tdehaeze/phd-nass-design
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

First complete rewrite

Browse Source
This commit is contained in:
Thomas Dehaeze 2025-04-21 22:55:21 +02:00
parent 05bd404570
commit 00e3871c42
3 changed files with 193 additions and 346 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

269
nass-design.org
View File

@ -246,49 +246,30 @@ CLOSED: [2025-04-21 Mon 14:13]
* Introduction :ignore:
The detailed mechanical design of the active platform, depicted in Figure ref:fig:detail_design_nano_hexapod_elements, is presented in this section.
Several primary objectives guided the mechanical design.
First, in order to have a well known Jacobian matrix (used in the control architecture), accurate positioning of rotation points of the top flexible joint and correct orientation of the struts were wanted.
Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of $120\,\text{mm}$ and a height of $95\,\text{mm}$.
Thirdly, because good performances were predicted by the multi-body model, the final design was intended to approximate the behavior of the "idealized" Stewart platform as closely as possible.
This objective implies that the frequencies of flexible modes potentially detrimental to control performance needed to be maximized.
Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure.
#+name: fig:detail_design_nano_hexapod_elements
#+caption: Obtained mechanical design of the Active platform, the "nano-hexapod"
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_design_nano_hexapod_elements.png]]
Detail design phase:
- key elements were optimized such as: actuator and flexible joints
- relative motion sensor (an encoder) was also selected
- 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.
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.
The main design objectives are:
- Well defined kinematics: Good positioning of the top flexible joint rotation point $\bm{b}_i$ and correct orientation of the struts $\hat{\bm{s}}_i$.
The goal is to have a well defined geometry such that the Jacobian matrix is well defined.
- Space constrains: it should fit within a cylinder with radius of $120\,\text{mm}$ and height of $95\,\text{mm}$
- As good performances were obtained with the multi-body model.
The final design should behave as close as possible to "perfect" stewart platform.
This means that the frequency of flexible modes that could be problematic for control must be made as high as possible.
- Easy mounting and alignment.
- Easy maintenance: the struts should be easily changed in case for failure.
* Mechanical Design
<<sec:detail_design_mechanics>>
**** 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
The strut design, illustrated in Figure ref:fig:detail_design_strut, was driven by several factors.
Stiff interfaces were required between the amplified piezoelectric actuator and the two flexible joints, as well as between the flexible joints and their respective mounting plates.
Due to the limited angular stroke of the flexible joints, it was important that the struts could be assembled in such a way that the two cylindrical interfaces were coaxial while the flexible joints were experiencing no stress (i.e. rest position).
To achieve this, cylindrical washers, shown in Figure ref:fig:detail_design_strut_without_enc, were integrated into the design to allow for poor flatness between the two interface planes of the APA, depicted in Figure ref:fig:detail_design_apa.
A dedicated mounting bench was also developed, such that each strut could be precisely aligned, even in the presence of machining inaccuracies.
The mounting procedure is described in Section ref:sec:test_struts_mounting.
Lastly, the design needed to permit the fixation of an encoder parallel to the strut axis, 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.
@ -308,28 +289,20 @@ The design of the struts was driven by:
#+end_subfigure
#+end_figure
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.
The flexible joints, shown in Figure ref:fig:detail_design_flexible_joint, were manufactured using wire-cut electrical discharge machining (EDM).
This manufacturing process was selected for few reasons.
First, because of the neck dimension of only $0.25\,\text{mm}$, the part is inherently fragile and is difficult to manufacture with classical machining as cutting forces may damage the part.
Also wire-cut EDM allows for very tight machining tolerances, which are critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure ref:fig:detail_design_flexible_joint) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties.
The material chosen for the flexible joints is a stainless steel designated /X5CrNiCuNb16-4/ (alternatively known as "F16Ph").
This selection was based on its high specified yield strength (exceeding $1\,\text{GPa}$ after appropriate heat treatment) and its high fatigue resistance.
The flexible joints are made from a stainless steel referenced as "X5CrNiCuNb16-4" (also called "F16Ph").
This material is chosen for:
- its high yield strength: specified >1GPa using heat treatment.
- its high fatigue resistance
As shown in Figure ref:fig:detail_design_flexible_joint, the interface designed to connect with the APA possesses a cylindrical shape, facilitating the use of the aforementioned cylindrical washers for alignment.
A slotted hole was incorporated to permit alignment of the flexible joint with the APA via a dowel pin.
Additionally, two threaded holes were included on the sides for mounting the encoder components.
The interface connecting the flexible joint to the platform plates will be described subsequently.
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.
Modifications to the standard mechanical interfaces of the APA300ML were requested from the manufacturer.
The modified design features two planar surfaces and a dowel hole for precise location and orientation, as illustrated in Figure ref:fig:detail_design_apa.
#+name: fig:detail_design_apa_joints
#+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}).
@ -349,36 +322,25 @@ The amplifying structure, is also made of stainless steel.
#+end_subfigure
#+end_figure
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.
Accurate measurement of the relative displacement within each strut requires the encoders to sense the motion between the rotational centers of the two associated flexible joints.
To achieve this, two interface parts, fabricated from aluminum, were designed.
These parts serve to fix the encoder head and the associated scale (ruler) to the two flexible joints, as depicted in Figure ref:fig:detail_design_strut_with_enc.
**** Plates
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.
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.
The design of the top and bottom plates of the active platform was governed by two main requirements: maximizing the frequency of flexible modes and ensuring accurate positioning of the top flexible joints and well-defined orientation of the struts.
To maximize the natural frequencies associated with plate flexibility, a simple network of reinforcing ribs was adopted, as shown for the top plate in Figure ref:fig:detail_design_top_plate.
While topology optimization methods could have been used, the presented designed was found to give high enough flexible modes.
#+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 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
Joints interfaces on the plate consist of "V-grooves".
The cylindrical portion of each flexible joint is constrained within its corresponding V-groove through two distinct line contacts, illustrated in Figure ref:fig:detail_design_fixation_flexible_joints.
These grooves consequently serve to define the nominal orientation of the struts.
High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the active platform is assembled.
#+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}).
@ -403,24 +365,24 @@ The two plates are made with a martensitic stainless steel "X30Cr13":
#+end_subfigure
#+end_figure
Furthermore, the flat interface surface of each top flexible joint is designed to be in direct contact with the top platform surface, as shown in Figure ref:fig:detail_design_location_top_flexible_joints.
This contact ensures that the centers of rotation of the top flexible joints, are precisely located relative to the top platform coordinate system.
The bottom flexible joints, however, are primarily oriented by the V-grooves without the same precise positional constraint against the bottom plate, as shown in Figure ref:fig:detail_design_location_bot_flex.
Both plates were specified to be manufactured from a martensitic stainless steel, X30Cr13.
This material was selected primarily for its high hardness, which minimizes the risk of deformation of the reference surfaces during the clamping of the flexible joints.
This characteristic is expected to permit repeated assembly and disassembly of the struts, should maintenance or reconfiguration be necessary.
**** 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]]
Finite element analysis of the complete active platform was performed to identify problematic modes (Figure ref:fig:detail_design_fem_nano_hexapod):
- 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)
- 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:
- 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)
A finite element analysis (FEA) of the complete active platform assembly was performed to identify modes that could potentially affect performance.
The analysis revealed that the first six modes correspond to "suspension" modes, where the top plate effectively moves as a rigid body, and motion primarily involves axial displacement of the six struts (an example is shown in Figure ref:fig:detail_design_fem_rigid_body_mode).
Following these suspension modes, numerous "local" modes associated with the struts themselves were observed in the frequency range between $205\,\text{Hz}$ and $420\,\text{Hz}$.
One such mode is represented in Figure ref:fig:detail_design_fem_strut_mode.
Although these modes do not appear to induce significant motion of the top platform, they do cause relative displacement between the encoder components (head and scale) mounted on the strut.
Consequently, such modes could potentially be problematic if the active platform's position is controlled based on the encoders.
The extent to which these modes might pose a problem is difficult to establish at this stage, as it depends on whether they are significantly excited by the APA actuation and their sensitivity to strut alignment.
Finally, the FEA indicated that flexible modes of the top plate itself begin to appear at frequencies above $650\,\text{Hz}$, with the first such mode shown in 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})
@ -448,13 +410,8 @@ Finite element analysis of the complete active platform was performed to identif
**** Alternative Encoder Placement
To anticipate potential issue with local modes of the struts, an alternative fixation for the encoder is planned:
- 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.
In anticipation of potential issues arising from the local modes of the struts affecting encoder measurements, an alternative fixation strategy for the encoders was envisaged.
In this configuration, the encoders are fixed directly to the top and bottom plates instead of the struts, as illustrated in Figure ref:fig:detail_design_enc_plates_design.
#+name: fig:detail_design_enc_plates_design
#+caption: Alternative way of using the encoders: they are fixed directly to the plates.
@ -474,41 +431,43 @@ To anticipate potential issue with local modes of the struts, an alternative fix
#+end_subfigure
#+end_figure
#+name: fig:detail_design_enc_support_modes
#+caption: Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.
#+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
Dedicated supports, machined from aluminum, were designed for this purpose.
It was verified through FEA that the natural modes of these supports occur at sufficiently high frequencies, with the first mode estimated at $1120\,\text{Hz}$.
Precise positioning of these encoder supports is achieved through machined pockets in both the top and bottom plates, visible in Figure ref:fig:detail_design_top_plate (indicated in green).
Although the encoders in this arrangement are aligned parallel to the nominal strut axes, they no longer measure the exact relative displacement along the strut between the flexible joint centers.
This geometric discrepancy implies that if the relative motion control of the active platform is based directly on these encoder readings, the kinematic calculations may be slightly inaccurate, potentially affecting the overall positioning accuracy of the platform.
# #+name: fig:detail_design_enc_support_modes
# #+caption: Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.
# #+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
<<sec:detail_design_model>>
**** Introduction :ignore:
Before all the mechanical parts were ordered, the multi-body model of the active platform was refined using the design parts.
Two configurations, displayed in Figure ref:fig:detail_design_simscape_encoder, were considered:
- Encoders fixed to the struts
- Encoders fixed to the plates
Plates were modelled as rigid bodies, with inertia computed from the 3D shape.
Prior to the procurement of mechanical components, the multi-body simulation model of the active platform was refined to incorporate the finalized design geometries.
Two distinct configurations, corresponding to the two encoder mounting strategies discussed previously, were considered in the model, as displayed in Figure ref:fig:detail_design_simscape_encoder: one with encoders fixed to the struts, and another with encoders fixed to the plates.
In these models, the top and bottom plates were represented as rigid bodies, with their inertial properties calculated directly from the 3D CAD geometry.
#+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}).
@ -530,13 +489,10 @@ Plates were modelled as rigid bodies, with inertia computed from the 3D shape.
**** 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.
Several levels of detail were considered for modeling the flexible joints within the multi-body model.
Models with two degrees of freedom incorporating only bending stiffnesses, models with three degrees of freedom adding torsional stiffness, and models with four degrees of freedom further adding axial stiffness were evaluated.
The multi-body representation corresponding to the 4DoF configuration is shown in Figure ref:fig:detail_design_simscape_model_flexible_joint.
This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from finite element analysis of the joint component.
#+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.
@ -545,28 +501,22 @@ It is composed of three solid bodies connected by joints whose stiffnesses are c
**** 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
The amplified piezoelectric actuators (APAs) were incorporated into the multi-body model following the methodology detailed in Section ref:sec:detail_fem_actuator.
Two distinct representations of the APA can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex "Reduced Order Flexible Body" model derived from a finite element model.
**** Encoders
Up to now, relative displacement sensors were implemented as a relative distance measurement between $\bm{a}_i$ and $\bm{b}_i$.
In earlier modeling stages, the relative displacement sensors (encoders) were implemented as a direct measurement of the relative distance between the joint connection points $\bm{a}_i$ and $\bm{b}_i$.
However, as indicated by the FEA results discussed previously, the flexible modes inherent to the struts could potentially affect the encoder measurement.
Therefore, a more sophisticated model of the optical encoder was necessary.
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 encoders operate based on the interaction between an encoder head and a graduated scale or ruler.
The optical encoder head contains a light source which is illuminating the ruler.
The position of the light on the ruler is represented by the reference frame $\{E\}$ in Figure ref:fig:detail_design_simscape_encoder_model.
The ruler features a precise grating pattern (in this case, with a $20\,\mu m$ pitch), and its position is associated with the reference frame $\{R\}$.
The displacement measured by the encoder corresponds to the relative position of the encoder frame $\{E\}$ (specifically, the point where the light interacts with the scale) with respect to the ruler frame $\{R\}$, projected along the measurement direction defined by the scale.
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.
An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure ref:fig:detail_design_simscape_encoder_disp, can induce a measured displacement.
#+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}).
@ -588,15 +538,12 @@ In that case, a rotation of the encoder, as shown in figure ref:fig:detail_desig
**** 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>>
Utilizing this refined multi-body model, several assessments were conducted.
The active platform model was integrated into the larger simulation model with the micro-station.
The dynamic behavior was evaluated and considered satisfactory.
Furthermore, simulations replicating tomography experiments were performed.
The performance metrics obtained from these simulations were found to be comparable to those achieved during the earlier conceptual design phase simulations.
Consequently, as the results closely mirror those presented previously in Section ref:ssec:test_id31_iff_hac_perf, they are not reiterated in detail here.
* Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

BIN
nass-design.pdf
View File

Binary file not shown.

270
nass-design.tex
View File

@ -1,4 +1,4 @@
% Created 2025-04-21 Mon 19:46
% Created 2025-04-21 Mon 22:54
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -23,55 +23,30 @@
\tableofcontents
\clearpage
The detailed mechanical design of the active platform, depicted in Figure \ref{fig:detail_design_nano_hexapod_elements}, is presented in this section.
Several primary objectives guided the mechanical design.
First, in order to have a well known Jacobian matrix (used in the control architecture), accurate positioning of rotation points of the top flexible joint and correct orientation of the struts were wanted.
Secondly, space constraints necessitated that the entire platform fit within a cylinder with a radius of \(120\,\text{mm}\) and a height of \(95\,\text{mm}\).
Thirdly, because good performances were predicted by the multi-body model, the final design was intended to approximate the behavior of the ``idealized'' Stewart platform as closely as possible.
This objective implies that the frequencies of flexible modes potentially detrimental to control performance needed to be maximized.
Finally, considerations for ease of mounting, alignment, and maintenance were incorporated, specifically ensuring that struts could be easily replaced in the event of failure.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_design_nano_hexapod_elements.png}
\caption{\label{fig:detail_design_nano_hexapod_elements}Obtained mechanical design of the Active platform, the ``nano-hexapod''}
\end{figure}
Detail design phase:
\begin{itemize}
\item key elements were optimized such as: actuator and flexible joints
\item relative motion sensor (an encoder) was also selected
\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.
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.
The main design objectives are:
\begin{itemize}
\item Well defined kinematics: Good positioning of the top flexible joint rotation point \(\bm{b}_i\) and correct orientation of the struts \(\hat{\bm{s}}_i\).
The goal is to have a well defined geometry such that the Jacobian matrix is well defined.
\item Space constrains: it should fit within a cylinder with radius of \(120\,\text{mm}\) and height of \(95\,\text{mm}\)
\item As good performances were obtained with the multi-body model.
The final design should behave as close as possible to ``perfect'' stewart platform.
This means that the frequency of flexible modes that could be problematic for control must be made as high as possible.
\item Easy mounting and alignment.
\item Easy maintenance: the struts should be easily changed in case for failure.
\end{itemize}
\chapter{Mechanical Design}
\label{sec:detail_design_mechanics}
\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}
The strut design, illustrated in Figure \ref{fig:detail_design_strut}, was driven by several factors.
Stiff interfaces were required between the amplified piezoelectric actuator and the two flexible joints, as well as between the flexible joints and their respective mounting plates.
Due to the limited angular stroke of the flexible joints, it was important that the struts could be assembled in such a way that the two cylindrical interfaces were coaxial while the flexible joints were experiencing no stress (i.e. rest position).
To achieve this, cylindrical washers, shown in Figure \ref{fig:detail_design_strut_without_enc}, were integrated into the design to allow for poor flatness between the two interface planes of the APA, depicted in Figure \ref{fig:detail_design_apa}.
A dedicated mounting bench was also developed, such that each strut could be precisely aligned, even in the presence of machining inaccuracies.
The mounting procedure is described in Section \ref{sec:test_struts_mounting}.
Lastly, the design needed to permit the fixation of an encoder parallel to the strut axis, as shown in Figure \ref{fig:detail_design_strut_with_enc}.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -89,36 +64,20 @@ With the added cylindrical washers and the mounting tool, it should be possible
\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}
The flexible joints are manufactured using wire-cut electrical discharge machining, allowing for:
\begin{itemize}
\item very tight tolerances:
\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, shown in Figure \ref{fig:detail_design_flexible_joint}, were manufactured using wire-cut electrical discharge machining (EDM).
This manufacturing process was selected for few reasons.
First, because of the neck dimension of only \(0.25\,\text{mm}\), the part is inherently fragile and is difficult to manufacture with classical machining as cutting forces may damage the part.
Also wire-cut EDM allows for very tight machining tolerances, which are critical for achieving accurate location of the center of rotation relative to the plate interfaces (indicated by red surfaces in Figure \ref{fig:detail_design_flexible_joint}) and for maintaining the correct neck dimensions necessary for the desired stiffness and angular stroke properties.
The material chosen for the flexible joints is a stainless steel designated \emph{X5CrNiCuNb16-4} (alternatively known as ``F16Ph'').
This selection was based on its high specified yield strength (exceeding \(1\,\text{GPa}\) after appropriate heat treatment) and its high fatigue resistance.
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}
As shown in Figure \ref{fig:detail_design_flexible_joint}, the interface designed to connect with the APA possesses a cylindrical shape, facilitating the use of the aforementioned cylindrical washers for alignment.
A slotted hole was incorporated to permit alignment of the flexible joint with the APA via a dowel pin.
Additionally, two threaded holes were included on the sides for mounting the encoder components.
The interface connecting the flexible joint to the platform plates will be described subsequently.
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.
Modifications to the standard mechanical interfaces of the APA300ML were requested from the manufacturer.
The modified design features two planar surfaces and a dowel hole for precise location and orientation, as illustrated in Figure \ref{fig:detail_design_apa}.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -136,18 +95,14 @@ The amplifying structure, is also made of stainless steel.
\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}
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}.
Accurate measurement of the relative displacement within each strut requires the encoders to sense the motion between the rotational centers of the two associated flexible joints.
To achieve this, two interface parts, fabricated from aluminum, were designed.
These parts serve to fix the encoder head and the associated scale (ruler) to the two flexible joints, as depicted in Figure \ref{fig:detail_design_strut_with_enc}.
\subsubsection{Plates}
The two plates of the active platform were designed to:
\begin{itemize}
\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.
While topology optimization could have been used, a network of reinforcing ribs was used as shown in Figure \ref{fig:detail_design_top_plate}.
The design of the top and bottom plates of the active platform was governed by two main requirements: maximizing the frequency of flexible modes and ensuring accurate positioning of the top flexible joints and well-defined orientation of the struts.
To maximize the natural frequencies associated with plate flexibility, a simple network of reinforcing ribs was adopted, as shown for the top plate in Figure \ref{fig:detail_design_top_plate}.
While topology optimization methods could have been used, the presented designed was found to give high enough flexible modes.
\begin{figure}[htbp]
\centering
@ -155,21 +110,10 @@ While topology optimization could have been used, a network of reinforcing ribs
\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 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}
Joints interfaces on the plate consist of ``V-grooves''.
The cylindrical portion of each flexible joint is constrained within its corresponding V-groove through two distinct line contacts, illustrated in Figure \ref{fig:detail_design_fixation_flexible_joints}.
These grooves consequently serve to define the nominal orientation of the struts.
High machining accuracy for these features is essential to ensure that the flexible joints are in their neutral, unstressed state when the active platform is assembled.
\begin{figure}
\begin{subfigure}{0.33\textwidth}
@ -192,22 +136,24 @@ The two plates are made with a martensitic stainless steel ``X30Cr13'':
\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}
Furthermore, the flat interface surface of each top flexible joint is designed to be in direct contact with the top platform surface, as shown in Figure \ref{fig:detail_design_location_top_flexible_joints}.
This contact ensures that the centers of rotation of the top flexible joints, are precisely located relative to the top platform coordinate system.
The bottom flexible joints, however, are primarily oriented by the V-grooves without the same precise positional constraint against the bottom plate, as shown in Figure \ref{fig:detail_design_location_bot_flex}.
Both plates were specified to be manufactured from a martensitic stainless steel, X30Cr13.
This material was selected primarily for its high hardness, which minimizes the risk of deformation of the reference surfaces during the clamping of the flexible joints.
This characteristic is expected to permit repeated assembly and disassembly of the struts, should maintenance or reconfiguration be necessary.
\subsubsection{Finite Element Analysis}
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 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 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}
A finite element analysis (FEA) of the complete active platform assembly was performed to identify modes that could potentially affect performance.
The analysis revealed that the first six modes correspond to ``suspension'' modes, where the top plate effectively moves as a rigid body, and motion primarily involves axial displacement of the six struts (an example is shown in Figure \ref{fig:detail_design_fem_rigid_body_mode}).
Following these suspension modes, numerous ``local'' modes associated with the struts themselves were observed in the frequency range between \(205\,\text{Hz}\) and \(420\,\text{Hz}\).
One such mode is represented in Figure \ref{fig:detail_design_fem_strut_mode}.
Although these modes do not appear to induce significant motion of the top platform, they do cause relative displacement between the encoder components (head and scale) mounted on the strut.
Consequently, such modes could potentially be problematic if the active platform's position is controlled based on the encoders.
The extent to which these modes might pose a problem is difficult to establish at this stage, as it depends on whether they are significantly excited by the APA actuation and their sensitivity to strut alignment.
Finally, the FEA indicated that flexible modes of the top plate itself begin to appear at frequencies above \(650\,\text{Hz}\), with the first such mode shown in Figure \ref{fig:detail_design_fem_plate_mode}.
\begin{figure}[htbp]
\begin{subfigure}{0.36\textwidth}
@ -232,15 +178,8 @@ Then, flexible modes of the top plate are appearing above \(650\,\text{Hz}\) (Fi
\end{figure}
\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 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 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}
In anticipation of potential issues arising from the local modes of the struts affecting encoder measurements, an alternative fixation strategy for the encoders was envisaged.
In this configuration, the encoders are fixed directly to the top and bottom plates instead of the struts, as illustrated in Figure \ref{fig:detail_design_enc_plates_design}.
\begin{figure}[htbp]
\begin{subfigure}{0.59\textwidth}
@ -258,38 +197,16 @@ The issue is that the Kinematics may not be correct.
\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]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\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}
Dedicated supports, machined from aluminum, were designed for this purpose.
It was verified through FEA that the natural modes of these supports occur at sufficiently high frequencies, with the first mode estimated at \(1120\,\text{Hz}\).
Precise positioning of these encoder supports is achieved through machined pockets in both the top and bottom plates, visible in Figure \ref{fig:detail_design_top_plate} (indicated in green).
Although the encoders in this arrangement are aligned parallel to the nominal strut axes, they no longer measure the exact relative displacement along the strut between the flexible joint centers.
This geometric discrepancy implies that if the relative motion control of the active platform is based directly on these encoder readings, the kinematic calculations may be slightly inaccurate, potentially affecting the overall positioning accuracy of the platform.
\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.
Two configurations, displayed in Figure \ref{fig:detail_design_simscape_encoder}, were considered:
\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.
Prior to the procurement of mechanical components, the multi-body simulation model of the active platform was refined to incorporate the finalized design geometries.
Two distinct configurations, corresponding to the two encoder mounting strategies discussed previously, were considered in the model, as displayed in Figure \ref{fig:detail_design_simscape_encoder}: one with encoders fixed to the struts, and another with encoders fixed to the plates.
In these models, the top and bottom plates were represented as rigid bodies, with their inertial properties calculated directly from the 3D CAD geometry.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -308,15 +225,10 @@ Plates were modelled as rigid bodies, with inertia computed from the 3D shape.
\end{figure}
\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.
Several levels of detail were considered for modeling the flexible joints within the multi-body model.
Models with two degrees of freedom incorporating only bending stiffnesses, models with three degrees of freedom adding torsional stiffness, and models with four degrees of freedom further adding axial stiffness were evaluated.
The multi-body representation corresponding to the 4DoF configuration is shown in Figure \ref{fig:detail_design_simscape_model_flexible_joint}.
This model is composed of three distinct solid bodies interconnected by joints, whose stiffness properties were derived from finite element analysis of the joint component.
\begin{figure}[htbp]
\centering
@ -325,30 +237,21 @@ It is composed of three solid bodies connected by joints whose stiffnesses are c
\end{figure}
\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}
The amplified piezoelectric actuators (APAs) were incorporated into the multi-body model following the methodology detailed in Section \ref{sec:detail_fem_actuator}.
Two distinct representations of the APA can be utilized within the simulation: a simplified 2DoF model capturing the axial behavior, or a more complex ``Reduced Order Flexible Body'' model derived from a finite element model.
\subsubsection{Encoders}
Up to now, relative displacement sensors were implemented as a relative distance measurement between \(\bm{a}_i\) and \(\bm{b}_i\).
In earlier modeling stages, the relative displacement sensors (encoders) were implemented as a direct measurement of the relative distance between the joint connection points \(\bm{a}_i\) and \(\bm{b}_i\).
However, as indicated by the FEA results discussed previously, the flexible modes inherent to the struts could potentially affect the encoder measurement.
Therefore, a more sophisticated model of the optical encoder was necessary.
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 encoders operate based on the interaction between an encoder head and a graduated scale or ruler.
The optical encoder head contains a light source which is illuminating the ruler.
The position of the light on the ruler is represented by the reference frame \(\{E\}\) in Figure \ref{fig:detail_design_simscape_encoder_model}.
The ruler features a precise grating pattern (in this case, with a \(20\,\mu m\) pitch), and its position is associated with the reference frame \(\{R\}\).
The displacement measured by the encoder corresponds to the relative position of the encoder frame \(\{E\}\) (specifically, the point where the light interacts with the scale) with respect to the ruler frame \(\{R\}\), projected along the measurement direction defined by the scale.
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.
An important consequence of this measurement principle is that a relative rotation between the encoder head and the ruler, as depicted conceptually in Figure \ref{fig:detail_design_simscape_encoder_disp}, can induce a measured displacement.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -367,14 +270,11 @@ In that case, a rotation of the encoder, as shown in figure \ref{fig:detail_desi
\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}
Utilizing this refined multi-body model, several assessments were conducted.
The active platform model was integrated into the larger simulation model with the micro-station.
The dynamic behavior was evaluated and considered satisfactory.
Furthermore, simulations replicating tomography experiments were performed.
The performance metrics obtained from these simulations were found to be comparable to those achieved during the earlier conceptual design phase simulations.
Consequently, as the results closely mirror those presented previously in Section \ref{ssec:test_id31_iff_hac_perf}, they are not reiterated in detail here.
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}