Finish draft

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nass-design.org

@@ -252,29 +252,43 @@ CLOSED: [2025-04-21 Mon 14:13]
 [[file:figs/detail_design_nano_hexapod_elements.png]]
 
 
-*Design goals*:
+Detail design phase:
-- Position =bi= and =si=
+- key elements were optimized such as: actuator and flexible joints
-- Maximum height of 95mm
+- relative motion sensor (an encoder) was also selected
-- As close as possible to "perfect" stewart platform: flexible modes at high frequency
+- specific kinematics of the Stewart platform (i.e. position of joints and orientation of struts) was not found to be too critical for this application.
-- Easy mounting, easy change of strut in case of failure
+  Yet, the geometry was fixed in Section [...]
 
+In this section, the mechanical design of the active platform, shown in Figure ref:fig:detail_design_nano_hexapod_elements, is detailed.
 
-Presentation of the obtained design:
+The main design objectives are:
-- Fixation
+- Well defined kinematics: Good positioning of the top flexible joint rotation point $\bm{b}_i$ and correct orientation of the struts $\hat{\bm{s}}_i$.
-- Section on: Complete strut
+  The goal is to have a well defined geometry such that the Jacobian matrix is well defined.
-- Cable management
+- Space constrains: it should fit within a cylinder with radius of $120\,\text{mm}$ and height of $95\,\text{mm}$
-- Plates design
+- As good performances were obtained with the multi-body model.
-- FEM results
+  The final design should behave as close as possible to "perfect" stewart platform.
-- Explain again the different specifications in terms of space, payload, etc..
+  This means that the frequency of flexible modes that could be problematic for control must be made as high as possible.
-- CAD view of the nano-hexapod
+- Easy mounting and alignment.
-- Chosen geometry, materials, ease of mounting, cabling, ...
+- Easy maintenance: the struts should be easily changed in case for failure.
-- Validation on Simscape with accurate model?
 
 * Mechanical Design
 <<sec:detail_design_mechanics>>
-
-** Struts
 **** Introduction                                                  :ignore:
+**** Struts
+
+The strut design is shown in Figure ref:fig:detail_design_strut.
+
+The design of the struts was driven by:
+- having stiff interface between the amplified piezoelectric actuator and the two flexible joints
+- having stiff interface between the flexible joints and the two places (discussed afterwards)
+- Because the angular stroke of the flexible joints is fairly limited, it is important to be able to mount the strut such that the two cylindrical interfaces are coaxial.
+  Do to so:
+  - A mounting bench was designed
+    The mounting procedure will be described in Section [...]
+    # TODO - Add link to section
+  - Cylindrical washers, shown in Figure ref:fig:detail_design_strut_without_enc, were integrated to allow for adjustments.
+    The issue was that the flatness between the two interface planes of the APA shown in Figure ref:fig:detail_design_apa could not be guaranteed.
+    With the added cylindrical washers and the mounting tool, it should be possible to well align the struts even in the presence of machining inaccuracies.
+- Possibility to fix the encoder parallel to the strut, as shown in Figure ref:fig:detail_design_strut_with_enc
 
 #+name: fig:detail_design_strut
 #+caption: Design of the Nano-Hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration.
@@ -283,65 +297,88 @@ Presentation of the obtained design:
 #+attr_latex: :caption \subcaption{\label{fig:detail_design_strut_without_enc}Before encoder integration}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.9
-[[file:figs/detail_design_strut_without_enc.jpg]]
+[[file:figs/detail_design_strut_without_enc.png]]
 #+end_subfigure
 #+attr_latex: :caption \subcaption{\label{fig:detail_design_strut_with_enc}With the mounted encoder}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
+#+attr_latex: :scale 0.9
-[[file:figs/detail_design_strut_with_enc.jpg]]
+[[file:figs/detail_design_strut_with_enc.png]]
 #+end_subfigure
 #+end_figure
 
-**** Flexible joints
+The flexible joints are manufactured using wire-cut electrical discharge machining, allowing for:
+- very tight tolerances:
+  - allowing good location of the center of rotation with respect to the plate interfaces (red surfaces shown in Figure ref:fig:detail_design_flexible_joint)
+  - allowing correct neck dimension to have the wanted properties (stiffness and angular stroke)
+- Such part is fragile, mainly due to its small "neck" dimension of only $0.25\,\text{mm}$
+  Such machining technique has little to no cutting forces.
 
-Flexible joints: X5CrNiCuNb16-4 (F16Ph)
+The flexible joints are made from a stainless steel referenced as "X5CrNiCuNb16-4" (also called "F16Ph").
-- high yield strength: specified >1GPa using heat treatment
+This material is chosen for:
-- high fatigue resistance
+- its high yield strength: specified >1GPa using heat treatment.
+- its high fatigue resistance
+
+Figure ref:fig:detail_design_flexible_joint
+- Interface with the APA has a cylindrical shape to allow the use of cylindrical washers
+  A slotted hole has been added to align the flexible joint with the APA using a dowel pin.
+- Two threaded holes on the sides can be used to mount the encoders
+- The interface with the plate will be latter described.
+
+The amplified piezoelectric actuators are APA300ML.
+Modification of the mechanical interfaces were asked to the manufacturer.
+Two planes surfaces and a dowel hole were used, as shown in Figure ref:fig:detail_design_apa.
+The amplifying structure, is also made of stainless steel.
 
 #+name: fig:detail_design_apa_joints
-#+caption: Two main components of the struts: the amplified piezoelectric actuator (\subref{fig:detail_design_apa}) and the flexible joint (\subref{fig:detail_design_flexible_joint}).
+#+caption: Two main components of the struts: the flexible joint (\subref{fig:detail_design_flexible_joint}) and the amplified piezoelectric actuator (\subref{fig:detail_design_apa}).
 #+attr_latex: :options [htbp]
 #+begin_figure
-#+attr_latex: :caption \subcaption{\label{fig:detail_design_apa}Amplified Piezoelectric Actuator}
-#+attr_latex: :options {0.49\textwidth}
-#+begin_subfigure
-#+attr_latex: :scale 1
-[[file:figs/detail_design_apa.png]]
-#+end_subfigure
 #+attr_latex: :caption \subcaption{\label{fig:detail_design_flexible_joint}Flexible joint}
 #+attr_latex: :options {0.49\textwidth}
 #+begin_subfigure
 #+attr_latex: :scale 1
 [[file:figs/detail_design_flexible_joint.png]]
 #+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_apa}Amplified Piezoelectric Actuator}
+#+attr_latex: :options {0.49\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 1
+[[file:figs/detail_design_apa.png]]
+#+end_subfigure
 #+end_figure
 
-**** Piezoelectric Amplified Actuators
+To correctly measure the relative motion of each strut, the encoders need to measure the relative motion between the two flexible joint's rotational centers.
+Two interface parts, made of aluminum, are used to fix the encoder and ruler to the two fleible joints as shown in Figure ref:fig:detail_design_strut_with_enc.
 
-APA: modification for better mounting
+**** Plates
 
-**** Encoder support
+The two plates of the active platform were designed to:
+- Maximize the frequency of flexible modes
+- have good positioning of the top flexible joints, and good/known orientation of the struts.
 
-All other parts are made of aluminum.
+To maximize the flexible joints, finite element analysis were used iteratively.
-
+While topology optimization could have been used, a network of reinforcing ribs was used as shown in Figure ref:fig:detail_design_top_plate.
-** Plates
-
-Plates: X30Cr13
-- high hardness to not deform
-
-
-- Maximize frequency of flexible modes (show FEM)
-- Good tolerances for interfaces with flexible joints
-  Positioning of =bi= and orientation =si=
 
 #+name: fig:detail_design_top_plate
 #+caption: The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is detailed later.
 #+attr_latex: :scale 1
 [[file:figs/detail_design_top_plate.png]]
 
-The cylindrical component is located (or constrained) within the V-groove via two distinct line contacts.
+
+The fixation interface for the joints and "V-grooves".
+The cylindrical part of the flexible joint is located (or constrained) within the V-groove via two distinct line contacts (Figure ref:fig:detail_design_fixation_flexible_joints).
+Therefore, these grooves are defining the initial strut orientation
+High machining accuracy is required, such that during the mounting of the active platform, the flexible joints are that "rest" position
+
+The "flat" interface of each top flexible joint is also in contact with the top platform, as shown in Figure ref:fig:detail_design_location_top_flexible_joints, such that the center of rotation of the top flexible joints $\bm{b}_i$ are well located with respect to the top platform.
+
+The bottom flexible joints are not Figure ref:fig:detail_design_location_bot_flex
+
+The two plates are made with a martensitic stainless steel "X30Cr13":
+- It has high hardness, such that the reference surfaces to not deform when fixing the flexible joints
+- This should allow to assemble and disassemble the struts many times if necessary
 
 #+name: fig:detail_design_fixation_flexible_joints
 #+caption: Fixation of the flexible points to the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joint is in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joint is just oriented (\subref{fig:detail_design_location_bot_flex}).
@@ -366,45 +403,58 @@ The cylindrical component is located (or constrained) within the V-groove via tw
 #+end_subfigure
 #+end_figure
 
-** Finite Element Analysis
+**** Finite Element Analysis
 
+# TODO - Maybe this picture is not necessary
+# #+name: fig:detail_design_enc_struts
+# #+caption: Obtained Nano-Hexapod design
+# #+attr_latex: :width 0.9\linewidth
+# [[file:figs/detail_design_enc_struts.jpg]]
 
-#+name: fig:detail_design_enc_struts
+Finite element analysis of the complete active platform was performed to identify problematic modes (Figure ref:fig:detail_design_fem_nano_hexapod):
-#+caption: Obtained Nano-Hexapod design
+- First six modes were found to be "suspension" modes were the top plate moves as a rigid body, and the six struts are only moving axially (Figure ref:fig:detail_design_fem_rigid_body_mode)
-#+attr_latex: :width 0.9\linewidth
+- Then, between $205\,\text{Hz}$ and $420\,\text{Hz}$ many "local" modes of the struts were observed.
-[[file:figs/detail_design_enc_struts.jpg]]
+  On is represented in Figure ref:fig:detail_design_fem_strut_mode.
-
+  While these modes seem not to induce any motion of the top platform, it induces a relative displacement of the encoder with respect to the ruler.
-- FEM of complete system
+  Therefore, when controlling the position of the active platform using the encoders, such modes could be problematic.
-- Show modes of the struts
+  Whether these modes are problematic is difficult to estimate at this point as:
+  - it is not known if the APA will "excite" these modes
+  - theoretically, if the struts are well aligned, these modes should not be observed
+  Then, flexible modes of the top plate are appearing above $650\,\text{Hz}$ (Figure ref:fig:detail_design_fem_plate_mode)
 
 #+name: fig:detail_design_fem_nano_hexapod
 #+caption: Measurement of strut flexible modes. First six modes are "suspension" modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from $205\,\text{Hz}$ to $420\,\text{Hz}$ (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at $650\,\text{Hz}$ (\subref{fig:detail_design_fem_plate_mode})
 #+attr_latex: :options [htbp]
 #+begin_figure
-#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension modes}
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension mode}
-#+attr_latex: :options {0.33\textwidth}
+#+attr_latex: :options {0.36\textwidth}
 #+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :width 0.95\linewidth
 [[file:figs/detail_design_fem_rigid_body_mode.jpg]]
 #+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_strut_mode}Strut - Local modes}
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_strut_mode}Strut - Local mode}
-#+attr_latex: :options {0.33\textwidth}
+#+attr_latex: :options {0.36\textwidth}
 #+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :width 0.95\linewidth
 [[file:figs/detail_design_fem_strut_mode.jpg]]
 #+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_plate_mode}Top plate modes}
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_plate_mode}Top plate mode}
-#+attr_latex: :options {0.33\textwidth}
+#+attr_latex: :options {0.26\textwidth}
 #+begin_subfigure
-#+attr_latex: :width 0.9\linewidth
+#+attr_latex: :width 0.95\linewidth
 [[file:figs/detail_design_fem_plate_mode.jpg]]
 #+end_subfigure
 #+end_figure
 
-** Obtained Design
+**** Alternative Encoder Placement
 
-- Alternative encoder position: on the plates
+To anticipate potential issue with local modes of the struts, an alternative fixation for the encoder is planned:
-- Support made of aluminum
+- Instead of being fixed to the struts, the encoders are fixed to the plates instead, as shown in Figure ref:fig:detail_design_enc_plates_design.
+- The support are made of aluminum, and it is verified that the natural modes are at high enough frequency (Figure ref:fig:detail_design_enc_support_modes).
+- The positioning of the encoders are made using pockets in both plates as shown in Figure ref:fig:detail_design_top_plate.
+- The encoders are aligned parallel to the struts, but yet they don't exactly measure the relative motion of each strut.
+- This means that if relative motion of the active platform is performed based on the encoders, the accuracy of the motion may be affected.
+  The issue is that the Kinematics may not be correct.
 
 #+name: fig:detail_design_enc_plates_design
 #+caption: Alternative way of using the encoders: they are fixed directly to the plates.
@@ -424,30 +474,42 @@ The cylindrical component is located (or constrained) within the V-groove via tw
 #+end_subfigure
 #+end_figure
 
-
+#+name: fig:detail_design_enc_support_modes
-#+name: fig:detail_design_fem_encoder_fix
 #+caption: Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.
-[[file:figs/detail_design_fem_encoder_fix.png]]
+#+attr_latex: :options [htbp]
+#+begin_figure
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_enc_support_mode_1}$1^{\text{st}}$ mode at $1120\,\text{Hz}$}
+#+attr_latex: :options {0.33\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 0.5
+[[file:figs/detail_design_enc_support_mode_1.jpg]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_enc_support_mode_2}$2^{\text{nd}}$ mode at $2020\,\text{Hz}$}
+#+attr_latex: :options {0.33\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 0.5
+[[file:figs/detail_design_enc_support_mode_2.jpg]]
+#+end_subfigure
+#+attr_latex: :caption \subcaption{\label{fig:detail_design_enc_support_mode_3}$3^{\text{rd}}$ mode at $2080\,\text{Hz}$}
+#+attr_latex: :options {0.33\textwidth}
+#+begin_subfigure
+#+attr_latex: :scale 0.5
+[[file:figs/detail_design_enc_support_mode_3.jpg]]
+#+end_subfigure
+#+end_figure
 
 * Multi-Body Model
-:PROPERTIES:
-:HEADER-ARGS:matlab+: :tangle matlab/detail_design_1_model.m
-:END:
 <<sec:detail_design_model>>
+**** Introduction                                                  :ignore:
 
-*Multi body Model*:
+Before all the mechanical parts were ordered, the multi-body model of the active platform was refined using the design parts.
-- Complete model: two plates, 6 joints, 6 actuators, 6 encoders
-- Joint Model
-- APA Model
-- Encoder model
-- Say that obtained dynamics was considered good + possible to perform simulations of tomography experiments with same performance as during the conceptual design
 
-** Introduction                                                      :ignore:
+Two configurations, displayed in Figure ref:fig:detail_design_simscape_encoder, were considered:
-
-Two configurations:
 - Encoders fixed to the struts
 - Encoders fixed to the plates
 
+Plates were modelled as rigid bodies, with inertia computed from the 3D shape.
+
 #+name: fig:detail_design_simscape_encoder
 #+caption: 3D representation of the multi-body model. There are two configurations: encoders fixed to the struts (\subref{fig:detail_design_simscape_encoder_struts}) and encoders fixed to the plates (\subref{fig:detail_design_simscape_encoder_plates}).
 #+attr_latex: :options [htbp]
@@ -466,19 +528,45 @@ Two configurations:
 #+end_subfigure
 #+end_figure
 
+**** Flexible Joints
 
-** Flexible Joints
+Different models of the flexible joints where considered:
+- 2DoF: only bending stiffnesses
+- 3DoF: added torsional stiffness
+- 4DoF: added axial stiffness
+
+The multi-body model for the 4DoF configuration is shown in Figure ref:fig:detail_design_simscape_model_flexible_joint.
+It is composed of three solid bodies connected by joints whose stiffnesses are computed from the finite element model.
 
 #+name: fig:detail_design_simscape_model_flexible_joint
 #+caption: Multi-Body (using the Simscape software) model of the flexible joints. A 4-DoFs model is shown.
 #+attr_latex: :scale 1
 [[file:figs/detail_design_simscape_model_flexible_joint.png]]
 
-** Amplified Piezoelectric Actuators
+**** Amplified Piezoelectric Actuators
+
+The amplified piezoelectric actuators are modelled as explained in Section [..].
+# Add link to section
+Two different models can be used in the multi-body model:
+- a 2DoF "axial" model
+- a "super-element" extracted from the finite element model
+
+**** Encoders
+
+Up to now, relative displacement sensors were implemented as a relative distance measurement between $\bm{a}_i$ and $\bm{b}_i$.
+
+As shown in the previous section, flexible modes of the struts may negatively impact the encoder signal.
+It was therefore necessary to better model the encoder.
+
+The optical encoder works:
+- Encoder heads contains a light source shine on the ruler, and a photo-diode.
+  This is represented by frame $\{E\}$ in Figure ref:fig:detail_design_simscape_encoder.
+- ruler or scale with a grating (here with a $20\,\mu m$ pitch). A reference frame is indicated by $\{R\}$
+
+Therefore, the measured displacement is the relative position of $\{E\}$ (i.e. there the light "hits" the scale) with respect to frame $\{R\}$, in the direction of the scale.
 
 
-
+In that case, a rotation of the encoder, as shown in figure ref:fig:detail_design_simscape_encoder_disp induces a measured displacement.
-** Encoders
 
 #+name: fig:detail_design_simscape_encoder_model
 #+caption: Representation of the encoder model in the multi-body model. Measurement $d_i$ corresponds to the $x$ position of the encoder frame $\{E\}$ expresssed in the ruller frame $\{R\}$ (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}).
@@ -498,6 +586,14 @@ Two configurations:
 #+end_subfigure
 #+end_figure
 
+**** Simulation
+
+Based on this refined model:
+- the active platform could be integrated on top of the micro-station's model.
+- the obtained dynamics was considered good
+- simulations of tomography experiments were performed, and similar performance were obtained as during the conceptual design
+- this is not presented here as results are very similar to the simulations performed in Section [...]
+  # Add link to section
 
 * Conclusion
 <<sec:detail_design_conclusion>>
@@ -505,24 +601,3 @@ Two configurations:
 * Bibliography                                                        :ignore:
 #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
 
-* Helping Functions                                                 :noexport:
-** Initialize Path
-#+NAME: m-init-path
-#+BEGIN_SRC matlab
-%% Path for functions, data and scripts
-addpath('./matlab/mat/'); % Path for data
-addpath('./matlab/');     % Path for scripts
-#+END_SRC
-
-#+NAME: m-init-path-tangle
-#+BEGIN_SRC matlab
-%% Path for functions, data and scripts
-addpath('./mat/'); % Path for data
-#+END_SRC
-
-** Initialize other elements
-#+NAME: m-init-other
-#+BEGIN_SRC matlab
-%% Colors for the figures
-colors = colororder;
-#+END_SRC

nass-design.tex

@@ -1,4 +1,4 @@
-% Created 2025-04-21 Mon 16:49
+% Created 2025-04-21 Mon 19:46
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -30,87 +30,124 @@
 \end{figure}
 
 
-\textbf{Design goals}:
+Detail design phase:
 \begin{itemize}
-\item Position \texttt{bi} and \texttt{si}
+\item key elements were optimized such as: actuator and flexible joints
-\item Maximum height of 95mm
+\item relative motion sensor (an encoder) was also selected
-\item As close as possible to ``perfect'' stewart platform: flexible modes at high frequency
+\item specific kinematics of the Stewart platform (i.e. position of joints and orientation of struts) was not found to be too critical for this application.
-\item Easy mounting, easy change of strut in case of failure
+Yet, the geometry was fixed in Section [\ldots{}]
 \end{itemize}
 
+In this section, the mechanical design of the active platform, shown in Figure \ref{fig:detail_design_nano_hexapod_elements}, is detailed.
 
-Presentation of the obtained design:
+The main design objectives are:
 \begin{itemize}
-\item Fixation
+\item Well defined kinematics: Good positioning of the top flexible joint rotation point \(\bm{b}_i\) and correct orientation of the struts \(\hat{\bm{s}}_i\).
-\item Section on: Complete strut
+The goal is to have a well defined geometry such that the Jacobian matrix is well defined.
-\item Cable management
+\item Space constrains: it should fit within a cylinder with radius of \(120\,\text{mm}\) and height of \(95\,\text{mm}\)
-\item Plates design
+\item As good performances were obtained with the multi-body model.
-\item FEM results
+The final design should behave as close as possible to ``perfect'' stewart platform.
-\item Explain again the different specifications in terms of space, payload, etc..
+This means that the frequency of flexible modes that could be problematic for control must be made as high as possible.
-\item CAD view of the nano-hexapod
+\item Easy mounting and alignment.
-\item Chosen geometry, materials, ease of mounting, cabling, \ldots{}
+\item Easy maintenance: the struts should be easily changed in case for failure.
-\item Validation on Simscape with accurate model?
 \end{itemize}
 \chapter{Mechanical Design}
 \label{sec:detail_design_mechanics}
-\section{Struts}
+\subsubsection{Struts}
+
+The strut design is shown in Figure \ref{fig:detail_design_strut}.
+
+The design of the struts was driven by:
+\begin{itemize}
+\item having stiff interface between the amplified piezoelectric actuator and the two flexible joints
+\item having stiff interface between the flexible joints and the two places (discussed afterwards)
+\item Because the angular stroke of the flexible joints is fairly limited, it is important to be able to mount the strut such that the two cylindrical interfaces are coaxial.
+Do to so:
+\begin{itemize}
+\item A mounting bench was designed
+The mounting procedure will be described in Section [\ldots{}]
+\item Cylindrical washers, shown in Figure \ref{fig:detail_design_strut_without_enc}, were integrated to allow for adjustments.
+The issue was that the flatness between the two interface planes of the APA shown in Figure \ref{fig:detail_design_apa} could not be guaranteed.
+With the added cylindrical washers and the mounting tool, it should be possible to well align the struts even in the presence of machining inaccuracies.
+\end{itemize}
+\item Possibility to fix the encoder parallel to the strut, as shown in Figure \ref{fig:detail_design_strut_with_enc}
+\end{itemize}
+
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_strut_without_enc.jpg}
+\includegraphics[scale=1,scale=0.9]{figs/detail_design_strut_without_enc.png}
 \end{center}
 \subcaption{\label{fig:detail_design_strut_without_enc}Before encoder integration}
 \end{subfigure}
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_strut_with_enc.jpg}
+\includegraphics[scale=1,scale=0.9]{figs/detail_design_strut_with_enc.png}
 \end{center}
 \subcaption{\label{fig:detail_design_strut_with_enc}With the mounted encoder}
 \end{subfigure}
 \caption{\label{fig:detail_design_strut}Design of the Nano-Hexapod struts. Before (\subref{fig:detail_design_strut_without_enc}) and after (\subref{fig:detail_design_strut_with_enc}) encoder integration.}
 \end{figure}
-\subsubsection{Flexible joints}
 
-Flexible joints: X5CrNiCuNb16-4 (F16Ph)
+The flexible joints are manufactured using wire-cut electrical discharge machining, allowing for:
 \begin{itemize}
-\item high yield strength: specified >1GPa using heat treatment
+\item very tight tolerances:
-\item high fatigue resistance
+\begin{itemize}
+\item allowing good location of the center of rotation with respect to the plate interfaces (red surfaces shown in Figure \ref{fig:detail_design_flexible_joint})
+\item allowing correct neck dimension to have the wanted properties (stiffness and angular stroke)
+\end{itemize}
+\item Such part is fragile, mainly due to its small ``neck'' dimension of only \(0.25\,\text{mm}\)
+Such machining technique has little to no cutting forces.
 \end{itemize}
 
+The flexible joints are made from a stainless steel referenced as ``X5CrNiCuNb16-4'' (also called ``F16Ph'').
+This material is chosen for:
+\begin{itemize}
+\item its high yield strength: specified >1GPa using heat treatment.
+\item its high fatigue resistance
+\end{itemize}
+
+Figure \ref{fig:detail_design_flexible_joint}
+\begin{itemize}
+\item Interface with the APA has a cylindrical shape to allow the use of cylindrical washers
+A slotted hole has been added to align the flexible joint with the APA using a dowel pin.
+\item Two threaded holes on the sides can be used to mount the encoders
+\item The interface with the plate will be latter described.
+\end{itemize}
+
+The amplified piezoelectric actuators are APA300ML.
+Modification of the mechanical interfaces were asked to the manufacturer.
+Two planes surfaces and a dowel hole were used, as shown in Figure \ref{fig:detail_design_apa}.
+The amplifying structure, is also made of stainless steel.
+
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
-\includegraphics[scale=1,scale=1]{figs/detail_design_apa.png}
-\end{center}
-\subcaption{\label{fig:detail_design_apa}Amplified Piezoelectric Actuator}
-\end{subfigure}
-\begin{subfigure}{0.49\textwidth}
-\begin{center}
 \includegraphics[scale=1,scale=1]{figs/detail_design_flexible_joint.png}
 \end{center}
 \subcaption{\label{fig:detail_design_flexible_joint}Flexible joint}
 \end{subfigure}
-\caption{\label{fig:detail_design_apa_joints}Two main components of the struts: the amplified piezoelectric actuator (\subref{fig:detail_design_apa}) and the flexible joint (\subref{fig:detail_design_flexible_joint}).}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_design_apa.png}
+\end{center}
+\subcaption{\label{fig:detail_design_apa}Amplified Piezoelectric Actuator}
+\end{subfigure}
+\caption{\label{fig:detail_design_apa_joints}Two main components of the struts: the flexible joint (\subref{fig:detail_design_flexible_joint}) and the amplified piezoelectric actuator (\subref{fig:detail_design_apa}).}
 \end{figure}
-\subsubsection{Piezoelectric Amplified Actuators}
 
-APA: modification for better mounting
+To correctly measure the relative motion of each strut, the encoders need to measure the relative motion between the two flexible joint's rotational centers.
-\subsubsection{Encoder support}
+Two interface parts, made of aluminum, are used to fix the encoder and ruler to the two fleible joints as shown in Figure \ref{fig:detail_design_strut_with_enc}.
+\subsubsection{Plates}
 
-All other parts are made of aluminum.
+The two plates of the active platform were designed to:
-\section{Plates}
-
-Plates: X30Cr13
 \begin{itemize}
-\item high hardness to not deform
+\item Maximize the frequency of flexible modes
+\item have good positioning of the top flexible joints, and good/known orientation of the struts.
 \end{itemize}
 
-
+To maximize the flexible joints, finite element analysis were used iteratively.
-\begin{itemize}
+While topology optimization could have been used, a network of reinforcing ribs was used as shown in Figure \ref{fig:detail_design_top_plate}.
-\item Maximize frequency of flexible modes (show FEM)
-\item Good tolerances for interfaces with flexible joints
-Positioning of \texttt{bi} and orientation \texttt{si}
-\end{itemize}
 
 \begin{figure}[htbp]
 \centering
@@ -118,7 +155,21 @@ Positioning of \texttt{bi} and orientation \texttt{si}
 \caption{\label{fig:detail_design_top_plate}The mechanical design for the top platform incorporates precisely positioned V-grooves for the joint interfaces (displayed in red). The purpose of the encoder interface (shown in green) is detailed later.}
 \end{figure}
 
-The cylindrical component is located (or constrained) within the V-groove via two distinct line contacts.
+
+The fixation interface for the joints and ``V-grooves''.
+The cylindrical part of the flexible joint is located (or constrained) within the V-groove via two distinct line contacts (Figure \ref{fig:detail_design_fixation_flexible_joints}).
+Therefore, these grooves are defining the initial strut orientation
+High machining accuracy is required, such that during the mounting of the active platform, the flexible joints are that ``rest'' position
+
+The ``flat'' interface of each top flexible joint is also in contact with the top platform, as shown in Figure \ref{fig:detail_design_location_top_flexible_joints}, such that the center of rotation of the top flexible joints \(\bm{b}_i\) are well located with respect to the top platform.
+
+The bottom flexible joints are not Figure \ref{fig:detail_design_location_bot_flex}
+
+The two plates are made with a martensitic stainless steel ``X30Cr13'':
+\begin{itemize}
+\item It has high hardness, such that the reference surfaces to not deform when fixing the flexible joints
+\item This should allow to assemble and disassemble the struts many times if necessary
+\end{itemize}
 
 \begin{figure}
 \begin{subfigure}{0.33\textwidth}
@@ -141,46 +192,54 @@ The cylindrical component is located (or constrained) within the V-groove via tw
 \end{subfigure}
 \caption{\label{fig:detail_design_fixation_flexible_joints}Fixation of the flexible points to the nano-hexapod plates. Both top and bottom flexible joints are clamped to the plates as shown in (\subref{fig:detail_design_fixation_flexible_joints}). While the top flexible joint is in contact with the top plate for precise positioning of its center of rotation (\subref{fig:detail_design_location_top_flexible_joints}), the bottom joint is just oriented (\subref{fig:detail_design_location_bot_flex}).}
 \end{figure}
-\section{Finite Element Analysis}
+\subsubsection{Finite Element Analysis}
-
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_design_enc_struts.jpg}
-\caption{\label{fig:detail_design_enc_struts}Obtained Nano-Hexapod design}
-\end{figure}
 
+Finite element analysis of the complete active platform was performed to identify problematic modes (Figure \ref{fig:detail_design_fem_nano_hexapod}):
 \begin{itemize}
-\item FEM of complete system
+\item First six modes were found to be ``suspension'' modes were the top plate moves as a rigid body, and the six struts are only moving axially (Figure \ref{fig:detail_design_fem_rigid_body_mode})
-\item Show modes of the struts
+\item Then, between \(205\,\text{Hz}\) and \(420\,\text{Hz}\) many ``local'' modes of the struts were observed.
+On is represented in Figure \ref{fig:detail_design_fem_strut_mode}.
+While these modes seem not to induce any motion of the top platform, it induces a relative displacement of the encoder with respect to the ruler.
+Therefore, when controlling the position of the active platform using the encoders, such modes could be problematic.
+Whether these modes are problematic is difficult to estimate at this point as:
+\begin{itemize}
+\item it is not known if the APA will ``excite'' these modes
+\item theoretically, if the struts are well aligned, these modes should not be observed
+\end{itemize}
+Then, flexible modes of the top plate are appearing above \(650\,\text{Hz}\) (Figure \ref{fig:detail_design_fem_plate_mode})
 \end{itemize}
 
 \begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.36\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_design_fem_rigid_body_mode.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_fem_rigid_body_mode.jpg}
 \end{center}
-\subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension modes}
+\subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension mode}
 \end{subfigure}
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.36\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_design_fem_strut_mode.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_fem_strut_mode.jpg}
 \end{center}
-\subcaption{\label{fig:detail_design_fem_strut_mode}Strut - Local modes}
+\subcaption{\label{fig:detail_design_fem_strut_mode}Strut - Local mode}
 \end{subfigure}
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.26\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.9\linewidth]{figs/detail_design_fem_plate_mode.jpg}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_fem_plate_mode.jpg}
 \end{center}
-\subcaption{\label{fig:detail_design_fem_plate_mode}Top plate modes}
+\subcaption{\label{fig:detail_design_fem_plate_mode}Top plate mode}
 \end{subfigure}
 \caption{\label{fig:detail_design_fem_nano_hexapod}Measurement of strut flexible modes. First six modes are ``suspension'' modes in which the top plate behaves as a rigid body (\subref{fig:detail_design_fem_rigid_body_mode}). Then modes of the struts have natural frequencies from \(205\,\text{Hz}\) to \(420\,\text{Hz}\) (\subref{fig:detail_design_fem_strut_mode}). Finally, the first flexible mode of the top plate is at \(650\,\text{Hz}\) (\subref{fig:detail_design_fem_plate_mode})}
 \end{figure}
-\section{Obtained Design}
+\subsubsection{Alternative Encoder Placement}
 
+To anticipate potential issue with local modes of the struts, an alternative fixation for the encoder is planned:
 \begin{itemize}
-\item Alternative encoder position: on the plates
+\item Instead of being fixed to the struts, the encoders are fixed to the plates instead, as shown in Figure \ref{fig:detail_design_enc_plates_design}.
-\item Support made of aluminum
+\item The support are made of aluminum, and it is verified that the natural modes are at high enough frequency (Figure \ref{fig:detail_design_enc_support_modes}).
+\item The positioning of the encoders are made using pockets in both plates as shown in Figure \ref{fig:detail_design_top_plate}.
+\item The encoders are aligned parallel to the struts, but yet they don't exactly measure the relative motion of each strut.
+\item This means that if relative motion of the active platform is performed based on the encoders, the accuracy of the motion may be affected.
+The issue is that the Kinematics may not be correct.
 \end{itemize}
 
 \begin{figure}[htbp]
@@ -199,30 +258,39 @@ The cylindrical component is located (or constrained) within the V-groove via tw
 \caption{\label{fig:detail_design_enc_plates_design}Alternative way of using the encoders: they are fixed directly to the plates.}
 \end{figure}
 
-
 \begin{figure}[htbp]
-\centering
+\begin{subfigure}{0.33\textwidth}
-\includegraphics[scale=1]{figs/detail_design_fem_encoder_fix.png}
+\begin{center}
-\caption{\label{fig:detail_design_fem_encoder_fix}Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.}
+\includegraphics[scale=1,scale=0.5]{figs/detail_design_enc_support_mode_1.jpg}
+\end{center}
+\subcaption{\label{fig:detail_design_enc_support_mode_1}$1^{\text{st}}$ mode at $1120\,\text{Hz}$}
+\end{subfigure}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=0.5]{figs/detail_design_enc_support_mode_2.jpg}
+\end{center}
+\subcaption{\label{fig:detail_design_enc_support_mode_2}$2^{\text{nd}}$ mode at $2020\,\text{Hz}$}
+\end{subfigure}
+\begin{subfigure}{0.33\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=0.5]{figs/detail_design_enc_support_mode_3.jpg}
+\end{center}
+\subcaption{\label{fig:detail_design_enc_support_mode_3}$3^{\text{rd}}$ mode at $2080\,\text{Hz}$}
+\end{subfigure}
+\caption{\label{fig:detail_design_enc_support_modes}Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.}
 \end{figure}
 \chapter{Multi-Body Model}
 \label{sec:detail_design_model}
+Before all the mechanical parts were ordered, the multi-body model of the active platform was refined using the design parts.
 
-\textbf{Multi body Model}:
+Two configurations, displayed in Figure \ref{fig:detail_design_simscape_encoder}, were considered:
-\begin{itemize}
-\item Complete model: two plates, 6 joints, 6 actuators, 6 encoders
-\item Joint Model
-\item APA Model
-\item Encoder model
-\item Say that obtained dynamics was considered good + possible to perform simulations of tomography experiments with same performance as during the conceptual design
-\end{itemize}
-
-Two configurations:
 \begin{itemize}
 \item Encoders fixed to the struts
 \item Encoders fixed to the plates
 \end{itemize}
 
+Plates were modelled as rigid bodies, with inertia computed from the 3D shape.
+
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
@@ -238,18 +306,49 @@ Two configurations:
 \end{subfigure}
 \caption{\label{fig:detail_design_simscape_encoder}3D representation of the multi-body model. There are two configurations: encoders fixed to the struts (\subref{fig:detail_design_simscape_encoder_struts}) and encoders fixed to the plates (\subref{fig:detail_design_simscape_encoder_plates}).}
 \end{figure}
-\section{Flexible Joints}
+\subsubsection{Flexible Joints}
+
+Different models of the flexible joints where considered:
+\begin{itemize}
+\item 2DoF: only bending stiffnesses
+\item 3DoF: added torsional stiffness
+\item 4DoF: added axial stiffness
+\end{itemize}
+
+The multi-body model for the 4DoF configuration is shown in Figure \ref{fig:detail_design_simscape_model_flexible_joint}.
+It is composed of three solid bodies connected by joints whose stiffnesses are computed from the finite element model.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,scale=1]{figs/detail_design_simscape_model_flexible_joint.png}
 \caption{\label{fig:detail_design_simscape_model_flexible_joint}Multi-Body (using the Simscape software) model of the flexible joints. A 4-DoFs model is shown.}
 \end{figure}
-\section{Amplified Piezoelectric Actuators}
+\subsubsection{Amplified Piezoelectric Actuators}
+
+The amplified piezoelectric actuators are modelled as explained in Section [..].
+Two different models can be used in the multi-body model:
+\begin{itemize}
+\item a 2DoF ``axial'' model
+\item a ``super-element'' extracted from the finite element model
+\end{itemize}
+\subsubsection{Encoders}
+
+Up to now, relative displacement sensors were implemented as a relative distance measurement between \(\bm{a}_i\) and \(\bm{b}_i\).
+
+As shown in the previous section, flexible modes of the struts may negatively impact the encoder signal.
+It was therefore necessary to better model the encoder.
+
+The optical encoder works:
+\begin{itemize}
+\item Encoder heads contains a light source shine on the ruler, and a photo-diode.
+This is represented by frame \(\{E\}\) in Figure \ref{fig:detail_design_simscape_encoder}.
+\item ruler or scale with a grating (here with a \(20\,\mu m\) pitch). A reference frame is indicated by \(\{R\}\)
+\end{itemize}
+
+Therefore, the measured displacement is the relative position of \(\{E\}\) (i.e. there the light ``hits'' the scale) with respect to frame \(\{R\}\), in the direction of the scale.
 
 
-
+In that case, a rotation of the encoder, as shown in figure \ref{fig:detail_design_simscape_encoder_disp} induces a measured displacement.
-\section{Encoders}
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -266,6 +365,15 @@ Two configurations:
 \end{subfigure}
 \caption{\label{fig:detail_design_simscape_encoder_model}Representation of the encoder model in the multi-body model. Measurement \(d_i\) corresponds to the \(x\) position of the encoder frame \(\{E\}\) expresssed in the ruller frame \(\{R\}\) (\subref{fig:detail_design_simscape_encoder}). A rotation of the encoder therefore induces a measured displacement (\subref{fig:detail_design_simscape_encoder_disp}).}
 \end{figure}
+\subsubsection{Simulation}
+
+Based on this refined model:
+\begin{itemize}
+\item the active platform could be integrated on top of the micro-station's model.
+\item the obtained dynamics was considered good
+\item simulations of tomography experiments were performed, and similar performance were obtained as during the conceptual design
+\item this is not presented here as results are very similar to the simulations performed in Section [\ldots{}]
+\end{itemize}
 \chapter{Conclusion}
 \label{sec:detail_design_conclusion}
 \printbibliography[heading=bibintoc,title={Bibliography}]