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@ -66,6 +66,8 @@
} }
@article{claeyssen07_amplif_piezoel_actuat, @article{claeyssen07_amplif_piezoel_actuat,
author = {Frank Claeyssen and R. Le Letty and F. Barillot and O. author = {Frank Claeyssen and R. Le Letty and F. Barillot and O.
Sosnicki}, Sosnicki},
@ -122,17 +124,6 @@
@phdthesis{hanieh03_activ_stewar,
author = {Hanieh, Ahmed Abu},
keywords = {parallel robot},
school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
title = {Active isolation and damping of vibrations via Stewart
platform},
year = 2003,
}
@article{souleille18_concep_activ_mount_space_applic, @article{souleille18_concep_activ_mount_space_applic,
author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and
Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues, Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues,
@ -149,6 +140,35 @@
@article{verma20_dynam_stabil_thin_apert_light,
author = {Mohit Verma and Adrien Pece and Sylvain Hellegouarch and
Jennifer Watchi and Gilles Durand and Simon Chesn{\'e} and
Christophe Collette},
title = {Dynamic Stabilization of Thin Aperture Light Collector
Space Telescope Using Active Rods},
journal = {Journal of Astronomical Telescopes, Instruments, and
Systems},
volume = 6,
number = 01,
pages = 1,
year = 2020,
doi = {10.1117/1.jatis.6.1.014002},
url = {http://dx.doi.org/10.1117/1.JATIS.6.1.014002},
DATE_ADDED = {Thu Apr 3 21:25:20 2025},
}
@phdthesis{hanieh03_activ_stewar,
author = {Hanieh, Ahmed Abu},
keywords = {parallel robot},
school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
title = {Active isolation and damping of vibrations via Stewart
platform},
year = 2003,
}
@book{schmidt20_desig_high_perfor_mechat_third_revis_edition, @book{schmidt20_desig_high_perfor_mechat_third_revis_edition,
author = {Schmidt, R Munnig and Schitter, Georg and Rankers, Adrian}, author = {Schmidt, R Munnig and Schitter, Georg and Rankers, Adrian},
title = {The Design of High Performance Mechatronics - Third Revised title = {The Design of High Performance Mechatronics - Third Revised
@ -191,7 +211,7 @@
url = {https://doi.org/10.1016/j.jsv.2018.10.007}, url = {https://doi.org/10.1016/j.jsv.2018.10.007},
issn = {0022-460X}, issn = {0022-460X},
keywords = {parallel robot, flexure, decoupled control}, keywords = {parallel robot, flexure, decoupled control},
month = {Jan}, month = 1,
publisher = {Elsevier BV}, publisher = {Elsevier BV},
} }

71
nass-fem.org
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@ -169,18 +169,12 @@ For Stewart platform:
* Introduction :ignore: * Introduction :ignore:
During the detailed design phase of the nano-hexapod, optimizing individual components while ensuring their dynamic compatibility with the complete system presents significant challenges. During the nano-hexapod's detailed design phase, a hybrid modeling approach combining finite element analysis with multi-body dynamics was developed.
While Finite Element Analysis (FEA) serves as a powerful tool for component-level optimization, understanding how the dynamics of each element interacts within the complete nano-active stabilization system (NASS) becomes crucial. This methodology, utilizing reduced-order flexible bodies, was created to enable both detailed component optimization and efficient system-level simulation, addressing the impracticality of a full FEM for real-time control scenarios.
A full Finite Element Model (FEM) of the NASS, while theoretically possible, would prove impractical for simulating real-time control scenarios due to its computational complexity.
This chapter presents a hybrid modeling approach that combines finite element analysis with multi-body dynamics, enabling both detailed component optimization and efficient system-level simulation. The theoretical foundations and implementation are presented in Section ref:sec:detail_fem_super_element, where experimental validation was performed using an Amplified Piezoelectric Actuator.
The methodology employs reduced-order flexible bodies, whereby components whose dynamic properties are determined through FEA can be effectively integrated into the multi-body framework. The framework was then applied to optimize two critical nano-hexapod elements: the actuators (Section ref:sec:detail_fem_actuator) and the flexible joints (Section ref:sec:detail_fem_joint).
The theoretical foundations and practical implementation of this approach are presented in Section ref:sec:detail_fem_super_element, where experimental validation using an Amplified Piezoelectric Actuator (APA) demonstrates the method's accuracy in predicting both open and closed-loop dynamic behavior. Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated.
This validated modeling framework is then applied to optimize two critical elements of the nano-hexapod: the actuators and the flexible joints.
Section ref:sec:detail_fem_actuator examines the selection and characterization of the actuators, developing both high-fidelity and computationally efficient models that capture essential dynamic characteristics.
Section ref:sec:detail_fem_joint addresses the design of flexible joints, where proper parasitic stiffness proves crucial for system performance.
In both cases, the hybrid modeling approach enables detailed component optimization while maintaining the ability to predict system-level dynamic behavior, particularly under closed-loop control conditions.
* Reduced order flexible bodies * Reduced order flexible bodies
:PROPERTIES: :PROPERTIES:
@ -244,10 +238,10 @@ Initially, the component is modeled in a finite element software with appropriat
Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component. Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component.
These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames. These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames.
Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method [[cite:&craig68_coupl_subst_dynam_analy]] (also known as the "fixed-interface method"), a technique that significantly reduce the number of DoF while while still presenting the main dynamical characteristics. Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method [[cite:&craig68_coupl_subst_dynam_analy]] (also known as the "fixed-interface method"), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics.
This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF. This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF.
The number of degrees of freedom in the reduced model is determined by eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled. The number of degrees of freedom in the reduced model is determined by eqref:eq:detail_fem_model_order where $n$ represents the number of defined frames and $p$ denotes the number of additional modes to be modeled.
The outcome of this procedure is an $m \times m$ set of reduced mass and stiffness matrices, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. The outcome of this procedure is an $m \times m$ set of reduced mass and stiffness matrices, $m$ being the total retained number of degrees of freedom, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior.
\begin{equation}\label{eq:detail_fem_model_order} \begin{equation}\label{eq:detail_fem_model_order}
m = 6 \times n + p m = 6 \times n + p
@ -261,7 +255,7 @@ The presented modeling framework was first applied to an Amplified Piezoelectric
Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section ref:sec:detail_fem_actuator. Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section ref:sec:detail_fem_actuator.
Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing. Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure ref:fig:detail_fem_apa95ml_picture) was available in the laboratory for experimental testing.
The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure ref:fig:detail_fem_apa95ml_picture) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement into the vertical direction [[cite:&claeyssen07_amplif_piezoel_actuat]]. The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure ref:fig:detail_fem_apa95ml_picture) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement in the vertical direction [[cite:&claeyssen07_amplif_piezoel_actuat]].
The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework.
The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation.
@ -288,7 +282,7 @@ The specific design of the APA allows for the simultaneous modeling of a mechani
**** Finite Element Model **** Finite Element Model
The development of the finite element model for the APA95ML necessitated the specification of appropriate material properties, as summarized in Table ref:tab:detail_fem_material_properties. The development of the finite element model for the APA95ML required the knowledge of the material properties, as summarized in Table ref:tab:detail_fem_material_properties.
The finite element mesh, shown in Figure ref:fig:detail_fem_apa95ml_mesh, was then generated. The finite element mesh, shown in Figure ref:fig:detail_fem_apa95ml_mesh, was then generated.
#+name: tab:detail_fem_material_properties #+name: tab:detail_fem_material_properties
@ -300,7 +294,7 @@ The finite element mesh, shown in Figure ref:fig:detail_fem_apa95ml_mesh, was th
| Stainless Steel | $190\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ | | Stainless Steel | $190\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ |
| Piezoelectric Ceramics (PZT) | $49.5\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ | | Piezoelectric Ceramics (PZT) | $49.5\,GPa$ | $0.31$ | $7800\,\text{kg}/m^3$ |
The definition of interface frames, or "remote points", constitute a critical aspect of the model preparation. The definition of interface frames constitutes a critical aspect of the model preparation.
Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation. Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation.
Six additional modes were considered, resulting in total model order of $48$. Six additional modes were considered, resulting in total model order of $48$.
@ -310,13 +304,13 @@ The modal reduction procedure was then executed, yielding the reduced mass and s
#+caption: Obtained mesh and defined interface frames (or "remote points") in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}). #+caption: Obtained mesh and defined interface frames (or "remote points") in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}).
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh and "remote points"} #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_mesh} }
#+attr_latex: :options {0.48\textwidth} #+attr_latex: :options {0.48\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :scale 1 #+attr_latex: :scale 1
[[file:figs/detail_fem_apa95ml_mesh.png]] [[file:figs/detail_fem_apa95ml_mesh.png]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa_model_schematic}Inclusion in multi-body model} #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa_model_schematic} }
#+attr_latex: :options {0.48\textwidth} #+attr_latex: :options {0.48\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :scale 1 #+attr_latex: :scale 1
@ -501,34 +495,20 @@ The high degree of concordance observed across multiple performance metrics prov
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation. Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element. The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
The test bench illustrated in Figure ref:fig:detail_fem_apa95ml_bench was used, which consists of a $5.7\,kg$ granite suspended on top of the APA95ML. The test bench illustrated in Figure ref:fig:detail_fem_apa95ml_bench_schematic was used, which consists of a $5.7\,kg$ granite suspended on top of the APA95ML.
The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$. The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement $y$.
A digital-to-analog converter (DAC) was used to generate the control signal $u$, which was subsequently conditioned through a voltage amplifier with a gain of $20$, ultimately yielding the effective voltage $V_a$ across the two piezoelectric stacks. A digital-to-analog converter (DAC) was used to generate the control signal $u$, which was subsequently conditioned through a voltage amplifier with a gain of $20$, ultimately yielding the effective voltage $V_a$ across the two piezoelectric stacks.
Measurement of the sensor stack voltage $V_s$ was performed using an analog-to-digital converter (ADC). Measurement of the sensor stack voltage $V_s$ was performed using an analog-to-digital converter (ADC).
#+name: fig:detail_fem_apa95ml_bench #+name: fig:detail_fem_apa95ml_bench_schematic
#+caption: Test bench used to validate "reduced order solid bodies" using an APA95ML. Picture of the bench is shown in (\subref{fig:detail_fem_apa95ml_bench_picture}). Schematic is shown in (\subref{fig:detail_fem_apa95ml_bench_schematic}). #+caption: Test bench used to validate "reduced order solid bodies" using an APA95ML.
#+attr_latex: :options [htbp] #+attr_latex: :width \linewidth
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_bench_picture}Picture of the test bench}
#+attr_latex: :options {0.34\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_fem_apa95ml_bench_picture.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_bench_schematic}Schematic with signals}
#+attr_latex: :options {0.72\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_fem_apa95ml_bench_schematic.png]] [[file:figs/detail_fem_apa95ml_bench_schematic.png]]
#+end_subfigure
#+end_figure
**** Comparison of the dynamics **** Comparison of the dynamics
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML. Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
The identification procedure necessitated careful choice of the excitation signal [[cite:&pintelon12_system_ident, chap. 5]]. The identification procedure required careful choice of the excitation signal [[cite:&pintelon12_system_ident, chap. 5]].
Commonly employed excitation signals include impulses (which are particularly effective for modal analysis), steps, random noise signals, and multi-sine excitations
During all this experimental work, random noise excitation was predominantly employed. During all this experimental work, random noise excitation was predominantly employed.
The designed excitation signal is then generated and both input and output signals are synchronously acquired. The designed excitation signal is then generated and both input and output signals are synchronously acquired.
@ -653,7 +633,7 @@ exportFig('figs/detail_fem_apa95ml_comp_plant_sensor.pdf', 'width', 'half', 'hei
#+end_src #+end_src
#+name: fig:detail_fem_apa95ml_comp_plant #+name: fig:detail_fem_apa95ml_comp_plant
#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA95ML. Both for the dynamics from $V_a$ to $y$ (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa95ml_comp_plant_sensor}) #+caption: Comparison of the measured frequency response functions and the finite element model of the APA95ML. Both for the dynamics from $V_a$ to $y$ (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from $V_a$ to $V_s$ (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$} #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_actuator}from $V_a$ to $y$}
@ -819,7 +799,7 @@ exportFig('figs/detail_fem_apa95ml_damped_plants.pdf', 'width', 'half', 'height'
#+end_src #+end_src
#+name: fig:detail_fem_apa95ml_iff_results #+name: fig:detail_fem_apa95ml_iff_results
#+caption: Obtained results using Integral Force Feedback with the APA95ML. Obtained closed-loop poles as a function of the controller gain $g$ are prediction by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains. #+caption: Results using Integral Force Feedback with the APA95ML. Closed-loop poles as a function of the controller gain $g$ are predicted by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot} #+attr_latex: :caption \subcaption{\label{fig:detail_fem_apa95ml_iff_root_locus}Root Locus plot}
@ -851,13 +831,6 @@ The approach will be especially beneficial for optimizing actuators (Section ref
:HEADER-ARGS:matlab+: :tangle matlab/detail_fem_2_actuators.m :HEADER-ARGS:matlab+: :tangle matlab/detail_fem_2_actuators.m
:END: :END:
<<sec:detail_fem_actuator>> <<sec:detail_fem_actuator>>
** Introduction :ignore:
The selection and modeling of actuators, that constitutes a critical step in the development of the nano-hexapod, is here presented.
First, specifications for the nano-hexapod actuators are derived from previous analyses, leading to the selection of the actuator type and ultimately to a specific model (Section ref:ssec:detail_fem_actuator_specifications).
Then, the chosen actuator is modeled using the reduced-order flexible body approach developed in the previous section, validating the choice of actuator through detailed dynamical analysis (Section ref:ssec:detail_fem_actuator_apa300ml).
Finally, a simplified two-degree-of-freedom model is developed to facilitate time-domain simulations while maintaining accurate representation of the actuator's essential characteristics (Section ref:ssec:detail_fem_actuator_apa300ml_2dof).
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>> <<matlab-dir>>
@ -909,7 +882,7 @@ Furthermore, the actuator stroke must exceed the micro-station positioning error
An actuator stroke of $\approx 100\,\mu m$ is therefore required. An actuator stroke of $\approx 100\,\mu m$ is therefore required.
Three actuator technologies were evaluated (examples of such actuators are shown in Figure ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. Three actuator technologies were evaluated (examples of such actuators are shown in Figure ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce unnecessary control complexity. Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity.
#+name: fig:detail_fem_actuator_pictures #+name: fig:detail_fem_actuator_pictures
#+caption: Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}). #+caption: Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).
@ -945,7 +918,7 @@ Additionally, their extremely high stiffness, typically around $100\,N/\mu m$, e
Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design. Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design.
The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile. The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile.
Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF. Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF [[cite:&souleille18_concep_activ_mount_space_applic;&verma20_dynam_stabil_thin_apert_light]].
Moreover, using APA for active damping has been successfully demonstrated in similar applications [[cite:&hanieh03_activ_stewar]]. Moreover, using APA for active damping has been successfully demonstrated in similar applications [[cite:&hanieh03_activ_stewar]].
Several specific APA models were evaluated against the established specifications (Table ref:tab:detail_fem_piezo_act_models). Several specific APA models were evaluated against the established specifications (Table ref:tab:detail_fem_piezo_act_models).
@ -1455,7 +1428,7 @@ For Stewart platforms requiring nanometric precision, numerous flexible joint de
For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts. For design simplicity and component standardization, identical joints are employed at both ends of the nano-hexapod struts.
#+name: fig:detail_fem_joints_examples #+name: fig:detail_fem_joints_examples
#+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in [[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in [[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having differnt "notch curves" are also used [[cite:&du14_piezo_actuat_high_precis_flexib]]. #+caption: Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical "universal" flexible joint used in [[cite:&preumont07_six_axis_singl_stage_activ]]. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in [[cite:&yang19_dynam_model_decoup_contr_flexib]]. (\subref{fig:detail_fem_joints_wire}) "Thin" flexible joints having "notch curves" are also used [[cite:&du14_piezo_actuat_high_precis_flexib]].
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_preumont}} #+attr_latex: :caption \subcaption{\label{fig:detail_fem_joints_preumont}}

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@ -1,4 +1,4 @@
% Created 2025-02-27 Thu 11:53 % Created 2025-04-03 Thu 21:39
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -8,13 +8,6 @@
\author{Dehaeze Thomas} \author{Dehaeze Thomas}
\date{\today} \date{\today}
\title{Optimization using Finite Element Models} \title{Optimization using Finite Element Models}
\hypersetup{
pdfauthor={Dehaeze Thomas},
pdftitle={Optimization using Finite Element Models},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 29.4 (Org mode 9.6)},
pdflang={English}}
\usepackage{biblatex} \usepackage{biblatex}
\begin{document} \begin{document}
@ -23,21 +16,14 @@
\tableofcontents \tableofcontents
\clearpage \clearpage
During the nano-hexapod's detailed design phase, a hybrid modeling approach combining finite element analysis with multi-body dynamics was developed.
This methodology, utilizing reduced-order flexible bodies, was created to enable both detailed component optimization and efficient system-level simulation, addressing the impracticality of a full FEM for real-time control scenarios.
During the detailed design phase of the nano-hexapod, optimizing individual components while ensuring their dynamic compatibility with the complete system presents significant challenges. The theoretical foundations and implementation are presented in Section \ref{sec:detail_fem_super_element}, where experimental validation was performed using an Amplified Piezoelectric Actuator.
While Finite Element Analysis (FEA) serves as a powerful tool for component-level optimization, understanding how the dynamics of each element interacts within the complete nano-active stabilization system (NASS) becomes crucial. The framework was then applied to optimize two critical nano-hexapod elements: the actuators (Section \ref{sec:detail_fem_actuator}) and the flexible joints (Section \ref{sec:detail_fem_joint}).
A full Finite Element Model (FEM) of the NASS, while theoretically possible, would prove impractical for simulating real-time control scenarios due to its computational complexity. Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated.
This chapter presents a hybrid modeling approach that combines finite element analysis with multi-body dynamics, enabling both detailed component optimization and efficient system-level simulation.
The methodology employs reduced-order flexible bodies, whereby components whose dynamic properties are determined through FEA can be effectively integrated into the multi-body framework.
The theoretical foundations and practical implementation of this approach are presented in Section \ref{sec:detail_fem_super_element}, where experimental validation using an Amplified Piezoelectric Actuator (APA) demonstrates the method's accuracy in predicting both open and closed-loop dynamic behavior.
This validated modeling framework is then applied to optimize two critical elements of the nano-hexapod: the actuators and the flexible joints.
Section \ref{sec:detail_fem_actuator} examines the selection and characterization of the actuators, developing both high-fidelity and computationally efficient models that capture essential dynamic characteristics.
Section \ref{sec:detail_fem_joint} addresses the design of flexible joints, where proper parasitic stiffness proves crucial for system performance.
In both cases, the hybrid modeling approach enables detailed component optimization while maintaining the ability to predict system-level dynamic behavior, particularly under closed-loop control conditions.
\chapter{Reduced order flexible bodies} \chapter{Reduced order flexible bodies}
\label{sec:orgcd4de1a}
\label{sec:detail_fem_super_element} \label{sec:detail_fem_super_element}
Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models. Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models.
These components are traditionally analyzed using Finite Element Analysis (FEA) software. These components are traditionally analyzed using Finite Element Analysis (FEA) software.
@ -49,6 +35,7 @@ First, the fundamental principles and methodological approaches of this modeling
It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section \ref{ssec:detail_fem_super_element_example}). It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section \ref{ssec:detail_fem_super_element_example}).
Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section \ref{ssec:detail_fem_super_element_validation}). Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section \ref{ssec:detail_fem_super_element_validation}).
\section{Procedure} \section{Procedure}
\label{sec:org89e4f49}
\label{ssec:detail_fem_super_element_theory} \label{ssec:detail_fem_super_element_theory}
In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models. In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models.
@ -62,22 +49,22 @@ Initially, the component is modeled in a finite element software with appropriat
Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component. Subsequently, interface frames are defined at locations where the multi-body model will establish connections with the component.
These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames. These frames serve multiple functions, including connecting to other parts, applying forces and torques, and measuring relative motion between defined frames.
Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method \cite{craig68_coupl_subst_dynam_analy} (also known as the ``fixed-interface method''), a technique that significantly reduce the number of DoF while while still presenting the main dynamical characteristics. Following the establishment of these interface parameters, modal reduction is performed using the Craig-Bampton method \cite{craig68_coupl_subst_dynam_analy} (also known as the ``fixed-interface method''), a technique that significantly reduces the number of DoF while while still presenting the main dynamical characteristics.
This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF. This transformation typically reduces the model complexity from hundreds of thousands to fewer than 100 DoF.
The number of degrees of freedom in the reduced model is determined by \eqref{eq:detail_fem_model_order} where \(n\) represents the number of defined frames and \(p\) denotes the number of additional modes to be modeled. The number of degrees of freedom in the reduced model is determined by \eqref{eq:detail_fem_model_order} where \(n\) represents the number of defined frames and \(p\) denotes the number of additional modes to be modeled.
The outcome of this procedure is an \(m \times m\) set of reduced mass and stiffness matrices, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior. The outcome of this procedure is an \(m \times m\) set of reduced mass and stiffness matrices, \(m\) being the total retained number of degrees of freedom, which can subsequently be incorporated into the multi-body model to represent the component's dynamic behavior.
\begin{equation}\label{eq:detail_fem_model_order} \begin{equation}\label{eq:detail_fem_model_order}
m = 6 \times n + p m = 6 \times n + p
\end{equation} \end{equation}
\section{Example with an Amplified Piezoelectric Actuator} \section{Example with an Amplified Piezoelectric Actuator}
\label{sec:org9338c69}
\label{ssec:detail_fem_super_element_example} \label{ssec:detail_fem_super_element_example}
The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons. The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons.
Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section \ref{sec:detail_fem_actuator}. Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section \ref{sec:detail_fem_actuator}.
Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure \ref{fig:detail_fem_apa95ml_picture}) was available in the laboratory for experimental testing. Additionally, an Amplified Piezoelectric Actuator (the APA95ML shown in Figure \ref{fig:detail_fem_apa95ml_picture}) was available in the laboratory for experimental testing.
The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure \ref{fig:detail_fem_apa95ml_picture}) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement into the vertical direction \cite{claeyssen07_amplif_piezoel_actuat}. The APA consists of multiple piezoelectric stacks arranged horizontally (depicted in blue in Figure \ref{fig:detail_fem_apa95ml_picture}) and of an amplifying shell structure (shown in red) that serves two purposes: the application of pre-stress to the piezoelectric elements and the amplification of their displacement in the vertical direction \cite{claeyssen07_amplif_piezoel_actuat}.
The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework. The selection of the APA for validation purposes was further justified by its capacity to simultaneously demonstrate multiple aspects of the modeling framework.
The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation. The specific design of the APA allows for the simultaneous modeling of a mechanical structure analogous to a flexible joint, piezoelectric actuation, and piezoelectric sensing, thereby encompassing the principal elements requiring validation.
@ -102,8 +89,9 @@ Stiffness & \(21\,N/\mu m\)\\
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications} \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage} \end{minipage}
\paragraph{Finite Element Model} \paragraph{Finite Element Model}
\label{sec:org1bd5502}
The development of the finite element model for the APA95ML necessitated the specification of appropriate material properties, as summarized in Table \ref{tab:detail_fem_material_properties}. The development of the finite element model for the APA95ML required the knowledge of the material properties, as summarized in Table \ref{tab:detail_fem_material_properties}.
The finite element mesh, shown in Figure \ref{fig:detail_fem_apa95ml_mesh}, was then generated. The finite element mesh, shown in Figure \ref{fig:detail_fem_apa95ml_mesh}, was then generated.
\begin{table}[htbp] \begin{table}[htbp]
@ -119,7 +107,7 @@ Piezoelectric Ceramics (PZT) & \(49.5\,GPa\) & \(0.31\) & \(7800\,\text{kg}/m^3\
\end{tabularx} \end{tabularx}
\end{table} \end{table}
The definition of interface frames, or ``remote points'', constitute a critical aspect of the model preparation. The definition of interface frames constitutes a critical aspect of the model preparation.
Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation. Seven frames were established: one frame at the two ends of each piezoelectric stack to facilitate strain measurement and force application, and additional frames at the top and bottom of the structure to enable connection with external elements in the multi-body simulation.
Six additional modes were considered, resulting in total model order of \(48\). Six additional modes were considered, resulting in total model order of \(48\).
@ -130,18 +118,18 @@ The modal reduction procedure was then executed, yielding the reduced mass and s
\begin{center} \begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa95ml_mesh.png} \includegraphics[scale=1,scale=1]{figs/detail_fem_apa95ml_mesh.png}
\end{center} \end{center}
\subcaption{\label{fig:detail_fem_apa95ml_mesh}Obtained mesh and "remote points"} \subcaption{\label{fig:detail_fem_apa95ml_mesh} }
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_fem_apa_modal_schematic.png} \includegraphics[scale=1,scale=1]{figs/detail_fem_apa_modal_schematic.png}
\end{center} \end{center}
\subcaption{\label{fig:detail_fem_apa_model_schematic}Inclusion in multi-body model} \subcaption{\label{fig:detail_fem_apa_model_schematic} }
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}).} \caption{\label{fig:detail_fem_apa95ml_model}Obtained mesh and defined interface frames (or ``remote points'') in the finite element model of the APA95ML (\subref{fig:detail_fem_apa95ml_mesh}). Interface with the multi-body model is shown in (\subref{fig:detail_fem_apa_model_schematic}).}
\end{figure} \end{figure}
\paragraph{Super Element in the Multi-Body Model} \paragraph{Super Element in the Multi-Body Model}
\label{sec:org5b98193}
Previously computed reduced order mass and stiffness matrices were imported in a multi-body model block called ``Reduced Order Flexible Solid''. Previously computed reduced order mass and stiffness matrices were imported in a multi-body model block called ``Reduced Order Flexible Solid''.
This block has several interface frames corresponding to the ones defined in the FEA software. This block has several interface frames corresponding to the ones defined in the FEA software.
@ -151,8 +139,8 @@ Therefore, a force source \(F_a\) operating between frames \(\{3\}\) and \(\{2\}
This is illustrated in Figure \ref{fig:detail_fem_apa_model_schematic}. This is illustrated in Figure \ref{fig:detail_fem_apa_model_schematic}.
However, to have access to the physical voltage input of the actuators stacks \(V_a\) and to the generated voltage by the force sensor \(V_s\), conversion between the electrical and mechanical domains need to be determined. However, to have access to the physical voltage input of the actuators stacks \(V_a\) and to the generated voltage by the force sensor \(V_s\), conversion between the electrical and mechanical domains need to be determined.
\paragraph{Sensor and Actuator ``constants''} \paragraph{Sensor and Actuator ``constants''}
\label{sec:org8148055}
To link the electrical domain to the mechanical domain, an ``actuator constant'' \(g_a\) and a ``sensor constant'' \(g_s\) were introduced as shown in Figure \ref{fig:detail_fem_apa_model_schematic}. To link the electrical domain to the mechanical domain, an ``actuator constant'' \(g_a\) and a ``sensor constant'' \(g_s\) were introduced as shown in Figure \ref{fig:detail_fem_apa_model_schematic}.
@ -209,8 +197,8 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\end{table} \end{table}
\paragraph{Identification of the APA Characteristics} \paragraph{Identification of the APA Characteristics}
\label{sec:orgcc15bce}
Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications. Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications.
@ -234,37 +222,27 @@ As three stacks are used, the horizontal displacement is \(60\,\mu m\).
Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\mu m\) which falls within the manufacturer-specified range of \(80\,\mu m\) and \(120\,\mu m\). Through the established amplification factor of 1.5, this translates to a predicted vertical stroke of \(90\,\mu m\) which falls within the manufacturer-specified range of \(80\,\mu m\) and \(120\,\mu m\).
The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model. The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model.
\section{Experimental Validation} \section{Experimental Validation}
\label{sec:org3f582e5}
\label{ssec:detail_fem_super_element_validation} \label{ssec:detail_fem_super_element_validation}
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation. Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element. The goal was to measure the dynamics of the APA95ML and to compare it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
The test bench illustrated in Figure \ref{fig:detail_fem_apa95ml_bench} was used, which consists of a \(5.7\,kg\) granite suspended on top of the APA95ML. The test bench illustrated in Figure \ref{fig:detail_fem_apa95ml_bench_schematic} was used, which consists of a \(5.7\,kg\) granite suspended on top of the APA95ML.
The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement \(y\). The granite's motion was vertically guided with an air bearing system, and a fibered interferometer was used to measured its vertical displacement \(y\).
A digital-to-analog converter (DAC) was used to generate the control signal \(u\), which was subsequently conditioned through a voltage amplifier with a gain of \(20\), ultimately yielding the effective voltage \(V_a\) across the two piezoelectric stacks. A digital-to-analog converter (DAC) was used to generate the control signal \(u\), which was subsequently conditioned through a voltage amplifier with a gain of \(20\), ultimately yielding the effective voltage \(V_a\) across the two piezoelectric stacks.
Measurement of the sensor stack voltage \(V_s\) was performed using an analog-to-digital converter (ADC). Measurement of the sensor stack voltage \(V_s\) was performed using an analog-to-digital converter (ADC).
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.34\textwidth} \centering
\begin{center} \includegraphics[scale=1,width=\linewidth]{figs/detail_fem_apa95ml_bench_schematic.png}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_bench_picture.png} \caption{\label{fig:detail_fem_apa95ml_bench_schematic}Test bench used to validate ``reduced order solid bodies'' using an APA95ML.}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_bench_picture}Picture of the test bench}
\end{subfigure}
\begin{subfigure}{0.72\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_fem_apa95ml_bench_schematic.png}
\end{center}
\subcaption{\label{fig:detail_fem_apa95ml_bench_schematic}Schematic with signals}
\end{subfigure}
\caption{\label{fig:detail_fem_apa95ml_bench}Test bench used to validate ``reduced order solid bodies'' using an APA95ML. Picture of the bench is shown in (\subref{fig:detail_fem_apa95ml_bench_picture}). Schematic is shown in (\subref{fig:detail_fem_apa95ml_bench_schematic}).}
\end{figure} \end{figure}
\paragraph{Comparison of the dynamics} \paragraph{Comparison of the dynamics}
\label{sec:org11ed21e}
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML. Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
The identification procedure necessitated careful choice of the excitation signal \cite[, chap. 5]{pintelon12_system_ident}. The identification procedure required careful choice of the excitation signal \cite[, chap. 5]{pintelon12_system_ident}.
Commonly employed excitation signals include impulses (which are particularly effective for modal analysis), steps, random noise signals, and multi-sine excitations
During all this experimental work, random noise excitation was predominantly employed. During all this experimental work, random noise excitation was predominantly employed.
The designed excitation signal is then generated and both input and output signals are synchronously acquired. The designed excitation signal is then generated and both input and output signals are synchronously acquired.
@ -291,10 +269,10 @@ Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could
\end{center} \end{center}
\subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$} \subcaption{\label{fig:detail_fem_apa95ml_comp_plant_sensor}from $V_a$ to $V_s$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_apa95ml_comp_plant}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA95ML. Both for the dynamics from \(V_a\) to \(y\) (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})} \caption{\label{fig:detail_fem_apa95ml_comp_plant}Comparison of the measured frequency response functions and the finite element model of the APA95ML. Both for the dynamics from \(V_a\) to \(y\) (\subref{fig:detail_fem_apa95ml_comp_plant_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa95ml_comp_plant_sensor})}
\end{figure} \end{figure}
\paragraph{Integral Force Feedback with APA} \paragraph{Integral Force Feedback with APA}
\label{sec:org0d2d636}
To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally. To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally.
Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains. Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains.
@ -324,22 +302,19 @@ The close agreement between experimental measurements and theoretical prediction
\end{center} \end{center}
\subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants} \subcaption{\label{fig:detail_fem_apa95ml_damped_plants}Damped plants}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_apa95ml_iff_results}Obtained results using Integral Force Feedback with the APA95ML. Obtained closed-loop poles as a function of the controller gain \(g\) are prediction by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.} \caption{\label{fig:detail_fem_apa95ml_iff_results}Results using Integral Force Feedback with the APA95ML. Closed-loop poles as a function of the controller gain \(g\) are predicted by root Locus plot (\subref{fig:detail_fem_apa95ml_iff_root_locus}). Circles are predictions from the model while crosses are poles estimated from the experimental data. Damped plants estimated from the model (dashed curves) and measured ones (solid curves) are compared in (\subref{fig:detail_fem_apa95ml_damped_plants}) for all tested controller gains.}
\end{figure} \end{figure}
\section*{Conclusion} \section*{Conclusion}
\label{sec:org8e4cad3}
The experimental validation with an Amplified Piezoelectric Actuator confirms that this methodology accurately predicts both open-loop and closed-loop dynamic behaviors. The experimental validation with an Amplified Piezoelectric Actuator confirms that this methodology accurately predicts both open-loop and closed-loop dynamic behaviors.
This verification establishes its effectiveness for component design and system analysis applications. This verification establishes its effectiveness for component design and system analysis applications.
The approach will be especially beneficial for optimizing actuators (Section \ref{sec:detail_fem_actuator}) and flexible joints (Section \ref{sec:detail_fem_joint}) for the nano-hexapod. The approach will be especially beneficial for optimizing actuators (Section \ref{sec:detail_fem_actuator}) and flexible joints (Section \ref{sec:detail_fem_joint}) for the nano-hexapod.
\chapter{Actuator Selection} \chapter{Actuator Selection}
\label{sec:orgaed2754}
\label{sec:detail_fem_actuator} \label{sec:detail_fem_actuator}
The selection and modeling of actuators, that constitutes a critical step in the development of the nano-hexapod, is here presented.
First, specifications for the nano-hexapod actuators are derived from previous analyses, leading to the selection of the actuator type and ultimately to a specific model (Section \ref{ssec:detail_fem_actuator_specifications}).
Then, the chosen actuator is modeled using the reduced-order flexible body approach developed in the previous section, validating the choice of actuator through detailed dynamical analysis (Section \ref{ssec:detail_fem_actuator_apa300ml}).
Finally, a simplified two-degree-of-freedom model is developed to facilitate time-domain simulations while maintaining accurate representation of the actuator's essential characteristics (Section \ref{ssec:detail_fem_actuator_apa300ml_2dof}).
\section{Choice of the Actuator based on Specifications} \section{Choice of the Actuator based on Specifications}
\label{sec:org55fd74f}
\label{ssec:detail_fem_actuator_specifications} \label{ssec:detail_fem_actuator_specifications}
The actuator selection process was driven by several critical requirements derived from previous dynamic analyses. The actuator selection process was driven by several critical requirements derived from previous dynamic analyses.
@ -355,7 +330,7 @@ Furthermore, the actuator stroke must exceed the micro-station positioning error
An actuator stroke of \(\approx 100\,\mu m\) is therefore required. An actuator stroke of \(\approx 100\,\mu m\) is therefore required.
Three actuator technologies were evaluated (examples of such actuators are shown in Figure \ref{fig:detail_fem_actuator_pictures}): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators. Three actuator technologies were evaluated (examples of such actuators are shown in Figure \ref{fig:detail_fem_actuator_pictures}): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce unnecessary control complexity. Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce control complexity.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.25\textwidth} \begin{subfigure}{0.25\textwidth}
@ -389,7 +364,7 @@ Additionally, their extremely high stiffness, typically around \(100\,N/\mu m\),
Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design. Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through a specific mechanical design.
The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile. The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile.
Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF. Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized IFF \cite{souleille18_concep_activ_mount_space_applic,verma20_dynam_stabil_thin_apert_light}.
Moreover, using APA for active damping has been successfully demonstrated in similar applications \cite{hanieh03_activ_stewar}. Moreover, using APA for active damping has been successfully demonstrated in similar applications \cite{hanieh03_activ_stewar}.
Several specific APA models were evaluated against the established specifications (Table \ref{tab:detail_fem_piezo_act_models}). Several specific APA models were evaluated against the established specifications (Table \ref{tab:detail_fem_piezo_act_models}).
@ -413,8 +388,8 @@ Height \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\end{table} \end{table}
\section{APA300ML - Reduced Order Flexible Body} \section{APA300ML - Reduced Order Flexible Body}
\label{sec:org596b422}
\label{ssec:detail_fem_actuator_apa300ml} \label{ssec:detail_fem_actuator_apa300ml}
The validation of the APA300ML started by incorporating a ``reduced order flexible body'' into the multi-body model as explained in Section \ref{sec:detail_fem_super_element}. The validation of the APA300ML started by incorporating a ``reduced order flexible body'' into the multi-body model as explained in Section \ref{sec:detail_fem_super_element}.
@ -440,8 +415,8 @@ While this high order provides excellent accuracy for validation purposes, it pr
\end{figure} \end{figure}
The sensor and actuator ``constants'' (\(g_s\) and \(g_a\)) derived in Section \ref{ssec:detail_fem_super_element_example} for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks. The sensor and actuator ``constants'' (\(g_s\) and \(g_a\)) derived in Section \ref{ssec:detail_fem_super_element_example} for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks.
\section{Simpler 2DoF Model of the APA300ML} \section{Simpler 2DoF Model of the APA300ML}
\label{sec:org8d0582b}
\label{ssec:detail_fem_actuator_apa300ml_2dof} \label{ssec:detail_fem_actuator_apa300ml_2dof}
To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from \cite{souleille18_concep_activ_mount_space_applic}, was developed. To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model, adapted from \cite{souleille18_concep_activ_mount_space_applic}, was developed.
@ -512,8 +487,8 @@ While higher-order modes and non-axial flexibility are not captured, the model a
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from \(V_a\) to \(d_i\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor})} \caption{\label{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}Comparison of the transfer functions extracted from the finite element model of the APA300ML and of the 2DoF model. Both for the dynamics from \(V_a\) to \(d_i\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_actuator}) and from \(V_a\) to \(V_s\) (\subref{fig:detail_fem_apa300ml_comp_fem_2dof_force_sensor})}
\end{figure} \end{figure}
\section{Electrical characteristics of the APA} \section{Electrical characteristics of the APA}
\label{sec:org68b0967}
\label{ssec:detail_fem_actuator_apa300ml_electrical} \label{ssec:detail_fem_actuator_apa300ml_electrical}
The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain) \cite[, chapter 5.5]{schmidt20_desig_high_perfor_mechat_third_revis_edition}. The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain) \cite[, chapter 5.5]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
@ -532,8 +507,8 @@ The developed models of the APA do not represent such behavior, but as this effe
However, the electrical characteristics of the APA remain crucial for instrumentation design. However, the electrical characteristics of the APA remain crucial for instrumentation design.
Proper consideration must be given to voltage amplifier specifications and force sensor signal conditioning requirements. Proper consideration must be given to voltage amplifier specifications and force sensor signal conditioning requirements.
These aspects will be addressed in the instrumentation chapter. These aspects will be addressed in the instrumentation chapter.
\section{Validation with the Nano-Hexapod} \section{Validation with the Nano-Hexapod}
\label{sec:org0bde7c1}
\label{ssec:detail_fem_actuator_apa300ml_validation} \label{ssec:detail_fem_actuator_apa300ml_validation}
The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation. The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation.
@ -564,8 +539,8 @@ These results validate both the selection of the APA300ML and the effectiveness
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a nano-hexapod having actuators modelled a 2DoF system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).} \caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a nano-hexapod having actuators modelled a 2DoF system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).}
\end{figure} \end{figure}
\chapter{Flexible Joint Design} \chapter{Flexible Joint Design}
\label{sec:orgf6572af}
\label{sec:detail_fem_joint} \label{sec:detail_fem_joint}
High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy. High-precision position control at the nanometer scale requires systems to be free from friction and backlash, as these nonlinear phenomena severely limit achievable positioning accuracy.
This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation. This fundamental requirement prevents the use of conventional joints, necessitating instead the implementation of flexible joints that achieve motion through elastic deformation.
@ -591,7 +566,7 @@ For design simplicity and component standardization, identical joints are employ
\end{center} \end{center}
\subcaption{\label{fig:detail_fem_joints_wire}} \subcaption{\label{fig:detail_fem_joints_wire}}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_joints_examples}Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical ``universal'' flexible joint used in \cite{preumont07_six_axis_singl_stage_activ}. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in \cite{yang19_dynam_model_decoup_contr_flexib}. (\subref{fig:detail_fem_joints_wire}) ``Thin'' flexible joints having differnt ``notch curves'' are also used \cite{du14_piezo_actuat_high_precis_flexib}.} \caption{\label{fig:detail_fem_joints_examples}Example of different flexible joints geometry used for Stewart platforms. (\subref{fig:detail_fem_joints_preumont}) Typical ``universal'' flexible joint used in \cite{preumont07_six_axis_singl_stage_activ}. (\subref{fig:detail_fem_joints_yang}) Torsional stiffness can be explicitely specified as done in \cite{yang19_dynam_model_decoup_contr_flexib}. (\subref{fig:detail_fem_joints_wire}) ``Thin'' flexible joints having ``notch curves'' are also used \cite{du14_piezo_actuat_high_precis_flexib}.}
\end{figure} \end{figure}
While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance \cite{mcinroy02_model_desig_flexur_joint_stewar}. While ideally these joints would permit free rotation about defined axes while maintaining infinite rigidity in other degrees of freedom, practical implementations exhibit parasitic stiffness that can impact control performance \cite{mcinroy02_model_desig_flexur_joint_stewar}.
@ -601,6 +576,7 @@ The analysis of bending and axial stiffness effects enables the establishment of
These specifications guide the development and optimization of a flexible joint design through finite element analysis (Section \ref{ssec:detail_fem_joint_specs}). These specifications guide the development and optimization of a flexible joint design through finite element analysis (Section \ref{ssec:detail_fem_joint_specs}).
The validation process, detailed in Section \ref{ssec:detail_fem_joint_validation}, begins with the integration of the joints as ``reduced order flexible bodies'' in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints. The validation process, detailed in Section \ref{ssec:detail_fem_joint_validation}, begins with the integration of the joints as ``reduced order flexible bodies'' in the nano-hexapod model, followed by the development of computationally efficient lower-order models that preserve the essential dynamic characteristics of the flexible joints.
\section{Bending and Torsional Stiffness} \section{Bending and Torsional Stiffness}
\label{sec:orgbb0d797}
\label{ssec:detail_fem_joint_bending} \label{ssec:detail_fem_joint_bending}
The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction \cite{mcinroy02_model_desig_flexur_joint_stewar} and can affect system dynamics. The presence of bending stiffness in flexible joints causes the forces applied by the struts to deviate from the strut direction \cite{mcinroy02_model_desig_flexur_joint_stewar} and can affect system dynamics.
@ -654,8 +630,8 @@ A parallel analysis of torsional stiffness revealed similar effects, though thes
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus}Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. When having an actuator modelled as 1DoF without parallel stiffness to the force sensor (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2DoF model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml})} \caption{\label{fig:detail_fem_joints_bending_stiffness_iff_locus}Effect of bending stiffness of the flexible joints on the attainable damping with decentralized IFF. When having an actuator modelled as 1DoF without parallel stiffness to the force sensor (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_1dof}), and with the 2DoF model of the APA300ML (\subref{fig:detail_fem_joints_bending_stiffness_iff_locus_apa300ml})}
\end{figure} \end{figure}
\section{Axial Stiffness} \section{Axial Stiffness}
\label{sec:orge681787}
\label{ssec:detail_fem_joint_axial} \label{ssec:detail_fem_joint_axial}
The limited axial stiffness (\(k_a\)) of flexible joints introduces an additional compliance between the actuation point and the measurement point. The limited axial stiffness (\(k_a\)) of flexible joints introduces an additional compliance between the actuation point and the measurement point.
@ -709,8 +685,8 @@ Based on this analysis, an axial stiffness specification of \(100\,N/\mu m\) was
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_joints_axial_stiffness_iff_results}Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant})} \caption{\label{fig:detail_fem_joints_axial_stiffness_iff_results}Effect of axial stiffness of the flexible joints on the attainable damping with decentralized IFF (\subref{fig:detail_fem_joints_axial_stiffness_iff_locus}). Estimation of the coupling of the damped plants using the RGA-number (\subref{fig:detail_fem_joints_axial_stiffness_rga_hac_plant})}
\end{figure} \end{figure}
\section{Specifications and Design flexible joints} \section{Specifications and Design flexible joints}
\label{sec:org1248d7e}
\label{ssec:detail_fem_joint_specs} \label{ssec:detail_fem_joint_specs}
The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics. The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics.
@ -757,8 +733,8 @@ The final design, featuring a neck dimension of 0.25mm, achieves mechanical prop
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_joints_design}Designed flexible joints.} \caption{\label{fig:detail_fem_joints_design}Designed flexible joints.}
\end{figure} \end{figure}
\section{Validation with the Nano-Hexapod} \section{Validation with the Nano-Hexapod}
\label{sec:org8e5fa20}
\label{ssec:detail_fem_joint_validation} \label{ssec:detail_fem_joint_validation}
The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis. The designed flexible joint was first validated through integration into the nano-hexapod model using reduced-order flexible bodies derived from finite element analysis.
@ -794,8 +770,8 @@ While additional degrees of freedom could potentially capture more dynamic featu
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_fem_joints_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a nano-hexapod having bottom joint modelled by bending stiffness \(k_f\) and axial stiffness \(k_a\) and top joints modelled by bending stiffness \(k_f\), torsion stiffness \(k_t\) and axial stiffness \(k_a\). Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}).} \caption{\label{fig:detail_fem_joints_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod including joints modelled with FEM and a nano-hexapod having bottom joint modelled by bending stiffness \(k_f\) and axial stiffness \(k_a\) and top joints modelled by bending stiffness \(k_f\), torsion stiffness \(k_t\) and axial stiffness \(k_a\). Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_joints_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_joints_fem_vs_perfect_hac_plant}).}
\end{figure} \end{figure}
\chapter*{Conclusion} \chapter*{Conclusion}
\label{sec:org3cd82fd}
\label{sec:detail_fem_conclusion} \label{sec:detail_fem_conclusion}
In this chapter, the methodology of combining finite element analysis with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design of nano-hexapod components. In this chapter, the methodology of combining finite element analysis with multi-body modeling has been demonstrated and validated, proving particularly valuable for the detailed design of nano-hexapod components.
@ -809,6 +785,5 @@ In both cases, the ability to seamlessly integrate finite element models into th
A key outcome of this work is the development of reduced-order models that maintain prediction accuracy while enabling efficient time-domain simulation. A key outcome of this work is the development of reduced-order models that maintain prediction accuracy while enabling efficient time-domain simulation.
Such model reduction, guided by detailed understanding of component behavior, provides the foundation for subsequent control system design and optimization. Such model reduction, guided by detailed understanding of component behavior, provides the foundation for subsequent control system design and optimization.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]
\end{document} \end{document}