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@ -252,29 +252,43 @@ CLOSED: [2025-04-21 Mon 14:13]
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[[file:figs/detail_design_nano_hexapod_elements.png]]
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*Design goals*:
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- Position =bi= and =si=
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- Maximum height of 95mm
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- As close as possible to "perfect" stewart platform: flexible modes at high frequency
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- Easy mounting, easy change of strut in case of failure
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Detail design phase:
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- key elements were optimized such as: actuator and flexible joints
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- relative motion sensor (an encoder) was also selected
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- 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.
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Yet, the geometry was fixed in Section [...]
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In this section, the mechanical design of the active platform, shown in Figure ref:fig:detail_design_nano_hexapod_elements, is detailed.
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Presentation of the obtained design:
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- Fixation
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- Section on: Complete strut
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- Cable management
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- Plates design
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- FEM results
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- Explain again the different specifications in terms of space, payload, etc..
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- CAD view of the nano-hexapod
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- Chosen geometry, materials, ease of mounting, cabling, ...
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- Validation on Simscape with accurate model?
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The main design objectives are:
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- Well defined kinematics: Good positioning of the top flexible joint rotation point $\bm{b}_i$ and correct orientation of the struts $\hat{\bm{s}}_i$.
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The goal is to have a well defined geometry such that the Jacobian matrix is well defined.
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- Space constrains: it should fit within a cylinder with radius of $120\,\text{mm}$ and height of $95\,\text{mm}$
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- As good performances were obtained with the multi-body model.
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The final design should behave as close as possible to "perfect" stewart platform.
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This means that the frequency of flexible modes that could be problematic for control must be made as high as possible.
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- Easy mounting and alignment.
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- Easy maintenance: the struts should be easily changed in case for failure.
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* Mechanical Design
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<<sec:detail_design_mechanics>>
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** Struts
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**** Introduction :ignore:
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**** Struts
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The strut design is shown in Figure ref:fig:detail_design_strut.
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The design of the struts was driven by:
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- having stiff interface between the amplified piezoelectric actuator and the two flexible joints
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- having stiff interface between the flexible joints and the two places (discussed afterwards)
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- 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.
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Do to so:
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- A mounting bench was designed
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The mounting procedure will be described in Section [...]
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# TODO - Add link to section
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- Cylindrical washers, shown in Figure ref:fig:detail_design_strut_without_enc, were integrated to allow for adjustments.
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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.
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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.
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- Possibility to fix the encoder parallel to the strut, as shown in Figure ref:fig:detail_design_strut_with_enc
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#+name: fig:detail_design_strut
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#+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.
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@ -283,65 +297,88 @@ Presentation of the obtained design:
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#+attr_latex: :caption \subcaption{\label{fig:detail_design_strut_without_enc}Before encoder integration}
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#+attr_latex: :options {0.49\textwidth}
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#+begin_subfigure
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#+attr_latex: :width 0.95\linewidth
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[[file:figs/detail_design_strut_without_enc.jpg]]
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#+attr_latex: :scale 0.9
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[[file:figs/detail_design_strut_without_enc.png]]
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#+end_subfigure
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#+attr_latex: :caption \subcaption{\label{fig:detail_design_strut_with_enc}With the mounted encoder}
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#+attr_latex: :options {0.49\textwidth}
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#+begin_subfigure
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#+attr_latex: :width 0.95\linewidth
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[[file:figs/detail_design_strut_with_enc.jpg]]
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#+attr_latex: :scale 0.9
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||||
[[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)
|
||||
- high yield strength: specified >1GPa using heat treatment
|
||||
- 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
|
||||
|
||||
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.
|
||||
|
||||
** 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=
|
||||
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.
|
||||
|
||||
#+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
|
||||
#+caption: Obtained Nano-Hexapod design
|
||||
#+attr_latex: :width 0.9\linewidth
|
||||
[[file:figs/detail_design_enc_struts.jpg]]
|
||||
|
||||
- FEM of complete system
|
||||
- Show modes of the struts
|
||||
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)
|
||||
|
||||
#+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: :options {0.33\textwidth}
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_rigid_body_mode}Suspension mode}
|
||||
#+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: :options {0.33\textwidth}
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_strut_mode}Strut - Local mode}
|
||||
#+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: :options {0.33\textwidth}
|
||||
#+attr_latex: :caption \subcaption{\label{fig:detail_design_fem_plate_mode}Top plate mode}
|
||||
#+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
|
||||
- Support made of aluminum
|
||||
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.
|
||||
|
||||
#+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_fem_encoder_fix
|
||||
#+name: fig:detail_design_enc_support_modes
|
||||
#+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*:
|
||||
- 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
|
||||
Before all the mechanical parts were ordered, the multi-body model of the active platform was refined using the design parts.
|
||||
|
||||
** Introduction :ignore:
|
||||
|
||||
Two configurations:
|
||||
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.
|
||||
|
||||
#+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.
|
||||
|
||||
|
||||
|
||||
** Encoders
|
||||
In that case, a rotation of the encoder, as shown in figure ref:fig:detail_design_simscape_encoder_disp induces 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}).
|
||||
@ -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
|
||||
|
||||
BIN
nass-design.pdf
BIN
nass-design.pdf
Binary file not shown.
278
nass-design.tex
278
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 Maximum height of 95mm
|
||||
\item As close as possible to ``perfect'' stewart platform: flexible modes at high frequency
|
||||
\item Easy mounting, easy change of strut in case of failure
|
||||
\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.
|
||||
|
||||
Presentation of the obtained design:
|
||||
The main design objectives are:
|
||||
\begin{itemize}
|
||||
\item Fixation
|
||||
\item Section on: Complete strut
|
||||
\item Cable management
|
||||
\item Plates design
|
||||
\item FEM results
|
||||
\item Explain again the different specifications in terms of space, payload, etc..
|
||||
\item CAD view of the nano-hexapod
|
||||
\item Chosen geometry, materials, ease of mounting, cabling, \ldots{}
|
||||
\item Validation on Simscape with accurate model?
|
||||
\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}
|
||||
\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 high fatigue resistance
|
||||
\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 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
|
||||
\subsubsection{Encoder support}
|
||||
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}.
|
||||
\subsubsection{Plates}
|
||||
|
||||
All other parts are made of aluminum.
|
||||
\section{Plates}
|
||||
|
||||
Plates: X30Cr13
|
||||
The two plates of the active platform were designed to:
|
||||
\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}
|
||||
|
||||
|
||||
\begin{itemize}
|
||||
\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}
|
||||
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}.
|
||||
|
||||
\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}
|
||||
|
||||
|
||||
\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}
|
||||
\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 FEM of complete system
|
||||
\item Show modes of the struts
|
||||
\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}
|
||||
|
||||
\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 Support made of aluminum
|
||||
\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}
|
||||
|
||||
\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
|
||||
\includegraphics[scale=1]{figs/detail_design_fem_encoder_fix.png}
|
||||
\caption{\label{fig:detail_design_fem_encoder_fix}Finite Element Analysis of the encoder supports. Encoder inertia was taken into account.}
|
||||
\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}
|
||||
\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}:
|
||||
\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:
|
||||
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.
|
||||
|
||||
\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.
|
||||
|
||||
|
||||
|
||||
\section{Encoders}
|
||||
In that case, a rotation of the encoder, as shown in figure \ref{fig:detail_design_simscape_encoder_disp} induces a measured displacement.
|
||||
|
||||
\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}).}
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\end{figure}
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\subsubsection{Simulation}
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Based on this refined model:
|
||||
\begin{itemize}
|
||||
\item the active platform could be integrated on top of the micro-station's model.
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||||
\item the obtained dynamics was considered good
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||||
\item simulations of tomography experiments were performed, and similar performance were obtained as during the conceptual design
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\item this is not presented here as results are very similar to the simulations performed in Section [\ldots{}]
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\end{itemize}
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\chapter{Conclusion}
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\label{sec:detail_design_conclusion}
|
||||
\printbibliography[heading=bibintoc,title={Bibliography}]
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|
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