itemize}
-\item Axial stiffness of the struts \(\approx 2\,\mu m/N\) such that the nano-hexapod dynamics is insensible to the rotation as well as decoupled from the micro-station dynamics
-\item Small bending stiffness and high axial stiffness of the flexible joints
-\item Precise positioning of the \(b_i\) and \(\hat{s}_i\)
-\item Flexible modes of the top-plate as high as possible
-\item Integration of a force sensor for active damping purposes (more in the next section)
-\end{itemize}
-
-\subsection{Parts' Optimization}
-\begin{itemize}
-\item APA / Flexible Joints / Plates
-\end{itemize}
-
-The flexible joints and the top plates have been optimize using a Finite Element Model combine with the multi-body model of the nano-hexapod.
-
-The actuators are APA300ML from Cedrat Technologies.
-Three stacks: two as actuator one as sensor
 
 \begin{figure*}
 \centering
-\includegraphics[scale=1,width=\linewidth]{figs/nano_hexapod_elements.pdf}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/nano_hexapod_elements.pdf}
 \caption{\label{fig:nano_hexapod_elements}CAD view of the nano-hexapod with key elements}
 \end{figure*}
 
-\subsection{Mounted Nano-Hexapod}
+Based on the models used throughout the mechatronic approach, several specifications was obtained in order to maximize the performances of the system:
 \begin{itemize}
-\item Mounting benches
+\item Actuator: axial stiffness \(\approx \SI{2}{\um}\)
+\item Flexible joints: bending stiffness \(< \SI{100}{Nm/rad}\) and axial stiffness \(> \SI{100}{N/\um}\)
+\item Precise positioning of the \(b_i\) and \(\hat{s}_i\)
+\item Flexible modes of the top-plate as high as possible
+\item Integration of a force sensor in each strut
 \end{itemize}
 
+\subsection{Parts' Optimization}
+The geometry of the flexible joint could be optimized using a finite element software.
+The obtained stiffnesses are compliance with the requirements and the model was updated.
+
+The top plate was manually optimized to maximize its flexible modes.
+Flexible modes at around \(\SI{700}{Hz}\) could be obtained.
+
+Amplified Piezoelectric Actuators (APA) were found to be the most suitable actuator for the nano-hexapod due to its compact size, large stroke and adequate stiffness.
+The chosen model was the APA300ML from Cedrat Technologies (shown in Fig.~\ref{fig:picture_nano_hexapod_strut}).
+It is composed of three piezoelectric stacks, a lever mechanism increasing the stroke up to \(\approx \SI{300}{\um}\) and decreasing the axial stiffness down to \(\approx \SI{1.8}{\um}\).
+One of the three stacks can be used as a force sensor, at the price of loosing \(1/3\) of the stroke.
+The main benefits is the good ``collocation'' of the sensor stack with the actuator stacks, meaning that the active damping controller will easily be made robust.
+
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.9\linewidth]{figs/picture_nano_hexapod_strut.pdf}
 \caption{\label{fig:picture_nano_hexapod_strut}Picture of a nano-hexapod's strut}
 \end{figure}
 
+\subsection{Nano-Hexapod Mounting}
+After each element
+
+The nano-hexapod mounted on top of the micro-station is shown in Fig.~\ref{fig:nano_hexapod_picture}.
+
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.9\linewidth]{figs/nano_hexapod_picture.jpg}
@@ -192,20 +190,28 @@ Three stacks: two as actuator one as sensor
 
 \section{TEST-BENCHES}
 \subsection{Flexible Joints and Instrumentation}
-\subsection{APA/Struts Dynamics}
+Several test benches were used to characterize the individual elements of the NASS.
-Several test benches were used for all the critical elements of the nano-hexapod.
-For instant, the bending stiffness of the flexible joints are measured, and the model is refined.
-The measurement noise of the encoders are also measured, and the input/output relationship and the output voltage noise of the voltage amplifiers are measured.
 
-Perhaps the most important test bench was the one used to identify the dynamics of the amplified piezoelectric actuator (shown in Fig.~\ref{fig:test_bench_apa_schematic}).
+The bending stiffness of the flexible joints was measured by applying a (measured) force to one end of the joint while measuring its deflection at the same time.
+This helped exclude the ones not compliant with the requirement and pair the remaining ones.
+
+The transfer function from input to output voltage of the voltage amplifier\footnote{PD200 from PiezoDrive} as well as its output noise was measured.
+Similarly, the measurement noise of the encoders\footnote{Vionic from Renishaw} was also measured.
+
+These simple measurements on individual elements are useful to refine their models, found any problem as early as possible, and will help analyzing the results once the nano-hexapod is mounted and all elements combined.
+
+\subsection{APA and Struts Dynamics}
+
+An other test bench schematically shown in Fig.~\ref{fig:test_bench_apa_schematic} was used to identify the dynamics of the APA.
 It consist of a \(5\,\text{kg}\) granite vertical guided with an air bearing and fixed on top of the APA.
 An excitation signal (low pass filtered white noise) is generated and applied to two of the piezoelectric stacks.
 Both the voltage generated by the third piezoelectric stack and the displacement measured by the encoder are recorded.
-The two obtained FRF can then be compared with the model and the piezoelectric constant are identified.
+The two obtained frequency response functions (FRF) can then be compared with the model (Fig.~\ref{fig:apa_test_bench_results}).
-These constants are used to do the conversion from the mechanical domain (force, strain) easily accessible on the model to the electrical domain (voltages, charges) easily measured.
-After identification of these constant, the match between the measured FRF and the model dynamics is quite good (Fig.~\ref{fig:apa_test_bench_results})
 
-The same bench was also used with the struts in order to study the effects of the flexible joints.
+The piezoelectric constants describing the conversion from the mechanical domain (force, strain) easily accessible on the model to the electrical domain (voltages, charges) easily measured can be estimated.
+With these constants, the match between the measured FRF and the model dynamics is very good (Fig.~\ref{fig:apa_test_bench_results})
+
+The same bench was also used with the struts in order to study the added effects of the flexible joints.
 
 \begin{figure}[htbp]
 \centering
@@ -231,31 +237,71 @@ The same bench was also used with the struts in order to study the effects of th
 
 \subsection{Nano-Hexapod}
 
+Once the nano-hexapod is mounted, its dynamics is identified.
+To do so, each actuator is individually excited and the six force sensors and six encoders signals are recorded each time.
+Two \(6\) by \(6\) FRF matrices are computed.
+The diagonal elements of these two matrices are shown in Fig.~\ref{fig:nano_hexapod_identification_comp_simscape} and compared with the model.
+
+From Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_de} one can observe the following modes:
+\begin{itemize}
+\item From \(\SI{100}{Hz}\) to \(\SI{200}{Hz}\): six suspension modes
+\item At \(\SI{230}{Hz}\) and \(\SI{340}{Hz}\): flexible modes of the APA, also modeled thanks to the flexible model of the APA
+\item At around \(\SI{700}{Hz}\): flexible modes of the top plate, not modeled (taken as a rigid body)
+\end{itemize}
+
+The transfer function from the actuator to the force sensors has alternating poles and zeros (Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_Vs}) which is confirming the good ``collocation'' between the stacks.
+
+IFF is then applied individually on each pair of actuator/force sensor in order to actively damp the modes shown in Fig.~\ref{fig:nano_hexapod_identification_comp_simscape_Vs}.
+The optimal gain of the IFF controller is determined from the model.
+After applying the active damping technique, the \(6\) by \(6\) FRF matrix from the actuator to the encoders is identified again and shown in Fig.~\ref{fig:nano_hexapod_identification_damp_comp_simscape}.
+
 \begin{figure}[htbp]
+  \begin{subfigure}[t]{0.48\linewidth}
+    \centering
+    \includegraphics[width=0.95\linewidth]{figs/nano_hexapod_identification_comp_simscape_de.pdf}
+    \caption{\label{fig:nano_hexapod_identification_comp_simscape_de} Encoder}
+  \end{subfigure}
+  \hfill
+  \begin{subfigure}[t]{0.48\linewidth}
+    \centering
+    \includegraphics[width=0.95\linewidth]{figs/nano_hexapod_identification_comp_simscape_Vs.pdf}
+    \caption{\label{fig:nano_hexapod_identification_comp_simscape_Vs} Force Sensor}
+  \end{subfigure}
+  \caption{\label{fig:nano_hexapod_identification_comp_simscape}Measured Frequency Response functions compared with the Simscape model. From the actuator stacks voltage to the encoder (\subref{fig:apa_test_bench_results_de}) and to the force sensor stack (\subref{fig:apa_test_bench_results_Vs}).}
   \centering
-\includegraphics[scale=1,width=\linewidth]{figs/nass_hac_lac_block_diagram_without_elec.pdf}
-\caption{\label{fig:nass_hac_lac_schematic_test}HAC-LAC Strategy - Block Diagram. The signals are: \(\bm{r}\) the wanted sample's position, \(\bm{X}\) the measured sample's position, \(\bm{\epsilon}_{\mathcal{X}}\) the sample's position error, \(\bm{\epsilon}_{\mathcal{L}}\) the sample position error expressed in the ``frame'' of the nano-hexapod struts, \(\bm{u}\) the generated DAC voltages applied to the voltage amplifiers and then to the piezoelectric actuator stacks, \(\bm{u}^\prime\) the new inputs corresponding to the damped plant, \(\bm{\tau}\) the measured sensor stack voltages. \(\bm{T}\) is . \(\bm{K}_{\tiny IFF}\) is the Low Authority Controller used for active damping. \(\bm{K}_{L}\) is the High Authority Controller.}
 \end{figure}
 
-
 \begin{figure}[htbp]
+  \begin{subfigure}[t]{0.49\linewidth}
     \centering
-\includegraphics[scale=1,width=\linewidth]{figs/nano_hexapod_identification_comp_simscape.pdf}
+    \includegraphics[height=5.5cm]{figs/nano_hexapod_identification_damp_comp_simscape_diag.pdf}
-\caption{\label{fig:nano_hexapod_identification_comp_simscape}Measured FRF and Simscape dynamics.}
+    \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape_diag} Diagonal}
-\end{figure}
+  \end{subfigure}
-
+  \hfill
-
+  \begin{subfigure}[t]{0.49\linewidth}
-\begin{figure}[htbp]
+    \centering
+    \includegraphics[height=5.5cm]{figs/nano_hexapod_identification_damp_comp_simscape_off_diag.pdf}
+    \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape_off_diag} Off-Diagonal}
+  \end{subfigure}
+  \caption{\label{fig:nano_hexapod_identification_damp_comp_simscape}}
   \centering
-\includegraphics[scale=1,width=\linewidth]{figs/nano_hexapod_identification_damp_comp_simscape.pdf}
-\caption{\label{fig:nano_hexapod_identification_damp_comp_simscape}Undamped and Damped plant using IFF (measured FRF and Simscape model).}
 \end{figure}
 
 \section{CONCLUSION}
 
+Future work:
+\begin{itemize}
+\item actively damp the top plate flexible modes
+\item make the controller robust to change of payload mass
+\item integrate it on top of the micro-station
+\end{itemize}
+
 \section{ACKNOWLEDGMENTS}
 This research was made possible by a grant from the FRIA.
-We thank the following people for their support, without whose help this work would never have been possible: V. Honkimaki, L. Ducotte and M. Lessourd and the whole team of the Precision Mechatronic Laboratory.
+Damien Coulomb
+Youness Benya
+Marc Lesourd
+Philipp Brumund
 
 \printbibliography{}
 \end{document}